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\section{Method}
\label{sec:method}
At every step of active learning, we divide the entire training data by three pools.
At first, the \textit{unlabeled pool} ($\mathcal{P}^o$) contains all the samples that are not labeled yet.
The \textit{labeled pool} ($\mathcal{P}$) consists of the other samples labeled by the users.
We further divide $\mathcal{P}$ by two pools: a labeled pool for VAE training ($\mathcal{P}_v$) and a labeled pool for uncertainty estimation ($\mathcal{P}_e$).
Thus, $\mathcal{P}=\mathcal{P}_v \bigcup \mathcal{P}_e$ is always satisfied, and $\mathcal{P}_v$ and $\mathcal{P}_e$ are utilized for training the VAE module and estimating the posterior probability, respectively.
\begin{algorithm}[t]
\small
\label{algorithm}
\caption{Posterior Active Learning}
Input: $\mathcal{P^o}$, $\mathcal{P}$, $N_s$\\
\While{not at the maximum sampling iteration}{
Train $\mathcal{M}$ using $\mathcal{P}$ and its own loss\\
Freeze $\mathcal{M}$\\
Divide $\mathcal{P}$ by $\mathcal{P}_v$ and $\mathcal{P}_e$
Train the VAE module using $\mathcal{P}_v$ and Eq.~\ref{eq:vaeLoss}\\
Estimate the label uncertainty probability using Eq.~\ref{eq:labelUncertaintyProb}\\
Estimate the prior probability using Eq.~\ref{eq:prior}\\
Estimate the likelihood probability using Eq.~\ref{eq:likelihood}\\
Estimate the posterior uncertainty using Eq.~\ref{eq:posteriorUncertainty}\\
$\mathbf{X}$ $\leftarrow$ Sample $N_s$ samples with the highest uncertainty\\
$\mathcal{P^o}$ $\leftarrow$ $\mathcal{P^o}$ - $\mathbf{X}$\\
$\mathcal{P}$ $\leftarrow$ $\mathcal{P}$ $\bigcup$ $\mathbf{X}$\\
}
\normalsize
\end{algorithm}
\subsection{Overall Structure}
As shown in Fig.~\jw{figure}, the entire architecture of the proposed algorithm contains two sub-networks: one baseline network and one VAE.
The structure of the baseline network follows the original architecture used by the target algorithm for the active learning, and the VAE is fed by the multiple features from the baseline network.
At the initial state where every training sample is included in $\mathcal{P}^o$, the defined number ($N_s$) of samples are randomly selected for the initial training.
After the initial samples are labeled by the users, the samples are excluded from $\mathcal{P}^o$ and added into $\mathcal{P}$ with the labels.
Then, by using the updated $\mathcal{P}$, we train the baseline network by the original training method of the target task.
After freezing the trained baseline network, we train the VAE to represent both of the overall and class-wise distributions by $\mathcal{P}$ and $\mathcal{P}_v$.
From the trained distributions of the VAE and $\mathcal{P}_e$, we estimate the posterior uncertainty of every sample in $\mathcal{P}^o$, and the $N_s$ unlabeled samples with the highest posterior uncertainty are chosen for the user labeling.
Finally, the labeled samples are moved from $\mathcal{P}^o$ to $\mathcal{P}$, and then we repeat the training of the baseline network by using the updated $\mathcal{P}$.
The details of the procedures are summarized in Algorithm~\ref{algorithm}.
\subsection{VAE Module}
The VAE utilizes the multiple features of the baseline network as its input, and thus the size of the VAE can be much smaller than that of the baseline network.
To represent the overall and class-wise feature distributions in the latent space of the VAE, we split the dimensions of the latent features in the VAE according to the number of classes.
Then, while the entire latent feature depicts the overall feature distribution, the class-wise feature distribution can be found when we consider only the split latent features.
Since the latent features are hypothetically divided, the architecture of the VAE is equivalent with the conventional VAE, but the training scheme is novel to train the class-wise feature distribution.
\subsubsection{Architecture}
The raw features extracted from the baseline network are pre-processed before being fed into the VAE.
When the extracted feature is a convolution feature, we reduce the feature size by applying a global average pooling layer (GAP).
Then, each of the features is respectively normalized by a batch normalization layer to align the various feature distributions from the multiple layers.
Finally, the normalized features are fed respectively into the fully-connected layers, and then the feature vector concatenated by the outputs of the fully-connected layers is fed into the input of the VAE.
Then, we denote the feature extraction and the pre-processing processes by $\mathcal{H}(\bullet)$ as following:
\begin{equation}
\mathbf{h}^{(i)} = \mathcal{H} (\mathbf{x}^{(i)}),
\end{equation}
where $\mathbf{h}^{(i)}$ is the input of the VAE when the image sample $\mathbf{x}^{(i)}$ is given.
As the conventional VAE, the VAE module consists of one probabilistic encoder and one decoder that are denoted by $q_{\phi}(\mathbf{z}|\mathbf{h})$ and $p_{\theta}(\mathbf{h}|\mathbf{z})$, respectively.
Both of the encoder and decoder is composed of multiple fully-connected layers followed by ReLU.
Then, the random variable $\tilde{\mathbf{z}}\sim q_{\phi}(\mathbf{z}|\mathbf{h})$ can be reformulated by using an auxiliary noise variable $\mathbf{\epsilon}$ as following:
\begin{equation}
\tilde{\mathbf{z}}^{(i,l)} = g_{\phi}(\epsilon^{(l)}, \mathbf{h}^{(i)}) \,\,\,\,\,\,\,\,with\,\,\,\epsilon^{(l)} \sim \mathcal{N}(0,1),
\end{equation}
where $\mathcal{N}(\mu,\sigma^2)$ is a Gaussian probability distribution with a average $\mu$ and a standard deviation $\sigma$.
From the sampled $\tilde{\mathbf{z}}$, the class-wise response probability $\mathbf{r}$ is estimated by the softmax probability as:
\begin{equation}
\begin{aligned}
\tilde{\mathbf{r}} &= softmax \bigg(\Big[\sum_{j\in C_1} -z_j^2, \sum_{j\in C_2} -z_j^2, \dots, \sum_{j\in C_{N_c}} -z_j^2\Big]^T\bigg),
\end{aligned}
\end{equation}
where $N_c$ means the number of classes, $z_j$ is the $j$-th dimension value of $\mathbf{z}$, and $C_n$ is a set of indices for the split dimensions of $\mathbf{z}$ that represent the $n$-th class.
\jw{
Thus, in the class-wise feature distribution, the samples of the target classes are located near $0$ on the split dimensions that represent the target class.
Then, the samples of the other classes are placed far from $0$.
- 다시 쓰기}
\jw{여기서 이렇게 학습된 latent feature의 특성을 보여주면 좋을듯. 각 클래스별 latent vector들을 보면 특정 클래스의 sample들이 0에 뭉쳐있고 나머지는 외부에 배치. 혹은 2-D로 보여주는 것도 재밌는 방법이 될수도 있음.}
\subsubsection{Training Method}
We define the samples from $\mathcal{P}_v$ by $\{\mathbf{x}_v^{(i)}\}$ and the samples from $\mathcal{P}^o$ by $\{\mathbf{x}_o^{(i)}\}$.
After finishing the training of the baseline network, the weights of the baseline network are frozen to avoid the degradation of its performance.
Thus, when the VAE is trained, only the preprocessing layers and the layers of the VAE modules are updated.
As a result, when the users want to stop the active learning, the baseline network can be served directly as the final model without any additional training sequence.
To let the latent features represent the overall and class-wise feature distribution simultaneously, we define the loss $\mathcal{L}$ by:
\begin{equation}\label{eq:vaeloss}
\mathcal{L} = \mathcal{L}_{overall} + \lambda_c \mathcal{L}_{class},
\end{equation}
where $\mathcal{L}_{overall}$ and $\mathcal{L}_{class}$ denotes the losses for the overall and class-wise feature distribution, respectively, with $\lambda_c$ that controls the ratio of the two losses.
The overall feature distribution is trained by using the loss terms that were proposed in~\jw{VAE refer}.
Thus,
\begin{equation} \label{eq:vaeLoss}
\mathcal{L}_{overall} = \mathbb{E}[\log p_{\theta}(\mathbf{h}|\mathbf{z})] - \mathbf{D}_{KL}\big(q_{\phi}(\mathbf{z}|\mathbf{h})\parallel p(\mathbf{z})\big).
\end{equation}
Since $\mathcal{L}_{overall}$ can be estimated even with the unlabeled data, every sample from both of $\mathcal{P}_v$ and $\mathcal{P}^o$ is utilized for learning the overall feature distribution.
$p(\mathbf{z})$ is the prior that is chosen by a unit Gaussian ($\mathcal{N}(0,1)$).
Referred by~\jw{VAE refer}, we use the reparameterization trick to calculate the gradients from the sampling-based loss.
Like $\mathcal{L}_{overall}$ considering all the training data, $\mathcal{L}_{class}$ contains the two loss terms acquired by $\mathcal{P}_v$ and $\mathcal{P}^o$, respectively.
\begin{equation}
\mathcal{L}_{class} = - \sum_{\mathbf{x}^{(i)}_v \in \mathcal{P}_v} \mathbf{y}^{(i)}_v \log \tilde{\mathbf{r}}^{(i)}_v - \sum_{\mathbf{x}^{(i)}_o \in \mathcal{P}_o} \tilde{\mathbf{r}}^{(i)}_o \log \tilde{\mathbf{r}}^{(i)}_o,
\end{equation}
where $\mathbf{y}^{(i)}_v$ is a one-hot vector for the labels paired by $\mathbf{x}^{(i)}_v$.
Among the two loss terms in $\mathcal{L}_{class}$, the first loss term is the conventional cross-entropy loss for the labeled samples, while the second loss term drives the unlabeled samples to be associated one of the multiple classes by the marginal entropy loss.
To update the VAE, we build a mini-batch with $N_b$ samples by randomly extracting $N_b/2$ samples from $\mathcal{P}_v$ and the other samples from $\mathcal{P}^o$.
The weights of the VAE are updated by Adam Optimizer~\jw{refer}.
\subsection{Class-wise Sampling Number Estimation}
Initial sampling number estimation (assume all the predicted labels are correct)
Sampling number refinement (consider the class간의 prediction ambiguity)
\subsubsection{Initial Sampling Number Estimation}
Introduction of Water-filling algorithm
0. $S={0,...,N_c}$
1. iterate for $i$-th class maximum sample number, $S \leftarrow S - \{index of i\}$
2. if $ \{index of i\}-th class sample number \times #(S) - \sum_{j \in S} j-th class sample number < N_{sampling}$
2-1. $eta = \frac{N_{sampling} - \sum_{j \in S} j-th class sample number}{N_c - i + 1} + $ $i$-th class maximum sample number
2-2. $j$-th result $=$ $eta - j-th class sample number$ for $S$, otherwise 0
2-3. break
\subsubsection{Sampling Number Refinement?}
Sampling Number Refinement를 위한 기본 식 정의
Posterior probability definition ($p(y_n|\widehat{y_m})$)
여기서 Initial sampling number가 ideal case이고, uncertainty?를 기반으로 계산한 sampling number를 얻고자 할 때, 아래와 같이 정의
\begin{equation}
\mathbf{N} = \mathbf{P} \mathbf{\widehat{N}}
\end{equation}
\begin{equation}
\begin{aligned}
\mathbf{P} &= \mathcal{R}^{N_c\times N_c}\\
\mathbf{P}_{n, m} &= p(y_n|\widehat{y_m})
\end{aligned}
\end{equation}
2. Bayes rule-based posterior probability estimation
\begin{equation}
p(y_n|\widehat{y_m}) = \frac{p(y_n) p(\widehat{y_m}|y_n)}{\sum_{n=1}^{N_c} p(\widehat{y_m}|y_n)}
\end{equation}
3. Real label prior probability estimation ($p(y_n)$): Approximated by the class-wise sample ratio at the initial random sampling.
4. Likelihood probability estimation ($p(\widehat{y_m}|y_n)$): Estimated by predicting the labels for labeled data''.
: N times sampling > probability estimation
이 후 $\mathbf{\widehat{N}}$을 추정하기 위해 Convex optimization 풀기.
5. Convex optimization for estimating the refined sampling number
\begin{equation}
\mathbf{N} = \mathbf{P} \mathbf{\widehat{N}}
\end{equation}
s.t. sum($\widehat{N}$)=$N_{sampling}$, \widehat{N} is integer
\begin{equation}
E = ||\mathbf{N}_{init} - \mathbf{P} \widehat{\mathbf{N}}||_2^2 + \lambda||\mathbf{N}_{init} - \widehat{\mathbf{N}}||_2^2 + \lambda_p (\mathbf{N}_{sampling} - \sum\widehat{\mathbf{N}})^2
\end{equation}
5-1. Loss term 1: $(N - P * \widehat{N})^2$. 추정되는 Sampling number개수가 Real 기준 개수와 유사해지도록 Posterior probability 곱.
5-2. Loss term 2: $(N - \widehat{N})^2$. 다만, 추정되는 sampling number와 Real sampling number가 너무 크게 차이나지 않게. 만약 없으면 1번 loss term에 의해 특정 클래스로 추정되는 샘플들만 선택하려고 하는 경향이 매우 강해짐. (특히 dominant class 상황일 때 dominant class를 선택하지 않으려는 특성이 강한데 이로인해 dominant class로 잘못 추정될 가능성이 가장 적은 class의 sample만 선택하도록 최적화 결과가 나오는 경우 많음.)
5-3. Penalty term 2: $\lambda_p(\sum\widehat{N} - N_{sampling})^2$ 클래스별 샘플링하는 샘플의 총합은 우리가 원하는 전체 샘플링하는 샘플의 개수와 같아야 함. 원래는 constraint지만 penalty term으로 넣어서 relaxation.
5-4. Optimization: Non-negative Least Square Optimization 알고리즘 적용. 개수는 Integer이기 때문에 Integer programming 등을 이용해야하지만, 효율적인 계산을 위해 Interger를 일반 Float으로 Relaxation시켜 계산한 뒤, Round시켜 최종 결과값 추정. 만약 round로 인해 총합이 사용자가 원하는 sample개수와 다르거나 원하는 sample 개수만큼의 sample이 해당 클래스에 존재하지 않는 경우, 가장 앞선 클래스부터 더하거나 하나씩 빼서 개수를 맞춘다.
\subsection{Active Sampling Algorithm}
Using the trained VAE, we estimate the posterior uncertainty of the unlabeled samples $\mathbf{x}_o^{(i)}$ from $\mathcal{P}^o$.
At first, we define the uncertainty of the training sample $\mathbf{x}$ by:
\begin{equation}
P( \mathbf{y} \neq \widehat{\mathbf{y}} | \mathbf{x} ),
\end{equation}
where $\mathbf{y}$ and $\widehat{\mathbf{y}}$ are the real label and the predicted label of $\mathbf{x}$, respectively.
$\widehat{\mathbf{y}}$ is predicted by the class-wise feature distribution trained in the VAE as:
\begin{equation}
\widehat{\mathbf{y}} = \arg \max [\tilde{r}_1, \tilde{r}_2, \dots, \tilde{r}_{N_c}].
\end{equation}
Then, we derive the posterior uncertainty as following:
\begin{equation}\label{eq:posteriorUncertainty}
\begin{aligned}
P(\mathbf{y} \neq \widehat{\mathbf{y}} | \mathbf{x})
&= 1 - P(\mathbf{y} = \widehat{\mathbf{y}} | \mathbf{x})\\
&= 1 - \sum_{n=1}^{N_c} P(\mathbf{y}_n, \widehat{\mathbf{y}}_n | \mathbf{x}),
\end{aligned}
\end{equation}
where $\mathbf{y}_n$ and $\widehat{\mathbf{y}}_n$ mean $\mathbf{y}=n$ and $\widehat{\mathbf{y}}=n$, respectively.
Unfortunately, it is infeasible to estimate the posterior probability of $P(\mathbf{y}_n, \widehat{\mathbf{y}}_n | \mathbf{x})$ directly, so we reformulate it through Bayes rule as:
\begin{equation}
\begin{aligned}
P(\mathbf{y}_n, \widehat{\mathbf{y}}_n | \mathbf{x})
= P(\mathbf{y}_n | \widehat{\mathbf{y}}_n, \mathbf{x}) P(\mathbf{x} | \widehat{\mathbf{y}}_n) P(\widehat{\mathbf{y}}_n) / P(\mathbf{x}).
\end{aligned}
\end{equation}
We denote $P(\mathbf{y}_n | \widehat{\mathbf{y}}_n, \mathbf{x})$, $P(\mathbf{x} | \widehat{\mathbf{y}}_n)$, and $P(\widehat{\mathbf{y}}_n)$ by a \textit{label uncertainty}, a \textit{likelihood}, and a \textit{prior}, respectively.
\ky{Here, $P(\mathbf{y}_n | \widehat{\mathbf{y}}_n, \mathbf{x})$ again involves $\mathbf{y}$ and $\mathbf{x}$, which cannot be evaluated until we observe the label.
Therefore, we instead do an informed guess, by ignoring $\mathbf{x}$ and approximating with $P(\mathbf{y}_n | \widehat{\mathbf{y}}_n)$, in other words, relying on our current estimate of our label.
We therefore write
}
\begin{equation}
\begin{aligned}
P(\mathbf{y}\neq\widehat{\mathbf{y}} | \mathbf{x}) \propto C - \sum_{n=1}^{N_c} P(\mathbf{y}_n | \widehat{\mathbf{y}}_n) P(\mathbf{x} | \widehat{\mathbf{y}}_n) P(\widehat{\mathbf{y}}_n),
\end{aligned}
\end{equation}
when $P(\mathbf{x})=C$ follows the uniform distribution.
Then, we estimate each of the three terms by using the variational classification results from the trained VAE.
At first, the label uncertainty can be estimated by using the labeled samples of $\mathcal{P}_c$ that is excluded when the VAE is trained. Thus,
\begin{equation}\label{eq:labelUncertaintyProb}
P(\mathbf{y}_n | \widehat{\mathbf{y}}_n) = \frac{ \sum_{l\in L, (\mathbf{x}_c^{(i)},\mathbf{y}_c^{(i)})\in \mathcal{P}_c} \delta (\widehat{\mathbf{y}}^{(i,l)}, \mathbf{y}_c^{(i)}, n)}{\sum_{l\in L, (\mathbf{x}_c^{(i)},\mathbf{y}_c^{(i)})\in \mathcal{P}_c} \delta (\widehat{\mathbf{y}}^{(i,l)}, n)}
\end{equation}
where $\delta(\bullet)$ is a function resulting $1$ if all the inputs are equivalent otherwise $0$, and $L$ is a set containing the latent noises. The size of $L$ is determined by a user-defined parameter $N_L$.
Next, the likelihood is estimated for each unlabeled sample of $\mathcal{P_o}$ as following:
\begin{equation}\label{eq:likelihood}
P(\mathbf{x}_o^{(i)} | \widehat{\mathbf{y}}_n) = \frac{\sum_{l\in L} \delta ( \widehat{\mathbf{y}}^{(i,l)}, n)}{N_L}.
\end{equation}
At last, the prior is acquired by using the unlabeled samples included in $\mathcal{P_o}$.
Thus,
\begin{equation}\label{eq:prior}
P(\widehat{\mathbf{y}}_n) = \frac{\sum_{l\in L, \mathbf{x}_o^{(i)}\in \mathcal{P}^o} \delta (\widehat{\mathbf{y}}^{(i,l)}, n)}{N_L N(\mathcal{P}^o)},
\end{equation}
where $N(\bullet)$ means the number of elements in the given pool.
Then, we can estimate the posterior uncertainty $P( \mathbf{y} \neq \widehat{\mathbf{y}} | \mathbf{x} )$ for every sample according to Eq.~\ref{eq:posteriorUncertainty}.
Then, the labeling query is found by selecting the $N_s$ samples with the highest posterior uncertainty as:
\begin{equation}
Q = \arg \max_{\mathbf{x}\in\mathcal{P}_o}\,_{N_s} P( \mathbf{y} \neq \widehat{\mathbf{y}} | \mathbf{x} ),
\end{equation}
where $\max\,_{N_s}(\bullet)$ results in the $N_s$ maximum values of the input vectors.\jw{??}
\section{Introduction}
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\begin{center}
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\medskip
\noindent
FAQ\medskip\\
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\begin{figure}[t]
\begin{center}
\fbox{\rule{0pt}{2in} \rule{0.9\linewidth}{0pt}}
\end{center}
\caption{Example of caption. It is set in Roman so that mathematics
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\label{fig:long}
\label{fig:onecol}
\end{figure}
\subsection{Miscellaneous}
\noindent
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\end{tabular}\\
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\begin{table}
\begin{center}
\begin{tabular}{|l|c|}
\hline
Method & Frobnability \\
\hline\hline
Theirs & Frumpy \\
Yours & Frobbly \\
Ours & Makes one's heart Frob\\
\hline
\end{tabular}
\end{center}
\caption{Results. Ours is better.}
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{\small
\bibliographystyle{ieee_fullname}
\section{Introduction}
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\section*{Acknowledgements}
This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant and Compute Canada.
\part*{Appendix}
\twocolumn[
\centering
\Large
\textbf{VaB-AL: Incorporating Class Imbalance and Difficulty \\ with Variational Bayes for Active Learning} \\
\vspace{0.5em}Supplementary Material \\
\vspace{1.0em}
]
\section{Conclusion}
We have proposed a novel active learning method that incorporates class imbalance and label difficulty.
Our method was derived through the Bayes' rule, which results in three types of probabilities -- data likelihood, label prior, label difficulty -- being considered together.
We implement our method via a VAE, that is regularised to behave as a conditional VAE.
We have shown that this creates a significant difference for a real-world dataset that exhibits data imbalance, as well as in cases when data imbalance is introduced to CIFAR-10 and CIFAR-100 datasets.
While we limit our experiments to classification in this work, our method is application agnostic. In the future, we plan to extend our work to other discriminative tasks, for example, object detection and segmentation.
\section{Introduction}
\label{sec:intro}
\input{figs/teaser/item.tex}
Active learning focuses on efficient labelling of data and has drawn much interest lately~\cite{Yoo19_ac,Sener18_ac,Sinha19_ac}, due to deep learning being attempted at new domains, such as biomedical imaging~\cite{Atzeni18,Yuankai18} and industrial imaging~\cite{zhao2017vision,lin2019automated}, where acquiring data can be costly~\cite{Settles12_ac}.
Even for cases where data is not scarce, the effective usage of data may reduce training time, therefore the computational cost, including carbon foot-prints required to train each model.
There have been various studies based on semi-supervised~\cite{Donahue13b,Kipf16} and unsupervised~\cite{Saito17,Yan17} learning schemes to improve the training efficiency of data.
However, with limited labelling budget, the performance of the studies are significantly worse to the supervised learning with the additionally labelled data~\cite{Rasmus15}.
In other words, their label efficiency could be improved.
Existing methods~\cite{Yoo19_ac,Gal17_ac,Beluch18_ac,Sener18_ac,Sinha19_ac}, regardless of how they are formulated, have a common underlying assumption that all classes are equal --
they do not consider that some classes might just be harder to learn compared to others, or some classes might be more prevalent in the dataset than others.
Instead, they focus on, given a data sample, how much error a trained model is expected to make, or the estimated uncertainties~\cite{Yoo19_ac,Gal17_ac}.
These assumptions could be harmful, as in practice, since data is often imbalanced and not all classes are of the same difficulty~\cite{zhu2017synthetic,al2016transfer}.
This can create a bias in the labelled data pool, leading to the trained classifier and active learning methods also being biased in deciding which samples to the label.
As we show in Fig.~\ref{fig:teaser}, this can damage the capabilities of an active learning method significantly, even for typical benchmark datasets~\cite{cifarDataset,stl10dataset}.
In this work, we present a novel formulation for active learning, based on the classical Bayes' rule that allows us to incorporate multiple factors of a classification network together.
Through derivation, we show that the probability of a classifier making mistakes can be decomposed into three terms; i) the probability of misclassification for a given predicted class, ii) the likelihood of a sample given predicted class, and iii) the prior probability of a class being predicted.
In other words, one needs to take into account i) the difficulty of a class, ii) the performance of the classifier and iii) the abundance of a certain class of data holistically when determining the potential of a classification error.
We take all of them into account and choose samples to be labelled by selecting those that have the highest misclassification probability.
While the task is discriminative, our method requires the estimation of likelihood, which could be intractable.
We, therefore, propose to use a Variational Auto Encoder (VAE)~\cite{Kingma14b} to model the lower bound of the likelihood of a sample.
To make VAE conditioned on a predicted label, a naive way would be applied to train multiple VAEs for each predicted class.
However, this quickly becomes impractical with a large number of classes.
We thus propose to train a single VAE, with regularisation that acts as conditioning based on the predicted label.
To further tie the VAE with the classifier, and for quick training of the VAE, we use the deep feature representations of the classifier as inputs to the VAE.
Being generative, this training of VAE does not involve any labels, and we utilise the vast amount of unlabelled data with their predicted labels while inferring probabilities that are independent of data samples on the labelled ones.
We show empirically in \Section{results} that our method allows a significant leap in performance for the benchmark dataset and the real application dataset, especially when the labelling budget is highly limited.
Furthermore, we posit that considering the prior, that is, the distribution of labels during training, is critical when analysing the uncertainty of estimates.
In summary, our contributions are four-fold:
\begin{itemize}
\setlength{\itemsep}{0pt}%
\setlength{\parskip}{0pt}%
\vspace{-.5em}
\item we derive a novel formulation for active learning based on the Bayes' rule and posterior probability;
\item we propose a framework based on VAE that realises this formulation;
\item we reveal that considering the differences between classes -- abundance and difficulty -- is important;
\item we outperform the state of the art in the various experiments.
\end{itemize}
\section{Methodology}
\label{sec:method}
We first formally describe the active learning problem and detail how we select new samples to be labelled with the estimated correctness of predictions.
We then derive our method via Bayes' rule and describe how our derivation can be implemented in practice through a VAE.
Afterwards, we provide a summary of how we tie every component together as an active learning algorithm and implementation details.
\subsection{Problem formulation}
Active learning can be formulated as a problem of increasing the pool of labelled data at every round by labelling subsets of the unlabelled pool.
Formally, given a pool of data $\mathcal{P}$, we keep a pool of labelled data $\calP_L$ and unlabelled data $\calP_U$, such that $\mathcal{P} = \calP_L \cup \calP_U$ and $\varnothing = \calP_L \cap \calP_U$.
Then, for each active learning round $r$, we select $\calP_S^{(r)}$ that is a subset of $\calP_U$ with $N_\round$ samples according to a criterion that defines the active learning method, which is moved from $\calP_U$ to $\calP_L$.
Thus, $\calP_U^{(r+1)} = \calP_U^{(r)} - \calP_S^{(r)}$ and $\calP_L^{(r+1)} = \calP_L^{(r)} + \calP_S^{(r)}$.
The core of active learning is how $\calP_S^{(r)}$ is selected, which in our case is based on the probability of a model providing a wrong answer for the given data.
\paragraph{Active learning based on the hardest examples.}
The underlying idea of our method is that when acquiring data with a limited labelling budget, one should acquire those that have the highest probability of making wrong predictions~\cite{Yoo19_ac,Li13_ac,Gal17_ac,Liu19_ac}.
Formally, if we let $y$ denote the real label and $\widehat{y}$ the label predicted by a model, we find samples $\mathbf{x}$ with large
\begin{equation}
p( y \neq \widehat{y} | \mathbf{x} ).
\end{equation}
Unfortunately, this is not a probability distribution that can be modelled directly, and we, therefore, estimate this probability via Bayes' rule and approximations.
\subsection{Bayesian active learning}
We now derive our Bayesian formulation.
To estimate $p( y \neq \widehat{y} | \mathbf{x} )$, we take an alternative route through Bayes' rule, instead of the direct estimation that could be given by a discriminative model.
We first represent this probability with its complement, which can then be written as the sum of joint probabilities.
We write
\begin{equation}\label{eq:posteriorUncertainty}
\begin{aligned}
p(y \neq \widehat{y} | \mathbf{x})
= 1 - p(y = \widehat{y} | \mathbf{x})
= 1 - \sum_{n=1}^{N_c} p(y_n, \widehat{y}_n | \mathbf{x})
,
\end{aligned}
\end{equation}
where $N_c$ is the number of classes and $p(y_n)$ and $p(\widehat{y}_n)$ is a shorthand for $p(y=n)$ and $p(\widehat{y}=n)$, respectively.
Then, each $p(y_n, \widehat{y}_n | \mathbf{x})$ within the summation can be written through Bayes' rule as following:
\begin{equation}
\begin{aligned}
p(y_n, \widehat{y}_n | \mathbf{x})
= \frac{p(y_n | \widehat{y}_n, \mathbf{x}) p(\mathbf{x} | \widehat{y}_n) p(\widehat{y}_n)}{\sum_{n=1}^{N_c}{p(\mathbf{x} | \widehat{y}_n)p(\widehat{y}_n)}}.
\end{aligned}
\end{equation}
Here, $p(y_n | \widehat{y}_n, \mathbf{x})$ corresponds to the probability of the real label being $n$, given that the predicted label is $n$ and the data being $\mathbf{x}$.
However, this is a probability that cannot be evaluated unless we have the paired true label.
Thus, we instead do an informed guess, by ignoring $\mathbf{x}$ and approximating with $p(y_n | \widehat{y}_n)$.
In other words, we assume that the probability of a model making a mistake is highly related to the label.
Thus we approximate by writing
\begin{equation}
p(y_n, \widehat{y}_n | \mathbf{x})
\approx
\frac{p(y_n | \widehat{y}_n) p(\mathbf{x} | \widehat{y}_n) p(\widehat{y}_n)}{\sum_{n=1}^{N_c}{p(\mathbf{x} | \widehat{y}_n)p(\widehat{y}_n)}}.
\end{equation}
Finally, with \Eq{posteriorUncertainty}, we have
\begin{equation}\label{eq:uncertainty}
p(y\neq\widehat{y} | \mathbf{x})
\approx
1 - \sum_{n=1}^{N_c}{
\frac{p(y_n | \widehat{y}_n) p(\mathbf{x} | \widehat{y}_n) p(\widehat{y}_n)}{\sum_{n=1}^{N_c}{p(\mathbf{x} | \widehat{y}_n)p(\widehat{y}_n)}}.
}
\end{equation}
Note that here, i) $p(y_n | \widehat{y}_n)$ is the probability of a model making mistake based on label, ii) $p(\mathbf{x} | \widehat{y}_n)$ is the \emph{likelihood} of a sample given predicted label, iii) $p(\widehat{y}_n)$ is the \emph{prior} on the distribution of predicted labels, which represents how imbalanced the predictions of a model are.
As mentioned earlier in \Section{intro}, the likelihood estimation is non-trivial and requires a generative model.
We now detail how we model these three probabilities.
\subsection{Estimating probabilities with regularized VAE} \label{sec:prob}
To estimate the probabilities, we use a VAE~\cite{Kingma14b}.
Before we discuss the details, let us first clarify that we train this VAE exclusively in the unlabelled data pool $\calP_U$, and associate it with the behaviour of the discriminative model on unseen data.
We do not use the labelled pool $\calP_L$, as it is likely that the classifier has overfitted to the data.
Note also that while we explain in terms of $\mathbf{x}$, in fact, we use the deep feature representations given by the classifier
to tie the VAE more with the classifier and to facilitate training by removing the need of learning the deep features.
See \Fig{framework} for an illustration of our framework.
We first detail why and how we estimate the likelihood with a VAE and then discuss the other two probabilities.
\input{figs/framework/item.tex}
\paragraph{Likelihood of a sample -- $p(\mathbf{x} | \widehat{y}_n)$.}
Estimating $p(\mathbf{x} | \widehat{y}_n)$ is not straightforward.
A naive idea would be to implement multiple generative models that model $p(\mathbf{x})$, each trained with labels predicted to be of a certain class.
However, this becomes quickly impractical as the number of classes grows.
Moreover, estimating $p(\mathbf{x})$ can be intractable in practice~\cite{Kingma14b}.
In our work, we use a \emph{single} VAE to estimate the lower bound of $p(\mathbf{x} | \widehat{y}_n)$ for all $\widehat{y}_n$.
We use a VAE, as it learns to reconstruct the data sample using the tractable lower bound for $p(\mathbf{x})$.
Distinct from existing work, to condition the $p(\mathbf{x})$ based on predicted labels, we propose to learn a latent space where the \emph{absence} of parts of the latent embeddings are related to the predicted label.
In other words, this is as if we are training multiple VAEs with shared weights and overlapping latent spaces.
Once such latent embeddings are learned, we compute the lower bound of $p(\mathbf{x} | \widehat{y}_n)$, by simply enforcing the absence manually via masking -- thus selecting a VAE dedicated to a certain predicted class among the multiple virtual VAEs -- and computing $p(\mathbf{x})$.
This strategy allows us to have a manageable latent space while still being able to deal with many classes.
In more detail, if we denote
the $j$-th embedding dimension of VAE as $z_j$ and write $j \in C_n$ to denote dimension $j$ is related to class $n$, we write this absence condition as
\begin{equation}
\begin{aligned}
\widehat{y} = \argmin_n \left[
\sum_{j\in C_1} z_j^2, \sum_{j\in C_2} z_j^2, \dots, \sum_{j\in C_{N_c}} z_j^2
\right]^\top .
\end{aligned}
\label{eq:condition}
\end{equation}
Notice how this condition is conceptually similar to disentangled representations~\cite{sohn2015learning,burgess2018understanding}.
In our earlier attempts, we have also tried forming this condition as disentangled representations or enforcing $\sum_{j\in C_n} z_j^2$ to be zero if $\widehat{y}_n$, which neither was successful.
We suspect that enforcing such constraints limit the capacity of the latent space and interferes with the training of the VAE too much.
We have also tried other ways of enforcing absence -- using the $\ell-1$ norm or the sigmoid -- but using the square worked best. We provide empirical results in \Section{ablation}.
We enforce this constraint as a form of regularisation.
Let $\mathbf{w}=\left[w_1, w_2, \dots, w_n\right]^\top$, where $w_n = \sum_{j\in C_n} z_j^2$, then we form an additional regularisation loss $\mathcal{L}_\text{Class}$ to be used during training of VAE as
\begin{equation}
\mathcal{L}_\text{Class} = \calH\left(
\text{softmax}\left(-\mathbf{w}\right), \mathbf{\widehat{y}}
\right),
\label{eq:l_class}
\end{equation}
where $\calH$ denotes the cross entropy, $\text{softmax}$ is the softmax, and $\mathbf{\widehat{y}}$ is the one-hot encoded vector representation of $\widehat{y}$.
With this regularisation term, recall from \cite{Kingma14b}, that the training loss for VAEs $\mathcal{L}_\text{VAE}$ is the inverse of the empirical lower bound (ELBO) of the likelihood, which is defined as
\begin{equation}\label{eq:likelihoodProb}
\begin{aligned}
\mathcal{L}_\text{VAE}
= -
\mathds{E}_{\mathbf{z}\sim q_{\phi}\left(\mathbf{z} | \mathbf{x}\right)}\left[
\log p_{\theta}(\mathbf{x} |\mathbf{z})
\right]
+
D_{KL}\left(
q_{\phi}\left(\mathbf{z} |\mathbf{x}\right)
\parallel
p(\mathbf{z})
\right)
\end{aligned}
\;,
\end{equation}
where $p_{\theta}(\mathbf{x} |\mathbf{z})$ is the decoder with parameters $\theta$ and $q_{\phi}\left(\mathbf{z} |\mathbf{x}\right)$ is the encoder with parameters $\phi$.
Therefore, the total loss to train our VAE is
\begin{equation}\label{eq:vaeLoss}
\mathcal{L} = \mathcal{L}_\text{VAE} + \lambda \mathcal{L}_\text{Class}
,
\end{equation}
where $\lambda$ is a hyperparameter that controls the regularisation strength.
Note that with this loss, the VAE now also tries to mimic the behaviour of the classifier.
Once the VAE is trained, this VAE -- without any conditioning -- is now able to estimate the lower bound of the likelihood of a given data $p(\mathbf{x})$~\cite{Kingma14b} by simply computing the inversed value of $\mathcal{L}_{VAE}$.
Furthermore, by masking the embedding space associated with $\widehat{y}$ with zero, we can compute the lower bound of $p(\mathbf{x}|\widehat{y})$.
To avoid the decoder going too far from the latent embeddings, it was trained for; we use \Eq{condition} to obtain $\widehat{y}$, instead of the one from the classifier.
\paragraph{Probability of labelling error -- $p(y_n | \widehat{y}_n)$.}
While the labelled pool $\calP_L$ is the only set of data that we have labels for, as the trained classifier is likely to have overfitted to $\calP_L$, we cannot use it for modelling this probability.
We, therefore, use the labels given by the VAE for the data samples in the labelled pool $\calP_L$.
Note that the VAE has never seen these data points during training.
Mathematically, if we denote the $i$-th sample as $\mathbf{x}^{(i)}$, its label as $y^{(i)}$, the predicted label from \Eq{condition} as $\widehat{y}^{(i)}$, and introduce an indicator function $\delta$ that is $1$ if all inputs are equal and $0$ otherwise, we write
\begin{equation}\label{eq:labelUncertaintyProb}
p(y_n | \widehat{y}_n) \approx \frac{
\mathds{E}_{\mathbf{x}\in\calP_L, \mathbf{z}\sim q_\phi\left(\mathbf{z}|\mathbf{x}\right)}
\left[
\delta\left(y^{(i)}, \widehat{y}^{(i)}, n\right)
\right]
}{
\mathds{E}_{\mathbf{x}\in\calP_L, \mathbf{z}\sim q_\phi\left(\mathbf{z}|\mathbf{x}\right)}
\left[
\delta\left(\widehat{y}^{(i)}, n\right)
\right]
}
.
\end{equation}
Here, we approximate the expectations with Monte Carlo estimates.
Note that we also take the expectation over $\mathbf{z}$, to take into account the stochastic nature of VAEs.
\paragraph{Prior -- $p(\widehat{y}_n)$.}
The prior is also acquired by using the labelled samples included in $\calP_L$ as this probability should be related to how the classifier is trained.
Same as in the case of $p(y_n | \widehat{y}_n)$, we cannot use the classifier predictions for the labelled pool, and we use the predictions from the VAE.
Thus, sharing the notations as in \Eq{labelUncertaintyProb}, we write
\begin{equation}\label{eq:prior}
p(\widehat{y}_n) \approx
\mathds{E}_{\mathbf{x}\in\calP_L, \mathbf{z}\sim q_\phi\left(\mathbf{z}|\mathbf{x}\right)}
\left[
\delta\left(\widehat{y}^{(i)}, n\right)
\right]
.
\end{equation}
\subsection{Summary and implementation details}
\label{sec:impl}
\input{figs/algo_method/item.tex}
We summarise our method in \Algorithm{method}.
For each Active Learning round $r$, we train a classifier $\mathcal{M}^{(r)}$ with the labelled pool $\calP_L^{(r)}$.
We then freeze the weights of the classifier, and train our VAE -- $p^{(r)}_\theta\left(\mathbf{x}|\mathbf{z}\right)$ and $q^{(r)}_\phi\left(\mathbf{z}|\mathbf{x}\right)$ -- with the unlabelled pool $\calP_U^{(r)}$.
We then estimate the three probabilities, which we use to construct the final estimate of $p(y\neq\widehat{y}|\mathbf{x})$, and sample those from $\calP_U^{(r)}$ that have the highest value.
As noted earlier, we use the deep features of the baseline network (the classifier) as input to the VAE.
Specifically, we extract deep features from the specific four layers of baseline.
If the feature representations are from fully-connected layers, we use it as is.
If it is from a convolutional layer, thus a feature map, we first apply global average pooling.
We then apply batch normalisation to each feature, so that they are roughly in the same range, and feed it to a fully connected layer with 128 neurons and Sigmoid activation.
Finally, the outputs from these fully-connected layers are concatenated to form a $512\times1$ vector and given as input to the VAE.
For the architecture of VAEs, we opt for a simple one as the deep features already have abstracted context.
We apply four layers of fully connected layers with again 128 neurons and ReLU activations, with the exception of the layer that outputs the latent embeddings.
For this layer, we use $10\times N_c$ number of neurons and dedicate 10 dimensions per each class.
We implement our method with PyTorch~\cite{paszke2017automatic}
To train our VAE, we set $\lambda{=}0.005$ in Eq.~\eqref{eq:vaeLoss}, for all tasks.
We use Adam optimiser~\cite{Kingma15} with a learning rate of $10^{-4}$ and default parameters.
We train for 20 epochs.
To remove randomness from experiments, we perform our training multiple times with different initial conditions.
We report both the average and the standard deviation of our experiments.
For inference, to perform the Monte Carlo estimation in Eqns.~\eqref{eq:likelihoodProb}, \eqref{eq:labelUncertaintyProb}, and \eqref{eq:prior}, we iterate the probabilistic inference 100 times for every sample.
\section{Related Works}
\paragraph{Traditional Active Learning.}
Increasing the label efficiency, thus reducing the cost associated with obtaining labels, has been of interest for decades.
Even before deep learning became popular, various methods were suggested towards this goal~\cite{Settles12_ac}.
Methods were proposed to estimate the uncertainty of the unlabelled samples through the probability of prediction~\cite{Lewis94_ac}, the difference between the best prediction and the second one~\cite{Li13_ac,Luo13_ac,Roth06_ac}, or the entropy covering all the possible classes~\cite{Settles08_ac,Joshi09_ac}.
For support vector machine classifiers the methods were suggested to utilise the distance from the decision boundary, for both the classification task~\cite{Li14_ac,Tong01_ac} and the detection task~\cite{Vijayanarasimhan14_ac}.
The algorithms clustering and finding representative samples were also suggested as another way~\cite{Nguyen04_ac,Hasan15_ac,Aodha14_ac}.
Discrete optimisation algorithms have been proposed to consider the relationship between the sampling result and the model performance~\cite{Elhamifar13_ac,Guo10_ac,Yang15_ac}.
In a voting scheme-based algorithm~\cite{McCallumzy14_ac}, multiple different models are trained by the labelled pool, which determines the next queries according to their disagreement.
Despite these efforts, the classical approaches are geared towards simple features and may hold limitations when applying to a large deep network with many nonlinear estimations.
\paragraph{Active Learning for deep networks.}
Active learning algorithms for deep networks can be categorised into uncertainty-based methods and representation-based methods.
The uncertainty-based methods aim to select the uncertain samples from the unlabelled data pool and annotate them to increase the labelled pool~\cite{Yoo19_ac,Gal17_ac,Beluch18_ac}.
Yoo and Kwon~\cite{Yoo19_ac} proposed to use a small auxiliary ``module'' network predicting the training loss of the baseline network that is being trained with the active learning scheme.
They then select the samples that are expected to give high losses.
While their method is similar to ours in that an additional network is trained, as they require a ground-truth loss value while training, the auxiliary network can only be trained with labelled data, creating yet another network that the performance depends on how data is sampled.
In contrast, we train our VAE with unlabelled data since we only rely on the predicted labels from the baseline network during training, and result in stable performance even with few labelled data.
Gal~\emph{et al}.~\cite{Gal17_ac} proposed a method based on the estimation of sample-wise posterior probability through a Bayesian deep learning~\cite{Gal16} framework.
The method can be implemented simply by locating several dropout layers in a deep network, but this increases training time significantly until the convergence.
Beluch~\emph{et al}.~\cite{Beluch18_ac} suggest an active sampling method that estimates the disagreement of the prediction by using multiple deep networks.
The downside of their method is that, as they use multiple networks, the memory and the computational requirement increases proportionally.
The representation-based methods target on finding representative samples within the high-dimensional space that deep networks learn~\cite{Sener18_ac,Sinha19_ac}.
Sener and Savarese~\cite{Sener18_ac} proposed the Coreset algorithm that determines representative samples by using the feature maps of the intermediate layers of a deep network, rather than the last layer.
However, the optimisation method in the Coreset algorithm does not scale well as the number of classes, and the number of unlabelled samples grows.
To improve the scalability, Sinha~\emph{et al}.~\cite{Sinha19_ac} proposed to map the high dimensional feature maps into a lower dimension through adversarial training.
Unfortunately, being based on adversarial training, the method requires a large amount of training data for the mapping to be reliable.
Beyond them, hybrid approaches combine the best of both uncertainty-based and representation-based methods~\cite{Zhou17_ac,Paul17_ac}.
Some works focused on a specific task: for example
person re-identification~\cite{Liu19_ac} and a human pose estimation~\cite{Liu17_ac}.
While our work is most similar to uncertainty-based methods, it falls into neither uncertainty-based nor representation-based methods.
Contrary to the previous uncertainty-based works, we take into account characteristics that are not restricted to a single sample -- we consider the class difficulty and class imbalance.
Also, unlike the representation-based methods, we are not aiming to find representative samples, but a global trend of samples that are predicted to belong to a certain class.
\paragraph{Semi-supervised learning with VAEs.}
As we utilise VAEs~\cite{Kingma14b}, we also briefly review works related to VAEs that associate them with labelled data.
Since VAEs model the likelihood of data, Lee~\emph{et al}.~\cite{Lee2018adv} used them to identify out-of-distribution samples for each class.
We are loosely inspired by them, as we also use conditioned VAEs.
However, unlike them, we estimate one portion of our conditional probabilities in estimating the label correctness.
M2 VAE models~\cite{kingma2014semi} and Conditional VAEs~\cite{sohn2015learning} have been proposed to model conditional distributions.
They directly add the condition as an additional latent dimension that is trained independently with the other latent dimensions for the reconstruction.
In contrast, we apply conditioning implicitly during training to represent the class information and the feature distribution in the same latent dimensions.
In our early attempts, we were not able to obtain successful modelling for our application with the former.
\section{Experimental Results}
\label{sec:results}
We first introduce the dataset, then the experimental setup, including the classification network and the compared baselines.
We then report our results and discuss the effect of the individual probability terms through an ablation study.
\subsection{Dataset}
To validate our experiments we apply our method to the standard \textit{CIFAR-10} and \textit{CIFAR-100}~\cite{cifarDataset} datasets as well as the Northeastern University surface defect classification dataset (NEU)~\cite{neudataset}.
We use the CIFAR datasets to be comparable with recent Active Learning studies~\cite{Sinha19_ac}, and NEU to demonstrate that class imbalance is important in real-world applications.
In more detail, the \textit{CIFAR-10} dataset contains 60,000 images sized by $32\times 32\times 3$ with 10 classes.
The \textit{CIFAR-100} dataset also consists of 60,000 images of the $32\times 32\times 3$ size with 100 classes.
Both the \textit{CIFAR-10} and the \textit{CIFAR-100} datasets are well balanced -- all included images are assigned exactly one class among the 10 or the 100, and the number of instances per each class is equal.
The datasets come with a pre-determined training and evaluation split of 50,000 and 10,000 images, respectively.
The goal of the NEU dataset is to classify the 9 defect types on steels.
The main feature of this dataset is that it is heavily imbalanced -- the number of instances per class varies from 200 to 1589.
In total, the NEU dataset contains 7226 defect images of size $64\times 64$ pixel.
This dataset does not provide pre-determined splits, and we thus randomly reserve 20\% of the dataset for evaluation.
This results in 5778 training samples and 1448 validation samples.
\paragraph{Introducing synthetic class imbalance.}
To demonstrate the importance of considering class-wise probabilities, we build additional variants of \textit{CIFAR} datasets by removing a part of the samples in \textit{CIFAR-10} and \textit{CIFAR-100}.
First, we build four datasets that have dominant classes within them -- we hereafter refer to them as \textit{dominant} datasets.
A real-world example would be when scraping data from the internet -- there will be many images of people, cats, and dogs, but not so many of platypus.
The first three dominant datasets are built from \textit{CIFAR-10} by randomly removing 90\% samples of every category except for the $\{1, 5, 10\}^{th}$ classes, respectively.
For \textit{CIFAR-100}, as there are many classes, there are only a few instances each -- 500 for each class in the training set.
We, therefore, build the last dominant dataset by removing 40\% of the original samples for all categories other than the middle ones from $45^{th}$ class to $55^{th}$ class, from the \textit{CIFAR-100} dataset.
We denote these dominant datasets as \textit{CIFAR-10$^{+[1]}$}, \textit{CIFAR-10$^{+[5]}$}, \textit{CIFAR-10$^{+[10]}$}, and \textit{CIFAR-100$^{+[45:55]}$}, respectively.
We further consider the case when some samples are rare.
This would be, for example, cases where there are rare events, such as accidents or defects that need to be discovered.
Similar to the \textit{dominant} datasets, we build \textit{rare} datasets by taking certain classes away from \textit{CIFAR} datasets.
Specifically, we use three variants, where we remove 90\% of the samples that correspond to the $\{1, 5, 10\}^{th}$ classes.
We denote these as \textit{CIFAR-10$^{-[1]}$}, \textit{CIFAR-10$^{-[5]}$}, and \textit{CIFAR-10$^{-[10]}$}, respectively.
With these datasets, we run different active learning setups, based on the size of the dataset.
For NEU, we run five active learning rounds, where each round samples 250 samples.
For CIFAR-10 based ones, we run six active learning round, which 500 samples each.
For CIFAR-100, we run five rounds with 1000 samples each.
\subsection{Experimental setup}
\paragraph{The baseline classification network. }
As the baseline classifier, we utilise ResNet-18~\cite{He16}.
This network is composed of an initial simple convolution layer, followed by four basic residual blocks.
As discussed previously in \Section{impl}, the output feature maps from these four residual blocks are used to form the input to our VAE.
At every Active Learning round, we train the network for 200 epochs, with a batch size of 128.
We train with a typical setup for classification on CIFAR-10: SGD with the momentum of 0.9, weight decay of 0.0005, the learning rate is also initialised as 0.1 and decreased to 0.01 after 160 epochs.
We repeat the same experiments with the different random seeds three times to remove the randomness from our experiments.
\paragraph{The competitors.}
To demonstrate the effectiveness of the proposed algorithm, we compare our method with the state-of-the-art for active learning.
We consider
\textit{VAAL}~\cite{Sinha19_ac}, \textit{MC-DROPOUT}~\cite{Gal17_ac}, and \textit{Core-set}~\cite{Sener18_ac}.
We utilise the author's implementation for \textit{VAAL} and \textit{Core-set}.
For \textit{MC-DROPOUT} we integrate the author's implementation to our framework to utilise our baseline classification network -- this method is architecture dependant.
In more detail, we utilise the authors' code for uncertainty estimation and embed it into ours.
Among the multiple methods to estimate the uncertainty in \textit{MC-DROPOUT}, we use the Bayesian Active Learning by Disagreement (BALD)~\cite{houlsby2011bayesian} as the estimation method -- it showed the best generality in our experiments.
For the dropout layers of \textit{MC-DROPOUT}, we place them before each the residual blocks of the ResNet and fix the dropout ratio to 0.25.
As suggested by the authors~\cite{Gal17_ac}, we further use 100 samples to estimate the uncertainty.
In addition,
we also consider uniform random sampling as a baseline.
Finally, for a fair comparison, after new samples are extracted by these methods, we train the networks with the same random seed to further exclude any random surprises.
\subsection{Comparison results}
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{figs/new_result/results.pdf}
\caption{Results for the dominant, rare, and full datasets.}
\vspace{-4mm}
\label{fig:allDataset}
\end{figure*}
\paragraph{NEU dataset -- a real-world imbalanced dataset.}
We first compare different methods on the NEU dataset, which is a typical case of an imbalanced real-world dataset.
We report the results in \Fig{allDataset}(a).
As shown, the proposed method delivers improved performance, especially for the early iterations.
Methods here mostly converge after 5 rounds, as the dataset is relatively small.
However, judging by the gap in early iterations, it could be expected that a similar gap would exist in a larger dataset.
Interestingly, for this small dataset, VAAL performs worse than simple random selection.
This is because the method requests a lot of labels to \emph{bake in}.
\paragraph{\textit{Dominant} datasets.}
We now consider the dominant variants.
In the dominant situation, a large part of the entire training samples is dedicated to a few specific categories -- in the case of \textit{CIFAR-10}, just one.
In \Fig{allDataset}(b), we average the results from \textit{CIFAR-10}$^{+[1]}$, \textit{CIFAR-10}$^{+[5]}$, and \textit{CIFAR-10}$^{+[10]}$ that we run three times each -- a total of nine experiments.
As shown, the proposed method outperforms the compared methods by a significant margin.
This gap comes from the fact that the dataset is biased, and this causes the active learning methods also to become biased.
However, our method successfully mitigates this class imbalance and provides improved performance.
A similar phenomenon happens when there are more classes.
In \Fig{allDataset}(c), we report results for \textit{CIFAR-100}$^{+[45:55]}$.
Here, it is worth noting that all compared methods perform worse than \textit{Random}.
This is because the dataset has 100 classes, and it is easy to start ignoring a certain class.
Nonetheless, our method is the only method that provides improved label efficiency.
To mitigate this, existing works have only studied when there is a sufficient number of labels from the beginning -- for example in \textit{VAAL}~\cite{Sinha19_ac}, 10\% of the entire set.
However, this greatly limits the applicability of active learning methods, as 10\% is sometimes already too much to afford.
\paragraph{\textit{Rare} datasets.}
We further test the case when some classes are rare.
In \Fig{allDataset}(d), we show the average performance of each method on the three rare datasets -- \textit{CIFAR-10}$^{-[1]}$, \textit{CIFAR-10}$^{-[5]}$, and \textit{CIFAR-10}$^{-[10]}$.
Similar to the dominant case, our method shows the best performance among the compared methods.
Interestingly, in this case, similar to the results with \textit{CIFAR-100}$^{+[45:55]}$, existing methods perform worse than the random baseline.
Again, this is because existing methods can break down easily without enough labelled samples from the beginning.
This is a limitation our method does not have.
\paragraph{On the full dataset.}
For completeness, we further study how methods perform with the full \textit{CIFAR-10} and \textit{CIFAR-100} datasets.
However note that, as we demonstrated previously, these datasets are with perfect data balance, which is not the case for real-world datasets.
We report a summary of these results in Figures~\Fig{allDataset}(e)~and~(f).
Our method performs comparable to existing methods for \textit{CIFAR-10} and outperforms existing methods for \textit{CIFAR-100}.
However, experiment for \textit{CIFAR-100} shows a limitation of active learning methods including ours, that when there are too many classes and very few examples, their performances are close to random, and sometimes even worse.
Nonetheless, compared to existing methods, our method performs best, even in this extreme situation.
\paragraph{Additional results.}
In the supplementary document, we further provide the original plots for all experiments before averaging.
From them, we can confirm that the proposed method outperforms existing methods consistently for various situations.
\input{tbl/ablation/item.tex
\input{tbl/ablation/wFunction.tex}
\subsection{Ablation study}
\label{sec:ablation}
\subsubsection{Effectiveness of $p(\widehat{y})$ and $p(y_n|\widehat{y})$.}
To validate their effectiveness, we perform an ablation study by excluding one or both of the prior ($p(\widehat{y})$) and the label difficulty ($p(y|\widehat{y})$) -- we artificially set either to 1.
We use \textit{CIFAR-10} and \textit{CIFAR-10}$^{+[1]}$ for these experiments to remove randomness.
To avoid tuning on the test set, we use the validation splits for these experiments.
We take the average performance over all active learning rounds and also the average final performance.
We summarise the results in \Table{ablation_prior}.
We report both the average performance over all six active learning rounds (avg.) and the performance of the final round (final).
We run each experiment three times and report the average.
As shown, all terms contribute to performance improvement.
We observe that all terms contribute to providing the best performance.
Among the two terms, the prior ($p(\widehat{y})$) provides more gain compared to the label difficulty ($p(y|\widehat{y})$), demonstrating that it is critical to take data imbalance into account.
We show an example of the two terms in action in \Fig{qual_dominant}.
Our method balances the training data even when the dataset is imbalanced.
\begin{figure}[t]
\input{figs/qual_dominant/item.tex}
\end{figure}
\subsubsection{Other choices for $w_n$}
To motivate our design decision, we further build two variants of our method, where we replace $w_n$ in \Eq{condition}, either by a Sigmoid function ($Sigmoid(z_j)$) or a $\ell1$-norm ($|z_j|$).
We summarise the results in \Table{ablation_cond}.
We observe that $\ell2$-norm outperforms the other two regularisation types for all compared cases.
This is unsurprising, considering that in the case of $Sigmoid(z_j)$, it forces the latent embedding $z_j$ to have extreme values, thus conflicting too much with the Gaussian distribution assumption that VAE aims to satisfy.
This leads to the worst performance among the three types that we tried.
In case of $|z_j|$, it would not suffer from this problem but would create constant gradients that are irrelevant to the magnitude of $z_j$, thus making it hard to enforce absence.
Our choice, $\ell2$-norm, on the other hand, does not suffer from these shortcomings, becoming a natural choice for enforcing absence.
\input{figs/new_result/ablationFig}
\subsubsection{Robustness for various budgets and $\lambda$ values}
To analyse the sensitivity of the proposed framework against the user-defined parameters, we report the result of our method with various sampling budgets and $\lambda$ values in \Eq{vaeLoss}: see~\Fig{ablation_graph}(a).
Regardless of the choice of the budget size, our method outperforms all compared methods.
Smaller budget size tends to allow methods to react faster to the training outcomes and increase the effectiveness of active learning.
In \Fig{ablation_graph}(b) we report the performance of our method with varying $\lambda$.
We test with various $\lambda$, which do affect the performance of the proposed method to some degree.
However, as before in the case of the budget, our method outperforms all compared methods regardless of the choice of $\lambda$.
This further hints that the two-loss terms that we train for in \Eq{vaeLoss} do not compete.
Note that $\lambda=0$ completely disables $p(\widehat{y})$ and $p(y|\widehat{y})$, as $\widehat{y}$ estimation becomes meaningless, which causes the method to perform significantly worse as shown earlier in \Table{ablation_prior}, and we have therefore left it out in this figure to reduce cluster; see supplementary appendix for full results.
|
1,108,101,565,616 | arxiv | \section{Introduction}
Numerous astrophysical observations, such as galaxy rotational curves \cite{1980ApJ...238..471R,1982ApJ...261..439R}, velocity dispersions \citep{1976ApJ...204..668F}, and gravitational lensing \citep{Massey:2010hh} reveal the existence of invisible matter, the so-called dark matter. In combination with observational evidence of the Universe’s accelerating expansion, the standard Lambda Cold Dark Matter ($\rm{\Lambda}$CDM) cosmological model has been established. Precision analyses of the cosmic microwave background show that dark matter constitutes $26\%$ of the total energy density of the present-day universe \citep{Planck:2018vyg}.
The cold dark matter paradigm has achieved great success in describing the structure of galaxies on large scales \citep{Roszkowski:2017nbc, Marsh:2015xka,Wantz:2009it}, but it is met with puzzling discrepancies between the predictions and observations of galaxies and their clustering on small scales. For example, the N-body simulations based on the cold dark matter model show a much steeper central density profile in the dark matter halos than that inferred from the galaxy rotational curves (the ``core-cusp problem" \cite{Gentile:2004tb,deBlok:2009sp}). The predicted number of subhalos with decreasing mass grows much more steeply than what is observed around galaxies (the ``missing-satellites problem" \cite{Moore:1999nt,Klypin:1999uc}).
Because of the difficulty in solving the small-scale problems as well as the null result in searching for traditional cold dark matter candidates, e.g., weakly interactive massive particles \citep{Schumann:2019eaa}, alternative paradigms for dark matter have been proposed.
These include the warm dark matter \citep{Bode:2000gq} and fuzzy dark matter \citep{Hu:2000ke}.
The term ``fuzzy dark matter” often refers to ultralight scalar particles with a mass around $m\sim \unit[10^{-22}]{\rm eV}$. Such a dark matter scenario can get the correct relic abundance through the misalignment mechanism similar to that of axions \citep{Fox:2004kb}; that is, when the initial value of the scalar field is away from its potential minimum, the field is condensed during inflation when its mass is smaller than the Hubble scale, and then starts a coherent oscillation as a non-relativistic matter at a later epoch. Fuzzy dark matter makes the same large-scale structure predictions as $\rm{\Lambda}$CDM, but the particle’s large de Broglie wavelength, $\lambda\sim \rm{kpc}$, suppresses the structure on small scales and thus explains well the corresponding smaller-scale observational phenomena \citep{Hui:2016ltb}.
Besides the scalar particle, a naturally light vector boson predicted in string-inspired models with compactified extra dimensions \citep{Goodsell:2009xc} can also act as a good fuzzy dark matter candidate. There are several mechanisms to produce vector dark matter with the correct relic abundance, such as the misalignment mechanism \citep{Nelson:2011sf,Nakayama:2019rhg}, quantum fluctuations during inflation \citep{Graham:2015rva, Nomura:2019cvc}, and decay of a network of global cosmic strings \citep{Long:2019lwl}. Because of their different spins, if the dark matter is assumed to have interaction with Standard Model, scalar and vector fields couple with Standard Model particles in ways which lead to different observable phenomena. For example, if the vector dark matter particle is a $U(1)_{B}$ (``$B$" refers to baryon ) or $U(1)_{B-L}$ (``$L$" refers to lepton) gauge boson, the so-called ``dark photon" would interact with ordinary matter \citep{Graham:2015ifn}, then it can be detected with gravitational-wave interferometers because it exerts forces on test masses and results in displacements \citep{Pierce:2018xmy,LIGOScientificCollaborationVirgoCollaboration:2021eyz}; it can also be detected with binary pulsar systems via its effects on the secular dynamics of binary systems \citep{Blas:2016ddr,LopezNacir:2018epg}. Such a gauge effect is not applicable to scalar dark matter \citep{Antypas:2022asj}.
In addition to unknown interaction with the Standard Model, pure gravitational effects of fuzzy dark matter can also lead to observable results and help distinguish the scalar and vector dark matter. The dark matter field with ultralight mass has a wave nature with the oscillating frequency of $f_{\rm{dm}}=mc^2/h=\unit[2.4\times 10^{-9}(mc^2/10^{-23}\rm{eV})]{{Hz}}$. Such a coherently oscillating field leads to periodic oscillations in the gravitational potentials and further induces periodic signals with the frequency on the order of nanohertz \citep{Khmelnitsky:2013lxt,Nomura:2019cvc}, which falls into the sensitive range of the pulsar timing arrays (PTAs).
A PTA consists of stable millisecond pulsars for which times of arrival (ToAs) of radio pulses are monitored with high precision over a course of years to decades \citep{1978SvA....22...36S,Detweiler:1979wn,1990ApJ...361..300F}. Any unmodelled signal will induce timing residues, which represent the difference between the measured and predicted ToAs.
In contrast to the ultralight scalar dark matter, the timing residuals caused by the ultralight vector dark matter are dependent on the oscillation direction of the vector fields \citep{Khmelnitsky:2013lxt, Nomura:2019cvc}.
Several previous works have used PTA data to search for ultralight scalar dark matter \citep{Porayko:2014rfa, Porayko:2018sfa, Kato:2019bqz}.
In a recent work \cite{PPTA:2021uzb},
a search was performed for the dark photon dark matter in the PPTA second data release (DR2) based on the gauge effect.
This resulted in upper limits on the coupling strength between dark photons and ordinary matter, assuming that all dark matter is composed of ultralight dark photons.
In this work, we search for ultralight vector dark matter in the mass range of $\unit[[2\times 10^{-24}, 2\times 10^{-22}]]{\rm{eV}}$ in the PPTA DR2 data set based on the gravitational effect without assuming its interaction with Standard Model particles.
This paper is organized as follows. In Sec. \ref{sec2} we describe the observable pulsar timing effects induced by vector dark matter.
In Sec. \ref{sec3} we provide details of our data analysis.
We present the results and conclusions in Sec. \ref{sec4}. In the following sections, we set $c=\hbar=1$.
\section{Gravitational Effect from vector dark matter}\label{sec2}
In this section, we first introduce the timing residuals caused by the gravitational effect from the ultralight vector dark matter in the Galaxy; a more detailed derivation can be found in Ref. \citep{Nomura:2019cvc}.
Assuming no coupling between the ultralight particles and any other fields, we take the action for a free vector field $A_{\mu}$ with mass $m$ as,
\begin{equation}
S = \int d^4x \sqrt{-g} \(-\frac{1}{4} F^{\mu\nu}F_{\mu\nu} - \frac{1}{2}m^2 A_{\mu}A^{\mu}\),
\end{equation}
where $g$ is the determinant of the metric $g_{\mu\nu}$ and $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$.
On Galactic scales, the cosmic expansion is negligible and the background is approximately Minkowski. The energy-momentum tensor carried by the vector dark matter induces perturbations into the metric which, in the Newtonian gauge, can be written as
\begin{equation}
\begin{split}
ds^2 = &\eta_{\mu\nu}dx^{\mu}dx^{\nu}-2\Phi(t,\bx)dt^2 + 2\Psi(t,\bx)\dt_{ij}dx^{i}dx^{j} \\
&+h_{ij}(t,\bx)dx^{i}dx^{j}\, ,
\end{split}
\label{metric}
\end{equation}
where $\eta_{\mu\nu}=\rm{diag}(-1,+1,+1,+1)$ is the background Minkowski metric, $\Phi$ and $\Psi$ are gravitational potentials, and $h_{ij}$ describes the traceless spatial metric perturbations. $h_{ij}$ is absent in the scalar-field case and demonstrates the anisotropy induced by additional degrees of freedom in vector fields.
With a huge occupation number, the vector field can be described as a classical wave with a monochromatic frequency determined by its mass.
This is a good approximation because the characteristic speed of the dark matter is non-relativistic $v\sim10^{-3}$. During inflation, only the longitudinal mode of the vector fields survives \citep{Graham:2015rva}, so the equation of motion of the vector field is given by the component in the oscillating direction $\hat{k}=(\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta)$,
\begin{equation}\label{A}
A_{\hat{k}}(t,\bx) = A(\bx) \cos(mt+\al(\bx)).
\end{equation}
The vector fields contribute a time-independent energy density
\begin{equation}\label{rho}
\rho_{\rm{\tiny{VF}}}(x)=\frac{1}{2}m^2 A^2(x),
\end{equation}
and an anisotropic oscillating pressure which leads to oscillating gravitational potentials. By solving the photon geodesic equation from the pulsar to the Earth under the metric \Eq{metric}, it is found that the metric perturbations that give rise to the observable effects in PTAs are from the spatial components (see the Appendix of \citep{Nomura:2019cvc}). Furthermore, splitting the potential $\Psi$ into a dominant time-independent part and an oscillating part and solving the linear Einstein equation by neglecting the spatial gradient of the oscillating part (which is suppressed by order of $v\sim 10^{-3}$), the spatial perturbations take the following form,
\begin{eqnarray}
\!\!\!\!\Psi(t,\bx)&\!=\!&\Psi_0(\bx)+\Posc(\bx)\cos(2mt+2\al(\bx)),\\
\!\!\!\!h_{ij}(t,\bx)&\!=\!&\hosc(\bx)\!\cos(2mt\!+\!2\al(\bx))(\hat{l}\otimes \hat{ l}\!+\! \hat{n}\otimes \hat{n}\!-\! 2\hat{k}\otimes \hat{k} ),
\end{eqnarray}
where $\Psi_0(x)$ is the potential independent of time determined by the local energy density of the vector field dark matter $\rho_{\rm{\tiny{VF}}}(x)$, and $\hat{l}$ and $\hat{n}$ are the unit vectors perpendicular to the propagation direction given by $\hat{l}=(\sin\phi, -\cos\phi, 0)$ and $\hat{n}=(\cos \theta \cos \phi, \cos\theta\sin\phi, -\sin\theta)$, respectively.
The amplitudes of potentials in the oscillation part can also be related to $\rho_{\rm{\tiny{VF}}}(x)$ through the relationship between $A(x)$ with $\rho_{\rm{\tiny{VF}}}(x)$ (\Eq{rho}),
\begin{eqnarray}
\Psi_{\mathrm{osc}}(\mathbf{x}) &=& -\frac{\pi G \rho_{\rm{\tiny{VF}}}(\mathbf{x})}{3 m^{2}} \notag\\
&=&-2.2 \times 10^{-16} \frac{\rho_{\rm{\tiny{VF}}}(\mathbf{x})}{\rho_0} \left(\frac{10^{-23} \mathrm{eV}}{m}\right)^{2}, \\
h_{\mathrm{osc}}(\mathbf{x}) &=&\frac{8 \pi G \rho_{\rm{\tiny{VF}}}(\mathbf{x})}{3 m^{2}} \notag\\
&=&1.7 \times 10^{-15} \frac{\rho_{\rm{\tiny{VF}}}(\mathbf{x})}{\rho_0} \left(\frac{10^{-23} \mathrm{eV}}{m}\right)^{2}.
\label{h_rho}
\end{eqnarray}
where we use the measured local energy density $\rho_0=\unit[0.4]{ \mathrm{GeV} / \mathrm{cm}^{3}}$ \citep{Salucci:2010qr} as the normalized factor,
and the oscillation frequency in potentials is given by:
\begin{equation}
f=\frac{2 m}{2 \pi}=4.8 \times 10^{-9} \mathrm{~Hz}\left(\frac{m}{10^{-23} \mathrm{eV}}\right).
\end{equation}
The oscillating part of spatial metric $\Psi$ and $h_{ij}$ induces a mearsurable redshift in the radio pulse propagating from a pulsar to the Earth
\begin{eqnarray}
z_{\Psi}(t)=&&\Psi_{\mathrm{osc}}(\bx_{e})\cos(2mt+2\al(\bx_{e}))\notag \\
&&-\Psi_{\rm{osc}}(\bx_{p})\cos[2m(t-|\bx_{p}|)+2\al(\bx_{p})],\\
z_h(t)=&&\frac{1}{2}\hat{p}^i \hat{p}^j [h_{ij}(t,\bx_{e})-h_{ij}(t-|\bx_{p}|,\bx_{p})],
\end{eqnarray}
where $\bx_{e}$ and $\bx_{p}$ respectively represent the location of the Earth and the pulsar, and $\hat{p}$ is the unit vector pointing to the pulsar. As the distance between most pulsars and the Earth is at the order of $\mathcal{O}(\mathrm{kpc})$, which is comparable to the de Broglie wavelength of the dark matter, it is legitimate to assume that they are in a region where the vector dark matter keeps its coherent oscillation direction, and that the Earth term and the pulsar term take the same amplitudes $\Psi_{\rm osc}$ and $h_{\rm osc}$.
Integrating the redshifts separately, the results combine into the total timing residuals,
\begin{eqnarray}
R_t=&&R_{\Psi}+R_{h}\notag\\
=&&\frac{h_{\rm osc}}{\pi f}\left\{\frac{1}{2}\left[(\hat{p}\cdot \hat{l})^2+(\hat{p} \cdot \hat{n})^2-2(\hat{p} \cdot \hat{k})^2\right]-\frac{1}{8}\right\}\notag \\
&& \times \sin (\alpha_{e}-\alpha_{p}) \cos \left(2 \pi f t+\alpha_{e}+\alpha_{p}\right),
\label{R_t}
\end{eqnarray}
where we have defined the phases in Earth term and pulsar term as $\alpha_{e}=\alpha(\bx_{e})$ and $\alpha_{p}=\alpha\left(\bx_{p}\right)-m\left|\mathbf{x}_{p}\right|$, respectively.
\Eq{R_t} shows that timing residuals induced by the coherent oscillation of vector dark matter is angle dependent, which is a distinctive feature in comparison to the scalar dark matter; see Ref. \citep{Nomura:2019cvc} for a more detailed comparison.
\section{Data analysis}\label{sec3}
Now we turn to search for ultralight vector dark matter in the PPTA DR2 data set, which includes observations for 26 pulsars with a timespan up to 15 years.
By balancing sensitivity and computational costs, we choose the six best pulsars, i.e., those with relatively long observational timespan and high timing precision in the array. A summary of the basic properties of these six pulsars is given in \Table{properties}.
\begin{table}[!htbp]
\caption{Basic properties of the 6 pulsars used in our analysis. RMS -- the weighted root-mean-square band-averaged post-fit timing residuals, $N_{\rm{obs}}$ -- the number of observations, $N_{\rm{ToA}}$ -- the number of ToAs, Span -- observational data span. See Ref. \citep{Kerr:2020qdo} for details.}
\label{properties}
\begin{tabular}{c c c c c}
\hline
\hline
Pulsar Name &RMS\,[$\mu$s] &$N_{\rm{obs}}$&$N_{\rm{ToA}}$ &Span\,[$\yr$] \,\\
\hline
J0437$-$4715 &0.59&4149&29262&15.0\\
\hline
J1600$-$3053&0.58&1096&7047&14.2\\
\hline
J1713+0747&0.32&1049&7804&14.2\\
\hline
J1744$-$1134&0.46&939&6717&14.2\\
\hline
J1909$-$3744&0.24&2223&14627&14.2\\
\hline
J2241$-$5236&0.26&821&5224&8.2\\
\hline
\end{tabular}
\vspace{2ex}
\end{table}
We process the data in the same way as Refs. \cite{Porayko:2018sfa,PPTA:2021uzb}. To extract the target signal from the ToAs, one needs to provide a comprehensive analysis on the noise that might be present in timing residuals. After subtracting the expected arrival times described by the timing model, the timing residuals can be decomposed into
\begin{equation}
\delta \boldsymbol{t}= M {\boldsymbol{\epsilon}} + \delta \boldsymbol{t}_{n}+ \delta \boldsymbol{t}_{s},
\end{equation}
where $M {\boldsymbol{\epsilon}}$ accounts for the inaccuracy of the timing model with $M$ being the design matrix and $\boldsymbol{\epsilon}$ being the vector of timing model parameter offsets, $\delta \boldsymbol{t}_{n}$ contains noise contributions, and $ \delta \boldsymbol{t}_{s}$, given by \Eq{R_t}, is the signal that we are searching for.
\begin{table*}[!htbp]
\footnotesize
\caption{Parameters and their prior distributions used in the analyses. Here U and log-U represent a uniform and log uniform distribution, respectively. Here ``one parameter for PTA" means the parameter is common in the whole data set, while ``one parameter per pulsar" indicates the parameter varies from pulsar to pulsar; the same goes for the case of ``one parameter per band/system" and ``one parameter per exponential-dip event".}
\label{prior}
\begin{tabular}{c c c c}
\hline
\textbf{Parameter} & \textbf{description} & \textbf{prior} & \textbf{comments} \\
\hline
\multicolumn{4}{c}{White noise}\,\\
$E_{k}$ & EFAC per backend/receiver system & $\uni[0, 10]$ & single-pulsar analysis only \\
$Q_{k}\,[\mrm{s}]$ & EQUAD per backend/receiver system & $\logu[-8.5, -5]$ & single-pulsar analysis only \\
$J_{k}\,[\mrm{s}]$ & ECORR per backend/receiver system & $\logu[-8.5, -5]$ & single-pulsar analysis only \\
\hline
\multicolumn{4}{c}{Red noise (including SN and DM)} \\
$\Am_{\RN}$ & Red-noise power-law amplitude &$\logu[-20, -8]$ & one parameter per pulsar\, \\
$\gamma_{\RN}$ &red-noise power-law index &$\uni[0,10]$ & one parameter per pulsar\, \\
\hline
\multicolumn{4}{c}{Band/System noise}\,\\
$\Am_{\BN,\GN}$ & band/group-noise power-law amplitude &$\logu[-20, -8]$ & one parameter per band/system\, \\
$\gamma_{\BN,\GN}$ &band/group-noise power-law index &$\uni[0,10]$ &one parameter per band/system\, \\
\hline
\multicolumn{4}{c}{Deterministic event}\,\\
$\Am_{\mathrm{E}}$ & exponential-dip amplitude &$\logu[-10, -2]$ & one parameter per exponential-dip event \, \\
$ t_{\mathrm{E}}\, [\mrm{MJD}]$ &time of the event &
$\uni[54500, 54900]$ for PSR J1713 & first exponential-dip event\\
$ $ & &$\uni[57500, 57520]$ for PSR J1713 & second exponential-dip event\\
$\mrm{log_{10}} \tau_{\mrm{E}}\, [\mrm{MJD}]$ & relaxation time for the dip &$\uni[\mrm{log_{10}}5, 2]$ &one parameter per exponential-dip event \, \\
\hline
\multicolumn{4}{c}{Common noise} \\
$\Am_{\rm{CN}}$ & common-noise power-law amplitude &$\logu[-18, -11]$ & one parameter for PTA\, \\
$\gamma_{\rm{CN}}$ &common noise power-law index &$\uni[0,7]$ & one parameter for PTA\, \\
\hline
\multicolumn{4}{c}{Ultralight vector dark matter signal}\,\\
$h_{\rm{osc}}$ & oscillation amplitude &$\logu[-19, -10]$ (search) & one parameter for PTA\, \\
&&$\uni[-19,-10]$ (limit)\,\\
$\al_e$ &oscillation phase on Earth &$\uni[0,2\pi]$ & one parameter for PTA\, \\
$\al_p$ & equivalent oscillation phase on pulsar &$\uni[0,2\pi]$ & one parameter per pulsar\, \\
$\cos\theta$ & polar angle of propagation direction&$\uni[-1, 1]$ & one parameter for PTA\, \\
$\phi$ & azimuth angle of propagation direction &$\uni[0,2\pi]$ & one parameter for PTA\, \\
$f\,[\mrm{Hz}]$ & oscillation frequency &$\logu[-9,-7]$ (search) & one parameter for PTA\, \\
& &delta function (limit) & fixed\,\\
\hline
\multicolumn{4}{c}{BayesEphem}\,\\
$z_{\mrm{drift}}$ & drift-rate of Earth’s orbit about ecliptic z-axis &$\uni[-10^{-9}, 10^{-9}]$ & one parameter for PTA\, \\
$\Dt M_{\mrm{jupiter}}\,[\Msun]$ &perturbation to Jupiter’s mass & $\mcN(0, 1.55\times10^{-11})$ &one parameter for PTA \, \\
$\Dt M_{\mrm{saturn}}\,[\Msun]$ &perturbation to Saturn’s mass & $\mcN(0, 8.17\times10^{-12})$ &one parameter for PTA \, \\
$\Dt M_{\mrm{uranus}}\,[\Msun]$ &perturbation to Uranus’s mass & $\mcN(0, 5.72\times10^{-11})$ &one parameter for PTA \, \\
$\Dt M_{\mrm{neptune}}\,[\Msun]$ &perturbation to Neptune’s mass & $\mcN(0, 7.96\times10^{-11})$ &one parameter for PTA \, \\
$\mrm{PCA}_i$ &principal components of Jupiter’s orbit&$\uni[-0.05,0.05]$ &six parameters for PTA \, \\
\hline
\end{tabular}
\end{table*}
The noise in all of these pulsars from possible stochastic and deterministic processes has been analyzed by Ref.~\cite{Goncharov:2020krd} in great detail. The stochastic noise processes contain white noise and time-correlated red noise. White noise accounts for measurement uncertainties; they are modeled by three parameters EFAC, EQUAD, ECORR \citep{NANOGrav:2015qfw}, with EFAC being the scale factor of ToA uncertainty, EQUAD being an extra component independent of uncertainty and ECORR being the excess variance for sub-banded observations. The red noise includes the spin noise (SN; \citep{Shannon:2010bv}) from rotational irregularities of the pulsar itself, the dispersion measure variations \citep{Keith:2012ht} due to the change in column density of ionized plasma in the interstellar medium, and the band noise (BN) and system (``group") noise (GN) that are only present in a specific band or system \citep{Lentati:2016ygu}. Red noise is modeled by a power-law spectrum with the amplitude parameter $\Am_{\RN}$ and spectral index $\gamma_{\RN}$. In our analysis, we set the number of Fourier frequencies $N_{\rm{mode}}=30$ following Ref.~\citep{Arzoumanian:2020vkk} in the calculation of the covariance matrix. For deterministic noise contributions, a typical example is the exponential dip that might be attributed to the sudden change in dispersion in the interstellar medium \citep{Lentati:2016ygu,Keith:2012ht} or change in pulse profile shape \citep{shannon1643}, and it can be described by an exponential function.
Meanwhile, some systematic errors should be taken into consideration. We use the \texttt{BayesEphem} module \citep{NANOGrav:2020tig} to account for potential uncertainties in the solar system ephemeris (SSE); we adopt JPL DE438 \citep{DE438} to project ToAs from the local observatory to the solar system barycenter. Moreover, the North American Nanohertz Observatory for Gravitational Waves (NANOGrav), PPTA, the European Pulsar Timing Array (EPTA), and the International Pulsar Timing Array (IPTA) collaborations all report evidence for an uncorrelated common process (UCP) which can be modeled by a power-law spectrum in their lastest data sets \citep{Arzoumanian:2020vkk, Goncharov:2021oub,Antoniadis:2022pcn,Chen:2021rqp}. Although no definite evidence was found for a Hellings-Downs correlation which is deemed to be necessary for the detection of stochastic gravitational-wave background, the presence of UCP is taken as a promising sign of the gravitational-wave background \citep{Arzoumanian:2020vkk}. Positive Bayesian evidence supporting a scalar-transverse correlation in the process was found in some publications \citep{Chen:2021ncc,Chen:2021wdo}. However, simulations based on the PPTA DR2 showed that even when no signal is present, the UCP pops out when pulsars have similar intrinsic timing noise \citep{Goncharov:2021oub}. Despite the continuing efforts \citep{Xue:2021gyq,Wu:2021kmd,Chen:2022azo,Bian:2022tju}, the nature of the UCP remains to be determined, and we hence treat it as a common noise in the analysis.
As \Eq{R_t} indicates, the vector dark matter signal that we are searching for is described by six parameters: the oscillation amplitude $h_{osc}$, the oscillation frequency $f$, the oscillation (propagation) direction described by the polar and azimuth angles ($\theta, \phi$) and the equivalent phase term in pulsar $\al_{p}$ and in the Earth $\al_{e}$. In the analyses, we first perform the parameter estimations by including the white noise, red noise, band/system noise, and deterministic noise following \citep{Goncharov:2020krd} for each single pulsar. Then we collect all the chosen pulsars as a whole PTA and allow noise parameters to vary simultaneously with the signal parameters. As the white noise parameters should have little or no correlation with the dark matter parameters, we fix the white noise parameters to their maximum-likelihood values from the single pulsar noise analyses. Fixing white noise parameters should have negligible impact on our results \citep{NANOGRAV:2018hou} but significantly reduce the computational cost. All the parameters and their priors are listed in \Table{prior}.
Similar to the procedure of searching for a gravitational wave background \citep{Arzoumanian:2020vkk, Goncharov:2021oub}, we perform Bayesian inference to extract information from the data.
First, we need to determine whether the data $\Dm$ supports the existence of the signal by calculating the Bayes factor between the noise-plus-signal hypothesis $\Hm_1$ and the noise-only hypothesis $\Hm_0$,
\begin{equation}
\rm{BF}=\frac{\rm{Pr}(\Dm|\Hm_1)}{\rm{Pr}(\Dm|\Hm_0)}\, .
\end{equation}
Here $\rm{Pr}(\Dm|\Hm)$ denotes the evidence given by the integral of the product of the likelihood $\mathcal{L}$ and the prior probability $\pi$ over the prior volume,
\begin{equation}
\rm{Pr}(\Dm|\Hm)=\int \mathcal{L}(\Dm|{\Theta})\pi({\Theta}) d^n{\Theta},
\end{equation}
where $n$ is the dimension of the parameters $\Theta$.
If we do not find significant evidence for the target signal, we place constrains on certain parameters. In the work, we derive the $95\%$ upper limit for the oscillation amplitude $\bar{h}_{\rm osc}$ from its marginal posterior probability distribution,
\begin{equation}
0.95=\frac{1}{\rm{Pr}(\Dm|\Hm)}\int_0^{\bar{h}_{\rm{osc}}} dh_{\rm{osc}}\int \mathcal{L}(\Dm|\Theta)\pi(\Theta)d^{n-1}\Theta' ,
\end{equation}
where $\Theta'$ denotes all the other parameters except $h_{\rm{osc}}$.
We use the \texttt{enterprise} \citep{enterprise} and \texttt{enterprise\_extension} \citep{enterprise_extensioins} packages to evaluate the likelihood and compute the Bayes factor using the product-space method \citep{10.2307/2346151,Hee:2015eba,Taylor:2020zpk}. For the Markov-chain Monte-Carlo sampling needed for parameter estimation, we employ the \texttt{PTMCMCSampler} package \citep{justin_ellis_2017_1037579}.
\section{results and discussion}\label{sec4}
We first determine whether there is a vector dark matter signal in the data by comparing the signal hypothesis $\mathcal{H}_1$ and the null hypothesis $\mathcal{H}_0$. The log Bayes factor, $\ln \rm{BF}$, is about 13.0 for the parameter range $\log_{10}(f/\mathrm{Hz}) \in [-7.2,-7.0]$, seemingly implying strong evidence for a signal. However, when we conduct the search in two separate logarithmic frequency bands $[-9.0,-7.0]$ and $[-7.2,-7.0]$ by excluding PSR J0437$-$4715, the $\ln \rm{BF}$ is found to be 0.5 and 1.5, respectively; these are small Bayes factors that indicates no preference for the signal hypothesis. Therefore, the suspected ``signal" is completely due to PSR J0437$-$4715, and we conclude that it is not a genuine signal. A similar result has been reported in Ref.~\cite{PPTA:2021uzb}.
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.5\textwidth]{uvdm_freefreq2.pdf}\caption{ \label{freefreq}The posterior distributions for the parameters of the vector dark matter signal, containing the oscillation amplitude $h_{\rm{osc}}$, oscillation frequency $\log_{10}(f/\mathrm{Hz})\in [-9.0,-7.2]$, and the angles of propagation direction $\theta$ and $\phi$. These posteriors have been marginalized over oscillation phase parameters $\alpha_{e}$ and $\alpha_p$.}
\end{figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.48\textwidth]{hosc_6p.pdf}\caption{\label{hosc_f}Upper limits on the signal amplitude $h_{\rm{osc}}$ (orange line), generated by the vector dark matter in the Galatic as a function of frequency (mass). The blue dashed line shows the model amplitude $h_{\rm{osc}}$, assuming $\rho_{\rm{\tiny{VF}}}=\unit[0.4]{ \rm{GeV/cm^{3}}}$, given by \Eq{h_rho}. The peak in the orange curve is due to the loss in sensitivity at the frequency $1\rm{yr}^{-1}$ caused by fitting for the pulsar's sky location and proper motion \citep{Zhu:2014rta}.}
\end{figure}
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.5\textwidth]{rho_f_vdm_sdm.pdf}\caption{\label{rho_f} Upper limits on the vector dark matter density $\rho$ in the vicinity of the Earth (orange line). The green dot-dashed line denotes the upper limits on the scalar dark matter density from PPTA 12-year data set \cite{Porayko:2018sfa} and the blue dashed line denotes the measured dark matter density of $\rho_0=\unit[0.4]{ \rm{GeV/cm^{3}}}$.}
\end{figure}
\Fig{freefreq} shows the posterior distributions of the vector dark matter signal parameters after marginalizing over oscillation phases $\alpha_{e}$ and $\alpha_p$.
Since we find no evidence of a vector dark matter signal, these posterior distributions resemble those of priors except that we are able to exclude large oscillation amplitudes, i.e., $h_{\rm{osc}} \lesssim 10^{-13.5}$.
In \Fig{hosc_f}, we show the $95\%$ upper limits on the oscillation amplitude
$h_{\rm osc}$ as a function of frequency. We also plot the predicted amplitude according to \Eq{h_rho} by taking the vector field dark matter density $\rho_{\rm{\tiny{VF}}}$ equal to the measured local dark matter density $\rho_0=\unit[0.4]{\rm{GeV/cm^{3}}}$, assuming that all the dark matter is composed of ultralight vector fields. The predicted amplitude is always below the upper limits, implying we cannot exclude the possibility that the vector fields with masses considered in this work constitute all the dark matter. A more intuitive picture can be obtained by translating the amplitude into the vector field dark matter energy density $\rho_{\rm{\tiny{VF}}}$ through \Eq{h_rho} and placing the $95\%$ upper limit on the energy density, as shown in \Fig{rho_f}.
As a comparison, we also plot the upper limits derived from a scalar dark matter search presented in Ref.~\cite{Porayko:2018sfa}.
We note that the lighter mass provides tighter constraints in our search range. The strongest bound on the dark matter energy density is $\unit[1]{ \rm{GeV/cm^3}}$ at the lowest frequency $10^{-9}$ Hz. This is above the local energy density of $\rho_0=\unit[0.4]{ \rm{GeV/cm^3}}$, so we cannot place effective constraints on the mass of vector dark matter from current PTA data sets.
The results from this work are consistent with the scalar dark matter search performed with an earlier version of the PPTA data published in 2018.
Specifically, our constraints on vector dark matter density $\rho_{\rm{\tiny{VF}}}<\unit[5]{\rm{GeV/cm^{3}}}$ for $m\lesssim \unit[10^{-23}]{\rm{eV}}$ is only slightly more stringent than the scalar dark matter density limit of $\unit[6]{\rm{GeV cm^{-3}}}$ given by \citep{Porayko:2018sfa}.
This is unsurprising because only when the vector field oscillates along a particular direction that is parallel to the line of sight to the pulsar, can we expect the timing residuals induced by the gravitational effect to be three times that caused by the scalar dark matter. However, we do not find a preferred oscillation direction of vector dark matter (see \Fig{freefreq}) in the data set.
Several cosmological and astrophysical probes can place constraints on the mass of ultralight dark matter when considering the model has only gravitational coupling. Planck cosmic microwave background data implies a bound on the ultralight dark matter energy density fraction for the $10^{-33} - \unit[10^{-24}]{\rm{eV}}$ mass range \cite{Hlozek:2017zzf}. The Lyman-$\alpha$ forest used to trace the underlying dark matter distribution at high redshifts excludes the possibility that the ultralight particles with mass lighter than $\unit[2 \times 10^{-20}]{ \rm{eV}}$ make up all the dark matter \citep{Rogers:2020ltq}. While the above constraints come from axions and their applicability to ultralight vector dark matter needs to be discussed, the black hole superradiance estimates from supermassive black hole spin measurements constrain the vector dark matter within the mass range $6\times 10^{-20}$--$\unit[2 \times 10^{-17}]{ \rm{eV}}$ \citep{Baryakhtar:2017ngi}. Although there is no consensus on the constraints on the mass of ultralight dark matter because most of the experiments are subject to their own uncertainties, it is important to note that PTA experiments can constrain the energy density of ultralight dark matter independently and determine whether dark matter is dominated by the ultralight vector particles and thus provide complementary tools to other experiments.
Looking into the future, PTAs based on the Five-hundred-meter Aperture Spherical Telescope \citep{Nan:2011um}, MeerKAT \cite{meerkat}, and Square Kilometer Array (SKA) \citep{Lazio_2013} with wider frequency bands and large collecting areas, will increase the sensitivity significantly. In the SKA era, a conservative estimate is that 10 years of observations for the 10 best pulsars with an observing cadence of once every 14 days have the potential to constrain the contribution of ultralight dark matter down to $10\%$ of the local dark matter density for $m<\unit[10^{-23}]{\rm{eV}}$ \citep{Porayko:2018sfa}.
In the shorter terms sensitivity improvments could be achieved through searches with the IPTA.
Combining efforts from PTAs and other experiments will greatly advance our understanding of the nature of dark matter by studying a wide range of dark matter models.
\begin{acknowledgments}
We thank the referee for very useful comments.
We acknowledge the use of HPC Cluster of ITP-CAS and HPC Cluster of Tianhe II in National Supercomputing Center in Guangzhou. QGH is supported by the National Key Research and Development Program of China Grant No.2020YFC2201502, grants from NSFC (grant No. 11975019, 11991052, 12047503), Key Research Program of Frontier Sciences, CAS, Grant NO. ZDBS-LY-7009, CAS Project for Young Scientists in Basic Research YSBR-006, the Key Research Program of the Chinese Academy of Sciences (Grant NO. XDPB15).
ZCC is supported by the fellowship of China Postdoctoral Science Foundation No. 2022M710429.
RMS acknowledges support through Australian Research Council Future Fellowship FT190100155.
This work has been carried out by the Parkes Pulsar Timing Array, which is part of the International Pulsar Timing Array. The Parkes radio telescope (``Murriyang'') is part of the Australia Telescope, which is funded by the Commonwealth Government for operation as a National Facility managed by CSIRO.
\end{acknowledgments}
|
1,108,101,565,617 | arxiv | \section*{Introduction}
Semiconductor avalanche photodiodes (APD's) are versatile for weak light detection, with applications from remote ranging\cite{Wehr1999,Schreiber1999}, quantum communication \cite{yuan2018} and fluorescence lifetime imaging\cite{Damalakiene2016} to optical time-domain reflectometry \cite{Healey1984,Eraerds2010}.
For practical fiber quantum key distribution (QKD), InGaAs/InP APD's are the detector of choice because they are compact and low cost, and allow cryogenic-free or even room-temperature operation \cite{yuan2018}.
However, they suffer from spurious afterpulsing arising from carrier trapping by defects in the multiplication layer, especially at high detection efficiencies \cite{Comandar2015,Tada2020}.
To minimise afterpulsing, an APD can be biased on for a sub-nanosecond duration only when a photon arrival is expected.
In doing so, charge per avalanche can be reduced to the order of 10~fC \cite{yuan2010,Restelli2013,Namekata2006}, corresponding to a transient current of less than 0.1~mA.
Such weak avalanches have to be discriminated through use of a readout circuit that removes the strong capacitive response to the applied gates. Gated InGaAs detectors are capable of counting photons at up to 60\% efficiencies \cite{Fang2020} and 1~GHz rate \cite{patel2012} and with photon number resolution \cite{kardynal2008}. Thanks to this success, gating approach has been applied to traditionally free-running Si devices for performance enhancement \cite{Thomas2010,wayne2021}.
Existing readout circuits include band stop \cite{Namekata2006,Namekata09,Tada2020} or low-pass\cite{walenta2012,He2017,Fang2020} filtering under sine-wave gating\cite{Namekata2006}, self-differencing \cite{Yuan2007,Comandar2015}, and transient reference cancellation \cite{Restelli2013,Liang2019}. While simple for implementation, frequency filtering distorts the avalanche signals due to its rejection of a sizeable portion of frequency components, thus increasing time jitter and temporal errors in photon registrations \cite{walenta2012}. Self-differencing \cite{Yuan2007} and reference cancellation methods \cite{Restelli2013} are able to maintain avalanche signal fidelity but may suffer operational complexities.
The former requires wideband performance for the entire circuitry and thus inconvenience an adjustable delayline \cite{yuan2010} for frequency alignment, while the latter can be unstable because the transient reference is derived separately from the capacitive response.
Here we propose and experimentally demonstrate a simple, low-distortion ultra-narrowband interference circuit (UNIC)
that can suppress the capacitive response for a 1.25~GHz gated InGaAs/InP APD single photon detector.
The circuit is an asymmetric radio-frequency (RF) interferometer. One of its arms contains a narrow band pass filter (BPF) based on surface acoustic wave resonator (SAW) to retrieve the fundamental wave of the gating signal.
The filtered wave then interferes destructively with the same frequency component transmitted via the other arm through a coupling module, thereby eliminating the capacitive response.
This interference occurs over the narrow band, so it can provide a broad
and continuous pass band in frequency domain to maintain the avalanche signal with little distortion.
This allows to achieve ultra-low afterpulsing probabilities and excellent jitter performance at high detection efficiencies from two InGaAs APD's that exhibit capacitive responses of very different amplitudes.
\section*{Detector characterisation setup}
\begin{figure*}[htb]
\centering
\includegraphics[width=.6\linewidth]{Figure1.pdf}
\caption{
\textbf{(a)} Single-photon characterisation setup for 1.25~GHz sinusoidally gated InGaAs/InP APDs using UNICs for avalanche impulse readout;
\textbf{(b)} Histogram of the photon detection events measured by the characterisation setup \textbf{(a)} on an InGaAs APD detector that was regulated at a temperature of 30~$^\circ$C. The photon detection peak exhibits a 30~dB width of 650~ps.
AMP: amplifier; APD: avalanche photodiode; BSF: band stop filter; DISC: discriminator;
SG: signal generator; TDC: time-to-digital converter; UNIC: ultra-narrowband interference circuit; VOA: variable optical attenuator.
}
\label{fig:001}
\newpage
\end{figure*}
Figure~\ref{fig:001}(a) shows our single photon characterisation setup for InGaAs APDs. A 1550~nm passively mode-locked laser serves as the light source and provides stable short pulses of 0.5~ps duration at a repetition rate of 10~MHz.
The laser output power is monitored by an optical power meter (EXFO FTB-1750) and its pulse intensity is set by a variable optical attenuator (VOA, EXFO FTB-3500) to 0.1 photon/pulse at the fiber input of APD under test. It provides a 10~MHz reference to a signal generator (SG) for synthesising a 1.25~GHz sinusoidal wave with up to 27 V voltage swing.
In combination of a suitable DC bias, this AC signal periodically gates the APD above its breakdown voltage ($60 - 70$~V) to achieve single photon sensitivity.
The APD output is processed by the readout module consisting of two identical 1.25~GHz UNIC's, one 2.5~GHz band stop filter (BSF), and three RF amplifiers (AMPs). Amplification of the raw APD signalis is useful as it prevents weak avalanche signals from falling below thermal noise by attenuation of the first UNIC.
The readout signal is discriminated by a discriminator for avalanches before feeding to a time-digital-converter (TDC) with a dead time of
2~ns for time-resolved photon counting.
Figure~\ref{fig:001}\textbf{(b)} is a typical histogram obtained with this setup.
APD under test is temperature-regulated using their integrated thermal-electric cooler, which is driven by a temperature controller (Thorlabs TED200C).
A source-measure unit (Keithley 2635B) provides the DC bias and simultaneously monitors the current flowing through the APD. In characterising the maximum count rate, we replace the 10~MHz laser with a continuous-wave distributed feedback laser (DFB) laser, the output of which is carved into 1.25~GHz, 50~ps pulse train using an intensity modulator.
We use a high speed digital oscilloscope to record the detector output and extract the count rate through digital discrimination in software. The oscilloscope method is carefully calibrated at low count rate regimes to be consistent with the hardware discriminated result using the photon counter (Stanford Research SR400).
The setup is able to measure dark count probability, afterpulsing probability, detection efficiency, maximum count rate, avalanche charge and time jitter. With no performance screening, two fiber-pigtailed APDs from different manufacturers were used in this study, named APD\#1 and APD\#2 respectively.
\section*{Ultra-narrowband interference circuit (UNIC)}
\begin{figure}[ht]
\centering
\includegraphics[width=.8\linewidth]{Figure2.pdf}
\caption{\textbf{(a)} Schematic for ultranarrow interference circuit (UNIC); \textbf{(b)} Transmission spectrum of a heroic UNIC PCB; Inset: Magnified view for region of 1.24 -- 1.26~GHz. \textbf{(c)} Raw capacitive responses from APD\#1 (top) and APD\#2 (bottom) under identical 27.0~V V$_{p-p}$ gating; \textbf{(d)} Recovered avalanche impulses. ATT: attenuator; SAW BPF: surface acoustic wave band pass filter.
}
\label{fig:002}
\end{figure}
With sub-nanosecond gating, a photon induced avalanche is an impulse and has a wide spectrum. On the other hand, the capacitive response is periodic and has its most energy concentrated at the gating frequency or its higher harmonics. This spectral difference allows frequency-dependent signal processing to remove the capacitive response and keep the wide-band impulses intact.
Figure~\ref{fig:002}\textbf{(a)} shows a circuit diagram of UNIC. It is an RF interferometer containing two couplers of 9:1 power splitting ratio, a $\pi$-resistive attenuator (ATT) and surface acoustic wave band pass filter.
Two of the ports are terminated by 50~$\Omega$ resistors.
The SAW BPF features a central frequency of 1.25~GHz, 20-dB passband of 35~MHz, insertion loss of 3~dB, and group delay of 34~ns. It filters out the fundamental wave of the gating frequency, which then interferes with the APD signal transmitted through the other arm.
The attenuation and differential delay are set to enable destructive interference for the 1.25~GHz frequency component at the UNIC output port. The UNIC differential delay ($\Delta t$) meets the condition below
\begin{equation}
\Delta t = T_g^{SAW} + \delta t
= (N+1/2)/f_g,
\end{equation}
\noindent where $T_g^{SAW}$ is the group delay of the SAW BPF, $\delta t$ the delay caused by the track length difference between two arms, $f_g = 1.25$~GHz the APD gating frequency, and $N$ is an integer number. For a compact circuit, we choose $\delta t$ to be less than the half-wave of the gating signal. With the SAW device used, $N = 42$ and $\delta t = 155$~ps. The resulting UNIC unit has a small footprint of $38 \times 15$~mm$^2$ on printed circuit boards (PCBs).
The large $T_g^{SAW}$ brings two additional benefits. Firstly, it substantially increases the PCB manufacturing tolerance, as a 0.5~mm deviation in the RF track length will just alter the circuit central frequency by less than $10^{-4}$. This eliminates the requirement of an adjustable delayline which is required in a self-differencing circuit for precise frequency alignment. Secondly, it helps to produce an ultra-narrow band rejection at its designed frequency. Figure~\ref{fig:002}(b) shows the measured transmission spectrum (S21 parameter) of our heroic UNIC PCB, and its inset expands the frequency section of 1.24 -- 1.26~GHz to show the narrowness of the insertion loss dip in the close proximity of the resonance frequency of 1.25~GHz. The dip of the heroic (typical) PCB features a loss of -95~dB (-80~dB), representing a suppression of 80~dB (65~dB) as compared with the background loss for other frequencies under 2~GHz.
The dip has a 30~dB linewidth of merely 30~kHz, thus ensuring crucial suppression of the APD gating signal without overly distorting the avalanche signals.
The background loss of about 14~dB is caused mainly by the 9:1 couplers and can be reduced in future with more balanced splitters.
Cascading two UNIC's enables a stable 100~dB suppression of the primary gating frequency and thus provides sufficient performance redundancy. Their attenuation to the avalanche signal is compensated by using RF amplifiers (Fig.~\ref{fig:001}\textbf{(a)}). Second order harmonics (2.5~GHz) is suppressed by a band stop filter of conventional LC design.
Figure~\ref{fig:002}\textbf{(c)} shows raw outputs from two different APD's under identical sinusoidal gating. Their respective capacitive responses are measured to be 0.42~V and 1.75~V. Despite their 4 times differences, UNIC's can successfully reject the sinusoidal responses and retrieve avalanches with excellent signal-to-background ratio, as shown in Fig.~\ref{fig:002}\textbf{(d)}. For APD\#2, we just adjusted the gain of the first AMP to avoid saturation and distortion.
\section*{Results and discussion}
Time-resolved photon counting allows precise extraction of the net photon detection efficiency ($\eta_{net}$) and the afterpulsing probability ($P_A$), which is defined as the ratio of the total afterpulses per photon counting event.
Figure~\ref{fig:001}\textbf{(b)} shows a histogram of avalanche events measured for APD\#1 under 10~MHz pulsed excitation of 0.1~photon/pulse.
The illuminated peak has a full-width of 1/1000 maximum (30~dB width) of just 650~ps, which is shorter than the gating period of 800~ps and thus allows low-error clock number assignment that is essential for high speed QKD. The count at non-illuminated gates arise from detector dark count and afterpulses and is more than 3 orders of magnitude lower than that of the illuminated gate.
We extract quantities of $P_I$ and $P_{NI}$, \textit{i.e.}, the respective counting probabilities for each illuminated and non-illuminated gate. With a separate measurement of the detector dark count probability ($P_D$), we calculate the afterpulsing probability using the standard method \cite{Yuan2007,Namekata09},
\begin{equation}
P_A = \frac{(P_{NI} - P_{D}) \cdot R}{P_I - P_{NI}},
\end{equation}
\noindent where $R = 125$ here is the ratio of the gating frequency (1.25~GHz) to the laser illumination (10~MHz).
Excluding the dark and afterpulse count probabilities, the net single photon detection efficiency is given by~\cite{Comandar2015}
\begin{equation}
\eta_{net} = \frac{1}{\mu}\mathrm{ln}\frac{1-P_{NI}}{1-P_{I}},
\end{equation}
\noindent where $\mu$ is the average incident photon number per illumination pulse.
\begin{figure}[t]
\centering
\includegraphics[width=.8\linewidth]{Figure3.pdf}
\caption{Dark count probability (top) and afterpulse probability (bottom) as a function of photon detection efficiency of \textbf{(a)} APD\#1 and \textbf{(b)} APD\#2 measured for several different temperatures.
}
\label{fig:003}
\newpage
\end{figure}
Figure~\ref{fig:003} shows the characterisation results for APD\#1 and APD\#2.
We fixed the amplitude of the 1.25~GHz sinusoidal signal at 27.0~V, and measured the relevant parameters as a function of the applied direct current (DC) bias, but for clarity the results are plotted as a function of the net detection efficiency ($\eta_{net}$). Each device was measured at several different temperatures, while APD\#2 could reach only a narrower temperature range due to its cooler compatibility with the temperature control driver.
Qualitatively, two devices behave similarly.
Both dark count and afterpulsing probabilities increase with photon detection efficiency, and exhibit opposite dependencies on temperature. For both APDs at $\eta_\mathrm{net} = 30~\%$, the afterpulsing probabilities are less than 2.3~\% at their lowest measurement temperatures with corresponding dark count probabilities of $1.25 \times 10^{-6}$ and $1.6 \times 10^{-6}$ for APD\#1 (-30~$^\circ$C) and APD\#2 (-20~$^\circ$C), respectively.
Moreover, our UNIC-APDs can offer record low afterpulsing probabilities, as summarised for APD\#1 in Figure~\ref{fig:004}. At -30~$^\circ$C, APD\#1 is able to achieve 5~\% and 21.2~\% detection efficiencies at 0.5~\% and 1.0~\% afterpulsing probabilities. At these afterpulsing probabilities, the maximum detection efficiency increases with temperature and reaches 25.3~\% and 34.2~\% at 30~$^\circ$C. At 5.9~\% $P_A$, APD\#2 has a detection efficiency of 50~\% efficiency at 30~$^\circ$C and dark count probability of $1.1 \times 10^{-4}$.
\begin{figure}[t]
\centering
\includegraphics[width=.8\linewidth]{Figure4.pdf}
\caption{Temperature dependencies of photon detection efficiency for APD\#1 at the given afterpulsing probabilities of 0.5 \% (blue) and 1 \% (red).}
\label{fig:004}
\newpage
\end{figure}
\begin{figure}[b]
\centering
\includegraphics[width=.8\linewidth]{Figure5.pdf}
\caption{Maximum count rate (blue) and photoncurrent (red) \textit{vs } incident flux for APD\#1.
}
\label{fig:005}
\newpage
\end{figure}
The maximum count rate is a crucial parameter for a number of applications, for example, high bit rate QKD \cite{yuan2018} and rapid phase tracking in twin-field QKD \cite{lucamarini18,zhou22}.
To determine their maximum count rates, we used a DFB laser transmitting at 1.25~GHz as the illumination source and measure the count rate as a function of photon flux.
Figure~\ref{fig:005} shows an exemplar result obtained from APD\#1 at a temperature of
30~$^\circ$C with its detection efficiency set to 25.3~\% in the low flux regime. The detector maintains a linear dependence with incident flux for count rates exceeding 100~MHz, while a maximum count rate of 700~MHz is obtained at the few photons/pulse regime.
We attribute the high count rate to the UNIC's ability of removing the capacitive response and thus allowing discrimination of faint avalanches. From the accompanying current measurement, we extract an avalanche charge of 38~fC, comparable to the best value of 35~fC \cite{yuan2010} obtained with the photocurrent measurement method. The ability to detect such weak avalanches ensures low afterpulsing probabilities in our UNIC-APDs.
APD\#2 was measured to have a similar avalanche charge as that of APD\#1. When setting its efficiency to 50~\%, APD\#2's avalanche charge rose to 65~fC due to stronger bias applied. Nevertheless, it was still able to achieve a maximum count rate of 600~MHz.
\begin{table*}[h]
\caption{\label{tab:Comparison} Performance comparison of sub-nanosecond gated InGaAs detectors using different types of readout circuits.}
\setlength{\tabcolsep}{0.4mm}{
\centering
\begin{tabular}{ccccccc}
\hline\hline
& $ P_{\mathrm{A}}$(\%)& $\eta_\mathrm{net}$ (\%)&$P _{\mathrm{D}}$ ($ \mathrm{gate}^{\mathrm{-1}}$)&T ($^{\circ}$C) & $f_g$ (GHz)& Readout Method
\\
\hline \hline
This work & 1.0 &21.2 & 5.4$\times10^{-7} $ & -30 &1.25 & UNIC \\
\hline
He \textit{et al}\cite{He2017} &1.0 &20.7 & 7.6$\times10^{-7}$ &-30 &1.00 &low-pass filter +\\ & & & & & & variable width discriminator \\
\hline
Tada \textit{et al}\cite{Tada2020} &1.8 &27.7 & 8$\times10^{-7}$ &-35 &1.27
& band stop filter \\
\hline
Fang \textit{et al}\cite{Fang2020} &2.5 &20 & 1.1$\times10^{-6}$ &-30 &1.25 &low-pass filter \\
\hline
Comandar \textit{et al}\cite{Comandar2015} &2.9 &20 & 1.0$\times10^{-6}$ &-30 &1.00 &self-differencing \\
\hline
Liang \textit{et al}\cite{Liang2019} &4.5 &20 & 3.2$\times10^{-6}$ &-30 &1.25 &reference subtraction \\
\hline\hline
\end{tabular}}
\end{table*}
Table~\ref{tab:Comparison} compares our results with those gigahertz-gated detectors equipped with different readout circuits. For impartiality, we list just data measured at a fixed temperature of -30~$^\circ$C whenever possible. Here, our UNIC-APD achieved an impressive 1\% afterpulsing probability at $\eta_\mathrm{net} =21.2$~\%, considerably outperforming most other methods among filtering\cite{Fang2020,Tada2020}, self-differencing \cite{Comandar2015} and reference subtraction \cite{Liang2019}.
In terms of detection efficiency, our result improves marginally over the previous best\cite{He2017}, but which was achieved with help of an uncommon variable width discriminator to mitigate signal distortion by excessive filtering. We attribute the outstanding performance of our detectors to low-distortion signal processing by UNIC's.
It is useful to compare our UNIC-APDs with detectors deployed in QKD systems. In the QKD system optimised for secure key rates (SKRs) \cite{yuan2018}, the room-temperature self-differencing detectors featured $f_g = 1$~GHz, $\eta_\mathrm{net} = 31$ \%, $P_A = 4.4$\% and $P_D = 2.25 \times 10^{-4}$ and a SKR of 13.72~Mb/s over a 2~dB channel was obtained. Our UNIC-APD could outperform in all these parameters. At 30~$^\circ$C and with $P_A =4.4$~\%, APD\#2 offers a higher efficiency of 49~\% efficiency and twice lower dark count probability of $9.4 \times 10^{-5}$, see Fig.~\ref{fig:003}\textbf{b}. Combined with its high count capability, UNIC detectors are expected to allow a SKR exceeding 25~Mb/s over the same channel loss. This provides an interesting technological path towards 100~Mb/s via wavelength multiplexing.
\section*{Conclusion}
To summarise, we have developed a novel approach of using UNICs for reading out avalanche signals from 1.25~GHz sinusoidally gated InGaAs APDs.
UNIC-APDs were characterised to exhibit excellent performance across the temperature range of $-30$ -- 30~$^\circ$C, and can offer \textgreater20~\% detection efficiency at an ultra low afterpulsing probability of 1~\%. This performance, together with the circuit's compactness and manufacturing tolerance, will allow UNIC-APDs a considerable potential in QKD applications.
\begin{backmatter}
\bmsection{Disclosures}
The authors declare that there are no conflicts of interest related to this article.
\bmsection{Data availability}
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
\end{backmatter}
|
1,108,101,565,618 | arxiv | \section{Introduction}
Perturbation theory for gauge theories at finite temperature suffers from
infrared singularities and gauge dependent results for physical quantities.
These problems are avoided by using an effective perturbation theory
(Braaten-Pisarski method \cite{ref1}) which is based on the resummation
of hard thermal loop (HTL) diagrams into effective Green functions. This
powerful method was derived within the imaginary time formalism (ITF).
Using resummed Green functions, medium effects of the heat bath, such as
Debye screening, collective plasma modes, and Landau damping, are taken
into account. The HTL resummation technique has been applied to a number of
interesting problems, in particular to the prediction of signatures and
properties of a quark-gluon plasma (QGP) expected to be produced in
relativistic heavy ion collisions (for a review see \cite{ref2}).
However, the use of thermal field theories for describing a QGP in
nucleus-nucleus collisions is restricted by the fact that at least the
early stage of such a collision leads to a fireball, which is not in
equilibrium. It is not clear if a complete thermal and chemical equilibrium will be
achieved later on. Hence, non-equilibrium effects in a
parton gas should be considered for predicting signatures of
QGP formation and for obtaining a consistent picture of the fireball.
This can be done in the case of a chemically non-equilibrated parton gas by
means of rate equations \cite{ref3} or more generally by using
transport models \cite{ref4}. However, these approaches are based
on a semiclassical approximation. In particular, infrared divergences have
to be removed phenomenologically. Therefore it is desirable to derive
a Green function approach including medium effects as in the case of
the HTL resummation. For this purpose one has to abandon the ITF, which
is restricted to equilibrium situations. The real time formalism (RTF), on
the other hand, can be extended to investigate non-equilibrium systems
\cite{ref5,ref6}.
The RTF involves choosing a contour in the complex energy plane which fulfills
the Kubo-Martin-Schwinger boundary condition and contains the real axis
\cite{ref5}. This leads to propagators and self energies which are given
by $2\times 2$ matrices. The choice of the contour is not unique.
We will
adopt the Keldysh or closed time path contour, which was invented
for the non-equilibrium case \cite{ref5}. In particular, we will demonstrate
the usefulness of the Keldysh representation \cite{ref7} based on
advanced and retarded propagators and self energies and show how potentially
dangerous terms (pinch singularities) \cite{ref8} in non-equilibrium are
treated easily within this representation.
In the next section we review the Keldysh representation. In section 3, we
discuss the equilibrium calculation. We consider QED and give the results of
the real time calculation, in the HTL approximation, of the photon self energy, the
resummed photon propagator, and the electron damping rate. The results are, of
course, identical to those of the ITF, which demonstrates that although the HTL
resummation scheme was derived within the ITF, the result is independent of the
choice of contour.
In section 4, we extend the HTL resummation technique to
off-equilibrium situations by following the equilibrium calculations outlined in section 3.
We show that no pinch singularities
appear in the non-equilibrium HTL effective propagator.
\section{Keldysh representation}
In this section we review the Keldysh representation of the RTF. The bare
propagator for bosons reads \cite{ref6}
\begin{equation}
D(K)=\left (\begin{array}{cc} \frac{1}{K^2-m^2+i\epsilon} & 0\\
0 & \frac{-1}{K^2-m^2-i\epsilon}\\
\end{array} \right )
-2\pi i\, \delta (K^2-m^2)\> \left (\begin{array}{cc}
n_B(k_0) & \theta (-k_0)+n_B(k_0)\\
\theta (k_0)+n_B(k_0) & n_B(k_0) \\ \end{array} \right ),
\label{e1}
\end{equation}
where $K=(k_0,{\bf k})$, $k=|{\bf k}|$, $\theta $ denotes the step
function, and the distribution function is given by $n_B(k_0)=
1/[\exp(|k_0|/T)-1]$ in the equilibrium case.
For fermions the bare propagator can be written as
\begin{eqnarray}
S(K)=(K \!\!\!\!/ \, +m)\> && \left [ \left (\begin{array}{cc}
\frac{1}{K^2-m^2+i\epsilon} & 0\\
0 & \frac{-1}{K^2-m^2-i\epsilon}\\
\end{array} \right )\right .\nonumber \\
&& +2\pi i\, \delta (K^2-m^2)\> \left .\left (\begin{array}{cc}
n_F(k_0) & -\theta (-k_0)+n_F(k_0)\\
-\theta (k_0)+n_F(k_0) & n_F(k_0) \\ \end{array} \right )\right ],
\label{e2}
\end{eqnarray}
where the Fermi distribution is given by $n_F(k_0)=1/[\exp(|k_0|)+1]$
in equilibrium.
The components of these propagators are not independent, but fulfill
the relation
\begin{equation}
G_{11}-G_{12}-G_{21}+G_{22}=0,
\label{e3}
\end{equation}
where $G$ stands for $D$ or $S$.
By an orthogonal transformation of these $2\times 2$ matrices we arrive
at a representation of the propagators in terms of advanced and retarded
propagators which was first introduced by Keldysh \cite{ref7}. The three independent
components of this representation are defined as \cite{ref6}
\begin{eqnarray}
G_R & = & G_{11}-G_{12},\nonumber \\
G_A & = & G_{11}-G_{21},\nonumber \\
G_F & = & G_{11}+G_{22}.
\label{e4}
\end{eqnarray}
The inverted relations read
\begin{eqnarray}
G_{11} & = & \frac{1}{2}\, (G_F+G_A+G_R),\nonumber \\
G_{12} & = & \frac{1}{2}\, (G_F+G_A-G_R),\nonumber \\
G_{21} & = & \frac{1}{2}\, (G_F-G_A+G_R),\nonumber \\
G_{22} & = & \frac{1}{2}\, (G_F-G_A-G_R).
\label{e5}
\end{eqnarray}
Similar relations to (\ref{e3}) and (\ref{e4}) hold for the self
energies \cite{ref10}:
\begin{equation}
\Pi_{11}+\Pi_{12}+\Pi_{21}+\Pi_{22}=0
\label{e9}
\end{equation}
and
\begin{eqnarray}
\Pi_R & = & \Pi_{11}+\Pi_{12},\nonumber \\
\Pi_A & = & \Pi_{11}+\Pi_{21},\nonumber \\
\Pi_F & = & \Pi_{11}+\Pi_{22},
\label{e10}
\end{eqnarray}
where $\Pi $ stands for the self energy of a boson or fermion.
Using (\ref{e1}) and (\ref{e2}) in (\ref{e4}) the bare propagators of
the Keldysh representation are given by
\begin{eqnarray}
D_R(K) & = & \frac{1}{K^2-m^2+i\, \mbox{sgn}(k_0) \epsilon},\nonumber \\
D_A(K) & = & \frac{1}{K^2-m^2-i\, \mbox{sgn}(k_0) \epsilon},\nonumber \\
D_F(K) & = & -2\pi i\, [1+2n_B(k_0)]\, \delta (K^2-m^2)
\label{e6}
\end{eqnarray}
for bosons and
\begin{eqnarray}
S_R(K) & = & \frac{K\!\!\!\!/ \, +m}{K^2-m^2+i\, \mbox{sgn}(k_0) \epsilon},\nonumber \\
S_A(K) & = & \frac{K\!\!\!\!/ \, +m}{K^2-m^2-i\, \mbox{sgn}(k_0) \epsilon},\nonumber \\
S_F(K) & = & -2\pi i\, (K\!\!\!\!/ \, +m)\, [1-2n_F(k_0)]\, \delta (K^2-m^2)
\label{e7}
\end{eqnarray}
for fermions.
The bare propagators $D_F$ and $S_F$ can be written also as
\begin{eqnarray}
D_F(K) & = & [1+2n_B(k_0)]\, \mbox{sgn}(k_0)\, [D_R(K)-D_A(K)],\nonumber \\
S_F(K) & = & [1-2n_F(k_0)]\, \mbox{sgn}(k_0)\, [S_R(K)-S_A(K)].
\label{e8}
\end{eqnarray}
In the non-equilibrium case, all of these equations are valid, with the equilibrium
distribution functions ($n_B,\,\,n_F$) replaced by non-equilibrium distribution
functions ($f_B,\,\,f_F$) which depend on the four momentum and the
space-time coordinate \cite{ref6}.
Now we consider the situation for full (resummed) propagators. In equilibrium,
(\ref{e8}) is valid for full propagators as a consequence of the dissipation-fluctuation
theorem \cite{ref10}. The polarization tensor satisfies,
\begin{equation}
\Pi_F(K)=[1+2n_B(k_0)]\, \mbox{sgn}(k_0)\, [\Pi _R(K)-\Pi _A(K)]
\label{e10a}
\end{equation}
for bosons, and for fermions we have to replace $n_B$ by $-n_F$.
Out of equilibrium however, the situation is more complicated.
Equations (\ref{e8}) and (\ref{e10a}) are not satisfied by resummed
propagators out of equilibrium. Additional terms occur which appear
to give rise to pinch singularities. In section 4 we will discuss
these terms in detail.
\section{Equilibrium}
In this section we consider the hot QED plasma in equilibrium. We discuss
the HTL resummation technique in the context of the Keldysh representation
of the RTF, as a starting point for our study of non-equilibrium situations.
\subsection{HTL photon self energy}
The first step of the Braaten-Pisarski method is to extract the HTL
diagrams which have to be resummed into effective Green functions. A
typical example is the HTL photon self energy.
It is given by the diagram of fig.1, where the
momenta of the internal electron lines are of the order of the temperature
or larger.
Applying standard Feynman rules one finds
\begin{equation}
\Pi^{\mu\nu}(P)=-ie^2\int \frac {d^4K}{(2\pi )^4} tr \left [\gamma ^\mu
S(Q)\gamma^ \nu S(K)\right ],
\label{e11}
\end{equation}
where $S$ denotes the electron propagator and $Q=K-P$.
The retarded self energy is defined in
(\ref{e10}),
\begin{eqnarray}
&& \Pi _R^{\mu \nu }(P)=\Pi _{11}^{\mu \nu }(P)+\Pi _{12}^{\mu \nu }(P)
\nonumber \\
&& =-ie^2\int \frac {d^4K}{(2\pi )^4}\left \{ tr\left [\gamma ^\mu S_{11}(Q)
\gamma ^\nu S_{11}(K)\right ]-tr\left [\gamma ^\mu S_{21}(Q)\gamma ^\nu
S_{12}(K)\right ]\right \},
\label{e12}
\end{eqnarray}
where the minus sign in front of the second term comes from the vertex
of the type 2 fields \cite{ref5}. In the following we will neglect the
electron mass assuming $m\ll T$ and write the electron propagator as
$S_{ij}(K)\equiv K\!\!\!\!/ \, \tilde \Delta _{ij}(K)$. For now we
will restrict
ourselves to the longitudinal component of the self energy $\Pi ^L\equiv
\Pi ^{00}$. Performing the trace over the
$\gamma $-matrices
and using (\ref{e5}) gives,
\begin{eqnarray}
\Pi _R^L(P)=-2ie^2\int \frac{d^4K}{(2\pi )^4} (q_0k_0+{\bf q}\cdot {\bf k})
&& \biggl [\tilde \Delta _F(Q)\tilde \Delta _R(K)+\tilde \Delta _A(Q)
\tilde \Delta _F(K)\nonumber \\
&& +\tilde \Delta _A(Q)\tilde \Delta _A(K)+\tilde
\Delta _R(Q)\tilde \Delta _R(K)\biggr ].
\label{e14}
\end{eqnarray}
Terms proportional to $(\tilde \Delta _F(Q))^2$ that contain products of
$\delta $-functions, which might cause
pinch singularities \cite{ref5}, do not appear. This cancellation is well
established in equilibrium calculations. A great advantage of the Keldysh
representation is that
the cancellation is immediately evident, before any momentum integrals are done.
To proceed further we do the integral using bare electron propagators and taking the HTL
approximation. This approximation is based on the assumption that we can
distinguish between soft momenta of the order $eT$ and hard ones of the order $T$,
which is possible in the weak coupling limit $e\ll 1$. We assume that the
external momentum $P$ is soft (because it is only for soft momenta that the HTL
self energies have to be resummed), and that the internal momentum
$K$ is hard.\footnote{In the ITF, i.e. in euclidean space, this assumption
corresponds to $|p_0|, p\ll k$ \cite{ref2}. In the RTF (Minkowski space) however,
the requirement $|P|\ll k$ is sufficient since the exact one-loop self energies
coincide with the HTL ones on the light cone $P^2=0$ \cite{ref11b}.} The resulting
integral can be done
analytically and gives the final result:
\begin{equation}
\Pi_R^L(P)=-3 m_\gamma^2 \left (1-\frac{p_0}{2 p}\ln \frac{p_0+p+i\epsilon}
{p_0-p+i\epsilon} \right),
\label{e17}
\end{equation}
where $m_\gamma=eT/3$ is the effective photon mass.
This result agrees with the result in the ITF \cite{ref1,ref2}
(found earlier by Weldon and Klimov using the high temperature
approximation \cite{ref11c}, which is equivalent to the HTL limit \cite{ref2}).
Analogously one obtains for the advanced photon self energy
\begin{eqnarray}
\Pi_A^L(P) & = & \Pi _{11}^L(P)+\Pi _{21}^L(P)\nonumber \\
& = & -3 m_\gamma^2 \left (1-\frac{p_0}{2 p}\ln \frac{p_0+p-i\epsilon}
{p_0-p-i\epsilon} \right).
\label{e18}
\end{eqnarray}
The transverse part of the HTL photon self energy, $\Pi _T(P)=(\delta _{ij}-
p_ip_j/p^2)\Pi _{ij}(P)/2$, is computed in a similar way yielding
\begin{equation}
\Pi_{R,A}^{T}(P)=\frac{3}{2}\, m_\gamma^2\, \frac{p_0^2}{p^2}
\left[ 1- \left( 1-\frac{p^2}{p_0^2} \right) \frac{p_0}{2 p}\ln
\frac{p_0+p\pm i\epsilon}{p_0-p\pm i\epsilon} \right].
\label{e20}
\end{equation}
Next we calculate $\Pi _F^L=-\Pi _{12}^L-\Pi _{21}^L$ (see (\ref{e9})
and (\ref{e10})) within the HTL approximation. As we will show in section 4,
this quantity is necessary to obtain the resummed propagator out
of equilibrium. Using (\ref{e5}) we obtain
\begin{eqnarray}
\Pi _F^L(P)=-2ie^2\int \frac{d^4k}{(2 \pi )^4}\, (q_0k_0+{\bf q}\cdot
{\bf k})\> && \{\tilde \Delta _F(Q)\tilde \Delta _F(K)-[\tilde \Delta _R(Q)
-\tilde \Delta _A(Q)]\nonumber \\
&& [\tilde \Delta _R(K)-\tilde \Delta _A(K)]\}.
\label{e38a}
\end{eqnarray}
Extracting $\tilde{\Delta}_F$ from (\ref{e7}), using $\tilde \Delta _R
(Q)-\tilde \Delta _A(Q)=-2\pi i\, \mbox{sgn}(q_0)\, \delta (Q^2)$, and taking the
HTL approximation we obtain,
\begin{equation}
\Pi _F^L(P)=-\frac{4ie^2}{\pi p}\theta (p^2-p_0^2)\, \int _0^\infty
dk\, k^2\, n_F(k)\, [1-n_F(k)] =-6\pi i\, m_\gamma ^2 \frac{T}{p} \theta (p^2-p_0^2).
\label{e40}
\end{equation}
The transverse part is given analogously by
\begin{equation}
\Pi _F^T(P)=-3\pi i\, m_\gamma ^2 \frac{T}{p}\left (1-\frac {p_0^2}{p^2}\right )
\theta (p^2-p_0^2).
\label{e40a}
\end{equation}
Note that the HTL expression for $\Pi _F$ is of higher order in the
coupling constant than $\Pi _{R,A}$ for soft momenta $p\sim eT$.
This observation also follows directly from
(\ref{e10a}) for soft $k_0$.
It is easy to show that these HTL results
satisfy (\ref{e10a}) for soft $p_0$.
\subsection{Resummed photon propagator}
The second step of the Braaten-Pisarski method is the construction of the
effective Green functions to be used in the effective perturbation theory.
The resummed photon propagator, for instance, describing the propagation of
a collective plasma mode,
is given by the Dyson-Schwinger equation of fig.2, where we adopt
the HTL result for the photon self energy. The equation
reads in Coulomb gauge ($D^{00}\equiv D^L$)
\begin{equation}
{D^*}^L=D^L+D^L\Pi ^L{D^*}^L,
\label{e21}
\end{equation}
where the propagators and self energy are $2\times 2$ matrices and
$*$ indicates a resummed propagator and not a complex conjugation.
Throughout this paper we use the Coulomb gauge, which is convenient for
later applications \cite{ref2}. Since the final results for physical
quantities are gauge independent using the HTL resummation method, we may
choose any gauge.
Using
the identities (\ref{e3}) for the bare and resummed propagators,
(\ref{e9}) for the self energies, and the definitions (\ref{e4}) for the
advanced and retarded propagators $D_{A,R}$ and $D^*_{A,R}$ it is easy
to show that
\begin{equation}
{D^*}_{R,A}^L=D_{R,A}^L+D_{R,A}^L\Pi _{R,A}^L{D^*}_{R,A}^L.
\label{e22}
\end{equation}
$\,$From this expression we find for the effective longitudinal retarded and
advanced photon propagators
\begin{equation}
{D^*}_{R,A}^L(P)=\left [p^2+3m_\gamma ^2\left (1-\frac {p_0}{2p}\ln
\frac{p_0+p\pm i\epsilon}{p_0-p\pm i\epsilon}\right )\right ]^{-1}.
\label{e23}
\end{equation}
$\,$From (\ref{e8}) we obtain,
\begin{equation}
{D^*}_F^L(P)=[1+2n_B(p_0)]\, \mbox{sgn}(p_0)\, \left [{D^*}_R^L(P)-{D^*}_A^L(P)
\right ].
\label{e24}
\end{equation}
Introducing
the spectral function \cite{ref11a}
\begin{equation}
\rho _L(P)\equiv -\frac{1}{\pi } Im {D^*}_R^L(P)
\label{e25}
\end{equation}
the propagator (\ref{e24}) can be written as
\begin{equation}
{D^*}_F^L(P)=-2\pi i\, [1+2n_B(p_0)]\, \mbox{sgn}(p_0)\, \rho _L(P).
\label{e26}
\end{equation}
Compared to the bare propagator we simply have to replace the bare
spectral function $\mbox{sgn}(p_0)\delta (P^2)$ in (\ref{e6}) by the spectral
function for the effective propagator.
For the effective transverse photon propagator in Coulomb gauge we obtain
analogously
\begin{equation}
{D^*}_{R,A}^T(P)=\left \{ p_0^2-p^2-\frac{3}{2}m_\gamma ^2\frac{p_0^2}{p^2}
\left [1-\left (1-\frac {p^2}{p_0^2}\right ) \ln \frac{p_0+p\pm i\epsilon}
{p_0-p\pm i\epsilon}\right ]\right \} ^{-1}
\label{e27}
\end{equation}
and
\begin{equation}
{D^*}_F^T(P)=-2\pi i\, [1+2n_B(p_0)]\, \mbox{sgn}(p_0)\, \rho _T(P)
\label{e28}
\end{equation}
with the transverse spectral function $\rho _T\equiv -\frac{1}{\pi} Im\, {D^*}_R^T $.
\subsection{Interaction rate of a hard electron}
The last step of the Braaten-Pisarski method is the use of the effective Green
functions for calculating observables of hot gauge theories in the weak
coupling limit $e\ll 1$. Famous and often discussed examples are damping or
interaction rates of
particles in hot relativistic plasmas (for references
see \cite{ref2}). In this section we discuss the interaction rate
of a hard electron ($p \sim T \gg eT$) in a QED plasma with zero chemical
potential.
The interaction rate of a massless fermion is defined by
\begin{equation}
\Gamma _{eq}(p)=-\frac{1}{2p}\, [1-n_F(p)]\> tr\, [P\!\!\!\!/ \, \, Im \,
\Sigma _R(p_0=p,{\bf p})].
\label{e29}
\end{equation}
The electron self energy $\Sigma $ is shown
in fig.3. The imaginary part of the diagram corresponds to the elastic scattering
of the hard electron off thermal
electrons in the QED plasma via the exchange of a collective plasma mode.
Since $p\gg eT$ we do not need effective
vertices. Also, the diagram containing an effective electron propagator
and a bare photon propagator, corresponding to Compton scattering,
can be neglected (it leads to a higher order contribution since the electron
propagator is less singular than the photon propagator). The integral
over the photon momentum $Q$ is
dominated by small photon momenta (the Rutherford singularity). The leading
order contribution to the interaction rate is obtained by
integrating over the entire momentum range of the exchanged photon using
a resummed propagator. The result is of order $e^2T$ which is greater by a
factor of $1/e^2$ than the result one would expect from the natural two loop
scale. This anomalously large rate occurs because of the presence of the
thermal photon mass in the denominator of the effective photon propagator, and
the fact that the integral is dominated by small photon momenta.
Using the Keldysh formalism and taking the hard thermal loop limit we find,
\begin{equation}
\Gamma _{eq}(p)=\frac{e^2T}{2\pi }\, [1-n_F(p)] \int _0^\infty dq\, q
\int _{-q}^q \frac{dq_0}{q_0}\, \left [\rho _L(Q)+\left (1-\frac{q_0^2}{q^2}
\right )\rho _T(Q)\right ],
\label{e33}
\end{equation}
in agreement with the result found in the ITF \cite{ref12}.
Using the static approximation $q_0\ll q$ for the spectral functions
which is accurate to about 10\% \cite{ref2} we end up with
\begin{equation}
\Gamma _{eq}(p)\simeq \frac{e^2T}{2\pi }\, [1-n_F(p)]\, \ln \frac{const}{e},
\label{e34}
\end{equation}
where the $const$ under the logarithm, which comes from a singularity in
the transverse photon propagator, cannot be determined within the
Braaten-Pisarski resummation scheme \cite{ref14}.
Assuming an infrared cutoff of the order $e^2T$, which could be provided by the
interaction rate itself \cite{ref14}, our result (\ref{e34}) is correct to order
$e^2\ln e$. In order to determine the order $e^2$ correction one has to go beyond
the HTL resummation scheme, which lies out of the scope of the present
investigation.
\section{Non-Equilibrium}
So far there are only a few investigations using HTL resummed Green functions
out of equilibrium. Baier et al. \cite{ref17} have studied the photon production
rate in chemical non-equilibrium and Le Bellac and Mabilat
have investigated off-equilibrium reaction rates of heavy fermions in the appendix of
Ref.\cite{ref9}.
In this section we want to consider a non-equilibrium situation within the
Keldysh representation by following the steps outlined in section 3. At this point we distinguish between two separate aspects of the non-equilibrium problem. The study of how a system that is initially out of equilibrium will relax towards equilibrium is beyond the scope of this work. We restrict ourselves to the study of microscopic processes which take place in an out of equilibrium background, under the implicit assumption that the time scale of this microscopic process is much smaller than the time scale of the relaxation of the background towards equilibrium. This assumption is consistent with the HTL expansion. HTL propagators and vertices, with quasistationary distribution functions, describe the physics of modes with momenta of the order $e$ times the hard momentum scale or larger. The damping rates which determine the relaxation time of the system are of order $e^2$ times the hard momentum scale. Equilibration is therefore slow, at least close to equilibrium.
In a relativistic heavy ion collision for example, we expect a fast thermalization \cite{ref4} which could not be described by our method, and a much slower chemical equilibration \cite{ref3} where our approach should be valid \cite{ref17}.
Out of equilibrium, difficulties arise because of the fact that (\ref{e8})
and (\ref{e10a}) do not hold for resummed propagators. In equilibrium these
relations lead to the a priori cancellation of the pinch singularities
associated with the product of an advanced and retarded propagator carrying
the same momentum. Out of equilibrium, where these relations do not hold,
the situation is more
involved and the cancellation of pinch singularities is not automatic. We will
discuss this problem in the remainder of
this section.
The derivation of the retarded and advanced HTL photon self energies are
completely analogous
to the equilibrium case, because the bare electron propagator has the same
structure as in equilibrium.
Note that the HTL approximation $|p_0|,p\ll k$ does not require the assumption
of the existence of a temperature.
We obtain the same results for the advanced and retarded
HTL self energies, (\ref{e17}),
(\ref{e18}) and (\ref{e20}), with the equilibrium thermal
photon mass
\begin{equation}
m_\gamma^2 = \frac{4e^2}{3\pi^2}\int _0^\infty dk\, k\, n_F(k)
=\frac{e^2T^2}{9}
\end{equation}
replaced by the expression
\begin{equation}
\tilde m_\gamma ^2=\frac{4e^2}{3\pi ^2}\int _0^\infty dk\, k\, f_F(k).
\label{e35}
\end{equation}
We note that there are no pinch singularities in the advanced and retarded
HTL self energies.
Since the Dyson-Schwinger equation (\ref{e21}) for the advanced and retarded
propagators is identical in equilibrium and non-equilibrium, the resummed
advanced and retarded propagators are given again by (\ref{e23})
and (\ref{e27}), using $\tilde m_\gamma$ for the thermal photon
mass.
We obtain the resummed symmetric propagator
${D^*}_F^L={D^*}_{11}^L+{D^*}_{22}^L$ from the Dyson-Schwinger equation
\begin{equation}
{D^*}_{11}^L+{D^*}_{22}^L=D_{11}^L+\sum _{i,j=1}^2 D_{1i}^L\Pi _{ij}^L
{D^*}_{j1}^L+D_{22}^L+\sum _{i,j=1}^2 D_{2i}^L\Pi _{ij}^L{D^*}_{j2}^L.
\label{e36}
\end{equation}
Using (\ref{e5}) for the bare and full propagators and (\ref{e9}) and
(\ref{e10}) for the self energies we have,
\begin{equation}
{D^*}_{F}^L=D_{F}^L+D_{R}^L\Pi _R^L{D^*}_{F}^L+D_F^L\Pi _{A}^L
{D^*}_{A}^L+D_{R}^L\Pi _{F}^L{D^*}_{A}^L.
\label{e37}
\end{equation}
It is easy to show that this equation is solved by the
following propagator:
\begin{eqnarray}
{D^*}_{F}^L(P)= && [1+2f_B(p_0)]\, \mbox{sgn}(p_0)\, [{D^*}_{R}^L(P)-{D^*}_{A}^L(P)]
\nonumber \\
&& +\{\Pi _F^L(P)-[1+2f_B(p_0)]\, \mbox{sgn}(p_0)\, [\Pi _R^L(P)-\Pi _A^L(P)]\}
\, {D^*}_{R}^L(P)\, {D^*}_A^L(P).
\label{e38}
\end{eqnarray}
In equilibrium the second term, which might lead to pinch singularities
(because it contains the product of an advanced and a retarded propagator
\cite{ref5}), vanishes due to (\ref{e10a}). Equation (\ref{e24}) is
recovered.
Out of equilibrium (\ref{e10a}) does not hold, and the second term in (\ref{e38}) does not
automatically give zero. We now consider this situation. A product of bare propagators in
this expression would contain the product of delta functions which is called a pinch
singularity:
\begin{equation}
D^L_R(P) D^L_A(P) = \frac{1}{P^2 + i {\rm sgn}(p_0)\epsilon} \,\, \frac{1}{P^2 - i {\rm sgn}
(p_0)\epsilon} \rightarrow [\delta(P^2)]^2.
\label{pinch}
\end{equation}
Consider, however, what happens when we use resummed propagators in (\ref{e38}). In this
case we have,
\begin{eqnarray}
{D^*}_R^L(P)-{D^*}_A^L(P)
&\equiv&-2\pi i \tilde \rho _L(P),\nonumber \\
{D^*}_R^L(P){D^*}_A^L(P) &=& -\pi \frac{\tilde \rho _L(P)}{Im\, \Pi _R^L(P)},
\label{xx}
\end{eqnarray}
where the
non-equilibrium spectral function $\tilde \rho _L$ defined in
(\ref{xx}) differs from the
equilibrium one (\ref{e25}) only by the thermal mass (\ref{e35}).
To calculate $\Pi _F^L$ we note that (\ref{e17})
and (\ref{e40}) hold also out of equilibrium if we use $\tilde m_\gamma ^2$
in (\ref{e17}) and replace $n_F$ by $f_F$ in (\ref{e40}). Then we can write $\Pi _F^L$ as
\begin{equation}
\Pi _F^L(P)=2iA\, \frac{Im\, \Pi _R^L(P)}{p_0},
\label{e41}
\end{equation}
where the constant $A$ is given by
\begin{equation}
A=\frac{\int _0^\infty dk\, k^2\, f_F(k)\, [1-f_F(k)]}{\int _0^\infty
dk\, k\, f_F(k)}.
\label{e42}
\end{equation}
Inserting (\ref{xx}) and (\ref{e41}) into (\ref{e38}) and using
\begin{equation}
\Pi _R^L(P)-\Pi _A^L(P) =2i Im\, \Pi _R^L(P)
\label{e42a}
\end{equation}
we obtain
\begin{equation}
{D^*}_F^L(P)=-2\pi i\, \frac{A}{p_0}\, \tilde \rho _L(P).
\label{e43}
\end{equation}
In spite of (\ref{xx}) this result holds also for a vanishing imaginary
part of the self energy, because $Im \Pi_R^L$ drops out of the second term
of the propagator (\ref{e38}) according to (\ref{xx}), (\ref{e41}), and
(\ref{e42a}).
In equilibrium $A$ reduces to $2T$. Consequently (\ref{e43})
agrees with (\ref{e26}) in the equilibrium case in the soft $p_0$ (HTL) limit. For the transverse
propagator we simply have to replace $\tilde \rho _L$ by $\tilde \rho _T$
in (\ref{e43}).
The conclusion is the following. When (\ref{e38}) is rewritten in the form (\ref{e43})
it is clear that the apparent pinch singularity in the QED HTL effective photon propagator
does not in fact occur. Physically we have found that this singularity is regulated by the
use of the resummed propagators. This result agrees with that obtained by Altherr
\cite{ref19} who found that finite results could be obtained in a scalar field theory by
resumming pinch terms. Since the scalar self energy has no imaginary part at one loop,
a finite width was inserted by hand to provide the regularization. The same result was
also found by Baier et al. \cite{ref17} for the fermion propagator in a chemically
non-equilibrated QCD plasma, in the HTL limit.
We now discuss higher order calculations. To begin we consider the `Altherr type' diagram
in the case of the electron self energy
shown in fig.4. When this diagram is calculated with bare lines, a pinch singularity appears
to occur because of the product of the two propagators with the same momentum dependence.
However, resumming diagrams with all possible numbers of self energy insertions yields a
finite result since this sum of diagrams is equivalent to calculating the one loop diagram
with the HTL effective propagator (in the case of soft external momenta), which we have shown
does not contain a pinch singularity. To next order we consider the same Altherr type diagram in
fig.4 where internal lines are HTL effective propagators, and the self energy insertion
($\bar{\Pi}$) are the one loop diagrams with HTL effective propagators on the internal lines
and HTL effective vertices shown in fig.5. At first glance it appears that above the light cone, where the
imaginary part of the HTL self energy is zero, this diagram will have a pinch singularity
that arises in the same way as for the diagram with bare propagators. Consider, however,
what happens when we resum diagrams with all possible numbers of self energy insertions
$\bar{\Pi}$. This procedure is equivalent to calculating the one loop diagram with an
effective propagator $D^{**}$ that is given by the Dyson-Schwinger equation,
\begin{eqnarray}
{D^{**}}_{F}^L(Q)= && \frac{A}{q_0}\, [{D^{**}}_{R}^L(Q)-{D^{**}}_{A}^L(Q)]
\nonumber \\
&& +\{\bar{\Pi} _F^L(Q)-\frac{A}{q_0}\, [\bar{\Pi} _R^L(Q)-\bar{\Pi} _A^L(Q)]
\} \, {D^{**}}_{R}^L(Q)\, {D^{**}}_A^L(Q).
\label{e38b}
\end{eqnarray}
The product of propagators $D_R^{**}D_A^{**}$ can be rewritten as proportional to a spectral
function divided by the imaginary part of $\bar{\Pi}_R$ in exactly the same way as before
(see (\ref{xx})). Thus, the regulation of the singularity will occur as before, if we can
write the symmetric self energy $\bar{\Pi}_F$ as proportional to the imaginary part of the
retarded self energy, as in equation (\ref{e41}). So far, this result has only been proven
for the HTL self-energy. If it is true in general, then the mechanism outlined above for the
HTL effective propagator will work at all orders, and all physical quantities will be free
of pinch singularities, as expected.
It should be noted that the self energy $\bar{\Pi}$ contains an imaginary part (damping)
also above the light cone. Hence the effective propagator $D^{**}$ has a finite width
and will regulate all pinch singularities according to Altherr \cite{ref19}.
Quantities that are logarithmically infrared divergent using bare propagators such as
the photon production rate in a QGP can be calculated consistently to leading order
by a decomposition into a soft and a hard part \cite{ref18}. For this purpose
a separation scale $eT\ll q^*\ll T$ for the momentum $Q$ of the exchanged particle
is introduced. The hard part then follows from a two-loop self energy containing only
bare propagators analogously to fig.4. However, due to the kinematical restriction
$-Q^2>{q^*}^2$
no pinch singularity $[\delta (Q^2)]^2$ (see (\ref{pinch})) occurs \cite{ref17}.
Hence there are no pinch singularities using the HTL resummation technique to leading order.
At higher orders a resummation beyond the HTL scheme leading to (\ref{e38b}) might
be necessary.
Lastly, we investigate the non-equilibrium electron damping rate. The equilibrium result
(\ref{e33}) is modified to become,
\begin{equation}
\Gamma _{neq}(p)=\frac{e^2}{4\pi }\, [1-f_F(p)] \int _0^\infty dq\, q
\int _{-q}^q dq_0\, \frac{A}{q_0}\, \left [\tilde \rho _L(Q)+
\left (1-\frac{q_0^2}{q^2} \right )\tilde \rho _T(Q)\right ],
\label{e44}
\end{equation}
leading to the final result
\begin{equation}
\Gamma _{neq}(p)\simeq \frac{e^2A}{4\pi }\, [1-f_F(p)]\, \ln \frac{const}{e}.
\label{e45}
\end{equation}
The deviation of the spectral function from the equilibrium one does not
matter here because the thermal photon mass drops out after integrating
over $q$ \cite{ref2}. Comparison with the equilibrium case (\ref{e34}) gives
\begin{equation}
\Gamma _{neq}(p)=\frac {A}{2T}\, \frac{1-f_F(p)}{1-n_F(p)}\, \Gamma _{eq}.
\label{e45a}
\end{equation}
Finally, we discuss the specific case of a chemically non-equilibrated
QED plasma. Numerical transport simulations of the QGP
in relativistic heavy ion collisions show that there is rapid
thermalization in a partonic fireball. However, chemical equilibration
takes much longer, if it is achieved at all during the lifetime of the QGP
\cite{ref3,ref4}. In order to describe this deviation from chemical
equilibrium, phase space suppression factors $\lambda _{B,F}$ depending on time
-- sometimes also called
fugacities -- are introduced \cite{ref3}. Assuming that the photons,
electrons and positrons in a QED plasma are not in chemical equilibrium,
the distributions are given by
\begin{eqnarray}
f_B(p_0) & = & \lambda _B\, n_B(p_0),\nonumber \\
f_F(p_0) & = & \lambda _F\, n_F(p_0),
\label{e46}
\end{eqnarray}
where $0<\lambda _{B,F}<1$ indicates undersaturation
and $\lambda _{B,F}>1$ oversaturation of the corresponding photons and fermions
compared to an equilibrated QED plasma. Using the distributions
(\ref{e46}) we find for the constant $A$ in (\ref{e42}) after numerical
integration $A=2T+0.192T(1-\lambda _F)$. Substitution in (\ref{e45a}) gives
\begin{equation}
\Gamma _{neq}(p)=\frac{1-\lambda _F\, n_F(p)}{1-n_F(p)}\, [1+0.096
(1-\lambda _F)]\Gamma _{eq}(p).
\label{e47}
\end{equation}
The non-equilibrium rate is independent of the photon fugacity $\lambda _B$ and
depends only weakly on $\lambda _F$. This observation is the result of a cancellation
of two efects: in an
undersaturated (oversaturated) plasma the number of scattering partners is reduced
(enhanced) and, at the same time, the Debye mass $m_D^3=3m_\gamma ^3$ is reduced
(enhanced) leading to less (more) screening. To a large extent, these two effects
cancel each other and lead to a rate that is approximately independent of
the fugacities. As a matter of fact, the non-equilibrium rate increases in an
undersaturated plasma a little bit, because there is less Pauli blocking in this case.
In equilibrium the cancellation between the number of scattering partners (flavors)
and the Debye screening is exact \cite{ref20}.
\section{Conclusions}
In the present paper we have studied explicitly the HTL resummation
technique in equilibrium and non-equilibrium within the RTF using
the Keldysh representation. We have considered the HTL photon self energy, the resummed
photon propagator, and the interaction rate of a hard electron in a
QED plasma. We have pointed out the convenience of the Keldysh representation,
where only the symmetric propagators $G_F$ depend on the distribution functions and
where possible pinch terms cancel automatically in equilibrium.
We have shown that the HTL resummation technique can be extended to
non-equilibrium situations assuming quasistationary distributions.
This assumption does not allow us to study the equilibration of the
system; it restricts us to the study of microscopic processes taking place in an out of equilibrium background under the assumption that the time scale of this microscopic process is much smaller than the time scale of the relaxation of the background towards equilibrium. This assumption is consistent with the HTL expansion. HTL propagators and vertices describe the physics of
modes with momenta of the order of $e$ times the hard momentum scale or larger. The damping rates which
determine the relaxation time of the system are of order $e^2$ times the hard momentum scale.
Equilibration is therefore slow, at least close to equilibrium, and quasistationary
distributions can be assumed. In relativistic heavy ion collisions, for example, we expect a fast
thermalization \cite{ref4}, which could not be described by our method, and a much slower chemical equilibration \cite{ref3} where our approach should be
applicable \cite{ref17}.
The retarded and advanced HTL photon self energies in
non-equilibrium are obtained from the equilibrium quantity by replacing the thermal mass
of the photon by a
non-equilibrium expression (\ref{e35}). The retarded and advanced resummed photon
propagators have the same structure as their equilibrium counterparts. However,
the resummed symmetric photon propagator
${D^*}_F^{L,T}$ (\ref{e38}) contains an additional term (pinch term)
compared to the
equilibrium expression (\ref{e24}). This singularity is regulated by the resummed
propagators in the pinch term in (\ref{e38}). One obtains an expression (\ref{e43})
for the HTL effective propagator that has the same structure as the
equilibrium result (\ref{e26}). Therefore, there are no additional pinch singularities
in HTL effective propagator in the non-equilibrium formalism, compared with the
equilibrium situation. We have discussed how to extend these results beyond leading order
by an additional resummation beyond the HTL one.
Higher n-point functions could also be calculated
efficiently using the Keldysh representation \cite{ref10,ref22}.
We expect that the absence of pinch singularities persists. Since the HTL
self energies are gauge invariant out of equilibrium (since they differ from the
equilibrium HTL's only by the definition of the thermal masses), we expect that
Ward identities will hold out of equilibrium \cite{ref23}, and thus the structure of all HTL
Green functions should be the same both in and out of equilibrium.
As an example we have discussed the interaction rate of a hard electron and showed
that the result has the same form out of equilibrium as in equilibrium. (We note
that this discussion of pinch singularities has no bearing on the infrared divergence
that occurs in the HTL calculation of this quantity). We have considered a chemical
non-equilibrium situation by
multiplying the equilibrium distribution functions by a fugacity factor.
The non-equilibrium interaction rate is approximately independent of the fugacities.
Using the formalism developed in this paper, it will be straightforward to
calculate observables in a non-equilibrium parton gas. Examples that have already been
considered in an equilibrated QGP include
parton damping and transport rates, the energy loss of partons,
transport coefficients, and production rates of partons, leptons,
and photons.
\acknowledgements
We would like to thank E. Braaten, P. Danielewicz, C. Greiner, U. Heinz, R. Kobes,
S. Leupold, and B. M\"uller for stimulating and helpful discussions.
|
1,108,101,565,619 | arxiv | \section{Introduction}
The discovery of the Higgs boson with a mass of about 125~GeV \cite{Atlas:2012gk,CMS:2012gu} has a strong impact on the parameter range of supersymmetric models. In particular in the Constrained Minimal Supersymmetric Standard Model (CMSSM) large regions of the parameter space are not consistent with this mass range and a large mass splitting in the stop sector is needed to push the tree level mass $m_h \leq M_Z$ to that level. The fine tuning needed to achieve this mass is large, requiring a cancellation between uncorrelated parameters of order 1 part in 300. In the more general context of the MSSM one still requires 1\% fine tuning even for an extremely low messenger scale of 10 TeV~\cite{Hall:2011aa}\footnote{Note that this definition of fine tuning differs from ours in the choice of the measure and the fact that the parameters are taken to be low-scale parameters.}. In addition, it has recently been pointed out that those regions in parameter space which could explain the Higgs mass by a rather light
SUSY spectrum together with maximal mixing in the stop sector have only a metastable electroweak vacuum while in the global vacuum charge and color are broken \cite{Camargo-Molina:2013sta,Blinov:2013fta,Chowdhury:2013dka}.
To accommodate a heavier Higgs while avoiding very large fine tuning and the need of radiative corrections of about 35~GeV requires new structure. The most widely studied solution is an enhancement of the Higgs mass already at tree level by new F- or D-term contributions \cite{Ellwanger:2009dp,Ellwanger:2006rm,Ma:2011ea,Zhang:2008jm,Hirsch:2011hg}. In singlet extensions the mass of the SM-like Higgs is at tree-level roughly given by $m^2_h \simeq M^2_Z \left(\cos^22\beta + \frac{\lambda^2}{g^2}\sin^22\beta\right)$ and becomes maximal for $\tan\beta \sim 2$ and for a large coupling $\lambda$ between the singlet and Higgs fields. The most common singlet extension is the Next-to-Minimal-Supersymmetric-Standard-Model (NMSSM, see e.g.~\cite{Ellwanger:2006rm} for a review) which assumes an underlying $\mathbbm{Z}_3$ symmetry. As expected the fine tuning in the NMSSM gets significantly reduced in comparison to the MSSM \cite{BasteroGil:2000bw,Dermisek:2005gg,Dermisek:2006py,Dermisek:2007yt,Ellwanger:2011mu}.
However, other singlet extensions fare even better in terms of fine tuning. Based on an operator analysis it could be expected that singlet extensions leading to certain operators are favoured \cite{Dine:2007xi, Cassel:2009ps}. One option to generate the necessary operators
is an underlying $R$ symmetry, $\Z{4}^R$ or $\Z{8}^R$. After supersymmetry breaking, both the singlet mass and the $\mu$ term are generated but both are constrained to be of order the supersymmetry breaking mass \cite{Lee:2010gv, Lee:2011dya}. The resulting model is therefore a generalised version of the NMSSM (GNMSSM)~\cite{Ross:2011xv}. Indeed it was found that the fine tuning in the GNMSSM becomes even better than in
the NMSSM \cite{Ross:2011xv,Ross:2012nr,Kaminska:2013mya}, see also \cite{Delgado:2010uj,Delgado:2012yd}. In addition, the symmetry underlying the GNMSSM has the appealing feature that it forbids dangerous dimension 5 proton decay operators and does not lead to the domain wall problem of the NMSSM \cite{Abel:1995wk}.
Other phenomenologically interesting aspects of the GNMSSM include a possible enhancement of the diphoton decay rate of the Higgs boson \cite{SchmidtHoberg:2012yy} as well as a potential simultaneous explanation of the Fermi line at 130~GeV \cite{SchmidtHoberg:2012ip}. Such signals however would require $\lambda$ to become non-perturbative well below the scale of a grand unified theory (GUT), making an interpretation in terms of an underlying GUT model difficult, see however \cite{Hardy:2012ef}.
In this article we will assume an underlying GUT structure and a fully perturbative extrapolation to the GUT scale.
An idea to reduce the fine tuning even further has recently been proposed in Ref.~\cite{Lu:2013cta},
where it has been argued that a very natural extension of the MSSM is the DiracNMSSM with two additional singlets.
The main motivation for the second singlet ${\bar{S}}$ was a possible mixed ('Dirac') mass term, $M_s S {\bar{S}}$ which allows for very heavy singlets without a suppression
of the tree-level F-term contribution to the Higgs mass while keeping the soft SUSY breaking terms small. As the soft mass squared of the NMSSM singlet
feeds into the soft Higgs masses, this was argued to eliminate this source of fine tuning.
The first study of the fine tuning in the DiracNMSSM was based on a rough fine tuning measure including only parameters at the electroweak scale which takes the impact of the RGEs only crudely into account. In addition, the estimate of the Higgs mass was subject to large theoretical uncertainties and the constraints from SUSY searches as well as dark matter abundance were not included. In this work we perform a full numerical study of the fine tuning in the DiracNMSSM using state of the art computer tools.
To this end we implemented the DiracNMSSM in {\tt SARAH}\xspace to produce a corresponding version
of {\tt SPheno}\xspace -- a state of the art spectrum calculator.
Our estimate of the fine tuning is based on a full two-loop running of the renormalisation group equations and we perform a precise mass calculation in the Higgs sector. The dark matter abundance is calculated with {\tt MicrOmegas}\xspace.
We proceed as follows: in sec.~\ref{sec:model} we introduce the DiracNMSSM and discuss the Higgs sector in some detail. In sec.~\ref{sec:FT} we give details about the fine tuning calculation and present our numerical results in sec.~\ref{sec:results}. We conclude in sec.~\ref{sec:conclusion}. In the appendix we present all renormalisation group equations, mass matrices and vertices which are changed in comparison to the MSSM and explain in great detail the renormalisation of the CP even Higgs sector in the DiracNMSSM.
\section{The DiracNMSSM }
\label{sec:model}
\subsection{The superpotential and soft-breaking terms}
In the DiracNMSSM one adds two chiral singlet superfields $S$ and $\bar{S}$ to the MSSM with superpotential
\begin{equation}
\mathcal{W} = \mathcal{W}_\text{MSSM} + \lambda S H_u H_d + M_s S \bar{S} + \xi_s S + \xi_{\bar{s}}\bar{S} \;.
\end{equation}
The general soft SUSY breaking terms associated with the Higgs and singlet sectors are
\begin{align}
V_\text{soft}
&= m_s^2 |s|^2 + m_{\bar{s}}^2 |{\bar{s}}|^2 + m_{h_u}^2 |h_u|^2+ m_{h_d}^2 |h_d|^2 \nonumber \\
&+ \left(b\mu \, h_u h_d + \lambda A_\lambda s h_u h_d + b_s s {\bar{s}} + t_s s + t_{\bar{s}} {\bar{s}} + h.c.\right) \;.
\label{soft}
\end{align}
Since the renormalisation group equations (RGEs) for the DiracNMSSM have not been
given in the literature before we list the $\beta$-functions for all superpotential and soft-breaking parameters
as well as all gauge couplings and vacuum expectation values (VEVs) up to two loop in Appendix~\ref{app:RGEs}.
\subsection{Particle content after EWSB}
After electroweak symmetry breaking (EWSB) the complex scalars in the Higgs sector acquire VEVs and they
are decomposed in their neutral components as
\begin{align}
\label{eq:EWSB1}
h_d^0 = & \, \frac{1}{\sqrt{2}} \left(v_d + \phi_{d} + i \sigma_{d}\right) \, , \hspace{1cm}
h_u^0 = \, \frac{1}{\sqrt{2}} \left(v_u + \phi_{u} + i \sigma_{u}\right) \, , \\
\label{eq:EWSB2}
s = & \, \frac{1}{\sqrt{2}} \left(v_s + \phi_s + i \sigma_s\right) \, , \hspace{1cm}
\bar{s} = \, \frac{1}{\sqrt{2}} \left( v_{\bar{s}} + \phi_{\bar{s}} + i \sigma_{\bar{s}}\right) \;.
\end{align}
As usual, the ratio of the two Higgs VEVs is given by $\frac{v_u}{v_d} = \tan\beta$ and $v=\sqrt{v_d^2+v_u^2} \simeq 246$~GeV.
The charged Higgs sector is very similar to the MSSM and contains one physical charged Higgs with mass
\begin{equation}
M_{H^+} = \frac{1}{4} v^2 (g_2^2 - 2 \lambda^2) + \frac{1+\tan^2\beta}{\tan\beta} B_\text{eff} \;.
\end{equation}
Here, we defined
\begin{equation}
B_\text{eff} = b\mu + \frac{\lambda}{\sqrt{2}}(M_s v_{\bar{s}} + v_s A_\lambda) \, .
\end{equation}
In the neutral Higgs sector there are four CP even states and three CP odd ones. These fields come together with in total six neutralinos.
The neutralino mass matrix is given in the basis $\left(\lambda_{\tilde{B}}, \tilde{W}^0, \tilde{H}_d^0, \tilde{H}_u^0, \tilde{S},\tilde{\bar{S}}\right)$ by
\begin{equation}
m_{\tilde{\chi}^0} = \left(
\begin{array}{cccccc}
M_1 &0 &-\frac{1}{2} g_1 v_d &\frac{1}{2} g_1 v_u &0 &0\\
0 &M_2 &\frac{1}{2} g_2 v_d &-\frac{1}{2} g_2 v_u &0 &0\\
-\frac{1}{2} g_1 v_d &\frac{1}{2} g_2 v_d &0 &-\mu_\text{eff} &- \frac{1}{\sqrt{2}} v_u \lambda &0\\
\frac{1}{2} g_1 v_u &-\frac{1}{2} g_2 v_u &- \mu_\text{eff} &0 &- \frac{1}{\sqrt{2}} v_d \lambda &0\\
0 &0 &- \frac{1}{\sqrt{2}} v_u \lambda &- \frac{1}{\sqrt{2}} v_d \lambda &0 &M_s\\
0 &0 &0 &0 &M_s &0\end{array}
\right)
\end{equation}
Here, we introduced
\begin{equation}
\mu_\text{eff} = \frac{1}{\sqrt{2}} v_{s} \lambda + \mu \;.
\end{equation}
The matrix which diagonalizes the neutralinos is called $N$ in the following
\begin{equation}
N^* m_{\tilde{\chi}^0} N^{\dagger} = m^{dia}_{\tilde{\chi}^0} \;.
\end{equation}
Generically the lightest neutralino which is a good dark matter candidate can be an admixture of the bino, the Higgsinos (in the case of non-universal gaugino masses also the Wino)
and the Singlinos. However, we will see that usually large $M_s$ is preferred so that the Singlino component is typically very suppressed.
In the following we will mostly concentrate on the CP even Higgs states. For completeness, all matrices and vertices which are different to the MSSM are listed in
Appendices~\ref{app:matrices}--\ref{app:vertices}.
\subsection{The Higgs sector}
\subsubsection{The Higgs mass at tree level}
\label{sec:HiggsTree}
The four minimum conditions with respect to the CP even scalars introduced in eqs.~(\ref{eq:EWSB1})--(\ref{eq:EWSB2}) read
\begin{align}
\Theta_d = \frac{\partial V}{\partial \phi_{d}} =& +\frac{1}{8} \Big(g_{1}^{2} + g_{2}^{2}\Big)v_d \Big(v^2_d- v^2_u\Big)+ v_d \Big(m_{h_d}^2 + |\mu|^2 + \sqrt{2} v_{s} \Re(\lambda \mu^*) + \Big(v_{s}^{2} + v_{u}^{2}\Big)\frac{|\lambda|^2}{2} \Big)\nonumber \\
&- v_u \Big(\Re(\lambda \xi_s^*) + {\Re\Big(b\mu\Big)} + \frac{\sqrt{2}}{2} \Big(v_{\bar{s}} \Re(\lambda M_s^* ) + v_{s} \Re(\lambda A_{\lambda})\Big)\Big)=0\\
\Theta_u = \frac{\partial V}{\partial \phi_{u}} =& +\frac{1}{8} \Big(g_{1}^{2} + g_{2}^{2}\Big)v_u \Big(v_{u}^{2}- v_{d}^{2}\Big)+v_u \Big(m_{h_u}^2 + |\mu|^2 + \sqrt{2} v_{s} \Re(\lambda \mu^*) + \Big(v_{d}^{2} + v_{s}^{2}\Big)\frac{|\lambda|^2}{2} \Big)\nonumber \\
&- v_d \Big(\Re(\lambda \xi_s^*) + {\Re\Big(b\mu\Big)} + \frac{\sqrt{2}}{2} \Big(v_{\bar{s}} \Re(\lambda M_s^*) + v_{s} \Re(\lambda A_{\lambda})\Big)\Big)=0\\
\Theta_s = \frac{\partial V}{\partial \phi_s} =& \Big(v_{d}^{2} + v_{u}^{2}\Big)\Big(v_{s} \frac{|\lambda|^2}{2} + \frac{\sqrt{2}}{2} \Re(\mu \lambda^*) \Big) + m_{s}^2 v_{s} + v_{\bar{s}} {\Re(b_s)} + |M_s|^2 v_{s} + \sqrt{2} \Re(M_s^* {\bar{\xi}}_s) \nonumber \\
& + \sqrt{2}\Re(t_s) - \frac{\sqrt{2}}{2} v_d v_u \Re(\lambda A_{\lambda})=0\\
\Theta_{\bar{s}} = \frac{\partial V}{\partial \phi_{\bar{s}}} =& |M_s|^2 v_{\bar{s}} + \sqrt{2} \Re(M_s^* \xi_s) -\frac{\sqrt{2}}{2} v_d v_u \Re(M_s^* \lambda) + \frac{2}{\sqrt{2}} \Re(t_{\bar{s}}) + m_{\bar{s}}^2 v_{\bar{s}} + v_{s} {\Re\Big(b_s\Big)} =0
\end{align}
where $\Re(a)$ refers to the real part of $a$.
For given input parameters these equations can now in principle be solved for the four vevs $v_d,v_u,v_s,v_{\bar{s}}$.
However, given that the electroweak vev is known, $v=\sqrt{v_u^2+v_d^2}\simeq 246 \:\text{Ge\eVdist V}$, it makes sense to use this information and
solve for some of the input parameters instead.
There are now many reasonable combinations of parameters which can be fixed by these equations. The easiest choice might be to choose the Higgs and singlet soft masses $m_{h_d}^2$, $m_{h_u}^2$, $m_s^2$, and $m_{\bar{s}}^2$. However, this assumes automatically that the Higgs soft-terms don't unify with the other scalar soft SUSY masses. A set of parameters which can be consistent with the unification of scalars is $\mu,b\mu,t_s, t_{\bar{s}}$. In the limit of real $\mu$ two distinct solutions are found
\begin{align}
\nonumber
t_s =& \frac{\mp 1}{8 \Big(v^2_d- v^2_u\Big)} \Big(\sqrt{2} \Big(4 v_{s} \Big(v^2_d- v^2_u\Big)(M_{s}^{2} + m_{s}^2)+\Big(v_{d}^{2} + v_{u}^{2}\Big)\sqrt{D} \lambda \Big)\Big)\\
& \hspace{1cm} +\frac{1}{2} \Big(v_d v_u \lambda A_{\lambda} - \sqrt{2} v_{\bar{s}} b_s \Big) \\
t_{\bar{s}} =& \frac{1}{2} \Big(M_s v_d v_u \lambda - \sqrt{2} \Big(m_{\bar{s}}^2 v_{\bar{s}} + v_{s} b_s \Big) - \sqrt{2} M_{s}^{2} v_{\bar{s}} \Big) \\
b\mu =& \frac{1}{4} \Big(2 \Big(m_{h_d}^2- m_{h_u}^2\Big) \frac{2 v_d v_u}{v^2_u- v^2_d } - v_d v_u \Big(g_{1}^{2} + g_{2}^{2} -2 \lambda^{2} \Big)-2 \sqrt{2} \Big(M_s v_{\bar{s}} \lambda + v_{s} \lambda A_{\lambda} \Big)\Big) \\
\label{eq:tadMu}
\mu =& -\frac{1}{\sqrt{2}} \lambda v_s \pm \frac{\sqrt{D}}{2\sqrt{2}\Big(v^2_d- v^2_u \Big)}
\end{align}
with
\begin{equation}
\sqrt{D} = \sqrt{\Big(v^2_u-v^2_d\Big) \Big(\Big(g_1^2+g_2^2\Big) \Big(v_d^4-v_u^4\Big)+8 m_{h_d}^2 v_d^2-8 m_{h_u}^2 v_u^2\Big)}
\end{equation}
The mass matrix of the CP even Higgs fields is given in the basis \( \left(\phi_{d}, \phi_{u}, \phi_s, \phi_{\bar{s}}\right)\) by
\begin{equation}
\label{eq:HiggsTreeMass}
m^2_{h} = \left(\begin{array}{cc} M_D & M_M \\ M^T_M & M_S \end{array}\right)
\end{equation}
with $2\times 2$ matrices containing the elements involving only doublets ($M_D$) or singlets ($M_S$) as well as the entries linking both sectors:
\begin{align}
M_D = & \left(\begin{array}{cc}
m_{\phi_{d}\phi_{d}} &m_{\phi_{u}\phi_{d}} \\
m_{\phi_{u}\phi_{d}} & m_{\phi_{u}\phi_{u}}
\end{array} \right) \\
M_S = & \left(\begin{array}{cc}
\frac{1}{2} \Big(v_{d}^{2} + v_{u}^{2}\Big)|\lambda|^2 + m_{s}^2 + |M_s|^2 &{\Re\Big(b_s\Big)}\\
{\Re\Big(b_s\Big)} &m_{\bar{s}}^2 + |M_s|^2
\end{array} \right) \\
M_M = & \left(\begin{array}{cc}
\frac{1}{\sqrt{2}} \Big(2 \Re(v_d \lambda \mu^*) - v_u {\Re(\lambda A_{\lambda})} \Big) + v_d v_{s} |\lambda|^2 & \quad - \frac{1}{\sqrt{2}} v_u {\Re\Big(\lambda M_s^* \Big)} \\
\frac{1}{\sqrt{2}} \Big(2 \Re(v_u \lambda \mu^*) - v_d {\Re(\lambda A_{\lambda})} \Big) + v_u v_{s} |\lambda|^2 &\quad - \frac{1}{\sqrt{2}} v_d {\Re\Big(\lambda M_s^* \Big)} \end{array} \right)
\end{align}
with
\begin{align}
m_{\phi_{d}\phi_{d}} &= |\mu|^2 + \Re(\sqrt{2} v_{s} \lambda \mu^*) + \frac{1}{2}\Big(v_{s}^{2} + v_{u}^{2}\Big)|\lambda|^2 + \frac{1}{8} \Big(g_{1}^{2} + g_{2}^{2}\Big)\Big(3 v_{d}^{2} - v_{u}^{2} \Big) + m_{h_d}^2\\
m_{\phi_{d}\phi_{u}} &= -\Re(\lambda \xi_s^* + b\mu)+ v_d v_u \Big(|\lambda|^2 -\frac{1}{4} (g_{1}^{2} + g_{2}^{2})\Big) - \frac{\sqrt{2}}{4}\Big(2 v_{\bar{s}} {\Re(\lambda M_s^*)}+2 v_{s} {\Re(\lambda A_{\lambda})} \Big) \\
m_{\phi_{u}\phi_{u}} &= \frac{1}{2} \Big(2 |\mu|^2 + 2 \Re(\sqrt{2} v_{s} \lambda\mu^*) + \Big(v_{d}^{2} + v_{s}^{2}\Big)|\lambda|^2 \Big) -\frac{1}{8} \Big(g_{1}^{2} + g_{2}^{2}\Big)\Big(v_{d}^{2}-3 v_{u}^{2}\Big) + m_{h_u}^2
\end{align}
This matrix is diagonalized by \(Z^H\):
\begin{equation}
Z^H m^2_{h} Z^{H,\dagger} = m^{dia}_{2,h}
\end{equation}
To gain some insight into the mass-dependence of the lightest, SM-like Higgs boson on the different parameters, we can perform a rotation of the tree level Higgs matrix to the basis $(h,H,S,\bar{S})$ via the rotation matrix
\begin{equation}
\left(\begin{array}{cc} \cos\beta & \sin\beta \\ -\sin\beta & \cos\beta \end{array} \right) \;.
\end{equation}
In first approximation, the $h$--$H$ mixing as well as the $h$--$\bar{S}$ mixing can be neglected and the remaining matrix in the basis $(h,S)$ reads
\begin{equation}
\label{eq:mhS}
m_{hS} = \left(\begin{array}{cc} M_{hh} & M_{hS} \\ M_{hS} & M_{SS} \end{array} \right)
\end{equation}
with
\begin{align}
M_{hh} =& M^2_Z \left(\cos^22\beta + \frac{\lambda^2}{g^2}\sin^22\beta\right)\\
M_{hS} =& -\frac{1}{2v}(\lambda \sqrt{D}\sec2\beta + \sqrt{2} v^2 \lambda A_{\lambda}) \sin2\beta \\
M_{SS} =& M_s^2 + m_s^2
\end{align}
$M_{hh}$ shows already the famous F-term enhancement of the Higgs mass at tree-level. Hence, if we want to make use of this effect to reduce the fine tuning, we have to concentrate on large $\lambda$ and small $\tan\beta$. However, this effect can easily be spoiled by the mixing with the singlet-sector coming from $M_{hS}$
which tends to reduce the smaller eigenvalue. Not relying on special cancellations, the natural size of $M_{hS}$ is $\lambda v M_{SUSY}$,
which implies that for small mixing $M_{SS}$ should be rather large. The expectation that for the range of interest $M_s$ is in the few TeV range
for the correct Higgs mass will be confirmed in our numerical analysis.
\subsubsection{Radiative corrections to the Higgs mass}
\label{sec:ModelHiggsLoop}
Of course, a tree-level calculation is not sufficient to have a reliable estimate for the Higgs mass.
The radiative corrections to the Higgs masses in the NMSSM are discussed in detail in the literature \cite{King:1995vk,Elliott:1993uc,Elliott:1993bs,Ellwanger:1993hn,Franke:1995xn,Degrassi:2009yq,Staub:2010ty,Graf:2012hh,Ender:2011qh}. In contrast, studies in the context of extensions of the NMSSM often include just the dominant stop corrections at the 1-loop level but neglect all other, potentially important, effects. However, we are not relying on these approximations but calculate also the Higgs mass in the DiracNMSSM with a precision comparable to the NMSSM: we include all corrections at the one-loop level and the dominant two-loop corrections stemming from (s)tops. Details of the calculation are given in Appendix~\ref{app:HiggsLoop}. \\
To give an impression of the size of the loop corrections and the dependence on the different input parameters we compare in Fig.~\ref{fig:comparisonLoops} the Higgs mass calculated (i) at tree level, (ii) at one-loop, (iii) at two-loop including stop corrections.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=0.40\linewidth]{Plots/DiracNMSSMloopA0} \quad
\includegraphics[width=0.41\linewidth]{Plots/DiracNMSSMloopMS} \\
\includegraphics[width=0.40\linewidth]{Plots/DiracNMSSMloopTB} \quad
\includegraphics[width=0.40\linewidth]{Plots/DiracNMSSMloopLam}
\end{center}
\caption{Red line: tree-level mass; blue line: tree-level and full one-loop corrections; green line: tree-level, full one-loop and dominant two loop corrections.}
\label{fig:comparisonLoops}
\end{figure}
For these plots we have solved the tadpole equations with respect to $\{\mu,b\mu,t_s, t_{\bar{s}}\}$ and have used in addition
\begin{eqnarray*}
&m_0 = 300~\text{GeV}\,,\hspace{0.3cm} m_{1/2} = 800~\text{GeV}\,,\hspace{0.3cm} \tan\beta=2.3\,,\hspace{0.3cm} A_0 = -2500~\text{GeV}\,,&\\
&\lambda = 1.6\,,\hspace{0.3cm} A_\lambda=-100~\text{GeV}\,,\hspace{0.3cm}v_s=8.3~\text{GeV}\,,\hspace{0.3cm} v_{\bar{s}}=1.5~\text{GeV}\,,&\\
&M_s=9000~\text{GeV}\,,\hspace{0.3cm}b_s=3\cdot 10^6~\text{GeV}^2\,,\hspace{0.3cm} m_{\bar{s}}^2 = 7\cdot 10^{12}\:\text{Ge\eVdist V}^2\,.&
\end{eqnarray*}
There are some features visible in these plots: (i) as expected, the Higgs mass becomes maximal for $\tan\beta \sim 2$ and large values of $\lambda$. (ii) since $m_{s}^2$ was chosen to unify with the other scalars, large $M_s$ is needed to get a sufficiently large $m_h$. The observation that $m_{h}$ grows with increasing $M_{s}$ might seem at first glance inconsistent with the approximate tree-level expression for the Higgs mass derived in Ref.~\cite{Lu:2013cta}. This derivation however did not include the mixing effects in the neutral scalar sector, which have a non-negligible impact on the light eigenvalues and are fully included in our numeric computation. (iii) the absolute shift coming from the 1- and 2-loop corrections are rather insensitive to $\lambda$, i.e. the radiative corrections for the given point are completely dominated by the (s)top loops.
However, there are also parameter points where other loop corrections beside the (s)top-loops can be very important. In Fig.~\ref{fig:loopContributions} we show the one-loop corrected mass with and without the corrections stemming from Higgs and neutralino/chargino loops. The other parameters have been chosen as
\begin{eqnarray*}
&m_0 = 500~\text{GeV}\,,\hspace{0.3cm} m_{1/2} = 600~\text{GeV}\,,\hspace{0.3cm} \tan\beta=2.3\,,\hspace{0.3cm} A_0 = 2200~\text{GeV}\,,&\\
&\lambda = 1.6\,,\hspace{0.3cm} A_\lambda=1300~\text{GeV}\,,\hspace{0.3cm}v_s=1.0~\text{GeV}\,,\hspace{0.3cm} v_{\bar{s}}=0.1~\text{GeV}\,,&\\
&M_s=6600~\text{GeV}\,,\hspace{0.3cm}b_s=1\cdot 10^4~\text{GeV}^2\,,\hspace{0.3cm} m_{\bar{s}}^2 = 1\cdot 10^{14} \:\text{Ge\eVdist V}^2\,.&
\end{eqnarray*}
Obviously, in the case of large $\lambda$, the additional loops can easy push the Higgs mass up by more than 5~GeV and the full calculation presented here is really necessary.
\begin{figure}[hbt]
\centering
\includegraphics[width=0.45\linewidth]{Plots/LoopContributions}
\caption{One-loop contributions to the Higgs mass as function of the value of $\lambda$ at the GUT scale. The color code is as follows: black, dashed line: pure tree-level; blue-line: one-loop mass without Higgs/Neutralino/Chargino corrections;
green line: full one-loop corrections including Higgs/Neutralino/Chargino contributions.}
\label{fig:loopContributions}
\end{figure}
\section{Fine Tuning and precision calculations}
\label{sec:FT}
\subsection{Fine Tuning measure}
In Ref.~\cite{Ellis:1986yg, Barbieri:1987fn} a quantitative estimate of the the fine tuning with respect to a set of independent parameters, $p$, was
introduced as
\begin{equation}
\label{eq:measure}
\Delta \equiv \max {\text{Abs}}\big[\Delta _{p}\big],\qquad \Delta _{p}\equiv \frac{\partial \ln
v^{2}}{\partial \ln p} = \frac{p}{v^2}\frac{\partial v^2}{\partial p} \;.
\end{equation}
The quantity $\Delta^{-1}$ gives a measure of the accuracy to which independent parameters must be tuned to get the correct electroweak breaking scale \cite{Ghilencea:2012qk}. The fine tuning of a model of course depends on what one takes to be the fundamental parameters and at which scale these are defined.
Given the success of gauge coupling unification in supersymmetric models it is natural to assume an underlying GUT structure and
to define the fundamental parameters at the GUT scale. We therefore assume in the following study that all sfermion soft terms unify at the GUT scale to $m_0$.
In addition, the mass scale of the gaugino is set by the parameter $m_{1/2}$. However, we do not necessarily impose the unification of all gaugino masses (which can still be consistent with an underlying GUT) by allowing for
$M_1 = a \cdot m_{1/2}$, $M_2 = b \cdot m_{1/2}$, $M_3 = m_{1/2}$ and studying the cases $a=b=1$ and deviations from it separately.
It has been observed that such a non-universality can reduce the hierarchy problem through the appearance of a new ``focus point'' that makes the Higgs mass less sensitive to the gaugino mass scale~\cite{Choi:2005hd,Choi:2006xb,Abe:2007kf,Lebedev:2005ge,Horton:2009ed,Asano:2012sv,Antusch:2012gv,Abe:2012xm,Badziak:2012yg,Gogoladze:2012yf,Yanagida:2013ah}.
We assume that $a$ and $b$ are fixed by the underlying theory such that contributions to the fine tuning are absent. As discussed in \cite{Kaminska:2013mya, Horton:2009ed} values of $a$ and $b$ in the low-focus-point region occur naturally in a variety of models. However, as discussed below, dropping this assumption does not greatly increase the minimum fine tuning.
We calculate the fine tuning with respect to all independent parameters in the DiracNMSSM, defined at the GUT scale,
\begin{equation}
p \in \{m_0, m_{1/2}, A_0, \mu, b\mu, \lambda, A_\lambda, M_s, b_s, t_s, t_{\bar{s}}, m_s^2, m_{\bar{s}}^2, m_{h_d}^2, m_{h_u}^2\} \;.
\end{equation}
In many fine tuning analyses it has become customary to consider the fine tuning in terms of electroweak scale parameters only.
Specifically for the DiracNMSSM the following measure has been used in a previous study \cite{Lu:2013cta}
\begin{equation}
\Delta_h \equiv \frac{2}{m_h^2} \max\{m_{h_d}^2, m_{h_u}^2, \beta^{(1)}_{m^2_{h_d}} L, \beta^{(1)}_{m^2_{h_u}} L, B_\text{eff}, \delta h \}
\end{equation}
with the one-loop $\beta$-functions for the Higgs soft terms, $L=\log\frac{M_{GUT}}{M_{SUSY}}\simeq 30$ and $\delta h = \frac{(\lambda M_s)^2}{(4\pi)^2}\log\frac{M_s^2 + m_s^2}{M^2}$. Here the factor $L$ is supposed to account for the fine tuning
from running, i.e.\ mimicking the effect of defining the parameters at the high scale.
We have compared both measures. The result is shown for a set of 100,000 points in Fig.~\ref{fig:FTmeasures}. Even if they usually predict a FT of the same order, the differences can be sizable, with a factor of more than an order of magnitude in both directions.
One feature that does not show up in the measure $\Delta_h$ is the appearance of focus point correlations, hence the fine tuning
can be overestimated. We believe however that it is interesting to study such correlations among parameters which
reduce the fine tuning as this might give valuable information about the desirable structure of the high energy theory.
On the other hand for large SUSY parameters which usually lead to very large $\Delta$, the FT in $\Delta_h$ appears to be much smaller. The reason is that for large SUSY parameters $\beta^{(1)}_{m^2_{h_{u,d}}} L$ is no longer a good approximation for the RGE dependence of the soft terms. Also the 'source' of fine tuning is less clear in the low scale
picture. For instance, the strong dependence on the gluino mass which just enters indirectly in the running of $m_{h_u}^2$ is completely missed \cite{Arvanitaki:2013yja}.
\begin{figure}[!h!]
\centering
\includegraphics[width=0.45\linewidth]{Plots/FT_ratio_nozoom.png}
\caption{Comparison of the two fine tuning measures.}
\label{fig:FTmeasures}
\end{figure}
\subsection{Fine tuning calculation}
Fine tuning studies have a long tradition. However, the precision of these studies has not necessarily improved
as other theoretical predictions have. In particular for models which go beyond the MSSM
the prediction of the fine tuning often suffers from large theoretical uncertainties which
are often not mentioned:
\begin{itemize}
\item Higgs mass prediction: in many BMSSM studies only the dominant radiative corrections from (s)tops
known from the MSSM are included. In addition, also the impact of external momenta is neglected. However,
there are two main issues with this approach: (i) potentially large, additional loop corrections which don't
exist in the MSSM are completely missed. The best example is the NMSSM with large $\lambda$. This has been
demonstrated for the NMSSM in Ref.~\cite{Degrassi:2009yq} and we also show an example for the DiracNMSSM
in sec.~\ref{sec:ModelHiggsLoop}. (ii) the
effective potential approach corresponding to $p^2=0$ can differ significantly from a full one-loop correction
demanding $p^2=m_h^2$. This has been shown for an NMSSM extension in Ref.~\cite{Benakli:2012cy}.
Both effects can be larger than the experimental uncertainty.
\item Dark matter prediction: in particular, in co-annihilation regions the relic density of a particle has to
be calculated numerically and {\tt MicrOmegas}\xspace \cite{Belanger:2006is,Belanger:2007zz,Belanger:2010pz} is often used for that.
There exist nowadays several
possibilities to create model files for {\tt MicrOmegas}\xspace for models which are not included in the public version \cite{Semenov:1998eb,Christensen:2008py,Staub:2009bi}. Using one of these codes should be strongly favored instead
of modifying existing files by hand which is very error prone.
However, even if one uses the correct model files one must keep in mind that the masses
used as input for {\tt MicrOmegas}\xspace suffer from an uncertainty. It has been pointed out in Ref.~\cite{Staub:2010ty}
that for instance the stau co-annihilation band in the NMSSM gets a sizable shift when going from tree
to the one-loop level.
\item RGE running: for phenomenological studies in the MSSM or NMSSM it has become standard that the full 2-loop
RGEs are solved numerically. Even if there exist now several public tools which can calculate the RGEs for a
given theory at the two-loop level \cite{Staub:2010jh,Fonseca:2011sy,Lyonnet:2013dna}, often one-loop approximations are used for FT studies. In particular for parameter
regions which need cancellations between different contributions to the RGE running for specific parameters like the
Higgs mass terms, those approximations can fail badly. Moreover, it is well known that the GUT scale defined by $g_1=g_2$
is shifted at the 2-loop level quite a bit. Already this effect causes a non-negligible difference between the running
parameters at the SUSY scale and changes in particular the masses of particles which depend on the strong interaction \cite{Jack:2004ch}.
\item SUSY thresholds: the running parameters entering the RGEs have to be derived from the measured observables. To get an accurate
set of gauge and Yukawa couplings at the SUSY scale the SUSY thresholds have to be included \cite{Pierce:1996zz}.
In particular, $g_3^{\overline{\text{DR}}}$ can differ by several percent from $g_3^{\overline{\text{MS}}} = \sqrt{4\pi\alpha_s(M_Z)}$ \cite{Carena:1993ag}. Depending on the size of these threshold corrections the perturbativity limit of $\lambda$ is shifted
in singlet extensions of the NMSSM \cite{Ellwanger:2009dp}.
\end{itemize}
In the current study we take care of all of these issues by using the public codes {\tt SARAH}\xspace and {\tt SPheno}\xspace: we have implemented
the DiracNMSSM in {\tt SARAH}\xspace \cite{Staub:2008uz,Staub:2009bi,Staub:2010jh,Staub:2012pb,Staub:2013tta} \footnote{The model files might become public with a future release of {\tt SARAH}\xspace. If you are interested in them beforehand, please contact the authors.}. {\tt SARAH}\xspace has then been used to create a {\tt SPheno}\xspace \cite{Porod:2011nf,Porod:2003um} module to get a full fledged spectrum generator for the DiracNMSSM which provides the following features:
\begin{itemize}
\item All masses of SUSY and Higgs particles are calculated at the one-loop level including all contributions present in the DiracNMSSM
and including the impact of the external momenta in the loop functions. Details about the calculation in the Higgs sector are given in
Appendix~\ref{app:HiggsLoop}.
\item The dominant 2-loop corrections ($\mathcal{O}(\alpha_t^2), \mathcal{O}(\alpha_t \alpha_s), \mathcal{O}(\alpha_t \alpha_b)$) known from the MSSM are included \cite{Degrassi:2001yf,Brignole:2001jy,Brignole:2002bz,Dedes:2002dy,Dedes:2003km}. By including these corrections, we have reduced the theoretical uncertainty of the Higgs mass. However, since two loop contributions involving $\lambda$ which are potentially important for large $\lambda$ are missing, the remaining uncertainty is still expected to be slightly larger in comparison to the MSSM with the Higgs mass calculated at the same level.
\item Prediction for precision observables like $b\to s\gamma$, $g-2$ or $B_{s,d}^0 \to \bar{\mu}\mu$ are derived at the one-loop level which can be used to further constrain models. Also all of these calculations are automatically adjusted to the present model by {\tt SARAH}\xspace as explained in Ref.~\cite{Dreiner:2012dh}.
\item All SUSY thresholds are included when calculating the gauge and Yukawa couplings in the $\overline{\text{DR}}$ scheme from the measured values of SM fermion and gauge boson masses, $G_F$ and $\alpha_s(M_Z)$.
\item The RGE running is performed at the 2-loop level without any approximation. To calculate the fine tuning, we vary each of the independent parameters at the GUT scale, run all the parameters down to the weak scale and evaluate the shift in the Z mass by consistently solving all tadpole equations with respect to all vevs numerically.
The last step, which is necessary to get the precise fine tuning is not yet part of the public version of {\tt SARAH}\xspace but will be included in the next update.
\end{itemize}
For the calculation of the relic density we have used {\tt MicrOmegas}\xspace and created the corresponding model files for the DiracNMSSM also with {\tt SARAH}\xspace. To perform
the parameter scans we made use of {\tt SSP}\xspace \cite{Staub:2011dp}. The exchange of parameter values between {\tt SPheno}\xspace modules written by {\tt SARAH}\xspace and {\tt MicrOmegas}\xspace
model files also written by {\tt SARAH}\xspace happens automatically by using the SLHAplus functionality \cite{Belanger:2010st} of {\tt CalcHep}\xspace \cite{Pukhov:2004ca,Belyaev:2012qa}.
We have to mention that there is one issue which is still hard to include with the same precision as the other aspects: the question if a parameter point is ruled out by LHC searches or not. To be sure one would have to make a collider study for each parameter point what is far beyond the scope of this work here.\footnote{However, also this situation is expected to be improved significantly since several tools are currently developed to test SUSY points against LHC results \cite{Drees:2013wra,atom,Kraml:2013mwa}.} As discussed below, the non-universal gaugino mass case in the GNMSSM often leads to a compressed SUSY spectrum with small mass differences between gauginos and the LSP that makes SUSY discovery more difficult. To account for this in a manner consistent with the non-observation of superpartners at the LHC we implemented a cut on the gluino mass as a function of the gluino-LSP mass difference as presented in \cite{ATLAS-CONF-2013-047,CMS-PAS-SUS-13-008}.
In Fig.~7 of \cite{ATLAS-CONF-2013-047} two bounds are shown, a weaker one for decoupled squarks and a stronger one for $m_\text{squark} \sim m_\text{gluino}$. Most parameter space points of interest to us are in the intermediate regime, but we will use the stronger bound here.\footnote{This of course assumes that the bound on $m_\text{gluino} \sim m_\text{squark}$ is at least as strong as say $m_\text{gluino} \sim 0.7 \cdot m_\text{squark}$. This is not quite clear as in the case of compressed spectra the gluino is still very close in mass to the lightest neutralino while some missing $E_T$ could be coming from the squarks. On the other hand the production cross section will be smaller in this case. As noted above a thorough study would need to examine every parameter point individually. Here we will content ourselves with
this approximate cut.}
We further require the chargino and slepton masses to be above $100\:\text{Ge\eVdist V}$.
We also require that the lightest supersymmetric particle (LSP) is a neutralino which is a good dark matter candidate and its
relic density does not exceed the $5\sigma$ PLANCK \cite{Ade:2013zuv} upper bound of $\Omega h^2 \le 0.1334$.
\section{Exploring the DiracNMSSM vs.\ the GNMSSM}
\label{sec:results}
In the following we will study the fine tuning of the DiracNMSSM and compare it with the GNMSSM.
The superpotential of the GNMSSM is given by
\begin{equation}
\mathcal{W} = \mathcal{W}_\text{MSSM} +\lambda S H_u H_d + \xi_s S + \frac{1}{2} \mu_s S^2+ \frac{1}{3}\kappa S^3 \;.
\end{equation}
and the corresponding soft-breaking terms read
\begin{align}
V_\text{soft} &= V_\text{soft,MSSM} + m_s^2 |s|^2 + \left(\frac{1}{3} \kappa A_\kappa s^3 + \lambda A_\lambda s h_u h_d + b_s s^2 + t_s s + h.c.\right) \;.
\end{align}
\subsection{Universal gaugino masses}
Let us first concentrate on the fine tuning in the DiracNMSSM in the case that all gaugino masses unify at the GUT scale. We solve the four conditions for correct EWSB for $\{ \mu, b\mu, t_s, t_{\bar{s}}\}$. This leaves us with 16 input parameters
\begin{equation}
m_0, m_{1/2}, A_0, \tan\beta, m_{h_u}^2, m_{h_d}^2, \lambda, A_\lambda, v_s, v_{\bar{s}}, M_s, b_s, m_s^2, m_{\bar{s}}^2, \xi_s, \xi_{\bar{s}}
\end{equation}
For the case of universal gaugino masses the lightest gaugino will always be a bino.
This potentially leaves a mixture of bino-, higgsino, and singlino-like neutralinos as LSP candidates.
For the bino it is difficult to achieve a small enough relic abundance except in the stau coannihilation or Higgs funnel regions
while the singlino turns out to be
typically rather heavy in the region of interest. Accordingly most of the viable points we find have a LSP with a sizable
higgsino fraction.
\begin{figure}[!h!]
\centering
\includegraphics[width=0.45\linewidth]{Plots/Plot_mh_FT_MZ}
\includegraphics[width=0.45\linewidth]{Plots/GNMSSMnuhm_mh_FT_MZ}
\caption{The fine tuning as a function of the SM like Higgs mass for the DiracNMSSM (left) and the GNMSSM (right).
The light blue points are before any cuts.
For the dark blue points we use appropriate SUSY and dark matter cuts. }
\label{fig:DiracNMSSMnuhm}
\end{figure}
The overall result is summarized in Fig.~\ref{fig:DiracNMSSMnuhm}. This plot shows the fine tuning as function of the SM-like Higgs mass for the DiracNMSSM (left) and the GNMSSM (right). Several different cuts are indicated: (i) light blue points correspond to no cuts, (ii) medium blue points include LHC limits on SUSY masses, and (iii) dark blue points include in addition the upper limit on the dark matter abundance. Note that these plots are combinations of several scans of different regions in the parameter space of the models. Since we have zoomed into interesting regions with a small or moderate fine tuning the density of points can't be interpreted as some probability measure.
Nevertheless, it is interesting that there is no big difference between the best fine tuning after the LHC and after the LHC and dark matter cut for the DiracNMSSM. The reason is that the lightest neutralino in the DiracNMSSM turns out to have easily a large Higgsino fraction which is sufficient to reduce the dark matter abundance to the allowed level. One can see that the best fine tuning consistent with all observables in the DiracNMSSM with unified gaugino masses is about 70. This is not improved if we dropped the upper limit on the neutralino relic density. This value is, of course, significantly better than in the MSSM where one expects $\Delta \gtrsim 300$ and also improves the situation compared to the NMSSM. However, it is of the same size as the fine tuning reported for the GNMSSM in a similar setup, see the right plot in Fig.~\ref{fig:DiracNMSSMnuhm}. Here we find as best fine tuning $\Delta \simeq 78$ including the DM cut and $\Delta \simeq 71$ without the DM cut.
To compare the dependence of the fine tuning in both models we have randomly picked 200,000 valid points for each model. The parameter values for all points are in the ranges
\begin{eqnarray*}
&m_0 \in [0,1]~\text{TeV}\,,\hspace{1cm} m_{1/2} \in [0,1]~\text{TeV}\,,\hspace{1cm} \tan\beta \in [1.5,4.0]&\\
&\lambda \in [1.0,1.7]\,,\hspace{1cm} A_\lambda \in [-1.5,1.5]~\text{TeV}\,,\hspace{1cm} A_0 \in [-2.5,2.5]~\text{TeV}&\\
&M_s,\mu_s \in [-10,10]~\text{TeV}\,,\hspace{1cm} b_s \in [-10,10]~\text{TeV}^2\,,\hspace{1cm} v_s \in [-2,2]~\text{TeV}& \\
&m_{h_d}^2 \in [-5,5]~\text{TeV}^2\,,\hspace{1cm} m_{h_u}^2 \in [-5,5]~\text{TeV}^2\,,\hspace{1cm} m_s^2 \in [-5,5]~\text{TeV}^2&
\end{eqnarray*}
in addition, the specific parameters for the DiracNMSSM have been chosen to be
\begin{eqnarray*}
&v_{\bar{s}} \in [-2,2]~\text{GeV} \,,\hspace{1cm} m_{\bar{s}}^2 \in [0,10^4]~\text{TeV}^2 &
\end{eqnarray*}
and those for the GNMSSM
\begin{eqnarray*}
\kappa \in [-1,1]\,,\hspace{1cm} A_\kappa \in [-1.5,1.5]~\text{TeV}
\end{eqnarray*}
Note, since the points shown in the following are chosen randomly from the full data-set they don't include the points with the best fine tuning which have been found for both models. They are just meant to demonstrate the general similarities and differences in the fine tuning of both models.
The effect of the different cuts can be summarized as follows:
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
& DiracNMSSM & GNMSSM \\
\hline
\hline
Valid points & 200,000 & 200,000 \\
After SUSY cuts & 52,623 & 64,514 \\
After Higgs cut & 5,527 & 23,783 \\
After DM cut & 943 & 204 \\
\hline
\end{tabular}
\end{center}
In both models the SUSY cuts rule out about 2/3 to 3/4 of the valid points. That's not surprising since the changes in the Higgs sector are expected to have only a small impact on the squark and gluino masses and for $m_{1/2} \lesssim 750\:\text{Ge\eVdist V}$ the gluino mass turns out to be too light. However, there are two obvious differences: while it seems to be much easier to accommodate a Higgs mass in the expected range in the GNMSSM than in the DiracNMSSM, the DiracNMSSM is doing better in satisfying the upper limit of the dark matter abundance. The reason for this is, as mentioned already above, that the lightest neutralino in the DiracNMSSM is much more often a Higgsino which annihilates very efficiently.
\begin{figure}[tb]
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_m0_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_M12_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_A0_FT_MZ} \\
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_m0_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_M12_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_A0_FT_MZ}
\caption{Fine tuning vs. $m_0$, $m_{1/2}$ and $A_0$ in the DiracNMSSM (first line)
and GNMSSM (second line). With increasing saturation the following cuts are applied:
(i) no cut, (ii) cut on SUSY masses, (iii) cut on the SM like Higgs mass, (iv) cut
on the upper limit of the dark matter abundance. }
\label{fig:compFT1}
\end{figure}
We turn now to the dependence of the fine tuning on the different parameters. For this purpose the fine tuning versus $m_0$, $m_{1/2}$ and $A_0$ is shown in Fig.~\ref{fig:compFT1}, versus $\lambda$, $A_\lambda$ and $M_s / \mu_s$ in Fig.~\ref{fig:compFT2}, versus $v_s$, $\mu$ and $b\mu$ in Fig.~\ref{fig:compFT4}, and versus $m_{h_d}^2$, $m_{h_u}^2$ and $m_s^2$ in Fig.~\ref{fig:compFT3}. For the parameters not shown here ($b_s$, $m_{\bar{s}}^2$, $v_{\bar{s}}$, $\kappa$, $A_{\kappa}$) there is no visible dependence of the fine tuning on those parameters.
In Fig.~\ref{fig:compFT1} there is a very strong dependence of the fine tuning in both models on the gaugino mass parameters $m_{1/2}$. Small values for $m_{1/2}$
are usually forbidden by the gluino searches at the LHC. That's one of the main reasons which pushes the fine tuning to larger values. The impact of $m_0$ is rather moderate in both models, as long as it is not too large (as we
don't assume the sfermion masses to unify with the soft Higgs masses, the focus point solution for large $m_0$ won't work here). Further the DiracNMSSM and the GNMSSM seem to slightly favour positive $A_0$.
\begin{figure}[tb]
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_Lambda_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_Alambda_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_MS_FT_MZ} \\
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_Lambda_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_Alambda_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_MS_FT_MZ}
\caption{Fine tuning vs. $\lambda$, $A_\lambda$ and $M_s$ in the DiracNMSSM (first line)
and GNMSSM (second line). The color code is the same as in Fig.~\ref{fig:compFT1}.}
\label{fig:compFT2}
\end{figure}
The fine tuning as function of the additional superpotential parameters $\lambda$ and $M_s$ or $\mu_s$ is shown in Fig.~\ref{fig:compFT2}. While $\lambda$ plays an important role in lifting the Higgs mass to the desired range, only a mild preference towards very large $\lambda$ in terms of the fine tuning is seen. On the other hand the dependence on $M_s$ is much more pronounced. First, there are hardly any valid points for $|M_s| < 1$~TeV. The reason for this in the DiracNMSSM has already been discussed in sec.~\ref{sec:HiggsTree}: usually, large values of $M_s^2$ are needed to get a large enough Higgs mass or even three positive eigenvalues of the Higgs mass matrix squared. Similarly, it is very difficult to find valid parameter points in the GNMSSM which don't suffer from tachyonic states in the Higgs sector\footnote{This seems at odds with a number of NMSSM studies, which of course have $\mu_s=0$. One should note however that the density of valid points in the NMSSM is significantly smaller than in the
MSSM, which
indirectly shows up here.}. Furthermore, even for $|M_s| > 4$~TeV there is a strong correlation between the fine tuning and the value of $M_s$ in the DiracNMSSM after the Higgs cut is applied. Usually very large values of $M_s$ are needed to reduce the fine tuning. The reason is that the tree-level mass of the SM like Higgs increases with $M_s$ and therefore for large values of $M_s$ the necessity of large loop corrections due to heavy stops is reduced.
That's different to the GNMSSM where the relation between the fine tuning and $\mu_s$ is roughly flat in this range even after the Higgs mass cut.
\begin{figure}[tb]
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_vS_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_mu_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_Bmu_FT_MZ} \\
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_vS_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_mu_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_Bmu_FT_MZ}
\caption{Fine tuning vs.\ $v_s$, $\mu$ and $b\mu$ in the DiracNMSSM (first line)
and GNMSSM (second line). Here $\mu$ and $b\mu$ are given at the SUSY scale. The color code is the same as in Fig.~\ref{fig:compFT1}.}
\label{fig:compFT4}
\end{figure}
In Fig.~\ref{fig:compFT4} we show the fine tuning as function of $v_s$. While the fine tuning in the GNMSSM shows hardly any dependence on the singlet VEV after all cuts, the fine tuning in the DiracNMSSM increases with increasing $|v_s|$. Thus, singlet VEVs below 1~TeV are preferred in the DiracNMSSM. One might be surprised that the fine tuning depends on $v_s$ at all since this parameter does not enter eq.~(\ref{eq:measure}). However, one can see from eq.~(\ref{eq:tadMu}) that large $|v_s|$ also leads in general to larger $|\mu|$. That the fine tuning in the MSSM strongly depends on $\mu$ is well known. As we can see in the middle of Fig.~\ref{fig:compFT4} this is also the case for the DiracNMSSM and the GNMSSM. $|\mu|$ should be not larger than a few hundred GeV to have good fine tuning.
\begin{figure}[tb]
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_mHd2_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_mHu2_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/DiracNMSSMnuhm_comp_mS2_FT_MZ} \\
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_mHd2_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_mHu2_FT_MZ}
\hfill
\includegraphics[width=0.3\linewidth]{Plots/GNMSSMnuhm_comp_mS2_FT_MZ}
\caption{Fine tuning vs. $m_{h_d}^2$, $m_{h_u}^2$ and $m_s^2$ in the DiracNMSSM (first line)
and GNMSSM (second line). The color code is the same as in Fig.~\ref{fig:compFT2}.}
\label{fig:compFT3}
\end{figure}
From the model building point of view, the main difference between the GNMSSM and DiracNMSSM is the extended singlet sector of the DiracNMSSM. A second singlet was introduced to allow for smaller values of $m_s^2$ without decoupling of the tree-level contribution to the Higgs mass for large singlet masses. This in turn was supposed to reduce the fine tuning in the DiracNMSSM. However, the plots in the last row in Fig.~\ref{fig:compFT3} show that the fine tuning with respect to $m_s^2$ is rather mild and also in the GNMSSM there are points after all cuts with $m_s^2=0$. This might be the main reason why there is not a significant improvement in the fine tuning in the DiracNMSSM compared to the GNMSSM. Actually, it turns out that the fine tuning in both models is roughly the same for universal gaugino masses as we have seen.
\subsection{Non-universal gaugino masses}
\label{sec:nonuniversal}
As we have mentioned before, non-universal gaugino masses tend to improve the fine tuning through the appearance of a new ``focus point'' that makes the Higgs mass less sensitive to the gaugino mass scale~\cite{Choi:2005hd,Choi:2006xb,Abe:2007kf,Lebedev:2005ge,Horton:2009ed,Asano:2012sv,Antusch:2012gv,Abe:2012xm,Badziak:2012yg,Gogoladze:2012yf,Yanagida:2013ah}.
We assume that $a$ and $b$ are fixed by the underlying theory such that their contributions to the fine tuning are absent. As discussed in \cite{Kaminska:2013mya, Horton:2009ed} values of $a$ and $b$ in the low-focus-point region occur naturally in a variety of models.
\begin{figure}[!h!]
\centering
\includegraphics[width=0.45\linewidth]{Plots/DiracNMSSMFTAll}
\includegraphics[width=0.45\linewidth]{Plots/GNMSSMFT}
\caption{The fine tuning as a function of the SM like Higgs mass allowing for non-universal gaugino masses
for the DiracNMSSM (left) and the GNMSSM (right).
The light blue points are before any cuts.
For the dark blue points we use appropriate SUSY and dark matter cuts. The minimal FT we find in the DiracNMSSM is $\Delta=32$, in the GNMSSM it is $\Delta=14$.}
\label{fig:4}
\end{figure}
We show in Fig.~\ref{fig:4} the overall fine tuning vs.\ the mass of the SM-like Higgs for this case. We see that in both models it improves significantly. The best fine tuning we find now for the DiracNMSSM fulfilling all experimental constraints and satisfying the upper limit of the neutralino abundance is $\Delta \simeq 32$, for the GNMSSM we even find points with $\Delta \simeq 14$ \footnote{However, a strict ratio of the wino to gluino masses is not necessary as long as both are comparable at the weak scale; in the case of the GNMSSM, including the fine tuning with respect to a and b, the minimum fine tuning is still only 35. Hence, dropping the assumption of definite a and b ratios does not greatly increase the fine tuning.}. The plot also shows that before SUSY cuts the fine tuning in both models is very similar.
We find the difference between the two models in the case of nonuniversal gaugino masses to be the following: The mass of the higgsino-like neutralino is set
by $\mu_\text{eff}=\mu + \tfrac{1}{\sqrt{2}} \lambda_{EW} v_s$, while the fine tuning is mainly dominated by $\mu$. In the GNMSSM it seems to be easier to have sizable $\mu_\text{eff} \gtrsim 700 \:\text{Ge\eVdist V}$ which would allow for a compressed spectrum and hence lighter gluinos. In the DiracNMSSM on the other hand $\mu_\text{eff}$ is typically closer to $\mu$.
This means that for very compressed spectra with $m_\text{higgsino} \sim m_\text{gluino} \sim 700 \:\text{Ge\eVdist V}$ the $\mu$ term is sizable, implying moderate fine tuning.
On the other hand for very small values of $\mu$ we typically have a rather light higgsino-like LSP. In this case a compressed spectrum is not possible and the gluino has to be
correspondingly heavier, implying larger fine tuning.
\begin{figure}[!h!]
\begin{center}
\includegraphics[width=0.49\linewidth]{Plots/DiracNMSSMFThiggssquarkgluino}
\includegraphics[width=0.49\linewidth]{Plots/DiracNMSSMFThiggslspgluino}
\includegraphics[width=0.49\linewidth]{Plots/GNMSSMFThiggssquarkgluino}
\includegraphics[width=0.49\linewidth]{Plots/GNMSSMFThiggslspgluino}
\caption{Smallest fine tuning in the gluino--squark and gluino--LSP plane for the DiracNMSSM (top)
and the GNMSSM (bottom).
It can be seen that low fine tuning often corresponds to compressed spectra.}
\label{GNMSSMfinetuningab}
\end{center}
\end{figure}
In Fig.~\ref{GNMSSMfinetuningab} we present the best fine tuning we find in the $(m_{\rm gluino},m_{\rm squark})$ and in the $(m_{\rm gluino}, m_{\rm LSP})$ plane. We see that a very good fine tuning is not only possible close to the parameter regions where the gluino is nearly degenerate with the LSP but also for heavy gluinos above 2~TeV. Also note that our scans were optimized to find the smallest fine tuning, so it is conceivable that the fine-tuning for large masses is overestimated. The heavy gluino case will be hard to exclude by the next LHC runs \cite{Ulmer:2013csa}. Thus, despite the excellent performance of the LHC experiments there is still the possibility of SUSY with a fine tuning less than 100 which can't be tested in the near future. However, to find models with this small fine tuning for gluinos above 2~TeV one has to give up the most constrained models with universal soft parameters for all scalars and all gauginos.
\section{Summary and Conclusions}
\label{sec:conclusion}
We have performed a careful evaluation of the level of fine tuning in the DiracNMSSM. For this purpose we have implemented this model in public computer tools to get a precise prediction for the Higgs mass, the dark matter relic abundance, and the SUSY mass spectrum. We have considered rather general high scale boundary conditions -- in particular we assumed that the Higgs and singlet soft terms are independent of $m_0$ and of each other. Also all the A-terms were taken to be independent. If we force the gaugino mass terms to unify at the GUT scale the minimal fine tuning allowed by all experimental constraints is about 70 for both the DiracNMSSM and the GNMSSM. If we relax the unification conditions in the gaugino sector the minimal fine tuning gets improved to 32 in the DiracNMSSM and 14 in the GNMSSM.
Hence, both models significantly improve the fine tuning situation compared to the MSSM and the level of fine tuning is comparable in both models, albeit slightly lower in the GNMSSM.
\section*{Acknowledgements}
The research presented here was partially supported by the EU ITN grant UNILHC 237920 (Unification in the LHC era).
FS is supported by the BMBF
PT DESY Verbundprojekt 05H2013-THEORIE 'Vergleich von LHC-Daten mit supersymmetrischen Modellen'.
\begin{appendix}
\section{Renormalisation Group Equations}
\label{app:RGEs}
For those parameters present in the MSSM, we give only the difference to the RGEs in comparison to the MSSM. The convention for the $\beta$ function and
anomalous dimensions are
\begin{equation}
\beta_x = \frac{1}{16\pi^2} \beta^{(1)}_x+\frac{1}{(16\pi^2)^2} \beta^{(2)}_x\,,\hspace{1cm}
\gamma_x = \frac{1}{16\pi^2} \gamma^{(1)}_x+\frac{1}{(16\pi^2)^2} \gamma^{(2)}_x
\end{equation}
The calculation of the $\beta$-functions is performed by {\tt SARAH}\xspace. The calculation of $\beta$-functions for SUSY models
up to two loop with {\tt SARAH}\xspace are based on the generic results given in Refs.~\cite{Martin:1993zk,Yamada:1994id,Fonseca:2011vn,Goodsell:2012fm,Sperling:2013eva,Sperling:2013xqa}.
\subsection{Anomalous Dimensions}
{\allowdisplaybreaks \begin{align}
\Delta \gamma_{\hat{q}}^{(2)} & =
- |\lambda|^2 \Big({Y_{d}^{\dagger} Y_d} + {Y_{u}^{\dagger} Y_u}\Big)\\
\Delta \gamma_{\hat{l}}^{(2)} & =
- |\lambda|^2 {Y_{e}^{\dagger} Y_e} \\
\Delta \gamma_{\hat{H}_d}^{(1)} & =
|\lambda|^2\\
\Delta \gamma_{\hat{H}_d}^{(2)} & =
-3 |\lambda|^2 \Big(|\lambda|^2 + \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big)\Big)\\
\Delta \gamma_{\hat{H}_u}^{(1)} & =
|\lambda|^2\\
\Delta \gamma_{\hat{H}_u}^{(2)} & =
- |\lambda|^2 \Big(3 |\lambda|^2 + 3 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big)\Big)\\
\Delta \gamma_{\hat{d}}^{(2)} & =
-2 |\lambda|^2 {Y_d^* Y_{d}^{T}} \\
\Delta \gamma_{\hat{u}}^{(2)} & =
-2 |\lambda|^2 {Y_u^* Y_{u}^{T}} \\
\Delta \gamma_{\hat{e}}^{(2)} & =
-2 |\lambda|^2 {Y_e^* Y_{e}^{T}} \\
\gamma_{\hat{s}}^{(1)} & =
2 |\lambda|^2 \\
\gamma_{\hat{s}}^{(2)} & =
-\frac{2}{5} |\lambda|^2 \Big(10 |\lambda|^2 -15 g_{2}^{2} + 15 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 15 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) -3 g_{1}^{2} + 5 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \Big)\\
\gamma_{\hat{\bar{s}}}^{(1)} & = 0 \\
\gamma_{\hat{\bar{s}}}^{(2)} & = 0
\end{align} }
\subsection{Gauge Couplings}
{\allowdisplaybreaks \begin{align}
\Delta \beta_{g_1}^{(2)} & =
-\frac{6}{5} g_{1}^{3} |\lambda|^2 \\
\Delta \beta_{g_2}^{(2)} & =
-2 g_{2}^{3} |\lambda|^2
\end{align}}
\subsection{Gaugino Mass Parameters}
{\allowdisplaybreaks \begin{align}
\Delta \beta_{M_1}^{(2)} & =
-\frac{12}{5} g_{1}^{2} \lambda^* \Big(M_1 \lambda - T_{\lambda} \Big)\\
\Delta \beta_{M_2}^{(2)} & =
4 g_{2}^{2} \lambda^* \Big(- M_2 \lambda + T_{\lambda}\Big)
\end{align}}
\subsection{Trilinear Superpotential Parameters}
{\allowdisplaybreaks \begin{align}
\Delta \beta_{Y_d}^{(1)} & =
Y_d |\lambda|^2 \\
\Delta \beta_{Y_d}^{(2)} & =
- |\lambda|^2 \Big(3 Y_d |\lambda|^2 + 3 Y_d \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + 3 {Y_d Y_{d}^{\dagger} Y_d} + {Y_d Y_{u}^{\dagger} Y_u}\Big)\\
\Delta \beta_{Y_e}^{(1)} & =
Y_e |\lambda|^2 \\
\Delta \beta_{Y_e}^{(2)} & =
-3 |\lambda|^2 \Big(Y_e |\lambda|^2 + Y_e \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + {Y_e Y_{e}^{\dagger} Y_e}\Big)\\
\Delta \beta_{Y_u}^{(1)} & =
Y_u |\lambda|^2 \\
\Delta \beta_{Y_u}^{(2)} & =
- |\lambda|^2 \Big(3 Y_u |\lambda|^2 + 3 Y_u \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 3 {Y_u Y_{u}^{\dagger} Y_u} + Y_u \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) + {Y_u Y_{d}^{\dagger} Y_d}\Big)\\
\beta_{\lambda}^{(1)} & =
-3 g_{2}^{2} \lambda + 3 \lambda \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 3 \lambda \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + 4 \lambda^{2} \lambda^* -\frac{3}{5} g_{1}^{2} \lambda + \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \\
\beta_{\lambda}^{(2)} & =
-\frac{1}{50} \lambda \Big(500 |\lambda|^4-207 g_{1}^{4} -90 g_{1}^{2} g_{2}^{2} -375 g_{2}^{4} +20 \Big(g_{1}^{2}-40 g_{3}^{2} \Big)\mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) \nonumber \\
&-20 g_{1}^{2} \mbox{Tr}\Big(3{Y_e Y_{e}^{\dagger}}+2{Y_u Y_{u}^{\dagger}}\Big) -30 |\lambda|^2 \Big(10 g_{2}^{2} -5\mbox{Tr}\Big(3{Y_d Y_{d}^{\dagger}}+3{Y_u Y_{u}^{\dagger}}+{Y_e Y_{e}^{\dagger}}\Big) + 2 g_{1}^{2} \Big) \nonumber \\
&-800 g_{3}^{2} \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) +15 \mbox{Tr}\Big(3{Y_d Y_{d}^{\dagger} Y_d Y_{d}^{\dagger}}+2{Y_d Y_{u}^{\dagger} Y_u Y_{d}^{\dagger}}+{Y_e Y_{e}^{\dagger} Y_e Y_{e}^{\dagger}}+3{Y_u Y_{u}^{\dagger} Y_u Y_{u}^{\dagger}}\Big) \Big)
\end{align}}
\subsection{Bilinear Superpotential Parameters}
{\allowdisplaybreaks \begin{align}
\Delta \beta_{\mu}^{(1)} & =
2 \mu |\lambda|^2 \\
\Delta \beta_{\mu}^{(2)} & =
- \mu |\lambda|^2 \Big(3 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 3 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + 6 |\lambda|^2 + \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big)\Big)\\
\beta_{M_s}^{(1)} & =
2 M_s |\lambda|^2 \\
\beta_{M_s}^{(2)} & =
-\frac{2}{5} M_s |\lambda|^2 \Big(10 |\lambda|^2 -15 g_{2}^{2} + 15 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 15 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) -3 g_{1}^{2} + 5 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \Big)
\end{align}}
\subsection{Trilinear Soft-Breaking Parameters}
{\allowdisplaybreaks \begin{align}
\Delta \beta_{T_d}^{(1)} & =
\lambda^* \Big(2 Y_d T_{\lambda} + \lambda T_d \Big)\\
\Delta \beta_{T_d}^{(2)} & =
- \lambda^* \Big(3 |\lambda|^2 \Big(4 Y_d T_{\lambda} + \lambda T_d \Big)+2 T_{\lambda} \Big(3 Y_d \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + 3 {Y_d Y_{d}^{\dagger} Y_d} + {Y_d Y_{u}^{\dagger} Y_u}\Big)\nonumber \\
&+\lambda \Big(2 {Y_d Y_{u}^{\dagger} T_u} + 3 T_d \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + 4 {Y_d Y_{d}^{\dagger} T_d} + 5 {T_d Y_{d}^{\dagger} Y_d} + 6 Y_d \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) + {T_d Y_{u}^{\dagger} Y_u}\Big)\Big)\\
\Delta \beta_{T_e}^{(1)} & =
\lambda^* \Big(2 Y_e T_{\lambda} + \lambda T_e \Big)\\
\Delta \beta_{T_e}^{(2)} & =
- \lambda^* \Big(3 |\lambda|^2 \Big(4 Y_e T_{\lambda} + \lambda T_e \Big)+6 T_{\lambda} \Big(Y_e \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + {Y_e Y_{e}^{\dagger} Y_e}\Big)\nonumber \\
&+\lambda \Big(3 T_e \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + 4 {Y_e Y_{e}^{\dagger} T_e} + 5 {T_e Y_{e}^{\dagger} Y_e} + 6 Y_e \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) \Big)\Big)\\
\Delta \beta_{T_u}^{(1)} & =
\lambda^* \Big(2 Y_u T_{\lambda} + \lambda T_u \Big)\\
\Delta \beta_{T_u}^{(2)} & =
- \lambda^* \Big(3 |\lambda|^2 \Big(4 Y_u T_{\lambda} + \lambda T_u \Big)+2 T_{\lambda} \Big(Y_u \mbox{Tr}\Big(3{Y_d Y_{d}^{\dagger}}+{Y_e Y_{e}^{\dagger}}\Big) + 3 {Y_u Y_{u}^{\dagger} Y_u} + {Y_u Y_{d}^{\dagger} Y_d}\Big)\nonumber \\
&+\lambda \Big(2 {Y_u Y_{d}^{\dagger} T_d} +4 {Y_u Y_{u}^{\dagger} T_u} +{T_u Y_{d}^{\dagger} Y_d}+5 {T_u Y_{u}^{\dagger} Y_u} +3 T_u \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) +T_u \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \nonumber \\
&+6 Y_u \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) +2 Y_u \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) \Big)\Big)\\
\beta_{T_{\lambda}}^{(1)} & =
+\frac{6}{5} g_{1}^{2} M_1 \lambda +6 g_{2}^{2} M_2 \lambda +T_{\lambda} \Big(12 |\lambda|^2 -3 g_{2}^{2} + \mbox{Tr}\Big(3{Y_d Y_{d}^{\dagger}}+{Y_e Y_{e}^{\dagger}}+ 3 {Y_u Y_{u}^{\dagger}}\Big) -\frac{3}{5} g_{1}^{2}\Big)\nonumber \\
&+6 \lambda \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) +2 \lambda \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) +6 \lambda \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) \\
\beta_{T_{\lambda}}^{(2)} & =
-50 |\lambda|^4 T_{\lambda} -\frac{3}{5} |\lambda|^2 \Big(T_{\lambda} \Big(15 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) -30 g_{2}^{2} + 45 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}+{Y_u Y_{u}^{\dagger}}\Big) -6 g_{1}^{2} \Big)\nonumber \\
&+2 \lambda \Big(10 g_{2}^{2} M_2 + 15 \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) + 15 \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) + 2 g_{1}^{2} M_1 + 5 \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) \Big)\Big)\nonumber \\
&+T_{\lambda} \Big(\frac{207}{50} g_{1}^{4} +\frac{9}{5} g_{1}^{2} g_{2}^{2} +\frac{15}{2} g_{2}^{4} -\frac{2}{5} \Big(g_{1}^{2}-40 g_{3}^{2}\Big)\mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) +\frac{2}{5} g_{1}^{2} \mbox{Tr}\Big(3{Y_e Y_{e}^{\dagger}}+2{Y_u Y_{u}^{\dagger}}\Big) \nonumber \\
&+16 g_{3}^{2} \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) -3\mbox{Tr}\Big(3{Y_d Y_{d}^{\dagger} Y_d Y_{d}^{\dagger}} +2 {Y_d Y_{u}^{\dagger} Y_u Y_{d}^{\dagger}}+{Y_e Y_{e}^{\dagger} Y_e Y_{e}^{\dagger}}+3{Y_u Y_{u}^{\dagger} Y_u Y_{u}^{\dagger}}\Big) \Big)\nonumber \\
&-\frac{2}{25} \lambda \Big(207 g_{1}^{4} M_1 +45 g_{1}^{2} g_{2}^{2} (M_1 +M_2) +375 g_{2}^{4} M_2 -10 \Big( g_{1}^{2} M_1 -40 g_{3}^{2} M_3\Big)\mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) \nonumber \\
&+30 g_{1}^{2} M_1 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) +20 g_{1}^{2} M_1 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) +400 g_{3}^{2} M_3 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) +10 g_{1}^{2} \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) \nonumber \\
&-400 g_{3}^{2} \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) -30 g_{1}^{2} \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) -20 g_{1}^{2} \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) -400 g_{3}^{2} \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) \nonumber \\
&+150 \mbox{Tr}\Big(3{Y_d Y_{d}^{\dagger} T_d Y_{d}^{\dagger}}+{Y_d Y_{u}^{\dagger} T_u Y_{d}^{\dagger}}+{Y_e Y_{e}^{\dagger} T_e Y_{e}^{\dagger}}+{Y_u Y_{d}^{\dagger} T_d Y_{u}^{\dagger}}+3{Y_u Y_{u}^{\dagger} T_u Y_{u}^{\dagger}}\Big) \Big)
\end{align}}
\subsection{Bilinear Soft-Breaking Parameters}
{\allowdisplaybreaks \begin{align}
\Delta \beta_{B{\mu}}^{(1)} & =
2 \lambda^* \Big(2 \mu T_{\lambda} + 3 \lambda B{\mu} \Big)\\
\Delta \beta_{B{\mu}}^{(2)} & =
-\frac{1}{5} \lambda^* \Big(\lambda B{\mu} \Big(25 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) -36 g_{1}^{2}-180 g_{2}^{2} + 70 |\lambda|^2 + 75 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}+{Y_u Y_{u}^{\dagger}}\Big) \Big)\nonumber \\
&+2 \mu \Big(80 |\lambda|^2 T_{\lambda} +5 T_{\lambda} \Big(3 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 3 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big)\Big)\nonumber \\
&+3 \lambda \Big(15 \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) + 15 \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) + 30 g_{2}^{2} M_2 + 5 \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) + 6 g_{1}^{2} M_1 \Big)\Big)\Big)\\
\beta_{b_s}^{(1)} & =
2 \lambda^* \Big(2 M_s T_{\lambda} + \lambda b_s \Big)\\
\beta_{b_s}^{(2)} & =
-\frac{2}{5} \lambda^* \Big(\lambda b_s \Big(10 |\lambda|^2 -15 g_{2}^{2} + 15 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 15 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) -3 g_{1}^{2} + 5 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \Big)\nonumber \\
&+2 M_s \Big(T_{\lambda} \Big(-15 g_{2}^{2} + 15 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 15 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + 20 |\lambda|^2 -3 g_{1}^{2} + 5 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \Big)\nonumber \\
&+\lambda \Big(15 g_{2}^{2} M_2 + 15 \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) + 15 \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) + 3 g_{1}^{2} M_1 + 5 \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) \Big)\Big)\Big)
\end{align}}
\subsection{Linear Soft-Breaking Parameters}
{\allowdisplaybreaks \begin{align}
\beta_{t_s}^{(1)} & =
2 \lambda^* \Big(2 \xi_s T_{\lambda} + \lambda t_s \Big) + 4 B{\mu}^* T_{\lambda} + 4 \Big(m_{h_d}^2 + m_{h_u}^2\Big)\lambda \mu^* \\
\beta_{t_s}^{(2)} & =
-\frac{2}{5} \Big(10 \lambda \lambda^{*,2} \Big(4 \xi_s T_{\lambda} + \lambda t_s \Big)+\lambda^* \Big(20 \Big(2 m_{h_d}^2 + 2 m_{h_u}^2 + m_{s}^2\Big)\lambda^{2} \mu^* \nonumber \\
&+\lambda t_s \Big(15 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) -3 \Big(5 g_{2}^{2} -5 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + g_{1}^{2}\Big) + 5 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \Big)\nonumber \\
&+2 \Big(T_{\lambda} \Big(20 \lambda B{\mu}^* + \xi_s \Big(15 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) -3 \Big(5 g_{2}^{2} -5 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + g_{1}^{2}\Big) + 5 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \Big)\Big)\nonumber \\
&+\xi_s \lambda \Big(15 \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) + 3 \Big(5 g_{2}^{2} M_2 + 5 \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) + g_{1}^{2} M_1 \Big) + 5 \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) \Big)\Big)\Big)\nonumber \\
&+2 \Big(B{\mu}^* \Big(T_{\lambda} \Big(15 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) -3 \Big(5 g_{2}^{2} -5 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + g_{1}^{2}\Big) + 5 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \Big)\nonumber \\
&+\lambda \Big(15 \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) + 3 \Big(5 g_{2}^{2} M_2 + 5 \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) + g_{1}^{2} M_1 \Big) + 5 \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) \Big)\Big)\nonumber \\
&+\mu^* \Big(-3 g_{1}^{2} m_{h_d}^2 \lambda -15 g_{2}^{2} m_{h_d}^2 \lambda -3 g_{1}^{2} m_{h_u}^2 \lambda -15 g_{2}^{2} m_{h_u}^2 \lambda -6 g_{1}^{2} \lambda |M_1|^2 -30 g_{2}^{2} \lambda |M_2|^2 \nonumber \\
&+20 \lambda |T_{\lambda}|^2 +3 g_{1}^{2} M_1 T_{\lambda} +15 g_{2}^{2} M_2 T_{\lambda} +30 m_{h_d}^2 \lambda \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) +15 m_{h_u}^2 \lambda \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) \nonumber \\
&+10 m_{h_d}^2 \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) +5 m_{h_u}^2 \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) +15 m_{h_d}^2 \lambda \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) +30 m_{h_u}^2 \lambda \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) \nonumber \\
&+15 T_{\lambda} \mbox{Tr}\Big({T_d^* Y_{d}^{T}}\Big) +15 \lambda \mbox{Tr}\Big({T_d^* T_{d}^{T}}\Big) +5 T_{\lambda} \mbox{Tr}\Big({T_e^* Y_{e}^{T}}\Big) +5 \lambda \mbox{Tr}\Big({T_e^* T_{e}^{T}}\Big) +15 T_{\lambda} \mbox{Tr}\Big({T_u^* Y_{u}^{T}}\Big) \nonumber \\
&+15 \lambda \mbox{Tr}\Big({T_u^* T_{u}^{T}}\Big) +15 \lambda \mbox{Tr}\Big({Y_d Y_{d}^{\dagger} m_d^{2 *}}\Big) +15 \lambda \mbox{Tr}\Big({Y_d m_q^{2 *} Y_{d}^{\dagger}}\Big) +5 \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger} m_e^{2 *}}\Big) \nonumber \\
&+5 \lambda \mbox{Tr}\Big({Y_e m_l^{2 *} Y_{e}^{\dagger}}\Big) +15 \lambda \mbox{Tr}\Big({Y_u Y_{u}^{\dagger} m_u^{2 *}}\Big) +15 \lambda \mbox{Tr}\Big({Y_u m_q^{2 *} Y_{u}^{\dagger}}\Big) \Big)\Big)\Big)\\
\beta_{t_{\bar{s}}}^{(1)} & =
4 M_s B{\mu} \lambda^* \\
\beta_{t_{\bar{s}}}^{(2)} & =
-\frac{4}{5} M_s \lambda^* \Big(B{\mu} \Big(10 |\lambda|^2 -15 g_{2}^{2} + 15 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 15 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) -3 g_{1}^{2} + 5 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \Big)\nonumber \\
&+\mu \Big(10 \lambda^* T_{\lambda} + 15 g_{2}^{2} M_2 + 15 \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) + 15 \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) + 3 g_{1}^{2} M_1 + 5 \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) \Big)\Big)
\end{align}}
\subsection{Soft-Breaking Scalar Masses}
{\allowdisplaybreaks \begin{align}
\Delta \beta_{m_q^2}^{(2)} & =
-2 T_{\lambda}^* \Big(\lambda \Big({Y_{d}^{\dagger} T_d} + {Y_{u}^{\dagger} T_u}\Big) + \Big({Y_{d}^{\dagger} Y_d} + {Y_{u}^{\dagger} Y_u}\Big)T_{\lambda} \Big)\nonumber \\
&- \lambda^* \Big(2 \Big(2 m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)\lambda {Y_{d}^{\dagger} Y_d} +2 \Big(2 m_{h_u}^2 + m_{h_d}^2 + m_{s}^2\Big)\lambda {Y_{u}^{\dagger} Y_u} \nonumber \\
&+\lambda {m_q^2 Y_{d}^{\dagger} Y_d} +\lambda {m_q^2 Y_{u}^{\dagger} Y_u} +2 \lambda {Y_{d}^{\dagger} m_d^2 Y_d} +\lambda {Y_{d}^{\dagger} Y_d m_q^2} +2 \lambda {Y_{u}^{\dagger} m_u^2 Y_u} \nonumber \\
&+2 \lambda {T_{d}^{\dagger} T_d} +2 \lambda {T_{u}^{\dagger} T_u} +\lambda {Y_{u}^{\dagger} Y_u m_q^2} +2 {T_{d}^{\dagger} Y_d} T_{\lambda} +2 {T_{u}^{\dagger} Y_u} T_{\lambda} \Big)\\
\Delta \beta_{m_l^2}^{(2)} & =
-2 T_{\lambda}^* \Big(\lambda {Y_{e}^{\dagger} T_e} + {Y_{e}^{\dagger} Y_e} T_{\lambda} \Big)- \lambda^* \Big(2 \Big(2 m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)\lambda {Y_{e}^{\dagger} Y_e} \nonumber \\
&+\lambda \Big(2 {T_{e}^{\dagger} T_e} + 2 {Y_{e}^{\dagger} m_e^2 Y_e} + {m_l^2 Y_{e}^{\dagger} Y_e} + {Y_{e}^{\dagger} Y_e m_l^2}\Big)+2 {T_{e}^{\dagger} Y_e} T_{\lambda} \Big)\\
\Delta \beta_{m_{h_d}^2}^{(1)} & =
2 \Big(\Big(m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)|\lambda|^2 + |T_{\lambda}|^2\Big)\\
\Delta \beta_{m_{h_d}^2}^{(2)} & =
-6 \Big(2 \Big(m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)|\lambda|^4 +T_{\lambda}^* \Big(\lambda \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) + T_{\lambda} \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) \Big)\nonumber \\
&+\lambda^* \Big(4 \lambda |T_{\lambda}|^2 +\Big(2 m_{h_u}^2 + m_{h_d}^2 + m_{s}^2\Big)\lambda \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) +T_{\lambda} \mbox{Tr}\Big({T_u^* Y_{u}^{T}}\Big) +\lambda \mbox{Tr}\Big({T_u^* T_{u}^{T}}\Big) \nonumber \\
& +\lambda \mbox{Tr}\Big({m_q^2 Y_{u}^{\dagger} Y_u}\Big)+\lambda \mbox{Tr}\Big({m_u^2 Y_u Y_{u}^{\dagger}}\Big) \Big)\Big)\\
\Delta \beta_{m_{h_u}^2}^{(1)} & =
2 \Big(\Big(m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)|\lambda|^2 + |T_{\lambda}|^2\Big)\\
\Delta \beta_{m_{h_u}^2}^{(2)} & =
-2 \Big(6 \Big(m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)|\lambda|^4 +T_{\lambda}^* \Big(\lambda \Big(3 \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) + \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big)\Big) \nonumber \\
& + T_{\lambda} \Big(3 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big)\Big)\Big)+\lambda^* \Big(12 \lambda |T_{\lambda}|^2 \nonumber \\
&+3 \Big(2 m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)\lambda \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) +2 m_{h_d}^2 \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) +m_{h_u}^2 \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \nonumber \\
&+m_{s}^2 \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) +3 T_{\lambda} \mbox{Tr}\Big({T_d^* Y_{d}^{T}}\Big) +3 \lambda \mbox{Tr}\Big({T_d^* T_{d}^{T}}\Big) +T_{\lambda} \mbox{Tr}\Big({T_e^* Y_{e}^{T}}\Big) +\lambda \mbox{Tr}\Big({T_e^* T_{e}^{T}}\Big) \nonumber \\
&+3 \lambda \mbox{Tr}\Big({m_d^2 Y_d Y_{d}^{\dagger}}\Big) +\lambda \mbox{Tr}\Big({m_e^2 Y_e Y_{e}^{\dagger}}\Big) +\lambda \mbox{Tr}\Big({m_l^2 Y_{e}^{\dagger} Y_e}\Big) +3 \lambda \mbox{Tr}\Big({m_q^2 Y_{d}^{\dagger} Y_d}\Big) \Big)\Big)\\
\Delta \beta_{m_d^2}^{(2)} & =
-2 \Big(2 T_{\lambda}^* \Big(\lambda {T_d Y_{d}^{\dagger}} + {Y_d Y_{d}^{\dagger}} T_{\lambda} \Big)+\lambda^* \Big(2 \Big(2 m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)\lambda {Y_d Y_{d}^{\dagger}} \nonumber \\
&+\lambda \Big(2 {T_d T_{d}^{\dagger}} + 2 {Y_d m_q^2 Y_{d}^{\dagger}} + {m_d^2 Y_d Y_{d}^{\dagger}} + {Y_d Y_{d}^{\dagger} m_d^2}\Big)+2 {Y_d T_{d}^{\dagger}} T_{\lambda} \Big)\Big)\\
\Delta \beta_{m_u^2}^{(2)} & =
-2 \Big(2 T_{\lambda}^* \Big(\lambda {T_u Y_{u}^{\dagger}} + {Y_u Y_{u}^{\dagger}} T_{\lambda} \Big)+\lambda^* \Big(2 \Big(2 m_{h_u}^2 + m_{h_d}^2 + m_{s}^2\Big)\lambda {Y_u Y_{u}^{\dagger}} \nonumber \\
&+\lambda \Big(2 {T_u T_{u}^{\dagger}} + 2 {Y_u m_q^2 Y_{u}^{\dagger}} + {m_u^2 Y_u Y_{u}^{\dagger}} + {Y_u Y_{u}^{\dagger} m_u^2}\Big)+2 {Y_u T_{u}^{\dagger}} T_{\lambda} \Big)\Big)\\
\Delta \beta_{m_e^2}^{(2)} & =
-2 \Big(2 T_{\lambda}^* \Big(\lambda {T_e Y_{e}^{\dagger}} + {Y_e Y_{e}^{\dagger}} T_{\lambda} \Big)+\lambda^* \Big(2 \Big(2 m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)\lambda {Y_e Y_{e}^{\dagger}} \nonumber \\
&+\lambda \Big(2 {T_e T_{e}^{\dagger}} + 2 {Y_e m_l^2 Y_{e}^{\dagger}} + {m_e^2 Y_e Y_{e}^{\dagger}} + {Y_e Y_{e}^{\dagger} m_e^2}\Big)+2 {Y_e T_{e}^{\dagger}} T_{\lambda} \Big)\Big)\\
\beta_{m_{s}^2}^{(1)} & =
4 \Big(\Big(m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)|\lambda|^2 + |T_{\lambda}|^2\Big)\\
\beta_{m_{s}^2}^{(2)} & =
-\frac{4}{5} \Big(20 \Big(m_{h_d}^2 + m_{h_u}^2 + m_{s}^2\Big)|\lambda|^4 \nonumber \\
&+T_{\lambda}^* \Big(T_{\lambda} \Big(15 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) -3 \Big(5 g_{2}^{2} -5 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + g_{1}^{2}\Big) + 5 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \Big)\nonumber \\
&+\lambda \Big(15 \mbox{Tr}\Big({Y_{d}^{\dagger} T_d}\Big) + 3 \Big(5 g_{2}^{2} M_2 + 5 \mbox{Tr}\Big({Y_{u}^{\dagger} T_u}\Big) + g_{1}^{2} M_1 \Big) + 5 \mbox{Tr}\Big({Y_{e}^{\dagger} T_e}\Big) \Big)\Big)\nonumber \\
&+\lambda^* \Big(-3 g_{1}^{2} m_{h_d}^2 \lambda -15 g_{2}^{2} m_{h_d}^2 \lambda -3 g_{1}^{2} m_{h_u}^2 \lambda -15 g_{2}^{2} m_{h_u}^2 \lambda -3 g_{1}^{2} m_{s}^2 \lambda -15 g_{2}^{2} m_{s}^2 \lambda \nonumber \\
&+40 \lambda |T_{\lambda}|^2 +3 g_{1}^{2} M_1^* \Big(T_{\lambda}-2 M_1 \lambda\Big)+15 g_{2}^{2} M_2^* \Big(T_{\lambda}-2 M_2 \lambda \Big)+30 m_{h_d}^2 \lambda \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) \nonumber \\
&+15 m_{h_u}^2 \lambda \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) +15 m_{s}^2 \lambda \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) +10 m_{h_d}^2 \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) +5 m_{h_u}^2 \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) \nonumber \\
&+5 m_{s}^2 \lambda \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) +15 m_{h_d}^2 \lambda \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) +30 m_{h_u}^2 \lambda \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) +15 m_{s}^2 \lambda \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) \nonumber \\
&+5 T_{\lambda} \mbox{Tr}\Big(3{T_d^* Y_{d}^{T}}+3{T_u^* Y_{u}^{T}}+{T_e^* Y_{e}^{T}}\Big) +5 \lambda \mbox{Tr}\Big(3{T_d^* T_{d}^{T}}+{T_e^* T_{e}^{T}}+3{T_u^* T_{u}^{T}}\Big)\nonumber \\
& +5 \lambda \mbox{Tr}\Big(3{m_d^2 Y_d Y_{d}^{\dagger}}+{m_e^2 Y_e Y_{e}^{\dagger}}+{m_l^2 Y_{e}^{\dagger} Y_e}+3{m_q^2 Y_{d}^{\dagger} Y_d}+3{m_q^2 Y_{u}^{\dagger} Y_u}+3{m_u^2 Y_u Y_{u}^{\dagger}}\Big) \Big)\Big)\\
\beta_{m_{\bar{s}}^2}^{(1)} & =
0\\
\beta_{m_{\bar{s}}^2}^{(2)} & =
0
\end{align}}
\subsection{Vacuum expectation values}
{\allowdisplaybreaks \begin{align}
\Delta \beta_{v_d}^{(1)} & =
- v_d |\lambda|^2 \\
\Delta \beta_{v_d}^{(2)} & =
\frac{3}{10} v_d |\lambda|^2 \Big(10 |\lambda|^2 + 10 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) - \Big(5 g_{2}^{2} + g_{1}^{2}\Big)\xi \Big)\\
\Delta \beta_{v_u}^{(1)} & =
- v_u |\lambda|^2 \\
\Delta \beta_{v_u}^{(2)} & =
\frac{1}{10} v_u |\lambda|^2 \Big(10 \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big) + 30 |\lambda|^2 + 30 \mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) -3( g_{1}^{2} + 5 g_{2}^{2}) \xi \Big)\\
\beta_{v_{s}}^{(1)} & =
-2 v_{s} |\lambda|^2 \\
\beta_{v_{s}}^{(2)} & =
\frac{2}{5} v_{s} |\lambda|^2 \Big(10 |\lambda|^2 -15 g_{2}^{2} + 5 \left(3\mbox{Tr}\Big({Y_d Y_{d}^{\dagger}}\Big) + 3 \mbox{Tr}\Big({Y_u Y_{u}^{\dagger}}\Big) + \mbox{Tr}\Big({Y_e Y_{e}^{\dagger}}\Big)\right) -3 g_{1}^{2} \Big)\\
\beta_{v_{\bar{s}}}^{(1)} & =
0\\
\beta_{v_{\bar{s}}}^{(2)} & =
0
\end{align}}
\section{Mass matrices}
\label{app:matrices}
\begin{itemize}
\item {\bf Mass matrix for Charginos}, Basis: \( \left(\tilde{W}^-, \tilde{H}_d^-\right), \left(\tilde{W}^+, \tilde{H}_u^+\right) \)
\begin{equation}
m_{\tilde{\chi}^-} = \left(
\begin{array}{cc}
M_2 &\frac{1}{\sqrt{2}} g_2 v_u \\
\frac{1}{\sqrt{2}} g_2 v_d &\frac{1}{\sqrt{2}} v_{s} \lambda + \mu\end{array}
\right)
\end{equation}
This matrix is diagonalized by \(U\) and \(V\)
\begin{equation}
U^* m_{\tilde{\chi}^-} V^{\dagger} = m^{dia}_{\tilde{\chi}^-}
\end{equation}
\item {\bf Mass matrix for CP odd Higgs}, Basis:$\left(\sigma_{d}, \sigma_{u}, \sigma_s, \sigma_{\bar{s}}\right)$
In Landau gauge the mass matrix is given by
\begin{equation}
m^2_{A^0} = \left(
\begin{array}{cccc}
m_{\sigma_{d}\sigma_{d}} &m_{\sigma_{u}\sigma_{d}} &\frac{1}{\sqrt{2}} v_u {\Re\Big(T_{\lambda}\Big)} &- \frac{1}{\sqrt{2}} v_u {\Re\Big(\lambda M_s^* \Big)} \\
m_{\sigma_{d}\sigma_{u}} &m_{\sigma_{u}\sigma_{u}} &\frac{1}{\sqrt{2}} v_d {\Re\Big(T_{\lambda}\Big)} &- \frac{1}{\sqrt{2}} v_d {\Re\Big(\lambda M_s^* \Big)} \\
\frac{1}{\sqrt{2}} v_u {\Re\Big(T_{\lambda}\Big)} &\frac{1}{\sqrt{2}} v_d {\Re\Big(T_{\lambda}\Big)} &m_{\sigma_s\sigma_s} &- {\Re\Big(b_s\Big)} \\
- \frac{1}{\sqrt{2}} v_u {\Re\Big(\lambda M_s^* \Big)} &- \frac{1}{\sqrt{2}} v_d {\Re\Big(\lambda M_s^* \Big)} &- {\Re\Big(b_s\Big)} &m_{\bar{s}}^2 + |M_s|^2\end{array}
\right)
\end{equation}
with
\begin{align}
m_{\sigma_{d}\sigma_{d}} &= |\mu|^2 + \sqrt{2} v_{s} \Re(\lambda \mu^*)+ \Big(v_{s}^{2} + v_{u}^{2}\Big)\frac{|\lambda|^2}{2}+ \frac{1}{8} \Big(g_{1}^{2} + g_{2}^{2}\Big)\Big(v_{d}^{2}- v_{u}^{2} \Big) + m_{h_d}^2\\
m_{\sigma_{d}\sigma_{u}} &= \frac{1}{4} \Big(4 {\Re\Big(B{\mu}\Big)} + 4 {\Re\Big(\lambda \xi_s^* \Big)} + \sqrt{2} \Big(2 v_{\bar{s}} {\Re\Big(\lambda M_s^* \Big)} + 2 v_{s} {\Re\Big(T_{\lambda}\Big)} \Big)\Big)\\
m_{\sigma_{u}\sigma_{u}} &= |\mu|^2 + \sqrt{2} v_{s} \Re(\lambda \mu^*) + \Big(v_{d}^{2} + v_{s}^{2}\Big)\frac{|\lambda|^2}{2} -\frac{1}{8} \Big(g_{1}^{2} + g_{2}^{2}\Big)\Big(v_{d}^{2}- v_{u}^{2} \Big) + m_{h_u}^2\\
m_{\sigma_s\sigma_s} &= \frac{1}{2} \Big(v_{d}^{2} + v_{u}^{2}\Big)|\lambda|^2 + m_{s}^2 + |M_s|^2
\end{align}
The gauge fixing part is the same as in the MSSM:
\begin{equation}
m^2 (\xi_{Z}) = \frac{1}{4} \left(
\begin{array}{cccc}
v_d^2 &-v_d v_u &0 &0\\
-v_d v_u & v_u^2 &0 &0\\
0 &0 &0 &0\\
0 &0 &0 &0\end{array}
\right) \Big(g_1 \sin\Theta_W + g_2 \cos\Theta_W \Big)^{2}
\end{equation}
This matrix is diagonalized by \(Z^A\):
\begin{equation}
Z^A m^2_{A^0} Z^{A,\dagger} = m^{dia}_{2,A^0}
\end{equation}
\item {\bf Mass matrix for Charged Higgs}, Basis: \( \left(H_d^-, H_u^{+,*}\right), \left(H_d^{-,*}, H_u^+\right) \)
\begin{equation}
m^2_{H^-} = \left(
\begin{array}{cc}
m_{H_d^-H_d^{-,*}} &m^*_{H_u^{+,*}H_d^{-,*}}\\
m_{H_d^-H_u^+} &m_{H_u^{+,*}H_u^+}\end{array}
\right)
\end{equation}
with
\begin{align}
m_{H_d^-H_d^{-,*}} &= |\mu|^2 + \sqrt{2} v_{s} \Re(\lambda \mu^*) + v^2_{s} \frac{|\lambda|^2}{2}+ \frac{1}{8} \Big(g_{1}^{2} \Big(v_{d}^{2}- v_{u}^{2} \Big) + g_{2}^{2} \Big(v_{d}^{2} + v_{u}^{2}\Big)\Big) + m_{h_d}^2\\
m_{H_d^-H_u^+} &= \frac{1}{4} g_{2}^{2} v_d v_u + \frac{1}{\sqrt{2}} v_{s} T_{\lambda} + \lambda \Big(-\frac{1}{2} v_d v_u \lambda^* + \frac{1}{\sqrt{2}} v_{\bar{s}} M_s^* + \xi_s^*\Big) + B{\mu}\\
m_{H_u^{+,*}H_u^+} &= |\mu|^2 + \sqrt{2} v_{s} \Re(\lambda \mu^*) + v^2_{s} \frac{|\lambda|^2}{2}+ \frac{1}{8} \Big(g_{1}^{2} \Big(v_{u}^{2}- v_{d}^{2} \Big) + g_{2}^{2} \Big(v_{d}^{2} + v_{u}^{2}\Big)\Big) + m_{h_u}^2
\end{align}
and
\begin{equation}
m^2 (\xi_{W^-}) = \frac{1}{4} g_{2}^{2}\left(
\begin{array}{cc}
v_{d}^{2} &-v_d v_u \\
- v_d v_u & v_{u}^{2} \end{array}
\right)
\end{equation}
This matrix is diagonalized by \(Z^+\):
\begin{equation}
Z^+ m^2_{H^-} Z^{+,\dagger} = m^{dia}_{2,H^-}
\end{equation}
\end{itemize}
\section{Vertices}
\label{app:vertices}
We give here the difference of Higgs and neutralino vertices in comparison to the MSSM. The shown expressions are understood as
\begin{equation}
\Delta \Gamma_{fields} = \Gamma_{fields}^{\text{DiracNMSSM}} - \Gamma_{fields}^{\text{MSSM}}
\end{equation}
Chiral vertices are parametrized by
\begin{equation}
\Gamma^L_{fields} P_L + \Gamma^R_{fields} P_R
\end{equation}
with the projection operators $P_L$ and $P_R$.
\subsection{Interactions with Fermions}
\begin{align}
\Delta& \Gamma^L_{\tilde{\chi}^+_{{i}}\tilde{\chi}^-_{{j}}A^0_{{k}}} = \,
\frac{1}{\sqrt{2}} \lambda U^*_{j 2} V^*_{i 2} Z_{{k 3}}^{A} \\
\Delta& \Gamma^R_{\tilde{\chi}^+_{{i}}\tilde{\chi}^-_{{j}}A^0_{{k}}} = \,- \frac{1}{\sqrt{2}} \lambda^* U_{{i 2}} V_{{j 2}} Z_{{k 3}}^{A}
\\
\Delta& \Gamma^L_{\tilde{\chi}^0_{{i}}\tilde{\chi}^0_{{j}}A^0_{{k}}} = \,
- \frac{1}{\sqrt{2}} \lambda \Big(N^*_{i 3} \Big(N^*_{j 4} Z_{{k 3}}^{A} + N^*_{j 5} Z_{{k 2}}^{A} \Big) + N^*_{i 4} \Big(N^*_{j 3} Z_{{k 3}}^{A} + N^*_{j 5} Z_{{k 1}}^{A} \Big) + N^*_{i 5} \Big(N^*_{j 3} Z_{{k 2}}^{A} + N^*_{j 4} Z_{{k 1}}^{A} \Big)\Big)\\
\Delta& \Gamma^R_{\tilde{\chi}^0_{{i}}\tilde{\chi}^0_{{j}}A^0_{{k}}} = \,\frac{1}{\sqrt{2}} \lambda^* \Big(Z_{{k 1}}^{A} \Big(N_{{i 4}} N_{{j 5}} + N_{{i 5}} N_{{j 4}} \Big) + Z_{{k 2}}^{A} \Big(N_{{i 3}} N_{{j 5}} + N_{{i 5}} N_{{j 3}} \Big) + Z_{{k 3}}^{A} \Big(N_{{i 3}} N_{{j 4}} + N_{{i 4}} N_{{j 3}} \Big)\Big)
\\
\Delta& \Gamma^L_{\tilde{\chi}^0_{{i}}\tilde{\chi}^-_{{j}}H^+_{{k}}} = \,
-i \lambda U^*_{j 2} N^*_{i 5} Z_{{k 2}}^{+} \\
\Delta& \Gamma^R_{\tilde{\chi}^0_{{i}}\tilde{\chi}^-_{{j}}H^+_{{k}}} = \,-i \lambda^* V_{{j 2}} N_{{i 5}} Z_{{k 1}}^{+}
\\
\Delta& \Gamma^L_{\tilde{\chi}^+_{{i}}\tilde{\chi}^-_{{j}}h_{{k}}} = \,
-i \frac{1}{\sqrt{2}} \lambda U^*_{j 2} V^*_{i 2} Z_{{k 3}}^{H} \\
\Delta& \Gamma^R_{\tilde{\chi}^+_{{i}}\tilde{\chi}^-_{{j}}h_{{k}}} = \,-i \frac{1}{\sqrt{2}} \lambda^* U_{{i 2}} V_{{j 2}} Z_{{k 3}}^{H} \\
\Delta& \Gamma^L_{\tilde{\chi}^0_{{i}}\tilde{\chi}^0_{{j}}h_{{k}}} = \,
i \frac{1}{\sqrt{2}} \lambda \Big(N^*_{i 3} \Big(N^*_{j 4} Z_{{k 3}}^{H} + N^*_{j 5} Z_{{k 2}}^{H} \Big) + N^*_{i 4} \Big(N^*_{j 3} Z_{{k 3}}^{H} + N^*_{j 5} Z_{{k 1}}^{H} \Big) + N^*_{i 5} \Big(N^*_{j 3} Z_{{k 2}}^{H} + N^*_{j 4} Z_{{k 1}}^{H} \Big)\Big)\\
\Delta& \Gamma^R_{\tilde{\chi}^0_{{i}}\tilde{\chi}^0_{{j}}h_{{k}}} = \,i \frac{1}{\sqrt{2}} \lambda^* \Big(Z_{{k 1}}^{H} \Big(N_{{i 4}} N_{{j 5}} + N_{{i 5}} N_{{j 4}} \Big) + Z_{{k 2}}^{H} \Big(N_{{i 3}} N_{{j 5}} + N_{{i 5}} N_{{j 3}} \Big) + Z_{{k 3}}^{H} \Big(N_{{i 3}} N_{{j 4}} + N_{{i 4}} N_{{j 3}} \Big)\Big)
\\
\Delta& \Gamma^L_{\tilde{\chi}^+_{{i}}\tilde{\chi}^0_{{j}}H^-_{{k}}} = \,-i \lambda V^*_{i 2} N^*_{j 5} Z^{+,*}_{k 1} \\
\Delta& \Gamma^R_{\tilde{\chi}^+_{{i}}\tilde{\chi}^0_{{j}}H^-_{{k}}} = \,-i \lambda^* Z^{+,*}_{k 2} U_{{i 2}} N_{{j 5}}
\end{align}
\subsection{Three scalar interactions}
\begin{align}
\Delta &\Gamma_{A^0_{{i}}A^0_{{j}}h_{{k}}} = \,
-\frac{i}{4} \Big(\lambda^* \Big(4 \lambda Z_{{i 3}}^{A} Z_{{j 3}}^{A} \Big(v_d Z_{{k 1}}^{H} + v_u Z_{{k 2}}^{H} \Big)+2 Z_{{i 1}}^{A} Z_{{j 1}}^{A} \Big(\Big(2 v_{s} \lambda + \sqrt{2} \mu \Big)Z_{{k 3}}^{H} + 2 v_u \lambda Z_{{k 2}}^{H} \Big)\nonumber \\
&+Z_{{i 2}}^{A} \Big(2 Z_{{j 2}}^{A} \Big(2 v_d \lambda Z_{{k 1}}^{H} + \Big(2 v_{s} \lambda + \sqrt{2} \mu \Big)Z_{{k 3}}^{H} \Big) + \sqrt{2} M_s \Big(Z_{{j 1}}^{A} Z_{{k 4}}^{H} - Z_{{j 4}}^{A} Z_{{k 1}}^{H} \Big)\Big)\nonumber \\
&- \sqrt{2} M_s \Big(Z_{{i 1}}^{A} \Big(- Z_{{j 2}}^{A} Z_{{k 4}}^{H} + Z_{{j 4}}^{A} Z_{{k 2}}^{H} \Big) + Z_{{i 4}}^{A} \Big(Z_{{j 1}}^{A} Z_{{k 2}}^{H} + Z_{{j 2}}^{A} Z_{{k 1}}^{H} \Big)\Big)\Big)\nonumber \\
&+\sqrt{2} \Big(T_{\lambda}^* \Big(\Big(Z_{{i 1}}^{A} Z_{{j 2}}^{A} + Z_{{i 2}}^{A} Z_{{j 1}}^{A} \Big)Z_{{k 3}}^{H} + Z_{{i 3}}^{A} \Big(Z_{{j 1}}^{A} Z_{{k 2}}^{H} + Z_{{j 2}}^{A} Z_{{k 1}}^{H} \Big) + Z_{{j 3}}^{A} \Big(Z_{{i 1}}^{A} Z_{{k 2}}^{H} + Z_{{i 2}}^{A} Z_{{k 1}}^{H} \Big)\Big)\nonumber \\
&+T_{\lambda} \Big(\Big(Z_{{i 1}}^{A} Z_{{j 2}}^{A} + Z_{{i 2}}^{A} Z_{{j 1}}^{A} \Big)Z_{{k 3}}^{H} + Z_{{i 3}}^{A} \Big(Z_{{j 1}}^{A} Z_{{k 2}}^{H} + Z_{{j 2}}^{A} Z_{{k 1}}^{H} \Big) + Z_{{j 3}}^{A} \Big(Z_{{i 1}}^{A} Z_{{k 2}}^{H} + Z_{{i 2}}^{A} Z_{{k 1}}^{H} \Big)\Big)\nonumber \\
&+\lambda \Big(2 \mu^* \Big(Z_{{i 1}}^{A} Z_{{j 1}}^{A} + Z_{{i 2}}^{A} Z_{{j 2}}^{A} \Big)Z_{{k 3}}^{H} +M_s^* \Big(\Big(Z_{{i 1}}^{A} Z_{{j 2}}^{A} + Z_{{i 2}}^{A} Z_{{j 1}}^{A} \Big)Z_{{k 4}}^{H} - \Big(Z_{{i 1}}^{A} Z_{{j 4}}^{A} + Z_{{i 4}}^{A} Z_{{j 1}}^{A} \Big)Z_{{k 2}}^{H} \nonumber \\
& - \Big(Z_{{i 2}}^{A} Z_{{j 4}}^{A} + Z_{{i 4}}^{A} Z_{{j 2}}^{A} \Big)Z_{{k 1}}^{H} \Big)\Big)\Big)\Big)
\\
\Delta &\Gamma_{A^0_{{i}}\tilde{d}_{{j \beta}}\tilde{d}^*_{{k \gamma}}} = \,
\frac{1}{2} \delta_{\beta \gamma} \Big(- \lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{a b}} Z^{D,*}_{j 3 + a} Z_{{k b}}^{D} + \lambda^* \sum_{b=1}^{3}Z^{D,*}_{j b} \sum_{a=1}^{3}Y_{d,{a b}} Z_{{k 3 + a}}^{D} \Big)\Big(v_{s} Z_{{i 2}}^{A} + v_u Z_{{i 3}}^{A} \Big)
\\
\Delta &\Gamma_{A^0_{{i}}\tilde{e}_{{j}}\tilde{e}^*_{{k}}} = \,
\frac{1}{2} \Big(- \lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{a b}} Z^{E,*}_{j 3 + a} Z_{{k b}}^{E} + \lambda^* \sum_{b=1}^{3}Z^{E,*}_{j b} \sum_{a=1}^{3}Y_{e,{a b}} Z_{{k 3 + a}}^{E} \Big)\Big(v_{s} Z_{{i 2}}^{A} + v_u Z_{{i 3}}^{A} \Big)
\\
\Delta &\Gamma_{A^0_{{i}}\tilde{u}_{{j \beta}}\tilde{u}^*_{{k \gamma}}} = \,
\frac{1}{2} \delta_{\beta \gamma} \Big(- \lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{u,{a b}} Z^{U,*}_{j 3 + a} Z_{{k b}}^{U} + \lambda^* \sum_{b=1}^{3}Z^{U,*}_{j b} \sum_{a=1}^{3}Y_{u,{a b}} Z_{{k 3 + a}}^{U} \Big)\Big(v_d Z_{{i 3}}^{A} + v_{s} Z_{{i 1}}^{A} \Big)
\\
\Delta &\Gamma_{h_{{i}}h_{{j}}h_{{k}}} = \,
\frac{i}{4} \Big(- \lambda^* \Big(2 Z_{{i 1}}^{H} \Big(2 \lambda Z_{{j 2}}^{H} \Big(v_d Z_{{k 2}}^{H} + v_u Z_{{k 1}}^{H} \Big)+Z_{{j 3}}^{H} \Big(2 v_d \lambda Z_{{k 3}}^{H} + \Big(2 v_{s} \lambda + \sqrt{2} \mu \Big)Z_{{k 1}}^{H} \Big)\nonumber \\
&+Z_{{j 1}}^{H} \Big(\Big(2 v_{s} \lambda + \sqrt{2} \mu \Big)Z_{{k 3}}^{H} + 2 v_u \lambda Z_{{k 2}}^{H} \Big)\Big)+2 Z_{{i 3}}^{H} \Big(\sqrt{2} \mu \Big(Z_{{j 1}}^{H} Z_{{k 1}}^{H} + Z_{{j 2}}^{H} Z_{{k 2}}^{H} \Big)\nonumber \\
&+2 \lambda \Big(Z_{{j 1}}^{H} \Big(v_d Z_{{k 3}}^{H} + v_{s} Z_{{k 1}}^{H} \Big) + Z_{{j 2}}^{H} \Big(v_{s} Z_{{k 2}}^{H} + v_u Z_{{k 3}}^{H} \Big) + Z_{{j 3}}^{H} \Big(v_d Z_{{k 1}}^{H} + v_u Z_{{k 2}}^{H} \Big)\Big)\Big)\nonumber \\
&+Z_{{i 2}}^{H} \Big(- \sqrt{2} M_s Z_{{j 4}}^{H} Z_{{k 1}}^{H} +2 Z_{{j 3}}^{H} \Big(\Big(2 v_{s} \lambda + \sqrt{2} \mu \Big)Z_{{k 2}}^{H} + 2 v_u \lambda Z_{{k 3}}^{H} \Big)\nonumber \\
&+2 Z_{{j 2}}^{H} \Big(2 v_d \lambda Z_{{k 1}}^{H} + \Big(2 v_{s} \lambda + \sqrt{2} \mu \Big)Z_{{k 3}}^{H} \Big)+Z_{{j 1}}^{H} \Big(4 v_d \lambda Z_{{k 2}}^{H} + 4 v_u \lambda Z_{{k 1}}^{H} - \sqrt{2} M_s Z_{{k 4}}^{H} \Big)\Big)\nonumber \\
&- \sqrt{2} M_s \Big(Z_{{i 1}}^{H} \Big(Z_{{j 2}}^{H} Z_{{k 4}}^{H} + Z_{{j 4}}^{H} Z_{{k 2}}^{H} \Big) + Z_{{i 4}}^{H} \Big(Z_{{j 1}}^{H} Z_{{k 2}}^{H} + Z_{{j 2}}^{H} Z_{{k 1}}^{H} \Big)\Big)\Big)\nonumber \\
&+\sqrt{2} \Big(\Big(T_{\lambda}^* + T_{\lambda}\Big)\Big(Z_{{i 1}}^{H} \Big(Z_{{j 2}}^{H} Z_{{k 3}}^{H} + Z_{{j 3}}^{H} Z_{{k 2}}^{H} \Big) + Z_{{i 2}}^{H} \Big(Z_{{j 1}}^{H} Z_{{k 3}}^{H} + Z_{{j 3}}^{H} Z_{{k 1}}^{H} \Big) + Z_{{i 3}}^{H} \Big(Z_{{j 1}}^{H} Z_{{k 2}}^{H} + Z_{{j 2}}^{H} Z_{{k 1}}^{H} \Big)\Big)\nonumber \\
&+\lambda \Big(-2 \mu^* \Big(\Big(Z_{{i 1}}^{H} Z_{{j 1}}^{H} + Z_{{i 2}}^{H} Z_{{j 2}}^{H} \Big)Z_{{k 3}}^{H} + Z_{{i 3}}^{H} \Big(Z_{{j 1}}^{H} Z_{{k 1}}^{H} + Z_{{j 2}}^{H} Z_{{k 2}}^{H} \Big) + Z_{{j 3}}^{H} \Big(Z_{{i 1}}^{H} Z_{{k 1}}^{H} + Z_{{i 2}}^{H} Z_{{k 2}}^{H} \Big)\Big)\nonumber \\
&+M_s^* \Big(\Big(Z_{{i 1}}^{H} Z_{{j 2}}^{H} + Z_{{i 2}}^{H} Z_{{j 1}}^{H} \Big)Z_{{k 4}}^{H} + Z_{{i 4}}^{H} \Big(Z_{{j 1}}^{H} Z_{{k 2}}^{H} + Z_{{j 2}}^{H} Z_{{k 1}}^{H} \Big) + Z_{{j 4}}^{H} \Big(Z_{{i 1}}^{H} Z_{{k 2}}^{H} + Z_{{i 2}}^{H} Z_{{k 1}}^{H} \Big)\Big)\Big)\Big)\Big)
\\
\Delta &\Gamma_{h_{{i}}H^-_{{j}}H^+_{{k}}} = \,
-\frac{i}{2} \Big(\sqrt{2} \Big(\lambda \mu^* Z_{{i 3}}^{H} \Big(Z^{+,*}_{j 1} Z_{{k 1}}^{+} + Z^{+,*}_{j 2} Z_{{k 2}}^{+} \Big) + T_{\lambda}^* Z^{+,*}_{j 2} Z_{{i 3}}^{H} Z_{{k 1}}^{+} \nonumber \\
& + Z^{+,*}_{j 1} \Big(\lambda M_s^* Z_{{i 4}}^{H} + T_{\lambda} Z_{{i 3}}^{H} \Big)Z_{{k 2}}^{+} \Big) +\lambda^* \Big(Z^{+,*}_{j 1} \Big(\Big(2 v_{s} \lambda + \sqrt{2} \mu \Big)Z_{{i 3}}^{H} Z_{{k 1}}^{+} - \lambda \Big(v_d Z_{{i 2}}^{H} + v_u Z_{{i 1}}^{H} \Big)Z_{{k 2}}^{+} \Big)\nonumber \\
&+Z^{+,*}_{j 2} \Big(2 v_{s} \lambda Z_{{i 3}}^{H} Z_{{k 2}}^{+} + \sqrt{2} M_s Z_{{i 4}}^{H} Z_{{k 1}}^{+} + \sqrt{2} \mu Z_{{i 3}}^{H} Z_{{k 2}}^{+} - v_d \lambda Z_{{i 2}}^{H} Z_{{k 1}}^{+} - v_u \lambda Z_{{i 1}}^{H} Z_{{k 1}}^{+} \Big)\Big)\Big)
\\
\Delta &\Gamma_{h_{{i}}\tilde{d}_{{j \beta}}\tilde{d}^*_{{k \gamma}}} = \,
\frac{i}{2} \delta_{\beta \gamma} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{a b}} Z^{D,*}_{j 3 + a} Z_{{k b}}^{D} + \lambda^* \sum_{b=1}^{3}Z^{D,*}_{j b} \sum_{a=1}^{3}Y_{d,{a b}} Z_{{k 3 + a}}^{D} \Big)\Big(v_{s} Z_{{i 2}}^{H} + v_u Z_{{i 3}}^{H} \Big)
\\
\Delta &\Gamma_{h_{{i}}\tilde{e}_{{j}}\tilde{e}^*_{{k}}} = \,
\frac{i}{2} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{a b}} Z^{E,*}_{j 3 + a} Z_{{k b}}^{E} + \lambda^* \sum_{b=1}^{3}Z^{E,*}_{j b} \sum_{a=1}^{3}Y_{e,{a b}} Z_{{k 3 + a}}^{E} \Big)\Big(v_{s} Z_{{i 2}}^{H} + v_u Z_{{i 3}}^{H} \Big)
\\
\Delta &\Gamma_{h_{{i}}\tilde{u}_{{j \beta}}\tilde{u}^*_{{k \gamma}}} = \,
\frac{i}{2} \delta_{\beta \gamma} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{u,{a b}} Z^{U,*}_{j 3 + a} Z_{{k b}}^{U} + \lambda^* \sum_{b=1}^{3}Z^{U,*}_{j b} \sum_{a=1}^{3}Y_{u,{a b}} Z_{{k 3 + a}}^{U} \Big)\Big(v_d Z_{{i 3}}^{H} + v_{s} Z_{{i 1}}^{H} \Big)
\\
\Delta &\Gamma_{H^-_{{i}}\tilde{u}_{{j \beta}}\tilde{d}^*_{{k \gamma}}} = \,
i \frac{1}{\sqrt{2}} v_{s} \delta_{\beta \gamma} \Big(\lambda Z^{+,*}_{i 1} \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{u,{a b}} Z^{U,*}_{j 3 + a} Z_{{k b}}^{D} + \lambda^* Z^{+,*}_{i 2} \sum_{b=1}^{3}Z^{U,*}_{j b} \sum_{a=1}^{3}Y_{d,{a b}} Z_{{k 3 + a}}^{D} \Big)
\\
\Delta &\Gamma_{H^-_{{i}}\tilde{\nu}_{{j}}\tilde{e}^*_{{k}}} = \,
i \frac{1}{\sqrt{2}} v_{s} \lambda^* Z^{+,*}_{i 2} \sum_{b=1}^{3}Z^{V,*}_{j b} \sum_{a=1}^{3}Y_{e,{a b}} Z_{{k 3 + a}}^{E}
\\
\Delta &\Gamma_{\tilde{d}_{{i \alpha}}H^+_{{j}}\tilde{u}^*_{{k \gamma}}} = \,
i \frac{1}{\sqrt{2}} v_{s} \delta_{\alpha \gamma} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{a b}} Z^{D,*}_{i 3 + a} Z_{{k b}}^{U} Z_{{j 2}}^{+} + \lambda^* \sum_{b=1}^{3}Z^{D,*}_{i b} \sum_{a=1}^{3}Y_{u,{a b}} Z_{{k 3 + a}}^{U} Z_{{j 1}}^{+} \Big)
\\
\Delta &\Gamma_{\tilde{e}_{{i}}H^+_{{j}}\tilde{\nu}^*_{{k}}} = \,
i \frac{1}{\sqrt{2}} v_{s} \lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{a b}} Z^{E,*}_{i 3 + a} Z_{{k b}}^{V} Z_{{j 2}}^{+}
\end{align}
\subsection{Four scalar interactions}
\begin{align}
\Delta &\Gamma_{A^0_{{i}}A^0_{{j}}A^0_{{k}}A^0_{{l}}} = \,
-i |\lambda|^2 \times \nonumber \\
& \Big(Z_{{i 3}}^{A} \Big(Z_{{j 1}}^{A} \Big(Z_{{k 1}}^{A} Z_{{l 3}}^{A} + Z_{{k 3}}^{A} Z_{{l 1}}^{A} \Big) + Z_{{j 2}}^{A} \Big(Z_{{k 2}}^{A} Z_{{l 3}}^{A} + Z_{{k 3}}^{A} Z_{{l 2}}^{A} \Big) + Z_{{j 3}}^{A} \Big(Z_{{k 1}}^{A} Z_{{l 1}}^{A} + Z_{{k 2}}^{A} Z_{{l 2}}^{A} \Big)\Big)\nonumber \\
&+Z_{{i 2}}^{A} \Big(Z_{{j 1}}^{A} \Big(Z_{{k 1}}^{A} Z_{{l 2}}^{A} + Z_{{k 2}}^{A} Z_{{l 1}}^{A} \Big) + Z_{{j 2}}^{A} \Big(Z_{{k 1}}^{A} Z_{{l 1}}^{A} + Z_{{k 3}}^{A} Z_{{l 3}}^{A} \Big) + Z_{{j 3}}^{A} \Big(Z_{{k 2}}^{A} Z_{{l 3}}^{A} + Z_{{k 3}}^{A} Z_{{l 2}}^{A} \Big)\Big)\nonumber \\
&+Z_{{i 1}}^{A} \Big(Z_{{j 1}}^{A} \Big(Z_{{k 2}}^{A} Z_{{l 2}}^{A} + Z_{{k 3}}^{A} Z_{{l 3}}^{A} \Big) + Z_{{j 2}}^{A} \Big(Z_{{k 1}}^{A} Z_{{l 2}}^{A} + Z_{{k 2}}^{A} Z_{{l 1}}^{A} \Big) + Z_{{j 3}}^{A} \Big(Z_{{k 1}}^{A} Z_{{l 3}}^{A} + Z_{{k 3}}^{A} Z_{{l 1}}^{A} \Big)\Big)\Big)
\\
\Delta &\Gamma_{A^0_{{i}}A^0_{{j}}h_{{k}}h_{{l}}} = \,
-i |\lambda|^2 \Big(Z_{{i 3}}^{A} Z_{{j 3}}^{A} \Big(Z_{{k 1}}^{H} Z_{{l 1}}^{H} + Z_{{k 2}}^{H} Z_{{l 2}}^{H} \Big)+Z_{{i 2}}^{A} Z_{{j 2}}^{A} \Big(Z_{{k 1}}^{H} Z_{{l 1}}^{H} + Z_{{k 3}}^{H} Z_{{l 3}}^{H} \Big)\nonumber \\
&\hspace{3cm} +Z_{{i 1}}^{A} Z_{{j 1}}^{A} \Big(Z_{{k 2}}^{H} Z_{{l 2}}^{H} + Z_{{k 3}}^{H} Z_{{l 3}}^{H} \Big)\Big)
\\
\Delta &\Gamma_{A^0_{{i}}A^0_{{j}}H^-_{{k}}H^+_{{l}}} = \,
-\frac{i}{2} |\lambda|^2 \Big(Z^{+,*}_{k 1} \Big(2 Z_{{i 3}}^{A} Z_{{j 3}}^{A} Z_{{l 1}}^{+} + \Big(Z_{{i 1}}^{A} Z_{{j 2}}^{A} + Z_{{i 2}}^{A} Z_{{j 1}}^{A} \Big)Z_{{l 2}}^{+} \Big)\nonumber \\
&\hspace{3cm} +Z^{+,*}_{k 2} \Big(2 Z_{{i 3}}^{A} Z_{{j 3}}^{A} Z_{{l 2}}^{+} + Z_{{i 1}}^{A} Z_{{j 2}}^{A} Z_{{l 1}}^{+} + Z_{{i 2}}^{A} Z_{{j 1}}^{A} Z_{{l 1}}^{+} \Big)\Big)
\\
\Delta &\Gamma_{A^0_{{i}}A^0_{{j}}\tilde{d}_{{k \gamma}}\tilde{d}^*_{{l \delta}}} = \,
-\frac{i}{2} \delta_{\gamma \delta} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{a b}} Z^{D,*}_{k 3 + a} Z_{{l b}}^{D} + \lambda^* \sum_{b=1}^{3}Z^{D,*}_{k b} \sum_{a=1}^{3}Y_{d,{a b}} Z_{{l 3 + a}}^{D} \Big)\Big(Z_{{i 2}}^{A} Z_{{j 3}}^{A} + Z_{{i 3}}^{A} Z_{{j 2}}^{A} \Big)
\\
\Delta &\Gamma_{A^0_{{i}}A^0_{{j}}\tilde{e}_{{k}}\tilde{e}^*_{{l}}} = \,
-\frac{i}{2} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{a b}} Z^{E,*}_{k 3 + a} Z_{{l b}}^{E} + \lambda^* \sum_{b=1}^{3}Z^{E,*}_{k b} \sum_{a=1}^{3}Y_{e,{a b}} Z_{{l 3 + a}}^{E} \Big)\Big(Z_{{i 2}}^{A} Z_{{j 3}}^{A} + Z_{{i 3}}^{A} Z_{{j 2}}^{A} \Big)
\\
\Delta &\Gamma_{A^0_{{i}}A^0_{{j}}\tilde{u}_{{k \gamma}}\tilde{u}^*_{{l \delta}}} = \,
-\frac{i}{2} \delta_{\gamma \delta} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{u,{a b}} Z^{U,*}_{k 3 + a} Z_{{l b}}^{U} + \lambda^* \sum_{b=1}^{3}Z^{U,*}_{k b} \sum_{a=1}^{3}Y_{u,{a b}} Z_{{l 3 + a}}^{U} \Big)\Big(Z_{{i 1}}^{A} Z_{{j 3}}^{A} + Z_{{i 3}}^{A} Z_{{j 1}}^{A} \Big)
\\
\Delta &\Gamma_{A^0_{{i}}h_{{j}}H^-_{{k}}H^+_{{l}}} = \,
-\frac{1}{2} |\lambda|^2 \Big(Z_{{i 1}}^{A} Z_{{j 2}}^{H} + Z_{{i 2}}^{A} Z_{{j 1}}^{H} \Big)\Big(- Z^{+,*}_{k 1} Z_{{l 2}}^{+} + Z^{+,*}_{k 2} Z_{{l 1}}^{+} \Big)
\\
\Delta &\Gamma_{A^0_{{i}}H^-_{{j}}\tilde{u}_{{k \gamma}}\tilde{d}^*_{{l \delta}}} = \,
\frac{1}{\sqrt{2}} \delta_{\gamma \delta} \Big(- \lambda Z^{+,*}_{j 1} \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{u,{a b}} Z^{U,*}_{k 3 + a} Z_{{l b}}^{D} + \lambda^* Z^{+,*}_{j 2} \sum_{b=1}^{3}Z^{U,*}_{k b} \sum_{a=1}^{3}Y_{d,{a b}} Z_{{l 3 + a}}^{D} \Big)Z_{{i 3}}^{A}
\\
\Delta &\Gamma_{A^0_{{i}}H^-_{{j}}\tilde{\nu}_{{k}}\tilde{e}^*_{{l}}} = \,
\frac{1}{\sqrt{2}} \lambda^* Z^{+,*}_{j 2} \sum_{b=1}^{3}Z^{V,*}_{k b} \sum_{a=1}^{3}Y_{e,{a b}} Z_{{l 3 + a}}^{E} Z_{{i 3}}^{A}
\\
\Delta &\Gamma_{A^0_{{i}}\tilde{d}_{{j \beta}}H^+_{{k}}\tilde{u}^*_{{l \delta}}} = \,
\frac{1}{\sqrt{2}} \delta_{\beta \delta} Z_{{i 3}}^{A} \Big(- \lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{a b}} Z^{D,*}_{j 3 + a} Z_{{l b}}^{U} Z_{{k 2}}^{+} + \lambda^* \sum_{b=1}^{3}Z^{D,*}_{j b} \sum_{a=1}^{3}Y_{u,{a b}} Z_{{l 3 + a}}^{U} Z_{{k 1}}^{+} \Big)
\\
\Delta &\Gamma_{A^0_{{i}}\tilde{e}_{{j}}H^+_{{k}}\tilde{\nu}^*_{{l}}} = \,
- \frac{1}{\sqrt{2}} \lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{a b}} Z^{E,*}_{j 3 + a} Z_{{l b}}^{V} Z_{{i 3}}^{A} Z_{{k 2}}^{+}
\\
\Delta &\Gamma_{h_{{i}}h_{{j}}h_{{k}}h_{{l}}} = \,
-i |\lambda|^2 \times \nonumber \\
& \Big(Z_{{i 3}}^{H} \Big(Z_{{j 1}}^{H} \Big(Z_{{k 1}}^{H} Z_{{l 3}}^{H} + Z_{{k 3}}^{H} Z_{{l 1}}^{H} \Big) + Z_{{j 2}}^{H} \Big(Z_{{k 2}}^{H} Z_{{l 3}}^{H} + Z_{{k 3}}^{H} Z_{{l 2}}^{H} \Big) + Z_{{j 3}}^{H} \Big(Z_{{k 1}}^{H} Z_{{l 1}}^{H} + Z_{{k 2}}^{H} Z_{{l 2}}^{H} \Big)\Big)\nonumber \\
&+Z_{{i 2}}^{H} \Big(Z_{{j 1}}^{H} \Big(Z_{{k 1}}^{H} Z_{{l 2}}^{H} + Z_{{k 2}}^{H} Z_{{l 1}}^{H} \Big) + Z_{{j 2}}^{H} \Big(Z_{{k 1}}^{H} Z_{{l 1}}^{H} + Z_{{k 3}}^{H} Z_{{l 3}}^{H} \Big) + Z_{{j 3}}^{H} \Big(Z_{{k 2}}^{H} Z_{{l 3}}^{H} + Z_{{k 3}}^{H} Z_{{l 2}}^{H} \Big)\Big)\nonumber \\
&+Z_{{i 1}}^{H} \Big(Z_{{j 1}}^{H} \Big(Z_{{k 2}}^{H} Z_{{l 2}}^{H} + Z_{{k 3}}^{H} Z_{{l 3}}^{H} \Big) + Z_{{j 2}}^{H} \Big(Z_{{k 1}}^{H} Z_{{l 2}}^{H} + Z_{{k 2}}^{H} Z_{{l 1}}^{H} \Big) + Z_{{j 3}}^{H} \Big(Z_{{k 1}}^{H} Z_{{l 3}}^{H} + Z_{{k 3}}^{H} Z_{{l 1}}^{H} \Big)\Big)\Big)
\\
\Delta &\Gamma_{h_{{i}}h_{{j}}H^-_{{k}}H^+_{{l}}} = \,
\frac{i}{2} |\lambda|^2 \Big(Z^{+,*}_{k 1} \Big(-2 Z_{{i 3}}^{H} Z_{{j 3}}^{H} Z_{{l 1}}^{+} + \Big(Z_{{i 1}}^{H} Z_{{j 2}}^{H} + Z_{{i 2}}^{H} Z_{{j 1}}^{H} \Big)Z_{{l 2}}^{+} \Big)\nonumber \\
& \hspace{3cm} +Z^{+,*}_{k 2} \Big(-2 Z_{{i 3}}^{H} Z_{{j 3}}^{H} Z_{{l 2}}^{+} + Z_{{i 1}}^{H} Z_{{j 2}}^{H} Z_{{l 1}}^{+} + Z_{{i 2}}^{H} Z_{{j 1}}^{H} Z_{{l 1}}^{+} \Big)\Big)
\\
\Delta &\Gamma_{h_{{i}}h_{{j}}\tilde{d}_{{k \gamma}}\tilde{d}^*_{{l \delta}}} = \,
\frac{i}{2} \delta_{\gamma \delta} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{a b}} Z^{D,*}_{k 3 + a} Z_{{l b}}^{D} + \lambda^* \sum_{b=1}^{3}Z^{D,*}_{k b} \sum_{a=1}^{3}Y_{d,{a b}} Z_{{l 3 + a}}^{D} \Big)\Big(Z_{{i 2}}^{H} Z_{{j 3}}^{H} + Z_{{i 3}}^{H} Z_{{j 2}}^{H} \Big)
\\
\Delta &\Gamma_{h_{{i}}h_{{j}}\tilde{e}_{{k}}\tilde{e}^*_{{l}}} = \,
\frac{i}{2} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{a b}} Z^{E,*}_{k 3 + a} Z_{{l b}}^{E} + \lambda^* \sum_{b=1}^{3}Z^{E,*}_{k b} \sum_{a=1}^{3}Y_{e,{a b}} Z_{{l 3 + a}}^{E} \Big)\Big(Z_{{i 2}}^{H} Z_{{j 3}}^{H} + Z_{{i 3}}^{H} Z_{{j 2}}^{H} \Big)
\\
\Delta &\Gamma_{h_{{i}}h_{{j}}\tilde{u}_{{k \gamma}}\tilde{u}^*_{{l \delta}}} = \,
\frac{i}{2} \delta_{\gamma \delta} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{u,{a b}} Z^{U,*}_{k 3 + a} Z_{{l b}}^{U} + \lambda^* \sum_{b=1}^{3}Z^{U,*}_{k b} \sum_{a=1}^{3}Y_{u,{a b}} Z_{{l 3 + a}}^{U} \Big)\Big(Z_{{i 1}}^{H} Z_{{j 3}}^{H} + Z_{{i 3}}^{H} Z_{{j 1}}^{H} \Big)
\\
\Delta &\Gamma_{h_{{i}}H^-_{{j}}\tilde{u}_{{k \gamma}}\tilde{d}^*_{{l \delta}}} = \,
i \frac{1}{\sqrt{2}} \delta_{\gamma \delta} \Big(\lambda Z^{+,*}_{j 1} \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{u,{a b}} Z^{U,*}_{k 3 + a} Z_{{l b}}^{D} + \lambda^* Z^{+,*}_{j 2} \sum_{b=1}^{3}Z^{U,*}_{k b} \sum_{a=1}^{3}Y_{d,{a b}} Z_{{l 3 + a}}^{D} \Big)Z_{{i 3}}^{H}
\\
\Delta &\Gamma_{h_{{i}}H^-_{{j}}\tilde{\nu}_{{k}}\tilde{e}^*_{{l}}} = \,
i \frac{1}{\sqrt{2}} \lambda^* Z^{+,*}_{j 2} \sum_{b=1}^{3}Z^{V,*}_{k b} \sum_{a=1}^{3}Y_{e,{a b}} Z_{{l 3 + a}}^{E} Z_{{i 3}}^{H}
\\
\Delta &\Gamma_{h_{{i}}\tilde{d}_{{j \beta}}H^+_{{k}}\tilde{u}^*_{{l \delta}}} = \,
i \frac{1}{\sqrt{2}} \delta_{\beta \delta} Z_{{i 3}}^{H} \Big(\lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{d,{a b}} Z^{D,*}_{j 3 + a} Z_{{l b}}^{U} Z_{{k 2}}^{+} + \lambda^* \sum_{b=1}^{3}Z^{D,*}_{j b} \sum_{a=1}^{3}Y_{u,{a b}} Z_{{l 3 + a}}^{U} Z_{{k 1}}^{+} \Big)
\\
\Delta &\Gamma_{h_{{i}}\tilde{e}_{{j}}H^+_{{k}}\tilde{\nu}^*_{{l}}} = \,
i \frac{1}{\sqrt{2}} \lambda \sum_{b=1}^{3}\sum_{a=1}^{3}Y^*_{e,{a b}} Z^{E,*}_{j 3 + a} Z_{{l b}}^{V} Z_{{i 3}}^{H} Z_{{k 2}}^{+}
\\
\Delta &\Gamma_{H^-_{{i}}H^-_{{j}}H^+_{{k}}H^+_{{l}}} = \,
-i |\lambda|^2 \Big(Z^{+,*}_{i 1} Z^{+,*}_{j 2} + Z^{+,*}_{i 2} Z^{+,*}_{j 1} \Big)\Big(Z_{{k 1}}^{+} Z_{{l 2}}^{+} + Z_{{k 2}}^{+} Z_{{l 1}}^{+} \Big)
\end{align}
\section{The Higgs sector of the DiracNMSSM at the loop level}
\label{app:HiggsLoop}
We give here some details of the calculation which are carried out by a combination of the public tools {\tt SARAH}\xspace and {\tt SPheno}\xspace.
To calculate the Higgs mass at the one-loop level, first the one-loop corrections to the tadpoles equations are needed. These are a sum of loops
involving massive vector bosons and ghosts, standard model fermions, SUSY fermions, sfermions and Higgs fields:
\begin{equation}
\delta \Theta^{(1)}_{i} = \delta \Theta^{(1),V}_{i} + \delta \Theta^{(1),f}_{i} + \delta \Theta^{(1),\tilde{f}}_{i} +\delta \Theta^{(1),\chi}_{i} + \delta \Theta^{(1),\phi}_{i}
\end{equation}
The explicit expression for all contributions read
\begin{align}
\delta \Theta^{(1),V}_{i} = & \, +2 {A_0\Big(m^2_{Z}\Big)}{\Gamma_{\check{h}_{{i}},Z,Z}} +{A_0\Big(m^2_{\eta^-}\Big)} {\Gamma_{\check{h}_{{i}},\bar{\eta^-},\eta^-}} +{A_0\Big(m^2_{\eta^+}\Big)} {\Gamma_{\check{h}_{{i}},\bar{\eta^+},\eta^+}} \nonumber \\
&+{A_0\Big(m^2_{\eta^Z}\Big)} {\Gamma_{\check{h}_{{i}},\bar{\eta^Z},\eta^Z}} +4 {A_0\Big(m^2_{W^-}\Big)}{\Gamma_{\check{h}_{{i}},W^+,W^-}} \\
\delta \Theta^{(1),\phi}_{i} = &- \sum_{a=1}^{2}{A_0\Big(m^2_{H^-_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},H^+_{{a}},H^-_{{a}}}} -\frac{1}{2} \sum_{a=1}^{4}{A_0\Big(m^2_{A^0_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},A^0_{{a}},A^0_{{a}}}} -\frac{1}{2} \sum_{a=1}^{4}{A_0\Big(m^2_{h_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},h_{{a}},h_{{a}}}} \\
\delta \Theta^{(1),\chi}_{i} = &+2 \sum_{a=1}^{2}{A_0\Big(m^2_{\tilde{\chi}^-_{{a}}}\Big)} m_{\tilde{\chi}^-_{{a}}} \Big({\Gamma^L_{\check{h}_{{i}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{a}}}} + {\Gamma^R_{\check{h}_{{i}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{a}}}}\Big) +\sum_{a=1}^{6}{A_0\Big(m^2_{\tilde{\chi}^0_{{a}}}\Big)} m_{\tilde{\chi}^0_{{a}}} \Big({\Gamma^L_{\check{h}_{{i}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{a}}}} + {\Gamma^R_{\check{h}_{{i}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{a}}}}\Big) \\
\delta \Theta^{(1),\tilde{f}}_{i} = & - \sum_{a=1}^{3}{A_0\Big(m^2_{\tilde{\nu}_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\tilde{\nu}^*_{{a}},\tilde{\nu}_{{a}}}} -3 \sum_{a=1}^{6}{A_0\Big(m^2_{\tilde{d}_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\tilde{d}^*_{{a}},\tilde{d}_{{a}}}} - \sum_{a=1}^{6}{A_0\Big(m^2_{\tilde{e}_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\tilde{e}^*_{{a}},\tilde{e}_{{a}}}} \nonumber \\
&-3 \sum_{a=1}^{6}{A_0\Big(m^2_{\tilde{u}_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\tilde{u}^*_{{a}},\tilde{u}_{{a}}}} \\
\delta \Theta^{(1),f}_{i} = &+6 \sum_{a=1}^{3}{A_0\Big(m^2_{d_{{a}}}\Big)} m_{d_{{a}}} \Big({\Gamma^L_{\check{h}_{{i}},\bar{d}_{{a}},d_{{a}}}} + {\Gamma^R_{\check{h}_{{i}},\bar{d}_{{a}},d_{{a}}}}\Big) +2 \sum_{a=1}^{3}{A_0\Big(m^2_{e_{{a}}}\Big)} m_{e_{{a}}} \Big({\Gamma^L_{\check{h}_{{i}},\bar{e}_{{a}},e_{{a}}}} + {\Gamma^R_{\check{h}_{{i}},\bar{e}_{{a}},e_{{a}}}}\Big) \nonumber \\
&+6 \sum_{a=1}^{3}{A_0\Big(m^2_{u_{{a}}}\Big)} m_{u_{{a}}} \Big({\Gamma^L_{\check{h}_{{i}},\bar{u}_{{a}},u_{{a}}}} + {\Gamma^R_{\check{h}_{{i}},\bar{u}_{{a}},u_{{a}}}}\Big)
\end{align}
with $\delta \Theta^{(1)}_1 = \delta \Theta^{(1)}_d$, $\delta \Theta^{(1)}_2 = \delta \Theta^{(1)}_u$, $\delta \Theta^{(1)}_3 = \delta \Theta^{(1)}_s$, and
$\delta \Theta^{(1)}_4 = \delta \Theta^{(1)}_{\bar{s}}$, and $\check{h} = (\phi_d, \phi_u, \phi_s, \phi_{\bar{s}})^T$. Here, chiral vertices are parametrized by $\Gamma^L P_L + \Gamma^R P_R$ with the projection operators $P_{L,R}$ and non-chiral vertices by $\Gamma$. The subscripts denote the involved particles.
All necessary vertices and mass matrices for the DiracNMSSM are given in appendix~\ref{app:matrices}--\ref{app:vertices}. Note, the rotation matrix corresponding to the external scalar has to be replaced by the identity matrix in these calculations. The finite part of the Passarino-Veltman integral $A_0$ is given by
\begin{equation}
A_0(m^2)\ = m^2\left(1 -\ln{\frac{m^2}{Q^2}}\right)
\end{equation}
with the renormalisation scale $Q$ which is chosen to be the average of the stop masses \\
The one-loop corrections to the tadpoles are applied to find the new values of $\mu, b\mu, t_s, t_{\bar{s}}$ which the vacuum conditions
\begin{equation}
\label{eq:onelooptad}
\Theta_i + \delta \Theta^{(1)}_i = 0 \, , \hspace{1cm} i=d,u,s,\bar{s}
\end{equation}
The second step is to calculate the self-energy matrix of the CP even Higgs
\begin{equation}
\Pi^{h}_{i,j}(p^2) = \Pi^{h,V}_{i,j}(p^2) + \Pi^{h,\phi}_{i,j}(p^2) + \Pi^{h,V\phi}_{i,j}(p^2) + \Pi^{h,\tilde{f}}_{i,j}(p^2) + \Pi^{h,f}_{i,j}(p^2) + \Pi^{h,\chi}_{i,j}(p^2)
\end{equation}
The corrections stemming from vector bosons and ghosts are given
\begin{align}
\Pi^{h,V}_{i,j}(p^2) &= +2 {B_0\Big(p^{2},m^2_{Z},m^2_{Z}\Big)}{\Gamma^*_{\check{h}_{{j}},Z,Z}} {\Gamma_{\check{h}_{{i}},Z,Z}} +4 + {B_0\Big(p^{2},m^2_{W^-},m^2_{W^-}\Big)}{\Gamma^*_{\check{h}_{{j}},W^+,W^-}} {\Gamma_{\check{h}_{{i}},W^+,W^-}} \nonumber \\
&- {B_0\Big(p^{2},m^2_{\eta^-},m^2_{\eta^-}\Big)} {\Gamma_{\check{h}_{{i}},\bar{\eta^-},\eta^-}} {\Gamma_{\check{h}_{{j}},\bar{\eta^-},\eta^-}} - {B_0\Big(p^{2},m^2_{\eta^+},m^2_{\eta^+}\Big)} {\Gamma_{\check{h}_{{i}},\bar{\eta^+},\eta^+}} {\Gamma_{\check{h}_{{j}},\bar{\eta^+},\eta^+}} \nonumber \\
&- {B_0\Big(p^{2},m^2_{\eta^Z},m^2_{\eta^Z}\Big)} {\Gamma_{\check{h}_{{i}},\bar{\eta^Z},\eta^Z}} {\Gamma_{\check{h}_{{j}},\bar{\eta^Z},\eta^Z}} +2 {A_0\Big(m^2_{Z}\Big)}{\Gamma_{\check{h}_{{i}},\check{h}_{{j}},Z,Z}} +4 {A_0\Big(m^2_{W^-}\Big)}{\Gamma_{\check{h}_{{i}},\check{h}_{{j}},W^+,W^-}}
\end{align}
The corrections from Higgs scalars in the loops read
\begin{align}
\Pi^{h,\phi}_{i,j}(p^2) &=- \sum_{a=1}^{2}{A_0\Big(m^2_{H^-_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\check{h}_{{j}},H^+_{{a}},H^-_{{a}}}} +\sum_{a=1}^{2}\sum_{b=1}^{2}{B_0\Big(p^{2},m^2_{H^-_{{a}}},m^2_{H^-_{{b}}}\Big)} {\Gamma^*_{\check{h}_{{j}},H^+_{{a}},H^-_{{b}}}} {\Gamma_{\check{h}_{{i}},H^+_{{a}},H^-_{{b}}}} \nonumber \\
&-\frac{1}{2} \sum_{a=1}^{4}{A_0\Big(m^2_{A^0_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\check{h}_{{j}},A^0_{{a}},A^0_{{a}}}} -\frac{1}{2} \sum_{a=1}^{4}{A_0\Big(m^2_{h_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\check{h}_{{j}},h_{{a}},h_{{a}}}} \nonumber \\
&+\frac{1}{2} \sum_{a=1}^{4}\sum_{b=1}^{4}{B_0\Big(p^{2},m^2_{A^0_{{a}}},m^2_{A^0_{{b}}}\Big)} {\Gamma^*_{\check{h}_{{j}},A^0_{{a}},A^0_{{b}}}} {\Gamma_{\check{h}_{{i}},A^0_{{a}},A^0_{{b}}}} \nonumber \\
&+\sum_{a=1}^{4}\sum_{b=1}^{4}{B_0\Big(p^{2},m^2_{h_{{a}}},m^2_{A^0_{{b}}}\Big)} {\Gamma^*_{\check{h}_{{j}},h_{{a}},A^0_{{b}}}} {\Gamma_{\check{h}_{{i}},h_{{a}},A^0_{{b}}}} +\frac{1}{2} \sum_{a=1}^{4}\sum_{b=1}^{4}{B_0\Big(p^{2},m^2_{h_{{a}}},m^2_{h_{{b}}}\Big)} {\Gamma^*_{\check{h}_{{j}},h_{{a}},h_{{b}}}} {\Gamma_{\check{h}_{{i}},h_{{a}},h_{{b}}}} \nonumber \\
\end{align}
The mixed contributions involving scalars and vector bosons are given by
\begin{align}
\Pi^{h,V\phi}_{i,j}(p^2) &=+2 \sum_{b=1}^{2}{\Gamma^*_{\check{h}_{{j}},W^+,H^-_{{b}}}} {\Gamma_{\check{h}_{{i}},W^+,H^-_{{b}}}} {F_0\Big(p^{2},m^2_{H^-_{{b}}},m^2_{W^-}\Big)} +\sum_{b=1}^{4}{\Gamma^*_{\check{h}_{{j}},Z,A^0_{{b}}}} {\Gamma_{\check{h}_{{i}},Z,A^0_{{b}}}} {F_0\Big(p^{2},m^2_{A^0_{{b}}},m^2_{Z}\Big)}
\end{align}
The corrections due to charginos and neutralinos read
\begin{align}
\Pi^{h,\chi}_{i,j}(p^2) &=-2 \sum_{a=1}^{2}m_{\tilde{\chi}^-_{{a}}} \sum_{b=1}^{2}{B_0\Big(p^{2},m^2_{\tilde{\chi}^-_{{a}}},m^2_{\tilde{\chi}^-_{{b}}}\Big)} m_{\tilde{\chi}^-_{{b}}} \Big({\Gamma^{L*}_{\check{h}_{{j}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{b}}}} \Big) \nonumber \\
&+\sum_{a=1}^{2}\sum_{b=1}^{2}{G_0\Big(p^{2},m^2_{\tilde{\chi}^-_{{a}}},m^2_{\tilde{\chi}^-_{{b}}}\Big)} \Big({\Gamma^{L*}_{\check{h}_{{j}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\tilde{\chi}^+_{{a}},\tilde{\chi}^-_{{b}}}} \Big) \nonumber \\
&- \sum_{a=1}^{6}m_{\tilde{\chi}^0_{{a}}} \sum_{b=1}^{6}{B_0\Big(p^{2},m^2_{\tilde{\chi}^0_{{a}}},m^2_{\tilde{\chi}^0_{{b}}}\Big)} m_{\tilde{\chi}^0_{{b}}} \Big({\Gamma^{L*}_{\check{h}_{{j}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{b}}}} \Big) \nonumber \\
&+\frac{1}{2} \sum_{a=1}^{6}\sum_{b=1}^{6}{G_0\Big(p^{2},m^2_{\tilde{\chi}^0_{{a}}},m^2_{\tilde{\chi}^0_{{b}}}\Big)} \Big({\Gamma^{L*}_{\check{h}_{{j}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\tilde{\chi}^0_{{a}},\tilde{\chi}^0_{{b}}}} \Big) \nonumber \\
\end{align}
The corrections due to Sfermions are
\begin{align}
\Pi^{h,\tilde{f}}_{i,j}(p^2) &=- \sum_{a=1}^{3}{A_0\Big(m^2_{\tilde{\nu}_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\check{h}_{{j}},\tilde{\nu}^*_{{a}},\tilde{\nu}_{{a}}}} +\sum_{a=1}^{3}\sum_{b=1}^{3}{B_0\Big(p^{2},m^2_{\tilde{\nu}_{{a}}},m^2_{\tilde{\nu}_{{b}}}\Big)} {\Gamma^*_{\check{h}_{{j}},\tilde{\nu}^*_{{a}},\tilde{\nu}_{{b}}}} {\Gamma_{\check{h}_{{i}},\tilde{\nu}^*_{{a}},\tilde{\nu}_{{b}}}} \nonumber \\
&-3 \sum_{a=1}^{6}{A_0\Big(m^2_{\tilde{d}_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\check{h}_{{j}},\tilde{d}^*_{{a}},\tilde{d}_{{a}}}} - \sum_{a=1}^{6}{A_0\Big(m^2_{\tilde{e}_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\check{h}_{{j}},\tilde{e}^*_{{a}},\tilde{e}_{{a}}}} \nonumber \\
&-3 \sum_{a=1}^{6}{A_0\Big(m^2_{\tilde{u}_{{a}}}\Big)} {\Gamma_{\check{h}_{{i}},\check{h}_{{j}},\tilde{u}^*_{{a}},\tilde{u}_{{a}}}} +3 \sum_{a=1}^{6}\sum_{b=1}^{6}{B_0\Big(p^{2},m^2_{\tilde{d}_{{a}}},m^2_{\tilde{d}_{{b}}}\Big)} {\Gamma^*_{\check{h}_{{j}},\tilde{d}^*_{{a}},\tilde{d}_{{b}}}} {\Gamma_{\check{h}_{{i}},\tilde{d}^*_{{a}},\tilde{d}_{{b}}}} \nonumber \\
&+\sum_{a=1}^{6}\sum_{b=1}^{6}{B_0\Big(p^{2},m^2_{\tilde{e}_{{a}}},m^2_{\tilde{e}_{{b}}}\Big)} {\Gamma^*_{\check{h}_{{j}},\tilde{e}^*_{{a}},\tilde{e}_{{b}}}} {\Gamma_{\check{h}_{{i}},\tilde{e}^*_{{a}},\tilde{e}_{{b}}}} +3 \sum_{a=1}^{6}\sum_{b=1}^{6}{B_0\Big(p^{2},m^2_{\tilde{u}_{{a}}},m^2_{\tilde{u}_{{b}}}\Big)} {\Gamma^*_{\check{h}_{{j}},\tilde{u}^*_{{a}},\tilde{u}_{{b}}}} {\Gamma_{\check{h}_{{i}},\tilde{u}^*_{{a}},\tilde{u}_{{b}}}} \nonumber \\
\end{align}
And those stemming from SM fermions are
\begin{align}
\Pi^{h,f}_{i,j}(p^2) &= -6 \sum_{a=1}^{3}m_{d_{{a}}} \sum_{b=1}^{3}{B_0\Big(p^{2},m^2_{d_{{a}}},m^2_{d_{{b}}}\Big)} m_{d_{{b}}} \Big({\Gamma^{L*}_{\check{h}_{{j}},\bar{d}_{{a}},d_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\bar{d}_{{a}},d_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\bar{d}_{{a}},d_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\bar{d}_{{a}},d_{{b}}}} \Big) \nonumber \\
& +3 \sum_{a=1}^{3}\sum_{b=1}^{3}{G_0\Big(p^{2},m^2_{d_{{a}}},m^2_{d_{{b}}}\Big)} \Big({\Gamma^{L*}_{\check{h}_{{j}},\bar{d}_{{a}},d_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\bar{d}_{{a}},d_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\bar{d}_{{a}},d_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\bar{d}_{{a}},d_{{b}}}} \Big) \nonumber \\
&-2 \sum_{a=1}^{3}m_{e_{{a}}} \sum_{b=1}^{3}{B_0\Big(p^{2},m^2_{e_{{a}}},m^2_{e_{{b}}}\Big)} m_{e_{{b}}} \Big({\Gamma^{L*}_{\check{h}_{{j}},\bar{e}_{{a}},e_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\bar{e}_{{a}},e_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\bar{e}_{{a}},e_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\bar{e}_{{a}},e_{{b}}}} \Big) \nonumber \\
&+\sum_{a=1}^{3}\sum_{b=1}^{3}{G_0\Big(p^{2},m^2_{e_{{a}}},m^2_{e_{{b}}}\Big)} \Big({\Gamma^{L*}_{\check{h}_{{j}},\bar{e}_{{a}},e_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\bar{e}_{{a}},e_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\bar{e}_{{a}},e_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\bar{e}_{{a}},e_{{b}}}} \Big)\nonumber \\
&-6 \sum_{a=1}^{3}m_{u_{{a}}} \sum_{b=1}^{3}{B_0\Big(p^{2},m^2_{u_{{a}}},m^2_{u_{{b}}}\Big)} m_{u_{{b}}} \Big({\Gamma^{L*}_{\check{h}_{{j}},\bar{u}_{{a}},u_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\bar{u}_{{a}},u_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\bar{u}_{{a}},u_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\bar{u}_{{a}},u_{{b}}}} \Big) \nonumber \\
&+3 \sum_{a=1}^{3}\sum_{b=1}^{3}{G_0\Big(p^{2},m^2_{u_{{a}}},m^2_{u_{{b}}}\Big)} \Big({\Gamma^{L*}_{\check{h}_{{j}},\bar{u}_{{a}},u_{{b}}}} {\Gamma^L_{\check{h}_{{i}},\bar{u}_{{a}},u_{{b}}}} + {\Gamma^{R*}_{\check{h}_{{j}},\bar{u}_{{a}},u_{{b}}}} {\Gamma^R_{\check{h}_{{i}},\bar{u}_{{a}},u_{{b}}}} \Big) \nonumber \\
\end{align}
The Passarino-Veltman integral $B_0$ can be expressed by
\begin{equation}
B_0(p, m_1, m_2) = - \ln\left(\frac{p^2}{Q^2}\right) - f_B(x_+) - f_B(x_-) \, ,
\end{equation}
with $f_B(x) = \ln(1-x) - x\ln(1-x^{-1})-1$,
$ x_{\pm}\ =\ \frac{s \pm \sqrt{s^2 - 4p^2(m_1^2-i\varepsilon)}}{2p^2}$,
and $s=p^2-m_2^2+m_1^2$.
All appearing integrals can be expressed in terms of $A_0$ and $B_0$
\begin{align}
B_1(p, m_1,m_2) =& {\frac{1}{2p^2}}\Big( A_0(m_2) - A_0(m_1) + (p^2 +m_1^2 -m_2^2) B_0(p, m_1, m_2)\Big) \\
B_{22}(p, m_1,m_2) =& \frac{1}{6} \Big(\frac{1}{2}\big(A_0(m_1)+A_0(m_2)\big)
+\left(m_1^2+m_2^2-\frac{1}{2}p^2\right)B_0(p,m_1,m_2)\nonumber \\
&+ \frac{m_2^2-m_1^2}{2p^2} \big[A_0(m_2)-A_0(m_1)-(m_2^2-m_1^2) B_0(p,m_1,m_2)\big] \nonumber\\
& + m_1^2 + m_2^2 -\frac{1}{3}p^2\,\Big) \\
F_0(p,m_1,m_2) =& A_0(m_1)-2A_0(m_2)- (2p^2+2m^2_1-m^2_2) B_0(p,m_1,m_2)\ , \\
G_0(p,m_1,m_2) =& (p^2-m_1^2-m_2^2)B_0(p,m_1,m_2)-A_0(m_1)-A_0(m_2)\ .
\end{align}
In our numerical analysis the one-loop scalar Higgs masses are then calculated by taking the real part of the poles
of the corresponding propagator matrices
\begin{equation}
\mathrm{Det}\left[ p^2_i \mathbf{1} - m^{2,h}_{1L}(p^2) \right] = 0,
\label{eq:propagator}
\end{equation}
where
\begin{equation}
m^{2,h}_{1L}(p^2) = \tilde{m}^{2,h}_T - \Pi_{hh}(p^2) .
\end{equation}
Here, \(\tilde{m}^{2,h}_T\) is the tree-level mass matrix from eq.\
(\ref{eq:HiggsTreeMass}) where the parameters obtained from the one-loop
tadpole equations are inserted. Equation (\ref{eq:propagator}) has to be solved
for each eigenvalue $p^2=m^2_i$. The same procedure is also applied for
the pseudo scalar Higgs bosons.
\subsubsection*{Dominant Two-loop corrections}
In addition to the full correction at one-loop we have also added the dominant two-loop corrections due to stops known from the MSSM
presented in \cite{Brignole:2001jy,Brignole:2002bz,Dedes:2002dy,Dedes:2003km}. Because of the presence of these corrections, the renormalisation conditions eq.~(\ref{eq:onelooptad}) are modified to
\begin{align}
\Theta_i + \delta \Theta^{(1)}_i + \delta \Theta^{(2)}_i = 0 \, , \hspace{1cm} i=d,u \\
\Theta_i + \delta \Theta^{(1)}_i = 0 \, , \hspace{1cm} i=s,\bar{s}
\end{align}
and the 2-loop self-energy of the CP even Higgs takes the form
\begin{equation}
\Pi^{(2L)} = \left(\begin{array}{cc} \Pi^{(2L),MSSM} & {\bf 0} \\ {\bf 0} & {\bf 0} \end{array}\right)
\end{equation}
Here, ${\bf 0}$ is a $2\times 2$ matrix carrying only 0's, and $\Pi^{(2L),MSSM}$ are the two-loop MSSM self-energy contributions to the Higgs.
\end{appendix}
|
1,108,101,565,620 | arxiv | \section{Introduction}\label{s1}
\hskip\parindent John and Nirenberg \cite{jn61} introduced the {\it
space ${\mathrm {BMO}\,}(\rn)$}, which is defined to be the space of
all $f\in L_{\rm loc}^1(\rn)$ such that
$$\|f\|_{\rm BMO\,(\rn)}\ev\sup_{{\rm ball}\ B\st\rn}\frac1{|B|}
\int_B|f(x)-f_B|\,dx<\fz,$$
where and in what follows, $f_B\ev\frac1{|B|}\int_Bf(x)\,dx$. The
space ${\mathrm {BMO}\,}(\rn)$ was proved to be the dual of the Hardy space
$H^1(\rn)$ by Fefferman and Stein in \cite{fs72}.
Sarason \cite{s75} introduced the {\it space ${\mathrm {VMO}\,}(\rn)$}, which
is defined to be the space of all $f\in{\mathrm {BMO}\,}(\rn)$ such that
$$\lim_{c\to0}\sup_{\gfz{\mathrm{ball}\,B\st\rn}{r_B\le c}}\frac1{|B|}
\int_B|f(x)-f_B|\,dx=0,$$
where $r_B$ denotes the radius of the ball $B$.
In order to represent $H^1(\rn)$ as a dual
space, Coifman and Weiss \cite{cw77} introduced the {\it space
${\mathrm {CMO}\,}(\rn)$}, which is defined to be the closure of all
infinitely differentiable functions with compact support in the
${\mathrm {BMO}\,}(\rn)$ norm and was originally denoted by the \emph{symbol}
${\mathrm {VMO}\,}(\rn)$ in \cite{cw77}, and proved that $({\mathrm
{CMO}\,}(\rn))^*=H^1(\rn)$. For more properties of ${\mathrm
{BMO}\,}(\rn)$, ${\mathrm {VMO}\,}(\rn)$ and ${\mathrm
{CMO}\,}(\rn)$, we refer the reader to Janson \cite{j80} and
Bourdaud \cite{b02}.
Let $L$ be a \emph{linear operator} in $L^2(\rn)$ that generates an
analytic semigroup $\{e^{-tL}\}_{t\ge0}$ with kernels satisfying an
\emph{upper bound of Poisson type}. The Hardy space $H_L^1(\rn)$,
the BMO space ${\mathrm {BMO}}_L(\rn)$ and Morrey spaces
associated with $L$ were
introduced and studied in \cite{adm,dy05b,dxy07}. Duong and Yan
\cite{dy05a} further proved that $(H_L^1(\rn))^*={\mathrm
{BMO}}_{L^*}(\rn)$, where and in what follows, $L^*$ denotes the
{\it adjoint operator} of $L$ in $L^2(\rn)$. Moreover, recently,
Deng et al. \cite{ddsty} introduced the \emph{space} ${\mathrm
{VMO}}_L(\rn)$, the space of vanishing mean oscillation associated
with operator $L$, and proved that $({\mathrm
{VMO}}_L(\rn))^*=H_{L^*}^1(\rn)$ and also
$${\mathrm {VMO}}_\Delta(\rn) ={\mathrm {CMO}}(\rn)={\mathrm
{VMO}}_{\sqrt\bdz}(\rn)$$
with equivalent norms. Let $\Phi$ on
$(0,\fz)$ be a continuous, strictly increasing, subadditive function
of upper type $1$ and of critical lower type $p_\Phi^-\le 1$
but near to $1$ (see Section \ref{s2.4} below for the definition).
Let $\rho(t)\equiv t^{-1}/\Phi^{-1}(t^{-1})$ for all $t\in(0,\fz)$.
A typical example of such Orlicz functions is $\Phi(t)\equiv t^p$
for all $t\in(0,\fz)$ and $p\le 1$ but near to $1$. Jiang and Yang
\cite{jya} introduced the VMO-type space ${\mathrm
{VMO}}_{\rho,L}(\rn)$ and proved that the dual space of ${\mathrm
{VMO}}_{\rho, L^*}(\rn)$ is the space $B_{\Phi, L}(\rn)$, where
$B_{\Phi,L}(\rn)$ denotes the \emph{Banach completion} of the Orlicz-Hardy space
$H_{\Phi,L}(\rn)$ in \cite{jyz09}.
Let $L$ be a \emph{second order divergence form elliptic operator}
with complex bounded measurable coefficients and $\Phi$ a \emph{continuous,
strictly increasing, concave function} of
critical lower type $p_\Phi^-\in (0,1]$.
Jiang and Yang \cite{jy10a} studied the VMO-type spaces
${\mathrm {VMO}}_{\rho,L}(\rn)$ and proved that
the dual space of ${\mathrm {VMO}}_{\rho, L^*}(\rn)$ is
the space $B_{\Phi, L}(\rn)$, where $B_{\Phi, L}(\rn)$
denotes the \emph{Banach completion} of the Orlicz-Hardy space
$H_{\Phi,L}(\rn)$ in \cite{jy10}. (We remark that
the \emph{assumptions on $p_\Phi$ in \cite{jy10, jy10a} can
be relaxed into the same assumptions on
$p_\Phi^-$}; see Remark \ref{r2.2}(ii) below.)
In particular, when $\Phi(t)
\equiv t$ for all $t\in (0,\fz)$, then $\rho(t)\ev1$ and
$({\mathrm {VMO}}_{1,L}(\rn))^*=H_{L^*}^1(\rn)$, which was also
independently obtained by Song and Xu \cite{sx},
where $H_{L^*}^1(\rn)$ denotes the Hardy space first
introduced by Hofmann and Mayboroda \cite{hm09} (see also
\cite{hm09c}).
Let $(\cx,d)$ be a \emph{metric space endowed with a doubling measure} $\mu$ and
$L$ a \emph{non-negative self-adjoint operator satisfying Davies-Gaffney
estimates}. Hofmann et al. \cite{hlmmy} introduced the Hardy space
$H_L^1(\cx)$ associated to $L$. Jiang and Yang \cite{jy} further
introduced the Orlicz-Hardy space $\hx$. Anh \cite{a10} studied the
VMO space ${\mathrm {VMO}}_L(\cx)$ associated to $L$ and proved that
the dual space of ${\mathrm {VMO}}_L(\cx)$ is the Hardy space
$H_L^1(\cx)$. Recently, Duong and Li \cite{dl} observed that the
assumption ``$L$ is a non-negative self-adjoint operator'' in
\cite{hlmmy} can be replaced by a weaker assumption that ``\emph{$L$ has a
bounded $H_\fz$ functional calculus on $L^2(\cx)$}'' and introduced
the Hardy space $H_L^p(\cx)$ with $p\in (0,1]$, which was further
generalized by Anh and Li \cite{al11} to the Orlicz-Hardy spaces
$\hx$.
From now on, we always assume that {\it $L$ is a linear operator
which has a bounded $H_\fz$ functional calculus and satisfies
Davies-Gaffney estimates} and that {\it $\Phi$ is a continuous,
strictly increasing, concave function of critical lower
type $p_\Phi^-\in (0,1]$}. In this paper, we introduce the generalized
VMO space $\vmo$ associated with $L$, and establish its characterization
via the tent space in \cite{jy}. Then, we further prove that $\vmox=\bxx$,
where $\bxx$ denotes the \emph{Banach completion} of the Orlicz-Hardy space
$\hxx$ in \cite{al11}. When $\Phi(t)\equiv t$ for all $t\in(0,\fz)$,
we denote $\vmo$ simply by $\mathrm{\,VMO}_{L}(\cx)$. As a special
case of the main results in this paper, we show that
$({\mathrm{\,VMO}_{L}(\cx)})^*=H_{L^*}^1(\cx)$, which,
when $L$ is nonnegative self-adjoint, was already obtained by Anh \cite{a10}.
Precisely, the paper is organized as follows. In Section \ref{s2},
we recall some known notions and notation concerning metric measure
spaces $\cx$, then describe some basic assumptions on the considered
operator $L$ and the Orlicz function $\Phi$ and present some
properties of the operator $L$ and the Orlicz function $\Phi$ considered in
this paper.
In Section \ref{s3}, we first obtain the $\rho$-Carleson measure
characterization (see Theorem \ref{t3.1} below) of the space $\bmo$
in \cite{al11} via first establishing a Calder\'on reproducing formula
(see Proposition \ref{p3.3} below). Differently from
the Calder\'on reproducing formula in \cite[Proposition\,4.6]{jy},
the Calder\'on reproducing formula in
Proposition \ref{p3.3} below holds for
all molecules instead of atoms in \cite{jy}, which
brings us some extra difficulty due to the lack of the
support of molecules.
Then we introduce the generalized VMO space $\vmo$
associated with $L$, and the tent space $\txv$ and establish some
basic properties of these spaces. In particular, we characterize the
space $\vmo$ via $\txv$; see Theorem \ref{t3.4} below. To this end,
we first need make clear the dual relation between $\hxx$ and $\bmo$
(see Theorem \ref{t3.2} below), which is deduced from a technical
result on the optimal representation of finite linear combinations
of molecules (see Theorem \ref{t3.1} below). We remark that variants
of Theorems \ref{t3.1} and \ref{t3.2} below have already been
given respectively in \cite[Theorems 3.15, 3.13 and 3.16]{al11}
without a detailed proof of \cite[Theorem 3.15]{al11}.
We give a detailed proof of Theorem \ref{t3.1} below
which induces \emph{more accurate indices} appearing in
Theorems \ref{t3.1} and \ref{t3.2} below, comparing with
\cite[Theorems 3.13 and 3.15]{al11} (see Remark
\ref{r3.2} below). Moreover, the proof of
Theorem \ref{t3.1} below simplifies the proof of
\cite[Theorem 5.4]{hlmmy} in a subtle way, the proof of
\cite[Theorem 5.4]{hlmmy} strongly depends
on the support of atoms; see Remark \ref{r3.1} below.
In Section \ref{s4}, we first obtain, in Theorem \ref{t4.1} below,
the dual space of the tent space $\txv$ in Definition \ref{d3.4}
below, from which, we further deduce that $\vmox=\bxx$
in Theorem \ref{t4.2} below, where $\bxx$
denotes the \emph{Banach completion} of $\hxx$. In particular, we obtain
$({\mathrm{VMO}_{L}(\cx)})^*=H_{L^*}^1(\cx)$.
Finally we make some conventions on notation. Throughout the whole
paper, we denote by $C$ a {\it positive constant} which is
independent of the main parameters, but it may vary from line to
line. The {\it constant with subscripts}, such as $C_1$, does not
change in different occurrences. We also use $C(\gz,\cdots)$ to
denote {\it a positive constant depending on the indicated
parameters $\gz,$ $\cdots$}. The {\it symbol} $A\ls B$ means that
$A\le CB$. If $A\ls B$ and $B\ls A$, then we write $A\sim B$. We
also set $\nn\equiv\{1,\, 2,\, \cdots\}$ and
$\zz_+\equiv\nn\cup\{0\}$. The {\it symbol $B(x,r)$} denotes the
ball $\{y\in\cx:\ d(x,y)<r\}$; moreover, let $CB(x,r)\equiv
B(x,Cr)$. For a measurable set $E$,
denote by $\chi_{E}$ the {\it characteristic function} of $E$
and by $E^\com$ the \emph{complement} of $E$ in $\cx$.
\section{Preliminaries\label{s2}}
\hskip\parindent In this section, we first recall some notions and
notation on metric measure spaces and then describe some basic
assumptions on the considered operator $L$ in this paper and its
functional calculus; finally, we also present some basic assumptions
and properties on Orlicz functions.
\subsection{Metric measure spaces\label{s2.1}}
\hskip\parindent Throughout the whole paper, let $\cx$ be a \emph{set}, $d$
a \emph{metric} on $\cx$ and $\mu$ a \emph{nonnegative Borel regular measure} on
$\cx$. Moreover, assume that there exists a constant $C_1\ge1$ such
that for all $x\in\cx$ and $r>0$,
\begin{equation}\label{2.1}
V(x,2r)\le C_1V(x,r)<\fz,
\end{equation}
where $B(x,r)\equiv\{y\in\cx:\ d(x,y)<r\}$ and
\begin{equation}\label{2.2}
V(x,r)\equiv \mu(B(x,r)).
\end{equation}
Observe that if $d$ is further assumed to be a quasi-metric, then
$(\cx,d,\mu)$ is called a \emph{space of homogeneous type} in the sense of
Coifman and Weiss \cite{cw71} (see also \cite{cw77}).
Notice that the doubling property \eqref{2.1} implies the following
\emph{strong homogeneity property}: there exist some positive constants $C$
and $n$, depending on $C_1$, such that
\begin{equation}\label{2.3}
V(x,\lz r)\le C\lz^nV(x,r)
\end{equation}
uniformly for all $\lz\ge1$, $x\in\cx$ and $r>0$. The parameter $n$
measures the \emph{dimension} of the space $\cx$ in some sense. Also, there
exist constants $C\in (0,\fz)$ and $N\in [0, n]$, depending on
$C_1$, such that
\begin{equation}\label{2.4}
V(x, r)\le C\lf(1+\frac{d(x,y)}r\r)^NV(y,r)
\end{equation}
uniformly for all $x,y\in\cx$ and $r>0$. Indeed, the property \eqref{2.4}
with $N=n$ is a simple corollary of the strong homogeneity property
\eqref{2.3}. In the cases of Euclidean spaces, Lie groups of polynomial
growth and, more generally, Ahlfors regular spaces, $N$ can be chosen
to be $0$.
In what follows, for any ball $B\subset\cx$, we set
\begin{equation}\label{2.5}
U_0(B)\equiv B\quad {\rm and}\quad U_j(B)\equiv 2^jB\backslash2^{j-1}B
\quad{\rm for\ }j\in\nn.
\end{equation}
The following covering lemma established in \cite[Lemma 2.1]{a10} plays
a key role in the sequel.
\begin{lemma}\label{l2.1}
For any $\ell>0$, there exists $N_\ell\in\nn$,
depending on $\ell$, such that for all balls $B(x_B,\ell r)$,
with $x_B\in\cx$ and $r>0$, there exists a family
$\{B(x_{B, i},r)\}_{i=1}^{N_\ell}$ of balls such that
{\rm i)} $B(x_B,\ell r)\st\cup_{i=1}^{N_\ell}B(x_{B, i},r)$;
{\rm ii)} $N_\ell\le C\ell^n$;
{\rm iii)} $\sum_{i=1}^{N_\ell}\chi_{B(x_{B, i},r)}\le C$.
\noindent Here $C$ is a positive constant independent of $x_B$, $r$ and $\ell$.
\end{lemma}
\subsection{Holomorphic functional calculi\label{s2.2}}
\hskip\parindent We now recall some basic notions of holomorphic functional
calculi introduced by McIntosh \cite{m86}.
Let $0<\nu<\gz<\pi$. Define the {\it closed sector $S_{\nu}$} in the
complex plane $\cc$ by setting $S_{\nu}\equiv\{z\in\cc:\
|\mathrm{arg}\,z|\le\nu\}\cup\{0\}$ and denote by $S^0_{\nu}$ its
{\it interior}. We employ the following \emph{subspaces},
$H_\fz(S_\nu^0)$ and $\Psi(S_\nu^0)$, of the {\it space
$H(S^0_{\nu})$} of all holomorphic functions on $S^0_{\nu}$:
$$H_{\fz}(S^0_{\nu})\equiv\lf\{b\in H(S^0_{\nu}):\ \|b\|_{L^\fz(S^0_{\nu})}\equiv
\sup_{z\in S^0_{\nu}}|b(z)|<\fz\r\}$$ and
\begin{eqnarray*}
&&\Psi(S^0_{\nu})\equiv\{\psi\in H(S^0_{\nu}):\ \text{there exist}\
s\in(0,\fz)\ \text{and}\ C\in(0,\fz)\ \text{such that}\noz\\
&&\hspace{5em}\text{for all}\ z\in S^0_{\nu},\,|\psi(z)|\le C|z|^s
(1+|z|^{2s})^{-1}\}.
\end{eqnarray*}
Given $\nu\in(0,\pi)$, a closed operator $L$ in $L^2 (\rn)$ is
called to be of {\it type $\nu$} if $\sz(L)\subset S_{\nu}$, where $\sz(L)$
denotes its {\it spectra}, and if for all $\gz>\nu$, there
exists a positive constant $C_{\gz}$ such that for all $\lz\not\in
S_{\gz}$, $\|(L-\lz I)^{-1}\|_{L^2 (\rn)\to L^2 (\rn)}\le
C_{\gz}|\lz|^{-1}$. Let $\mathscr{X}$ and $\mathscr{Y}$ be two
\emph{linear normed spaces} and $T$ be a \emph{continuous linear operator} from
$\mathscr{X}$ to $\mathscr{Y}$. Here and in what follows,
$\|T\|_{\mathscr{X}\to\mathscr{Y}}$ denotes the {\it operator norm
of $T$ from $\mathscr{X}$ to $\mathscr{Y}$}. Let $\tz\in(\nu,\gz)$
and $\Gamma$ be the {\it contour} $\{\xi=re^{\pm i\tz}:\ r\ge0\}$
parameterized clockwise around $S_{\nu}$. Then if $L$ is of type
$\nu$ and $\psi\in\Psi(S^0_{\nu})$, the {\it operator} $\psi(L)$ is
defined by
$$\psi(L)\equiv\frac{1}{2\pi i}\int_{\Gamma}(L-\lz I)^{-1}\psi(\lz)\,d\lz,$$
where the integral is absolutely convergent in $\mathfrak{L}(L^2
(\rn), L^2 (\rn))$ (the {\it class of all bounded linear operators
in $L^2 (\rn)$}). By the Cauchy theorem, we know that
$\psi(L)$ is independent of the choices of $\nu$ and $\gz$ such
that $\tz\in(\nu,\gz)$. Moreover, if $L$ is one-to-one and has dense
range, and $b\in H_{\fz}(S^0_{\gz})$, then $b(L)$ is defined by
setting $b(L)\equiv[\psi(L)]^{-1}(b\psi)(L)$, where $\psi(z)\equiv
z(1+z)^{-2}$ for all $z\in S^0_{\gz}$. It was proved by McIntosh
\cite{m86} that $b(L)$ is a well-defined linear operator in $L^2 (\rn)$.
Moreover, the operator $L$ is said to have a {\it bounded
$H_{\fz}$-calculus} in $L^2 (\rn)$, if for all $\gz\in(\nu,\pi)$,
there exists a positive constant $\wz{C}_{\gz}$
such that for all $b\in H_{\fz}(S^0_{\gz})$, $b(L)\in
\mathfrak{L}(L^2 (\rn), L^2 (\rn))$ and
\begin{equation}\label{2.6}
\|b\|_{L^2 (\rn)\to L^2 (\rn)}\le
\wz{C}_{\gz}\|b\|_{L^\fz(S^0_{\gz})}.
\end{equation}
\subsection{Assumptions on the operator $L$\label{s2.3}}
\hskip\parindent Throughout the whole paper,
we always suppose that the considered operators
$L$ satisfy the following \emph{assumptions}.
\medskip
\noindent{\bf Assumption $(L)_1$.} The operator $L$ has a bounded
$H_\fz$-calculus in $L^2(\cx)$.
\medskip
\noindent{\bf Assumption $(L)_2$.} The semigroup $\{e^{-tL}\}_{t>0}$
generated by $L$ is analytic on $L^2(\cx)$ and satisfies the \emph{Davies-Gaffney
estimate}, namely, there exist positive constants $C_2$ and $C_3$ such that
for all closed sets $E$ and $F$ in $\cx$, $t\in(0,\fz)$ and $f\in L^2(E)$,
\begin{equation}\label{2.7}
\|e^{-tL}f\|_{L^2(F)}\le
C_2\exp\lf\{-\frac{[\dist(E,F)]^2}{C_3t}\r\}\|f\|_{L^2(E)},
\end{equation}
where and in what follows, $\dist(E,F)\equiv\inf_{x\in E,\,y\in
F}d(x,y)$ and the \emph{space} $L^2(E)$ denotes the set of all $\mu$-measurable
functions on $E$ such that
$\|f\|_{L^2(E)}\equiv\{\int_E|f(x)|^2\,d\mu(x)\}^{1/2}<\fz$.
\medskip
\begin{remark}\rm\label{r2.1}
By the functional calculus of $L$ on $L^2(\cx)$,
it is easy to see that if an operator $L$ satisfies Assumptions
$(L)_1$ and $(L)_2$, the adjoint operator $L^*$ also satisfies Assumptions
$(L)_1$ and $(L)_2$ and, therefore, the following Lemmas \ref{l2.2}
and \ref{l2.3} also hold for $L^*$.
\end{remark}
By Assumptions $(L)_1$ and $(L)_2$, we have the following
technical result which was obtained by Anh and Li
in \cite[Proposition 2.2]{al11}.
\begin{lemma}\label{l2.2}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$. Then for any
fixed $k\in\zz_+$ (resp. $j,k\in\zz_+$ with $j\le k$), the family
$\{(t^2L)^ke^{-t^2L}\}_{t>0}$ (resp.
$\{(t^2L)^j(I+t^2L)^{-k}\}_{t>0}$) of operators also satisfies the
Davies-Gaffney estimate \eqref{2.7} with positive constants $C_2$,
$C_3$ depending only on $n$ and $k$ (resp. $n$, $j$ and $k$).
\end{lemma}
By \eqref{2.6}, we have the following useful lemma.
\begin{lemma}\label{l2.3}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$. Then for any
fixed $k\in\nn$, the operator given by setting, for all
$f\in{L^2(\cx)}$ and $x\in\cx$,
$$S_L^kf(x)\ev\lf(\iint_{\bgz(x)}\lf|(t^2L)^ke^{-t^2L}f(y)\r|^2\,
\frac{d\mu(y)}{V(x,t)}\,\frac{dt}t\r)^{1/2},$$
is bounded on ${L^2(\cx)}$.
\end{lemma}
\subsection{Orlicz functions\label{s2.4}}
\hskip\parindent Let $\Phi$ be a positive function on
$\rr_{+}\equiv(0,\fz)$. The function $\Phi$ is called of {\it
upper \emph{(resp.} {\it lower}\emph{)} type $p$} for some $p\in[0,\fz)$,
if there exists a positive constant $C$ such that for all
$t\in[1,\fz)$ (resp. $t\in(0,1]$) and $s\in(0,\fz)$,
\begin{equation}\label{2.8}
\Phi(st)\le Ct^p \Phi(s).
\end{equation}
Obviously, if $\Phi$ is of lower type $p$ for some $p\in(0,\fz)$,
then $\lim_{t\to0_{+}}\Phi(t)=0$. So for the sake of convenience,
if it is necessary, we may \emph{assume} that $\Phi(0)=0$. If $\Phi$ is
of both upper type $p_1$ and lower type $p_0$, then $\Phi$ is called
of \emph{type} $(p_0,\,p_1)$. Let
\begin{eqnarray}\label{2.9}
&&p_{\Phi}^{+}\equiv\inf\{p\in(0,\fz):\ \text{there exists a
positive constant}\,
\,C\,\,\\
&&\hspace{4.5 em}\text{such that}\ \eqref{2.8} \,\, \text{holds for
all} \,\,t\in[1,\fz)\,\, \text{and}\,\,s\in(0,\fz)\}\noz
\end{eqnarray}
and
\begin{eqnarray}\label{2.10}
&&p_{\Phi}^-\equiv\sup\{p\in(0,\fz):\ \text{there exists a
positive constant}\,\,C\,\,\\
&&\hspace{4.8 em}\text{such that}\,\,\eqref{2.8} \
\text{holds for all}\ t\in(0,1)\ \text{and}\ s\in(0,\fz)\}.\noz
\end{eqnarray}
It is easy to see that $p_{\Phi}^-\le p_{\Phi}^{+}$
for all $\Phi$. In what follows, $p_{\Phi}^-$ and
$p_{\Phi}^{+}$ are respectively called the {\it critical
lower type index} and the {\it critical upper type index} of $\Phi$.
Throughout the whole paper, we always assume that $\Phi$ satisfies
the following \emph{assumption}.
\medskip
\noindent{\bf Assumption $(\Phi)$.} {Let $\Phi$ be a
positive, continuous, strictly increasing function on $(0,\fz)$
which is of critical lower type $p_{\Phi}^-
\in(0,1]$. Also assume that $\Phi$ is concave.}
\begin{remark}\rm\label{r2.2}
(i) Recall that the function $\Phi$ is called of \emph{strictly lower type} $p$
if \eqref{2.8} holds with $C\ev1$ for all $t\in(0,1)$ and $s\in(0,\fz)$.
Then the \emph{strictly critical lower type index} $p_{\Phi}$
of $\Phi$ is defined by
\begin{equation*}
p_{\Phi}\equiv\sup\{p\in(0,\fz):\,\Phi(st)\le t^p\Phi(s) \
\text{holds for all}\ t\in(0,1)\ \text{and}\ s\in(0,\fz)\}.
\end{equation*}
Obviously, $p_\Phi\le p_\Phi^-\le p_\Phi^+$. Moreover, it was proved in
\cite[Remark 2.1]{jy10} that $\Phi$ is also of strictly lower type
$p_\Phi$. In other words, $p_\Phi$ is \emph{attainable}.
However, $p_\Phi^-$ and $p_\Phi^+$ may not be attainable.
For example, for $p\in(0,1]$, if $\Phi(t)\equiv t^p$ for all $t\in (0,\fz)$,
then $\Phi$ satisfies Assumption $(\Phi)$ and $p_\Phi=p_\Phi^-=p_\Phi^+=p$;
for $p\in[1/2,1]$, if $\Phi(t)\equiv t^p/\ln(e+t)$ for all $t\in (0,\fz)$,
then $\Phi$ satisfies Assumption $(\Phi)$ and
$p_\Phi^-=p=p_\Phi^+$, $p_\Phi^-$ is not attainable,
but $p_\Phi^+$ is attainable;
for $p\in(0,1/2]$, if $\Phi(t)\equiv t^p\ln(e+t)$ for all $t\in (0,\fz)$, then
then $\Phi$ satisfies Assumption $(\Phi)$ and
$p_\Phi^-=p=p_\Phi^+$, $p_\Phi^-$ is attainable,
but $p_\Phi^+$ is not attainable.
(ii) We observe that, via the Aoki-Rolewicz theorem in \cite{ao,Ro57},
all results in \cite{al11,jy10,jy10a,jy} are still
true if the \emph{assumptions on $p_\Phi$
are replaced by the same assumptions on $p_\Phi^-$}.
\end{remark}
Notice that if $\Phi$ satisfies Assumption $(\Phi)$, then
$\Phi(0)=0$. For any positive function $\wz\Phi$ of critical lower
type $p_{\wz\Phi}^-$, if we set $\Phi(t)\equiv\int_0^t
\frac{\wz\Phi(s)}{s}\,ds$ for $t\in[0,\fz)$, then by
\cite[Proposition 3.1]{vi87}, $\Phi$ is equivalent to $\wz\Phi$,
namely, there exists a positive constant $C$ such that
$C^{-1}\wz\Phi(t)\le\Phi(t)\le C\wz\Phi(t)$ for all $t\in[0,\fz)$;
moreover, $\Phi$ is a positive, strictly increasing, concave and
continuous function of critical lower type $p_{\wz\Phi}^-$. Notice
that all our results of this paper are invariant on equivalent Orlicz
functions. From this, we deduce that {\it all results with $\Phi$ as
in Assumption $(\Phi)$ also hold for all positive functions
$\wz\Phi$ of the same critical lower type $p_\Phi^-$ as
$\Phi$}.
Let $\Phi$ satisfy Assumption $(\Phi)$. A
measurable function $f$ on $\cx$ is said to be in the {\it space
$L^{\Phi}(\cx)$} if $\int_{\cx}\Phi(|f(x)|)\,d\mu(x)<\fz$. Moreover, for
any $f\in L^{\Phi}(\cx)$, define
$$\|f\|_{L^{\Phi}(\cx)}\equiv\inf\left\{\lz\in(0,\fz):
\ \int_{\cx}\Phi\lf(\frac{|f(x)|}{\lz}\r)\,d\mu(x)\le1\right\}.$$
Since $\Phi$ is strictly increasing, we define the function $\rho(t)$
on $(0,\fz)$ by
\begin{equation}\label{2.11}
\rho(t)\ev \frac{t^{-1}}{\Phi^{-1}(t^{-1})}
\end{equation}
for all $t\in (0,\fz)$, where $\Phi^{-1}$ is the \emph{inverse function} of
$\Phi$. Then the types of $\Phi$ and $\rho$ have the following relation.
\emph{If $0<p_0\le p_1\le1$ and $\Phi$ is an increasing function, then $\Phi$
is of type $(p_0,p_1)$ if and only if $\rho$ is of type
$(p_1^{-1}-1,p_0^{-1}-1);$} see \cite{vi87} for its proof.
\section{The Space $\vmo${\label{s3}}}
\hskip\parindent In this section, we introduce the generalized
vanishing mean oscillation spaces associated with $L$. Throughout
this section, we \emph{always assume} that $L$ satisfies Assumptions
$(L)_1$ and $(L)_2$.
We first recall the
notion of tent spaces in \cite{r07}, which when $\cx\ev\rn$ were
first introduced by Coifman, Meyer and Stein \cite{cms85}.
For any $\nu>0$ and $x\in\cx$, let
$\bgz_\nu(x)\equiv\{(y,t)\in\cx\times (0,\fz):\,d(x,y)<\nu t\}$
denote the {\it cone of aperture $\nu$ with vertex} $x\in\cx$. For
any closed set $F$ of $\cx$, denote by $\mr_\nu F$ the {\it union of
all cones with vertices in $F$}, namely, $\mr_\nu
F\equiv\bigcup_{x\in F}\bgz_\nu(x)$; and for any open set $O$ in
$\cx$, denote the {\it tent over $O$} by $T_\nu(O)$, which is
defined as $T_\nu(O)\ev[\mr_\nu(O^\com)]^\com$. It is easy to see
that $T_\nu(O)=\{(x,t)\in\cx\times(0,\fz):\,d(x,O^\com)\ge\nu t\}$.
In what follows, we denote $\mr_1(F)$, $\bgz_1(x)$ and $T_1(O)$
simply by $\mr(F)$, $\bgz(x)$ and $\wh O$, respectively.
For all measurable functions $g$ on $\xt$ and $x\in\cx$, define
$$\ca_\nu(g)(x)\ev\lf(\iint_{\bgz_\nu(x)}|g(y,t)|^2\,\frac{d\mu(y)}
{V(x,t)}\,\frac{dt}t\r)^{1/2}$$
and
$$\cro(g)(x)\ev\sup_{B\ni x}\frac{1}{\rho(\mu(B))}\lf(\frac{1}{\mu(B)}
\iint_{\wh B}|g(y,t)|^2\,\frac{d\mu(y)\,dt}t\r)^{1/2},$$
where the supremum is taken over all balls $B$ containing $x$.
We denote $\ca_1(g)$ simply by $\ca(g)$.
Recall that for $p\in(0,\fz)$, the {\it tent space $T_2^p(\cx)$} is
defined to be the space of all measurable functions $g$ on
$\xt$ such that $\|g\|_{T_2^p(\cx)}\ev\|\ca(g)\|_\lp<\fz$, which
when $\cx\equiv\rn$ was introduced by Coifman, Meyer and Stein
\cite{cms85} and when $\cx$ is a space of homogeneous type by Russ
in \cite{r07}. Let $\Phi$ satisfy Assumption $(\Phi)$. In what
follows, we denote by $\tx$ the {\it space of all measurable
functions $g$ on $\xt$ such that $\ca(g)\in L^\Phi(\cx)$}, and for
any $g\in\tx$, define its {\it norm} by
$$\|g\|_{\tx}\ev\|\ca(g)\|_\lx=\inf\lf\{\lz>0:\,\int_\cx\Phi\lf(\frac
{\ca(g)(x)}\lz\r)\,d\mu(x)\le1\r\};$$
the {\it space} $\txz$ is defined to be the space of
all measurable functions $g$ on $\xt$
satisfying $\|g\|_\txz\ev\|\cro(g)\|_{L^\fz(\cx)}<\fz$.
Recall that a function $a$ on $\xt$ is called a {\it $\tx$-atom} if
{\rm (i)} there exists a ball $B\st\cx$ such that $\supp a\st\wh B$;
{\rm(ii)} $\iint_{\wh B}|a(x,t)|^2\,\frac{d\mu(x)\,dt}t\le[\mu(B)]^{-1}
[\rho(\mu(B))]^{-2}$.
Since $\Phi$ is concave, from Jensen's inequality and
H\"older's inequality, we deduce that for all $\tx$-atoms
$a$, $\|a\|_\tx\le1$; see \cite{jy} for the details.
Moreover, the following atomic decomposition for elements in
$\tx$ is just \cite[Theorem 3.1]{jy}.
\begin{lemma}\label{l3.1}
Let $\Phi$ satisfy Assumption $(\Phi)$. Then for any $f\in\tx$,
there exist $\tx$-atoms $\{a_j\}_{j=1}^\fz$ and
$\{\lz_j\}_{j=1}^\fz\st\cc$ such that for almost every
$(x,t)\in\xt$,
\begin{equation}\label{3.1}
f(x,t)=\sum_{j=1}^\fz\lz_j a_j(x,t),
\end{equation}
and the series converges in $\tx$. Moreover, there exists
a positive constant $C$ such that for all $f\in\tx$,
\begin{equation}\label{3.2}
\blz(\{\lz_j a_j\}_{j=1}^\fz)\ev\inf\lf\{\lz>0:\,\sum_{j=1}^\fz \mu(B_j)\Phi
\lf(\frac{|\lz_j|}{\lz \mu(B_j)\rho(\mu(B_j))}\r)\le1\r\}\le C\|f\|_\tx,
\end{equation}
where $\wh B_j$ appears as the support of $a_j$.
\end{lemma}
\begin{definition}\rm\label{d3.1}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$, $\Phi$ satisfy
Assumption $(\Phi)$, $\rho$ be as in \eqref{2.11},
$M\in\nn$, $\ez\in(0,\fz)$ and $B$
be a ball. A function $\bz\in L^2(\cx)$ is called a
\emph{$(\Phi,\,M,\,\ez)_L$-molecule adapted to the ball $B$} if there
exists a function $b\in\cd(L^M)$ such that
{\rm (i)} $\bz=L^Mb$;
{\rm(ii)} For every $k\in\{0,1,\cdots,M\}$ and $j\in\zz_+$, there holds
$$\|(r_B^2L)^kb\|_{L^2(U_j(B))}\le r_B^{2M}2^{-j\ez}[\mu(2^jB)]^{-1/2}
[\rho(\mu(2^jB))]^{-1},$$
where $U_j(B)$ for $j\in\zz_+$ is as in \eqref{2.5}.
\end{definition}
Let $\phi=L^M\nu$ be a function in $L^2(\cx)$, where
$\nu\in\cd(L^M)$. Following \cite{hlmmy, hm09}, for $\ez>0$,
$M\in\nn$ and a fixed $x_0\in\cx$, we introduce the {\it space}
\begin{equation}\label{3.3}
\cm^{M,\ez}_\Phi(L)\ev\lf\{\phi=L^M\nu\in L^2(\cx):\ \|\phi\|_
{\cm^{M,\ez}_\Phi(L)}<\fz\r\},
\end{equation}
where
$$\|\phi\|_{\cm^{M,\ez}_\Phi(L)}\ev\sup_{j\in\zz_+}\lf\{2^{j\ez}[V(x_0,
2^j)]^{1/2}\rho(V(x_0,2^j))\sum^M_{k=0}\|L^k\nu\|_{L^2(U_j(B(x_0,1)))}\r\};$$
see also \cite{al11}.
Notice that if $\phi\in\ml$ for some $\ez>0$ with norm $1$, then
$\phi$ is a $\pme$-molecule adapted to the ball
$B(x_0,1)$. Conversely, if $\bz$ is a $\pme$-molecule
adapted to any ball, then $\bz\in\ml$.
Let $A_t$ denote either $(I+t^2L)^{-1}$ or $e^{-t^2L}$ and
$A_t^*$ either $(I+t^2L^*)^{-1}$ or $e^{-t^2L^*}$.
For any $f\in(\mlx)^*$,
the {\it dual space } of $\mlx$, we claim that
{\it $(I-A_t)^Mf\in L^2_{\rm loc}(\cx)$
in the sense of distributions}.
Indeed, for any ball $B$, if $\psi\in L^2(B)$,
then it follows form the Davies-Gaffney estimate \eqref{2.7}
and Remark \ref{r2.1} that
$(I-A_t^*)^M\psi\in\mlx$ for every $\ez>0$. Thus, there exists
a non-negative constant $C(t,r_B,\dist(B,x_0))$, depending
on $t$, $r_B$ and $\dist(B,x_0)$, such that for all $\psi\in L^2(B)$,
$$|\la(I-A_t)^Mf,\psi\ra|\equiv|\la f,(I-A_t^*)^M\psi\ra|\le C(t,r_B,
\dist(B,x_0))\|f\|_{(\mlx)^*}\|\psi\|_{L^2(B)},$$ which implies that
$(I-A_t)^Mf\in L^2_{\rm loc}(\cx)$ in the sense of distributions.
Finally, for any $M\in\nn$, define
\begin{equation}\label{3.4}
\cm^M_{\Phi,L}(\cx)\equiv\bigcap_{\ez>n(1/p_\Phi^--1/p_\Phi^+)}
(\cm_\Phi^{M,\ez}(L^*))^*,
\end{equation}
where $p_\Phi^+$ and $p_\Phi^-$ are, respectively, as in \eqref{2.9}
and \eqref{2.10}.
\begin{definition}\rm\label{d3.2}
Let $L$, $\Phi$ and $\rho$ be as in Definition \ref{d3.1}
and $M>\frac n2(\frac1{p_\Phi^-}-\frac12)$.
A function $f\in\cm^M_{\Phi,L}(\cx)$ is said to be in the \emph{space} $\bmol$ if
$$\|f\|_{\bmol}\equiv\sup_{B\subset\cx}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\int_B|(I-e^{-r_B^2L})^Mf(x)|^2\,d\mu(x)\r]^{1/2}<\fz,$$
where the supremum is taken over all balls $B$ of $\cx$.
\end{definition}
Now, let us recall some notions on the Orlicz-Hardy spaces associated with
$L$. For all $f\in L^2(\cx)$ and $x\in\cx$, define
$$\cs_Lf(x)\ev\lf(\iint_{\bgz(x)}\lf|\tl f(y)\r|^2\dyt\r)^{1/2}.$$
The {\it Orlicz-Hardy space} $\hx$ is defined to be the completion of the set $\{f\in
L^2(\cx):\,\cs_Lf\in\lx\}$ with respect to the quasi-norm $\|f\|_\hx
\ev\|\cs_Lf\|_\lx$.
The Orlicz-Hardy space $\hx$ was introduced and studied in \cite{al11}
(see also \cite{jy}).
If $\Phi(t)\ev t^p$ for $p\in(0,1]$ and all $t\in(0,\fz)$, then the space $\hx$
coincides with the Hardy space $H_L^p(\cx)$, which was introduced
and studied by Duong and Li \cite{dl}.
Let the \emph{space} $\hmfl$ denote the spaces of finite
linear combinations of $\pme$-molecules.
By \cite[Corollary 3.8]{al11}, we obtain that $\hmfl$ is dense
in $\hx$; see also \cite[Corollary 4.2]{jy}.
In what follows, for $M\in\nn$, let $C(M)$ be the
\emph{positive constant} such that
\begin{equation}\label{3.5}
C(M)\int_0^\fz t^{2(M+1)}e^{-2t^2}\dt=1.
\end{equation}
Recall that a variant of the following representation of finite linear
combinations of molecules was gives by \cite[Theorem 3.15]{al11} without
a detailed proof. The following Theorem \ref{t3.1} gives more \emph{accurate
ranges} of $\ez$ and $M$, comparing with \cite[Theorem 3.15]{al11}.
\begin{theorem}\label{t3.1}
Let $L$, $\Phi$ and $M$ be as in Definition \ref{d3.2}, and $\ez\in(0,M-\mz)$.
Assume that $f=\sum_{i=0}^N\lz_ia_i$, where $N\in\nn$, $\{a_i\}_{i=0}^N$ is a
family of $(\Phi,2M,\ez)_L$-molecules, $\{\lz_i\}_{i=0}^N\subset\cc$
and $\sum_{i=0}^N|\lz_i|<\fz$.
Then there exists a representation of $f=\sum_{i=0}^{2N}\mu_im_i$, where
$\{m_i\}_{i=1}^{2N}$ are $\pme$-molecules, $\{\mu_i\}_{i=0}^{2N}\subset\cc$ and
$\sum_{i=0}^{2N}|\mu_i|\le C\|f\|_\hx,$
where $C$ is a positive constant, depending only on $\cx,L,M,\ez$ and $n$.
\end{theorem}
\begin{proof} Throughout this proof, we choose
$\wz p_\Phi\in(0, p_\Phi^-)$ such that $M>\mzx$ and $\ez\in(0,M-\mzx)$.
Therefore, $\Phi$ is of \emph{lower type} $\wz p_\Phi$ and hence $\rho$
of \emph{upper type} $1/\wz p_\Phi-1$.
Since $\{a_i\}_{i=0}^N$ is a family of $(\Phi,2M,\ez)_L$-molecules,
by definition there exist a family $\{b_i\}_{i=0}^N$ of functions
and a family $\{B_i\}_{i=0}^N$ of balls such that for every $i\in
\{0,1,\cdots,N\}$, $a_i=L^{2M}b_i$ satisfies
Definition \ref{d3.1}(ii). Fix a point $x_0\in\cx$.
Let $\wz C(M)\ev\frac{2C(M)}{M+1}$, where $C(M)$ is as in \eqref{3.5}.
Then $\wz C(M)\int_0^\fz t^{2(M+2)}e^{-2t^2}\dt=1$.
By this and the $L^2$-functional calculus, for
$f=\sum_{i=0}^N\lz_ia_i\in L^2(\cx)$, we have
\begin{eqnarray*}
f&&=\wz C(M)\int_0^\fz (t^2L)^{M+2}e^{-2t^2L}f\dt\\
&&=\wz C(M)\int_{K_1}^\fz (t^2L)^{M+2}e^{-2t^2L}f\dt
+\wz C(M)\int_0^{K_1}\cdots\ev f_1+f_2,
\end{eqnarray*}
where $K_1$ is a \emph{positive constant} which is determined later.
Let us start with the term $f_1$. Set $\mu\ev N^{-1}\|f\|_\hx$.
Substituting $f=\sum_{i=0}^N\lz_ia_i$ into $f_1$, we have
\begin{equation*}
f_1=\wz C(M)\sum_{i=0}^N\lz_i\int_{K_1}^\fz (t^2L)^{M+2}e^{-2t^2L}a_i\dt
=\sum_{i=0}^N\mu_im_{i,K_1},
\end{equation*}
where $\mu_i\ev\wz C(M)\mu$, $m_{i,K_1}\ev L^Mf_{i,K_1}$, and
$$f_{i,K_1}\ev\mu^{-1}\lz_i\int_{K_1}^\fz t^{2(M+2)}L^2
e^{-2t^2L}a_i\dt.$$
Then, obviously, $\sum_{i=0}^N|\mu_i|=\sum_{i=0}^N\mu_i=C(M)\|f\|_\hx$.
We now claim that
for an appropriate choice of $K_1$ and $i\in\{0,1,\cdots,N\}$,
$m_{i,K_1}$ is a $\pme$-molecule adapted to the
ball $B_i$. Observe that $a_i=L^{2M}b_i$, for $i\in\{0,1,\cdots,N\}$.
By Minkowski's inequality, for $k\in\{0,1,\cdots,M\}$,
$i\in\{0,1,\cdots,N\}$ and $j\in \zz_+$,
\begin{eqnarray*}
&&\lf\|(r_{B_i}^2L)^kf_{i,K_1}\r\|_{L^2(U_j(B_i))}\\
&&\hs\le\mu^{-1}|\lz_i|\int_{K_1}^\fz t^{-2M}
\lf\|(t^2L)^{2(M+1)}e^{-2t^2L}(r_{B_i}^2L)^kb_i\r\|_{L^2(U_j(B_i))}\dt\\
&&\hs\le\mu^{-1}|\lz_i|\sum_{l=0}^\fz\int_{K_1}^\fz t^{-2M}
\lf\|(t^2L)^{2(M+1)}e^{-2t^2L}\lf(\chi_{U_l(B_i)}
\lf[(r_{B_i}^2L)^kb_i\r]\r)\r\|_{L^2(U_j(B_i))}\dt\\
&&\hs\ev\mu^{-1}|\lz_i|\sum_{l=0}^\fz\hl,
\end{eqnarray*}
where $U_l(B_i)$ for $l\in\zz_+$ is as in \eqref{2.5}.
When $l<j-1$, by Lemma \ref{l2.2},
$\mu(2^jB_i)\ls2^{n(j-l)}\mu(2^lB_i)$,
$\rho(\mu(2^jB_i))\ls2^{n(j-l)(1/\wz p_\Phi-1)}\rho(\mu(2^lB_i))$
and Definition \ref{d3.1}(ii),
we conclude that
\begin{eqnarray*}
\hl&&\ls\int_{K_1}^\fz t^{-2M}\lf\|(r_{B_i}^2L)^kb_i\r\|_{L^2(U_l(B_i))}
\lf(\frac t{2^jr_{B_i}}\r)^{\ez+n(1/\wz p_\Phi-1/2)}\dt\\
&&\ls\int_{K_1}^\fz
t^{-2M}r_{B_i}^{4M}2^{-l\ez}[\mu(2^lB_i)]^{-1/2}[\rho(\mu(2^lB_i))]^{-1}
\lf(\frac t{2^jr_{B_i}}\r)^{\ez+n(1/\wz p_\Phi-1/2)}\dt\\
&&\ls
r_{B_i}^{2M}2^{-j\ez}[\mu(2^jB_i)]^{-1/2}[\rho(\mu(2^jB_i))]^{-1}2^{-l(\ez+\mzx)}
\lf(\frac {r_{B_i}}{K_1}\r)^{2[M-\frac\ez2-\mzx]}.
\end{eqnarray*}
When $l\in\{j-1,j,j+1\}$, from Lemma \ref{l2.2}
and Definition \ref{d3.1}(ii), it follows that
\begin{eqnarray*}
\hl&&\ls\int_{K_1}^\fz t^{-2M}\lf\|(r_{B_i}^2L)^kb_i\r\|_{L^2(U_j(B_i))}\dt\\
&&\ls r_{B_i}^{2M}2^{-j\ez}[\mu(2^jB_i)]^{-1/2}[\rho(\mu(2^jB_i))]^{-1}
\lf(\frac {r_{B_i}}{K_1}\r)^{2M}.
\end{eqnarray*}
When $l>j+1$, by Lemma \ref{l2.2},
$\mu(2^jB_i)\ls \mu(2^lB_i)$, $\rho(\mu(2^jB_i))\ls\rho(\mu(2^lB_i))$
and Definition \ref{d3.1}(ii), we obtain
\begin{eqnarray*}
\hl&&\ls\int_{K_1}^\fz t^{-2M}\lf\|(r_{B_i}^2L)^kb_i\r\|_{L^2(U_l(B_i))}
\lf(\frac t{2^lr_{B_i}}\r)^{\ez}\dt\\
&&\ls r_{B_i}^{2M}2^{-j\ez}[\mu(2^jB_i)]^{-1/2}[\rho(\mu(2^jB_i))]^{-1}2^{-l\ez}
\lf(\frac {r_{B_i}}{K_1}\r)^{2M-\ez}.
\end{eqnarray*}
Combining these estimates, by choosing $K_1>\max\{r_{B_1},\cdots,r_{B_N}\}$,
we further conclude that there exists a positive constant $\wz C$,
independent of $i$, such that
\begin{eqnarray*}
&&\lf\|(r_{B_i}^2L)^kf_{i,K_1}\r\|_{L^2(U_j(B_i))}\\
&&\hs\le \wz Cr_{B_i}^{2M}2^{-j\ez}[\mu(2^jB_i)]^{-1/2}[\rho(\mu(2^jB_i))]^{-1}
\mu^{-1}|\lz_i|\lf(\frac {r_{B_i}}{K_1}\r)^{2[M-\frac\ez2-\mzx]}.
\end{eqnarray*}
Then, by choosing
$$K_1\ev
\max_{0\le i\le N}\lf\{r_{B_i}\lf[\wz C\mu^{-1}\max_{0\le i\le N}|\lz_i|\r]
^{\frac1{2[M-\frac\ez2-\mzx]}}\r\},$$
we see that for $i\in\{0,1,\cdots,N\}$, $m_{i,K_1}$ is a $\pme$-molecule
adapted to the ball $B_i$, which shows the claim.
We now consider the term $f_2$. Set $\mu\ev N^{-1}\|f\|_\hx$.
Substituting $f=\sum_{i=0}^N\lz_ia_i$ into $f_2$, we have
\begin{equation*}
f_2=\wz C(M)\sum_{i=0}^N\lz_i\int_0^{K_1}(t^2L)^{M+1}e^{-t^2L}(\tl a_i)\dt
=\sum_{i=0}^N\mu_im_{i,K_1},
\end{equation*}
where $\mu_i\ev C(M)\mu$, $m_{i,K_1}\ev L^Mf_{i,K_1}$, and
$$f_{i,K_1}\ev\mu^{-1}\lz_i\int_0^{K_1} t^{2(M+1)}L
e^{-t^2L}(\tl a_i)\dt.$$
Then, obviously, $\sum_{i=0}^N|\mu_i|=\sum_{i=0}^N\mu_i=C(M)\|f\|_\hx$.
We now claim that for $K_1$ as above and $i\in\{0,1,\cdots,N\}$,
$m_{i,K_1}$ is a $\pme$-molecule adapted to the
ball $2^{K_0}B_i$, where $K_0\in(0,\fz)$ is determined later.
To show the claim, for $i\in\{0,1,\cdots,N\}$ and $j\in \zz_+$,
set
$\boz_{j,K_0}\ev 2^{j+K_0+2}B_i\bh2^{j+K_0-2}B_i$
and write
\begin{eqnarray*}
f_{i,K_1}=&&\mu^{-1}\lz_i\int_0^{K_1} t^{2(M+1)}L
e^{-t^2L}\lf([\tl a_i]\chi_{\boz_{j,K_0}}\r)\dt\\
&&+\mu^{-1}\lz_i\int_0^{K_1} t^{2(M+1)}L
e^{-t^2L}\lf([\tl a_i]\chi_{\boz_{j,K_0}^\com}\r)\dt
\ev g_{i,K_1,K_0}+h_{i,K_1,K_0}.
\end{eqnarray*}
Then, by Minkowski's inequality, for $k\in\{0,1,\cdots,M\}$,
$i\in\{0,1,\cdots,N\}$ and $j\in \zz_+$,
\begin{eqnarray*}
&&\lf\|(2^{2K_0}r_{B_i}^2L)^kg_{i,K_1,K_0}\r\|_{L^2(U_j(2^{K_0}B_i))}\\
&&\hs\le\mu^{-1}|\lz_i|r_{B_i}^{2M}\lf\|\int_0^{K_1}
\lf(\frac t{r_{B_i}}\r)^{2M-2k}2^{2kK_0}\r.\\
&&\hs\hs\times\lf.
(t^2L)^{k+1}e^{-t^2L}\lf(\lf[\tl a_i\r]\chi_{\boz_{j,K_0}}\r)
\dt\r\|_{L^2(U_j(2^{K_0}B_i))}\\
&&\hs\le
\mu^{-1}|\lz_i|\sum_{l=0}^\fz\int_0^{K_1}\lf(\frac t{r_{B_i}}\r)^{2M-2k}2^{2kK_0}
\lf\|\chi_{U_l(2^{K_0}B_i)}
\tl a_i\r\|_{L^2(\boz_{j,K_0})}\dt\\
&&\hs\ev\mu^{-1}|\lz_i|\sum_{l=0}^\fz\hl.
\end{eqnarray*}
When $l<j-2$, from Lemma \ref{l2.2},
$\mu(2^{j+K_0}B_i)\ls2^{n(j-l)}\mu(2^{l+K_0}B_i)$,
$\rho(\mu(2^{j+K_0}B_i))\ls2^{n(j-l)
(1/\wz p_\Phi-1)}\rho(\mu(2^{l+K_0}B_i))$ and Definition \ref{d3.1}(ii),
it follows that
\begin{eqnarray*}
\hl&&\ls\int_0^{K_1}
\lf(\frac t{r_{B_i}}\r)^{2M-2k}2^{2kK_0}\lf\|a_i\r\|_{L^2(U_l(2^{K_0}B_i))}
\lf(\frac t{2^{j+K_0}r_{B_i}}\r)^{\ez+n(1/\wz p_\Phi-1/2)}\dt\\
&&\ls\int_0^{K_1} \lf(\frac t{r_{B_i}}\r)^{2M-2k}2^{2kK_0}r_{B_i}^{4M}
2^{-(l+K_0)\ez}[\mu(2^{l+K_0}B_i)]^{-1/2}[\rho(\mu(2^{l+K_0}B_i))]^{-1}\\
&&\hs\times\lf(\frac t{2^{j+K_0}r_{B_i}}\r)^{\ez+n(1/\wz p_\Phi-1/2)}\dt\\
&&\ls (2^{K_0}r_{B_i})^{2M}2^{-j\ez}[\mu(2^{j+K_0}B_i)]^{-1/2}
[\rho(\mu(2^{j+K_0}B_i))]^{-1}2^{-l[\ez+\mzx]}\\
&&\hs\times2^{-2K_0[M-k+\ez+\mzx]}
K_1^{2M-2k+\ez+n(1/\wz p_\Phi-1/2)}r_{B_i}^{2M+2k-\ez-n(1/\wz p_\Phi-1/2)}.
\end{eqnarray*}
When $l\in\{j-2,\cdots,j+2\}$, by Lemma \ref{l2.2} and Definition \ref{d3.1}(ii),
we see that
\begin{eqnarray*}
\hl&&\ls\int_0^{K_1} \lf(\frac t{r_{B_i}}\r)^{2M-2k}2^{2kK_0}
\lf\|a_i\r\|_{L^2(U_j(2^{K_0}B_i))}\dt\\
&&\ls (2^{K_0}r_{B_i})^{2M}2^{-j\ez}[\mu(2^{j+K_0}B_i)]^{-1/2}
[\rho(\mu(2^{j+K_0}B_i))]^{-1}2^{-2K_0(M-k+\ez/2)}
K_1^{2M-2k}r_{B_i}^{2M+2k}.
\end{eqnarray*}
When $l>j+2$, from Lemma \ref{l2.2},
$\mu(2^jB_i)\ls \mu(2^lB_i)$, $\rho(\mu(2^{j+K_0}B_i))\ls\rho(\mu(2^{l+K_0}B_i))$
and Definition \ref{d3.1}(ii), we infer that
\begin{eqnarray*}
\hl&&\ls\int_0^{K_1}
\lf(\frac t{r_{B_i}}\r)^{2M-2k}2^{2kK_0}\lf\|a_i\r\|_{L^2(U_l(2^{K_0}B_i))}
\lf(\frac t{2^{l+K_0}r_{B_i}}\r)^{\ez}\dt\\
&&\ls\int_0^{K_1} \lf(\frac t{r_{B_i}}\r)^{2M-2k}2^{2kK_0}r_{B_i}^{4M}
2^{-(l+K_0)\ez}[\mu(2^{l+K_0}B_i)]^{-1/2}[\rho(\mu(2^{l+K_0}B_i))]^{-1}\\
&&\hs\times\lf(\frac t{2^{l+K_0}r_{B_i}}\r)^{\ez}\dt\\
&&\ls (2^{K_0}r_{B_i})^{2M}2^{-j\ez}[\mu(2^{j+K_0}B_i)]^{-1/2}
[\rho(\mu(2^{j+K_0}B_i))]^{-1}2^{-l\ez}\\
&&\hs\times2^{-2K_0(M-k+\ez)}K_1^{2M-2k+\ez}r_{B_i}^{2M+2k-\ez}.
\end{eqnarray*}
Then we estimate $h_{i,K_1,K_0}$. By Minkowski's inequality
and Definition \ref{d3.1}(ii), for $k\in\{0,1,\cdots,M\}$,
$i\in\{0,1,\cdots,N\}$ and $j\in \zz_+$, we conclude that
\begin{eqnarray*}
&&\lf\|(2^{2K_0}r_{B_i}^2L)^kh_{i,K_1,K_0}\r\|_{L^2(U_j(2^{K_0}B_i))}\\
&&\hs\le\mu^{-1}|\lz_i|r_{B_i}^{2M}\lf\|\int_0^{K_1}
\lf(\frac t{r_{B_i}}\r)^{2M-2k}2^{2kK_0}\r.\\
&&\hs\hs\times\lf.
(t^2L)^{k+1}e^{-t^2L}\lf(\lf[\tl a_i\r]
\chi_{\boz_{j,K_0}^\com}\r)\dt\r\|_{L^2(U_j(2^{K_0}B_i))}\\
&&\hs\le\mu^{-1}|\lz_i|\int_0^{K_1}\lf(\frac t{r_{B_i}}\r)^{2M-2k}2^{2kK_0}
\lf(\frac t{2^{j+K_0}r_{B_i}}\r)^{\ez+n(1/\wz p_\Phi-1/2)}
\lf\|\tl a_i\r\|_{L^2(\cx)}\dt\\
&&\hs\ls (2^{K_0}r_{B_i})^{2M}2^{-j\ez}[\mu(2^{j+K_0}B_i)]^{-1/2}
[\rho(\mu(2^{j+K_0}B_i))]^{-1}\\
&&\hs\hs\times2^{-2K_0[M-k+\ez+\mzx]}
K_1^{2M-2k+\ez+n(1/\wz p_\Phi-1/2)}r_{B_i}^{2M+2k-\ez-n(1/\wz p_\Phi-1/2)}.
\end{eqnarray*}
Combining these estimates, by choosing $K_1>\max\{r_{B_1},\cdots,r_{B_N}\}$,
we further see that
\begin{eqnarray*}
\lf\|(2^{2K_0}r_{B_i}^2L)^kf_{i,K_1}\r\|_{L^2(U_j(2^{K_0}B_i))}
&&\ls(2^{K_0}r_{B_i})^{2M}2^{-j\ez}[\mu(2^{j+K_0}B_i)]^{-1/2}
[\rho(\mu(2^{j+K_0}B_i))]^{-1}\\
&&\hs\hs\times2^{-2K_0(M-k+\ez/2)}K_1^{2M-2k+\ez+\mzx}r_{B_i}^{2M+2k}.
\end{eqnarray*}
Then, by choosing
$$K_0\ev\max_{0\le k\le M}\lf(\frac {\ln\Big(K_1^{2M-2k+\ez+\mzx}
{\max_{0\le i\le N}}\{r_{B_i}^{2M+2k}\}\Big)}{2\ln2(M-k+\ez/2)}\r),$$
we conclude that for $i\in\{0,1,\cdots,N\}$, $m_{i,K_1}$ is
a $\pme$-molecule adapted to the ball $2^{K_0}B_i$, which shows
the claim, and hence completes the proof of Theorem \ref{t3.1}.
\end{proof}
\begin{remark}\rm\label{r3.1} We point out that the proof of
Theorem \ref{t3.1} also works for \cite[Theorem 5.4]{hlmmy}.
Moreover, due to the lack of the support of molecules,
we show that $m_{i, K_1}$ for $i\in\{1,\cdots, N\}$
is a $\pme$-molecule adapted to
the ball $2^{K_0}B_i$, instead of $B_i$ as in the proof of
\cite[Theorem 5.4]{hlmmy}, which also simplifies the proof
of \cite[Theorem 5.4]{hlmmy}.
\end{remark}
By Theorem \ref{t3.1}, the argument same as the proofs
of \cite[Theorems 3.13 and 3.16]{al11}, we obtain the following
dual theorem. We omit the details.
\begin{theorem}\label{t3.2}
Let $L$, $\Phi$, $\rho$ and $M$ be as in Definition \ref{d3.2}. Then for any
function $f\in\bmol$, the linear functional $\ell$, defined by
$\ell(g)\ev\la f,g\ra$
initially on $H_{\Phi,{\rm fin},L^*}^{\rm{mol},\ez,2\wz M}(\cx)$ with $\wz M>M$
and $\ez\in(0,\wz M-\mz)$, has a unique extension to $\hxx$ and, moreover,
$\|\ell\|_{(\hxx)^*}\le C\|f\|_\bmol$
for some nonnegative constant $C$ independent of $f$.
Conversely, for any $\ell\in(\hxx)^*$, there exists $f\in\bmol$
such that
$\ell(g)\ev\la f,g\ra$
for all $g\in H_{\Phi,fin,L^*}^{mol,\ez, M}(\cx)$ and $\|f\|_\bmol\le C
\|\ell\|_{(\hxx)^*}$, where $C$ is a nonnegative constant independent of $\ell$.
\end{theorem}
\begin{remark}\rm\label{r3.2}
(i) Theorem \ref{t3.1} is just \cite[Theorems 3.15]{al11} but with the ranges
of indices $M$ and $\ez$ replaced, respectively, by $M>\mz$ and $\ez\in(0,M-\mz)$.
(ii) By Theorem \ref{t3.2}, we see that for all $M>\mz$, the spaces $\bmol$ for
different $M$ coincide with equivalent norms; thus, in what follows,
we denote $\bmol$ simply by $\bmo$.
\end{remark}
The following two propositions are just \cite[Propositions 3.11 and 3.12]{al11}
(see also \cite[Propositions 4.4 and 4.5]{jy}).
\begin{proposition}\label{p3.1}
Let $L$, $\Phi$, $\rho$ and $M$ be as in Definition \ref{d3.2}. Then
$f\in\bmo$ if and only if $f\in\cm^M_{\Phi,L}(\cx)$ and
$$\sup_{B\subset\cx}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\int_B\lf|\lf[I-(I+r_B^2L)^{-1}\r]^Mf(x)\r|^2\,d\mu(x)\r]^{1/2}<\fz.$$
Moreover, the quantity appeared in the left-hand side of the above
formula is equivalent to $\|f\|_{\bmol}$.
\end{proposition}
\begin{proposition}\label{p3.2}
Let $L$, $\Phi$, $\rho$ and $M$ be as in Definition \ref{d3.2}. Then there
exists a positive constant $C$ such that for all $f\in\bmo$,
$$\sup_{B\subset\cx}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}\iint_{\wh B}
|(t^2L)^Me^{-t^2L}f(x)|^2\,\frac{d\mu(x)\,dt}t\r]^{1/2}\le C\|f\|_{\bmol}.$$
\end{proposition}
The following Proposition \ref{p3.3} and Lemma \ref{l3.2}
are a kind of Calder\'on reproducing formulae.
\begin{proposition}\label{p3.3}
Let $L$, $\Phi$, $\rho$ and $M$ be as in Definition \ref{d3.2},
$\ez, \ez_1\in(0,\fz)$
and $\wz M>M+\ez_1+\frac n4+\frac N2(\frac 1{p_\Phi^-}-1)$, where
$N$ is as in \eqref{2.4}. Fix $x_0\in\cx$. Assume that
$f\in\cm_{\Phi,L}^{M}(\cx)$ satisfies that
\begin{equation}\label{3.6}
\int_\cx
\frac{|(I-(I+L)^{-1})^Mf(x)|^2}{1+[d(x,x_0)]^{n+\ez_1+2N(1/p_\Phi^--1)}}
\,d\mu(x)<\fz.
\end{equation}
Then for all $\pml$-molecules $\az$,
$$\la f,\az\ra=C(M)\iint_\xt\tml f(x) \ov{\tlx\az(x)}\dxt,$$
where $C(M)$ is as in \eqref{3.5}.
\end{proposition}
\begin{proof}
For $R>\dz>0$, write
\begin{eqnarray*}
&&C(M)\int_\dz^R\int_\cx\tml f(x) \ov{\tlx\az(x)}\dxt\\
&&\hs=\lf\la f,C(M)\int_\dz^R (t^2L^*)^{M+1}e^{-2t^2L^*}\az\dt\r\ra\\
&&\hs=\la f,\az\ra-\lf\la f,\az-C(M)\int_\dz^R (t^2L^*)^{M+1}e^{-2t^2L^*}\az\dt\r\ra.
\end{eqnarray*}
Since $\az$ is a $\pml$-molecule, by Definition \ref{d3.1},
there exists $b\in L^2(\cx)$ such that $\az=(L^*)^{\wz M}b$.
Notice that
\begin{eqnarray*}
f&&=\lf[I-(I+L)^{-1}+(I+L)^{-1}\r]^Mf\\
&&=\sum_{k=0}^M\binom M k\lf[I-(I+L)^{-1}\r]^{M-k}(I+L)^{-k}f
=\sum_{k=0}^M\binom M k\lf[I-(I+L)^{-1}\r]^{M}L^{-k}f,
\end{eqnarray*}
where $\binom M k$ denotes the \emph{binomial coefficient},
which, together with $H_\fz$-functional calculus, further implies that
\begin{eqnarray*}
&&\lf\la f,\az-C(M)\int_\dz^R (t^2L^*)^{M+1}e^{-2t^2L^*}\az\dt\r\ra\\
&&\hs=\sum_{k=0}^M\binom M k
\lf\la \lf[I-(I+L)^{-1}\r]^{M}f,
L^{\wz M-k}b-C(M)\int_\dz^R (t^2L^*)^{M+1}e^{-2t^2L^*}(L^*)^{\wz M-k}b\dt\r\ra\\
&&\hs=\sum_{k=0}^M\binom M k
\lf\la \lf[I-(I+L)^{-1}\r]^{M}f,
C(M)\int_0^\dz (t^2L^*)^{M+1}e^{-2t^2L^*}(L^*)^{\wz M-k}b\dt\r\ra\\
&&\hs\hs+\sum_{k=0}^M\binom M k
\lf\la \lf[I-(I+L)^{-1}\r]^{M}f,
C(M)\int_R^\fz (t^2L^*)^{M+1}e^{-2t^2L^*}(L^*)^{\wz M-k}b\dt\r\ra\\
&&\hs\ev
\sum_{k=0}^M\binom M k(\mh+\mj).
\end{eqnarray*}
For $\mj$, by \eqref{3.6} and H\"older's inequality, we conclude that
\begin{eqnarray*}
|\mj|&&\ls
\lf\{\int_\cx\frac{|(I-(I+L)^{-1})^Mf(x)|^2}{1+[d(x,x_0)]^{n+\ez_1+2N(1/p_\Phi^--1)}}
\,d\mu(x)\r\}^{1/2}\\
&&\hs\times
\lf\{\int_\cx\lf|
\int_R^\fz(t^2L^*)^{M+\wz M-k+1}e^{-2t^2L^*}b(x)\frac1{t^{2(\wz M-k)+1}}\,dt\r|^2
\r.\\
&&\hs\times\lf(1+[d(x,x_0)]^{n+\ez_1+2N(1/p_\Phi^--1)}\r)\,d\mu(x)\Bigg\}^{1/2}\\
&&\ls
\int_R^\fz\lf\|(t^2L^*)^{M+\wz M-k+1}e^{-2t^2L^*}b
\lf(1+[d(\cdot,x_0)]^{n+\ez_1+2N(1/p_\Phi^--1)}\r)^{1/2}\r\|_{L^2(\cx)}\\
&&\hs\times\frac 1{t^{2(\wz M-k)+1}}\,dt.
\end{eqnarray*}
Let $B_0\ev B(x_0, 1)$. Notice that there exist $\wz N,\,d\in\nn$
such that for all $j\in\nn$, $j\ge \wz N$,
$$U_j(B_0)\st\bigcup_{i=-d}^{d}U_{j+i}(B),$$
where $B$ is the ball adapted to $\az$ and $U_j(B)$ for $j\in\zz_+$
is as in \eqref{2.5}.
By choosing $j_0\ge\wz N$, we conclude that
\begin{eqnarray*}
|\mj|
&&\ls
\int_R^\fz\lf\|(t^2L^*)^{M+\wz M-k+1}e^{-2t^2L^*}b\r.\\
&&\hs\lf.\times
(1+[d(\cdot,x_0)]^{n+\ez_1+2N(1/p_\Phi^--1)})^{1/2}\r\|_{L^2(2^{j_0}B_0)}
\frac1{t^{2(\wz M-k)+1}}\,dt\\
&&\hs+\sum_{j=j_0+1}^\fz
\int_R^\fz\lf\|(t^2L^*)^{M+\wz M-k+1}e^{-2t^2L^*}b\r.\\
&&\hs\lf.\times
(1+[d(\cdot,x_0)]^{n+\ez_1+2N(1/p_\Phi^--1)})^{1/2}\r\|_{L^2(U_j(B_0))}
\frac1{t^{2(\wz M-k)+1}}\,dt\ev\mj_1+\mj_2.
\end{eqnarray*}
For all $\wz\ez>0$, let
$C_1\ev2^{\frac {j_0}2(n+\ez_1+2N(1/p_\Phi^--1))}\|b\|_{L^2(\cx)}$ and
$R_1\ev(\frac {C_1}{\wz\ez})^{\frac1{2(\wz M-k)}}$, then for all
$R>R_1$, we obtain
\begin{eqnarray*}
\mj_1
\ls2^{\frac {j_0}2(n+\ez_1+2N(1/p_\Phi^--1))}
\int_R^\fz\frac{dt}{t^{2(\wz M-k)+1}}\|b\|_{L^2(\cx)}\ls\wz\ez.
\end{eqnarray*}
Letting $C_2\ev r_B^{\mz +2\wz M}$ and
$R_1\ev(\frac {C_2}{\wz\ez})^{\frac1{2(\wz M-k)}}$, then for all
$R>R_1$, we know that
\begin{eqnarray*}
\mj_2&&
\ls\sum_{j=j_0+1}^\fz 2^{\frac j2(n+\ez_1+2N(1/p_\Phi^--1))}\\
&&\hs\times\sum_{i=-d}^d\Bigg\{
\int_R^\fz\lf\|(t^2L^*)^{M+\wz M-k+1}e^{-2t^2L^*}
(\chi_{\wz U_{j+i}(B)} b)
\r\|_{L^2(U_{j+i}(B))}
\frac1{t^{2(\wz M-k)+1}}\,dt\\
&&\hs+\int_R^\fz\lf\|(t^2L^*)^{M+\wz M-k+1}e^{-2t^2L^*}
(\chi_{(\wz U_{j+i}(B))^\com} b)
\r\|_{L^2(U_{j+i}(B))}
\frac1{t^{2(\wz M-k)+1}}\,dt\Bigg\},
\end{eqnarray*}
where $\wz U_{j+i}(B)\ev 2^{j+i+1}B\bh2^{j+i-1}B$. Then, since
\begin{eqnarray*}
&&\int_R^\fz\lf\|(t^2L^*)^{M+\wz M-k+1}e^{-2t^2L^*}
(\chi_{\wz U_{j+i}(B)} b)
\r\|_{L^2(U_{j+i}(B))}
\frac1{t^{2(\wz M-k)+1}}\,dt\\
&&\hs\ls\frac1{R^{2(\wz M-k)}}\|b\|_{L^2(\wz U_{j+i}(B))}
\ls2^{-\frac{j}{2}(n+\ez_1+2N(1/p_\Phi^--1))}\wz \ez,
\end{eqnarray*}
and $\int_R^\fz\lf\|(t^2L^*)^{M+\wz M-k+1}e^{-2t^2L^*}
(\chi_{(\wz U_{j+i}(B))^\com} b)
\r\|_{L^2(U_{j+i}(B))}
\frac1{t^{2(\wz M-k)+1}}\,dt$
satisfies the same estimate, we see that $\mj_2\ls\wz\ez$.
Thus, $\lim_{R\to\fz}\mj=0$.
To consider $\mh$, let $\wz f\ev[I-(I+L)^{-1}]^Mf$. Then
\begin{eqnarray*}
S_{M+1}&&\ev\lf\la \wz f,
\int_0^\dz (t^2L^*)^{M+1}e^{-2t^2L^*}(L^*)^{\wz M-k}b\dt\r\ra\\
&&=-\frac14\lf\la \wz f,
\int_0^\dz (t^2L^*)^{M}\frac{\pa}{\pa t}( e^{-2t^2L^*})(L^*)^{\wz M-k}b\dt\r\ra\\
&&=-\frac14\lf\la \wz f,
(\dz^2L^*)^{M} e^{-2\dz^2L^*}(L^*)^{\wz M-k}b\r\ra
+\frac M2\lf\la \wz f,
\int_0^\dz (t^2L^*)^{M} e^{-2t^2L^*}(L^*)^{\wz M-k}b\dt\r\ra.
\end{eqnarray*}
Thus,
\begin{eqnarray*}
S_{M+1}&&=
-\frac14\lf\la \wz f,(\dz^2L^*)^{M} e^{-2\dz^2L^*}(L^*)^{\wz M-k}b\r\ra
+\frac M2S_M\\
&&=\sum_{\ell=1}^M\frac{-M!}{2^{\ell+1}(M-\ell+1)!}
\lf\la \wz f,(\dz^2L^*)^{M-\ell+1} e^{-2\dz^2L^*}(L^*)^{\wz M-k}b\r\ra
+\frac{M!}{2^M}S_1.
\end{eqnarray*}
For all $\ell\in\{1,\cdots,M\}$, from H\"older's inequality, we infer that
\begin{eqnarray*}
&&\lf|\lf\la \wz f,(\dz^2L^*)^{M-\ell+1} e^{-2\dz^2L^*}(L^*)^{\wz M-k}b\r\ra\r|\\
&&\hs\ls
\lf\{\int_\cx\frac{|(I-(I+L)^{-1})^Mf(x)|^2}{1+[d(x,x_0)]^{n+\ez_1+2N(1/p_\Phi^--1)}}
\,d\mu(x)\r\}^{1/2}\\
&&\hs\hs\times
\lf\{\int_\cx\lf|
(\dz^2L^*)^{M-\ell+1}e^{-2\dz^2L^*}(L^*)^{\wz M-k}b(x)\r|^2
\lf(1+[d(x,x_0)]^{n+\ez_1+2N(1/p_\Phi^--1)}\r)\,d\mu(x)\r\}^{1/2}\\
&&\hs\ls
2^{\frac{j_0}2[n+\ez_1+2N(1/p_\Phi^--1)]}
\lf\|(\dz^2L^*)^{M-\ell+1}e^{-2\dz^2L^*}(L^*)^{\wz M-k}b\r\|_{L^2(2^{j_0}B_0)}\\
&&\hs\hs+\sum_{j=j_0+1}^\fz
2^{\frac{j}2[n+\ez_1+2N(1/p_\Phi^--1)]}\\
&&\hs\hs\times\Bigg\{
\lf\|(\dz^2L^*)^{M-\ell+1}e^{-2\dz^2L^*}
\lf(\chi_{\bigcup_{i=j-d-1}^{j+d+1}U_i(B)}
(L^*)^{\wz M-k}b\r)\r\|_{L^2(U_j(B_0))}\\
&&\hs\hs+\lf\|(\dz^2L^*)^{M-\ell+1}e^{-2\dz^2L^*}
\lf(\chi_{(\bigcup_{i=j-d-1}^{j+d+1}U_i(B))^\com}
(L^*)^{\wz M-k}b\r)\r\|_{L^2(U_j(B_0))}\Bigg\}.
\end{eqnarray*}
By the $L^2$-functional calculus, we see that
$\lim_{\dz\to0}(\dz^2L^*)^{M-\ell+1}e^{-2\dz^2L^*}(L^*)^{\wz M-k}b=0$ in
$L^2(\cx)$ and, by Lemma \ref{l2.2}, we know that
\begin{eqnarray*}
&&\sum_{j=j_0+1}^\fz
2^{\frac{j}2[n+\ez_1+2N(1/p_\Phi^--1)]}\Bigg\{
\lf\|(\dz^2L^*)^{M-\ell+1}e^{-2\dz^2L^*}
\lf(\chi_{\bigcup_{i=j-d-1}^{j+d+1}U_i(B)}(L^*)^{\wz M-k}b\r)\r\|_{L^2(U_j(B_0))}\\
&&\hs\hs+\lf\|(\dz^2L^*)^{M-\ell+1}e^{-2\dz^2L^*}
\lf(\chi_{(\bigcup_{i=j-d-1}^{j+d+1}
U_i(B))^\com}(L^*)^{\wz M-k}b\r)\r\|_{L^2(U_j(B_0))}\Bigg\}\\
&&\hs\ls\sum_{j=j_0+1}^\fz
2^{\frac{j}2[n+\ez_1+2N(1/p_\Phi^--1)]}
\lf[\|(L^*)^{\wz M-k}b\|_{L^2(\bigcup_{i=j-d-1}^{j+d+1}U_i(B))}
+e^{-\frac{2^jr_B}{\dz}}\|(L^*)^{\wz M-k}b\|_{L^2(\cx)}\r]\\
&&\hs\ls\wz\ez.
\end{eqnarray*}
From
\begin{eqnarray*}
S_1=\lf\la \wz f,
\int_0^\dz (t^2L^*)e^{-2t^2L^*}(L^*)^{\wz M-k}b\dt\r\ra
=\lf\la \wz f, \lf(e^{-2\dz^2L^*}-I\r)(L^*)^{\wz M-k}b\r\ra,
\end{eqnarray*}
and
$$\lim_{\dz\to0}\lf\|\lf(e^{-2\dz^2L^*}-I\r)(L^*)^{\wz M-k}b\r\|_{L^2(\cx)}=0,$$
it follows that $\lim_{\dz\to0}H=0$, which completes the proof of
Proposition \ref{p3.3}.
\end{proof}
Instead of \cite[Proposition 4.6]{jy} by Proposition \ref{3.3} here,
repeating the proof of \cite[Corollary 4.3]{jy}, we obtain the following
Lemma \ref{3.2}. The details are omitted.
\begin{lemma}\label{l3.2}
Let $L$, $\Phi$, $\rho$ and $M$ be as in Definition \ref{d3.2} and $\ez\in(0,\fz)$.
If $f\in\bmo$, then for any $\pmx$-molecule $\az$,
there holds
$$\la f, \az\ra=C(M)\iint_\xt\tml f(x) \ov{\tlx\az(x)}\dxt.$$
\end{lemma}
Recall that a measure $d\mu$ on $\xt$ is called a \emph{$\rho$-Carleson
measure} if
\begin{equation*}
\|d\mu\|_\rho\ev
\sup_{B\st \cx}\lf\{\frac1{\mu(B)[\rho(\mu(B))]^2}
\iint_{\wh B}|d\mu|\r\}^{1/2}<\fz,
\end{equation*}
where the supremum is taken over all balls $B$ of $\cx$.
Using Theorem \ref{t3.2} and Proposition \ref{p3.2}, similar to
the proof of \cite[Theorem 4.2]{jy}, we obtain
the following $\rho$-Carleson measure characterization of $\bmo$.
\begin{theorem}\label{t3.3}
Let $L$, $\Phi$, $\rho$ and $M$ be as in Definition \ref{d3.2}.
Fix $x_0\in\cx$. Then the following are equivalent:
{\rm (i)} $f\in\bmo$;
{\rm (ii)} $f\in\cm_{\Phi,L}^{M}(\cx)$ satisfies that
$$\int_\cx
\frac{|(I-(I+L)^{-1})^Mf(x)|^2}{1+[d(x,x_0)]^{n+\ez_1+2N(1/p_\Phi^--1)}}
\,d\mu(x)<\fz$$
for some $\ez_1\in(0,\fz)$, and $d\mu_f$ is a $\rho$-Carleson measure,
where $d\mu_f$ is defined by $d\mu_f(x,t)\ev|(t^2L)^{M}e^{-t^2L}f(x)|^2\dxt$
for all $(x,t)\in\xt$.
Moreover, $\|d\mu_f\|_\rho$ is equivalent to $\|f\|_\bmo$.
\end{theorem}
\begin{proof}
It follows from Proposition \ref{p3.1} and the proof of Lemma \ref{l3.2}
that (i) implies (ii).
To show that (ii) implies (i),
let $\wz M>M+\ez_1+\frac n4+\frac N2(\frac1{p_\Phi^-}-1)$.
From Proposition \ref{p3.3}, we deduce that
$$\la f,g\ra=C(M)\iint_\xt\tml f(x) \ov{\tlx g(x)}\dxt,$$
where $g$ is any finite combination of $\pmx$-molecules.
Then $t^2L^*e^{-t^2L^*}g\in\tx$. By Lemma \ref{l3.1}, there exist
$\{\lz_j\}_{j=1}^\fz\st\cc$ and $\tx$-atoms $\{a_j\}_{j=1}^\fz$
supported in $\{\wh B_j\}_{j=1}^\fz$ such that \eqref{3.1} and
\eqref{3.2} hold. This, together with Fatou's lemma and H\"older's
inequality, implies that
\begin{eqnarray*}
|\la f,g\ra|&&=\lf|C(M)\iint_\xt\tml f(x) \ov{\tlx g(x)}\dxt\r|\\
&&\ls\sum_j|\lz_j|\int_0^\fz\int_\cx|\tml f(x)\ov{a_j(x,t)}|\dxt\\
&&\ls\sum_j|\lz_j|\|a_j\|_{T_2^2(\cx)}
\lf(\iint_{\wh B_j}|\tml f(x)|^2\dxt\r)^{1/2}\\
&&\ls\sum_j|\lz_j|\|d\mu_f\|_\rho\ls\|\tmlx g\|_\tx\|d\mu_f\|_\rho
\sim\|g\|_\hxx\|d\mu_f\|_\rho.
\end{eqnarray*}
By this and Theorem \ref{t3.2}, we conclude that $f\in(\hxx)^*=\bmo$,
which completes the proof of Theorem \ref{t3.3}.
\end{proof}
Now we introduce the space $\vmo$.
\begin{definition}\rm\label{d3.3}
Let $L$, $\Phi$, $\rho$ and $M$ be as in Definition \ref{d3.2}. An
element $f\in\bmo$ is said to be in the \emph{space} $\vmol$ if it satisfies the
following limiting conditions $\gz_1(f) =\gz_2(f)=\gz_3(f)=0$, where
$x_0\in\cx$ is a fixed point, $c\in(0,\fz)$,
$$\gz_1(f)\equiv
\lim_{c\to0}\sup_{B:\,r_B\le c}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\int_B|(I-e^{-r_B^2L})^Mf(x)|^2\,d\mu(x)\r]^{1/2},$$
$$\gz_2(f)\equiv
\lim_{c\to\fz}\sup_{B:\,r_B\ge c}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\int_B|(I-e^{-r_B^2L})^Mf(x)|^2\,d\mu(x)\r]^{1/2},$$
and
$$\gz_3(f)\equiv\lim_{c\to\fz}\sup_{B:\,B\subset[B(x_0,c)]^
\complement}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\int_B|(I-e^{-r_B^2L})^Mf(x)|^2\,d\mu(x)\r]^{1/2}.$$
For any $f\in\vmol$, define $\|f\|_{\vmol}\equiv\|f\|_{\bmo}$.
\end{definition}
\begin{definition}\rm\label{d3.4}
Let $\Phi$ satisfy Assumption $(\Phi)$ and
$\rho$ be as in \eqref{2.11}. The \emph{space}
$\txv$ is defined to be the space of all $f\in\txz$
satisfying $\eta_1(f)=\eta_2(f)=\eta_3(f)=0$ with the same norm as
the space $\txz$, where $x_0\in\cx$ is a fixed point, $c\in(0,\fz)$,
$$\eta_1(f)\equiv
\lim_{c\to0}\sup_{B:\,r_B\le c}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\iint_{\wh B}|f(y,t)|^2\,\frac{d\mu(y)\,dt}t\r]^{1/2},$$
$$\eta_2(f)\equiv
\lim_{c\to\fz}\sup_{B:\,r_B\ge c}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\iint_{\wh B}|f(y,t)|^2\,\frac{d\mu(y)\,dt}t\r]^{1/2},$$
and
$$\eta_3(f)\equiv\lim_{c\to\fz}\sup_{B:\,B\subset[B(x_0,c)]^
\complement}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\iint_{\wh B}|f(y,t)|^2\,\frac{d\mu(y)\,dt}t\r]^{1/2}.$$
\end{definition}
It is easy to see that $\txv$ is a \emph{closed linear subspace} of $\txz$.
Further, denote by $\txy$ the {\it space of all $f\in\txz$ with
$\eta_1(f)=0$}, and $\txb$ the {\it space of all $f\in T_2^2(\cx)$
with bounded support}. Obviously, we have $\txb\st\txv\st\txy$.
Finally, denote by $\txl$ the {\it closure of $\txb$ in the space
$\txy$}.
\begin{lemma}\label{l3.3}
Let $L$ and $\Phi$ be as in Definition \ref{d3.1}, and $\txv$ and
$\txl$ defined as above. Then $\txv$ and $\txl$ coincide with
equivalent norms.
\end{lemma}
\begin{proof}
Since $\txb\st\txv$ and $\txv$ is a closed linear subspace
of $\txz$, we conclude that $\txl= {\txb}\st\txv$.
Conversely, for any $f\in\txv$, by the definition of $\txv$, for any $\ez>0$,
there exist positive constants $a_0$, $b_0$ and $c_0$ such that
\begin{equation}\label{3.7}
\sup_{B:\,r_B\le a_0}\frac1{\mu(B)[\rho(\mu(B))]^2}
\iint_{\wh B}|f(y,t)|^2\,\frac{d\mu(y)\,dt}t<\ez,
\end{equation}
\begin{equation}\label{3.8}
\sup_{B:\,r_B\ge b_0}\frac1{\mu(B)[\rho(\mu(B))]^2}
\iint_{\wh B}|f(y,t)|^2\,\frac{d\mu(y)\,dt}t<\ez,
\end{equation}
and
\begin{equation}\label{3.9}
\sup_{B:\,B\subset[B(x_0,c_0)]^
\complement}\frac1{\mu(B)[\rho(\mu(B))]^2}
\iint_{\wh B}|f(y,t)|^2\,\frac{d\mu(y)\,dt}t<\ez.
\end{equation}
Let $K_0\ev\max\{a_0^{-1},b_0,c_0\}$ and, for all $(y,t)\in\xt$,
\begin{equation*}
g(y,t)\ev f(y,t)\chi_{B(x_0,2K_0)\times((2K_0)^{-1},2K_0)}(y,t).
\end{equation*}
Obviously, $g\in\txb$. To complete the proof of Lemma \ref{l3.3},
we need show that
$$\|f-g\|_\txz^2\ls \ez.$$
We consider the following three cases
for all balls $B$ in \eqref{3.7}, \eqref{3.8} and \eqref{3.9}.
Case (i) $r_B<a_0$ or $r_B>b_0$. In this case,
from \eqref{3.7} and \eqref{3.8}, we deduce that
\begin{equation*}
\|f-g\|_\txz^2\le\frac2{\mu(B)[\rho(\mu(B))]^2}
\iint_{\wh B}|f(y,t)|^2\,\frac{d\mu(y)\,dt}t\le2\ez.
\end{equation*}
Case (ii) $a_0\le r_B\le b_0$ and $B\st [B(x_0,c_0)]^\com$.
In this case, by \eqref{3.9}, we conclude that
\begin{equation*}
\|f-g\|_\txz^2\le\frac2{\mu(B)[\rho(\mu(B))]^2}
\iint_{\wh B}|f(y,t)|^2\,\frac{d\mu(y)\,dt}t\le2\ez.
\end{equation*}
Case (iii) $a_0\le r_B\le b_0$ and $B\cap B(x_0,c_0)\neq\emptyset$.
In this case, we have
\begin{eqnarray*}
\iint_{\wh B}|f(y,t)-g(y,t)|^2\,\frac{d\mu(y)\,dt}t
&&\le
\int_0^{(2K_0)^{-1}}\int_B|f(y,t)|^2\dyt\\
&&\le
\int_0^{(2K_0)^{-1}}\int_{B(x_B,2^ka_0)}|f(y,t)|^2\dyt,
\end{eqnarray*}
where $x_B$ is the \emph{center} of $B$ and $k$
the \emph{smallest integer} such that $2^ka_0>r_B$.
Then, by Lemma \ref{l2.1}, we pick a family of balls with
the same radius $a_0$, $\{B(x_{B,i},a_0)\}_{i=1}^{N_k}$, such that
$B(x_B,2^ka_0)\st\cup_{i=1}^{N_k}B(x_{B,i},a_0)$, ${N_k}\ls 2^{kn}$ and
$\sum_{i=1}^{N_k}\chi_{B(x_{B,i},a_0)}\ls1$. Therefore, combining the fact that
$\rho$ is an increasing function, we obtain
\begin{eqnarray*}
\iint_{\wh B}|f(y,t)-g(y,t)|^2\,\frac{d\mu(y)\,dt}t
&&\le
\int_0^{(2K_0)^{-1}}\int_{\cup_{i=1}^{N_k}B(x_{B,i},a_0)}|f(y,t)|^2\dyt\\
&&\le
\sum_{i=1}^{N_k}\iint_{\wh B(x_{B,i},a_0)}|f(y,t)|^2\dyt\\
&&\ls
\ez\sum_{i=1}^{N_k}\mu(B(x_{B,i},a_0))[\rho(\mu(B(x_{B,i},a_0)))]^2\\
&&\ls
\ez[\rho(\mu(B))]^2\sum_{i=1}^{N_k}\mu(B(x_{B,i},a_0))
\ls\ez\mu(B)[\rho(\mu(B))]^2,
\end{eqnarray*}
which completes the proof of Lemma \ref{l3.3}.
\end{proof}
\begin{definition}\rm\label{d3.5}
Let $L$, $\Phi$, $\rho$ and $M$ be as in Definition \ref{d3.2}.
The \emph{space} $\tvmol$ is defined to be the space of all elements
$f\in\bmol$ that satisfy the following limiting conditions
$\wz\gz_1(f)=\wz\gz_2(f)=\wz\gz_3(f)=0$, where $c\in(0,\fz)$,
$$\wz\gz_1(f)\equiv
\lim_{c\to0}\sup_{B:\,r_B\le c}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\int_B|(I-[I+r_B^2L]^{-1})^Mf(x)|^2\,d\mu(x)\r]^{1/2},$$
$$\wz\gz_1(f)\equiv
\lim_{c\to\fz}\sup_{B:\,r_B\ge c}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\int_B|(I-[I+r_B^2L]^{-1})^Mf(x)|^2\,d\mu(x)\r]^{1/2},$$
and
$$\wz\gz_1(f)\equiv\lim_{c\to\fz}\sup_{B:\,B\subset[B(0,c)
]^\complement}\frac1{\rho(\mu(B))}\lf[\frac1{\mu(B)}
\int_B|(I-[I+r_B^2L]^{-1})^Mf(x)|^2\,d\mu(x)\r]^{1/2}.$$
\end{definition}
\begin{proposition}\label{p3.4}
Let $L$, $\Phi$, $\rho$ and $M$ be as in Definition \ref{d3.2}. Then
$f\in\vmol$ if and only if $f\in\tvmol$.
\end{proposition}
\begin{proof}
Suppose that $f\in\tvmol$. To see $f\in\vmol$, it suffices to
show that
\begin{equation}\label{3.10}
\frac1{\rho(\mu(B))[\mu(B)]^{1/2}}\lf[\int_B\lf|(I-e^{-r_B^2L})^Mf(x)\r|^2
\,d\mu(x)\r]^{1/2}\ls\sum_{k=0}^\fz2^{-k}\dz_k(f,B),
\end{equation}
where
\begin{eqnarray}\label{3.11}
\dz_k(f,B)\ev&&\sup_{\{B'\st2^{k+1}B:\,r_{B'}\in[2^{-1}r_B,r_B]\}}
\frac1{\rho(\mu(B))[\mu(B)]^{1/2}}\\
&&\times\lf[\int_B\lf|(I-[I+r_B^2L]^{-1})^Mf(x)\r|^2\,d\mu(x)\r]^{1/2}.\noz
\end{eqnarray}
Indeed, since $f\in\tvmol$, by Definition \ref{d3.5} and Proposition
\ref{p3.1}, we conclude that $\dfb\ls\|f\|_\bmo$ and for all $k\in\zz_+$,
$$\lim_{c\to0}\sup_{B:\,r_B\le c}\dfb=\lim_{c\to\fz}\sup_{B:\,r_B\ge c}
\dfb=\lim_{c\to\fz}\sup_{B:\,B\subset[B(x_0,c)]^\complement}\dfb=0.$$
Then by the dominated convergence theorem for series, we have
\begin{eqnarray*}
\gz_1(f)&&=\lim_{c\to0}\sup_{B:\,r_B\le c}\frac1{\rho(\mu(B))[\mu(B)]^{1/2}}
\lf[\int_B\lf|\lf(I-e^{-r_B^2L}\r)^Mf(x)\r|^2\,d\mu(x)\r]^{1/2}\\
&&\ls\sum_{k=1}^\fz2^{-k}\lim_{c\to0}\sup_{B:\,r_B\le c}\dfb=0.
\end{eqnarray*}
Similarly we see that $\gz_2(f)=\gz_3(f)=0$, and hence $f\in\vmol$.
Let us now prove \eqref{3.10}. Write
\begin{equation}\label{3.12}
f=\lf(I-[I+r_B^2L]^{-1}\r)^Mf+\lf\{I-\lf(I-[I+r_B^2L]^{-1}\r)^M\r\}f\ev f_1+f_2.
\end{equation}
By Lemma \ref{l2.2}, we have
\begin{eqnarray}\label{3.13}
&&\lf\|\Iem f_1\r\|_{L^2(B)}\\
&&\hs\le\sum_{k=0}^\fz\lf\|\Iem(f_1\chi_{U_k(B)})\r\|_{L^2(B)}\ls
\sum_{k=0}^\fz e^{-c2^{2k}}\lf\|f_1\chi_{U_k(B)}\r\|_{L^2(\cx)}\noz\\
&&\hs\ls\rho(\mu(B))[\mu(B)]^{1/2}\sum_{k=0}^\fz e^{-c2^{2k}}2^{kn}\dfb\noz\\
&&\hs\ls\rho(\mu(B))[\mu(B)]^{1/2}\sum_{k=0}^\fz 2^{-k}\dfb,\noz
\end{eqnarray}
where $U_k(B)$ for all $k\in\zz_+$ is as in \eqref{2.5},
$c$ is a positive constant and the third inequality follows
from Lemma \ref{l2.1} that there exists a collection, $\{B_{k,1},B_{k,2},
\cdots,B_{k,N_k}\}$, of balls such that each ball $B_{k,i}$ is of
radius $r_B$, $B(x_B,2^{k+1}r_B)\st\cup_{i=1}^{N_k}B_{k,i}$ and
$N_k\ls2^{nk}$.
To estimate the remaining term, by the formula that
\begin{equation}\label{3.14}
I-\Ilm=\sum_{j=1}^M\frac{M!}{j!(M-j)!}(r_B^2L)^{-j}\Ilm
\end{equation}
(which relies on the fact that $(I-(I+r^2L)^{-1})(r^2L)^{-1}=
(I+r^2L)^{-1}$ for all $r\in(0,\fz)$), and Minkowski's
inequality, we obtain
\begin{eqnarray}\label{3.15}
&&\lf\|\Iem f_2\r\|_{L^2(B)}\\
&&\hs\ls\sum_{j=1}^M\lf\{\int_B\lf|\lf(I-e^{-r_B^2L}\r)^{M-j}
\lf[-\int_0^{r_B}\frac s{r_B^2}e^{-s^2L}\,ds\r]^jf_1(x)\r|^2
\,d\mu(x)\r\}^{1/2}\noz\\
&&\hs\ls\sum_{j=1}^M\sum_{i=0}^{M-j}\int_0^{r_B}\cdots\int_0^{r_B}
\frac{s_1}{r_B^2}\cdots\frac{s_j}{r_B^2}\|e^{-(ir_B^2+s_1^2+
\cdots+s_j^2)L}f_1\|_{L^2(B)}\,ds_1\cdots ds_j\noz\\
&&\hs\ls\sum_{j=1}^M\sum_{i=0}^{M-j}\int_0^{r_B}\!\!\cdots\!\!\int_0^{r_B}
\frac{s_1}{r_B^2}\!\cdots\!\frac{s_j}{r_B^2}\sum_{k=0}^\fz e^{-\frac{c
(2^kr_B)^2}{ir_B^2+s_1^2+\cdots+s_j^2}}\|f_1\chi_{U_k(B)}\|_{L^2(\cx)}
\,ds_1\cdots ds_j\noz\\
&&\hs\ls\rho(\mu(B))[\mu(B)]^{1/2}\sum_{k=0}^\fz
e^{-\frac{c2^{2k}}M}2^{kn}\dfb\noz\\
&&\hs\ls\rho(\mu(B))[\mu(B)]^{1/2}\sum_{k=0}^\fz 2^{-k}\dfb,\noz
\end{eqnarray}
where $c$ is a positive constant and in the penultimate inequality,
we used the fact that $\int_0^{r_B}\cdots\int_0^{r_B}
\frac{s_1}{r_B^2}\cdots\frac{s_j}{r_B^2}\,ds_1\cdots ds_j\sim1$.
Combining the estimates \eqref{3.13} and \eqref{3.15}, we obtain
\eqref{3.10}, which further implies that $\tvmol\st\vmol$.
By borrowing some ideas from the proof of \cite[Lemma 8.1]{hm09},
then similar to the proof above, we conclude that $\vmol\st\tvmol$
and the details are omitted. This finishes the proof of Proposition
\ref{p3.4}.
\end{proof}
We now characterize the space $\vmol$ via the tent space.
\begin{theorem}\label{t3.4}
Let $L$, $\Phi$ and $\rho$ be as in Definition \ref{d3.1}, $M$, $M_1\in\nn$
and $M_1\ge M>\mz$. Then the following are equivalent:
{\rm (i)} $f\in\vmol$;
{\rm (ii)} $f\in\cm_{\Phi,L}^{M_1}(\cx)$ and $(t^2L)^{M_1}e^{-t^2L}f\in\txv$.
Moreover, $\|(t^2L)^{M_1}e^{-t^2L}f\|_\txz$ is equivalent to $\|f\|_\bmo$.
\end{theorem}
\begin{proof}
We first show that (i) implies (ii).
Let $f\in\vmol$. By Proposition \ref{p3.2}, we know that $\tmy f\in\txz$.
To see that $\tmy f\in\txv$, we claim that it suffices to
show that for all balls $B$,
\begin{equation}\label{3.16}
\frac1{\rho(\mu(B))[\mu(B)]^{1/2}}\lf[\iint_{\wh B}|\tmy f(x)|^2
\,\frac{d\mu(x)\,dt}t\r]^{1/2}\ls\sum_{k=0}^\fz2^{-k}\dz_k(f,B),
\end{equation}
where $\dfb$ is as in \eqref{3.11}.
Indeed, since $f\in\vmol=\tvmol$, we conclude that for each $k\in\nn$,
$\dfb\ls\|f\|_\bmo$ and
$$\lim_{c\to0}\sup_{B:\,r_B\le c}\dfb=\lim_{c\to\fz}\sup_{B:\,r_B\ge c}
\dfb=\lim_{c\to\fz}\sup_{B:\,B\subset[B(x_0,c)]^\complement}\dfb=0.$$
Then from the dominated convergence theorem for series, we infer that
\begin{eqnarray*}
\eta_1(f)&&=\lim_{c\to0}\sup_{B:\,r_B\le c}\frac1{\rho(\mu(B))[\mu(B)]^{1/2}}
\lf[\iint_{\wh B}\lf|\tmy f(x)\r|^2\,\frac{d\mu(x)\,dt}t\r]^{1/2}\\
&&\ls\sum_{k=1}^\fz2^{-k}\lim_{c\to0}\sup_{B:\,r_B\le c}\dfb=0.
\end{eqnarray*}
Similarly we see that $\eta_2(f)=\eta_3(f)=0$, and hence $\tmy f\in\txv$.
Let us now prove \eqref{3.16}. Write $f\ev f_1+f_2$ as in \eqref{3.12}.
Then by Lemmas \ref{l2.2} and \ref{l2.3},
similar to the estimate of \eqref{3.13}, we have
\begin{eqnarray}\label{3.17}
&&\lf\{\iint_{\wh B}\lf|\tmy f_1(x)\r|^2\dxt\r\}^{1/2}\\
&&\hs\le
\sum_{k=0}^\fz\lf\{\iint_{\wh B}\lf|\tmy (f_1\cub)(x)\r|^2\dxt\r\}^{1/2}\noz\\
&&\hs\ls\|f_1\|_{L^2(4B)}+\sum_{k=3}^\fz\lf[\int_0^{r_B}\exp\lf\{
-\frac{(2^kr_B)^2}{ct^2}\r\}\,\frac{dt}t\r]^{1/2}\lf\|f_1\cub\r\|_{L^2(\cx)}\noz\\
&&\hs\ls\|f_1\|_{L^2(4B)}+\sum_{k=3}^\fz\lf\{\int_0^{r_B}\lf[
\frac{t^2}{(2^kr_B)^2}\r]^{n+1}
\,\frac{dt}t\r\}^{1/2}\lf\|f_1\cub\r\|_{L^2(\cx)}\noz\\
&&\hs\ls\rho(\mu(B))[\mu(B)]^{1/2}\sum_{k=0}^\fz 2^{-k}\dfb,\noz
\end{eqnarray}
where $U_k(B)$ for all $k\in\zz_+$ is as in \eqref{2.5} and
$c$ is a positive constant. Applying \eqref{3.14}, Lemma \ref{l2.2}
and $M_1>M$ to $f_2$, we see that
\begin{eqnarray}\label{3.18}
&&\lf\{\iint_{\wh B}\lf|\tmy f_2(x)\r|^2\dxt\r\}^{1/2}\\
&&\hs\ls
\sum_{j=1}^M\lf\{\iint_{\wh B}\lf|\tmy (r_B^2L)^{-j}f_1(x)\r|^2\dxt\r\}^{1/2}\noz\\
&&\hs\ls\sum_{j=1}^M\sum_{k=0}^\fz\lf\{\iint_{\wh B}\lf[\frac{t^2}
{r_B^2}\r]^{2j}\lf|(t^2L)^{M_1-j}e^{-t^2L} (f_1\cub)(x)\r|^2\dxt\r\}^{1/2}\noz\\
&&\hs\ls\sum_{j=1}^M\lf\{\sum_{k=0}^2\lf[\int_0^{r_B}\lf(\frac{t^2}
{r_B^2}\r)^{2j}\dt\r]^{1/2}\|f_1\|_{L^2(4B)}\r.\noz\\
&&\hs\hs\lf.+\sum_{k=3}^\fz\lf[\int_0^{r_B}\exp\lf\{
-\frac{(2^kr_B)^2}{ct^2}\r\}
\,\frac{dt}t\r]^{1/2}\lf\|f_1\cub\r\|_{L^2(\cx)}\r\}\noz\\
&&\hs\ls\|f_1\|_{L^2(4B)}+\sum_{k=3}^\fz\lf\{\int_0^{r_B}\lf[
\frac{t^2}{(2^kr_B)^2}\r]^{n+1}
\,\frac{dt}t\r\}^{1/2}\lf\|f_1\cub\r\|_{L^2(\cx)}\noz\\
&&\hs\ls\rho(\mu(B))[\mu(B)]^{1/2}\sum_{k=0}^\fz 2^{-k}\dfb.\noz
\end{eqnarray}
The estimates \eqref{3.17} and \eqref{3.18} imply \eqref{3.16},
which completes the proof that (i) implies (ii).
Conversely, let $f\in\cm_{\Phi,L}^{M_1}(\cx)$ and $\tmy f\in\txv$.
By Proposition \ref{p3.2}, we conclude that $f\in\bmo$. For any ball $B$,
write
\begin{eqnarray*}
\lf(\int_B\lf|\Iem f(x)\r|^2\,d\mu(x)\r)^{1/2}&&=\sup_{\|g\|_{L^2(B)}\le1}
\lf|\int_B\Iem f(x)\ov{g(x)}\,d\mu(x)\r|\\
&&=\sup_{\|g\|_{L^2(B)}\le1}\lf|\int_B f(x)\ov{\Iemx g(x)}\,d\mu(x)\r|.
\end{eqnarray*}
Notice that for any $g\in L^2(B)$, $\iemx g$ is a multiple of a
$(\Phi,\,M,\,\ez)_{L^*}$-molecule; see \cite[p. 43]{hm09}.
Then by Lemma \ref{l3.2}
and H\"older's inequality, we obtain
\begin{eqnarray*}
&&\lf[\int_B\lf|\Iem f(x)\r|^2\,d\mu(x)\r]^{1/2}\\
&&\hs\sim\sup_{\|g\|_{L^2(B)}\le1}
\lf|\iint_\xt\tmy f(x)t^2L^*e^{-t^2L^*}\ov{\Iemx g(x)}\dxt\r|\\
&&\hs\sim\sum_{k=0}^\fz\lf\{\iint_{V_k(B)}\lf|\tmy f(x)\r|^2\dxt\r\}^{1/2}\\
&&\hs\hs\times\sup_{\|g\|_{L^2(B)}\le1}\lf\{\iint_{V_k(B)}\lf|t^2L^*e^{-t^2L^*}
\Iemx g(x)\r|^2\dxt\r\}^{1/2}\\
&&\hs\ev\sum_{k=0}^\fz\sigma_k(f,B)\mi_k,
\end{eqnarray*}
where $V_0(B)\ev\wh B$ and $V_k(B)\ev(\wh{2^kB})\bh(\wh{2^{k-1}B})$ for
$k\in\nn$. In what follows, for $k\ge2$, let $V_{k,1}\ev(\wh{2^kB})\bh
(2^{k-2}B\times(0,\fz))$ and $V_{k,2}\ev V_k(B)\bh V_{k,1}(B)$.
For $k\in\{0,1,2\}$, by Lemmas \ref{l2.2} and \ref{l2.3}, we conclude that
\begin{eqnarray*}
\mi_k&&=\sup_{\|g\|_{L^2(B)}\le1}\lf\{\iint_{V_k(B)}\lf|t^2L^*e^{-t^2L^*}
\Iemx g(x)\r|^2\dxt\r\}^{1/2}\\
&&\ls\sup_{\|g\|_{L^2(B)}\le1}\lf\|\Iemx g\r\|_{L^2(\cx)}\ls1.
\end{eqnarray*}
Now for $k\ge3$, write
\begin{eqnarray*}
\mi_k&&\ls\sup_{\|g\|_{L^2(B)}\le1}\lf\{\iint_{V_{k,1}(B)}\lf|t^2L^*e^{-t^2L^*}
\Iemx g(x)\r|^2\dxt\r\}^{1/2}\\
&&\hs+
\sup_{\|g\|_{L^2(B)}\le1}\lf\{\iint_{V_{k,2}(B)}\cdots\r\}^{1/2}
\ev\mi_{k,1}+\mi_{k,2}.
\end{eqnarray*}
Since for any $(y,t)\in\ve$, $t\ge2^{k-2}r_B$, from Minkowski's inequality
and Lemmas \ref{l2.2} and \ref{l2.3}, it follows that
\begin{eqnarray*}
\mi_{k,2}&&=\sup_{\|g\|_{L^2(B)}\le1}\lf\{\iint_{V_{k,2}(B)}\lf|t^2L^*e^{-t^2L^*}
\Iemx g(x)\r|^2\dxt\r\}^{1/2}\\
&&=\sup_{\|g\|_{L^2(B)}\le1}\lf\{\iint_{V_{k,2}(B)}\lf|t^2L^*e^{-t^2L^*}
\lf[-\int_0^{r_B^2}L^*e^{-sL^*}\,ds\r]^M
g(x)\r|^2\dxt\r\}^{1/2}\\
&&\ls\sup_{\|g\|_{L^2(B)}\le1}\int_0^{r_B^2}\cdots\int_0^{r_B^2}\lf\{
\iint_\ve\lf|t^2(L^*)^{M+1}\r.\r.\\
&&\hs\times\lf.\lf.
e^{-(t^2+s_1+\cdots+s_M)L^*}g(x)\r|^2\dxt\r\}^{1/2}ds_1\cdots ds_M\\
&&\ls\sup_{\|g\|_{L^2(B)}\le1}\int_0^{r_B^2}\cdots\int_0^{r_B^2}\lf\{
\int_{2^{k-2}r_B}^{2^{k}r_B}\frac{t^4\|g\|_{L^2(B)}^2}{(t^2+s_1+
\cdots+s_M)^{2(M+1)}}\dt\r\}^{1/2}ds_1\cdots ds_M\\&&\ls2^{-2kM}.
\end{eqnarray*}
Similarly, we see that $\mi_{k,1}\ls2^{-2kM}$.
Let $\wz p_\Phi\in (0, p_\Phi^-)$ such that
$M>\frac n2(\frac1{\wz p_\Phi}-\frac12)$. Combining the above
estimates and the fact that $\rho$ is of upper type $1/\wz p_\Phi-1$,
we finally conclude that
\begin{eqnarray*}
&&\frac1{\rho(\mu(B))[\mu(B)]^{1/2}}\lf[\int_{ B}\lf|\Iem f(x)\r|^2\,d\mu(x)\r]^{1/2}\\
&&\hs\ls\sum_{k=0}^\fz2^{-2kM}\frac1{\rho(\mu(B))[\mu(B)]^{1/2}}\sz_k(f,B)\\
&&\hs\ls\sum_{k=0}^\fz2^{-k[2M-n(1/\wz p_\Phi-1/2)]}\frac{\sz_k(f,B)}
{\rho(\mu(2^kB))[\mu(2^kB)]^{1/2}}.
\end{eqnarray*}
Since $\tmy f\in\txv\st\tx,$
from $M>\frac n2(\frac1{\wz p_\Phi}-\frac12)$
and the dominated convergence theorem for series, we infer that
\begin{eqnarray*}
\gz_1(f)&&=\lim_{c\to0}\sup_{B:\,r_B\le c}\frac1{\rho(\mu(B))[\mu(B)]^{1/2}}
\lf[\int_B\lf|\lf(I-e^{-r_B^2L}\r)^Mf(x)\r|^2\,d\mu(x)\r]^{1/2}\\
&&\ls\sum_{k=1}^\fz2^{-k[2M-n( 1/\wz p_\Phi-1/2)]}\lim_{c\to0}\sup_{B:\,r_B\le c}
\frac{\sz_k(f,B)}{\rho(\mu(2^kB))[\mu(2^kB)]^{1/2}}=0.
\end{eqnarray*}
Similarly, $\gz_2(f)=\gz_3(f)=0$, which implies that $f\in\vmol$,
and hence completes the proof of Theorem \ref{t3.4}.
\end{proof}
\begin{remark}\rm\label{r3.3}
It follows from Theorem \ref{t3.4} that for all $M\in\nn$ and $M>\mz$, the spaces
$\vmol$ coincide with equivalent norms. Thus, in what follows, we
denote the $\vmol$ simply by $\vmo$.
\end{remark}
\section{The Dual Space of $\vmo${\label{s4}}}
\hskip\parindent In this section, we show that the dual space of
$\vmo$ is $\bxx$, where the \emph{space} $\bxx$
denotes the Banach completion of the space
$\hxx$; see Definition \ref{d4.3} and Theorem \ref{t4.2} below.
The proof of the following proposition is similar to that of
\cite[Proposition 4.1]{jyz09}; we omit the details here.
\begin{proposition}\label{p4.1}
Let $\Phi$ satisfy Assumption $(\Phi)$. Then the dual space of $\tx$
is $\txz$. Moreover, the pairing
$$\la f,g\ra\to\int_\xt f(y,t)g(y,t)\dyt$$
for all $f\in\ttx$ and $g\in\txz$
realizes $\txz$ being equivalent to the dual of $\tx$.
\end{proposition}
We now introduce a new tent space $\ttx$ and present some properties.
\begin{definition}\rm\label{d4.1}
Let $p\in(0,1)$. The \emph{space} $\ttx$ is defined to be the space of
all $f=\sum_{j=1}^\fz\lz_j a_j$ in $(\txz)^*$, where $\{a_j\}_{j=1}^\fz$
are $\tx$-atoms and $\{\lz_j\}_{j=1}^\fz\subset\cc$ such
that $\sum_{j=1}^\fz|\lz_j|<\fz$. If $f\in\ttx$,
then define $\|f\|_\ttx\ev\inf\{\sum_{j=1}^\fz|\lz_j|\}$, where
the infimum is taken over all the possible decompositions of $f$
as above.
\end{definition}
By \cite[Lemma 3.1]{hm09},
$\ttx$ is a Banach space. Moreover, from Definition \ref{d4.1},
it is easy to deduce that $\tx$ is dense in $\ttx$; in other
words, $\ttx$ is a \emph{Banach completion} of $\tx$.
\begin{lemma}\label{l4.1}
Let $\Phi$ satisfy Assumption $(\Phi)$. Then $\tx$ is a dense
subspace of $\ttx$ and there exists a positive constant $C$ such
that for all $f\in\tx$, $\|f\|_\ttx\le C\|f\|_\tx$.
\end{lemma}
\begin{proof}
Let $f\in\tx$. By Theorem \ref{l3.1}, there exist
$\tx$-atoms $\{a_j\}_{j=1}^\fz$ and
$\{\lz_j\}_{j=1}^\fz\subset\cc$ such that \eqref{3.1} and
\eqref{3.2} hold.
For any $L\in\nn$, set $\sz_L\ev\sum_{j=1}^L|\lz_j|$. Since $\Phi$
is of upper type $1$, by this together with $\rho(t)=t^{-1}
/\Phi^{-1}(t^{-1})$ for all $t\in(0,\fz)$, we obtain
$$\sum_{j=1}^\fz \mu(B_j)\Phi\lf(\frac{|\lz_j|}{\sz_L \mu(B_j)\rho(\mu(B_j))}\r)
\ge\sum_{j=1}^L\mu(B_j)\Phi\lf(\frac{1}{\sz_L \mu(B_j)\rho(\mu(B_j))}\r)
\frac{|\lz_j|}{\sz_L}\gs 1,$$
which implies that
$$\sum_{j=1}^L|\lz_j|\ls\blz(\{\lz_ja_j\}_{j=1}^\fz)\ls\|f\|_\tx.$$
Letting $L\to\fz$, we further conclude that $\sum_{j=1}^\fz|\lz_j|\ls\|f\|_\tx$.
Since $f\in\tx$ and $\txx=\txz$, we see that
$$f\in\tx\st(\txx)^*=(\txz)^*.$$
Thus, $f\in(\txz)^*$ and $\|f\|_{(\txz)^*}\ls\|f\|_\tx$. Recall that for
any $\ell\in(\txz)^*$, its $(\txz)^*$ norm is defined by
$$\|\ell\|_{(\txz)^*}=\sup_{\|g\|_\txz\le1}|\ell(g)|.$$
Observe also that $a_j\in(\txz)^*$ for all $j\in\nn$. Now, from these
observations, the monotone convergence theorem and H\"older's inequality,
it follows that
\begin{eqnarray*}
\lf\|f-\sum_{j=1}^L\lz_ja_j\r\|_{(\txz)^*}
&&\!=\!\sup_{\|g\|_\txz\le1}\lf|\int_\xt\lf[f(x,t)-\sum_{j=1}^L\lz_ja_j
(x,t)\r]g(x,t)\dxt\r|\\
&&\le\sup_{\|g\|_\txz\le1}\int_\xt\sum_{j= L+1}^\fz|\lz_j||a_j(x,t)g(x,t)|\dxt\\
&&=\sup_{\|g\|_\txz\le1}\sum_{j= L+1}^\fz|\lz_j|\int_{\wh{B_j}}|a_j(x,t)g(x,t)|\dxt\\
&&\le
\sup_{\|g\|_\txz\le1}\sum_{j= L+1}^\fz|\lz_j|\|a_j\|_\txe\|g\chi_{\wh{B_j}}\|_\txe
\le\sum_{j= L+1}^\fz|\lz_j|\to0,
\end{eqnarray*}
as $L\to\fz$. Thus, the series in \eqref{3.1} converges in $(\txz)^*$, which
further implies that $f\in\ttx$ and
$\|f\|_\ttx\le\sum_{j=1}^\fz|\lz_j|\ls\|f\|_\tx.$
This finishes the proof of Lemma \ref{l4.1}.
\end{proof}
\begin{lemma}\label{l4.2}
Let $\Phi$ satisfy Assumption $(\Phi)$. Then $\txb$ is dense in $\ttx$.
\end{lemma}
\begin{proof}
Since $\tx$ is dense in $\ttx$, to prove this lemma, it suffices to prove that
$\txb$ is dense in $\tx$ in the norm $\|\cdot\|_\ttx$.
Fix $x_0\in\cx$. For any $g\in\tx$ and $k\in\nn$, let $g_k\ev g\chi_{O_k}$, where
$$O_k\ev\{(x,t)\in\xt:\ \dist(x,x_0)<k,\ t\in(1/k,k)\}.$$
By the dominated convergence theorem and the continuity of $\Phi$,
we conclude that for any $\lz>0$,
$$\lim_{k\to\fz}\int_\cx\Phi\lf(\frac{\ca(g-g_k)(x)}\lz\r)\,d\mu(x)
=\int_\cx\lim_{k\to\fz}\Phi\lf(\frac{\ca(g-g_k)(x)}\lz\r)\,d\mu(x)=0,$$
which implies that $\lim_{k\to\fz}\|g-g_k\|_\ttx=0$. Then, by Lemma
\ref{l4.1}, we see that
$$\|g-g_k\|_\ttx\ls\|g-g_k\|_\tx\to0,$$
as $k\to\fz$, which completes the proof of Lemma \ref{l4.2}.
\end{proof}
\begin{lemma}\label{l4.3}
Let $\Phi$ satisfy Assumption $(\Phi)$. Then $(\ttx)^*=\txz$
via the pairing
$$\la f,g\ra\to\int_\xt f(y,t)g(y,t)\dyt$$
for all $f\in\ttx$ and $g\in\txz$.
\end{lemma}
\begin{proof}
By Proposition \ref{p4.1} and the definition of $\ttx$, we see that
$(\tx)^*=\txz$ and $\tx\st\ttx$, which further implies that $(\ttx)^*\st\txz$.
Conversely, let $g\in\txz$. Then for any $f\in\ttx$, choose a sequence
$\tx$-atoms $\{a_j\}_{j=1}^\fz$ and $\{\lz_j\}_{j=1}^\fz\subset\cc$
such that $f=\sum_j\lz_ja_j$ in $(\txz)^*$ and $\sum_j|\lz_j|\ls \|f\|_\ttx$.
Thus, by H\"older's inequality, we obtain
\begin{eqnarray*}
|\la f,g\ra|&&\le\sum_j\int_\xt|a_j(x,t)g(x,t)|\dxt\\
&&\le \|g\|_\txz\sum_j|\lz_j|\ls\|g\|_\txz\|f\|_\ttx,
\end{eqnarray*}
which implies that $g\in(\ttx)^*$, and hence completes the proof of
Lemma \ref{l4.3}.
\end{proof}
\begin{lemma}\label{l4.4}
Let $\Phi$ satisfy Assumption $(\Phi)$. If $f\in\ttx$, then
\begin{equation}\label{4.1}
\|f\|_\ttx=\sup_{g\in\txb,\,\|g\|_\txz\le1}\lf|\int_\xt f(x,t)g(x,t)\dxt\r|.
\end{equation}
\end{lemma}
\begin{proof}
Let $f\in\ttx$. From Lemma \ref{4.2}, we deduce that
$$\|f\|_\ttx=\sup_{\|g\|_\txz\le1}\lf|\int_\xt f(x,t)g(x,t)\dxt\r|.$$
Thus, for any $\bz>0$, there exists $g\in\txz$
such that $\|g\|_\txb\le1$ and
$$\lf|\int_\xt f(x,t)g(x,t)\dxt\r|\ge\|f\|_\ttx-\frac\bz2.$$
Observe here that $fg\in L^1(\xt)$. Fix $x_0\in\cx$. Let
$$O_k\ev\{(x,t)\in\xt:\ \dist(x,x_0)<k,\ 1/k<t<k\}.$$
Then there exists $k\in\nn$ such that
$$\lf|\int_\xt f(x,t)g(x,t)\chi_{O_k}\dxt\r|\ge\|f\|_\ttx-\bz.$$
Obviously, $g\chi_{O_k}\in\txb$. Thus, \eqref{4.1} holds, which
completes the proof of Lemma \ref{l4.4}.
\end{proof}
The following lemma is a slight modification of \cite[Lemma 4.2]{cw77};
see also \cite{jya}. We omit the details here.
\begin{lemma}\label{l4.5}
Let $\Phi$ satisfy Assumption $(\Phi)$. Suppose that $\{f_k\}_{k=1}^\fz$
is a bounded family of functions in $\ttx$. Then there exist $f\in\ttx$ and a
subsequence $\{f_{k_j}\}_{j=1}^\fz$
of $\{f_{k}\}_{k=1}^\fz$ such that for all $g\in\txb$,
$$\lim_{j\to\fz}\int_\xt f_{k_j}(x,t)g(x,t)\dxt=\int_\xt f(x,t)g(x,t)\dxt.$$
\end{lemma}
\begin{theorem}\label{t4.1}
Let $\Phi$ satisfy Assumption $(\Phi)$. Then $(\txv)^*$, the
dual space of the space $\txv$, coincides with $\ttx$ in the following
sense:
For any $g\in\ttx$, define the linear function $\ell$ by setting,
for all $f\in\txz$,
\begin{equation}\label{4.2}
\ell(f)\ev\int_\xt f(x,t)g(x,t)\dxt.
\end{equation}
Then there exists a positive constant $C$, independent of $g$, such that
$$\|\ell\|_{(\txz)^*}\le C\|g\|_\ttx.$$
Conversely, for any $\ell\in(\txz)^*$, there exists $g\in\ttx$ such that
\eqref{4.2} holds for all $f\in\txz$ and $\|g\|_\ttx\le C\|\ell\|_{(\txz)^*}$,
where $C$ is a positive constant independent of $\ell$.
\end{theorem}
\begin{proof}
From Lemma \ref{4.2}, we infer that $\txv\st\txz=(\ttx)^*$, which further
implies that $\ttx\st(\ttx)^*\st(\txv)^*$.
Conversely, let $\ell\in(\txv)^*$. Notice that for any $f\in\txb$,
without loss of generality,
we may assume that $\supp f\st K$, where $K$ is a bounded set in $\xt$.
Then we have
$\|f\|_\txv=\|f\|_\txz\le C(K)\|f\|_\txb.$
Thus, $\ell$ induces a bounded linear functional on $\txb$. Let $O_k$ be
as in the proof of Lemma \ref{l4.4}.
By the Riesz representation theorem, there exists
a unique $g_k\in L^2(O_k)$ such that for all $f\in L^2(O_k)$,
$$\ell(f)=\int_\xt f(x,t)g_k(x,t)\dxt.$$
Obviously, $g_{k+1}O_k=g_k$ for all $k\in\nn$. Let $g\ev g_1\chi_{O_1}
+\sum_{k=2}^\fz g_k\chi_{O_k\bh O_{k-1}}$. Then $g\in L_{\rm loc}^2(\xt)$
and for any $f\in\txb$, we have
$$\ell(f)=\int_\xt f(y,t)g(y,t)\dyt.$$
Set $\wz g_k\ev g\chi_{O_k}$. Then for each $k\in\nn$, obviously, we
see that $\wz g_k\in\txb\st\tx\st\ttx$. Then from Lemma \ref{l4.4},
it follows that
\begin{eqnarray*}
\|\wz g_k\|_\ttx&&=\sup_{f\in\txb,\,\|f\|_\txz\le1}
\lf|\int_\xt f(x,t)g(x,t)\chi_{O_k}(x,t)\dxt\r|\\
&&=\sup_{f\in\txb,\,\|f\|_\txz\le1}\lf|\ell(f\chi_{O_k})\r|\\
&&\le\sup_{f\in\txb,\,\|f\|_\txz\le1}\|\ell\|_{(\txv)^*}\|f\|_\txz
\le \|\ell\|_{(\txv)^*}.
\end{eqnarray*}
Thus, by Lemma \ref{l4.5}, there exist $\wz g\in\ttx$ and
$\{\wz g_{k_j}\}_{j=1}^\fz\st\{\wz g_k\}_{k=1}^\fz$ such that for all $f\in\txb$,
$$\lim_{j\to\fz}\int_\xt f(x,t)\wz g_{k_j}(x,t)\dxt=
\int_\xt f(x,t)\wz{g}(x,t)\dxt.$$
On the other hand, notice that for sufficient large $k_j$, we have
\begin{eqnarray*}
\ell(f)&&=\int_\xt f(x,t)g(x,t)\dxt\\
&&=\int_\xt f(x,t)\wz g_{k_j}(x,t)\dxt=\int_\xt f(x,t)\wz{g}(x,t)\dxt,
\end{eqnarray*}
which implies that $g=\wz g$ almost everywhere, and hence $g\in\ttx$.
By a density argument, we conclude that \eqref{4.2} also holds for
$g$ and all $f\in\txz$, which completes the proof of Theorem \ref{t4.1}.
\end{proof}
\begin{definition}\rm\label{d4.2}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$, $\Phi$ satisfy
Assumption $(\Phi)$, $M\in\nn$, $M>\frac n2(\frac1{p_\Phi^-}-\frac12)$
and $\ez\in(n(1/p_\Phi^-- 1/p_\Phi^+),\fz)$. An element
$f\in(\bmox)^*$ is said to be in the \emph{space} $\hmx$ if there exist
$\{\lz_j\}_{j=1}^\fz\st\cc$ and $(\Phi,\,M,\,\ez)_L$-molecules
$\{\az_j\}_{j=1}^\fz$ such that $f=\sum_{j=1}^\fz\lz_j\az_j$ in
$(\bmox)^*$ and
$$\blz(\{\lz_j\az_j\}_{j=1}^\fz)\ev\inf\lf\{\lz>0:\, \sum_{j=1}^\fz\mu(B_j)
\Phi\lf(\frac{|\lz_j|}{\lz \mu(B_j)\rho(\mu(B_j))}\r)\le1\r\}<\fz,$$
where for each $j$, $\az_j$ is adapted to the ball $B_j$.
If $f\in\hmx$, then its \emph{norm} is defined by $\|f\|_\hmx\ev\inf\{\blz
(\{\lz_j\az_j\}_{j=1}^\fz)\}$, where the infimum is taken over all the
possible decompositions of $f$ as above.
\end{definition}
By \cite[Theorem 5.1]{jy}, we see that for all $M>\mz$ and
$\ez\in(n(1/p_\Phi^-- 1/p_\Phi^+),\fz)$, the \emph{spaces $\hx$ and $\hmx$
coincide with equivalent norms}.
Let us introduce the Banach completion of the space $\hx$.
\begin{definition}\rm\label{d4.3}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$, $\Phi$ satisfy
Assumption $(\Phi)$, $\ez\in(n(1/p_\Phi^--1/p_\Phi^+),\fz)$ and
$M>\mz$. The \emph{space} $\bmx$ is defined to be the space of all
$f=\sum_{j=1}^\fz\lz_j\az_j$ in $(\bmox)^*$, where
$\{\lz_j\}_{j=1}^\fz\subset\cc$ with $\sum_{j=1}^\fz|\lz_j|<\fz$
and $\{\az_j\}_{j=1}^\fz$ are
$(\Phi,\,M,\,\ez)_L$-molecules. If $f\in\bmx$, define
$\|f\|_\bmx\ev\inf\{\sum_{j=1}^\fz|\lz_j|\}$, where the infimum is
taken over all the possible decomposition of $f$ as above.
\end{definition}
By \cite[Lemma 3.1]{hm09}, we know that $\bmx$ is a Banach space.
Moreover, from Definition \ref{d4.2}, it is easy to deduce that
$\hx$ is dense in $\bmx$. More precisely, we have the following
lemma.
\begin{lemma}\label{l4.6}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$, $\Phi$ satisfy
Assumption $(\Phi)$, $\ez\in(n(1/p_\Phi^--1/p_\Phi^+),\fz)$ and
$M>\mz$. Then
{\rm i)} $\hx\st\bmx$ and the inclusion is continuous.
{\rm ii)} For any $\ez_1\in(n(1/p_\Phi^--1/p_\Phi^+),\fz)$ and $M_1>\frac
n2(\frac1{p_\Phi^-}-\frac12)$, the spaces $\bmx$ and
$B_{\Phi,L}^{M_1,\ez_1}(\cx)$ coincide with equivalent norms.
\end{lemma}
\begin{proof}
From Definition \ref{d4.3} and the molecular characterization of $\hx$,
it is easy to deduce i).
Let us prove ii). By symmetry, it suffices to
show that $\bmx\st\byx$.
Let $f\in\bmx$. By Definition \ref{d4.3}, there exist $\pme$-molecules
$\{\az_j\}_{j=1}^\fz$ and $\{\lz_j\}_{j=1}^\fz\st\cc$ such that
$f=\sum_{j=1}^\fz\lz_j\az_j$ in $\bmoxx$ and $\sum_{j=1}^\fz|\lz_j|\ls
\|f\|_\bmx$.
By i), for each $j\in\nn$, we see that $\az_j\in\hx\st\byx$ and
$\|\az_j\|_\byx\ls\|\az_j\|_\hx\ls1$. Since $\byx$ is a Banach space, we
see that $f\in\byx$ and $\|f\|_\byx\le\sum_{j=1}^\fz|\lz_j|\|\az_j\|
_\byx\ls\|f\|_\bmx$. Thus, $\bmx\st\byx$, which completes the proof of
Lemma \ref{l4.6}.
\end{proof}
Since the spaces $\bmx$ coincide for all
$\ez\in(n(1/p_\Phi^--1/p_\Phi^+), \fz)$ and $M>\frac
n2(\frac1{p_\Phi^-}-\frac12)$, in what follows, we denote $\bmx$ simple
by $\bx$.
\begin{lemma}\label{l4.7}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$, and $\Phi$ satisfy
Assumption $(\Phi)$. Then $(\bx)^*=\bmox$.
\end{lemma}
\begin{proof}
Since $(\hx)^*=\bmox$ and $\hx\st\bx$, by duality, we conclude that
$(\bx)^*\st\bmox$.
Conversely, let $\ez\in(n(1/p_\Phi^--1/p_\Phi^+),\fz)$, $M>\mz$ and
$f\in\bmox$. For any $g\in\bx$, by Definition \ref{d4.3}, there exist
$\pme$-molecules $\{\az_j\}_{j=1}^\fz$ and $\{\lz_j\}_{j=1}^\fz\st\cc$ such that
$g=\sum_{j=1}^\fz\lz_j\az_j$ in $\bmoxx$ and $\sum_{j=1}^\fz|\lz_j|\ls
\|g\|_\bx$. Thus,
\begin{eqnarray*}
|\la f,g\ra|&&\le\sum_{j=1}^\fz|\lz_j||\la f,\az_j\ra|
\ls\sum_{j=1}^\fz|\lz_j|\|f\|_\bmox\|\az_j\|_\hx\\
&&\ls\|f\|_\bmox\|g\|_\bx,
\end{eqnarray*}
which implies that $f\in(\bx)^*$, and hence completes the proof of
Lemma \ref{l4.7}.
\end{proof}
Let $M\in\nn$. For all $F\in L^2(\xt)$ with bounded support, define
\begin{equation}\label{4.3}
\plm F\ev C(M)\int_0^\fz\tml F(\cdot,t)\dt,
\end{equation}
where $C(M)$ is as in \eqref{3.5}.
\begin{proposition}\label{p4.2}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$, $\Phi$ satisfy
Assumption $(\Phi)$ and $M\in\nn$. Then the operator $\plm$,
initially defined on $\txb$, extends to a bounded linear operator
{\rm i)} from $T_2^2(\cx)$ to $L^2(\cx)$;
{\rm ii)} from $\tx$ to $\hx$, if $M>\frac n2(\frac1{p_\Phi^-}-\frac12)$;
{\rm iii)} from $\ttx$ to $\bx$, if $M>\frac n2(\frac1{p_\Phi^-}-\frac12)$;
{\rm iv)} from $\txv$ to $\vmo$.
\end{proposition}
\begin{proof}
i) and ii) were established in \cite[Proposition 3.6]{al11} (see also
\cite[Lemma 3.1]{jy}).
By Lemma \ref{l4.2}, we know that $\txb$ is dense in $\ttx$. Let $f\in\txb$.
From ii) and Lemma \ref{l4.6}, we deduce that $\plm f\in\hx\st\bx$.
Moreover, by Definition \ref{d4.1}, there exist $\tx$-atoms
$\{a_j\}_{j=1}^\fz$ and $\{\lz_j\}_{j=1}^\fz\st\cc$ such that
$f=\sum_{j=1}^\fz\lz_ja_j$ in $(\txz)^*$ and $\sum_j|\lz_j|\ls
\|f\|_\ttx$. In addition, for any $g\in\bmox$, we have
$\tmlx g\in\txz$. Thus, by $(\tx)^*=\txz$, we conclude that
\begin{eqnarray*}
\la\plm(f),g\ra&&=C(M)\int_\xt f(x,t) \ov{\tmlx g(x)}\dxt\\
&&=\sum_{j=1}^\fz\lz_jC(M)\int_\xt a_j(x,t) \ov{\tmlx g(x)}\dxt\\
&&=\sum_{j=1}^\fz\lz_j\la\plm(a_j),g\ra,
\end{eqnarray*}
which implies that $\plm(f)=\sum_{j=1}^\fz\lz_j\plm(a_j)$ in $(\bmox)^*$.
By ii), we further conclude that
\begin{eqnarray*}
\|\plm(f)\|_\bx&&\le\sum_{j=1}^\fz|\lz_j|\|\plm(a_j)\|_\bx\\
&&\ls\sum_{j=1}^\fz|\lz_j|\|\plm(a_j)\|_\hx\ls\|f\|_\ttx.
\end{eqnarray*}
Since $\txb$ is dense in $\ttx$, we see that $\plm$ extends to a
bounded linear operator from $\ttx$ to $\bx$, which completes the
proof of iii).
Let us now prove iv). From Lemma \ref{l3.3}, we infer that $\txb$ is
dense in $\txv$. Thus, to prove iv), it suffices to show that $\plm$
maps $\txb$ continuously into $\vmo$.
Let $f\in\txb$. By i), we see that $\plm f\in L^2(\cx)$. Notice that
\eqref{3.3} and \eqref{3.4} with $L$ and $L^*$ exchanged implies
that $L^2(\cx)\st\mly$, when $M_1\in\nn$ and $M_1>\mz$. Thus, $\plm f
\in\mly$. To show $\plm f\in\vmo$, by Theorem \ref{t3.4}, we still need
show that $\tmy\plm f\in\txv$.
For any ball $B\ev B(x_B,r_B)$, let $V_0(B)\ev\wh B$ and $V_k(B)\ev
(\wh{2^kB})\bh(\wh{2^{k-1}B})$ for any $k\in\nn$. For all $k\in\zz_+$,
let $f_k\ev f\chi_{V_k(B)}$. Thus, for $k\in\{0,1,2\}$, by Lemma \ref{l2.2}
and i), we see that
\begin{eqnarray*}
\lf[\iint_{\wh B}\lf|\tmy\plm f_k(x)\r|^2\dxt\r]^{1/2}\ls\|\plm f_k\|_{L^2(\cx)}
\ls\|f_k\|_{T_2^2(\cx)}.
\end{eqnarray*}
For $k\ge3$, let $\vy\ev(\wh{2^kB})\bh(2^{k-2}B\times(0,\fz))$ and
$\ve\ev V_k(B)\bh\vy$. We further write
$f_k=f_k\chi_\vy+f_k\chi_\ve\ev f_{k,1}+f_{k,2}$. From Minkowski's
inequality, Lemma \ref{l2.3} and H\"older's inequality, we deduce that
\begin{eqnarray*}
&&\lf[\iint_{\wh B}\lf|\tmy\plm f_{k,2}(x)\r|^2\dxt\r]^{1/2}\\
&&\hs\sim\lf[\iint_{\wh B}\lf|\int_{2^{k-2}r_B}^{2^kr_B}\tmy
(s^2L)^Me^{-s^2L}(f_{k,2}(\cdot,s))(x)\ds\r|^2\dxt\r]^{1/2}\\
&&\hs\ls\int_{2^{k-2}r_B}^{2^kr_B}\lf[\iint_{\wh B}\lf|t^{2M_1}
s^{2M}L^{M+M_1}e^{-(s^2+t^2)L}(f_{k,2}(\cdot,s))(x)\r|^2\dxt\r]^{1/2}\ds\\
&&\hs\ls\int_{2^{k-2}r_B}^{2^kr_B}\lf[\int_0^{r_B}\lf|\frac{t^{2M_1}
s^{2M}}{(s^2+t^2)^{M+M_1}}\r|^2\|f_{k,2}(\cdot,s)\|_{L^2(\cx)}^2\dt\r]^{1/2}\ds\\
&&\hs\ls2^{-2kM_1}\int_{2^{k-2}r_B}^{2^kr_B}\|f_{k,2}(\cdot,s)\|_{L^2(\cx)}\ds
\ls2^{-2kM_1}\|f_{k,2}\|_{T_2^2(\cx)}.
\end{eqnarray*}
Similarly, we have
$$\lf[\iint_{\wh B}\lf|\tmy\plm f_{k,1}(x)\r|^2\dxt\r]^{1/2}
\ls2^{-2kM_1}\|f_{k,1}\|_{T_2^2(\cx)}.$$
Let $\wz p_\Phi\in (0,p_\Phi^-)$
such that $M>\mzx$ and $M_1>\mzx$.
Combining the above estimates, since $\Phi$ is of lower type $\wz p_\Phi$,
we finally conclude that
\begin{eqnarray*}
&&\frac1\rb\lf[\iint_{\wh B}\lf|\tmy\plm f(x)\r|^2\dxt\r]^{1/2}\\
&&\hs\ls
\sum_{k=0}^2\frac1\rb\lf[\iint_{\wh B}\lf|\tmy\plm f_k(x)\r|^2\dxt\r]^{1/2}\\
&&\hs\hs+\sum_{k=3}^\fz\sum_{i=1}^2\frac1\rb\lf[\iint_{\wh B}
\lf|\tmy\plm f_{k,i}(x)\r|^2\dxt\r]^{1/2}\\
&&\hs\ls\sum_{k=0}^2\frac1\rb\|f_k\|_{T_2^2(\cx)}
+\sum_{k=3}^\fz\sum_{i=1}^2\frac{2^{-2kM_1}}\rb\|f_{k,i}\|_{T_2^2(\cx)}\\
&&\hs\ls\sum_{k=0}^\fz2^{-2k[M_1-\mzx]}\frac1\rkb\|f_k\|_{T_2^2(\cx)}.
\end{eqnarray*}
Since $f\in\txv\st\txz$, we have
$$\frac1\rkb\|f_k\|_{T_2^2(\cx)}\ls\|f\|_\txz$$
and, for all fixed $k\in\nn$,
\begin{eqnarray*}
\lim_{c\to0}\sup_{B:\,r_B\le c}\frac{\|f_k\|_{T_2^2(\cx)}}\rkb&&
=\lim_{c\to\fz}\sup_{B:\,r_B\ge c}\frac{\|f_k\|_{T_2^2(\cx)}}\rkb\\
&&=\lim_{c\to\fz}\sup_{B:\,B\st [B(0,c)]^\com}\frac{\|f_k\|_{T_2^2(\cx)}}\rkb=0.
\end{eqnarray*}
Thus, by the dominated convergence theorem for series, we further conclude that
\begin{eqnarray*}
&&\eta_1(\tmy\plm f)\\
&&\hs=\lim_{c\to0}\sup_{B:\,r_B\le c}\frac1\rb
\lf[\iint_{\wh B}\lf|\tmy\plm f(x)\r|^2\dxt\r]^{1/2}\\
&&\hs\ls\sum_{k=0}^\fz2^{-2k[M_1-\mzx]}
\lim_{c\to0}\sup_{B:\,r_B\le c}\frac{\|f_k\|_{T_2^2(\cx)}}\rkb=0.
\end{eqnarray*}
Similarly, we have $\eta_2(\tmy\plm f)=\eta_3(\tmy\plm f)=0$, and
hence $\tmy\plm f\in\txv$, which completes the proof of Proposition \ref{p4.2}.
\end{proof}
\begin{lemma}\label{l4.8}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$, and $\Phi$ satisfy
Assumption $(\Phi)$. Then $\vmo\cap L^2(\cx)$ is dense in $\vmo$.
\end{lemma}
\begin{proof}
Let $f\in\vmo$ and $M>\mz$. Then by Theorem \ref{t3.4}, we have
$h\ev\tml f \in\txv$. Similarly to the proof of Proposition \ref{p4.2},
by Lemma \ref{l3.3}, there exist $\{h_k\}_{k\in\nn}\st\txb\st\txv$
such that $\|h-h_k\|_\txz\to0$, as $k\to\fz$. Thus, by i) and iv)
of Proposition \ref{p4.2}, we see that $\ply h_k\in L^2(\cx)\cap\vmo$ and
\begin{equation}\label{4.4}
\|\ply(h-h_k)\|_\bmo\ls\|h-h_k\|_\txz\to0,
\end{equation}
as $k\to\fz$.
Let $\az$ be a $\pme$-molecule. Then by the definition of $\hx$, we
know that $e^{-t^2L}\az\in\tx$, which, together with Lemma \ref{l3.2},
the fact that $(\tx)^*=\txz$ and $(\hx)^*=\bmo$, further implies that
\begin{eqnarray*}
\la f,\az\ra&&=C(M)\iint_\xt\tml f(x) \ov{t^2L^*e^{-t^2L^*}\az(x)}\dxt\\
&&=\lim_{k\to\fz} C(M)\iint_\xt h_k(x) \ov{t^2L^*e^{-t^2L^*}\az(x)}\dxt\\
&&=\frac{C(M)}{C_1}\lim_{k\to\fz}\int_\cx(\ply h_k(x)) \ov{\az(x)}\,d\mu(x)
=\frac{C(M)}{C_1}\la\ply h,\az\ra.
\end{eqnarray*}
Since the set of finite combinations of molecules is dense in $\hx$,
we then see that $f=\frac{C(M)}{C_1}\ply h$ in $\bmo$.
Now, for each $k\in\nn$, let $f_k\ev\frac{C(M)}{C_1}\ply h_k$. Then $f_k
\in\vmo\cap{L^2(\cx)}$ and, moreover, by \eqref{4.4},
we have $\|f-f_k\|_\bmo\to0$,
as $k\to\fz$, which completes the proof of Lemma \ref{l4.8}.
\end{proof}
In what follows, the \emph{symbol} $\la\cdot,\cdot\ra$ in the following
theorem means the duality between the space $\bmo$ and the space
$\bxx$ in the sense of Lemma \ref{l4.7} with $L$ and $L^*$ exchanged.
\begin{theorem}\label{t4.2}
Let $L$ satisfy Assumptions $(L)_1$ and $(L)_2$, and $\Phi$ satisfy
Assumption $(\Phi)$. Then the dual space of $\vmo$, $(\vmo)^*$,
coincides with the space $\bxx$ in the following sense:
For any $g\in\bxx$, define the linear functional $\ell$ by setting,
for all $f\in\vmo$,
\begin{equation}\label{4.5}
\ell(f)\ev\la f,g\ra.
\end{equation}
Then there exists a positive constant $C$ independent of $g$ such that
$$\|\ell\|_\vmox\le C\|g\|_\bxx.$$
Conversely, for any $\ell\in\vmox$, there exist $g\in\bxx$ such
that \eqref{4.5} holds and a positive constant $C$,
independent of $\ell$, such that
$$\|g\|_\bxx\le C\|\ell\|_\vmox.$$
\end{theorem}
\begin{proof}
By Lemma \ref{l4.7}, we have $(\bxx)^*=\bmo$. Definition \ref{d3.3}
implies that $\vmo\st\bmo$, which further implies that $\bxx\st(\vmo)^*$.
Conversely, let $M>\mz$ and $\ell\in(\vmo)^*$. By Proposition
\ref{p4.2}, $\ply$ is bounded from $\txv$ to $\vmo$, which implies
that $\ell\circ\ply$ is a bounded linear functional on $\txv$. Thus,
by Theorem \ref{t4.1}, there exists $g\in\ttx$ such that for all
$g\in\txv$, $\ell\circ\ply(f)=\la f,g\ra$.
Now, suppose that $f\in\vmo\cap L^2(\cx)$. By Theorem \ref{t3.4}, we conclude
that $\tml f\in\txv$. Moreover, from the proof of Lemma \ref{l4.8}, we deduce
that $f=\frac{C(M)}{C_1}\ply(\tml f)$ in $\bmo$. Thus
\begin{eqnarray}\label{4.6}
\ell(f)&&=\frac{C(M)}{C_1}\ell\circ\ply(\tml f)\\
&&=\frac{C(M)}{C_1}\iint_\xt\tml f(x)g(x,t)\dxt.\noz
\end{eqnarray}
By Lemma \ref{l4.2}, $\txb$ is dense in $\ttx$. Since $g\in\ttx$, we choose
$\{g_k\}_{k\in\nn}\st\txb$ such that $g_k\to g$ in $\ttx$. By iii) of Proposition
\ref{p4.2}, we see that $\pi_{L^*,M}(g)$, $\pi_{L^*,M}(g_k)\in
B_{\Phi,L^*}(\cx)$ and
$$\|\pi_{L^*,M}(g-g_k)\|_{B_{\Phi,L^*}(\cx)}\ls\|g-g_k\|_\ttx\to0,$$
as $k\to\fz$. This, together with \eqref{4.6}, Theorem \ref{t4.1},
the dominated convergence theorem and Lemma \ref{l4.7}, implies that
\begin{eqnarray}\label{4.7}
\ell(f)&&=\frac{C(M)}{C_1}\lim_{k\to\fz}\iint_\xt\tml f(x)g_k(x,t)\dxt\\
&&=\frac{C(M)}{C_1}\lim_{k\to\fz}\int_\cx f(x)\int_0^\fz\tmlx
(g_k(\cdot,t))(x)\dt\,d\mu(x)\noz\\
&&=
\frac1{C_1}\lim_{k\to\fz}\la f,\pi_{L^*,M}(g_k)\ra
=\frac1{C_1}\la f,\pi_{L^*,M}(g)\ra.\noz
\end{eqnarray}
Since $\vmo\cap{L^2(\cx)}$ is dense in $\vmo$, we finally conclude
that \eqref{4.7} holds for all $f\in\vmo$, and
$\|\ell\|_\vmox=\frac1{C_1}\|\pi_{L^*,M}g\|_\bxx$. In this sense, we
have $\vmox\st\bxx$, which completes the proof of Theorem
\ref{t4.2}.
\end{proof}
\noindent{\bf Acknowledgements.} The authors would like to
thank Dr. Bui The Anh and Dr. Renjin Jiang
for some helpful discussions on the subject of this paper
and, especially, for Dr. Bui The Anh to give us his preprint \cite{a10}.
The authors would also like to thank the referee for his/her several
valuable remarks which made this article more readable.
|
1,108,101,565,621 | arxiv | \section*{Theory of cascaded-mode resonances}
The functionality of electromagnetic resonators can be understood from the constructive interference of waves---creating resonant modes. A crucial parameter that determines these modes is the round-trip phase $\Delta \phi$, accumulated by the field after completing one round trip in the resonator~\cite{yariv2007photonics}. Waves that pick up a round-trip phase equal to a multiple of $2\pi$ constructively interfere with themselves and become resonant modes of the resonator (Fig.~1a). In the case of a Fabry-Perot geometry, the resonance condition is then given by
\begin{equation}
2Ln\frac{2\pi \nu}{c} + 2\phi_{r}= 2\pi m,
\end{equation}
where $\nu$ is the frequency of light, $m$ is an integer number representing the index of the resonant modes of frequency $\nu_m$, $c$ is the speed of light in vacuum, $L$ is the length of the resonator, $n$ is the refractive index of the material inside the resonator, and $\phi_r$ is the reflection phase at the mirrors.
This simple equation explains two essential properties of resonators: the existence of the fundamental mode and the appearance of a spectrum with only a discrete number of modes. The resulting frequency spectrum from Eq.~(1) is then given by $\nu_m = c (m -\phi_r/\pi)/(2nL)$.
Above, we ignore the properties of the mode in the transversal plane. Typically, a discrete number of orthogonal transverse modes exist for each frequency, e.g., $\mathrm{TE}_i$ and $\mathrm{TM}_i$ waves, where each transverse mode experiences a different effective index ($n_{\mathrm{eff},i})$. As a result, the resonant modes of a resonator generally consist of a superposition of spectra, corresponding to the various families of transverse modes (Fig.~1b). The spectra are given by
\begin{equation}
\nu_{i,m} = \frac{c (m - \phi_{r,i}/\pi)}{2 n_{\mathrm{eff},i} L}.
\end{equation}
We now introduce a new type of resonator based on cascaded-mode coupling. We illustrate this principle in Fig.~1c,d: upon reflection on the rightmost mode-converting mirror, an incident wave with a particular transverse mode profile is converted into another transverse mode. When this mode returns to the leftmost mirror, another mode conversion occurs upon reflection. This cascade of mode conversions can be repeated as many times as the number of transverse modes supported by the waveguide. Finally, a ``supermode'' emerges when the wave is converted back to the original configuration of the incident mode. For resonators with $N$ different transverse modes, the round-trip phase is given by (Supplementary Materials):
\begin{equation}
\Delta \phi = k_0L \xi \sum_{i=1}^N n_{\mathrm{eff},i} + \phi_{r,\mathrm{tot}}.
\end{equation}
Here, $k_0$ equals $2\pi/\lambda_0$ with $\lambda_0$ the vacuum wavelength, $\phi_{r,\mathrm{tot}}$ is the sum of all reflection phases, and $\xi$ is the parameter that encodes whether the contributing transverse modes appear once ($\xi=1$) or twice ($\xi=2$) in the chain. The round-trip phase is thus no longer merely determined by the length of the resonator and the refractive index but also by the number of coupled transverse modes. The corresponding resonance condition is
\begin{equation}
\nu_m = \frac{c\left[m - \phi_{r,\mathrm{tot}}/(2\pi)\right]}{L\xi\sum_{i=1}^N n_{\mathrm{eff},i}}.
\end{equation}
The free spectral range is thus set by the sum of the round-trip optical paths of the different cascaded modes $L\xi\sum_{i=1}^N n_{\mathrm{eff},i}$ rather than by $2 n_{\mathrm{eff},i} L$ as in a conventional resonator. Next, whereas traditional resonators feature an incoherent superposition of different spectra, each corresponding to a different transverse mode, cascaded-mode resonators exhibit just one superspectrum (Fig.~1c,d).
This analysis is independent of how the mode conversions are realized. For instance, in the context of transverse modes in waveguides, a mode converter can be implemented using a specific refractive index variation (blue regions in Fig.~1a-d). The last column of Fig.~1 presents a useful abstraction to visualize and study cascaded-mode resonances using directed graphs. In this picture, cascade-mode resonances appear as cyclic graphs, which allows for studying the resonators using the properties of their associated adjacency matrix. (Supplementary Materials).
Above, we only consider the round-trip phase resonance condition to get insights into the spectrum of cascaded-mode resonators. To obtain a more accurate picture of this spectrum, we need to account for both the phase and the amplitude of the different waves. The transmission spectrum of a cascaded-mode resonator, where $N$ different forward-propagating modes are coupled with each other, is given by (Supplemental Material):
\begin{equation}
\vec{E}_\mathrm{out} = \sum_{i=1}^N \frac{t_{\mathrm{pt}_i} \mathrm{e}^{\mathrm{i}\phi_i}}{1-r_\mathrm{rt} \mathrm{e}^{\mathrm{i}\Delta \phi}}E_\mathrm{in} \vec{u}_{\mathrm{f}_i}.
\end{equation}
Here, $r_\mathrm{rt}$, $t_{\mathrm{pt}_i}$, $\phi_{i}$, and $\vec{u}_{\mathrm{f}_i}$ are respectively the round-trip reflection coefficient, the pass-through transmission amplitude, the transmission phase, and the unit vector of the forward propagating mode $i$ (Supplemental Materials).
The first striking feature of the spectrum is the modified fundamental mode wavelength. The largest wavelength $\lambda_0$ that can be confined in a traditional resonator is approximately that for which $\lambda_0/n_{\mathrm{eff}} = 2 L$. In the case of cascaded-mode resonances the largest wavelength is given by:
\begin{equation}
\lambda_0 = L\xi\sum_{i=1}^N n_{\mathrm{eff},i}\left(1-\frac{\mathrm{mod}(\phi_{r,\mathrm{tot}},2\pi)}{2\pi}\right)^{-1}.
\end{equation}
This wavelength can be much larger than the resonator?s dimensions if a significant number $N$ of transverse modes are coupled. Indeed, compared with a traditional resonance, a supermode acquires a larger propagation phase in combination with a larger reflection phase. Both effects contribute to a larger round-trip phase. In Fig.~2a we visualize the ratio of the largest wavelength in a cascaded-mode resonator to that in a traditional resonator as a function of the two preceding parameters. In the Supplementary Materials, we compare this mechanism with the mechanism underlying other subwavelength resonator designs~\cite{kuznetsov2016optically,hill2007lasing,koshelev2020subwavelength,shaltout2018ultrathin}. Here, it is important to note that the local refractive index remains unchanged in a cascaded-mode resonator. The confinement occurs through the cascading of transverse modes, which increases the round-trip phase, i.e., $2L n_{\mathrm{eff},i}$ is being replaced by $L\xi\Sigma_{i=1}^N n_{\mathrm{eff},i}$.
A second interesting feature of cascaded-mode resonances, in agreement with the geometrical model described above, is the modification of the free spectral range $\Delta \nu$, given by
\begin{equation}
\Delta \nu = \frac{c}{\xi \sum_{i=1}^N n_{\mathrm{g},i} L},
\label{eq:fsr}
\end{equation}
where $n_{\mathrm{g},i}$ is the group index of transverse mode $i$ at frequency $\nu$.
Finally, two other crucial, spectral parameters can be engineered in a cascaded-mode resonator by controlling the round-trip phase: the linewidth $\gamma$ and the quality factor $Q$ (Fig.~2b). Unlike the previous two parameters ($\lambda_{0,\mathrm{max}}$ and $\Delta \nu$), the linewidth and the quality factor depend on the round-trip losses (Supplementary Materials).
Not only the spectral properties but also the temporal and spatial properties of these modes can be engineered by using cascaded-mode coupling. The intracavity power build-up and the intracavity power build-up time both scale proportionally to the number of coupled modes.
While the intensity of longitudinal modes in traditional resonators exhibits a simple standing-wave profile, the intensity profile in a cascaded-mode resonator will have a more irregular profile, potentially with many different local minima and maxima.
A unique spatial property of cascaded-mode resonators is that the propagation constant of a supermode depends on the propagation direction. This phenomenon is shown in its most straightforward implementation in Fig.~1c. When a field with transverse profile of mode~1 is incident on the left side of this resonator, a cascaded mode will exist with wave vector $k = k_0n_{\mathrm{eff},1}$ propagating from left to right, and a wave vector $k = k_0n_{\mathrm{eff},2}$ propagating from right to left. Due to the distinct propagation constants in opposite directions, directional nonlinear optical effects can occur in the resonator since the phase-matching conditions may only be satisfied in one direction~\cite{boyd2020nonlinear}. The directionality could also give additional control over chiral, optomechanical, or quantum mechanical interactions inside the resonator.
A final property of cascaded-mode resonances that deserves special attention is the existence of mode-independent spectra. Indeed, different transverse modes at the input may excite the same resonance, i.e., a mode-independent resonance. As an example, in the resonators of Fig.~1c-d the transmission spectrum (third column) is the same for the two incident transverse modes (1 and 2). We show in Supplementary Materials that the different modes that excite the same resonance in a cascaded-mode resonator can be extracted from the adjacency matrix of the graph that encodes the different mode conversions in the resonator. The mode-independent behavior of cascaded-mode resonators is a unique transmission characteristic, a feature verified experimentally in Fig.~4. This is in contrast to traditional resonators, where different transverse modes exhibit different transmission spectra. Based on this property, it becomes possible to manipulate modes with different spatial profiles in an identical way using only one resonator.
\section*{Experiments}
We experimentally realize the proposed cascaded-mode resonators using the silicon-on-insulator (SOI) platform at telecom wavelengths (1550~nm). In our on-chip implementation, the cascaded modes have distinct transverse profiles $\mathrm{TE}_i$, an in-plane polarization, and propagate along waveguides rather than in free space. The SOI platform offers design flexibility in engineering the properties of the mode converters (reflection phase and magnitude), as well as the propagation properties of all modes participating in the cascade, such as their effective indices $n_{\mathrm{eff},i}$.
The device geometry is shown in Fig.~3a,b together with scanning electron microscope (SEM) pictures of the fabricated structures. Further details are provided in the Supplementary Materials. In general, each device consists of three main optical components: input/output waveguides that couple and guide light of chosen transverse modes to and away from the mode-converting resonators; a multi-mode waveguide section of length $L_\mathrm{wg}$ in which the cascaded modes are confined; specialized corrugated Bragg reflectors located on either side of the multi-mode waveguide that reflect one transverse mode into another. While, as described theoretically above, the number of conversions in a cascaded mode is only limited by the number of available transverse modes, we restrict our experimental demonstration to cascaded-mode resonators of the type shown in Fig.~1c that couple the two distinct transverse modes $\mathrm{TE}_0$ and $\mathrm{TE}_2$. Their transverse mode profiles are shown in the inset of Fig.~3b. Consequently, the width of the waveguide in the cavity region ($w_\mathrm{wg} = 1.07~\mathrm{\mu m}$) was chosen such that it cuts off all transverse modes of a higher order than $\mathrm{TE}_2$. (See Supplementary Materials for details on the design of the individual photonic structures and their transmission/filter performance.) In addition, the grating period $\Lambda$ of the mode converters is chosen as to satisfy the phase-matching condition and provide the necessary momentum for the mode conversion to occur on the reflected wave: $2\pi/\Lambda = \Delta \beta_{12} = \beta_1 + \beta_2$, with $\beta_1 = \beta_{\mathrm{TE}_0} = n_{\mathrm{eff,TE}_0} \omega_0/c$ and $\beta_2 = \beta_{\mathrm{TE}_2} = n_{\mathrm{eff,TE}_2} \omega_0/c$ the propagation constants of the two coupled modes. This type of coupling is typically referred to as contra-directional coupling. In the Methods, we outline in more detail the strategy for designing the cascaded-mode resonators in the SOI platform.
We now demonstrate in experiments and simulations the most evident signatures of cascaded-mode resonanators: the mode-independent spectrum with modified spectral parameters. The symmetric cascaded-mode resonator of Fig.~1c provides resonant confinement to input modes that correspond to either $\mathrm{TE}_0$ or $\mathrm{TE}_2$ transverse modes and has the same transmission spectrum for either input. We confirm this computationally in Fig.~3c, where we report the simulated field profile of the same cascaded-mode resonator for the two possible inputs and find a locally enhanced field inside the resonator in both cases. Moreover, the hybrid nature of the near-infrared cascaded mode inside the resonator becomes apparent in the zoom-in of the spatial profile shown in Fig.~3d. The field profile can be decomposed into a superposition of counter-propagating $\mathrm{TE}_0$ and $\mathrm{TE}_2$ waveguide modes that exhibit, as expected, the same beating length for both inputs (marked by the white arrow). We demonstrate this property experimentally by transmission spectroscopy and contrast it with two test Fabry-Perot resonators that employ standard mirrors and provide cavity confinement to only one of $\mathrm{TE}_0$ or $\mathrm{TE}_2$ modes. The experimental results are shown for the three cases in Fig.~4a-c and Supplemental Fig.~S11: We find that cavity modes appear, as expected, for both $\mathrm{TE}_0$ and $\mathrm{TE}_2$ modes in the case of the cascaded-mode resonator only. Moreover, the experimental results are well-reproduced by our simulations. Cavity modes appear only for one of the two modes for the conventional Fabry-Perot resonators, while light is simply transmitted for the other modes. In the Methods, we describe the spectroscopic technique used in these measurements.
Next, we analyze the resonator properties of the cavity modes associated with the cascaded-mode resonators compared to the conventional Fabry-Perot modes in Fig.~4d-f. Firstly, we show in Fig.~4d that the intra-cavity modes of the cascaded-mode resonators excited by the two inputs ($\mathrm{TE}_0$ or $\mathrm{TE}_2$) coincide in frequency. We experimentally find a negligible relative deviation between the two sets of resonant wavelengths of ($\lambda_{\mathrm{TE}_0} - \lambda_{\mathrm{TE},2})/ \lambda_{\mathrm{TE}_0} \approx 4 \mathrm{ x } 10^{-5}$. Furthermore, the quality factors of the two sets are approximately equal, as shown in Fig.~4e. Finally, the group index of the cascaded modes is approximately equal to $n_\mathrm{g} = 4.3$, regardless of whether they are excited by $\mathrm{TE}_0$ or $\mathrm{TE}_2$. In contrast, the group index of the Fabry-Perot modes are equal to $n_{\mathrm{g,TE}_0} = 3.75$ and $n_{\mathrm{g,TE}_2} = 4.85$ (Fig.~4f). This result confirms once more the cascaded-mode character of the measured spectra, particularly because the group index is approximately the arithmetic mean of the group indices of the participating transverse modes, in agreement with Eq.~(7).
\section*{Discussion}
This work shows how electromagnetic resonators can be generalized to cascaded-mode resonators, where the spectrum of supermodes reflects the generalized round-trip phase condition of a cascade of different transverse modes propagating in different directions. The theory is generally valid for any cascade of orthogonal modes inside cavities of arbitrary shape and is thus not only applicable to a cascade of transverse mode profiles of an integrated waveguide~\cite{bahari2017nonreciprocal}. Indeed, for the round trip to occur after $N$ conversions, the $N+1$-th mode in the chain needs to be indistinguishable from the first, i.e., with identical frequency, temporal shape, $k$-vector, polarization, and phase profile~\cite{limonov2017fano,shiri2020hybrid, piccardo2021roadmap}. Therefore, the theory can be applied equally well for a cascade of modes with, e.g., different spin or orbital angular momenta.
The spectral, temporal, and spatial properties are no longer solely determined by the length and refractive index of the medium, but also by the number of coupled modes. This insight allows to circumvent existing trade-offs and, for example, design resonators smaller than the local wavelength. In addition, these resonators exhibit completely new properties not found in their traditional counterparts, e.g., mode-independent resonances and directionally dependent propagation properties.
We anticipate that the concept of cascaded-mode resonators will be further exploited in a broad class of technological devices and scientific experiments since the underlying principles of cascaded-mode resonances can be extended even beyond optics.
\vspace{1cm}
\textbf{Data and Code Availability}
All data and codes associated with this manuscript will be uploaded on the zenodo database prior to publication.
\medskip
\textbf{Acknowledgements}
We acknowledge support from AFOSR grants FA550-19-1-0352 and FA95550-19-1-0135. This work was performed in part at the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Coordinated Infrastructure Network (NNCI), which is supported by the National Science Foundation under NSF Award no. 1541959.
\medskip
\textbf{Author contributions}
V.G. initiated the project and conceived the concept of cascaded-mode resonators. V.G. developed the theory with inputs from I.C.B.C., J.L., M.P., and F.C.; I.C.B.C, J.L, V.G., and M.P. designed the experiment; I.C.B.C. fabricated the devices, carried out the measurements, and analysed the experimental data; J.L. carried out the numerical simulations. All authors contributed to the analysis, discussion and
writing of the manuscript.
\medskip
\textbf{Competing interests}
A provisional patent application has been filed on the subject of this work by the President and Fellows of Harvard College.
\section{\large{T\lowercase{heory}}}
\subsection{Maximal round-trip phase in general cascaded-mode resonators}
We calculate the maximal round-trip phase that can be obtained in a cavity of length $L$ with $N$ different transverse modes, each experiencing an effective index $n_{\mathrm{eff},i}$. A central observation in this derivation is the maximum number of times a transverse mode can be excited in one round trip. Each transverse mode can occur only two times in the round trip: once in each direction. Indeed, each mode can only be coupled with another mode in the left and right ends of the resonator. In general, mode $i$ will be coupled to mode $j$ on the left end through the conversion $\Delta ij$, and $i$ will be coupled to $k$ on the right end through the conversion $\Delta ik$. Adding an extra conversion on either side will split the cascade of modes into sub-chains but not increase the round-trip phase of either chain.
With this knowledge, we can easily calculate that the maximum round-trip phase is obtained by having each mode occur two times - once in both directions - in one large cascade. The total propagation round-trip phase $\Delta \phi_{p}$ that maximally can be obtained is thus given by the sum of all propagation phases:
\begin{equation}
\Delta \phi_{p} = 2k_0L \sum_{i=1}^N n_{\mathrm{eff},i}.
\end{equation}
Additionally, at each reflection, the wave may experience a nontrivial phase shift. The total maximum round-trip phase is then given by:
\begin{equation}
\Delta \phi = 2k_0L \sum_{i=1}^N n_{\mathrm{eff},i} + 2\sum_{i=1}^N \phi_{r,i,i+1},
\end{equation}
where $\phi_{r,i,i+1}$ is the reflection phase upon conversion from mode $i$ to $i+1$. We number the transversal modes in order of appearance in the cascade. In this notation, we implicitly assume that mode $N+1$ is the first mode again.
\subsection{The transmission spectrum of cascaded-mode resonators}
In agreement with the traditional derivation of a Fabry-Perot transmission spectrum, we now calculate the transmission spectrum of a general cascaded-mode resonator. We explicitly calculate the partially transmitted fields through the resonator, as the field inside the resonator travels back and forth. We do this in the most general case where $N$ different forward modes are coupled with each other.
In our analysis, we number the modes $1$ to $N$. Because the cyclic nature of the mode conversions, we can rename the first node to coincide with the incident mode and subsequently rename all the other nodes in the order that they appear in the loop.
An incident field is partially transmitted through the resonator, after being transmitted through the first and second reflector and propagating through the cavity. Assuming that the product of the two transmissions through the first and second reflector is given by $t_\mathrm{pt}$, we can thus write that:
\begin{equation}
\vec{E}_{\mathrm{out},1}=t_{\mathrm{pt}_1}\mathrm{e}^{\mathrm{i} \phi_1}E_\mathrm{in}\vec{u}_{\mathrm{f}_1}.
\end{equation}
The parameter $t_\mathrm{pt_1}$ encodes the transmission efficiency through both reflectors. The unit vector $\vec{u}_{\mathrm{f1}}$ keeps track to the vectorial nature of the field, i.e., the specific mode that is transmitted---in this case forward mode $\mathrm{f1}$.
A significant portion of the field stays inside the resonator and converts to the another mode at the second reflector. After one one backward and one forward propagation of the wave, the second partially transmitted field will be given by
\begin{equation}
\vec{E}_{\mathrm{out},2}=t_{\mathrm{pt}_2} \mathrm{e}^{\mathrm{i}\phi_2} E_{\mathrm{in}}\vec{u}_{\mathrm{f}_2}.
\end{equation}
\begin{figure}[t!]
\centering
\includegraphics[width=16 cm]{Nature_FigS1.pdf} \\
\caption{\textbf{The derivation of the transmission spectrum of a general cascaded-mode resonator.} \textbf{a,}~A compact representation of the mode conversions in the resonator. The modes are renumbered in the order that they appear in the chain, where we distinguish $N$ forward propagating modes ($\mathrm{f}_i$) and $N$ backward propagating modes ($\mathrm{b}_i$). According to this nomenclature, the left and the right reflector implement $\Sigma_{i=1}^N \Delta f_{i+1} b_i$ and $\Sigma_{i=1}^N \Delta f_i b_i$, respectively. Here, we also assume $\mathrm{f}_{N+1} = \mathrm{f}_1$. \textbf{b,}~The graph representation of the general resonator, defined in \textbf{a}. \textbf{c,}~A visualization of the partially transmitted fields at the right-hand side of the resonator. The total transmitted field is the infinite sum of all these partially transmitted fields. This sum is given by Eq.~(S12).}
\end{figure}
This picture continues until we have reached the last forward propagating mode in the system, which gives rise to the $N$th partially transmitted field:
\begin{equation}
\vec{E}_{\mathrm{out},N}=t_{\mathrm{pt}_N} \mathrm{e}^{\mathrm{i}\phi_N} E_{\mathrm{in}}\vec{u}_{\mathrm{f}_N}.
\end{equation}
After the $N$th wave has been partially transmitted, the whole cascade has been completed and the cascade starts again. It is interesting to note that, for each mode transmitted, the following law is now fulfilled:
\begin{equation}
\vec{E}_{\mathrm{out},kN+i} = (r_\mathrm{rt}\mathrm{e}^{\mathrm{i}\Delta \phi})^k \vec{E}_{\mathrm{out},i},
\end{equation}
where $i$ varies between $1$ and $N$ and $k$ is an integer number keeping track of the number of full round trips that have been completed.
Here, we retrieve the round-trip phase shift $\Delta \phi$, as defined in the previous section, and we introduce the parameter $r_\mathrm{rt}$:
\begin{equation}
r_\mathrm{rt} = \prod_{i=1}^{M} r_{i,i+1},
\end{equation}
where $M$ is the total number of mode conversions in one full round trip.
The total transmitted field through the resonator, for a given incident field $\vec{E}_\mathrm{in}$ is the sum of all the partially transmitted fields, mathematically taking the sum up to infinity:
\begin{align}
\vec{E}_\mathrm{out}&=\lim_{m\to \infty}\sum_{j=1}^m \vec{E}_{\mathrm{out},j}\\
& = \lim_{n\to \infty}\sum_{k=0}^n\sum_{i=1}^N \vec{E}_{\mathrm{out},kN+i}\\
& = \lim_{n\to \infty}\sum_{i=1}^N \sum_{k=0}^n \vec{E}_{\mathrm{out},kN+i}\\
&=\sum_{i=1}^N t_\mathrm{pt_i}\vec{u}_{\mathrm{f}_i} \mathrm{e}^{\mathrm{i}\phi_i} E_{\mathrm{in}}\lim_{n\to \infty}\sum_{k=0}^n (r_\mathrm{rt} \mathrm{e}^{\mathrm{i}\Delta \phi})^k.
\end{align}
This last sum is a geometric series. In closed form it can be rewritten as $(1-r_\mathrm{rt} \mathrm{e}^{\mathrm{i}\Delta \phi})^{-1}$. The total transmitted field thus equals:
\begin{equation}
\vec{E}_\mathrm{out} = \sum_{i=1}^N \frac{t_{\mathrm{pt}_i} \mathrm{e}^{\mathrm{i}\phi_i}}{1-r_\mathrm{rt} \mathrm{e}^{\mathrm{i}\Delta \phi}}E_\mathrm{in} \vec{u}_{\mathrm{f}_i}
\end{equation}
In general, for a given incident field, the output will will be a sum of different forward propagating modes, whose amplitudes are given by the different $t_{\mathrm{pt},i}$. The different modes follow the same spectrum, defined by the round-trip reflection efficiency $r_\mathrm{rt}$ and the round-trip loss $\Delta \phi$.
For each of the output modes, we can calculate the transmitted intensity. Defining $t_i = E_{\mathrm{out},i}/E_\mathrm{in}$ and $\vec{t}_{\mathrm{pt},i} = t_\mathrm{pt_i}\vec{u}_i$, with $\vec{u}_i$ the unit vector of output mode $i$, we get:
\begin{align}
T_i &= t_i^* t_i\\
&= \frac{t_{\mathrm{pt}_i}^* t_{\mathrm{pt}_i}}{(1-r_\mathrm{rt}\mathrm{e}^{\mathrm{i}\Delta \phi})(1-r_\mathrm{rt}\mathrm{e}^{-\mathrm{i}\Delta \phi})}\\
&= \frac{|t_{\mathrm{pt},i}|^2}{1-2r_\mathrm{rt}\cos{(\Delta \phi)}+r_\mathrm{rt}^2}\\
&= \frac{|t_{\mathrm{pt},i}|^2}{(1-r_\mathrm{rt})^2+4r_\mathrm{rt}\sin^2{(\Delta \phi/2)}}\\
&= \frac{\alpha_i}{1+F\sin^2{(\Delta \phi/2)}},
\end{align}
where we retrieve a generalized definition of the finesse $F = 4r_\mathrm{rt}/(1-r_\mathrm{rt})^2$ and define the normalized outgoing intensity amplitude for mode $i$: $\alpha_i = |t_{tp_i}|^2/(1-r_\mathrm{rt})^2$.
\subsection{The directed graph representation of cascaded-mode resonators}
We now discuss the relationship between the resonators, the mode converters, and the directed graphs in more detail. As shown in Fig.~S2a, cascaded-mode resonators consist of two sets of converters that convert forward propagating modes into backward propagating modes and vice-versa. One can construct the directed graph of a cascaded-mode resonator by associating each mode with a node and each mode-converter with an edge between the nodes. In doing so, it is essential to disambiguate the forward and backward propagating modes. Indeed, since there are no conversions between forward propagating modes or backward propagating modes, this procedure results in constructing a bipartite graph. The graph is directed since the mode conversions occur at one end of the resonator, and both ends of the resonator are not necessarily identical.
The general form of the adjacency matrix is shown in Fig.~S2b. The action of the converters at both ends of the resonator is visible as separate submatrices in this matrix. Here, the color of the submatrices corresponds to the color of the converters. It is now worth mentioning a subtlety about the internal symmetry of the adjacency matrix. Since both mode converters can be significantly different for one another, the adjacency matrix is not symmetric. However, the mode converters themselves are generally symmetric. For example, if a converter converts mode i to mode -j, it generally also converts mode j to mode -i. Therefore, the submatrices that implement the two converters are, in turn, symmetric matrices.
In Fig.~S2c-d, we show, by way of illustration, the graph of a specific cascaded-mode resonator where the right-hand converter consists of $\Delta 12 + \Delta 34$ and the left-hand converter consists of $\Delta 23 + \Delta 14$. The graph is shown in Fig.~S2c, where each mode conversion has a different color. In Fig.~S2d, we show the corresponding adjacency matrix where each element is circled by the color of the corresponding edge in the graph.
In Fig.~S3, we show the directed graph and the adjacency matrix of three different implementations of cascaded-mode resonators. In (a-c) we present an implementation with $N=4$ and $\xi=1$, in (d-f) an implementation with $N=4$ and $\xi=2$ and in (g-i) we have a resonator that simultaneously contains two different types of resonances: $N=4, \xi=1$ and $N=1, \xi=2$.
\begin{figure}[h!]
\centering
\includegraphics[width=12 cm]{Nature_FigS2.pdf} \\
\caption{\textbf{The directed graph representation of a general cascaded-mode resonator.} \textbf{a,}~A schematic of the mode conversions in a cascaded-mode resonator. \textbf{b,}~The adjacency matrix of this cascaded mode resonator contains two symmetric submatrices that correspond to the left and right mode converters. \textbf{c,}~An example of the graph in a four-mode system where the left and right converter implement $\Delta 23 + \Delta 14$ and $\Delta 12 + \Delta 34$, respectively. \textbf{d,}~The adjacency matrix of the system shown in \textbf{c}.}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=12 cm]{Nature_FigS3.pdf} \\
\caption{\textbf{Several examples of graphs and adjacency matrices related to cascaded-mode resonators} \textbf{a-c,}~A cascaded-mode resonator where $N=4, \xi=1$. \textbf{d-f,}~A cascaded-mode resonator where $N=4, \xi=2$. \textbf{g-i,}~A mixed-resonances cascaded-mode resonator, where two cascaded-mode resonances co-exist: $N=4,\xi=1$ and $N=1,\xi=2$.}
\end{figure}
\newpage
\subsection{Summary of scaling of spectral properties in cascaded-mode resonators}
\begin{center}
\begin{tabular}{ c c c }
\textbf{Spectral parameter} & \textbf{traditional resonator (tr)} & \textbf{cascaded-mode resonator (CM)} \\
Fundamental mode wavelength & $\lambda_\mathrm{max,tr}$ & $\lambda_\mathrm{max,CM} = \lambda_\mathrm{max,tr}N$\\
Free spectral range & $\Delta \nu_\mathrm{tr}$ & $\Delta \nu_\mathrm{CM}=\Delta \nu_\mathrm{tr}/N$ \\
Resonance linewidth & $\Delta \gamma_\mathrm{tr}$ & $\Delta \gamma_\mathrm{CM}/N$\\
Quality factor & $Q_\mathrm{tr}$ & $Q_\mathrm{CM} = Q_\mathrm{tr}N$ \\
Cavity ring-down time & $\tau_\mathrm{tr}$ & $\tau_\mathrm{CM} = \tau_\mathrm{tr}N$ \\
Intracavity power build-up & $\kappa_\mathrm{tr} $& $\kappa_\mathrm{CM} = \kappa_\mathrm{tr}N $
\end{tabular}
\end{center}
\subsection{Comparison with other subwavelength resonator designs}
The largest wavelength that can be confined in a cascaded-mode resonator is given by:
\begin{equation}
\lambda_0 = L\xi\sum_{i=1}^N n_{\mathrm{eff},i}\left(1-\frac{\mathrm{mod}(\phi_{r,\mathrm{tot}},2\pi)}{2\pi}\right)^{-1}.
\end{equation}
Here, $\mathrm{mod}$ refers to the modulo operation, comparing the phase to the nearest multiple of $2\pi$.
The largest wavelength that can be confined effectively can be much larger than the resonator?s dimensions if a significant number $N$ of transverse modes are coupled together. This occurs through a combination of two effects, as visualized in Fig.~\ref{Fig_subwavelength}, where we show the frequency of the longitudinal modes versus mode index $m$. We compare a traditional resonator with a maximum effective index $n_\mathrm{eff,max}$ with a cascaded-mode resonator with the same $n_\mathrm{eff,max}$. The fundamental frequency of a cascaded-mode resonator is lower than that of the traditional resonator because of, on the one hand, the smaller gradient of the line connecting the resonance frequencies, given by $(\xi/2 \Sigma_{i=1}^N n_\mathrm{eff,i}/n_\mathrm{eff,max})^{-1}$, and, on the other hand, a lower intercept on the $y$ axis, determined by $- \phi_{r,\mathrm{tot}}/(2\pi)(\xi/2 \Sigma_{i=1}^N n_\mathrm{eff,i}/n_\mathrm{eff,max})^{-1}$. Physically, this can be understood as follows: compared with a traditional resonance, a supermode acquires a larger propagation phase (smaller gradient) in combination with a larger reflection phase (lower intercept). Both effects contribute to a larger round-trip phase for a fixed resonator length.
\begin{figure}[h!]
\centering
\includegraphics[width=6 cm]{Nature_FigS5.pdf} \\
\caption{The resonance condition versus longitudinal mode number $m$, as defined in Eq.~(4) in the main paper, comparing a traditional resonator (blue) and a cascaded-mode resonator with identical $n_\mathrm{eff,max}$ (red). The sum of the coupled effective indices ($n_\mathrm{eff,c} = \xi/2 \Sigma_{i=1}^N n_{\mathrm{eff},i}$) determines the slope of the cascaded-mode spectrum ($\theta = \arctan{[(\xi/2 \Sigma_{i=1}^N n_{\mathrm{eff},i}/n_\mathrm{eff,max})^{-1}]}$. The cascaded reflection phase shift of the mode conversions ($\phi_{r,\mathrm{tot}}$) determines the intercept with the $y$ axis ($\tilde{\phi}_r = -\phi_{r,\mathrm{tot}}/(2\pi)(\xi/2 \Sigma_{i=1}^N n_{\mathrm{eff},i}/n_\mathrm{eff,max})^{-1}$. The fundamental frequency can be much lower than in a traditional resonator.}
\label{Fig_subwavelength}
\end{figure}
It is interesting to compare this mechanism with other resonator designs that confine waves with a free-space wavelength much larger than the dimensions of the resonators. A first example is the set of devices in which the wavelength in the resonator is adjusted, e.g., using metals or dielectrics with a large refractive index~\cite{kuznetsov2016optically,hill2007lasing,koshelev2020subwavelength}. However, there is a fundamentally different mechanism at work here. In a cascaded-mode resonator, the local refractive index remains unchanged, but the confinement occurs through the cascading of transverse modes that all combine into one mode and increase the effective phase after a round trip of that mode. In other words, $2L n_{\mathrm{eff},i}$ is being replaced by $L\xi\Sigma_{i=1}^N n_{\mathrm{eff},i}$. Second, researchers have created subwavelength resonators by adjusting the reflection phase of the mirrors, e.g., by implementing metasurface mirrors~\cite{shaltout2018ultrathin}. This mechanism is equivalent to the role played by $\phi_{r,\mathrm{tot}}$ in the above analysis. In this type of resonator, the slope of the frequency versus mode number remains along the bisector in Fig.~\ref{Fig_subwavelength}, but the line is shifted downwards.
\section{\large{M\lowercase{aterials and } \uppercase{M}\lowercase{ethods}}}
\subsection{Fabrication}
The integrated photonic circuit structures discussed in this work are fabricated using standard silicon-on-insulator~(SOI) fabrication techniques. Starting from an SOI wafer with a $2~\mathrm{\mu m}$ buried oxide layer and a $220~\mathrm{nm}$ silicon layer, we perform electron-beam lithography with an 125~keV Elionix system using ZEP520A positive resist (spin-coating at 3000~rpm, pre-exposure bake 3~min at 90$^\circ$C and 3~min at 180$^\circ$C). After exposure, we develop the photoresist in cold Oxylene for 60~s, and perform an oxygen plasma for 15~s at 40~sccm, 100~W. In a second step, the silicon layer is etched using the resist as an etch mask by single-step reactive ion etching with fluorine chemistry~($\mathrm{SF_6}$ and $\mathrm{C_4F_8}$). The buried oxide layer works as an etch stop layer. The remaining resist layer is then removed by leaving the samples overnight in Remover PG at 80$^\circ$C. A final cleaning process is performed using Piranha etch for 15~s. Finally, a 700~nm thick cladding layer of silicon dioxide is deposited via chemical vapor deposition.
A set of fabricated samples are shown in Fig.~\ref{fig_SOIresonatorSEM} by optical microscope~(a) and scanning electron microscope~(b-d). Fig~\ref{fig_SOIresonatorSEM}a illustrates one resonator structure together with waveguides that guide light of a well-defined optical mode~($\mathrm{TE_0}$ or $\mathrm{TE_2}$) into and out of the resonator, which is located in the center of the chip~(marked by the square rectangle). This allows to investigate the spectral response of the resonator for a transverse mode of choice. The SEM images in Fig~~\ref{fig_SOIresonatorSEM}b-d illustrate the marked area in the optical microscope image prior to the deposition of the silicon dioxide cladding layer for the three different types of resonators discussed in the main text: (b) the mode converting resonator where upon each reflection at the Bragg mirror, $\mathrm{TE_0}$ transverse modes are transformed into $\mathrm{TE_2}$ modes, and vice-versa; (c) a standard Fabry-Perot resonator which provides selective reflection to the $\mathrm{TE_0}$ tranverse mode, and no mode conversion occurs; and (d) a second Fabry-Perot resonator which provides selective reflection to the $\mathrm{TE_2}$ tranverse mode, and no mode conversion occurs. Each SEM figure contains an inset that shows a close-up view of the mode converter alone. For each resonator, the two Bragg gratings are identical in the cases discussed here. Fig.~\ref{fig_SOIresonatorSEM}E illustrates a top-view and a side-view schematic of the mode converters as well as the main dimensions of the chip~(per = period). The periods are equal to $\mathrm{per = 313~(b),~304~(c),~480~nm~(d)}$, the duty cycle of all gratings is $40\%$, the width of the multimode waveguide is $\mathrm{w_{wg}=1.07~\mu m}$, and the depth of the corrugations is $\mathrm{D = 506~(b),~296~(c),~149~nm~(d)}$.
\begin{figure}[t!]
\centering
\includegraphics[width=14 cm]{Nature_Suppl_FabricatedDevices.pdf} \\
\caption{\textbf{Images of the fabricated resonators.} \textbf{a,} Optical microscope picture of a fabricated resonator features three waveguide inputs on the left and three waveguide outputs on the right. From these, two sets are used to probe the resonator for input $\mathrm{TE_0}$ and $\mathrm{TE_2}$ modes and to analyze the output of the resonator also in these two transverse modes. \textbf{b-d,} SEM figures are provided for all three types of resonators we investigate in this work, and a close-up of the Bragg gratings is provided in each figure as an inset. \textbf{e,} Top-view and side-view schematics are provided for both types of Bragg reflectors. CVD = chemical vapor deposition, per = period, $\mathrm{SiO_2}$ = silicon dioxide, $\mathrm{\Delta \beta_{11}}$ = Bragg mirror that reflects selectively mode $\mathrm{TE_0}$ into $\mathrm{TE_0}$, $\mathrm{\Delta \beta_{12}}$ = Bragg mirror that reflects mode $\mathrm{TE_0}$ into $\mathrm{TE_2}$, and vice-versa, $\mathrm{\Delta \beta_{22}}$ = Bragg mirror that reflects selectively mode $\mathrm{TE_2}$ into $\mathrm{TE_2}$.}
\label{fig_SOIresonatorSEM}
\end{figure}
\subsection{Characterization setup}
All resonators were characterized by transmission spectroscopy using a tunable Santec TSL-550 laser, having a linewidth of 200~kHz, much below the linewidth of the resonances considered here. The polarization of the incident light is adjusted using fiber polarizers to maximize the power transmitted through the chip. Cleaved fiber probes from Lightwave at an angle of $\mathrm{10^{\circ}}$ are placed above the grating couplers located at the ends of the on-chip waveguides and couple light from the fiber to on-chip waveguides into $\mathrm{TE_0}$ mode~(in-plane polarization). The transmitted power is measured with an InGaAs photodiode with adjustable gain.
\subsection{Extraction of experimental resonator parameters}
In Fig.~5 of the main text, we report resonator parameters~(resonant wavelength, quality factor, group index) that were extracted from the experimentally measured transmission spectra. In this section, we elaborate our procedure to extract these parameters, which was applied to all data. We start by fitting Lorentzian lineshapes to each longitudinal mode and use the Lorentzian fit to determine the resonant wavelength and the Q-factor. These are reported in panels a and b. For the computation of the group index from the transmission spectra, we need an accurate estimation of the effective cavity length for each resonator. The effective cavity length $\mathrm{L_{eff} = L_{wg} +2L_{Bragg}}$ accounts for the penetration of optical fields into the Bragg reflectors~(which we summarize into an effective Bragg length $\mathrm{L_{Bragg}}$), which adds to the geometrical length of the multimode waveguide $\mathrm{L_{wg}}$. To determine $\mathrm{L_{Bragg}}$ experimentally, we fabricated in each case three separate resonators for each type of mode-coverters, each with a length of the multimode waveguide of $\mathrm{L_{wg} =100~\mu m}$, $150~\mu m$ and $200~\mu m$. Since in all three cases, $\mathrm{L_{Bragg}}$ is constant, it is possible to extract $\mathrm{L_{Bragg}}$ from the free spectral range of the longitudinal modes at different waveguide lengths. We find that $\mathrm{L_{Bragg} = 4.3~\mu m, 6.15~\mu m~and~0.5~\mu m}$ for the mode-converting grating, the $\mathrm{TE_0}$ standard Fabry Perot resonator and the $\mathrm{TE_2}$ standard Fabry Perot resonator. By knowing the effective total cavity length, the group indices can be extracted from the spacing of adjacent longitudinal modes.
\section{\large{S\lowercase{imulations and } \uppercase{E}\lowercase{xperiments}}}
The multimode waveguide preceding the resonators, located in between the Bragg gratings and after the resonator has a width of $\mathrm{w_{wg} = 1.07~\mu m}$. This width was chosen as to maximize the difference between the effective indices of the $\mathrm{TE_0}$ and $\mathrm{TE_2}$, as to ensure a high selectivity of the three different Bragg mirrors we consider. The simulated effective indices of the modes are reported for various waveguide widths in Fig.~\ref{fig_SOIproperties_1}a. Additionally, the $\mathrm{TE_3}$ mode and all subsequent higher order modes are not supported at this waveguide width. Furthermore, we use adiabatic tapers connected to single-mode waveguides of width 440~nm to filter out any $\mathrm{TE_2}$ mode from the $\mathrm{TE_0}$ analyzer port. The efficiency of the taper and single-mode waveguide to achieve this filtering effect is simulated using finite-time-domain methods and the resulting field distribution is shown in Fig.~\ref{fig_SOIproperties_1}b. As visible from the inset, the transmission is below $10^{-5}$.
\begin{figure}[tbh]
\centering
\includegraphics[width=13 cm]{Nature_Figure5_1_1.pdf} \\
\caption{\textbf{Multimode waveguide properties.} \textbf{a,} Dispersion curves of the effective indices of transverse modes $\mathrm{TE}_i$ as a function of waveguide width. \textbf{b,} Simulation of transmitted electric field of mode $\mathrm{TE}_2$ through the adiabatic taper reveals a transmission below $10^{-5}$. }
\label{fig_SOIproperties_1}
\end{figure}
\subsection{D\lowercase{esign of co-directional waveguide coupler}}
In order to probe and analyze the resonator properties under an incident $\mathrm{TE_2}$, we design a co-directional mode converter based on the evanescent coupling of a nanowaveguide and our multimode waveguide shown in Fig.~\ref{fig_SOIproperties_2}. For the conversion to be efficient, the effective index of the $\mathrm{TE_0}$ mode in the narrow waveguide needs to match the effective index of the $\mathrm{TE_2}$ mode in the multimode waveguide. In this situation, the propagation constants of the two modes are equal and coherent injection of mode $\mathrm{TE_2}$ from $\mathrm{TE_0}$ can be ensured. Graphically, this corresponds to horizontal lines in the plot of Fig.~\ref{fig_SOIproperties_1}a, and to a width of the nanowaveguide of 335~nm. We find from full-wave simulations that a coupling length of $\mathrm{70~\mu m}$ as was used in the experiments, is sufficient to couple $70\%$ of the power into mode $\mathrm{TE_2}$ around a central wavelength of 1550~nm.
\begin{figure}[tbh]
\centering
\includegraphics[width=14 cm]{Nature_Figure5_1_2.pdf} \\
\caption{\textbf{Co-directional waveguide coupler properties.} Simulation of the adiabatic forward mode coupler shows efficient energy transfer from the narrow single-mode waveguide populated with $\mathrm{TE}_0$ to the wide multimode waveguide where $\mathrm{TE}_2$ is parametrically generated. Inset: A maximal transmission of $70\%$ is achieved.}
\label{fig_SOIproperties_2}
\end{figure}
\subsection{D\lowercase{esign of contra-directional mode-converting and non-mode converting} B\lowercase{ragg gratings}}
In this section we discuss the design strategy of mode-converting and non-mode converting Bragg gratings. For these gratings to fulfill their intended purpose, several conditions need to be satisfied concomitantly: 1. all gratings provide selective and efficient reflection for a pre-selected transverse mode profile, 2. all gratings provide only minimal reflection to all other transverse mode profiles, and 3. all gratings effect only negligible co-directional mode conversion. As we anticipate above, these conditions can be fulfilled by careful choice of the effective refractive index of the transverse modes supported by the waveguide. In general, a Bragg grating provides an additional momentum that can be leveraged to achieve phase matching for a given set of modes. For contra-directional coupling of modes with propagation constants $\mathrm{\beta_1 = n_{eff,1}\frac{\omega}{c_0}}$, and $\mathrm{\beta_2 = n_{eff,2}\frac{\omega}{c_0}}$~(corresponding to mode-converting Bragg mirrors), the grating period $\mathrm{\Lambda}$ needs to be chosen such that $\mathrm{\frac{2\pi}{\Lambda} = \beta_1 + \beta_2}$. For co-directional coupling, the grating period $\mathrm{\Lambda}$ needs to be chosen such that $\mathrm{\frac{2\pi}{\Lambda} = \beta_1 - \beta_2}$. We recognize here already that the corresponding grating periods will be considerably distinct and that thereby, in general, the co-directional coupling will be negligible when contra-directional coupling is achieved. For contra-directional coupling of modes with equal propagation constants $\mathrm{\beta_1 = n_{eff,1}\frac{\omega}{c_0}}$, or $\mathrm{\beta_2 = n_{eff,2}\frac{\omega}{c_0}}$~(corresponding to mode-converting Bragg mirrors), the grating period $\mathrm{\Lambda}$ needs to be chosen such that $\mathrm{\frac{2\pi}{\Lambda} = 2\beta_1~or~2\beta_2}$~(corresponding to standard non-mode converting Bragg mirrors). Having chosen the two modes $\mathrm{TE_0}$ and $\mathrm{TE_2}$ to have maximally different effective refractive indices, it is possible to choose a grating period that achieves only mode-converting reflection and negligible standard reflection, or vice-versa.
\begin{figure}[tbh]
\centering
\includegraphics[width=14 cm]{Nature_Suppl_FigGratingperiod_k1in.pdf} \\
\caption{\textbf{Contra-directional mode-converting grating properties under an incident $\mathrm{TE_0}$ mode.} \textbf{a-b,} Transmitted power into mode $\mathrm{TE_0}$ and $\mathrm{TE_2}$. (\textbf{c-d,}) Reflected power into mode $\mathrm{TE_0}$ and $\mathrm{TE_2}$. Overall, these graphs demonstrate an efficient and selective conversion from input mode $\mathrm{TE_0}$ into an output mode with transverse profile $\mathrm{TE_2}$, and that this conversion is efficient in a reflection geometry around 1560~nm. Furthermore, the Bragg grating reflects back into the same mode $\mathrm{TE_0}$ only minimally, and also the co-directional conversion is negligible. Furthermore, as the number of grating periods increases, the reflected power also increases. We chose a number of periods $\mathrm{N_{per}= 36}$.}
\label{fig_NperK1IN}
\end{figure}
In the Fig.~\ref{fig_NperK1IN}, we report the simulated reflection and transmission properties of the mode converting grating with parameters as described above, under an incident $\mathrm{TE_0}$ mode. We find that an efficient mode conversion is provided by the grating around 1560~nm, and that reflection into the same mode is negligible. Furthermore, with an increasing number of grating periods, the reflected power at the grating increases.
In the Fig.~\ref{fig_NperK3IN}, we report the simulated reflection and transmission properties of the mode converting grating with parameters as described above, under an incident $\mathrm{TE_2}$ mode. We find that an efficient mode conversion is provided by the grating around 1560~nm, and that reflection into the same mode is negligible. Furthermore, with an increasing number of grating periods, the reflected power at the grating increases. By comparing panel~c of Fig.~\ref{fig_NperK1IN} with panel~d of Fig.~\ref{fig_NperK3IN}, we observe the reciprocal behavior of the mode converters.
\begin{figure}[tbh]
\centering
\includegraphics[width=14 cm]{Nature_Suppl_FigGratingperiod_k3in.pdf} \\
\caption{\textbf{Contra-directional mode converting grating properties under an incident $\mathrm{TE_2}$ mode.} \textbf{a-b,} Transmitted power into mode $\mathrm{TE_0}$ and $\mathrm{TE_2}$. \textbf{c-d,} Reflected power into mode $\mathrm{TE_0}$ and $\mathrm{TE_2}$. Overall, these graphs demonstrate an efficient and selective conversion from input mode $\mathrm{TE_2}$ into the output mode with transverse profile $\mathrm{TE_0}$, and that this conversion is efficient in a reflection geometry around 1560~nm. Furthermore, the Bragg grating reflects back into the same mode $\mathrm{TE_2}$ only minimally, and also the co-directional conversion is negligible. Furthermore, as the number of grating periods increases, the reflected power also increases. We chose a number of periods $\mathrm{N_{per}= 36}$.}
\label{fig_NperK3IN}
\end{figure}
Finally, we report in Fig.~\ref{fig_SOIproperties_3} the reflection curves for all types of Bragg grating, mode-converting~(a) and standard non-mode converting~(b,c).
\begin{figure}[tbh]
\centering
\includegraphics[width=14 cm]{Nature_Figure5_1_3.pdf} \\
\caption{\textbf{Reflection properties of mode-converting and non-mode converting gratings.} \textbf{a-c,} The reflection curves of the various Bragg gratings used in the resonators shown in the left panels are reported under different input and output conditions. }
\label{fig_SOIproperties_3}
\end{figure}
\subsection{Cascaded-mode resonators and Fabry-Perot resonators}
Cascaded-mode resonators as discussed in the main text provide resonant confinement to input modes that correspond to either $\mathrm{TE}_0$ or $\mathrm{TE}_2$ transverse modes and have the same transmission spectrum for either input. We contrast here this property with two test Fabry-Perot resonators that employ standard mirrors and provide cavity confinement to only one of $\mathrm{TE}_0$ or $\mathrm{TE}_2$ modes. The experimental results are shown for the three cases in Fig.~\ref{fig_modeparameters}: cascaded-mode resonator (measurements d-e, simulations f-g), Fabry-Perot resonator operating on the $\mathrm{TE}_0$ mode (h-i) and Fabry-Perot resonator operating on the $\mathrm{TE}_2$ mode (j-k). By comparing the three cases, we find that cavity modes appear, as expected, for both $\mathrm{TE}_0$ and $\mathrm{TE}_2$ modes in the case of the cascaded-mode resonator only. Moreover, the experimental results are well-reproduced by our simulations. Cavity modes appear only for one of the two modes for the conventional Fabry-Perot resonators, while light is simply transmitted for the other modes.
\newgeometry{textwidth=18cm}
\begin{figure}[t!]
\centering
\includegraphics[width=18 cm]{Nature_Fig5_final.pdf} \\
\caption{\textbf{Transmission spectroscopy of cascaded-mode resonators versus conventional Fabry-Perot resonators.} \textbf{a,} A cascaded-mode resonator, where a reflection at both the left and the right Bragg mirror results into a conversion of the transverse mode from $\mathrm{TE}_0$ to $\mathrm{TE}_2$ and vice-versa, is compared to conventional Fabry-Perot resonators, where no mode conversion occurs upon reflection. Two types of Fabry-Perot resonators are considered, where the Bragg mirror provides selective reflection to either mode $\mathrm{TE}_0$ \textbf{b,} or mode $\mathrm{TE}_2$ \textbf{c}. \textbf{d-e,} The measured transmission spectra of the cascaded-mode resonator exhibit resonances regardless of whether $\mathrm{TE}_0$ or $\mathrm{TE}_2$ is incident onto the resonator. These mode-independent resonances are a defining feature of cascaded-mode resonators. \textbf{f-g,} Simulated transmission spectra reproduce well the measurements. \textbf{h-i,} Measured transmission of a Fabry-Perot resonator for the case that either $\mathrm{TE}_0$ or $\mathrm{TE}_2$ is incident onto the resonator. The Bragg mirrors reflect selectively only $\mathrm{TE}_0$ and no mode conversion occurs. Cavity modes appear only for the $\mathrm{TE}_0$ input. \textbf{j-k,} In this case, the Bragg mirrors only reflect $\mathrm{TE}_2$ and no mode conversion occurs. Cavity modes appear only for the $\mathrm{TE}_2$ input. In all measurements and simulations, the analyzed mode (output mode) is the same as the probe mode (input mode) and the transmission curves are normalized to the transmitted intensity of a bare waveguide without any resonator. Blue curves are transmission measurements under a $\mathrm{TE}_0$ probe/analyzed mode, whereas red curves are transmission measurements under a $\mathrm{TE}_2$ probe/analyzed mode.}
\label{fig_modeparameters}
\end{figure}
|
1,108,101,565,622 | arxiv | \section{Introduction}\label{introduction}
Let $f = \sum_{j = 0}^d a_j X^j\in \mathbb{R}[X]$ be a univariate polynomial
of degree $d \in \mathbb{Z}^+$. It is a classical result due to
Newton (see \cite{HLP}, \S2.22 and \S4.3 for two proofs) that
whenever all the roots of $f$ are real, then the coefficients of $f$
satisfy the following log-concavity condition:
\begin{equation} a_i^2 \geq \frac{d-i+1}{d-i} \frac{i+1}{i}\, a_{i-1} a_{i+1}
{\rm \ for \ all \ } i \in
\{1,\ldots,d-1\}.\label{newton}\end{equation} Moreover, if the roots
of $f$ are not all equal, these inequalities are strict. When $d =
2$, condition (\ref{newton}) becomes $a_1 \geq 4 a_0 a_2$, which is
well known to be a necessary and sufficient condition for all the
roots of $f$ to be real. Nevertheless, for $d \geq 3$, the converse
of Newton's result does not hold any more~\cite{Kurtz}.
\medskip
When $f \in \mathbb{R}^+[X]$, i.e., when $f = \sum_{j = 0}^d a_j X^j$ with
$a_j \geq 0$ for all $j \in \{0,\ldots,d\}$, a weak converse of
Newton's result holds true. Namely, a sufficient condition for $f$
to only have real (and distinct) roots is that
$$a_i^2
> 4 a_{i-1} a_{i+1} {\rm \ for\ all}\ i \in \{1,\ldots,d-1\}.$$
Whenever a polynomial fulfills this condition, we
say that it satisfies the {\it Kurtz condition} since this converse
result is often attributed to Kurtz~\cite{Kurtz}.
Note however that it was obtained some 70 years earlier by Hutchinson~\cite{Hutchinson}.
\medskip
If $f$ satisfies the Kurtz condition, all of its $d+1$ coefficients
are nonzero except possibly the constant term. Such a polynomial is
therefore very far from being sparse (recall that a polynomial is
informally called {\em sparse} if the number of its nonzero coefficients
is small compared to its degree).
One question that we investigate in this paper is: how can we
construct polynomials satisfying the Kurtz condition using sparse
polynomials as building blocks?
More precisely, consider $f$ a polynomial of the form
\begin{equation}\label{sumprod} f = \sum_{i = 1}^k \prod_{j = 1}^m
f_{i,j}\end{equation} where $f_{i,j}$ are polynomials with at most
$t$ monomials each. By expanding the products in~(\ref{sumprod}) we
see that $f$ has at most $k t^m$ monomials.
As a result, $d \leq k t^m$ if $f$ satisfies the Kurtz condition.
Our goal is to improve this very coarse bound. For the case of
polynomials $f_{i,j}$ with nonnegative coefficients, we obtain the
following result.
\begin{theorem}\label{bound}
Consider a polynomial $f \in \mathbb{R}^+[X]$ of degree $d$ of the form $$f
= \sum_{i = 1}^k \prod_{j = 1}^m f_{i,j},$$ where $m \geq 2$ and the
$f_{i,j} \in \mathbb{R}^+[X]$ have at most $t$ monomials. If $f$ satisfies
the Kurtz condition, then $d = \mathcal O(k m^{2/3} t^{2m/3} {\rm
log^{2/3}}(kt))$.
\end{theorem}
We prove this result in Section \ref{kurtzsection}. After that, in
Section \ref{strongsection}, we study the following stronger
log-concavity condition
\begin{equation} \label{stronglogconcave}a_i^2
> d^{2d} a_{i-1} a_{i+1} {\rm \ for\ all}\ i \in \{1,\ldots,d-1\}.\end{equation} In
this setting we prove the following improved analogue of Theorem
\ref{bound}.
\begin{theorem}\label{bound2} Consider a
polynomial $f \in \mathbb{R}^+[X]$ of degree $d$ of the form $$f = \sum_{i =
1}^k \prod_{j = 1}^m f_{i,j},$$ where $m \geq 2$ and the $f_{i,j}
\in \mathbb{R}^+[X]$ have at most $t$ monomials. If $f$
satisfies~$(\ref{stronglogconcave})$, then $d \leq k m t$.
\end{theorem}
This
investigation has a complexity-theoretic motivation: we show in
Section~\ref{complexity} that a suitable
extension of
Theorem~\ref{bound2} (allowing negative coefficients for the
polynomials $f_{ij}$) would imply
a separation of the algebraic complexity classes $\vp$ and $\vnp$.
The classes $\vp$ of ``easily computable polynomial families'' and
$\vnp$ of ``easily definable polynomial families'' were proposed by
Valiant~\cite{Val79} as algebraic analogues of $\p$ and $\np$. As
shown in Theorem~\ref{monotone}, Theorem~\ref{bound2} as it now
stands is strong enough to provide a new example of a family of
polynomials in $\vnp$ which cannot be computed by monotone
arithmetic circuits of polynomial size.
\section{The Kurtz log-concavity condition}\label{kurtzsection}
Our main tool in this section is a result of convex geometry
\cite{EPRS}. To state this result, we need to introduce some
definitions and notations. For a pair of planar finite sets $R, S
\subset \mathbb{R}^2$, the {\it Minkowski sum} of $R$ and $S$ is the set $R
+ S := \{y + z \, \vert \, y \in R, z \in S\} \subset \mathbb{R}^2$. A
finite set $C \subset \mathbb{R}^2$ is {\it convexly independent} if and
only if its elements are vertices of a convex polygon. The following
result provides an upper bound for the number of elements of a
convexly independent set contained in the Minkowski sum of two other
sets.
\begin{theorem}\cite[Theorem 1]{EPRS}\label{convex} Let $R$ and $S$ be two planar point sets with $\vert
R \vert = r$ and $\vert S \vert = s$. Let $C$ be a subset of the
Minkowski sum $R + S$. If $C$ is convexly independent we have that
$\vert C \vert = \mathcal O(r^{2/3} s^{2/3} + r + s)$.
\end{theorem}
\medskip From this result the following corollary follows easily.
\begin{corollary}\label{maxconvex}Let $R_1,\ldots,R_k,S_1,\ldots,S_k,Q_1,Q_2$ be planar point sets with $\vert
R_i \vert = r, \ \vert S_i \vert = s$ for all $i \in \{1,\ldots,k\}$,
$\lvert Q_1\rvert = q_1$
and $\lvert Q_2 \rvert = q_2$. Let $C$ be a subset of $\cup_{i = 1}^k (R_i
+ S_i) + Q_1+Q_2$. If $C$ is convexly independent, then $\vert C \vert =
\mathcal O(k r^{2/3} s^{2/3} q_1^{2/3}q_2^{2/3} + k r q_1 + k s q_2)$.
\end{corollary}
\begin{proof}We observe that $\cup_{i = 1}^k (R_i + S_i) + Q_1+Q_2 =
\cup_{i = 1}^k ((R_i+Q_1) + (S_i + Q_2))$. Therefore, we partition $C$ into
$k$ convexly independent disjoint sets $C_1,\ldots,C_k$ such that
$C_i \subset (R_i+Q_1) + (S_i + Q_2)$ for all $i \in \{1,\ldots,k\}$. Since
$\vert R_i +Q_1\vert = rq_1$ and $\vert S_i + Q_2 \vert \leq sq_2$, by Theorem
\ref{convex}, we get that $\vert C_i \vert = \mathcal O(r^{2/3}
s^{2/3} q_1^{2/3}q_2^{2/3}+ rq_1 + sq_2)$ and the result follows.
\end{proof}
\medskip
\begin{theorem}\label{bound2summands}Consider a polynomial $f \in \mathbb{R}^+[X]$ of degree $d$ of the form $$f =
\sum_{i = 1}^k g_i h_i,$$ where $g_i,h_i \in \mathbb{R}^+[X]$, the $g_i$
have at most $r$ monomials and the $h_i$ have at most $s$ monomials.
If $f$ satisfies the Kurtz condition, then $d = \mathcal O(k
r^{2/3}s^{2/3}\,{\rm log}^{2/3}(k r)+ k(r+s)\log^{1/2}(kr))$.
\end{theorem}
\begin{proof}We write $f = \sum_{i = 0}^d c_i X^i$, where $c_i > 0$ for all $i \in \{1,\ldots,d\}$ and $c_0 \geq 0$.
Since $f$ satisfies the Kurtz condition, setting $\epsilon := {\rm
log}(4)/2$ we get that
\begin{equation}\label{ineq} 2 {\rm log}(c_i) > {\rm log}(c_{i-1}) + {\rm log}(c_{i+1})
+ 2 \epsilon. \end{equation} for every $i \geq 2$. For every
$\delta_1,\ldots,\delta_{d} \in \mathbb{R}$, we set
$C_{(\delta_1,\ldots,\delta_d)} := \{(i,{\rm log}(c_i) + \delta_i)
\, \vert \, 1 \leq i \leq d\}$. We observe that (\ref{ineq}) implies
that $C_{(\delta_1,\ldots,\delta_d)}$ is convexly independent
whenever $0 \leq \delta_i < \epsilon$ for all $i \in
\{1,\ldots,d\}$.
\medskip
We write $g_i = \sum_{j = 1}^{r_i} a_{i,j} X^{\alpha_{i,j}}$ and
$h_i = \sum_{j = 1}^{s_i} b_{i,j} X^{\beta_{i,j}}$, with $r_i \leq
r$, $s_i \leq s$ and $a_{i,j}, b_{i,j}
> 0$ for all $i,j$. Then, $c_l = \sum_{i = 1}^k (\sum_{\alpha_{i,j_1} + \beta_{i,j_2} =
l} a_{i,j_1} b_{i,j_2})$. So, setting $M_l := {\rm max} \{a_{i,j_1}
b_{i,j_2} \, \vert \, i \in \{1,\ldots,k\}, \alpha_{i,j_1} +
\beta_{i,j_2} = l\}$ for all $l \in \{1,\ldots,d\}$, we have that
$M_l \leq c_l \leq k r M_l$, so ${\rm log}(M_l) \leq {\rm log}(c_l)
\leq {\rm log}(M_l) + {\rm log}(k r)$.
\medskip For every $l \in \{1,\ldots,d\}$, we set
\begin{equation}\label{lambdaepsilon} \lambda_l := \left\lceil \frac{{\rm log}(c_l) - {\rm
log}(M_l)}{\epsilon} \right\rceil {\rm \ and \ } \delta_l := {\rm
log}(M_l) + \lambda_l \epsilon - {\rm log}(c_l),\end{equation} and
have that $0 \leq \lambda_l \leq \lceil ({\rm log}(k r))/\epsilon
\rceil$ and that $0 \leq \delta_l < \epsilon$.
\medskip
Now, we consider the sets
\begin{itemize}
\item $R_i :=
\{(\alpha_{i,j}, {\rm log}(a_{i,j}))\, \vert \, 1 \leq j \leq r_i\}$
for $i = 1,\ldots,k$,
\item $S_i := \{(\beta_{i,j}, {\rm
log}(b_{i,j}))\, \vert \, 1 \leq j \leq s_i\}$ for $i = 1,\ldots,k$,
\item $Q := \{(0, \lambda \epsilon) \, \vert \, 0 \leq \lambda
\leq \lceil {\rm log}(k r) / \epsilon \rceil \}$,
\item $Q_1 := \{(0, \mu \epsilon) \, \vert \, 0 \leq \mu
\leq \lceil \sqrt{\log(k r) / \epsilon} \rceil \}$, and
\item $Q_2 := \{(0, \nu \lceil \sqrt{\log(k r) / \epsilon} \rceil \epsilon) \, \vert \, 0 \leq \nu
\leq \lceil \sqrt{\log(k r) / \epsilon} \rceil \}$.
\end{itemize}
If $(0,\lambda\epsilon)\in Q$, then there exist $\mu$ and $\nu$ such that
$\lambda=\nu\lceil\sqrt{\log(kr) / \epsilon}\rceil + \mu$ where
$\mu,\nu\leq\lceil\sqrt{\log(kr) / \epsilon}\rceil$. We have,
\begin{align*}
(0,\lambda\epsilon)=
(0,\nu\lceil\sqrt{\log(kr) / \epsilon}\rceil\epsilon)+(0,\mu\epsilon)\in Q_1+Q_2,
\end{align*}
so $Q\subset Q_1+Q_2$.
Then, we claim that $C_{(\delta_1,\ldots,\delta_d)} \subset \cup_{i = 1}^k
(R_i + S_i) + Q$. Indeed, for all $l \in \{1,\ldots,d\}$, by
(\ref{lambdaepsilon}), $${\rm log}(c_l) + \delta_l = {\rm log}(M_l)
+ \lambda_l \epsilon = {\rm log}(a_{i,j_1}) + {\rm log}(b_{i,j_2}) +
\lambda_l \epsilon$$ for some $i \in \{1,\ldots,k\}$ and some
$j_1,j_2$ such that $\alpha_{i,j_1} + \beta_{i,j_2} = l$; thus
$$(l,{\rm log}(c_l) + \delta_l) = (\alpha_{i,j_1}, {\rm log}(a_{i,j_1})) +
(\beta_{i,j_2}, {\rm log}(b_{i,j_1})) + (0, \lambda_l \epsilon) \in
\cup_{i = 1}^k (R_i + S_i) + Q.$$ Since
$C_{(\delta_1,\ldots,\delta_d)}$ is a convexly independent set of
$d$ elements contained in $\cup_{i = 1}^k (R_i + S_i) + Q_1+Q_2$, a direct
application of Corollary \ref{maxconvex} yields the result.
\end{proof}
\medskip
From this result it is easy to derive an upper bound for the
general case, where we have the products of $m \geq 2$ polynomials.
If suffices to divide the $m$ factors into two groups of
approximately $m/2$ factors, and in each group we expand the product
by brute force.
\medskip
\begin{proof}[Proof of Theorem~\ref{bound}]
We write each of the $k$ products as a product of two
polynomials $G_i := \prod_{j = 1}^{\lfloor m/2 \rfloor} f_{i,j}$ and
$H_i := \prod_{j = \lfloor m/2 \rfloor + 1}^{m} f_{i,j}$. We can now
apply Theorem \ref{bound2summands} to $f = \sum_{i = 1}^k G_i H_i$
with $r = t^{\lfloor m/2 \rfloor}$ and $s = t^{m - \lfloor m/2
\rfloor}$ and we get the result.
\end{proof}
\bigskip
\begin{remark}We observe that the role of the constant $4$ in the
Kurtz condition can be played by any other constant $\tau > 1$ in
order to obtain the conclusion of Theorem \ref{bound}, i.e., we
obtain the same result for $f = \sum_{i = 0}^d a_i X^i$
satisfying that $a_i^2 > \tau a_{i-1} a_{i+1}$ for all $i \in
\{1,\ldots,d-1\}$. For proving this it suffices to replace the value
$\epsilon = {\rm log}(4) / 2$ by $\epsilon = {\rm log}(\tau) / 2$ in
the proof of Theorem \ref{bound2summands} to conclude this more
general result.
\end{remark}
\bigskip
For $f = g h$ with $g,h \in \mathbb{R}^+[X]$ with at most $t$ monomials,
whenever $f$ satisfies the Kurtz condition, then $f$ has only real
(and distinct) roots and so do $g$ and $h$. As a consequence, both
$g$ and $h$ satisfy (\ref{newton}) with strict inequalities and we
derive that $d \leq 2t$. Nevertheless, in the similar setting where
$f = g h + x^i$ for some $i > 0$, the same argument does not apply
and a direct application of Theorem \ref{bound} yields $d = \mathcal
O(t^{4/3}\, {\rm log^{2/3}}(t))$, a bound
which seems to be very far from optimal.
\subsection*{Comparison with the setting of Newton polygons}
A result similar to Theorem~\ref{bound} was obtained in~\cite{KPTT}
for the Newton polygons
of bivariate polynomials. Recall that the Newton polygon of a
polynomial $f(X,Y)$ is the convex hull of the points $(i,j)$ such
that the monomial $X^iY^j$ appears in $f$ with a nonzero coefficient.
\begin{theorem}[Koiran-Portier-Tavenas-Thomass\'e] \label{mpolys}
Consider a bivariate polynomial
of the form
\begin{equation} \label{bivariateSPS}
f(X,Y)=\sum_{i=1}^k \prod_{j=1}^m f_{i,j}(X,Y)
\end{equation}
where $m \geq 2$ and the $f_{i,j}$ have at most $t$ monomials. The
Newton polygon of $f$ has $O(k t^{2m/3})$ edges.
\end{theorem}
In the setting of Newton polygons, the main issue is how to deal with
the cancellations arising from the addition of the $k$ products
in~(\ref{bivariateSPS}).
Two monomials of the form $cX^iY^j$ with the
same pair $(i,j)$ of exponents but oppositive values of the
coefficient $c$ will cancel, thereby deleting the point $(i,j)$
from the Newton polygon.
In the present paper we associate to the monomial $cX^i$ with $c>0$
the point $(i,\log c)$. There are no cancellations since we only
consider polynomials $f_{i,j}$ with nonnegative coefficients in
Theorems~\ref{bound} and~\ref{bound2summands}. However, the addition
of two monomials $cX^i, c'X^i$ with the same exponent will ``move''
the corresponding point along the coefficient axis. By contrast, in
the setting of Newton polygons points can be deleted but cannot
move. In the proof of Theorem~\ref{bound2summands} we deal with the
issue of ``movable points'' by an approximation argument, using the
fact that the constant $\epsilon=\log(4)/2>0$ gives us a little bit
of slack.
\section{A stronger log-concavity condition}\label{strongsection}
The objective of this section is to improve the bound provided in
Theorem \ref{bound} when $f = \sum_{i = 0}^d a_i X^i \in \mathbb{R}^+[x]$
satisfies a stronger log-concavity condition, namely, when
$a_i^2 > d^{2d} a_{i-1} a_{i+1}$ for all $i \in \{1,\ldots,d-1\}$.
To prove this bound, we make use of the following
well-known lemma (a reference and similar results for polytopes in
higher dimension can be found in~\cite{karavelas2012}).
For completeness, we provide a short proof.
\begin{lemma}\label{convexhull} If $R_1,\ldots,R_s$ are planar sets and $\vert R_i
\vert = r_i$ for all $i \in \{1,\ldots,s\}$, then the convex hull of
$R_1 + \cdots + R_s$ has at most $r_1 + \cdots + r_s$ vertices.
\end{lemma}
\begin{proof}We denote by $k_i$ the number of vertices of the
convex hull of $R_i$. Clearly $k_i \leq r_i$. Let us prove that the
convex hull of $R_1 + \cdots + R_s$ has at most $k_1 + \cdots + k_s$
vertices. Assume that $s = 2$. We write $R_1 =
\{a_1,\ldots,a_{r_1}\}$, then $a_i \in R_1$ is a vertex of the
convex hull of $R_1$ if and only if there exists $w \in S^1$ (the
unit Euclidean sphere) such that $w \cdot a_i
> w \cdot a_j$ for all $j \in \{1,\ldots,r_1\} \setminus \{i\}$.
Thus, $R_1$ induces a partition of $S^1$ into $k_1$ half-closed
intervals. Similarly, $R_2$ induces a partition of $S^1$ into $k_2$
half-closed intervals. Moreover, these two partitions induce a new
one on $S^1$ with at most $k_1 + k_2$ half-closed intervals; these
intervals correspond to the vertices of $R_1 + R_2$ and; thus, there
are at most $k_1 + k_2$. By induction we get the result for any
value of $s$.
\end{proof}
\begin{proposition}\label{SPS}
Consider a polynomial $f=\sum_{i=0}^d a_i X^i \in \mathbb{R}^+[X]$ of the form
\begin{align*}
f=\sum_{i = 1}^k \prod_{j = 1}^m f_{i,j}
\end{align*}
where the $f_{i,j} \in \mathbb{R}^+[x]$. If $f$ satisfies the
condition
\begin{align*}
a_i^2 > k^2 d^{2m} a_{i-1} a_{i+1} ,
\end{align*}
then there exists a polynomial $f_{i,j}$ with at least $d / km$ monomials.
\end{proposition}
\begin{proof}
Every polynomial $f_{i,j} := \sum_{l = 0}^{d_{i,j}} c_{i,j,l}\,
X^l$, where $d_{i,j}$ is the degree of $f_{i,j}$, corresponds to a
planar set
\begin{align*} R_{i,j} := \{(l, {\rm log}(c_{i,j,l}))\, \vert \,
c_{i,j,l} > 0\} \subset \mathbb{R}^2.
\end{align*} We set,
$C_{i,l} := {\rm max} \{0, \prod_{r = 1}^m c_{i,r,l_r} \, \vert \,
l_1 + \cdots + l_m = l \},$ for all $i \in \{1,\ldots,k\}$, $l \in
\{0,\ldots,d\}$, and $ C_l := {\rm max}\{C_{i,l} \, \vert \, 1
\leq i \leq k\}$ for all $l \in \{0,\ldots,d\}$. Since the
polynomials $f_{i,j} \in \mathbb{R}^+[X]$ and
$$a_l = \sum_{i = 1}^k \left(\sum_{l_1 + \cdots + l_m = l}\
\prod_{r= 1}^m c_{i,r,l_r}\right)$$ for all $l \in \{0,\ldots,d\}$,
we derive the following two properties:
\begin{itemize}
\item $C_l \leq a_l \leq k d^m C_l$ for all $l \in \{0,\ldots,d\}$,
\item either $C_{i,l} = 0$ or $(l,
{\rm log}(C_{i,l})) \in R_{i,1} + \cdots + R_{i,m}$ for all $i \in
\{1,\ldots,k\}, \, l \in \{0,\ldots,d\}$. Since $a_l > 0$ for all $l
\in \{1,\ldots,d\}$, we have that $C_l > 0$ and $(l, {\rm log}(C_l))
\in \bigcup_{i = 1}^k \left(R_{i,1} + \cdots + R_{i,m}\right)$
\end{itemize}
We claim that the points in the set $\{(l, {\rm log}(C_l)) \, \vert
\, 1 \leq l \leq d\}$ belong to the upper convex envelope of
$\bigcup_{i = 1}^k (R_{i,1} + \cdots + R_{i,m})$. Indeed, if
$(a,\log(b)) \in \bigcup_{i = 1}^k (R_{i,1} + \cdots + R_{i,m})$,
then $a \in \{0,\ldots,d\}$ and $b \leq C_{a}$; moreover, for all $l
\in \{1,\ldots,d-1\}$, we have that $$C_l^2 \geq a_l^2 / (k^2
d^{2m}) > a_{l-1} \, a_{l+1} \geq C_{l-1} C_{l+1}.$$
Hence, there exist $i_0 \in \{1,\ldots,k\}$ and $L \subset
\{1,\ldots,d\}$ such that $\vert L\vert \geq d/k$ and $C_l =
C_{i_0,l}$ for all $l \in L$. Since the points in $\{(l,{\rm
log}(C_{l}))\, \vert \, 1 \leq l \leq d\}$ belong to the upper
convex envelope of $\bigcup_{i = 1}^k (R_{i,1} + \cdots +
R_{i,m})$ we easily get that the set $\{(l, {\rm
log}(C_{i_0,l})) \, \vert \, l \in L\}$ is a subset of the vertices in the convex hull
of $R_{i_0,1} + \cdots + R_{i_0,m}$. By Lemma \ref{convexhull}, we
get that there exists $j_0$ such that $\vert R_{i_0,j_0} \vert \geq
\lvert L \rvert / m \geq d / km$ points. Finally, we conclude that
$f_{i_0,j_0}$ involves at least $d / km$ monomials.
\end{proof}
\bigskip
\begin{proof}[Proof of Theorem \ref{bound2}]If $d \leq k$ or $d \leq m$, then $d \leq kmt$.
Otherwise, $d^{2d} > k^2 d^{2(d-1)} \geq k^2 d^{2m}$ and, thus, $f$
satisfies (\ref{stronglogconcave}). A direct application of
Proposition \ref{SPS} yields the result.
\end{proof}
\section{Applications to Complexity Theory}\label{complexity}
We first recall some standard definitions from algebraic complexity
theory (see e.g.~\cite{Burgi} or~\cite{Val79} for more
details). Fix a field $K$.
The elements of the complexity class $\vp$ are sequences $(f_n)$ of
multivariate polynomials with coefficients from $K$. By definition,
such a sequence belongs to $\vp$ if the degree of $f_n$ is bounded by
a polynomial function of $n$ and if $f_n$ can be evaluated in a
polynomial number of arithmetic operations (additions and
multiplications) starting from variables and from constants in $K$.
This can be formalized with the familiar model of {\em arithmetic
circuits}.
In such a circuit, input gates are labeled by a constant or a
variable and the other gates are labeled by an arithmetic operation
(addition or multiplication). In this paper we take $K = \mathbb{R}$ since
there is a focus on polynomials with nonnegative coefficients. An
arithmetic circuit is {\em monotone} if input gates are labeled by
nonnegative constants only.
A family of polynomials
belongs to the complexity class $\vnp$ if it can be
obtained by summation from a family in $\vp$.
More precisely, $f_n(\overline{x})$ belongs to $\vnp$ if there exists
a family $(g_n(\overline{x},\overline{y}))$ in $\vp$ and a polynomial $p$
such that the tuple of variables
$\overline{y}$ is of length $l(n) \leq p(n)$ and
$$f_n(\overline{x})=\sum_{\overline{y} \in \{0,1\}^{l(n)}} g_n(\overline{x},\overline{y}).$$
Note that this summation over all boolean values of $\overline{y}$
may be of exponential size.
Whether the inclusion $\vp \subseteq \vnp$ is strict is a major open
problem in algebraic complexity.
Valiant's criterion~\cite{Burgi,Val79} shows that ``explicit''
polynomial families belong to $\vnp$. One version of it is as follows.
\begin{lemma}
Suppose that the function $\phi:\{0,1\}^* \rightarrow \{0,1\}$ is
computable in polynomial time. Then the family $(f_n)$ of multilinear
polynomials defined by
$$f_n=\sum_{e \in \{0,1\}^n} \phi(e)x_1^{e_1} \cdots x_n^{e_n}$$
belongs to $\vnp$.
\end{lemma}
Note that more general versions of Valiant's criterion are know. One
may allow polynomials with integer rather than
0/1 coefficients~\cite{Burgi}, but in Theorem~\ref{monotone}
below we will only have to deal with 0/1 coefficients.
Also, one may allow $f_n$ to depend on any (polynomially bounded)
number of variables rather than exactly $n$ variables and in this case, one may
allow the algorithm for computing the coefficients of $f_n$ to take as
input the index $n$ in addition to the tuple $e$ of exponents
(see~\cite{Koi04}, Theorem~2.3).
Reduction of arithmetic circuits to depth~4 is an important
ingredient in the proof of the forthcoming results. This phenomenon
was discovered by Agrawal and Vinay \cite{AV}. Here we will use it
under the form of \cite{Tavenas}, which is an improvement of
\cite{Koiran2012}. We will also need the fact that if the original
circuit is monotone, then the resulting depth~4 circuit is also
monotone (this is clear by inspection of the proof
in~\cite{Tavenas}). Recall that a depth~4 circuit is a sum of
products of sums of products of inputs; sum gates appear on layers
2 and 4 and product gates on layers 1
and 3. All gates may have arbitrary fan-in.
\begin{lemma}\label{redprof4}Let $C$ be an arithmetic circuit of size $s > 1$
computing a $v$-variate polynomial of degree $d$. Then, there is an
equivalent depth $4$ circuit $\Gamma$ of size $2^{\, \mathcal
O\left(\sqrt{d \log (ds) \log (v)} \right)}$ with multiplication
gates at layer $3$ of fan-in $\mathcal O(\sqrt{d})$. Moreover, if
$C$ is monotone, then $\Gamma$ can also be chosen to be monotone.
\end{lemma}
We will use this result under the additional hypothesis that $d$ is
polynomially bounded by the number of variables $v$. In this
setting, since $v \leq s$, we get that the resulting depth $4$
circuit $\Gamma$ provided by Lemma \ref{redprof4} has size
$s^{\mathcal O(\sqrt{d})}$.
\medskip
Before stating the main results of this section, we construct an explicit family of log-concave polynomials.
\begin{lemma}\label{lem_concavVn}Let $n, s \in \mathbb{Z}^+$ and
consider $g_{n,s}(X) := \sum_{i=0}^{2^n-1} a_i X^i$, with
\begin{align*} a_i := 2^{si(2^n-i-1)} {\text \ for \ all \ } i \in \{0,\ldots,2^n
- 1\}. \end{align*}
Then, $a_i^2
> 2^s \, a_{i-1} \, a_{i+1}$.
\end{lemma}
\begin{proof}Take $i \in \{1,\ldots,2^n - 2\}$, we have that
\begin{align*}
\log\left(2^s a_{i-1}a_{i+1}\right) & = s + s2^n(i-1) - s(i-1)i
+s2^n(i+1) - s(i+1)(i+2) \\
& = 2s2^n i - 2s i(i+1)- s \\
& < 2s 2^n i -2s i(i+1) \\
& = \log(a_i^2).
\end{align*}
\end{proof}
In the next theorem we start from the family $g_{n,s}$
of Lemma~\ref{lem_concavVn} and we set $s=n2^{n+1}$.
\begin{theorem}\label{ifvpvnp}
Let $(f_n) \in \mathbb{N}[X]$ be the family of polynomials $f_n(x)=g_{n,n2^{n+1}}(x)$.
\begin{itemize}
\item[(i)] $f_n$ has degree $2^n-1$ and satisfies the log-concavity condition {\rm (\ref{stronglogconcave})}.
\item[(ii)] If $\vp=\vnp$, $f_n$ can be written under form~{\rm
(\ref{sumprod})} with $k=n^{O(\sqrt{n})}$,
$m=O(\sqrt{n})$ and $t=n^{O(\sqrt{n})}$.
\end{itemize}
\end{theorem}
\begin{proof}
It is clear that $f_n \in
\mathbb{N}[X]$ has degree $2^n - 1$ and, by Lemma \ref{lem_concavVn}, $f_n$
satisfies (\ref{stronglogconcave}).
Consider now the related family of bivariate polynomials
$g_n(X,Y)=\sum_{i=0}^{2^n-1}X^i Y^{e(n,i)},$ where $e(n,i) = s i (2^n - i - 1)$. One can check in time polynomial in $n$ whether
a given monomial $X^iY^j$ occurs in $g_n$: we just need to check
that $i<2^n$ and that $j=e(n,i)$. By mimicking the proof of Theorem
1 in \cite{KPTT} and taking into account Lemma \ref{redprof4} we get
that, if $\vp = \vnp$, one can write
\begin{equation} \label{sumprod2}
g_n(X,Y)=\sum_{i=1}^k\prod_{j=1}^m g_{i,j,n}(X,Y)
\end{equation}
where the bivariate polynomials $g_{i,j,n}$ have $n^{O(\sqrt{n})}$
monomials, $k=n^{O(\sqrt{n})}$ and $m=O(\sqrt{n})$. Performing the
substitution $Y=2$ in~(\ref{sumprod2}) yields the required
expression for $f_n$.
\end{proof}
We believe that there is in fact no way to write $f_n$ under
form~(\ref{sumprod}) so that the parameters $k,m,t$ satisfy the
constraints $k=n^{O(\sqrt{n})}$,
$m=O(\sqrt{n})$ and $t=n^{O(\sqrt{n})}$.
By part (ii) of Theorem~\ref{ifvpvnp}, a proof of this would
separate $\vp$ from $\vnp$. The proof of Theorem~\ref{monotone} below
shows that our belief is actually correct in the special case where
the polynomials $f_{i,j}$ in~(\ref{sumprod}) have nonnegative
coefficients.
\medskip
The main point of Theorem \ref{monotone} is to present an
unconditional lower bound for a polynomial family $(h_n)$ in $\vnp$
derived from $(f_n)$. Note that $(f_n)$ itself is not in $\vnp$
since its degree is too high. Recall that
\begin{equation} \label{fneq}
f_n(X) := \sum_{i=0}^{2^n-1} 2^{2n2^ni(2^n-i-1)} X^i.
\end{equation}
To construct $h_n$ we write down in base 2 the exponents of ``2'' and
``$X$'' in~(\ref{fneq}).
More precisely, we take $h_n$ of the form:
\begin{equation} \label{hneq}
h_n :=\sum_{\alpha \in \{0,1\}^{n} \atop \beta \in \{0,1\}^{4n}}
\lambda(n, \alpha, \beta)\, X_0^{\alpha_0} \cdots
X_{n-1}^{\alpha_{n-1}} Y_0^{\beta_0} \cdots
Y_{4n-1}^{\beta_{4n-1}},
\end{equation}
where $\alpha =
(\alpha_0,\ldots,\alpha_{n-1}),\, \beta =
(\beta_0,\ldots,\beta_{4n-1})$ and $\lambda(n,\alpha,\beta) \in
\{0,1\}$; we set $\lambda(n,\alpha,\beta) = 1$ if and only
if $\sum_{j = 0}^{4n-1} \beta_j 2^j = 2n2^n i (2^n - i - 1) <
2^{4n},$ where $i := \sum_{k = 0}^{n-1} \alpha_{i,k} 2^k$.
By construction, we have:
\begin{equation} \label{transfor} f_n(X) = h_n(X^{2^0}, X^{2^1},\ldots,X^{2^{n-1}},2^{2^0},
2^{2^1},\ldots,2^{2^{4n-1}}). \end{equation}
This relation will be useful in the proof of the following lower bound theorem.
\begin{theorem}\label{monotone}
The family $(h_n)$ in~{\rm(\ref{hneq})} is in $\vnp$. If $(h_n)$ is
computed by depth $4$ monotone arithmetic circuits of size $s(n)$,
then $s(n) = 2^{\,\Omega(n)}$. If $(h_n)$ is computed by monotone
arithmetic circuits of size $s(n)$, then $s(n) =
2^{\,\Omega(\sqrt{n})}$. In particular, $(h_n)$ cannot be computed
by monotone arithmetic circuits of polynomial size.
\end{theorem}
\begin{proof}
Note that $h_n$ is a polynomial in $5n$ variables, of degree at most
$5n$, and its coefficients $\lambda(n, \alpha, \beta)$ can be
computed in polynomial time. Thus, by Valiant's criterion
we conclude that $(h_n)\in \vnp$.
Assume that $(h_n)$ can be computed by depth $4$ monotone arithmetic
circuits of size $s(n)$. Using (\ref{transfor}), we get that $f_n =
\sum_{i = 1}^k \prod_{j = 1}^m f_{i,j}$ where $f_{i,j} \in \mathbb{R}^+[X]$
have at most $t$ monomials and $k,m,t$ are $\mathcal
O(s(n))$. Since the degree of $f_n$ is $2^n - 1$, by Theorem
\ref{bound2}, we get that $2^n - 1 \leq kmt$. We conclude that $s(n)
= 2^{\,\Omega(n)}$.
To complete the proof of the theorem, assume that $(h_n)$ can be
computed by monotone arithmetic circuits of size $s(n)$. By
Lemma~\ref{redprof4}, it follows that the polynomials $h_n$ are
computable by depth~4 monotone circuits of size $s'(n) :=
s(n)^{\mathcal O(\sqrt{n})}$. Therefore $s'(n) = 2^{\,\Omega(n)}$
and we finally get that $s(n) = 2^{\,\Omega(\sqrt{n})}$.
\end{proof}
Lower bounds for monotone arithmetic circuits have been known for a
long time (see for instance~\cite{jerrum82,valiant79negation}).
Theorem~\ref{monotone} provides yet another example of a polynomial
family which is hard for monotone arithmetic circuits, with an
apparently new proof method.
\section{Discussion}
As explained in the introduction, log-concavity plays a role
in the study of real roots of polynomials.
In~\cite{Koi10a} bounding the number of real roots of sums of products
of sparse polynomials was suggested as an approach for separating
$\vp$ from $\vnp$. Hrube\v{s}~\cite{Hrubes13} suggested to bound the
multiplicities of roots, and~\cite{KPTT} to bound the number of edges
of Newton polygons of bivariate polynomials.
Theorem~\ref{ifvpvnp} provides another
plausible approach to $\vp \neq \vnp$: it
suffices to show that if a polynomial $f \in \mathbb{R}^+[X]$ under
form~(\ref{sumprod}) satisfies the Kurtz condition or the stronger
log-concavity condition (\ref{stronglogconcave}) then its degree is
bounded by a ``small'' function of the parameters $k,m,t$. A degree
bound which is polynomial bound in $k,t$ and $2^m$ would be good
enough to separate $\vp$ from $\vnp$. Theorem~\ref{bound} improves
on the trivial $kt^m$ upper bound when $f$ satisfies the Kurtz
condition, but certainly falls short of this goal: not only is the
bound on $\deg(f)$ too coarse, but we would also need to allow
negative coefficients in the polynomials $f_{i,j}$.
Theorem~\ref{bound2} provides a polynomial bound on $k,m$ and $t$
under a stronger log-concavity condition, but still needs the extra
assumption that the coefficients in the polynomials $f_{i,j}$ are
nonnegative. The unconditional lower bound in Theorem~\ref{monotone}
provides a ``proof of concept'' of this approach for the easier
setting of monotone arithmetic circuits.
\bibliographystyle{plain}
|
1,108,101,565,623 | arxiv | \section{Introduction}
\label{sec:introduction}
The COVID-19 pandemic is an ongoing epidemic of the coronavirus disease~\cite{cao2020covid}. The epidemic was declared a public health emergency of international concern by the World Health Organization (WHO) and it is impacting all aspects of life. At the same time, the COVID-19 epidemic is also generating a large amount of data that could be used to effectively guide decision making in the COVID-19 response~\cite{Thomas2021o}. These emergency responses include the detection, prevention and control of the disease, and the recovery from global economic disruption, which is potentially the largest global recession since the Great Depression \cite{verschuur2021observed,schwab2021real}.
Important decisions to be made include postponing or cancellation of sporting, religious, political, cultural activities, and those related to severe stock shortages caused by panic buying \cite{heymann2020covid}. It has been reported that schools, universities, and colleges in more than 180 countries have either closed on a national or local basis, impacting most of the student population worldwide \cite{viner2020school}. Due to the epidemic, vast amounts of data are being generated, which could be used in data-driven decision making to cope with this disease and to respond to relevant emergencies.
Data-driven decision making is the process of using evidences and insights derived from data to guide the decision making process and to verify a plan of actions before it is committed~\cite{cippa2021data,zhang2020data}. It plays a vital role in the success of businesses, industries, governments, and even an individual's life. We continually make decisions during every aspect of our lives~\cite{nisioti2021data,catelani2020optimizing}. Indeed, success is based on these decisions, with correct decisions helping to achieve goals meanwhile wrong decisions resulting in failure. Data is the key element, playing the most significant role in effective decision making. It is the primary input for many fields of science like data mining, big data, machine learning, and data science~\cite{liu2019data,bedru2020big,liu2019shifu2}. The objective of all these fields is to analyse data and to provide insights that actually assist to perform effective decision making. Data-driven decision making is changing the field in all professional endeavours, including business, medicine, and education~\cite{dolgin2020core}. Here, experts gather and analyse data to effectively prompt their decision making~\cite{Leung2020,Gheluwe9239038}. There has been broad adoption in industry, with businesses using data analytics at every stage in their operation, from the management of supply chains to the creation of competitive advantage.
\begin{figure*}[!t]
\centering
\includegraphics[width=0.8\textwidth]{3.pdf}
\caption{Big data processing stages.}
\label{data-decision}
\end{figure*}
Data play a vital role in effective decision making \cite{ellul2015big}. In the response to the COVID-19 epidemic, a lot of important decisions need to be taken to save communities and economies worldwide. In this paper, we review the different decisions made in response to the COVID-19 epidemic from the perspective of data and subsequent insights. This includes the use of data for the epidemic prevention and control, psychological counselling, the protection of livelihoods, and the financial aid efforts to recover the economy~\cite{Xin9209939}. This survey also includes the data-driven decisions made related to resumption of work and schools. We also describe the challenges and open issues which hinder the development of data-driven decisions based on insights and analytics, and these challenges will open new directions for future research.
The paper is organized as follows. The literature and techniques of prevention and control are introduced in Section~\ref{sec:2}. The Psychological effects of the disease and related counselling is described in Section~\ref{sec:3}. The needs for financial aid for the protection of lives and recovery of economy is described in Section~\ref{sec:4}. Data-driven decisions in the return to workplaces and schools are described in Section~\ref{sec:5} and Section~\ref{sec:6}, respectively. The key challenges and issues for data-driven decision making specifically related to the COVID-19 epidemic are explained in Section~\ref{sec:7}. Section~\ref{sec:8} concludes the paper.
\section{COVID-19 Prevention and Control}
\label{sec:2}
\begin{table*}[!t]
\centering
\caption{COVID-19 Dataset}
\label{tabdatasets}
\resizebox{1.0\linewidth}{!}{
\begin{threeparttable}
\begin{spacing}{1.19}
\begin{tabular}{p{3cm}p{4.5cm}p{10cm}p{3.3cm}}
\toprule[2pt]
\textbf {Category} & \textbf {Title} &\textbf {Main Content} & \textbf {Provider} \\
\midrule[1pt]
Epidemic & Worldometer COVID-19 Data & Confirmed cases or deaths in different countries and regions & Worldometer \\
Scientific Research & COVID-19 Open Research Dataset (CORD-19) & More than 45,000 academic articles are freely available, including 33,000 full texts on COVID-19 and coronavirus family viruses & Allen Institute for AI\\
Knowledge Graph & COKG-19 & The COKG-19 contains 505 concepts, 393 attributes, 26,282 entities and 32,352 knowledge triples, covering medical treatment, health, materials, prevention and control, scientific research and people, etc & AMiner \\
Migration & Baidu migration & Population migration during the Spring Festival in China & Baidu Map\\
Social media & COVID-19-TweetIDs & The repository contains an ongoing collection of tweets IDs associated with the novel coronavirus COVID-19, which commenced on January 28, 2020 & Github (Emily Chen)\\
Medical & COVID-chestxray-dataset & It an open database of COVID-19 cases with chest X-ray or CT images & Github (Joseph Paul Cohen)\\
Traffic & TSA: Airport confirmed case data & American airports have tested for confirmed cases in the past 14 days & Transportation Security Administration\\
\bottomrule[2pt]
\end{tabular}%
\end{spacing} %
\end{threeparttable}
}
\end{table*}%
The global spread of COVID-19, which has affected hundreds of countries and regions, is a huge challenge to the ability of governments to manage crises~\cite{de2020initial}. Early response and timely action are extremely important in the fight against COVID-19. Hern{\'a}ndez-Orallo et al.~\cite{hernandez2020evaluating} have proven that using data analysis to track the disease source, isolating the infected person in time, and controlling social distance are effective methods in preventing further spread of COVID-19. There are some studies that illustrate the response of different countries in the initial period of the outbreak and the later control measures~\cite{kraemer2020effect,atk2020G20,tomar2020prediction}. These works show that despite different countries having specific medical conditions, there are still several decisions that can be made to cope with disease outbreaks. If certain measures of prevention and control are adopted according to the situation in the early stages, the spread of the epidemic can be slowed down. The development of science and technology has made it possible for us to mine knowledge from big data, which can guide decision making in epidemic prevention and control as well as policy formulation~\cite{anastassopoulou2020data}.
\subsection{Data-Driven Decisions}
With the continuous spread of the epidemic, many types of information and data related to the epidemic are continually updated on the Internet. There is a huge amount of information, including global epidemic data, COVID-19 related research statistics, medical equipment details, transportation information and so on. The integration and classification of these forms of information are important for epidemic prevention and control. We have categorized some commonly-used datasets by information type, as shown in Table~\ref{tabdatasets}. By using this, researchers can select appropriate datasets for analysis according to their needs.
To make full use of the data, it is necessary to process the data. The big data processing stages include data collection, data pre-processing, data storage, data analysis, data visualization, data application, etc, which are shown in Fig.~\ref{data-decision}. Through data analysis, technicians can effectively process raw data and aggregate related information. Subsequently, valuable information can be derived and then used to interpret and predict the development of the epidemic. Based on more comprehensive information or stronger evidences, decision-makers can make more scientific decisions~\cite{lytras2017big}.
Policies are generally made dynamically. This is because policies need to be adjusted in time according to specific situations. In modern society, decisions are no longer made based on empirical judgements, but are generally made based on big data~\cite{mcafee2012big}. Big data statistics are used in order to permit policy makers to rapidly assess the epidemic risk. With respect to COVID-19, related decisions and relevant plans are also adjusted correspondingly based on big data analysis~\cite{power2014using}. Big data technologies provide accurate information for the decision making and analysis of the headquarters for the epidemic prevention. With the help of big data technologies, epidemic prevention and control can be carried out in an orderly manner, so that the epidemic can be effectively controlled within a determined period of time.
\subsection{Tracing Close Contacts}
Close contact tracing is a common intervention to control outbreaks of infectious diseases~\cite{hellewell2020feasibility}. In the early stages of the epidemic, traditional door-to-door surveys and paper form filling were still adopted in most areas, which was organized and implemented step by step~\cite{li2020early}. If the execution was not strong enough or the provided information was wrong, it would affect the judgement and prevention of the epidemic. After that, each person's action trajectory was generated by analyzing people's location information. The responsible persons were isolated and additionally the close contacts, who were identified based on the action trajectory and reported information. If some of these people are found to have abnormal health, the authorities can provide them with timely help~\cite{fisher2020global}.
The ``Big data and Grid" method is adapted to improve the screening efficiency in inspecting the general trend of people mobility~\cite{li2020early,xia2018commag,xia2018tii}. Additionally, the government release information about ``epidemic community", ``infection routes" and ``patients on the same journey". People can see if there are new cases within their community on the web via HealthMap\footnote{\url{https://www.healthmap.org/covid-19/}} at any time. The travel information of suspected infected persons can also be queried, to quickly determine whether they are close contacts. To some extent, this also reduces for the relevant departments the pressure of screening. National Health Insurance data and the database from the Immigration Department can be merged for better decision making. After analyzing the data, medical staff can make judgements based on the patient's travel records and clinical symptoms~\cite{wang2020res}. Smart card data can also be employed to identify suspected patients and isolate them at an early stage, which will effectively reduce the spread of the epidemic.
Smartphone solutions have also been developed to enhance tracing close contacts. Location-based technologies are employed in a range of applications that utilize location sensors and capabilities of smartphones~\cite{9117157,xia2014exploiting,xia2014community}. Proximity details, identity, location data, as well as other condition information can be captured by apps on the smartphone. Apps such as TraceTogether employ a data-driven community-based approach to share position data and time stamps. Meanwhile, to protect privacy of users, TraceTogether will overwrite its data after it collects newer data every 21 days. This is to say, this app only saves position and time data for 21 days. We can see the list of corresponding apps shown in Table~\ref{tab:app}. However, some non-specialized apps such as QR-codes and electronic cards in WeChat and Alipay are not included in the Table.
Tracing methods can be further enhanced by optimizing testing and tracing coverage. More mobile application technologies need to be developed to improve contact tracing effectiveness. Bradshaw et al.~\cite{bradshaw2021bidirectional} found that contact tracing plays a critical role in controlling and preventing COVID-19, but most tracing protocols rely on forward-trace instead of bidirectional tracing protocols. They utilized the bidirectional tracing method, resulting in significant improvement in COVID-19 control. Ng et al.~\cite{9373368} presented a smart contact tracing system named SCT, which employs a smartphone's Bluetooth low energy signals together with machine learning classifiers to detect potential contacts to confirmed cases. The privacy problem should also be considered when releasing the confirmed cases. Publicizing close contact data can help the public better avoid causing COVID-19 infections, but also increases the potential for data breach problems at the same time. The technological limitations and the balance between privacy and tracing effectiveness needs to be taken into full consideration.
\begin{table*}[!t]
\centering
\caption{Smartphone applications}
\label{tab:app}
\resizebox{1.0\linewidth}{!}{
\begin{threeparttable}
\begin{spacing}{1.19}
\begin{tabular}{p{1.5cm}p{2 cm}p{1.5cm}p{2cm}p{3cm}p{3cm}p{9cm}}
\toprule[2pt]
\textbf {Number} & \textbf {App Aame} &\textbf {Country} & \textbf {Operating System} & \textbf {Developer} & \textbf {Size} & \textbf {Official Website}\\
\midrule[1pt]
1 & Aarogya Setu & India & iOS/Android & National Information Center & iOS:37.1M Android:4.2M & \url{https://www.aarogyasetu.gov.in/}\\
2 & PrivateKit & USA & iOS/Android & Massachusetts Institute of Technology & iOS:34M Android:8.4M & \url{http://privatekit.mit.edu/}
\\
3 & Rakning C-19 & Iceland & iOS/Android & Icelandic Government & iOS:20M Android:28M & \url{https://www.covid.is/app/is}
\\
4 & Stopp Corona & Austria & iOS/Android & Austrian Red Cross & iOS:15.6M Android:6.2M & \url{https://www.stopp-corona.at/}
\\
5 & TousAntiCovid & France & iOS/Android & Government Technical Institutions & iOS:76.1M Android:23M & \url{https://www.frandroid.com/telecharger/apps/stopcovid-france}
\\
6 & TraceTogether & Singapore & iOS/Android & Government Technical Institutions & iOS:71.4M Android:26.1M & \url{https://www.tracetogether.gov.sg/}
\\
\bottomrule[2pt]
\end{tabular}%
\end{spacing} %
\end{threeparttable}
}
\end{table*}
To better solve this problem, Aarogya Setu uses contact information to track details. If one of the contacts is confirmed to be COVID-19 positive, then the user is notified to implement aggressive medical interventions. Rakning C-19 runs in the background and stores GPS positions many times per hour for 14 days. These position data will only be saved in the user's phone and no-one else has the access to these data. Stopp Corona transfers data with encryption and anonymous processing. TousAntiCovid requires the user to have an open Bluetooth connection, providing anonymous record services. If a certain user is infected, TousAntiCovid will inform this user and close contacts based on anonymous records.
\subsection{Trend Forecasting}
Forecasting the trend of development of the epidemic is great significant to the prevention and control of the situation, the distribution of medical resources, economic development, and the arrangement of production activities~\cite{li2020trend,liu2020cri}. By releasing significant amounts of data and statistics, policy makers and the public can understand the development of the epidemic and the effectiveness of its control. For policy makers, if the scale or deterioration of the epidemic is known as early as possible, they can take corresponding measures to disrupt the chain of transmission and more reasonably allocate medical resources, which can save lives~\cite{fanelli2020analysis}.
Many researchers have proposed different propagation prediction models in combination with their own research directions. Zhou et al.~\cite{9338466} used Autoregressive Integrated Moving Average (ARIMA) model, logistic regression, Susceptible Infective Removed (SIR) model, and improved Susceptible Exposed Infective Removed (SEIR) model to predict the global pandemic. The benefits and drawbacks of all these models are respectively discussed. The global cumulative number of confirmed, cured, and cases of death of both COVID-19 and SARS are employed to better predict the global pandemic trend. The theoretical basis is also provided for prevention and control policies. To accurately forecast the development trend of COVID-19, Kumar and Susan~\cite{9225319} used temporal data from cumulative cases of the 10 most affected countries. Based on ARIMA and Prophet time series forecasting models, the evolution of the COVID-19 outbreak is then modeled and evaluated by strict mathematical metrics such as mean absolute error. It is verified that ARIMA is more effective in forecasting COVID-19 prevalence. The forecasting results will provide potential assistance for governments when planning policies for containing the spread of COVID-19.
\subsection{Hospital Relevant Policies}
COVID-19 has extremely challenged medical processes such as hospital admissions. Data-driven hospital decision making has become more and more important. To better mitigate the epidemic and meanwhile limit collateral economic damage, proper hospital policies should be undertaken. Duque et al.~\cite{Duque19873} presented a strategy to minimize the number of days of disruption. When hospital admissions exceed a certain threshold, short-term in-place placement orders can be triggered. Random optimization is employed to export the triggers to make sure that the hospital admission capacity will not be insufficient. Based on COVID-19 hospitalization data, this work provides a flexible framework, which allows optimization using a relatively small discrete grid. Similar epidemics can be simulated with this proposed model.
The medical examination process can also be improved by using data analysis. Cohort-level clinical data, patient-level hospital data, and census-level epidemiological data can be analysed to develop an integrated optimization model. In the city of Daegu, the South Korean government reorganized the health system along with hospital-level interventions. Equipment are concentrated to relieve the shortage of medical resources. This policy ultimately protected patients and health care staff. The Iranian government added 3,000 hospital beds to the Iran Mall Exhibition based on data analysis. Meanwhile, a 2000-bed army hospital has been opened to accommodate more patients.
While optimizing the medical configuration and treating confirmed cases, scientists from all over the world are working towards effective vaccines for COVID-19. The United States, Australia, Russia, UK, China, South Africa, Korea, and many other countries have made progress towards the COVID-19 vaccine. With the progress of vaccine development, many hospitals have provided vaccine injections.
\begin{figure*}[!t]
\begin{center}
\includegraphics[width=0.8\textwidth]{4.pdf}
\caption{Impact of COVID-19 (Data source: National Bureau of Statistics of China, World Tourism Alliance, Ministry of Transport of China, and General Administration of Customs of China)}
\label{industry}
\end{center}
\end{figure*}
\subsection{Influence of the Pandemic}
The outbreak and rapid spread of COVID-19 has caused an enormous impact on every country's economy in a short period of time. The impact on key industries such as accommodation and catering, transportation, tourism, and foreign trade is significant. To prevent the spread of the epidemic, many cities have closed roads, airports, and high-speed trains. Major tourist attractions have been closed and major cultural events cancelled. There has been a sharp decrease in all kinds of dinner parties and a large number of restaurants have been closed, with the stock of prepared dishes being sold at low prices. Workers cannot work normally, and companies have no way to produce and deliver as planned, so the supply chain is interrupted, and many factories face the crisis of closure~\cite{baldwin2020thinking}. The specific impact of the epidemic on these five industries is shown in Fig.~\ref{industry}.
Therefore, taking measures as early as possible to control the epidemic is key to reducing the impact of the epidemic on multiple industries. At present, the epidemic situation has been effectively controlled in many countries.
\section{Psychological Counselling}
\label{sec:3}
The outbreak of COVID-19 has made people face many changes in lifestyle and environment~\cite{kong2018human}. Many places have adopted mandatory isolation measures to prevent the further spread of the virus. Isolation increases the possibility of depression and even makes people ``desperate"~\cite{horton2020offline}. Moreover, the negative reports about COVID-19 illnesses are constantly updated every day. The number of dead cases is increasing, and the number of confirmed cases is increasing daily, resulting in incalculable loss. These series of problems increase people's mental stress and negative psychological states. The large-scale pandemic disease causes panic in people, worrying about the possibility of becoming sick, and meanwhile increasing the fear of death, helplessness, irrespective of being a patient~\cite{North2013}. These cases involve the pain of treatment, the risk of relapse of mental diseases, and several uncontrollable behaviours.
\begin{table*}[!t]
\begin{threeparttable}
\caption{Some Decision Making Issued by International Organization}
\centering
\label{table_example}
\begin{tabular}{p{1.5cm}p{4cm}p{2cm}p{9cm}}
\toprule[2pt]
\textbf {Number\tnote{*}} & \textbf {Institution}&\textbf {Date issued} & \textbf {Decision making} \\
\midrule[1pt]
1&World Health Organization (WHO)&Mar 18, 2020& Mental health and psychosocial considerations during the COVID-19 outbreak\\
2&Centers for Disease Control and Prevention (CDC) &Apr 30, 2020& Coping with Stress\\
3&World Federation for Mental Health (WFMH) & Apr 22, 2020& Appeal for National Plans for Mental Health during the Coronavirus Global Emergency\\
4&National Alliance on Mental Illness (NAMI) & May 12, 2020& COVID-19 Resource and Information Guide\\
5&Office for the Coordination of Humanitarian Affairs (OCHA) &Mar 17, 2020& Interim Briefing Note Addressing Mental Health and Psychosocial Aspects of COVID-19 Outbreak\\
6&Bureau of Disease Control and Prevention & Mar 18, 2020& Notice on Issuing the Work Plan for Psychological Counseling of New Crown Pneumonia Epidemic\\
7&Bureau of Disease Control and Prevention &Feb 7, 2020& Guidelines for the Work of Psychological Aid Hotlines\\
8&National Health Commission &Feb 15, 2020& Implement Several Measures to Improve Medical Staff's Working Conditions and Physical and Mental Health\\
9&Ministry of Education &Feb 16, 2020& Suggestions on Guiding Children to Study and Live at Home during Epidemic Prevention and Control\\
10&National Food Safety Risk Assessment Center & Mar 11, 2020&Advice on Public Mental Health Guidance during New Crown Pneumonia Epidemic\\
\bottomrule[2pt]
\end{tabular}
\begin{tablenotes}
\footnotesize
\item[*]Number does not represent the ranking order of the institutions.\\
\end{tablenotes}
\end{threeparttable}
\end{table*}
More than half of the people were seriously affected by the outbreak of the disease. According to reports, during the SARS outbreak, there were many mental health diseases, such as depression, anxiety, panic, and additionally people developed suicidal tendencies~\cite{ben2004traumatic}. Among patients, 59\% of those hospitalized suffer from the impact of depression and trauma. The importance of psychological protection for medical staff should also be emphasized. What is more, with the expansion of the epidemic, the economic downturn, unemployment, and financial difficulties also affect people's mental health, and its serious results are immeasurable. Therefore, during the prevalence of COVID-19, besides the need for active and effective treatment measures, mental and psychological health guidance is also indispensable.
\subsection{Official Policies}
The WHO issued customized mental health guidance for individuals, which aims to immediately consider people with mental health problems~\cite{world2020mental}. Similar guidelines have also been issued. A large-scale survey which covers 58 countries and more than 100,000 participants was conducted to study the public reaction of government policies\footnote{\url{https://covid19-survey.org/}}. Cross-country data analysis shows that strong government policies can reduce public worries and depression. Moreover, it is also suggested that policymakers should also consider how their decisions affect the mental health of their population. Some international health organizations have issued suggestions on mental health problems during the COVID-19 epidemic, as shown in Table~\ref{table_example}.
Guidelines for Emergency Psychological Crisis Intervention in Pneumonia Infected in COVID-19 has been issued. Such guidelines put forward psychological intervention points for confirmed patients, suspected patients, medical care, and related personnel such as those who are in close contact with patients (family members, friends, colleagues, etc.), those who do not want to see a doctor publicly, and vulnerable people from other different groups. Li et al.~\cite{li2020progression} collated the documents of specific intervention and guidance measures for different groups in the severe outbreak period. However, with the continuous changes during the epidemic situation, the corresponding psychological guidance measures should be improved for different populations. Duan and Zhu~\cite{duan2020psychological} pointed out the shortcomings of the existing system and proposed corresponding improvement measures for specific problems. The accurate publication of the government's daily disease data-although making people feel panicked about the increase in number of cases-reduces people's fear of the unknown because the transparency of this information is also crucial to reduce the spread of rumours.
Policy documents from the Interagency Standing Committee entitled ``Addressing Mental Health and Psychosocial Aspects of the COVID-19 Outbreak" have also been released. Wherein, this document summarizes the global definitions of mental health and psychosocial support and introduces mental health and psychosocial responses to COVID-19. The British Psychological Society released information handouts entitled ``Talking to Children about Coronavirus", aiming to provide parents with better guidance when talking about COVID-19 with children. The University of Reading and University of Oxford also gives advice to parents about how to reduce anxiety of children or young people about COVID-19.
Psychological issues during the epidemic can be caused by various reasons, including social distancing, the worldwide lockdown creating economic recession, social boycott, and discrimination, etc. Medical healthcare professionals and staff also suffer from stress, anxiety, and pressure, which also potentially cause psychological issues. To better offer psychological support and advice to front-line medical staff and the public, many organizations and institutions provided remote help. Oxford Centre for Anxiety Disorders and Trauma offers remote treatment of Post Traumatic Stress Disorder (PTSD) and social anxiety with cognitive therapy. Panic disorder can also be helped through remote delivery or teleworking. The British Psychological Society, Division of Clinical Psychology, Digital Healthcare Sub-Committee also provide therapy via video. A variety of websites have been developed to provide support as well. Examples include Support The Workers, COVID Trauma Response Working Group, Intensive Care Society: Wellbeing Resource Library, Coping With Coronavirus, Maintaining health and wellbeing during the COVID-19 pandemic, Psychosocial responses to COVID-19, Second Victim, Just Listening, American Psychological Association-Pandemics, etc.
\subsection{Enterprise Policies}
Significantly, most psychological intervention guiding measures have been taken up by hospitals and psychological counselling agencies. The decreed isolation policies also prevent enterprises from resuming work. Whether it is a private enterprise or a state-owned enterprise, employees are also isolated to their homes like most people. Due to the isolation policies, face-to-face communication is not possible, so hospitals and consulting agencies have jointly built many online consultation platforms~\cite {liu2020online}. Zhang et al.~\cite{Zhang2020} proposed a mental health intervention model, which uses an Internet technology platform to provide timely psychological counselling to patients and their families. Doctors need the courage to overcome disease meanwhile, they need to deal with certain extreme emotions of patients due to illness, who can be physically and mentally exhausted. Therefore, their psychological guidance is also a top priority. Chen et al.~\cite{chen2020mental} detailed the psychological guide measures for medical staff, including the establishment of a psychological intervention medical team, the establishment of a psychological assistance hotline team and psychological skills training.
\subsection{Education Policies}
To cope with the disease, some countries has explicitly closed schools. However, long-term suspension of classes at home will certainly have a serious of negative impacts on students' psychologies \cite{Brooks2020}. Children are more likely to feel pressure after isolation, which is four times higher than those who are not isolated \cite{Sprang2013Posttraumatic}. Research shows that people with higher education are more likely to feel uncomfortable because of their higher awareness of health \cite{Tessa2018Factors}. Cao et al. \cite{CAO2020112934} analysed the psychological status of college students during the new outbreak by questionnaires and found that they all had different degrees of anxiety.
In addition to researching effective measures for the treatment of COVID-19~\cite{jrfm13020036}, many universities also provide online one-to-one psychological counselling services. A public welfare project: ``Combating Epidemic Disease and Psychological Assistance" has been carried out by the jointly organization of many institutions. This provides free psychological assistance to the public and face-to-face psychological counselling services \cite{Tsinghua2020}. Shaanxi Normal University has also published the first national ``Anti-epidemic Psychological Guidance Manual", which is published in paper, electronic versions and audiobook formats. Some university professors even spontaneously organized and recorded videos of exercise at home, which can effectively regulate the physical and mental health of people at home, and help maintain social stability.
\subsection{Technological Innovation}
During the SARS outbreak, because the development of the Internet was not yet mature, and smartphones were not as popular as today. Online psychological counselling services were not available \cite{chan2007improving}. Today, with the prevalence of mobile phones, the development of Internet services, and the emergence of the 5G era, major medical platforms have provided online psychological counselling services, allowing people to conduct psychological counselling even if they are isolated at home. For example, the Structured Letter Therapy method \cite{articleXiao} and Health Intervention Model \cite{Zhang2020}. Liu et al. \cite{liu2020online} discussed the specific online service counselling during this period. Through psychological questionnaires and counselling, doctors can better understand one's tendencies and guide them in a timely manner through their psychological disorders. According to research, Artificial Intelligence (AI) technology \cite{liu2018artificial,xia2019random} can also help identify people with suicidal tendencies, which also provided a strong assistance for professionals supporting mental health \cite{Just2017}. The WHO and the CDC, to help people cope with their emotions arising from the epidemic of COVID-19, haved employed chat robots to communicate with people and provide emotional communication. Miner et al. \cite{Miner2020} also analysed the advantages and disadvantages of chat robots in mental health guidance, indicating future prevention work.
In general, with the gradual stabilization of the domestic epidemic situation, the psychological health guidance for the public also needs to be given priority consideration. Although government agencies have issued many related guidance programs and achieved initial results, there are still some deficiencies. At present, people are still facing many problems such as long-term shutdown, which makes many enterprises and individuals face the risk of bankruptcy and unemployment. All industries, including education, economy, and tourism, are bearing the consequences of the loss brought about by COVID-19. Domestic epidemic control is relatively stable, but foreign situations are still relatively severe. The most serious problem is that, according to the current research situation, the prevention and control of COVID-19 will be a protracted effort, which will coexist with human beings for a long time. The existence of these problems makes people feel unstable and originates countless psychological problems. Therefore, psychological health guidance is essential.
\section{Financial Aid}
\label{sec:4}
The WHO and the International Monetary Fund (IMF) have never been more important in supporting the global emergency response. Funding needs to be made available in tracking the spread of the virus, ensuring patients receive assistance and primary staff receive essential supplies and information \cite{wang2020combating}. All these decisions about financial support are based on the data of the affected communities and economies. The Governments, agencies, industries, markets, and individuals all have come together to fight the COVID-19 pandemic and to help respond to this global outbreak.
\subsection{Financial Aid for the Protection of Lives and Livelihood}
The world is facing an immense challenge, with the rising pandemic of COVID-19 impacting populations and economies everywhere. A tremendous amount of financial assistance is required to support the communities and economies of the world. Financial assistance should be provided on two priorities: 1) the protection of lives; and, 2) the protection of livelihoods. The protection of life means that countries should put health spending at the top of the priority list. This includes funding health systems receiving monies for doctors, nurses, and hospitals, purchasing medical equipment, and helping the most in need \cite{world2020considerations}. The protection of livelihoods means providing financial support for the provision of lifelines for households and companies during this time of economic problems. This includes cash incentives, wage increases, and tax cuts, helping people meet their needs and helping companies keep operating. For those who have been laid off, unemployment compensation may be temporarily improved by extending its length, increasing the benefits, or relaxing the requirements for eligibility. As far as monetary policy is concerned, providing sufficient liquidity to banks and non-bank finance companies, particularly those lending to small and medium-sized enterprises that may be less able to withstand severe disruption, is crucial at this point \cite{Mckibbin2020}. It is also reported that United Nations International Children's Emergency Fund (UNICEF) has launched an emergency response that will help provide food for children in the United Kingdom who are affected by COVID-19. It is reported that UNICEF will provide grants to 30 local organizations. For example, one of the organizations directly provided 18,000 nutritious breakfasts to 25 schools\footnote{\url{https://news.sky.com/story/covid-19-for-the-first-time-in-its-history-unicef-will-help-feed-kids-in-the-uk-12163515}}.
Though COVID-19 began as a health crisis, it has triggered a grave and unfolding economic crisis with unpredictable loss. People on low-income and those who barely survive on precarious livelihoods have suffered the hardest. According to the United Nations statistics, over 300 million children who rely on school meals might now be at risk of acute hunger. Additionally, social distancing might not work in urban slums and rural households since toilets are shared by multiple families. To make real-time financial response under such circumstances, we need not just appropriate isolation policies, but also normal lives. It is suggested that countries should let people continue their normal lives, that is, to work, earn money, and feed their families. Shop keepers should open and provide services with effective protection such as gloves and masks. Widespread and strict lockdowns might unintentionally lead to more deaths, which affects more the poor community. Otherwise, the global financial institutions will have to write off debts from low-income countries and provide enough resources for recovering economies~\cite{2020Has}.
\subsection{Financial Aid for Economic Recovery}
The global economic crisis triggered by COVID-19 has never before been seen in history, and there is significant concern as to its effect on people's lives and livelihoods \cite{bethune2020covid}. Most of the situations depends on the epidemiology of the virus and the effectiveness of control steps, all of which are hard to anticipate. In addition, several countries are now facing compounded crises - a health crisis, a financial crisis, and a fall in stock prices, all that interact in complex ways. The estimated loss to global
Gross Domestic Product (GDP) between 2020 and 2021 from the pandemic may be about \$9 trillion, more than the economies of Japan and Germany combined\footnote{\url{https://openknowledge.worldbank.org/handle/10986/33488}}.
A large amount of financial assistance is needed for the recovery of the economy worldwide. When the economic crisis is over, countries will face high rates of debt, bankruptcies, unemployment, and falling wages \cite{nicola2020socio}. The pace of economic recovery will depend on the policies pursued during this crisis. When policies ensure that employees will not lose their jobs, tenants and homeowners are not disadvantaged, businesses prevent bankruptcies, and market and trade networks are preserved, then recovery can take place faster and more smoothly \cite{fernandes2020economic}. In the case of countries that do not have a fiscal environment to implement these initiatives, it is possible that the IMF can assist these countries through its lending facilities. Support for emerging markets and developing countries is an important priority of the IMF \cite{loayza2020macroeconomic}. Developed countries are still more economically unstable than industrialized economies and these are now especially hard hit by a shortage of medical supplies, a sudden impact on the world economy, capital flight, and, for some, a dramatic drop in commodity prices.
The European Union has implemented many data-driven strategies to face the formidable challenge brought by COVID-19. Despite the closure of borders and limitations of mobility, financial support has been provided to find effective vaccines, promote treatment therapies, protect salaries, etc. According to statistics from 2,061 adults, more than 82\% respondents think that the government should first consider the health and wellbeing of citizens. About 61\% of respondents are in favour of promoting social and environmental outcomes when the COVID-19 pandemic is over. Based on the report published by the Office for National Statistics, the economic growth will be significantly affected. A report entitled ``Tragedy of Growth" published by Positive Money on Monday points out that more attention should be paid on social and environmental indicators to save lives and improve the environment. A series of policies that can promote wellbeing without first increasing GDP are also released in this report, including cancellations or reductions of household debt. It is inferred that the macroeconomic shocks of the COVID-19 pandemic include financial stability and risk. With the emergence of economic adversities, designing and implementing innovative policies are necessary for a long-term view.
\section{Work Resumption}
\label{sec:5}
To handle the high infectivity of COVID-19, many countries have experienced several months of quarantine, social distancing, and travel restrictions. Although these measures can be effective, there is a corresponding influence upon the economy. Governments in different countries must resume work to restore their economy affected by COVID-19~\cite{gentilini2020social,liu2020china}. However, work resumption is accompanied by the risk of epidemic rebound without social isolation, and the current long-term continuation of COVID-19 restrictions will lead to economic downturn and unpredictable social problems~\cite{mansilla2020manage}. Therefore, it is necessary to make appropriate decisions to balance the two contradictory forces of work resumption and epidemic rebound. In this section, we first discuss the recovery strategy in areas where the epidemic is under control. Then we introduce some related research on data-driven decision making for work resumption. Here, we give the analytics framework of work resumption from two perspectives, i.e., microscope and macroscope as shown in Fig.~\ref{fig0}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{1.pdf}
\caption{Data-driven analytics of work resumption.}
\label{fig0}
\end{figure}
Zhao et al.~\cite{zhao2020negligible} assessed the risks of COVID-19 rebounding caused by resumption of work. It was concluded that the probability of COVID-19 rebounding during resumption of work is very limited or even negligible. However, this work ignored the imported cases which are inconsistent with the current situation of COVID-19. Also, this work assumed that secondary attack rate of COVID-19 in enterprise clusters from an unidentified infected case is the same as that rate in family clusters, which is not correct for different countries~\cite{leung2020first,wang2020risk}.
Wang et al.~\cite{xia2020will} proposed a data-driven network modelling analysis of the work resumption. Epidemiological report data, and Baidu's population migration trends and distribution data are integrated to estimate the actual cumulative number of cases with the reproductive matrix method. The primary conclusions revealed that the risk of a second outbreak will be negligible under strong prevention and self-protection measures. This work aimed to assess the risk of recovery work from a micro perspective.
The work resumption of is related to the economic impact of COVID-19. Some studies aim to quantify the impact of COVID-19 to provide decision making supports for work resumption from an indirect perspective. Baidu has used the Baidu map data to quantify the economic impact of COVID-19~\cite{huang2020quantifying}. The results show that travel-dependent industries (e.g. hotels, public transportation, etc.) have not yet recovered compared to same period in the past. Financial support should be provided to these sectors to strengthen the recovery. On the other hand, the sectors that are essential to human life such as workplaces, restaurants, and shopping venues have been recovering rapidly during the current period of work resumption.
Work resumption for migrant workers, which occupies a great proportion of the labour force, should also be noticed. Due to the social isolation and travel restrictions, these workers fail to return to urban areas to work. Furthermore, these workers usually exist in the industrial sectors such as export-oriented industry, hotels, tourism, and the entertainment industry. Many governments have taken specific measures to avoid unemployment for this group. For example, local governments have enhanced infrastructure building in rural areas to provide job opportunities for migrant workers. In addition, many countries attempt to start the progress of work resumption to recover their economies to some extent. Gentilini et al.~\cite{gentilini2020social} pointed out that labour market intervention is an effective way for the government to support work in the formal and informal sectors. Many countries, including Bosnia, Herzegovina, Romania, etc., have considered activation measures (worker training) to prepare for work resumption.
A community detection algorithm is utilized to identify local job markets in~\cite{bonato2020mobile}. The important conclusion reveals that the reduced mobility caused by social isolation and travel restrictions force the establishment of smaller local job markets, which has some impacts on the future work resumption. Barbieri et al.~\cite{barbieri2020italian} utilized the Italian Sample Survey on Professions data to assess the risk of infection of workers in 600 sectors. It showed that social isolation has effectively reduced the possibility of workers at risk of contagion. This work aims to provide data-driven decision support for policy makers who plan to adopt work resumption strategies.
Williams and Kayaoglu~\cite{williams2020covid} evaluated the undeclared worker group without access to financial support affected by COVID-19 and provided the possible recommendations to the government for the worker group in Europe. They analysed data from 27,565 interviews in 28 European countries and point out that the government policy needs adjustment during the period of COVID-19. Greater financial support to undeclared workers should be established by European governments. Bailey and West~\cite{bailey2020covid19} analysed the mortality and work resumption. Their analysis showed that the strict restrictions lead a total number of deaths at least eight times fewer than those who immediately resumed work. The conclusion may provide some suggestions while considering the work resumption strategies.
Rio-Chanona et al.~\cite{del2020supply} analysed the supply and demand shocks of the COVID-19 pandemic, which indicates that transportation and other industries are more vulnerable to demand-side shocks and entertainment. Aside from these impacts, hotels and tourism are facing constraints from both supply and demand. Furthermore, the study finds that the high-wage occupations are relatively immune from both supply-side and demand-side shocks while many low-wage occupations are much more economically vulnerable to these two shocks. For policymakers, this study suggests two important implications. The first is that it is vitally important to resume work as soon as possible without increasing the risk of epidemic rebound. The second is related to the issue of fairness emphasized in this study. The negative impact of COVID-19 on higher income knowledge and service workers can be negligible, while lower income workers receive greater consequence from this epidemic.
\section{Re-opening of Schools}
\label{sec:6}
\par Isolation measures can effectively reduce the infection rate and the speed of transmission. Therefore, policy makers in schools should first pay attention to non-pharmacological interventions, including delaying school attendance. Furthermore, the purpose of these measures is to establish social segregation and isolation. In the context of the epidemic, many studies focus on building models to analyse the impact of epidemic transmission and school closures. For example, Bayham and Fenichel \cite{bayham2020impact} studied the impact of school closures on the medical system.
However, the epidemiological benefits come with economic damage. Lempel et al. \cite{lempel2009economic} estimated the direct impact on the economy and health care of closing schools for 2, 4, 6, and 12 weeks. They discovered that the economic cost of closing all schools in for 4 weeks might be between $\$$$1$ billion and $\$$$47$ billion (0.1$\%$-0.3$\%$ of GDP), and it reduces primary healthcare personnel by 6$\%$ to 19$\%$. Therefore, after analyzing the school closures, when to reopen schools has become an important social issue.
\subsection{Re-opening Strategies}
\par In the context of the SARS outbreak, schools in the severely affected areas also chose to close. Normal teaching progress at many schools had been disrupted. The specific length of closure depended on the situation of different schools. After the spread of the SARS epidemic had weakened, the school re-opening strategies were different according to diverse circumstances.
\begin{figure*}[!t]
\small
\centering
\includegraphics[width=\textwidth]{2.pdf}
\caption{Student study managed in batches depending on the priority.} \label{fig1}
\end{figure*}
When the status of COVID-19 transmission in some areas is under control, schools should adopted a similar strategy to SARS. Specifically, for students who have urgent tasks and examinations, they are regarded as the first batch of students back to school. The rest of students are divided into different batches according to the relative level of urgency. For example, most local governments stipulate that the graduates of high schools are the first batch of students, university graduates are the second batch of students, primary school students are the third batch of students, and non-graduates from university are the fourth batch of students.
The time of re-opening school for each batch of students is reasonably divided. When the previous batch of students returns to school completely, the next batch of students is permitted to start returning to school. The entire process is roughly illustrated in Fig.~\ref{fig1}, in which nucleic acid test (NAT) and isolation may be required. Moreover, students can voluntarily choose whether to resume school. Significant recovery strategies guarantee schools not gather large numbers of students at the same time and avoid large numbers of people moving nationwide.
Wang et al. \cite{wang2020risk} proposed some solutions for, teaching, medical treatment, and other aspects. Singh and Adhikari \cite{singh2020age} calculated the basic reproduction rate $R_{0}$ and its time-related changes. On this basis, they study the impact of the duration of social alienation measures (i.e., closed schools). In the end, they concluded that a short period of social isolation after the outbreak cannot effectively prevent a second outbreak. They recommended a continuous lock-in program that relaxes regularly. Lee et al. \cite{lee2010simulating} simulated school activities and showed the significant effectiveness of closing schools. They found that it may be necessary to maintain some type of school suspension for at least 8 weeks to affect the overall serological seizure rate. They believed that the relatively short school suspension time (i.e. 2 weeks or less) can increase the overall seizure rate by returning susceptible students to school in the middle of the epidemic. Based on this, they concluded that individual and short suspensions may not quell the epidemic, but if they are maintained for at least 8 weeks, the spread of the epidemic can at least be delayed by one week. This provides more time to implement additional interventions and make existing interventions more effective.
\par For improvement of the decision making and decision support of re-opening schools, Duan et al. \cite{duan2013acp} used the largest H1N1 influenza outbreak as an example to establish an artificial society to simulate the epidemic in universities. They further proposed the Artificial societies, Computational experiments, and Parallel execution (ACP) method to more effectively control the outbreak of H1N1 influenza. The ACP model can be effectively applied in the improvement of intervention strategies. Moreover, in the application of strategy improvement, they concluded that the ACP method is useful for public health emergency management. In addition, Popa \cite{popa2020decision} found that there are no automated tools to suggest which decisions to make. Based on this, they first designed an algorithm that abstracts decisions into a combinatorial optimization problem. Finally, they show the integer linear program formula of the problem.
As for the decision support for the re-opening of school, Araz et al. \cite{araz2011simulation} modeled the spread of pandemic influenza in local universities and evaluate university mitigation policies. Beaton et al. \cite{beaton2007pandemic} provided a table-top exercise to evaluate the plans and policies used by the University of Washington (UW) to deal with influenza pandemics. This work reveals gaps in university pandemic influenza plans and policies. Issues analysed in this work include quarantine, decision making for re-entry, mental health services, and tracking the travel of personnel. Finally, they provide policy and planning suggestions on these issues. Based on the trade-off between the cost and benefit of the intervention strategy, Cao et al. \cite{cao2014evaluating} also analysed the various decisions of the epidemic situation, and their research can also be used to support the relevant decision making for re-opening of schools.
\subsection{Online Education}
\par A large number of schools have used the Internet for remote learning to ensure the continuity of teaching during the epidemic. In related research in educational psychology, multitasking has a negative impact on students' academic performance. This theory has been effectively proven by investigating the self-efficacy of self-regulated learning (SESRL) \cite{zuffiano2013academic}. But in online education, SESRL eases the contradictory relationship between multitasking and academic performance \cite{alghamdi2020online}. The moderated mediation effect of self-efficacy is only found in online classrooms. Based on the above findings, research in \cite{alghamdi2020online} shows that when students with high SESRL levels study online, the negative impact of multitasking on the score is reduced. As for the public acceptance of online education, Samad and Khalid\cite{samad2019acceptance} survey 125 elementary school science teachers using a 10-item questionnaire and analyse the questionnaire data obtained. Their research results show that primary school science teachers have a high acceptance of using online learning. And based on an online survey of 607 valid responses gathered from 896 online respondents, Vate \cite{vate2020psychological} discovered that there is a significant positive correlation between students' attitudes towards online education and students' life satisfaction. However, there are still many problems in online education, and these problems pose challenges to the quality of learning, for example, academic integrity issues. In response to these issues, Ohio State University \cite{bane2019academic} has adopted a dual approach to ensure the academic integrity of its online education products. However, to meet the needs of the changing educational situation, a complete solution is still needed.
\section{Challenges and Open Issues}
\label{sec:7}
Many strict policies have been taken to contain the COVID-19 pandemic, whereas the impact of the enforcement and subsequent loosening of these policies have not been very well analysed or understood. Assumptions, premises, and the context are necessary in the decision making process. All of these are required to be verified through data in data-driven policymaking. Data-driven research can better guide policymaking if the data are appropriately analysed. It is generally believed that data-based decision making will lead to better decisions compared to decisions based on observation or informed guesswork. However, data-driven policy making during COVID-19 is quite special. For one thing, traditional data-driven policy making is extrapolated from key data sets that contain historical data. Nevertheless, we have no historical data for COVID-19, which will definitely restrain the ability and adaptability of decision making.
How can we best make decisions with regard to available data? It appears that consistent, comparable, and traceable data are becoming more important under these circumstances. Various heterogeneous data sources can provide abundant data but bring about privacy problems at the same time. Further study is needed about how to mine these data and develop efficient policies that can meanwhile protect public privacy as much as possible. Also, how to evaluate and balance the importance of different measurements in relation to data-driven policy making. Since not enough historical data can be employed, there is the possibility of using intuition-driven decisions. Here, we list three fundamental challenges and open issues that are considered important to tackle first.
\textbf{Data Collection and Quality: }Effective data collection and high-quality data are decisive to data processing. Data-driven decision making is based on both well-collected and well-processed data. The quality of data is extremely significant, especially when data is used to guide decision making such as financial aid, resumption of work or school, etc. Data quality is generally evaluated by accuracy, precision, correctness, timeliness, etc. It is significantly difficult to collect data and at the same time ensure the quality of data. Current methods and tools can be applied to collect COVID-19 data, but COVID-19 data will also need some specialized methods to process the data to ensure quality. Therefore, further work should focus on how to collect COVID-19 data effectively and at the same time ensure the quality of data.
\textbf{Complex Data Analysis: }The rapid spread of COVID-19 has brought about diverse and abundant data, which are represented in nearly all areas of life. Under these exceptional circumstances, all the relevant data should be integrated to guide COVID-19 decision making. However, COVID-19 related data include basic information, transportation data, diagnosis data, scientific research data, and other kinds of data from multiple-sources, which makes it especially difficult to process. High dynamism, heterogeneity, as well as unpredictability are also properties of COVID-19 data. Analyzing such data requires both hardware equipment and highly efficient algorithms, and currently there are no such specialized ones. Thus, further studies should pay additional attention to data analysis of highly dynamic, heterogeneous, multi-sources, as well as unpredictable sources, including new processing methods and analytic tools.
\textbf{Fairness of Decision Making: }Fairness is always one of the most important problems in decision making. For some time, decision making has been believed to be more reliable when data driven rather than human based. However, even well designed and correctly implemented algorithms may still make decisions with prejudice. Such prejudice can be reduced by improving the fairness and interpretability of machine learning algorithms. It is generally acknowledged that the judgement of humans is also an important perspective to ensure fairness of decision making. Therefore, there is a need for more research when implementing data-driven decision making generally, and not just limited to the COVID-19 response.
\section{Conclusion}
\label{sec:8}
Data-driven decision making has been demonstrated to be both important and effective for the COVID-19 response. Different countries and regions have implemented many policies that are data-driven, such as prevention and control policies, psychological counselling policies, financial aid policies, and resumption/re-opening policies. In this paper we have summarized these policies including examples from all around the world. From different kinds of data related to COVID-19, policy makers have developed large volumes of useful information, and based on this information and machine learning algorithms, data are guiding us to better and more appropriate choices. We first discussed COVID-19 related data and some prevention and control policies and then described the psychological counselling policies driven by COVID-19 data. In the financial aid section, we have discussed the protection of lives as well as economic recovery. Following on, we introduced in detail policies in the resumption of both work and school. Finally, we listed the current challenges and open issues, including data collection and quality, complex data analysis, and fairness of decision making.
\bibliographystyle{ACM-Reference-Format}
|
1,108,101,565,624 | arxiv | \section{Preliminaries}
The configuration space $\Ga :=\Ga _{\R^d}$ over $\R^d$,
$d\in\N$, is defined as the set of all locally finite subsets of
$\R^d$,
\begin{equation}
\Ga :=\left\{ \ga \subset \R^d:\left| \ga_\La\right|
<\infty \text{ for every compact }\La\subset \R^d\right\} ,
\end{equation}
where $\left| \cdot \right|$ denotes the cardinality of a set and
$\ga_\La := \ga \cap \La$. As usual we identify each
$\ga \in \Ga $ with the non-negative Radon measure $\sum_{x\in
\ga }\delta_x\in \mathcal{M}(\R^d)$, where $\delta_x$ is
the Dirac measure with unit mass at $x$,
$\sum_{x\in\varnothing}\delta_x$ is, by definition, the zero measure,
and $\mathcal{M}(\R^d)$ denotes the space of all
non-negative Radon measures on the Borel $\sigma$-algebra
$\mathcal{B}(\R^d)$. This identification allows to endow
$\Ga $ with the topology induced by the vague topology on
$\mathcal{M}(\R^d)$, i.e., the weakest topology on $\Ga$
with respect to which all mappings
\begin{equation*}
\Ga \ni \ga \longmapsto \langle f,\ga\rangle :=
\int_{\R^d}f(x)d\ga(x)=\sum_{x\in \ga }f(x),\quad
f\in C_0(\R^d),
\end{equation*}
are continuous. Here $C_0(\R^d)$ denotes the set of all continuous functions
on $\R^d$ with compact support. We denote by $\mathcal{B}(\Ga )$ the
corresponding Borel $\sigma$-algebra on $\Ga$.
Let us now consider the space of finite configurations
\begin{equation*}
\Ga_0 := \bigsqcup_{n=0}^\infty \Ga^{(n)},
\end{equation*}
where $\Ga^{(n)} := \Ga^{(n)}_{\R^d} := \{ \ga\in \Ga:
\vert \ga\vert = n\}$ for $n\in \N$ and $\Ga^{(0)} :=
\{\varnothing\}$. For $n\in \N$, there is a natural bijection between
the space $\Ga^{(n)}$ and the symmetrization
$\widetilde{(\R^d)^n}\diagup S_n$ of the set $\widetilde{(\R^d)^n}:=
\{(x_1,...,x_n)\in (\R^d)^n: x_i\not= x_j \hbox{ if } i\not= j\}$
under the permutation group $S_n$ over $\{1,...,n\}$ acting on
$\widetilde{(\R^d)^n}$ by permuting the coordinate indexes. This
bijection induces a metrizable topology on $\Ga^{(n)}$, and we
endow $\Ga_0$ with the topology of disjoint union of topological
spaces. By $\mathcal{B}(\Ga^{(n)})$ and $\mathcal{B}(\Ga_0)$
we denote the corresponding Borel $\sigma$-algebras on
$\Ga^{(n)}$ and $\Ga_0$, respectively.
Given a constant $z>0$, let $\la_z$ be the
Lebesgue-Poisson measure
$$
\la_z:=\sum_{n=0}^\infty \frac{z^n}{n!} m^{(n)},
$$
where each $m^{(n)}$, $n\in \N$, is the image measure on $\Gamma^{(n)}$ of
the product measure $dx_1...dx_n$ under the mapping
$\widetilde{(\R^d)^n}\ni (x_1,...,x_n)\mapsto\{x_1,...,x_n\}\in \Gamma^{(n)}$.
For $n=0$ we set $m^{(0)}(\{\varnothing\}):=1$.
We proceed to consider the $K$-transform \cite{Le73}, \cite{Le75a},
\cite{Le75b}, \cite{KoKu99}, that is, a mapping which maps functions
defined on $\Ga_0$ into functions defined on the space $\Ga$. Let
$\mathcal{B}_c(\R^d)$ denote the set of all bounded Borel sets in
$\R^d$, and for any $\La\in \mathcal{B}_c(\R^d)$ let $\Ga_\La :=
\{\eta\in \Ga: \eta\subset \La\}$. Evidently $\Ga_\La =
\bigsqcup_{n=0}^\infty \Ga_\La^{(n)}$, where $\Ga_\La^{(n)}:=
\Ga_\La \cap \Ga^{(n)}$ for each $n\in \N_0$, leading to a situation
similar to the one for $\Ga_0$, described above. We endow $\Ga_\La$
with the topology of the disjoint union of topological spaces and
with the corresponding Borel $\sigma$-algebra
$\mathcal{B}(\Ga_\La)$.
Given a $\mathcal{B}(\Ga_0)$-measurable function $G$ with local
support, that is, $G\!\!\upharpoonright
_{\Ga\setminus\Ga_\La}\equiv 0$ for some $\La \in
\mathcal{B}_c(\R^d)$, the $K$-transform of $G$ is a mapping
$KG:\Ga\to\R$ defined at each $\ga\in\Ga$ by
\begin{equation}
(KG)(\ga ):=\sum_{\eta \Subset \ga }
G(\eta ),
\label{Eq2.9}
\end{equation}
where $\eta\Subset\ga$ means that $\eta\subset\ga$ and $\vert\eta\vert < \infty$. Note that for every such function $G$ the sum in (\ref{Eq2.9}) has only a
finite number of summands different from zero, and thus $KG$ is a well-defined
function on $\Ga$. Moreover, if $G$ has support described as before, then
the restriction $(KG)\!\!\upharpoonright _{\Ga _\La }$ is a
$\mathcal{B}(\Ga_\La)$-measurable function and
$(KG)(\ga)=(KG)\!\!\upharpoonright _{\Ga _\La }\!\!(\ga_\La)$
for all $\ga\in\Ga$, i.e., $KG$ is a cylinder function.
Let now $G$ be a bounded $\mathcal{B}(\Ga_0)$-measurable function
with bounded support, that is, $G\!\!\upharpoonright _{\Ga
_0\backslash \left(\bigsqcup_{n=0}^N\Ga _\La ^{(n)}\right)
}\equiv 0$ for some $N\in\N_0, \La \in \mathcal{B}_c(\R^d)$. In
this situation, for each $C\geq \vert G\vert$ one finds $\vert
(KG)(\ga)\vert\leq C(1+\vert\ga_\La\vert)^N$ for all
$\ga\in\Ga$. As a result, besides the cylindricity property,
$KG$ is also polynomially bounded. In the sequel we denote the space
of all bounded $\mathcal{B}(\Ga_0)$-measurable functions with
bounded support by $B_{bs}(\Ga_0)$. It has been shown in
\cite{KoKu99} that the $K$-transform is a linear isomorphism which
inverse mapping is defined on cylinder functions by
\begin{equation}
\left( K^{-1}F\right) (\eta ):=\sum_{\xi \subset \eta }(-1)^{|\eta
\backslash \xi |}F(\xi ),\quad \eta \in \Ga _0.
\end{equation}
\section{The description of problem and main results}
\subsection{Basic facts and notations}
Two-component contact process in $\R^d$ describes
a birth-and-death stochastic dynamics of
a infinite system of two type particles.
Such system may be interpreted as pair of
configurations in $\R^d$ as well as one configuration of marked particles
that means that each particle has mark (spin) $+1$ or $-1$. The first interpretation sometimes
is more useful but we should additionally
assume that these two configurations don't
interact.
Let us give the rigorous definitions. Consider
two copies of the space $\Ga$: $\Ga^+$
and $\Ga^-$. Let
\begin{equation}
\Ga^2:=\Big\{(\ga^+,\ga^-)\in\Ga^+\times\Ga^-
: \ga^+\cap\ga^-=\varnothing\Big\}.
\end{equation}
Any configuration $\ga:=(\ga^+,\ga^-)\in\Ga^2$
may be identified with marked configuration
\begin{equation*}
\hat{\ga}= \big\{(x,\sigma_x) : x\in\ga^+\cup\ga^-,
\sigma_x=\1_{x\in\ga^+}-\1_{x\in\ga^-}\big\}\in\hat{\Ga},
\end{equation*}
since $\ga^+\sqcup\ga^-\in\Ga$. Here $\hat{\Ga}$
is the space of all marked configurations
in $\R^d$ with marks equal to $\pm 1$. One can induce topology on $\Ga^2$
from the weakest topology on $\hat{\Ga}$ such that all functions
\begin{equation*}
\hat{\Ga}\ni\hat{\ga}\longmapsto \sum_{(x,\sigma_x)\in\hat{\ga}}\hat{
f}((x,\sigma_x))\in\R
\end{equation*}
are continuous for all $\hat{f}\in C_0(\R^d\times\{-1;1\})$.
Clearly, in this induced topology on $\Ga^2$
all functions
\begin{equation*}
\Ga^2\ni\ga=(\ga^+,\ga^-)\longmapsto \sum_{x\in\ga^+}f(x)
+ \sum_{y\in\ga^-}g(y)\in\R
\end{equation*}
will be continuous for any $f,g\in C_0(\R^d)$.
On the other hand this topology may be induced from the topology on product
$\Ga^+\times\Ga^-$. Let $\B(\Ga^2):=\B(\Ga^+)\times\B(\Ga^-)$ be the~corresponding $\sigma$-algebra.
Let us now consider the space of finite configurations. Consider
two copies of the space $\Ga_0$: $\Ga^+_0$
and $\Ga^-_0$. Let
\begin{equation}
\Ga^2_0:=\Big\{(\eta^+,\eta^-)\in\Ga^+_0\times\Ga^-_0 : \eta^+\cap\eta^-=\varnothing\Big\}.
\end{equation}
Again one can consider the topology on $\Ga_0^2$ induced by the product-topology. By $\B(\Ga^2_0):=\B(\Ga^+_0)\times\B(\Ga^-_0)$ we denote
the~corresponding $\sigma$-algebra.
We will say that a function $G:\Ga_0^2\to\R$ is a bounded function
with bounded support if for any $(\eta^+,\eta^-)\in\Ga_0^2$
\begin{equation*}
G(\cdot,\eta^-)\in B_{bs}(\Ga_0^+), \quad
G(\eta^+,\cdot)\in B_{bs}(\Ga_0^-).
\end{equation*}
Class of all such functions we denote by $B_{bs}(\Ga_0^2)$.
For any $G\in B_{bs}(\Ga_0^2)$ one can define the $\K$-transform of $G$ as mapping $\K G:\Ga^2\to\R$ defined at each $\ga=(\ga^+,\ga^-)\in\Ga^2$ by
\begin{equation}
(\K G)(\ga)=\sum_{\substack{\eta^+\Subset\ga^+ \\ \eta^-\Subset\ga^-}}
G(\eta^+,\eta^-).
\end{equation}
On the other hand if $\1^\pm$ are unit operators on functions on $\Ga^\pm_0$
and $K^+:=K\otimes\1^-$, $K^-:=\1^+\otimes K$ then
\begin{equation*}
\K=K^+K^-=K^-K^+.
\end{equation*}
Hence, $\K G <\infty$ and $\K G$ is cylinder function on both variables.
Moreover,
$\K G$ is polynomially bounded: for the proper $C>0$, $\La\in\B_c(\R^d)$, $N\in\N$
\begin{equation*}
\vert (\K G)(\ga) \vert \leq C (1+\vert \ga^+_\La \vert)^N (1+\vert \ga^-_\La \vert)^N.
\end{equation*}
The inverse mapping is defined
on cylinder (on both variables) functions by
\begin{equation}
(\K^{-1} F)(\eta):= \sum_{\substack{\xi^+\subset\eta^+ \\ \xi^-\subset\eta^-}}
(-1)^{\vert \eta^+\setminus\xi^+\vert + \vert \eta^-\setminus\xi^-\vert}F(\xi^+,\xi^-), \quad \eta=(\eta^+,\eta^-)\in
\Ga^2_0.
\end{equation}
Let $\mu$ be a probability measure on
$\Big(\Ga^2,\B\bigl(\Ga^2\bigr)\Big)$ (we denote class of the all such measures by $\M^1\bigl(\Ga^2\bigr)$).
The function $k_\mu:\Ga^2_0\to\R$ is called a~{\it correlation
function} of~the~measure $\mu$ if for~any $G\in B_{bs}(\Ga^2_0)$
\begin{equation}
\int_{\Ga^2} (\K G)(\ga) d\mu(\ga) =
\int_{\Ga^2_0} G(\eta^+,\eta^-) k_\mu(\eta^+,\eta^-)
d\la_1(\eta^+) d\la_1(\eta^-).
\end{equation}
\subsection{Description of model}
Let us consider the generator $L$ of two-component contact process with one independent component. This generator is well-defined at least on cylindric
functions on $\Ga^2$ and has the~following form:
\begin{equation}
L=\LC ^{+}+\LC ^{-}+\Lint ^{+}.
\end{equation}
Here $\LC ^{+}$ is the generator of the one-component contact model of
($+$)-system, $\LC ^{-}$ is the analogous generator of ($-$)-system, $\Lint ^{+}$ is interaction term that describes birth of ($+$%
)-particles under influence of ($-$)-particles. Namely,
\begin{align*}
\left( \LC ^{+} F\right) ( \ga ^{+},\ga ^{-}) &=\sum_{x\in \ga
^{+}}\left[ F\left( \ga ^{+}\setminus x,\ga ^{-}\right)
-F( \ga ^{+},\ga ^{-}) \right] \\
&\quad+\la ^{+}\int_{\R^{d}}\left( \sum_{x' \in \ga ^{+}}a^{+}\left(
x-x' \right) \right) \left[ F\left(
\ga ^{+}\cup x,\ga ^{-}\right) -F( \ga ^{+},\ga ^{-}) \right] dx,\\
\left( \LC ^{-} F\right) ( \ga ^{+},\ga ^{-}) &=\sum_{y\in \ga
^{-}}\left[ F\left( \ga ^{+},\ga ^{-}\setminus y\right)
-F( \ga ^{+},\ga ^{-}) \right] \\
&\quad +\la ^{-}\int_{\R^{d}}\left( \sum_{y' \in \ga
^{-}}a^{-}\left( y-y' \right) \right) \left[ F\left(
\ga ^{+},\ga ^{-}\cup y\right) -F( \ga ^{+},\ga ^{-}) \right] dy,\\
\left( \Lint ^{+} F\right) ( \ga ^{+},\ga ^{-}) &=\la
\int_{\R^{d}}\left( \sum_{y\in \ga ^{-}}a\left( x-y\right) \right)
\left[ F\left( \ga ^{+}\cup x,\ga ^{-}\right) -F( \ga ^{+},\ga ^{-})
\right] dx.
\end{align*}
Constants $\la^+,\la^-,\la$ are positive, functions $a^+,a^-,a$ are
non-negative, even, integrable and normalised:
\begin{equation*}
\langle a^{+}\rangle = \langle a^{-}\rangle = \langle a \rangle = 1.
\end{equation*}
Here and in the sequel we use the following notation
\begin{equation*}
\langle f \rangle :=\int_{\R^{d}}f(x)dx, \quad f\in L^1(\R^d).
\end{equation*}
We also denote the Fourier
transform of such $f$ as $\hat{f}$:
\begin{equation*}
\hat{f}(p)=\int_\X e^{-i(p,x)}f(x)dx,
\end{equation*}
where $(\cdot,\cdot)$ is a scalar product in $\X$.
Next theorem is the partial case of the results obtained in \cite{FKS}.
\begin{theorem}
Let $d\geq 2$ and there exists constants
$A>0, \delta>2d$ such that
\begin{equation}\label{decayofa}
a^+(x) + a^-(x) + a(x) \leq \frac{A}{(1+\vert x \vert)^\delta}.
\end{equation}
Then there exists a Markov process $X_t$ on $\Ga^2$ with generator $L$.
\end{theorem}
We will always suppose also that
\begin{equation}\label{intoffour}
\hat{a}, \hat{a}^+, \hat{a}^- \in L^1(\R^d).
\end{equation}
Hence, one has stochastic dynamics of configurations that implies
dynamics of measures, namely $\M^1\bigl(\Ga^2\bigr)\ni\mu_0 \mapsto
\mu_t\in\M^1\bigl(\Ga^2\bigr)$ such that
for any measurable bounded $F:\Ga^2\rightarrow\R$
\begin{equation*}
\int_{\Ga^2}F(\ga)d\mu_t(\ga):= \E \Biggl[\int_{\Ga^2}F(X_t^\ga)d\mu_0(\ga)\Biggr],
\end{equation*}
where process $X_t$ starts from $\ga\in\Ga^2$ (more precisely, $\ga$
belongs to proper support set, see \cite{FKS}).
This dynamics of measures implies dynamics of corresponding
correlation functions (if they exist). For obtain explicit
differential equations for this dynamics we should calculate so-called
descent operator $\hat{L}$ which defined on functions
$G\in B_{bs}(\Ga^2_0)$ by
\begin{equation}
\bigl( \hat{L}G \bigr) ( \eta ) = \bigl( (\K^{-1} L \K) G \bigr) ( \eta ),
\quad \eta\in\Ga^2_0.
\end{equation}
Next we should obtain the adjoint operator $\hat{L}^{\ast}$ (with respect
to measure $d\la_1d\la_1$):
\begin{multline}\label{adjoint}
\int_{\Ga _{0}^{2}}\hat{L}G ( \eta ^{+},\eta ^{-})
k ( \eta ^{+},\eta ^{-}) d\la_1( \eta
^{+}) d\la _1( \eta ^{-}) \\=\int_{\Ga
_{0}^{2}}G ( \eta ^{+},\eta ^{-}) \hat{L}^{\ast } k( \eta
^{+},\eta ^{-}) d\la_1 ( \eta ^{+} ) d\la_1
( \eta ^{-} ) .
\end{multline}
Then equations for time evolution of correlation function
will be following:
\begin{equation}\label{geneq}
\frac{\partial k_{t} ( \eta ^{+},\eta ^{-} ) }{\partial t}=\bigl(
\hat{L}^{\ast }k_{t}\bigr) ( \eta ^{+},\eta ^{-} ) .
\end{equation}
In the present article we concentrate our attention
on the correlation functions of the first and second orders:
\begin{equation}\label{cf12}
\begin{aligned}
k_t^+(x)&:=k_t(\{x\},\varnothing), &\quad x&\in\R^d;\\
k_t^-(y)&:=k_t(\varnothing,\{y\}), &\quad y&\in\R^d;\\
k_t^{++}(x_1,x_2)&:=k_t(\{x_1, x_2\},\varnothing), &\quad x_1, x_2&\in\R^d;\\
k_t^{+-}(x,y)&:=k_t(\{x\},\{y\}), &\quad x, y&\in\R^d;\\
k_t^{--}(y_1,y_2)&:=k_t(\varnothing, \{y_1, y_2\}), &\quad y_1, y_2&\in\R^d.
\end{aligned}
\end{equation}
The main subject for our studying will be explicit
expression for correlation functions of the first and second orders
and their asymptotic at $t\to\infty$.
\subsection{Problems and results}
In this subsection we state main problems and formulate results.
All proofs are presented in the next section.
First two results give explicit forms of~the~equation~\eqref{geneq}
for the~first and second order correlation functions~\eqref{cf12}.
\begin{proposition}\label{propeq1}
For any $x,y\in\R^d$
\begin{align*}
\frac{\partial k_{t}^{-}(y)}{\partial t} &=-k_{t}^{-}(y)+\la
^{-}\int_{\R^{d}}a^{-}(y-y' )k_{t}^{-}(y' )dy' ,
\\
\frac{\partial k_{t}^{+}(x)}{\partial t} &=-k_{t}^{+}(x)+\la
^{+}\int_{\R^{d}}a^{+}(x-x' )k_{t}^{+}(x' )dx' +\la
\int_{\R^{d}}a(x-y)k_{t}^{-}(y)dy
\end{align*}%
\end{proposition}
\begin{proposition}\label{propeq2}
For any $x,y,x_1,x_2,y_1,y_2\in\R^d$
\begin{align*}
\frac{\partial k_{t}^{--}(y_{1},y_{2})}{\partial t} &= \la
^{-}\int_{\R^{d}}a^{-}(y_{2}-y' )k_{t}^{--}(y_{1},y' )dy' +\la
^{-}\int_{\R^{d}}a^{-}(y_{1}-y' )k_{t}^{--}(y_{2},y' )dy' \\
&\quad -2k_{t}^{--}(y_{1},y_{2}) +\la ^{-}a^{-}(y_{1}-y_{2})[k_{t}^{-}(y_{1})+k_{t}^{-}(y_{2})], \\
\frac{\partial k_{t}^{+-}(x,y)}{\partial t} &=\la
^{+}\int_{\R^{d}}a^{+}(x-x' )k_{t}^{+-}(x' ,y)dx' +\la
^{-}\int_{\R^{d}}a^{-}(y-y' )k_{t}^{+-}(x,y' )dy' \\
&\quad -2k_{t}^{+-}(x,y) +\la a(x-y)k_{t}^{-}(y) + \la
\int_{\R^{d}}a(x-y' )k_{t}^{--}(y,y' )dy' , \\
\frac{\partial k_{t}^{++}(x_{1},x_{2})}{\partial t} &=\la
^{+}\int_{\R^{d}}a^{+}(x_{1}-x' )k_{t}^{++}(x_{2},x' )dx' +\la
^{+}\int_{\R^{d}}a^{+}(x_{2}-x' )k_{t}^{++}(x_{1},x' )dx' \\
&\quad -2k_{t}^{++}(x_{1},x_{2}) +\la ^{+}a^{+}(x_{1}-x_{2})[k_{t}^{+}(x_{1})+k_{t}^{+}(x_{2})] \\
&\quad+\la \int_{\R^{d}}a(x_{1}-y)k_{t}^{+-}(x_{2},y)dy +\la
\int_{\R^{d}}a(x_{2}-y)k_{t}^{+-}(x_{1},y)dy .
\end{align*}%
\end{proposition}
Obviously, equations for ($-$)-system are independent. Recall that
such equations were studied in \cite{KoKuPi07}.
Let us formulate the main problem
for the first order correlation functions.
\begin{problem}
We should to study the asymptotic properties of the solutions
of equations from Proposition~\ref{propeq1} under following
initial conditions:
\begin{equation}\label{initval1}
k_{0}^{+}(x) =c^{+}+\psi ^{+}(x)\geq 0, \qquad
k_{0}^{-}(y) =c^{-}+\psi ^{-}(y)\geq \alpha^->0,
\end{equation}
where constants $c^{+}, c^{-}$ are positive, functions $\psi ^{+}, \psi ^{-}$ and their Fourier
transforms $\hat{\psi}^{+}, \hat{\psi}^{-}$ are
integrable on $\R^{d}$.
\end{problem}
Explicit expressions for solutions are in the next section.
The answer of the Problem~1 may be found in the next theorem.
\begin{theorem}\label{thm_as_1}
Let $d\geq 3$ and \eqref{decayofa}, \eqref{intoffour} hold. The
first correlation functions have the following asymptotic at $t\to
\infty$: \step{1} for any $y\in\R^d$
\begin{equation*}
k_{t}^{-}(y)\rightarrow \left\{
\begin{array}{@{\,}r@{\quad}l@{}}
0, & \text{if } \ \la ^{-}<1 \\
\infty , & \text{if } \ \la ^{-}>1
\end{array}
\right. ,
\end{equation*}
and in the case $\la ^{-}=1$
\begin{equation*}
k_{t}^{-}(y)\rightarrow c^{-};
\end{equation*}
\step{2} for any $x\in\R^d$
\begin{equation*}
k_{t}^{+}(x)\rightarrow \left\{
\begin{array}{@{\,}r@{\quad}l@{}}
0, & \text{if } \ \max \{\la ^{+},\la ^{-}\}<1 \\
\infty , & \text{if } \ \min \{\la ^{+},\la ^{-}\}\geq 1%
\end{array}%
\right. ,
\end{equation*}
next, in the case $1=\la ^{+}>\la ^{-}$
\begin{equation*}
k_{t}^{+}(x)\rightarrow c^{+}+\frac{\la c^{-}}{1-\la ^{-}},
\end{equation*}
and in the case $\la ^{+}<\la ^{-}=1$
\begin{equation*}
k_{t}^{+}(x)\rightarrow \frac{\la c^{-}}{1-\la ^{+}}.
\end{equation*}
\end{theorem}
Let us discuss this result. Of course, first part about the
independent $(-)$-system is the same as in \cite{KoKuPi07, KoSk06}.
It state that $\la^-=1$ is critical value; below of this value
$(-)$-system will degenerate at infinity, above of this value
$(-)$-system will grow (exponentially, see next section for
details). At this critical value $(-)$-system continues to be
stable.
$(+)$-system consists of two parts: independent
contact and influence from the side of~
$(-)$-system. If~$\max \{\la ^{+},\la ^{-}\}<1$
it means that independent part of $(+)$-system is sub-critical
(and~should disappear at infinity)
and additionally it has influence of disappearing
$(-)$-system; naturally, such $(+)$-system will
disappear. If $\min \{\la ^{+},\la ^{-}\}\geq
1$ it means that growing or stable independent part of $(+)$-system has
influence by stable or growing \mbox{$(-)$-system}, hence,
$(+)$-system will grow.
Let us concentrate our attention on two other
cases. If $\la ^{+}=1, \la ^{-}<1$ it means
that independent part of $(+)$-system is
stable and has influence by degenerating
$(-)$-system. As a result, $(+)$-system will
keep stability property but the limiting
value will have the initial value
of $(-)$-system which will disappearing at
infinity. Hence, $(+)$-system will have memory
about vanished $(-)$-system.
If $\la ^{+}<1, \la ^{-}=1$ it means that
degenerating independent part of $(+)$-system
has influence by stable $(-)$-system. In
result, $(+)$-system will stop disappearing
and become stable. But ``fare'' for this will
be absence of the initial value of
$(+)$-system in limit. Therefore, $(+)$-system
``will lost memory'' about its origin and ``remember''
only about origin of ``donor''.
In studying asymptotic of the second correlation
functions we concentrate our attention only
on this two cases when $(+)$-system will
be stable. For simplicity of computations
we consider translation invariant case only:
\begin{equation}\label{tr_inv}
\psi ^{+}=\psi ^{-}\equiv 0.
\end{equation}
\begin{problem}
We should to study the asymptotic properties of the solutions of equations from Proposition~\ref{propeq2}
under following initial conditions:
\begin{equation}\label{initval2}
\begin{aligned}
k_{0}^{++}(x_{1},x_{2}) &=c^{++}+\varphi ^{++}(x_{1}-x_{2})\geq
0, \\
k_{0}^{+-}(x,y) &=c^{+-}+\varphi ^{+-}(x-y)\geq 0, \\
k_{0}^{--}(y_{1},y_{2}) &=c^{--}+\varphi ^{--}(y_{1}-y_{2})\geq 0,
\end{aligned}
\end{equation}
where $c^{--},~c^{+-},~c^{++}$ are positive
constants and
and functions $\varphi ^{--},~\varphi ^{+-},~\varphi ^{++}$ are
even functions which are integrable on $\R^{d}$ together with
their Fourier transforms
$\hat{\varphi}^{--},~\hat{\varphi}^{+-},~\hat{\varphi}^{++}$.
\end{problem}
Explicit expressions for solutions are also in the next section.
The answer of the Problem~2 may be found in the next theorem.
\begin{theorem} \label{thm_as_2} Let $d\geq 3$ and \eqref{decayofa}, \eqref{intoffour},
\eqref{tr_inv} hold. The second correlation functions have the
following asymptotic at $t\to \infty$: \step{1} let $\la
^{+}=1,~0<\la ^{-}<1$, then for any $x,y,x_1,x_2,y_1,y_2\in\R^d$
\begin{equation*}
\left\{
\begin{aligned}
k_{t}^{--}(y_{1},y_{2})&\rightarrow0, \\
k_{t}^{+-}(x,y)&\rightarrow0, \\
k_{t}^{++}(x_{1},x_{2})&\rightarrow \left( c^{++}-\frac{%
2\la c^{+-}}{\la ^{-}-1}+\frac{\la ^{2}c^{--}}{(\la
^{-}-1)^{2}}\right) +\Omega ^{++}(x_{1}-x_{2})<\infty;
\end{aligned}%
\right.
\end{equation*}
\step{2} let $\la ^{-}=1,~0<\la ^{+}<1$,
then for any $x,y,x_1,x_2,y_1,y_2\in\R^d$
\begin{equation*}
\left\{
\begin{aligned}
k_{t}^{--}(y_{1},y_{2})&\rightarrow c^{--}+\Xi
^{--}(y_{1}-y_{2})<\infty, \\
k_{t}^{+-}(x,y)&\rightarrow\frac{\la c^{--}}{1-\la
^{+}}+\Xi ^{+-}(x-y)<\infty, \\
k_{t}^{++}(x_{1},x_{2})&\rightarrow\frac{\la ^{2}c^{--}}{%
(1-\la ^{+})^{2}}+\Xi ^{++}(x_{1}-x_{2})<\infty;
\end{aligned}
\right.
\end{equation*}%
here functions $\Xi ^{--},\Xi ^{+-},\Xi ^{++}$ depend on initial
value $c^{-}$ only and function $\Omega ^{++}$ depends on initial
value $c^{+}$ only
(of course,
they also depend on $\la,\la^\pm, a, a^\pm$).
\end{theorem}
The explicit expressions for limits will be presented in the next
section.
As we see, the situation with ``memory'' which we had for the first
correlation functions is the same for the second one: in the first
case $(+)$-system will obtain additional memory about vanished
$(-)$-system; in the second case $(+)$-system will have memory about
$(-)$-system only.
\begin{remark}
Note that if $c^{++}=(c^+)^2$, $c^{+-}=c^+c^-$,
$c^{--}=(c^-)^2$ then the previous theorems
show, in fact, that there exist
finite limits of so-called second order Ursell
functions $k_t^{++}-(k_t^{+})^2$, $k_t^{+-}-k_t^{+}k_t^{-}$,
$k_t^{--}-(k_t^{-})^2$.
\end{remark}
\section{Proofs}
In this section we present proofs of all
our results.
\subsection{Equations for time evolution of the correlation functions}
First of all we show how to obtain the equations from the Propositions~\ref{propeq1} and \ref{propeq2}.
We start from the explicit form of the descent
operator $\hat{L}$.
\begin{proposition}
Let $G\in B_{bs}(\Ga^2_0)$. Then for any
$\eta=(\eta^+,\eta^-)\in\Ga^2_0$
\begin{align*}
\left( \hat{L}G\right) ( \eta ^{+},\eta ^{-})
&=-\left( \left\vert \eta ^{+}\right\vert +\left\vert \eta
^{-}\right\vert
\right) G( \eta ^{+},\eta ^{-}) \\
&\quad+\la ^{+}\int_{\X}G\left( \eta ^{+}\cup x,\eta ^{-}\right)
\left(
\sum_{x' \in \eta ^{+}}a^{+}\left( x-x' \right) \right) dx \\
&\quad+\la ^{+}\int_{\X}\sum_{x' \in \eta ^{+}}G\left( \eta
^{+}\setminus x' \cup x,\eta ^{-}\right) a^{+}\left( x-x' \right) dx
\\
&\quad+\la ^{-}\int_{\X}G\left( \eta ^{+},\eta ^{-}\cup y\right)
\left(
\sum_{y' \in \eta ^{-}}a^{-}\left( y-y' \right) \right) dy \\
&\quad+\la ^{-}\int_{\X}\sum_{y' \in \eta ^{-}}G( \eta ^{+},\eta
^{-}\setminus y' \cup y) a^{-}\left( y-y' \right) dy \\
&\quad+\la \int_{\X}G\left( \eta ^{+}\cup x,\eta ^{-}\right) \left(
\sum_{y' \in \eta ^{-}}a\left( x-y' \right) \right) dx \\
&\quad+\la \int_{\X}\sum_{y' \in \eta ^{-}}G\left( \eta ^{+}\cup
x,\eta ^{-}\setminus y' \right) a\left( x-y' \right) dx
\end{align*}
\end{proposition}
\begin{proof}
Let us denote death and birth parts of
the operator $\LC^+$ by
\begin{align*}
(L_{d}^{+}F)(\ga^+,\ga^-) &:=\sum_{x\in \ga ^{+}}\left[ F\left( \ga
^{+}\setminus
x,\ga ^{-}\right) -F( \ga ^{+},\ga ^{-}) \right], \\
(L_{b}^{+}F)(\ga^+,\ga^-) &:=\la ^{+}\int_{\R^{d}}\left( \sum_{x'
\in \ga ^{+}}a^{+}\left( x-x' \right) \right) \left[ F\left( \ga
^{+}\cup x,\ga ^{-}\right) -F( \ga ^{+},\ga ^{-}) \right] dx.
\end{align*}
In the same way we denote death and birth parts of the operator $\LC^-$: $\LC^-=L_{d}^{-}+L_{b}^{-}$.
As a result,
\begin{equation*}
L =L_{d}^{+}+L_{b}^{+}+L_{d}^{-}+L_{b}^{-}+\Lint ^{+}.
\end{equation*}
Now we calculate image under $\K$-transform
of all this operators.
One has for any $\eta=(\eta^+,\eta^-)\in\Ga^2_0$
\begin{align*}
\left( \hat{L}_{b}^{+}G\right) ( \eta) &=\left( \K ^{-1} L_{b}^{+}\K^{+} G\right)
( \eta ) \\
&=\sum_{\xi ^{+}\subset \eta ^{+}}(-1)^{|\eta ^{+}\setminus \xi
^{+}|}\sum_{\xi ^{-}\subset \eta ^{-}}(-1)^{|\eta ^{-}\setminus \xi
^{-}|}\la ^{+}\int_{\R^{d}}\sum_{x' \in \xi
^{+}}a^{+}(x-x' ) \\
&\qquad \times \left( \sum_{\zeta ^{+}\subset \xi ^{+}\cup x}\sum_{\zeta
^{-}\subset \xi ^{-}}G(\zeta ^{+},\zeta ^{-})-\sum_{\zeta
^{+}\subset \xi
^{+}}\sum_{\zeta ^{-}\subset \xi ^{-}}G(\zeta ^{+},\zeta ^{-})\right) dx \\
&=\la ^{+}\int_{\X}\sum_{x' \in \eta ^{+}}G\left( \eta ^{+}\cup
x,\eta ^{-}\right) a^{+}\left( x-x' \right) dx \\
&\quad+\la ^{+}\int_{\X}\sum_{x' \in \eta ^{+}}G\left( \eta
^{+}\setminus x' \cup x,\eta ^{-}\right) a^{+}\left( x-x' \right)
dx,
\end{align*}%
analogously, we have that%
\begin{align*}
\left( \hat{L}_{b}^{-}G\right) ( \eta ^{+},\eta ^{-}) &=\la
^{-}\int_{\X}\sum_{y' \in \eta ^{-}}G\left(
\eta ^{+},\eta ^{-}\cup y\right) a^{-}\left( y-y' \right) dy \\
&\quad\quad+\la ^{-}\int_{\X}\sum_{y' \in \eta ^{-}}G\left( \eta
^{+},\eta ^{-}\setminus y' \cup y\right) a^{-}\left( y-y' \right)
dy.
\end{align*}
Next,
\begin{align*}
\left( \hLint^{+}G\right) ( \eta )
&=\left( \K ^{-1} \Lint^{+}\K^{+} G\right) ( \eta ) \\
&=\sum_{\xi ^{+}\subset \eta ^{+}}(-1)^{|\eta ^{+}\setminus \xi
^{+}|}\sum_{\xi ^{-}\subset \eta ^{-}}(-1)^{|\eta ^{-}\setminus \xi
^{-}|}\la \int_{\R^{d}}\sum_{y\in \xi ^{-}}a(x-y) \\
&\qquad\times \left( \sum_{\zeta ^{+}\subset \xi ^{+}\cup x}\sum_{\zeta
^{-}\subset \xi ^{-}}G(\zeta ^{+},\zeta ^{-})-\sum_{\zeta
^{+}\subset \xi
^{+}}\sum_{\zeta ^{-}\subset \xi ^{-}}G(\zeta ^{+},\zeta ^{-})\right) dx \\
&=\la \int_{\X}\sum_{y' \in \eta ^{-}}G\left( \eta ^{+}\cup
x,\eta ^{-}\right) a\left( x-y' \right) dx \\
&\quad+\la \int_{\X}\sum_{y' \in \eta ^{-}}G\left( \eta ^{+}\cup
x,\eta ^{-}\setminus y' \right) a\left( x-y' \right) dx.
\end{align*}%
Finally,
\begin{align*}
\left( \hat{L}_{d}^{-}G\right) ( \eta ) &=\left( \K ^{-1} L_{d}^{-}\K^{+} G\right)
( \eta ) \\
&=\sum_{\xi ^{+}\subset \eta ^{+}}(-1)^{|\eta ^{+}\setminus \xi
^{+}|}\sum_{\xi ^{-}\subset \eta ^{-}}(-1)^{|\eta ^{-}\setminus \xi ^{-}|} \\
&\qquad\times\sum_{y\in \xi ^{-}}\left( \sum_{\zeta ^{+}\subset \xi
^{+}}\sum_{\zeta ^{-}\subset \xi ^{-}\setminus y}G(\zeta ^{+},\zeta
^{-})-\sum_{\zeta ^{+}\subset \xi ^{+}}\sum_{\zeta ^{-}\subset \xi
^{-}}G(\zeta ^{+},\zeta
^{-})\right) \\
&=-\left\vert \eta ^{-}\right\vert G( \eta ^{+},\eta ^{-}),
\end{align*}%
and, analogously,
\begin{equation*}
\left( \hat{L}_{d}^{+}G\right) ( \eta ^{+},\eta ^{-}) =-\left\vert
\eta ^{+}\right\vert G( \eta ^{+},\eta ^{-}).
\end{equation*}%
The statement is proved.
\end{proof}
Now we should calculate the adjoint operator
$\hat{L}^{\ast }$.
\begin{proposition}\label{prop_adj_oper}
The adjoint operator $\hat{L}^{\ast }$ has the following form:
\begin{align*}
\left( \hat{L}^{\ast }k\right) ( \eta ^{+},\eta ^{-}) &=-\left(\left\vert \eta ^{+}\right\vert +\left\vert \eta ^{-}\right\vert
\right)k( \eta ^{+},\eta ^{-}) \\
&\quad+\la ^{+}\sum_{x\in \eta ^{+}}\sum_{x' \in \eta ^{+}\setminus
x}a^{+}(x-x' )k\left( \eta ^{+}\setminus x,\eta ^{-}\right) \\
&\quad+\la ^{+}\sum_{x\in \eta ^{+}}\int_{\R^{d}}a^{+}(x-x' )k\left(
\eta ^{+}\setminus x\cup x' ,\eta ^{-}\right) dx'
\\
&\quad+\la ^{-}\sum_{y\in \eta ^{-}}\sum_{y' \in \eta ^{-}\setminus
y}a^{-}(y-y' )k\left( \eta ^{+},\eta ^{-}\setminus y\right) \\
&\quad+\la ^{-}\sum_{y\in \eta ^{-}}\int_{\R^{d}}a^{-}(y-y' )k\left(
\eta ^{+},\eta ^{-}\setminus y\cup y' \right) dy'
\\
&\quad+\la \sum_{x\in \eta ^{+}}\sum_{y\in \eta ^{-}}a(x-y)k (\eta
^{+}\setminus x,\eta ^{-}) \\
&\quad+\la \sum_{x\in \eta
^{+}}\int_{\R^{d}}a(x-y)k\left( \eta ^{+}\setminus
x,\eta ^{-}\cup y\right) dy
\end{align*}
\end{proposition}
\begin{proof}
We may use
the following corollaries of the classical Mecke formula (see, e.g., \cite{AKR1}):
\begin{multline*}
\int_{\Ga _{0}^{2}}\sum_{x\in \eta ^{+}}h_{+}(x,\eta
^{+},\eta ^{-})d\la_{1}( \eta ^{+}) d\la
_1( \eta ^{-}) \\
\shoveright{
=\int_{\Ga
_{0}^{2}}\int_{\R^{d}}h_{+}(x,\eta ^{+}\cup x,\eta
^{-})dxd\la_{1}( \eta ^{+}) d\la
_1( \eta ^{-}),}
\\
\shoveleft{
\int_{\Ga _{0}^{2}}\sum_{y\in \eta ^{-}}h_{-}(y,\eta
^{+},\eta ^{-})d\la_{1}( \eta ^{+}) d\la
_1( \eta ^{-})} \\
\shoveright{
=\int_{\Ga
_{0}^{2}}\int_{\R^{d}}h_{-}(y,\eta ^{+},\eta ^{-}\cup
y)dyd\la_{1}( \eta ^{+}) d\la
_1( \eta ^{-}),}
\\
\shoveleft{
\int_{\Ga _{0}^{2}}\sum_{x\in \eta ^{+}}\sum_{y\in \eta
^{-}}h(x,\eta ^{+},\eta ^{-})d\la_{1}( \eta ^{+}) d\la
_1( \eta ^{-})}\\
=\int_{\Ga
_{0}^{2}}\int_{\R^{d}}\int_{\R^{d}}h(x,\eta
^{+}\cup x,\eta ^{-}\cup y)dxdyd\la_{1}( \eta ^{+}) d\la
_1( \eta ^{-}).
\end{multline*}
Then one can obtain the explicit formula for the operator $\hat{L}^{\ast }$ directly
from definition~\eqref{adjoint}.
\end{proof}
As a result, the statements of the Propositions~\ref{propeq1} and
\ref{propeq2} are directly follow from the
Proposition~\ref{prop_adj_oper} and \eqref{geneq}--\eqref{cf12}.
\subsection{Solution of the equations for time evolution of the correlation
functions}
To solve the equations from the Propositions~\ref{propeq1}
and \ref{propeq2} using classical
perturbation method we rewrite these equations in the following
forms:
\begin{align}
\frac{\partial k_{t}^{-}(y)}{\partial t} &=(\la
^{-}-1)k_{t}^{-}(y)+\la ^{-}(L^{-}k_{t}^{-})(y), \label{eq1-}\\
\frac{\partial k_{t}^{+}(x)}{\partial t} &=(\la
^{+}-1)k_{t}^{+}(x)+\la ^{+}(L^{+}k_{t}^{+})(x)+\la
\int_{\R^{d}}a(x-y)k_{t}^{-}(y)dy,\label{eq1+}
\end{align}
where Markov-type generators $L^\pm$ are
defined on functions on $\X$ by
\begin{align*}
(L^{-}f)(y) &=\int_{\R^{d}}a^{-}(y-y' )[f(y' )-f(y)]dy', \\
(L^{+}f)(x) &=\int_{\R^{d}}a^{+}(x-x' )[f(x' )-f(x)]dx';
\end{align*}
and for the second order correlation functions:
\begin{align}
\frac{\partial k_{t}^{--}(y_{1},y_{2})}{\partial t}
&=2k_{t}^{--}(y_{1},y_{2})(\la ^{-}-1)+\la
^{-}(L_{1}^{--}k_{t}^{--})(y_{1},y_{2}) \notag\\
&\quad+\la ^{-}(L_{2}^{--}k_{t}^{--})(y_{1},y_{2})+\la
^{-}a^{-}(y_{1}-y_{2})[k_{t}^{-}(y_{1})+k_{t}^{-}(y_{2})],
\label{eq2--}\\[3ex] \frac{\partial
k_{t}^{+-}(x,y)}{\partial t} &=(\la ^{+}+\la
^{-}-2)k_{t}^{+-}(x,y)+\la ^{+}L_{1}^{+-}k_{t}^{+-}(x,y)+\la
^{-}L_{2}^{+-}k_{t}^{+-}(x,y) \notag\\
&\quad+\la a(x-y)k_{t}^{-}(y)+\la \int_{\R^{d}}a(x-y'
)k_{t}^{--}(y,y' )dy', \label{eq2+-}\\[3ex]
\frac{\partial k_{t}^{++}(x_{1},x_{2})}{\partial t}
&=2k_{t}^{++}(x_{1},x_{2})(\la ^{+}-1)+\la
^{+}L_{1}^{++}k_{t}^{++}(x_{1},x_{2})+\la
^{+}L_{2}^{++}k_{t}^{++}(x_{1},x_{2}) \notag\\
&\quad+\{\la ^{+}a^{+}(x_{1}-x_{2})[k_{t}^{+}(x_{1})+k_{t}^{+}(x_{2})] \notag\\
&\quad+\la
\int_{\R^{d}}a(x_{1}-y)k_{t}^{+-}(x_{2},y)dy+\la
\int_{\R^{d}}a(x_{2}-y)k_{t}^{+-}(x_{1},y)dy\},\label{eq2++}
\end{align}
where Markov-type generators $L_{i}^{\pm\pm}$, $i=1,2$ are defined
on functions on $\X\times\X$ by
\begin{align*}
(L_{1}^{--}f)(y_{1},y_{2})
&=\int_{\R^{d}}a^{-}(y_{1}-y' )[f(y_{2},y' )-f(y_{2},y_{1})]dy' , \\
(L_{2}^{--}f)(y_{1},y_{2}) &=\int_{\R^{d}}a^{-}(y_{2}-y'
)[f(y_{1},y' )-f(y_{1},y_{2})]dy',\\
(L_{1}^{+-}f)(x,y) &=\int_{\R^{d}}a^{+}(x-x' )[f(x' ,y)-f(x,y)]dx' , \\
(L_{2}^{+-}f)(x,y) &=\int_{\R^{d}}a^{-}(y-y' )[f(x,y' )-f(x,y)]dy',\\
(L_{1}^{++}f)(x_{1},x_{2})
&=\int_{\R^{d}}a^{+}(x_{1}-x' )[f(x_{2},x' )-f(x_{2},x_{1})]dx', \\
(L_{2}^{++}f)(x_{1},x_{2}) &=\int_{\R^{d}}a^{+}(x_{2}-x'
)[f(x_{1},x' )-f(x_{1},x_{2})]dx'.
\end{align*}%
Next propositions are direct corollaries of the perturbation method
(note also that any Markov semigroup preserves constants).
\begin{proposition}
The solutions of \eqref{eq1-}--\eqref{eq1+}
with initial values \eqref{initval1} have
the following forms:
\begin{align}
k_{t}^{-}(y) &=c^{-}e^{t(\la ^{-}-1)}+e^{t(\la
^{-}-1)}e^{t\la ^{-}L^{-}}\psi ^{-}(y), \label{expr_c1-}\\
k_{t}^{+}(x) &=c^{+}e^{t(\la ^{+}-1)}+e^{t(\la
^{+}-1)}e^{t\la ^{+}L^{+}}\psi ^{+}(x)+\la
c^{-}e^{t(\la^{+}-1)}\int_{0}^{t}e^{\tau(\la^{-}-
\la^{+})}d\tau \label{expr_c1+}\\
&\phantom{{}=c^{+}e^{t(\la ^{+}-1)}}+\la
e^{t(\la^{+}-1)}\int_{0}^{t}e^{\tau(\la^{-}-%
\la^{+})}e^{(t-\tau)\la^{+}L^{+}}(a\ast (e^{\tau
\la^{-}L^{-}}\psi^{-}))(x)d\tau.\notag
\end{align}
\end{proposition}
\begin{proposition} Let \eqref{tr_inv} holds.
Then the solutions
of~\eqref{eq2--}--\eqref{eq2++} with initial values \eqref{initval2} have
the following forms:
\begin{align}
k_{t}^{--}&(y_{1},y_{2}) =e^{t2(\la ^{-}-1)}e^{t\la
^{-}L_{1}^{--}}e^{t\la ^{-}L_{2}^{--}}(c^{--}+\varphi
^{--}(y_{1}-y_{2}))\notag
\\
&+\int_{0}^{t}e^{(t-\tau )2(\la ^{-}-1)}e^{(t-\tau )\la
^{-}L_{1}^{--}}e^{(t-\tau )\la ^{-}L_{2}^{--}}\la
^{-}a^{-}(y_{1}-y_{2})[k_{\tau }^{-}(y_{1})+k_{\tau
}^{-}(y_{2})]d\tau, \label{expr_c2--}\\[3ex]
k_{t}^{+-}&(x,y) =e^{t(\la ^{+}+\la ^{-}-2)}e^{t\la
^{+}L_{1}^{+-}}e^{t\la ^{-}L_{2}^{+-}}(c^{+-}+\varphi ^{+-}(x-y)) \notag\\
&+\int_{0}^{t}e^{(t-\tau )(\la ^{+}+\la
^{-}-2)}e^{(t-\tau
)\la ^{+}L_{1}^{+-}}e^{(t-\tau )\la ^{-}L_{2}^{+-}}\notag\\
&\qquad\times \{\la a(x-y)k_{\tau }^{-}(y)+\la \int_{\R^{d}}a(x-y'
)k_{\tau }^{--}(y,y' )dy' \}d\tau,
\label{expr_c2+-}\\[3ex] k_{t}^{++}&(x_{1},x_{2})
=e^{t2(\la ^{+}-1)}e^{t\la ^{+}L_{1}^{++}}e^{t\la
^{+}L_{2}^{++}}(c^{++}+\varphi
^{++}(x_{1}-x_{2}))\notag\\
&+\int_{0}^{t}e^{(t-\tau )2(\la ^{+}-1)}e^{(t-\tau
)\la ^{+}L_{1}^{++}}e^{(t-\tau )\la ^{+}L_{2}^{++}}\{\la
^{+}a^{+}(x_{1}-x_{2})[k_{\tau }^{+}(x_{1})+k_{\tau }^{+}(x_{2})] \notag\\
&+\la \int_{\R^{d}}a(x_{1}-y)k_{\tau
}^{+-}(x_{2},y)dy+\la
\int_{\R^{d}}a(x_{2}-y)k_{\tau
}^{+-}(x_{1},y)dy\}d\tau \label{expr_c2++}
\end{align}%
\end{proposition}
\subsection{Technical lemmas} In this subsection
we present several useful notations and notes
and prove
technical lemmas needed in the sequel. Let us define\begin{alignat}{2}
\mu ^{+} &:=\la ^{+}-1,
&\qquad \mu ^{-} &:=\la ^{-}-1, \label{def_mu}\\
f^{+}(p)& :=\la ^{+}\hat{a}^{+}(p)-1,
&\qquad f^{-}(p)& :=\la ^{-}\hat{a}^{-}(p)-1.\label{def_f}
\end{alignat}
Note that conditions $0<\la^\pm\leq1$ equivalent
to $ -1<\mu ^{\pm }\leq 0$
and $\mu ^{\pm }=0$ only if $\la ^{\pm }=1$. Recall that $a^\pm$ are positive, even and
normalized. Then
\begin{equation}\label{apmineq}
\hat{a}^\pm(p)=\int_{\R^{d}}\cos (p,x)a^\pm(x)dx,
\qquad |\hat{a}^\pm(p)|\leq 1,
\end{equation}
and $\hat{a}^\pm(p)=1$
only at $p=0$.
Hence, the conditions $0<\la^\pm\leq1$ imply
\begin{equation}
-\la^{\pm}-1\leq f^{\pm }(p)\leq \mu ^{\pm }\leq0,\label{flessmu}
\end{equation}
and $f^{\pm }(p)=\mu ^{\pm }$ only at point
$p=0$.
Let $C^-(\X)$ be a set of non-positive continuous functions on $\X$ which equal to $0$ only
on
countable sets. Since Fourier image of
integrable function is continuous one has
$f^\pm\in C^-(\X)$. For any $f\in C^-(\X)$
define two closed sets
\begin{equation}
\D[\pm]_f:=\{x\in\X :f(x)=f^\pm(x)\}.
\end{equation}
Note that that set $\X\setminus\D[+]_{f^-}=\X\setminus\D[-]_{f^+}$
has zero Lebesgue measure only if
$\la^+\hat{a}^+\equiv\la^-\hat{a}^{-}$ and, hence, $\la^+=\la^-$.
\begin{lemma}\label{int_a} Let $d\geq 3$ and $b\in L^1(\X)\cap
L^\infty(\X)$.Then
\begin{equation*}
c^\pm(p)=\frac{b(p)}{\hat{a}^\pm (p)-1}
\end{equation*}
are integrable functions on $\X$.
\end{lemma}
\begin{proof}By \eqref{apmineq}, $\hat{a}^\pm (0)=1$. Due to \eqref{decayofa}, $a^\pm$
has at least first and second finite moments.
Then using~\eqref{apmineq} one has
in some neighbourhood of the origin
\begin{align*}
\hat{a}^\pm (p)-1 =\int_{\R^{d}}[\cos (p,x)-1]a^\pm(x)dx
\sim -\frac{1}{2}\int_{\R^{d}}( p,x)^{2}a^{\pm}(x)dx
\sim -\frac{1}{2} \vert p\vert^2
\end{align*}
and outside of this neighbourhood $\vert
\hat{a}^\pm (p)-1 \vert$ are bounded from
below.
Hence, $c^\pm$ are integrable in this neighbourhood
since $b$ is bounded and $\dfrac{1}{\vert p\vert^2}\in
L^1(\X)$ for $d\geq 3$; and $c^\pm$ are integrable outside of this neighbourhood since $b$ is
integrable.
\end{proof}
\begin{lemma}\label{boundint} Let $d\geq 3$, $0<\la^\pm\leq1$,
and $b\in L^1(\X)\cap
L^\infty(\X)$.Then
for any $f\in C^-(\X)$
\begin{equation*}
d^\pm(p)= b(p)\sup_{t\geq0} \frac{e^{tf^{}(p)}-e^{tf^\pm(p)}}
{f(p)-f^\pm(p)}
\end{equation*}
are integrable functions on $\X\setminus\D[\pm]_f$.
\end{lemma}
\begin{proof}
Let $p\in\X\setminus\D[+]_f$ for example.
Without loss of generality assume that $p\neq0$
and $f(p)\neq0$. Set $a=f(p), b=f^+(p)$.
Then $a<0$, $b<0$, $a\neq b$. Let us define
\begin{equation*}
h(t):=\dfrac{e^{ta}-e^{tb}}{a-b}, \quad
t\geq0.
\end{equation*}
Clearly, $h(t)\geq 0$ and $h(t)=0$ only at
$t=0$. One has
\begin{equation*}
h'(t):=\dfrac{be^{ta}\Bigl(\dfrac{a}{b}-e^{t(b-a)}\Bigr)}{a-b}.
\end{equation*}
Set $t_0=\dfrac{1}{b-a}\ln\dfrac{a}{b}$.
If $0>a>b$ then $t_0>0$ and for $0<t<t_0$
we have $e^{t(b-a)}>\dfrac{a}{b}$, hence,
$h'(t)>0$; for $t>t_0$ one has $h'(t)<0$.
If $0>b>a$ then $t_0>0$ also and for $0<t<t_0$
we obtain $e^{t(b-a)}<\dfrac{a}{b}$, therefore,
$h'(t)>0$; for $t>t_0$ again $h'(t)<0$. As~a~result,
\begin{equation*}
\max_{[0;\infty)}h(t)=h(t_0)=\dfrac{e^{t_0a}(1-e^{t_0(b-a)})}{a-b}=\frac{e^{t_0a}\Bigl(1-\dfrac{a}{b}\Bigr)}{a-b}=-\frac{1}{b}e^{t_0a}<-\frac{1}{b},
\end{equation*}
since $-b>0$, $a<0$.
Hence, for any $p\in\X\setminus\D[+]_f$,
$t\geq0$
\begin{equation*}
0\leq\frac{e^{tf^{}(p)}-e^{tf^+(p)}}
{f(p)-f^+(p)}<-\frac{1}{f^+(p)}.
\end{equation*}
Then using \eqref{flessmu}, \eqref{def_mu} for
$\la^+<1$ one has $\mu^+<0$ and
$d^+(p)< \dfrac{b(p)}{-\mu^+}$ that imply
the statement of this Lemma. For $\la^+=1$
the result is followed from Lemma~\ref{int_a}.
\end{proof}
\subsection{Asymptotic behaviour of the first
order correlation functions}
In this subsection we prove the Theorem~\ref{thm_as_1}.
\step{1} We should use~\eqref{expr_c1-}.
Note that $\psi^{-}\in L^1(\X)$ and Markov
semigroup maps $L^1(\X)$ into $L^1(\X)$.
Then using inverse Fourier transform one
has
\begin{equation}\label{invF1}
\bigl(e^{t\la ^{-}L^{-}}\psi^{-}\bigr)(y
) =
c_{d}\int_{\R^{d}}e^{i(p,y)}e^{t\la^{-}(\hat{a}^{-}(p)-1)}\hat{\psi}
^{-}\left( p\right) dp,
\end{equation}
where $c_d:=\dfrac{1}{(2\pi n)^d}$. Using
\eqref{apmineq}, the expression in the integral
in~\eqref{invF1} goes to $0$ for any $y$
and a.a. $p$. Since $\hat{\psi}^{-}\in L^1(\X)$
and $\left\vert e^{i(p,y)}e^{t\la^{-}(\hat{a}^{-}(p)-1)}\right\vert\leq
1$ one has that the~integral
also goes to $0$ for any $y$. Then the statement
is directly followed from~\eqref{expr_c1-}.
\step{2} We will use~\eqref{expr_c1+}. Note that similarly to the
first step $e^{t \la^{+}L^{+}}\psi^{+}\to 0$ point-wisely.
\step{2.1} If $\la ^{+}>1$ then for any
$\la^{-}>0$
\begin{equation*}
k_{t}^{+}\left( x\right) \rightarrow \infty ,
\end{equation*}%
since $\psi^{-}\geq \alpha^--c^{-}>-c^-$, hence, the last
term in~\eqref{expr_c1+} is bigger than
\begin{equation*}
-\la c^{-}
e^{t(\la^{+}-1)}\int_{0}^{t}e^{\tau(\la^{-}-\la^{+})}d\tau
\end{equation*}
and, therefore,
\begin{equation*}
k_{t}^{+}(x) > c^{+}e^{t(\la ^{+}-1)}+e^{t(\la
^{+}-1)}e^{t\la ^{+}L^{+}}\psi ^{+}(x)\rightarrow\infty
\end{equation*}
\step{2.2} Let now $\la ^{+}\leq 1$. Divide proof on several
sub-steps. \step{2.2.1} Suppose $\la ^{+}=\la ^{-}=\nu$ then using~\eqref{expr_c1+} one
has
\begin{equation}\label{dop22}
k_{t}^{+}\left( x\right) =e^{t\left( \nu -1\right)
}c^{+}+e^{t\left( \nu
-1\right) }e^{t\nu L^{+}}\psi ^{+}\left( x\right) +\la e^{t\left( \nu -1\right) }c^{-}t + u_{t}(x)
\end{equation}
where
\begin{align*}
u_{t}(x)&=\la e^{t\left( \nu -1\right) }\int_{0}^{t}e^{\left( t-\tau
\right) \nu L^{+}}\left( a\ast (e^{\tau \nu L^{-}}\psi ^{-})\right)
\left( x\right) d\tau.
\end{align*}
Let us find $\lim\limits_{t\rightarrow\infty}u_{t}(x)$, for $\nu
\leq 1$. Note that $u_{t}\in L^1(\X)$ since semigroup and
convolution preserve integrability. Hence, we may compute the
Fourier transform of $u_t$:
\begin{equation}\label{hatut}
\hat{u}_{t}(p)= \left\{
\begin{array}{ll}
\la \hat{a}(p)\hat{\psi}^{-}(p)e^{tf^+(p)}t, & p\in\D[+]_{f^-},\\[2ex]
\la \hat{a}(p)\hat{\psi}^{-}(p)\dfrac{e^{tf^-(p)}-
e^{tf^+(p)}}
{f^{-}(p)-f^{+}(p)}, & p\in\X\setminus\D[+]_{f^-}.
\end{array}
\right.
\end{equation}
Since $\hat{\psi}^-$ is bounded and $\hat{a}$ is bounded and integrable due to \eqref{decayofa}
one can apply Lemma~\ref{boundint}, hence,
$\hat{u}_{t}(p)$ has integrable majorant
on $\X\setminus\D[+]_{f^-}$. Since $e^{ta}t<-\dfrac{e^{-1}}{a}$
for any $t\geq0$, $a<0$ one has for any $p\in\D[+]_{f^-}\setminus\{0\}$
\[
\bigl\vert\hat{u}_{t}(p)\bigr\vert\leq c_1\Biggl\vert
\frac{\hat{a}(p)}{f^+(p)}\Biggr\vert.
\]
Again if $\nu<1$ then denominator is separated
from zero, otherwise one can apply Lemma~\ref{int_a}.
As a result, $\hat{u}_{t}(p)$ has integrable majorant on whole $\X$ and pointwisely goes
to $0$ as $t\to\infty$ (except case $\nu=1$,
$p=0$).
Therefore, using majorized convergence theorem
the inverse Fourier transform of $\hat{u}_{t}(p)$ converges to zero, i.e. pointwisely $u_{t}(x)\to0$
as $t\to\infty$.
Thus, using~\eqref{dop22} one has that
$k_{t}^{+}\to\infty$ if $\nu=1$ and
$k_{t}^{+}\to0$ if $\nu<1$.
\step{2.2.2} Let now $\la ^{+}\neq \la
^{-}$. Using~\eqref{expr_c1+} obtain
\begin{align}
k_{t}^{+}\left( x\right) &= c^{+}e^{t\left( \la ^{+}-1\right)
}+e^{t\left( \la ^{+}-1\right) }e^{t\la ^{+}L^{+}}\psi
^{+}\left(
x\right) \notag\\
&\quad+\la c^{-}\frac{1}{\la ^{-}-\la ^{+}}\left( e^{t\left(
\la
^{-}-1\right) }-e^{t\left( \la ^{+}-1\right) }\right) \label{aaa}\\
&\quad+\la e^{t\left( \la ^{+}-1\right) }\int_{0}^{t}e^{\tau
\left( \la ^{-}-\la ^{+}\right) }e^{\left( t-\tau \right)
\la ^{+}L^{+}}\left( a\ast e^{\tau \la ^{-}L^{-}}\psi
^{-}\right) \left( x\right) d\tau .\notag
\end{align}
\step{2.2.2.1} Suppose that $\la ^{-}>1$.
Then since $\la ^{+}\leq 1$ and $\psi^-\geq\alpha^--c^-
>0$ we obtain that
\begin{equation*}
k_{t}^{+}\left( x\right) \rightarrow \infty ,\text{~~~}t\rightarrow \infty
\end{equation*}
\step{2.2.2.2} Next, let $\la ^{-}<1,\ \la ^{+}<1$. Since $\hat{\psi}^-$ is bounded
one has for $M=\sup_\X\vert\hat{\psi}^-\vert$
that the last term in~\eqref{aaa} is not bigger (by absolute value) than
\begin{equation*}
\frac{M}{\la^{-}-\la^{+}}\left(e^{t(%
\la^{-}-1)}-e^{t(\la^{+}-1)}\right)\to 0.
\end{equation*}
Then due to~\eqref{aaa} $k_{t}^{+}\left( x\right) \to 0$.
\step{2.2.2.3} Finally, let $\la ^{-}<1,~\la ^{+}=1$ or
$\la ^{-}=1,~\la ^{+}<1$. The last term in~\eqref{aaa}
is integrable function since semigroup and
convolution preserve integrability.
By direct computation its Fourier transform
has form~\eqref{hatut}. Hence, this last
term pointwisely goes to $0$.
As a result, by~\eqref{aaa} we obtain that
if $\la ^{+}=1$, $\la ^{-}<1$
\begin{equation*}
k_{t}^{+}\left( x\right) \rightarrow c^{+}+\frac{\la c^{-}}{1-\la
^{-}},\quad t\rightarrow \infty;
\end{equation*}
and if $\la ^{+}<1$, $\la ^{-}=1$%
\begin{equation*}
k_{t}^{+}\left( x\right) \rightarrow \frac{\la c^{-}}{1-\la ^{+}},\quad t\rightarrow \infty.
\end{equation*}
Theorem~\ref{thm_as_1} is proved.
\subsection{Asymptotic behaviour of the second
order correlation functions}
In this subsection we prove the Theorem~\ref{thm_as_2}.
First of all we present explicit expressions for $\Omega ^{++},~\Xi
^{--},~\Xi ^{+-},~\Xi ^{++}$, and after that we prove the Theorem. These functions are
inverse Fourier transforms of the following
\begin{align}
\omega ^{++}(p) &=\frac{\la ^{-}+\la -1}{\la ^{-}-1}\cdot \frac{%
c^{+}\hat{a}^{+}\left( p\right) }{1-\hat{a}^{+}\left( p\right) } ,\label{a1}\\
\xi ^{--}(p) &=\frac{c^{-}\hat{a}^{-}\left( p\right)
}{1-\hat{a}^{-}\left(
p\right) } ,\label{a2}\\
\xi ^{+-}(p) &=\frac{1}{2}\cdot \frac{\mu ^{-}+2}{2-\la ^{+}\hat{a}%
^{+}\left( p\right) -\hat{a}^{-}\left( p\right) }\cdot \frac{c^{-}\la
\hat{a}\left( p\right) }{1-\hat{a}^{-}\left( p\right) } ,\label{a3}\\
\xi ^{++}(p) &=\frac{\la }{1-\la ^{+}\hat{a}^{+}(p)}\left( \frac{%
\la ^{+}c^{-}\hat{a}^{+}(p)}{1-\la ^{+}}+\frac{\la c^{-}}{%
2-\la ^{+}\hat{a}^{+}(p)-\hat{a}^{+}(p)}\cdot \frac{\hat{a}^{2}(p)}{1-%
\hat{a}^{-}(p)}\right),\label{a4}
\end{align}
correspondingly.
Let us introduce the following denotations for the Markov semigroups
\begin{equation*}
T_{t}^{11}=e^{t\la ^{+}L_{1}^{++}},~~T_{t}^{12}=e^{t\la
^{+}L_{2}^{++}},~~T_{t}^{13}=e^{t\la ^{+}L_{1}^{+-}},
\end{equation*}%
\begin{equation*}
T_{t}^{21}=e^{t\la ^{-}L_{1}^{--}},~~T_{t}^{22}=e^{t\la
^{-}L_{2}^{--}},~~T_{t}^{23}=e^{t\la ^{-}L_{1}^{+-}}.
\end{equation*}
We start with trivial remark that for any
even functions $c,g\in L^1(\X)$
\begin{equation*}
\left( L_{1}g\right) \left( x_{1}-x_{2}\right) =\left( L_{2}g\right) \left(
x_{1}-x_{2}\right),
\end{equation*}
where
\begin{align*}
(L_{1}f)(x_{1},x_{2})
&:=\int_{\R^{d}}c(x_{1}-x' )[f(x_{2},x' )-f(x_{2},x_{1})]dx', \\
(L_{2}f)(x_{1},x_{2}) &:=\int_{\R^{d}}c(x_{2}-x' )[f(x_{1},x'
)-f(x_{1},x_{2})]dx'.
\end{align*}
After transformations, substitutions and simplifying we obtain
for~\eqref{expr_c2--}--\eqref{expr_c2++}
the~following representations:
\begin{align*}
k_{t}^{--}(y_{1},y_{2})&=c^{--}e^{2\mu^{-}t}+e^{2%
\mu^{-}t}T_{t}^{21}T_{t}^{22}\varphi
^{--}(y_{1}-y_{2})+U_{t}^{--}(y_{1}-y_{2}),\\
k_{t}^{+-}(x,y) &=\left( c^{+-}-\frac{\la c^{--}}{\mu ^{-}-\mu ^{+}}%
\right) e^{(\mu^{+}+\mu^{-})t}+\frac{\la c^{--}}{\mu ^{-}-\mu ^{+}}%
e^{2\mu^{-}t} \\
&\quad+e^{(\mu^{+}+\mu^{-})t}T_{t}^{13}T_{t}^{23}\varphi
^{+-}(x-y)+U_{t}^{+-}(x-y),\\
k_{t}^{++}(x_{1},x_{2}) &=\left( c^{++}-\frac{2\la c^{+-}}{\mu
^{-}-\mu ^{+}}+\frac{\la ^{2}c^{--}}{(\mu ^{-}-\mu
^{+})^{2}}\right) e^{2\mu^{+}t}
\\
&\quad+\left( \frac{2\la c^{+-}}{\mu ^{-}-\mu ^{+}}-\frac{2\la ^{2}c^{--}%
}{(\mu ^{-}-\mu ^{+})^{2}}\right) e^{(\mu^{+}+\mu^{-})t} \\
&\quad+\frac{\la ^{2}c^{--}}{(\mu ^{-}-\mu^{+})^{2}}e^{2\mu^{-}t} \\
&\quad+e^{2\mu^{+}t}T_{t}^{11}T_{t}^{12}\varphi
^{++}(x_{1}-x_{2})+U_{t}^{++}(x_{1}-x_{2}).
\end{align*}
Here
\begin{equation*}
U_{t}^{--}(y_{1}-y_{2})=2\la ^{-}c^{-}\int_{0}^{t}e^{\mu ^{-}\tau
}e^{2\mu ^{-}(t-\tau )}T_{t-\tau }^{21}T_{t-\tau
}^{22}a^{-}(y_{1}-y_{2})d\tau,
\end{equation*}\\*[-2\parindent]
\begin{align*}
&U_{t}^{+-}(x-y) \\
&=\la c^{-}\int_{0}^{t}e^{\mu ^{-}\tau }e^{(\mu ^{+}+\mu
^{-})(t-\tau )}T_{t-\tau }^{13}T_{t-\tau }^{23}a(x-y)d\tau \\
&\quad+\la \int_{0}^{t}e^{2\mu ^{-}\tau }e^{(\mu ^{+}+\mu
^{-})(t-\tau )}T_{t-\tau }^{13}T_{t-\tau }^{23}\int_{\R^{d}}a(x-y'
)T_{\tau }^{21}T_{\tau }^{22}\varphi
^{--}(y-y' )dy' d\tau \\
&\quad+2c^{-}\la \la ^{-}\int_{0}^{t}e^{(\mu ^{+}+\mu ^{-})(t-\tau
)}T_{t-\tau }^{13}T_{t-\tau }^{23}\\&\qquad\times\int_{\R^{d}}a(x-y'
)\int_{0}^{\tau }e^{\mu ^{-}s}e^{2\mu ^{-}(\tau -s)}T_{\tau
-s}^{21}T_{\tau -s}^{22}a^{-}(y-y' )dsdy' d\tau,
\end{align*}\\*[-2\parindent]
\begin{align*}
&U_{t}^{++}(x_{1}-x_{2}) \\
&=2\la ^{+}c^{+}\int_{0}^{t}e^{\mu ^{+}\tau }e^{2\mu
^{+}(t-\tau
)}T_{t-\tau }^{11}T_{t-\tau }^{12}a^{+}(x_{1}-x_{2})d\tau \\
&\quad+2\la \la ^{+}c^{-}\int_{0}^{t}e^{2\mu ^{+}(t-\tau
)}T_{t-\tau }^{11}T_{t-\tau
}^{12}a^{+}(x_{1}-x_{2})\int_{0}^{\tau
}e^{\mu ^{-}s}e^{\mu ^{+}(\tau -s)}dsd\tau \\
&\quad+2\la \int_{0}^{t}e^{(\mu ^{+}+\mu ^{-})\tau }e^{2\mu
^{+}(t-\tau )}T_{t-\tau }^{11}T_{t-\tau
}^{12}\int_{\R^{d}}a(x_{1}-y)T_{\tau }^{13}T_{\tau
}^{23}\varphi
^{+-}(x_{2}-y)dyd\tau \\
&\quad+2\la ^{2}c^{-}\int_{0}^{t}e^{2\mu ^{+}(t-\tau
)}T_{t-\tau }^{11}T_{t-\tau
}^{12}\int_{\R^{d}}a(x_{1}-y)\\&\qquad\times\int_{0}^{\tau
}e^{\mu ^{-}s}e^{(\mu ^{+}+\mu ^{-})(\tau -s)}T_{\tau
-s}^{13}T_{\tau
-s}^{23}a(x_{2}-y)dydsd\tau \\
&\quad+2\la ^{2}\int_{0}^{t}e^{2\mu ^{+}(t-\tau )}T_{t-\tau
}^{11}T_{t-\tau
}^{12}\int_{\R^{d}}a(x_{1}-y)\int_{0}^{\tau
}e^{2\mu ^{-}s}e^{(\mu ^{+}+\mu ^{-})(\tau -s)}T_{\tau
-s}^{13}T_{\tau
-s}^{23} \\
&\qquad \times \int_{\R^{d}}a(x_{2}-y' )T_{s}^{21}T_{s}^{22}\varphi ^{--}(y-y' )dy' dsdyd\tau \\
&\quad+4\la ^{-}c^{-}\la ^{2}\int_{0}^{t}e^{2\mu
^{+}(t-\tau )}T_{t-\tau }^{11}T_{t-\tau
}^{12}\int_{\R^{d}}a(x_{1}-y)\\&\qquad\times\int_{0}^{\tau
}e^{(\mu ^{+}+\mu ^{-})(\tau -s)}T_{\tau -s}^{13}T_{\tau
-s}^{23}\int_{\R^{d}}a(x_{2}-y' ) \\
&\qquad \times \int_{0}^{s}e^{\mu ^{-}\theta }e^{2\mu ^{-}(s-\theta
)}T_{s-\theta }^{21}T_{s-\theta }^{22}a^{-}(y-y' )d\theta dy'
dsdyd\tau.
\end{align*}
Since semigroups and convolutions preserve integrability we have that $T_{t}^{21}T_{t}^{22}\varphi^{--}$,
$T_{t}^{13}T_{t}^{23}\varphi^{+-}$, $T_{t}^{11}T_{t}^{12}\varphi^{++}$
as well as $U_{t}^{--},~U_{t}^{+-}$ and $U_{t}^{++}$ are integrable on $\R^{d}$ functions. So, to find their limits as $t\to\infty$ we may use the Fourier transforms.
Namely,\begin{align*}
T_{t}^{21}T_{t}^{22}\varphi
^{--}(y_1-y_2)&=c_{d}\int%
\limits_{\R^{d}}e^{ip(y_1-y_2)}e^{2(f^{-}(p)-\mu^-)t}\hat{\varphi
}^{--}(p)dp,
\\
T_{t}^{13}T_{t}^{23}\varphi ^{+-}(x-y)&=c_{d}\int%
\limits_{\R^{d}}e^{ip(x-y)}e^{(f^{+}(p)-\mu^+)t}
e^{(f^{-}(p)-\mu^-)t}\hat{\varphi} ^{+-}(p)dp,
\\
T_{t}^{11}T_{t}^{12}\varphi ^{++}(x_1-x_2)&=c_{d}\int%
\limits_{\R^{d}}e^{ip(x_1-x_2)}e^{2(f^{+}(p)-\mu^+)t}
\hat{\varphi }^{++}(p)dp.
\end{align*}
Since $\hat{\varphi}^{--}, \hat{\varphi}^{+-},
\hat{\varphi}^{++}$ are integrable we have
using \eqref{flessmu} and dominated convergence theorem that these three terms go to $0$.
Let us introduce for further simplicity of notations the following functions
\begin{align*}
h_1(p)&:=\mu^+-2f^+(p)\geq 0, \\
h_2(p)&:=\mu^--2f^-(p)\geq 0, \\
h_3(p)&:=f^+(p)+f^-(p)<0, \\
h_4(p)&:=\mu^--f^+(p)-f^-(p)\geq 0.
\end{align*}
These inequalities are followed from~\eqref{def_mu},
\eqref{def_f} and \eqref{flessmu} as well
as the fact that equalities are possible
only at $p=0$.
Consider also the following two functions $g_1$ and $g_2$
\begin{align*}
g_1(p)&=f^-(p)-f^+(p), \\
g_2(p)&=\mu^--2f^+(p).
\end{align*}
They can be equal zero on a~set of non-zero
measure.
We have in the new notations:
\begin{align*}
\widehat{U_{t}}^{{--}}(p)&=2c^-\la^-\hat{a}^-(p)e^{2f^-(p)t}\int%
\limits_0^te^{h_2(p)\tau}d\tau,
\\
\widehat{U_{t}}^{{+-}}(p)&=c^-\la\hat{a}(p)e^{h_3(p)t}\int%
\limits_0^te^{h_4(p)\tau}d\tau \\
&\quad+\la\hat{a}(p)\hat{\varphi}^{--}(p)e^{h_3(p)t}\int_0^te^{g_1(p)%
\tau}d\tau \\
&\quad+2c^-\la\hat{a}(p)\la^-\hat{a}^-(p)e^{h_3(p)t}\int%
\limits_0^te^{g_1(p)\tau}\int_0^\tau e^{h_2(p)s}dsd\tau,
\\
\widehat{U_{t}}^{{++}}(p)&=2c^+\la^+\hat{a}^+(p)e^{2f^+(p)t}\int%
\limits_0^te^{h_1(p)\tau}d\tau \\
&\quad+\frac{2\la c^-\la^+\hat{a}^+(p)}{\mu^--\mu^+}e^{2f^+(p)t}\left(%
\int_0^te^{g_2(p)\tau}d\tau-\int_0^te^{h_1(p)\tau}d\tau\right)
\\
&\quad+2\la\hat{a}(p)\hat{\varphi}^{+-}(p)e^{2f^+(p)t}\int%
\limits_0^te^{g_1(p)\tau}d\tau \\
&\quad+2c^-\la^2\hat{a}^2(p)e^{2f^+(p)t}\int_0^te^{g_1(p)\tau}\int%
\limits_0^{\tau}e^{h_4(p)s}dsd\tau \\
&\quad+2\la^2\hat{a}^2(p)\hat{\varphi}^{+-}(p)e^{2f^+(p)t}\int%
\limits_0^te^{g_1(p)\tau}\int_0^{\tau}e^{g_1(p)s}dsd\tau \\
&\quad+4c^-\la^-\hat{a}^-(p)\la^2\hat{a}^2(p)e^{2f^+(p)t}\int%
\limits_0^te^{g_1(p)\tau}\int_0^{\tau}e^{g_1(p)s}\int%
\limits_0^se^{h_2(p)\theta}d\theta dsd\tau.
\end{align*}
Let us consider the following closed set $\D=\D_1\cup\D_2$,
where $\D_1:=\{p:\ g_1(p)=0\}=\D[+]_{f^-},\ \mathfrak{D}_2=\{p:\ g_2(p)=0\}$.
It's easy to see that $\D_1\cap\D_2=\varnothing$.
Indeed, by~\eqref{flessmu} for any $p\in\D_1\cap\D_2$
\[
\mu^-=2f^+(p)=2f^-(p)\leq 2\mu^-.
\]
But $\mu^-\leq0$, hence, it should be equality
that implies
$f^-(p)=\mu^-$, and with necessity $p=0$.
But if $0\in\D_1\cap\D_2$, then $f^+(0)=f^-(0)$,
i.e., $\mu^+=\mu^-$, that contradicts to the condition of the theorem.
Next we note that the functions $\widehat{U_{t}}^{{+-}}(p)$ and $%
\widehat{U_{t}}^{{++}}(p)$ have different
explicit expressions for $p\in\D$ and for
$p\in\DC:=\R^d\setminus\D$. Note also that these functions are continuous functions of $p$ as compositions of the integrals of
the continuous functions of $t$ with continuous dependence on a parameter $p$.
Hence, for calculate these expressions for $p\in\D$ we may calculate their for $p\in\DC$
and take limits
as $\dist(p,\D)\to 0$.
By direct calculations for any $p\in\DC\setminus\{0\}$
we obtain
\begin{align*}
\widehat{U}_{t}^{--}(p)&=2\lambda
^{-}c^{-}\hat{a}^{-}(p)\frac{e^{\mu ^{-}t}-e^{2f^{-}(p)t}}{\mu
^{-}-2f^{-}(p)},
\\
\widehat{U}_{t}^{+-}(p)&= \lambda c^{-}\hat{a}(p)\cdot \frac{\mu ^{-}+2}{\mu
^{-}-2f^{-}(p)}\cdot \frac{e^{\mu ^{-}t}-e^{[f^{+}(p)+f^{-}(p)]t}}{\mu
^{-}-[f^{+}(p)+f^{-}(p)]} \\
& \quad+\left( \lambda \hat{a}(p)\hat{\varphi}^{--}(p)-\frac{2c^{-}\lambda
\lambda ^{-}\hat{a}(p)\hat{a}^{-}(p)}{\mu ^{-}-2f^{-}(p)}\right)
G_{t}^{(1)}(p) e^{2f^{-}(p)t},
\\
\widehat{U}_{t}^{++}(p)&=\left(\frac{2\lambda c^{-}\lambda^{+}\hat{a}^{+}(p)%
}{\mu^{-}-\mu^{+}}+\frac{2c^{-}\lambda^{2}\hat{a}^{2}(p)}{%
\mu^{-}-f^{+}(p)-f^{-}(p)}\cdot\frac{\mu^{-}+2}{\mu^{-}-2f^{-}(p)}%
\right)G_t^{(2)}(p) e^{2f^{+}(p)t} \\
&\quad+2c^{+}\lambda^{+}\hat{a}^{+}(p)\cdot\frac{\mu^{-}-\mu^{+}+\lambda}{%
\mu^{-}-\mu^{+}}\cdot\frac{e^{\mu^{+}t}-e^{2f^{+}(p)t}}{\mu^{+}-2f^{+}(p)} \\
&\quad+\left(\lambda^{2}\hat{a}^{2}(p)\hat{\varphi}^{--}(p)-\frac{%
2c^{-}\lambda^{-}\hat{a}^{-}(p)\lambda^{2}\hat{a}^{2}(p)}{\mu^{-}-2f^{-}(p)}%
\right)\left(G_t^{(1^{})}(p)\right)^{2} e^{2f^{-}(p)t} \\
&\quad+\left(2\lambda\hat{a}(p)\hat{\varphi}^{+-}(p)-\frac{2c^{-}\lambda^{2}\hat{a%
}^{2}(p)}{\mu^{-}-f^{+}(p)-f^{-}(p)}\cdot\frac{\mu^{-}+2}{\mu^{-}-2f^{-}(p)}%
\right)\\ &\qquad \times G_t^{(1^{})}(p) e^{[f^{+}(p)+f^{-}(p)]t},
\end{align*}
where we denote objects which are not defined
for $p\in\D$ by
\begin{align*}
&&G_t^{(1)}(p)&=\frac{e^{[f^+(p)-f^-(p)]t}-1}{f^+(p)-f^-(p)},
&p&\in\DC_1:=\X\setminus\D_1,\\
&&G_t^{(2)}(p)&=\frac{e^{[\mu^--2f^+(p)]t}-1}{\mu^--2f^+(p)},
&p&\in\DC_2:=\X\setminus\D_2.
\end{align*}
Obviously $\dist(p,\D_1)\to 0$ implies $g_1(p)\to0$
and, hence, $G_{t}^{(1)}(p)\to t$. In the
same manner $\dist(p,\D_2)\to 0$ provides $G_{t}^{(2)}(p)\to t$. Therefore, for obtain
the explicit expressions for $\widehat{U_{t}}^{{+-}}(p)$ and $\widehat{U_{t}}^{{++}}(p)$ on $\D\setminus\{0\}$
it's enough to define
\[
G_t^{(1)}(p):=t, \quad
p\in\D_1; \qquad G_t^{(2)}(p):=t, \quad
p\in\D_2.
\]
Then we have for any $b\in L^1(\X)\cap L^\infty(\X)$
\begin{equation*}
\vert b(p) \vert G_{t}^{(1)}(p) e^{f^{-}(p)t}\leq
\begin{cases}
\vert b(p)\vert \dfrac{e^{f^+(p)t}-e^{f^-(p)t}}%
{f^+(p)-f^-(p)},\qquad p\in\DC_1\setminus\{0\},\\
\vert b(p)\vert\dfrac{ e^{-1}}{-2f^-(p)}, \qquad\qquad\quad\, p\in\D_1.
\end{cases}
\end{equation*}
And by result and proof of Lemma~\ref{boundint}
this function has integrable majorante (which
doesn't depend on $t$) on
whole $\X$.
Note also that $e^{f^\pm(p)t}\leq 1$, hence,
all terms with $G_t^{(1)}$ have this property.
Next,
\begin{equation*}
\vert b(p) \vert G_{t}^{(2)}(p) e^{2f^{+}(p)t}\leq
\begin{cases}
\vert b(p)\vert \dfrac{e^{\mu^-t}- e^{2f^{+}(p)t}}{\mu^--2f^+(p)}, \quad p\in\DC_2\setminus\{0\},\\
\vert b(p)\vert \dfrac{e^{-1}}{-2f^+(p)},\qquad\quad\
\,\,
p\in\D_2.
\end{cases}
\end{equation*}
If $\mu^-<0$ then may apply the previous
considerations ($\mu^-\in C^-$). Otherwise, we may use that
a function $u(t)=\dfrac{1-e^{at}}{-a}$ ($a<0$)
is increasing and, hence, bounded by $u(+\infty)=-
\dfrac{1}{a}.$
Note also that other numerators depended on $t$ in the expressions for $\widehat{U}_t^{--}$,
$\widehat{U}_t^{+-}$, $\widehat{U}_t^{++}$
may be estimated by $2$ (recall that corresponding
denominators are not equal to $0$ if $p\neq0$).
Therefore, for prove that functions $\widehat{U}_t^{--}$,
$\widehat{U}_t^{+-}$, $\widehat{U}_t^{++}$
have integrable majorants it's enough to
show that all terms which independent on $t$ are
integrable.
Recall that $\hat{\varphi}^{--}$, $\hat{%
\varphi}^{+-}$ and $\hat{\varphi}^{++}$ are bounded, $\hat{a%
},\ \hat{a}^{+}$ and $\hat{a}^{-}$ are bounded and integrable. Thus, we should prove integrability
of two terms:
\begin{equation}\label{lala}
\frac{b(p)}{\mu^{\pm}-2f^{\pm}(p)}
\quad \text{and}\quad \frac{%
b(p)}{\mu^{-}-f^{-}(p)-f^{+}(p)}\cdot
\frac{1}{\mu^{-}-2f^{-}(p)},
\end{equation}
where $b\in L^1(\X)\cap L^\infty(\X)$.
If $\mu^{\pm}=0$ then we have
\begin{align*}
\frac{b(p)}{\mu^{\pm}-2f^{\pm}(p)}=-\frac{1}{2}\frac{b(p)}{\hat{a}^{\pm}(p)-1}
\end{align*}
and due to Lemma~\ref{int_a} these functions
are integrable. If $\mu^{\pm}< 0$ then using
\eqref{flessmu} we obtain
\begin{align*}
0<-\mu^{\pm} \leq \mu^{\pm}-2f^{\pm}(p),
\end{align*}
that implies
\[
\frac{\vert b(p)\vert}{\mu^{\pm}-2f^{\pm}(p)} \leq\frac{\vert
b(p)\vert}{-\mu^\pm}
\]
which are also integrable functions.
Next, if $\mu^{-}=0$ then $\mu^{+}<0$ and
using \eqref{flessmu}
\[
(\mu^{-}-f^{-}(p)-f^{+}(p))(\mu^{-}-2f^{-}(p))\geq
-2\mu^+(1-\hat{a}^{-}(p)),
\]
and we again may use Lemma~\ref{int_a}.
Finally, if $\mu^{-}<0$ then $\mu^+=0$ and
\[
\bigl( (\mu^{-}-f^{-}(p))+(-f^{+}(p))\bigr) \cdot
\bigl(\mu^{-}-2f^{-}(p)\bigr) \geq -\mu^-
(1-\hat{a}^{+}(p)),
\]
and we also may use Lemma~\ref{int_a}.
As a result, the functions $\widehat{U}_t^{--}$,
$\widehat{U}_t^{+-}$, $\widehat{U}_t^{++}$ have integrable majorants
and by dominated convergence theorem for obtain limits of
${U}_t^{--}$, ${U}_t^{+-}$, ${U}_t^{++}$ as $t\to\infty$ we may
calculate limits of the Fourier transforms and after apply the
inverse Fourier transforms. Hence, taking $t\to\infty$ in the
expressions for $\widehat{U}_t^{--}$, $\widehat{U}_t^{+-}$,
$\widehat{U}_t^{++}$ we immediately obtain the statement of the
Theorem~\ref{thm_as_2} with functions $\Omega ^{++},~\Xi ^{--},~\Xi
^{+-},~\Xi ^{++}$ which are inverse Fourier transforms of
\eqref{a1}--\eqref{a4}.
\subsection*{Acknowledgments}
The authors acknowledge the financial support of the DFG through SFB
701 ``Spectral structures and topological methods in mathematics'',
Bielefeld University.
|
1,108,101,565,625 | arxiv | \section{Introduction}
\input{text/1.introduction}
\section{Materials and Methods}
\input{text/2.method}
\section{Results}
\input{text/3.results}
\section{Analysis and Discussion}
\input{text/4.discussion}
\section{Conclusion}
\input{text/5.conclusion}
\section{Acknowledgements}
This study was supported in part by the National Institutes of Health (NIH) under award numbers R01EB031849, R01NS106030, and R01EB032366; and in part by the Office of the Director of the NIH under award number S10OD0250111. The content of this paper is solely the responsibility of the authors and does not necessarily represent the official views of the NIH.
\clearpage
\bibliographystyle{splncs04}
\subsection{Related Work}
While several studies have addressed 3D fetal brain segmentation on stack-of-slices or reconstructed fetal MRI scans~\cite{anquez2009automatic,salehi2017auto,khalili2017automatic,ebner2018automated}, only a few studies have addressed the more challenging task of segmenting the fetal brain on every slice. Keraudren et al.~\cite{keraudren2014automated} developed a method based on support vector machines and random forests. More recent works have almost exclusively been based on deep learning (DL), and in particular convolutional neural networks (CNNs). These methods are more suitable for real-time applications because they can harness the parallel computation capabilities of Graphical Processing Units (GPUs)~\cite{rampun2019automated}. Salehi et al.~\cite{salehi2018real} used a DL method based on the U-Net architecture~\cite{ronneberger2015u}. Wang et al.~\cite{wang2019aleatoric} computed aleatoric uncertainty and used test time augmentation to improve the accuracy of fetal brain segmentation on 2D slices. While these works focused on improving segmentation accuracy, none of them addressed the accuracy-speed trade-off. To address this gap, in this paper we focused on improving inference speed as well as accuracy.
Many applications demand real-time image processing. This demand has given rise to a growing body of real-time DL-based methods~\cite{papadeas2021real}. The majority of these works have aimed at reducing the computation time by devising lighter or specialized network architectures. A typical example of architectural innovations is the depthwise-separable convolution, which breaks down a 3D convolution operation into a succession of 2D and 1D convolutions~\cite{howard2017mobilenets}. Another approach to reducing the computational cost is channel shuffling as used in ShuffleNets~\cite{zhang2018shufflenet,ma2018shufflenet}. Gamal et al.~\cite{gamal2018shuffleseg} proposed ShuffleSeg based on ShuffleNet by using ShuffleNet with grouped convolutions, channel shuffling as encoder, and FCN8s~\cite{long2015fully} as decoder. ENet~\cite{paszke2016enet} uses early downsampling of the input to extract relevant image features while reducing the image size. ENet also uses a much smaller decoder module than in typical symmetric encoder–decoder architectures~\cite{badrinarayanan2017segnet,ronneberger2015u}.
Two-branch networks~\cite{poudel2018contextnet,yu2018bisenet} are another way to design faster models and are among the fastest existing methods. Unlike standard models where the entire network learns low-level and high-level details, in two-branch networks these two tasks are performed by two separate branches. A shallow branch captures spatial details and generates high-resolution feature representation, while a deeper but lightweight branch learns high-level semantic context. ContextNet~\cite{poudel2018contextnet}, Fast-SCNN~\cite{poudel2019fast}, and BiseNet~\cite{yu2018bisenet} are examples of two-branch networks. An important consideration in designing these architectures is to ensure proper integration of high-level and low-level context information. For example, ICNet~\cite{zhao2018icnet} computes a multi-resolution set of feature maps and employs a cascade feature fusion unit to fuse these feature maps, whereas DFANet~\cite{li2019dfanet} uses several interconnected encoding paths to add high-level context into the encoded features.
In this work we aimed to design a network with proper, efficient integration of high-level and low-level information to achieve high accuracy and very fast inference in fetal brain MRI segmentation. To achieve this, we developed a new, efficient CNN-based network, which we term Real-time Fetal Brain Segmentation Network (RFBSNet). RFBSNet uses an encoder-decoder architecture with forward connections to retain accuracy; and a two-branch architecture with an input downsampling module to achieve fast inference. We compared RFBSNet with eight alternative state-of-the-art DL models. In the following sections, we provide a detailed description of our methods, data, results, and analysis.
\subsection{Proposed Network Architecture}
We designed RFBSNet to strike a balance between inference speed and accuracy. Fig.~\ref{fig:RFBSNet} shows the layout of RFBSNet. It contains an input downsampling module, a feature extractor, a decoder, and a classification module. In the following, we describe each structure module in more detail.
\subsubsection{Input downsampling module.}
The first module in our proposed network is a downsampling module that reduces the size of the input image while also providing high resolution spatial information into the classifier module using a forward path. Input downsampling can greatly speed up the network by significantly reducing the amount of computation performed by all down-stream network layers. When this down-sampling is not excessive and is carried out using learnable functions, such as a convolution layer, the loss in segmentation accuracy can be very small. However, excessive down-sampling can result in a loss of important detail such as fine object boundaries~\cite{paszke2016enet}. Besides, downsampling the input image by a factor of $m$ would require upsampling by the same factor in order to obtain a segmentation map with the same size as the input image. Although upsampling can also be accomplished using learnable transposed convolutions, it can result in further loss of fine detail if it is excessive. To avoid these negative effects, we used a down-sampling module, shown in Fig.~\ref{fig:init}, that consists of two paths: (1) a max-pooling path with non-overlapping $2 \times 2$ windows, and (2) a convolutional layer with $3 \times 3$ kernels. The outputs of these two paths are concatenated.
\subsubsection{Feature Extractor.}
This module is responsible for learning multi-resolution image features for accurate segmentation. Our feature extractor module shares its computation of the first few layers with the input downsampling module. This parameter sharing not only reduces the computational complexity of the network, it also improves the segmentation accuracy.
In this architecture, we used one convolutional layer followed by ReLU in the shallow branch to encode detailed spatial information.
The deep feature extractor branch of RFBSNet provides sufficient receptive field.
We deployed U-Net style~\cite{ronneberger2015u} forward skip connections to fuse multi-resolution features into the decoder module.
\begin{figure}[t!]
\includegraphics[width=0.95\textwidth]{figures/network-crop.pdf}
\caption{Overview of the proposed architecture (RFBSNet). It consists of an input down-sampling module, a feature extractor, a decoder, and a classifier, with two branches and forward connections from the feature extractor to the decoder. All modules are built using classical convolution layers using operations shown in the figure legend. The detail of the down sampling module is shown in Fig.~\ref{fig:init}. Numbers next to each block show the number of channels, while the length indicates the spatial size considering the input size of I.}
\label{fig:RFBSNet}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=0.35\textwidth]{figures/init-crop.pdf}
\caption{The input downsampling module in RFBSNet consists of a convolution and a max pooling path. The outputs are concatenated to build a feature map.} \label{fig:init}
\end{figure}
\subsubsection{Decoder and Classification modules.}
A U-Net-type decoder with skip connections upsamples the features learned by the different sections of the feature extractor module to the size of the feature maps generated by the input downsampling module. These feature maps are finally fused together using a simple addition.
In the classification module, an upsampling layer and a pointwise convolution layer are applied to the fused feature maps. A softmax operation is applied to the final layer to generate class probability maps.
\subsection{Alternative Methods and Evaluation Metrics}
We compare the proposed RFBSNet to two standard networks for medical image segmentation (U-Net~\cite{ronneberger2015u} and SegNet~\cite{badrinarayanan2017segnet}), as well as several recent architectures that have been proposed for real-time segmentation (see Table~\ref{table:results}). We also introduce ShuffleSeg V2 following the design of ShuffleSeg~\cite{gamal2018shuffleseg}. It employs ShuffleNet V2~\cite{ma2018shufflenet} as encoder and FCN8s~\cite{long2015fully} as decoder.
The accuracy of all methods are evaluated and compared using the Dice similarity coefficient and Intersection-over-Union IoU, also known as Jaccard index, metrics defined as $Dice(P, R) = \frac{2|P\cap R|}{|P|+|R|} = \frac{2TP}{2TP+FP+FN}$ and $IoU(P,R) = \frac{|P\cap R|}{|P\cup R|} = \frac{TP}{TP+FP+FN}$ respectively. where P is predicted brain mask, R is a ground truth mask and TP, FP, and FN are the true positive, false positive, and false negative rates, respectively.
We assess segmentation speed in terms of the average inference time and standard deviation with 100 iterations for each method while using batch size of 1. In addition, we report the number of floating point operations (FLOPs) and the number of trainable parameters for each network.
\subsection{Data, Implementation, and Training}
The fetal MRI data used in this study were acquired using 3T Siemens scanners. The study was approved by the institutional review board; and written informed consent was obtained from all research MRI participants. For each subject, multiple half-Fourier single shot turbo spin echo (HASTE) images were acquired with in-plane resolution of 1 to 1.25 mm, and slice thickness of 2 to 4 mm. The gestational ages of the fetuses at the time of scans were between 22 to 38 weeks (mean=29, stdev=5).
In total, 3496 2D fetal MRI slices (of 131 stacks from 23 fetal MRI sessions) were included in the training and validation procedure (80\% train, 20\% validation). A set of 840 2D slices (17 stacks) of two normal fetuses without severe artifacts was used as normal test set, and a set of 136 2D slices of a fetal MRI scan with artifacts (from 4 stacks) was used as the challenging test set. An experienced annotator carefully segmented the fetal brain in every slice of all these stacks. We used these manual segmentations as the ground truth for model training and evaluation.
All experiments were conducted with an NVIDIA GeForce RTX 2080 Ti, using TensorFlow and Keras 2.6.0. All models were trained with a batch size of 8 and input image size of $256 \times 256$. We used Dice similarity coefficient between the network predictions and the ground truth as the training loss function. The learning rate for each of the compared networks was tuned separately. For our model we used an initial learning rate of $1\times 10^{-4}$, which we multiplied by 0.9 after every 2000 training steps. We trained each model for 100 epochs using Adam optimization~\cite{kingma2014adam} of stochastic gradient descent.
\subsection{Data and Implementation Details}
Table~\ref{table:results} summarizes the performance of our proposed RFBSNet compared to other methods.
In terms of almost all evaluation criteria,
RFBSNet outperformed the standard methods as
well as other state-of-the-art real-time segmentation models. It reached 97.99\% aDice (average Dice of all test images), 96.12\% aIoU on normal and 86.04\% aDice, 75.50\% aIoU on challenging test sets with outstanding inference time of 3.36 ms. Indeed, our network can run on a single GPU in real time, i.e., it runs as soon as a single MRI slice is acquired and reconstructed. We note that RFBSNet performed better than the standard medical image segmentation network U-Net in terms of both accuracy and speed while having $\approx 14$ times less number of parameters and FLOPs. Our method also outperformed other real-time segmentation methods in both accuracy and speed while representing comparable number of parameters and FLOPs.
We performed paired t-tests with a $p$ value threshold of 0.001 to test if the segmentation accuracy, in terms of Dice and IoU on the test sets, for our model was higher than other models. These tests showed that our model was significantly more accurate than all competing real-time segmentation models on the normal and challenging test images in terms of both Dice and IoU. Our model was also significantly more accurate than SegNet. However, the differences with the U-Net were not statistically significant ($p \approx 0.3 $). We note that in addition to computation time, both UNet and SegNet require high GPU memory which may limit their use with larger images on standard GPUs.
Example segmentation results can be seen in Fig.~\ref{fig:mask}. In addition to our own method, in this figure we have shown the results of those competing methods that were proposed originally for real-time segmentation. As these representative examples show, compared to those other methods, which showed large segmentation errors and often completely failed to segment the fetal brain on challenging images, our method accurately segmented both normal and challenging images.
\begin{figure}[t!]
\centering
\includegraphics[width=0.85\textwidth]{figures/seg-crop.pdf}
\caption{Representative examples of predicted brain masks overlaid on original fetal MRI slices for normal cases (top 3 rows) and challenging cases (bottom 3 rows). Note that RFBSNet correctly segmented the brain in all slices of this challenging test case which was not segmented by the other methods.} \label{fig:mask}
\end{figure}
|
1,108,101,565,626 | arxiv | \section{Introduction}
For time to event outcomes, hazard models are convenient because they allow for straightforward regressions on covariates. Nevertheless, it is well-known that estimates on the hazard scale, e.g.\ hazard ratios (HRs), are hard to interpret causally \citep{robins1989probability, greenland1996absence, hernan2010hazards, aalen2015does, stensrud2017exploring}. A particular issue arise due to left truncation: An exposure may be introduced at time $t_0$, but hazard estimates at a later time $t > t_0$ are calculated from subjects alive at $t$. Hence, at any $t > t_0$ the HRs are derived from left truncated samples, and not from the baseline population. Standard HR models, e.g.\ the Cox model, are based on multiplying likelihood functions for each event time, and thereby they are sequentially calculated from left truncated samples. Thus, such HRs do not have an immediate relevance for individual subjects \citep{robins1989probability}, even if the subjects are followed from the onset of an exposure. This issue has been denoted the inbuilt selection bias of the HR \citep{hernan2010hazards}, truncation bias \citep{vansteelandt2017survivor} or survival bias \citep{ aalen2015understanding}.
The issue of left truncation is particularly severe if the onset of follow-up is delayed. In observational studies, exposures are often present before the subjects are recruited to the study, and the study sample is not necessarily representative of the pre-exposure population. Even in randomised controlled trials, the same issue arise when the treatment effect is assessed in individuals at a time $t$ later than randomisation at $t_0$ \citep{hernan2010hazards, greenland1996absence, aalen2015does, stensrud2017exploring,mcnamee2017serious}. In such scenarios, causal inference is not straightforward.
Intuitively, individuals who die before $t_0$ are expected to be the more \textit{frail}; the hazard of an event may be heterogeneously distributed across individuals, and the subjects who died before $t_0$ were expected to have higher average hazard \citep{vaupel1979impact, hougaard1995frailty, aalen2015understanding}. In some scenarios we have a good understanding of the factors that determine the individual risk, and then we may include these factors as covariates in our model. However, the heterogeneity may often be due to unobserved factors, and adjusting for measured factors will not be sufficient. In such situations, frailty models have been used to account for the variation in susceptibility to disease \citep{vaupel1979impact,lancaster1979econometric, moger2004frailty, haugen2009frailty, moger2008regression, moger2004analysis}. The frailty models may be seen as extensions of the Cox model, allowing for an unknown heterogeneity parameter. Unfortunately, specifying the parameters of the frailty distribution is not obvious, in particular when analysing data on independent individuals.
This article suggests an approach to learn about the frailty distribution using published summary data. After estimating the frailty distribution, it is possible to explore the causal interpretation of HRs. In particular, I describe a method to adjust for survival bias in proportional hazards models. The strategy consists of two steps: 1) a standard Cox proportional hazards model is fitted to obtain a marginal HR, and 2) this HR is adjusted to account for unmeasured heterogeneity, using frailty theory. The method relies on strong parametric assumptions, but under these conditions we may use published summary data from e.g.\ twin studies to find the frailty distribution, and thereby obtain frailty adjusted HRs, assuming that the frailty is shared among identical twins.
I will illustrate that left truncation is an issue in genetic epidemiology, e.g.\ Mendelian Randomisation studies \citep{vansteelandt2017survivor,boef2015mendelian}. These studies rely on genetic variants that are carried from conception, but subjects are often included into these studies in late adulthood. In particular, I will use published summary data on the relation between mortality and a gene associated with alcohol consumption to highlight the magnitude of bias.
\section{Issues with conditioning on survival.}
Before the modeling framework is formally defined, it is helpful to consider truncation bias in a simple causal graph (Figure \ref{Figure:DAGbasic}). We are interested in the effect of a binary exposure $X$ (taking values 0 and 1) on our survival outcome $Y$. That is, we aim to assess how the rate of $Y$ at a time $t$ may differ under (hypothetical) interventions on $X$. We observe individuals conditioning on survival until time $t_1$ (Hence the node $S$ in Figure \ref{Figure:DAGbasic}). There is an unknown factor $U$ that also influences the event times.
The graph provides some immediate insights. The truncation bias due to conditioning on $S$ is described by the non-causal path $X \rightarrow S \leftarrow U \rightarrow Y$. Under the null hypothesis of no causal effect of $X$ on $Y$, $X\rightarrow Y$ is absent, i.e.\ there is no truncation bias, and standard hypothesis tests are therefore null-calibrated even if we condition on $S$. Similarly, if $X \rightarrow S$ is absent, truncation bias is neither an issue. This scenario e.g.\ arise if the exposure effect is delayed such that it does not influence entry into the study \citep{vansteelandt2017survivor}. There may also be particular parameterisations of th data generating mechanism that do not lead to bias, which e.g.\ may be derived from additive hazards models \citep{vansteelandt2017survivor}.
For predictions \textit{per se}, causal effects are not always the primary interests, and the considerations above are not necessarily relevant; indeed, standard models may sometimes be immediately applicable.
\section{Notation and motivation}
Similar to \citet{vansteelandt2017survivor}, who recently studied survival bias in the additive hazards framework, let $T$ denote lifetimes, $T_0$ indicate the time of left truncation and $C$ denotes the censoring time. Let $X$ be a binary exposure, $L$ is a vector of measured covariates, and $U$ is an unmeasured time-fixed variable which we also denote the frailty, such that $U \mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}} L$, $X \mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}} U$ and $X \mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}} L$. We use superscript notation for counterfactual outcomes such that $T^{X=x}$ denotes the lifetime if, possibly contrary to fact, $X$ is set to $x$. We will assume that event times are generated from the hazard
\begin{align}
h^{X=x}(t | L,U) = h(t | X=x, L,U) = h_0(t) \exp(\beta_xX + \beta^T_lL) U,
\label{eq: new basic frailty}
\end{align}
where $L$ is a vector of measured covariates and $r(t)=e^{\beta_x}$ is the counterfactual HR under $X$ conditional on $U$ and $L$. Equation \eqref{eq: new basic frailty} is coherent with the classical frailty model, suggested by e.g.\ \citet{vaupel1979impact} and \citet{lancaster1979econometric}. The model in Equation \eqref{eq: new basic frailty} lends on the untestable assumptions that $U$ is time constant, and that $r(t)$ is the same for all levels of $U$. These assumptions allow us to consider $r(t)$ as a marginalised estimand because
\begin{align}
\frac{E_U(h^{X=1}(t |L, U))}{E_U(h^{X=0}(t | L, U))} = r(t)
\label{eq: marginalised hazard ratio}.
\end{align}
Such ratios are often considered in the statistics and economics literature \textit{individual hazard rates} \citep{hougaard1995frailty, van2001duration, van2016inference}.
In a deterministic counterfactual framework individual hazards may not be interesting \textit{per se}: At the event time, the individual hazard will be infinity, and at any other time the hazard will be 0, meaning that the individual HR is either 0 or undefined. Nevertheless, $r(t)$ does have a meaning under the data generating mechanism in Equation \eqref{eq: new basic frailty}, which may be interpreted as a particular stochastic counterfactual model: Under these particular assumptions, it may be interpreted as the probability of causation due to $X$ \citep{robins1989probability}, which e.g.\ seems to be used in compensation issues: Assume that a subject exposed to an environmental toxin $X$ experienced an event at $t$, and an insurance company will give compansation based on the probability that the event was caused by $X$. Then, $\frac{r(t)-1}{r(t)}$ may denote the probability of causation. Intuitively, $r(t)$ is the relative effect that an individual cares about: It is the rate ratio under the counterfactual scenarios of being exposed or unexposed. Since $U$ is unknown, $r(t)$ cannot yield precise information about the absolute increase in hazard or risk.
The frailty model in Equation \ref{eq: new basic frailty} is compatible with a Cox model with hazard ratio $r^{\star}$ of $X$ only if
\begin{align}
r^{\star} = \frac{E_U(h(t |X=1,L, U))}{E_U(h(t | X=0,L, U))}
\label{eq: comparing cox frailty}
\end{align}
is valid for all $t$. Importantly, under a constant $r(t) = r$, Equation \eqref{eq: comparing cox frailty} will only hold under very special conditions, such as $r=1$ or $\text{VAR}(U)=0$. However, even if the Cox proportional hazard assumption is valid, there is still no simple relation between $r^{\star}$ and $r(t)$ \citep{hernan2010hazards}. In contrast to $r(t)$, it is therefore hard to interpret $r^{\star}$, and it is not a counterfactual rate ratio with an immediate relevance to an individual.
The Cox PH assumption can obviously be assessed from the data during follow-up, which is clearly an advantage of the method. Before the onset of follow-up ($T_0$), however, we have no way to justify the proportional hazards assumption. Hence, not only the frailty model, but also the standard Cox model will rely on untestable assumptions in left truncated samples. We will hereby assume that data are generated from models in which $r(t)=r$ for all $t$, which is analogous to the proportional hazards assumption of the Cox model.
\subsection{Study populations}
Assume that we got a random sample $S_1$ of subjects $1,2,...,n$ from a population $P$ with event times generated by Equation \eqref{eq: new basic frailty}. We observe left truncated lifetimes, i.e. all study subjects got $T > t_0$, where $t_0$ is the time of left truncation. The event times may be censored at $C$ such that for each individual we observe $V=\text{min}(T,C)$, and we assume independent censoring. For the ease of presentation, there are no measured covariates $L$ in the following considerations, but the approach is valid when $L$ is present. Based on the observed event times we aim to estimate $r$. The frailty $U$ is unmeasured, and left truncation is obviously an issue \citep{vansteelandt2017survivor}.
Assume also that we got summary results from another random sample $S_2$ of $P$, consisting of pairs of individuals sharing the same $U$, and each subject is unexposed to $X$. If $U$ was solely determined by genetic factors, $S_2$ could e.g.\ be a population of monozygotic twin pairs. This sample only contains information on whether each subject survived until $T_0$. In practice, these data could e.g.\ be derived from a twin birth registry. Suppose also that we know the parametric distribution of $U$. This information may allow us to estimate $r(t)$ in a left truncated sample.
\section{Learning the frailty distribution from the data}
\label{sec: using family}
Let $U$ be a random variable with $\text{E}(U)=1$, i.e.\ any standardised (frailty) distribution with a finite mean. Let $S(t)=\text{P}(T>t)$, i.e. the survival probability unconditional on $U$ and $X$. The cumulative baseline hazard is $H_0(t) = \int_{o}^{t} h_0(t) dt$. Let $T_i$ and $T_j$ denote the time of event for individuals $i$ and $j$ who are monozygotic co-twins. Assuming everybody is unexposed, and let the twin recurrence risk at time $t$ be TRR($t$), which is the relative risk of surviving until time $t$, given a co-twin lived longer than $t$,
\begin{align}
\text{TRR(t)} &= \frac{\text{P}(T^{X=0}_i>t |T^{X=0}_j>t)}{\text{P}(T^{X=0}_i>t)} \nonumber \\
&= \frac{\text{P}(T^{X=0}_i>t, T^{X=0}_j>t )}{\text{P}(T^{X=0}_i>t)^{2}} \quad \text{ because } \text{P}(T^{X=0}_i>t)=\text{P}(T^{X=0}_j>t)\nonumber \\
&= \frac{E[S(t^{X=0}|U)^2]}{S(t^{X=0})^{2}} \nonumber \\
&= \frac{E[e^{-2H_0(t)U}]}{E[e^{-H_0(t)U}]^{2}} \nonumber \\
&= \frac{L[2H_0(t)]}{L[H_0(t)]^2},
\label{eq:TRR}
\end{align}
where $L$ denotes the Laplace transform of $U$.
\subsection{Analytic results for the power variance function distributions}
\label{sec:PVFderivations}
Assume that the distribution of $U$ belongs to the large class of power variance function (PVF) distributions, which e.g.\ includes the gamma distribution, the inverse Gaussian distribution, the (compound) and the Poisson distribution. The PVF family also includes the Hougaard distributions\citep{hougaard2012analysis} that are continuous and unimodal and cover the inverse Gaussian distribution as a special case. Similar to \citet{aalen2008survival}, we consider PVF distributions with $\text{E}(U)=1$, and we express the expected value, variance and Laplace transform of the PVF family as
\begin{align}
\frac{m\rho}{\nu} = 1, \nonumber \\
\text{VAR}(U ) & = \frac{(m+1)}{\nu}, \nonumber \\
\qquad L(c|\nu,m) &= e^{-\frac{\nu}{m}(1-(\frac{\nu}{\nu+c})^m)},
\label{generalProperties E1}
\end{align}
where $\nu > 0$, $m > -1$ and $m\rho > 0$. The survival function under a PVF distribution is
\begin{align}
\label{survivalFunction}
S(t^ {X=0}) &= \text{E}(e^{-H_0(t)U}) \nonumber \\
&=L(H_0(t)|\nu,m) \nonumber \\
&= e^{-\frac{\nu}{m}\left(1-\left(\frac{\nu}{\nu+H_0(t)}\right)^m\right)}.
\end{align}
We will use Equations \eqref{generalProperties E1} and \eqref{survivalFunction} to find $\text{VAR}(U)$ under a particular PVF distribution.
\subsubsection{Calculations for the gamma distribution}
\label{sec:gammaExample}
To illustrate, we consider the gamma distribution, which is probably the most frequently used distribution in frailty theory. The gamma distribution is mathematically tractable, and it may be theoretically appealing because the heterogeneity of frailty distributions converges to a gamma distribution among survivors \citep{abbring2007unobserved}. However, assuming a gamma distributed $U$ at $t_0$ can typically not be tested, and we must heuristically justify the distribution, e.g.\ because we believe that $U$ consists of a sum of factors, making it continuous in the population. Using the notation in Equations \eqref{generalProperties E1} and \eqref{survivalFunction}, the gamma distribution arises when $\rho \rightarrow \infty$ and $m \rightarrow 0$ such that $\rho m \rightarrow \eta > 0$, where $\eta$. since $\text{E}(U) = \frac{\eta}{\nu} = 1$, the gamma frailty reduces to a one parameter distribution, and the Laplace transform becomes
\begin{align}
L(c | \nu) = \left(\frac{\nu}{\nu + c}\right)^\nu.
\label{eq gamma laplace}
\end{align}
We may simplify Equation \eqref{survivalFunction} to
\begin{align}
\label{survivalFunctiongamma}
S(t^{X=0}) = \text{E}(e^{-H_0(t)U}) =L(H_0(t) | \nu) = \left(\frac{\nu}{\nu + H_0(t)}\right)^\nu,
\end{align}
and from Equation \eqref{eq gamma laplace} we simplify TRR(t) to
\begin{align}
\label{TRRfunctiongamma}
\text{TRR(t)} = \frac{\left(\frac{\nu}{\nu + 2H_0(t)}\right)^\nu}{S(t^{X=0})^2},
\end{align}
We use Equation \eqref{survivalFunctiongamma} to express $H_0(t)$ as a function of $S(t | X=0)$ and $\nu$
\begin{align}
\label{hazardFunctiongamma}
H_0(t) = \frac{\nu(1-S(t^{X=0})^{\frac{1}{\nu}})}{S(t^{X=0})^{\frac{1}{\nu}}}.
\end{align}
Finally, we replace $H_0(t)$ in Equation \eqref{TRRfunctiongamma} by the right side of Equation \eqref{hazardFunctiongamma}, and to find $\nu$ we can numerically solve
\begin{align}
\label{eq: nu solver}
S(t^{X=0})^2 & \text{TRR(t)} - \left(\frac{1}{1+2\left(\frac{1- S(t^{X=0})^{\frac{1}{\nu}}}{ S(t^{X=0})^{\frac{1}{\nu}}}\right)}\right)^{\nu} = 0 \nonumber \\
\end{align}
for any time point $t=t_1$. When $\nu$ is derived, we are able to specify the variance of a gamma distributed $U$. The same logic can be used for other PVF distributions (see the Appendix for R scripts that implement these methods numerically). We consider a counterfactual population of unexposed individuals, and this population is generally unobserved. In practice, we will estimate the $TRR$ from observed quantities.
In the next section, I will describe how the information on $U$ can be used to obtain estimates of HRs $r$ with a causal interpretation. This requires an estimate of the marginal HR among survivors at a specific time $t_1$, which is approximated by a Cox proportional hazards estimate in an interval $[t_1, t_1 + \delta]$.
\subsection{Using the marginal HR to estimate the causal HR}
\label{sec: observed to causal}
We continue to study the PVF family. Assume that our data are generated by a proportional hazards model as in Equation \eqref{eq: new basic frailty}, and let $r_{mar}(t)$ denote the marginal HR among survivors at time $t$. Let $r$ denote the (constant) causal HR conditional on $U$, such that \citep{aalen2008survival}
\begin{align}
r_{mar}(t) = r \Bigg( \frac{1+\frac{H_0(t)}{\nu}}{1+\frac{rH_0(t)}{\nu}} \Bigg)^{m+1},
\label{equation:observedHazardExpressedWithCausal}
\end{align}
which means that
\begin{align}
\label{eq:observedRiskSolver}
r_{mar}(t)^{\frac{1}{m+1}} \left(1+\frac{H_0(t)r}{\nu}\right) = r^{\frac{1}{m+1}}\left(1+\frac{H_0(t)}{\nu}\right).
\end{align}
Assume that $\nu$ is derived by the approach in Section \ref{sec:PVFderivations}, e.g. using twin data. The marginal HR at a particular time $t_1$ is approximately derived from a Cox proportional hazards model in an interval $[t_1, t_1 + \delta]$. For the gamma distribution, $m\rightarrow 0$, and we find $r$ analytically by solving
\begin{align}
r = r_{mar}(t) \left(1+\frac{H_0(t)}{\nu} - \frac{H_0(t)r_{mar}(t) }{\nu}\right)^{-1}
\label{eq:gammaRiskSolver}
\end{align}
for any time point $t=t_1$, where we use Equation \eqref{hazardFunctiongamma} to find $H_0(t)$. For the inverse Gaussian distribution, $m=-0.5$, and we obtain $r$ analytically by solving a quadratic equation
\begin{align*}
r^{2}\left(1+\frac{H_0(t)}{\nu}\right) - r\left(r_{mar}(t)^{2}\frac{H_0(t)}{\nu}\right) - r_{mar}(t)^{2} = 0. \\
\end{align*}
For other distributions in the PVF class, we may solve Equation \eqref{eq:observedRiskSolver} with respect to $r$ numerically. Confidence intervals of $r$ can be found numerically, as suggested in Appendix \ref{appendix: confidence intervals}. In Appendix \ref{appendix: simulations to verify the approach}, a simulation study of plausible scenarios was performed, in which the adjusted 95\% confidence intervals obtained approximately $95\%$ coverage of the true $r$ in all scenarios.
In applied settings, we may expect a slight bias towards a null effect because because the marginal $HR$ at $t_1$ is approximated by a Cox proportional hazards estimate in $[t_1, t_1 + \delta]$, as suggested in Section \ref{sec:gammaExample}. This means that we adjust for the impact of $U$ until $t_1$, but we do not adjust for the effect of $U$ during follow up $[t_1, t_1 + \delta]$. In the simulations in Appendix \ref{appendix: simulations to verify the approach}, $t_1=50$ was much larger than $\delta =2$, and the bias was negligible.
Nevertheless, \citet{mcnamee2017serious} has recently suggested a heuristic approach to deal with this model misspecification if the frailty is gamma distributed: Assume that we follow a population from $t_0$, and let $t_{median}$ be the median of all event times in the population. Under a gamma distributed frailty, replacing $t$ by $t_{median}$ in Equation \eqref{eq:gammaRiskSolver}, will yield adequate estimates of $r$. This approach was shown to perform satisfactory in simulations \citep{mcnamee2017serious}. Hence, if we follow subjects from $t_0$, we may plug in frailty estimates from twin data into the expression suggested by \citet{mcnamee2017serious}, and thereby explore the truncation bias, e.g.\ in RCTs.
\subsection{When is survival bias an issue?}
The magnitude of survival bias varies with (i) time, (ii) the size of the observed TRR and (iii) the parameterisation of the frailty distribution. We shall consider some scenarios, using the derivations in Sections \ref{sec: using family}-\ref{sec: observed to causal}.
First, Equation \eqref{equation:observedHazardExpressedWithCausal} shows that the bias increases with $t$. In particular, when $t \rightarrow \infty$, we have that $r_{mar}(t) \rightarrow r^{-m}$. For the gamma distribution, $m \rightarrow 0$, and $r_{mar}(t) \rightarrow 1$. Hence, the marginal HR will be attenuated towards a null-effect \citep{van2001duration}. For some conditions, it may be that only a fraction of the population is susceptible, e.g. for a particular disease. In these scenarios, the compound Poisson model is convenient, because it allows for a non-susceptible fraction. For the compound Poisson distribution, $m > 0$, and
\begin{align*}
\lim_{t\to\infty} r_{mar}(t) = r^{-m} < 1 \qquad \text{ if } r > 1 \nonumber \\
\lim_{t\to\infty} r_{mar}(t) = r^{-m} > 1 \qquad \text{ if } r < 1,
\end{align*}
which means that we eventually will observe $r_{mar}$ in the opposite direction of $r$. This may have important implications: The marginal HR is not only a biased estimate of $r$, but it may also be invalid for hypothesis testing of an effect of $r$. Even though this relation is well-known in the methodological literature \citep{robins1989probability,van2001duration}, it may be under-appreciated in the applied literature \citep{burgess2015commentary}. In Figure \ref{figure: TRR over time}, the relation between the unconditional HR and left truncation is displayed for some PVF distributions with variance equal to 1. In all scenarios the conditional HR is 1.2, but the unadjusted HR declines a function of the population fraction lost to left truncation.
Second, a large TRR$(t)$ yields a large variation in risk between individuals \citep{valberg2017surprising}. Intuitively, we may also think that a large variation in risk leads to larger survival bias. To perform a numerical evaluation of the bias, let the population be assessed at time $t_1$ such that $S(t_1)=0.9$. In Figure \ref{figure: TRR bias}, we display $r$ as a function of the TRR$(t_1)$ when $r_{mar}(t_1)= 1.2$. For a fixed $r_{mar}(t_1)$, the conditional $r$ increases TRR$(t_1)$.
\section{Extension to IV estimates}
\label{sec: extension to IV estimates}
Instrumental variable (IV) approaches may be useful to identify causal relations. To find the effect of an exposure on an outcome, these techniques rely on an additional variable, the instrument. Let $L$ be measured covariates, and let $U$ be a possibly unmeasured confounder. Given $L$, an instrument must satisfy the following assumptions to obtain unbiased effect estimates:
\begin{itemize}
\item The instrument $G$ is associated with the exposure $A$, i.e.\ $G \centernot{\bigCI} A | L$.
\item The only path leading from $G$ to the outcome $Y$ goes through the exposure, $ G \mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}} Y | A, U, L $.
\item $G$ is independent of any unmeasured factor $U$ that confounds the exposure-outcome relation $ G \mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}} U | L $.
\end{itemize}
For more information on the IV assumptions, see e.g. \citet{didelez2007mendelian}. In the biomedical literature, the number of analyses based on IVs are increasing. Mendelian Randomisation (MR) studies are particularly popular, and such analyses rely on genetic variants as instruments. There is, however, a temporal aspect of MR studies. The genetic variants are allocated at conception ($t=0$), but the follow-up starts in adult life. Hence, survival bias is potentially a major issue because (i) there is often a large time-lag between perceived randomization at $t_0$ and follow-up, and (ii) MR holds the promise to reveal causal effects.
A causal structure of an IV setting with survival outcomes is shown in Figure \ref{Figure:DAGsurvival}. Here, $S$ is survival until time $t_1$. Until now we have considered a simple binary exposure, but we will hereby let the exposure $A(t)$ be a time-varying continuous variable expressed as a function of the binary instrument $G$ and the unknown component $U$. Let the data generating mechanism of $A(t)$ be
\begin{equation}
A(t) = b_0(t) + b_g G + b^T_l L + f(U) + e(t),
\label{eq: initial A and G}
\end{equation}
where $f(U)$ is a random variable such that $E( f (U | G)) = E( f (U ))$, $L$ denotes the measured covariates assuming that $L \mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}} G$ and $L \mathrel{\text{\scalebox{1.07}{$\perp\mkern-10mu\perp$}}} U$, and $e(t)$ is a residual error independent of $U$ which we do not specify further. If $\text{E}(f(U))=z$, we substitute $b_0(t)$ with $b_0^*(t) = b_0(t) + z$ and $f(U)$ by $f^*(U) = f(U)-z$, such that $\text{E}(f^*(U))=0$. In Equation \eqref{eq: initial A and G} we assume that $b_g$ is time constant for each subject, which is not trivial to
justify in practice (see e.g.\ \citet{abbring2007econometric} who discuss this in an economics context).
\citet{tchetgen2015instrumental} suggested a proportional hazards strategy for IV analyses, which is only valid for rare outcomes due to the non-collapsibility of the hazard ratio. By modelling the relation between $U$ and survivial ($T$), I will suggest a proportional hazards approach that is not only valid under the rare events assumptions. Similar to \citet{tchetgen2015instrumental}, I will consider a proportional hazards model for the outcome $Y$,
\begin{align}
h(t | U, A(t), G) &= h(t | U, A(t)) \nonumber \\
&= h_0(t) e^{\beta_aA(t) + \beta^T_l L} \psi(U),
\label{Assumptions equality 1}
\end{align}
where $\psi(U)$ is a function of $U$ independent of $A$. In Equation \eqref{Assumptions equality 1}, the first line is justified by the graph in Figure \ref{Figure:DAGsurvival}, and the second line displays the assumed causal hazard function. Hence, $\beta_a$ denotes the causal effect of $A(t)$ on $Y$ conditional on $U$, which is our estimand. Different from \citet{tchetgen2015instrumental}, we allow $A(t)$ be time-varying, but we restrict $\psi(U)$ to be time-constant.
To estimate $\beta_a$, we must rely on information about $G$ and $U$, and we will use the results derived in Section \ref{sec:PVFderivations}. It may indeed be unappealing that this IV analysis will require explicit assumptions about $U$, because the motivation for using an instrument is to avoid modeling $U$. Nevertheless, in the Mendelian randomization context, one may also argue that the IV assumptions are not met, because $G \centernot{\bigCI} U$ under left truncation. To deal with this issue, I will model the relation between $U$ and survival $T$, but I will still be agnostic about the relation $U \rightarrow A$.
\subsection{Causal estimates in Mendelian randomisation studies.}
We aim to estimate $\beta_a$. Writing out Equation \eqref{Assumptions equality 1} gives
\begin{align}
h(t | U, A(t), G) &= h_0(t) e^{\beta_a(b_0(t) + b_g g + b_v V + f(U) + e(t)) + \beta^T_l L} \psi(U).
\label{Causal hazard 1 g}
\end{align}
We introduce the parameter $U^{\star}$ such that $U^{\star}=e^{\beta_af(U)}\psi(U)$. Similar to Section \ref{sec:PVFderivations}, we assume that $U^{\star}$ has a distribution with finite mean in the population, and we assume that $U^{\star}$ is standardised such that $\text{E}(U^{\star})=1$. Inserting $U^{\star}$ in Equation \eqref{Causal hazard 1 g}, we obtain
\begin{align}
h(t | U, A(t), G)
&= h_0(t) e^{\beta_a(b_0(t) + b_v V + b_g g + e(t))+ \beta^T_l L}U^{\star}.
\label{IvHazard}
\end{align}
Let $g_1$ and $g_2$ be two variants of the instrument $G$. We can then consider the genetic HR
\begin{align}
\text{HR}_{G} &= \frac{h(t | U^{\star}, G=g_1) }{h(t | U^{\star},G=g_2)} \nonumber \\
&= e^{\beta_ab_g(g_1 - g_2)}. \nonumber \\
\label{expression:analogueToStd}
\end{align}
We can find an estimate $\hat{b}_{g}$ of $b_{g}$, e.g.\ by fitting a linear regression based on Equation \eqref{eq: initial A and G}. Furthermore, $\text{HR}_{G}$ can be estimated with the strategy in Section \ref{sec: using family} such that
\begin{align}
\hat{\beta}_a = \frac{\log[\hat{\text{HR}}_{G}]}{\hat{b}_g(g_1 - g_2)}.
\label{expression:identification}
\end{align}
In contrast to \citet{tchetgen2015instrumental}, in Equation \eqref{expression:identification} it is not assumed that $S(t | U, A(t), G) \approx 1$. \citet{tchetgen2015instrumental} required the rare events assumption, because they used standard Cox proportional hazards regression to find the marginal HR under a data generating mechanism similar to Equation \eqref{Assumptions equality 1}. Hence, they assumed that the causal effect of the exposure conditional on $U$ is proportional on the hazard scale, but they fitted a Cox proportional hazards model unconditional on $U$. Due to the non-collapsibility of the HR, the unconditional model is approximately correct only if the event is rare. In this article, we do use the estimates from a (mis-specified) marginal proportional hazards model in an interval $[t_1,t_1 + \delta]$ as an approximation to $r_{mar}(t_1)$. However, we do not require that the true, marginal HR can be correctly estimated by a Cox model at other times $t$, and we therefore do not need the rare events assumption.
Until now the instrument $G$ has been considered to be binary. A binary $G$ may be reasonable when the instrument is a single gene, which initially was the standard approach for MR analyses. Today, however, most analysts use instruments that are combinations of multiple variants, usually quantified in a continuous genetic risk score \citep{burgess2013use}. This approach will often increase the power of the study, but it also requires all the single variants to satisfy the IV assumptions. Our approach is readily applicable to continuous instruments. Specifically, we then consider $G$ to be a continuous variable, and Equation \eqref{eq: initial A and G} thereby require $G$ to have an additive effect on the Exposure $A(t)$. The remaining derivations in Section \ref{sec: extension to IV estimates} will all be valid for both a continuous and a binary $G$.
\section{An illustrative example}
\label{sec: full example section}
\subsection{The effect of alcohol on all-cause mortality}
\label{sec: standard alcohol example}
\citet{almeida2015excessive} assessed the impact of alcohol consumption on all-cause mortality in a sample with 3496 old men (aged 70-89 years at baseline). To do this, they used genetic information on the Alcohol Dehydrogenase 1B (ADH1B) gene. A mutation in the gene, which was carried by 225 of the subjects, leads to abnormal metabolism of ethanol, and carriers experience an unpleasant flushing while drinking. It is well-known that carriers consume less alcohol \citep{holmes2014association}, and their reduced consumption is thought to be independent of confounding factors.
Following \citet{almeida2015excessive}, we will consider the effect of (ADH1B) gene on all-cause mortality, and not the effect of alcohol per se. At baseline there should not be confounders that affects all-cause mortality and carrying ADH1B. However, in a left truncated sample the mortality HR between carriers and non-carriers of the genetic variant ADH1B (now, considered to be the exposure $X$ in Figure \ref{Figure:DAGbasic}) is not easy to interpret; at least it cannot be interpreted as the counterfactual rate ratio of carrying vs not carrying the ADH1B in a subject. To illustrate, we consider the marginal HR $$r_{Almeida} = 0.68 \text{ }[0.54,0.87]$$ in carriers of the mutation, which was derived by \citet{almeida2014alcohol}. With the frailty methods, we may explore how this marginal estimate may deviate from a counterfactual rate ratio under explicitly defined model assumptions.
First, we consider the age of 75 years, which is approximately the mean age at baseline in the study by \citet{almeida2014alcohol}, such that $r_{mar}(75) = r_{Almeida}$. Analyses of a Scandinavian registry of 9272 male twins suggest that the relative probability of surviving 75 years, given a co-twin survived 75 years, is 1.27 in men \citep{iachine2006genetic}. Therefore we let $\hat{\text{TRR}}(75)=1.27$, assuming that the probability of surviving until 75 years is shared among monozygotic twins. Here, we should ideally have assessed the TRR(75) in a (counterfactual) population of non-carriers of the ADH1B mutation, but we used the whole population as a crude estimate (in Scandinavians less than 4\% carry the mutation, which is even rarer than in Australians \citep{linneberg2010genetic}).
According to Australian cohort life tables of subjects, 0.56 of Australian men born 1931-1936 survive until age 75 years \citep{rowland1997demography}. We assume a gamma distribution of $U$; heuristically we then believe that everybody has a hazard larger than 0 for dying at any time, and the individual hazards vary continously in the population. Intuitively, these hazards will arise due to a combination of several unmeasured factors denoted $U$. We will use the values $\text{TRR(75)}=1.27 \text{ }[1.20,1.34]$, $S(75)=0.56$ and $r_{mar}(75)=0.68 \text{ }[0.54,0.87]$ to estimate the causal HR conditional on $U$. To highlight the magnitude of bias due to $U$, we first assume that $\text{TRR(75)}$ and $S(75)$ are true values without uncertainty, and we numerically solve Equation \eqref{eq: nu solver} to find $\nu=0.846$. Then, we use Equation \eqref{eq:gammaRiskSolver} to estimate the causal HR by
\begin{align*}
\hat{r} = r_{mar}(75) \Big(1+\frac{H_0(75)}{0.846} - \frac{H_0(75)r_{mar}(75) }{0.846}\Big)^{-1} = 0.52 \quad [0.37, 0.77].
\label{eq:gammaRiskSolver2}
\end{align*}
The 95\% confidence interval is simply derived by plugging in the confidence limits of $r_{mar}(75)$ into Equation \eqref{eq:gammaRiskSolver}, and these estimates are valid due to the monotonic relation between $r$ and $r_{mar}$. We thereby assume that $TRR(75)$ and $S(75)$ are the true values in the population.
The frailty analysis suggests that $r_{mar}(75)$ is a conservative estimate of $r$. To derive this estimate, we have required that a weighted average of $r(t)$ before follow up is equal to the the causal HR during follow up. Allthough this is a straight-forward extension of the standard Cox proportional hazards assumption, it is an untestable assumption, and interpreting $\hat{r}$ causally requires very strong structural assumptions. Rather, $\hat{r}$ gives us an impression on how the crude HRs from Cox models may differ substantially from HRs that actually have a causal interpretation.
We may also obtain estimates of $r$ by assuming other parameterisations of $U$. In Table \ref{tab: different distributions example}, we have shown results for 4 PVF distributions, using plug-in confidence intervals ($95\% \text{ CI plug-in}$). In this example, the results are robust to the parameterisation of $U$. Intuitively, this is due to the relatively small $TRR(t_1)$. However, this robustness does not necessarily apply to other scenarios \citep{stensrud2017exploring}, e.g. for larger $TRR(t_1)$, as also seen in Figure \ref{figure: TRR bias}.
In applications, it is not sufficient to account for the uncertainty in $r_{mar}$. We must also consider the uncertainty in the other summary estimates that are used, i.e.\ $\text{TRR}(t_1)$ and $S(t_1)$. A numeric approach to derive such confidence intervals is described in Appendix \ref{appendix: confidence intervals}. We use this approach to account for the uncertainty in $\text{TRR(75)}=1.27 \text{ } [1.20, 1.34]$ in the 4 frailty distributions, and these results are displayed in Table \ref{tab: different distributions example} ($95\% \text{ CI numeric}$). In this example, these estimates are similar to the plug-in estimates.
\begin{table}[ht]
\centering
\begin{tabular}{lrrrr}
Distribution of $U$ & $\hat{r}_{[t_1, t_1 + \delta]}$ & $95\% \text{ CI plug-in}$ & $95\% \text{ CI numeric}$ \\
\hline
Gamma & 0.52 & [0.37, 0.77] & [0.35, 0.74]\\
Inverse Gaussian & 0.53 & [0.37, 0.79] & [0.36, 0.77]\\
Hougaard ($m = -0.125$) & 0.52 & [0.37, 0.77] & [0.35, 0.77]\\
Compound Poisson (10\% nonsusceptible) & 0.52 & [0.38, 0.77] & [0.36, 0.76]\\
\end{tabular}
\caption{Estimates of the causal hazard ratio of the ADH1B variant on all-cause mortality, conditional on the frailty. Results from 4 different PVF distributions are displayed.}
\label{tab: different distributions example}
\end{table}
\section{Discussion}
Conventional HRs are difficult to interpret \citep{stensrud2017exploring, aalen2015does, hernan2010hazards,robins1989probability}. This article explores the causal understanding of HRs. By modelling the unobserved heterogeneity in disease risk, conventional hazard ratios are compared to conditional HRs with a causal interpretation. This approach may be useful for sensitivity analysis, e.g.\ to explore how a HR from Cox model may approximately be relevant to individual patients.
More generally, this article highlights a link between frailty models in survival analysis and causal inference \citep{stensrud2017exploring}: Interpreting estimates from Cox regressions may be challenging, e.g. due to non-collapsibility \citep{martinussen2013collapsibility} and left truncation. However, by using frailty models with strong parametric assumptions, we find estimates that are easier to interpret: We identify the effect of the exposure conditional on the unobserved variable $U$, and intuitively this is the effect of the exposure on the individual level
When measured covariates are able to explain most of the heterogeneity in risk, the frailty approach may be less desirable than alternative approaches, in which covariates are included in the model. For many conditions, however, unmeasured factors may have considerable impact. In such situations, the frailty approach seems to be particularly attractive, e.g.\ in sensitivity analyses.
I have suggested an approach that only relies on summary data of the risk heterogeneity, e.g.\ derived from twin registries. This is desirable, because individual level data are often unavailable. However, If we got access to individual level data, it may be better to perform a joint analysis (see e.g. \citet{van2001duration}).
By using twin data, I attempt to adjust for heterogeneity due to (unmeasured) genetic factors and shared environment. Using data from monozygotic twins is desirable, because such co-twins are genetically identical and expected to share several environmental factors. Nevertheless, non-shared environmental factors will not be captured unless they are included as measured covariates $L$. If such factors are unmeasured and have large (multiplicative) effect on the hazard function, we may underestimate the magnitude of survival bias. Furthermore, assuming a time-constant $U$ is simplistic, in particular because co-twins may influence each other over time, e.g.\ in health seeking behaviour. We have also implicitly assumed that twins are representative of the general population. In particular, unmeasured factors that influence survival until follow-up at $t_1$ should be similarly distributed in twins as in the general population. Recently, it has been suggested that monozygotic twins live longer than the general population, but the difference was found to be modest \citep{sharrow2016twin}.
This article has considered proportional hazards models with time to death as outcome. The approach can also be used to other time to event outcomes. e.g.\ time to progression of a disease. For many diseases we will expect the survival bias to be larger \citep{stensrud2017exploring}; we have shown that a larger variance in the unobserved heterogeneity yields a larger survival bias, and the familial clustering of several diseases is considerably larger than the familial clustering of longevity. For such outcomes, we may deal with survival bias by introducing correlated unknown factors ($U_1$ and $U_2$) that influence the time-to-event, respectively, and thereafter use theory for separate, correlated frailty variables (see e.g.\ \citet{aalen2008survival} chapter 6.6).
The survival bias is a particular issue in MR studies, due to the long time lag between randomisation and follow up \citep{boef2015mendelian,martinussen2017instrumental}. Interpreting IV estimates for time-to-event outcomes is not straightforward \citep{burgess2015commentary}, but such methods have recently been developed, under explicitly defined assumptions. In particular, estimates from the Aalen additive model do not suffer from survival bias under some explicit parameterisations of $U$ \citep{tchetgen2015instrumental, martinussen2016instrumental, li2015instrumental}. \citet{mackenzie2014using} considered a Cox proportional hazards model with additive unobserved confounding, but it relies on some unrealistic assumptions \citep{tchetgen2015instrumental}. Very recently, estimation under a IV structural Cox model for the treatment effect of the treated was suggested \citep{martinussen2017instrumental}. This approach allows us to estimate a causal HR when subjects are follow from $t_0$, but it may still be hard to handle left truncated samples.
\section*{Acknowledgements}
I would like to thank Odd O. Aalen, Kjetil R\o ysland, Morten Valberg and Susanne Strohmaier for their support and discussions on this manuscript.
\clearpage
|
1,108,101,565,627 | arxiv | \section{Introduction}\label{s1}
\IEEEPARstart{G}raph-based representations have been widely employed to model and analyze data that lies on high-dimensional non-Euclidean domains and that is naturally described in terms of pairwise relationships between its parts~\cite{DBLP:conf/nips/DefferrardBV16,DBLP:journals/tnn/ZambonAL18}. Typical instances where data can be represented using graphs include a) classifying proteins or chemical compounds~\cite{DBLP:conf/icml/KriegeM12,DBLP:journals/tnn/WuPZZY18}, b) recognizing objects from digital images~\cite{DBLP:conf/cvpr/HarchaouiB07}, c) visualizing social networks~\cite{DBLP:conf/kdd/WangC016}. A fundamental challenge arising in the analysis of real-world data represented as graphs is the lack of a clear and accurate way to represent discrete graph structures as numeric features that can be directly analyzed by standard machine learning techniques~\cite{DBLP:journals/tsmc/RiesenB09}. This paper aims to develop a new graph convolutional neural network using quantum vertex saliency, for the purpose of graph classification. Our method is based on identifying the transitive alignment information between vertices of all different graphs. That is, given three vertices $v$, $w$ and $x$ from three sample graphs, suppose $v$ and $x$ are aligned, and $w$ and $x$ are aligned, the proposed model can guarantee that $v$ and $w$ are also aligned. The alignment procedure not only provides a way of mapping each graph into a fixed-sized vertex grid structure, but also bridges the gap between the graph convolution layer and the traditional convolutional neural network layer.
\subsection{Literature Review}
There have been a large number of methods aimed at converting graph structures into numeric representations, thus providing a way of directly applying standard machine learning algorithm to problems of graph classification or clustering. Generally speaking, in the last three decades, most classical state-of-the-art approaches to the analysis of graph structures can be divided into two classes, namely 1) graph embedding methods and 2) graph kernels. The methods from the first class aim to represent graphs as vectors of permutation invariant features, so that one can directly employ standard vectorial machine learning algorithms~\cite{DBLP:journals/pr/GibertVB12,DBLP:journals/pami/WilsonHL05,DBLP:conf/icml/KondorB08,DBLP:journals/tnn/RenWH11}. All of the previous approaches are based on the computation of explicit embeddings into low dimensional vector spaces, which inevitably leads to the loss of structural information. Graph kernels, on the other hand, try to soften this limitation by (implicitly) mapping graphs to a high dimensional Hilbert space where the structural information is better preserved~\cite{DBLP:series/smpai/NeuhausB07,DBLP:journals/tnn/OnetoNDRSAA18}. The majority of state-of the-art graph kernels are instances of the R-convolution kernel originally proposed by Haussler~\cite{haussler99convolution}. The main idea underpinning R-convolution kernels is that of decomposing graphs into substructures (e.g, walks, paths, subtrees, and subgraphs) and then to measure the similarity between a pair of input graphs in terms of the similarity between their constituent substructures. Representative R-convolution graph kernels include the Weisfeiler-Lehman subtree kernel~\cite{shervashidze2010weisfeiler}, the subgraph matching kernel~\cite{DBLP:conf/icml/KriegeM12}, the backtracless path kernel~\cite{DBLP:journals/tnn/AzizWH13}, the tree-based continuous attributed kernel~\cite{DBLP:journals/tnn/MartinoNS18}, and the aligned subtree kernel~\cite{DBLP:conf/icml/Bai0ZH15}. A common limitation shared by both graph embedding methods and kernels is that of ignoring information from multiple graphs. This is because graph embedding methods usually capture structural features of individual graphs, while graph kernels reflect structural characteristics for pairs of graphs.
Recently, deep learning networks have emerged as an effective way to extract highly meaningful statistical patterns in large-scale and high-dimensional data~\cite{DBLP:series/acvpr/978-3-319-42998-4}. As evidenced by their recent successes in computer vision problems, convolutional neural networks (CNNs)~\cite{DBLP:conf/cvpr/VinyalsTBE15,DBLP:journals/cacm/KrizhevskySH17} are one of the most popular class of deep learning architectures and many researchers have devoted their efforts to generalizing CNNs to the graph domain~\cite{DBLP:journals/corr/abs-1708-02218}. Unfortunately, applying CNNs for graphs in a straightforward way is not trivial, since these networks are designed to operate on regular grids~\cite{DBLP:conf/nips/DefferrardBV16} and the associated operations of convolution, pooling and weight-sharing cannot be easily extended to graphs.
To address the aforementioned problem, two popular strategies have been proposed and employed to extend convolutional neural networks to graph domains, i.e., the spectral and the spatial strategies. Specifically, approaches using the spectral strategy utilise the property of the convolution operator from the graph Fourier
domain, and relate to the graph Laplacian~\cite{DBLP:journals/corr/BrunaZSL13,DBLP:conf/nips/RippelSA15,DBLP:journals/corr/HenaffBL15}. By transforming the graph into the spectral domain through the Laplacian matrix eigenvectors, the filter operation is performed by multiplying the graph by a series of filter coefficients. Unfortunately, most spectral-based approaches demand the size of the graph structures to be the same and cannot be performed on graphs with different sizes and Fourier bases. As a result, approaches based on the spectral strategy are usually applied to vertex classification tasks. By contrast, methods based on the spatial strategy are not restricted to the same graph structure. These methods generalize the convolution operation to the spatial structure of a graph by propagating features between neighboring vertices~\cite{DBLP:journals/corr/VialatteGM16}. For instance, Duvenaud et al.~\cite{DBLP:conf/nips/DuvenaudMABHAA15} have proposed a Neural Graph Fingerprint Network by propagating vertex features between their $1$-layer neighbors to simulate the traditional circular fingerprint. Atwood and Towsley~\cite{DBLP:conf/nips/AtwoodT16} have proposed a Diffusion Convolution Neural Network by propagating vertex features between neighbors of different layers rooted at a vertex. Although spatially based approaches can be directly applied to real-world graph classification problems, most existing methods have fairly poor performance on graph classification. This is because these methods tend to directly sum up the extracted local-level vertex features from the convolution operation as global-level graph features through a SumPooling layer. It is then difficult to learn the topological information residing in a graph through these global features.
To overcome the shortcoming of the graph convolutional neural networks associated with SumPooling, unlike the works in~\cite{DBLP:conf/nips/DuvenaudMABHAA15} and~\cite{DBLP:conf/nips/AtwoodT16}, Nieper et al.~\cite{DBLP:conf/icml/NiepertAK16} have developed a different graph convolutional neural network by constructing a fixed-sized local neighborhood for each vertex and re-ordering the vertices based on graph labeling methods and graph canonization tools. This procedure naturally forms a fixed-sized vertex grid structure for each graph, and the graph convolution operation can be performed by sliding a fixed-sized filter over spatially neighboring vertices. This operation is similar to that performed on images with standard convolutional neural networks. Zhang et al.~\cite{DBLP:conf/aaai/ZhangCNC18} have developed a novel Deep Graph Convolutional Neural Network model that can preserve more vertex information and learn from the global graph topology. Specifically, this model utilizes a newly developed SortPooling layer, that can transform the extracted vertex features of unordered vertices from spatial graph convolution layers into a fixed-sized vertex grid structure. Then a traditional convolutional neural networks can be applied to the grid structures to further learn the graph topological information.
Although both methods of Nieper et al.~\cite{DBLP:conf/icml/NiepertAK16} and Zhang et al.~\cite{DBLP:conf/aaai/ZhangCNC18} outperform state-of-the-art graph convolutional neural network models and graph kernels on graph classification tasks, these approaches suffer from the drawback of ignoring structural correspondence information between graphs, or rely on simple but inaccurate heuristics to align the vertices of the graphs, i.e., they sort the vertex orders based on the local structure descriptor of each individual graph and ignore the vertex correspondence information between different graphs. As a result, both the methods cannot reflect the precise topological correspondence information for graph structures. Moreover, these approaches also lead to significant information loss. This usually occurs when these approaches form the fixed-sized vertex grid structure and some vertices associated with lower ranking may be discarded. In summary, developing effective methods to preserve the structural information residing in graphs still remains a significant challenge.
\subsection{Contribution}
The aim in this paper is to overcome the shortcomings of the aforementioned methods by developing a new spatial graph convolutional neural network model. One key innovation of the new model is the identification of the transitive vertex alignment information between graphs. Specifically, the new model can employ the transitive alignment information to map different sized graphs into fixed-sized aligned representations, i.e., it can transform different graphs into fixed-sized aligned grid structures with consistent vertex orders. Note that the aligned grid structure can precisely integrate the structural correspondence information and preserve both the original graph topology and the vertex feature information without any information loss, since all the original vertex information will be mapped into the grid structure through the transitive alignment. Thus, it not only bridges the gap between the spatial graph convolution layer and the traditional convolutional neural network layer, but also addresses the shortcomings of information loss and imprecise information representation arising in most state-of-the-art graph convolutional neural networks associated with SortPooling or SumPooling layers. Overall, the main contributions of this work are threefold.
First, we develop a new framework for transitively aligning the vertices of a family of graphs in terms of vertex point matching. This framework can establish reliable vertex correspondence information between graphs, by gradually minimizing the inner-vertex-cluster sum of squares over the vertices of all graphs. We show that this framework can be further employed to map graphs of arbitrary sizes into fixed-sized aligned vertex grid structures, integrating precise structural correspondence information and thus minimising the loss of structural information. The resulting grid structures can bridge the gap between the spatial graph convolution layer and the traditional convolutional neural network layer.
Second, with the aligned vertex grid structures and their associated adjacency matrices to hand, we propose a novel quantum spatial graph convolution layer to extract multi-scale vertex features in terms of the quantum vertex information propagation. More specifically, we use the average mixing matrix associated with continuous-time quantum walks. We show that the new convolution layer theoretically overcomes the shortcoming of popular graph convolutional neural networks and graph kernels, supporting the empirical evidence collected in our experimental validation. Moreover, since the proposed convolution layer does not change the original spatial sequence of vertices, it allows us to directly employ the traditional convolutional neural network to further learn from the global graph topology, providing an end-to-end deep learning architecture that integrates the graph representation and learning into both the quantum spatial graph convolution and the traditional convolutional layers for graph classifications.
Third, we empirically evaluate the proposed Quantum Spatial Graph Convolutional Neural Network (QSGCNN). Experimental results on benchmark graph classification datasets demonstrate that our proposed QSGCNN significantly outperforms state-of-the-art graph kernels and deep graph convolutional network models for graph classifications.
\section{Preliminary Concepts}\label{s2}
\subsection{Continuous-time Quantum Walks}\label{s2.1}
One main objective of this work is to develop a new spatial graph convolution layer to extract multi-scale vertex features by gradually propagating information for each vertex to its neighboring vertices as well as the vertex itself. This usually requires connection information between each vertex and its neighboring vertices. Most existing methods employ the vertex adjacency matrix of each graph in the formulation of the information propagation framework~\cite{DBLP:conf/nips/DuvenaudMABHAA15,DBLP:conf/nips/AtwoodT16,DBLP:conf/icml/NiepertAK16,DBLP:conf/aaai/ZhangCNC18}. In order to capture richer vertex features from the proposed graph convolutional layer, in this work we propose to employ the vertex information propagation process of the continuous-time quantum walk. This is the quantum analogue of the classical continuous-time random walk~\cite{farhi1998quantum}.
The main reason for relying on quantum walks is that, unlike classical random walks, whose state is described by a real-valued vector
and where the evolution is governed by a doubly stochastic matrix, the state
vector of the quantum walks is complex-valued and its evolution is governed by a time-varying
unitary matrix. Thus, the quantum walk evolution is reversible, implying that it is non-ergodic and does not possess a limiting distribution. As a result, the behaviour of quantum walks is significantly different from their classical counterpart and possesses a number of important properties, e.g., it allows interference to take place. This interference, in turn, helps to reduce the tottering problem of random walks, as a quantum walkers backtracking on an edge does so with opposite phase. Furthermore, since the evolution of the quantum walk is not dominated by the low frequency components of the Laplacian spectrum, it has better ability to distinguish different graph structures. In Section~\ref{s3}, we will show that the proposed graph convolutional layer associated with the continuous-time quantum can not only reduce the tottering problem arising in some state-of-the-art graph kernels and graph convolutional network models, but also better discriminates between different graph structures.
\begin{figure*}
\vspace{-0pt}
\centering
\includegraphics[width=1.0\linewidth]{GCN_TNN.pdf}
\vspace{-20pt}
\caption{The architecture of the proposed QSGCNN model. An input graph $G_p(V_p,E_p)\in \mathbf{G}$ of arbitrary size is first aligned to the prototype graph $G_R(V_R,E_R)$. Then, $G_p$ is mapped into a fixed-sized aligned vertex grid structure, where the vertex orders follows that of $G_R$. The grid structure of $G_p$ is passed through multiple quantum spatial graph convolution layers to extract multi-scale vertex features, where the vertex information is propagated between specified vertices following the average mixing matrix. Since the graph convolution layers preserve the original vertex orders of the input grid structure, the concatenated vertex features through the graph convolution layers form a new vertex grid structure for $G_p$. This vertex grid structure is then passed to a traditional CNN layer to learn a classification function. Note, vertex features are visualized as different colors.}\label{f:QSGCNN}
\vspace{-20pt}
\end{figure*}
In this subsection, we briefly review the concept of continuous-time quantum walks. Specifically, we use the average mixing matrix to capture the time-averaged behaviour of the quantum walk and to measure the quantum information being transmitted between the graph vertices. The continuous-time quantum walk is the quantum analogue of the continuous-time classical random walk~\cite{farhi1998quantum}, where the latter models a Markovian diffusion process over the vertices of a graph through the transitions between adjacent vertices. Let a sample graph be denoted as $G(V,E)$ with vertex set $V$ and edge set $E$. Like the classical random walk, the state space of the quantum walk is the vertex set $V$. Its state at time $t$ is a complex linear combination of the basis states $\Ket{u}$, i.e., $\Ket{\psi(t)} = \sum_{u\in V} \alpha_u(t) \Ket{u}$, where $\alpha_u (t) \in \mathbb{C}$ and $\Ket{\psi(t)} \in \mathbb{C}^{|V|}$ are the amplitude and both complex. Furthermore, $\alpha_u (t) \alpha_u^* (t)$ indicates the probability of the walker visiting vertex $u$ at time $t$, $\sum_{u \in V} \alpha_u (t) \alpha^{*}_u(t) = 1$, and $\alpha_u (t) \alpha^{*}_u(t) \in [0,1]$, for all $u \in V$, $t \in \mathbb{R}^{+}$. Unlike the classical counterpart, the continuous-time quantum walk evolves based on the Schr\"{o}dinger equation
\begin{equation}
\partial/\partial t \Ket{\psi_t} = -iH\Ket{\psi_t},
\end{equation}
where $H$ represents the system Hamiltonian. In this work, we use the adjacency matrix as the Hamiltonian. The behaviour of a quantum walk over the graph $G(V,E)$ at time $t$ can be summarized using the mixing matrix~\cite{godsil2013average}
\begin{align}\label{}
Q_M(t) = U(t) \circ U(-t) = e^{iHt} \circ e^{-iHt},
\end{align}
where the operation symbol $\circ$ represents the Schur-Hadamard product of $e^{iHt}$ and $e^{-iHt}$. Because $U$ is unitary, $Q_M(t)$ is a doubly stochastic matrix and each entry $Q_M(t)_{uv}$ indicates the probability of the walk visiting vertex $v$ at time $t$ when the walk initially starts from vertex $u$. However, $M(t)$ cannot converge, because $U(t)$ is also norm-preserving. To overcome this problem, we can enforce convergence by taking a time average. Specifically, we take the Ces\`{a}ro mean and define the average mixing matrix as $Q = \lim_{T \rightarrow \infty} \int_{0}^{T} Q_M(t) dt$,
where each entry $Q_{v_iv_j}$ of the average mixing matrix $Q$ represents the average probability for a quantum walk to visit vertex $v_j$ starting from vertex $v_i$, and $Q$ is still a doubly stochastic matrix. Furthermore, Godsil~\cite{godsil2013average} has indicated that the entries of $Q$ are rational numbers. We can easily compute $Q$ from the spectrum of the Hamiltonian. Specifically, let the adjacency matrix $A$ of $G$ be the Hamiltonian $H$. Let $\lambda_1,\ldots,\lambda_{|V|}$ represent the $|V|$ distinct eigenvalues of $H$ and $\mathbb{P}_j$ is the matrix representation of the orthogonal projection on the eigenspace associated with the $\lambda_j$, i.e., $H = \sum_{j = 1}^{|V|} \lambda_j \mathbb{P}_j.$ Then, we can re-write the average mixing matrix $Q$ as
\begin{equation}
Q = \sum_{j = 1}^{|V|} \mathbb{P}_j \circ \mathbb{P}_j.
\end{equation}
\subsection{Transitive Alignment Between Vertices of Graphs}
We introduce a new transitive vertex alignment method. To this end, we commence by identifying a family of prototype representations that reflect the main characteristics of the vectorial vertex representations over a set of graphs $\mathbf{G}$. Assume there are $n$ vertices over all graphs in $\mathbf{G}$, and the associated $K$-dimensional vectorial representations of these vertices are $\mathbf{{R}}^K =(\mathrm{R}_1^K,\mathrm{R}_2^K,\ldots,\mathrm{R}_n^K)$. We use $k$-means~\cite{witten2011data} to identify $M$ centroids over all representations in $\mathbf{{R}}^K$. Specifically, given $M$ clusters $\Omega=(c_1,c_2,\ldots,c_M)$, the aim of $k$-means is to minimize the objective function
\begin{equation}
\arg\min_{\Omega} \sum_{i=1}^M \sum_{\mathrm{R}_j^K \in c_i^K} \|\mathrm{R}_j^K- \mu_i^K\|^2,\label{kmeans}
\end{equation}
where $\mu_i^K$ is the mean of the vectorial vertex representations belonging to the $i$-th cluster $c_i$. Since Eq.(\ref{kmeans}) minimizes the sum of the square Euclidean distances between the vertex points $\mathrm{R}_j^K$ and the centroid point of cluster $c_i^K$, the $M$ centroid points $\{\mu_1^K,\cdots,\mu_i^K,\cdots,\mu_M^K\}$ can be seen as a family of $K$-dimensional \textbf{prototype representations} that encapsulate representative characteristics over all graphs in $\mathbf{G}$.
Let $\mathbf{G}=\{G_1,\cdots,G_p,\cdots,G_q,\cdots,G_N\}$ be a set of graphs. For each graph $G_p(V_p,E_p)\in {\mathbf{G}}$ and each vertex $v_i\in V_p$ associated with its $K$-dimensional vectorial representation $\mathrm{{R}}_{p;i}^K$, we commence by identifying the set of $K$-dimensional prototype representations as $\mathbf{PR}^K=\{\mu_1^K,\ldots,\mu_j^K,\ldots,\mu_M^K \}$ for the graph set $\mathbf{G}$. To establish a set of correspondences between the graph vertices, we align the vectorial vertex representations of each graph $G_p$ to the family of prototype representations $\mathbf{PR}^K$. The alignment process is similar to that introduced in~\cite{DBLP:conf/icml/Bai0ZH15} for point matching in a pattern space. Specifically, we compute a $K$-level affinity matrix in terms of the Euclidean distances between the two sets of points
\begin{align}
A^K_p(i,j)=\|\mathrm{{DB}}_{p;i}^K - \mu_j^K\|_2.\label{AffinityM}
\end{align}
where $A^K_p$ is a ${|V_p|}\times {M}$ matrix, and each element $R^K_p(i,j)$ represents the distance between the vectrial representation $\mathrm{{R}}_{p;i}^K$ of $v_\in V_p$ and the $j$-prototype representation $\mu_j^K\in \mathbf{PR}^K$. If the value of $A^K_p(i,j)$ is the smallest in row $i$, we say that $\mathrm{{R}}_{p;i}^K$ is aligned to $\mu_j^K$, i.e., the vertex $v_i$ is aligned to the $j$-th prototype representation. Note that for each graph there may be two or more vertices aligned to the same prototype representation. We record the correspondence information using the $K$-level correspondence matrix
$C^K_p\in \{0,1\}^{|V_p|\times M}$
\begin{equation}
C^K_p(i,j)=\left\{
\begin{array}{cl}
1 & \small{\mathrm{if} \ A^K_p(i,j) \ \mathrm{is \ the \ smallest \ in \ row } \ i} \\
0 & \small{\mathrm{otherwise}}.
\end{array} \right.
\label{CoMatrix}
\end{equation}
For a pair of graphs $G_p$ and $G_q$, if their vertices $v_p$ and $v_q$ are aligned to the same prototype representation $\mathrm{PR}_j^K$, we say that $v_p$ and $v_q$ are also aligned. Thus, we can identify the transitive alignment information between the vertices of all graphs in $\mathbf{G}$, by matching their vertices to a common set of reference points, i.e., the prototype representations.
\noindent\textbf{Discussion:} The alignment process illustrated by Eq.(\ref{AffinityM}) and Eq.(\ref{CoMatrix}) can be explained by the objective function of $k$-means defined by Eq.(\ref{kmeans}). This is because identifying the smallest element $A^K_p(i,j)$ in the $i$-row of $A^K_p$ is equivalent to assigning the vectorial representation $\mathrm{{R}}_{p;i}^K$ of $v_i\in V_p$ to the cluster $c_i^K$ whose mean vector is $\mu_i^K$. As a result, the proposed alignment procedure can be seen as an optimization process that gradually minimizes the inner-vertex-cluster sum of squares over the vertices of all graphS, and can establish reliable vertex correspondence information over all graphs.
\section{Quantum Spatial Graph Convolutional Neural Network}\label{s3}
In this section, we develop a new Quantum Spatial Graph Convolutional Neural Network (QSGCNN) model. The architecture of the proposed model is shown in Fig.\ref{f:QSGCNN}. Specifically, the architecture is composed of three sequential stages, i.e., 1) the grid structure construction and input layer, 2) the quantum spatial graph convolution layer, and 3) the traditional convolutional neural network and Softmax layers. Specifically, the grid structure construction and input layer a) first maps graphs of arbitrary sizes into fixed-sized grid structures with consistent vertex orders, and b) inputs the grid structures into the proposed QSGCNN model. With the input graph grid structures to hand, the quantum spatial graph convolution layer further extracts multi-scale vertex features by propagating vertex feature information between the aligned grid vertices. Since the extracted vertex features from the graph convolution layer preserve the original vertex orders of the input grid structures, the traditional convolutional neural network and Softmax layer can read the extracted vertex features and predict the graph class.
\subsection{Aligned Vertex Grid Structures of Graphs}
In this subsection, we show how to map graphs of different sizes onto fixed-sized aligned vertex grid structures and associated corresponding fixed-sized aligned grid vertex adjacency matrices. For the set of graphs $\mathbf{G}$ defined earlier, suppose $G_p(V_p,E_p,A_p)\in \mathbf{G}$ is a sample graph, with $V_p$ representing the vertex set, $E_p$ representing the edge set, and $A_p$ representing the vertex adjacency matrix. Suppose each vertex $v_p\in V_p$ is represented as a $c$-dimensional feature vector. Then the features of all the $n$ vertices can be encoded using the $n\times c$ matrix $X_p$, i.e., $X_p\in \mathbb{R}^{n\times c}$. Note that the row of $X_p$ follows the same vertex order of $A_p$. If the graphs in $\mathbf{G}$ are vertex attributed graphs, $X_p$ can be the one-hot encoding matrix of the vertex labels. For unattributed graphs, we propose to use the vertex degree as the vertex label. Based on the transitive vertex alignment method introduced in Section~\ref{s2}, for each graph $G_p\in \mathbf{G}$, we commence by computing the $K$-level vertex correspondence matrix $C^K_p$ that records the correspondence information between the $K$-dimensional vectorial vertex representation of $G_p$ and the $K$-dimensional prototype representations in $\mathbf{PR}^K=\{\mu_1^K,\ldots,\mu_j^K,\ldots,\mu_M^K \}$ of $\mathbf{G}$. The row and column of $C^K_p$ are indexed by the vertices in $V_p$ and the prototype representations in $\mathbf{PR}^K$, respectively. With $C^K_p$ to hand, we compute the $K$-level aligned vertex feature matrix for $G_p$ as
\begin{equation}
\widehat{X}_{p}^{K}= (C^K_p)^T X_p,\label{alignDB}
\end{equation}
where $\widehat{X}_{p}^{K}$ is a $M\times c$ matrix and each row of $\widehat{X}_{p}^{K}$ represents the feature of a corresponding aligned vertex. Moreover, we also compute the associated $K$-level aligned vertex adjacency matrix for $G_p$ as
\begin{equation}
\widehat{A}_{p}^{K}= (C^K_p)^T (A_{p}) (C^K_p),\label{alignA}
\end{equation}
where $\widehat{A}_{p}^{K}$ is a $M\times M$ matrix. With the correspondence matrix $C^K_p$ to hand, $\widehat{X}_{p}^{K}$ and $\widehat{A}_{p}^{K}$ are computed from the original vertex feature matrix and adjacency matrix, respectively, by mapping the original feature and adjacency information of each vertex $v_p\in V_p$ to that of the new aligned vertices indexed by the corresponding prototypes in $\mathbf{PR}^K$. In other words $\widehat{X}_{p}^{K}$ and $\widehat{A}_{p}^{K}$ encapsulate the original feature and structural information of $G_p$. Note also that according to Eq.~\ref{CoMatrix} each vertex $v_p\in V_p$ can be aligned to more than one prototype, and thus in general $\widehat{A}_{p}^{K}$ is a weighted adjacency matrix.
In order to construct the fixed-sized aligned grid structure for each graph $G_p\in \mathbf{G}$, we need to establish a consistent order for the vertices of the graphs in $\mathbf{G}$. Since the vertices of all the graphs are aligned to the same prototype representations, we determine the vertex orders by reordering the prototype representations. To this end, we construct a prototype graph that captures the pairwise similarity between the prototype representations. Given this graph, one approach could be to sort the prototype representations based on their degree. This would be equivalent to sorting the prototypes in orders of average similarity to the remaining ones. Specifically, we compute the prototype graph $G_{\mathrm{R}}(V_{\mathrm{R}},E_{\mathrm{R}})$ that characterizes the relationship information between the $K$-dimensional prototype representations in $\mathbf{PR}^K$, with each vertex $v_j\in V_{\mathrm{R}}$ representing the prototype representation $\mu_j^K\in \mathbf{PR}^K$ and each edge $(v_j,v_k)\in E_{\mathrm{R}} $ representing the similarity between $\mu_j^K\in \mathbf{PR}^K$ and $\mu_k^K\in \mathbf{PR}^K$. The similarity between two vertices of $G_{\mathrm{R}}$ is computed as \begin{equation}
s(\mu_j^K,\mu_k^K)=\exp (-\frac{\| \mu_j^K-\mu_k^K \|_2}{K}).
\end{equation}
The degree of each prototype representation $\mu_j^K$ is $D_R(\mu_j^K)=\sum_{k=1}^{M}s(\mu_j^K,\mu_k^K)$. We sort the $K$-dimensional prototype representations in $\mathbf{PR}^K$ according to their degree $D_R(\mu_j^K)$. Then, we rearrange $\widehat{X}_{p}^{K}$ and $\widehat{A}_{p}^{K}$ accordingly.
\begin{figure*}
\vspace{-0pt}
\centering
\includegraphics[width=0.85\linewidth]{Align_TNN.pdf}
\vspace{-10pt}
\caption{The procedure of computing the correspondence matrix. Given a set of graphs, for each graph $G_p$: (1) we compute the $K$-dimensional depth-based (DB) representation $\mathrm{{DB}}_{p;v}^K$ rooted at each vertex (e.g., vertex 2) as the $K$-dimensional vectorial vertex representation, where each element $H_s(G_{p;2}^K)$ represents the Shannon entropy of the $K$-layer expansion subgraph rooted at vertex $v_2$ of $G_p$ associated with steady state random walk~\cite{DBLP:journals/prl/Bai18}; (2) we identify a family of $K$-dimensional prototype representations $\mathbf{PR}^K = \{\mu_1^K,\ldots,\mu_j^K,\ldots,\mu_M^K \}$ using k-means on the $K$-dimensional DB representations of all graphs; (3) we align the $K$-dimensional DB representations to the $K$-dimensional prototype representations and compute a $K$-level correspondence matrix $C_p^K$.}\label{f:alignment}
\vspace{-10pt}
\end{figure*}
Finally, note that, to construct reliable grid structures for graphs, in this work we employ the depth-based representations as the vectorial vertex representations to compute the required $K$-level vertex correspondence matrix $C_p^K$. Specifically, the depth-based representation of each vertex is computed by measuring the entropies on a family of $k$-layer expansion subgraphs rooted at the vertex~\cite{DBLP:journals/prl/Bai18}, where the parameter $k$ varies from $1$ to $K$. Moreover, it has been shown that such a $K$-dimensional depth-based representation of a vertex can be seen as \emph{\textbf{a nested vertex representation}} that encapsulates rich nested entropy-based information content flow from each local vertex to the global graph structure~\cite{DBLP:journals/prl/Bai18}, as a function of depth. The process of computing the correspondence matrix $C_p^K$ associated with depth-based representations is
shown in Fig.\ref{f:alignment}. When we vary the largest layer $K$ of the expansion subgraphs from $1$ to $L$ (i.e., $K\leq L$), we compute the final \textbf{aligned vertex grid structure} for each graph $G_p\in \mathbf{G}$ as
\begin{equation}
\widehat{X}_{p}= \sum_{K=1}^L \frac{\widehat{X}_{p}^{K}}{L},\label{AlignV}
\end{equation}
and the associated \textbf{aligned grid vertex adjacency matrix} as
\begin{equation}
\widehat{A}_{p}= \sum_{K=1}^L \frac{\widehat{A}_{p}^{K}}{L},\label{AlignA}
\end{equation}
where $\widehat{X}_{p}$ is a $M\times c$ matrix, and $\widehat{A}_{p}$ is a $M\times M$ matrix.
\\
\noindent\textbf{Discussion:} Eq.(\ref{AlignV}) and Eq.(\ref{AlignA}) transform the original graphs $G_p\in \mathbf{G}$ with varying number of nodes $|V_p|$ into a new aligned grid graph structure with the same number of vertices, where $\widehat{X}_{p}$ is the corresponding aligned grid vertex feature matrix and $\widehat{A}_{p}$ is the corresponding aligned grid vertex adjacency matrix. Since for any graph $G_p\in \mathbf{G}$ the rows of $\widehat{X}_{p}$ are consistently indexed by the same prototype representations, the fixed-sized vertex grid structure $\widehat{X}_{p}$ can be directly employed as the input of a traditional convolutional neural network. In other words, one can apply a fixed sized classical convolutional filter to slide over the rows of $\widehat{X}_{p}$ and learn the feature for $G_p\in \mathbf{G}$. Finally, note that $\widehat{X}_{p}$ and $\widehat{A}_{p}$ accurately encapsulate the original feature and structural information of $G_p$, respectively.
\subsection{The Quantum Spatial Graph Convolution Layer}
In this subsection, we propose a new quantum spatial graph convolution layer to further extract the features of the vertices of each graph. This is defined by quantum information propagation between aligned grid vertices. To this end, we employ the average mixing matrix of the continuous-time quantum walk on the associated aligned grid vertex adjacency matrix. For the sample graph $G_p(V_p,E_p)$, we pass the aligned vertex grid structure $\widehat{X}_{p}\in \mathbb{R}^{M\times c}$ and the associated aligned grid vertex adjacency matrix $\widehat{A}_{p}\in \mathbb{R}^{M\times M}$ of $G_p$ as the input of the quantum spatial graph convolution layer. The proposed spatial graph convolution layer takes the following form, i.e.,
\begin{equation}
Z= \mathrm{Relu}(Q \widehat{X}_{p} W),\label{GCN_EQ}
\end{equation}
where $\mathrm{Relu}$ is the rectified linear units function (i.e., a nonlinear activation function), $Q$ is the average mixing matrix of the continuous-time quantum walk on $\widehat{A}_{p}$ of $G_p$ defined in Section~\ref{s2.1}, $W\in \mathbb{R}^{c\times c^{'}}$ is the matrix of trainable parameters of the proposed graph convolutional layer, and $Z\in \mathbb{R}^{M\times c^{'}}$ is the output activation matrix.
The proposed quantum spatial graph convolution layer defined by Eq.(\ref{GCN_EQ}) consists of three steps. In the first step the operation $\widehat{X}_{p} W$ is applied to transform the aligned grid vertex information matrix into a new aligned grid vertex information matrix. This in turn maps the $c$-dimensional features of each aligned grid vertex into new $c^{'}$-dimensional features, i.e., $\widehat{X}_{p} W$ maps the $c$ feature channels to $c^{'}$ channels in the next layer. The weights $W$ are shared among all aligned grid vertices. The second step computes $Q Y$, where $Y:= \widehat{X}_{p} W$. This propagates the feature information of each aligned grid vertex to the remaining vertices as well as the vertex itself, in terms of the vertex visiting information of quantum walks. Specifically, we note that ${Q}_{ij}$ encapsulates the average probability for a continuous-time quantum walk to visit the $j$-th aligned grid vertex starting from the $i$-th aligned grid vertex, and $(Q \widehat{X}_{p}^{'})_i=\sum_j {Q}_{ij} {Y}_{j}$. Here, $i$ can be equal to $j$, i.e., $Q$ includes the self-loop information for each vertex. Thus, the $i$-th row of the resulting matrix of $Q \widehat{X}_{p}^{'}$ is the feature summation of the $i$-th aligned grid vertex and the remaining aligned grid vertices associated with the average visiting probability of quantum walks from the $i$-th vertex to the remaining vertices as well as the $i$-th vertex itself. The final step applies the rectified linear unit function to $Q \widehat{X}_{p} W)$ and outputs the graph convolution result.
The proposed quantum spatial graph convolution propagates the aligned grid vertex information in terms of the vertex visiting information associated with the continuous-time quantum walk between vertices. To further extract the multi-scale features of the aligned grid vertices, we stack multiple graph convolution layers defined by Eq.(\ref{GCN_EQ}) as follows
\begin{equation}
Z_{t+1}= \mathrm{Relu}(Q Z_t W_t),\label{GCN_EQM}
\end{equation}
where $Z_0$ is the input aligned vertex grid structure $\widehat{X}_{p}$, $Z_t \in \mathbb{R}^{M\times c_t}$ is the output of the $t$-th spatial graph convolution layer, and $W_t \in \mathbb{R}^{c_t\times c_{t+1}}$ is the trainable parameter matrix mapping $c_t$ channels to $c_{t+1}$ channels.
After each $t$-th quantum spatial graph convolutional layer, we also add a layer to horizontally concatenate the output $Z^t$ associated with the outputs of the previous $1$ to $t-1$ spatial graph convolutional layers as well as the original input $Z^0$ as $Z_{0:t}$, i.e., $Z_{0:t}=[Z_0,Z_1,\ldots,Z^t]$ and $Z_{0:t}\in \mathbb{R}^{M\times \sum_{z=0}^t c_z}$. As a result, for the concatenated output $Z_{0:t}$, each of its row can be seen as the new multi-scale features for the corresponding grid vertex.\\
\noindent\textbf{Discussion:} Note that the proposed quantum spatial graph convolution only extracts new features for the grid vertex and does not change the orders of the vertices. As a result, both the output $Z^t$ and the concatenated output $Z_{0:t}$ preserve the grid structure property of the original input $Z_0=\widehat{X}_{p}$, and can be directly employed as the input of the traditional convolutional neural network. This provides an elegant way of bridging the gap between the proposed quantum spatial graph convolution layer and the traditional convolutional neural network, making an end-to-end deep learning architecture that integrates the graph representation and
learning in both the quantum spatial graph convolution layer and the traditional convolution layer for graph classification problems.
\subsection{The Traditional Convolutional Neural Network Layers}
After the $t$-th proposed quantum spatial graph convolution layers, we get a concatenated vertex grid structure $Z_{0:t}\in \mathbb{R}^{M\times \sum_{z=0}^t c_z}$, where each row of $Z_{0:t}$ represents the multi-scale feature for a corresponding grid vertex. As we mentioned above, each grid structure $Z_{0:t}$ can be directly employed as the input to the traditional convolutional neural network (CNN). Specifically, the Classical One-dimensional CNN part of Fig.\ref{f:QSGCNN} exhibits the architecture of the traditional CNN layers associated with each $Z_{0:t}$. Here, each concatenated vertex grid structure $Z_{0:t}$ is seen as a $M \times 1$ (in Fig.\ref{f:QSGCNN} $M = 5$) vertex grid structure and each vertex is represented by a $\sum_{z=0}^t c_z$-dimensional feature, i.e., the channel of each grid vertex is $\sum_{z=0}^t c_z$. Then, we add a one-dimensional convolutional layer. The convolutional operation can be performed by sliding a fixed-sized filter of size $k\times 1$ (in Fig.\ref{f:QSGCNN} $k = 3$) over the spatially neighboring vertices. After this, several MaxPooling layers and remaining one-dimensional convolutional layers can be added to learn the local patterns on the aligned grid vertex sequence. Finally, when we vary $t$ from $0$ to $T$ (in Fig.\ref{f:QSGCNN} $T = 2$), we will obtain $T+1$ extracted pattern representations. We concatenate the extracted patterns of each $Z_{0:t}$ and add a fully-connected layer followed by a Softmax layer.
\subsection{Discussion of the Proposed QSGCNN Model}\label{s3.4}
The proposed QSGCNN model is related to some existing state-of-the-art graph convolution network models and graph kernels. However, there are a number of significant theoretical differences between the proposed QSGCNN model and these state-of-the-art methods, explaining the effectiveness of the proposed model. In this subsection, we discuss the relationships between these methods and demonstrate the advantages of the proposed model.
\textbf{First}, similar to the quantum spatial graph convolution of the proposed QSGCNN model, the associated graph convolution of the Deep Graph Convolutional Neural Network (DGCNN)~\cite{DBLP:conf/aaai/ZhangCNC18} and the spectral graph convolution of the Fast Approximate Graph Convolutional Neural Network (FAGCNN)~\cite{DBLP:journals/corr/KipfW16} also propagate the features between the graph vertices. Specifically, the graph convolutions of the DGCNN and FAGCNN models use the graph adjacency matrix or the normalized Laplacian matrix to determine how to pass the information among the vertices. In contrast, our quantum spatial graph convolution utilizes the average mixing matrix of the continuous-time quantum walk associated with the graph. As we mentioned in Section~\ref{s2.1}, the quantum walk is not dominated by the low frequency values of the Laplacian spectrum and thus has a better ability to distinguish different graph structures. As a result, the proposed method can extract more discriminative vertex features.
\textbf{Second}, in order to maintain the scale of the vertex features after each graph convolution layer, the graph convolution of the DGCNN model~\cite{DBLP:conf/aaai/ZhangCNC18} and the spectral graph convolution of the FAGCNN model~\cite{DBLP:journals/corr/KipfW16} need to perform a multiplication by the inverse of the vertex degree matrix. For instance, the graph convolution layer of the DGCNN model associated with a graph having $n$ vertices is
\begin{equation}
Z= \mathrm{f}(\widetilde{D}^{-1} \widetilde{A} X W),\label{GCN_EAM}
\end{equation}
where $\widetilde{A}=A+I$ is the adjacency matrix of the graph with added self-loops, $\widetilde{D}$ is the degree matrix of $\widetilde{A}$, $X^{n\times c}$ is the vertex feature matrix with each row representing the $c$-dimensional features of a vertex, $W^{c\times c^{'}}$ is the matrix of trainable parameters, $\mathrm{f}$ is a nonlinear activation function (e.g., the Relu function), and $Z^{n\times c^{'}}$ is the output. In a manner similar to the proposed quantum spatial graph convolution defined in Eq.(\ref{GCN_EQ}), $X W$ maps the $c$-dimensional features of each vertex into a set of new $c^{'}$-dimensional features. Moreover, $\widetilde{A} Y$ ($Y:= \widehat{X}_{p} W$) propagates the feature information of each vertex to its neighboring vertices as well as the vertex itself. The $i$-th row $(\widetilde{A} Y)_i$ of the resulting matrix $\widetilde{A} Y$ represents the extracted features of the $i$-th vertex, and corresponds to the summation of $Y_i$ itself and $Y_j$ from the neighbor vertices of the $i$-th vertex. Multiplying by the inverse of $\widetilde{D}$ (i.e., $\widetilde{D}^{-1}$) can be seen as the process of normalizing and assigning equal weights between the $i$-th vertex and each of its neighbours. In other words, the graph convolution of the DGCNN model considers the mutual-influences between specified vertices for the convolution operation as the same. In contrast, the quantum spatial graph convolution of the proposed QSGCNN model defined in Eq.(\ref{GCN_EQ}) assigns an average quantum walk visiting probability distribution to specified vertices with each vertex having a different visiting probability as the weight. Therefore, the extracted vertex feature is the weighted summation of the specified vertex features. As a result, the quantum spatial graph convolution of the proposed QSGCNN model not only maintains the feature scale, but also discriminates the mutual-influences between specified vertices in terms of the different visiting probabilities during the convolution operation.
\textbf{Third}, similar to the proposed QSGCNN model, both the PATCHY-SAN based Graph Convolution Neural Network (PSGCNN) model~\cite{DBLP:conf/icml/NiepertAK16} and the DGCNN model~\cite{DBLP:conf/aaai/ZhangCNC18} need to rearrange the vertex order of each graph structure and transform each graph into the fixed-sized vertex grid structure. Specifically, the PSGCNN model first forms the grid structures and then performs the standard classical CNN on the grid structures. The DGCNN model sorts the vertices through a SortPooling associated with the extracted vertex features from multiple spatial graph convolution layers. Unfortunately, both the PSGCNN model and the DGCNN model sort the vertices of each graph based on the local structural descriptor, ignoring consistent vertex correspondence information between different graphs. By contrast, the proposed QSGCNN model associates with a transitive vertex alignment procedure to transform each graph into an aligned fixed-sized vertex grid structure. As a result, only the proposed QSGCNN model can integrate the precise structural correspondence information over all graphs under investigations.
\textbf{Fourth}, when the PSGCNN model~\cite{DBLP:conf/icml/NiepertAK16} and the DGCNN model~\cite{DBLP:conf/aaai/ZhangCNC18} form fixed-sized vertex grid structures, some vertices with lower ranking will be discarded. Moreover, the Neural Graph Fingerprint Network (NGFN)~\cite{DBLP:conf/nips/DuvenaudMABHAA15} and the Diffusion Convolution Neural Network (DCNN)~\cite{DBLP:conf/nips/AtwoodT16} tend to capture global-level graph features by summing up the extracted local-level vertex features through a SumPooling layer, since both the NGFN model and the DCNN model cannot directly form vertex grid structures. This leads to significant information loss for local-level vertex features. By contrast, the required aligned vertex grid structures and the associated grid vertex adjacency matrices for the proposed QSGCNN model can accurately encapsulate both the original vertex features and the topological structure information of the original graphs. As a result, the proposed QSGCNN overcomes the shortcoming of information loss arising in the mentioned state-of-the-art graph convolutional neural network models.
\textbf{Fifth}, similar to the DGCNN model~\cite{DBLP:conf/aaai/ZhangCNC18}, the quantum spatial graph convolution of the proposed QSGCNN model is also related to the Weisfeiler-Lehman subtree kernel (WLSK)~\cite{shervashidze2010weisfeiler} Specifically, the WLSK kernel employs the classical Weisfeiler-Lehman (WL) algorithm as a canonical labeling method to extract multi-scale vertex features corresponding to subtrees for graph classification. The key idea of the WL method is to concatenate a vertex label with the labels of its neighbor vertices, and then sort the concatenated label lexicographically to assign each vertex a new label. The procedure repeats until a maximum iteration $h$, and each vertex label at an iteration $h$ corresponds to a subtree of height $t$ rooted at the vertex. If the concatenated label of two vertices are the same, the subtree rooted at the two vertices are isomorphic, i.e., the two vertices are seen to share the same structural characteristics within the graph. The WLSK kernel uses this idea to measure the similarity between two graphs. It uses the WL method to update the vertex labels, and then counts the number of identical vertex labels (i.e. counting the number of the isomorphic subtrees) until the maximum of the iteration $h$ in order to compare two graphs at multiple scales. To exhibit the relationship between the proposed quantum spatial graph convolution defined in Eq.(\ref{GCN_EQ}) and the WLSK kernel, we decompose Eq.(\ref{GCN_EQ}) in a row-wise manner, i.e.,
\begin{equation}
Z_i= \mathrm{Relu}(Q_i Y)= \mathrm{Relu} (Q_{ii}Y_i+ \sum_j Q_{ij}),\label{GCN_WL}
\end{equation}
where $Y=\widehat{X}_{p} W$. For Eq.(\ref{GCN_WL}), $Y_i$ can be seen as the continuous valued vectorial vertex label of the $i$-th vertex. Moreover, if $Q_{ij}>0$, the quantum walk starting from the $i$-th vertex can visit the $j$-th vertex, and the visiting probability is $Q_{ij}$. In a manner similar to the WL methods, Eq.(\ref{GCN_WL}) aggregates the continuous label $Y_i$ of the $i$-th vertex and the continuous labels $Y_j$ of the vertices, that can be visited by the quantum walk starting from the $i$-th vertex, as a new signature vector $Q_{ii}Y_i+ \sum_j Q_{ij}$ for the $i$-th vertex. The $\mathrm{Relu}$ function maps $Q_{ii}Y_i+ \sum_j Q_{ij}$ to a new continuous vectorial label. As a result, the the quantum spatial graph convolution of the proposed QSGCNN model can be seen as \textbf{a quantum version of the WL algorithm}, in terms of the quantum vertex information propagation formulated by the quantum walk. As we mentioned in Section~\ref{s2.1}, the quantum walk can significantly reduce the effect of the tottering problem. On the other hand, the classical WL method also suffers from tottering problem~\cite{DBLP:conf/icml/Bai0ZH15}. As a result, the quantum spatial graph convolution can address the tottering problem arising in the classical WL method, and the graph convolution of the DGCNN model is similar to the clasical WL method. In other words, the quantum spatial graph convolution of the proposed QSGCNN model can learn better vertex features of graphs
\textbf{Finally}, note that the proposed QSGCNN model for each graph is invariant with respect to the permutation of the vertices, indicating that the activations of a pair of isomorphic graphs will be the same. As we mentioned, the proposed QSGCNN model consists of three stages, i.e., a) the grid structure construction and input layer, b) the quantum spatial graph convolution layer, and c) the traditional CNN layer. For the first layer, the construction of grid structures relies on the vertex features and adjacency matrix, and is invariant to vertex permutations. As a result, the grid structures for a pair of isomorphic graphs are the same. For the second layer, the input grid structures of different graphs share the same parameter weights, thus the quantum spatial graph convolutions will produce the same extracted vertex features for a pair of isomorphic graphs associated with the same grid structures. Consequently, the subsequent classical CNN layer will correctly identify the isomorphic graphs. As a result, the proposed QSGCNN model can correctly identify pairs of isomorphic graphs.
These observations reveal the advantages of the proposed QSGCNN model, explaining the effectiveness of the proposed model. The proposed QSGCNN model not only overcomes the shortcomings of existing state-of-the-art methods, but also bridges the theoretical and computational gaps between these methods.
\begin{table*}
\centering {
\scriptsize
\vspace{-0pt}
\caption{Information of the Graph Datasets}\label{T:GraphInformation} \vspace{0pt}
\begin{tabular}{|c||c||c||c||c||c||c||c||c||c|}
\hline
~Datasets ~ & ~MUTAG ~ & ~NCI1~ & ~PROTEINS~& ~D\&D~ & ~PTC(MR)~ & ~COLLAB ~ & ~IMDB-B~ & ~IMDB-M~ & ~RED-B~\\ \hline \hline
~Max \# vertices~ & ~$28$~ & ~$111$~ & ~$620$~ & ~$5748$~ & ~$109$~ & ~$492$~ & ~$136$~ & ~$89$~ & ~$3783$~\\ \hline
~Mean \# vertices~ & ~$17.93$~ & ~$29.87$~ & ~$39.06$~ & ~$284.30$~ & ~$25.60$~ & ~$74.49$~ & ~$19.77$~ & ~$13.00$~ & ~$429.61$~\\ \hline
~Mean \# edges~ & ~$19.79$~ & ~$32.30$~ & ~$72.82$~ & ~$715.65$~ & ~$14.69$~ & ~$4914.99$~ & ~$193.06$~ & ~$131.87$~ & ~$497.80$~\\ \hline
~\# graphs~ & ~$188$~ & ~$4110$~ & ~$1113$~ & ~$1178$~ & ~$344$~ & ~$5000$~ & ~$1000$~ & ~$1500$~ & ~$2000$~ \\ \hline
~\# vertex labels~ & ~$7$~ & ~$37$~ & ~$61$~ & ~$82$~ & ~$19$~ & ~$-$~ & ~$-$~ & ~$-$~ & ~$-$~ \\ \hline
~\# classes~ & ~$2$~ & ~$2$~ & ~$2$~ & ~$2$~ & ~$2$~ & ~$3$~ & ~$2$~ & ~$3$~ & ~$2$~ \\ \hline
~Description~ & ~Chemical~ & ~Chemical~ & ~Chemical~ & ~Chemical~ & ~Chemical~ & ~Social~ & ~Social~ & ~Social~ & ~Social~ \\ \hline
\end{tabular}
} \vspace{-0pt}
\end{table*}
\begin{table*}
\centering {
\tiny
\caption{Classification Accuracy (In $\%$ $\pm$ Standard Error) for Comparisons with Graph Kernels.}\label{T:ClassificationGK}
\vspace{0pt}
\begin{tabular}{|c||c||c||c||c||c||c||c||c||c|}
\hline
~Datasets~& ~MUTAG ~ & ~NCI1~ & ~PROTEINS~ & ~D\&D~ & ~PTC(MR)~ & ~COLLAB~ & ~IBDM-B~ & ~IBDM-M~ & ~RED-B~\\ \hline \hline
~\textbf{QSGCNN}~ & ~$\textbf{90.52}\pm0.95$~& ~$77.50\pm0.91$~ & ~$\textbf{75.90}\pm0.79$~ & ~$\textbf{81.70}\pm0.92$~ & ~$\textbf{63.37}\pm1.15$~ & ~$78.80\pm0.89$ ~& ~$\textbf{73.62}\pm1.12$ & ~$\textbf{51.60}\pm1.15$ & ~$\textbf{91.50}\pm0.24$\\ \hline
~JTQK~ & ~$85.50 \pm0.55$~& ~$\textbf{85.32}\pm0.14$~ & ~$72.86\pm0.41$~ & ~$79.89\pm0.32$~ & ~$58.50\pm0.39$~ &~$76.85\pm0.40$~ &~$72.45\pm0.81$~ & ~$50.33\pm0.49$~ & ~$77.60\pm0.35$\\ \hline
~WLSK~ & ~$82.88\pm0.57 $~ &~$84.77\pm0.13$~ & ~$73.52\pm0.43$~ & ~$79.78\pm0.36$~ & ~$58.26\pm0.47 $~ &~$77.39\pm0.35$~ &~$71.88\pm0.77$~ & ~$49.50\pm0.49$~ & ~$76.56\pm0.30$\\ \hline
~WL-OA~ & ~$82.88\pm0.57 $~ &~$84.77\pm0.13$~ & ~$73.52\pm0.43$~ & ~$79.78\pm0.36$~ & ~$58.26\pm0.47 $~ &~$\textbf{80.70}\pm0.10$~ &~$71.88\pm0.77$~ & ~$49.50\pm0.49$~ & ~$76.56\pm0.30$\\ \hline
~SPGK~ & ~$83.38\pm0.81 $~ &~$74.21\pm0.30$~ & ~$75.10\pm0.50$~ & ~$78.45\pm0.26$~ & ~$55.52\pm0.46 $~&~$58.80\pm0.2$~ &~$71.26\pm1.04$~ & ~$51.33\pm0.57$~ & ~$84.20\pm0.70$\\ \hline
~CORE SP~ & ~$88.29\pm1.55 $~ &~$73.46\pm0.32$~ & ~$-$~ & ~$77.30\pm0.80$~ & ~$59.06\pm0.93 $~&~$-$~ &~$72.62\pm0.59$~ & ~$49.43\pm0.42$~ & ~$90.84\pm0.14$\\ \hline
~PIGK~ & ~$76.00\pm2.69 $~ &~$82.54\pm0.47$~ & ~$73.68\pm0.69$~ & ~$78.25\pm0.51$~ & ~$59.50\pm2.44 $~ &~$-$~ &~$-$~ & ~$-$~ & ~$-$\\ \hline
~ GK~ & ~$81.66\pm2.11 $~ &~$62.28\pm0.29$~ & ~$71.67\pm0.55$~ & ~$78.45\pm0.26$~ & ~$52.26\pm1.41 $~ &~$72.83\pm0.28$~ &~$65.87\pm0.98$~ & ~$45.42\pm0.87$~ & ~$77.34\pm0.18$\\ \hline
~RWGK~ & ~$80.77\pm0.72 $~ &~$63.34\pm0.27$~ &~$74.20\pm0.40$~ & ~$71.70\pm0.47$~ & ~$55.91\pm0.37 $~ &~$-$~ &~$67.94\pm0.77$~ & ~$46.72\pm0.30$~ & ~$-$\\ \hline
\end{tabular}
} \vspace{-10pt}
\end{table*}
\begin{table*}
\centering {
\tiny
\caption{Classification Accuracy (In $\%$ $\pm$ Standard Error) for Comparisons with Graph Convolutional Neural Networks.}\label{T:ClassificationGCNN}
\vspace{0pt}
\begin{tabular}{|c||c||c||c||c||c||c||c||c||c|}
\hline
~Datasets~& ~MUTAG ~ & ~NCI1~ & ~PROTEINS~ & ~D\&D~ & ~PTC(MR)~ & ~COLLAB~ & ~IBDM-B~ & ~IMDB-M~ & ~RED-B~ \\ \hline \hline
~\textbf{QSGCNN}~ & ~$\textbf{90.52}\pm0.95$~&~$77.50\pm0.91$~& ~$75.90\pm0.79$~& ~$\textbf{81.70}\pm0.92$~& ~$63.37\pm1.15$~ & ~$\textbf{78.80}\pm0.89$ ~& ~$\textbf{73.62}\pm1.12$ & ~$\textbf{51.60}\pm1.15$ & ~$\textbf{91.50}\pm0.24$\\ \hline
~DGCNN~ & ~$85.83\pm1.66$~&~$74.44\pm0.47$~& ~$75.54\pm0.94$~& ~$9.37\pm0.94$~& ~$58.59\pm2.47$~ & ~$73.76\pm0.49$ ~& ~$70.03\pm0.86$ & ~$47.83\pm0.85$ & ~$76.02\pm1.73$\\ \hline
~PSGCNN~ & ~$88.95\pm4.37$~&~$76.34\pm1.68$~& ~$75.00\pm2.51$~& ~$76.27\pm2.64$~& ~$62.29\pm5.68$~ & ~$72.60\pm2.15$ ~& ~$71.00\pm2.29$ & ~$45.23\pm2.84$ & ~$86.30\pm1.58$\\ \hline
~DCNN~ & ~$66.98$~ &~$56.61\pm1.04$~& ~$61.29\pm1.60$~& ~$58.09\pm0.53$~& ~$56.60$~ & ~$52.11\pm0.71$ ~& ~$49.06\pm1.37$ & ~$33.49\pm1.42$ & ~$-$\\ \hline
~ECC~ & ~$76.11$~ &~$76.82$~ & ~$72.65$~ & ~$74.10$~ & ~$-$~ & ~$67.79$~ & ~$-$ & ~$-$ & ~$-$\\ \hline
~GCCNN~& ~$-$~ &~$\textbf{82.72}\pm2.38$~ & ~$\textbf{76.40}\pm4.71$~ & ~$77.62\pm4.99$~ & ~$\textbf{66.01}\pm5.91$~ & ~$77.71\pm2.51$~ & ~$71.69\pm3.40$ & ~$48.50\pm4.10$ & ~$87.61\pm2.51$\\ \hline
~DGK~ & ~$82.66\pm1.45$~ &~$62.48\pm0.25$~& ~$71.68\pm0.50$~& ~$78.50\pm0.22$~ & ~$57.32\pm1.13$~ & ~$73.09\pm0.25$~& ~$66.96\pm0.56$ & ~$44.55\pm0.52$ & ~$78.30\pm0.30$\\ \hline
~AWE~& ~$87.87\pm9.76$~ &~$-$~ & ~$-$~ & ~$71.51\pm4.02$~ & ~$-$~ & ~$70.99\pm1.49$~& ~$73.13\pm3.28$ & ~$51.58\pm4.66$ & ~$82.97\pm2.86$\\ \hline
\end{tabular}
} \vspace{-10pt}
\end{table*}
\section{Experiments}\label{s4}
In this section, we empirically compare the performance of the proposed QSGCNN model to state-of-the-art graph kernels and deep learning methods on graph classification problems.
\subsection{Comparisons with Graph Kernels}
\noindent\textbf{Datasets:} In this subsection, we utilize nine standard graph datasets from bioinformatics~\cite{DBLP:journals/nar/SchomburgCEGHHS04,doi:10.1021/jm00106a046,bioinformatics2003,DBLP:journals/kais/WaleWK08} and social networks~\cite{socialnetworks2018} to evaluate the performance of the proposed QSGCNN model. These datasets include MUTAG, PTC, NCI1, PROTEINS, D\&D, COLLAB, IMDB-B, IMDB-M and RED-B. A selection of statistics of these datasets are shown in Table.\ref{T:GraphInformation}.\\
\noindent\textbf{Experimental Setup:} We evaluate the performance of the proposed QSGCNN model on graph classification problems against five alternative state-of-the-art graph kernels. These graph kernels include 1) Jensen-Tsallis q-difference kernel (JTQK) with $q=2$~\cite{DBLP:conf/pkdd/Bai0BH14}, 2) the Weisfeiler-Lehman subtree kernel (WLSK)~\cite{shervashidze2010weisfeiler}, 3) optima assignment Weisfeiler-Lehman kernel (WL-OA)~\cite{DBLP:conf/nips/KriegeGW16}, 4) the shortest path graph kernel (SPGK) \cite{DBLP:conf/icdm/BorgwardtK05}, 5) the shortest path kernel based on core variants (CORE SP)~\cite{DBLP:conf/ijcai/NikolentzosMLV18}, 6) the random walk graph kernel (RWGK)~\cite{DBLP:conf/icml/KashimaTI03}, 7) the graphlet count kernel (GK)~\cite{DBLP:journals/jmlr/ShervashidzeVPMB09}, and 8) the propagated information graph kernel (PIGK)~\cite{DBLP:journals/ml/NeumannGBK16}.
For the evaluation, \textbf{the proposed QSGCNN model uses the same network structure on all graph datasets}. Specifically, we set the number of the prototype representations as $M=64$, the number of the quantum spatial graph convolution layers as $5$ (note that, including the original input grid structures, the spatial graph convolution produces $6$ concatenated outputs), and the channels of each quantum spatial graph convolution as $32$. Following each of the concatenated outputs after the quantum graph convolution layers, we add a traditional CNN layer with the architecture as C$64$-P$2$-C$64$-P$2$-C$64$-F$64$ to learn the extracted patterns, where C$k$ denotes a traditional convolutional layer with $k$ channels, $P$k denotes a classical MaxPooling layer of size and stride $k$, and FC$k$ denotes a fully-connected layer consisting of $k$ hidden units. The filter size and stride of each C$k$ are all $5$ and $1$. With the six sets of extracted patterns after the CNN layers to hand, we concatenate them and add a new fully-connected layer followed by a Softmax layer with a dropout rate of $0.5$. We use the rectified linear units (ReLU) in either the graph convolution or the traditional convolution layer. The learning rate of the proposed model is $0.00005$ for all datasets. The only hyperparameter we optimized is the number of epochs and the batch size for the mini-batch gradient decent algorithm. To optimize the proposed QSGCNN model, we use the Stochastic Gradient Descent with the Adam updating rules. Finally, note that, the proposed QSGCNN model needs to construct the prototype representations to identify the transitive vertex alignment information over all graphs. The prototype representations can be computed from the training graphs or both the training and testing graphs. We observe that the proposed model associated with the two variants dose not influence the final performance. Thus, in our evaluation we proposed to compute the prototype representations from both the training and testing graphs. In this sense, our model can be seen as an instance of transductive learning~\cite{DBLP:conf/uai/GammermanAV98}, where all the graphs are used to compute the prototype representations, and the class labels of the test graphs are not observed during the training phase. For the proposed QSGCNN model, we perform $10$-fold cross-validation to compute the classification accuracies, with nine folds for training and one folds for testing. For each dataset, we repeat the experiment 10 times and report the average classification accuracies and standard errors in Table.\ref{T:ClassificationGK}.
We set the parameters controlling the maximum height of the subtrees for the Weisfeiler-Lehman isomorphism test (WLSK kernel) and for the tree-index method (JTQK kernel) to $10$. This is based on the previous empirical studies of Shervashidze et al.~\cite{shervashidze2010weisfeiler} and Bai et al.~\cite{DBLP:conf/pkdd/Bai0BH14}. For each graph kernel, we perform $10$-fold cross-validation using the LIBSVM implementation of C-Support Vector Machines (C-SVM) and we compute the classification accuracies. We perform cross-validation on the training data to select the optimal parameters for each kernel and fold. We repeat the experiment 10 times for each kernel and dataset and we report the average classification accuracies and standard errors in Table.\ref{T:ClassificationGK}. Note that for some kernels we directly report the best results from the original corresponding papers, since the evaluation of these kernels followed the same setting of ours.\\
\noindent\textbf{Experimental Results and Discussion:} Table.\ref{T:ClassificationGK} shows that the proposed QSGCNN model significantly outperforms the alternative state-of-the-art graph kernels in this study. Although, the proposed model cannot achieve the best classification accuracy on the NCI1 and COLLAB datasets, but the proposed model is still competitive and the accuracy on the COLLAB dataset is only a little lower than the WL-OA kernel. On the other hand, the accuracy of the proposed model on the NCI1 dataset is still higher than the SPGK, CORE SP, GK and RWGK kernels. The reasons for the effectiveness are twofold. First, the state-of-the-art graph kernels for comparisons are typical examples of R-convolution kernels. Specifically, these kernels are based on the isomorphism measure between any pair of substructures, ignoring the structure correspondence information between the substructures. By contrast, the associated aligned vertex grid structure for the proposed QSGCNN model incorporates the transitive alignment information between vertex over all graphs. Thus, the proposed model can better reflect the precise characteristics of graphs. Second, the C-SVM classifier associated with graph kernels can only be seen as a shallow learning framework~\cite{DBLP:conf/icassp/ZhangLYG15}. By contrast, the proposed QSGCNN model can provide an end-to-end deep learning architecture for graph classification, and can better learn the graph characteristics. The experiments demonstrate the advantages of the proposed QSGCNN model, compared to the shallow learning framework. Third, some alternative kernels are related to the Weisfeiler-Lehman method. As we have stated in Section~\ref{s3.4}, the kernels based on the Weisfeiler-Lehman method may suffer from the tottering problem. By contrast, the proposed model based on quantum walk can significantly reduce the effect of tottering walks. The experiments also demonstrate the effectiveness.
\subsection{Comparisons with Deep Learning Mthods}
\noindent\textbf{Datasets:} In this subsection, we further compare the performance of the proposed QSGCNN model with state-of-the-art deep learning methods for graph classifications. The datasets for the evaluations include the mentioned five datsets from bioinformatics, as well as three social network datasets. The social network datasets include COLLAB, IMDB-B, and IMDB-M. Details of these social network datasets can be found in Table.\ref{T:GraphInformation}.\\
\noindent\textbf{Experimental Setup:} We evaluate the performance of the proposed QSGCNN model on graph classification problems against five alternative state-of-the-art deep learning methods for graphs. These methods include 1) the deep graph convolutional neural network (DGCNN)~\cite{DBLP:conf/aaai/ZhangCNC18}, 2) the PATCHY-SAN based convolutional neural network for graphs (PSGCNN)~\cite{DBLP:conf/icml/NiepertAK16}, 3) the diffusion convolutional neural network (DCNN)~\cite{DBLP:conf/nips/AtwoodT16}, 4) the edge-conditioned convolutional networks (ECC)~\cite{DBLP:conf/cvpr/SimonovskyK17}, 5) the deep graphlet kernel (DGK)~\cite{DBLP:conf/kdd/YanardagV15}, 6) the graph capsule convolutional neural network (GCCNN)~\cite{DBLP:journals/corr/abs-1805-08090}, and 7) the anonymous walk embeddings based on feature driven (AWE)~\cite{DBLP:conf/icml/IvanovB18}. For the proposed QSGCNN model, we use the same experimental setups when we compare the proposed model to graph kernels. For the PSGCNN, ECC, and DGK model, we report the best results from the original papers~\cite{DBLP:conf/icml/NiepertAK16,DBLP:conf/cvpr/SimonovskyK17,DBLP:conf/kdd/YanardagV15}. Note that, these methods follow the same setting with the proposed QSGCNN model. For the DCNN model, we report the best results from the work of Zhang et al.,~\cite{DBLP:conf/aaai/ZhangCNC18}, following the same setting of ours. For the AWE model, we report the classification accuracies of the feature-driven AWE, since the author have stated that this kind of AWE model can achieve competitive performance on label dataset. Finally, the PSCN and ECC models can leverage additional edge features. Since most graph datasets and all the alternative methods to used for comparisons do not leverage edge features, in this work we do not report the results associated with edge features. The classification accuracies and standard errors for each deep learning method are shown in Table.\ref{T:ClassificationGCNN}.\\
\noindent\textbf{Experimental Results and Discussion:} Table~\ref{T:ClassificationGCNN} indicates that the proposed QSGCNN model significantly outperforms state-of-the-art deep learning methods for graph classifications, on the MUTAG, D\&D, COLLAB, IBDM-B, IBDM-M and RET-B datasets. On the other hand, only the accuracy of the GCCNN model on the NCI1 and PTC datasets and that of the DGCNN model on the PROTEINS dataset are a higher than the proposed QSGCNN model. But the proposed QSGCNN is still competitive and outperform the remaining methods on the three datasets. The reasons of the effectiveness are fivefold.
First, similar to the state-of-the-art graph kernels, all the alternative deep learning methods (i.e., the DGCNN, PSGCNN, DCNN, ECC, GCCNN, DGK and AWE models) for comparisons also cannot integrate the correspondence information between graphs into the learning architecture. Especially, the PSGCNN, DGCNN and ECC models need to reorder the vertices, but these methods rely on simple but inaccurate heuristics to align the vertices of the graphs, i.e., they sort the vertex orders based on the local structure descriptor of each individual graph and ignore the vertex correspondence information between different graphs. Thus, only the proposed QSDCNN model can precisely reflect the graph characteristics through the layer-wise learning.
Second, the PSGCNN and DGCNN models need to form a fixed-sized vertex grid structure for each graph. Since the vertex numbers of different graphs are different, forming such sixed-sized grid structures means some vertices of each graph may be discarded, leading to information loss. By contrast, as we have mentioned in Section~\ref{s2} and Section~\ref{s3}, the associated aligned vertex grid structures can completely preserve the information of original graphs. As a result, only the proposed QSGCNN model can completely integrate the original graph characteristics into the learning process.
Third, unlike the proposed model, the DCNN model needs to sum up the extracted local-level vertex features from the convolution operation as global-level graph features through a SumPooling layer. Thus, only the QSGCNN model can learn the graph topological information through the local vertex features.
Forth, unlike the PSGCNN, DGCNN, GCCNN and ECC models that are based on the original vertex adjacency matrix to formulate vertex connection information of the graph convolution operation, the graph convolution operation of the proposed QSGCNN model formulates the vertex connection information in terms of the average mixing matrix of continuous-time quantum walk. As we have stated in Section~\ref{s2}, the quantum walk is not dominated by the low frequency of the Laplacian spectrum and can better distinguish different graph structures. Thus, the proposed QSDCNN model has better ability to identify the difference between different graphs.
Fifth, similar to the DGCNN, PSGCNN and DGK models, the proposed QSGCNN model is also related to the classical Weisfeiler-Lehman (WL) method. Since the classical WL method suffers from tottering problem, the related DGCNN, PSGCNN and DGK models also process the same drawback. By contrast, the graph convolution operation of the proposed QSGCNN model can be seen as the quantum version of the classical WL algorithm. Since the quantum walk can reduce the problem of tottering problem, the proposed QSGCNN model overcomes the shortcoming of tottering problem arising in the DGCNN, PSGCNN and DGK models. Sixth, the AWE model is based on the classical random walk. By contrast, the proposed QSGCNN model is based on the quantum random walk, that has been proven powerful to better distinguish different graph structures. The evaluation demonstrates the advantages of the proposed QSGCNN model, compared to the state-of-the-art deep learning methods.
\section{Conclusion}\label{s6}
In this paper we have developed a new Quantum Spatial Graph Convolutional Neural Network (QSGCNN) model, that can directly learn an end-to-end deep learning architecture for classifying graphs of arbitrary sizes. The main idea of the proposed QSGCNN model is to transform each graph into a fixed-sized vertex grid structure through transitive alignment between graphs and propagate the grid vertex features using the proposed quantum spatial graph convolution operation. Compared to state-of-the-are deep learning methods and graph kernels, the proposed QSGCNN model cannot only preserve the original graph characteristics, but also bridge the gap between the spatial graph convolution layer and the traditional convolutional neural network layer. Moreover, the proposed QSGCNN can better distinguish different structures, and the experimental evaluations demonstrate the effectiveness of the proposed QSGCNN model on graph classification problems.
In this work, we used the same network architecture for all datasets. In future works, we aim to learning the optimal structure for each dataset, which in turn should lead to improved performance. Furthermore, in future works, we also aim to extend the proposed QSGCNN model and develop a new quantum graph neural network drawing on edge-based grid structures. In previous works~\cite{DBLP:conf/iciap/BaiZR0H15,DBLP:journals/prl/BaiRCZRBH17,DBLP:journals/pr/BaiEH16} we have shown how to characterize the edge information of the original graphs through the directed line graphs, where each vertex of the line graph represents an edge of original graphs. Moreover, we have illustrated the relationship between the discrete-time quantum walks and the directed line graphs. It will be interesting to develop a novel quantum edge-based convolutional network associated with the discrete-time quantum walks and the directed line graphs.
Finally, note that, Xu et al.,~\cite{DBLP:journals/corr/abs-1810-00826} have recently indicated that the convolutional operation of most existing graph convolutional neural networks associated with the adjacency matrix can be seen as directly employing a $1$-layer perceptron followed by a non-linear activation function such as a ReLU. Moreover, they developed a new graph isomorphism network model based on a new vertex information aggregation layer followed by multi layer perceptrons. They demonstrated that this can significantly improve the performance of state-of-the-art graph convolutional networks. This work enlightens our future work, and we will further extend the proposed QSGCNN model into a new quantum isomorphism network.
\section*{Acknowledgments}
This work is supported by the National Natural Science Foundation of China (Grant no.61503422 and 61602535), the Open Projects Program of National Laboratory of Pattern Recognition (NLPR), and the program for innovation research in Central University of Finance and Economics.
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\bibliographystyle{IEEEtran}
|
1,108,101,565,628 | arxiv | \section*{Executive Summary}
The document at hand describes the physics case, the key
experiments proposed and the technical solution for the project ``WASA at
COSY'', which concerns the transfer of the WASA pellet-target and
detector system from CELSIUS (TSL, Uppsala, Sweden) to COSY (FZJ,
J\"ulich, Germany) and its installation at an internal target position.
WASA at COSY will provide unique scientific possibilities for
research in hadron physics with hadronic probes, since it combines:
\begin{itemize}
\item COSY with its (phase-space cooled, polarized) proton and deuteron
beams with energies sufficient to cover the strange quark sector
including $\phi$-mesons
\item WASA, a close to $4\pi$ detector for both photons and charged
particles
\end{itemize}
Since both COSY and WASA are operational devices, the
time needed to install and commission it is short: it is
anticipated that the physics program will start at the end of 2006.\\
\noindent
The physics that will be investigated comprises:
\begin{description}
\item {\bf Symmetries and symmetry breaking:} \\
{\em The fundamental theories of physics are based on symmetry, yet:
our world is filled with asymmetry} (T.D. Lee).
Finding and studying symmetries and symmetry violations in hadronic
reactions will help
to better understand the strong interaction.
Hadron beams are particularly well suited for such investigations due to
the spin- and isospin filter mechanism. In addition, it is even conceivable
to search for physics beyond the Standard Model.
\item{\bf Hadron structure and interactions:} \\
{\em That [intermediate distance] scale is the richest
phenomenologically, and is certainly the crux region to understand
\ldots what QCD is really about. And at the heart of the subject is
the hadron spectrum, in particular the spectrum built from light
quarks. (\ldots) Without question, there is a great need \ldots for
a new round of experiments, especially utilizing hadron beams}
(J.D. Bjorken).
Finding and further
investigating specific hadronic bound systems will --- together with
corresponding progress in theory --- provide a more fundamental
insight into how nature makes hadrons.
\end{description}
The proposal concentrates on the study of
symmetry breaking as the primary objective, in particular in $\eta$
and $\eta{}'$-decays and
meson production. In addition, the potential of
further studies is discussed.
An experienced and enthusiastic community of more than one hundred scientists
from all over Europe and abroad has started to work for ''WASA at COSY''
to become a reality and is looking forward to exploit its scientific
potential.
\cleardoublepage
{\bf
\tableofcontents
}
\cleardoublepage
\pagenumbering{arabic}
\setcounter{page}{1}
\section{Introduction}
Hadron physics is concerned with one of the most difficult but
also most fascinating problems in contemporary physics.
It is commonly accepted that the underlying theory,
Quantum Chromo Dynamics (QCD), is correct. However, very little is
known about the solution of this non--linear quantum field theory in
the regime of small
and moderate energy scales pertinent to the matter surrounding us.
QCD is characterized by two particular phenomena,
confinement and chiral symmetry breaking. QCD is formulated in terms
of colored quarks and gluons. These fundamental particles have
never been observed as free states but only in composite systems,
so called hadrons, i.e. they are confined within colorless mesons,
baryons and, possibly, exotics.
Furthermore, in the sector of the light up, down and strange
quarks, QCD possesses an approximate chiral symmetry, which is
not observed in nature but rather broken spontaneously with the
appearance of almost massless Goldstone bosons. These can be
identified with the lowest octet of pseudoscalar mesons. Such broken
symmetries also play an important role in phenomenon of superconductivity
or the Higgs mechanism, which is believed to lead to the creation masses of the
quarks and leptons. The consequences of the dynamical and also the
explicit chiral symmetry violation through the current quark masses
in QCD can be systematically
explored
with
effective field theory. Presumably
related to these phenomena is the generation of hadron masses in
QCD, e.g. the
proton and the neutron are much heavier than the current quarks they
are made of. In the energy regime considered here, the interaction
between the quarks and gluons is highly non-linear and also
non-perturbative, excluding the use of standard perturbative methods
as applied e.g. in Quantum Electro Dynamics or QCD at high energies,
where asymptotic freedom is the dominant feature of the
theory. A deeper understanding of the structure and dynamics of
hadrons, in particular their excitation spectrum, as derived from
QCD would explore
one of the last {\it terrae} {\it incognitae} of the Standard Model.
\noindent
To understand complex systems such as hadrons, a variety of
complementary experimental and theoretical studies are mandatory.
The WASA detector at COSY offers a unique possibility to
deepen our understanding of aspects connected with QCD in the
non-perturbative regime through a precise study of symmetry breakings
and very specific investigations of hadron structure.
The $\eta$ and $\eta'$ decays, that vanish in the limit of equal quark masses,
allow us to explore the explicit isospin symmetry
breaking
in QCD.
It is well established that isospin violating quark mass
effects are often masked by electromagnetic corrections and thus
such studies require
the detection of neutral decay particles.
A textbook example for this is the threshold
photoproduction of neutral pions off protons which has so clearly
revealed the chiral loops inherent
to the spontaneous chiral symmetry breaking of QCD.
Through the same quark mass term of QCD, such quark mass effects also
appear in systems of a few nucleons, so that $dd$ collisions
producing $^4$He plus neutral Goldstone bosons at COSY energies
offer yet another tool to explore this area - also because only now
theoretical tools are becoming available that allow a model-independent
analysis of these processes. Furthermore, precision measuremetns of
rare $\eta$ and $\eta'$
decays can be used to get new
limits on the breaking of the fundamental $C$, $P$ and $T$ symmetries, or
combinations thereof. In addition, reactions involving the $\eta '$
might also offer insight into the elusive glue through the
anomaly.
\noindent
The spontaneous chiral symmetry breaking is believed
to be at the heart of the mass generation in QCD. Still, this issue cannot
be separated from the question: what kind of hadrons are really generated
in QCD? In the course of the last year, the paradigm of the constituent quark model,
i.e. the hadrons are either quark-antiquark or three quark states, has been
challenged by many experiments. However, a definite picture concerning the
nature of e.g. the light scalar mesons, the $a_0 (980)$, $f_0 (980)$, etc.,
the $\Lambda (1405)$ or the
pentaquark, the $\Theta^+ (1540)$, has not yet emerged. Evidently, if one
does not understand the nature of these states and others, a true comprehension
of confinement can not be achieved. WASA at COSY is capable to contribute
significantly to test the models which are offered to explain
these states through the precise measurements of decay chains or the
coupling to other hadrons. It should
be stressed that all these proposed experiments
will be {\em precision measurements}, because only accurate data
will allow to test and constrain the theory.
With the advent of effective field theory methods, supplemented
by coupled channel analyses, the theoretical tools of the required precision
are available to analyze the data in a model-independent way, reflecting the
change of paradigm in the foundations in this area of theoretical physics,
that in the past has been dominated by model building.
Moreover, lattice
gauge theory offers prospects to confront certain observables, in particular
spectroscopic issues, with ab initio calculations.
A very close contact between
experiment and theory is foreseen and indispensable to achieve progress in
the field of hadron physics.
\noindent
In this document it is proposed to perform high precision
measurements
of several hadronic reactions to confront the theoretical
predictions
sketched above. In particular, investigations of symmetry breaking
in the decay of the $\eta$ and $\eta'$ mesons and isospin violation
in the
reaction $\vec{d}d\to\alpha\pi^0$ are seen to be very
promising for this purpose. The combination of the detector system
WASA
and the COSY accelerator facility perfectly matches the requirements
needed to measure these reactions. The beam energy range, the phase
space cooling and the availability of (polarized) proton and
deuteron beams at COSY are essential aspects for high luminosity
measurements with low background of these processes. The WASA
detector provides nearly full solid angle coverage for both charged
and neutral particles, which is needed to for kinematically complete
measurements of the multiparticle final states. Furthermore, the
usage of frozen pellets of liquid hydrogen and deuterium as the
target minimizes background from secondary reactions but at the same
time allows high luminosity conditions. Since the developement and
construction of a detector system fulfilling these requirements
requires a number of years, the transfer of an existing detector
like WASA to COSY for these measurements will greatly reduce the
lead-time until physics output can be expected.
\label{WASAintro}
\noindent
To summarize: The challenge of hadron physics is not only
to gain an understanding
of QCD in the non--perturbative regime, e.g. to explain the mechanism
underlying confinement, but also the possibility of unhinging the foundations
of the Standard Model at low energies. The understanding of the structure
of matter is intimately linked to the structure and dynamics of
hadrons. Thus, hadron physics plays a central role in the whole building
of physics. This is the area of research where WASA at COSY
can contribute significantly as will be demonstrated in the following
sections.
\clearpage
\section{Physics case}
\label{sec:physics}
\subsection{Primary objective:\\ Symmetries and symmetry breaking}
\label{sec:symmetries}
The degrees of freedom within the Standard Model are leptons and
quarks for the matter fields and gluons, photons and the weak gauge bosons.
However, at low and intermediate energies
only the leptons and photons play a role as dynamical degrees of
freedom: $W^\pm$ and $Z^0$ are too heavy and quarks and gluons are
trapped inside composite objects (hadrons, glueballs, and possibly
exotics). The mechanisms behind this process are confinement and
spontaneous breakdown of chiral symmetry. However, through its
symmetries the underlying theory is still visible.
If a symmetry is spontaneously broken, there is no obvious
connection between the excitation spectrum and the symmetry.
The currents are still conserved and, as a consequence, the
interactions of the hadrons at low energies are largely constrained.
The same holds true in the presence of a small explicit symmetry
breaking: in nature the chiral symmetry is broken by the non-vanishing
quark masses. We will discuss in the following
examples, where a study of symmetries and symmetry breaking patterns
in hadronic systems allows to get insights on symmetries and symmetry
breaking patterns of QCD.
\subsubsection{\boldmath Decays of $\eta$ and $\eta^\prime$ mesons}
\label{subsubetap}
Isospin symmetry forbidden decays $\eta \left(\eta^\prime\right)
\rightarrow 3\pi$ are sensitive to isospin symmetry breaking (ISB) due to
the light quark mass difference $\Delta \mbox{m} =
\mbox{m}_d - \mbox{m}_u$, and provide an approach complementary to ISB
estimates from pseudoscalar meson masses and production processes (see
section~\ref{subsubalphapi} and \cite{SWeinberg77,JGasser82,miller}).
ISB effects in these decay channels can be associated directly with
$\Delta \mbox{m}$, since electromagnetic effects are expected to be
small~\cite{Sutherland:1966pl,Baur:1995gc}.
Moreover, the ratio of symmetry breaking and allowed $\eta^\prime$ decays
to $3\pi$ and $\eta\,2\pi$, respectively, is directly proportional to the
square of $\Delta \mbox{m}$ (see Ref.~\cite{Gross:1979ur} and
section~\ref{subsubetapday1}):
Mixing between isospin eigenstates $\mid\tilde{\pi}^0>$ and
$\mid\tilde{\eta}>$ occurs due to the non--vanishing matrix element of the
quark mass contribution to the QCD Hamiltonian density ${\cal H}_m =
\mbox{m}_u u \bar{u} + \mbox{m}_d d \bar{d} + \mbox{m}_s s \bar{s}$
\begin{equation}
<\tilde{\pi}^0\mid {\cal H}_m \mid\tilde{\eta}> =
\frac{1}{\sqrt{6}} \left< \left( u \bar{u} - d \bar{d} \right) \mid
{\cal H}_m \mid
\left( u \bar{u} + d \bar{d} - s \bar{s} \right) \right> =
- \frac{1}{\sqrt{6}} \Delta\mbox{m} \;,
\label{eq_pi_eta_delta_m}
\end{equation}
where SU(3) meson wave functions and a pseudoscalar singlet--octet mixing
angle $\Theta_{ps} \approx -19.5^\circ$ have been employed~\cite{Nefkens:2002sa}.
The mixing angle $\Theta_{\pi\eta}$ for the physical states $\pi^0$ and
$\eta$ is related to $\Delta \mbox{m}$ by~\cite{Gross:1979ur,Kaiser:2000, Feldmann:1998}:
\begin{equation}
\left( \begin{array}{l}
\pi^0 \\ \eta
\end{array} \right) =
\left( \begin{array}{rr}
\cos{\Theta_{\pi\eta}} & \sin{\Theta_{\pi\eta}}\\
-\sin{\Theta_{\pi\eta}} & \cos{\Theta_{\pi\eta}}
\end{array} \right)\;
\left( \begin{array}{l}
\tilde{\pi}^0 \\ \tilde{\eta}
\end{array} \right) \; ; \;
\sin{\Theta_{\pi\eta}} =
\frac{\sqrt{3}\,\Delta \mbox{m}}
{4\left(\mbox{m}_s - \hat{\mbox{m}} \right)}
\label{eq_pi_eta_mix}
\end{equation}
with the average light quark mass $\hat{\mbox{m}} =
(\mbox{m}_u + \mbox{m}_d)/2$.
Measuring the ratios of isospin symmetry breaking $\eta^\prime \rightarrow
3\pi$ decays with respect to the allowed $\eta\,2\pi$ modes
\begin{equation}
{\cal R}_1 =
\frac{\Gamma\left( \eta^\prime \rightarrow \pi^0 \pi^0 \pi^0 \right)}
{\Gamma\left( \eta^\prime \rightarrow \eta \pi^0 \pi^0 \right)} \; ;\;
{\cal R}_2 =
\frac{\Gamma\left( \eta^\prime \rightarrow \pi^0 \pi^+ \pi^- \right)}
{\Gamma\left( \eta^\prime \rightarrow \eta \pi^+ \pi^- \right)} \; ,
\label{eq_pi_eta_ratio}
\end{equation}
the $\pi$--$\eta$ mixing angle in this approach can be determined in
$\eta^\prime$ decays from
\begin{equation}
{\cal R}_i = \mbox{P}_i \sin^2{\Theta_{\pi\eta}},
\end{equation}
with $\mbox{P}_i$ denoting phase space factors~\cite{Gross:1979ur}.
The energy dependence of hadronic decays of $\eta$ and $\eta^\prime$ to
pseudoscalars, i.e.\ $\eta\left(\eta^\prime\right) \rightarrow 3\,\pi$ and
$\eta^\prime \rightarrow \eta\pi\pi$, is contained in the Dalitz plot
distribution of the decay products.
Some of the published results obtained for slope parameters of the Dalitz
plots for $\eta \rightarrow 3\pi$~\cite{Gormley:1970qz,Layter:1973ti,
Alde:1984wj,Amsler:1995sy,Amsler:1997up,Abele:1998yi,Abele:1998yj,
Tippens:2001fm} are not in agreement with each other and with theoretical
predictions~\cite{Kambor:1995yc,Beisert:2003zs}.
In particular, the predicted slope parameter for the neutral decay is
excluded by the most recent published data~\cite{Tippens:2001fm}.
Preliminary results on both charged and neutral $3\pi$ decay modes from a
high statistics measurement at the KLOE facility have recently been
reported~\cite{DiMicco:2003cd}.
The energy dependence of hadronic decays $\eta^\prime \rightarrow
\eta\,2\pi$ and $\eta^\prime \rightarrow 3\pi$ is discussed in the ChPT
framework in more detail in~\cite{Beisert:2002ad,Beisert:2003zs}.
The presently available data with highest statistics on the $\eta\,2\pi$
decay mode are still inconclusive, whether Dalitz plot distributions
deviate from phase space.
For the $\eta^\prime \rightarrow 3\pi$ decay, no experimental data on the
Dalitz plot parameters are presently available.
Hadronic decay modes also constrain the interpretation of the scalar meson
nonet:
Isovector $a_0(980)$ exchange is found to dominate $\eta^\prime \rightarrow
\eta\,2\pi$~\cite{Fariborz:1999gr,Beisert:2003zs}, and the experimental
data are used to fix unknown scalar--pseudoscalar--pseudoscalar
couplings~\cite{Fariborz:1999gr}.
In comparison to $\eta \rightarrow 3\pi$ decays, exchange diagrams
involving $f_0(980)$ and $a_0(980)$ are expected to have a stronger
influence in $\eta^\prime$ decays to the $3\pi$ channel, since the exchange
particles are much closer to the mass shell~\cite{Abdel-Rehim:2002an}.
The axial U(1) anomaly prevents the pseudoscalar singlet $\eta_0$ from being
a
Goldstone boson.
The dominant contribution to the mass of the singlet state arises from the
divergence of the singlet axial current, that acquires an additional term
with the gluonic field strength tensor and, consequently, does not vanish
in the chiral limit.
Since the physical states $\eta$ and $\eta^\prime$ are mixtures of the
octet and singlet fields, $\eta_8$ and $\eta_0$, respectively, the
$\eta^\prime$ cannot be disentangled from the $\eta$ and a proper study of
$\eta^\prime$ physics requires to account for the $\eta$ as well.
Precision data on decays of $\eta$ and $\eta^\prime$ allow to probe a
variety of aspects of low--energy hadron dynamics.
Experimental results can be compared with model--independent calculations
of chiral perturbation theory (ChPT) with a clear theoretical connection to
QCD.
Two-photon decays in the $\eta$--$\eta^\prime$ system allows ChPT predictions
to be tested,
particularly its extension to the U(3) framework including the axial U(1)
anomaly.
Radiative decays $\eta\left(\eta^\prime\right) \rightarrow 2\gamma$ and
$\eta\left(\eta^\prime\right) \rightarrow \pi^+ \pi^- \gamma$ test
parameters of the QCD triangle and box anomalies.
Two--photon decays $\eta\left(\eta^\prime\right) \rightarrow
\gamma \gamma$ provide insight into the axial $\mbox{U}_A(1)$ anomaly of
QCD, since they are determined by the pseudoscalar singlet and octet
couplings to the divergences of the relevant axial--vector currents and the
singlet--octet mixing angle $\Theta_{ps}$ (see
section~\ref{subsubetapmedium} and~\cite{Ball:1996zv}).
The latter can be constrained from radiative decays of vector ($V$) and
pseudoscalar ($P$) mesons in $V\,(P) \rightarrow P\,(V) \gamma$ processes,
i.e.\ from $\eta^\prime$ by the $\rho \gamma$ and $\omega \gamma$ decay
modes.
A model framework to conclude about the glue content of the $\eta^\prime$
is discussed in~\cite{Kou:1999tt}.
The two--photon decays of $\eta$ and $\eta^\prime$ are furthermore well
suited to confirm the number of colors in the low--energy hadronic sector
to be $\mbox{N}_c = 3$~\cite{Baer:2001,Borasoy:2004ua}.
Within a few weeks, the available statistics for radiative $\eta^\prime$
decays could be increased by two orders of magnitude at the WASA facility
(Table~\ref{tab_etap_rates}).
Theoretically, the anomalous behavior of the effective action under chiral
transformations is reproduced by the Wess--Zumino--Witten term of the
effective Lagrangian~\cite{Wess:1971yu,Witten:1983tw} which yields the
major contribution to the decays $\eta\left(\eta^\prime\right) \rightarrow
\gamma \gamma$.
The $\pi^+ \pi^- \gamma$ mode is dominated by $\eta^\prime \rightarrow
\rho \gamma$.
However, the amplitude of the non--resonant $\eta^\prime \rightarrow
\pi^+ \pi^- \gamma$ decay is determined both by parameters of the AVV
(axialvector--vector--vector) triangle anomaly, that can be derived from
radiative decays as discussed above, and by the AAAV box
anomaly~\cite{Wess:1971yu,Chanowitz:1975jm,Chanowitz:1980ma,
Benayoun:2003we,Borasoy:2003yb,Borasoy:2004qj}.
On the basis of the presently available statistics, non--resonant
contributions have not been unequivocally extracted (see
Ref.~\cite{Acciarri:1998yx,Abele:1997yi} and section~\ref{subsubetapday1}).
Dalitz decays $\eta\left(\eta^\prime\right) \rightarrow \gamma \gamma^*
\rightarrow \gamma l^+ l^-$ with $l= e,\,\mu$, where a time--like virtual
photon converts to a lepton pair, probe the transition form factor, i.e.\
the electromagnetic properties of $\eta$ and $\eta^\prime$ in terms of the
spatial distribution of meson charge with the $l^+l^-$ invariant mass
corresponding to the four--momentum squared of the virtual photon
(theoretical approaches for $\eta$ decays are discussed
in~\cite{Stepaniak:2002ad,Borasoy:2003yb}).
The available data for $\eta^\prime \rightarrow \gamma l^+ l^-$ consist of
$33 \pm 7$ $\mu^+ \mu^- \gamma$
events~\cite{Dzhelyadin:1980ki,Landsberg:1986fd}.
Thus, a detailed analysis of the leptonic mass spectrum is presently not
possible due to the lack of statistics.
Significant progress could be made using the WASA facility at COSY
(section~\ref{subsubetapmedium}).
The decays $\eta\left(\eta^\prime\right) \rightarrow
l^+ l^- l^+ l^-$ into lepton pairs address decays via two off--shell
photons and indicate whether double vector meson dominance is realized in
nature.
The coupling to two virtual photons is substantial for the real part of the
decay amplitude to a single lepton pair, but also an important issue in
kaon decays, and for the hadronic contribution to the anomalous magnetic
moment of the muon~\cite{Hayakawa:1997rq,Bijnens:1997ac}.
The corresponding $\eta \gamma^* \gamma^*$ form factor has neither been
measured in the time--like nor in the space--like region (see
section~\ref{subsubetapday1}), a recent theoretical investigation is
presented in~\cite{Borasoy:2003yb}.
The dominant mechanism in the leptonic decay $\eta \rightarrow e^+ e^-$,
which is forbidden to proceed via a single photon intermediate state, is a
fourth order electromagnetic process with two virtual
photons~\cite{Bergstrom:1982zq,Landsberg:1986fd}.
The decay is additionally suppressed by helicity factors $\mbox{m}_e /
\mbox{m}_{\eta}$ at the $\gamma e^+ e^-$ vertices.
A very low branching ratio
$5 \cdot 10^{-9}$~\cite{Savage:1992ac,Ametller:1993we}, predicted by the
Standard Model, makes the decay sensitive to contributions from
non--conventional effects that would significantly increase the branching
ratio (section~\ref{subsubetapmedium}).
\renewcommand{\arraystretch}{1.1}
\begin{table}[h]
\caption{Counting rate estimates including detector efficiencies for
$\eta$ ($\eta^\prime$) decays with WASA at COSY based on a luminosity of
$10^{32}\,\mbox{cm}^{-2}\,\mbox{s}^{-1}$, with a $20\,\mu\mbox{b}$
($300\,\mbox{nb}$) cross section in $pp \rightarrow
pp \eta\left(\eta^\prime\right)$ at $2.250\,\mbox{GeV/c}$
($3.350\,\mbox{GeV/c}$) beam momentum.}
\vspace{1ex}
{\begin{tabular}{@{}llr@{.}l@{$\,\pm\,$}r@{.}lrr@{\hspace{0.5ex}}l@{}}
\hline
& Decay & \multicolumn{4}{c}{Branching fraction} &
Existing & Counting & \\
& mode & \multicolumn{4}{c}{$\Gamma_i / \Gamma_{tot}$} & data & rate & \\
& & \multicolumn{4}{c}{\cite{PDBook}} & [events] &
[evts/day] & \\ \hline\hline
$\eta$ & $e^+ e^- \pi^+ \pi^-$ & 4 &
\multicolumn{3}{l}{$\mbox{}^{+14.0}_{-2.7}$ $\cdot 10^{-4}$} & 5 &
7000 & \footnotemark\\
{\it (semi)--leptonic} & $e^+ e^- e^+ e^-$ & $<\,6$ &
\multicolumn{3}{l}{\hspace{-2ex} 9 $\cdot 10^{-5}$} & - &
450 & \footnotemark\\
& $e^+ e^-$ & $<\,7$ & \multicolumn{3}{l}{\hspace{-2ex} 7 $\cdot 10^{-5}$}
& - & $1/6$ & \footnotemark\\
& $\pi^0 e^+ e^-$ & $<\,4$ & \multicolumn{3}{l}{$\cdot 10^{-5}$} & - &
$1/15 - 1/2$ & \footnotemark\\
\hline\hline
$\eta^\prime$ & $\pi^+ \pi^- \eta$ & 44 & 3 & 1 & 5 $\%$ &
8200 & 18000 & \\
{\it hadronic} & $\pi^0 \pi^0 \eta$ & 20 & 9 & 1 & 2 $\%$ & 5400 & 14500 & \\
& $3\,\pi^0$ & 1 & 56 & 0 & 26 $\cdot 10^{-3}$ & 130 & 145 & \\
& $\pi^+ \pi^- \pi^0$ & $<\,5$ & \multicolumn{3}{l}{$\%$} & - & 85 &
\footnotemark \\
\hline
$\eta^\prime$ & $\rho^0 \gamma$ & 29 & 5 & 1 & 0 $\%$ &
9550 & 44000 & \footnotemark \\
{\it radiative} & $\omega \gamma$ & 3 & 03 & 0 & 31 $\%$ & 160 & 1200 & \\
& $\gamma \gamma$ & 2 & 12 & 0 & 14 $\%$ & 2767 & 17100 & \\
\hline
$\eta^\prime$ & $\mu^+ \mu^- \gamma$ & 1 & 04 & 0 &
26 $\cdot 10^{-4}$ & 33 & 15 & \\
{\it semi--leptonic} & $e^+ e^- \gamma$ & $<\,9$ & \multicolumn{3}{l}{$\cdot 10^{-4}$} &
- & 45 & \footnotemark \\ \hline\hline
\end{tabular}}
\label{tab_etap_rates}
\end{table}
The observation of CP or C violating rare decays might hint at effects from
new physics beyond the Standard Model:
The single conversion decay $\eta \rightarrow \pi^+ \pi^- e^+ e^-$ allows
to search for CP violation in flavor--conserving processes beyond the
Cabbibo--Kobayashi--Maskawa (CKM)
mechanism~\cite{Cabibbo:1963yz,Kobayashi:1973fv}, which are not constrained
by limits on the neutron electric dipole moment.
A recently proposed CP violating mechanism from an interference between
magnetic and
electric decay amplitudes can induce a sizable linear photon polarization
in the $\eta \rightarrow \pi^+ \pi^- \gamma$
decay~\cite{Gao:2002gq,Geng:2002ua}.
The photon polarization is accessible in the associated conversion decay
$\eta \rightarrow \pi^+ \pi^- e^+ e^-$ as an asymmetry in the angular
distribution between the $\pi^+ \pi^-$ and $e^+ e^-$ production planes.
Similarly, this method was used to measure CP violating effects in the
$K^0$ system in the decay $K_L \rightarrow
\pi^+ \pi^- e^+ e^-$~\cite{Lai:2003ad}, with an asymmetry reported as large
as $14\,\%$.
The corresponding asymmetry for $\eta \rightarrow \pi^+ \pi^- e^+ e^-$ is
expected to be at a level of $10^{-3}$--$10^{-2}$ (see
section~\ref{subsubetapday1} and~\cite{Gao:2002gq,Geng:2002ua}).
\addtocounter{footnote}{-6}
\footnotetext{assuming the theoretical estimate of $3\cdot 10^{-4}$ for the
branching ratio~\cite{Jarlskog:1967np}}
\stepcounter{footnote}
\footnotetext{assuming $\Gamma(\eta \rightarrow e^+ e^- e^+ e^-) /
\Gamma_{tot} = 2.52 \cdot 10^{-5}$~\cite{Jarlskog:1967np}}
\stepcounter{footnote}
\footnotetext{Standard Model estimate $\Gamma(\eta \rightarrow e^+ e^-) /
\Gamma_{tot} = 5 \cdot 10^{-9}$ from $\eta \rightarrow
\mu^+ \mu^-$~\cite{Savage:1992ac,Ametller:1993we}}
\stepcounter{footnote}
\footnotetext{Standard Model estimates $\Gamma(\eta \rightarrow
\pi^0 e^+ e^-) / \Gamma_{tot} =
0.2-1.3 \cdot 10^{-8}$~\cite{Cheng:1967pr,Ng:1993sc,Jarlskog:2002zz}}
\stepcounter{footnote}
\footnotetext{assuming $\Gamma(\eta^\prime \rightarrow \pi^+ \pi^- \pi^0)
\approx \Gamma(\eta^\prime \rightarrow 3\,\pi^0)$,
see~\cite{Beisert:2003zs}}
\stepcounter{footnote}
\footnotetext{including $\eta^\prime \rightarrow \pi^+ \pi^- \gamma$}
\stepcounter{footnote}
\footnotetext{assuming $\Gamma(\eta^\prime \rightarrow e^+ e^- \gamma) /
\Gamma_{tot} \approx 3 \cdot 10^{-4}$~\cite{Briere:1999bp}}
\stepcounter{footnote}
A variety of decays, e.g.\ $\eta\left(\eta^\prime\right) \rightarrow
\pi^0 \pi^0 \gamma$, $\eta\left(\eta^\prime\right) \rightarrow
3 \pi^0 \gamma$, $\eta\left(\eta^\prime\right) \rightarrow 3 \gamma$,
and $\eta^\prime \rightarrow e^+ e^- \pi^0 \left(\eta\right)$ or $\eta
\rightarrow e^+ e^- \pi^0$ probe C--invariance, for which only moderate
experimental limits exist.
For the latter decays to the $e^+ e^- \pi^0$ mode the dominant
Standard Model --- i.e.\ C conserving --- mechanism is an intermediate
state with two photons $\gamma^* \gamma^* \pi^0$ resulting in a branching
ratio of
$0.2 - 1.3 \cdot 10^{-8}$~\cite{Cheng:1967pr,Ng:1993sc,Jarlskog:2002zz} for
the $\eta$ decay.
If C invariance is violated, the intermediate $\gamma^* \pi^0$ state with
one virtual photon can contribute, increasing the branching ratio with
respect to the Standard Model prediction (section~\ref{subsubetapmedium}).
The move of the WASA facility to COSY offers the possibility to
extend the programs both on $\eta$ decays at CELSIUS and on $\eta^\prime$
production dynamics at COSY.
Until the end of the present data taking period at KLOE in the middle of
2005, an
integral luminosity of $2000\,\mbox{pb}^{-1}$ is anticipated (presently
$500\,\mbox{pb}^{-1}$), corresponding to $8 \cdot 10^7$ and $4 \cdot 10^5$
fully reconstructed $\eta$ and $\eta^\prime$ events, respectively, from
$\phi \rightarrow \eta\left(\eta^\prime\right) \gamma$
events~\cite{Kluge:2004pc}.
The KLOE data will significantly improve the presently available data set.
However, the total anticipated statistics from KLOE can be reached
with WASA at COSY in a period of 1 to 10 days, according to the rate
estimates given in table~\ref{tab_etap_rates}, which take into account
acceptance and reconstruction efficiency of the existing experimental setup
of the WASA facility. Considering these future developments we
concentrate on specific rare and not-so-rare decays.
Together with the decay data, new and important information will be gained
on the $\eta'$--nucleon interaction. Because of the special features
of the $\eta'$ outlined above, this is a system of interest by itself.
In addition, it is well known that final state interactions that occur
subsequent to short range processes are universal. As a consequence the
$\eta'$--nucleon interaction is the same in $pp\to pp\eta'$ as in
$B\to \eta'pX$. Thus, the measurement of the former reaction will provide
necessary information to interpret the latter one
that will eventually be measured at the B factories
and might shed light on the dynamics of charmless
B decays~\cite{Piccinini:2001ja}.
\subsubsection{\boldmath Isospin violation in $\vec{d}d\,\to\,\alpha\pi^0$}
\label{sec:dd2alphapi0}
\label{subsubalphapi}
Within the standard model there are only two sources of isospin
violation, namely the electro-magnetic interaction and the differences
in the masses of the lightest quarks~\cite{SWeinberg77,JGasser82}.
Especially in situations where we are able to disentangle these two
sources, the observation of isospin violation in hadronic reactions is a
direct window to quark mass ratios~\cite{JGasser82,miller} --- quark masses
themselves are not directly observable and additional information is
necessary to assign a scale to these fundamental parameters of the standard
model (see e.g.~\cite{Leutwyler:1996qg}).
Already in 1977 Weinberg predicted a huge effect (up to 30\%
difference in the scattering lengths for $p\pi^0$ and $n\pi^0$) of
isospin violation in $\pi^0 N$ scattering~\cite{SWeinberg77}. Also the impact
of soft photons was studied
systematically~\cite{Meissner:1997ii,GMueller99,NFettes01,Gasser:2002am}.
Since scattering experiments with neutral pions are not feasible, in
Ref.~\cite{jouni} it was suggested to use $NN$ induced pion production
instead\footnote{The $\pi^0p$ scattering length might also be measurable
in polarized neutral pion photoproduction at
threshold~\cite{AMBernstein98}.}.
The authors demonstrated that a charge symmetry breaking
(CSB)\footnote{Charge symmetry is fulfilled, if the amplitudes are
invariant under a 180 degree rotation in isospin space.
CSB is thus a subclass of isospin breaking effects.}
$\pi N$ seagull term (the Weinberg term), required by chiral symmetry,
should be even more relevant for $A_{fb}(pn\to d\pi^0)$ --- the
forward-backward asymmetry in $pn\to{}d\pi^0$ --- than $\pi -\eta$ mixing,
previously believed to completely dominate this CSB
observable~\cite{Niskanen:1998yi}.
\begin{SCfigure}[1.0][tb]
\resizebox{0.5\textwidth}{!}{\includegraphics{figures/csbfig2.eps}}
\caption{Experimental missing-mass spectra ($\mathrm{dd}\to\alpha$X) from
Ref.~\cite{Stephenson:2003dv} revealing a clean pion peak. The
smooth curves show a reproduction of the data using a Gaussian
peak and a continuum. The corresponding cross sections are
$\sigma = (12.7 \pm 2.2) \mathrm{pb}$ and
$\sigma = (15.1 \pm 3.1) \mathrm{pb}$
for the lower and higher energy, respectively. The
data are consistent with pure $s$-waves
contributions. \label{fig:iucf} }
\end{SCfigure}
Interest in CSB in hadronic reactions was revived recently with the
successful completion of experiments for $A_{fb}(pn\to
d\pi^0)$~\cite{Opper:2003sb} as well as $dd\to \alpha \pi^0$
(\cite{Stephenson:2003dv}, see Fig.~\ref{fig:iucf}) close to the pion
threshold. A large collaboration has formed to perform
the corresponding calculations within effective field theory. First
results are presented in~\cite{Gardestig:2004hs}. The studies show
that the relative importance of the various charge symmetry breaking
effects is very different for the two reactions and a consistent
investigation of both should help to further disentangle the leading
CSB matrix elements.
However, open questions will remain. It is the main result of
Ref.~\cite{Gardestig:2004hs} that, within a plane wave calculation,
the Weinberg term is suppressed due to selection rules in the reaction
$dd\to \alpha \pi^0$. Thus, the lowest order that gives finite
contributions is next-to-next-to leading order (NNLO).
At this order there are several diagrams contributing and only a NNNLO
contribution will really prove that there is a convergence in the series
for this particular reaction.
However, there is one more important test for the approach that will
be used for the analysis of the recent CSB experiments, namely, once
the parameters are fixed the $p$-waves in $dd\to \alpha \pi^0$ can be
predicted parameter free to leading and next-to-leading order. The
IUCF experiment~\cite{Stephenson:2003dv} was consistent with purely
$s$-waves contributing. Thus, the same experiment at somewhat higher
energies (but still well below the $\Delta$ region, e.g. at $Q\approx 60$
MeV) is urgently called for.
The Bose symmetry of the initial wave function strongly limits the
number of allowed partial waves. As a consequence, for $s$-wave as
well as $p$-wave production only a single partial wave is allowed,
namely $^3P_0\to s$ and $^5D_1\to p$. Thus, they do not interfere in
any unpolarized measurement. It would therefore be very important to
measure the unpolarized total cross section as well as
polarized differential observables. In addition, the analyzing
powers are given by the imaginary parts of interference terms and thus
would provide an important test of the initial state interaction,
the ingredient to the calculations that is controlled to the smallest
extent. In a plane wave treatment the analyzing powers would vanish
identically.
In a further step, CSB effects involving the $\Delta$ isobar and
purely nucleonic ones can be disentangled taking data in the $\Delta$
regime as well ($Q\approx 160$ MeV). This will not only be an additional
important cross check of the analysis carried out previously, but
gives in addition quite direct access to matrix elements difficult to
get at from other experiments. However, in order to study the
different partial waves polarized differential observables are as
important as for the lower energy.
It is important to stress that CSB studies for few-nucleon systems can be
based on recent developments in effective field
theories~\cite{EEpelbaum04,EEpelbaum02a,EEpelbaum02b,EEpelbaum04a,JLFriar}, thus allowing
for a model-independent analysis of the data.
\clearpage
\subsection{Additional objectives:\\ Exotic and cryptoexotic hadron resonances}
The study of hadron resonances and their decays is one important
approach to obtain information about quark dynamics in the
non-perturbative regime of QCD. The experiments performed at the
proton-antiproton storage ring LEAR (Crystal Barrel)
and at the new generation of experiments in
electron-positron colliders (BaBar, Belle, BES, CLEO) have found
new mesons which challenge the traditional interpretation of mesons as
quark-antiquark states\cite{Amsler:2004ps,Belle03}. In particular, new
meson candidates with the quantum numbers $J^{PC}=1^{-+}$ cannot be
generated by $q\bar{q}$ states\cite{Abele98,Kuhn04}. Because of this feature
these states
are called exotics.
With the recent evidence for the existence of the so called pentaquark
$\Theta^+$ a first baryon candidate was added to this list.
In addition, in the hadron spectrum there are
several states with quantum numbers consistent with the quark model.
However, some of their other properties are not. These states are called
cryptoexotics. Both mesons, like the well known $a_0$
and $f_0$, as well as baryons like the $\Lambda (1405)$, are discussed
in the literature as candidates for this class of states. However, it
is clear that additional experimental information is needed to verify
or falsify these conjectures.
Hadronic probes provide an experimental tool complementary to
electromagnetic excitations because of different spin- and isospin
selectivities.
In the following we elaborate why WASA is particularly
suited to investigate those exotic and cryptoexotic resonances.
In the {\it meson sector} the controversial states that are within the
COSY energy range are the scalar--isoscalar $f_0(980)$ and the
scalar--isovector $a_0(980)$. Although
QCD can in principle be treated explicitly in the low
Mo-men\-tum-transfer regime using lattice
techniques~\cite{Kunihiro:2003xe}, those are not yet in the
position to make quantitative statements about light scalar states
($J^P{=}0^+$). Alternatively, QCD inspired models, which use effective
degrees of freedom, are to be used. The constituent quark model is one
of the most successful in this respect (see e.g.\
Ref.~\cite{qmod1,qmod2,qmod3,qmod4,qmod5,qmod6}). This approach treats
the lightest scalar
resonances $a_0/f_0$(980) as conventional $q\bar{q}$ states. However,
they have also been identified with $K\bar{K}$
molecules~\cite{KK_1,KK_2,KK_3,KK_4,KK_5} or compact $qq$-$\bar{q}\bar{q}$ states
\cite{4q_1,4q_2,4q_3}. It has even been suggested that at masses below 1.0
GeV/c$^2$ a full nonet of 4-quark states might exist
\cite{Close:2002zu}.
As in the reaction $dd\to \alpha \pi^0$, with WASA at COSY we can use the isospin
selectivity in the $dd\to \alpha X$ transition, to model independently
access
the isospin-violating
$a_0$-$f_0$ mixing: the total cross section for the reaction
$dd\to\alpha(\pi^0\eta)$ was shown to be directly proportional to the
square of the mixing matrix element with the influence of other
isospin violating effects being suppressed by one order of magnitude
\cite{Hanhart:2003pg}.
There is the additional possibility to extract the phase of the mixing matrix
element from
the forward-backward asymmetry in
$pn\to da_0^0/f_0$. Both measurements will reveal independent
new information about the structure
of the light scalar mesons, in particular about their $K\bar K$-meson
content.
In the {\it baryon sector}, several facilities have obtained evidence for a
new baryon resonance, the $\Theta^+$, which has achieved a three-star
status in the Particle Data Group~\cite{PDBook}. If ultimately
confirmed with convincing statistics, this resonance will have a large
impact on the field. The $\Theta^+$ has positive strangeness which can
only be generated by an antistrange quark as a building block of the
resonance and therefore cannot be accommodated by a three quark
structure. The width of the $\Theta^+$ is unexpectedly small: while
direct determinations of the width can only give upper limits of the
order of 10~MeV/c$^2$ due to limited energy resolution, indirect analysis
point to a width of less than
1~MeV/c$^2$~\cite{Arndt03,Haid03,cahn04,sib04}.
Jaffe and Jain have recently pointed out the limitations such a small
width imposes on the possibilities to generate the $\Theta^+$ dynamically in the
K N channel or as a Castillejo-Dalitz-Dyson pole\cite{Jaffe:2004at}.
Theorists have pointed out the possible existence of pentaquark states
with narrow widths due to the formation of colored clusters about
thirty years ago~\cite{Hogaasen:1978jw}. Recent theoretical work has
focussed on the diquark degree of freedom~\cite{Jaffe03}. Though the
clustering of colored diquarks provides a qualitative explanation of
the small width of the $\Theta^+$, explicit dynamical calculation
require fine-tuning of the model~\cite{Stech04}. Combining the colored
diquark degree of freedom with chiral symmetry, however, produces
hadrons which are stable and do not couple to the kaon-nucleon degree
of freedom in the chiral limit~\cite{Beane,Ioffe,Melikhov}.
The production of the $\Theta^+$ via the reaction
$ p d \rightarrow p \Lambda \Theta^+$
allows access to a pure isospin $I=0$ combination $\Lambda \Theta^+$
in the final state. This allows to obtain information
complementary to the proton-proton reaction which produces the
$\Theta^+$ starting from a pure isospin $I=1$ state.
It is well established that if there is one exotic there should be
many and it is the corresponding multiplet structure that encodes
essential information on the underlying substructures.
The $N^*(1710)$ has been assumed to be a member of the
pentaquark antidecuplet and predictions for the branching into
various decay channels have been published\cite{Diakonov}
awaiting experimental tests.
The Roper resonance, $N(1440)$, has been studied already with WASA at
CELSIUS. The Roper is not a manifestly exotic state, but its broad
width and small mass have caused many experimental and theoretical
investigations of its structure. In the Jaffe-Wilczek diquark scenario,
the Roper may be related to the $\Theta^+$.
In addition,
the $\Lambda(1405)$ has been
a candidate for a non-triplet quark structure and thus for a crypto
exotic for a long time.
Recent work within unitarized chiral perturbation theory
suggests that the $\Lambda(1405)$ is made of two overlapping
resonances with different flavor structure. This calls for detailed
investigations of the decay channels as a function of the excitation
energy.
Recently, the ANKE collaboration has found evidence for a hyperon
$Y(1475)$ (see Fig.\ref{fig:Y1475} in section~\ref{sec:hyperons})
which is not predicted
by any known quark model. The PDG gives a one-star status to a surmised
$\Sigma(1480)$.
\subsubsection{\boldmath Mixing of the scalar mesons $a_0/f_0$(980)}
\label{sec:a0_f0}
Both, the (isospin $I{=}1$) $a_0$- and the ($I{=}0$) $f_0$-resonances
can decay into $K\bar K$, whereas in the non-strange sector the decays
are into different final states according to their isospin,
$a_0^\pm{\to} (\pi^\pm\eta)_{I=1}$, $a_0^0{\to} (\pi^0\eta)_{I=1}$ and
$f_0{\to} (\pi^0\pi^0)_{I=0}$ or $(\pi^+\pi^-)_{I=0}$. Thus, only
the non-strange decay channels have defined isospin and allow one to
(model independently) discriminate between the two mesons. It is also only by measuring the non-strange
decay channels that isospin violating effects can be investigated.
Such measurements
can be carried out with WASA at COSY for $\pi^0$- or $\eta$-meson
identification, while the strange decay channels $a_0/f_0{\to} K_SK_S$
might be measured in parallel. Measurements of the $K\bar K$ final
state with at least one charged kaon have already been performed at
COSY using magnetic spectrometers. The main results of these
experiments and their implications on the proposed measurements are
described below.
In 1979 Achasov and collaborators~\cite{Achasov:1979xc} pointed out
that between the charged and neutral kaon thresholds the leading term
to the $a_0$-$f_0$ mixing amplitude is dominated by the unitary cut of
the intermediate two-kaon system. It was demonstrated, that the
leading piece of the $a_0$-$f_0$ mixing amplitude can be written as
\begin{equation}
\Lambda = \langle f_0 |T| a_0\rangle = ig_{f_0K\bar K}g_{a_0K\bar
K}\sqrt{s}\left( p_{K^0}-p_{K^+} \right) \ + {\cal
O}\left(\frac{p_{K^0}^2-p_{K^+}^2}{s}\right) \ ,
\label{eq:llam}
\end{equation}
where the effective coupling constants are defined through
$\Gamma_{xK\bar K}=g_{xK\bar K}^2p_K$. This $\sqrt{s}$ dependence of
$\Lambda$ is depicted in Fig.~\ref{fig:mixsdep}. Here, electromagnetic
effects between the kaons were neglected as in~\cite{Achasov:1979xc}.
\begin{figure}[tb]
\centering
\resizebox{7.5cm}{!}{\includegraphics[scale=1]{figures/Tafelberg.eps}}
\caption{Modulus of the leading term of the mixing amplitude
$\Lambda$ defined in Eq.\ (\protect\ref{eq:llam}). The two kinks occur
at the $K^+ K^-$ (at 987.35 MeV) and the $\bar K^0 K^0$ (995.34 MeV)
threshold respectively.}
\label{fig:mixsdep}
\end{figure}
As demonstrated in a recent compilation~\cite{Baru:2003qq}, the values
for $g_{a_0K\bar K}$ and $g_{f_0K\bar K}$ range from 0.224 to 0.834
and from 1.31 to 2.84, respectively --- depending on the Flatt\'e
parameterization of measured $\pi\pi$ and $\pi\eta$ spectra. It has
also been shown in Ref.~\cite{Baru:2003qq} that the coupling of a
physical particle to mesons carries information about the nature of
that particle, as was shown by Weinberg for the deuteron case
\cite{weinberg}. Based on the existing data, the authors of
Ref.~\cite{Baru:2003qq} conclude that both the $a_0$ and $f_0$ have
significant mesonic ($K\bar K$) components, however, more quantitative
statements require a better knowledge of $g_{a_0/f_0K\bar K}$, which
could be obtained from an accurate measurement of $\Lambda$.
In addition, since $\Lambda$ (as defined in eq.~\ref{eq:llam}) is essentially
the overlap of the
wavefunctions of $a_0$ and $f_0$, it is expected that also that part
of
the mixing amplitude
that does not stem from kaons will reveal new insights into the
structure of the light scalar mesons.
A $pp\to dX$ reaction must lead to $a_0^+$ ($I{=}1$) production, a
$pn\to dX$ interaction is not isospin selective and both the $a_0^0$
and $f_0$ may be produced ($I{=}0,\, 1$), whereas the $dd\to\alpha X$
reaction --- neglecting the small isospin violating contributions which
are the final
goal of the proposed experimental program --- is a filter for the
$f_0$ ($I{=}0$) resonance, since the initial deuterons and the
$\alpha$ particle in the final state have isospin $I{=}0$ (``isospin
filter''). Since at COSY it is possible to manipulate the initial
isospin one can thus selectively produce the $a_0$ or $f_0$ resonances
and can identify observables that vanish in the absence of
isospin violation~\cite{miller,ANKE_WS}. The idea behind the proposed
experiments is
the same as behind recent measurements of isospin violation in
the reactions
$np{\to} d\pi^0$~\cite{Opper:2003sb} and $dd{\to}
\alpha\pi^0$~\cite{Stephenson:2003dv}. However, the interpretation of the
signal from the scalar mesons is much simpler as compared to the pion
case. Since the $a_0$ and the $f_0$ are rather narrow overlapping
resonances, the $a_0$-$f_0$ mixing in the final state is enhanced by
more than an order of magnitude compared to isospin violation in the production
operator (i.e.\ ``direct'' isospin violating $dd{\to}\alpha a_0$ production) and
should, e.g., give the dominant contribution to the isospin violating effect
via the
reaction chain $dd{\to} \alpha f_0(I{=}0) {\to}
\alpha a_0^0(I{=}1) {\to} \alpha (\pi^0\eta)$~\cite{Hanhart:2003pg}.
The $dd{\to} \alpha (\pi^0\eta)$ reaction seems to be most promising
for the extraction of isospin violating effects. Any observation of $\pi^0\eta$
production in this particular channel is a direct indication of isospin
violation,
however, the cross section $\mathrm{d}\sigma/\mathrm{d}m$ will be
given by the product of the mixing amplitude $\Lambda(m)$ and the
$dd{\to}\alpha f_0$ production operator. It is therefore compulsory to
determine the latter in an independent measurement in order to extract
the mixing amplitude. A corresponding experiment aiming at the
measurement of the $dd\to\alpha (K^+K^-)_{I{=}l{=}0}$ cross section is
foreseen for winter 2004/05 at ANKE~\cite{dd_proposal}. These data,
together with the information on the $dd{\to} \alpha (\pi^0\eta)$
reaction from WASA will allow one to determine $\Lambda$
independently.
In analogy to the measurement of isospin violation in the
reaction $np{\to}
d\pi^0$, it has been predicted that the measurement of angular
asymmetries (i.e.\ forward-backward asymmetry in the $da_0$ c.m.s.)
also allows to extract the $a_0$-$f_0$ mixing amplitude
\cite{Grishina:2001zj,Kudryavtsev:2001ee,Kudryavtsev:2002uu}. It was
stressed in Ref.\ \cite{Kudryavtsev:2001ee} that --- in contrast to
the $np{\to} d\pi^0$ experiment where the forward-backward asymmetry
was found to be as small as 0.17\% \cite{Opper:2003sb} --- the
reaction $pn{\to} d\pi^0\eta$ is subject to a kinematical enhancement.
As a consequence, the effect is predicted to be significantly larger
in the $a_0$/$f_0$ case. The numbers range from some
10\%~\cite{Kudryavtsev:2001ee} to a few factors
\cite{Grishina:2001zj} and, thus, should easily be observable. It has
been pointed out in Ref.~\cite{Kudryavtsev:2002uu} that the analyzing
power of the reaction $\vec p n{\to} d \pi^0 \eta$ also carries
information about the $a_0$-$f_0$ mixing amplitude. This quantity can
be measured at COSY as well.
An experimental program has already been started at COSY which aims at
exclusive data on $a_0/f_0$ production close to the $K\bar{K}$
threshold from $pp$~\cite{Quentmeier:2001ec,Moskal:2002jd,a+_proposal},
$pn$ \cite{a0f0_proposal}, $pd$~\cite{a0f0_proposal,momo-KK} and $dd$
\cite{css2002,dd_proposal} interactions. The reactions $pp {\to}
ppK^+K^-$ and $pd {\to} ^3\mathrm{He}\, K^+K^-$ have been measured at
COSY-11 \cite{Quentmeier:2001ec,Moskal:2002jd} and
MOMO~\cite{momo-KK}, respectively, at excitation energies up to
$Q=56$~MeV above the $K\bar K$ threshold. However, mainly due to the
lack of precise angular distributions, the contribution of the
$a_0/f_0$ to $K\bar K$ production remains unclear for these reactions.
At ANKE, the reaction $pp {\to} dK^+\bar{K^0}$ has been measured
exclusively (by reconstructing the $\bar{K^0}$ from the measured
$dK^+$ missing mass) at beam momenta of $p{=}3.46$ and 3.65 GeV/c
($Q{=}46$ and 103 MeV). The differential spectra for the lower beam
momentum are shown in Fig.\
\ref{fig:pp2dKKbar}~\cite{Kleber:2003kx}. The background of
misidentified events at ANKE is less than 10\% which is crucial for
the partial-wave analysis. This analysis reveals that the
$K^+\bar{K}^0$ pairs are mainly (83\%) produced in a relative $S$-wave
(dashed line in Fig.\ \ref{fig:pp2dKKbar}), presumably via the $a_0^+$
channel~\cite{Kleber:2003kx}. Based on model calculations for the $pp
{\to} dK^+\bar{K^0}$ reaction the authors of
Ref.~\cite{Grishina:2004rd} draw the conclusion that for $Q{<}100$ MeV
$K\bar K$ pair production proceeds dominantly via the
$a_0^+$-resonance.
Limits on the $\overline{K^0}d$ scattering length have been obtained in
Ref.~\cite{ASibirtsev04}.
For a further discussion of scalar mesons in these reactions see
e.g. Ref.~\cite{EOset96}.
\begin{figure}[tb]
\resizebox{7.0cm}{6.5cm}
{\includegraphics[scale=1]{figures/a0+_differential_spectra.eps}}
\resizebox{5cm}{!}{\includegraphics[scale=1]{figures/a+_vectors.eps}}
\caption{ANKE data for the reaction $p(3.46\, \mathrm{GeV/c})p\to
dK^+\bar{K}^0$~\cite{Kleber:2003kx}. The shaded areas correspond to the
systematic uncertainties of the acceptance correction. The dashed
(dotted) line corresponds to $K^+\bar{K}^0$-production in a relative
$S$-($P$-) wave and the solid line is the sum of both
contributions. For definition of the vectors $p$, $q$ and $k$ in the
cms of the reaction $pp {\to} dK^+\bar{K^0}$ see right hand part of
the figure. Angular distributions with respect to the beam
direction $\vec{p}$ have to be symmetric around $90^\circ$ since the
two protons in the entrance channel are indistinguishable.
}
\label{fig:pp2dKKbar}
\end{figure}
Data on the reaction $p(3.46\, \mathrm{GeV/c}) p\to d \pi^+X$ have
been obtained at ANKE in parallel to the kaon data. In contrast to
the latter, where the spectra are almost background free, the $pp\to
d\pi^+\eta$ signal is on top of a huge multi-pion
background~\cite{a+_pieta}. This demonstrates the need for a photon
detector supplying active $\pi^0$- or $\eta$-meson
identification. With such a detector one may expect data of a quality
comparable to what is shown in Fig.\ \ref{fig:pp2dKKbar} also for the
non-strange decay channels.
\subsubsection{Hyperon resonances}
\label{sec:hyperons}
Compared to the spectrum of nucleon resonances, the excitation modes
of hyperons ($\Lambda$,$\Sigma$) are much less known.
The lowest excitations listed in Ref.~\cite{PDG} are the $\Lambda$(1405)
$J^P=\frac{1}{2}^-$ isospin singlet and the $\Sigma$(1385)
$J^P=\frac{3}{2}^+$ isospin triplet states, well accessible in
proton-nucleon collisions at COSY. The study of the $\Lambda$(1405)
state is particularly interesting since this resonance has still not
been understood in its nature, its spin and isospin assignment is
based on indirect arguments only. In the quark model the
$\Lambda$(1405) is interpreted as a state with orbital excitation.
However, three well established quark models with entirely different
residual interaction (gluon exchange~\cite{Isgur:1978xj}, meson
exchange~\cite{Glozman:1999vd}, and t'Hooft instanton induced
interaction~\cite{Loring:2001kx}) have difficulties in reproducing the
mass of the $\Lambda$(1405). All
models~\cite{Isgur:1978xj,Glozman:1999vd,Loring:2001kx} find a
degeneracy of the computed $\Lambda$(1405) with the
$J^P=\frac{3}{2}^-$ $\Lambda$(1520) resonance in contrast to the
observation. A similar problem is also manifest in very recent
lattice QCD calculations where an extrapolation to the physical pion
mass results in a too high value for the $\Lambda$(1405)
mass~\cite{Melnitchouk:2002eg}.
On the other hand, since long time the $\Lambda$(1405) has been
interpreted as a $\bar{K}N$ quasi-bound
state~\cite{RCArnold62,RHDalitz67,Jones:1977yk}, later on substantiated by
the consideration of chiral symmetry~\cite{Kaiser:1995eg,Kaiser:1996js}.
Recent work in the framework of unitarized chiral effective field
theories has pointed out that the structure of the $\Lambda$(1405) is
extremely sensitive to the breaking of the $SU(3)$
symmetry~\cite{Oset:1997it,JAOller01,Oset:2001cn,Jido:2003cb}.
Ref.~\cite{Jido:2003cb} finds two poles of the scattering matrix close to
the nominal $\Lambda$(1405) resonance, one of which couples more
strongly to $\pi\Sigma$ states and the other one mostly to
$\bar{K}N$ states. As an experimentally observable consequence,
peak structures with different invariant mass distributions are
expected in photon and hadron induced reactions that populate the
$\Lambda$(1405) resonance~\cite{Jido:2003cb}.
The experimental information available on the $\Lambda$(1405) and
$\Sigma$(1385) resonances is primarily based on studies of $K^-$
proton collisions in bubble
chambers~\cite{Thomas:1973uh,Hemingway:1984pz,MAguilar81,MBaubillier84}.
In this entrance channel the resonances are only
observable through the decay of higher-lying $\Lambda^{\star}$ and
$\Sigma^{\star}$ resonances.
Recently photon-induced hyperon resonance production with a statistics
of $\approx$100 events in the $\Lambda$(1405)/$\Sigma$(1385) region was
observed at SPring-8~\cite{Ahn:2003mv}, showing different spectral
shapes in $\Sigma^+\pi^-$ and $\Sigma^-\pi^+$ final states. Based on
arguments given in Ref.~\cite{Nacher:1998mi}, this was interpreted as
an indication for a meson-baryon nature of the
$\Lambda$(1405)~\cite{Ahn:2003mv} (see Fig.~\ref{fig:SPring-8}).
\begin{figure}[tb]
\begin{center}
\epsfig{file=figures/JKAhn_fig4.eps, width=7.5cm}
\caption{Invariant mass spectra for $\pi^+\Sigma^-$ and $\pi^-\Sigma^+$
as observed in the $p(\gamma{},K^+\pi)\Sigma$ reaction at
SPring-8/LEPS~\cite{Ahn:2003mv}.}
\label{fig:SPring-8}
\end{center}
\end{figure}
In view of the limited quality of the existing data and the lack of
proton induced studies, it is important to investigate
the nature of the $\Lambda$(1405) resonance in proton-proton collisions by
measuring its spectral shape with good resolution and statistics.
This will allow recent theoretical predictions based on the chiral
unitarity approach to be tested. In particular, a comparison of the
$\Sigma^+\pi^-$ and $\Sigma^-\pi^+$ spectral distribution will reveal
a possible resonant behavior of the the $I=1$ amplitude in this mass
region, whereas the $\Sigma^0\pi^0$ distribution provides the true
shape of the $I=0$ resonance according to Ref.~\cite{Nacher:1998mi}.
A comparison with the shape of the spectral distribution measured in
photo-induced reactions is predicted to reveal the two-pole structure
of the resonance~\cite{Jido:2003cb}.
\begin{figure}[tb]
\begin{center}
\epsfig{file=figures/Y1475.eps, width=9cm}
\caption{$K^+p$ missing mass spectrum for the reaction
$p(\mathrm{3.60\ GeV/c})p \rightarrow pK^+\,\Sigma^-\pi^+$ measured
at ANKE in comparison with a simulation (upper panel).
The simulated $K^+p$ missing mass spectrum decomposes into background
from misidentified events and non-resonant $\Sigma^-\pi^+$ production,
as well as the known hyperons $\Sigma(1385)$, $\Lambda(1405)$ and
$\Lambda(1520)$ decaying into $\Sigma^-\pi^+$ (lower panel).}
\label{fig:Y1475}
\end{center}
\end{figure}
An experimental study of the $\Lambda$(1405) resonance also delivers
information on the $\Sigma$(1385) resonance as a by-product, since due
to their similar pole masses and widths the production of these two
resonances can only be disentangled by the complete identification of
the final states populated in their decays ($\Lambda{\rm{}(1405)}
\rightarrow\Sigma\pi$ (100\%); $\Sigma{\rm{}(1385)} \rightarrow
\Lambda\pi$ (88$\pm$2\%), $\Sigma\pi$ (12$\pm$2\%)). This involves
the detection of both charged and neutral pions, and thus a large
acceptance detector with charged and neutral particle detection
capability like the WASA detector. In particular, photon detection is
required for the observation of the decay channel
$\Lambda(1405) \rightarrow\Sigma^0\pi^0\rightarrow
(\Lambda\gamma)(\gamma\gamma)$. The identification of this final
state is particularly important since it is not populated in the decay
of the $\Sigma^0(1385)$ resonance and thus provides direct
access to the spectral shape of the $\Lambda$(1405) resonance.
The $\Sigma^0$(1385) contribution in the charged $\Sigma\pi$ final states
($\Sigma^+\pi^-$,$\Sigma^-\pi^+$) can be deduced from a measurement of the
$\Lambda\pi^0$ final state which is not populated in the $\Lambda$(1405)
decay.
Recent data from ANKE on the reaction
$pp\rightarrow pK^+\,Y^*\rightarrow pK^+\,\Sigma(1190)^-\pi^+$
measured at a beam momentum of 3.60 GeV/c are consistent with a
hyperon resonance $Y^*$ with a mass of 1475~MeV/c$^2$ and a width of
45~MeV/c$^2$ (see \cite{koptev02} and Fig.~\ref{fig:Y1475}).
This state is also seen with comparable strength in the charge-conjugate
decay channel $Y^*\rightarrow \Sigma(1190)^+\pi^-$ and its production cross
section is of the same order as for the $\Lambda(1405)$.
Within the above mentioned quark
models~\cite{Isgur:1978xj,Glozman:1999vd,Loring:2001kx} there is no
room for an additional $\Lambda$ or $\Sigma$ hyperon below $\approx$1600
MeV/c$^2$.
However, a $\Sigma(1480)$ with a $4q1\bar q$ structure has been predicted
in Ref.~\cite{Hogaasen:1978jw} --- as the $I=1$ partner of the
$\Lambda(1405)$ in a common multiplet.
Further studies are needed in order to confirm the conjecture that the
observed enhancement in the $K^+p$ missing mass spectrum originates from a
new hyperon resonance.
If this interpretation will be supported after further analysis or
new measurements, the next step is to determine the isospin of the new
hyperon state.
This is not possible from the ANKE data but requires a comparison of the
possible decay channels $\Sigma^{\pm}\pi^{\mp}$, $\Sigma^0\pi^0$, and
$\Lambda\pi^0$.
\label{Y1475ref}
As an additional note, it should be mentioned that the different nature of
the $\Lambda$(1405) and $\Sigma$(1385) states is also expected to manifest
itself in different in-medium properties of these hyperon resonances.
In particular, if the meson-baryon picture of the $\Lambda$(1405) is
correct, its medium properties are intimately related to the behavior of
antikaons ($K^-$,$\bar{K}^0$) in nuclear matter, which is a still unsolved
problem, of importance in both hadron physics and astrophysics.
A comparative study of $\Lambda$(1405) and $\Sigma$(1385) resonance
production on nuclear targets allows to investigate the propagation of
these hyperon resonances inside the nuclear medium.
A better knowledge of the in-medium properties of the $\Lambda$(1405)
(absorption cross section, modification of the spectral shape) will also
help to understand the dynamics of antikaons in nuclear matter.
\subsubsection{Pentaquarks}
\label{sec:pentaquark}
Early in 2003, the LEPS Collaboration at the SPring-8 facility in
Japan observed a sharp resonance, $\Theta^{+}$, at $(1.54\pm 0.01)$~GeV/c$^2$
with a width smaller than 25~MeV/c$^2$ and a statistical significance of
$4.6 \sigma$ in the reaction $\gamma n\rightarrow
K^{+}K^{-}n$~\cite{Nakano:2003qx}. This resonance decays into
$K^{+}n$, hence carries strangeness $S= +1$. Later, many other groups
have claimed the observation of this
state~\cite{pos_1,pos_2,pos_3,pos_4,pos_5,pos_6,pos_7,pos_8,pos_9,pos_10,pos_11}.
In particular, evidence for the $\Theta^+$ has also been seen at the
COSY-TOF experiment in the reaction
$pp\rightarrow\Sigma^+pK_s$~\cite{MAbdel-Bary04} in the $pK_s$ invariant
mass spectrum.
While in some experiments detecting a $pK_s$ in the final state the
strangeness of the resonance in this system is undefined, in this experiment
it is uniquely determined that the observed $pK_s$ system originates from
a $S=+1$ state since the full final state with an associated $\Sigma^+$ is
identified.
Since all known baryons with $B= +1$ carry negative or zero strangeness,
such a resonance must have a minimum quark content $uudd\bar{s}$, a
structure which is clearly beyond the conventional quark model. This
experimental discovery triggered a lot of theoretical activity and up
to now over 250 papers appeared trying to interpret this exotic
state~\cite{review}.
There is preliminary evidence that the $\Theta^{+}$ is an iso-scalar
because no enhancement was observed in the $pK^{+}$ invariant mass
distribution~\cite{pos_3,pos_5,pos_6,pos_11}. All the other quantum numbers
including
its angular momentum and parity remain undetermined. Most of
theoretical work postulated its angular momentum to be one half,
because of its low mass, but the possibility of $J = 3/2$ still cannot
be excluded completely.
It is important to point out that many other experimental groups
reported negative results~\cite{neg_1,neg_2,neg_3}. A long list of experiments
yielding negative results (still unpublished at the time of the writing) can be found
in Ref.~\cite{morenegative}.
With both positive and null evidence for the $\Theta^{+}$ from a
variety of experiments, it is difficult to conclude whether the
$\Theta^{+}$ exists or not. Furthermore, the theoretical difficulties
to explain a possible narrow width, perhaps as small as 1~MeV/c$^2$, suggest
that if the $\Theta^{+}$ exists, it is very unusual indeed. Guidance
from lattice gauge theory is not helpful, as some calculations show
evidence for a negative parity resonance, while another indicates positive
parity. Furthermore, one does not see a resonance in either parity. So the
confirmation of the $\Theta^{+}$ is of highest importance for which
the COSY-TOF collaboration will start an extensive experimental
investigation in October 2004.
One should note that the $\Theta^+$ is a member of a multiplet. This calls
for the existence for additional unusual states.
WASA at COSY with its very good mass resolution and large acceptance
can make a decisive contribution to this field by providing high
statistics data on the differential cross sections of exotic baryon
production in elementary reactions.
While there are several elementary reactions with the $\Theta^{+}$ in the
final state within the COSY energy range, the
$pd\rightarrow p\Lambda\Theta^{+}$ reaction is of special interest.
Currently this reaction is investigated by the CELSIUS/WASA collaboration
close to threshold.
This reaction provides access to the pure isospin zero $\Theta^+$
production channel via the subsystem $pn\to \Theta^+ \Lambda$ and thus
allows to obtain further and independent insight into the physics of the
$\Theta^+$ pentaquark as compared to elementary transitions like
$pp\to \Theta^+ \Sigma^+$ (which is pure isospin one).
For example, one can obtain additional and complementary information on the
parity of the $\Theta^+$ \cite{CHanhart04} but explore also the $\Theta^+$
production mechanism, which is so far completely unknown.
\clearpage
\section{Key experiments}
\label{keyintro}
In this chapter several key experiments are presented to address the
issues discussed in the previous chapter. After the commissioning
phase of WASA at COSY, in which the performance of the detector will
be demonstrated, the intention is to carry out the experiments
discussed below. The proposed immediate program consists of
experiments that are feasible shortly after the initial
commissioning of WASA at COSY, and will produce physics results
soon. The medium term plans require either extended data taking or
improvements of the experimental conditions depending on the
experience gained in the initial experiments. The order of the
experiments listed below presents a reasonable balance between the
physics interest and timeliness of the experimental feasibility.
\subsection{\boldmath Not-so-rare $\eta^\prime$ decays}
\label{sec:day1_eta}
\label{subsubetapday1}
\paragraph{Isospin symmetry breaking in $\eta^\prime$ decays.}
The experiment aims at a precise determination of the ratios of isospin
symmetry breaking decays $\eta^\prime \rightarrow 3\,\pi$ with respect to
the isospin allowed $\eta\,2\pi$ modes as defined in
relation~(\ref{eq_pi_eta_ratio}) of section~\ref{subsubetap} to extract
the mixing angle
$\Theta_{\pi\eta}$, which is proportional to the light quark mass
difference $\Delta\,\mbox{m}$ (eq.~\ref{eq_pi_eta_mix}).
While only an upper limit is known for $\eta^\prime \rightarrow
\pi^0 \pi^+ \pi^-$~\cite{PDBook}, in the neutral decay modes in eq.
(\ref{eq_pi_eta_ratio}) the ratio ${\cal R}_1$ has been measured with a
statistics of $\approx 130$ events, leading to $\sin{\Theta_{\pi\eta}} =
0.023 \pm 0.002$~\cite{Binon:1984fe}.
However, there is considerable uncertainty in the literature concerning
$\sin{\Theta_{\pi\eta}}$:
a value of $\sin{\Theta_{\pi\eta}} = 0.010$ was extracted using PCAC
relations for meson masses and removing electromagnetic
contributions~\cite{Gross:1979ur}, in agreement with the estimate
in~\cite{Gasser:1985gg}.
Further results given in the literature cover a range between 0.013 and
0.014~\cite{Piekarewicz:1993ad,Chan:1994kg,Maltman:1994uu}, except for
$\sin{\Theta_{\pi\eta}} = 0.034 \pm 0.013$ reported in~\cite{Bagchi:1990ah}.
With the WASA facility at COSY, in the tagging reaction $p p \rightarrow
p p \eta^\prime$ a statistical accuracy of $1\,\%$ can be achieved for
the lowest $ \sin{\Theta_{\pi\eta}}$ estimate within a running time of 12
weeks, improving the precision of the existing experimental value by an
order of magnitude (see Table~\ref{tab_etap_rates}).
Background influencing the precision of the extracted value is primarily
expected from non--resonant $3\,\pi$ production.
Although the background will be distributed smoothly in the vicinity of an
$\eta^\prime$ signal in the two proton missing mass, the
final precision of the result will depend on the four--momentum resolution
achieved for both protons in the forward detector and the decay system.
Modifications of the WASA detector setup to ensure a required performance
at higher energies are discussed in section~\ref{subsubdetmodi}.
With selective central detector trigger conditions on either six neutral
particles from the decays $\eta(\pi^0) \rightarrow 2\gamma$ or two neutral
and two charged particles, both ratios ${\cal R}_i$ can be determined
simultaneously.
Since the Dalitz plots of the hadronic decay channels are measured
completely, the energy dependence of the decays is inherent to the data,
and can be used both to test predictions from ChPT, and as a constraint on
the the scalar meson sector as discussed in section~\ref{subsubetap}.
The present performance of the WASA detector for the detection of
multipionic final states is demonstrated in Fig.~\ref{fig_etap_multipi}
for a data sample obtained at CELSIUS at a kinetic beam energy of
$1360\,\mbox{MeV}$, i.e.\ at an excess energy of $\approx 40\,\mbox{MeV}$
with respect to the $\eta$ production threshold in proton--proton
scattering~\cite{Pauly:2004me}.
The experimental $pp$ missing mass spectrum is well reproduced by
contributions from $pp \rightarrow p p \eta \rightarrow pp 3\pi^0$ and
prompt $3\pi^0$ production (fig.~\ref{fig_etap_multipi}a).
The complete kinematical information is contained in the Dalitz plot for
the $\eta \rightarrow 3\,\pi^0$ decay, that is shown in its symmetrized
form ($<\mbox{T}_{\pi}> - \mbox{T}_{\pi\,3}$ vs.\ $\mbox{T}_{\pi\,2} -
\mbox{T}_{\pi\,1}/\sqrt{3}$ ) in Fig.~\ref{fig_etap_multipi}b.
The slope parameter $\alpha$ (fig.~\ref{fig_etap_multipi}c), which reflects
the strong and energy dependent $\pi \pi$ interaction, is extracted from
a linear fit of the normalized radial density
distribution~\cite{Nefkens:2002sa}.
The preliminary value $\alpha = -0.03 \pm 0.025$ is limited by the
statistics of 11700 events, but agrees in sign with the result
$\alpha = -0.031(4)$ of the Crystal Ball analysis based on
$10^6\,\mbox{events}$~\cite{Tippens:2001fm}.
\begin{figure}[tb]
\parbox{0.49\textwidth}{\epsfig{file=figures/eta_3pi.eps,width=0.49\textwidth}}
\hfill
\parbox{0.49\textwidth}{\vskip-7ex\caption{\label{fig_etap_multipi}
\small a) Measured $pp$ missing mass (crosses) and Monte Carlo event
mixture (shaded area) of prompt $3\,\pi^0$ (dashed line) and resonant
$\eta \rightarrow 3\,\pi^0$ production (solid line).
b) Symmetrized experimental Dalitz plot for the decay $\eta \rightarrow
3\,\pi^0$.
c) Efficiency corrected, normalized radial density distribution of the
Dalitz plot. Crosses denote data, the solid line corresponds to the fit
of the slope parameter $\alpha$ (figures from~\cite{Pauly:2004me}).}}
\parbox{0.04\textwidth}{\raisebox{1ex}[0ex][0ex]{\mbox{}}}\hfill
\parbox{0.95\textwidth}{\raisebox{1ex}[0ex][0ex]{a)}}\hfill
\parbox{0.49\textwidth}
{\epsfig{file=figures/eta_3pi_dalitz.eps,width=0.49\textwidth}}
\hfill
\parbox{0.49\textwidth}
{\epsfig{file=figures/eta_3pi_dal_slope.eps,width=0.49\textwidth}}
\parbox{0.04\textwidth}{\raisebox{1ex}[0ex][0ex]{\mbox{}}}\hfill
\parbox{0.50\textwidth}{\raisebox{1ex}[0ex][0ex]{b)}}\hfill
\parbox{0.45\textwidth}{\raisebox{1ex}[0ex][0ex]{c)}}
\end{figure}
\paragraph{Search for evidence of the box anomaly of QCD in $\eta^\prime
\rightarrow \pi^+ \pi^- \gamma$ decays.}
The experimental signature for the detection of the $\pi^+ \pi^- \gamma$
decay mode of the $\eta^\prime$ consists of two charged and one neutral
particle in the central detector, with the $\eta^\prime$ being tagged from
the missing mass with respect to the two protons identified in the forward
detector.
This signature can be used for a selective trigger condition.
Moreover, with the requirement of one $\gamma$ in the central detector,
the experiment trigger for a measurement of the hadronic decays
$\eta^\prime \rightarrow \eta (\pi^0) \pi^+ \pi^-$ discussed above is a
mere subset, i.e.\ both experiments could run simultaneously.
The experiment aims at measuring the $\pi^+ \pi^-$ invariant mass
distribution in the decay $\eta^\prime \rightarrow \pi^+ \pi^- \gamma$ with
high accuracy.
One (Two) order(s) of magnitude higher statistics compared to the presently
available data can be achieved within a running time of two days (three
weeks).
Following the experimental technique used in~\cite{Abele:1997yi} the number
of $\eta^\prime \rightarrow \pi^+ \pi^- \gamma$ can be determined in each
bin of the $\pi^+ \pi^-$ invariant mass separately.
Consequently, the $\pi^+ \pi^-$ spectrum will be background free
(Fig.~\ref{fig_cb_box}).
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.64\textwidth]{figures/cb_rho.eps}
\caption{\label{fig_cb_box}
Distribution of the $\pi^+ \pi^-$ invariant mass for $\eta^\prime
\rightarrow \pi^+ \pi^- \gamma$ decays from~\cite{Abele:1997yi}.
Crosses denote experimental data, the dashed line is the result of the
fit used to extract box anomaly parameters with a statistical significance
of $4\,\sigma$.}
\end{center}
\end{figure}
The quantity $E_{\eta^\prime}$ associated with the matrix element of a
non--resonant contribution as a consequence of the box anomaly is related
at the chiral point to the parameters of the pseudoscalar nonet by one of
the Chanowitz relations~\cite{Chanowitz:1975jm,Chanowitz:1980ma}:
\begin{equation}
\label{eq_chanowitz}
E_{\eta^\prime} \left(0\right) =
- \frac{e}{4\,\pi^2\,\sqrt{3}} \frac{1}{f_{\pi}^2}
\left[
\frac{\sin{\Theta_{ps}}}{f_8} + \sqrt{2}\,\frac{\cos{\Theta_{ps}}}{f_0}
\right] \, .
\end{equation}
$\Theta_{ps}$ denotes the pseudoscalar octet--singlet mixing angle,
$f_{\pi}$ is the pion leptonic decay constant, and, in analogy, $f_0$ and
$f_8$ are defined as couplings of the pseudoscalar singlet and octet
states $\eta_0$ and $\eta_8$ to the divergences of the singlet and octet
axial--vector currents, respectively.
$E_{\eta^\prime}$ can be derived from a fit to the spectrum of the
$\pi^+ \pi^-$ mass $\mbox{m}_{\pi\pi}$ (Fig.~\ref{fig_cb_box}
and~\cite{Benayoun:1993ty,Benayoun:1995mr,Abele:1997yi}) in the
$\pi^+ \pi^- \gamma$ decay mode of the $\eta^\prime$ using
(for a more sophisticated analysis of this decay, see~\cite{Borasoy:2004qj},
which should be used in the final analysis)
\begin{equation}
\label{eq_pipi_box}
\frac{d\,\Gamma_{\eta^\prime}}{d\,\mbox{m}_{\pi\pi}} =
\frac{1}{48\,\pi^3} \left|
\frac{2\,\mbox{G}_{\rho}\left(\mbox{m}_{\pi\pi}\right)\,
\mbox{g}_{\eta^\prime\rho\gamma}}
{\mbox{D}_{\rho} \left(\mbox{m}_{\pi\pi}\right)} +
E_{\eta^\prime} \right|
\mbox{k}_{\gamma}^3\,\mbox{q}_{\pi}^3 \, ,
\end{equation}
where $\mbox{k}_{\gamma}$ and $\mbox{q}_{\pi}$ are the four--momenta of the
outgoing $\gamma$ and $\pi^\pm$, $\mbox{g}_{\eta^\prime\rho\gamma}$ denotes
the coupling constant for the $\rho$ contribution to the
$\pi^+ \pi^- \gamma$ decay mode, and
$\mbox{D}_{\rho} \left(\mbox{m}_{\pi\pi}\right)$
and
$\mbox{G}_{\rho}\left(\mbox{m}_{\pi\pi}\right)$
are the $\rho$ propagator and its coupling to the $\pi^+ \pi^-$ channel.
On the one hand, the extraction of the box anomaly contribution is strongly
dependent on the model employed in the description of the $\rho$.
On the other hand, from the two--photon widths of $\eta$ and $\eta^\prime$
as described by the Chanowitz
equations~\cite{Chanowitz:1975jm,Chanowitz:1980ma} using additional
information from radiative $J/\Psi \rightarrow \eta(\eta^\prime) \gamma$
decays (AFN relation~\cite{Novikov:1979uy,Akhoury:1987ed}), the parameters
of the pseudoscalar nonet are completely determined, and can be used to
predict the box anomaly constant $E_{\eta^\prime}$.
Consequently, the $\rho$ shape and the box anomaly parameters are closely
related to each other --- the approach might allow for an alternative approach
to extract the $\rho$ line shape in a clean way.
So far, experimentally no consistent picture has been obtained:
The box anomaly contribution extracted from the Crystal Barrel
data~\cite{Abele:1997yi} is rather disfavoured (confidence level $3\,\%$)
by the L3 result (Fig.~\ref{fig_l3_box} and~\cite{Acciarri:1998yx}),
which is consistent with a pure $\rho$ line shape (C.L.\ $37\,\%$).
On the other hand, the Crystal Barrel data would require a mass
$\mbox{m}_\rho = 790\,\mbox{MeV}$ when neglecting the anomaly related
non--resonant contribution, i.e.\ $20\,\mbox{MeV}$ above the presently
accepted value~\cite{PDBook}.
In this respect, these two results with the up to now highest statistics
are inconsistent.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.5\textwidth]{figures/l3_rho.eps}
\caption{\label{fig_l3_box}
$\pi^+ \pi^-$ effective mass distribution in the decay
$\eta^\prime \rightarrow \pi^+ \pi^- \gamma$ from
L3~\cite{Acciarri:1998yx}. Points denote experimental data, the solid
line is the best fit result with $\mbox{m}_{\rho} = 766\,\mbox{MeV}$ and
$\Gamma_{\rho} = 150\,\mbox{MeV}$. The estimated background (shaded area)
is mainly attributed to decays of the $a_2(1320)$.}
\end{center}
\end{figure}
\subsection{\boldmath $a_0^+$(980) production in $pp{\to} d\pi^+\eta$}
\label{subsec:a0f0}
$a_0^+$-production has first been investigated at COSY by measuring
the reaction $pp \to dK^+ \bar{K}^0$ at a beam momentum of
$p=3.46$~GeV/c ($Q=46$ MeV) with ANKE~\cite{Kleber:2003kx}. The
measured total cross section for the $pp\to dK^+\bar{K}^0$ reaction is
$(38\pm 2\pm 14)$~nb and, thus, $\approx$1000 events could be collected
within five days of beam time using a cluster-jet target ($L= 2.7\cdot
10^{31}\, \mathrm{cm}^{-2} \mathrm{s}^{-1}$ during these
measurements). According to the partial-wave analysis the $K^+ \bar
{K}^0$ pairs are mainly (about 83\%) produced in a relative $S$-wave,
i.e.\ via the $a_0^+$ channel~\cite{Kleber:2003kx}. This observation
is in good agreement with a model prediction~\cite{Grishina:2004rd}
for the total $a_0^+$-production cross section in $pp$
collisions. This model has also been used to estimate the
cross-section ratio for resonant (via the
$a_0^+$)~\cite{Grishina:2001zj,Grishina:2000xp} and
non-resonant~\cite{Grishina:2001zj,pieta00} $\pi^+\eta$ production:
\begin{equation}
R^{{\mathrm{res/nres}}}_{d} \left|_{p=3.46{\mathrm{GeV/c}}}
\right. =
\frac{\sigma (pp \to d a_0^+)}{\sigma (pp \to d \pi^+ \eta)} \approx
0.3 \ldots 0.5.
\label{ratio_d}
\end{equation}
This prediction for $R^{{\mathrm{res/nres}}}_{d}$ is in line with data
from ANKE for the reaction $pp\to d\pi^+X$~\cite{Fedorets}. Since most
of the non-resonant $\pi^+\eta$ pairs are expected at lower invariant
masses, we anticipate that the resonant signal can well be identified in
case of the $pp\to d\pi^+\eta$ reaction.
The model can also be used to estimate $R^{{\mathrm{res/nres}}}_{pn}$
for the $pp \to pn \pi^+ \eta $ reaction (see
Refs.~\cite{Bratkovskaya:2001ts,Kondratyuk:2002yf} for $a_0^+$- and
Ref.~\cite{pieta00} for non-resonant $\pi^+ \eta$ production):
\begin{equation}
R^{{\mathrm{res/nres}}}_{pn} \left|_{p=3.46{\mathrm{GeV/c}}}
\right. = \frac{\sigma (pp \to pn a_0^+)}
{\sigma (pp \to pn \pi^+ \eta)}
\approx 0.015 \ldots 0.03.
\label{ratio_pn}
\end{equation}
Thus $R^{{\mathrm{res/nres}}}_{pn}$ is expected to be about one order
of magnitude smaller than the corresponding ratio
$R^{{\mathrm{res/nres}}}_{d}$ with deuteron formation in the final
state. Therefore, an experimental study of the $a_0^+$ in $pp$
reactions with WASA requires the identification of the $pp\to
d\pi^+\eta$ reaction, and the $pp \to pn
\pi^+ \eta$ reaction will be the most ``dangerous'' source of background.
Consequently, the following simulation calculations focus on the
problem of deuteron-vs.-proton discrimination.
The reaction $pp{\to}da_0^+{\to} d\pi^+\eta$ with subsequent decay
$\eta{\to}\gamma\gamma$ can be measured with WASA by detecting
deuterons in the forward detector (FD) in coincidence with the $\pi^+$
and photons in the central detector (CD). The reaction is identified
by reconstructing the masses $m(\eta)=m(\gamma\gamma)$ and
$m(d)=m.m.(\pi^+\gamma\gamma)$. In order to investigate the acceptance
and background suppression the simulations were performed for
$a_0^+$(980) production with a deuteron in the final state and both
background reactions using the cross section ratios
$\sigma(a_0^+){\colon}\sigma(d\pi^+\eta){\colon}\sigma(pn\pi^+\eta)=
1.1\,\mu\mathrm{b} {\colon} 3.5\,\mu\mathrm{b} {\colon}
96\,\mu\mathrm{b}$~\cite{Grishina:2000xp,Grishina:2001zj,pieta00}.
Fig.~\ref{fig:invmass} shows the initial $(\pi^+\eta)$ invariant
mass distributions for these reactions: a Flatte distribution for the
$a_0^+$(980)~\cite{Grishina:2000xp,Kondratyuk:2002yf}, a distribution
according to the model in
Refs.~\cite{pieta00,Kondratyuk:2002yf,Kudryavtsev:2003au} for the
non-resonant $d\pi^+\eta$, and a phase-space distribution for the
non-resonant $pn\pi^+\eta$ production.
\begin{figure}[tb]
\begin{center}
\resizebox{0.5\textwidth}{5.5cm}{\includegraphics{figures/inv_a0_non_pn_init.eps}}
\caption{Invariant mass $(\pi^+\eta)$ for $a_0^+$ production in
$pp{\to}da_0^+{\to} d\pi^+\eta$ and for the two background processes,
$pp \to d\pi^+\eta$ and $pp \to pn\pi^+\eta$ (downscaled by factor 10).}
\label{fig:invmass}
\end{center}
\end{figure}
The WASA acceptance for forward going particles is $\theta \approx
3^\circ \ldots 18^\circ$. This acceptance covers about $93\%$ of
deuterons from $pp{\to}da_0^+$, which are distributed in the range
$\theta = 0^\circ \ldots 17^\circ$. The acceptance for pions and
photons is $\theta \approx 20^\circ \ldots 169^\circ$ (for the
simulations angles $\theta \approx 20^\circ \ldots 143^\circ$ ---
SEC and SEF, the central and forward part of the calorimeter --- were used).
The result of the acceptance estimate is shown
in the Table~\ref{accep}. For the background reactions the acceptance
is smaller due to the wider angular distributions. We have only
considered the decay channel $\eta{\to}\gamma\gamma$ since for the
other two main decay channels $\eta{\to}\pi^+\pi^-\pi^0$ and
$\eta{\to}3\pi^0$ the acceptance is two times smaller.
\begin{table}[h]
\begin{center}
\begin{tabular}{c|c}
Reaction & Acceptance \\
& ($\eta{\to}\gamma\gamma$ decay only)\\
\hline
$pp{\to}da_0^+{\to} d\pi^+\eta$ & 0.42 \\
$pp \to d\pi^+\eta$ & 0.34 \\
$pp \to pn\pi^+\eta$ & 0.32 \\
\end{tabular}
\caption{Simulated acceptance of WASA for $a_0^+$(980) production
and the two background processes. Deuterons/protons are detected in
the FD ($\theta\approx 3^\circ \ldots 18^\circ$), pions and photons
in the CD ($\theta \approx 20^\circ \ldots 143^\circ$).}
\label{accep}
\end{center}
\end{table}
In the simulations the $\pi^+$ detected in the CD have angular resolution
better than $\sigma(\theta)= 0.4^\circ$ (for angles larger than
30$^\circ$) and momentum resolution between $\sigma(p)/p=
3{\ldots}6\%$.
For photons in the CD the resolution is
$\approx 5^\circ$(FWHM), due to the crystal sizes in the SEC. The energy
resolution varies from 10~MeV (FWHM) for large angles and low initial
energies up to 40~MeV (FWHM) for angles less than 30$^\circ$ and
initial energies close to 1~GeV. Such resolutions yield
$\gamma\gamma$ invariant mass distributions with a FWHM of
43~MeV/c$^2$ in the $\eta$ mass region, which is in agreement with
WASA at CELSIUS data~\cite{Koch}.
\begin{figure}[tb]
\begin{center}
\resizebox{1.0\textwidth}{!}{\includegraphics{figures/a0_init_rec_2_12.eps}}
\caption{Initial (dotted line) and reconstructed (solid) invariant
mass $(\pi^+ \gamma\gamma)$ (left) and missing mass
$(\pi^+\gamma\gamma)$ (right) for the reaction $pp{\to}da_0^+{\to}
d\pi^+\eta$.}
\label{fig:rec_init}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\textwidth]{figures/main_cuts.eps}
\caption{Reconstructed missing mass $(\pi^+\gamma\gamma)$ for the
reactions $pp{\to}da_0^+{\to} d\pi^+\eta$ and $pp \to pn\pi^+\eta$
(left). The cut is indicated by the arrow. Energy losses in the FD
vs.\ kinetic energy of the deuterons/protons (calculated from the
detected $\pi^+$ and the two photons assuming that the forward
particle is a deuteron). The cut is indicated by the two lines.}
\label{fig:mx_td}
\end{center}
\end{figure}
Fig.~\ref{fig:rec_init} shows the initial and reconstructed
invariant mass $(\pi^+ \gamma\gamma)$ and missing mass
$(\pi^+\gamma\gamma)$ for the reaction $pp{\to}da_0^+{\to}
d\pi^+\eta{\to} d\pi^+\gamma\gamma$. Pions and photons detected in
coincidence in CD were used for the reconstruction, which provide a
$(\pi^+\gamma\gamma)$ missing mass resolution of $\sigma=40$
MeV/c$^2$.
The expected background from the reaction $pp \to pn\pi^+\eta$ is two
order higher than the $a^+_0$ signal. Due to the large momenta of the
protons and deuterons they cannot be stopped in the FD and their
initial kinetic energy cannot be reconstructed. Moreover, their energy
losses are close to minimum ionizing and the standard WASA at CELSIUS
$\Delta{E}/E$ method cannot be used for $p/d$ discrimination.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\textwidth]{figures/2d_angle.eps}
\caption{(a,b) Measured vs.\ calculated forward azimuthal and polar
angles assuming that the forward particle is a deuteron. (c,d)
Difference between measured and calculated angles of the forward
particles. The background $pn\pi^+\eta$ events (dotted line) is
downscaled by a factor 100. The cuts are indicated by the arrows.
\label{fig:angles}}
\end{center}
\end{figure}
In order to suppress the proton background a set of criteria has been
applied. The first cut was applied on the reconstructed
$(\pi^+\gamma\gamma)$ missing mass, see Fig.~\ref{fig:mx_td}. The cut
at 1.95~GeV suppresses protons by a factor $\approx 1.6$. The
two-dimensional distribution energy loss vs.\ kinetic energy of the
forward particle shows a correlation for $a_0^+$ events, see
Fig.~\ref{fig:mx_td}. Applying a gate with ${\pm}20$~MeV around these
events, background protons can be suppressed by a factor of $\approx
3.9$.
Another strong criterion for proton suppression is the difference
between the measured azimuthal and polar angles of the forward
particles and the expected deuteron angles calculated from the $\pi^+$
and two photons. For the $a_0^+$ events these angles coincide within
the resolutions of the detector and the reconstruction procedure. For
protons from the $pn\pi^+\eta$ events the correlation is much
weaker. This is clearly seen in two dimensional plots of measured vs.\
calculated angles. If one assumes that all forward particles are
deuterons, then the real deuterons are seen as lines whereas the
protons are smeared out (Fig.~\ref{fig:angles} (a,b)). Cuts were applied
to the difference between measured and calculated azimuthal and polar
angles of forward particles (Fig.~\ref{fig:angles} (c,d)). An
azimuthal angle cut of ${\pm}20^\circ$ and a polar angle cut of
${\pm}2^\circ$ suppresses protons by a factor $\approx 21$.
Taking into account all mentioned cuts and the difference in
acceptances the proton suppression factor is
$1.3{\times}1.6{\times}3.9{\times}21{\approx}170$. We therefore expect
that the non-resonant $pn\pi^+\eta$ events can sufficiently be
suppressed without modifications of the existing FD for the higher
particle momenta at COSY.
Assuming a luminosity of $L=10^{31}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}$,
an overall efficiency due to detector acceptance, reconstruction algorithms
and $d/p$ cuts of $\approx 0.07$, the effect of dead-time and detector
efficiencies of $\approx 0.5$ and using the cross section estimate from
Refs.~\cite{Grishina:2001zj,Grishina:2000xp},
$\sigma(pp{\to}da_0^+{\to}d\pi^+\eta){\times}BR(\eta\to2\gamma)=
1.1\,\mu\mathrm{b} {\times} 0.393$, the final count rate is
$\approx 0.15\,\mathrm{s}^{-1} \approx 90000\,\mathrm{week}^{-1}$.
\subsection{\boldmath Pentaquarks}
\label{sec:day2_pentaquark}
At present, several experimental groups, including TOF at COSY, are
performing high statistics experiments to confirm the existence of the
$\Theta^{+}$.
Provided the existence of the $\Theta^+$ is confirmed, we
will be interested in basic properties of this hadron and
details of the production mechanism.
In the following we concentrate on the $pd\rightarrow p\Lambda \Theta ^{+}$
reaction as one example to illustrate the feasibility of high statistics
experiments involving both neutral and charged decay products.
Out of the two decay modes of the $\Theta^{+}$, the $K^{0}p$ channel
gives the best experimental conditions for being measured in the WASA
detector, so we shall here concentrate on this decay branch.
For the $\Lambda$ there are two important decay modes, $p\pi^{-}$
and $n\pi^{0}$ with branching ratios of 63.9\% and 35.8\%,
respectively. The choice of using the charged decay mode to identify
the $\Lambda$ is advantageous due to the following reasons:
\begin{itemize}
\item the energy resolution of WASA for charged particles is
generally better than for photons,
\item the combination of the forward tracker and the MDC provides a
clean identification of the $\Lambda$ by reconstruction of its
decay vertex,
\item both decay products are measured,
\item higher branching ratios as compared to neutral decay modes.
\end{itemize}
Regarding the kaon decay, half of the events are lost due to
$K_{L}^{0}$ having $c\tau=15.5$~m and thus escaping detection. For the
$K_{S}^{0}$ two modes are important, $\pi^{+}\pi^{-}$ ($BR = 68.6$\%)
and $\pi^{0}\pi^{0}$ ($BR = 31.4$\%). Selecting the charged mode has the
advantage of higher rate. However, it leads to a final state with six
charged particles. It might be difficult to distinguish between the
$\pi^{-}$ produced in the $\Lambda$ decay and the $K_{S}^{0}$ decay,
which could impair the reconstruction of the $\Lambda$. Therefore,
the neutral decay mode will be used.
Taking all branching ratios together gives the overall branching
ratio, or efficiency of detecting $\Theta^{+}$ using this particular
final state, of 0.5$\times$0.639$\times$0.314 = 0.1.
This final state with four charged particles, $\pi^{-}$ and three
protons, and the four photons from $\pi^{0}\pi^{0}$ provides a well
defined trigger as well as favorable analysis conditions.
A rough estimate of the WASA acceptance for the $pd\rightarrow
p+\Lambda+\Theta^{+}\rightarrow p+p\pi^{-}+p\gamma\gamma\gamma\gamma$
reaction has been obtained using phase space Monte Carlo
simulation. The first things needed are the energies and angles of all
charged final state particles. They are shown in
Fig.~\ref{fig:pentaquark_f2} below. The proton produced directly is
denoted $p$ while those produced in the decays of the $\Lambda$ and
$\Theta^{+}$ are denoted $p_{\Lambda}$ and $p_{\Theta}$,
respectively. It is seen that the maximum energies for $p$ and
$p_{\Lambda}$ are below 300~MeV, which is the highest energy deposited
by protons in the FD. For such protons one may expect a very good
energy resolution.
\begin{figure}[tb]
\begin{center}
\resizebox{10cm}{!}{\includegraphics[scale=1]{figures/Pentaquark_f2.eps}}
\caption{Energy and angular ranges of the
four final state charged particles in the reaction $pd\rightarrow
p+\Lambda+\Theta^{+}\rightarrow p+p\pi^{-}+p\gamma\gamma\gamma\gamma$ at
1360 MeV proton beam energy.}
\label{fig:pentaquark_f2}
\end{center}
\end{figure}
Assuming that reconstruction of the $\Lambda$ requires the
$p_{\Lambda}$ proton to be detected in the FD
($2.5^{\circ}<\theta_{p\Lambda}<18^{\circ}$) and the $\pi^{-}$ either
in the FD or MDC ($22^{\circ}<\theta_{\pi}<158^{\circ}$) one finds
that this requirement is fulfilled in $\approx 85$\% of cases. This is
roughly the efficiency of the $\Lambda$ reconstruction,
$\epsilon_{\Lambda}$. It should be noted that the resolution of the
MDC is limited at the forward and backward angles due to a reduced
number of wire planes being crossed by the particles. All 17 layers
are crossed only for angles between 44$^{0}$ and 134$^{0}$. For the
$\pi^{-}$ going into the FD or into this limited angular range of the
MDC, the $\epsilon_{\Lambda}$ is reduced to 55\%.
The requirement that the proton $p_{\Theta}$ from $\Theta^{+}$ enters
the FD or the plastic barrel is fulfilled in $\epsilon_{p\Theta}=87$\%
of cases.
Finally, the requirement of the four photons (from the $K_{S}^{0}$
decay into $\pi^{0}\pi^{0}$) to fall into the angular range of the
electromagnetic calorimeter is fulfilled with an efficiency
$\epsilon_{4\gamma}=62$\%. This gives the overall acceptance of $\approx
45$\% for $\epsilon_{\Lambda}=85$\% and 29\% for
$\epsilon_{\Lambda}=55$\%.
Remembering that the resolution of the WASA detector is far better for
charged particles than for photons one would like the $\Theta^{+}$
mass to be calculated as the missing mass to the reconstructed $p$,
$p_{\Lambda}$ and $\pi^{-}$ (for $\pi^{-}$ only angles will be used,
see below). In such a case one has to take into account losses of
protons due to nuclear interactions with the detector material, which
is expected to reduce the acceptance by roughly a factor of two.
As already mentioned, the $\Theta^{+}$ mass will be found as the
missing mass calculated for the $p$ and $p_{\Lambda}$ reconstructed in
the FD and $\pi^{-}$ either in the FD or MDC. Measurement of the
energy and angle of the $p$ proton is straightforward since the
reaction vertex is at the known target position. This is not the case
for the $p_{\Lambda}$ because of the $\Lambda$'s $c\tau $=7.89 cm. In
order to reconstruct the $\Lambda$ momentum one has to find the
$\Lambda$ decay vertex using either the forward tracker, when both
$p_{\Lambda}$ and $\pi^{-}$ enter the FD, or the forward tracker and
the MDC. Reconstruction of the vertex requires finding two tracks
which, together with the target point, are in the same plane. When two
tracks fulfilling the above criteria are found, one can calculate the
absolute value of the $\pi^{-}$ momentum, $k$, from the condition:
$(\sqrt{m_{\pi}^{2}+k^{2}}+E)^{2}-(k\hat{u}+\vec{p})^{2}=m_{\Lambda}^{2}$,
\noindent where $E$, $\vec{p}$ represent the proton total energy and
momentum,\ $\hat{u}$\ is the unit vector representing direction of the
presumed $\pi^{-}$ and $m_{\Lambda}$\ is the $\Lambda$ mass. One
should verify that the reconstructed $\Lambda$-momentum points to the
target area.
A further check is provided by the $\Delta E$-$E$ method when the pion
enters the FD (see Fig.~\ref{fig:pentaquark_f3}) and extracting the
momentum from the measurement of the track curvature in the magnetic
field, when the pion enters the MDC.
\begin{figure}[tb]
\begin{center}
\resizebox{10cm}{!}{\includegraphics[scale=1]{figures/Pentaquark_f3.eps}}
\caption{$\Delta E$-$E$ plots for three consecutive layers of the
range hodoscope in the FD as obtained in the M-C simulation of the
reaction $pd\rightarrow p+\Lambda+\Theta^{+}\rightarrow
p+p\pi^{-}+p\gamma\gamma\gamma\gamma$ at 1360 MeV proton beam
energy.}
\label{fig:pentaquark_f3}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.7\textwidth]{figures/Pentaquark_f4.eps}
\caption{Phase space Monte Carlo simulation of the reaction
$p+d\rightarrow p+\Lambda+\Theta^{+}$ at 1360 MeV beam energy. The
$\Theta^{+}$ peaks are seen on a wide background from non-resonant
$p+d\rightarrow p+\Lambda +K^{0}+p$ reaction, which has been assumed
to be 10 stronger than the $\Theta^{+}$ production. For both peaks
the energy resolution for protons has been assumed to be equal to
3\% and the $\pi^{-}$ angular resolution to amount to
$3^{\circ}$. The narrower peak, FWHM=4.6 MeV/c$^2$, corresponds to the
proton angular resolution of $1^{\circ}$ and for the wider one the
resolution of $3^{\circ}$ has been assumed. The width obtained in
this case amounts to 9.6~MeV/c$^2$.}
\label{fig:pentaquark_f4}
\end{center}
\end{figure}
At present the position resolution of the forward tracker amounts to
$\approx 2$~cm in the $x$-$y$ plane (perpendicular to the beam)
\cite{Dyring} so one can estimate the resolution in angle to be of the
order of $\arctan(2/140)\approx 1^{\circ}$, where 140~cm is the
distance between the target position and the FD tracker. For the
proton $p$, the resolution in angle is given by the size of the
beam-target overlap and accuracy of the FD tracker. For the $\pi^{-}$,
which in most cases enters the MDC, one can expect the resolution in
$\theta$ to be of the order of $\arctan(1/22) = 2.6^{\circ}$.
A rough estimate of the resolution in $M_{\Theta^{+}}$ can be obtained
by smearing the energies of the $p$, $p_{\Lambda}$ and angles of the
$p$, $p_{\Lambda}$ and $\pi^{-}$ with the experimental
resolutions. Taking for the energy resolution 3\%, and $1.0^{\circ}$,
and $3.0^{\circ}$ for the angular resolution for protons and $\pi^{-}$
respectively, leads to the results shown in the
Fig.~\ref{fig:pentaquark_f4} (solid line). The width of the peak at
half maximum amounts to 4.5~MeV/c$^2$, which is nearly five times better
than the resolution quoted in Ref.~\cite{Nakano:2003qx}. This is a
valuable feature which could improve our knowledge of the true width
of the $\Theta^{+}$ resonance expected to be much smaller than the
20~MeV/c$^2$ measured in the experiments so far. The wide background seen in
the figure corresponds to the phase space simulation of the
non-resonant reaction $p+d\rightarrow p+\Lambda+K^{0}+p$.
For the $\Theta^{+}$ cross section of 30 nb, the beam-target
luminosity of 10$^{31}\,$/cm$^{-2}$s$^{-1}$ and the acceptance of 1.5\% (with
the overall branching ratio of 10\% included) one would expect the
$\Theta^{+}$ event rate of $30 \times 10^{-33}\times 10 ^{31}\times
0.015 \mathrm{s}^{-1} = 0.0045\mathrm{s}^{-1}$ at a beam energy of
1360 MeV. Simulation shows that the detector acceptance depends rather
weakly on the beam energy. Thus increasing the beam energy to 1450 MeV
would increase the event rate by a factor of $\approx$7 due to the change
of the cross section~\cite{Goeran}. With a duty factor of 50\%
obtaining 10$^{4}$ events would require less than 200 hours.
\subsection{\boldmath Isospin violation in $\vec{d}d\,\to\,\alpha\pi^0$}
\label{sec:mediumterm_dd2alphapi0}
Due to the small cross section (a few pb) and the high background from
the reaction $dd \to \alpha\mathrm{X}$ the detection and
unambiguous identification of the $\alpha\pi^0$ channel is rather
difficult. Hence, first attempts to measure this reaction have
produced only upper limits (see Ref.~\cite{Banaigs:1987mx}). The only
positive report at a deuteron energy of 1.1~GeV~\cite{Goldzahl:1991ge}
has been questioned, because the background from the double radiative
capture process $dd \to \alpha \gamma\gamma$ could have been
misinterpreted~\cite{Dobrokhotov:1999cs}. Consequently, in the recent
experiment performed at IUCF~\cite{Stephenson:2003dv} the $\alpha$ and
the two $\gamma$'s have been measured in coincidence to provide a
clean signal of $dd \to \alpha\pi^0$ channel.
At COSY, studies of this reaction were initially proposed for
BIG KARL~\cite{MAG97}. However, a sufficient background
suppression without photon detection would have been hard to
achieve. The idea was resubmitted within a proposal for an
electromagnetic calorimeter at ANKE~\cite{PD00}. This combination
--- like WASA --- would have provided a clean identification of the
forward-going
$\alpha$ particle and the $\pi^0$ decaying into two
photons.
Compared with the latter proposal, the WASA detector provides a
similar acceptance for photons ($\Theta = 20^\circ \ldots 169^\circ$)
but a significantly wider angular acceptance for $\alpha$ particles
($\Theta \approx 3^\circ \ldots 18^\circ$). This will allow the
extraction of angular distributions starting from $Q\approx60$~MeV up
to and beyond the $\eta$ threshold (see Fig.~\ref{fig:alphaacc}). In
addition, the fully symmetric WASA detector covers the entire azimuthal
angular range of 2$\pi$ and, thus, is well suited to measure
polarization observables.
The simulations are done for the measurement at $Q=60$~MeV (studying
the development of $p$-waves).
As indicated previously, the reaction $dd \to\alpha\pi^0$ will
be identified by detecting the $\alpha$ and the decay
$\pi^0\to\gamma\gamma$ in coincidence. Here, the high energy loss of
the $\alpha$ in combination with two neutral hits in the calorimeter
provides a very efficient first level trigger. For $\alpha$ momenta
above 1~GeV/c all particles pass the (currently used) Forward Window
Counter (FWC) of WASA and are stopped in one of the layers of the
Forward Trigger Hodoscope (FTH). Subsequent layers (e.g. of the
Forward Range Hodoscope, FRH) can be used as veto layers for
particles, which are not stopped. Angles are reconstructed by means of
the Forward Proportional Chambers (FPC).
\begin{figure}[tb]
\begin{center}
\resizebox{1.0\textwidth}{!}{\includegraphics{figures/aplhapi0_all.eps}}
\caption{Kinematics of the reaction $\mathrm{dd}\to\alpha\pi^0$ for
various $Q$ values. The transverse momentum of the $\alpha$ versus
its longitudinal momentum in laboratory is plotted. The lines
indicate the azimuthal angle covered by WASA.
\label{fig:alphaacc}}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\resizebox{1.0\textwidth}{5.5cm}{\includegraphics{figures/e_loss.eps}}
\caption{Energy losses of $\alpha$ particles from the reaction
$dd\to\alpha\pi^0$. Left: Reconstructed angle versus
total energy loss. Right: Energy loss in the three single layers
of the Forward Tracking Hodoscope versus total energy loss.
\label{fig:e_loss}}
\end{center}
\end{figure}
\begin{figure}[tb]
\begin{center}
\resizebox{1.0\textwidth}{5.5cm}{\includegraphics{figures/e_loss_other.eps}}
\caption{Energy loss in the individual layers versus total energy loss.
On the left $^3$He, on the right tritons. While the energy loss
pattern for $^3$He is rather similar to $^4$He, the one for
tritons can be used efficiently for particle discrimination.}
\label{fig:e_loss_other}
\end{center}
\end{figure}
Fig.~\ref{fig:e_loss} (right side) and Fig.~\ref{fig:e_loss_other}
show the energy losses in the individual layers of the tracking
hodoscope for $\alpha$ particles as well as for $^3$He and
tritons. While tritons can already be discriminated by these energy
loss patterns, $^3$He is quite similar to $\alpha$
particles\footnote{Protons and deuterons are not shown. Their energy
loss is --- compared with tritons --- again smaller and the bands for
stopped particles are shifted further to lower energy
losses.}. However, a further reduction of the $^3$He content can
already be done by requiring the correct kinematics as shown in
Fig.~\ref{fig:e_loss} (left side).
Together with the measured $\pi^0$, the
reaction is kinematically over-constrained. Tests on the energy and
momentum conservation prevents further particles from being missing.
While the standard procedure would be an overall kinematic
fit, Fig.~\ref{fig:pions} shows --- as an example --- the correlation
between the measured polar angle of the $\alpha$ and the one
reconstructed from the pion kinematics as well as the missing mass
with only the pion detected.
\begin{figure}[tb]
\begin{center}
\resizebox{1.0\textwidth}{5.5cm}{\includegraphics{figures/pion_kinematics.eps}}
\caption{$\alpha$ kinematics reconstructed from measured $\pi^0$.
Left: Reconstructed polar angle versus measured polar
angle. Right: Reconstructed $\alpha$ missing mass.
\label{fig:pions}}
\end{center}
\end{figure}
Taking into account the current geometry of WASA, the combined
acceptance for $\alpha$ and $\pi^0$ does not vary much and will be
$\epsilon_\mathrm{acc}\approx0.5$ for all beam momenta from 1.2~GeV/c
up to $\eta$ threshold. For the following estimates an additional
factor of 0.5 is considered representing dead-time corrections,
detector- and analysis efficiencies.
Assuming a luminosity of $L = 10^{32}\,\mathrm{cm}^{-2}\mathrm{s}^{-1}
= 10^{-4}\,\mathrm{s}^{-1}\mathrm{pb}^{-1}$ the final count rate will
be 15 events per week and pb. Since the goal of this experimental program is
to extract differential cross sections, the total yield has to be at
least a few hundred events for each beam energy. Using a total cross
section of $\sigma\approx75\mathrm{pb}$ at $Q=60$~MeV (i.e.\ scaling
the IUCF result by $s$-wave phase space) leads to beam times of about
1 or 2 weeks per measurement. Consequently, although a lower
luminosity would still allow to measure one data point within a longer
(but still reasonable) beam time, it would hardly be possible to carry
out the full experimental program. This program
would cover measurements in two different energy ranges, namely at
$Q\approx60$~MeV and in the $\Delta$ region ($Q\approx160$~MeV).
However, initially one data
point at the lowest beam energy should be measured. It is aimed to run
with polarized beam to disentangle the contribution of different
partial waves (especially $p$-waves) as discussed in
section~\ref{sec:dd2alphapi0}. Further runs will then depend
on the analysis of these data.
As shown above the reaction $dd\to\alpha\pi^0$ can be
measured with the existing WASA setup without any modifications on the
detector system itself. The electronics has to be upgraded: the
trigger logic has to be adopted in order to use an efficient energy loss
trigger and the data acquisition systems has to be replaced to be able
to run at high luminosities (see section~\ref{subsubdaq}).
\subsection{\boldmath $a_0^0$-$f_0$ mixing in $pn{\to} d\pi^0\eta$ and
$dd{\to}\alpha \pi^0\eta$}
\label{subsuba0f0mixing}
At a later stage of the experimental program with WASA at COSY the
reactions $pn\to d\pi^0\eta$ and $dd\to \alpha\pi^0\eta$ will be
measured. The main challenges for the identification of the isospin
violating effects in these reactions will be:
\begin{description}
\item[\boldmath $pn\to d\pi^0\eta$:] The measurement of this reaction
is similar to the $pp$ experiment described in Section~\ref{subsec:a0f0}. However, an
additional complication comes from the fact that deuterium has to
be used as an effective neutron target. The
signal for isospin violation in this reaction would be provided from the
measurement of
the angular forward-backward asymmetry of the $(\pi^0\eta)_{l=0}$
system.
\item[\boldmath $dd\to \alpha\pi^0\eta$:] This reaction is only
possible due to isospin violation and the yield will be rather
small. First estimates show that the cross section should be of the order of
100~pb~\cite{dd_proposal}. A better prediction can be obtained from
data on the reaction $dd{\to}\alpha K^+K^-$ which are expected from
ANKE in winter 2004/05. It is
thus suggested to perform these measurements (at lower beam
energies, see Sect.~\ref{sec:mediumterm_dd2alphapi0}) before the
$dd\to \alpha\pi^0\eta$ reaction will be studied at maximum COSY
momentum ($\approx 3.7$ GeV/c).
\end{description}
\subsection{\boldmath Study of hyperon resonances}
\label{sec:mediumterm_lambda1405}
In order to understand the nature of the $\Lambda$(1405) hyperon
resonance it is proposed to study $\Lambda$(1405) production in
proton-proton collisions, and in a second step also in proton-nucleus
collisions. The primary goal of this study is to measure the
$\Lambda$(1405) production cross section and, in particular, its
spectral shape. As a by-product also information on the
$\Sigma$(1385) resonance is obtained.
Due to its large acceptance for both charged particles and photons,
the WASA detector allows to investigate all isospin combinations of
$\Sigma\pi$ and $\Lambda\pi$ that are populated in hyperon resonance
decay subsequent to the production reactions
$pp\rightarrow\Lambda(1405)K^+p$ and
$pp\rightarrow{}[\Sigma(1385)K]^+p$. With the use of a deuterium
target the implementation of a proton spectator detector close to the
interaction point would be required, in order to allow the full
measurement of all particles in the final state (equivalent to $pp$
collisions) also in $pn\rightarrow\Lambda(1405)K^0p$ and
$pn\rightarrow{}[\Sigma(1385)K]^0p$ reactions. Note that the latter
reaction also allows to excite the $\Sigma^-$(1385) state.
The photon detection capability of WASA matches particularly well to
the study of the $pp\rightarrow\Lambda(1405)K^+p$ reaction with the
decay channel $\Lambda(1405)\rightarrow\Sigma^0\pi^0\rightarrow{}
(\Lambda\gamma{})(\gamma\gamma{})$ which is not populated in
$\Sigma^0$(1385) decay, and thus should reflect the $\Lambda$(1405)
spectral distribution in an undisturbed way. Therefore special
emphasis should be given to this channel, but as discussed in
Sect.~\ref{sec:hyperons} also the $\Sigma^+\pi^-$ and $\Sigma^-\pi^+$
channels deserve investigation in order to extract a possible
isovector contribution. For the observed final state in the decay
channel $\Lambda(1405)\rightarrow{}\Sigma^0\pi^0\rightarrow{}
(\Lambda\gamma{})(\gamma\gamma{})\rightarrow{}(p\pi^-\gamma{})
(\gamma\gamma{})$ the expected branching ratio is rather large, and
amounts to 21\%.
Fig.~\ref{fig:l1405-sim} shows the simulated transverse versus
longitudinal momentum distribution for the particles in the final
state in this reaction induced by 3.6~GeV/c protons, assuming a pure
phase space distribution of the events, and taking into account a
$\Lambda$(1405) Lorentz mass distribution of a full width
$\Gamma{}=50$~MeV/c$^2$. It demonstrates that both protons in the final
state, from the primary vertex and from the $\Lambda$ decay, have a
high probability to be emitted into the acceptance of the forward
detector, whereas the dominant fraction of the pions will be detected
by the central detector of WASA. The $K^+$ mesons populate both
regions of the phase space seen by the central and by the forward
detector. Only a small fraction of photons is not covered by the CsI
calorimeter acceptance.
\begin{figure}[tb]
\begin{center}
\resizebox{10cm}{!}{\includegraphics{figures/l1405b.eps}}
\caption{Simulated transverse vs.\ longitudinal momentum distribution
for the final state particles in the reaction
$pp\rightarrow\Lambda(1405)pK^+\rightarrow\Sigma^0\pi^0pK^+
\rightarrow\Lambda\gamma\pi^0pK^+\rightarrow{}
p^{(e)}\pi^{-(b)}\gamma^{(c)}\gamma\gamma^{(d)}p^{(a)}K^{+(f)}$.
The superscripts denote the corresponding panels. The ellipses in
panels (a) and (f) show the kinematic limits with $\Lambda$(1405)
production at its pole mass as well as 25~MeV/c$^2$ below and above. The
straight lines separate the acceptance of the central and the
forward spectrometer of WASA.}
\label{fig:l1405-sim}
\end{center}
\end{figure}
In order to add new information to the experimental knowledge of the
spectral shapes of the $\Lambda$(1405) and $\Sigma$(1385) resonances,
the mass resolution has to be clearly better than the widths of these
resonances of $\approx{}50$~MeV/c$^2$ and $~\approx{}40$~MeV/c$^2$, respectively. The
resolution of better than $\approx{}10$~MeV/c$^2$ allowed by the WASA detector
is considered to be sufficient for this study. Further improvement of
the resolution is possible if new tracking detectors, such as
$\mu$-strip silicon arrays, are implemented close to the target. This
would allow, in addition to the reduction of non-strange background,
to make use of additional kinematic constraints by a precise
measurement of tracks including displaced vertex information.
The optimum proton beam energy is given by the requirement that the
kinematic limit be at sufficient distance from the pole of the
$\Lambda$(1405) resonance in order to guarantee the undisturbed
measurement of its spectral distribution. At energies too close to
threshold it may be difficult to disentangle the $\Lambda$(1405)
spectral function and the effect of an energy dependent final state
interaction. A reasonable beam momentum is $p_p=3.6$~GeV/c, well
within the COSY momentum range for internal beams, at which the
kinematic limit in the $\Lambda^{\star}$ or $\Sigma^{\star}$ mass is
$M_{max}=1.53$~GeV, more than 100~MeV/c$^2$ above the $\Lambda$(1405) pole.
The expected hyperon resonance production cross sections are of the
order of $1\,\mu$b~\cite{AnkeWS}. With a luminosity
$L=10^{31}\,{\rm{}cm}^{-2}{\rm{}s}^{-1}$, assuming 10\% overall
efficiency, one thus expects approximately 18000 counts per day for
the $pp\rightarrow\Lambda(1405)K^+p$ reaction with a
$(p\pi^-\gamma{})_{\Sigma^0}(\gamma\gamma{})_{\pi^0}K^+p$ final state
discussed above. In case the cross section and/or efficiencies should
be smaller, the maximum design luminosity
$L=2\cdot{}10^{32}\,{\rm{}cm}^{-2}{\rm{}s}^{-1}$ of WASA gives room for
an increase of the luminosity.
The second part of the program related to comparative studies of the
in-medium properties of the $\Lambda$(1405) and $\Sigma$(1385) hyperon
resonances requires the usage of nuclear targets, and thus an
extension of the existing pellet target to an operation with gases
like nitrogen, argon and xenon which still needs to be developed.
Furthermore, for these studies additional tracking information close
to the target allowing hyperon and $K_s$ identification will be
necessary, since a full reconstruction of the final state is not
possible with a nuclear target, which results in a loss of kinematic
constraints. For this purpose the WASA detector would have to be
extended by the implementation of $\mu$-strip silicon detectors in
close geometry to the interaction point.
\subsection{\boldmath Rare and very rare $\eta$ and $\eta^\prime$ decays}
\label{sec:mediumterm_eta_etaprime}
\label{subsubetapmedium}
\paragraph{High precision branching ratio for $\eta^\prime \rightarrow
\gamma \gamma$.}
In spite of the large branching ratio this experiment will be
performed at a later time since it requires technical modifications to
achieve high precision as discussed below.
The radiative decay widths of $\eta (\eta^\prime) \rightarrow
\gamma \gamma$, together with additional information from radiative
$J/\Psi$ decays to $\eta (\eta^\prime) \gamma$ or from radiative
transitions of vector ($V$) and pseudoscalar ($P$) mesons in $V
\rightarrow P \gamma$ and $P \rightarrow V \gamma$ processes, allow to
extract the fundamental constants of the pseudoscalar nonet (see
e.g.~\cite{Ball:1996zv}).
The unprecedented statistical accuracy, that is expected to be feasible
for radiative decays using the WASA facility at COSY (see
table~\ref{tab_etap_rates}) suggests to increase the precision on branching
ratios of radiative decays presently available.
For the decay $\eta^\prime \rightarrow \gamma \gamma$, the statistical
accuracy of the branching ratio might be increased by an order of
magnitude in a dedicated run of two to three weeks, yielding a statistical
error below the percent level.
However, to obtain precision values for the branching ratio requires
to control the systematic error with comparable accuracy.
A suitable experimental approach is described for the decay $\eta
\rightarrow \gamma \gamma$ in~\cite{Abegg:1996wz}, with a direct
measurement of the branching ratio.
Thus, all systematic uncertainties related to the ''normalization'',
i.e.\ the simultaneous measurement of a known branching ratio, are
eliminated, and, analogously to~\cite{Abegg:1996wz} the branching ratio
is given by
\begin{equation}
\label{eq_etap_2g_direct}
\frac{\Gamma\left(\eta^\prime \rightarrow \gamma \gamma\right)}
{\Gamma_{tot}} =
\frac{\mbox{N}\left(\eta^\prime \rightarrow \gamma \gamma\right)}
{\mbox{N}\left(p p \rightarrow p p \eta^\prime \right)} \times
\left( A_{\eta^\prime \rightarrow \gamma \gamma}\,
\epsilon_{\eta^\prime \rightarrow \gamma \gamma}^{analysis}\,
\epsilon_{\eta^\prime \rightarrow \gamma \gamma}^{electronics}
\right) \, ,
\end{equation}
where $A_{\eta^\prime \rightarrow \gamma \gamma}$ denotes the detector
acceptance and $\epsilon_{\eta^\prime \rightarrow \gamma \gamma}$ the
efficiencies in the analysis and in the electronics to reconstruct and
reliably digitize an $\eta^\prime \rightarrow \gamma \gamma$ decay.
The precise number of events, background subtraction and (different)
efficiency corrections only have to be considered for the $\gamma \gamma$
decay mode.
Experimentally, the approach requires to trigger on the $pp$ system
associated with $\eta^\prime$ production, i.e.\ on two protons in the
forward detector.
With an estimate of a $50\,\mbox{kHz}$ rate from a trigger on two charged
particles in the forward detector at luminosities of
$10^{32}\,\mbox{cm}^{-2}\mbox{s}^{-1}$, and in view of the attainable event
rates for the planned data acquisition system (see section~\ref{subsubdaq})
in the order of $10\,\mbox{kHz}$ a high precision measurement of the
$\eta^\prime \rightarrow \gamma \gamma$ branching ratio seems feasible
after all, but will require further improvements of the selectivity of
the forward detector trigger.
Developments have already started for WASA at CELSIUS to implement a
missing mass trigger for the $\Delta \mbox{E}/\mbox{E}$ technique by
implementing energy reconstruction and angular correction of the energy
loss information on the trigger level, and direct branching ratio
measurements should become feasible after some experience and development
with WASA at COSY.
\paragraph{Double vector meson dominance in $\eta \rightarrow
e^+ e^- e^+ e^-$.}
Experimentally, the $\eta$ will be tagged by using the missing mass of the
two protons detected in the forward detector.
Tracks of electrons and positrons are measured in the Mini Drift Chamber
(MDC) inside the super-conducting solenoid, and their energy is determined
by means of the CsI calorimeter.
Background from $2\,\pi^0$ production via $\pi^0 \pi^0 \rightarrow
e^+ e^- e^+ e^- \gamma \gamma$ with two undetected $\gamma$s can be
effectively removed by kinematic cuts.
The $\eta$ decay modes to $e^+ e^- \gamma$ ($\Gamma(\eta \rightarrow
e^+ e^- \gamma)/\Gamma_{tot} = 6.0 \pm 0.8 \cdot 10^{-3}$~\cite{PDBook})
and to $2\,\gamma$ ($\Gamma(\eta \rightarrow \gamma \gamma)/\Gamma_{tot} =
39.43 \pm 0.26\,\%$~\cite{PDBook}) contribute via photon conversion in the
beam tube.
Both channels are effectively reduced to a level below $1\,\%$ by
reconstructing the vertices of both $e^+ e^-$ pairs~\cite{Bosch:1996wr}.
Based on the QED prediction for the branching ratio of
$2.52 \cdot 10^{-5}$~\cite{Jarlskog:1967np} and the acceptance of the WASA
setup, at a luminosity of $10^{32}\,\mbox{cm}^{-2}\mbox{s}^{-1}$ we expect a
rate of about $3 \cdot 10^3$ events per week.
In a 3-months run one will collect a data sample of 30000 events which
will allow for a high statistics measurement of the form factor dependence
on the virtual photon masses. In a second phase the data for the decay
$\eta\rightarrow e^+ e^- e^+ e^-$ can be collected together with the
$\eta\rightarrow e^+ e^-$ experiments.
\paragraph{Search for CP violation in the rare decay $\eta \rightarrow
\pi^+ \pi^- e^+ e^-$.}
$\eta$ tagging as well as electron and positron detection will be performed
as described in Section~\ref{subsubetapday1}.
Electrons are separated from pions using the momentum over energy ratio
with an accuracy of $5 \cdot 10^{-3}$.
The fraction of misidentified charged leptons has been estimated to be
less than $2.5 \cdot 10^{-5}$.
From the complete four--momentum information for all four charged decay
products, the angle between the $\pi^+ \pi^-$ and $e^+ e^-$ production
planes can be reconstructed.
The major source of background arises from the conversion of photons from
the $\eta \rightarrow \pi^+ \pi^- \gamma$ decay in the beam tube.
With an overall conversion probability of $0.28\,\%$ the background is
important only in the region of small $e^+ e^-$ invariant masses below
$40\,\mbox{MeV/c$^2$}$.
In comparison, background from other sources like direct production is
negligible.
The experiment can be carried out together with studies of the
$\eta\rightarrow e^+ e^- e^+ e^-$ decay channel (see paragraph
above) and statistics of nearly $4\cdot 10^5$
$\eta\rightarrow \pi^+ \pi^- e^+ e^-$ events could be obtained,
corresponding to an accuracy of $2\cdot 10^{-3}$ for the
asymmetry.
\paragraph{Search for physics beyond the Standard Model in the very rare $\eta
\rightarrow e^+ e^-$ decay.}
The experiment aims at reaching a sensitivity that allows to test whether
the $\eta$ decay to the $e^+ e^-$ mode occurs at a rate compatible with the
Standard Model prediction of
$5 \cdot 10^{-9}$~\cite{Savage:1992ac,Ametller:1993we}, which makes the
decay susceptible to contributions from physics beyond the Standard Model, with a
present experimental upper limit at $7.7 \cdot 10^{-5}$~\cite{PDBook}.
The experimental signature will consist of four charged tracks, with two
protons in the reaction $p p \rightarrow p p \eta \rightarrow p p e^+ e^-$
tagged in the forward detector and the $e^+ e^-$, with an isotropic
distribution in the $\eta$ rest frame measured in the central detector with
magnetic field.
In view of the low branching ratio the experiment requires high acceptance
and high luminosity, charged particle tracking, and the possibility to
reject events with photons by means of an electromagnetic calorimeter,
which makes the WASA facility at COSY a unique device for this rare decay
channel.
The most important physics processes that can contribute to a possible
background are the $\eta \rightarrow \gamma e^+ e^-$ Dalitz decay, the
radiative decay mode $\eta \rightarrow \gamma \gamma$, and prompt
$e^+ e^-$ production.
The continuous spectrum of $e^+ e^-$ pairs up to the $\eta$ mass from the
$\gamma e^+ e^-$ decay mode ($\Gamma(\eta \rightarrow \gamma e^+ e^-) /
\Gamma_{tot} = 6.0 \pm 0.8 \cdot 10^{-3}$~\cite{PDBook}) is effectively
reduced to a level of less than $10\,\%$ compared to the $\eta \rightarrow
e^+ e^-$ signal by rejecting photons in the central detector
(Fig.~\ref{fig_eta_ee_bg}).
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.6\textwidth]{figures/conv_dal_eta.eps}
\caption{\label{fig_eta_ee_bg}
Background contribution from $\eta \rightarrow \gamma e^+ e^-$
(black), $\eta \rightarrow \gamma \gamma$ (dark grey) and sum of the
$\eta \rightarrow e^+ e^-$ signal and the two background sources (light
grey). The simulation assumes the Standard Model prediction for the
branching ratio of $5 \cdot 10^{-9}$ for $\eta \rightarrow e^+ e^-$ and
the PDG values for the background channels, scaled to a total of
$10^{10}$ $\eta$ events produced.}
\end{center}
\end{figure}
$\gamma$ conversion in the material between the interaction point and the
plastic barrel (PSB) leads to a background contribution from the $\eta
\rightarrow \gamma \gamma$ mode ($\Gamma(\eta \rightarrow \gamma \gamma) /
\Gamma_{tot} = 39.43 \pm 0.26\,\%$~\cite{PDBook}), if both $\gamma$s
convert asymmetrically in energy.
Most of the relevant material is concentrated by the $1.2\,\mbox{mm}$
Be beam--pipe, giving rise to a signal--to--background ratio\footnote{The
background is expected to be reduced further by vertex reconstruction,
which is not implemented in the simulations.} of 1--2 to 1 in the $\eta$
mass range (Fig.~\ref{fig_eta_ee_bg}).
A statistically significant signal from the rare $\eta \rightarrow e^+ e^-$
decay on the level of the Standard Model prediction, as indicated in
figure~\ref{fig_eta_ee_bg}, could be obtained in a running time of 2 months.
It should be noted, that an appropriate trigger setting will allow to
run the experiment in parallel with other $\eta$ decay studies, and, that
already after one hour of beamtime, the experiment can increase the
level of sensitivity in comparison to the present upper limit by an order
of magnitude.
Furthermore, it should be emphasized, that these investigations are only
feasible due to the optimization of the WASA setup with respect to the
amount of material between the interaction point and the central detector
(see section~\ref{subwasa}), to fully exploit the advantages of a
windowless internal target.
However, the theoretical and experimental uncertainties for the
background from prompt $e^+ e^-$ production in the $\eta$ mass range are large.
Calculations~\cite{Stepaniak:1998wm} based on the approach described
in~\cite{Titov:1994vg} lead to a signal--to--background ratio for the decay
$\eta \rightarrow e^+ e^-$ of 1 to 30 with respect to prompt production.
Thus, in order to have a reliable estimate for the prompt production
background, it is suggested to prepare the $\eta \rightarrow e^+ e^-$
search by measuring the prompt contribution $pp \rightarrow p p e^+ e^-$
first.
At the same time, since WASA allows for exclusive measurements, this
preparatory study can shed light on both the relative strength of
different sources of $e^+ e^-$ pairs, and meson and baryon isobar form
factors.
The first aspect is intimately motivated by the expectation that dileptons
carry information from the early stage of heavy ion collisions.
Discrepancies between theoretical calculations (see
e.g.~\cite{Zetenyi:2001fu,Shekhter:2003xd}) and the presently available
data for dilepton production in proton--proton and proton--deuteron
collisions measured with rather small geometrical acceptance at the DLS
spectrometer~\cite{Wilson:1997sr} --- especially in the mass range between
0.2 and $0.5\,\mbox{GeV/c$^2$}$ --- could be verified experimentally using WASA
at COSY.
Assuming a luminosity of $10^{32}\,\mbox{cm}^{-2}\mbox{s}^{-1}$ reasonable
statistics of about $10^4$ events can be achieved in five days of data
taking, i.e.\ a program of four kinetic beam energies ranging from
1.04 to $2.5\,\mbox{GeV}$ to overlap both with the DLS data and theoretical
calculations is feasible within three weeks, and will provide the
necessary information for a reliable background estimate from prompt
production concerning the search for the rare $\eta \rightarrow e^+ e^-$
decay.
\paragraph{C violation in $\eta \rightarrow \pi^0 e^+ e^-$.}
The experiment aims at a sensitivity down to the Standard Model prediction
for the branching ratio of
$0.2 - 1.3 \cdot 10^{-8}$~\cite{Cheng:1967pr,Ng:1993sc,Jarlskog:2002zz},
to search for new C--violating effects in the electromagnetic interaction.
With a value of $\Gamma(\eta \rightarrow \pi^0 e^+ e^-) / \Gamma_{tot} <
4 \cdot 10^{-5}$~\cite{PDBook} the present upper limit is three orders of
magnitude weaker than the theoretical estimate.
Monte Carlo simulations have been carried out to investigate
the feasibility of the experiment and background conditions.
In conclusion, a search sensitivity of $10^{-9}$ is possible considering
the full background.
A sample of ten events from C--conserving mechanisms for the rare decay
can be expected after a period of three weeks to five months, depending on
the actual value of the branching ratio.
A C--violating admixture at the level of $10^{-3}$ would already double
the $\eta \rightarrow \pi^0 e^+ e^-$ decay rate.
However, at present this effect is small compared to the uncertainty in
theoretical estimates for the C conserving decay mechanism, which is due to
insufficient knowledge of value and structure of the $\eta \rightarrow
\pi^0 \gamma \gamma$ amplitude.
Consequently, an improved precision in the $\eta \rightarrow
\pi^0 \gamma \gamma$ decay rate is important for the interpretation of the
experimental result of the proposed study of the $\pi^0 e^+ e^-$ decay
mode, and a high statistics study of $\eta \rightarrow \pi^0 \gamma \gamma$
should be carried out simultaneously.
\paragraph{Measurement of the $\eta^{\prime}$ transition form factor.}
With $\eta^\prime$ tagging by means of two detected protons in the forward
detector and the reconstruction of the decay system of the
Dalitz conversion decay $\eta^\prime \rightarrow \gamma \gamma^*
\rightarrow \gamma e^+ e^-$ in the central detector, the WASA setup is well
suited for measuring the $\eta^\prime$ transition form factor.
With the tracking information from the Mini Drift Chamber, the momenta
of electron and positron are determined.
Thus, the differential cross section
\begin{equation}
\frac{d\,\sigma}{d\,\mbox{q}^2} =
\left[ \frac{d\,\sigma}{d\,\mbox{q}^2} \right]_{pointlike} \,
\left[ \mbox{F}(\mbox{q}^2)\right]^2
\end{equation}
with respect to the four--momentum transfer squared $\mbox{q}^2$ of the
time--like virtual photon, which is equal to the invariant mass squared
of the $e^+ e^-$ pair, can be measured, and the transition form factor
$\mbox{F}(\mbox{q}^2)$ can be extracted.
The form factor is usually fitted with the simplest pole--type formula
\begin{equation}
\label{eq_dalitz_slope}
\mbox{F}(\mbox{q}^2) =
\frac{1}{1-\frac{\mbox{\footnotesize q}^2}{\Lambda^2}} \approx
1 + \frac{\mbox{q}^2}{\Lambda^2}
\end{equation}
with the slope parameter $\Lambda$.
At present, only an upper limit exists for the $\gamma e^+ e^-$ decay mode
of the $\eta^\prime$ ($\Gamma(\eta^\prime \rightarrow \gamma e^+ e^-) /
\Gamma_{tot} < 9 \cdot 10^{-4}$).
With an estimated branching ratio of $3 \cdot 10^{-4}$, as discussed
in~\cite{Briere:1999bp}, we expect 45 fully reconstructed $\eta^\prime$
Dalitz conversion decays per day.
Following the discussion for the $\eta^\prime \rightarrow
\gamma \pi^+ \pi^-$ decay in section~\ref{subsubetapday1} we only
consider the radiative decay $\eta^\prime \rightarrow \gamma \gamma$ with
subsequent conversion of one $\gamma$ as a possibly important source of
background.
With fits to the $\eta^\prime$ signal in each bin of the $e^+ e^-$
invariant mass scale, the differential cross section can be extracted
background free, unless there is a background process for the $eta^\prime
\rightarrow \gamma e^+ e^-$ transition.
However, since for $\eta$ decays the $\gamma \gamma$ mode was found to
be negligible above very low invariant lepton pair
masses~\cite{Stepaniak:2002ad}, and since the ratios
$\Gamma(\eta(\eta^\prime) \rightarrow \gamma e^+ e^-) /
\Gamma(\eta(\eta^\prime) \rightarrow \gamma \gamma)$ are approximately the
same for $\eta$ and $\eta^\prime$, and since the photon conversion
probability is not changing drastically, it can safely be neglected as a
background contribution.
The beam time estimate depends on the desired accuracy for the slope
parameter $\Lambda$ in eq.~\ref{eq_dalitz_slope}.
From the analogous discussion of the $\eta$ transition form factor
in~\cite{Stepaniak:2002ad}, a running time of two months would allow a
determination of the slope parameter with a $10\,\%$ error.
A further improvement by a factor of two in accuracy would require a nine
months period of data taking.
It should be noted, that due to the trigger condition of two charged and
one neutral particle in the central detector data taking could be done
in parallel with other $\eta^\prime$ decay studies.
For example, the specific trigger requirement for the $\eta^\prime$
Dalitz conversion decay exactly matches the trigger setup for the
$\pi^+ \pi^- \gamma$ mode discussed in section~\ref{subsubetapday1} in
terms of particle multiplicities.
\clearpage
\newcommand{\emparag}[1]{\paragraph{#1}~\\}
\section{Experimental facility}
\subsection{Cooler Synchrotron COSY }
\label{subcosy}
\begin{figure}[b]
\begin{center}
\includegraphics[clip,width=0.7\textwidth]{figures/cosy.eps}
\caption{Floorplan of the accelerator complex. The WASA detector
will be installed at the place of the old cavity in
front of the electron cooler.}
\label{fig:cosy}
\end{center}
\end{figure}
COSY is a cooler synchrotron and storage ring operated at the IKP of FZJ.
The accelerator complex (see Fig.~\ref{fig:cosy}) comprises an isochronous
cyclotron (JULIC), used as an injector, a race track shaped cooler
synchrotron with a circumference of 184~m, and
internal and external target stations~\cite{RMaier97}. COSY delivers
beams of polarized and unpolarized protons and deuterons in the momentum
range between 0.3~GeV/c and 3.7~GeV/c. The ring can be filled with up
to $10^{11}$~particles leading to typical luminosities of
$10^{31}\,\mathrm{cm^{-2}s^{-1}}$ when using an internal cluster target.
Beams can be phase-space cooled by
means of electron cooling at injection energy as well as stochastic cooling
at high energies. Typical beam preparation times, including injection,
accumulation and acceleration, are of the order of a few
seconds, while the beam lifetime with a cluster target is between several
minutes and an hour.
Currently, four internal experiments (ANKE, COSY-11, EDDA, PISA) and three
external detector systems (BIG KARL, JESSICA, TOF) are operated by large
international collaborations.
On the average COSY is running for more than 7000~hours per year.
Typically, it delivers
beams for experiments with a reliability of 94\%.
\emparag{Implementation of WASA at COSY}
The Cooler Synchrotron has two 40~m long straight sections joining
the arcs. After removal of the old RF cavity at the beginning of
2004 space for the implementation of WASA at an internal target
position in front of the electron cooler is available (see
Fig.~\ref{fig:cosy}). This location has a number of advantages: most
importantly it minimizes both the interference with the existing COSY detectors
and the modifications needed to the WASA detector and the accelerator.
Operating WASA with the pellet target with an effective thickness of
$2.5\cdot 10^{15}\,\mathrm{cm^{-2}}$ average luminosities of
$10^{32}\,\mathrm{cm^{-2}s^{-1}}$ and beam lifetimes of a couple of
minutes are expected~\cite{lehrach03}. Compared to the present
operation at CELSIUS the experimental conditions will be improved
due to the different accelerator characteristics: fast magnet ramping,
dispersion-free target position, stochastic cooling and
a smooth microscopic time structure of the beam.
\subsection{The WASA detector}
\label{sec:setup}
\label{subwasa}
The 4$\pi$ detector facility WASA \cite{CWwww,Zabi02} was designed for
studies of
production and decays of light mesons at CELSIUS.
Pions and eta mesons are produced in proton-proton and proton-deuteron
interactions. The highest beam-proton kinetic energy reachable at CELSIUS is 1.5 GeV.
The pellet-target system is integrated in the setup and it provides small spheres of
frozen hydrogen and deuterium as internal targets. This allows high luminosity and
high detection coverage for meson decay products like photons, electrons
and charged pions.
WASA consists of a forward part for measurements of charged target-recoil particles
and scattered projectiles, and a central part designed for measurements of
the meson decay products. The forward part consists of eleven
planes of plastic scintillators and of proportional counter drift tubes.
The central part consists of an electromagnetic calorimeter of CsI(Na) crystals
surrounding a superconducting solenoid.
Inside of the solenoid a cylindrical chamber of drift tubes and a barrel
of plastic scintillators are placed. A vertical cross section of the WASA detector is shown in
Fig.~\ref{fig:wasa}.
\begin{figure}[!htb]
\includegraphics[width=\textwidth,clip]{figures/wasa_setup2.ps}
\caption{\label{fig:wasa} Cross section of the WASA detector. The
central detector built around the interaction point (at the left) is
surrounded by an iron yoke. The layers of the forward detector are
visible on the right-hand side. The
individual components are described in the text.}
\end{figure}
\subsubsection{Pellet target}
\label{sec:pellet}
The pellet target system was a unique development for the
CELSIUS/WASA experiment \cite{Tro95,Eks96}.
The main components of the system are shown in Fig.~\ref{fig:pellet1}.
\begin{figure}[!htbp]
\centering
\includegraphics[width=\textwidth]{figures/pellet1.eps}
\caption{\label{fig:pellet1}Layout of the pellet target system.}
\end{figure}
The heart of the setup is the pellet generator where a jet of liquid hydrogen
is broken up into droplets with a diameter around 35 $\mu$m
by a vibrating nozzle. The droplets freeze by evaporation in a droplet
chamber and form a beam of pellets that enter a 7\ cm long
vacuum-injection capillary. After
collimation, the pellets are directed through a thin 2~m long pipe into
the scattering chamber and further down to a pellet beam dump.
The inner diameter of the pipe is 5\ mm at its entrance
to the scattering chamber. This arrangement provides the necessary space
to put the 4$\pi$ detection system around the interaction region.
Pellet target thicknesses of up to $3\cdot 10^{15}$ atoms/cm$^2$ give
acceptable half-lives of the circulating ion beam as well as
acceptable vacuum conditions. Some of the parameters of the pellet
target at the present stage of operation are listed in table \ref{tab:pellet}.
The pellet target system operates regularly with pellets of normal hydrogen
and
deuterium.
\begin{table}[!htbp]
\centering
\small
\begin{tabular}{|l|c|}
\hline
{\bf Pellet target parameters} & {\bf Present performance}\\
\hline
Pellet diameter & 25 - 35 $\mu$m \\
Pellet frequency & 5-12 kHz \\
Pellet-pellet distance & 9-20 mm \\
Effective target thickness & $> 10^{15}$ atoms/cm$^2$\\
Beam diameter & $2-4$ mm \\
\hline
\end{tabular}
\normalsize
\caption{\label{tab:pellet} Present performance of the
pellet target system.}
\end{table}
\subsubsection{Forward detector}
The forward detector (FD) is designed mainly
for detection and identification of scattered projectiles and charged recoil particles
like protons, deuterons and He nuclei in $\pi$ and $\eta$ production reactions.
Also neutrons and charged pions can be measured.
All FD plastic scintillators may supply information for the first level trigger logic.
This part of the setup has already been used in a previous experiment at CELSIUS,
WASA/PROMICE, which is described more in detail in \cite{Calen96}.
A summary of the most important features of the forward detector (FD) is given in
table \ref{tab:FD}. The individual components are described in some
detail in the following.
\begin{table}[!h]
\centering
\begin{tabular}{|l|c|}
\hline
\multicolumn{2}{|l|}{\large {\bf Forward detector}} \\
\hline
Total number of scintillator elements & 280\\
Scattering angle coverage & $3^{\circ}$ - $17^{\circ}$ \\
Scattering angle resolution & $0.2^{\circ}$\\
Amount of sensitive material [g/$cm^{2}$] & 50\\
~~~~[radiation lengths] & $\approx$ 1 \\
~~~~[nuclear interaction lengths] & $\approx$ 0.6 \\
Thickness of vacuum window (st. steel) [mm] & $\approx$ 0.4\\
Maximum kinetic energy ($T_{stop}$) for stopping: & \\
~~~~$\pi^{\pm}$/proton/deuteron/alpha\qquad [MeV] & 170/300/400/900\\
Time resolution & $<$ 3 ns \\
Energy resolution for: & \\
~~~~ stopped particles & $\approx$ $3\%$\\
~~~~ particles with $T_{stop}$ $<T<$ 2$T_{stop}$ & 4 - $8\%$\\
Particle identification & $\Delta$E-E\\
\hline
\end{tabular}
\normalsize
\caption{\label{tab:FD} Some features of the Forward Detector.}
\end{table}
\emparag{The Forward Window Counters (FWC)}
\noindent The FWC is the first detector layer in the FD
(along the beam direction) and consists of 12, 5 mm thick plastic scintillators
(Fig.~\ref{fig:fwc}).
It is mounted tightly on the paraboloidal stainless steel vacuum window.
Therefore, the elements
are inclined by approximately
10$^{\circ}$ with respect to the plane perpendicular to the beam direction.
The FWC signals are used in the first level trigger logic to reduce the
background caused by particles scattered in the downstream beam pipe and in the flange
at the entrance to the FD. The signals are also used to select He ejectiles
on the trigger level.
\begin{figure}[ht]
\hspace{3cm}\includegraphics[width=0.5\textwidth,clip]{figures/inken_fig1_7_cable_fwc_p.eps}
\caption{\label{fig:fwc} Schematic view of the FWC.}
\end{figure}
\emparag{The Forward Proportional Chambers (FPC)}
\noindent Immediately downstream of the FWC, there is a tracking device.
It is composed of
4 modules, each with 4 staggered layers of 122 proportional drift
tubes (so called straws) of 8 mm diameter
(Fig.~\ref{fig:fpc}).
The modules are rotated by 45$^o$ with
respect to each other (in the plane perpendicular
to the beam axis). They are used for accurate reconstruction
of track coordinates and provide precise
angular information of the particles originating from the target region.
\begin{figure}[ht]
\hspace{3cm}\includegraphics[width=0.5\textwidth,clip]{figures/inken_fig1_8_tracker2.ps}
\caption{\label{fig:fpc} Schematic view of one module of the FPC.}
\end{figure}
\emparag{The Forward Trigger Hodoscope (FTH)}
\noindent The FTH, consisting
of three layers of 5mm thick plastic scintillators, is placed
next to the FPC. There are 24 Archimedian spiral shaped elements in the
first two planes and 48 radial elements in the third.
The FTH is mainly used in the first level trigger logic. However, the
special geometry, combining all three layers, results in a pixel
structure, which is useful for resolving multi-hit ambiguities. In
Fig.~\ref{fig:fth}
the structure of the FTH is shown with hits from two passing
charged particles.
\begin{figure}[ht]
\vspace{-2cm}
\includegraphics[width=0.5\textwidth,origin=b,angle=55.5,clip]{figures/fth_3d.eps}%
\includegraphics[width=0.3\textwidth,origin=t,clip]{figures/fth_2d.eps}
\vspace{-2cm}
\caption{\label{fig:fth} Schematic view of the three forward trigger
hodoscope layers.}
\end{figure}
\newpage
\emparag{The Forward Range Hodoscope (FRH)}
\noindent Behind the FTH, the four layers of the FRH are positioned (Fig.~\ref{fig:frh}). Each plane is made
of 24, 11 cm thick, plastic scintillator modules. The FRH, together with FTH,
is used for energy determination of charged particles and for particle
identification by $\Delta$E-E technique. Fig.~\ref{fig:FDdep} shows how
protons and deuterons can be identified.
\begin{figure}[ht]
\hspace{3cm}\includegraphics[width=0.5\textwidth,clip]{figures/frh.eps}
\caption{\label{fig:frh} Schematic view of the four forward range
hodoscope layers. Dimensions are given in mm.}
\end{figure}
\begin{figure}[!hbt]
\centering
\includegraphics[width=0.7\textwidth,angle=-90,clip]{figures/deeFD.ps}
\caption{\label{fig:FDdep} Example of particle identification in the
forward detector for data collected at 640~MeV. Energy deposited
in the 1$^{st}$ layer of FRH is plotted versus total
energy deposited in FTH and FRH. The lines indicate cuts to be used for selecting
protons (lower) and deuterons (upper).
}
\end{figure}
The identity and initial kinetic energy of a charged particle is reconstructed from the
pattern of deposited energy in the different detector planes.
Fig.~\ref{fig:fdedep} shows how the initial kinetic energy of a proton is
related to the deposited energy in the forward detector scintillators.
Even for particles that are stopped in a detector,
the total deposited energy is different from the initial kinetic energy.
This is because some of the energy is lost in inactive material between
the detectors elements, in the scattering chamber windows, etc.
Fig.~\ref{fig:fdedep} also shows that the variation of the deposited
energy is strong enough to be useful for energy reconstruction also for
high energy particles not stopping in the detector material. For protons,
this can be used in the kinetic energy range 300~MeV to 800~MeV and for
deuterons in a similar energy range starting from 400~MeV. Identification
of punch-through particles can be done using either the veto hodoscope (FVH)
or the $\Delta$E information from the last FRH planes.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.6\textwidth,clip]{figures/stina_ahfig4_2.eps}
\caption{\label{fig:fdedep} Energy absorbed in the forward detector
scintillator material for protons as a function of their initial
kinetic energy. The upper curve gives the total energy
deposited in the plastic scintillator planes and the lower curves show
the energy distributed among the different FRH planes.}
\end{center}
\end{figure}
\emparag{The Forward Range Interleaving Hodoscope (FRI)}
\noindent Between the third and fourth layers of the FRH there are two interleaving
layers of 5.2 mm thick plastic scintillator bars (Fig.~\ref{fig:fri}).
Each layer has 32 bars, oriented horizontally in one and vertically in the other.
The main purpose of this addition to the FRH is to provide a two-dimensional position
sensitivity inside the FRH
necessary for measurement of scattering angles for neutrons. The efficiency
for detection of neutrons of a few hundred MeV kinetic energy by the FRH is around 35 \%.
In addition, the FRI can help in vertex reconstruction and to discriminate against background tracks
due to secondary interactions in the beam pipe and other structural material.
The FRI was recently commissioned; more information
about the design and performance of the FRI can be found in \cite{Pauly04}.
\begin{figure}[ht]
\hspace{3cm}\includegraphics[width=0.5\textwidth,clip]{figures/inken_fig1_11_fridesign.eps}
\caption{\label{fig:fri} Schematic view of the FRI detector (upper picture) and its
two planes with orthogonally oriented scintillator bars (lower
picture).}
\end{figure}
\emparag{The Forward Veto Hodoscope (FVH)}
\noindent The last detector layer of the FD is a wall of plastic
scintillators (Fig.~\ref{fig:fvh}).
It consists of 12 horizontal plastic scintillator bars,
equipped with photomultipliers on both sides. The hit position along a bar may
be reconstructed from signal time information.
In the first level trigger the signals are used for rejection
(or selection) of particles punching through the FRH.
\begin{figure}[ht]
\hspace{3cm}\includegraphics[width=0.5\textwidth,clip]{figures/inken_fig1_12_cable_fvh_p.ps}
\caption{\label{fig:fvh} Schematic view of the Forward Veto Hodoscope.}
\end{figure}
\emparag{The Forward Absorber (FRA)}
\noindent
Optionally, a passive absorber layer made of iron can be introduced between the last layer of the FRH
and the FVH. The thickness of the absorber can be chosen from 5~mm up to
100~mm.
The absorber has been used for stopping the protons from the
$pp\rightarrow pp\eta$ reaction at a beam proton energy around 1360 MeV.
In this case the faster protons from elastic scattering and from pion production penetrate
the FRA and induce signals in the FVH which can be used for veto purposes in the trigger.
\subsubsection{Central detector}
The central detector (CD) is built around the interaction point and
is designed mainly
for detection and identification of the decay products of $\pi^\circ$ and
$\eta$ mesons: photons,
electrons and charged pions. It consists
of an inner drift chamber (MDC), a solenoid (SCS) providing magnetic field for
momentum measurements, thin plastic scintillators in a cylinder
geometry (PSB) and a CsI calorimeter (SEC).
The amount of structural material is kept minimal to reduce the
disturbances on the particles.
The beam pipe is made of 1.2~mm thick
beryllium and the total thickness of the solenoid corresponds to 0.18 radiation
lengths only.
For the design the main requirements were the
following:
\begin{itemize}
\item it has to handle high particle fluxes at luminosities around ${\rm
10^{32}\,cm^{-2}s^{-1}}$.
\item it should be able to measure photons with energies from a few MeV
up to 800~MeV.
\item it should be able to measure, in a magnetic field of about 1 T,
the momenta of electrons and positrons in the range
${\rm p \approx 20-600~MeV/c}$ with an accuracy ${\rm \sigma_p/p < 2\%}$.
\end{itemize}
The momenta of heavier charged particles can also be measured in a
similar momentum
ranges, but with lower accuracy:\\
for pions and muons with ${\rm p \approx 100-600~MeV/c}$, ${\rm \sigma_p/p < 4\%}$ \\
for protons with ${\rm p \approx 200-800~MeV/c}$, ${\rm \sigma_p/p < 6\%}$ \\
The main components of the Central Detector, shown in Fig.~\ref{fig:wasa},
are presented below in some detail.
\emparag{The Superconducting Solenoid - (SCS)}
\noindent The SCS provides an axial magnetic field for the momentum
measurements
for the tracks measured by the MDC. It also protects the CD from low-energy
delta electrons produced in the interactions of beam particles with the pellets.
The wall thickness of the SCS is minimized in order to allow high
accuracy of the energy measurements in the calorimeter.
The return path for the magnetic flux is provided by a yoke
made out of 5 tons of
soft iron with low carbon content. The yoke shields the
readout electronics from the magnetic field and serves also as support for the calorimeter
crystals. The main parameters of the SCS are given in table
\ref{tab:scs}.
In order to map the magnetic field inside the volume enclosed by the
SCS, the magnetic field strength inside the MDC was measured with Hall
probes and, in addition, the field distribution was calculated with simulation
programs. The calculated values were fitted to the measured
ones with an error of $\pm$1\% of B$_{total}$ \cite{Ruber01}.
The magnetic flux density distribution inside of the iron yoke
calculated for a current of 667~A is given in Fig.~\ref{fig:sol}.
The SCS is described in detail in the Ph.D. thesis of
Roger Ruber \cite{Ruber99}.
\begin{table}[!h]
\centering
\begin{tabular}{|l|c|}
\hline
\multicolumn{2}{|l|}{\large {\bf Superconducting coil}} \\
\hline
Inner/outer radius [mm] & 267.8 / 288.8\\
Superconductor (stabilizer) & NbTi/Cu (pure Al)\\
Total winding length & 465 mm\\
Maximum central magnetic flux density, {\bf B$_{c}$} & 1.3 T\\
Field uniformity in the MDC & 1.22 T $\pm$$20\%$\\
Cooling & Liquid He, $4.5^{\circ}$K\\
\hline
\multicolumn{2}{|l|}{\large {\bf Cryostat}}\\
\hline
Material & Aluminium\\
Inner / outer radius [mm] & 245 / 325\\
Overall length [mm] & 555\\
\hline
{\bf SCS wall thickness} (coil+cryostat) [radl] & {\bf 0.18}\\
\hline
\end{tabular}
\caption{\label{tab:scs} Main parameters of the superconducting
coil and its cryostat.}
\end{table}
\begin{figure}[!h]
\includegraphics[width=\textwidth,clip]{figures/mfield.ps}
\caption{\label{fig:sol}Calculated distribution of the magnetic flux
density for a coil current of 667~A \cite{Ruber01}. Contour maxima are indicated by lines
marked A-H, where: A=0.10T, B=0.25T, C=0.50T,
D=0.75T, E=1.00T, F=1.20T, G=1.30T, H=1.50T. }
\end{figure}
\emparag{The Mini Drift Chamber - (MDC)}
\noindent The MDC is placed around the beam pipe and is used for
determination of particle
momenta and reaction vertex. It is a cylindrical chamber covering
scattering angles from 24$^o$ to
159$^o$. For large angle scattered protons from elastic proton-proton
scattering, a vertex
resolution ($\sigma$) of 0.2~mm perpendicular and 3~mm along the beam
axis can be reached.
A detailed description of the MDC
can be found in the Ph.D. thesis of Marek Jacewicz~\cite{Jacewicz04}.
The MDC consists of 1738 drift tubes, so called straws, arranged in 17 cylindrical layers.
The diameter of the straws in the 5 inner most layers is 4~mm, 6~mm in the
next 6 layers and 8~mm in the outer 6 layers.
The straws are made of a thin (25~$\mu$m) mylar foil coated with
0.1~$\mu$m aluminum on the inner side only.
In the center of each straw there is a 20~$\mu m$ diameter sensing wire
made of gold plated tungsten (W(Re)), stretched
with a tension of 40~g.
The wires are aligned with a precision of $\pm 20\;\mu$m.
This design was chosen in order to cope with the expected high
particle flux allowing a maximum deposited energy of approximately 70
$\frac{MeV}{mm\cdot s}$ for the straws exposed most at the inner part of the chamber.
The layers are located between radii of 41 and
203~mm. The straws in nine
layers are parallel to the beam axis (z-axis) and the other eight layers have small
skew angles (6$^{o}$-9$^{o}$) with respect to the z-axis. These ``stereo'' layers form a
hyperboloidal shape.
The straws in the five inner layers are divided unequally by
the center of the pellet pipe, while the other layers are
symmetrical. The straws in each layer are inter-spaced by small gaps in order to
prevent the mechanical deformation by neighboring tubes.
The MDC is fitted inside a cylindrical cover made of 1~mm Al-Be and
is placed inside
the solenoid (Fig.~\ref{fig:mdccyl}).
\begin{figure}[!h]
\includegraphics[width=0.425\textwidth,]{figures/mdc_all1.ps}%
\includegraphics[width=0.575\textwidth,]{figures/assem2bw.ps}
\caption{\label{fig:mdccyl} (Left) The fully assembled MDC inside
the Al-Be cylinder. (Right) The MDC surrounded by PSB elements.}
\end{figure}
\begin{figure}[!hbt]
\centering
\includegraphics[width=0.6\textwidth]{figures/MDC.40.ps}%
\includegraphics[width=0.4\textwidth]{figures/MDC.41.ps}
\caption{\label{fig:mdcpipe} (Left) Drift tubes secured in the
end-plates. Note the ``stereo'' layers interleaved with parallel
layers. (Right) Be beam-pipe.}
\end{figure}
\begin{figure}
\includegraphics[width=\textwidth]{figures/MDC_schematics1.ps}
\caption{\label{fig:mdcschem1} Beryllium beam-pipe including the crossing
pellet target tube.}
\end{figure}
The straws in each (half) layer are mounted between $\approx$5~mm thick
Al-Be end-plates. The layers are assembled around the Be beam-pipe
(Fig.~\ref{fig:mdcpipe}) and the attached pipe for the pellets.
The beam-pipe has a diameter of 60~mm and a wall thickness of 1.2~mm.
The design drawing of the Be beam pipe is shown in Fig.~\ref{fig:mdcschem1}.
\newpage
\emparag{The Plastic Scintillator Barrel - (PSB)}
\noindent The PSB is located inside the SCS coil and surrounds the MDC.
It provides fast signals for the first level trigger logic and,
together with the mini drift chamber and the CsI calorimeter, it is
employed for charged particle identification by the $\Delta$E-p
and $\Delta$E-E methods and serves as a veto for $\gamma$ identification.
The performance of the PSB has been studied using proton-proton elastic scattering events.
Fig.~\ref{fig:pscal} (left plot) shows the result of a Monte Carlo simulation
of the angular dependence on the energy deposited in
the PSB. Fig.~\ref{fig:pscal} (right plots) shows typical experimental spectra
after a correction for nonuniform signal response has been applied.
\begin{figure}[!hbt]
\centering
\framebox{
\includegraphics[width=0.25\textwidth,angle=-90,clip]{figures/ps_mc_cal2.ps}
}
\framebox{
\includegraphics[width=0.4\textwidth,angle=-90,clip]{figures/ps_data_after_cal.ps}
}
\caption{\label{fig:pscal} (Left)
Angular dependence of the deposited energy in
the PSB for simulated elastically scattered protons.
The energy deposition increases with
increasing polar angle (corresponding to a decrease of the kinetic
energy of the proton), until particles begin to stop in the plastic
scintillator material (at around $\theta$=77$^o$).
(Right) Experimental spectra corrected for light attenuation
for four of the PSB central elements.}
\end{figure}
In the initial experiments the momentum and energy resolution allowed
reasonable discrimination
between pions and protons, which is illustrated in Fig.~\ref{fig:CDdep}.
For high energy charged particles also SEC information is available and
can be used for the identification.
\begin{figure}[!hbt]
\centering
\includegraphics[width=0.4\textwidth,angle=-90,clip]{figures/de-p_d325.ps}
\caption{\label{fig:CDdep} Example of particle identification in
the central detector for a raw data sample collected at 1360
MeV. Energy deposited in the PSB is given as a
function of signed momentum from the MDC. The regions for
protons and pions are marked.
}
\end{figure}
The PSB consists of a cylindrical part and two end caps and contains in
total 146 elements shaped as strips of 8mm thickness. In the cylindrical part there are
48(+2) elements of 550~mm length and 38~mm width, forming 2 layers
with a small (on average
6~mm) overlap between neighboring elements to avoid that particles
pass without registration.
The end caps with an outer diameter of approximately 42~cm in the backward and
51~cm in the forward part contain 48
``cake-piece'' shaped elements each. The front end cap is flat while
the rear cap forms a conical surface. Both end caps have a central hole for
the beam pipe (19~cm diameter in the forward and 12~cm diameter in the
backward part).
The PSB as modeled in the detector
simulation program is shown in Fig.~\ref{fig:ps}.
One sector of the PSB is shown in Fig.~\ref{fig:psbsection}
and \ref{fig:psbmeas}.
\begin{figure}[!hbt]
\includegraphics[width=0.32\textwidth,clip]{figures/ps_forward.ps}
\includegraphics[width=0.32\textwidth,clip]{figures/ps_central.ps}
\includegraphics[width=0.32\textwidth,clip]{figures/ps_backward.ps}
\caption{\label{fig:ps} Forward, central and backward parts of the
PSB. In the central part, the gaps for the pellet target pipe are visible.}
\end{figure}
\begin{figure}[!hbt]
\begin{minipage}[t]{0.35\linewidth}
\vspace{0.5cm}
\includegraphics[width=\textwidth,clip]{figures/psb_section.ps}
\end{minipage}
\begin{minipage}[t]{0.65\linewidth}
\includegraphics[width=0.45\textwidth,angle=-90,clip]{figures/ps_trapez.ps}
\end{minipage}
\caption{\label{fig:psbsection} (Left) Layout of one section of the PSB
detector. {\bf A} denotes the rectangular counters of the barrel wall and
{\bf B } and {\bf C} are trapezoidal elements in the forward and in the
backward caps respectively. {\bf D} are bended light guides. (Middle)
Two shapes of the trapezoidal forward elements with dimensions marked
in mm. (Right) Shape and dimensions in mm of the trapezoidal backward
element.
}
\end{figure}
\begin{figure}[!hbt]
\centering
\includegraphics[width=\textwidth,clip]{figures/psb_meas.ps}
\caption{\label{fig:psbmeas} Cross section of the PSB scintillators with
dimensions marked in mm. No structural material is shown.
}
\end{figure}
Each scintillator is glued to an acrylic light guide coupled to the photomultiplier tube (PMT).
The PMTs are placed outside of the iron yoke to shield them from the
magnetic field. For this purpose, approximately 50~cm long light guides are
used.
\emparag{The Scintillator Electromagnetic Calorimeter - (SEC)}
The CD calorimeter SEC is able to measure photons, electrons and positrons
with energies up to 800~MeV. The energy threshold for detection of photons is
about 2~MeV. SEC
consists of 1012 sodium-doped CsI scintillating crystals placed
between the super-conducting solenoid and the iron yoke. The scattering
angles covered by the SEC are between 20$^{\circ}$ and
169$^{\circ}$.
The crystals are shaped as
truncated pyramids and are placed in 24 layers along the beam
(Fig.~\ref{fig:wasa} and \ref{se}). The
lengths of the crystals vary from 30~cm (16.2
radiation lengths) in the central part to 25~cm in
the forward and 20~cm in the backward part.
Fig.~\ref{secross} shows the angular coverage together with the
thickness of SEC. As a measure of the anticipated photon fluxes, the
center of mass (CM) system solid angle vs.\,the laboratory (LAB)
scattering angle is shown for some experimental conditions at WASA.
\begin{figure}
\begin{center}
\epsfysize7cm
\epsfclipon
\epsffile{figures/inken_fig4_5_solid_angle.eps}
\epsfclipoff
\end{center}
\caption[The angular coverage of the SEC]
{\label{secross}The angular coverage of the SEC. The CM system
solid angle vs.\,the LAB scattering angle is shown for pp and pd
interactions at 1500~MeV and 895~MeV.}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\epsfxsize10cm
\epsfclipon
\epsffile[33 155 555 683]{figures/inken_fig4_1_se.ps}
\epsfclipoff
\end{center}
\vspace*{-10mm}
\caption[A schematic view of the SEC]
{\label{se}Schematic view of the SEC. It consists of the forward
part (shadowed
area to the left), the central part (not shadowed area in the middle)
and the backward part (shadowed area to the right). The beam is
coming from the right side.}
\vspace*{5mm}
\begin{center}
\epsfysize6cm
\epsfclipon
\epsffile{figures/inken_fig4_2_planar_map_sec.eps}
\epsfclipoff
\end{center}
\caption[Planar map of the SEC]
{\label{planarmap}Planar map of the SEC.}
\end{figure}
A planar map of the crystals is given in Fig.~\ref{planarmap}. One can
clearly distinguish the three different main parts of the calorimeter:
the forward, central and backward parts. The forward part consists of
4 layers with 36 elements each. It covers scattering angles from
nearly 20$^{\circ}$ to 36$^{\circ}$. The central part consists of 17
layers each having 48 elements, and covers scattering angles from
36$^{\circ}$ to 150$^{\circ}$. The backward part consists of three
layers. Two layers have 24 elements and one layer closest to the
CELSIUS beam pipe has only 12 elements.
The small spaces between the forward-central and central-backward parts are
occupied by PSB light guides and mechanical support for the solenoid
(back end only). The calorimeter covers nearly
360$^{\circ}$ in ${\rm \Phi}$ angle. Holes for the pellet pipe (2+2 crystals)
and the solenoid chimney (4 crystals) are not shown in the figure. Some
design parameters of the calorimeter are given in table~\ref{sec2}.
\begin{table}[bht]
\begin{center}
\vspace{\baselineskip}
\begin{tabular}[b]{|l|l|} \hline
\multicolumn{2}{|c|}{\bf Scintillator Electromagnetic Calorimeter} \\ \hline
Amount of sensitive material & 135~g/cm$^2$ \\ \hline
\hspace*{3mm} [radiation lengths] & ${\rm \approx 16}$ \\ \hline
\hspace*{3mm} [nuclear interaction length] & ${\rm \approx 0.8}$ \\ \hline
Geometric acceptance: & 96\% \\ \hline
\hspace*{3mm} polar angle & ${\rm \approx 20^{\circ}-169^{\circ}}$\\ \hline
\hspace*{3mm} azimuth angle & ${\rm \approx 0^{\circ}-360^{\circ}}$\\ \hline
Max kinetic energy for stopping & \\ \hline
\hspace*{3mm} ${\rm \pi^{\pm}}$/proton/deuteron & 190/400/500 \\ \hline
Scattering angle resolution & ${\rm \approx 5^{\circ}}$(FWHM) \\ \hline
Time resolution & \\ \hline
\hspace*{3mm}charged particles & 5~ns(FWHM) \\ \hline
\hspace*{3mm}photons & ${\rm \approx}$40~ns(FWHM)\\ \hline
Energy resolution & \\ \hline
\hspace*{3mm}charged particles & ${\rm \approx 3\%}$(FWHM) \\ \hline
\hspace*{3mm}photons & ${\rm \approx 8\%}$(FWHM) \\ \hline
\end{tabular}
\end{center}
\caption[SEC design parameters]
{\label{sec2}SEC design parameters.}
\end{table}
The SEC is composed of sodium-doped CsI scintillating crystals. This
type of scintillator material provides a large light yield, has short
radiation length and good mechanical properties.
CsI(Na) was chosen instead of the more commonly used CsI(Tl)
scintillators for the following reasons \cite{rub:90,sch:95}:
\begin{itemize}
{\item Its emission peak at 420~nm matches well the bi-alkali S11
photocatode of the selected PM tubes, giving good photon statistics and
sufficiently fast response.
\item Its shorter scintillation decay time is preferable in high-rate
applications.
\item CsI(Na) gives much less afterglow than CsI(Tl).
\item CsI(Na) seems more resistant against radiation damage. When irradiated
by a proton
beam corresponding to 10 years of operation a test crystal did not
show any visible change in its structure. The CsI(Tl) test crystal, on
the contrary, lost its transparency.}
\end{itemize}
The crystals are connected
by plastic light guides, 120~mm to 180~mm long, with the photomultipliers
placed the outside of the iron yoke.
In Fig.~\ref{csimodule}, a fully equipped single calorimeter module
consisting of a CsI crystal, a light guide, a PM tube and a high
voltage unit, enclosed inside a special housing, is shown.
The SEC and its performance is described in more detail in the
Ph.D. thesis of Inken Koch \cite{Koch}.
\begin{figure}[htbp]
\begin{center}
\epsfysize5cm
\epsfclipon
\epsffile{figures/inken_fig4_3_csi_module.eps}
\epsfclipoff
\end{center}
\caption[A fully equipped CsI module]
{\label{csimodule}A fully equipped module with CsI crystal,
light guide, PM tube and housing.}
\end{figure}
\emparag{The light pulser system (LPS)}
\noindent The LPS delivers reference light pulses via light fibers to all
scintillation counters in order to monitor their gain during the experiment.
Since both organic and inorganic
scintillators are used, two types of light sources were designed. A
xenon flash tube from Hamamatsu is used for the CsI elements of the calorimeter and three
LED-based light sources for all plastic scintillators. From those four sources the light signals are
transmitted to individual elements via a network of light
fibers~\cite{Zabi94}.
\subsection{Modifications}
\label{subsubdetmodi}
\label{sec:modifications}
\subsubsection{Forward detector (FD)}
The FD was designed mainly for measurements of protons of kinetic energies
up to 500 MeV. An energy resolution $\sigma_E/E$ around 3\% is obtained
at the highest energy.
The track coordinates are measured by the FPC straw chambers. At CELSIUS, three
modules with 4 layers of straws each, are installed.
To improve track coordinate measurements, energy measurements and particle
identification for the higher energies anticipated at COSY, the following
modifications are planned:
\begin{itemize}
\item The first FRH layer will be removed. It gives the few centimeters
of extra space, needed for the
installation of one more (existing) FPC module. By moving the remaining
FRH layers a few centimeters upstream, one gets a better energy measurement
performance at the largest scattering angles ($17^{\circ}$) covered by the FD.
\item Repair of FTH. The FTH and FRH have served at CELSIUS for twelve
years and signs of severe aging effects have appeared. Most detectors
still work well, but in one of the spiral planes of FTH some elements
do not give acceptable signals and the whole plane should be replaced.
\item Two new FRH planes should be installed to improve particle energy
reconstruction. This would mean that protons of about 350 MeV kinetic
energy, instead of 300 MeV at present, would be stopped in the FD
(figure \ref{fig:fdetot}). The energy resolution at high energy would
consequently also be improved. A $\sigma_E/E$ around 10\% could be
reached for protons at 900 MeV.
\item Downstream of the FRH, we consider the installation of a
10~cm thick water Cherenkov detector, that is
composed of $\approx$2~m long and 10~cm wide bars. For
protons it would have a range of sensitivity from 500 MeV
($\beta$=0.75) to 1200 MeV ($\beta$=0.9), which well covers
the region where the energy
determination based on the FRH depositions deteriorates. The
combination of the FRH and the Cherenkov information
will improve both the particle identification power and the
kinetic energy resolution.
As an example, the Cherenkov light output is plotted as a function of
kinetic energy for protons and pions from the
reaction $pp\to pp{\eta }'$ where ${\eta }'\to \eta \pi^{+} \pi^{-}$
at a beam momentum of 3.3 GeV/c (figure \ref{fig:fdcerenkov}). The
proton kinetic energy energy resolution is about 5\% for protons above
500 MeV. Studies are underway to obtain more quantitative information
on the performance of the proposed Cerenkov detector.
\end{itemize}
\begin{figure}[ht]
\begin{center}
\includegraphics[clip=1,width=0.5\textwidth]{figures/fd_edeptot_p_pip.eps}
\caption{\label{fig:fdetot} Energy absorbed in the (upgraded) FRH planes
for protons (upper curve) and pions (lower curve)
as a function of their kinetic energy at the target.}
\end{center}
\end{figure}
\begin{figure}[ht]
\hspace{1.3cm}
\rotatebox{-90}{
\includegraphics[width=0.5\textwidth,clip]{figures/EvsLO.eps}
}
\caption{\label{fig:fdcerenkov} Cherenkov light output for charged particles
from the
reaction $pp\to pp\eta'$ at 3.3 GeV/c as a function of kinetic energy.}
\end{figure}
\subsubsection{Central detector (CD)}
The CD was originally designed for measurements of gammas with energy up
to 600 MeV and of electrons and positrons with momenta up to 400 MeV/c.
The charged particle momenta are determined by the track curvature in
a magnetic field of about 1 Tesla. The maximum lever arm is 160~mm for
these measurements and the design accuracy in the position
measurements in each of the 17 MDC layers is 100 $\mu$m. The accuracy
of the field map should be better than 1\%. The design momentum
resolution for electrons and positrons, scattered at large angles is
$<$~2\% (${\rm \sigma_p/p}$). With the present implementation of the
MDC in the CELSIUS/WASA experiment an optimal resolution, ${\rm
\sigma_p/p} \approx$3\% for $e^{\pm}$, $\approx$4\% for $\pi^{\pm}$
and $\approx$8\% for protons could be obtained. In real experiments,
the momentum resolution has been evaluated for $\pi^{\pm}$ and protons
and ${\rm \sigma_p/p}$ values of $\approx$4.5\% and $\approx$12\% were
obtained \cite{Jacewicz04}. To improve the momentum measurements for
work at higher energies the following actions are planned:
\begin{itemize}
\item An increase of the maximum magnetic field to 1.3 Tesla. This puts great
demands on some parts of the magnet system, since there will be a
70\% increase in the stored energy and in the mechanical forces
involved, but it would not require any major hardware upgrades
\cite{Ruber99}. However, it needs careful tests and "training" of
the magnet by expert personnel.
\item The MDC electronics have to be upgraded in order to improve the
efficiency for detection of pions and electrons and to reach the
design time resolution of $\approx$0.5~ns. The extent of this task
depends on the actual electrical noise environment at the WASA site
at COSY.
\end{itemize}
The design energy resolution for photons measured in the SE
calorimeter is about 5 \% ($\sigma_E/E$). At present an effective
energy resolution of 7-10 \% is obtained \cite{Koch}, and the main
limitation is a non-linear signal response. This is the product of
some different effects and the situation will be improved by
modification of PM bases, by usage of new QDCs and, if neccessary, by
changing some 150 PM tubes.
One source of background in CELSIUS/WASA is interactions in the
restgas in the scattering chamber. At present there are three big
cryopumps at WASA, two at the forward cone of the scattering chamber
and one at the backward cone. The vacuum in the scattering chamber
will be improved by adding one more big cryopump in the backward
direction. This needs a careful redesign of the backward part of WASA.
Another source of background is due to scattering events originating
from the vacuum chamber walls. Most of these events are caused by beam
halo particles. It is suggested to put a ring-shaped detector inside
of the beampipe just at the entrance to WASA. Signals from this
detector should be used in the trigger logic to "veto" events caused
by beam halo. Further prestudies will be done to design such a
detector and evaluate its anticipated performance in more detail.
\subsubsection{Cost estimates}
\label{sec:cost}
\begin{table}[!htbp]
\centering
\begin{tabular}{|l|c|}
\hline
{\bf Cost estimate} & {\bf kEuro} \\
\hline
FRH upgrade with two additional layers & 200\\
FTH repair / replacement of one layer & 50\\
FD Cerenkov detector & 100\\
SEC new PM bases / change of PM tubes & 50\\
MDC system refurbishments & 130\\
\hline
Additional upstream vacuum pump & 40 \\
Upstream internal veto detector & 30 \\
\hline
\end{tabular}
\caption[modcost]
{\label{modcost}Cost estimates for some planned detector modifications.}
\end{table}
\subsubsection{DAQ- and Trigger-System}
\label{subsubdaq}
The WASA DAQ-System was designed more than 10 years ago and
implemented with electronics from the same period.
Most of the digitizing modules were
delivered by the company LeCroy, which is no longer active on the market
of nuclear electronics. Furthermore, many parts of the detector
electronics have a very low level of integration,
leading to a high cabling effort at the detector side,
especially for the straws in the MDC and the Forward Detector.
In addition, spare parts of the used components cannot be acquired
anymore.
Due to the progress in technology the readout speed of the
existing system is much slower than possible using a
modern, state-of-the-art system. Since the event rate
at COSY must be significantly higher than the one currently
reached at CELSIUS, this becomes a serious problem. Considering also
the maintenance problems and the age of the digitizing modules,
it is obvious that successful operation of WASA at COSY can only
be achieved by a major upgrade of the existing DAQ system.
Currently, all experiments running at COSY are using the same
DAQ system following a uniform approach developed, implemented
and maintained by permanent staff of FZ J\"ulich. In view of
the time scale for ''WASA at COSY'' the upgrade of the
WASA DAQ should be based on the third generation of the DAQ at
COSY. The main advantages are:
\begin{itemize}
\item it implements the desired functionality, including a modern
synchronization technique, avoiding the -- inherently slow -- sequential
readout of the digitizing modules,
\item it is almost completely developed, thus fitting into the limited
time scale,
\item it has a good on-site support by the original developers, which
is essential for a smooth and successful operation,
\item and it is compatible to all existing DAQ systems at COSY.
\end{itemize}
\emparag{The current DAQ system}
The existing DAQ system of WASA at CELSIUS is mainly based on
four FASTBUS crates containing the digitizing modules (see
Fig.~\ref{fig:zel1}). The readout is done by PCs connected to
the FASTBUS crates via a proprietary parallel link based on
RS485. Since there is no synchronization system, all digitizing
modules have to be read out sequentially when a trigger
occurs. The readout of the (about) 1500 PMTs is done with
LeCroy LRS1881 QDCs. The time information from the straws
(2000 channels from the FPC and 1700 channels from the MDC) and
the $\approx500$ fast plastic scintillators is digitized using
LeCroy LRS1876 TDCs~\cite{zel1}.
The trigger system of WASA at CELSIUS is based on dedicated hardware
that has been developed by TSL. Due to the high background
the trigger system is still evolving to reach higher performance.
The first level trigger uses the discriminator outputs of the
fast plastic scintillators. It is based on multiplicity and generates
the gates for the QDCs and the stop for the TDCs. The second level
trigger uses the discriminator outputs from the calorimeter. It is based
on cluster multiplicity and energy deposition and generates the FastClear
for the digitizing modules.
\begin{figure}[hbt]
\begin{center}
\resizebox{0.7\textwidth}{!}{\includegraphics{figures/zel1.eps}}
\caption{Structure of the existing WASA DAQ and trigger
system~\cite{zel1,zel1a}.}
\label{fig:zel1}
\end{center}
\end{figure}
\emparag{The new DAQ system}
Starting with the first experiments at COSY permanent staff from ZEL
was responsible for a common DAQ system and its further development.
In the meantime two evolutionary and elementary steps have been made
to adapt it to the rapidly changing technologies~\cite{zel2}.
Now, the implementation of a third generation of DAQ has been started
aiming for the highest possible event rates~\cite{zel3}. Therefore,
state-of-the-art FPGA technologies are combined with fast communication
paths to achieve system latencies as low as possible. Commonly used
in all three generations is a DAQ software called EMS~\cite{zel2}, which
guarantees the software compatibility of all implemented systems. EMS
follows a well proven client/server architecture and has been developed
during a period of more than 10 years (corresponding to a manpower effort
of nearly 30 man-years).
As in other DAQ systems (e.g.\ at CERN) this third generation of DAQ at
COSY is based on proprietary solutions in order to guarantee
implementations optimized for speed. However, also CAMAC and VME can
be integrated despite of the resulting loss of performance. The whole system
is designed to run exclusively in a ``common stop mode'' to avoid many
expensive and hard to handle delay units. Here, ``common stop mode'' means
that the trigger is available some 100~ns to 1 $\mu$s after the signal
has been digitized. Therefore, the acquisition boards must run in a
self-triggering mode digitizing all interesting signals and storing them
together with a time stamp. When the -- also time-stamped -- trigger arrives
an FPGA selects the digitized signals inside a predefined time interval.
\begin{figure}[hbt]
\begin{center}
\resizebox{0.7\textwidth}{!}{\includegraphics{figures/zel2.eps}}
\caption{Structure of the third generation DAQ system at COSY.}
\label{fig:zel2}
\end{center}
\end{figure}
An essential key component required by this operation mode is a
Synchronization System (SYS) to reliably control and synchronize the
fast event flow~\cite{zel4}. The SYS delivers a global time base
relative to the trigger time. The control of the synchronization
system as well as the readout of the digitization modules is done
by a farm of embedded industrial PCs. On this farm, EMS-servers
controlling the front end modules are implemented. As a future extension
also the cancellation of events (software trigger) is considered.
The third generation of DAQ at COSY is based on a bus
controller (refer to Fig.~\ref{fig:zel2}) connected by a fiber
optical link to the PC farm and the synchronization system. The
acquisition modules are connected by a high speed LVDS bus.
This bus can be implemented as a backplane or as a cable bus
covering a distance of 15~meters and serving for control as well as
for transmission of event data. The physical layer of the bus complies
to the SCSI bus with a maximum transmission speed of 80 or
160~Mbytes/s~\cite{zel5}. In general, acquisition, trigger processing,
data reduction, buffer management, and sub-event building can be done by
means of FPGAs --- either on the acquisition boards or in the bus
controller. All these functions are realized without the intervention
of a general purpose processor and, thus, lead to the envisaged
performance.
As shown in Fig.~\ref{fig:zel3} a new computing layer and a new layer
of digitization electronics will be implemented. The readout will be done
by a farm of 15 PCs connected via an optical link (a common development
of ZEL and the company SIS, with a physical layer identical to Gigabit
Ethernet). The overall experiment control and storage is connected
via Gigabit Ethernet. The whole system runs under control of the EMS
software as used at all other experiments at COSY.
\begin{figure}[hbt]
\begin{center}
\resizebox{1.0\textwidth}{!}{\includegraphics{figures/zel3.eps}}
\caption{Structure of the new DAQ and trigger system of WASA at COSY.}
\label{fig:zel3}
\end{center}
\end{figure}
The digitization layer consists of crates with readout controllers,
QDC and TDC modules and the synchronization system. The event buffering
capabilities of the digitization modules in combination with the
synchronization system allow the operation in common stop mode, thus
avoiding delay lines. It is expected that with the new system
the event rate will be increased at least by a factor of 10. Because of
performance requirements and costs the use of commercially available
digitizing modules is not possible.
The existing 64 channel TDC module (being used for the proportional
chambers of ANKE and the straws at TOF) can be used without modifications
for the straws at WASA, provided the discriminator electronics will be
changed to a CMP16 based system. It is a 6U board
for the LVDS bus based on the F1 chip system developed by ZEL
compatible with the third generation of DAQ at COSY~\cite{zel6}.
Optionally, also a VME board developed at TSL may be used, although
some modifications and a further test phase are necessary~\cite{zel7}.
This board has 64 channels, too, and is based on the CERN TDC32.
The decision which TDC to use will have only minor impact on the overall
cost estimate and project timing and will be done within the next year.
For the calorimeter (pulse duration about 2 $\mu$s) the QDC will
be implemented by sampling the analog signal with a 100 MHz FlashADC
and subsequent integration in an FPGA.
Considering
connector requirements and layout restrictions, it is planned to have 16
channels on one 6U base board conforming to the standards of the
LVDS bus system developed by ZEL.
The latter technique is not possible for the signals from the fast plastic
scintillators, because the pulse duration is only $\approx 20$~ns. Here,
additional shaping or even the use of a QAC chip is required. The
technical issues are intensively discussed with people from the University
of Giessen (TAPS collaboration), the KFKI Budapest and the TSL.
Independent of the final decision, a mezzanine board will be manufactured
fitting to the FlashADC board described above and performing the analog
preprocessing for 16 channels.
The time stamps required by the free-running mode will be provided
by the FPGA on the motherboard for the ''slow'' QDC using the global
time information and by additional discriminators and TDCs for the
''fast'' QDC.
As indicated in Fig.~\ref{fig:zel3}, the discriminators and the
trigger electronics from the existing system shall remain.
Only with regard to the straws it is intended to replace the discrete
preamplifier and discriminator electronics with new modules based
on the CMP16 chip. The main advantage will be the increased sensitivity
and the compact layout, which allows much shorter cable lengths between
straws and preamplifier, thus reducing noise.
\begin{table}[t]
\begin{center}
\begin{tabular}[b]{|lr@{~channels\hspace*{4cm}}r@{~k\euro}r|}
\hline
\multicolumn{4}{|l|}{Electronics for MDC and
FPC based on the CMP16 ASIC\hspace*{1cm}}\\
~ & 3700 &95&\\
\hline
\multicolumn{4}{|l|}{Digitizing modules}\\
TDCs (MDC, FPC) & 3700 & 140&\\
QDCs (slow) & 1000 & 100&\\
QDCs (fast) & 400 & 80 &\\
add. TDCs & 400 & 16 &\\
\hline
\multicolumn{4}{|l|}{System components}\\
\multicolumn{2}{|l}{17 Crates (incl.\ crate controllers,
interfaces)} & 140 &\\
\multicolumn{2}{|l}{Industrial PC farm (embedded)} & 30& \\
\multicolumn{2}{|l}{Communication equipment} & 20& \\
\hline
\multicolumn{2}{|l}{Synchronization system} & 70& \\
\hline
\multicolumn{4}{|l|}{Miscellaneous}\\
\multicolumn{2}{|l}{Racks, infrastructure test
and measuring equipment} & 80& \\
\multicolumn{2}{|l}{Experiment control, visualization
and storage} & 20& \\
\multicolumn{2}{|l}{Additional cables and connectors} & 70& \\
\multicolumn{2}{|l}{External developments} & 50& \\
\multicolumn{2}{|l}{6\% risk reserve} & 60& \\
\hline
\multicolumn{2}{|l}{{\bf Total}} & {\bf 1000}& \\
\hline
\end{tabular}
\caption{Cost estimate for the development of a new data acquisition
system for WASA at COSY.\label{tab:daq}}
\end{center}
\end{table}
A further improvement of the trigger system can be achieved by
implementing a third level trigger in the context of the new
DAQ system. While this is not possible in the limited time scale
of about one year, it should be considered as a future extension.
At the moment two basic hardware concepts are discussed:
\begin{itemize}
\item Implementation as a processor- or
FPGA-farm that operates directly on the DAQ data stream.
\item Splitting the data from the digitizing modules into two
independent branches, one for the trigger and one going to
the DAQ system~\cite{zel8}.
\end{itemize}
In a further future-oriented development project
it is planned to integrate the embedded PC
functionality in the LVDS bus controller itself
using ``system on chip'' (SOC) technology with a general purpose FPGA
and processor on the same chip. However, it is quite uncertain, whether
this project will be finished next year. Therefore it has not been
considered for the DAQ system of WASA at COSY.
\emparag{Cost estimate}
The cost estimate shown in table~\ref{tab:daq} does not include manpower,
since all developments
concerning the new DAQ system and improvements of the existing
WASA electronics can be made by existing permanent staff.
One exception could be the development of the analog part of the fast
QDCs. Most of the existing cables should be reused. However, additional
shielded cables are required in some cases and all connectors
will be replaced.
\clearpage
\section{Project plan}
\label{sec:fin_man}
\emparag{General remarks}
The project ``WASA at COSY'' deals with the relocation of an existing
detector system --- the Wide Angle Shower Array (WASA) from The
Svedberg Laboratory (TSL) at Uppsala university (Sweden) to the
Research Center J\"ulich (Forschungszentrum J\"ulich, FZJ, Germany), more
specifically to an internal target position of COSY (Cooler
Synchrotron) at IKP.
WASA at COSY will be the experimental activity with the highest priority
at the IKP in J\"ulich for the next several years.
The approach of adapting an existing detector has a number of advantages
compared to building a new one: It is faster, provided the detector
is available on short notice. Given the necessary lead-time for
preparations at COSY, the
date that WASA will be available
(end of 2005) is regarded as optimal. It is
cost-effective if the detector is ``in good shape''
which is the case for WASA.
However, the movement and setting up of a detector as complex as WASA
is not free of charge --- even neglecting basic costs such as
dismounting, transport, and installation. We anticipate the following
investments for necessary repairs and changes to the detector:
Adaptation to the higher COSY energy, in particular the
forward detectors of WASA. Exchange of electronics for which no spare
parts exist and no replacement can be purchased any longer.
Replacement of the data acquisition system to improve data taking
capabilities and to adapt the WASA-DAQ to COSY standards. These will
allow one to carry out the initial experiments described above.
\emparag{Manpower}
Well over one hundred scientists have committed themselves to actively
persue the physics program laid out in this proposal. This includes
members of the current WASA at CELSIUS collaboration as well as a
wide representation from the COSY users. As the host institute, IKP
will have a special role in providing the scientific and technical
infrastructure for WASA at COSY. Consequently, IKP will redirect
significant manpower (about 10 F.T.E.) to this project, which implies
a reduction of the other activities at IKP.
\emparag{Finances}
The financing of this project is not yet finalized. In view of the high
priority WASA at COSY has for IKP about one half of the needed investments
summarized below could be contributed from the running budget of IKP.
This would require a one-time reduction of the number of COSY running hours by
50\% for one year in either 2005 or 2006. The remaining investment must be
provided by other sources in either Sweden, the EU, the German BMBF, or
the Research Center J\"ulich.
\begin{table}[h]
\begin{tabular}[b]{lr@{~k}l}
Dismounting, transfer, set-up\hspace*{2cm}& 100 &\euro\\
Preparations at COSY & 300 &\euro\\
Changes at WASA (``adaptation'') & 600 &\euro\\
New trigger- and readout electronics & 1000&\euro\\
Sum: & {\bf 2000} &{\bf \euro}\\
\end{tabular}
\end{table}
\emparag{Time schedule}
\begin{table}[h]
\begin{tabular}[b]{r@{~200}ll}
June &5\hspace*{1cm}&Termination of CELSIUS operations at TSL\\
Oct.\ &5 &Finish dismounting of WASA at CELSIUS\\
Nov.\ &5 &Transfer of WASA to J\"ulich\\
June &6 &Finish set-up of WASA at COSY\\
July &6 &Start of commissioning\\
Jan.\ &7 &Start of experiments
\end{tabular}
\end{table}
\clearpage
\bibliographystyle{prsty}
|
1,108,101,565,629 | arxiv | \section{Analytical Insights from $2 \times 2$ MIMO}
In this section, we provide detailed insight regarding quantization via the $2 \times 2$ system. By analysis of a limiting noiseless regime, we prove that phase-only quantization cannot yield unique decodability, while amplitude-phase quantization can.
\subsection{Phase-only Quantization}\label{sec:phaseOnly}
The following lemma establishes a negative result for phase-only quantization.
Consider a pair of possible transmitted symbol vectors $\mathbf{X}^{(1)} = (X_1^{(1)}, X_2^{(1)})$ and $\mathbf{X}^{(2)} = (X_1^{(2)}, X_2^{(2)})$. The corresponding noise-free received samples are given by
$Y_1^{(i)} \triangleq e^{-j\phi}(X_1^{(i)} + e^{-j\theta} X_2^{(i)} )/\sqrt{2}$ and $Y_2^{(i)} \triangleq e^{-j\phi}(e^{-j\theta} X_1^{(i)} + X_2^{(i)} )/\sqrt{2}$ for $i\in\{1,2\}$.
The lemma specifies choices for $\mathbf{X}^{1}$ and $\mathbf{X}^{2}$ for which the noise-free received samples prior to quantization
have the same phase. Thus, these pairs cannot be distinguished based on phase-only quantization.
\textbf{Lemma 1:}
\textit{For $(X_1^{(1)}, X_2^{(1)}) = (e^{j\pi (2i-1)/4}, e^{j\pi(2i+1)/4})$ where $i\in\{1,\ldots, 4\}$, the following statements hold:}
\begin{enumerate}
\item[] \textit{(i) For $\theta \in (-\pi/2, \pi/2)$ and $(X_1^{(2)}, X_2^{(2)}) = (X_2^{(1)}, X_1^{(1)})$,}
\begin{align}\label{eq:EqualPhase}
\angle Y_1^{(1)} = \angle Y_2^{(1)},\,
\angle Y_1^{(2)} = \angle Y_2^{(2)},\,
\angle Y_1^{(1)} = \angle Y_1^{(2)}
\end{align}
\item[] \textit{(ii) For $\theta \in (\pi/2, 3\pi/2)$ and $(X_1^{(2)}, X_2^{(2)}) = (X_2^{(1)}, X_1^{(1)})$,}
\begin{align}\label{eq:EqualPhase2}
\angle Y_1^{(1)} = \angle Y_2^{(1)} + \pi ,\,
\angle Y_1^{(2)} = \angle Y_2^{(2)} + \pi ,\,
\angle Y_1^{(1)} = \angle Y_1^{(2)} + \pi
\end{align}
\item[] \textit{(iii) For $\theta \in (-\pi/2, \pi/2)$ and $(X_1^{(2)}, X_2^{(2)}) = (e^{j\pi}X_2^{(1)}, e^{j\pi}X_1^{(1)})$,}
\begin{align}\label{eq:EqualPhase3}
\angle Y_1^{(1)} = \angle Y_2^{(1)} ,\,
\angle Y_1^{(2)} = \angle Y_2^{(2)} ,\,
\angle Y_1^{(1)} = \angle Y_1^{(2)} + \pi
\end{align}
\item[] \textit{(iv) For $\theta \in (\pi/2, 3\pi/2)$ and $(X_1^{(2)}, X_2^{(2)}) = (e^{j\pi}X_2^{(1)}, e^{j\pi}X_1^{(1)})$,}
\begin{align}\label{eq:EqualPhase4}
\angle Y_1^{(1)} = \angle Y_2^{(1)} + \pi ,\,
\angle Y_1^{(2)} = \angle Y_2^{(2)} + \pi ,\,
\angle Y_1^{(1)} = \angle Y_1^{(2)}
\end{align}
\end{enumerate}
\indent\indent\textit{Proof:} The result in the lemma can simply be shown by using Euler's formula and Pythagorean trigonometric identity.
\hfill $\blacksquare$
This results in the following proposition stating that phase-only quantization cannot achieve the unquantized benchmark.
\textbf{Proposition 1:} \textit{For any phase-only quantization scheme with any number of bins, $D(\mathbf{X}, \mathbf{Y}_Q, \theta) > 0$ for all $\theta \in [0, 2\pi)$ as $\sigma \to 0$.}
\indent\indent\textit{Proof:} In order to show that $D(\mathbf{X}, \mathbf{Y}_Q, \theta) > 0$ for all $\theta \in [0, 2\pi)$ as $\sigma \to 0$, $\mathop{\mathbb{E}_\Phi}\{I(\mathbf{X}; \mathbf{Y}_Q \mid \Phi, \theta)\} < 4$ should be proved for $\theta \in [0, 2\pi)$ as $\sigma \to 0$ since $ \mathop{\mathbb{E}_\Phi}\{I(\mathbf{X}; \mathbf{Y} \mid \Phi, \theta) \to 4$ as $\sigma \to 0$. Before proving that, first, it is shown that $p(\mathbf{X} = \mathbf{x} \mid \mathbf{Y}_Q = \mathbf{y}_Q, \Phi = \phi, \theta)$ satisfies for some $\mathbf{Y}_Q = \mathbf{y}_Q$ that $0 < p(\mathbf{X} = \mathbf{x} \mid \mathbf{Y}_Q = \mathbf{y}_Q, \Phi = \phi, \theta)< 1$ as $\sigma \to 0$. Based on the statement in Lemma~1, it can be stated that the noise-free outputs corresponding to inputs $(X_1^{(1)}, X_2^{(1)}) = (e^{j\pi (2i-1)/4}, e^{j\pi(2i+1)/4})$ for $i\in\{1,\ldots, 4\}$ and $(X_1^{(2)}, X_2^{(2)}) = (X_2^{(1)}, X_1^{(1)})$ have the same phase and fall in the same quantization bin for $\theta \in (-\pi/2, \pi/2)$. Similarly, for $(X_1^{(1)}, X_2^{(1)}) = (e^{j\pi (2i-1)/4}, e^{j\pi(2i+1)/4})$ for $i\in\{1,\ldots, 4\}$ and $(X_1^{(2)}, X_2^{(2)}) = (e^{j\pi}X_2^{(1)}, e^{j\pi}X_1^{(1)})$, the outputs without additive noise stay in the same bin of any given phase-only quantization mapping since they have the same phase for $\theta \in (\pi/2, 3\pi/2)$. In addition, for $\theta = \pi/2$ and $\theta = 3\pi/2$, the amplitude of one of the noise-free outputs is zero and the same ambiguity occurs for those cases as well. Without loss of generality, say that those noiseless outputs (i.e., the outputs with additive Gaussian noise as $\sigma \to 0$) after quantization is $ \mathbf{Y}_Q = \bar{\mathbf{y}}_Q$ for the inputs $\mathbf{X} = \mathbf{x}_1$ and $\mathbf{X} = \mathbf{x}_2$. Then, the following statements hold for $i\in\{1, 2\}$ based on Bayes' theorem:
\begin{align}\label{eq:CondProb}
0 < p(\mathbf{x}_i \mid \mathbf{Y}_Q = \bar{\mathbf{y}}_Q, \phi, \theta) = \frac{p(\mathbf{Y}_Q = \bar{\mathbf{y}}_Q \mid \mathbf{x}_{i}, \phi, \theta)}{\sum_{\bar{\mathbf{x}}}~p(\mathbf{Y}_Q = \bar{\mathbf{y}}_Q \mid \bar{\mathbf{x}}, \phi, \theta)} < 1
\end{align}
since $p(\mathbf{Y}_Q = \bar{\mathbf{y}}_Q \mid \mathbf{x}_{1}, \phi, \theta) \to 1$ and $p(\mathbf{Y}_Q = \bar{\mathbf{y}}_Q \mid \mathbf{x}_{2}, \phi, \theta) \to 1$ as $\sigma \to 0$. Then, as $\sigma \to 0$, $H(\mathbf{X} \mid \mathbf{Y}_Q, \Phi = \phi, \theta)>0$ for all $\theta \in [0, 2\pi)$ and consequently $I(\mathbf{X}; \mathbf{Y}_Q\mid \Phi = \phi, \theta)<4$ for all $\phi\in [0, 2\pi)$ based on the definition of mutual information and $\mathop{\mathbb{E}_\Phi}\{I(\mathbf{X}; \mathbf{Y}_Q \mid \Phi, \theta)\} < 4$ is satisfied for all $\theta \in [0, 2\pi)$.
\hfill $\blacksquare$
While the unquantized benchmark cannot be achieved, it is still of interest to ask how many phase quantization bins are enough to reach the high-SNR asymptote for phase-only quantization.
We now establish that, for our system, 8 phase quantization bins suffice. We begin with the following lemma.
\textbf{Lemma 2:} \textit{For any possible $(X_1^{(1)}, X_2^{(1)})$ and $(X_1^{(2)}, X_2^{(2)})$ input pairs, $\angle Y_1^{(1)} - \angle Y_1^{(2)} = 0~\pmod{\pi/4}$ and $\angle Y_2^{(1)} - \angle Y_2^{(2)} = 0~\pmod{\pi/4}$, where $Y_1^{(i)}$ and $Y_2^{(i)}$ are as defined in Lemma~1. Also, $\angle Y_1^{(1)} - \angle Y_1^{(2)}$ and $\angle Y_2^{(1)} - \angle Y_2^{(2)}$ can take $8$ different values.}
\indent\indent\textit{Proof:} By using $\arctan(x) - \arctan(y) = \arctan(\frac{x-y}{1+xy})$ and Euler's formula, the proof is straightforward.
\hfill $\blacksquare$
Based on Lemma~2, we can derive the following proposition stating that 8 phase quantization bins suffice.
\textbf{Proposition 2:} \textit{As $\sigma \to 0$, any phase-only quantization schemes with more than 8 regions cannot achieve higher data rate than phase-only quantization scheme with 8 equally partitioned sectors.}
\indent\indent\textit{Proof:} Consider a phase-only quantization scheme having more than 8 bins and let $L>8$ denote the number of bins of that scheme. Lemma~2 implies that the noise-free outputs (i.e., the outputs as $\sigma \to 0$) of the all possible inputs can take 8 different phase values for given $\phi$ and $\theta$. Then, based on the pigeonhole principle, at least $L-8$ bins of the phase-only quantizer do not contain any outputs for given $\phi$ and $\theta$ as $\sigma \to 0$. In other words, none of the outputs corresponding to all possible inputs fall into those bins as $\sigma \to 0$. Let $\mathcal{\bar{S}}$ denote the index set of those empty bins. Then, define a new set, $\mathcal{S}_E$ as $\mathcal{S}_E = \{\mathbf{y} = [i,j] \mid ( i\in \mathcal{\bar{S}} \land j \in \{1,\ldots, L\})\, \lor\, (j\in \mathcal{\bar{S}} \land i \in \{1,\ldots, L\}) \}$. It can be stated that $p(\mathbf{Y}_Q = \mathbf{y}_Q) = 0$ for all $\mathbf{y}_Q \in \mathcal{S}_E$ as $\sigma \to 0$. Now, two cases should be analyzed separately. First, if exactly $L-8$ bins of the phase-only quantizer are empty as $\sigma \to 0$; then, any two different noise-free outputs having different phases cannot be in the same bin due to the result in Lemma~2, which also holds for the phase-only quantization scheme with equally divided 8 regions. Let $\mathcal{S}^L_Q$ and $\mathcal{S}_Q$ denote the sets of all possible quantized outputs for the quantization scheme having $L>8$ bins and 8 bins, respectively. There exists a one-to-one correspondence between $\mathcal{S}^L_Q\backslash \mathcal{S}_E$ and $\mathcal{S}_Q$ and if $\mathbf{m} \in \mathcal{S}^L_Q\backslash \mathcal{S}_E$ and $\mathbf{n} \in \mathcal{S}_Q$ are the paired elements, it is stated that the input producing quantized output $\mathbf{m}$ in the quantization scheme with more than 8 bins produces $\mathbf{n}$ in the quantization scheme with equally partitioned 8 regions as $\sigma \to 0$. Thus,
\begin{align}\label{eq:DiscardEmptySet}
\sum_{\mathbf{y}_Q \in\mathcal{S}^L_Q} H(\mathbf{X} \mid \mathbf{y}_Q) p(\mathbf{y}_Q) &= \sum_{\mathbf{y}_Q \in\mathcal{S}^L_Q\backslash \mathcal{S}_E} H(\mathbf{X} \mid \mathbf{y}_Q) p(\mathbf{y}_Q) \\\label{eq:SameMutu}
& = \sum_{\mathbf{y}_Q \in\mathcal{S}_Q} H(\mathbf{X} \mid \mathbf{y}_Q) p(\mathbf{y}_Q)\,,
\end{align}
where \eqref{eq:DiscardEmptySet} is due to $p(\mathbf{Y}_Q = \mathbf{y}_Q) = 0$ for all $\mathbf{y}_Q \in \mathcal{S}_E$. Therefore, both of the schemes achieve the same data rate if $L-8$ bins of the phase-only quantizer are empty as $\sigma \to 0$. Next, consider the case that more than $L-8$ bins are empty. Since the noise-free outputs can have 8 different phase values for given $\phi$ and $\theta$, some of those outputs having different phases are in the same bin, which is not a possible case for the phase-only quantization scheme with equally sized regions. For that reason, it can be calculated that $H(\mathbf{X} \mid \mathbf{Y}_Q^{(1)}, \Phi = \phi, \theta) - H(\mathbf{X} \mid \mathbf{Y}_Q^{(2)}, \Phi = \phi, \theta) \geq 0$ as $\sigma \to 0$, where $\mathbf{Y}_Q^{(1)}$ and $\mathbf{Y}_Q^{(2)}$ denote the quantized outputs under the quantization schemes with more than 8 regions and equally partitioned exactly 8 regions, respectively. Therefore, $I(\mathbf{X}; \mathbf{Y}_Q^{(1)} \mid \Phi = \phi, \theta) \leq I(\mathbf{X}; \mathbf{Y}_Q^{(2)} \mid \Phi = \phi, \theta)$.
\hfill $\blacksquare$
\subsection{Amplitude-Phase Quantization}\label{sec:AmpPhaseQuant}
For $K$-ary amplitude and $M$-ary phase quantization, the quantization set of $(m+M(k-1))$th-bin of a quantizer can be written as
\begin{multline}\label{eq:Amp_Quant_Region}
\bar{\Gamma}_{m+M(k-1)} = \{ \bar{Y} \mid A_{k-1} \leq |\bar{Y}| < A_{k} \,, \frac{2\pi}{M}(m-1) \leq \angle \bar{Y} < \frac{2\pi}{M}m \}\,,
\end{multline}
for $m\in\{1,\ldots, M\}$ and $k\in\{1,\ldots, K\}$, where $A_1,\ldots,A_{K-1}$ are the amplitude thresholds (we set $A_0 = 0$ and $A_K = \infty$ to maintain a unified notation across quantization bins).
The following proposition states that $K=2$ and $M=8$ suffices to attain the unquantized benchmark.
\textbf{Proposition 3:} \textit{As $\sigma \to 0$, circularly symmetric quantization with $2$-level amplitude and $8$-level phase quantization
attains $D(\mathbf{X}, \mathbf{Y}_Q, \theta) \rightarrow 0$ for $\theta \in [0, 2\pi)$.}
\indent\indent\textit{Proof:} Proposition~1 is based on the observation that the outputs of some input pairs have the same phase at both of the antennas as $\sigma \to 0$, so that those outputs cannot be differentiated by employing any phase-only quantization scheme. On the other hand, the proof of Proposition~2 shows that a phase-only scheme with equally partitioned $8$ regions can distinguish noise-free outputs having two different phases, due to the result in Lemma~2. In this proof, the aim is to show that considering a 2-level amplitude quantization together with phase quantization resolves the ambiguities leading to the result in Proposition~1. First, it can be shown that only the outputs corresponding to the input pairs discussed in Lemma~1 cannot be distinguished via phase-only scheme having equally partitioned $8$ regions. For that reason, consider the input pairs in Lemma~1. For $(X_1^{(1)}, X_2^{(1)}) = (e^{j\pi (2i-1)/4}, e^{j\pi(2i+1)/4})$ and $(X_1^{(2)}, X_2^{(2)}) = (X_2^{(1)}, X_1^{(1)})$, where $i\in\{1,\ldots, 4\}$, $|Y_1^{(2)}| < 1 < |Y_1^{(1)}|$ and $|Y_2^{(1)}| < 1 < |Y_2^{(2)}|$ for $\theta \in (0, \pi/2]$, $|Y_1^{(1)}| = |Y_1^{(2)}| = 1$ and $|Y_2^{(1)}| = |Y_2^{(2)}| = 1$ for $\theta = 0$, and $|Y_1^{(1)}| < 1 < |Y_1^{(2)}|$ and $|Y_2^{(2)}| < 1 < |Y_2^{(1)}|$ for $\theta \in [-\pi/2, 0)$. Due to the symmetry, the same approach can be applied for other input pairs (i.e., $(X_1^{(1)}, X_2^{(1)}) = (e^{j\pi (2i-1)/4}, e^{j\pi(2i+1)/4})$ and $(X_1^{(2)}, X_2^{(2)}) = (e^{j\pi}X_2^{(1)}, e^{j\pi}X_1^{(1)})$ for $i\in\{1,\ldots, 4\}$) when $\theta \in (\pi/2, 3\pi/2)$. Since the amplitude of the outputs does not depend on $\Phi = \phi$ and a circularly symmetric quantization scheme is employed, a phase quantization scheme including a 2-level amplitude quantization with $A_0 = 0$, $A_1 = 1$, and $A_2 = \infty$ resolves the ambiguity between those outputs. It is easy to now conclude that $D(\mathbf{X}, \mathbf{Y}_Q, \theta) \to 0$ for $\theta \in [0, 2\pi)$ as $\sigma \to 0$.
\hfill $\blacksquare$
\subsection{Numerical Results}\label{sec:NumRes_2by2_Quantization}
In this section, numerical examples are provided to illustrate the theoretical results. We first illustrate the statements in the lemmas and the propositions
via example noise-free outputs prior to quantization. We then compute and compare Shannon limits for different quantization schemes.
\begin{figure}
\vspace{-0.2cm}
\begin{center}
\includegraphics[width=0.70\columnwidth,draft=false]{Cons_theta_75_phi_45_final.pdf}
\vspace{-0.3cm}
\caption{All possible noise-free outputs before the quantization, $e^{-j\phi}(X_1 + e^{-j\theta} X_2)/\sqrt{2}$ (Left) and $e^{-j\phi}(e^{-j\theta} X_1 + X_2 )/\sqrt{2}$ (Right), for $\theta = 5\pi/12 $ and $\phi = \pi/4$.}\label{fig:Cons_theta_75_phi_45}
\end{center}
\vspace{-0.6cm}
\end{figure}
\begin{figure}
\vspace{-0.2cm}
\begin{center}
\includegraphics[width=0.70\columnwidth,draft=false]{Cons_theta_170_phi_10_final.pdf}
\vspace{-0.3cm}
\caption{All possible noise-free outputs before the quantization, $e^{-j\phi}(X_1 + e^{-j\theta} X_2)/\sqrt{2}$ (Left) and $e^{-j\phi}(e^{-j\theta} X_1 + X_2 )/\sqrt{2}$ (Right), for $\theta = 17\pi/18 $ and $\phi = \pi/18$.}\label{fig:Cons_theta_10_phi_10}
\end{center}
\vspace{-0.6cm}
\end{figure}
We illustrate the geometry behind the proofs by presenting noiseless outputs prior to quantization for a well-conditioned and a poorly conditioned channel in Fig.~\ref{fig:Cons_theta_75_phi_45} and Fig.~\ref{fig:Cons_theta_10_phi_10}, respectively. We see that some output pairs (e.g., $(b, e)$, $(d, m)$, $(g, j)$ and $(l, o)$ in Fig.~\ref{fig:Cons_theta_75_phi_45} and $(b, o)$, $(d, g)$, $(e, l)$ and $(j, m)$ in Fig.~\ref{fig:Cons_theta_10_phi_10}) have the same phase at {\it both} receive antennas, and hence cannot be distinguished based on phase-only quantization, as stated in Lemma~1. On the other hand, the other outputs can indeed be distinguished based on phase-only quantization. In addition, for given $\theta$ and $\phi$, the phase of noise-free outputs can have $8$ different values, and two different outputs having two different phase values cannot be in the same bin for phase-only quantization with 8 equal sectors. This is the intuitive basis for Proposition~2. Lastly, noise-free output pairs having the same phase at both receive antennas, such as $(b, e)$ in Fig.~\ref{fig:Cons_theta_75_phi_45} can be separated by employing an amplitude quantization scheme with 2 regions as illustrated in Fig.~\ref{fig:Cons_theta_75_phi_45} and Fig.~\ref{fig:Cons_theta_10_phi_10}. This is the intuition behind Proposition~3.
\begin{figure}
\vspace{-0.2cm}
\begin{center}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth,draft=false]{CapVsTheta_phase_only_TWC-eps-converted-to.pdf}
\caption{}\label{fig:Phase_Only_Cap_VS_Theta}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\textwidth,draft=false]{CapVsTheta_Amp_Phase_Final_TWC-eps-converted-to.pdf}
\caption{}\label{fig:Amp_Phase_Cap_VS_Theta}
\end{subfigure}
\caption{Data rate versus $\theta$ for various scenarios including (a) the phase-only quantization schemes and (b) the amplitude and phase quantization schemes with different number of regions, where SNR is $15\,$dB.}
\end{center}
\vspace{-0.6cm}
\end{figure}
\begin{figure}
\vspace{-0.2cm}
\begin{center}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\columnwidth,draft=false]{CapVsSNR_phase_only_TWC-eps-converted-to.pdf}
\caption{}\label{fig:Phase_Only_Cap_VS_SNR}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\columnwidth,draft=false]{CapVsSNR_Amp_Phase_Final_TWC-eps-converted-to.pdf}
\caption{}\label{fig:Amp_Phase_Cap_VS_SNR}
\end{subfigure}
\caption{Data rate versus SNR for various scenarios including (a) the phase-only quantization schemes and (b) the amplitude and phase quantization schemes with different number of regions, where $\theta = \pi/2$.}
\end{center}
\vspace{-0.6cm}
\end{figure}
Next, we plot the data rate (mutual information) attained by different quantization schemes. Two benchmarks are considered: an unquantized system, and a quantizer based on Voronoi regions separating the outputs at each antenna. We may view the latter as an ML decision rule at each antenna, where input-pairs that fall on top of each other are interpreted as a single point, and it is easy to see that it attains the unquantized benchmark at high SNR. However, it depends on $\theta$ and $\Phi$, and is an irregular quantizer, which is unattractive in practice.
For a $2\times2$ MIMO system, Fig.~\ref{fig:Phase_Only_Cap_VS_Theta} and Fig.~\ref{fig:Amp_Phase_Cap_VS_Theta} plot data rate versus $\theta \in [0, 2\pi)$ at 15 dB SNR for phase-only and amplitude-phase quantization, respectively. Similarly, Fig.~\ref{fig:Phase_Only_Cap_VS_SNR} and Fig.~\ref{fig:Amp_Phase_Cap_VS_SNR} plot data rates versus SNR, fixing $\theta = \pi/2$ (the best conditioned channel). For 2-level amplitude and 8-level phase quantization, the amplitude threshold is set to $A_1 = 1$, whereas the thresholds are $A_1 = 0.75$ and $A_2 = 1.25$ for 3-level amplitude and 8-level phase quantization. The plots illustrate the trends predicted by our theoretical results: phase-only quantization does not attain the unquantized or ML benchmarks, while amplitude-phase quantization does attain these at high enough SNR. However, the performance at moderate SNR can benefit from a larger number of quantization
bins than those indicated by high-SNR asymptotics.
For example, while 8 phase quantization bins are as good as any other phase-only quantization scheme asymptotically, using 16 quantization bins does provide better performance at moderate SNRs (Fig.~\ref{fig:Phase_Only_Cap_VS_Theta} and Fig.~\ref{fig:Phase_Only_Cap_VS_SNR}). Similarly, while 2-level amplitude quantization suffices, there is a gain at moderate SNRs with 3-level quantization (Fig.~\ref{fig:Amp_Phase_Cap_VS_Theta} and Fig.~\ref{fig:Amp_Phase_Cap_VS_SNR}). In particular, Fig.~\ref{fig:Amp_Phase_Cap_VS_SNR} shows that for a well-conditioned channel, while 2-level amplitude quantization attains the unquantized benchmark at high enough SNR, 3-level amplitude quantization has a significant advantage at moderate SNRs, reaching unquantized performance at around 12.5 dB.
Armed with these insights, we consider quantizer design for a $4 \times 4$ system in the next section.
\section{Quantization for $4\times4$ LoS MIMO}\label{sec:Quantizationfor4x4}
In this section, we investigate design of regular quantizers for a $4\times4$ LoS MIMO system in which the transmit and receive antennas are
configured in a two-dimensional (2D) planar array as in Fig.~\ref{fig:SystemConf}.
\subsection{Quantizer Design}\label{sec:QuantizerDesignfor4x4}
We seek to design regular quantizers which are identical for each receive antenna.
For per-stream QPSK modulation, there are $4^4$ possible noise-free values for the received sample at each antenna, and detailed analysis as in the $2 \times 2$ MIMO
system is no longer feasible. However, as we shall see, a relatively simple approximation for the distribution of the received samples provides an effective approach
for quantizer design.
The distribution functions of $\{Y_i\}_{i=1}^{4}$ can be expressed as
\begin{align}\label{eq:Distribution_Y}
f_{Y_i}(y_i) = \int_0^{2\pi} \frac{1}{\left|\mathcal{S}\right|^4} \mathop{\sum_{\{x_i\}_{i=1}^{4}}}_{x_i \in \mathcal{S}} f(y_i \mid \phi, \{X_i=x_i\}_{i=1}^{4})\,p(\phi)\,d\phi
\end{align}
for all $i\in\{1,\ldots, 4\}$ with
\begin{align}\label{eq:Distribution_cond_Y}
f(y_i \mid \phi, \{X_i=x_i\}_{i=1}^{4})= \frac{1}{\pi \sigma^2}e^{-\frac{\|y_i - (\mathbf{H} \mathbf{x})_i\|^2}{\sigma^2}}
\end{align}
where $\mathcal{S} = \{e^{j\pi/4}, e^{j3\pi/4}, e^{j5\pi/4}, e^{j7\pi/4}\}$, $p(\phi) = 1/(2\pi)$ by the assumption, $\mathbf{x} \triangleq \left[x_1 \, x_2 \, x_3 \, x_4 \right]^\intercal$, $\mathbf{H}$ is as in \eqref{eq:ChannelMatrix_4x4} with $\Phi = \phi$ and $\theta$, and $(\mathbf{H} \mathbf{x})_i$ selects the $i$th element of $\mathbf{H} \mathbf{x}$.
Due to the symmetry, it is clear that the outputs before quantization (i.e., $\{Y_i\}_{i=1}^{4}$) have the same probability density function. Hence, without loss of generality, we focus on one of the outputs before quantization (e.g., say $Y_1$) to design the corresponding quantizer and employ the same quantizer for all outputs. As seen in \eqref{eq:Distribution_Y}, $Y_1$ has a complex and intractable distribution: conditioned on the common phase $\phi$, it is a mixture of $4^4$ Gaussians, and this conditional density then needs to be averaged over the continuum
$[0, 2 \pi )$ of values taken by $\phi$. We therefore approximate this distribution by
a circularly-symmetric complex Gaussian distribution, $\tilde{Y} \sim\mathcal{CN}(\tilde{\mu}, \tilde{\sigma}^2)$, with parameters chosen to minimize the Kullback-Leibler (KL) divergence between the distributions of $Y_1$ and $\tilde{Y}$, which is given by
\begin{align}\label{eq:KL_Divergence}
D_{KL} \left( Y_1\, \middle\|\, \tilde{Y} \right) = \int_{y \in \mathbb{C}} f_{Y_1}(y) \log\left(\frac{f_{Y_1}(y)}{f_{\tilde{Y}}(y)}\right)\,dy\,.
\end{align}
The optimal $\tilde{Y}$ that minimizes the KL divergence in \eqref{eq:KL_Divergence} can be found by \emph{moment matching} \cite{Rasmussen_2004}: the first and second moments of $\tilde{Y}$ and $Y_1$ are matched to obtain the optimal Gaussian approximation. Therefore, $\tilde{\mu}$ and $\tilde{\sigma}^2$ can be calculated, respectively, as
\begin{align}\label{eq:meanmatching}
\tilde{\mu} = \mathop{\mathbb{E}}\{Y_1\} = 0
\end{align}
and
\begin{align}\label{eq:variancematching}
\tilde{\sigma}^2 &= \mathop{\mathbb{E}}\{Y_1\,Y_{1}^{\dagger}\}\\
&= \sum_{i=1}^{4} \frac{1}{4}\mathop{\mathbb{E}}\{X_{i}\,X_{i}^{\dagger}\} + \mathop{\mathbb{E}}\{N_{1}\,N_{1}^{\dagger}\}\\
&= 1 + \sigma^2
\end{align}
where $\mathop{\mathbb{E}}\{X_{i}\,X_{i}^{\dagger}\} = 1$ for all $i\in\{1,\ldots, 4\}$ by definition. As a result, we consider the complex Gaussian approximation with $\tilde{\mu} = 0$ and $\tilde{\sigma}^2 = 1 + \sigma^2$ to design quantizers for $4\times4$ LoS MIMO.
\begin{figure}
\vspace{-0.2cm}
\begin{center}
\includegraphics[width=0.70\columnwidth,draft=false]{Quantizers_v3.pdf}
\vspace{-0.3cm}
\caption{Quantization schemes for $4\times4$ LoS MIMO at 10 dB SNR}\label{fig:Quantization_4x4}
\end{center}
\vspace{-0.6cm}
\end{figure}
We consider two regular quantization schemes: I/Q quantization and amplitude/phase quantization.
\begin{itemize}
\item \textbf{I/Q quantization:} For I/Q quantization scheme with $S^2$ regions, the quantization set of $(j+S(i-1))$th-bin of a quantizer can be written as
\begin{multline}\label{eq:I_Q_Region}
\bar{\Gamma}_{j+S(i-1)} = \{ \bar{Y} \mid I_{i-1} \leq \Re(\bar{Y}) < I_{i} \,, Q_{j-1} \leq \Im(\bar{Y}) < Q_{j} \}\,,
\end{multline}
for $i,j\in\{1,\ldots, S\}$, where $I_1,\ldots,I_{S-1}$ and $Q_1,\ldots,Q_{S-1}$ are the thresholds for in-phase and quadrature, respectively. We set $I_0 = -\infty$, $I_S = \infty$, $Q_0 = -\infty$, and $Q_S = \infty$ in order to go along with the unified notation.
\item \textbf{Amplitude/phase quantization:} As for $2\times2$ LoS MIMO, the quantization set for this scheme is specified as in \eqref{eq:Amp_Quant_Region}.
\end{itemize}
We determine the quantizer regions (i.e., the thresholds in \eqref{eq:Amp_Quant_Region} and \eqref{eq:I_Q_Region}) based on the following two different metrics:
\begin{itemize}
\item \textbf{Minimum mean squared quantization error (MMSQE)-based regions:} This is the conventional approach to quantizer design based on minimization of the mean squared error given by
\begin{align}\label{eq:MMSQE_design}
\mathop{\mathbb{E}}\{(\tilde{Y} - Q(\tilde{Y}))^2\}
\end{align}
where $Q(\cdot)$ is the quantizer function, whose set is defined as either \eqref{eq:Amp_Quant_Region} or \eqref{eq:I_Q_Region}. The optimal decision boundaries in \eqref{eq:Amp_Quant_Region} and \eqref{eq:I_Q_Region} are obtained as usual, by applying the Lloyd-Max algorithm \cite{Lloyd_82,Max_60}.
\item \textbf{Equal probability-based regions:} The quantizer regions here are obtained by partitioning the fitted complex Gaussian distribution into equal probability regions.
In other words, the quantizer boundaries maximize the entropy of $\tilde{Y}$; that is, $H(\tilde{Y})$. For the circular Gaussian distribution, the boundaries can
be specified analytically.
For I/Q quantization, the entropy-maximizer thresholds can be calculated as
\begin{align}\label{eq:IQ_Thresholds}
I_i = Q_i = \tilde{\mu} + \frac{\tilde{\sigma}}{\sqrt{2}}\inv{\Phi}\left(\frac{i}{S}\right) = \sqrt{\frac{1 + \sigma^2}{2}}\inv{\Phi}\left(\frac{i}{S}\right)
\end{align}
for $i \in\{1,\ldots, S-1\}$, where $\inv{\Phi}$ is the inverse distribution function (i.e., the quantile function) for the standard Gaussian distribution with a mean of $0$ and a standard deviation of $1$.
For amplitude/phase quantization, the phase quantization is uniform, and the amplitude thresholds that maximize the entropy can be found as
\begin{align}\label{eq:AP_Thresholds}
A_i = \sqrt{(1 + \sigma^2)\log{\left(\frac{K}{K-i}\right)}}
\end{align}
for $i \in\{1,\ldots, K-1\}$.
\end{itemize}
For those two different metrics, Fig.~\ref{fig:Quantization_4x4} shows the I/Q and amplitude/phase quantizers at 10 dB SNR, each having a total of $16$ regions (i.e., $K=2$, $M = 8$, and $S = 4$).
\subsection{Numerical Results}\label{sec:NumRes_4by4_Quantization}
\begin{figure}[htbp]
\vspace{-0.2cm}
\begin{center}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\columnwidth,draft=false]{IQ_and_AP_MMSQE_CapVsSNR_TWC-eps-converted-to.pdf}
\caption{}\label{fig:IQ_and_AP_MMSQE}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\columnwidth,draft=false]{IQ_and_AP_EqualProb_CapVsSNR_TWC-eps-converted-to.pdf}
\caption{}\label{fig:IQ_and_AP_EqualProb}
\end{subfigure}
\caption{Data rate versus SNR for (a) MMSQE-based quantization and (b) equal probability-based quantization for a $4\times4$ system.}
\end{center}
\vspace{-0.6cm}
\end{figure}
We now investigate Shannon limits for different quantizer designs; in each case, there are 16 quantization regions in the complex plane.
Fig.~\ref{fig:IQ_and_AP_MMSQE} shows that, for conventional MMSQE-based quantization, amplitude-phase and I/Q quantization are both unable to
reach the maximum data rate ($8$ bits per channel use) even at SNR as high as 20 dB, compared to the unquantized system, which saturates at 10 dB SNR.
On the other hand, Fig.~\ref{fig:IQ_and_AP_EqualProb} shows that, for equal probability regions, I/Q quantization attains the maximum data rate of $8$ bits per channel use
at around 15 dB, while amplitude/phase quantization continues to exhibit a gap to the unquantized limit even at SNR of 20 dB. We conclude that 2 bit quantization on I and Q based on equal probability regions should suffice to attain acceptable performance at moderate SNR, and focus on
this setting for investigation of spatial demultiplexing algorithms in the next section.
\section{Conclusion} \label{sec:conclusion}
Our study of ideal $4 \times 4$ LoS MIMO system at high SNR yields fundamental insight into the impact of severe quantization. We show that equal probability quantization, which maximizes per-antenna output entropy, outperforms
standard MMSQE quantization in such regimes. For spatial demultiplexing, we introduce the novel concept of virtual quantization, which may be viewed as approximate joint estimation of the transmitted symbols and the unquantized received signal, and show that linear detection with virtual quantization is an effective low-complexity alternative to maximum likelihood detection, which requires complexity exponential in the number of transmitted bits. It remains an open issue as to whether the gap to maximum likelihood detection at higher SNR can be further reduced, and the error floor eliminated, while maintaining reasonable complexity.
An important direction for future research is to investigate quantization-constrained LoS MIMO is more complex settings, including understanding the impact of dispersion due to geometric misalignments and potential performance advantages
of spatial oversampling \cite{Sawaby_Asilomar2016, Raviteja_Spawc2018}. While we do not consider transmit precoding here, joint transmit-receive optimization subject to dynamic range constraints and nonlinearities at both ends is of great interest.
At a fundamental level, the concept of virtual quantization, which treats the unquantized output as a hidden variable, may be worth exploring for other system models.
\section{Spatial Demultiplexing under Severe Quantization}\label{sec:SpatialDemultiplexing}
We begin with spatial demultiplexing for $4 \times 4$ LoS MIMO system with QPSK modulation with 2 bit I/Q quantization as designed in the previous section.
In order to highlight the impact of quantization, consider an ideally conditioned channel with $\theta = \frac{\pi}{2}$ in (\ref{eq:ChannelMatrix_4x4}), for which
the received antenna responses for different transmitted streams are orthogonal, so that matched filter, linear ZF and maximum likelihood detection all yield the same
performance with {\it unquantized} observations. We shall see, however, that drastic quantization can have a severe impact on the performance of linear detection even in such an ideal setting because of the common channel phase $\Phi$ in (\ref{eq:ChannelMatrix_4x4}), which can move the observations close to quantization boundaries. On the other hand, the mutual information plot in
Figure \ref{fig:IQ_and_AP_EqualProb} shows that reliable communication at the maximum data rate of 8 bits per channel use should be possible at an SNR of about 15 dB in this setting.
Our goal, therefore, is to devise spatial demultiplexing with reasonable complexity that can approach this performance.
We consider an uncoded BER target of about $10^{-3}$, which yields reliable communication with high-rate bit interleaved
coded modulation, since a binary symmetric channel with this cross-over probability has capacity close to 0.99 bits per channel use.
From the point of view of minimizing the probability of error, the optimal detector based on the quantized output is the maximum likelihood (ML) detector \cite{Yang_Survey_2015}, given by
\begin{align}\label{eq:ML_detector}
\mathbf{\hat{X}}(\mathbf{Y}_Q) = \mathop{\underset{\mathbf{X} \in \mathcal{S}^{N_R}}{\mathrm{argmax}}}\,p(\mathbf{Y}_Q \mid \mathbf{X})\,.
\end{align}
Since the minimization in \eqref{eq:ML_detector} is over all possible transmitted vectors and it is difficult to calculate $p(\mathbf{Y}_Q \mid \mathbf{X})$ for a given $\mathbf{X}$, the problem in \eqref{eq:ML_detector} has prohibitive complexity. As a low-complexity alternative, we consider linear ZF detection.
We have verified by simulations that other standard multiuser detection techniques such as linear minimum mean squared error (MMSE) detector \cite{Honig_Book_2009} and sphere decoding \cite{Damen_2003} achieve the same performance as linear ZF for our system.
Recall that the ZF solution minimizes
\begin{align}\label{eq:ZF_problem}
\tilde{\mathbf{X}}(\mathbf{Y}_Q) = \underset{\mathbf{X}}{\mathrm{argmin}} \norm{\mathbf{Y}_Q - \mathbf{H}\mathbf{X}}
\end{align}
and can be obtained by computing
\begin{align}\label{eq:ZF_step1}
\tilde{\mathbf{X}}(\mathbf{Y}_Q) = \inv{(\mathbf{H}^\dagger \mathbf{H})}\mathbf{H}^\dagger\mathbf{Y}_Q
\end{align}
first, and then finding $i$th element of $\mathbf{\hat{X}}(\mathbf{Y}_Q)$ as
\begin{align}\label{eq:ZF_step2}
\hat{X}_i(\mathbf{Y}_Q) = \underset{X_i \in \mathcal{S}}{\mathrm{argmin}} |\tilde{X}_i(\mathbf{Y}_Q) - X_i|
\end{align}
for all $i\in\{1,\ldots, 4\}$, where $\tilde{X}_i(\cdot)$ is the $i$th element of $\tilde{\mathbf{X}}(\cdot)$. Note that $\mathbf{Y}_Q$ in \eqref{eq:ZF_step1} represents the quantized output at the receiver and the quantizer outputs are set to the centroids of the quantizer regions, which are obtained based on the complex Gaussian approximation. Mathematically, the quantized output of $i$th antenna for the received signal $Y_i$ is equal to
\begin{align}\label{eq:Centroids}
\bar{Y}_i = \frac{\int_{\tilde{\Gamma}_i} y f_{\tilde{Y}}(y)\,dy}{\int_{\tilde{\Gamma}_i} f_{\tilde{Y}}(y)\,dy}\,,
\end{align}
where $\tilde{\Gamma}_i$ denotes the quantizer region that $Y_i$ falls into; that is, $Y_i \in \tilde{\Gamma}_i$.
\subsection{Virtual Quantization}\label{sec:VirQuantization}
Linear ZF detection based on the centroids codebook performs poorly when the unquantized outputs are far from the centroids of the regions that they belong to. On the other hand,
we know that linear ZF detection with unquantized outputs yields excellent performance for a well-conditioned MIMO channel. This motivates viewing
the unquantized output vector as a hidden variable, or nuisance parameter, for our hypothesis testing problem of estimating the transmitted symbols.
We can now apply any of the standard tools of composite hypothesis testing to this problem. We choose here a Generalized Likelihood Ratio Test (GLRT) approach,
in which we jointly estimate the unquantized output and the transmitted symbols given the quantized outputs.
That is, the transmitted symbols are estimated by jointly maximizing the probability of unquantized output and transmitted symbols given the quantized output:
\begin{align}\label{eq:Joint_Estimation}
(\mathbf{\hat{X}}(\mathbf{Y}_Q), \mathbf{Y}(\mathbf{Y}_Q)) = \mathop{\underset{\mathbf{X} \in \mathcal{S}^{N_R}, \mathbf{Y}}{\mathrm{argmax}}}\,p(\mathbf{X}, \mathbf{Y} \mid \mathbf{Y}_Q)\,.
\end{align}
A key difficulty in this optimization problem is that we need to search over a continuum of values for the hidden unquantized output $\mathbf{Y}$.
We therefore consider a grid-based approximation of $\mathbf{Y}$, where the grid is finer than that provided by the quantizer. We term this approach
{\it virtual quantization.}
\begin{figure}[htbp]
\vspace{-0.2cm}
\begin{center}
\includegraphics[width=0.7\columnwidth,draft=false]{Virtual_quantizer_final_v1.pdf}
\vspace{-0.3cm}
\caption{Physical and virtual quantizers with the outputs corresponding to the centroids of the regions for $10\,$dB SNR.}\label{fig:Virtual_quantizer}
\end{center}
\vspace{-0.6cm}
\end{figure}
\begin{figure*}
\vspace{-0.2cm}
\begin{center}
\includegraphics[width=0.7\columnwidth,draft=false]{block_diag_virtual_quantization.pdf}
\vspace{-0.3cm}
\caption{Flow diagram of virtual quantization based spatial demultiplexing.}\label{fig:Flow_Virtual_quantizer}
\end{center}
\vspace{-0.6cm}
\end{figure*}
Let $Q_v(\cdot)$ denote the element-wise virtual quantizer function. The virtual quantized hidden variable can be obtained as $\mathbf{Y}_V = Q_v(\mathbf{Y})$. Then,
the quantized output $\mathbf{Y}_Q$ can be written as $\mathbf{Y}_Q = Q_c(\mathbf{Y}_V)$ as a coarsening of virtual quantized hidden variable, where $Q_c(\cdot)$ is the coarsening function. Thus, $\mathbf{Y}_Q = Q_c(Q_v(\mathbf{Y})) = Q(\mathbf{Y})$, where $Q(\cdot)$ is the element-wise actual quantizer function defined in \eqref{eq:Quantizer} for the $i$th receive antenna and $Q_c(\cdot)$ can be considered as $Q_c(\cdot) = Q(\cdot)$.
Thus, $\mathbf{X} \rightarrow \mathbf{Y} \rightarrow \mathbf{Y}_V \rightarrow \mathbf{Y}_Q$ is a Markov chain. Figure \ref{fig:Virtual_quantizer} shows an example of physical and virtual quantizers used in our numerical results (detailed description is provided later in this section).
Quantizing the nuisance parameter $\mathbf{Y}$ in the joint estimation problem in \eqref{eq:Joint_Estimation} using the virtual quantizer, we seek to jointly estimate
the transmitted symbols and the virtual quantizer outputs:
\begin{align}\label{eq:Joint_Estimation_Approx}
(\mathbf{\hat{X}}(\mathbf{Y}_Q), \mathbf{Y}(\mathbf{Y}_Q)) &= \mathop{\underset{\mathbf{X} \in \mathcal{S}^{N_R}, \mathbf{Y}}{\mathrm{argmax}}}\,p(\mathbf{X}, Q_v(\mathbf{Y}) \mid \mathbf{Y}_Q) \\\label{eq:Joint_Estimation_Approx_2}
& = \mathop{\underset{\mathbf{X} \in \mathcal{S}^{N_R}, \mathbf{Y}_V \in \mathcal{T}}{\mathrm{argmax}}}\,p(\mathbf{X}, \mathbf{Y}_V \mid \mathbf{Y}_Q)
\end{align}
where \eqref{eq:Joint_Estimation_Approx_2} follows from $\mathbf{Y}_V = Q_v(Y)$ and $\mathcal{T} = \mathcal{T} \left( \mathbf{Y}_Q \right)$ is a discrete set including all possible combinations of virtual quantized output for observed $\mathbf{Y}_Q$:
$$
\mathcal{T} = \{\mathbf{Y}_V = Q_v(\mathbf{Y}) \mid Q(\mathbf{Y}_V) = \mathbf{Y}_Q \}
$$
In order to solve the problem in \eqref{eq:Joint_Estimation_Approx_2} and estimate the transmitted symbols, we maximize the objective function in \eqref{eq:Joint_Estimation_Approx_2} with respect to $\mathbf{X}$ for a given $\mathbf{Y}_V \in \mathcal{T}$ first, then substitute the obtained $\mathbf{X}$ to the objective function to solve for $\mathbf{Y}_V$. To begin with, for a given $\mathbf{Y}_V \in \mathcal{T}$, the maximization of the optimization problem in \eqref{eq:Joint_Estimation_Approx_2} over $\mathbf{X}$ can be expressed as follows:
\begin{align}\label{eq:Joint_prob_max_over_X_1}
\mathbf{\hat{X}}(\mathbf{Y}_V, \mathbf{Y}_Q) &= \mathop{\underset{\mathbf{X} \in \mathcal{S}^{N_R}}{\mathrm{argmax}}}\,p(\mathbf{X}, \mathbf{Y}_V \mid \mathbf{Y}_Q) \\\label{eq:Joint_prob_max_over_X_2}
& = \mathop{\underset{\mathbf{X} \in \mathcal{S}^{N_R}}{\mathrm{argmax}}}\,p(\mathbf{X} \mid \mathbf{Y}_V, \mathbf{Y}_Q)\,p(\mathbf{Y}_V \mid \mathbf{Y}_Q) \\\label{eq:Joint_prob_max_over_X_3}
& = \mathop{\underset{\mathbf{X} \in \mathcal{S}^{N_R}}{\mathrm{argmax}}}\,p(\mathbf{X} \mid \mathbf{Y}_V)\,p(\mathbf{Y}_V \mid \mathbf{Y}_Q) \\\label{eq:Joint_prob_max_over_X_4}
& = \mathop{\underset{\mathbf{X} \in \mathcal{S}^{N_R}}{\mathrm{argmax}}}\,p(\mathbf{X} \mid \mathbf{Y}_V) \\\label{eq:Joint_prob_max_over_X_5}
& = \mathop{\underset{\mathbf{X} \in \mathcal{S}^{N_R}}{\mathrm{argmax}}}\,p(\mathbf{Y}_V \mid \mathbf{X})
\end{align}
where \eqref{eq:Joint_prob_max_over_X_2} is a standard conditional probability computation, \eqref{eq:Joint_prob_max_over_X_3}
is based on the Markov property $\mathbf{X} \rightarrow \mathbf{Y}_V \rightarrow \mathbf{Y}_Q$, \eqref{eq:Joint_prob_max_over_X_4} is obtained since $p(\mathbf{Y}_V \mid \mathbf{Y}_Q)$ does not depend on $\mathbf{X}$ for given $\mathbf{Y}_V$ and $\mathbf{Y}_Q$, and \eqref{eq:Joint_prob_max_over_X_5} follows from Bayes' theorem and equally likely transmitted symbols.
The optimization problem in \eqref{eq:Joint_prob_max_over_X_5} is in the form of \eqref{eq:ML_detector}, except that it considers virtual quantized hidden variable instead of actual quantized output. As described in \eqref{eq:ZF_step1} and \eqref{eq:ZF_step2}, \eqref{eq:Joint_prob_max_over_X_5} can be approximated by linear ZF detection.
Therefore, for given $\mathbf{Y}_V$ and observed quantized output $\mathbf{Y}_Q$, $\mathbf{\hat{X}}(\mathbf{Y}_V, \mathbf{Y}_Q)$ can be obtained as in \eqref{eq:ZF_step1} and \eqref{eq:ZF_step2}.
Next, we substitute $\mathbf{\hat{X}}(\mathbf{Y}_V, \mathbf{Y}_Q)$ to the optimization problem in \eqref{eq:Joint_Estimation_Approx_2} and maximize it over $\mathbf{Y}_V$. Substituting $\mathbf{\hat{X}}(\mathbf{Y}_V, \mathbf{Y}_Q)$ to \eqref{eq:Joint_Estimation_Approx_2},
\begin{align}\label{eq:Joint_prob_max_over_Y_v_1}
\hat{Y}_v(\mathbf{\hat{X}}, \mathbf{Y}_Q) &= \mathop{\underset{\mathbf{Y}_V \in \mathcal{T}}{\mathrm{argmax}}} \,p(\mathbf{\hat{X}}, \mathbf{Y}_V \mid \mathbf{Y}_Q) \\\label{eq:Joint_prob_max_over_Y_v_2}
& = \mathop{\underset{\mathbf{Y}_V \in \mathcal{T}}{\mathrm{argmax}}}\,p(\mathbf{Y}_V \mid \mathbf{\hat{X}}, \mathbf{Y}_Q)\,p(\mathbf{\hat{X}} \mid \mathbf{Y}_Q) \\\label{eq:Joint_prob_max_over_Y_v_3}
& \approx \mathop{\underset{\mathbf{Y}_V \in \mathcal{T}}{\mathrm{argmin}}}\,\norm{\mathbf{Y}_V - \mathbf{\bar{Y}}_V}^2
\end{align}
where \eqref{eq:Joint_prob_max_over_Y_v_2} follows from Bayes' theorem and \eqref{eq:Joint_prob_max_over_Y_v_3} is obtained by approximating the impact of quantization and noise as Gaussian and modeling $p(\mathbf{Y}_V \mid \mathbf{\hat{X}}, \mathbf{Y}_Q)$ as Gaussian with mean $\mathbf{\bar{Y}}_V$, where $\mathbf{\bar{Y}}_V$ denotes the noiseless reconstruction and is equal to $\mathbf{\bar{Y}}_V = Q_v(\mathbf{H}\mathbf{\hat{X}})$.
Note that we ignore $p(\mathbf{\hat{X}} \mid \mathbf{Y}_Q)$ term in \eqref{eq:Joint_prob_max_over_Y_v_3} even though $\mathbf{\hat{X}}$ depends on $\mathbf{Y}_V$, which means that we are not necessarily
attaining the true maximum for (\ref{eq:Joint_prob_max_over_Y_v_1}). We expect the impact of this approximation on \eqref{eq:Joint_prob_max_over_Y_v_3} to be small
compared to the term $p(\mathbf{Y}_V \mid \mathbf{\hat{X}}, \mathbf{Y}_Q)$, which decays exponentially according to our Gaussian approximation for the sum of the virtual quantization noise and thermal noise.
The proposed virtual quantization method can now be summarized as follows:
\begin{enumerate}
\item Generate set $\mathcal{T}$ based on the physical quantized observation and the virtual quantizer function such that $\mathcal{T} = \{\mathbf{Y}_V = Q_v(\mathbf{Y}) \mid Q(\mathbf{Y}_V) = \mathbf{Y}_Q \}$.
\item For each $\mathbf{Y}_V \in \mathcal{T}$, calculate the ZF solution to obtain $\mathbf{\hat{X}}(\mathbf{Y}_V, \mathbf{Y}_Q)$ by treating $\mathbf{Y}_V$ as if it is the observed output at the receiver.
\item Find $\mathbf{Y}_V \in \mathcal{T}$ that minimizes the Euclidean distance between $\mathbf{Y}_V$ and $Q_v(\mathbf{H}\mathbf{\hat{X}})$; that is, $\lVert \mathbf{Y}_V -Q_v(\mathbf{H}\mathbf{\hat{X}})\rVert^2$.
\item Declare the corresponding $\mathbf{\hat{X}}(\mathbf{Y}_V, \mathbf{Y}_Q)$ as the estimated symbol vector.
\end{enumerate}
Fig.~\ref{fig:Flow_Virtual_quantizer} illustrates this procedure via a flow diagram.
The result of the estimation depends on the virtual quantization function (i.e., $Q_v(\cdot)$) and the virtual quantization output (i.e., $\mathbf{Y}_V$), both of which determine the elements in $\mathcal{T}$ and contribute to the cost function used to specify the estimated symbols. Different virtual quantization functions can be designed for our proposed method. However, in this paper, we consider the same method that we use for the design of the actual physical quantizer. Considering the Gaussian approximation discussed in the design of the actual quantizer, we design an I/Q quantization-based virtual quantizer having equal-probability regions. For the 2-bit per I/Q physical quantizer having $16$ regions, we consider a virtual quantizer with $64$ regions (i.e., $S = 8$) whose outputs are decided based on the centroids of the corresponding regions similar to those of the physical quantizer, as shown in Fig.~\ref{fig:Virtual_quantizer}. In this case, the virtual quantizer divides each physical quantizer bin into $4$ regions, which can be considered as a virtually created 1-bit quantizer per each I/Q for each physical quantization bin.
\subsection{Numerical Results}\label{sec:NumRes_Demodulation}
\begin{figure}[htbp]
\vspace{-0.2cm}
\begin{center}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\columnwidth,draft=false]{Detection_Results_phi_zero_v3_TWC-eps-converted-to.pdf}
\caption{}\label{fig:Detection_phi_zero}
\end{subfigure}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width=\columnwidth,draft=false]{Detection_Results_phi_average_v3_TWC-eps-converted-to.pdf}
\caption{}\label{fig:Detection_phi_average}
\end{subfigure}
\caption{Bit error rate versus SNR for different detection methods when the common phase is (a) fixed to $\Phi = 0$; (b) uniformly distributed over $[0, 2\pi)$, for a $4\times4$ MIMO system with QPSK.}
\end{center}
\vspace{-0.6cm}
\end{figure}
We consider two benchmarks: ML with no quantization and ML with quantization. The former assumes there is no quantization in the system; that is $\mathbf{Y}_Q = \mathbf{Y}$ and provides an unquantized benchmark as if ADCs at the receiver have infinite precision. The latter considers the quantized outputs at the receiver and is obtained based on \eqref{eq:ML_detector}.
For a $4\times4$ MIMO system, Fig.~\ref{fig:Detection_phi_zero} plots bit error rate (BER) versus SNR, setting $\theta = \pi/2$, for different detection methods when the common phase is fixed to $\Phi = 0$.
We may also view this as equivalent to a hybrid analog-digital processing scheme in which the common channel phase $\Phi$ is removed by analog derotation {\it prior} to quantization.
The plot indicates that linear ZF detection with equal probability quantization and the centroids codebook remains near-optimal, achieving the same performance as ML reception,
as long as the common channel phase is removed prior to quantization. On the other hand, as expected from our mutual information computations, MMSQE-based quantization performs
significantly worse than the equal probability-based quantizer. Note that we have simulated other detection methods such as the linear MMSE detector and the sphere decoder based on the quantized outputs at the receiver, and have verified that they achieve the same performance as linear ZF detection.
We now turn to the scenario of interest for us: fully digital processing without derotation of the common phase $\Phi$ prior to quantization. The receiver processing with quantized observations does employ
knowledge of $\Phi$, but performance is adversely affected by points being rotated close to quantization boundaries. Fig.~\ref{fig:Detection_phi_average} plots BER averaged over the common phase $\Phi$ versus SNR, setting $\theta = \pi/2$, for different detection methods, where $\Phi$ is uniformly distributed over $[0, 2\pi)$. We now see that linear ZF detection with the centroids codebook performs significantly worse, with an error floor that stays higher than our target $10^{-3}$ BER. The proposed virtual quantization approach performs significantly better: its performance is close to that of ML detection at BER of $10^{-2}$, while being 5 dB worse at the target BER of $10^{-3}$. It still exhibits an error floor at $10^{-4}$, motivating additional effort in devising low-complexity strategies for approaching maximum likelihood performance.
\noindent
{\bf Beyond the ideal model:} Since severe quantization destroys the orthogonality of the received signals for different data streams, we expect that our all-digital receiver should
be robust to changes in link distance $R$ around the nominal range $R_N$. Fig.~\ref{fig:BER_vs_RangeChanges} plots BER vs $R/R_N$ at $40\,$dB SNR for our proposed virtual quantization approach and linear ZF detection with the centroids codebook, where $R \in [0.8R_N, 1.2R_N]$. The poor performance of linear ZF detection also persists as we vary $R$.
\begin{figure}[htbp]
\vspace{-0.2cm}
\begin{center}
\includegraphics[width=0.7\columnwidth,draft=false]{HorizontalDistance_VirtualQuantization_vs_ZF_Final_v2_TWC-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{Bit error rate averaged over $\Phi$ versus $R/R_N$ for different detection methods for a $4\times4$ MIMO system with QPSK, where SNR is $40\,$dB.}\label{fig:BER_vs_RangeChanges}
\end{center}
\vspace{-0.6cm}
\end{figure}
\noindent
{\bf Scaling to larger constellations:} A key advantage of virtual quantization is that its complexity does not scale with constellation size.
We verify this by evaluating performance for 16QAM modulation. Given the higher dynamic range of 16QAM, we consider 3 bit and 4 bit I/Q physical quantization, and then
add 1 bit virtual quantization as before. Fig.~\ref{fig:Detection_phi_average_16QAM} plots BER averaged over the common phase $\Phi$ versus SNR, setting $\theta = \pi/2$, for different detection methods. We see that neither 4 bit physical quantization (without virtual quantization) nor 3 bit physical quantization with 1 bit virtual quantization achieve our
target $10^{-3}$ BER even at very high SNR. However, 4 bit physical quantization with 1 bit virtual quantization does achieve our BER target at approximately $20\,$dB SNR.
\begin{figure}[htbp]
\vspace{-0.2cm}
\begin{center}
\includegraphics[width=0.7\columnwidth,draft=false]{16QAM_Detection_phi_average_v1_TWC-eps-converted-to.pdf}
\vspace{-0.3cm}
\caption{Bit error rate averaged over $\Phi$ versus SNR for different detection methods including our proposed virtual quantization approach when the common phase is uniformly distributed over $[0, 2\pi)$ for a $4\times4$ MIMO system with 16QAM.}\label{fig:Detection_phi_average_16QAM}
\end{center}
\vspace{-0.6cm}
\end{figure}
\section{Introduction}\label{sec:intro}
\IEEEPARstart{L}{ine} of sight (LoS) multi-input multi-output (MIMO) communication is well-matched to higher carrier frequencies in the millimeter (mm) wave and THz bands because of the attractive scaling of spatial degrees of freedom (DoF) and bandwidth. For 1D apertures with a horizontal distance, or range, $R$ between transmit and receive arrays having lengths of $L_T$ and $L_R$, the number of spatial DoF based on information-theoretic considerations, given by \cite{Torkildson}
\begin{gather}\label{eq:DoF_1D}
DoF \approx \frac{L_T L_R}{R \lambda } + 1 \,
\end{gather}
scales inversely with the carrier wavelength $\lambda$, and therefore linearly with the carrier frequency $f_c = c/\lambda$, where $c$ is the speed of light. The result in \eqref{eq:DoF_1D} can be extended for 2D apertures and rewritten as \cite{Torkildson}
\begin{gather}\label{eq:DoF_2D}
DoF \approx \frac{A_T A_R}{R^2 \lambda^2 } + 1
\end{gather}
where $A_T$ and $A_R$ are the areas occupied by the 2D arrays at the transmitter and the receiver, respectively, so that the scaling with $f_c$ becomes quadratic. Since transmission bandwidth typically scales linearly with carrier frequency, the overall data rates can potentially scale cubically with carrier frequency.
Advances in mmWave radio frequency integrated circuits (RFIC) in low-cost silicon semiconductor processes open up the possibility of deploying LoS MIMO at scale, for example, to boost link capacities in wireless backhaul mesh networks for urban picocells \cite{Rasekh_TWC2019}. Consider $4\times4$ LoS MIMO with a link distance of $100\,$m. At a carrier frequency of 140 GHz,
the form factor required for a well-conditioned spatial channel is small enough to permit opportunistic deployment (e.g., on lampposts): the inter-antenna spacing for orthogonal eigenmodes is 33 cm. With QPSK modulation and $10-20\,$GHz bandwidth, we can achieve $80-160\,$Gbps uncoded data rates. Therefore, with lightweight channel coding, $100\,$Gbps becomes a feasible target. However, can we leverage the economies of scale in digital computation to realize such transceivers at reasonable cost and power consumption, using all-digital
baseband signal processing? While this is standard in modern communication receivers operating at lower bandwidths (typically below 1 GHz), as signaling bandwidths increase,
realizing high-precision analog-to-digital converters (ADCs) is a challenge \cite{Murmann, Walden}. Motivated by these considerations, we investigate in this paper
whether it is possible to use all-digital processing in LoS MIMO receivers with severely quantized samples.
\noindent
{\bf Contributions:}
We investigate design of low-precision quantizers and of spatial demultiplexing with heavily quantized observations. \\
{\it Quantizer Design:} Rather than trying to design optimal quantizers, our first goal is to design quantizers with regular structure which approach the same Shannon limit as an unquantized system at high SNR. \\
$\bullet$ Our first result is negative. As bandwidth increases, a particularly attractive approach is phase-only quantization: this can be implemented by passing linear combinations of the real and imaginary parts of the sample through sign detectors (one-bit ADCs), and therefore
does not require automatic gain control (see \cite{Singh_2009}). However, we prove for the $2 \times 2$ QPSK system that phase-only quantizers cannot meet the unquantized benchmark at high SNR.\\
$\bullet$ Our second result shows that amplitude-phase quantization with a relatively small number of bins does attain the unquantized benchmark.
Specifically, we prove for the $2 \times 2$ QPSK system that 2-level amplitude and 8-level phase quantization works.\\
$\bullet$ For the $4 \times 4$ system, we obtain practical guidelines and design prescriptions for quantizer design. We show via mutual information computations that per-antenna quantization into equal probability regions (which maximizes per-antenna output entropy) performs better than conventional MMSQE quantization, and that I/Q quantization performs better than amplitude/phase quantization. In particular, we show that equal probability I/Q quantization with 2 bits per real dimension, designed using a Gaussian approximation for the received samples, achieves the unquantized benchmark at high SNR for QPSK modulation, attaining a maximum data rate of $8$ bits per channel use.\\
{\it Spatial demultiplexing:} For a $4 \times 4$ QPSK system, we investigate spatial demultiplexing with observations quantized using the 2 bit I/Q quantizer that we have designed.
We considered well-conditioned LoS MIMO channels for which linear zero-forcing detection provides near-optimal performance with unquantized observations. Our goal is to attain uncoded error probabilities of $10^{-3}$ or better,
for which reliable communication can be obtained with lightweight, high-rate error correcting codes.\\
$\bullet$ We show that linear detection with quantized observations leads to an error floor. Since mutual information computations show that the maximum rate of 8 bits per channel use is attainable with moderate SNR penalty for the quantizer design used, we expect maximum likelihood detection not to exhibit an error floor. We show that this is indeed the case, but the prohibitive complexity (exponential in the number
of transmitted bits) motivates design of lower-complexity spatial demultiplexing schemes. \\
$\bullet$ We introduce the concept of {\it virtual quantization,} modeling the uncertainty created by quantization as a nuisance parameter, so that the task of estimating
the transmitted symbols can be approached using the tools of composite hypothesis testing. We employ a Generalized Likelihood Ratio Test (GLRT) approach which leverages the efficacy of linear detection for the well-conditioned MIMO channels
considered here. A key computational advantage of the proposed approach is that, unlike maximum likelihood detection, its complexity does not scale with constellation size. In addition, we show that our proposed virtual quantization concept is resistant to the changes in channel condition due to suboptimal separation of the transmitter and the receiver (up to $\pm 20\%$ variations of the nominal range). \\
$\bullet$ While the bulk of our numerical examples are for QPSK, we also present results for 16QAM, demonstrating that our prescriptions for quantizer design and our proposed virtual quantization approach extend to larger constellations.
\textit{Notation:} Throughout the paper, random variables are denoted by capital letters and small letters are used for the specific value that the random variables take. Bold letters are used to denote vectors and matrices. $\mathop{\mathbb{E}_Z}$ denotes the expectation operator over the random variable $Z$. $|Z|$ and $\angle Z$ represent the amplitude and the phase of $Z$, respectively. $\Re(Z)$ and $\Im(Z)$ denote real and imaginary part of complex number $Z$, respectively. $\mathbf{X}^\intercal$ and $\mathbf{X}^\dagger$ are the transpose and Hermitian transpose of $\mathbf{X}$, respectively. $\mathbf{I}_n$ is the identity matrix of size $n$.
\section{Related Work}
The DoF for LoS MIMO as a function of transceiver form factor and antenna placement, range and carrier frequency are by now well known \cite{Bohagen2007, Torkildson}.
It is worth contrasting the motivation for our work with a recently developed LoS MIMO system \cite{LoSMIMO_Ericsson} which employs $2.5\,$GHz bandwidth in E-band ($70-80\,$GHz carrier frequency), and achieves $100+\,$Gbps at a distance of 1.5 km using 8-fold multiplexing (spatial degrees of freedom along with dual polarization) and alphabets as large as 64QAM. The optimal antenna separation is $1.72\,$m, requiring bulky antenna structures and careful installation.
We envision higher frequencies and shorter ranges to reduce form factor to enable opportunistic deployment.
The goal of our investigation of severely quantized LoS MIMO, therefore, is to examine how far we can push the paradigm of using larger available bandwidths (which limits
the precision of available ADCs) to reduce the required constellation size (which potentially enables reduction in ADC precision).
Our work also contrasts with recent efforts in the research literature based on analog-centric \cite{Sawaby_Asilomar2016, Mamandipoor, Sheldon_APS2010,Yan_2018} or hybrid analog-digital \cite{Wadhwa_2016, Khalili_ISIT2018, Raviteja_Spawc2018, Zhu_2019} processing in an attempt to sidestep the ADC bottleneck.
Since LoS MIMO is often envisioned for quasi-static links (e.g., wireless backhaul), it is natural to consider precoding with channel state information at the transmitter, as in a number of
theoretical studies \cite{Kobayashi_2007, Zhou_2014, Zhu_2019}. Transmit precoding can also significantly reduce the dynamic range at the receiver, easing the task of
analog-to-digital conversion. Indeed, prior studies of MIMO capacity with low-precision ADC assume
transmit precoding \cite{Mo_TSP, Khalili_ISIT2018, Khalili_2020}. Channel capacity with 1-bit ADC is studied in \cite{Mo_TSP}, which provides capacity bounds and a convex optimization based algorithm to obtain capacity-achieving constellations. In \cite{Khalili_ISIT2018}, the capacity with transmit precoding, together with hybrid analog-digital processing at the receiver, where analog linear combinations of the signals received at different antennas are quantized, is studied. In \cite{Khalili_2020}, joint transmit power and ADC allocation problem is studied for throughput maximization, which results in that using few one-bit ADCs with the adaptive threshold receiver is enough to achieve near optimal performance.
Transmit precoding leads to increased dynamic range at the transmitter, which aggravates the already difficult problem of producing power at higher frequencies, such as
the millimeter wave or THz bands. In this paper, therefore, we explore LoS MIMO {\it without} transmit precoding, in contrast to the cited
prior work. We assume that the receiver has ideal channel estimates. Channel estimation with low-precision ADC is not as challenging as demodulation: \cite{Chan_Est_Dabeer} is an early example for a SISO dispersive channel, while \cite{Nossek} and \cite{Mo_2014} propose effective estimation techniques for massive MIMO with 1-bit quantization at the receive antennas.
There have been prior studies of demodulation \cite{Mezghani_2008, Mezghani_2010} based on quantized samples for MIMO systems without precoding, but these consider
Rayleigh faded channel models associated with rich scattering environments, unlike the LoS MIMO setting considered here. The computational intractability of maximum likelihood
detection is pointed out in \cite{Mezghani_2008}, while large system analysis for suboptimal loopy belief propagation is considered in \cite{Mezghani_2010}.
Shannon limits for an ideal SISO discrete-time additive white Gaussian noise (AWGN) channel with low-precision ADC are studied in \cite{TCOM_Madhow}. It is shown that the optimal input distribution is discrete and can be computed numerically, but standard constellations are near-optimal. Further, the use of ADCs with 2-3 bits precision results in only a small reduction in channel capacity even at moderately high SNR. Our model is perhaps the simplest possible extension of this framework to MIMO systems.
This paper builds on our preliminary results on quantizer design in an earlier conference paper \cite{ADS_UM_2019}. We provide proofs and technical details, as well as more
detailed insights and numerical results, for quantizer design here. The results on spatial demultiplexing, including the proposed virtual quantization concept, are entirely new.
\section{System Model and Problem Formulation}\label{sec:probForm}
We consider a symmetric $4 \times 4$ LoS MIMO communication system, with equal inter-antenna spacings at transmitter and receiver, as shown in Fig.~\ref{fig:SystemModel}. Each transmit/receive antenna may be a fixed beam antenna \cite{LoSMIMO_Ericsson}, or an electronically steerable ``subarray'' \cite{Torkildson}, with a directive beam along the LoS, and multipath is ignored. The received signal vector $\mathbf{Y} \triangleq \left[Y_1 \, \cdots \, Y_{4} \right]^\intercal \in \mathbb{C}^{4 \times 1}$ is given by
\begin{gather}\label{eq:ReceivedSignal}
\mathbf{Y} = \mathbf{H}\,\mathbf{X} + \mathbf{N} \,,
\end{gather}
where $\mathbf{X} \triangleq \left[X_1 \, \cdots \, X_{4} \right]^\intercal \in \mathbb{C}^{4 \times 1}$ is the transmitted symbol vector, $\mathbf{H} \in \mathbb{C}^{4 \times 4}$ is the normalized channel matrix (with each column normalized to unit norm), and $\mathbf{N} \sim\mathcal{CN}(0, \sigma^2\,\mathbf{I}_{4})$ is AWGN. Under this normalization, the SNR for the $k$th data stream is given by $SNR = \mathop{\mathbb{E}}\{ \lvert X_k \rvert^2 \}/\sigma^2$.
\begin{figure}
\vspace{-0.2cm}
\begin{center}
\begin{subfigure}[b]{0.70\textwidth}
\includegraphics[width=\textwidth,draft=false]{LoSMIMO_math_model_TWC.pdf}
\caption{Mathematical model for LoS MIMO communication system}\label{fig:MathModel}
\end{subfigure}
\\
\begin{subfigure}[b]{0.70\textwidth}
\includegraphics[width=\textwidth,draft=false]{SystemModel_4x4_TWC.pdf}
\caption{Geometric configuration for the $4\times 4$ LoS MIMO system (with blue and red colored antennas) and the $2\times 2$ LoS MIMO system (with only red colored antennas)}\label{fig:SystemConf}
\end{subfigure}
\caption{LoS MIMO communication system model}\label{fig:SystemModel}
\end{center}
\vspace{-0.6cm}
\end{figure}
\noindent
{\bf Input:} We consider QPSK modulation unless otherwise stated (results for 16QAM are included in Section \ref{sec:SpatialDemultiplexing}). For QPSK modulation, $\{X_i\}_{i=1}^{4}$ are independent and identically distributed symbols taking values $\{e^{j\pi/4}, e^{j3\pi/4}, e^{j5\pi/4}, e^{j7\pi/4}\}$ with equal probability. Thus, $SNR =\mathop{\mathbb{E}}\{ \lvert X_k \rvert^2 \}/\sigma^2 = 1/\sigma^2$ where $\mathop{\mathbb{E}}\{ \lvert X_k \rvert^2 \} = 1$ for all $k\in\{1,\ldots,4\}$.
\noindent
{{\bf Channel:}
For the pure LoS channel we consider, the elements of $\mathbf{H}$ in \eqref{eq:ReceivedSignal} are calculated by employing {\it ray-tracing} in consideration of the spherical nature of the wave propagation \cite{Driessen_1999, Bohagen_2009} and the columns of $\mathbf{H}$ are normalized to unit norm. Since the path loss differences among different transmit-receive antenna pairs are negligible, the normalized channel matrix for the symmetric $4 \times 4$ LoS MIMO communication system is given by \cite{Mamandipoor}
\begin{gather}\label{eq:ChannelMatrix_4x4}
\mathbf{H} = \frac{1}{2} e^{-j\Phi}
\begin{bmatrix}
1 & e^{-j\theta} & e^{-j2\theta} & e^{-j\theta} \\
e^{-j\theta} & 1 & e^{-j\theta} & e^{-j2\theta} \\
e^{-j2\theta} & e^{-j\theta} & 1 & e^{-j\theta} \\
e^{-j\theta} & e^{-j2\theta} & e^{-j\theta} & 1
\end{bmatrix}\,,
\end{gather}
where the random variable $\Phi$ denotes the common phase change along the path between the transmitter and the receiver, and the ``cross-over phase'' depends on the inter-antenna spacing $d$ and link distance $R$ as follows \cite{Torkildson, Garcia_2018}:
\begin{gather} \label{eq:theta}
\theta = \frac{2\pi}{\lambda}(\sqrt{R^2+ d^2} - R) \approx \frac{\pi d^2}{\lambda R} ~{\rm for}~ R \gg d
\end{gather}
where $\lambda$ denotes the carrier wavelength. We would like our quantizer designs to be robust to variations in the common phase $\Phi$, which is assumed to be uniformly distributed over $[0, 2\pi)$.
\noindent
{\bf Quantizer:} We consider identical quantizers at each receive antenna. The quantized output of the $i$th receive antenna can be expressed as
\begin{gather}\label{eq:QuantizedOutput}
\bar{Y}_i = Q(Y_i) \,,
\end{gather}
for $i\in\{1,\ldots,4\}$. $Q(\cdot)$ in \eqref{eq:QuantizedOutput} represents the quantizer function at each receive antenna and for a given input $y$, $Q(y)$ can be characterized as
\begin{gather}\label{eq:Quantizer}
Q(y) = \tilde{y}_j\,,~\textrm{if}\;~y\in \Gamma_j\,,
\end{gather}
for $j\in\{1,\ldots,T\}$, where $\tilde{y}_j$ for $j\in\{1,\ldots,T\}$ is a design parameter and $\Gamma_1$, $\ldots$, $\Gamma_{T}$ denote the decision regions for the quantizer, with $T$ denoting the number of quantizer bins at each receive antenna.
\noindent
{\bf Spatial demultiplexer:} Based on the quantized observations; that is, $\mathbf{Y}_Q \triangleq \left[\bar{Y}_1 \, \cdots \, \bar{Y}_{4} \right]^\intercal \in \mathbb{C}^{4 \times 1}$, the receiver performs the spatial demultiplexing and provides the estimate of the transmitted symbol as $\mathbf{\hat{X}} \triangleq \left[\hat{X}_1 \, \cdots \, \hat{X}_{4} \right]^\intercal \in \mathbb{C}^{4 \times 1}$.
We begin with the $2 \times 2$ system depicted in Fig.~\ref{fig:SystemConf} with red colored antennas in order to make some fundamental theoretical observations regarding quantization. For this scheme, $\mathbf{X}$ and $\mathbf{Y}$ in \eqref{eq:ReceivedSignal} are defined as $\mathbf{X} \triangleq \left[X_1 \, X_2 \right]^\intercal \in \mathbb{C}^{2 \times 1}$ and $\mathbf{Y} \triangleq \left[Y_1 \, Y_2 \right]^\intercal \in \mathbb{C}^{2 \times 1}$, respectively. Also, $\mathbf{N} \sim\mathcal{CN}(0, \sigma^2\,\mathbf{I}_2)$. The channel matrix for this scheme corresponds to the renormalized version of the upper-left $2 \times 2$ submatrix of \eqref{eq:ChannelMatrix_4x4} and is given by
\begin{gather}\label{eq:ChannelMatrix}
\mathbf{H} = \frac{1}{\sqrt{2}} e^{-j\Phi}
\begin{bmatrix}
1 & e^{-j\theta} \\
e^{-j\theta} & 1
\end{bmatrix}\,,
\end{gather}
where $\theta \approx \frac{\pi d^2}{\lambda R}$ for $R \gg d$ as in (\ref{eq:theta}).
One possible formulation of optimal quantization is to minimize
\begin{gather}\label{eq:Difference}
D(\mathbf{X}, \mathbf{Y}_Q, \theta) \triangleq \mathop{\mathbb{E}_\Phi}\{I(\mathbf{X}; \mathbf{Y} \mid \Phi, \theta) - I(\mathbf{X}; \mathbf{Y}_Q \mid \Phi, \theta)\}\,
\end{gather}
where $\mathbf{Y}_Q \triangleq \left[\bar{Y}_1 \, \bar{Y}_2 \right]^\intercal$ and the function $I(\bar{\mathbf{X}}; \bar{\mathbf{Y}}\mid \Phi, \theta)$ represents the mutual information between the random variables $\bar{\mathbf{X}}$ and $\bar{\mathbf{Y}}$ for given $\Phi$ and $\theta$. Based on the data processing equality, $D(\mathbf{X}, \mathbf{Y}_Q, \theta) \geq 0$ since $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Y}_Q$ form a Markov chain; that is, $\mathbf{X} \rightarrow \mathbf{Y} \rightarrow \mathbf{Y}_Q$. Also, $I(\mathbf{X}; \mathbf{Y} \mid \Phi, \theta)$ in \eqref{eq:Difference} does not depend on any parameter related to quantizer. For that reason, the problem of minimizing $D(\mathbf{X}, \mathbf{Y}_Q, \theta)$ in \eqref{eq:Difference} is equivalent to
\begin{gather}\label{eq:OptProb}
\underset{\{\Gamma_j\}_{j=1}^{T}}\max~ \mathop{\mathbb{E}_\Phi}\{I(\mathbf{X}; \mathbf{Y}_Q \mid \Phi, \theta)\}\,.
\end{gather}
In the optimization problem in \eqref{eq:OptProb}, the mutual information between $\mathbf{X}$ and $\mathbf{Y}_Q$ must be maximized over the set of all possible quantization regions of the quantizer at the receive antennas. The number of quantization bins for the quantizer is not fixed in \eqref{eq:OptProb}, and must also be optimized. Thus, it is difficult to solve \eqref{eq:OptProb}. Furthermore, the optimal
quantizers may correspond to irregular regions, leading to implementation difficulties. In this paper, therefore,
we opt for designing regular quantizers with the goal of ensuring that $D(\mathbf{X}, \mathbf{Y}_Q, \theta) \rightarrow 0$ at high SNR. |
1,108,101,565,630 | arxiv | \section{Introduction}
\label{sec:introduction}
Let $\Gr = (\Gr,+)$ be an abelian group.
For two sets $A,B\subseteq \Gr$ define the {\it sumset} as
$$A+B := \{ x\in \Gr ~:~ x=a+b \,,a\in A\,,b\in B \}$$
and, similarly, the {\it difference} set
$$A-B := \{ x\in \Gr ~:~ x=a-b \,,a\in A\,,b\in B \} \,.$$
Also denote the {\it additive energy} of a set $A$ by
$$
\E(A) = \E_2 (A) = |\{ a_1-a'_1 = a_2-a'_2 ~:~ a_1,a'_1,a_2,a'_2 \in A \}| \,,
$$
and
{\it $\E_k (A)$ energy} as
\begin{equation}\label{f:start_E_k}
\E_k (A) = |\{ a_1-a'_1 = a_2-a'_2 = \dots = a_k-a'_k ~:~ a_1,a'_1,\dots, a_k,a'_k \in A \}| \,.
\end{equation}
The special case $k=1$
gives us $\E_1 (A) = |A|^2$
because of there is no any restriction in the set from (\ref{f:start_E_k}).
So, the cardinality of a set can be considered as a degenerate sort of energy.
Note that a trivial upper bound for $\E_k (A)$ is $|A|^{k+1}$.
Now recall a well--known Balog--Szemer\'{e}di--Gowers Theorem \cite{TV}.
\begin{theorem}
Let $A\subseteq \Gr$ be a set, and $K\ge 1$ be a real number.
Suppose that $\E(A) \ge |A|^3 / K$.
Then there is $A' \subseteq A$ such that $|A'| \gg |A|/K^C$ and
\begin{equation}\label{f:BSzG_introduction}
|A'-A'| \ll K^C |A'| \,,
\end{equation}
where $C>0$ is an absolute constant.
\label{t:BSzG_introduction}
\end{theorem}
So, Balog--Szemer\'{e}di--Gowers Theorem can be considered as a result
about
the structure
of sets $A$ having the extremal (in terms of its cardinality or $\E_1(A)$ in other words) value of $\E(A)$.
Namely,
any of such a set has a subset $A'$ with the extremal value
of the cardinality of its difference set.
These sets $A'$ are called sets with {\it small doubling}.
On the other hand, it is easy to obtain, using the Cauchy--Schwarz inequality,
that any $A$ having subset $A'$
such that
(\ref{f:BSzG_introduction}) holds, automatically has polynomially large energy
$\E(A) \gg_K |A|^3$ (see e.g. \cite{TV}).
Moreover, the structure of sets with small doubling is known more or less thanks to a well--known
Freiman's theorem (see \cite{TV} or \cite{Sanders_2A-2A}).
Thus,
Theorem \ref{t:BSzG_introduction}
finds subsets of $A$
with rather rigid structure
and,
actually,
it is a criterium for a set $A$ to be a set with large (in terms of $\E_1(A)$) the additive energy:
$\E(A) \sim_K |A|^3 \sim_K (\E_1 (A))^{3/2}$.
In the paper we consider another extremal relations between different energies and describe the structure
of sets having these critical relations.
Such kind of theorems have plenty of applications.
It is obvious for Balog--Szemer\'{e}di--Gowers theorem, see
e.g. \cite{TV}, \cite{BK_AP3}, \cite{Gow_4}, \cite{Gow_m}, \cite{KSh}, \cite{M_R-N_S} and so on;
for recent
applications using
critical relations between energies $\E_2 (A)$ and $\E_3 (A)$,
see e.g. \cite{s_ineq}, \cite{s_mixed} and others.
Before formulate our main results let us recall a beautiful theorem of Bateman--Katz
\cite{BK_AP3}, \cite{BK_struct}
which is another example of theorems are called "structural"\, by us.
\begin{theorem}
Let $A \subseteq \Gr$ be a symmetric set, $\tau_0$, $\sigma_0$ be nonnegative real numbers
and $A$ has the property that for any $A_*\subseteq A$, $|A_*| \gg |A|$ the following holds
$\E(A_*) \gg \E (A) = |A|^{2+\tau_0}$.
Suppose that $\T_4 (A) \ll |A|^{4+3\tau_0+\sigma_0}$.
Then there exists a function $f_{\tau_0} : (0,1) \to (0,\infty)$ with $f_{\tau_0} (\eta) \to 0$ as $\eta \to 0$
and a number $0\le \a \le \frac{1-\tau_0}{2}$ such that there are sets $X_j,H_j\subseteq \Gr$, $B_j\subseteq A$,
$j\in [|A|^{\a-f_{\tau_0} (\sigma_0)}]$ with
\begin{equation}\label{f:BK_1}
|H_j| \ll |A|^{\tau_0+\a+ f_{\tau_0} (\sigma_0)} \,,\quad \quad |X_j| \ll |A|^{1-\tau_0-2\a+ f_{\tau_0} (\sigma_0)} \,,
\end{equation}
\begin{equation}\label{f:BK_2}
|H_j-H_j| \ll |H_j|^{1+f_{\tau_0} (\sigma_0)} \,,
\end{equation}
\begin{equation}\label{f:BK_3}
|(X_j+H_j) \cap B_j| \gg |A|^{1-\a-f_{\tau_0} (\sigma_0)} \,,
\end{equation}
and $B_i \cap B_j = \emptyset$ for all $i\neq j$.
\label{t:BK_structural}
\end{theorem}
Here $\T_4 (A)$ is the number of solutions of the equation $a_1+a_2+a_3+a_4=a'_1+a'_2+a'_3+a'_4$,
$a_1,a_2,a_3,a_4,a'_1,a'_2,a'_3,a'_4 \in A$
and this characteristic is another sort of energy.
One can check that any set satisfying (\ref{f:BK_1})---(\ref{f:BK_3}), $\E(A) = |A|^{2+\tau_0}$
and all another conditions of the theorem is an example of a set having $\T_4 (A) \ll |A|^{4+3\tau_0+\sigma_0}$.
Note that if $\E (A) = |A|^{2+\tau_0}$ then by the H\"{o}lder inequality one has $\T_4 (A) \ge |A|^{4+3\tau_0}$.
Thus, Theorem \ref{t:BK_structural} gives us a full description of sets having critical relations between
a pair of two energies:
$\E(A)$ and $\T_4 (A)$.
There are two opposite extremal cases in Theorem \ref{t:BK_structural} : $\a = 0$ and $\a = \frac{1-\tau_0}{2}$.
For simplicity consider the situation when $\Gr = \f_2^n$.
In the case $\a = 0$ by Bateman--Katz result our set $A$, roughly speaking, is close to a set of the form
$H \dotplus \Lambda$, where $\dotplus$ means the direct sum,
$H\subseteq \mathbf{F}_2^n$ is a subspace, and $\Lambda \subseteq \mathbf{F}_2^n$ is a dissociated set (basis),
$|\L| \sim |A| /|H| \sim |A|^{1-\tau_0}$.
These sets are interesting in its own right being counterexamples in many problems of additive combinatorics.
The reason for this is that they have mixed properties : on the one hand they contain translations $H+\la$, $\la \in \L$
of really structured set $H$ but on the other hand they have also some random properties, for example, its Fourier
coefficients (see the definition in section \ref{sec:definitions}) are small.
Our first result says that a set $A$ is close to a set of the form $H \dotplus \Lambda$ iff there is
the critical relation between $\E_3 (A)$ and $\E(A)$, that is $\E_3(A) \gg |A|\E(A)$, more precisely,
see Theorem \ref{t:H+L_description}.
In the situation $\a = \frac{1-\tau_0}{2}$, $\Gr = \f_2^n$ our set $A$ looks like a union
of (additively) disjoint subspaces $H_1,\dots,H_k$ (see example (iii) from \cite{Sanders_survey2})
with $k = |A|^{\frac{1+\tau_0}{2}}$.
Such sets
can be
called {\it self--dual} sets, see \cite{s_mixed}.
In our second result we show that, roughly speaking,
any such a set has critical relation between $\E_3 (A)$
(more precisely
$\E(A) \cdot \E_4(A)$)
and so--called
Gowers $U^3$--norm of the set $A$ (see the definition in section \ref{sec:Gowers}) and vice versa.
Theorem \ref{t:self-dual} contains the
exact
formulation.
These two structural results on sets having critical relations between a pair of its energies are the hearth of our paper.
In the opposite of Theorem \ref{t:BK_structural}
almost
all bounds of the paper are polynomial, excluding, of course,
the dependence on the number $k$ of the considered energies
$\E_k$, $\T_k$ or $U^k$ if its appear.
Moreover
the first
structural theorem
hints us a partial answer to the following
important question.
Consider the difference set $D=A-A$ or the sumset $S=A+A$ of an arbitrary set $A$.
What can we say nontrivial about the energies of $D$, $S$ in terms of the energies of $A$?
In view of the first of the main examples
above,
that is $\Gr = \f_2^n$, $A=H\dotplus \L$,
$|\L| = K$, $\E(A) \sim |A|^3 / K$
we cannot hope to obtain a nontrivial bound for the additive energy of $D$ or $S$ because in the case
$D=S=H \dotplus (\L+\L)$, and so it has a similar structure to $A$ with $\L$ replacing by $\L+\L$.
On the other hand, we know that the sets of the form $H\dotplus \L$ have large $\E_3$ energy.
Thus, one can hope to
obtain
a good lower bound for $\E_3 (D)$ and $\E_3 (S)$.
It turns out to be the case and we prove it in section \ref{sec:sumsets1}.
Roughly speaking, our result asserts that if $|D|=K|A|$, $\E(A) \ll |A|^3 /K$ then
\begin{equation}\label{f:start_E3D}
\E_3 (D) \gg K^{7/4} |A|^4 \,,
\end{equation}
and a similar inequality for $A+A$.
The paper is organized as follows.
We start with definitions and notations used in the paper.
In the next section we give several characterisations of sets of the form $A=H\dotplus \L$, where
$H$ is a set with small doubling, $\L$ is a "dissociated"\, set.
Also we consider a "dual"\, question on sets having critical relations between $\T_4$ and $\E$ energies,
that is the situation when $\T_4 (A)$ is large in terms on $\E(A)$.
It was proved that, roughly, $A$ contains a large subset $A'$ such that the sequence $A', 2A', 3A', \dots$
is stabilized at the second step, namely, $A'+A'$ is a set with small doubling
and, besides, $|A'+A'| \approx |A|^4 / \E (A)$
in the
only case when
$\T_4 (A) \gg |A|^2 \E(A)$, see Theorem \ref{t:T_3_and_E_critical}.
Section \ref{sec:sumsets1} contains the proof of inequality (\ref{f:start_E3D})
and we make some preliminaries to this in section \ref{sec:sumsets}.
For example, we obtain in the section an interesting characterisation of sumsets $S=A+A$ or difference sets $D=A-A$ with
extremal cardinalities of intersections
$$|A| \le |D\cap (D+x_1) \cap \dots \cap (D+x_s)| \le |A|^{1+o(1)} \,,$$
and, similarly, for $S$,
see Theorem \ref{t:dichotomy_DS}.
It turns out that for such sets $D$, $S$ the set $A$
should have either very small $O(|A|^{k+o(1)})$
the energy $\E_k (A)$
or very large $\gg |A|^{3-o(1)}$
the additive energy.
In other words either $A$ has "random behaviour"\,
or, in contrary, is very structured.
Clearly, both situations are realized: the first one in the situation when
$A$ is a fair random set (and hence $A\pm A$ has almost no structure) and the second one
if $A$ is a set with small doubling, say.
In section \ref{sec:Gowers} we consider some simple properties of Gowers norms of
{\it the characteristic function of a set $A$} and
prove a preliminary result on the connection of $\E(A)$ with $\E(A\cap (A+s))$, $s\in A-A$,
see Theorem \ref{t:E(A_s)}.
It gives a partial counterexample to a famous construction of Gowers \cite{Gow_4}, \cite{Gow_m} of uniform sets
with non-uniform intersections $\E(A\cap (A+s))$
(see the definitions in \cite{Gow_4}, \cite{Gow_m} or \cite{TV}).
We show that although all sets $A\cap (A+s)$ can be non-uniform but there is always $s\neq 0$
such that $\E(A\cap (A+s)) \ll |A\cap (A+s)|^{3-c}$, $c>0$, provided by some weak conditions take place.
This question was asked to the author by T. Schoen.
In the next section we develop the investigation from the previous one and
characterize all sets with critical relation between Gowers $U^3$--norm and the energies $\E,\E_4$ or $\E_3$.
Also we consider some questions on finding in $A$ a family of disjoint sets $A\cap (A+s)$ or its large disjoint subsets.
A lot of results of the paper such as Bateman--Katz theorem are
proved under
some regular conditions
on $A$.
For example, the
assumption
from Theorem \ref{t:BK_structural} require that
for all $A_*\subseteq A$, $|A_*| \gg |A|$ the following holds $\E(A_*) \gg \E (A)$.
We call the conditions as connectedness of our set $A$
(see the definitions from sections \ref{sec:preliminaries}, \ref{sec:Gowers})
and prove in the appendix that any set contains some large connected subset.
Basically, we
generalize the method from \cite{s_doubling}.
Thus, we have characterized two extremal
situations
of Theorem \ref{t:BK_structural} in terms of energies.
Is there some similar characterisation for other cases of the result?
Do exist criteria in terms of
energies for another families of sets?
Finally,
are there
further
characteristics of sumsets/difference sets which separate it from arbitrary sets?
The author is grateful to Vsevolod F. Lev and Tomasz Schoen
for useful discussions.
\section{Definitions}
\label{sec:definitions}
Let $\Gr$ be an abelian group.
If $\Gr$ is finite then denote by $N$ the cardinality of $\Gr$.
It is well--known~\cite{Rudin_book} that the dual group $\FF{\Gr}$ is isomorphic to $\Gr$ in the case.
Let $f$ be a function from $\Gr$ to $\mathbb{C}.$ We denote the Fourier transform of $f$ by~$\FF{f},$
\begin{equation}\label{F:Fourier}
\FF{f}(\xi) = \sum_{x \in \Gr} f(x) e( -\xi \cdot x) \,,
\end{equation}
where $e(x) = e^{2\pi i x}$
and $\xi$ is a homomorphism from $\FF{\Gr}$ to $\R/\Z$ acting as $\xi : x \to \xi \cdot x$.
We rely on the following basic identities
\begin{equation}\label{F_Par}
\sum_{x\in \Gr} |f(x)|^2
=
\frac{1}{N} \sum_{\xi \in \FF{\Gr}} \big|\widehat{f} (\xi)\big|^2 \,,
\end{equation}
\begin{equation}\label{svertka}
\sum_{y\in \Gr} \Big|\sum_{x\in \Gr} f(x) g(y-x) \Big|^2
= \frac{1}{N} \sum_{\xi \in \FF{\Gr}} \big|\widehat{f} (\xi)\big|^2 \big|\widehat{g} (\xi)\big|^2 \,,
\end{equation}
and
\begin{equation}\label{f:inverse}
f(x) = \frac{1}{N} \sum_{\xi \in \FF{\Gr}} \FF{f}(\xi) e(\xi \cdot x) \,.
\end{equation}
If
$$
(f*g) (x) := \sum_{y\in \Gr} f(y) g(x-y) \quad \mbox{ and } \quad
(f\circ g) (x) := \sum_{y\in \Gr} f(y) g(y+x)
$$
then
\begin{equation}\label{f:F_svertka}
\FF{f*g} = \FF{f} \FF{g} \quad \mbox{ and } \quad \FF{f \circ g} = \FF{f^c} \FF{g} = \ov{\FF{\ov{f}}} \FF{g} \,,
\end{equation}
where for a function $f:\Gr \to \mathbb{C}$ we put $f^c (x):= f(-x)$.
Clearly, $(f*g) (x) = (g*f) (x)$ and $(f\c g)(x) = (g \c f) (-x)$, $x\in \Gr$.
The $k$--fold convolution, $k\in \N$ we denote by $*_k$,
so $*_k := *(*_{k-1})$.
We use in the paper the same letter to denote a set
$S\subseteq \Gr$ and its characteristic function $S:\Gr\rightarrow \{0,1\}.$
Clearly, $S$ is the characteristic function of a set iff
\begin{equation}\label{f:char_char}
\FF{S} (x) = N^{-1} (\ov{\FF{S}} \c \FF{S}) (x) \,.
\end{equation}
Write $\E(A,B)$ for the {\it additive energy} of two sets $A,B \subseteq \Gr$
(see e.g. \cite{TV}), that is
$$
\E(A,B) = |\{ a_1+b_1 = a_2+b_2 ~:~ a_1,a_2 \in A,\, b_1,b_2 \in B \}| \,.
$$
If $A=B$ we simply write $\E(A)$ instead of $\E(A,A).$
Clearly,
\begin{equation}\label{f:energy_convolution}
\E(A,B) = \sum_x (A*B) (x)^2 = \sum_x (A \circ B) (x)^2 = \sum_x (A \circ A) (x) (B \circ B) (x)
\,.
\end{equation}
and by (\ref{svertka}),
\begin{equation}\label{f:energy_Fourier}
\E(A,B) = \frac{1}{N} \sum_{\xi} |\FF{A} (\xi)|^2 |\FF{B} (\xi)|^2 \,.
\end{equation}
Let
$$
\T_k (A) := | \{ a_1 + \dots + a_k = a'_1 + \dots + a'_k ~:~ a_1, \dots, a_k, a'_1,\dots,a'_k \in A \} |
=
\frac{1}{N} \sum_{\xi} |\FF{A} (\xi)|^{2k}
$$
and more generally
$$
\T_k (A_1,\dots,A_k) :=
| \{ a_1 + \dots + a_k = a'_1 + \dots + a'_k ~:~ a_1,a'_1 \in A_1, \dots, a_k, a'_k \in A_k \} | \,.
$$
Let also
$$
\sigma_k (A) := (A*_k A)(0)=| \{ a_1 + \dots + a_k = 0 ~:~ a_1, \dots, a_k \in A \} | \,.
$$
Notice that for a symmetric set $A$ that is $A=-A$ one has $\sigma_2
(A) = |A|$ and $\sigma_{2k} (A) = \T_k (A)$.
Having a set $P\subseteq A-A$ we write $\sigma_P (A) := \sum_{x\in P} (A\c A) (x)$.
For a sequence $s=(s_1,\dots, s_{k-1})$ put
$A^B_s = B \cap (A-s_1)\dots \cap (A-s_{k-1}).$
If $B=A$ then write $A_s$ for $A^A_s$.
Let
\begin{equation}\label{f:E_k_preliminalies}
\E_k(A)=\sum_{x\in \Gr} (A\c A)(x)^k = \sum_{s_1,\dots,s_{k-1} \in \Gr} |A_s|^2
\end{equation}
and
\begin{equation}\label{f:E_k_preliminalies_B}
\E_k(A,B)=\sum_{x\in \Gr} (A\c A)(x) (B\c B)(x)^{k-1} = \sum_{s_1,\dots,s_{k-1} \in \Gr} |B^A_s|^2
\end{equation}
be the higher energies of $A$ and $B$.
The second formulas in (\ref{f:E_k_preliminalies}), (\ref{f:E_k_preliminalies_B})
can be considered as the definitions of $\E_k(A)$, $\E_k(A,B)$ for non integer $k$, $k\ge 1$.
Similarly, we write $\E_k(f,g)$ for any complex functions $f$, $g$
and more generally
$$
\E_k (f_1,\dots,f_{k}) = \sum_x (f_1 \c f_1) (x) \dots (f_k \c f_k) (x) \,.
$$
Putting $\E_1 (A) = |A|^2$.
For a set $P\subseteq \Gr$ write $\E^P_k (A) := \sum_{s\in P} |A_s|^k$, $\E^P (A) := \E^P_2 (A)$.
We
put
$\E^*_k (A)$ for $\E^*_k (A) = \sum_{s\neq 0} |A_s|^k$.
Clearly,
\begin{eqnarray}\label{f:energy-B^k-Delta}
\E_{k+1}(A,
B)&=&\sum_x(A\c A)(x)(B\c B)(x)^{k}\nonumber \\
&=&\sum_{x_1,\dots, x_{k-1}}\Big (\sum_y A(y)B(y+x_1)\dots
B(y+x_{k})\Big )^2 =\E(\Delta_k (A),B^{k}) \,,
\end{eqnarray}
where
$$
\Delta (A) = \Delta_k (A) := \{ (a,a, \dots, a)\in A^k \}\,.
$$
We also put $\Delta(x) = \Delta (\{ x \})$, $x\in \Gr$.
\bigskip
Quantities $\E_k (A,B)$ can be written in terms of generalized convolutions.
\begin{definition}
Let $k\ge 2$ be a positive number, and $f_0,\dots,f_{k-1} : \Gr \to \C$ be functions.
Denote by
$${\mathcal C}_k (f_0,\dots,f_{k-1}) (x_1,\dots, x_{k-1})$$
the function
$$
\Cf_k (f_0,\dots,f_{k-1}) (x_1,\dots, x_{k-1}) = \sum_z f_0 (z) f_1 (z+x_1) \dots f_{k-1} (z+x_{k-1}) \,.
$$
Thus, $\Cf_2 (f_1,f_2) (x) = (f_1 \circ f_2) (x)$.
If $f_1=\dots=f_k=f$ then write
$\Cf_k (f) (x_1,\dots, x_{k-1})$ for $\Cf_k (f_1,\dots,f_{k}) (x_1,\dots, x_{k-1})$.
\end{definition}
In particular, $(\Delta_k (B) \c A^k) (x_1,\dots,x_k) = \Cf_{k+1} (B,A,\dots,A) (x_1,\dots,x_k)$, $k\ge 1$.
Quantities $\E_k (A)$ and $\T_k (A)$ are "dual"\, in some sense.
For example in \cite{s_mixed}, Note 6.6 (see also \cite{SS1}) it was proved that
$$
\left( \frac{\E_{3/2} (A)}{|A|} \right)^{2k}
\le
\E_k (A) \T_k (A) \,,
$$
provided by $k$ is even.
Moreover, from (\ref{F:Fourier})---(\ref{f:inverse}), (\ref{f:char_char}) it follows that
$\t{\E}_{2k} (\FF{A}) := \sum_{x} ( \ov{\FF{A}} \c \FF{A})^{k} (x) (\FF{A} \c \ov{\FF{A}})^{k} (x)
= N^{2k+1} \T_k (A)$ and $\T_k (|\FF{A}|^2) = N^{2k-1} \E_{2k} (A)$.
\bigskip
For a positive integer $n,$ we set $[n]=\{1,\ldots,n\}.$
Let $x$ be a vector. By $\| x \|$ denote the number of components of $x$.
All logarithms are to base $2.$ Signs $\ll$ and $\gg$ are the usual Vinogradov's symbols
and if the bounds depend on some parameter $M$ {\it polynomially} then we write $\ll_M$, $\gg_M$.
If for two numbers $a$, $b$ the following holds $a \ll_M b$, $b \ll_M a$ then we write $a \sim_M b$.
In particular, $a \sim b$ means $a\ll b$ and $b\ll a$.
All polynomial bounds in the paper can be obtained in explicit way.
\newpage
\section{Preliminaries}
\label{sec:preliminaries}
Let us begin with the famous Pl\"{u}nnecke--Ruzsa inequality (see \cite{petridis} or \cite{TV}, e.g.).
\begin{lemma}
Let $A\subseteq \Gr$ be a set.
Then for all positive integers $n,m$ the following holds
\begin{equation}\label{f:Plunnecke}
|nA-mA| \le K^{n+m} |A| \,.
\end{equation}
\label{l:Plunnecke}
\end{lemma}
We need in several quantitative versions
of the Balog--Szemer\'{e}di--Gowers Theorem.
The first symmetric variant
is due to T. Schoen \cite{schoen_BSzG}.
\begin{theorem}
Let $A\subseteq \Gr$ be a set, $K\geq{1}$ and $\E(A)\geq{\frac{|A|^{3}}{K}}$.
Then there is $A'\subseteq A$ such that
$$
|A'| \gg{\frac{|A|}{K}}\,,
$$
and
$$
|A'-A'| \ll K^4 |A'| \,.
$$
\label{BSG}
\end{theorem}
\bigskip
Also we need in
a version
of Balog--Szemer\'{e}di--Gowers theorem in the asymmetric form, see \cite{TV}, Theorem 2.35.
\begin{theorem}
Let $A,B\subseteq \Gr$ be two sets, $|B| \le |A|$, and $M\ge 1$ be a real number.
Let also $L=|A|/|B|$ and $\eps \in (0,1]$ be a real parameter.
Suppose that
\begin{equation}\label{cond:BSzG_as}
\E (A,B) \ge \frac{|A| |B|^2}{M} \,.
\end{equation}
Then there are two sets $H\subseteq \Gr$, $\L \subseteq \Gr$ and $z\in \Gr$ such that
\begin{equation}\label{f:BSzG_as_1}
|(H+z) \cap B| \gg_\eps M^{-O_\eps (1)} L^{-\eps} |B| \,,
\quad \quad
|\L| \ll_{\eps} M^{O_\eps (1)} L^\eps \frac{|A|}{|H|} \,,
\end{equation}
\begin{equation}\label{f:BSzG_as_2}
|H - H| \ll_{\eps} M^{O_\eps (1)} L^\eps \cdot |H| \,,
\end{equation}
and
\begin{equation}\label{f:BSzG_as_3}
|A\cap (H+\L)| \gg_{\eps} M^{-O_\eps (1)} L^{-\eps} |A| \,.
\end{equation}
\label{t:BSzG_as}
\end{theorem}
The next lemma is a special case of Lemma 2.8 from \cite{SV}.
In particular, it gives us a connection between $\E_3 (A)$
and $\E (A,A_s)$, see e.g \cite{SS1}.
\begin{lemma}
Let $A\subseteq \Gr$ be a set.
Then for every $k,l\in \N$
$$
\sum_{s,t:\atop \|s\|=k-1,\,\, \|t\|=l-1} \E(A_s,A_t)=\E_{k+l}(A)
\,.
$$
In particular,
$$
\E_3 (A) = \sum_s \E (A,A_s) \,.
$$
\label{l:E_3_A_s}
\end{lemma}
Now recall a lemma from \cite{SS3}, \cite{s_mixed}.
\begin{lemma}
\label{corpop}
Let $A$ be a subset of an abelian group, $P_* \subseteq A-A$.
Then
\begin{equation*}
\sum_{s\in P_*} |A\pm A_s| \geq \frac{\sigma^2_{P_*} (A) |A|^2}{\E_3(A) }
\end{equation*}
and
\begin{equation}\label{f:corpop2}
\E_3 (P_*,A,A) \cdot \E_3 (A) \ge \frac{\E^2 (A) \sigma^4_{P_*} (A)}{|A|^{6}}
\,.
\end{equation}
\end{lemma}
\bigskip
Let also give a simple
Corollary 18 from \cite{s_ineq}.
\begin{lemma}
Let $A\subseteq \Gr$ be a set.
Then
$$
\sum_{s} \frac{|A_s|^2}{|A\pm A_s|} \le \frac{\E_3 (A)}{|A|^2}
\,.
$$
\label{l:E_3_weight}
\end{lemma}
We give a
small
generalization of Proposition 11 from \cite{SS1},
see also \cite{M_R-N_S}.
\begin{lemma}
Let $A\subseteq \Gr$ be a set, $n,m\ge 1$ be positive integers.
Then
\begin{equation}\label{tmp:12.05.2014_1}
|A^{n+m} - \D(A)| \ge |A|^m |A^{n} - \D(A)| \,,
\end{equation}
and
\begin{equation}\label{tmp:12.05.2014_2}
|A^{n+m} + \D(A)| \ge |A|^m \max\{ |A^{n} + \D(A)|, |A^{n} - \D(A)| \} \,.
\end{equation}
In particular,
\begin{equation}\label{f:A^2_pm_p1}
|A^2 - \D(A)| = \sum_{s\in A-A} |A-A_s| \ge |A| |A-A| \,,
\end{equation}
and
\begin{equation}\label{f:A^2_pm_p2}
|A^2 + \D(A)| = \sum_{s\in A-A} |A+A_s| \ge |A| \max\{ |A+A|,|A-A| \} \,.
\end{equation}
\label{l:A^2_pm}
\end{lemma}
\begin{proof}
In view of \cite{SS1}, Proposition 11 it remains to prove
the second bound from (\ref{tmp:12.05.2014_2})
in the case $m=1$ only, namely, that
$|A^{n+1} + \D(A)| \ge |A| |A^{n} - \D(A)|$, $n\ge 1$.
But $(a_1+a,\dots,a_{n}+a,a_{n+1}+a) \in A^{n+1} + \D(A)$ iff $a_{n+1}\in A_{s_1,\dots,s_n}$,
where $s_j = a_j-a_{n+1}$, $(s_1,\dots,s_n) \in A^n - \D(A)$.
Thus
$$
|A^{n+1} + \D(A)| = \sum_{(s_1,\dots,s_n) \in A^n - \D(A)} |A+A_{s_1,\dots,s_n}|
\ge
|A| |A^{n} - \D(A)|
$$
and the result follows.
$\hfill\Box$
\end{proof}
\bigskip
We will use very often the Katz--Koester trick \cite{kk}
\begin{equation}\label{f:KK_trick}
A - A_s \subseteq (A-A)_{-s}\,, \quad \quad \quad A + A_s \subseteq (A+A)_s \,,
\end{equation}
and its generalization (see e.g. \cite{SV})
\begin{equation}\label{f:KK_trick_new}
A - A_{\v{x}} \subseteq (A-A)_{-\v{x}} \,, \quad \quad \quad A + A_{\v{x}} \subseteq (A+A)_{\v{x}} \,.
\end{equation}
\bigskip
Finally, recall some results from \cite{s_mixed}.
We begin with an analog of a definition from \cite{s_doubling}.
\begin{definition}
Let $\a > 1$ be a real number, $\beta,\gamma \in [0,1]$.
A set $A\subseteq \Gr$ is called $(\a,\beta,\gamma)$--connected
if for any $B \subseteq A$, $|B| \ge \beta|A|$
the following holds
$$
\E_\a (B) \ge \gamma \left( \frac{|B|}{|A|} \right)^{2\a} \E_\a (A) \,.
$$
\label{def:conn}
\end{definition}
Thus, a set from Theorem \ref{t:BK_structural} is a $(2,\beta,\gamma)$--connected set
with $\beta,\gamma \gg 1$.
The H\"{o}lder inequality implies that if
$\E_\a (A) \le \gamma^{-1} |A|^{2\a} |A-A|^{1-\a}$ then $A$ is $(\a,\beta,\gamma)$--connected for any $\beta$.
As was proved in \cite{s_doubling} that for $\a=2$ {\it every} set $A$ always contains large connected subset.
For
integers $\a>2$, see the Appendix.
\bigskip
Our first lemma from \cite{s_mixed} (where some operators were used in the proof)
is about a nontrivial lower bound for $\E_s (A)$, $s\in [1,2]$
in terms of $\E(A)$.
\begin{lemma}
Let $A\subseteq \Gr$ be a set, and $\beta,\gamma \in [0,1]$.
Suppose that $A$ is $(2,\beta,\gamma)$--connected with $\beta \le 1/2$.
Then for any $s\in [1,2]$ the following holds
\begin{equation}\label{f:connected}
\E_s (A) \ge 2^{-5} \gamma |A|^{1-s/2} \E^{s/2} (A) \,.
\end{equation}
\label{l:connected}
\end{lemma}
The second lemma from \cite{s_mixed} is about an upper bound for eigenvalues of some operators.
To avoid of using the operators notation we formulate the result in the following way.
\begin{lemma}
Let $A\subseteq \Gr$ be a set.
Then for an arbitrary function $f : A \to \C$ one has
\begin{equation}\label{f:eigen_A}
\E(A,f) \le \E^{1/2}_3 (A) \| f\|_2^2 \,.
\end{equation}
Further, there is a set $A'\subseteq A$, $|A'| \ge |A|/2$, namely,
\begin{equation}\label{f:A'_def}
A' := \{ x ~:~ ((A*A) \c A) (x) \le 2 \E(A) |A|^{-1} \}
\end{equation}
such that for any
function $f : A' \to \C$ the following holds
\begin{equation}\label{f:eigen_A'}
\E(A,f) \le \frac{2\E(A)}{|A|} \cdot \| f\|_2^2 \,.
\end{equation}
Moreover for any even real function $g$ there is a set $A'\subseteq A$, $|A'| \ge |A|/2$
such that for any
function $f : A' \to \C$ the following holds
\begin{equation}\label{f:eigen_A''}
\sum_x g(x) (\ov{f} \c f)(x) = \sum_x g(x) (f\c \ov{f})(x)
\le 2 |A|^{-1} \sum_x g(x) (A\c A) (x) \cdot \| f\|_2^2 \,.
\end{equation}
\label{l:eigen_A'}
\end{lemma}
Note that
for the characteristic functions $f$ of
sets from
$A$
bound
(\ref{f:eigen_A}) can be obtained using the Cauchy--Schwarz inequality.
Further, estimate (\ref{f:eigen_A''}) is a generalization of (\ref{f:eigen_A'})
which was
proved
in \cite{s_mixed}, see Lemma 44. Bound (\ref{f:eigen_A''}) can be obtained in a similar way.
\bigskip
We finish the section noting a generalization of formula (\ref{f:corpop2}) of Lemma \ref{corpop}.
That is just a part of Lemma 4.2 from \cite{s_mixed}.
\begin{lemma}
Let $A,B\subseteq \Gr$ be finite sets, $S\subseteq \Gr$ be a set such that
$A+B \subseteq S$.
Suppose that $\psi$ is a
function on $\Gr$.
Then
\begin{equation}\label{f:T_A,B}
|B|^2 \cdot \left( \sum_{x} \psi(x) (A\c A)(x) \right)^2
\le
\E_3(B,A) \sum_{x} \psi^2 (x) (S\c S)(x) \,.
\end{equation}
\label{l:T_A,B}
\end{lemma}
\section{Structural results}
\label{sec:structural}
In this section we obtain several general structural results,
some of which have applications to sum--products phenomenon, for example.
These results are closely related to the Balog-Szemer\'{e}di-Gowers Theorem, and we adopt the convention of writing $\Gr$ as an additive group.
The proofs follow the arguments from \cite{SS1} and \cite{SS2}.
Now we formulate the first result of the section.
\begin{proposition}
Let $A\subseteq \Gr$ be a finite set, and $M\ge 1$, $\eta \in (0,1]$ be real numbers.
Let $\E(A) = |A|^3/K$.
Suppose that for some set $P\subseteq A-A$ the following holds
\begin{equation}\label{cond:eta}
\sum_{s\in P} (A\c A) (s) = \eta |A|^2 \,,
\end{equation}
and
\begin{equation}\label{cond:M}
\sum_{s\in P} |A \pm A_s| \le M K |A|^2 \,.
\end{equation}
Then
for any $\eps \in (0,1)$,
there are two sets $H\subseteq \Gr$, $\L \subseteq \Gr$ and $z\in \Gr$ such that
\begin{equation}\label{f:E_3_and_E_critical_1'}
|(H+z) \cap A| \gg_{M,\eta^{-1},K^\eps} \frac{\E(A)}{|A|^2} \,,
\quad \quad
|\L| \ll_{M,\eta^{-1},K^\eps} \frac{|A|}{|H|} \,,
\end{equation}
\begin{equation}\label{f:E_3_and_E_critical_2'}
|H - H| \ll_{M,\eta^{-1},K^\eps} |H| \,,
\end{equation}
and
\begin{equation}\label{f:E_3_and_E_critical_3'}
|A\bigcap (H+\L)| \gg_{M,\eta^{-1},K^\eps} |A| \,.
\end{equation}
\label{p:A^2-D(A)}
\end{proposition}
\begin{proof}
Using Lemma \ref{corpop} with $P_*=P$, we see that
\begin{equation}\label{tmp:08.05.2014_1}
\E_3 (A) \ge \frac{\eta^2 |A|^4}{M K} \,.
\end{equation}
Note that
\begin{align*}
\sum_{s ~:~ |A_s|<\frac{|A|\eta^2}{2KM}} \E(A,A_s)&\leq{\left(\frac{|A|\eta^2}{2KM}\right)\sum_s|A_s||A|}
\\&=\frac{|A|^4\eta^2}{2KM} \,.
\end{align*}
Applying Lemma \ref{l:E_3_A_s}, that is the formula $\E_3 (A) = \sum_s \E(A,A_s)$,
combining with (\ref{tmp:08.05.2014_1}), we get
\begin{equation}\label{TMP:05.10.2013}
\sum_{s ~:~ |A_s| \ge 2^{-1} \eta^2 M^{-1} K^{-1} |A| } \E(A,A_s) \ge \frac{\eta^2 |A|^4}{2M K} \,.
\end{equation}
Put
$$
\mu := \max_{s ~:~ |A_s| \ge 2^{-1} \eta^2 M^{-1} K^{-1} |A| } \frac{\E(A,A_s)}{|A| |A_s|^2 } \,.
$$
Using (\ref{TMP:05.10.2013}), we have
$$
\mu |A| \E (A) \ge \mu |A| \cdot \sum_{s ~:~ |A_s| \ge 2^{-1} \eta^2 M^{-1} K^{-1} |A| } |A_s|^2
\ge
\frac{\eta^2 |A|^4}{2M K} \,.
$$
Thus, $\mu \ge \frac{\eta^2}{2M}$.
Hence there is an $s$
with
$|A_s| \ge 2^{-1} \eta^2 M^{-1} K^{-1} |A|$
and
such that\\
${\E(A,A_s) \ge 2^{-1} M^{-1} \eta^2|A||A_s|^2}$.
Applying the asymmetric version of Balog--Szemer\'{e}di--Gowers Theorem \ref{t:BSzG_as},
we find two sets $\L,H$ such that
(\ref{f:E_3_and_E_critical_1'})---(\ref{f:E_3_and_E_critical_3'}) take place.
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
We write the fact that sets $A,H,\L$ satisfy (\ref{f:E_3_and_E_critical_1'})---(\ref{f:E_3_and_E_critical_3'})
with $\eta \gg 1$ as
\begin{equation}
A \approx_{M,K^\eps} \L \dotplus H.
\label{approxdefn}
\end{equation}
Note that the degree of polynomial dependence in formula (\ref{approxdefn}) is a function on $\eps$.
\begin{example}
Let $H\subseteq \mathbf{F}_2^n$ be a subspace and $\Lambda \subseteq \mathbf{F}_2^n$ be a dissociated set (basis).
Put $A=H \dotplus \Lambda$, where $\dotplus$ means the direct sum, and $|\Lambda| = K$.
Detailed discussion of the example can be found, e.g. in \cite{s_mixed}.
If $s\in H$ then $A_s = A$ and hence $A+A_s = A+A$.
If $s\in (A+A) \setminus H$ then $A_s$ is the disjoint union of two shifts of $H$
and thus $|A+A_s| \le 2|A|$.
Whence
$$
\sum_{s\in A+A} |A+A_s| \le |H| |A+A| + 2|A+A||A| \ll K|A|^2 \,,
$$
and $\E(A) \sim |A|^3 /K$.
It means that condition (\ref{cond:M}) takes place in the case
$A=H \dotplus \Lambda$.
\end{example}
Taking $P=A-A$ and applying Proposition \ref{p:A^2-D(A)}
as well as formulas (\ref{f:A^2_pm_p1}), (\ref{f:A^2_pm_p2}) of Lemma \ref{l:A^2_pm},
we obtain the following consequence.
\begin{corollary}
Let $A\subseteq \Gr$ be a set, $M\in{\mathbb{R}}$, $\eps \in (0,1)$ and $\E(A) = |A|^3/K$.
Then either
$$
|A^2 \pm \D(A)| \ge M K |A|^2
$$
or
$A \approx_{M,K^{\eps}} \L \dotplus H$.
\label{c:A^2-D(A)}
\end{corollary}
Note that for any set $A\subseteq \Gr$ with $\E(A) = |A|^3/K$
the inequality $|A^2 \pm \D(A)| \ge K |A|^2$ follows from Lemma \ref{corpop}
and a trivial
estimate
$\E_3(A) \le |A| \E(A)$.
We will deal
with the reverse condition $\E_3(A) \gg |A| \E(A)$
in Proposition \ref{p:E_3_and_E_critical}
and Theorem \ref{t:E_3_and_E_critical} below.
The next corollary shows that if a set $A$ is not close
to a set of the form $\L \dotplus H$ then there is some imbalance
(in view of Pl\"{u}nnecke--Ruzsa inequality (\ref{f:Plunnecke}))
between the doubling constant
and the additive energy of $A$ or $A-A$.
\begin{corollary}
Let $A\subseteq \Gr$ be a set, $M$, $\eps \in (0,1)$ be real numbers and
$|(A-A) \pm (A-A)| \gg |A-A|^3 / |A|^2$.
Then either
$$
\E(A) \gg \frac{M^{1/2} |A|^4}{|A-A|}
\quad \mbox{ or } \quad \E(A-A) \gg \frac{M |A-A|^4}{|(A-A) \pm (A-A)|}
$$
or
$A \approx_{M,K^{\eps}} \L \dotplus H$, where
$K = |A-A|/|A|$.
\label{c:3A}
\end{corollary}
\begin{proof}
Put $D=A-A$.
Suppose that $\E(A) \ll \frac{M^{1/2} |A|^4}{|D|}$ because otherwise we are done.
In view of Corollary \ref{c:A^2-D(A)} one can assume that $\sum_{s} |A-A_s| \ge M^{1/2} |A| |D|$.
Thus, by the Katz--Koester trick (\ref{f:KK_trick})
\begin{equation}\label{f:KK_trick'}
A - A_s \subseteq (A-A)_{-s}\,, \quad \quad \quad A + A_s \subseteq (A+A)_s \,,
\end{equation}
and the Cauchy--Schwarz inequality, we get
$$
|D| \E(D) \ge \left( \sum_{s\in D} (D\c D) (s) \right)^2 \ge \left( \sum_{s\in D} |A-A_s| \right)^2
\ge (M^{1/2} |A| |D|)^2 \,.
$$
Hence
$$
\E(D) \ge M^{} |D|^{} |A|^2 \gg \frac{M^{} |D|^4}{|(A-A) \pm (A-A)|}
$$
where the assumption of the corollary has been used.
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
The quantities $\E(A\pm A)$ (and hence $|A^2 \pm \D(A)|$ in view of Lemma \ref{l:A^2_pm},
see also Proposition \ref{pr:E_k(D)_simple} below)
appear in sum--products results (in multiplicative form).
For example, in \cite{M_R-N_S} the following
theorem was proved.
\begin{theorem}
Let $A,B\subseteq \R$ be finite sets.
Then
\begin{equation*}\label{f:main_intr_2_new}
|B+AA|^3 \gg \frac{|B| \E^{\times} (AA)}{\log |A|} \,.
\end{equation*}
\end{theorem}
Here $\E^\times (A) := |\{ a_1 a_2 = a_3 a_4 ~:~ a_1, a_2, a_3, a_4 \in A \}|$.
Thus, by the obtained results, we have, roughly, that either $\E^\times (A)$, $\E^\times (AA)$
can be estimated better then
by Lemma \ref{l:A^2_pm}, formulas (\ref{f:A^2_pm_p1}), (\ref{f:A^2_pm_p2})
or $A$ has the rigid structure $A \approx \L \cdot H$.
Usually the last case is easy to deal with.
Similar methods were used in \cite{M_R-N_S}.
\bigskip
Now we obtain another structural result.
Using Lemma \ref{corpop} as well as a trivial estimate $\E_3 (A) \le |A| \E(A)$
one can derive Proposition \ref{p:A^2-D(A)} from Proposition \ref{p:E_3_and_E_critical}
below.
\begin{proposition}
Let $A\subseteq \Gr$ be a set, and $M\ge 1$ be a real number.
Suppose that
\begin{equation}\label{cond:E_3_and_E_critical}
\E_3 (A) \ge \frac{|A| \E(A)}{M} \,.
\end{equation}
Then there is $A' \subseteq A$ such that
\begin{equation}\label{f:E_3_and_E_critical_1}
|A'| \gg \frac{\E(A)}{|A|^2 (M \log M)^5}
\end{equation}
and
\begin{equation}\label{f:E_3_and_E_critical_2}
|A' - A'| \ll M^{15} \log^{16} M \cdot |A'| \,.
\end{equation}
Further, take any $\eps \in (0,1)$ and
put $K:=\frac{|A|^3}{\E (A)} $.
Then $A \approx_{M,K^\eps} \L \dotplus H$.
\label{p:E_3_and_E_critical}
\end{proposition}
\begin{proof}
First of all prove (\ref{f:E_3_and_E_critical_1}),
(\ref{f:E_3_and_E_critical_2}). Let
$$
P_j = \{ x ~:~ 2^{j-1} |A| / (2^{2} M)
< |A_x| \le 2^{j} |A| / (2^{2} M) \} \,, \quad j\in [L] \,,
$$
where $L=[\log(4M)]$. By the pigeonhole principle there is $j\in [L]$
such that
$$
\frac{|A| \E(A)}{2M L} \le \frac{\E_3 (A)}{2L} \le \sum_{x\in P_j} |A_x|^3 \,.
$$
Put $P=P_j$ and $\D = 2^{j} |A| / (2^{2} M)$.
Thus
\begin{equation}\label{tmp:04.02.2013_2}
\frac{8M \E(A)}{2^{2j} |A|L} \le \sum_x P(x) (A\c A) (x) = \sum_x A(x) (A\c P) (x) \,.
\end{equation}
Hence, by the Cauchy--Schwarz inequality
\begin{equation}\label{tmp:05.02.2013_1}
\frac{2^{6} M^2 \E^2 (A)}{2^{4j} |A|^3 L^2} \le \E(A,P) \le (\E (A))^{1/2} (\E (P))^{1/2} \,.
\end{equation}
Note that
$$\E(A)\geq{\sum_{x\in{P}}|A_x|^2}\geq{\frac{|P||A|^22^{2j-2}}{2^4M^2}},$$
and therefore
\begin{equation}
|P| \le 2^{6} 2^{-2j} M^{2} \E(A) |A|^{-2}.
\label{Pbound}
\end{equation}
It follows that
\begin{equation}\label{tmp:08.02.2013_4}
\E(P) \ge \frac{2^{12} M^4 (\E (A))^3}{2^{8j} |A|^6 L^4} \ge \frac{|P|^3}{2^{6} M^2 2^{2j} L^4}
\ge
\frac{|P|^3}{2^{10} M^4 L^4} := \mu |P|^3 \,.
\end{equation}
By Theorem \ref{BSG} there is $P'\subseteq P$ such that
$|P'| \gg \mu |P|$ and $|P'-P'| \ll \mu^{-4} |P'|$.
Note that
$$\sum_{x\in{A}} (A\c P') (x)=\sum_{x\in{P'}} (A\c A) (x)\geq{\frac{|P'|2^{j-3}|A|}{M}},$$
and so there exists $x\in A$ such that the set $A' := A\cap (P'+x)$ has the size at
least $|P'| 2^{j-3} M^{-1}$. We have
\begin{equation}\label{tmp:08.02.2013_5}
|A'-A'| \le |P' - P'| \ll \mu^{-4} |P'| \ll \mu^{-4} 2^{-j} M |A'| \ll M^{15} L^{16} |A'| \,.
\end{equation}
Finally, from (\ref{tmp:04.02.2013_2}), say, one has
\begin{equation}\label{tmp:06.02.2013_1}
|P| \gg \frac{M^2 \E(A)}{2^{3j} |A|^2 L} \gg \frac{\E(A)}{M |A|^2 L}
\end{equation}
and because of
$$
|A'| \ge |P'| 2^{j-3} M^{-1} \gg \mu |P| \cdot 2^{j-3} M^{-1}
$$
the result follows.
To obtain
(\ref{f:E_3_and_E_critical_1'})---(\ref{f:E_3_and_E_critical_3'}),
that is $A \approx_{M,K^\eps} \L \dotplus H$, $K = |A|^3 \E^{-1} (A)$,
note that by the first inequality of (\ref{tmp:05.02.2013_1}) and the bound
$|P| \le 2^{6} 2^{-2j} M^{2} \E(A) |A|^{-2}$, we have
$$
\E(A,P) \ge \frac{2^{6} M^2 \E^2 (A)}{2^{4j} |A|^3 L^2}
\ge
\frac{|A| |P|^2}{2^6 L^2 M^2} \,.
$$
Also, by the definition of the number $K$, and inequality \eqref{tmp:06.02.2013_1} the following holds
$$|A|/|P| \ll ML K \ll_M K.$$
Applying the asymmetric version of
Balog--Szemer\'{e}di--Gowers Theorem \ref{t:BSzG_as} with $A=A$, $B=P$, and
recalling (\ref{tmp:06.02.2013_1}), we obtain the required inequalities,
excepting the first inequality of (\ref{f:E_3_and_E_critical_1'}),
where it remains to
replace $P$ by $A$.
Put $H' = (H+z) \cap P$.
We have $|H'| \gg_{M,K^\eps} |P|$.
Thus, by the definition of the number $\D$ and estimate (\ref{tmp:06.02.2013_1}), we obtain
\begin{equation}\label{tmp:04.05.2014_1}
\sum_{x\in A} (A \c H') (x) = \sum_{x\in H'} (A\c A) (x) \ge 2^{-1} \D |H'|
\gg_{M,K^\eps} \D |P|
\gg_{M,K^\eps} \frac{\E (A)}{|A|} \,.
\end{equation}
Hence there is $w\in A$ such that
\begin{equation}\label{tmp:04.05.2014_2}
|(H+w) \cap A| \ge |(H'+w) \cap A| \gg_{M,K^\eps} \frac{\E (A)}{|A|^2} \,.
\end{equation}
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Assumption (\ref{cond:E_3_and_E_critical}) of the Proposition \ref{p:E_3_and_E_critical}
is a generalisation of the usual condition $\E (A) \ge \frac{|A|^3}{M}$
(because of $\E(A) |A|^3 M^{-1} \le \E^2 (A) \le \E_3 (A) |A|^2$)
and $\E_3 (A) \ge \frac{|A|^4}{M}$
(because of $\E_3 (A) \ge |A|^4 M^{-1} \ge |A| \E(A) M^{-1}$).
Further, one can check that the same consequences
(\ref{f:E_3_and_E_critical_1'})---(\ref{f:E_3_and_E_critical_3'}) hold
if we replace condition (\ref{cond:E_3_and_E_critical}) by
$\E_s (A) \ge |A| \E_{s-1} (A) / M$, $s\ge 3$.
Let us write the correspondent result.
\begin{theorem}
Let $A\subseteq \Gr$ be a set, $s\ge 3$ be a positive integer, and $M\ge 1$ be a real number.
Suppose that
\begin{equation}\label{cond:E_3_and_E_criticalt}
\E_s (A) \ge \frac{|A| \E_{s-1} (A)}{M} \,.
\end{equation}
Take any $\eps \in (0,1)$ and put $K:=\frac{|A|^s}{\E_{s-1} (A)} $.
Then $A \approx_{s,\, M,\, K^\eps} \L \dotplus H$, $|H| \gg_{s,\, M,\, K^\eps} |A|/K$.
\label{t:E_3_and_E_critical}
\end{theorem}
\begin{proof}
The arguments almost repeat the proof of Proposition \ref{p:E_3_and_E_critical}, so we
skip some details.
Using dyadic pigeonholing and the assumption, we find $P\subseteq A-A$, $\D < |A_x|\le 2\D$, $x\in P$
with
$$
M^{-1} |A| \E_{s-1} (A) \le \E_{s} (A) \ll_{\log M} \sum_{x\in P} |A_x|^s \ll_{\log M} \D^{s-1} \sigma_P (A) \,.
$$
Thus, by the Cauchy--Schwarz inequality
$$
M^{-2} |A| \E^2_{s-1} (A) \D^{-2(s-1)} \ll_{\log M} \E(A,P) \,.
$$
On the other hand $|P| \D^{s-1} \le \E_{s-1} (A)$ and hence
$$
\E(A,P) \gg_{\log M} M^{-2} |A| |P|^2 \,.
$$
Note that by (\ref{cond:E_3_and_E_criticalt}) and our choice of the set $P$, we have
$$
\frac{|A|}{|P|} \ll_{M} \frac{\D^{s}}{\E_{s-1} (A)} \le \frac{|A|^s}{\E_{s-1} (A)} \,,
$$
and, again,
$$
|P| \sim_{\log M} \E_s (A) \D^{-s} \ge M^{-1} |A| \E_{s-1} (A) \D^{-s} = \frac{|A|^{s+1}}{MK \D^{s}}
\ge
\frac{|A|}{MK} \,.
$$
After that apply the asymmetric version of Balog--Szemer\'{e}di--Gowers Theorem \ref{t:BSzG_as}
and an analog of the arguments from (\ref{tmp:04.05.2014_1})---(\ref{tmp:04.05.2014_2}).
This concludes
the proof.
\end{proof}
$\hfill\Box$
\bigskip
The more general assumption
$\E_s (A) \ge |A|^k \E_{s-k} (A) / M^k$
implies that for some $j \in [k]$ one has $\E_{s-j+1} (A) \ge |A| \E_{s-j}(A) / M$.
Thus, we have considered
the common
case.
Note, finally, that estimates (\ref{f:E_3_and_E_critical_1}), (\ref{f:E_3_and_E_critical_2})
are the best possible.
Indeed, take $\Gr = \mathbf{F}^n_2$, $A=H\dotplus \L$, where $H\le \mathbf{F}^n_2$ is a linear subspace
and $\L$ is a dissociated set (basis).
\bigskip
Now we can prove a criterium for sets having critical relation between $\E (A) $ and $\E_3 (A)$ energies.
\begin{theorem}
Let $A\subseteq \Gr$ be a set, and $M\ge 1, \eps \in (0,1)$ be real numbers.
Put $K:=\frac{|A|^3}{\E (A)}$.
Then
\begin{equation*}\label{cond:E_3_and_E_critical_cor}
\E_3 (A) \gg_{M,K^\eps} |A| \E(A)
\end{equation*}
iff
$$
A \approx_{M,K^\eps} \L \dotplus H \,.
$$
\label{t:H+L_description}
\end{theorem}
\begin{proof}
In view of Proposition \ref{p:E_3_and_E_critical} it remains to prove that if
$A \approx_{M,K^\eps} \L \dotplus H$ then $\E_3 (A) \gg_{M,K^\eps} |A| \E(A)$.
Put $A_1 = A\cap (H+\L)$.
We have $|A_1| \gg_{M,K^\eps} |A|$.
Then
$$
|A|^2 |H|^2 \ll_{M,K^\eps} |A_1|^2 |H|^2 \le \E(A_1,H) |A_1-H| \le \E(A,H) |\L| |H-H| \ll_{M,K^\eps} \E(A,H) |A| \,.
$$
Thus
$$
(|A| |H|^2)^3 \ll_{M,K^\eps} \left( \sum_x (A\c A)(x) (H\c H) (x) \right)^3
\le
\E_3 (A) \E^2_{3/2} (H)
\le
\E_3 (A) |H|^5 \,.
$$
In other words
$$
\E_3 (A) \gg_{M,K^\eps} |A|^3 |H| \gg_{M,K^\eps} \E(A) |A| \,.
$$
To get the last estimate
we have used
the fact $|H| \gg_{M,K^\eps} \E (A) |A|^{-2}$.
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Recall that
$$
\T_k (A) :=| \{ a_1 + \dots + a_k = a'_1 + \dots + a'_k ~:~ a_1, \dots, a_k, a'_1,\dots,a'_k \in A \} |
\,.
$$
We conclude the section proving
a ``dual"\, analogue of Proposition \ref{p:E_3_and_E_critical},
that is we replace the condition on $\E_3 (A)$ with a similar condition for $\T_4 (A)$
and moreover for $\T_s (A)$.
Again, the proof follows the arguments from \cite{SS1}.
\begin{theorem}
Let $A\subseteq \Gr$ be a set, and $M\ge 1$ be a real number.
Suppose that
\begin{equation}\label{cond:T_3_and_E_critical}
\T_4 (A) \ge \frac{|A|^4 \E(A)}{M} \,.
\end{equation}
Then there is $A' \subseteq A$ such that
\begin{equation}\label{f:T_3_and_E_critical_1}
|A'| \gg \frac{|A|}{M^3 \log^{\frac{16}{3}} |A|} \,,
\end{equation}
and
\begin{equation}\label{f:T_3_and_E_critical_2}
|nA' - mA'| \ll (M^3 \log^4 |A|)^{4(n+m)} M |A'| \cdot \frac{|A|^3}{\E(A)}
\end{equation}
for every $n,m\in \N$.
Moreover, if
\begin{equation}\label{cond:T_3_and_E_critical_s}
\T_{2s} (A) \ge \frac{|A|^{2s} \T_s (A)}{M} \,,
\end{equation}
$s\ge 2$ then formulas (\ref{f:T_3_and_E_critical_1}),
(\ref{f:T_3_and_E_critical_2}) take place.
Conversely, bounds (\ref{f:T_3_and_E_critical_1}), (\ref{f:T_3_and_E_critical_2}) imply that
$\T_{2s} (A) \gg_{M,\,\log |A|,\,s} |A|^{2s} \T_s (A)$.
\label{t:T_3_and_E_critical}
\end{theorem}
\begin{proof}
Put
$\T_s = \T_s (A)$, $\T_{2s} = \T_{2s} (A)$, $a=|A|$,
$L_s = [\log (16Ma^{2s-1}/\T_s)] \ll_s \log a$.
Let
$$
P_j = \{ x ~:~ 2^{j-1} \T_s / (2^{4} M a^s)
< (A *_{s-1} A) (x) \le 2^{j} \T_s / (2^{4} M a^s) \} \,, \quad j\in [L_s] \,.
$$
Put $f_j (x) = P_j (x) (A *_{s-1} A) (x)$.
Thus,
\begin{equation}\label{tmp:04.05.2014_3}
(A *_{s-1} A) (x) = \sum_{j=1}^{L_s} f_j (x) + \Omega (x) (A *_{s-1} A)(x) \,,
\end{equation}
where
$\Omega = \{ x ~:~ (A *_{s-1} A) (x) \le 2^{-4} M^{-1} \T_s a^{-s} \}$.
Substituting formula (\ref{tmp:04.05.2014_3}) into the identity
$$
\T_{2s} (A) = \sum_{x} ((A *_{s-1} A) \c (A *_{s-1} A))^2 (x)
$$
and using assumptions (\ref{cond:T_3_and_E_critical}), (\ref{cond:T_3_and_E_critical_s}),
combining with the definition of sets $P_j$, $\Omega$,
we have
$$
2^{-1} \T_{2s} (A) \le \sum_{j_1,j_2,j_3,j_4 = 1}^{L_s} \sum_x ( f_{j_1} \c f_{j_2} ) (x) ( f_{j_3} \c f_{j_4} ) (x) \,.
$$
Applying the H\"{o}lder inequality, we get
$$
2^{-1} L_s^{-3} \T_{2s} (A) \le \sum_{j=1}^{L_s} \sum_x ( f_{j} \c f_{j} ) (x) ( f_{j} \c f_{j} ) (x) \,.
$$
By the pigeonhole principle there is $j\in [L_s]$ such that
\begin{equation}\label{tmp:08.02.2013_6}
\frac{a^{2s} \T_s}{2M L_s^4} \le \frac{\T_{2s}}{2L_s^{4}}
\le \sum_x ( f_{j} \c f_{j} ) (x) ( f_{j} \c f_{j} ) (x) \,.
\end{equation}
Put $P=P_j$, $f=f_j$ and $\D = 2^{j} \T_s / (2^{4} M a^s)$.
Thus
\begin{equation}\label{tmp:08.02.2013_3}
\frac{a^{2s} \T_s}{2M L_s^4 \D^4} \le \E (P) \,.
\end{equation}
Clearly, $|P| \le 4\T_s \D^{-2}$.
Using the last inequality, the definition of the number $\D$
and bound (\ref{tmp:08.02.2013_3}), we obtain
$$
\E (P) \ge \frac{a^{2s} \T_s}{2M L_s^4 \D^4} \ge |P|^3 \frac{a^{2s} \D^2 }{2^7 M L_s^4 \T_s^2}
\ge
|P|^3 \frac{2^{2j}}{2^{15} M^3 L_s^4}
\ge
\frac{|P|^3}{2^{15} M^3 L_s^4} := \mu |P|^3 \,.
$$
To estimate the size of $P$ we note by (\ref{tmp:08.02.2013_3}) that
\begin{equation}\label{tmp:08.02.2013_7}
|P|^3 \ge \frac{a^{2s} \T_s}{2M L_s^4 \D^4} \,.
\end{equation}
After that use
arguments (\ref{tmp:08.02.2013_4})---(\ref{tmp:08.02.2013_5})
of the proof of Proposition \ref{p:E_3_and_E_critical}.
By Theorem \ref{BSG} there is $P'\subseteq P$ such that
$|P'| \gg \mu |P|$ and $|P'-P'| \ll \mu^{-4} |P'|$.
Applying Pl\"{u}nnecke--Ruzsa inequality (\ref{f:Plunnecke}), we obtain
\begin{equation}\label{tmp:08.02.2013_8}
|nP'-mP'| \ll \mu^{-4(n+m)} |P'|
\end{equation}
for every $n,m\in \N$.
We have
$$
\D |P'| \le \sum_x (A *_{s-1} A) (x) P'(x) = \sum_{x_1,\dots,x_{s-1}\in A} (A \c P') (x_1+\dots+x_{s-1}) \,.
$$
By (\ref{tmp:08.02.2013_7})
and the definition of the number $\D$
there is $x\in (s-1)A$ such that the set $A' := A\cap (P'-x)$ has the size at
least
$$
|A'|
\gg
|P'| \D a^{-(s-1)}
\ge
\mu |P| \D a^{-(s-1)}
\gg
\mu \left( \frac{\T_s}{M L_s^4 \D a^{s-3}} \right)^{1/3}
\gg
\frac{2^{5j/3} a}{M^3 L_s^{16/3}}
\gg
\frac{a}{M^3 L_s^{16/3}} \,.
$$
We have by (\ref{tmp:08.02.2013_8}) that
\begin{equation*}\label{tmp:08.02.2013_5'}
|nA'-mA'| \le |nP' - mP'| \ll \mu^{-4(n+m)} |P'|
\ll \mu^{-4(n+m)} |A'| a^{s-1} \D^{-1} \ll \mu^{-4(n+m)} M |A'| \cdot \frac{a^{2s-1}}{\T_s}
\end{equation*}
for every $n,m\in \N$.
Conversely, applying bound (\ref{f:T_3_and_E_critical_2}) with $n=m=s$, combining with the Cauchy--Schwarz inequality,
we obtain
$$
|A'|^{4s} \le \left( \sum_x ( (A' *_{s-1} A') \c (A' *_{s-1} A') ) (x) \right)^2
\le
\T_{2s} (A') |sA'-sA'|
\ll_s
$$
$$
\ll_s
(M^{3} \log^{4} |A|)^{8s} M \cdot |A'| \frac{|A|^{2s-1}}{\T_s (A)} \T_{2s} (A) \,.
$$
Using (\ref{f:T_3_and_E_critical_1}), we get
$$
\T_{2s} (A) \gg_s \T_s (A) |A'|^{4s-1} |A|^{-(2s-1)} (M^{3} \log^{4} |A|)^{-8s} M^{-1}
\gg_s
\T_s (A) |A|^{2s} M^{-36s+2} \log^{-54s} |A| \,.
$$
In other words, $\T_{2s} (A) \gg_{M,\,\log |A|,\,s} |A|^{2s} \T_s (A)$.
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
So, we have proved in Theorem \ref{t:T_3_and_E_critical} that, roughly speaking, $A'-A'$
is a set with small (in terms of the parameter $M$) doubling and vice versa.
Note,
that we need in multiple $|A|^3 \E^{-1} (A)$ in (\ref{f:T_3_and_E_critical_2}),
because by (\ref{f:T_3_and_E_critical_1}) and the Cauchy--Schwarz inequality,
we have the same lower bound for $|A'-A'|$.
Thus, $A'$ does not equal to a set with small doubling but $A'-A'$ does.
Results of such a sort were obtained in \cite{SS1}, \cite{s_ineq} and \cite{s_mixed}.
It is easy to see that an analog of Theorem \ref{t:T_3_and_E_critical}
takes place if one replace (\ref{cond:T_3_and_E_critical})
onto condition
$\T_s (A) \ge |A|^{2(s-2)} \E(A) / M$,
where $s$ is an even number, $s\ge 4$,
and, further, even more general relations between $\T_k$ energies
can be reduced to the last case and Theorem \ref{t:T_3_and_E_critical}
via a trivial estimate
$ \T_s (A) \le |A|^2 \T_{s-1} (A)$.
We do not need in such generalizations in the paper.
\section{Sumsets: preliminaries}
\label{sec:sumsets}
Let $A \subseteq \Gr$ be a set.
Before studying the energies of sumsets or difference sets we concentrate on a following related question,
which was asked to the author by Tomasz Schoen.
Namely, what can be proved nontrivial concerning lower bounds for $|A\pm A_s|$, $s\neq 0$?
The connection with $A\pm A$ is obvious in view of Katz--Koester trick (\ref{f:KK_trick}).
We start with a result in the direction.
\begin{theorem}
Let $A\subseteq \Gr$ be a set.
If $\E_3 (A) \ge 2|A|^3$ then
\begin{equation}\label{f:E_3(D)_0}
\left( \max_{s\neq 0} |A\pm A_s| \right)^3 \gg \frac{|A|^{10}}{|A-A| \E^2 (A)} \,.
\end{equation}
Now suppose that $A$ is $(3,\beta,\gamma)$--connected with $\beta \le 1/2$,
and $\E_3 (A) \ge 2^4 \gamma^{-1} |A|^3$.
Then
\begin{equation}\label{f:E_3(D)_0'}
\left( \max_{s\neq 0} |A\pm A_s| \right)^2 \gg \gamma \frac{|A|^{5}}{\E (A)} \,.
\end{equation}
\label{t:E_3(D)-}
\end{theorem}
\begin{proof}
Write $\E = \E(A)$, $\E_3 = \E_3 (A)$, and $a=|A|$.
Let us begin with (\ref{f:E_3(D)_0}).
Denote by $\o$ the maximum in (\ref{f:E_3(D)_0}).
By the Cauchy--Schwarz inequality and formula (\ref{f:eigen_A}) of Lemma \ref{l:eigen_A'}, we obtain
\begin{equation}\label{f:E_3(D)_first}
a^2 |A_s|^2 \le \E(A,A_s) |A \pm A_s| \le \E^{1/2}_3 |A_s| |A \pm A_s| \,.
\end{equation}
Multiplying the last inequality by $|A_s|$, summing over $s\neq 0$
and using the assumption
$\E_3 (A) \ge 2|A|^3$,
we get
\begin{equation}\label{tmp:22.01.2014_1}
a^2 \E^{1/2}_3 \ll \o \E \,.
\end{equation}
On the other hand, by Lemma \ref{corpop}, we have
\begin{equation}\label{tmp:22.01.2014_2}
a^6 \ll \E_3 \sum_{s\in A-A,\, s\neq 0} |A \pm A_s| \le \E_3 |A-A| \o \,.
\end{equation}
Combining (\ref{tmp:22.01.2014_1}) with (\ref{tmp:22.01.2014_2}), we obtain
$$
\o^3 \gg \frac{a^{10}}{|A-A| \E^2}
$$
as required.
Now let us obtain (\ref{f:E_3(D)_0'}).
Using Lemma \ref{l:eigen_A'}, we find $A'$, $|A'| \ge |A|/2$ such that estimate (\ref{f:eigen_A'}) takes place.
As in (\ref{f:E_3(D)_first}), we get
\begin{equation}\label{f:E_3(D)_first'-}
a^2 |A'_s|^2 \le \E(A,A'_s) |A \pm A'_s| \le \frac{2\E}{a} |A'_s| |A \pm A_s| \,.
\end{equation}
By assumption $\E_3 (A) \ge 2^4 \gamma^{-1} |A|^3$.
Using the connectedness, we obtain
\begin{equation}\label{tmp:09.04.2014_I''}
\E_3 (A') \ge \gamma \frac{|A'|^6}{|A|^6} \E_3 (A) \ge 2 |A'|^3 \,.
\end{equation}
Multiplying
inequality (\ref{f:E_3(D)_first'-})
by $|A'_s|$, summing over $s \neq 0$, we have in view of (\ref{tmp:09.04.2014_I''}) that
\begin{equation}\label{tmp:22.01.2014_3}
a^2 \E_3 (A') \ll a^{-1} \E \E^* (A') \o \,.
\end{equation}
Combining the last formula with first inequality from (\ref{tmp:09.04.2014_I''}), we get
\begin{equation}\label{tmp:09.04.2014_I'}
a^2 \gamma \E_3 (A) \ll a^{-1} \E \E^* (A') \o \,.
\end{equation}
On the other hand, summing the first estimate from (\ref{f:E_3(D)_first'-}) over $s\neq 0$
and applying Lemma \ref{l:E_3_A_s}, we see that
\begin{equation}\label{tmp:09.04.2014_I'''}
a^2 \E^* (A') \le \o \E_3 \,.
\end{equation}
Combining (\ref{tmp:09.04.2014_I'}), (\ref{tmp:09.04.2014_I'''}), we obtain
$$
\o^2 \gg \gamma \frac{a^5}{\E} \,.
$$
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
From (\ref{f:E_3(D)_0}) it follows that if $|A-A| = K|A|$, $\E(A) \ll |A|^3 /K$, $\E_3 (A) \ge 2|A|^3$
then there is
$s\neq 0$ such that $|A-A_s| \gg K^{1/3} |A|$ as well as there exists
$s\neq 0$ with $|A+A_s| \gg K^{1/3} |A|$.
It improves a trivial lower bound $|A\pm A_s| \ge |A|$.
Using bound
(\ref{f:E_3(D)_0'})
one can show that there is $s\neq 0$ such that $|A-A_s| \gg K^{1/2} |A|$
(here $\E(A) \ll |A|^3 /K$),
provided by some connectedness assumptions take place.
We need in lower bounds on $\E_k (A)$ in Theorem \ref{t:E_3(D)-} and Proposition \ref{p:f:E_4(D,A)} below
to be separated from a very natural simple example, which can be called a "random sumset"\, case.
Namely, take a random $A\subseteq \Gr$ and consider $A\pm A$.
This "random sumset"\, has almost no structure (provided by $A\pm A$ is not a whole group, of course)
and we cannot say something useful in the situation.
It does not contradict Theorem \ref{t:E_3(D)-} and Proposition \ref{p:f:E_4(D,A)} (see also the results of the next section)
because the energies $\E_k (A)$ are really small in the case.
\bigskip
Now we give another prove of estimate (\ref{f:E_3(D)_0'}) which can be derived from
inequality (\ref{f:E_4(D,A)}), case $k=2$ below.
Actually, it gives us even stronger inequality, namely, $|A^2 - \D (A_s)| \gg |A|^5 \E^{-1} (A)$
for some $s\neq 0$, or, more generally, (see formulas (\ref{tmp:06.05.2014_1}), (\ref{tmp:06.05.2014_2}) below)
\begin{equation}\label{f:max_s,k}
\max_{s\neq 0} |A^k \pm \D (A_s)|
\gg
\max_{r\ge 1} \left\{ \frac{|A|^{2k+1} \E_{r+1} (A)}{\E_r (A) \E_{k+1} (A)} \right\}
\,,
\end{equation}
provided by some connectedness assumptions take place.
In particular, taking $r=k$ and $r=1$ in the previous formula, we get
$$
\max_{s\neq 0} |A^k \pm \D (A_s)|
\gg
\left\{ \frac{|A|^{2k+1}}{\E_{k} (A)} \,, \frac{|A|^{2k-1} \E(A)}{\E_{k+1} (A)} \right\} \,.
$$
\begin{proposition}
Let $A\subseteq \Gr$ be a set, $k\ge 2$ be a positive integer.
Take two sets $D,S$ such that $A-A \subseteq D$, $A+A \subseteq S$.
Then
\begin{equation}\label{f:E_4(D,A)}
\gamma |A|^{2k+1} \ll_k \sum_{x\neq 0} (A\c A)^k (x) d_k (x) \le \sum_{x\neq 0} (A\c A)^k (x) (D\c D)^k (x) \,,
\end{equation}
\begin{equation}\label{f:E_4(D,A)'}
\gamma |A|^{2k+1} \ll_k \sum_{x\neq 0} (A\c A)^k (x) s_k (x) \le \sum_{x\neq 0} (A\c A)^k (x) (S\c S)^k (x) \,,
\end{equation}
provided by $A$ is $(k+1,\beta,\gamma)$--connected with $\beta \le 1/2$
and
$\E_{k+1} (A) \ge 2^{k+2} \gamma^{-1} |A|^{k+1}$.
Here $d_k (x) = \sum_\a D_x (\a) (D \c D_x)^{k-1} (\a)$, $s_k (x) = \sum_\a S_x(\a) (S_x * D)^{k-1} (\a)$.
\label{p:f:E_4(D,A)}
\end{proposition}
\begin{proof}
Using Lemma \ref{l:eigen_A'}, we find $A'$, $|A'| \ge |A|/2$ such that estimate (\ref{f:eigen_A''}) takes place
with $g(z) = (A \c A)^k (z)$.
It follows that
\begin{equation}\label{tmp:06.05.2014_1}
|A|^{2k} |A'_x|^2 \le |A^k \pm \D(A'_x)| \E_{k+1} (A'_x, A)
\le |A^k \pm \D(A_x)| \frac{2\E_{k+1} (A)}{|A|} |A_x| \,.
\end{equation}
Multiplying the last inequality by $|A'_x|^{k-1}$ and
summing
over $x\neq 0$ (to obtain (\ref{f:max_s,k}) multiply by $|A'_x|^{r-1}$), we get
\begin{equation}\label{tmp:06.05.2014_2}
\gamma |A|^{2k+1} \E_{k+1} (A) \ll |A|^{2k+1} \E^*_{k+1} (A')
\ll \E_{k+1} (A) \sum_{x \neq 0} |A^k \pm \D(A_x)| |A_x|^k \,.
\end{equation}
Here we have used the fact
$$
\E_{k+1} (A') \ge \gamma \frac{|A'|^{2(k+1)}}{|A|^{2(k+1)}} \E_{k+1} (A) \ge 2 |A'|^{k+1}
$$
and the assumption $\E_{k+1} (A) \ge 2^{k+2} \gamma^{-1} |A|^{k+1}$.
Note that by Katz--Koester trick (\ref{f:KK_trick_new}), one has $|A^k - \D(A_x)| \le d_k (x)$, $|A^k + \D(A_x)| \le s_k (x)$
(or just see the proof of Proposition \ref{pr:E_k(D)_simple} below).
Finally, $d_k (x) \le |D_x|^k$, $s_k (x) \le |S_x|^k$
and we obtain (\ref{f:E_4(D,A)}).
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Using Katz--Koester trick or just the Cauchy--Schwarz inequality
one can show that (\ref{f:E_4(D,A)}) trivially takes place in the case $k=1$
without any conditions on connectedness of $A$ or lower bounds for any sort of energy.
Under the assumptions of Proposition \ref{p:f:E_4(D,A)} from (\ref{f:E_4(D,A)}) it follows that
$$
\gamma^2 |A|^{4k+2} \ll_k \E^*_{2k} (A) \E^*_{2k} (D)
$$
and similarly for $S$.
Some weaker results
but without any conditions on $A$
were obtained in \cite{SS1}.
\bigskip
Results above give an interesting
corollary on non--random sumsets/difference sets.
Put $D=A-A$, $S=A+A$.
Then for an arbitrary positive integer $k$ and any elements $a_1,\dots,a_k\in A$, we have
\begin{equation}\label{f:A,D_inclusion}
A \subseteq (D + a_1) \bigcap (D + a_2) \dots \bigcap (D+a_k) \,, \quad
A \subseteq (S - a_1) \bigcap (S - a_2) \dots \bigcap (S-a_k)
\,.
\end{equation}
In particular, there is $x\in D$, $x\neq 0$ such that $|D_x|, |S_x| \ge |A|$
(also it follows from Katz--Koester inclusion (\ref{f:KK_trick_new})).
By Corollary \ref{c:A^2-D(A)} (see also Lemma \ref{l:A^2_pm}) one can improve it to
$|D_x|, |S_x| \ge K^\eps |A|$,
where $K=|A|^3 \E^{-1} (A)$.
Theorem \ref{t:E_3(D)-} gives us
even stronger result.
\begin{corollary}
Let $A \subseteq \Gr$ be a set, $D=A-A$, $S=A+A$, $\E (A) = |A|^3/K$,
Suppose that $A$ is $(3,\beta,\gamma)$--connected, $\beta \le 1/2$,
and $\E_3 (A) \ge \gamma^{-1} 2^{4} |A|^3$.
Then there is $x\neq 0$ such that $|D_x|,|S_x| \gg \gamma^{1/2} K^{1/2} |A|$.
\label{c:Dx}
\end{corollary}
One can
get
an analog of Corollary \ref{c:Dx} for multiple intersections (\ref{f:A,D_inclusion})
but another types of energies will require in the case.
Nevertheless, some weaker inequality of the form $|D_{\v{x}}| \ge K^{\eps} |A|$ can be obtained,
using Proposition \ref{p:E_3_and_E_critical} and
Theorem \ref{t:E_3_and_E_critical}.
Here $K=|A|^{k+1} \E^{-1}_k (A)$, $k\ge 2$.
Interestingly, we do not even need in any connectedness in this weaker result.
\begin{proposition}
Let $A\subseteq \Gr$ be a set, $k\ge 2$ be a positive integer, $c\in (0,1]$ be a real number,
$\E_{k} (A) \ge k^2 \E_{k-1} (A)$,
and $K=|A|^{k+1} \E^{-1}_k (A) \le |A|^{1-c}$, $K_1 = |A|^3 \E^{-1} (A)$.
Then for all sufficiently small $\eps = \eps (c) >0$
there is $\v{x} = (x_1,\dots,x_{k-1})$ with distinct $x_j$, $j\in [k]$ such that
\begin{equation}\label{}
|D_{\v{x}}|\,, |S_{\v{x}}| \gg_k |A| \cdot \min\{ K^{\eps}, c_{K^{\eps}} K_1 \} \,,
\end{equation}
where the constant $c_{K^{\eps}}$ satisfies $c_{K^{\eps}} \gg_{K^{\eps}} 1$
and, again, the degree of the polynomial dependence is a function on $c$.
\label{p:DxSx}
\end{proposition}
\begin{proof}
Suppose not.
Take any set $\mathcal{P} \subseteq A^{k-1} - \D(A)$.
Applying Lemma \ref{l:E_3_A_s}, one has
$$
|A|^2 \left( \sum_{\v{x} \in \mathcal{P}} |A_{\v{x}}| \right)^2
=
\left( \sum_{\v{x} \in \mathcal{P}} \sum_z (A\c A_{\v{x}}) (z) \right)^2
\le
\E_{k+1} (A) \sum_{\v{x} \in \mathcal{P}} |A \pm A_{\v{x}}|
\le
$$
\begin{equation}\label{tmp:09.04.2014_1}
\le
\E_{k+1} (A) |\mathcal{P}| \cdot \max_{\v{x} \in \mathcal{P}} |A \pm A_{\v{x}}| \,.
\end{equation}
Note that the assumption $\E_{k} (A) \ge k^2 \E_{k-1} (A)$ implies
\begin{equation}\label{tmp:09.04.2014_2}
\E_k (A) = \sum_{\v{x}} |A_{\v{x}}|^2 \le \sum'_{\v{x}} |A_{\v{x}}|^2 + \binom{k-1}{2} \E_{k-1} (A)
\le
2 \sum'_{\v{x}} |A_{\v{x}}|^2 \,,
\end{equation}
where the sum $\sum'$ above is taken over $\v{x} = (x_1,\dots,x_{k-1})$ with distinct $x_j$.
Now take $\mathcal{P}$ such that $\D < |A_{\v{x}}| \le 2 \D$ for $\v{x} = (x_1,\dots,x_{k-1}) \in \mathcal{P}$,
where all $x_j$, $j\in [k]$ are distinct
and
$$
\sum_{\v{x} \in \mathcal{P}} |A_{\v{x}}|^2 \gg \frac{\E_{k} (A)}{\log K} \,.
$$
Of course, such $\mathcal{P}$ exists by the pigeonhole principle and bound (\ref{tmp:09.04.2014_2}).
Using the last inequality, and recalling (\ref{tmp:09.04.2014_1}), we obtain
$$
\frac{|A|^2 \E_{k} (A)}{\log K}
\ll
|A|^2 |\mathcal{P}| \D^2
\ll
\E_{k+1} (A) K^{\eps} |A| \,.
$$
In other words, $\E_{k+1} (A) \gg_{K^\eps} |A| \E_k (A)$.
Put $M=\max\{ K^\eps, k\}$.
Applying Proposition \ref{p:E_3_and_E_critical} as well as
Theorem \ref{t:E_3_and_E_critical},
we see that
$A \approx_{M} \L \dotplus H$.
After that put $A' = A\cap (H+\L)$.
Then $|A'| \gg_{M} |A|$ and $|H| \gg_M |A| /K$.
Further, as in the proof of Theorem \ref{t:H+L_description}, we get
$$
\sum_{\v{x},\,\|x\|=k-1} \Cf_{k} (A') (\v{x}) \Cf_{k} (H) (\v{x})
=
\sum_s (A' \c H)^k (s)
\ge \frac{|H|^{k} |A'|^{k}}{|A'-H|^{k-1}} \gg_{M} \frac{|H|^k |A|^k}{|\L+H-H|^{k-1}}
$$
$$
\gg_{M} \frac{|H|^k |A|^k}{(|\L||H-H|)^{k-1}}
\gg_{M}
|A| |H|^k
$$
and hence
\begin{equation}\label{tmp:09.04.2014_3}
\sum_{\v{x},\,\|x\|=k-1} |A'_{\v{x}}| |H_{\v{x}}| \gg_{M} |A| |H|^k \,.
\end{equation}
In particular, there are at least $\gg_{M} |H|^{k-1}$ elements
$\v{x} \in H^{k-1}- \D(H)$
such that
$|A'_{\v{x}}| \gg_{M} |A|$.
We can suppose that the summation in (\ref{tmp:09.04.2014_3}) is taken over
$\v{x} = (x_1,\dots,x_{k-1})$ with distinct $x_j$ because of the rest is bounded by
$$
\binom{k-1}{2} \sum_{x} (A\c H)^{k-1} (x) \le \binom{k-1}{2} |H|^{k-1} |A| \ll_{M} |A| |H|^k \,.
$$
The last estimate follows from the assumption $K \le |A|^{1-c}$.
Choosing any such $\v{x}$ and using the Cauchy--Schwarz inequality, we obtain
$$
|A \pm A_{\v{x}}| \ge |A' \pm A'_{\v{x}}| \ge \frac{|A'|^2 |A'_{\v{x}}|^2}{\E(A',A'_{\v{x}})}
\gg_{M}
\frac{|A|^4}{\E(A)} = K_1 |A|
$$
and in view of Katz--Koester trick (\ref{f:KK_trick_new}),
we see that $|D_{\v{x}}|\,, |S_{\v{x}}|$ are huge for large $K_1$.
This concludes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Using Proposition \ref{p:DxSx} one can derive an interesting dichotomy.
\begin{theorem}
Let $A\subseteq \Gr$ be a set, $D=A-A$, $S=A+A$, $k\ge 2$, and
$M\ge 1$, $\eps \in (0,1)$ be real numbers.
Put $K=|A|^{k+1} \E^{-1}_k (A)$.
Suppose that for any vector $\v{x} = (x_1,\dots,x_{k-1})$ with distinct $x_j$, $j\in [k]$
the following holds
\begin{equation*}\label{}
|D_{\v{x}}| \le M|A| \quad \mbox{or, similarly, } \quad |S_{\v{x}}| \le M|A| \,.
\end{equation*}
Then either $\E_k (A) \ll_{M,\, |A|^\eps,\, k} |A|^k$ or $\E(A) \gg_{M,\, |A|^\eps,\, k} |A|^3$.
Again, the degree of the polynomial dependence is a function on $\eps$.
\label{t:dichotomy_DS}
\end{theorem}
\begin{proof}
Put $K_1 = |A|^3 \E^{-1} (A)$.
Suppose, in contrary, that $\E_k (A) \gg_{M,\, |A|^\eps,\, k} |A|^k$ and\\ $\E(A) \ll_{M,\, |A|^\eps,\, k} |A|^3$.
Then $K \ll_{M,\, |A|^\eps,\, k} |A|^{}$ and $K_1 \gg_{M,\, |A|^\eps,\, k} 1$.
Trivially, $\E_k (A) \le |A|^{k-2} \E(A)$
and hence
$$|A| \gg_{M,\, |A|^\eps,\, k} K \ge K_1 \gg_{M,\, |A|^\eps,\, k} 1 \,.$$
Finally, by the upper bound for the parameter $K$ the number $c$
which is defined as $K = |A|^{1-c}$ can by taken depends on $\eps$ only.
Thus,
everything follows from Proposition \ref{p:DxSx}, the only thing we need to consider is the situation when
$\E_k (A) \le k^2 \E_{k-1} (A)$.
But in the case
$$
|A|^k \le \E_k (A) \le k^2 \E_{k-1} (A) \le k^2 |A|^{k-3} \E(A) \,.
$$
In other words $\E(A) \gg_k |A|^3$ and this concludes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Thus, if $|D_{\v{x}}|\,, |S_{\v{x}}|$ are not much larger than $|A|$
then either $A$ is close to what we called a "random sumset"\, or, on the contrary, is very structured.
Clearly, the both situations are realized.
\section{Energies of sumsets}
\label{sec:sumsets1}
Let $A \subseteq \Gr$ be a set.
Throughout the section we put $D=A-A$ and $S=A+A$.
As was explained in the introduction that one can hope to prove a good lower bound for $\E_3 (D)$.
It will be done in Theorem \ref{t:E_3(D)} below but before this we formulate a simple preliminary
lower bound for $\E^D_k (D)$, $\E^D_k (S)$.
Similar
lower bounds for $\E^D_2 (D)$, $\E^D_2 (S)$
were given
in Corollary 5.6 of paper \cite{M_R-N_S}.
Further, it was proved in \cite{SS1} (see Remark 8) that $\sigma_{k+1} (D) \ge |A^k - \D(A)|$.
Now we obtain a similar lower bound for $\E^{D}_k (D)$.
Recall that by $\E^{D}_1 (D)$ we mean $\sigma_{3} (D) = \sigma_D (D)$, that is $\sum_{x\in D} (D \c D) (x)$.
\begin{proposition}
Let $A\subseteq \Gr$ be a set.
Put $D=A-A$, $S=A+A$.
Then for all $k\ge 1$ one has
\begin{equation}\label{f:E_k(D)_simple}
\E^D_k (D) \ge |A^{k+1} - \D(A)|
\ge |A-A| |A|^k \,,
\end{equation}
and, similarly,
$$
\sum_{x} S(x) (S * D)^{k} (x) \ge |A^{k+1} + \D(A)| \ge |A|^k \max\{ |A-A|, |A+A| \} \,,
$$
\begin{equation}\label{f:E_k(D)_simple'}
\E^D_k (S) \ge |A|^{k-1} |A^2 + \D(A)| \ge |A|^k \max\{ |A-A|, |A+A| \} \,.
\end{equation}
\label{pr:E_k(D)_simple}
\end{proposition}
\begin{proof}
The second estimates in (\ref{f:E_k(D)_simple}), (\ref{f:E_k(D)_simple'})
follow from Lemma \ref{l:A^2_pm}.
Further, it is easy to get (or see e.g. \cite{SS1}) that
$$
|A^{k+1} - \D(A)| \le \sum_{x_1,\dots,x_{k+1}} D(x_1) \dots D(x_{k+1}) \prod_{i\neq j} D(x_i - x_j)
\le
$$
$$
\le
\sum_{x_1,\dots,x_{k+1}} D(x_1) \dots D(x_{k+1}) D(x_1-x_2) \dots D(x_1-x_{k+1})
=
\sum_{x} D(x) (D\c D)^{k} (x)
=
\E^D_{k} (D)
$$
as required.
Similarly,
$$
|A^{k+1} + \D(A)| \le \sum_{x_1,\dots,x_{k+1}} S(x_1) \dots S(x_{k+1}) \prod_{i\neq j} D(x_i - x_j)
\le
$$
$$
\le
\sum_{x_1,\dots,x_{k+1}} S(x_1) \dots S(x_{k+1}) D(x_1-x_2) \dots D(x_1-x_{k+1})
=
\sum_{x} S(x) (S* D)^{k} (x)
\,.
$$
Finally, by Lemma \ref{l:A^2_pm}, we get
$$
\E^D_k (S) = \sum_{s\in D} (S\c S)^k (s) \ge \sum_{s\in D} |A+A_s|^k
\ge |A|^{k-1} \sum_{s\in D} |A+A_s|
=
$$
$$
=
|A|^{k-1} |A^2 + \D(A)| \ge |A|^k \max\{ |A-A|, |A+A| \} \,.
$$
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Interestingly, that some sort of sumset, namely, $A^n \pm \D(A)$
gives a lower bound for an energy, although, usually, the energy
provides lower bounds for the cardinality sumsets via the Cauchy--Schwarz inequality.
The trick allows to obtain a series of results in \cite{SS1}---\cite{SS3}.
Although bounds (\ref{f:E_k(D)_simple}), (\ref{f:E_k(D)_simple'}) are very simple they can be tight in some cases.
For example, take $A$ to be a dissociated set
or, in contrary, a very structural set as a subspace.
\bigskip
Now we formulate the main result of the section concerning lower bounds for some energies of sumsets/difference sets.
Again we need in lower bounds on $\E_k (A)$ in Theorem \ref{t:E_3(D)} below
to be separated from the "random sumset"\, case.
\begin{theorem}
Let $A\subseteq \Gr$ be a set.
Take two sets $D,S$ such that $D=A-A$, $S=A+A$.
Then
\begin{equation}\label{f:E_3(D)_1}
\E^2_3 (D,A,A),\,~ \E^2_3 (S,A,A)
\ge \frac{|A|^{13}}{|A-A|^2 \E (A)} \,,
\end{equation}
and
\begin{equation}\label{f:E_3(D)_2}
(\E^D_3 (D))^4 \ge \max \left\{ |D|^{12}, \frac{|A|^{45}}{\E^9 (A) |D|^2} \right\} \,,
\quad \quad
(\E^D_3 (S))^4 \ge \max \left\{ |S|^{12}, \frac{|A|^{45}}{\E^9 (A) |D|^2} \right\} \,.
\end{equation}
Further, let $\beta,\gamma \in [0,1]$ be real numbers, $\beta \le 1/2$.
If $A$ is $(2,\beta,\gamma)$--connected then
\begin{equation}\label{f:E_3(D)_2.5}
\E^2_3 (D,A,A),\,~ \E^2_3 (S,A,A)
\gg
\gamma |A|^5 \E (A) \,.
\end{equation}
Suppose that $A$ is $(3/2,\beta,\gamma)$ and $(2,\beta,\gamma)$--connected, correspondingly.
Then
\begin{equation}\label{f:E_3(D)_3}
\E^D_3 (D),\, \E^D_3 (S)
\gg
\gamma \frac{|A|^{33/4} \E_{3/2} (A)}{\E^{9/4} (A) \log |A|} \,,
\quad \quad \quad
\E^D_3 (D),\, \E^D_3 (S)
\gg
\gamma \frac{|A|^{17/2}}{\E^{3/2} (A) \log |A|} \,,
\end{equation}
correspondingly.
\label{t:E_3(D)}
\end{theorem}
\begin{proof}
Write $\E = \E(A)$, $\E_3 = \E_3 (A)$, and $a=|A|$.
Let us obtain bounds (\ref{f:E_3(D)_1}), (\ref{f:E_3(D)_2.5}).
Using Lemma \ref{l:eigen_A'}, we find $A'$, $|A'| \ge |A|/2$ such that estimate (\ref{f:eigen_A'}) takes place.
As in the proof of inequalities (\ref{f:E_3(D)_first}), (\ref{f:E_3(D)_first'-}), we get
\begin{equation}\label{f:E_3(D)_first'}
a^2 |A'_s|^2 \le \E(A,A'_s) |A \pm A'_s| \le \frac{2\E}{a} |A'_s| |A \pm A_s| \,.
\end{equation}
Multiplying the last inequality by $|A'_s|$, summing over $s$ and using Katz--Koester trick, we have
\begin{equation}\label{tmp:22.01.2014_3}
a^2 \E_3 (A') \ll a^{-1} \E \cdot \E_3 (D,A)
\end{equation}
and similar for $S$.
On the other hand by the second part of Lemma \ref{corpop}, we obtain
\begin{equation}\label{tmp:22.01.2014_4}
\left( \frac{a^4}{|A-A|} \right)^2 a^2 \ll \E^2 (A') a^2 \ll \E_3 (A') \cdot \E_3 (D,A)
\end{equation}
and using the first part of the lemma, we have the same bound for $S$
\begin{equation}\label{tmp:22.01.2014_4+}
\left( \frac{a^2}{|A-A|} \right)^2 a^6 \ll \sigma^2_{\t{D}} (A') (a')^2 \left( \frac{(a')^2}{2|D|} \right)^2
\le \E_3 (A') \sum_{x \in \t{D}} |S_x| |A_x|^2
\le \E_3 (A') \cdot \E_3 (S,A) \,,
\end{equation}
where $\t{D} := \{ x\in D ~:~ |A'_x| \ge (a')^2 /2|D| \}$, $\sigma_{\t{D}} (A') \ge (a')^2 /2$
(for details, see \cite{SS3}).
Another way to prove the same is to use Lemma \ref{l:T_A,B} with $A=B=A'$, $\psi (x) = (A'\c A') (x)$.
Combining (\ref{tmp:22.01.2014_3}) and (\ref{tmp:22.01.2014_4}), (\ref{tmp:22.01.2014_4+}), we get
$$
\E^2_3 (D,A)\,,~ \E^2_3 (S,A)
\gg \frac{|A|^{13}}{|A-A|^2 \E (A)}
$$
Using the tensor trick (see \cite{TV} or \cite{s_mixed}), we have (\ref{f:E_3(D)_1}).
If $A$ is $(2,\beta,\gamma)$--connected then
$$
\E(A') \gg \gamma \E(A)
$$
and combining the last inequality with (\ref{tmp:22.01.2014_3})
and the second bound from (\ref{tmp:22.01.2014_4}),
we get (\ref{f:E_3(D)_2.5})
(to obtain lower bound for $\E^2_3 (S,A)$
one should use
Lemma \ref{l:T_A,B}).
It remains to prove (\ref{f:E_3(D)_2}) and (\ref{f:E_3(D)_3}).
Returning to (\ref{f:E_3(D)_first'})---(\ref{tmp:22.01.2014_3}), we obtain
$$
a^3 \E_3 (A') \ll \E \sum_s |A'_s|^2 |A' \pm A'_s|
$$
or, by the H\"{o}lder inequality
\begin{equation}\label{tmp:22.01.2014_5}
a^9 \E_3 (A') \ll \E^3 \E^D_3 (D) \,.
\end{equation}
On the other hand, by Lemma \ref{corpop} for any $P\subseteq A'-A'$, we have
$$
a^2 \sigma^2_P (A') \ll \E_3 (A') \sum_{s\in P} |A' \pm A'_s| \,.
$$
Applying the H\"{o}lder inequality, we get
\begin{equation}\label{tmp:22.01.2014_5'}
a^6 \sigma^6_P (A') \ll \E^3_3 (A') |P|^2 \E^D_3 (D)
\end{equation}
and similarly for $S$.
Now choose $P\subseteq A'-A'$ such that $P = \{ s\in A'-A' ~:~ \rho < |A'_s| \le 2 \rho \}$
for some positive number $\rho$ and such that
$$
\sum_{s\in P} |A'_s|^{3/2} \gg \frac{\E_{3/2} (A')}{\log a} \,.
$$
Of course, the set $P$ exists by Dirichlet principle.
Combining the last inequality with (\ref{tmp:22.01.2014_5'}), we obtain
\begin{equation}\label{tmp:22.01.2014_6}
a^6 \E^4_{3/2} (A') \log^{-4} a \ll a^6 ( |P| \rho^{3/2} )^4 \ll \E^3_3 (A') \E^D_3 (D)
\,.
\end{equation}
Using estimates (\ref{tmp:22.01.2014_5}), (\ref{tmp:22.01.2014_6}), we have
\begin{equation}\label{tmp:22.01.2014_7}
(\E^D_3 (D))^4 \E^9 \gg a^{33} \E^4_{3/2} (A') \log^{-4} a \,.
\end{equation}
Thus
$$
(\E^D_3 (D))^4 \,,~ (\E^D_3 (S))^4 \gg \frac{a^{45}}{\E^9 |D|^2 \log^4 a} \,.
$$
Applying the tensor trick again, we get (\ref{f:E_3(D)_2}).
To obtain (\ref{f:E_3(D)_3}) recall that $A$ is $(3/2,\beta,\gamma)$--connected set.
Hence by (\ref{tmp:22.01.2014_7}), we obtain
$$
(\E^D_3 (D))^4 \E^9 \gg \gamma^4 a^{33} \E^4_{3/2} (A) \log^{-4} a
$$
and the first formula of (\ref{f:E_3(D)_3}) follows.
Further, because of $A$ is $(2,\beta,\gamma)$--connected set then
using Lemma \ref{l:connected} for $A'$ as well as (\ref{tmp:22.01.2014_7}), we have
$$
(\E^D_3 (D))^2 \E^3 \gg \gamma^2 a^{17} \log^{-2} a
$$
and the last estimate coincide with the second inequality in (\ref{f:E_3(D)_3}).
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
From (\ref{f:E_3(D)_2}), the definition of the number $K$ as $|D|=K|A|$
and the assumption $\E(A) \ll |A|^3 /K$, we get
\begin{equation}\label{f:E_3_74}
\E_3 (D) \gg K^{7/4} |A|^4 \,.
\end{equation}
An upper bound here is $K^{2} |A|^4$ and it follows from the main example of section \ref{sec:structural},
that is $\Gr = \f_2^n$, $A=H\dotplus \L$.
Note also that the second inequality in formula (\ref{f:E_3(D)_3}) is weak but do not depends on the size of $A-A$
or on the energy $\E_{3/2} (A)$.
\bigskip
As
for dual quantities $\T_k (D)$, $\T_k (S)$,
our example $A=H\dotplus \L$ shows that
there are not nontrivial lower bounds for $\T_k (A\pm A)$ in general,
which is better than
a simple consequence of Katz--Koester
$$
\T_k (D) \ge |A|^2 \T_{k-1} (D) \ge \dots \ge |A|^{2(k-2)} \E (D) \ge |A|^{2k-2} |D| \,,
$$
$$
\T_k (S) \ge |A|^2 \T_{k-1} (S,\dots,S,D) \ge \dots \ge |A|^{2(k-2)} \E (D) \ge |A|^{2k-2} |D|
$$
(just use $(D\c D)(x) \ge |A| D(x)$ and $(S\c S) (x) \ge |A| D(x)$).
The reason is that the structure of $A\pm A$ is similar
to the structure of $A$ in the case, of course.
Nevertheless, it was proved in \cite{SS1}, Lemma 3 that
\begin{equation}\label{f:Lev-}
|A|^{4k} \le \E_{2k} (A) \T_k (A\pm A) \,,
\end{equation}
and also in \cite{s_mixed}, Note 6.6 that
\begin{equation}\label{f:Lev}
\left( \frac{\sum_{x\in P} (A\c A) (x)}{|A|} \right)^{4k} \le \E_{2k} (A) \T_k (P) \,,
\end{equation}
where $P\subseteq D$ is any set.
So, if we know something on $\E_{2k} (A)$ then it gives us a new information about $\T_k (A\pm A)$.
Trivially, formula (\ref{f:Lev-}) implies that $\T_k (A\pm A) \ge |A|^{2k+2} / \E(A)$.
Again, the last inequality is sharp as our main example $A=H\dotplus \L$ shows.
\bigskip
Vsevolod F. Lev asked the author about an analog of (\ref{f:Lev}) for different sets $A$ and $B$.
Proposition \ref{p:Lev_question} below is our result in the direction.
The proof is in spirit of \cite{s_mixed}.
For simplicity we consider the case $k=2$ only.
The case of greater powers of two is considered similarly if one take $M^2,M^4,\dots$
or just see the proof of Theorem 6.3 from \cite{s_mixed} (the case of any $k$).
We do not insert the full proof because we avoid to use the operators from \cite{s_ineq}, \cite{s_mixed}
in the paper which is considered to be elementary.
\begin{proposition}
Let $k\ge 2$ be a power of two, $A,B\subseteq \Gr$ be two sets, and $P\subseteq A-B$.
Then
$$
\left( \frac{\sum_{x\in P} (A \c B) (x)}{|A|^{1/2} |B|^{1/2}} \right)^{4k}
\le
\E_{2k} (A,\dots,A,B,\dots,B) \T_k (P) \,.
$$
\label{p:Lev_question}
\end{proposition}
\begin{proof}
Let $k=2$.
Define
the matrix
$$
M(x,y) = P(x-y) A(x) B(y)
$$
and calculate its rectangular norm
$$
\la^4_1 (M) \le \sum_j \la^4_j (M)
=
\sum_{x,y,x',y'} M(x,y) M(x',y) M(x,y') M(x',y')
=
$$
$$
=
\sum_{x,x'\in A}\, \sum_{y,y'\in B} P(x-y) P(x'-y) P(x-y') P(x'-y')
=
$$
$$
=
\sum_{\a,\beta,\gamma} \Cf_4 (B,A,A,B) (\a,\beta,\gamma) P(\a) P(\beta) P(\a-\gamma) P(\beta-\gamma) \,,
$$
where $\la_j (M)$ the singular numbers of $M$.
Clearly,
$$
\la_1 (M) \ge \frac{\sum_{x\in P} (A \c B) (x)}{|A|^{1/2} |B|^{1/2}} \,.
$$
Thus, by the Cauchy--Schwarz inequality, we get
$$
\left( \frac{\sum_{x\in P} (A \c B) (x)}{|A|^{1/2} |B|^{1/2}} \right)^8
\le
\E_4 (A,A,B,B) \E (P)
$$
as required.
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
We end this section showing that there is a different way to prove our Theorem \ref{t:E_3(D)} using
slightly bigger sets $D_x$ or $S_x$ not $A\pm A_x$.
The proof based on a lemma,
which demonstrates,
in particular, that $A\pm A$ contains approximately $|A|^3 \E^{-1} (A)$
almost disjoint translates of $A$, roughly.
In the proof we use arguments from \cite{ALON}.
\begin{lemma}
Let $A,B \subseteq \Gr$ be two sets.
Then there are
\begin{equation}\label{f:s_bound}
s \ge 2^{-4} |A| |B|^2 \E^{-1} (A,B)
\end{equation}
disjoint sets $A_j \subseteq A+b_j$, $|A_j|\ge |A|/2$, $b_j \in B$, $j\in [s]$.
Moreover, for any set $S\subseteq A+B$ put $\sigma = \sum_{x\in S} (A * B) (x)$.
Suppose that $\sigma \ge 16 |B|$.
Then there are
\begin{equation}\label{f:s_bound+}
s \ge 2^{-8} \sigma^3 |A|^{-2} |B|^{-1} \E^{-1} (A,B)
\end{equation}
disjoint sets $S_j \subseteq S \cap (A+b_j)$,
$|S_j|\ge 2^{-3} \sigma |B|^{-1}$, $b_j \in B$, $j\in [s]$.
\label{l:sumsets_disjoint}
\end{lemma}
\begin{proof}
Let us begin with (\ref{f:s_bound}).
Put $S=A+B$.
Our arguments is a sort of an algorithm.
At the first step of the algorithm take $A_1 = A+b$, where $b\in B$ is any element of $B$.
Suppose that we have constructed disjoint sets $A_1,\dots,A_k$.
If there is $b\in B$ such that $|(A+b) \setminus \bigsqcup_{j=1}^k A_j| \ge |A|/2$
then put $A_{k+1} = (A+b) \setminus \bigsqcup_{j=1}^k A_j$ and take $b_{k+1} = b$.
Suppose that our algorithm stops after $s$ steps.
If $s\ge |B|/2$ then we are done.
Put $U = \bigsqcup_{j=1}^s A_j$ and $B_* = B\setminus \{ b_1,\dots,b_s \}$.
Then $s|A|/2 < |U| \le s |A| $ and $|B_*| \ge |B|/2$.
We have
$$
2^{-2} |A| |B| \le 2^{-1} |A| |B_*| \le \sum_x (A \c U) (x) B_* (x) \,.
$$
Using the Cauchy--Schwarz inequality, we obtain
$$
2^{-4} |A|^2 |B|^2 \le \E(A,B) |U| \le \E(A,B) s |A|
$$
and the required lower bound for $s$ follows.
Let us prove the second part of the lemma.
Put $a=|A|$, $b=|B|$.
First of all note that
$$
\sigma = \sum_{x\in S} (A * B) (x) = \sum_{x\in B} |S \cap (A+x)|
$$
and hence there is $x\in B$ such that $|S\cap (A+x)| \ge \sigma b^{-1}$.
Put $b_1 = x$, and let $S_1 \subseteq S\cap (A+b_1)$ be an arbitrary set of size $\lceil \eps \sigma b^{-1} \rceil$,
where $\eps = 1/8$.
After that using the arguments as above, we construct a family of disjoint sets
$S_j \subseteq S \cap (A+b_j)$, $|S_j|\ge \eps \sigma b^{-1}$, $b_j \in B$, $j\in [s]$.
If $s \ge \sigma/(4a)$ then we are done.
If not then
put $U=\bigsqcup_{j=1}^s S_j$ and $B_* = B\setminus \{ b_1,\dots,b_s \}$.
We have $B_* \subseteq B'_* \bigcup B''_*$, where
$$B'_* = \{ x\in B_* ~:~ (A \c U) (x) \ge \eps \sigma b^{-1} \} \,,$$
$$B''_* = \{ x\in B_* ~:~ |(A+x) \setminus S| \ge a-2\eps \sigma b^{-1} \} \,.$$
Further, because of
$$
(a-2\eps \sigma b^{-1}) |B''_*| \le \sum_{x\in B''_*} |(A+b) \setminus S|
\le
\sum_{x\in B} |(A+x) \setminus S|
\le
ab - \sum_{x\in B} |(A+x) \bigcap S|
=
ab - \sigma
$$
we see that
$
|B''_*| \le (b-\sigma/a)(1+4\eps \sigma/(a b)).
$
Thus,
$$
|B'_*| \ge b-s -(b-\sigma/a)(1+4\eps \sigma/(a b))
\ge
\sigma/a - 4\eps \sigma/a - s
\ge
\sigma/4a \,.
$$
Finally, we obtain
$$
\eps \sigma b^{-1} \cdot \sigma/4a \le \sum_x (A \c U) (x) B'_* (x) \le \sum_x (A \c U) (x) B (x)
$$
and hence, in view of the condition $\sigma \ge 16 b$ the following holds
$$
(\eps \sigma b^{-1})^2 \cdot (\sigma/4a)^2
\le
|U| \E(A,B)
\le
(2 \eps \sigma b^{-1}) s \E(A,B) \,.
$$
Whence, $s\ge 2^{-8} \sigma^3 a^{-2} b^{-1} \E^{-1} (A,B)$.
This concludes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Now let us show how to get (\ref{f:E_3(D)_2.5}), for example.
Applying Lemma \ref{l:sumsets_disjoint} with $A=A$, $B=-A$, we obtain
\begin{equation}\label{tmp:11.05.2014_1}
\E_3 (D,A,A) \ge \sum_{j=1}^s \E_3 (A_j,A,A) \gg |A|^3 \E^{-1} (A) \E_3 (A)
\end{equation}
provided by $(3,\beta,\gamma)$--connectedness assumptions, $\beta,\gamma \gg 1$
(by the way bound (\ref{tmp:11.05.2014_1}) is tight as our main example $H\dotplus \Lambda$ shows).
On the other hand, we have by Lemma \ref{corpop} that
$$
\E^2 (A) |A|^2 \le \E_3 (A) \E_3 (D,A,A) \,.
$$
Combining the last two bounds, we get (\ref{f:E_3(D)_2.5}).
\bigskip
Using similar arguments and the second part of Lemma \ref{l:sumsets_disjoint}, we obtain
the following consequence, which shows, in particular,
that the popular difference sets \cite{Gow_m}, \cite{TV}
have some structure in the sense that they have large
energy
of some sort.
\begin{corollary}
Let $A\subseteq \Gr$ be a set, $P\subseteq A-A$.
Suppose that $A$ is a $(3,\beta,\gamma)$--connected set,
$\beta \le 2^{-3} \sigma_P (A) |A|^{-1}$.
Then
\begin{equation}\label{f:E_3(P,A,A)}
\E_3 (P,A,A) \ge 2^{-9} \gamma^{1/2} \sigma^5_P (A) \E(A) |A|^{-9} \,.
\end{equation}
\end{corollary}
\begin{proof}
Let
$\sigma = \sigma_P (A)$.
On the one hand, using Lemma \ref{l:sumsets_disjoint} with $A=A$, $B=-A$, we construct the family
of disjoint sets $P_j \subseteq P \cap (A-a_j)$, $a_j \in A$, $|P_j| \ge 2^{-3} \sigma |A|^{-1}$,
the number $s$ satisfies (\ref{f:s_bound+}).
Put $A_j = P_j + a_j \subseteq A$.
Thus, by connectedness of our set $A$, we have
$$
\E_3 (P,A) \ge \sum_{j=1}^s \E_3 (P_j,A) = \sum_{j=1}^s \E_3 (A_j,A)
\ge
\gamma \left( \frac{|A_j|}{|A|} \right)^6 \E_3 (A)
\ge
2^{-18} \gamma \frac{\sigma^6}{|A|^{12}} \E_3 (A) \,.
$$
On the other hand, applying Lemma \ref{corpop}, we get
$$
\sigma^4 \E^2 (A) |A|^{-6}
\le
\E_3 (A) \E_3 (P,A) \,.
$$
Combining the last two inequalities, we obtain
bound (\ref{f:E_3(P,A,A)}).
This concludes the proof.
$\hfill\Box$
\end{proof}
\section{On Gowers norms}
\label{sec:Gowers}
The notion of Gowers norms was introduced in papers \cite{Gow_4,Gow_m}.
At the moment it is a very important tool of investigation in wide class
of problems of additive combinatorics
(see e.g. \cite{GT_great}---\cite{Gow_m}, \cite{Samorod_false}, \cite{Samorod})
as well as in ergodic theory (see e.g. \cite{Austin}, \cite{Fu}, \cite{HK1}, \cite{HK2},
\cite{Tao1}, \cite{Tao2}, \cite{TZ}, \cite{Z1}, \cite{Z2}).
Recall the definitions.
Let
$G$
be a finite
set,
and $N=|G|$.
Let also $d$ be a positive integer, and
$$\{ 0,1 \}^d = \{ \omega = (\omega_1,\dots, \omega_d) ~:~ \omega_j \in \{0,1\},\, j=1,2,\dots,d \}$$
be the ordinary $d$---dimensional cube.
For $\omega \in \{0,1\}^d$ denote by $|\omega|$
the sum
$\omega_1 + \dots + \omega_d$.
Let also $\mathcal{C}$ be the operator of complex conjugation.
Let $\v{x} = (x_1,\dots,x_d), \v{x}'=(x'_1,\dots,x'_d)$ be two arbitrary vectors from $G^d$.
By $\v{x}^\o = (\v{x}^\o_1, \dots, \v{x}^\o_d)$ denote the vector
\begin{displaymath}
\v{x}^\o_i = \left\{ \begin{array}{ll}
x_i & \mbox{ if } \o_i = 0, \\
x'_i & \mbox{ if } \o_i = 1. \\
\end{array} \right.
\end{displaymath}
Thus $\v{x}^\o$ depends on $\v{x}$ and $\v{x}'$.
Let $f : G^d \to \C$ be an arbitrary function.
We will write $f(\v{x})$ for $f(x_1,\dots,x_d)$.
\begin{definition}
{\it Gowers $U^d$--norm} (or $d$--uniformity norm) of the function $f$ is the following expression
\begin{equation}\label{f:G_norm_d'}
\| f \|_{U^d}
=
\left( N^{-2d} \sum_{\v{x}\in G^d}\, \sum_{\v{x}' \in G^d}
\prod_{\o \in \{ 0,1 \}^d} \mathcal{C}^{|\o|} f (\v{x}^{\o}) \right)^{1/2^d} \,.
\end{equation}
\end{definition}
A sequence of $2^d$ points $\v{x}^\o \in G^d$, $\o \in \{ 0,1 \}^d$
is called {\it $d$--dimensional cube in $G^d$}
or just a $d$--dimensional cube.
Thus the summation in formula (\ref{f:G_norm_d'})
is taken over all cubes of $G^d$.
For example, $\{ (x,y), (x',y), (x,y'), (x',y') \}$, where $x,x',y,y'\in G$
is a two--dimensional cube in $G \m G$.
In the case Gowers norm is called rectangular norm.
For $d=1$ the expression above gives a semi--norm but for $d\ge 2$
Gowers norm is a norm.
In particular, the triangle inequality holds \cite{Gow_m}
\begin{equation}\label{e:triangle}
\| f+g \|_{U^d} \le \| f \|_{U^d} + \| g \|_{U^d} \,.
\end{equation}
One can prove also (see \cite{Gow_m}) the following monotonicity relation.
Let $f_{x_d} (x_1, \dots, x_{d-1}) := f (x_1, \dots, x_{d})$. Then
\begin{equation}\label{e:Gowers_mon}
N^{-1} \sum_{x_d \in G} \| f_{x_d} \|^{2^{d-1}}_{U^{d-1}} \le \| f \|^{2^{d-1}}_{U^d}
\end{equation}
for all $d\ge 2$.
If $\Gr = (\Gr,+)$ is a finite
Abelian group with additive group operation $+$, $N=|\Gr|$
then one can "project"\, the norm above onto the group
$\Gr$ and obtain the ordinary ("one--dimensional") Gowers norm.
In other words, we put the function $f(x_1,\dots,x_d)$ in formula
(\ref{f:G_norm_d'})
equals "one--dimensional"\, function $f(x_1,\dots,x_d) := f({\rm pr} (x_1,\dots,x_d))$,
where ${\rm pr} (x_1,\dots,x_d) = x_1+\dots+x_d$.
Denoting the obtained norm as $U^d$,
we
have
an analog of (\ref{e:Gowers_mon}), see \cite{Gow_m}, \cite{TV}
\begin{equation}\label{e:Gowers_mon_1}
\| f \|_{U^{d-1}} \le \| f \|_{U^d}
\end{equation}
for all $d\ge 2$.
It is convenient to write
\begin{equation}\label{f:G_norm_d'_UU-}
\| f \|_{\U^d} =
N^{-d+1}
\sum_{\v{x}\in \Gr^d}\, \sum_{\v{x}' \in \Gr^d}
\prod_{\o \in \{ 0,1 \}^d} \mathcal{C}^{|\o|} f ( {\rm pr} (\v{x}^{\o}))
=
\end{equation}
\begin{equation}\label{f:G_norm_d'_UU}
=
\sum_x \sum_{h_1,\dots,h_{d}}\, \prod_{\o \in \{ 0,1 \}^{d}}
\mathcal{C}^{|\o|} f(x + \o\cdot \v{h}) \,.
\end{equation}
In the case $f=A$, where $A\subseteq \Gr$ is a set, we have by formula (\ref{f:G_norm_d'_UU}) that
$$
\| A \|_{\U^d} = \sum_{s_1,\dots,s_d} |A_{\pi(s_1,\dots,s_d)}| \,,
$$
where $\pi(s_1,\dots,s_d)$ is a vector with $2^d$ components, namely,
$$
\pi(s_1,\dots,s_d) = \left( \sum_{j=1}^d s_j \eps_j \right)\,, \quad \quad \eps_j \in \{ 0,1 \}^d \,.
$$
Note also
\begin{equation}\label{f:Gowers_sq_A}
\| A \|_{\U^{d+1}} = \sum_{s_1,\dots,s_d} |A_{\pi(s_1,\dots,s_d)}|^2 \,.
\end{equation}
Further, $\| A \|_{\U^1} = \E_1 (A) = |A|^2$ and $\| A \|_{\U^2} = \E (A)$.
\bigskip
In definitions (\ref{f:G_norm_d'}), (\ref{f:G_norm_d'_UU-})
we have used the size of the set G/group $\Gr$.
The results of the paper are local, in the sense that they do not use the
cardinality of the
container group $\Gr$.
Thus it is natural to ask about the possibility to obtain an analog of (\ref{e:Gowers_mon_1}), say,
without any $N$ in the definition.
That is our simple result in the direction.
\begin{proposition}
Let $A \subseteq \Gr$ be a set.
Then for any integer $k\ge 2$ one has
\begin{equation}\label{f:Gowers_A}
\| A \|_{\U^{k+1}} \ge \frac{\| A \|^{(3k-2)/(k-1)}_{\U^{k}}}{\| A \|^{2k/(k-1)}_{\U^{k-1}}} \,.
\end{equation}
In particular,
\begin{equation}\label{f:Gowers_A_0}
\| A \|_{\U^3} \ge \frac{\E^4 (A)}{|A|^8} \,.
\end{equation}
\label{p:Gowers_A}
\end{proposition}
\begin{proof}
We have
$$
\| A \|_{\U^k} = \sum_{s_1,\dots,s_k} |A_{\pi(s_1,\dots,s_k)}|
=
\sum_{s_1,\dots,s_{k-1}} \sum_{s_k} \sum_z A_{\pi(s_1,\dots,s_{k-1})} (z) A_{\pi(s_1,\dots,s_{k-1})} (z+s_k)
=
$$
\begin{equation}\label{tmp:01.02.2014_1}
=
\sum_{s_1,\dots,s_{k-1}} \sum_z A_{\pi(s_1,\dots,s_{k-1})} (z) \cdot |A_{\pi(s_1,\dots,s_{k-1})}| \,.
\end{equation}
Thus, if the summation in (\ref{tmp:01.02.2014_1}) is taken over the set
\begin{equation}\label{tmp:Q_def}
Q_k := \{ (s_1,\dots,s_{k-1}) ~:~ |A_{\pi(s_1,\dots,s_{k-1})}| \ge \| A \|_{\U^k} (2k \| A \|_{\U^{k-1}} )^{-1} \}
\end{equation}
then it gives us $(1-1/2k)$ proportion of the norm $\| A \|_{\U^k}$.
Let us estimate the size of $Q_k$.
Clearly,
$$
|Q_k| \| A \|_{\U^k} (2k \| A \|_{\U^{k-1}} )^{-1}
\le
\sum_{s_1,\dots,s_{k-1}} |A_{\pi(s_1,\dots,s_{k-1})}| = \| A \|_{\U^{k-1}}
$$
and whence $|Q_k| \le 2k \| A \|^2_{\U^{k-1}} \| A \|^{-1}_{\U^k}$.
Certainly, the same bound holds for the cardinality of any set of tuples $(s_{i_1},\dots,s_{i_{k-1}})$
defined similar to (\ref{tmp:Q_def}) having the size $k-1$.
Hence, by the projection results, see e.g. \cite {Bol_Th}, we see that the summation
in (\ref{tmp:01.02.2014_1}) is taken over a set $\mathcal{S}$ of vectors $(s_1,\dots,s_k)$ of size at most
$
( 2k \| A \|^2_{\U^{k-1}} \| A \|^{-1}_{\U^k} )^{k/(k-1)}
$.
Returning to (\ref{tmp:01.02.2014_1}) and using the Cauchy--Schwarz inequality
as well as formula (\ref{f:Gowers_sq_A}), we obtain
$$
2^{-2} \| A \|^2_{\U^k}
\le
\left( \sum_{(s_1,\dots,s_k) \in \mathcal{S}} |A_{\pi(s_1,\dots,s_k)}| \right)^2
\le
|\mathcal{S}| \sum_{s_1,\dots,s_k} |A_{\pi(s_1,\dots,s_k)}|
\le
$$
$$
\le
( 2k \| A \|^2_{\U^{k-1}} \| A \|^{-1}_{\U^k} )^{k/(k-1)} \| A \|_{\U^{k+1}} \,.
$$
The last inequality implies that
$$
\| A \|_{\U^{k+1}} \ge C_k \frac{\| A \|^{(3k-2)/(k-1)}_{\U^{k}}}{\| A \|^{2k/(k-1)}_{\U^{k-1}}} \,,
$$
where $0<C_k<1$ depends on $k$ only.
Using the tensor trick we obtain the result.
This completes the proof.
$\hfill\Box$
\end{proof}
\begin{remark}
Estimate (\ref{f:Gowers_A}) is sharp as an example of a sufficiently dense random subset of a group $\Gr$ shows.
For higher Gowers norms one can show by induction a similar sharp inequality
$\| A\|_{\U^k} \ge \E(A)^{2^k-k-1} |A|^{-(3\cdot2^k-4k-4)}$.
It demonstrates expected exponential (in terms of $\E(A)$) growth of the norms.
\end{remark}
In the next section we will need in a statement, which is generalizes lower bound for $U^3$--norm (\ref{f:Gowers_A_0}).
\begin{lemma}
Let $A,B\subseteq \Gr$ be two sets.
Then
\begin{equation}\label{f:U^3(A,B)}
\sum_{s_1,s_2} \left( \sum_{x} A(x) B(x+s_1) A(x+s_2) B(x+s_1+s_2) \right)^2
\ge
\frac{\E^4 (A,B)}{|A|^4 |B|^4} \,.
\end{equation}
\end{lemma}
\begin{proof}
We use the same arguments as in the proof of Proposition \ref{p:Gowers_A}.
One has
$$
\E (A,B) = \sum_{s_1,s_2} \sum_{x} A(x) B(x+s_1) A(x+s_2) B(x+s_1+s_2)
=
$$
$$
=
\sum_{x} \sum_{s_1} B^A_{s_1} (x) |B^A_{s_1}| = \sum_x \sum_{s_2} A_{s_2} (x) |B_{s_2}| \,.
$$
Because of $\sum_s |A_s| = |A|^2$, $\sum_s |B_s| = |B|^2$, we get
for the set $\mathcal{S}$ above that\\ $|\mathcal{S}| \ll (|A|^2 |B|^2 \E^{-1} (A,B))^2$.
Thus
$$
\E^2 (A,B)
\ll
|\mathcal{S}| \sum_{s_1,s_2} \left( \sum_{x} A(x) B(x+s_1) A(x+s_2) B(x+s_1+s_2) \right)^2
$$
and the result follows.
$\hfill\Box$
\end{proof}
\bigskip
Using (\ref{f:Gowers_A_0}) and the Cauchy--Schwarz inequality, we have a consequence.
\begin{corollary}
Let $A\subseteq \Gr$ be a set and $|A - A|\le K|A|$ or $|A + A|\le K|A|$.
Then
$$
\| A \|_{\U^3} \ge \frac{|A|^4}{K^4} \,.
$$
\label{c:U^3&doubling}
\end{corollary}
Inequality (\ref{f:Gowers_A_0}) gives us a relation between $\| A\|_{\U^3} = \sum_s \E (A_s)$ and $\E(A)$.
W.T. Gowers (see \cite{Gow_m}) constructed a set $A$ having a random behavior in terms of $\E(A)$
(more precisely, he constructed a uniform set, that is having small Fourier coefficients, see \cite{Gow_m})
such that for all $s$ the sets $A_s$ have
non--random (non--uniform) behavior in terms of $\E(A_s)$.
Nevertheless, it is natural to ask about the possibility to find an $s\neq 0$ with a weaker notion of randomness,
that is $\E (A_s) \ll |A_s|^{3-c}$, $c>0$.
This question was asked to the author by T. Schoen.
We give an affirmative answer on it.
\begin{theorem}
Let $A\subseteq \Gr$ be a set, $\E(A) = |A|^3 /K$.
Suppose that for all $s\neq 0$ the
following holds
\begin{equation}\label{cond:A_s_bounded}
|A_s| \le \frac{M|A|}{K} \,,
\end{equation}
where $M\ge 1$ is a real number.
Let $K^4 \le M |A|$.
Then there is $s\neq 0$ such that $|A_s| \ge |A|/2K$ and
\begin{equation}\label{f:E(A_s)}
\E (A_s) \ll \frac{M^{93/79}}{K^{1/198}} \cdot |A_s|^3 \,.
\end{equation}
\label{t:E(A_s)}
\end{theorem}
\begin{proof}
Let
$$
P
:= \{ s ~:~ |A_s| \ge |A|/2K \} \,.
$$
Find the number $L$ satisfying $L := \max_{s\in P} |A_s|^3 \E^{-1} (A_s)$.
In other words for all $s \in P$, one has $\E(A_s) \ge |A_s|^3 / L$.
Our task is to find a lower bound for $L$.
Put
$$
\Cf(x,y) = \Cf_3 (A) (x,y) := |A \cap (A-x) \cap (A-y)|
$$
and
$$
\t{\Cf} (x,y) = \Cf_4 (A) (x,y,x-y) := |A \cap (A-x) \cap (A-y) \cap (A-x + y)|
\,.
$$
Clearly,
\begin{equation}\label{f:C_A_s}
\t{\Cf} (x,y) \le \Cf(x,y) \le \min\{ |A_x|, |A_y| \} \,.
\end{equation}
for any $x,y$.
Put also
$$
\mathcal{P} := \{ (x,y) ~:~ \t{\Cf} (x,y) \ge |A|/(4KL) \} \,.
$$
We will write $\mathcal{P}_x := \mathcal{P} \cap (\{x\} \m \Gr)$,
and $\mathcal{P}_y := \mathcal{P} \cap (\Gr \m \{y\})$.
Put also
$$
\mathcal{P}^\la := \mathcal{P} \cap \{(x,y) ~:~ x - y = \lambda \} \,.
$$
Our first lemma says that the size of $\mathcal{P}$ and some characteristics of the set
can be
estimated
in terms of $L$ and $M$.
\begin{lemma}
We have
\begin{equation}\label{f:P_size1}
\frac{|A|^2}{4LM^2} \le |\mathcal{P}| \le 4L|A|^2 \,.
\end{equation}
Further, for any nonzero $y$ and $\la$ the following holds
\begin{equation}\label{f:P_size2}
|\mathcal{P}_y|\,, |\mathcal{P}^\la| \le \frac{4M^2 L|A|}{K} \,.
\end{equation}
\label{l:P_size}
\end{lemma}
\begin{proof}
By the Cauchy--Schwarz inequality, we have
$$
\sum_{s\in P} |A_s|^3 \ge 2^{-1} \E_3 (A) \ge 2^{-1} |A|^4 / K^2 \,.
$$
Hence
$$
\frac{|A|^4}{2K^2 L}
\le
\frac{1}{L} \sum_{s\in P} |A_s|^3
\le
\sum_{s\in P} \E(A_s) \le \sum_s \E (A_s)
=
$$
$$
=
\sum_s \sum_{z,x,y} A_s (z) A_s (z+x) A_s (z+y) A_s (z+x-y) = \sum_{x,y} \t{\Cf}^2 (x,y) \,.
$$
We can assume that $|A| \ge 4 K L^{1/2}$ because otherwise the result is trivial
in view of the condition $K^4 \le M |A|$.
Since
$$
\sum_{x,y} \t{\Cf} (x,y) = \E (A) = \frac{|A|^3}{K} \,,
$$
it follows
by (\ref{f:C_A_s}) and the assumption $|A| \ge 4 K L^{1/2}$ that
$$
\frac{|A|^4}{4K^2 L} \le \sum_{0\neq (x,y) \in \mathcal{P}} \t{\Cf}^2 (x,y)
\le
\left( \frac{M|A|}{K} \right)^2 |\mathcal{P}| \,.
$$
In other words $\frac{|A|^2}{4LM^2} \le |\mathcal{P}|$.
On the other hand
$$
\frac{|A|}{4KL} |\mathcal{P}| \le \sum_{(x,y) \in \mathcal{P}} \t{\Cf} (x,y) \le \sum_{x,y} \t{\Cf} (x,y)
=
\E(A) = \frac{|A|^3}{K}
$$
and we obtain the required upper bound for the size of $\mathcal{P}$.
Further, for any fixed $y \neq 0$ the following holds
$$
|\mathcal{P}_y| \le \frac{4KL}{|A|} \sum_x \t{\Cf} (x,y) = \frac{4KL}{|A|} |A_{- y}| |A_y| \le \frac{4M^2 L|A|}{K} \,.
$$
Finally,
$$
|\mathcal{P}^\la| = \sum_{(x,y) ~:~ x - y = \lambda} \mathcal{P} (x,y)
\le
\frac{4KL}{|A|} \sum_{(x,y) ~:~ x - y = \lambda}\, \sum_x \t{\Cf} (x,y)
=
\frac{4KL}{|A|} |A_\la|^2
\le \frac{4M^2 L|A|}{K}
$$
as required.
$\hfill\Box$
\end{proof}
\bigskip
Now, we show that some norm of $\mathcal{P}$ is huge.
Actually, we use the function $\Cf$ not $\t{\Cf}$ in the proof.
\begin{lemma}
One has
\begin{equation}\label{f:P_norm}
\frac{|A|^{5}}{2^9 K L^4 M^6}
\le
\sum_{y,x',y'} |\sum_x \mathcal{P} (x,y+x) \mathcal{P} (x'+x,y'+x) |^2 \,.
\end{equation}
\label{l:P_norm}
\end{lemma}
\begin{proof}
As in the proof of Lemma \ref{l:P_size}, we get
$$
\frac{|A|^3}{4KLM} \le \sum_{(x,y) \in \mathcal{P}} \Cf (x,y)
=
\sum_z A(z) \sum_{x,y} \mathcal{P} (x,y) A(z+x) A(z+y) \,.
$$
Using the Cauchy--Schwarz inequality, we obtain
$$
\frac{|A|^5}{16 K^2 L^2 M^2}
\le
\sum_z A(z) \sum_{x,y}
\sum_{x',y'} \mathcal{P} (x,y) A(z+x) A(z+y) \mathcal{P} (x',y') A(z+x') A(z+y')
\le
$$
$$
\le
\sum_{x,y,x',y'} \mathcal{P} (x,y+x) \mathcal{P} (x'+x,y'+x) \Cf_4 (A) (y,x',y')
\,.
$$
Applying the Cauchy--Schwarz inequality again, we have
$$
\frac{|A|^{10}}{2^8 K^4 L^4 M^4}
\le
\sum_{y,x',y'} |\sum_x \mathcal{P} (x,y+x) \mathcal{P} (x'+x,y'+x) |^2
\m
\E_4 (A)
\le
$$
$$
\le
\sum_{y,x',y'} |\sum_x \mathcal{P} (x,y+x) \mathcal{P} (x'+x,y'+x) |^2
\m
\frac{2M^2 |A|^5}{K^3}
$$
because of $K^3\le K^4 \le M |A| \le M^2 |A|$ and hence
\begin{equation}\label{f:E_4_tmp'}
\E_4 (A)
\le
|A|^4 + \frac{M^2 |A|^2}{K^2} \E (A)
\le
\frac{2M^2 |A|^5}{K^3} \,.
\end{equation}
This concludes the proof of the lemma.
$\hfill\Box$
\end{proof}
\bigskip
In terms of the sets $\mathcal{P}^\la$ we can rewrite expression (\ref{f:P_norm}) as
$$
\sum_{y,x',y'} |\sum_x \mathcal{P} (x,y+x) \mathcal{P} (x'+x,y'+x) |^2
=
\sum_{y,x',y'} | \sum_x \mathcal{P}^{-y} (x) \mathcal{P}^{x'-y'} (x'+x) |^2
=
$$
$$
=
\sum_{y,x',y'} | \sum_x \mathcal{P}^{y} (x) \mathcal{P}^{y'} (x'+x) |^2
=
\sum_{\la,\mu} \E (\mathcal{P}^{\la}, \mathcal{P}^{\mu}) \,.
$$
In view of estimate (\ref{f:P_norm}), a trivial inequality
$$
\sum_{\la,\mu} \E (\mathcal{P}^{\la}, \mathcal{P}^{\mu})
\le
\sum_{\la,\mu} |\mathcal{P}^{\la}|^2 |\mathcal{P}^{\mu}|
\ll
|\mathcal{P}|^2 \cdot \max_{\la \neq 0} |\mathcal{P}^{\la}|
$$
and bounds for size of $\mathcal{P}^{\la}$ of Lemma \ref{l:P_size}
the sets $\mathcal{P}^\la$ should look, firstly, like some sets with small doubling
(more precisely as sets with large additive energy) and,
secondly, the large proportion of such sets must correlate to each other.
A model example here is $\mathcal{P}^\la (x) = Q(x+\a(\la))$, where $\a$ is an arbitrary function
and $Q$ is an arithmetic progression of size approximately $\Theta (|A|/K)$.
Now we prove that the example is the only case, in some sense.
\bigskip
Put $l = \log (LM)$.
Note that $l\le \log (KM)$ because if $L\ge K$ then the result is trivial.
We can suppose that
\begin{equation}\label{cond:L_tmp}
2^{17} M^6 K^3 L^7 \le |A|
\end{equation}
because otherwise the result follows
immediately
in view of the condition $M |A| \ge K^4$.
Reducing zero terms, we have
$$
\frac{|A|^{5}}{2^{10} K L^4 M^6}
\le
\sum_{\la,\mu \neq 0} \E (\mathcal{P}^{\la}, \mathcal{P}^{\mu})
$$
because of
Lemma \ref{l:P_size}, estimate (\ref{cond:L_tmp}), a bound
$$
|\mathcal{P}^{0}| \le 4KL |A|^{-1} \sum_{x} |A_x| = 4KL |A|
$$
and a calculation
$$
2(4KL|A|)^2 4L |A|^2 \le \frac{|A|^{5}}{2^{10} K L^4 M^6} \,.
$$
Below we will assume that any summation is taken over
nonzero indices
$\la$, $\mu$.
By Lemma \ref{l:P_size} and
a trivial estimate
$$
\sum_{\la} |\mathcal{P}^{\la}| = |\mathcal{P}| \le 4 L |A|^2 \,,
\quad \quad
\E (\mathcal{P}^{\la}, \mathcal{P}^{\mu})
\le
|\mathcal{P}^{\la}| |\mathcal{P}^{\mu}|^2
$$
the following holds
$$
\frac{|A|^{5}}{2^{11} K L^4 M^6}
\le
\sum_{\la,\mu ~:~ |\mathcal{P}^{\la}|,\, |\mathcal{P}^{\mu}| \ge \D_*} \E (\mathcal{P}^{\la}, \mathcal{P}^{\mu}) \,,
$$
where $\D_* = \frac{|A|}{2^{16} K L^6 M^6}$.
Using the pigeonhole principle and Lemma \ref{l:P_size}, we find a number $\D$
such that $\D_* \le \D \le \frac{4M^2 L|A|}{K}$ and
\begin{equation}\label{tmp:10.06.2013_1}
\frac{|A|^{5}}{lK L^4 M^6}
\ll
\sum_{\mu ~:~ \D < |\mathcal{P}^{\mu}| \le 2\D} \,\,
\sum_{\la :~ |\mathcal{P}^{\la}| \ge \D_*}
\E (\mathcal{P}^{\la}, \mathcal{P}^{\mu}) \,.
\end{equation}
From (\ref{tmp:10.06.2013_1}) and the Cauchy--Schwarz inequality one can see
that the summation in the
formula is taken over
$$
\E(\mathcal{P}^{\mu})
\gg
\frac{|\mathcal{P}^{\mu}|^3}{l^2 L^{14} M^{16}}
:= \eps |\mathcal{P}^{\mu}|^3 \,.
$$
By
inequality
(\ref{f:P_size1}) of Lemma \ref{l:P_size} there is $\mu$ with
\begin{equation}\label{tmp:11.03.2014_1}
\zeta \frac{|A|^{3} \D}{K}
:=
\frac{|A|^{3} \D}{lK L^5 M^6}
\ll
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*}
\E (\mathcal{P}^{\la}, \mathcal{P}^{\mu}) \,.
\end{equation}
Put $Q=\mathcal{P}^{\mu}$.
We have $\E(Q) \ge \eps |Q|^3$.
Applying a trivial general
bound
$$
\E(A,B) \le |A| |B| \m \max_{x} |A \cap (B-x)| \,,
$$
we get by Lemma \ref{l:P_size}
$$
\frac{|A|^{3} \D}{lK L^5 M^6}
\ll
\D \frac{4M^2 L|A|}{K} \m \sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*} \max_{x} |\mathcal{P}^{\la} \cap (Q-x)| \,.
$$
Given
an arbitrary $\la$ let the maximum in the last formula is attained at point $x:=\a(\la)$.
Thus, we have
$$
\frac{|A|^{2}}{l L^6 M^8}
\ll
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*} |\mathcal{P}^\la \cap (Q-\a(\la))|
=
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*}
\sum_{x} \mathcal{P} (x,x-\la) Q(x+\a(\la)) \,.
$$
Hence
we find a set $Q$ of the required form, that is having the large additive energy and which is
correlates with sets $\mathcal{P}^{\la}$.
Now, we transform the obtained information into some knowledge about the original set $A$.
\bigskip
Using the definition of the set $\mathcal{P}$, we obtain
$$
\frac{|A|^{3}}{l L^7 M^8 K}
\ll
\sum_{x,\la} Q(x+\a(\la)) \sum_z A (z) A (z+x) A (z+x-\la) A (z+\la)
=
$$
\begin{equation}\label{tmp:10.06.2013_2}
=
\sum_{z,\la} (Q\circ A_{-\la}) (z-\a(\la)) A_\la (z) \,.
\end{equation}
We know that $\E(Q) \ge \eps |Q|^3$.
By Balog--Szemer\'{e}di--Gowers Theorem \ref{BSG} we find $Q'\subseteq Q$, $|Q'| \gg \eps |Q|$
such that $|Q'-Q'| \ll {\eps}^{-4} |Q'|$.
We
will prove shortly
that the set $Q$ in (\ref{tmp:10.06.2013_2}) can be replaced by a set $\t{Q}$, namely
\begin{equation}\label{tmp:10.06.2013_3}
c(\eps) \frac{|A|^{3}}{K}
\ll
\sum_{z,\la} (\t{Q} \circ A_{-\la}) (z-\a(\la)) A_\la (z) \,,
\end{equation}
where $c(\eps)>0$ is some constant depends on $L$ and $M$ only
and $\t{Q}$ has small doubling.
Indeed, starting with (\ref{tmp:11.03.2014_1}),
put $Q'_1 = Q'$, and define inductively disjoint sets $Q'_j \subseteq Q$,
$B_j = \bigsqcup_{i=1}^j Q'_j$, $\bar{B}_j = Q\setminus B_j$, $j\in [s]$, applying Theorem \ref{BSG} to $\bar{B}_j$.
Put also $\E(\bar{B}_j) = \nu_j |Q|^3 \le 8 \nu_j \D^3$.
If at some stage $j$
\begin{equation}\label{tmp:11.03.2014_2}
\zeta \frac{|A|^{3} \D}{2K}
>
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*}
\E (\mathcal{P}^{\la}, \bar{B}_j)
\end{equation}
then we stop the algorithm.
Let the procedure works exactly $s$ steps
and put $B= B_s$, $\bar{B} = Q\setminus B$.
We claim that $s\ll \eps^{-1}$.
To prove this note that if (\ref{tmp:11.03.2014_2}) does not hold then
$$
\zeta \frac{|A|^{3} \D}{K} \ll \E^{1/2} (\bar{B}_j) \sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*} |\mathcal{P}^{\la}|^{3/2}
\ll
\nu^{1/2}_j \D \frac{M^2L |A|}{K} L |A|^2 \,.
$$
In other words,
$\E (\bar{B}_j) \gg \eps |Q|^3$ and, hence,
$|Q'_j| \gg \eps |Q|$, $|Q'_j-Q'_j| \ll {\eps}^{-4} |Q|$.
It means, in particular, that after $s\ll \eps^{-1}$ number of steps our algorithm
stops indeed.
At the last step, we get by the
construction
that
$$
\zeta \frac{|A|^{3} \D}{2K}
\le
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*} \E (\mathcal{P}^{\la},Q)
- \sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*} \E (\mathcal{P}^{\la}, \bar{B}, \bar{B})
\le
$$
$$
\le
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*} \E (\mathcal{P}^{\la}, B, B)
+
2\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*}
2\E (\mathcal{P}^{\la}, B, \bar{B}) ) \,.
$$
Let us prove
the following estimate
$$
\zeta \frac{|A|^{3} \D}{K}
\ll
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*}
\E (\mathcal{P}^{\la}, B) \,.
$$
If not then by the Cauchy--Schwarz inequality and the choice of $\bar{B}$,
we obtain
$$
\left( \zeta \frac{|A|^{3} \D}{K} \right)^2
\ll
\left( \sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*} \E (\mathcal{P}^{\la}, B, \bar{B}) \right)^2
\le
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*}
\E (\mathcal{P}^{\la}, B) \cdot \zeta \frac{|A|^{3} \D}{2K}
$$
and we
get a contradiction.
Hence
the following holds
$$
\zeta \frac{|A|^{3} \D}{K}
\ll
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*}
\E (\mathcal{P}^{\la}, B) \,.
$$
Applying
the H\"{o}lder inequality, we find a set $Q'_j$ such that
$$
\zeta \frac{|A|^{3} \D}{s^2 K}
\ll
\sum_{\la ~:~ |\mathcal{P}^{\la}| \ge \D_*}
\E (\mathcal{P}^{\la}, Q'_j) \,.
$$
So, putting $Q':=Q'_j$ we get (\ref{tmp:10.06.2013_3}) with $c(\eps) \gg \frac{\zeta \eps^2}{lL^2 M^2}$.
Of course, the summation in the
obtained
formula can be taken just over $\la$ with $|A_\la| \gg c(\eps) \frac{|A|}{K}$
and we will assume this.
\bigskip
Denote by $\Omega$ the set
$$
\Omega := \{ (z,\la) ~:~ A_\la (z) =1 \,, \mbox{ and } (\t{Q} \circ A_{-\la}) (z-\a(\la)) \ge \frac{c(\eps)|A|}{2K} \} \,.
$$
From (\ref{tmp:10.06.2013_3}) and our assumption (\ref{cond:A_s_bounded})
we have $|\Omega| \gg c(\eps) M^{-1} |A|^2$.
On the other hand, considering $\Omega_\la := \{ z~:~ (z,\la) \in \Omega \}$ for any fixed $\la$, one has
$$
|\Omega_{\la}| \frac{|A|}{K} c(\eps) \ll \sum_z A_\la (z) (\t{Q} \circ A_{-\la}) (z-\a(\la))
\le
|A_\la| |Q|
\ll
|A_\la| \D
\ll \frac{M^3 L |A|^2}{K^2} \,.
$$
Hence there are at least $\gg c^2 (\eps) K M^{-4} L^{-1} |A|$ sets $A_\la$
such that there exists some $z=z(\la)$ with $(\t{Q} \circ A_{\la}) (z-\a(\la)) \gg c(\eps) |A|/2K$.
Denote the set of these $\la$ by $T$.
For any such $A_\la$ there exists a shift of the set $\t{Q}$ such that
$|A_\la \cap (\t{Q} + w(\la))| \gg c(\eps) |A|/2K \gg_{\eps,M} |A_\la|$.
Put $A'_\la := A_\la \cap (\t{Q} + w(\la))$.
We have by Lemma \ref{l:Plunnecke} that for any $\la_1,\la_2 \in T$ the following holds
$$
|A'_{\la_1}+A'_{\la_2}| \le |\t{Q}+\t{Q}| \ll \eps^{-8} \frac{M^2 L|A|}{K} \,.
$$
In particular,
$$
\E(A'_{\la_1}, A'_{\la_2}) \gg \frac{c^4(\eps) \eps^8 |A|^3}{M^2 L K^3} \,.
$$
Finally, using (\ref{f:E_4_tmp'}) as well as Lemma \ref{l:E_3_A_s} with $k=l=2$, we obtain
$$
\frac{|A|^5}{l^{64} M^{458} L^{395} K}
\ll
\frac{\zeta^8 \eps^{24} |A|^5}{l^8 M^{26} L^{19} K}
\ll
\frac{c^8(\eps) \eps^8 |A|^5}{M^{10} L^3 K}
\ll
|T|^2 \cdot \frac{c^4(\eps) \eps^8 |A|^3}{M^2 L K^3}
\ll
\sum_{\la_1, \la_2 \in T} \E(A'_{\la_1}, A'_{\la_2})
\le
$$
$$
\le
\sum_{\la_1, \la_2 \in T} \E(A_{\la_1}, A_{\la_2}) \le \E_4 (A) \le \frac{2M^2 |A|^5}{K^3}
$$
with
the required lower bound for $L$.
This completes the proof of Theorem \ref{t:E(A_s)}.
$\hfill\Box$
\end{proof}
\bigskip
We finish the section by analog of Definition \ref{def:conn},
which we will use in the next section.
\begin{definition}
For $\beta,\gamma \in [0,1]$ a set $A$ is called $U^k (\beta,\gamma)$--connected
if for any $B \subseteq A$, $|B| \ge \beta|A|$
the following holds
$$
\| B \|_{\U^k} \ge \gamma \left( \frac{|B|}{|A|} \right)^{2^k} \| A \|_{\U^k} \,.
$$
\label{def:Uk-conn}
\end{definition}
Again, if, say, $\gamma^{-1} |A|^8 / |A\pm A|^4 \ge \|A \|_{\U^3}$ then by inequality (\ref{f:Gowers_A_0})
one can see that $A$ is $U^3 (\beta,\gamma)$--connected for any $\beta$.
The existence of $U^k (\beta,\gamma)$--connected subsets in an arbitrary set
is discussed in the Appendix.
\section{Self--dual sets}
\label{sec:structural2}
Inequality (\ref{f:Gowers_A_0}) gives us a relation between $\| A\|_{\U^3}$ and $\E(A)$.
It attaints at a random subset $A$ of $\Gr$,
where by randomness we mean that each element of $A$ belongs to the set with probability $\E(A)/|A|^3$.
On the other hand,
it is easy to see that an upper bound takes place
\begin{equation}\label{f:U_3_E_3}
\| A \|_{\U^3} = \sum_s \E(A_s) \le \sum_s |A_s|^3 = \E_3 (A) \,.
\end{equation}
A weaker estimate follows from (\ref{f:U_3_E_3}) combining with the Cauchy--Schwarz inequality
\begin{equation}\label{f:U_3_E_3'}
\| A \|^2_{\U^3} \le \E_4 (A) \E(A) \,.
\end{equation}
In the section we consider sets having critical relations between $\| A \|_{\U^3}$ and $\E_4 (A)$, $\E(A)$
that is the sets satisfying the reverse inequality to (\ref{f:U_3_E_3'})
(actually, we use a slightly stronger estimate then reverse to (\ref{f:U_3_E_3})).
It turns out that they are exactly which we called self--dual sets.
\bigskip
Let us recall a result on large deviations.
The following variant
can be found in \cite{GreenA+A}.
\begin{lemma}
Let $X_1,\dots,X_n$ be independent random variables with $\mathbb{E} X_j = 0$
and $\mathbb{E} |X_j|^2 = \sigma_j^2$.
Let $\sigma^2 = \sigma_1^2 + \dots + \sigma_n^2$.
Suppose that for all $j\in [n]$, we have $|X_j| \le 1$.
Let also $a$ be a real number such that $\sigma^2 \ge 6 na$.
Then
$$
\mathbb{P} \left( \left| \frac{X_1+\dots + X_n}{n} \right| \ge a \right) \le 4 e^{-n^2 a^2 / 8\sigma^2} \,.
$$
\label{l:large_deviations}
\end{lemma}
We need in a combinatorial lemma.
\begin{lemma}
Let $\D$, $\sigma$, $C>1$ are positive numbers, $t$ be a positive integer,
and $M_1, \dots, M_t$ be sets, $\D \le |M_j| \le C\D$, $j\in [t]$,
$\sigma \le 10^{-4} t^2 \D$,
where
$$
\sigma:= \sum_{i,j=1}^t |M_i \bigcap M_j| \,.
$$
Then
there are at least $\frac{t^2 \D}{16(2C+1) \sigma}$ disjoint sets
$\t{M}_l \subseteq M_{i_l}$
such that
$|\t{M}_l| \ge \frac{\D}{4(2C+1)}$.
\label{l:disjoint_probab}
\end{lemma}
\begin{proof}
We will choose our sets $\t{M}_i$ randomly with probability at least $1/4$.
First of all, we note that
\begin{equation}\label{tmp:25.03.2014_1}
10^{-4} t^2 \D \ge \sigma \ge \sum_{i=1}^t |M_i| \ge t \D \,.
\end{equation}
Put $p=t\D 2^{-1} \sigma^{-1}$.
In view of (\ref{tmp:25.03.2014_1}), we get $p\in (0,1/2]$.
Let us form a new family of sets taking a set $M_i$ from $M_1,\dots,M_t$ uniformly and independently
with probability $p$.
Denote the obtained family as $M'_1,\dots, M'_s$.
By Lemma \ref{l:large_deviations} and bound (\ref{tmp:25.03.2014_1}), we have after some calculations that
\begin{equation}\label{tmp:25.03.2014_2}
2^{-1} pt \le s \le 2pt
\end{equation}
with probability at least $3/4$.
Further the expectation of $\sigma$ equals
$$
\mathbb{E} \sum_{i,j=1}^t |M_i \bigcap M_j|
=
\sum_x \sum_{i=1}^t \mathbb{E} M_i (x) + \sum_x \sum_{i,j=1,\,i\neq j}^t \mathbb{E} M_i (x) M_j (x)
=
$$
$$
=
p \sum_{i=1}^t |M_i| + p^2 \sum_{i,j=1,\,i\neq j}^t |M_i \bigcap M_j|
\le
Cpt\D + p^2 \sigma
\le
(2C+1)p^2 \sigma
$$
by our choice of $p$.
Hence, by Markov inequality, with probability at least $1/2$ one has
$$
\sum_{i,j=1}^s |M'_i \bigcap M'_j| \le (4C+2)p^2 \sigma
$$
and by the Cauchy--Schwarz inequality, we get
$$
|\bigcup_{i=1}^s M'_i| \ge \frac{(\sum_{i=1}^s |M'_i| )^2}{\sum_{i,j=1}^s |M'_i \cap M'_j|}
\ge
\frac{s^2 \D^2}{(4C+2)p^2 \sigma} \ge \frac{2s^2 \sigma}{(2C+1)t^2} := q \,.
$$
Now we take disjoint subsets $M''_i \subseteq M'_i$, $i\in [s]$.
Thus
\begin{equation}\label{tmp:25.03.2014_3}
2 \sum_{i ~:~ |M''_i| \ge q (2s)^{-1}} |M''_i| \ge \sum_{i} |M''_i| \ge q \,.
\end{equation}
Finally, we put $\t{M}_i = M''_i$.
By our choice of parameters and estimates (\ref{tmp:25.03.2014_2}) the following holds
$$
|\t{M}_i| \ge \frac{q}{2s} = \frac{s\sigma}{(2C+1)t^2} \ge \frac{p\sigma}{(4C+2) t} = \frac{\D}{(8C+4)} \,.
$$
Similarly, the number $n$ of the sets $\t{M}_i$ can be estimated from (\ref{tmp:25.03.2014_3})
$$
n \ge \frac{q}{2\D} = \frac{s^2 \sigma}{(2C+1)t^2 \D} \ge \frac{p^2 \sigma}{(8C+4) \D} = \frac{t^2 \D}{(32C+16) \sigma} \,.
$$
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Let us remark an interesting consequence of Lemma \ref{l:disjoint_probab}.
\bigskip
\begin{corollary}
Let $A\subseteq \Gr$ be a
$(2,\beta,\gamma)$--connected set, $\beta\le 0.5$ be a constant.
Then there is a set
$P \subseteq \{ x~:~ (A\c A) (x) \ge \D \}$
satisfies
$\E^P (A) \gg \E (A) \log^{-1} |A|$,
and there are
$k \gg \gamma |A| \D^{-1} \log^{-1} |A|$ disjoint sets
$\t{A}_j \subseteq A_{s_j}$ with $|\t{A}_j| \gg \D$.
\label{c:disjoint_P}
\end{corollary}
\begin{proof}
Using Lemma \ref{l:eigen_A'}, we find $A'$, $|A'| \ge |A|/2$ such that estimate (\ref{f:eigen_A'}) takes place.
We want to apply Lemma \ref{l:disjoint_probab} to the sets $A'_s \subseteq A_s$, $s\in P'$,
where $P'=\{ x~:~ (A'\c A') (x) \sim \Delta \}$, $\E^{P'} (A') \gg \E (A') \log^{-1} |A|$.
Of course, such set $P'$ exists by the pigeonhole principle.
To apply Lemma \ref{l:disjoint_probab}, we need to calculate the quantity $\sigma$
$$
\sigma := \sum_{s,t\in P'} |A'_s \bigcap A'_t| = \sum_{x,y} \Cf_3 (A') (x,y) P'(x) P'(y) \,.
$$
By the last identity and estimate (\ref{f:eigen_A'}) (for details, see \cite{s_mixed}), we get
$$
\sigma \ll \frac{\E(A)}{|A|} \cdot |P'| \,.
$$
Applying Lemma \ref{l:disjoint_probab}, we find disjoint sets $\t{A}_j \subseteq A'_{s_j} \subseteq A_{s_j}$,
$s_j \in P'$, $j\in [k]$ such that
$$
k \gg \frac{|P'|^2 \D |A|}{\E(A) |P'|} \gg \gamma \frac{|A|}{\D \log |A|} \,.
$$
In the last inequality we have used $(2,\beta,\gamma)$--connectedness of $A$.
To complete the proof note that $P' \subseteq \{ x~:~ (A\c A) (x) \ge \D \}$.
$\hfill\Box$
\end{proof}
\bigskip
Clearly, the bound on $k$ in Corollary \ref{c:disjoint_P} is the best possible up to logarithms.
Calculating $\E(A,A_j)/|A_j|$ and comparing its with $\E_3$ (see \cite{s_mixed}) one can obtain an alternative
proof of lower bounds for $|A\pm A_s|$ as of section \ref{sec:sumsets}.
Another result on a family with disjoint $A_s$ is proved in Proposition \ref{p:disjoint_via_A-A_s} below.
\bigskip
Now we are able to
obtain
the main result of the section.
\begin{theorem}
Let $A\subseteq \Gr$ be a set, and $M\ge 1$ be a real number.
Put $l=\log |A|$.
Suppose that $A$ is
$U^3(\beta,\gamma)$ and
$(2,\beta,\gamma)$--connected with $\beta \le 0.5$.
Then inequality
\begin{equation}\label{cond:E_3_and_E_critical_cor}
\| A \|^2_{\U^3} \gg_{M} \E_4 (A) \E(A)
\end{equation}
takes place iff there is a positive real $\D \sim_{M,\,l} \E_3(A) \E(A)^{-1}$ and a set
$$
P \subseteq \{ s\in A-A ~:~ \D < |A_s| \} \,,
$$
such that $|P| \gg_{M,\,l} |A|$, $P=-P$, further,
\begin{equation}\label{f:self-dual0}
\E^P (A) \gg_{M,\,l} \E(A) \,,\quad \quad \E^P_3 (A) \gg_{M,\,l} \E_3 (A)
\,,\quad \quad \E^P_4 (A) \gg_{M,\,l} \E_4 (A) \,.
\end{equation}
and
such that for any $s\in P$ there is $H^s \subseteq A_s$,
$|H^s| \gg_{M,\,l} \D$,
with
\begin{equation}\label{f:self-dual1}
|H^s - H^s| \ll_{M,\,l} |H^s| \,,
\end{equation}
and $\E(A,H^s) \ll_{M,\,l} |H^s|^3$.\\
Moreover there are disjoint sets $H_j \subseteq A_{s_j}$, $|H_j| \gg_{M,\,l} \D$,
$s_j \in P$, $j\in [k]$
such that all $H_j$ have small doubling property (\ref{f:self-dual1}),
$\E(A,H_j) \ll_{M,\,l} |H_j|^3$ and $k\gg_{M,\,l} |A| \D^{-1}$.
\label{t:self-dual}
\end{theorem}
\begin{proof}
Put $a=|A|$, $\E = \E(A)$, $\E_3 = \E_3 (A)$, $\E_4 = \E_4 (A)$.
Let us begin with the necessary condition.
Using Lemma \ref{l:eigen_A'}, we find $A'$, $|A'| \ge |A|/2$ such that estimate (\ref{f:eigen_A'}) takes place.
Because of $A$ is
$U^3(\beta,\gamma)$ and
$(2,\beta,\gamma)$--connected with $\beta \le 0.5$,
we have $\| A' \|_{\U^3} \sim \| A \|_{\U^3}$ and $\E(A') \sim \E(A)$.
Combining assumption (\ref{cond:E_3_and_E_critical_cor})
with the Cauchy--Schwarz inequality, we get
\begin{equation}\label{f:E_4_begin}
\| A \|^2_{\U^3} \gg_{M} \E_4 (A) \E(A) \ge \E^2_3 (A)
\end{equation}
In particular, by the last inequality and (\ref{f:U_3_E_3}), (\ref{f:U_3_E_3'}), we obtain
$$
\E^2_3 (A') \ge \| A' \|^2_{\U^3} \gg \| A \|^2_{\U^3} \ge_M \E^2_3 (A) \,,
$$
$$
\E_4 (A') \E(A') \ge \| A' \|^2_{\U^3} \gg \| A \|^2_{\U^3} \ge_M \E_4 (A) \E(A) \ge \E_4 (A) \E(A')
$$
and, hence, $\E_3 (A') \sim_M \E_3 (A)$, $\E_4 (A') \sim_M \E_4 (A)$.
With some abuse of the notation we will use the same letter $A$ for $A'$ below.
By Lemma \ref{l:E_3_A_s}, we have
\begin{equation}\label{f:23.03.2014_1}
\| A \|_{\U^3} = \sum_s \E (A_s) = \sum_s \sum_t (A_s \c A_s)^2 (t)
\gg_{M} \sum_s |A_s|^3 = \sum_s \E(A,A_s) = \E_3 \,.
\end{equation}
One can assume that the summation in the last formula is taken over $s$ such that
$|A_s| \gg_M \E_3 \E^{-1}$
and $\E (A_{s}) \gg_M |A_{s}|^3$, $|A_{s}|^3 \gg_M \E(A,A_s)$.
Let us consider the condition $\E (A_{s}) \gg_M |A_{s}|^3$.
By Balog--Szemer\'{e}di--Gowers Theorem we can find $H^s \subseteq A_{s}$ with
$|H^s| \gg_M |A_{s}|$, and $|H^s - H^s| \ll_M |H^s|$.
Loosing a logarithm $l=\log a$ we can assume that the summation in (\ref{f:23.03.2014_1}) is taken over
$|A_s|$, $\D < |A_s| \le 2 \D$,
$\Delta \gg_{M,\,l} \E_3 \E^{-1}$ and $\E (A_{s}) \gg_{M,\,l} |A_{s}|^3$, $|A_{s}|^3 \gg_{M,\,l} \E(A,A_s)$.
By $P$ denote the set of such $s$.
Thus, $|P| \D^3 \gg_{M,\,l} \E_3$ and
it is easy to check that
$P=-P$.
Note also that $\E(A,H^s) \ll_{M,\,l} |H^s|^3$ for any $s\in P$.
We have
\begin{equation}\label{f:27.04.2014_1}
\sum_{s,t\in P} (A_s \c A_s)^2 (t) \gg_M \E_3 \,.
\end{equation}
Returning to (\ref{f:E_4_begin}), we obtain
$$
(|P| \D^3)^2 \gg_{M,\,l} \max \{ |P| \D^4 \E, \E_4 |P| \D^2 \}
$$
and hence $|P| \D^2 \gg_{M,\,l} \E$, $|P| \D^4 \gg_{M,\,l} \E_4$.
Thus, $\D \sim_{M,\,l} \E_3 \E^{-1}_2$ and
$$
\E_4 \sim_{M,\,l} \D \E_3 \sim_{M,\,l} \D^2 \E \sim_{M,\,l} \D^4 |P| \,.
$$
So, we know all energies $\E$, $\E_3$, $\E_4$ if we know $|P|$ and $\D$.
Let us estimate
the size of the $P$.
Taking any $s\in P$, we get by Lemma \ref{l:eigen_A'} that
$$
\D^3 \ll_{M,\,l} \E(A_s) \le \E(A,A_s) \ll \E a^{-1} \D \ll_{M,\,l} |P| \D^3 a^{-1}
$$
or
$|P| \gg_M a$.
So, we have proved (\ref{f:self-dual0})---(\ref{f:self-dual1}).
Further
$$
\sum_{s,t\in P} |H^s \cap H^t| \le \sum_{s,t\in P} |A_s \cap A_t| := \sigma \,.
$$
Applying Lemma \ref{l:disjoint_probab}, we find disjoint sets
$H_j \subseteq A_{s_j}$, $|H_j| \gg_M |A_{s_j}|$, $|H_j-H_j| \le |H_j-H_j| \ll_{M,\,l} |H_j|$,
$|H_j| \gg \D \sim_{M,\,l} \E_3(A) \E(A)^{-1}$,
$j\in [k]$ and $k\gg |P|^2 \D \sigma^{-1}$.
Arguing as in Corollary \ref{c:disjoint_P}, we get $\sigma \ll \E |P| |A|^{-1}$
and hence $k \gg |P| \D |A| \E^{-1} \gg_{M,\,l} |A| \D^{-1}$.
Of course the
last
bound on $k$ is the best possible up to constants depending on $M$, $l$.
We have obtained the necessary condition.
Let us prove the sufficient condition.
Using the Cauchy--Schwarz inequality and formulas (\ref{f:self-dual0})---(\ref{f:self-dual1}), we have
$$
\| A \|^2_{\U^3} \ge \left( \sum_{s\in P} \E(A_s) \right)^2 \ge
\left( \sum_{s\in P} \E(H_s) \right)^2
\gg_{M,\,l}
\left( \sum_{s\in P} |H_s|^3 \right)^2
\gg_{M,\,l}
$$
$$
\gg_{M,\,l}
|P|^2 \D^6
\gg_{M,\,l}
\E(A) \E_4 (A)
$$
as required.
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
\begin{remark}
In the statement of Theorem \ref{t:self-dual}
there is
the set of popular differences $P$
and the structure of $A$ is described in terms of the set $P$.
Although, we have obtained a criterium it can be named as a weak structural result.
Perhaps, a stronger version avoiding using of the set $P$ takes place, namely,
under the hypothesis of
Theorem \ref{t:self-dual} there are disjoint
sets $H_j \subseteq A_{s_j}$, $|H_j| \gg_{M,\,l} \D$,
$\D \sim_{M,\,l} \E_3(A) \E(A)^{-1}$, $j\in [k]$
such that (\ref{f:self-dual1})
holds
and
\begin{equation}\label{f:self-dual2_conj}
\sum_{j=1}^k |H_j|^4 \gg_{M,\,l} \E_3 (A) \,,
\quad \quad \sum_{j=1}^k |H_j|^3 \gg_{M,\,l} \E(A) \,,
\quad \quad \sum_{j=1}^k |H_j|^5 \gg_{M,\,l} \E_4(A)\,.
\end{equation}
It is easy to see that it is a sufficient condition.
Indeed, because of the sets $H_j \subseteq A$ are disjoint, we have
$$
\| A \|_{\U^3} \ge \sum_{j=1}^k \| H_j \|_{\U^3} \,.
$$
Using the assumption $|H_j-H_j| \ll_{M,\,l} |H_j|$,
the first bound from (\ref{f:self-dual2_conj}), as well as Corollary \ref{c:U^3&doubling}, we obtain
$$
\| A \|_{\U^3} \gg_{M,\,l} \sum_{j=1}^k |H_j|^4 \gg_{M,\,l} \E_3 (A)
$$
and, similarly, by the second and the third inequality of (\ref{f:self-dual2_conj}), we get
$$
\| A \|^2_{\U^3} \gg_{M,\,l} \E (A) \E_4 (A)
$$
as required.
\end{remark}
\bigskip
\begin{example}
Let $A\subseteq \Gr$ be a set having small Wiener norm, that is
the following quantity $\| A \|_W := N^{-1} \sum_{\xi} |\FF{A} (\xi)| := M$
is small.
Then for any $B\subseteq A$, applying the Parseval identity, one has
$$
|B| = \sum_{x} B(x) A(x) = N^{-1} \sum_{\xi} \FF{B} (\xi) \ov{\FF{A} (\xi)} \,.
$$
Using the H\"{o}lder inequality twice (see also \cite{KSh}), we get
$$
|B|^4 \le \left( N^{-1} M \sum_{\xi} |\FF{B} (\xi)|^2 |\FF{A} (\xi)| \right)^2
\le
M^2 |B| \E(A,B)
$$
or, in other words,
\begin{equation}\label{f:E(A,B)_Wiener}
\E(A,B) \ge \frac{|B|^3}{M^2} \,.
\end{equation}
By multiplicative property of Wiener norm, we have $\| A_s \|_W \le M^2$.
In particular, $\E(A_s) \ge \frac{|A_s|^3}{M^4}$.
Hence $\| A \|_{\U^3} \ge M^{-4} \E_3 (A)$.
Further, $\E(A) \ge M^{-2} |A|^3$, $\E_3 (A) \ge M^{-4} |A|^4$
and hence
$$
\| A \|^2_{\U^3} \ge M^{-8} \E^2_3 (A) \ge M^{-16} \E(A) \E_4 (A) \,.
$$
Thus, an application of Theorem \ref{t:self-dual}
gives us
that $A$ has very explicit structure
(2--connectedness follows from (\ref{f:E(A,B)_Wiener}) and $U^3$--connectedness can be obtained
via formula (\ref{f:U^3(A,B)}) in a similar way).
Another structural result on sets from $\F_p$ with small Wiener norm was given in \cite{KSh}.
\end{example}
\bigskip
If Theorem \ref{t:E(A_s)} does not hold that is $\E(A_s) \gg |A_s|^3$ for all $s$ then, clearly,
$\| A \|_{\U^3} \gg \E_3 (A)$
and we can try to apply our structural Theorem \ref{t:self-dual}.
On the other hand, if $A$ is a self--dual set, that is a disjoint union of sets with small doubling
then for any $s\neq 0$ one has exactly $\E(A_s) \gg |A_s|^3$.
It does not contradict to Theorem \ref{t:E(A_s)} because of condition (\ref{cond:A_s_bounded}).
\bigskip
Roughly speaking, in the proof of Theorem \ref{t:self-dual} we found disjoint subsets of $A_s$,
containing huge amount of
the
energy (see also Corollary \ref{c:disjoint_P}).
One can ask about the possibility to find some number of disjoint $A_s$ (and not its subsets)
in general situation.
Our next statement answer the question affirmatively.
\begin{proposition}
Let $A\subseteq \Gr$ be a set, $D\subseteq A-A$.
Put
\begin{equation}\label{cond:A^2-D(A)}
\sigma := \sum_{s \in D} |A-A_s| \,.
\end{equation}
Then there are at least $l \ge |D|^2/(4\sigma)$ disjoint sets $A_{s_1}, \dots, A_{s_l}$.
In particular, if
\begin{equation}\label{cond:A^2-D(A)'}
|A^2 - \Delta (A)| \le \frac{|A-A|^2}{M}
\end{equation}
then there are at least $l \ge M/4$ disjoint sets $A_{s_1}, \dots, A_{s_l}$.
\label{p:disjoint_via_A-A_s}
\end{proposition}
\begin{proof}
Our arguments is a sort of an algorithm.
By (\ref{cond:A^2-D(A)}) there is $s_1 \in D$ such that $|A-A_{s_1}| \le \sigma/|D|$.
Put $D_1 = D\setminus (A-A_{s_1})$.
If $|D_1| < |D|/2$ then terminate the algorithm.
If not then by an obvious estimate
$$
\sum_{s\in D_1} |A-A_s| \le \sigma
$$
we find $s_2 \in D_1$ such that
$$
|A-A_{s_2}| \le \frac{\sigma}{|D_1|} \le \frac{2\sigma}{|D|} \,.
$$
Put $D_2 = D_1 \setminus (A-A_{s_2})$.
If $|D_1| < |D|/2$ then terminate the algorithm.
And so on.
At the last step, we obtain the set $D_l = D\setminus \bigcup_{j=1}^l (A-A_{s_j})$, $|D_l|<|D|/2$.
It follows that
$$
\frac{|D|}{2} \le |\bigcup_{j=1}^l (A-A_{s_j})| \le \sum_{j=1}^l |A-A_{s_j}| \le l \frac{2\sigma}{|D|} \,.
$$
Thus $l \ge |D|^2/(4\sigma)$.
Finally, recall that
\begin{equation}\label{f:A-A_s}
t\in A-A_s \quad \mbox{ iff } \quad A_t \cap A_s \neq \emptyset \,.
\end{equation}
Thus all constructed sets $A_{s_1}, \dots, A_{s_l}$ are disjoint.
To get (\ref{cond:A^2-D(A)'}) put $D=A-A$ and recall that by Lemma \ref{l:A^2_pm}
the following holds $|A^2 - \D(A)| = \sum_{s\in A-A} |A-A_s|$.
This completes the proof.
$\hfill\Box$
\end{proof}
\bigskip
One can ask is it true that not only $\E_3$ energy but $U^3$--norm of sumsets or difference sets is large?
It is easy to see that the answer is no, because of our basic example $A=H\dotplus \L$, $|\L| = K$.
In the case $\E(A) \sim |A|^3/K$, $\E_3 (A) \sim |A|^4 /K$ but $\| A\|_{\U^3} \sim |A|^4 /K^2$
and similar for $A\pm A$.
\section{Appendix}
\label{sec:appendix}
In the section we prove that any set contains a relatively large connected subset.
The case $k=2$ of Proposition \ref{p:connected_k} below was proved in \cite{s_doubling}
(with slightly worse constants)
and we begin with
a wide generalization.
\begin{definition}
Let $X$, $Y$ be two nonempty sets, $|X|=|Y|$.
A nonnegative symmetric function $q(x,y)$, $x\in X$, $y\in Y$
is called {\bf weight} if the correspondent
matrix $q(x,y)$ is nonnegatively defined.
\end{definition}
Having two sets $A$ and $B$ put $\E_q (A,B) := \sum_{x,y} q(x,y) A(x) B(y)$, $\E_q (A) := \E_q (A,A)$.
Clearly, $\E_q (A,B) \le |A| |B| \| q \|_\infty$.
The main property of any weight is the following.
\begin{lemma}
Let $q$ be a weight.
Then for any sets $A,B$, one has
$$
\E^2_q (A,B) \le \E_q (A) \E_q (B) \,.
$$
\label{l:weight_property}
\end{lemma}
\begin{example}
Clearly, the function $q(x,y)=(B\c B)^k (x-y)$ for any set $B$
and an arbitrary positive integer $k$
is a weight.
Further, by the construction of Gowers $U^d$--norms it follows that
\begin{equation}\label{f:q_Gowers}
q_d (x_1,x'_1) = \sum_{x_2,\dots,x_d \in \Gr}\, \sum_{x'_2,\dots,x'_d \in \Gr}\,
\prod_{\o \in \{ 0,1 \}^d} f ({\rm pr} (\v{x}^{\o}))
\end{equation}
is also a weight for any nonnegative function $f$.
In formula (\ref{f:q_Gowers}), we have $\v{x} = (x_1,\dots,x_d)$, $\v{x}' = (x'_1,\dots,x'_d)$,
${\rm pr} (y_1,\dots,y_d) := y_1 + \dots + y_d$.
Another example of a weight is
\begin{equation}\label{f:q_Gowers+}
q^*_d (x,y)
=
\sum_{h_1,\dots,h_{d-1}}\, \prod_{\o \in \{ 0,1 \}^{d-1},\, \o \neq 0}
f(x + \o\cdot \v{h}) f(y + \o\cdot \v{h}) \,,
\end{equation}
where $f$ is an arbitrary nonnegative function again and $\v{h} = (h_1,\dots,h_{d-1})$.
\label{ex:weights}
\end{example}
For two sets $S,T \subseteq \Gr$, $S\neq \emptyset$, $T\subseteq S$ put $\mu_{S} (T) = |T|/|S|$.
Now we prove a general lemma on connected sets and quantities $\E_q$, where $q$ is a weight.
\begin{lemma}
Let $A,B \subseteq \Gr$ be two sets, $\beta_1,\beta_2,\rho \in (0,1]$ be real numbers,
$\beta_1 \le \beta_2$, $\rho < \beta_1 / \beta_2$.
Let $q$ be a weight.
Suppose that $\E_q (A) \ge c |A|^2 \| q \|_\infty$, $c\in (0,1]$.
Then there is $A'\subseteq A$ such that for any subset $\t{A} \subseteq A'$,
$\beta_1 |A'| \le |\t{A}| \le \beta_2 |A'|$ one has
\begin{equation}\label{f:connected_AB+}
\E_q (\t{A}) \ge \rho^2 \mu^2_{A'} (\t{A}) \cdot \E_q (A')
\,,
\end{equation}
and besides
\begin{equation}\label{f:connected_AB_E+}
\E_q (A') > (1-\beta_2 \rho)^{2s} \E_q (A) \,,
\end{equation}
where $s\le \log (1/c) ( 2 \log (\frac{1-\beta_2 \rho}{1-\beta_1} ))^{-1}$.
\label{l:connected_AB+}
\end{lemma}
\begin{proof}
Put $b=\| q \|_\infty$.
We use an inductive procedure
in the proof.
Let us describe the first step of our algorithm.
Suppose that (\ref{f:connected_AB+}) does not hold for some set $C \subseteq A$,
$\beta_1 |A| \le |C| \le \beta_2 |A|$.
Put $A^1 = A\setminus C$.
Then $|A^1| \le (1-\beta_1) |A|$.
Using
Lemma \ref{l:weight_property}, we get
$$
\E_q (A) = \E_q (C,A) + \E_q (A^1,A) < \rho \mu_A (C) \E_q (A) + \E^{1/2}_q (A^1) \E^{1/2}_q (A) \,.
$$
Hence
$$
\E_q (A^1) > \E_q (A) (1-\mu_A (C) \rho)^2 \ge \E_q (A) (1- \beta_2 \rho)^2 \,.
$$
After that applying the same arguments to the set $A^1$, find a subset $C \subseteq A^1$ such that (\ref{f:connected_AB+}) does not hold (if it exists) and so on.
We obtain a sequence of sets $A\supseteq A^1 \supseteq \dots \supseteq A^s$,
and $|A^s| \le (1-\beta_1)^s |A|$.
So, at the step $s$, we have
\begin{equation}\label{tmp:02.04.2014_1}
c|A|^2 b (1-\beta_2 \rho)^{2s} \le \E_q (A) (1-\beta_2 \rho)^{2s} < \E_q (A^s) \le |A^s|^2 b
\le
(1-\beta_1)^{2s} |A|^2 b \,.
\end{equation}
Thus, our algorithm must stop after at most $s\le \log (1/c) ( 2 \log (\frac{1-\beta_2 \rho}{1-\beta_1} ))^{-1}$ number of steps.
Putting $A' = A^s$, we see that inequality (\ref{f:connected_AB+}) takes place for any $\t{A} \subseteq A'$ with
$\beta_1 |A'| \le |\t{A}| \le \beta_2 |A'|$.
Finally, by the second estimate in (\ref{tmp:02.04.2014_1}), we obtain (\ref{f:connected_AB_E+}).
This concludes the proof.
$\hfill\Box$
\end{proof}
\bigskip
Let us formulate a useful particular case of Lemma \ref{l:connected_AB+}.
\begin{lemma}
Let $A,B \subseteq \Gr$ be two sets, $\beta_1,\beta_2,\rho \in (0,1]$ be real numbers,
$\beta_1 \le \beta_2$, $\rho < \beta_1 / \beta_2$.
Suppose that $\E(A,B) \ge c|A|^2 |B|$, $c\in (0,1]$.
Then there is $A'\subseteq A$ such that for any subset $\t{A} \subseteq A'$,
$\beta_1 |A'| \le |\t{A}| \le \beta_2 |A'|$ one has
\begin{equation}\label{f:connected_AB}
\E (\t{A},B) \ge \rho^2 \mu^2_{A'} (\t{A}) \cdot \E(A',B)
\,,
\end{equation}
and besides
\begin{equation}\label{f:connected_AB_E}
\E(A',B) > (1-\beta_2 \rho)^{2s} \E(A,B) \,,
\end{equation}
where $s\le \log (1/c) ( 2 \log (\frac{1-\beta_2 \rho}{1-\beta_1} ))^{-1}$.
\label{l:connected_AB}
\end{lemma}
Lemma \ref{l:connected_AB} implies the required statement, generalizing the result from \cite{s_doubling}.
\begin{proposition}
Let $A \subseteq \Gr$ be a set, $\beta \in (0,1)$ be real numbers, and $k\ge 2$ be an integer.
Put $c=\E_k(A) |A|^{-(k+1)}$.
Then there is $A'\subseteq A$ such that
\begin{equation}\label{p:connected_k_energy}
\E_k (A',A) > (1- 2^{-1} \beta)^{2s} \E_k (A) \,,
\end{equation}
where $s\le \log (1/c) ( 2 \log (\frac{2-\beta}{2-2\beta} ))^{-1}$,
and $A'$ is $(k,\beta,\gamma)$--connected with
\begin{equation}\label{p:connected_k_gamma}
\gamma \ge 2^{-(2sk+2k-2s)} \beta^{2k} (2-\beta)^{2s(k-1)} \,.
\end{equation}
In particular,
\begin{equation}\label{p:connected_k_card}
|A'| \ge (1-2^{-1} \beta)^s c^{1/2} |A| \,.
\end{equation}
\label{p:connected_k}
\end{proposition}
\begin{proof}
Note
that $T\subseteq S$ iff $\D (T) \subseteq \D (S)$.
Applying Lemma \ref{l:connected_AB} with $A=\D(A)$, $B=A^{k-1}$, $\beta_1 = \beta$, $\beta_2 = 1$, $\rho = \beta_1/(2\beta_2) = \beta/2$,
and using formula (\ref{f:energy-B^k-Delta}), we find a set $A'\subseteq A$ such that
for any subset $\t{A} \subseteq A'$, $\beta |A'| \le |\t{A}|$ one has
\begin{equation}\label{tmp:connected_AB}
\E_k (\t{A},A) \ge \rho^2 \mu^2_{A'} (\t{A}) \cdot \E_k (A',A)
\,,
\end{equation}
and
\begin{equation}\label{tmp:connected_AB_E}
\E_k (A',A) > (1- 2^{-1} \beta)^{2s} \E_k (A) \,,
\end{equation}
where $s\le \log (1/c) ( 2 \log (\frac{1-\rho}{1-\beta} ))^{-1}$.
We have obtained inequality (\ref{p:connected_k_energy}).
From (\ref{tmp:connected_AB}), (\ref{tmp:connected_AB_E}) and the H\"{o}lder inequality, we get
$$
\E_k (\t{A}) \ge \rho^{2k} \mu^{2k}_{A'} (\t{A}) \E^k_k (A',A) \E^{-(k-1)}_k (A)
\ge
(2^{-1} \beta)^{2k} (1- 2^{-1} \beta)^{2s(k-1)} \mu^{2k}_{A'} (\t{A}) \E_k (A',A)
\ge
$$
$$
\ge
(2^{-1} \beta)^{2k} (1- 2^{-1} \beta)^{2s(k-1)} \mu^{2k}_{A'} (\t{A}) \E_k (A')
\,.
$$
Thus, the set $A'$ is $(k,\beta,\gamma)$--connected with $\gamma$ satisfying (\ref{p:connected_k_gamma}).
To obtain (\ref{p:connected_k_card}) just apply (\ref{tmp:connected_AB_E}) and a trivial upper bound
for $\E_k (A',A)$
$$
|A'|^2 |A|^{k-1} \ge \E_k (A',A) > (1- 2^{-1} \beta)^{2s} \E_k (A) = (1- 2^{-1} \beta)^{2s} c|A|^{k+1}
$$
as required.
This completes the proof.
$\hfill\Box$
\end{proof}
In view of Lemma \ref{l:connected_AB+} and Example \ref{ex:weights} one can obtain an analog of Proposition \ref{p:connected_k} for $U^k (\beta,\gamma)$--connected sets, see Definition \ref{def:Uk-conn}.
We leave the details to an interested reader.
|
1,108,101,565,631 | arxiv | \section{Introduction}
The classical Ericksen-Leslie
theory (\cite {Er}, \cite
{Le}) successfully describes the dynamic
flow of uniaxial nematic liquid crystals. In \cite{BE94}, Beris-Edwards pointed out that the Ericksen-Leslie
flow theory has a limited domain of applications to liquid crystals. Therefore, based on the celebrated Landau-de Gennes $Q$-tensor theory,
Beris-Edwards \cite{BE94} proposed a general hydrodynamic theory to describe flows of liquid crystals in modeling both uniaxial and biaxial nematic liquid crystals.
In 1971, de Gennes \cite{De} introduced a $Q$-tensor order parameter to establish the Landau-de Gennes theory, which has been one of the successful continuum theories in modeling both uniaxial and biaxial nematic liquid crystals (c.f. \cite{DP}, \cite{Ba}). Mathematically, the Landau-de Gennes theory is described by a Landau-de Gennes functional in
the space of symmetric and traceless $3\times3$ matrices
\begin{equation*}
S_0:=\left\{Q\in \mathbb M^{3\times 3}:\quad Q^T=Q, \, \mbox{tr }Q =0\right\},
\end{equation*}
where $\mathbb M^{3\times 3}$ denotes the space of $3\times 3$ matrices. Let $S_*$ be the space of all uniaxial $Q$-tensors defined by
\[S_*:=\left\{Q \in S_0:\quad Q =s_+ (u\otimes u-\frac 13 I),\quad u\in S^2, \quad s_+:=\frac{b+\sqrt{b^2+24ac}}{4c} \right\},\]
where the constants $a$, $b$, $c$ correspond to a lower temperature regime in liquid crystals and we assume that $a$, $b$, $c$ are positive.
Let $U$ be a domain in $\ensuremath{\mathbb{R}}^3$. For a tensor $Q\in W^{1,2}(U; S_0)$,
the original Landau-de Gennes energy is defined by
\begin{equation}\label{LG energy}
E_{LG}(Q; U):=\int_{U}f_{LG}\,dx=\int_{U}( \tilde f_E + \tilde f_B)\,dx.
\end{equation}
Here $\tilde f_E$ is the elastic energy density with elastic constants $L_1,...,L_4$ of the form
\begin{equation}\label{LG}
\tilde f_E(Q,\nabla Q):=\frac{ L_1}{2}|\nabla Q|^2+\frac{ L_2}{2} \frac{\partial Q_{ij}}{\partial x_j} \frac{\partial Q_{ik}}{\partial x_k} +\frac{ L_3}{2}\frac{\partial
Q_{ik}}{\partial x_j}\frac{\partial Q_{ij}}{\partial x_k}+\frac{ L_4}{2}Q_{lk}\frac{\partial Q_{ij}}{\partial x_l}\frac{\partial Q_{ij}}{\partial x_k}
\end{equation}
in which and in the sequel, we adopt the Einstein summation convention for repeated indices and $\tilde f_B(Q)$ is a bulk energy density defined by
\begin{equation}\label{BE}
\tilde f_B(Q):=-\frac{ a}{2}\tr( Q^2)-\frac{ b}{3}\tr( Q^3)+\frac{ c}{4}\left[\tr( Q^2)\right]^2
\end{equation}
with three positive material constants $a,b,c$.
In \cite{De}, de Gennes discovered first two terms of the elastic energy density in (\ref{LG}) with $L_3=L_4=0$.
Later, combining the work of Schiele-Trimper \cite {ST} with the effect of Berreman-Meiboom \cite{BM84}, Dickmann \cite {Di} completed the full density \eqref{LG} with two additional terms (c.f. \cite{MGKB}, \cite{Ba}) in which the elastic energy density is consistent with
the Oseen-Frank density in for uniaxial nematic liquid crystals. However, for the case of $L_4\neq 0$, Ball-Majumdar \cite {BM} found an example that the Landau-de Gennes energy density \eqref{LG} does not satisfy the coercivity condition. In fact, Golovaty et al. \cite{GNS20} said that ``From the standpoint of energy minimization, unfortunately, such a version of Landau-de Gennes becomes problematic, since the inclusion of the cubic term leads to an energy which is unbounded from below''. Therefore, there is a problem between mathematical and physical theory on nematic liquid crystals in the case of $L_4\neq 0$. In their book \cite{DP}, de Gennes and Prost said that {\it``the bending constant is much larger than others''}; i.e. $k_3>\max\{{k_1, k_2\}}$ at different temperatures. For example, for p-azoxyanisole (PAA) at $134^\circ C$, $k_1=4.05$, $k_2=2.1$, $k_3=5.77$, $k_4=3.08$ (see \cite{ST}). By the physical experiments on liquid crystals, the elastic constant $ L_4=\frac{1}{2s_+^3}(k_3-k_1)$ is not zero in general.
To solve the above coercivity problem on the Landau-de Gennes energy density in the case of $L_4\neq 0$, it was observed in \cite {FH} that for uniaxial tensors,
the original
third order term on $L_4$ in (\ref {LG}), proposed by
Schiele and Trimper \cite{ST}*{p.~268} in physics, is a linear combination of a fourth order term and a second order term; i.e. for $Q\in S_*$, we have
\begin{align*}\alabel{third}
Q_{lk}\frac{\partial Q_{ij}}{\partial x_l}\frac{\partial Q_{ij}}{\partial x_k}=
\frac{3}{s_+}(Q_{ln}\frac{\partial Q_{ij}}{\partial x_l})(Q_{kn}\frac{\partial Q_{ij}}{\partial x_k})-\frac {2s_+}3|\ensuremath{\nabla} Q|^2.
\end{align*}
Therefore, Feng and Hong
\cite {FH} introduced
a new Landau-de Gennes energy given by
\begin{equation}\label{LDG}
E_{LG}(Q; U):=\int_{U} f(Q,\nabla Q)\,dx =\int_{U} \(f_E(Q,\nabla Q)+\frac 1L f_B(Q) \)\,dx,
\end{equation}
where
\begin{align*}
f_E(Q,\nabla Q)=&\frac{\tilde L_1}{2}|\nabla Q|^2
+\frac{L_2}{2} \frac{\partial Q_{ij}}{\partial x_j}\frac{\partial Q_{ik}}{\partial x_k}
+\frac{L_3}{2} \frac{\partial Q_{ik}}{\partial x_j}\frac{\partial Q_{ij}}{\partial x_k}+\frac{3}{2s_+} L^{(4)}(Q,\nabla Q)
\alabel{f_E}
\end{align*}
with $\tilde L_1=L_1-\frac{2}{3s_+} L_4$ for $L_4\geq0$, $\tilde L_1=L_1+\frac{4}{3s_+} L_4$ for $ L_4<0$,
\begin{align*}
L^{(4)}(Q,\nabla Q):=\begin{cases} L_4 Q_{ln}Q_{kn}\frac{\partial Q_{ij}}{\partial x_l}\frac{\partial Q_{ij}}{\partial x_k}&\mbox{ for } L_4\geq0,
\\
L_4 [ Q_{ln}Q_{kn}\frac{\partial Q_{ij}}{\partial x_l}\frac{\partial Q_{ij}}{\partial x_k}-|Q|^2|\nabla Q|^2]&\mbox{ for } L_4<0
\end{cases}
\end{align*}
and
\[ f_B(Q):= \tilde f_B(Q)-\min_{Q\in S_0} \tilde f_B(Q)\geq 0.\]
The new elastic energy density \eqref{f_E} keeps three physical terms of the original Landau-de Gennes density \eqref{LG} and is equivalent to the original density \eqref{LG} for $Q\in S_*$. In \eqref{LDG}, the constant $L$ is a rescaled dimensionless parameter, which drives four elastic constants
$\tilde L_1,\cdots,L_4$ to zero simultaneously as $L$ tends to zero. This corresponds to the large body limit which is of great importance in physics (c.f. \cite{Ba},\cite{Ga}).
We always assume that the constants $\tilde L_1,L_2$ and $L_3$ satisfy
\begin{align}\label{L cond}
& \tilde L_1+L_3>0,\quad 2\tilde L_1-L_3>0,
\quad \tilde L_1+\frac53L_2+\frac16L_3>0.
\end{align}
Under the condition \eqref{L cond}, the new Landau-de Gennes elastic energy density in \eqref{LDG} satisfies the coercivity condition; i.e. $f(Q,\nabla Q)$ for any $Q\in S_0$ is bounded from below by $\frac \alpha 2|\nabla Q|^2$ with some $\alpha >0$ (c.f. \cite{KRSZ},\cite{FH}).
In this paper, we investigate the Beris-Edwards system for the Landau-de Gennes energy \eqref{LDG} with $L_4\neq 0$. The Beris-Edwards system with $L_2=L_3=L_4=0$ has been extensively studied by many authors (see \cite{PZ11},\cite{PZ12},\cite{WWZ}). The Beris-Edwards system introduced in \cite{BE94} is a system of coupling Navier-Stokes equations with the gradient flow for the Landau-de Gennes energy. More precisely, let $v:\ensuremath{\mathbb{R}}^3\to \ensuremath{\mathbb{R}}^3$ be the velocity of the fluid and let $Q:\ensuremath{\mathbb{R}}^3 \to S_0$ be a $Q$-tensor order parameter, which depends on the director of the molecular field. The symmetric and skew-symmetric parts of the tensor $\nabla v$ are
\[ D =\frac 12(\nabla v +(\nabla v)^T ),\quad\Omega =\frac 12(\nabla v -(\nabla v)^T).\]
Define $ [Q,\Omega] :=Q \Omega -\Omega Q $ to be the Lie bracket product and set
\[S(Q,v)=\xi\Big(D(Q+\frac13 I)+(Q+\frac13 I)D-2(Q+\frac13 I)(Q\cdot D)\Big)- [Q,\Omega].\]
Then the Beris-Edwards system (c.f. \cite{BE94}, \cite{PZ12}) is given by
\begin{align*}
\partial_t v+v\cdot\nabla v- \nu \Delta v +\nabla P=&\nabla\cdot\Big( \tau (Q,\nabla Q)+ \sigma (Q,\nabla Q)\Big),\alabel{OBE1}
\\
\nabla\cdot v=&0,\alabel{OBE2}
\\
\partial_t Q +v\cdot \nabla Q -S(Q,v)=& \Gamma H(Q,\nabla Q),
\alabel{OBE3}
\end{align*}
where $H(Q,\nabla Q)$ is the molecular field, $P$ is the pressure, the antisymmetric part of the distortion stress $\tau (Q,\nabla Q)=[Q, H]$ and $\sigma (Q,\nabla Q)$ is the distortion stress (c.f. \cite{QS}) given by
\[\sigma_{ij}(Q, \nabla Q)=-\xi (QH+HQ+\frac 23 H)_{ij}+2\xi (Q\cdot H)(Q+\frac I 3)_{ij}-\partial_{p_{kl}^j} f(Q,\nabla Q)\nabla_iQ_{kl}.\]
Here and in the sequel, we denote $ \partial_{p_{kl}^j} f(Q,\nabla Q):=\frac{\partial f(Q,\nabla Q)}{\partial (\nabla_j Q_{kl})}$ with $p=\nabla Q$.
For simplicity, we assume that $\xi =0, \,\Gamma=\nu =1$ in the Beris-Edwards system.
For biaxial $Q$-tensors, we use the Landau-de Gennes energy
density \eqref{LDG} to formulate the Beris-Edwards system with $L_4\neq 0$. The rescaled Beris-Edwards system for $Q_L\in S_0$ and $v_L\in \ensuremath{\mathbb{R}}^3$ is:
\begin{align*}
\partial_t v_L+v_L\cdot\nabla v_L- \Delta v_L +\nabla P_L=&\nabla\cdot\Big([Q_L,\mathcal{H}(Q_L,\nabla Q_L)]+\sigma (Q_L,\nabla Q_L)\Big),\alabel{RBE1}
\\
\nabla\cdot v_L=&0,\alabel{RBE2}
\\
\partial_t Q_L+v_L\cdot \nabla Q_L +[Q_L, \Omega_L]=& \mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B(Q_L).\alabel{RBE3}
\end{align*}
The molecular field $H_L(Q_L,\nabla Q_L)$ is then given by
\begin{equation*}
H_L(Q_L,\nabla Q_L)=:\mathcal{H}(Q_L,\nabla Q_L)+\frac{1}{L}g_B(Q_L),
\end{equation*}
where
\begin{align*}
\mathcal{H}(Q_L,\nabla Q_L)_{ij}=&\frac12\(\nabla_k[\partial_{p_{ij}^k} f_E(Q_L,\nabla Q_L)]+\nabla_k[\partial_{p_{ji}^k} f_E(Q_L,\nabla Q_L)]\)
\\
&- \frac12\(\partial_{Q_{ij}} f_E(Q_L,\nabla Q_L)+\partial_{Q_{ji}} f_E(Q_L,\nabla Q_L)\)
\\
&-\frac{\delta_{ij}}3\sum_{l=1}^3\(\nabla_k[\partial_{p_{ll}^k} f_E(Q_L,\nabla Q_L)]-\partial_{Q_{ll}} f_E(Q_L,\nabla Q_L)\),
\alabel{Mol}
\end{align*}
the term $g_B(Q_L)$ is
\begin{align*}\alabel{gB}
g_B(Q_L) :=&aQ_L +b \big( Q_L Q_L -\frac {1}3\tr(Q_L^2)I \big)-cQ_L\tr(Q_L ^2)
\end{align*}
and $\sigma (Q_L,\nabla Q_L)$ is the distortion stress tensor with
\begin{equation*}\label{Mo0}
\nabla_j\sigma_{ij}(Q_L,\nabla Q_L)=-\nabla_j\(\nabla_i(Q_L)_{kl}\partial_{p_{kl}^j} f_E(Q_L,\nabla Q_L)\).
\end{equation*}
Set
\[H^2_{Q_e} (\ensuremath{\mathbb{R}}^3; S_0)=\{Q\in S_0: Q-Q_e\in H^2(\ensuremath{\mathbb{R}}^3)\},\]
where $Q_e=s_+ ( e \otimes e -\frac13 I)\in S_*$ and $e \in S^2$ is a constant vector.
We call $(Q_L,v_L)$ to be a strong solution to the system \eqref{RBE1}-\eqref{RBE3} in $\ensuremath{\mathbb{R}}^3\times (0,T)$ for some $T>0$ if it satisfies the system a.e. in $(x,t)\in\ensuremath{\mathbb{R}}^3 \times (0,T)$ and
\begin{align*}
&Q_L\in L^2(0,T;H^3_{Q_e}(\ensuremath{\mathbb{R}}^3))\cap L^\infty(0,T;H^2_{Q_e}(\ensuremath{\mathbb{R}}^3)),
\quad \partial_tQ_L\in L^2(0,T;H^1(\ensuremath{\mathbb{R}}^3)),
\\
&v_L\in L^2(0,T;H^2(\ensuremath{\mathbb{R}}^3))\cap L^\infty(0,T;H^1(\ensuremath{\mathbb{R}}^3)).
\end{align*}
Then we have
\begin{theorem}[Local Existence]\label{thm loc}
Let $(Q_{L,0},v_{L,0})\in H^{2}_{Q_e}(\ensuremath{\mathbb{R}}^3; S_0)\times H^{1}(\ensuremath{\mathbb{R}}^3;\ensuremath{\mathbb{R}}^3)$ be the initial values satisfying $\div v_{L,0}=0$
and $\|Q_{L,0}\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}\leq K$ for a constant $K>0$.
Then there is a unique strong
solution $(Q_L, v_L)$ to the system \eqref{RBE1}-\eqref{RBE3} in $\ensuremath{\mathbb{R}}^3\times[0, T)$ with initial data $(Q_{L,0},v_{L,0})$ for some
$T>0$.
\end{theorem}
\noindent Although Theorem \ref{thm loc} might be known for some experts, see \cite{PZ12} for the case $L_2=L_3=L_4=0$.
However, since there exists some new difficulty on $f_E$ with $L_4\neq 0$, we give a detailed proof in Section 5 for completeness.
\medskip
Next, we will formulate the Beris-Edwards system for uniaxial Q-tensors.
In their book \cite{BE94}, Beris-Edwards suggested the hydrodynamic theory to describe flows of liquid
crystals for uniaxial Q-tensors $Q\in S_*$, but they could not write an explicit form of molecular field $H(Q,\nabla Q)$ for $Q\in S_*$
with nonzero elastic constants $L_2$, $L_3$, $L_4$. Recently, the explicit form of the molecular field $H(Q,\nabla Q)$ for $Q\in S_*$ with general elastic constants was given in \cite{FH}, so we can apply the form to formulate the Beris-Edwards system for uniaxial Q-tensors.
For any two matrix $A, B\in S_0$, we denote the standard product by $\<A,B\>:=\sum_{i,j}A_{ij}B_{ij}$.
Then, the molecular field for $Q\in S_*$ is
\begin{align*}
&H(Q, \nabla Q)= \nabla_k \(
\partial_{p^k} f_E (Q +\frac {s_+}3 I)
+(Q +\frac {s_+}3 I)(\partial_{p^k} f_E)^T\)
\\
&-2s_+^{-1}\nabla_k\( (Q +\frac {s_+}3 I)\<Q+\frac {s_+}3 I,\, \partial_{p^k} f_E\>\) - \partial_{p^k} f_E\ensuremath{\nabla}_kQ
- \ensuremath{\nabla}_k Q (\partial_{p^k} f_E)^T
\\
&+2s_+^{-1}\left[\<\partial_{p^k} f_E, \nabla_k Q\>( Q +\frac {s_+}3 I)+ \<\partial_{p^k} f_E, ( Q+\frac {s_+}3 I)\>\nabla_k Q\right]
\\
&- \partial_{Q} f_E (Q +\frac {s_+}3 I)-(Q +\frac {s_+}3 I) (\partial_{Q} f_E)^T
+2s_+^{-1}\<\partial_{Q} f_E, Q +\frac {s_+}3 I\> (Q +\frac {s_+}3 I)
.\alabel{EL}
\end{align*}
Through the new molecular field \eqref{EL} with a uniaxial Q-tensors $Q\in S_* $, the Beris-Edwards system for nonzero elastic constants $L_1,\cdots, L_4$ is:
\begin{align}
(\partial_t+v\cdot\nabla- \Delta )v +\nabla P=&\nabla\cdot\Big([Q,H]+\sigma(Q,\nabla Q)\Big),\label{BE1}\\
\nabla\cdot v=&0,\label{BE2}\\
(\partial_t +v\cdot \nabla )Q+[Q, \Omega]=& H(Q,\nabla Q) \label{BE3}.
\end{align}
Then we prove the existence of strong solutions to \eqref{BE1}-\eqref{BE3} in the following:
\begin{theorem} \label{thm1}
Assume that $(Q_0,v_0)\in H^2_{Q_e}(\ensuremath{\mathbb{R}}^3;S_*)\times H^{1}(\ensuremath{\mathbb{R}}^3;\ensuremath{\mathbb{R}}^3)$ and $\div v_0=0$. Then there is a unique strong
solution $(Q, v)$ to the system \eqref{BE1}-\eqref{BE3} in $\ensuremath{\mathbb{R}}^3\times[0, T^*)$ with initial data $(Q_0,v_0)$.
Moreover, there are two positive constants $\varepsilon_0$ and $R_0$ such that at a singular point $x_i$, the maximal existence time $T^*$ satisfies
\begin{equation*}
\operatornamewithlimits{lim\,sup}_{t\rightarrow T^*}\int_{B_R(x_i)}|\nabla Q(x,t)|^3+|v(x,t)|^3\,dx\geq \varepsilon_0^3
\end{equation*}
for any $R>0$ with $R\leq R_0$.
\end{theorem}
For the proof of Theorem \ref{thm1}, one of the key steps is to establish Proposition \ref{prop Extension} and obtain that for a short time $T_1>0$, the strong solution to the system \eqref{RBE1}-\eqref{RBE3} with initial data $(Q_0, v_0)$ satisfies the uniform estimate:
\begin{align*}
&\sup_{0\leq s\leq T_1}\left(\|\nabla Q_L(s)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\|v_L(s)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\frac 1 L\| Q_L(s)-\pi(Q_L(s))\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2\right)
\\
&+\|\nabla^2 Q_L\|_{L^2(0,T_1;H^1(\ensuremath{\mathbb{R}}^3))}^2+\|\partial_t Q_L\|_{L^2(0,T_1;H^1(\ensuremath{\mathbb{R}}^3))}^2
\\
&+\|\nabla v_L\|_{L^2(0,T_1;H^1(\ensuremath{\mathbb{R}}^3))}^2 +\frac 1 L\|\nabla( Q_L-\pi(Q_L))\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2\leq C.
\end{align*}
Here $\pi(Q_L)$ is the projection of $Q_L$ defined below in the proof of Theorem \ref{thm2}.
The proof of Proposition \ref{prop Extension} is sophisticated and it will also play a crucial role in the proof of Theorem \ref{thm2} below. We will outline more details about it later.
\begin{remark}
It was pointed out in \cite{BE94} that \eqref{BE1}-\eqref{BE3} can be reduced to the hydrodynamic flow of the Oseen-Franks energy, known as the Ericksen-Leslie system. In fact, multiplying $u_j$ to \eqref{BE3} and employing $|u|^2=1$, one can check that \[\sigma(Q,\nabla Q)=-\ensuremath{\nabla} u^T\frac{\partial W(u,\nabla u)}{\partial (\ensuremath{\nabla} u)}, \quad \partial_{p_{kl}^j} f_E(Q_L,\nabla Q_L)=s_+^{-1}u_l\frac{\partial W(u,\ensuremath{\nabla} u)}{\partial(\nabla_j u_k)}.\]
\end{remark}
Ericksen \cite {Er} and Leslie \cite{Le} in the 1960s proposed the celebrated hydrodynamic theory to describe the behavior of liquid crystal flows.
The question on the
global existence of weak solutions to the Ericksen-Leslie system is very challenging.
Similarly to the result of Chen and Struwe \cite{CS} on the heat flow for harmonic maps, Lin and Liu \cite {LL2} introduced the Ginzburg-Landau approximation for the Ericksen-Leslie system to solve the existence problem, but they could not show that the solutions of the Ginzburg-Landau approximate systems approach the solution of the Ericksen-Leslie system.
In $\ensuremath{\mathbb{R}}^2$, Hong \cite{Ho} and Hong-Xin \cite {HX} proved that the solutions of the Ginzburg-Landau approximate systems with unequal Frank constants approach the solution of the Ericksen-Leslie system in a short time and showed the global existence of weak solutions to the Ericksen-Leslie system in $\ensuremath{\mathbb{R}}^2$ by using the idea of Struwe \cite{St} on the harmonic map flow.
In $\ensuremath{\mathbb{R}}^3$, Hong, Li and Xin \cite{HLX} showed the strong convergence of the Ginzburg-Landau approximate system with unequal Frank constants before the blow-up time of the Ericksen-Leslie system. Recently, we \cite{FHM} improved the result in \cite{HLX} by proving the smooth convergence of the Ginzburg-Landau approximate systems for a general Ericksen-Leslie system with Leslie tensors before the blow-up time.
By comparing with the convergence of
Ginzburg-Landau models for superconductivity theory, Gartland \cite {Ga} emphasised importance of the convergence on Landau-de Gennes solutions. In physics, both the Ericksen-Leslie
theory and the Beris-Edwards theory should unify in modelling uniaxial state of nematic liquid crystals, so it is very
interesting to give a mathematical proof to verify that the solutions of the Beris-Edwards system \eqref{RBE1}-\eqref{RBE3}
can approach a solution of the Ericksen-Leslie $Q$-tensor system as $L\to 0$.
We solve the convergence problem of the Beris-Edwards system in the following:
\begin{theorem}\label{thm2} Assume that $(Q_0, v_0)\in H_{Q_e}^{2}(\ensuremath{\mathbb{R}}^3;S_*)\times H^{1}(\ensuremath{\mathbb{R}}^3;\ensuremath{\mathbb{R}}^3)$ with $\div v_{0}=0$.
For each $L>0$, let $(Q_L, v_L)$ be the unique strong solution to the system \eqref{RBE1}-\eqref{RBE3} in
$\ensuremath{\mathbb{R}}^3\times[0,T_L)$ with initial data $(Q_0, v_0)$ for the maximal existence time $T_L$.
Let $(Q,v)$ be the strong solution to the system \eqref{BE1}-\eqref{BE3} in $\ensuremath{\mathbb{R}}^3\times[0,T^*)$ with the same initial data $(Q_0, v_0)$ and the maximal existence time $T^*$ in Theorem \ref{thm1}. Then, for any $T\in(0, T^*)$, there exists a sufficiently small $L_T>0$ such that $T\leq T_L$ for any $L\leq L_T$.
Moreover, as $L\to 0$, we have
\begin{equation}\label{con1}
(\nabla Q_L,v_L)\rightarrow(\nabla Q,v) \qquad\text{in }~~L^\infty(0,T;L^2(\ensuremath{\mathbb{R}}^3))\cap L^2(0,T;H^1(\ensuremath{\mathbb{R}}^3))
\end{equation}
and
\begin{equation}\label{con2}
(\nabla Q_L,v_L)\rightarrow(\nabla Q,v) \qquad\text{in }~~C^\infty(\tau,T;C_{loc}^\infty(\ensuremath{\mathbb{R}}^3)) \quad\text{for any }\tau>0.
\end{equation}
\end{theorem}
For the proof of Theorem \ref{thm2}, the main ideas are to establish uniform estimates on higher order derivatives of $(Q_L,v_L)$ in $L$. Using similar methods in \cite{HX},\cite{FHM}, we can handle all terms involving $f_E(Q_L)$, but the main difficulty is to obtain the uniform estimate of the terms involving $\frac 1Lg_B(Q_L)$ when $L\to 0$. To handle those difficult terms, we use a concept of a projection near $S_*$, which was first introduced on Riemannian manifolds by Schoen and Uhlenbeck \cite{ScU}. Denote
\begin{align}\alabel{S delta}
S_\delta:=\left \{Q\in S_0:\quad \dist(Q;S_*)\leq \delta\right\}.
\end{align}
Let $\pi: S_{\delta}\to S_*$ be the smooth projection map for a small $\delta>0$ so that $\pi (Q)$ is the nearest point; i.e. $|Q-\pi (Q)|=\dist(Q; S_*)$ for
$Q\in S_{\delta}$.
For each smooth $Q_L(x)\in S_{\delta}$, there is a rotation $R(Q_L(x))\in SO(3)$ such that $R^T(Q_L(x))Q_L(x)R(Q_L(x))$ is diagonal. However, $R(Q_L(x))$ is not always smooth; i.e there exists a measure zero set $\Sigma_L$ such that $R(Q_L(x))$ is differentiable in $U$ except for the singular set $\Sigma_L$. To study the convergence of solutions of the Landau-de Gennes system, Feng-Hong \cite {FH} proved a geometric identity $\nabla \( R^T(Q) Q R (Q)\)_{ii} =(R^T(Q)\nabla Q R (Q))_{ii}$ with $i=1, 2, 3$ away from the singular set for handling the difficulty on $\frac 1Lg_B(Q_L)$. Due to the singular set $\Sigma_L$ of the rotation $R(Q_L(x))\in SO(3)$, the proof in \cite {FH} is much complicated. In this paper,
we find a new approach to avoid the main difficulty arising from the singular set $\Sigma_L$ of the above rotation $R(Q_L(x))$ in \cite {FH}. We outline main steps of the new approach as follows:
The first key step is to establish some new estimates to overcome the difficulty arising from the term $f_B(Q_L)$.
For each smooth $Q\in S_\delta$, $\pi(Q)\in S_*$ has a constant number of distinct eigenvalues, so there exists a smooth rotation $R_Q:=R (\pi (Q))\in SO(3)$ such that
\begin{align}
R^T_Q \pi (Q )R_Q =\begin{pmatrix}
\frac{-s_+}{3}&0&0\\0&\frac{-s_+}{3}&0\\0&0&\frac{2s_+}{3}
\end{pmatrix}=:Q^+.
\end{align}
Since $\pi(Q)$ commutes with $Q$ for any $Q\in S_\delta$ (c.f. \cite{NZ}), one can see
\begin{align*}
\tilde Q =R^T_Q Q R_Q =\begin{pmatrix}
\tilde Q_{11} & \tilde Q_{12}&0
\\ \tilde Q_{21}& \tilde Q_{22}&0
\\0&0&\tilde Q_{33}\alabel{Q1}
\end{pmatrix}.
\end{align*}
For any $Q\in S_\delta$, we can derive an estimate
\begin{align}
\frac{\lambda}{2} |\nabla (Q-\pi(Q))|^2\leq \partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B(\tilde Q)\nabla_{x_\beta}\tilde Q_{ij}\nabla_{x_\beta} \tilde Q_{kl}
+C|\nabla Q|^2|Q-\pi(Q)|^2\alabel{2f1}
\end{align}
with some constant $\lambda >0$, which improves a result of diagonal matrices in \cite {FH}.
The second key step is to establish the uniform estimate on $(\nabla^2 Q_L,\nabla v_L)$. To overcome the difficulty arising from the term $g_B(Q_L)$,
we rotate the equation \eqref{RBE3} by $R_{Q_L}\in SO(3)$ such that $g_B(\tilde Q_L)$ has the same matrix form of $\tilde Q$ in \eqref{Q1}. For any $Q\in S_\delta$, we find an extension of the geometric identity of $\tilde Q$ in \eqref{Q1} that
\begin{align*}
&\<\nabla g_B(\tilde Q) ,\nabla R_Q^T \pi(Q)R_Q-R_Q^T \pi(Q )\nabla R_Q\>=0.\alabel{GG}
\end{align*}
Combining the above identity \eqref{2f1} with \eqref{GG}, we establish the uniform estimate in Lemmas \ref{lem 1ord}-\ref{lem 2ord}.
By a local $L^3$-type of estimate
\begin{align}
\sup_{T_0\leq t\leq T_L,x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(x_0)}|\nabla Q_L|^3+|v_L|^3+\frac{|Q_L-\pi(Q_L)|^3}{L^{\frac32}}\,dx\leq\varepsilon_0^3,
\end{align}
we establish a key Proposition \ref{prop Extension} and then prove Theorem \ref{thm1}, which is similar to the idea in \cite{FHM}. Finally, by an induction method, we establish the sophisticated uniform estimate of $(\nabla^{k+1} Q_L, \nabla^{k} v_L)$ in $L$ for any integer $k\geq 2$ to prove Theorem \ref{thm2}. We would point out that our proof on high order uniform estimates is new and different from one used for Ginzburg-Landau approximations in \cite{FHM}.
\begin{remark}
When $L_4=0$, Wang-Zhang-Zhang \cite{WZZ} proved some related convergence of \eqref{RBE1}-\eqref{RBE3} with smooth initial values
to the Ericksen-Leslie system in $\ensuremath{\mathbb{R}}^3$, but not to
the uniaxial Q-tensor Beris-Edwards system \eqref{BE1}-\eqref{BE3}. It seems that their method only works for smooth initial values.
Recently, Xin-Zhang \cite{XZ} proved that the weak convergence also holds in $\ensuremath{\mathbb{R}}^2$ for \eqref{RBE1}-\eqref{RBE3} with $L_2=L_3=L_4=0$.
\end{remark}
The paper is organized as follows. In Section 2, we derive some a-priori estimates on the strong solution $(Q_L,v_L)$ of the system \eqref{RBE1}-\eqref{RBE3} in
$\ensuremath{\mathbb{R}}^3\times[0,T_L]$. In Section 3, we prove Theorem \ref{thm1}. In Section 4, we prove Theorem \ref{thm2}. In Section 5, we prove Theorem
\ref{thm loc}.
\section{a-priori estimates}
In this section, we will derive some a-priori estimates on the strong solution $(Q_L,v_L)$ of the system \eqref{RBE1}-\eqref{RBE3} in
$\ensuremath{\mathbb{R}}^3\times[0,T_L]$.
\subsection{Property on the density}
In order to obtain a-priori energy estimates, we need to establish some key properties on the density.
Under the condition \eqref{L cond}, one can verify from a result in \cite{KRSZ} that there are two uniform constants $\alpha>0$ and $\Lambda >0$ such that for any $Q\in \mathbb{M}^{3\times 3}$ and $p \in\mathbb{M}^{3\times 3}\times \ensuremath{\mathbb{R}}^3$, $f_E(Q,p)$ also satisfies
\begin{align*}
\frac \alpha 2 |p|^2\leq f_E(Q,p)\leq& \Lambda(1+|Q|^2)|p|^2, \quad|\partial_Qf_E(Q,p)|\leq \Lambda (1+|Q|)|p|^2,
\\
|\partial^2_{Qp}f_E(Q,p)|\leq& \Lambda (1+|Q|)|p|,\qquad |\partial^2_{pp}f_E(Q,p)|\leq \Lambda(1+|Q|^2).\alabel{sec2 f_1}
\end{align*}
Noting that $f_E(Q,p)$ is
quadratic in $p$ and satisfies \eqref{sec2 f_1}, one has (c.f. \cite{HM})
\begin{align}\label{sec2 f_E}
\frac \alpha 2|\xi|^2\leq \partial^2_{p^i_{kl}p^j_{mn}} f_{E}(Q,p)\xi^i_{kl}\xi^j_{mn}\leq \Lambda (1+|Q|^2)|\xi|^2, \quad \forall \xi\in\mathbb{M}^{3\times 3}\times \ensuremath{\mathbb{R}}^3.
\end{align}
Recall that
\begin{align}
S_\delta=\left \{Q\in S_0:\quad \dist(Q;S_*)\leq \delta\right\}.
\end{align}
We assume that $\delta>0$ is sufficiently small through out this paper. Let $\pi(Q )$ be a smooth projection from $S_{\delta}$ to $S_*$. Then $ f_B(Q )$ satisfies (c.f. \cite{NZ}, \cite{FH})
\begin{align}
\frac{\lambda}2|Q -\pi(Q )|^2\leq f_B&(Q)\leq C |Q -\pi(Q )|^2
\label{f_B and dist}
\end{align} for some $C>0$.
Since each smooth $\pi(Q)\in S_*$ has a constant number of distinct eigenvalues, there exists a smooth matrix $R_Q:=R (\pi (Q))\in SO(3)$ such that $R^T_Q \pi (Q )R_Q$ is diagonal (c.f.\cite{No}). Since $S_*$ has only three elements of diagonal forms, we can assume without loss of generality that
\begin{align}\label{Q+}
R^T_Q \pi (Q )R_Q =\begin{pmatrix}
\frac{-s_+}{3}&0&0\\0&\frac{-s_+}{3}&0\\0&0&\frac{2s_+}{3}
\end{pmatrix}=:Q^+.
\end{align}
Since $\pi(Q)$ commutes with $Q$ (c.f. \cite{NZ}), we have
\begin{align*}
R^T_Q Q R_Q Q^+=Q^+ R^T_Q Q R_Q. \alabel{Q+ com}
\end{align*}
Then for any $Q\in S_\delta$, it follows from using \eqref{Q+}-\eqref{Q+ com} that
\begin{align*}
\tilde Q =R^T_Q Q R_Q =\begin{pmatrix}
\tilde Q_{11} & \tilde Q_{12}&0
\\ \tilde Q_{21}& \tilde Q_{22}&0
\\0&0&\tilde Q_{33}
\end{pmatrix}.\alabel{Q13}
\end{align*}
\begin{lemma}
\label{lem f_B 1st}
For any $Q\in S_\delta$, let $\tilde Q$ be defined in \eqref{Q13}. Then the Hessian of the bulk density $f_B(Q)$ satisfies
\begin{align*}
&\frac{\lambda}{2} |\xi|^2\leq\partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B(\tilde Q)\xi_{ij}\xi_{kl}\alabel{eq pos}
\end{align*}
for all $\xi\in S_0$ of the form
\begin{align*}
\xi =\begin{pmatrix}
\xi_{11} & \xi_{12}&0
\\ \xi_{21}& \xi_{22}&0
\\0&0&\xi_{33}
\end{pmatrix},\alabel{xi}\end{align*}
where $\lambda=\min\{{s_+b},{3a}\}>0$.
\end{lemma}
\begin{proof}
Calculating second partial derivatives of $f_B(\tilde Q)$ with respect to $\tilde Q$, we have
\begin{align*}
\partial_{\tilde Q_{\tilde i\tilde j}}\partial_{\tilde Q_{ij}} f_B(\tilde Q)=& -a\delta_{i\tilde i}\delta_{j\tilde j}
-b(\delta_{\tilde ij}\tilde Q_{\tilde j i}+\delta_{\tilde ji}\tilde Q_{j\tilde i})+c(\delta_{i\tilde i}\delta_{j\tilde j}|\tilde
Q|^2+2\tilde Q_{ij}\tilde Q_{\tilde i\tilde j}).
\end{align*}
Note from \cite{MN} that
\begin{align*}
|Q^+|^2=\frac23s_+^2,\qquad 2cs_+^2=3a+bs_+.\alabel{abc}
\end{align*}
Then we have
\begin{align*}
\partial_{\tilde Q_{\tilde i\tilde j}}\partial_{\tilde Q_{ij}} f_B(\tilde Q)\Big|_{\tilde Q =Q^+}=&\(b\Big(\frac13s_+\delta_{i\tilde i}\delta_{j\tilde j}-(\delta_{\tilde ij}\tilde Q_{\tilde j i}+\delta_{\tilde ji}\tilde Q_{j\tilde i})\Big)+2c\tilde Q_{ij}\tilde Q_{\tilde i\tilde j}\)\Big|_{\tilde Q =Q^+}
.\alabel{fB 2}
\end{align*}
In the case of $i=j=\tilde i=\tilde j$ in \eqref{fB 2}, we apply the relation \eqref{abc} to obtain
\begin{align}\label{f_B p1.0}
\partial_{\tilde Q_{11}}\partial_{\tilde Q_{11}} f_B(Q^+) =&\(s_+b+\frac{2s_+^2}{9}c\)=\frac 13a+\frac{10s_+}{9}b=\partial_{\tilde
Q_{22}}\partial_{\tilde Q_{22}} f_B(Q^+),
\\
\partial_{\tilde Q_{33}}\partial_{\tilde Q_{33}} f_B(Q^+)=&-s_+ b+\frac{8s_+^2}{9}c=\frac43a-\frac{5s_+}{9}b.
\end{align}
For the case of $i=j\neq\tilde i=\tilde j$ in \eqref{fB 2}, we compute
\begin{align}
\partial_{\tilde Q_{11}}\partial_{\tilde Q_{22}} f_B(Q^+) =&2cQ^+_{11}Q^+_{22}=\frac{2s_+^2}{9}c=\frac13a+\frac{s_+}{9}b,
\\
\partial_{\tilde Q_{11}}\partial_{\tilde Q_{33}} f_B(Q^+) =&2cQ^+_{11}Q^+_{33}=-\(\frac23a+\frac{2s_+}{9}b\)=\partial_{\tilde Q_{22}}\partial_{\tilde
Q_{33}} f_B(Q^+).\label{f_B p1.1}
\end{align}
For the remaining cases of either $i\neq j$ or $ \tilde i\neq \tilde j$ in \eqref{fB 2}, we find
\begin{align*}
&\left.\(\sum_{i\neq j}\sum_{\tilde i,\tilde j}+\sum_ {\tilde i\neq \tilde j}\sum_{i,j}\)\partial_{\tilde Q_{\tilde i\tilde j}}\partial_{\tilde Q_{ij}}f_B(\tilde Q)\xi_{\tilde i\tilde j}\xi_{ij}\right|_{\tilde Q=Q^+}
\\=& \(\sum_{i\neq j}\sum_{\tilde i,\tilde j}+\sum_ {\tilde i\neq \tilde j}\sum_{i,j}\)\(2cQ^+_{ij}Q^+_{\tilde i\tilde j}+b(\frac 13s_+\delta_{i\tilde i}\delta_{j\tilde j}-\delta_{\tilde ij}Q^+_{\tilde j i}-\delta_{\tilde ji}Q^+_{j\tilde i})\)\xi_{\tilde i\tilde j}\xi_{ij}
\\
=&\sum_{i\neq j}b\(\frac 13s_+-Q^+_{ii}- Q^+_{jj}\)\xi_{ij}^2=s_+b(\xi_{12}^2+\xi_{21}^2),\alabel{Second derivative case 3}
\end{align*}
where we employed \eqref{abc} in the last step.
Using the relations \eqref{f_B p1.0}-\eqref{Second derivative case 3} with the fact that $\tr(\xi)=0$ and \eqref{xi}, we have
\begin{align*}
&\partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B(Q^+)\xi_{ij}\xi_{kl}
\\=& \(\frac 13a+\frac{10s_+}{9}b\)( \xi_{11}^2 + \xi_{22}^2)
+\(\frac23a+\frac{2s_+}{9}b\)(
\xi_{11} \xi_{22})+2s_+b\xi_{12}^2
\\
&+\(\frac43a-\frac{5s_+}{9}b \) \xi_{33}^2 -\(\frac43a+\frac{4s_+}{9}b\) \xi_{33}(
\xi_{11}+ \xi_{22})
\\
=& s_+b( \xi_{11}^2 + \xi_{22}^2)+({\frac 83}a-\frac{s_+}{9}b) \xi_{33}^2+2s_+b\xi_{12}^2
+\(\frac13a+\frac{s_+}{9}b\)( \xi_{11}+ \xi_{22})^2
\\
=& s_+b( \xi_{11}^2 + \xi_{22}^2+\xi_{12}^2+\xi_{21}^2)+3a \xi_{33}^2 \geq \lambda |\xi|^2,
\alabel{f_B p1.2}
\end{align*}
where $\lambda=\min\{{s_+b},{3a}\}>0$.
Due to the continuity of second derivatives of $f_B(\tilde Q)$ and the fact that $|\tilde Q-Q^+|=\dist(Q;S_*)\leq \delta$ with sufficiently small $\delta$, the claim \eqref{eq pos} follows from using \eqref{f_B p1.2}.
\end{proof}
\begin{cor}\label{cor fB}
For any $Q \in S_\delta$, we have
\begin{align*}
\frac{\lambda}{2}|\nabla (Q-\pi(Q))|^2\leq &\partial^2_{\tilde Q_{ij} \tilde Q_{kl}} f_B(\tilde Q)\nabla \tilde Q _{ij}\nabla\tilde Q_{kl}
+C|Q-\pi(Q)|^2|\nabla Q|^2 .
\alabel{D2 tilde Q}
\end{align*}
Moreover, for any $k\geq 2$, we have
\begin{align*}
\frac\lambda 2 |\nabla^k (Q -\pi(Q ))|^2
&\leq \partial^2_{ \tilde Q_{ij} \tilde Q_{kl}} f_B(\tilde Q )\nabla^k \tilde Q _{ij}\nabla^k\tilde Q_{kl}
\\
&+C \sum_{\substack{\mu_1\leq k-1\\\mu_1+\cdots+\mu_{k+1}=k}}|\nabla^{\mu_1}(Q-\pi(Q))|^2|\nabla^{\mu_2} Q|^2\cdots |\nabla^{\mu_{k+1}} Q|^2
.
\alabel{f_B kth rule}
\end{align*}
\end{cor}
\begin{proof}
Taking $\xi = \nabla^k \tilde Q$ for $k\geq 1$ in Lemma \ref{lem f_B 1st}, we have
\begin{align*}
\frac{3\lambda}{4}|\nabla^k \tilde Q|^2\leq \partial^2_{ \tilde Q_{ij} \tilde Q_{kl}} f_B(\tilde Q)\nabla^k \tilde Q _{ij}\nabla^k\tilde Q_{kl}
\alabel{kth fQQ}.
\end{align*}
We note
\begin{align*}
&|\nabla \tilde Q|^2= |\nabla ( \tilde Q-Q^+)|^2= |\nabla (R_Q^T (Q-\pi(Q))R_Q)|^2
\\
&\geq \frac 23 |\nabla (Q-\pi(Q))|^2
-C|Q-\pi(Q))|^2|\nabla R_Q|^2.\alabel{kth2}
\end{align*}
Since $R_Q$ is smooth for $Q\in S_{\delta}$, the inequality \eqref{D2 tilde Q} follows from \eqref{kth fQQ}-\eqref{kth2}.
For any $k\geq 2$, we expand the left-hand side of \eqref{kth fQQ} to show
\begin{align*}
&\frac{3\lambda}{4}|\nabla^k \tilde Q|^2=\frac{3\lambda}{4}|\nabla^k (R_Q^T (Q-\pi(Q))R_Q)|^2
\\
\geq&\frac{\lambda}{2}|\nabla^k (Q-\pi(Q))|^2-C\sum_{\substack{\mu_1+\mu_2+\mu_3=k\\\mu_2< k}}|\nabla^{\mu_1}R_Q^T|^2|\nabla^{\mu_2}(Q-\pi(Q))|^2|\nabla^{\mu_3}R_Q|^2
.\alabel{kth tilde Q}
\end{align*}
Since $R_Q$ is smooth for $Q\in S_{\delta}$, we find
\begin{align*}
|\nabla^k R_Q|
\leq&C\(\sum_{\mu_1+\cdots+\mu_{k}=k}|\nabla^{\mu_1} Q|\cdots |\nabla^{\mu_{k}} Q| \)
.\alabel{kth R}
\end{align*}
Then applying \eqref{kth R} and \eqref{kth tilde Q} to \eqref{kth fQQ}, we prove \eqref{f_B kth rule}.
\end{proof}
\begin{lemma}\label{lemgb}For any $Q\in S_\delta$, let $\tilde Q$ be defined in \eqref{Q13}. Then
\begin{align*}
&\<\nabla^k g_B(\tilde Q) ,\nabla R_Q^T \pi(Q)R_Q-R_Q^T \pi(Q )\nabla R_Q\>=0.\alabel{gB pi}
\end{align*}
\end{lemma}
\begin{proof}
For a fixed $Q_0\in S_\delta$, there exists $R_0=R(\pi(Q_0))\in SO(3)$ such that $R_0^T \pi(Q_0 )R_0= Q^+ $ is a diagonal matrix. Set $\tilde R(Q) = R_0^TR_Q$. Since $\tilde R(Q_0) = I$, a direct calculation shows that
\begin{align*}
\nabla\tilde R_{ij}(Q_0)+ \nabla\tilde R_{ji}(Q_0) =0,\quad\forall i,j=1,2,3.\alabel{Rot}
\end{align*}
Then we obtain
\begin{align*}
&\left .\<\nabla^k g_B(\tilde Q) ,\nabla R_Q^T \pi(Q )R_Q-R_Q^T \pi(Q )\nabla R_Q\>\right |_{Q=Q_0}
\\
=&\<\nabla^k g_B(\tilde Q_0) , \nabla \tilde R^T(Q_0) (R_0^T\pi ( Q_0) R_0)\tilde R(Q_0)-\tilde R^T(Q) (R_0^T \pi ( Q_0) R_0)\nabla \tilde R(Q_0)\>
\\
=&\<\nabla^k g_B(\tilde Q_0) , \nabla \tilde R^T(Q_0) Q^+- Q^+\nabla \tilde R(Q_0)\>
\\
=&
\sum_{i,j =1}^3(\nabla^k g_B(\tilde Q_0))_{ij}\( \nabla \tilde R_{ji} (Q_0) Q^+_{jj}+Q^+_{ii}\nabla \tilde R_{ij}(Q_0)\).
\end{align*}
Using \eqref{Rot} and the fact that $Q^+_{11}=Q^+_{22}$, we have
\begin{align*}
\sum_{i,j =1}^2(\nabla^k g_B(\tilde Q_0))_{ij}\( \nabla \tilde R_{ji} (Q_0) Q^+_{jj} + Q^+_{ii}\nabla \tilde R_{ij}(Q_0)\)=0.\alabel{gB pi1}
\end{align*}
It follows from \eqref{gB} and \eqref{Q13} that $\tilde Q_{13}=\tilde Q_{23}=0$ for each $Q\in S_\delta$. Thus
\begin{align*}
(g_B(\tilde Q))_{13}=&a(\tilde Q)_{13}+b \sum_{k=1}^3\tilde Q_{1k}\tilde Q_{k3}-c(\tilde Q)_{13}\tr(\tilde Q ^2)=0.
\end{align*}
Then $\nabla^k (g_B(\tilde Q))_{13}=0$ for any $Q\in S_\delta$. Similarly, $\nabla^k(g_B(\tilde Q))_{23}=0$. For the case of $i=3$, we apply \eqref{Rot} to obtain
\begin{align*}
& \sum_{j =1}^3(\nabla^k g_B(\tilde Q_0))_{3j} \(\nabla \tilde R_{ji} (Q_0) Q^+_{jj} + Q^+_{33}\nabla \tilde R_{3j}(Q)\)\ \\
&=2(\nabla^k g_B(\tilde Q_0))_{33} \nabla \tilde R_{33} (Q_0) Q^+_{33}=0.\alabel{gB pi2}
\end{align*}
Similarly, we can prove it for $j=3$. In view of \eqref{gB pi1} and \eqref{gB pi2}, we prove the claim \eqref{gB pi} for any $Q=Q_0\in S_\delta$.
\end{proof}
\subsection{Some a-priori estimates}For simplicity of notations, we denote $f_E(Q,\nabla Q)$ by $f_E$ and only write the subscript $L$ in the statement
of each lemma, but omit it in all proofs in this section.
\begin{lemma}\label{Lie}
Let $F$ be a $3\times 3$ matrix. For any symmetric $A,B$ matrices, we have
\begin{align*}
\<[A, F],B\> =\<F, [A,B]\>=-\<F^T, [A,B]\>
.\alabel{energy p2.1}
\end{align*}
\end{lemma}
\begin{proof}
Note the following identity
\begin{align*}
\<[A,F],B \> =& \<(AF-FA),B\>=\tr\((AF)^TB-(FA)^TB\)
\\
=&\tr\(F^TA^TB-A^T(F^TB)\)=\tr\(F^TA^TB-(F^TB)A^T\)
\\
=&\<F,[A^T,B]\>=\<F,[A,B]\>.
\end{align*}
For the second identity in \eqref{energy p2.1}, we observe that
\[\<F,[A,B]\>=\<F^T,[A,B]^T\>=-\<F^T,(A^TB^T-B^TA^T)\>=-\<F^T,[A,B]\>.\]
Here we used the fact that $A,B$ are symmetric in the last step.
\end{proof}
Now, we show the following energy identity:
\begin{lemma}\label{lem energy}Let $(Q_L, v_L)$ be a strong solution to the system \eqref{RBE1}-\eqref{RBE3} in $\ensuremath{\mathbb{R}}^ 3 \times (0, T_L
)$ with the initial condition $(Q_{L,0},v_{L,0})\in H^2_{Q_e}(\ensuremath{\mathbb{R}}^3;S_*)\times H^1(\ensuremath{\mathbb{R}}^3;\ensuremath{\mathbb{R}}^3)$ and $\div v_{L,0}=0$. Then, for any $s\in (0, T_L)$, we have
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\(f_E(Q_L,\nabla Q_L)+\frac1L f_B(Q_L)+\frac{|v_L|^2}{2}\)(x,s)\,dx\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\left|\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B(Q_L)\right|^2\,dxdt
+ \int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla v_L|^2\,dxdt
\\
&=\int_{\ensuremath{\mathbb{R}}^3}\Big(f_E(Q_{L,0},\nabla Q_{L,0})+\frac1L f_B(Q_{L,0})+\frac{|v_{L,0}|^2}{2}\Big)\,dx.
\alabel{energy eq}
\end{align*}
\end{lemma}
\begin{proof}
Taking $L^2$ inner product of \eqref{RBE1} with $v$ and using integration by part yield
\begin{align*}
&\frac{1}{2}\frac{d}{\,dt}\int_{\ensuremath{\mathbb{R}}^3}|v|^2\,dx+\int_{\ensuremath{\mathbb{R}}^3}|\nabla v|^2\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3} \partial_{p_{kl}^j}
f_E\nabla_iQ_{kl} \nabla_j v_i\,dx-\int_{\ensuremath{\mathbb{R}}^3}[Q, \mathcal{H}(Q ,\nabla Q )]_{ij}\nabla_j v_i\,dx.\alabel{kinetic energy}
\end{align*}
Next, multiplying \eqref{RBE3} with $(\mathcal{H}(Q ,\nabla Q ) +\frac 1Lg_B(Q ))$ gives
\begin{align*}
&-\int_{\ensuremath{\mathbb{R}}^3} \<\partial_tQ ,\mathcal{H}(Q ,\nabla Q )+\frac 1Lg_B(Q )\>\,dx+\int_{\ensuremath{\mathbb{R}}^3}|\mathcal{H}(Q ,\nabla Q )+\frac 1Lg_B(Q )|^2\,dx\\
=&\int_{\ensuremath{\mathbb{R}}^3}\<(v\cdot \nabla) Q+[Q , \Omega ],\mathcal{H}(Q ,\nabla Q )+\frac 1Lg_B(Q )\>\,dx.
\alabel{energy p2}
\end{align*}
In view of \eqref{Mol} and the relation that $g_B(Q)=-\nabla_{Q }f_B(Q)-\frac{b}{3}\tr(Q^2)I$, we have
\begin{align}\label{energy p2.00}
&-\int_{\ensuremath{\mathbb{R}}^3} \<\partial_tQ,\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q)\>\,dx=\frac{d}{\,dt}\int_{\ensuremath{\mathbb{R}}^3} (f_E(Q,\nabla Q)+\frac1L f_B(Q))\,dx.
\end{align}
Utilizing \eqref{Mol}, \eqref{gB} and integrating by parts, we have
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3} \<(v\cdot\nabla) Q,\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q)\>\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3} \<(v\cdot\nabla)Q, \nabla_j\big(\partial_{p^j}
f_E\big)- \partial _{Q}f_E \> \,dx -\int_{\ensuremath{\mathbb{R}}^3} \<(v\cdot\nabla)Q,\frac1L\partial_{Q} f_B(Q) \>\,dx
\\
=&-\int_{\ensuremath{\mathbb{R}}^3}\nabla_jv_i\nabla_iQ _{kl}\partial_{p_{kl}^j}
f_E + v_i\( \nabla^2_{ij}Q _{kl}\partial_{p_{kl}^j}
f_E-\nabla_iQ_{kl} \partial _{Q_{kl}}f_E -\frac{1}{L}\nabla_i f_B(Q)\)\,dx
\\
=&-\int_{\ensuremath{\mathbb{R}}^3} \partial_{p_{kl}^j}
f_E\nabla_iQ_{kl} \nabla_j v_i\,dx-\int_{\ensuremath{\mathbb{R}}^3} v_i\nabla_i f\,dx=-\int_{\ensuremath{\mathbb{R}}^3}\partial_{p^j_{kl}} f_E\nabla_iQ _{kl}\nabla_jv_i \,dx
,\alabel{energy p2.0}
\end{align*}
where we have used $\tr Q=0$ in the second equality.
Choosing $A=Q,B= \mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q), F=\nabla v$ in Lemma \ref{Lie} and using the fact that $[Q,g_B]=0$, we have
\begin{align*}
& \<[Q , \Omega ],\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q)\> = \nabla_jv_i[Q,\mathcal{H}(Q,\nabla Q)]_{ij} .\alabel{energy p2.1 +}
\end{align*}
Integrating \eqref{energy p2.1 +} in $x$ and
substituting \eqref{energy p2.00}-\eqref{energy p2.0} into \eqref{energy p2} give
\begin{align*}
&\frac{d}{\,dt}\int_{\ensuremath{\mathbb{R}}^3} f(Q,\nabla Q)\,dx+\int_{\ensuremath{\mathbb{R}}^3}|\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q)|^2\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\nabla_lv_k[Q,\mathcal{H}(Q,\nabla Q)]_{kl}\,dx- \int_{\ensuremath{\mathbb{R}}^3} \partial_{p_{kl}^j}
f_E\nabla_iQ_{kl} \nabla_j v_i\,dx
.\alabel{free energy}
\end{align*}
Therefore, the energy identity \eqref{energy eq} follows from taking the sum of \eqref{kinetic energy} and \eqref{free energy} and integrating over the
time interval $[0,s]$.
\end{proof}
We rotate the equation \eqref{RBE3} by $R_{Q_L}=R(\pi (Q_L))$; i.e.
\begin{align*}\alabel{ROT RBE}
&R^T_{Q_L}\big(\partial_t Q_L+(v_L\cdot \nabla Q_L) +[Q_L, \Omega_L]\big)R_{Q_L}= R^T_{Q_L}\mathcal{H}(Q_L,\nabla Q_L) R_{Q_L}+\frac 1Lg_B(\tilde Q_L),
\end{align*}
where we use the fact that $R_{Q_L}^Tg_B(Q_L)R_{Q_L}=g_B(\tilde Q_L)$.
The strong solutions also admit the following local energy inequality:
\begin{lemma}\label{lem 1ord}
Let $(Q_L, v_L)$ be a strong solution to the system \eqref{RBE1}-\eqref{RBE3} in $\ensuremath{\mathbb{R}}^ 3 \times (0, T_L
)$. Assume that $Q\in S_\delta$ for sufficiently small $\delta$ on $\ensuremath{\mathbb{R}}^3\times(0, T_L)$. Then, for any $\phi \in C^\infty_0 (\ensuremath{\mathbb{R}}^3 ) $ and $s \in (0, T_L )$, we have
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla Q_L|^2+ |v_L |^2+\frac{|Q_L -\pi(Q_L )|^2}{L}\)(x,s)\phi^2\,dx
\\
&+\int_0^{s }\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla^2 Q_L|^2+|\nabla v_L|^2+|\partial_t Q_L|^2+\frac{|\nabla(Q_L -\pi(Q_L ))|^2}{L}\)\phi^2\,dxdt
\\
\leq& C\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla Q_{L,0}|^2+ |v_{L,0}|^2+\frac{|Q_{L,0}-\pi(Q_{L,0})|^2}{L}\)\phi^2\,dx
\\
&+C\int_0^{s }\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_L|^2\(|\nabla Q_L|^2+|v_L|^2+\frac{|Q_L-\pi(Q_L)|^2}{L}\)\phi^2\,dxdt
\\
&+C\int_0^{s }\int_{\ensuremath{\mathbb{R}}^3}|P_L-c_L^*(t)||v_L||\nabla \phi||\phi|+(|\nabla Q_L|^2+|v_L|^2)|\nabla\phi|^2\,dxdt
.\alabel{1ord eq}
\end{align*}
\end{lemma}
\begin{proof}
Differentiating \eqref{ROT RBE} and multiplying by $R_Q^T\nabla QR_Q\phi^2$ yield
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla\(R^T_Q(\partial_t Q+v\cdot \nabla Q +[Q, \Omega])R_Q\),R_Q^T\nabla_\beta QR_Q\>\phi^2\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla(R^T_Q \mathcal{H}(Q,\nabla Q)R_Q)+\frac 1L \nabla g_B(\tilde Q),R_Q^T\nabla_\beta QR_Q\>\phi^2\,dx
.\alabel{Delta Q eq}
\end{align*}
We observe that
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla(R^T_Q \mathcal{H}(Q,\nabla Q)R_Q),R_Q^T\nabla_\beta QR_Q\>\phi^2\,dx
\\
\leq &\int_{\ensuremath{\mathbb{R}}^3}\<\nabla \mathcal{H}(Q,\nabla Q) , \nabla_\beta Q \>\phi^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla R_Q||\nabla Q|| |\mathcal{H}(Q,\nabla Q) |\phi^2\,dx
\\
\leq&\int_{\ensuremath{\mathbb{R}}^3}\nabla_\beta\nabla_k (\partial_{p_{ij}^k} f_E)\nabla_\beta Q_{ij}\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}( |\mathcal{H}(Q,\nabla Q) ||\nabla Q|^2+| \partial_{Q} f_E(Q,\nabla Q)||\nabla^2 Q|)\phi^2\,dx.\alabel{fE 1st}
\end{align*}
Noting the condition \eqref{sec2 f_E} on $f_E$ and integrating by parts, we have
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\nabla_\beta\nabla_k (\partial_{p_{ij}^k} f_E)\nabla_\beta Q_{ij}\phi^2\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\nabla_k\( \partial^2_{p_{ij}^kp_{mn}^l} f_E \nabla^2_{\beta l}Q_{mn}+ \partial^2_{p_{ij}^kQ_{mn}} f_E \nabla_\beta Q_{mn}\)\nabla_\beta
Q_{ij}\phi^2\,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3}\nabla_k (\partial_{p_{ij}^k} f_E)\nabla_\beta Q_{ij}\nabla_\beta \phi^2\,dx
\\
\leq&\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p_{ij}^kp_{mn}^l} f_E \nabla^2_{\beta l}Q_{mn}\nabla^2_{k\beta}Q_{ij}\phi^2\,dx +C\int_{\ensuremath{\mathbb{R}}^3}|\partial^2_{p p } f_E| |\nabla^2Q||\nabla Q||\nabla\phi||\phi|\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}\Big(|\nabla(\partial^2_{p Q} f_E)| |\nabla Q|+|\partial^2_{p Q} f_E| |\nabla^2 Q|\Big)|\nabla Q|\phi^2+|\nabla (\partial_{p} f_E)||\nabla Q||\nabla\phi||\phi|\,dx
\\
\leq& -\int_{\ensuremath{\mathbb{R}}^3}\frac{3\alpha}{8}|\nabla^2 Q|^2\phi^2+C\(|\nabla Q|^4\phi^2+|\nabla Q|^2|\nabla \phi|^2\)\,dx
.\alabel{fE second}
\end{align*}
Here we used that $|\nabla (\partial^2_{pQ} f_E)|\leq C(|\nabla^2 Q|+|\nabla Q|^2)$ due to $Q\in S_\delta$.
In view of Corollary \ref{cor fB}, Lemma \ref{lemgb} with $k=1$ and $g_B(\tilde Q)=-\nabla_{\tilde Q }f_B(\tilde Q)-\frac{b}{3}\tr(\tilde Q^2)I$, we obtain
\begin{align*}
& \int_{\ensuremath{\mathbb{R}}^3}\<\frac 1L \nabla g_B(\tilde Q)) ,R_Q^T\nabla_\beta QR_Q\>\phi^2\,dx\\
=&\frac 1L \int_{\ensuremath{\mathbb{R}}^3}\<\nabla g_B(\tilde Q) ,\nabla\tilde Q - \nabla R_Q^T QR_Q-R_Q^T Q\nabla R_Q\>\phi^2\,dx\\
=& \frac 1L \int_{\ensuremath{\mathbb{R}}^3}\<-\nabla (\partial_{\tilde Q}f_B(\tilde Q)),\nabla\tilde Q \> \phi^2\,dx
\\
&+ \frac 1L \int_{\ensuremath{\mathbb{R}}^3}\<\nabla g_B(\tilde Q) , - \nabla R_Q^T QR_Q-R_Q^T Q\nabla R_Q\>\phi^2\,dx\\
=&-\frac1L\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{\tilde Q_{ij} \tilde Q_{kl}} f_B(\tilde Q)\nabla \tilde Q _{ij}\nabla\tilde Q_{kl}\phi^2\,dx
\\
&+ \frac1L\int_{\ensuremath{\mathbb{R}}^3}\<\nabla g_B(\tilde Q) , \nabla R_Q^T (\pi (Q)-Q)R_Q+ R_Q^T (\pi (Q)-Q)\nabla R_Q\>\phi^2\,dx
\\
\leq&-\frac\lambda4\int_{\ensuremath{\mathbb{R}}^3} \frac{|\nabla(Q-\pi(Q))|^2}{L}\phi^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2\frac{|Q-\pi(Q)|^2}{L}\phi^2\,dx,\alabel{1st gb}
\end{align*}
where we also used that
\[|\nabla g_B(\tilde Q)|\leq C |\nabla (\tilde Q-Q^+ )|\leq C |\nabla ( Q-\pi (Q) ) |+ C|Q-\pi(Q)| |\nabla Q|.\]
Combining \eqref{fE second}, \eqref{fE 1st} with \eqref{1st gb} yields
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla(R^T_Q(\mathcal{H}(Q,\nabla Q)R_Q)+\frac 1L \nabla g_B(\tilde Q),R_Q^T\nabla_\beta QR_Q\>\phi^2\,dx
\\
\leq&-\int_{\ensuremath{\mathbb{R}}^3} \(\frac{\alpha}{4} |\nabla^2 Q|^2+ \frac{\lambda }4\frac{|\nabla (Q-\pi(Q))|^2}{L}\)\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2|\nabla\phi|^2+|\nabla Q|^2\(|\nabla Q|^2+|v|^2+\frac{|Q-\pi(Q)|^2}{L} \)\phi^2\,dx
.\alabel{1ord p4.1}
\end{align*}
Integrating by parts, we estimate the left-hand side of \eqref{Delta Q eq} to obtain
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\(R^T_Q(\partial_t Q+v\cdot \nabla Q +[Q, \Omega])R_Q\),R_Q^T\nabla_\beta QR_Q\>\phi^2\,dx
\\
\geq& \int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\partial_tQ+(v\cdot \nabla) Q ,\nabla_\beta Q \>\phi^2\,dx- \int_{\ensuremath{\mathbb{R}}^3}\<[Q, \Omega],\nabla_\beta(\nabla_\beta Q\phi^2)\>\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2\(| \partial_tQ|^2 +|v|^2| \nabla Q|^2 +|\nabla v|^2 \)\phi^2\,dx
\\
\geq&\frac{1}{2}\frac{d}{\,dt}\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2\phi^2\,dx-\int_{\ensuremath{\mathbb{R}}^3}\Big(\frac{\alpha}{8}|\nabla^2
Q|^2+\frac{1}{4}|\partial_t
Q|^2+C|\nabla v|^2\Big)\phi^2\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2|v|^2\phi^2+|\nabla Q|^2|\nabla\phi|^2\,dx
.\alabel{1ord p4}
\end{align*}
Adding \eqref{1ord p4.1} to \eqref{1ord p4}, we have
\begin{align*}
&\frac{1}{2}\frac{d}{\,dt}\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2\phi^2\,dx+\int_{\ensuremath{\mathbb{R}}^3} \(\frac{\alpha}4 |\nabla^2 Q|^2+ \frac{\lambda }2\frac{|\nabla (Q-\pi(Q))|^2}{L}\)\phi^2\,dx
\\
\leq&\int_{\ensuremath{\mathbb{R}}^3}\Big(\frac{1}{2}|\partial_tQ|^2+C|\nabla v|^2\Big)\phi^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2|\nabla\phi|^2\,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2\(|\nabla Q|^2+|v|^2+\frac{|Q-\pi(Q)|^2}{L} \)\phi^2\,dx
.\alabel{1ord Delta Q}
\end{align*}
Multiplying \eqref{RBE3} by $\partial_t Q\phi^2$ and using \eqref{energy p2.00} in Lemma \ref{lem energy} yield
\begin{align*}
&\frac{d}{\,dt}\int_{\ensuremath{\mathbb{R}}^3}(f_E(Q,\nabla Q)+\frac1L f_B(Q))\phi^2\,dx+\int_{\ensuremath{\mathbb{R}}^3}|\partial_tQ|^2\phi^2\,dx\\
=&-2\int_{\ensuremath{\mathbb{R}}^3}\partial_tQ _{ij}\partial_{p^k_{ij}} f_E \nabla_k\phi\phi\,dx-\int_{\ensuremath{\mathbb{R}}^3}\<(v\cdot\nabla)
Q +[Q,\Omega] ,\partial_tQ\>\phi^2\,dx\\
\leq& \int_{\ensuremath{\mathbb{R}}^3}\( \frac14|\partial_tQ|^2+C|\nabla
v|^2\)\phi^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2|v|^2\phi^2+|\nabla Q|^2|\nabla\phi|^2\,dx
.\alabel{1ord Q_t}
\end{align*}
Adding \eqref{1ord Delta Q} to \eqref{1ord Q_t}, integrating in $t$ and using \eqref{f_B and dist}, we see
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla Q|^2+\frac{|Q-\pi(Q)|^2}{L}\)(x,s)\phi^2\,dx
\\
&+\int^s_0\int_{\ensuremath{\mathbb{R}}^3} \( |\nabla^2 Q|^2+ |\partial_tQ|^2+ \frac{|\nabla (Q-\pi(Q))|^2}{L}\)\phi^2\,dxdt
\\
\leq
&C\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla Q_0|^2+\frac{|Q_0-\pi(Q_0)|^2}{L}\)\phi^2\,dx
\\
&+C\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla v|^2\phi^2\,dxdt+C\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2|\nabla\phi|^2\,dxdt
\\
&+\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2\(|\nabla Q|^2+|v|^2+\frac{|Q-\pi(Q)|^2}{L} \)\phi^2\,dxdt
.\alabel{p3}
\end{align*}
Estimating the term $\nabla v$ on the right-hand side of \eqref{p3},
we multiply \eqref{RBE1} by $v\phi^2$ and \eqref{RBE3} by $\Big(\mathcal{H}(Q,\nabla Q)+\frac{1}{L}g_B(Q)\Big)\phi^2$. Then it follows from using the same argument in \eqref{kinetic energy}-\eqref{free energy} that
\begin{align*}
&\frac{d}{\,dt}\int_{\ensuremath{\mathbb{R}}^3}\left(\frac{1}{2}|v|^2+f_E(Q,\nabla Q)+\frac1L f_B(Q)\right)\phi^2\,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3}\(|\mathcal{H}(Q,\nabla Q)+\frac{1}{L}g_B(Q)|^2+|\nabla
v|^2\)\phi^2\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\(|v|^2+2\(P-c^\ast(t)\)\)v\cdot\nabla \phi\phi-2\nabla_k v_i v_i\nabla_k\phi\phi \,dx
\\
&-2\int_{\ensuremath{\mathbb{R}}^3}[Q, \mathcal{H}(Q,\nabla Q)]_{ij}v_i\nabla_j
\phi\phi\,dx+2\int_{\ensuremath{\mathbb{R}}^3}\partial_{p_{kl}^j}
f_E\nabla_i Q_{kl}v_i\nabla_j \phi\phi\,dx
\\
&-2\int_{\ensuremath{\mathbb{R}}^3}\partial_tQ _{kl}\partial_{p^j_{kl}} f_E
\nabla_j\phi\phi\,dx-\int_{\ensuremath{\mathbb{R}}^3}v\cdot\nabla f\phi^2\,dx-2\int_{\ensuremath{\mathbb{R}}^3}v_i\nabla_i Q_{kl}\partial_{p_{kl}^j}
f_E\nabla_j\phi\phi\,dx \\
\leq& \eta\int_{\ensuremath{\mathbb{R}}^3}(|\partial_tQ|^2+ |\nabla^2Q|^2)\phi^2\,dx+\frac{1}{2}\int_{\ensuremath{\mathbb{R}}^3}|\nabla v|^2\phi^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}(|\nabla Q|^4+|v|^4)\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}(|\nabla Q|^2+|v|^2)|\nabla\phi|^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}|P_L-c_L^*(t)||v_L||\nabla \phi||\phi|\,dx
\alabel{p1}.
\end{align*}
Integrating \eqref{p1} in $t$, employing \eqref{p3} and choosing sufficiently small $\eta$, we obtain
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3} |v(x,s)|^2\phi^2\,dx+\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla
v|^2\phi^2\,dxdt
\\
\leq&C\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla Q_0|^2+|v_0|^2+\frac{|Q_0-\pi(Q_0)|^2}{L}\)\phi^2\,dx+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}(|\nabla Q|^4+|v|^4)\phi^2\,dx
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}(|\nabla Q|^2+|v|^2)|\nabla\phi|^2\,dx+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|P_L-c_L^*(t)||v_L||\nabla \phi||\phi|\,dx
\alabel{v}.
\end{align*}
Applying \eqref{v} to \eqref{p3}, we prove \eqref{1ord eq}.
\end{proof}
Through Corollary \ref{cor fB} and the equation \eqref{ROT RBE}, we obtain second order estimates of $(\nabla Q_L,v_L)$ in the following:
\begin{lemma}\label{lem 2ord}
Let $(Q_L, v_L)$ be a strong solution to the system \eqref{RBE1}-\eqref{RBE3} in $\ensuremath{\mathbb{R}}^ 3 \times (0, T_L )$. Assume that $Q\in S_\delta$ for sufficiently small $\delta$ on $\ensuremath{\mathbb{R}}^3\times(0, T_L)$. Then for any $\phi \in C^\infty_0 (\ensuremath{\mathbb{R}}^3 ) $ and $s \in (0, T_L )$, we have the following local estimate
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla^2 Q_L|^2+ |\nabla v_L |^2+\frac{|\nabla (Q_{L}-\pi(Q_{L}))|^2}{L}\)(x,s)\phi^2\,dx
\\
&+\int_0^{s}\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla^3 Q_L|^2+|\nabla^2 v_L|^2+|\nabla\partial_t Q_L|^2+\frac{|\nabla^2 (Q_{L}-\pi(Q_{L}))|^2}{L}\)\phi^2\,dxdt
\\
\leq& C\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla^2 Q_{L,0}|^2+ |\nabla v_{L,0}|^2+\frac{|\nabla (Q_{L,0}-\pi(Q_{L,0}))|^2}{L}\)\phi^2\,dx
\\
& +C\int_{\ensuremath{\mathbb{R}}^3} \frac{|Q_{L,0}-\pi(Q_{L,0})|^2}{L}|\nabla Q_{L,0}|^2\phi^2+ \(\frac{|Q-\pi(Q)|^2}{L}|\nabla Q|^2\)(x,s)\phi^2\,dx
\\
&+C\int_0^{s}\int_{\ensuremath{\mathbb{R}}^3}e(Q_L,v_L)\(|\nabla^2 Q_L|^2+|\nabla v_L|^2+|\partial_t Q_L|^2\)\phi^2\,dxdt
\\
&+C\int_0^{s}\int_{\ensuremath{\mathbb{R}}^3}e(Q_L,v_L)\(\frac{|\nabla (Q_L-\pi(Q_L))|^2}{L}+e^2(Q_L,v_L)\)\phi^2\,dxdt
\\
&+C\int_0^{s}\int_{\ensuremath{\mathbb{R}}^3}\(e^2(Q_L,v_L)+|P_L-c_L^*(t)|^2\)
(|\nabla \phi|^2+|\nabla^2 \phi||\phi|)\,dxdt
\\
&+C\int_0^{s}\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla^2 Q_L|^2+|\nabla v_L|^2+|\partial_t Q_L|^2\)(|\nabla \phi|^2+|\nabla^2 \phi||\phi|)\,dxdt
.\alabel{2ord eq}
\end{align*}
Here we denote
\[e(Q_L,v_L):=|\nabla Q_L|^2+|v_L|^2+\frac{|Q_L-\pi(Q_L)|^2}{L}.\]
\end{lemma}
\begin{proof}
Differentiating \eqref{ROT RBE} with respect to $x_\beta$ and$x_\gamma$, we multiply by $\nabla_{\beta}(R^T_Q \nabla_\gamma QR_Q)\phi^2$ to obtain
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^2_{\beta_\gamma} \Big(R^T_Q\big(\partial_t Q+v\cdot \nabla Q +[Q, \Omega]\big)R_Q\Big), \nabla_{\beta}(R^T_Q \nabla_\gamma QR_Q) \>\phi^2\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^2_{\beta_\gamma}\(R^T_Q\mathcal{H}(Q,\nabla Q) R_Q+\frac1Lg_B(\tilde Q)\),\nabla_{\beta}(R^T_Q \nabla_\gamma QR_Q)\>\phi^2\,dx
.\alabel{D Delta Q eq}
\end{align*}
Integrating by parts twice and using \eqref{sec2 f_E}, we estimate
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^2_{\beta\gamma}\nabla_k\( \partial^2_{p^k} f_E\), \nabla^2_{\beta\gamma} Q \>\phi^2\,dx
\\
=&-\int_{\ensuremath{\mathbb{R}}^3}\nabla_\gamma\(\partial^2_{p_{ij}^kp_{mn}^l} f_E \nabla^2_{\beta l}Q_{mn}+\partial^2_{p_{ij}^k Q_{mn}} f_E \nabla_{\beta}Q_{mn}\) \nabla_k(\nabla^2_{\beta\gamma} Q_{ij}\phi^2)\,dx
\\
\leq&-\int_{\ensuremath{\mathbb{R}}^3}\nabla_\gamma\(\partial^2_{p_{ij}^kp_{mn}^l} f_E \nabla^2_{\beta l}Q_{mn}\)\nabla^3_{\beta \gamma k} Q_{ij}\phi^2\,dx
-C\int_{\ensuremath{\mathbb{R}}^3}|\partial^2_{pp} f_E|| \nabla^2 Q||\nabla^3 Q ||\nabla\phi||\phi|\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}(|\nabla\partial^2_{p Q} f_E||\nabla Q|+ |\partial^2_{p Q} f_E|\nabla^2 Q|)|\nabla^3 Q|\phi^2\,dx
\\
\leq&-\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p_{ij}^kp_{mn}^l} f_E \nabla^3_{\beta l \gamma}Q_{mn}\nabla^3_{\beta \gamma k} Q_{ij}\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}(| \nabla^2 Q||\nabla Q|+|\nabla Q|^3)|\nabla^3 Q |\phi^2+| \nabla^2 Q||\nabla^3 Q ||\nabla\phi||\phi|\,dx
\\
\leq&-\int_{\ensuremath{\mathbb{R}}^3}\frac{3\alpha}{8}|\nabla^3 Q|^2\phi^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2(|\nabla^2 Q|^2+|\nabla Q|^4)\phi^2+|\nabla^2 Q|^2|\nabla \phi|^2 \,dx
.\alabel{fE third}
\end{align*}
Then using \eqref{fE third} and integrating by parts, we find
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^2_{\beta\gamma}\(R^T_Q\mathcal{H}(Q,\nabla Q) R_Q\),\nabla_{\beta}(R^T_Q \nabla_\gamma QR_Q)\>\phi^2\,dx
\\
\leq&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^2_{\beta\gamma}\(\nabla_k \partial_{p^k} f_E-\partial_{ Q} f_E\), \nabla^2_{\beta\gamma} Q \>\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla \mathcal{H}(Q,\nabla Q)|(|\nabla R_Q||\nabla^2 Q|+|\nabla^2 R_Q||\nabla Q|+|\nabla R_Q|^2|\nabla Q|)\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla R_Q||\mathcal{H}(Q,\nabla Q)||\nabla^2(R^T_Q \nabla QR_Q)|\phi^2\,dx
\\
\leq&-\int_{\ensuremath{\mathbb{R}}^3}\frac{\alpha}{4}|\nabla^3 Q|^2\phi^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}(|\nabla^2 Q|^2+|\nabla Q|^4)(|\nabla Q|^2\phi^2+|\nabla\phi|^2)\,dx
.\alabel{2ord p4}
\end{align*}
Here we used that $|\nabla^2(R^T_Q \nabla QR_Q)|+|\nabla \mathcal{H}|\leq C(|\nabla^3 Q|+|\nabla^2 Q||\nabla Q|+|\nabla Q|^3)$.
Applying Corollary \ref{cor fB}, Lemma \ref{lemgb}, we obtain
\begin{align*}
& \int_{\ensuremath{\mathbb{R}}^3}\< \nabla^2_{\beta\gamma}\frac{g_B(\tilde Q)}L ,\nabla_{\beta}(R^T_Q \nabla_\gamma QR_Q)\>\phi^2\,dx
\\
=&-\int_{\ensuremath{\mathbb{R}}^3}\nabla_{\beta}\(\phi^2\nabla^2_{\beta\gamma} \frac{(g_B(\tilde Q))_{ij}}L \) \nabla_{\gamma}\tilde Q_{ij}\,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3}\nabla_{\beta}\(\phi^2\nabla^2_{\beta\gamma} \frac{(g_B(\tilde Q))_{ij}}L\)\(\nabla_{\gamma} R_Q^T (Q-\pi(Q))R_Q+ R_Q^T (Q-\pi(Q))\nabla_\gamma R_Q\)_{ij}\,dx
\\
\leq&-\frac 1L\int_{\ensuremath{\mathbb{R}}^3} \partial^2_{\tilde Q_{ij} \tilde Q_{kl}} f_B(\tilde Q)\nabla^2_{\beta\gamma} \tilde Q_{kl}\nabla^2_{\beta\gamma}\tilde Q_{ij}\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3} |\partial^3_{\tilde Q} f_B(\tilde Q)||\nabla Q|\frac{|\nabla\tilde Q|}{L^{\frac12}} \frac{| \nabla^2 \tilde Q|}{L^{\frac12}}\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}\frac{|\nabla^2 (g_B(\tilde Q))|}L\((|\nabla^2Q|+|\nabla Q|^2)|Q-\pi(Q)|+|\nabla Q||\nabla (Q-\pi(Q))| \)\phi^2\,dx
\\
\leq&-\frac{\lambda}{4}\int_{\ensuremath{\mathbb{R}}^3} \frac{|\nabla^2(Q-\pi(Q))|^2}{L}\phi^2
\,dx+C\int_{\ensuremath{\mathbb{R}}^3}\frac{| \nabla (Q-\pi(Q))|^2}{L}|\nabla Q|^2\phi^2 \,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}\frac{|Q-\pi(Q)|^2}{L}(|\nabla^2 Q|^2+|\nabla Q|^4)\phi^2 \,dx
,\alabel{2ord g_B D^2 Q}
\end{align*}
where we used that
\[|\nabla^2 (g_B(\tilde Q))|\leq C(|\nabla^2(Q-\pi(Q))|+|\nabla Q||\nabla (Q-\pi(Q))|+(|\nabla^2Q|+|\nabla Q|^2)|Q-\pi(Q)|.\]
We compute the left-hand side of \eqref{D Delta Q eq} to get
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\< \nabla^2_{\beta\gamma}\Big(R^T_Q\big(\partial_t Q+v\cdot \nabla Q +[Q, \Omega]\big)R_Q\Big),\nabla_{\beta} (R^T_Q \nabla_\gamma QR_Q)\>\phi^2\,dx
\\
\geq&\int_{\ensuremath{\mathbb{R}}^3} \<\nabla^2_{\beta\gamma}\partial_t Q,\nabla^2_{\beta\gamma} Q\>\phi^2+\<\nabla_{\beta}\partial_t Q,\nabla^2_{\beta\gamma} Q\>\nabla_\gamma(\phi^2)\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}|\nabla\partial_t Q|(|\nabla^2 Q||\nabla R_Q|+|\nabla Q||\nabla^2 R_Q|+|\nabla Q||\nabla R_Q|^2)\phi^2\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}\Big(|\partial_t Q||\nabla Q|+|v|(|\nabla^2 Q|+|\nabla Q|^2)\Big)|\nabla^2(R^T_Q \nabla QR_Q)|\phi^2\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}\Big(|\nabla v||\nabla Q|+|\nabla^2 v|\Big)|\nabla^2(R^T_Q \nabla QR_Q)|\phi^2\,dx
\\
\geq& \frac 12\frac{d}{dt}\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 Q|^2 \phi^2\,dx-\int_{\ensuremath{\mathbb{R}}^3}\(\frac14|\nabla \partial_t Q|^2+\frac{\alpha}8|\nabla^3 Q|^2+C|\nabla^2 v|^2\)\phi^2\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3} (|\nabla^2 Q|^2+|\nabla v|^2+|\partial_t Q|^2+|\nabla Q|^4)(|\nabla Q|^2+|v|^2)\phi^2+|\nabla^2 Q|^2|\nabla \phi|^2\,dx
.\alabel{2ord p4.1}
\end{align*}
In view of \eqref{D Delta Q eq}-\eqref{2ord p4.1}, we have
\begin{align*}
&\frac12\frac{d}{dt}\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 Q|^2 \phi^2\,dx+ \int_{\ensuremath{\mathbb{R}}^3} \(\frac{\alpha}{8}|\nabla^3 Q|^2+\frac{\lambda}{4}\frac{|\nabla^2( Q-\pi(Q))|^2}{L}\)\phi^2
\,dx
\\
\leq& \int_{\ensuremath{\mathbb{R}}^3}\(\frac14|\nabla \partial_t Q|^2+C|\nabla^2 v|^2\)\phi^2\,dx
\\
&+C \int_{\ensuremath{\mathbb{R}}^3}e(Q,v)\(|\nabla^2 Q|^2+|\nabla v|^2+|\partial_t Q|^2+|\nabla Q|^4+\frac{|\nabla (Q-\pi(Q))|^2}{L}\)\phi^2\,dx
\\
&+C \int_{\ensuremath{\mathbb{R}}^3}\(e^2(Q,v)+|\nabla^2 Q|^2\)|\nabla \phi|^2\,dx.
\alabel{2nd Delta Q}
\end{align*}
Estimating the term on $\nabla\partial_t Q$ in \eqref{2nd Delta Q}, we differentiate \eqref{ROT RBE} in $x_\beta$ and multiply it by $\nabla_\beta(R^T_Q\partial_t QR_Q)\phi^2$ to obtain
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\( R^T_Q(\partial_t Q +v\cdot \nabla Q +[Q, \Omega])R_Q\), \nabla_\beta (R^T_Q\partial_t QR_Q)\>\phi^2\,dxdt
\\
=&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\((R^T_Q\mathcal{H}(Q,\nabla Q) R_Q)+\frac1Lg_B(\tilde Q)\),\nabla_\beta(R^T_Q\partial_t QR_Q)\>\phi^2\,dxdt
.\alabel{2nd pt Q}
\end{align*}
Using \eqref{sec2 f_E}, we derive
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta(R^T_Q\mathcal{H}(Q,\nabla Q) R_Q),\nabla_\beta(R^T_Q\partial_t QR_Q)\>\phi^2\,dx dt
\\
\leq&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta(\nabla_k\partial_{p^k}f_E-\partial_Qf_E),\nabla_\beta\partial_t Q\>\phi^2\,dx dt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}(|\nabla\mathcal{H}||\nabla R_Q||\partial_t Q|+|\nabla R_Q||\mathcal{H}||\nabla(R^T_Q \partial_t QR_Q)|)\phi^2\,dx dt
\\
\leq& -\int_0^s\int_{\ensuremath{\mathbb{R}}^3} \frac12\partial_t\Big(\partial^2_{p_{ij}^kp_{mn}^l} f_E \nabla^2_{\beta
l}Q_{mn}\nabla^2_{\beta k} Q_{ij}\Big)\phi^2\,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\frac 12\partial_t\partial^2_{p_{ij}^kp_{mn}^l} f_E\nabla^2_{\beta l}Q_{mn}\nabla^2_{\beta k} Q_{ij}\phi^2+C|\partial^2_{pp}f_E||\nabla^2 Q||\nabla \partial_t Q||\nabla\phi||\phi|\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} (|\partial^2_{pQ}f_E||\nabla Q|+|\nabla \partial_Qf_E|)|\nabla \partial_t Q|\phi^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\Big(|\nabla \mathcal{H}(Q,\nabla Q)||\nabla Q||\partial_tQ|\Big)\phi^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\mathcal{H}(Q,\nabla Q)||\nabla Q|(|\nabla \partial_t Q|+|\nabla Q||\partial_t Q|)\ \phi^2\,dxdt
\\
\leq& \int_{\ensuremath{\mathbb{R}}^3}(C|\nabla^2 Q_{0}|^2-\frac {\alpha}4|\nabla^2 Q(x,s)|^2) \phi^2\,dx+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}(\frac{\alpha}{16}|\nabla^3 Q|^2+\frac 18|\nabla\partial_t Q|^2)\phi^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q|^2(|\nabla^2 Q|^2+|\partial_t Q|^2+|\nabla Q|^4)\phi^2+(|\nabla^2 Q|^2+|\nabla Q|^4)|\nabla \phi|^2\,dxdt
.\alabel{2ord H Q_t}
\end{align*}
By integrating by parts and using the fact that \[|\nabla(R^T_Q \partial_t QR_Q)|\leq C(|\nabla \partial_t Q|+|\partial_t Q||\nabla Q|),\] we have
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\frac1{2L}\partial_t \partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B(\tilde Q)\nabla_\beta \tilde Q_{kl}\nabla_\beta \tilde Q_{ij}\phi^2\,dx
\\
\leq&\frac C{L}\int_{\ensuremath{\mathbb{R}}^3}|Q-\pi(Q)|\( |\nabla\partial_t Q||\nabla \tilde Q|+|\partial_t Q||\nabla Q||\nabla \tilde Q|+|\partial_t Q||\nabla^2 \tilde Q|\)\phi^2\,dx
\\
&+\frac C{L}\int_{\ensuremath{\mathbb{R}}^3}|Q-\pi(Q)|\(|\partial_t Q||\nabla \tilde Q|\)|\nabla\phi||\phi|\,dx
\\
\leq&\int_{\ensuremath{\mathbb{R}}^3}\(\frac14|\nabla\partial_t Q|^2+\frac\lambda {16}\frac{|\nabla^2 (Q-\pi(Q))|^2}{L}\)\phi^2+C|\partial_t Q|^2|\nabla Q|^2+C|\partial_t Q|^2|\nabla\phi|^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}\frac{|\nabla (Q-\pi(Q))|^2}{L}\(|\nabla Q|^2+\frac{|Q-\pi(Q)|^2}{L}\)\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}\frac{|Q-\pi(Q)|^2}{L}\(\frac{|Q-\pi(Q)|^2}{L}|\nabla Q|^2+|\nabla^2 Q|^2+|\nabla Q|^4+|\partial_t Q|^2\)\phi^2\,dx
.\alabel{2nd pt fB}
\end{align*}
Here in the last inequality, we have used
\begin{align*}
|\nabla^2\tilde Q|^2 \leq & C(|\nabla^2(Q-\pi(Q))|^2+|\nabla(Q-\pi(Q))|^2|\nabla Q|^2
+|Q-\pi(Q))|^2(|\nabla^2 Q|^2+|\nabla Q|^4)).
\end{align*}
Utilizing a similar argument to one in Corollary \ref{cor fB} and \eqref{2nd pt fB}, we have
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\frac{\nabla_\beta g_B(\tilde Q)}{L},\nabla_\beta(R^T_Q\partial_t QR_Q)\>\phi^2\,dxdt
\\
=&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\frac{\nabla_\beta (g_B(\tilde Q))_{ij}}{L}\nabla_\beta\partial_t \tilde Q_{ij}\phi^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\frac{|\nabla^2 g_B(\tilde Q)|}{L^\frac12}|\partial_tR_Q|\frac{|Q-\pi(Q)|}{L^\frac12}\phi^2\,dxdt
\\
=&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\frac1{2L}\partial_t\(\partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B(\tilde Q)\nabla_\beta \tilde Q_{ij}\nabla_\beta \tilde Q_{kl}\)\phi^2\,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\frac1{2L}\partial_t \partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B(\tilde Q)\nabla_\beta \tilde Q_{kl}\nabla_\beta \tilde Q_{ij}\phi^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\frac{|\nabla^2 g_B(\tilde Q)|}{L^\frac12}|\partial_tQ|\frac{|Q-\pi(Q)|}{L^\frac12}\phi^2\,dxdt
\\
\leq&\int_{\ensuremath{\mathbb{R}}^3} \( -\frac \lambda4 \frac{|\nabla (Q(x,s)-\pi(Q(x,s)))|^2 }{L}+C\frac{|\nabla ( Q_0-\pi(Q_0))|^2}{L}\) \phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3} \(\frac{|Q_{L,0}-\pi(Q_{L,0})|^2}{L}|\nabla Q_{L,0}|^2+\frac{|Q-\pi(Q)|^2}{L}|\nabla Q|^2\)(x,s)\phi^2\,dx
\\
&+ \int_0^s\int_{\ensuremath{\mathbb{R}}^3}\(\frac14|\nabla\partial_t Q|^2+\frac\lambda 8\frac{|\nabla^2(Q-\pi(Q))|^2}{L}\)\phi^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}e(Q,v)\(|\nabla^2 Q|^2+|\partial_t Q|^2+\frac{|\nabla( Q-\pi(Q))|^2}{L}+e^2(Q,v)\)\phi^2\,dxdt
.\alabel{2ord g_B Q_t}
\end{align*}
Applying Young's inequality to the left-hand side of \eqref{2nd pt Q}, we obtain
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\( R^T_Q(\partial_t Q +v\cdot \nabla Q +[Q, \Omega])R_Q\), \nabla_\beta (R^T_Q\partial_t QR_Q)\>\phi^2\,dx
\\
\geq&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\partial_t Q , \nabla_\beta \partial_t Q\>\phi^2\,dx-\int_{\ensuremath{\mathbb{R}}^3}|\nabla\partial_t Q||\nabla R_Q||\partial_tQ|\phi^2\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}\Big(|\partial_t Q||\nabla Q|+|v|(|\nabla^2 Q|+|\nabla Q|^2)+|\nabla v||\nabla Q|+|\nabla^2 v|\Big)|\nabla(R^T_Q \partial_t QR_Q)|\phi^2\,dx
\\
\geq& \int_{\ensuremath{\mathbb{R}}^3}\frac34|\nabla\partial_t Q|^2\phi^2-C |\nabla^2 v|^2\phi^2-C\(|\nabla Q|^2(|\partial_t Q|^2+|\nabla v|^2)+|v|^2|\nabla^2 Q|^2\)\phi^2 \,dx
.\alabel{lem 2nd order p3}
\end{align*}
Substituting \eqref{2ord H Q_t}, \eqref{2ord g_B Q_t} and \eqref{lem 2nd order p3} into \eqref{2nd pt Q} yields
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3} \( \frac {\alpha}4|\nabla^2 Q |^2+\frac \lambda8 \frac{|\nabla (Q -\pi(Q ))|^2 }{L}\)(x,s) \phi^2\,dx
+\frac 38\int^s_0\int_{\ensuremath{\mathbb{R}}^3} |\nabla\partial_t Q|^2\phi^2\,dxdt
\\
\leq&C\int_{\ensuremath{\mathbb{R}}^3} \(|\nabla^2 Q_{0}|^2+\frac{|\nabla ( Q_0-\pi(Q_0))|^2}{L}+\frac{|Q_{L,0}-\pi(Q_{L,0})|^2}{L}|\nabla Q_{L,0}|^2\)\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3} \(\frac{|Q-\pi(Q)|^2}{L}|\nabla Q|^2\)(x,s)\phi^2\,dx+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v|^2\phi^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\(\frac{\alpha}{16}|\nabla^3 Q|^2+\frac\lambda 8\frac{|\nabla^2(Q-\pi(Q))|^2}{L}\)\phi^2\,dxdt
\\
&+C \int_0^s\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla^2 Q|^2+|\nabla v|^2+|\partial_t Q|^2\)(e(Q,v)\phi^2+|\nabla \phi|^2)\,dxdt
\\
&+C \int_0^s\int_{\ensuremath{\mathbb{R}}^3}\(\frac{|\nabla( Q-\pi(Q))|^2}{L}+e^2(Q,v)\)(e(Q,v)\phi^2+|\nabla \phi|^2)\,dxdt
.\alabel{2nd Qt}
\end{align*}
Integrating \eqref{2nd Delta Q} in $t$ then adding it to \eqref{2nd Qt}, we derive
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3} \( |\nabla^2 Q|^2+\frac{|\nabla (Q-\pi(Q))|^2 }{L}\)(x,s) \phi^2\,dx
\\
&+\int^s_0\int_{\ensuremath{\mathbb{R}}^3} \(|\nabla^3 Q|^2+|\nabla\partial_t Q|^2+\frac{|\nabla^2 (Q-\pi(Q))|^2}{L}\)\phi^2\,dxdt
\\
\leq&C\int_{\ensuremath{\mathbb{R}}^3} \(|\nabla^2 Q_{0}|^2+\frac{|\nabla ( Q_0-\pi(Q_0))|^2}{L}+\frac{|Q_{L,0}-\pi(Q_{L,0})|^2}{L}|\nabla Q_{L,0}|^2\)\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3} \(\frac{|Q-\pi(Q)|^2}{L}|\nabla Q|^2\)(x,s)\phi^2\,dx+\int_0^s\int_{\ensuremath{\mathbb{R}}^3} C|\nabla^2 v|^2\phi^2\,dxdt
\\
&+C \int_0^s\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla^2 Q|^2+|\nabla v|^2+|\partial_t Q|^2\)(e(Q,v)\phi^2+|\nabla \phi|^2)\,dxdt
\\
&+C \int_0^s\int_{\ensuremath{\mathbb{R}}^3}\(\frac{|\nabla( Q-\pi(Q))|^2}{L}+e^2(Q,v)\)(e(Q,v)\phi^2+|\nabla \phi|^2)\,dxdt
.\alabel{2ord p5}
\end{align*}
To estimate the term $\nabla^2 v$ in \eqref{2ord p5}, we take $L^2$ inner product of \eqref{RBE1} with $-\Delta v\phi^2$ and calculate
\begin{align*}
&\frac{1}{2}\frac{d}{\,dt}\int_{\ensuremath{\mathbb{R}}^3}|\nabla v|^2\phi^2\,dx+\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 v|^2\phi^2 \,dx
\\
=&-\int_{\ensuremath{\mathbb{R}}^3}2\partial_t v_i \nabla_j v_i\nabla_j \phi\phi\,dx+\int_{\ensuremath{\mathbb{R}}^3}2(\nabla_j v_i\Delta v_i-\nabla_k v_i\nabla_{kj}^2v_i)\nabla_j \phi\phi\,dx
\\
&-\int_{\ensuremath{\mathbb{R}}^3}(P-c^\ast)\Delta v_i\nabla_i\phi \phi\,dx-\int_{\ensuremath{\mathbb{R}}^3}\nabla_j\sigma_{ij}\Delta v_i\phi^2\,dx-\int_{\ensuremath{\mathbb{R}}^3}\nabla_j[Q,\mathcal{H}(Q,\nabla Q)]_{ij}\Delta v_i
\phi^2\,dx
\\
\leq& -\int_{\ensuremath{\mathbb{R}}^3}\nabla_j[Q,\mathcal{H}(Q,\nabla Q)]_{ij}\Delta v_i \phi^2\,dx-\int_{\ensuremath{\mathbb{R}}^3}2\partial_t v_i \nabla_j v_i\nabla_j \phi\phi\,dx+\frac{1}{4}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 v|^2\phi^2 \,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}(|\nabla v|^2+|P-c^\ast|^2)|\nabla\phi|^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}(|\nabla^2Q|^2+|\nabla Q|^4)|\nabla Q|^2\phi^2\,dx,
\end{align*}
where we have used the fact that $|\nabla\sigma(Q,\nabla Q)|\leq C(|\nabla^2Q|+|\nabla Q|^2)|\nabla Q|$.
By using \eqref{RBE1} and integrating by parts, we have
\begin{align*}
&-2\int_{\ensuremath{\mathbb{R}}^3}\partial_t v_i \nabla_j v_i\nabla_j \phi\phi\,dx
\\
=&2\int_{\ensuremath{\mathbb{R}}^3}(v_k\nabla_k v_i-\Delta v_i+\nabla_k\sigma_{ik})\nabla_j v_i\nabla_j\phi\phi\,dx+2\int_{\ensuremath{\mathbb{R}}^3}(P-c^\ast)\nabla_jv_i\nabla_i(\nabla_j\phi\phi)\,dx
\\
&+2\int_{\ensuremath{\mathbb{R}}^3}[Q,\mathcal{H}(Q,\nabla Q)]_{ik}\Big(\nabla_{kj}^2v_i\nabla_j\phi\phi+\nabla_jv_i\nabla_k(\nabla_j\phi\phi)\Big)\,dx
\\
\leq& \frac{1}{4}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 v|^2\phi^2
\,dx+C\int_{\ensuremath{\mathbb{R}}^3}(|v|^2|\nabla v|^2+(|\nabla^2Q|^2+|\nabla Q|^4)|\nabla Q|^2)\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}(|P-c^\ast|^2+|\nabla v|^2+|\nabla^2 Q|^2+|\nabla Q|^4)(|\nabla^2\phi||\phi|+|\nabla\phi|^2)\,dx,
\end{align*}
which, plugging into the previous inequality, yields
\begin{align*}
&\frac{1}{2}\frac{d}{\,dt}\int_{\ensuremath{\mathbb{R}}^3}|\nabla v|^2\,dx+\frac 12\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 v|^2\phi^2 \,dx\\
\leq&-\int_{\ensuremath{\mathbb{R}}^3}\nabla_j[Q,\mathcal{H}(Q,\nabla Q)]_{ij}\Delta v_i \phi^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3}|P-c^\ast|^2(|\nabla^2\phi||\phi|+|\nabla\phi|^2)\,dx\\
&+C\int_{\ensuremath{\mathbb{R}}^3}(|\nabla^2 Q|^2+|\nabla v|^2+|\nabla Q|^4)(|\nabla^2\phi||\phi|+|\nabla\phi|^2)\,dx\\
&+C\int_{\ensuremath{\mathbb{R}}^3}(|\nabla Q|^2+|v|^2)(|\nabla^2 Q|^2+|\nabla v|^2 +|\nabla Q|^4)\phi^2\,dx
.\alabel{2ord v}
\end{align*}
Choosing $A=Q, B=\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q), F=\Delta\nabla v$ in Lemma \ref{Lie}, we observe
\begin{align*}
\<[Q , \Delta\Omega ],\mathcal{H}+\frac 1Lg_B\>= \Delta\nabla_jv_i[Q,\mathcal{H}]_{ij}
.\alabel{Lie 1}
\end{align*}
Then, integrating by parts and using \eqref{Lie 1} and \eqref{RBE3} on the term $(\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q))$, we have
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\(R^T[Q, \Omega]R\),\nabla_\beta\(R^T_Q(\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q))R_Q\)\>\phi^2\,dx
\\
\leq& -\int_{\ensuremath{\mathbb{R}}^3}\<[\Delta Q , \Omega ]+2[\nabla_\beta Q , \nabla_\beta\Omega ]+[ Q , \Delta\Omega ],\mathcal{H}(Q,\nabla Q) +\frac 1Lg_B(Q)\>\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla[Q,\Omega]|\Big|\mathcal{H}(Q,\nabla Q) +\frac 1Lg_B(Q)\Big|(|\nabla Q|\phi^2+|\nabla \phi||\phi|)\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q||\nabla v|\Big|\nabla\(R^T_Q\big(\mathcal{H}(Q,\nabla Q) +\frac 1Lg_B(Q)\big)R_Q\)\Big| \phi^2\,dx
\\
\leq& \int_{\ensuremath{\mathbb{R}}^3}\Big(\Delta v_i\nabla_j[ Q ,\mathcal{H}(Q,\nabla Q) ]_{ij}+\eta|\nabla\partial_t Q|^2+\frac 14|\nabla^2 v|^2\Big) \phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3} (|\nabla^2 Q|^2+|\nabla v|^2+|\partial_t Q|^2+|\nabla Q|^4+|v|^4)((|\nabla Q|^2+|v|^2)\phi^2+|\nabla \phi|^2)\,dx
\\
&+\frac 14\int_{\ensuremath{\mathbb{R}}^3}\Big|\nabla\(R^T_Q\big(\mathcal{H}(Q,\nabla Q) +\frac 1Lg_B(Q)\big)R_Q\)\Big|^2 \phi^2\,dx
.
\alabel{2ord Lie}
\end{align*}
Here we used that $|[Q,\mathcal{H}(Q,\nabla Q)]|^2|\nabla\phi|^2\leq C(|\nabla^2 Q|^2+|\nabla Q|^4)|\nabla\phi|^2$.
We differentiate \eqref{ROT RBE} with respect to $x_\beta$, multiply by $\nabla_\beta \(R^T_Q\big(\mathcal{H}(Q,\nabla Q) +\frac 1Lg_B(Q)\big)R_Q\)$ and substitute \eqref{2ord Lie} to find
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\left|\nabla \(R^T_Q\big(\mathcal{H}(Q,\nabla Q) +\frac 1Lg_B(Q)\big)R_Q\)\right|^2\phi^2\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\( R_Q^T(\partial_t Q +v\cdot \nabla Q +[Q, \Omega])R\), \nabla_\beta \(R^T_Q(\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q))R_Q\)\>\phi^2\,dx
\\
\leq& \int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\( R^T_Q\partial_t QR_Q\), \nabla_\beta \(R^T_Q(\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q))R_Q\)\>\phi^2\,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3}\Big(\Delta v_i\nabla_j[ Q ,\mathcal{H}(Q,\nabla Q) ]_{ij}+\eta|\nabla\partial_t Q|^2+\frac 14|\nabla^2 v|^2\Big) \phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3} (|\nabla^2 Q|^2+|\nabla v|^2+|\partial_t Q|^2+|\nabla Q|^4+|v|^4)((|\nabla Q|^2+|v|^2)\phi^2+|\nabla \phi|^2)\,dx
\\
&+\frac12\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\(R^T[Q, \Omega]R\),\nabla_\beta\(R^T_Q(\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q))R_Q\)\>\phi^2\,dx
.\alabel{2ord can}
\end{align*}
Combining \eqref{2ord v} with \eqref{2ord can} and integrating in $t$, we then apply the arguments in \eqref{2ord H Q_t}-\eqref{2ord g_B Q_t} to derive
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\(\frac {\alpha}4|\nabla^2 Q |^2+\frac 12 |v |^2+\frac \lambda8 \frac{|\nabla ( Q-\pi(Q)|^2}{L}\) (x,s)\phi^2\,dx
+\frac14\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v|^2 \phi^2 \,dxdt
\\
\leq& C\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla^2 Q_{0}|^2+|v_0|^2+\frac{|\nabla ( Q_0-\pi(Q_0))|^2}{L}\)\phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3} \(\frac{|Q_{L,0}-\pi(Q_{L,0})|^2}{L}|\nabla Q_{L,0}|^2+\frac{|Q-\pi(Q)|^2}{L}|\nabla Q|^2(x,s)\)\phi^2\,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3}2\eta\Big(|\nabla^3 Q|^2+|\nabla\partial_t Q|^2+\frac{|\nabla^2 (Q-\pi(Q))|^2}{L}\Big) \phi^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3} (|\nabla^2 Q|^2+|\nabla v|^2+|\partial_t Q|^2)(e(Q,v)\phi^2+|\nabla \phi|^2)\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\(\frac{|\nabla( Q-\pi(Q))|^2}{L}+e^2(Q,v)\)(e(Q,v)\phi^2+|\nabla \phi|^2)\,dxdt
.\alabel{2ord p2.2}
\end{align*}
Substituting \eqref{2ord p5} and choosing suitable $\eta$, we obtain the estimate for $\nabla v$. Combining the resulting expression with \eqref{2ord p5}, we proved \eqref{2ord eq}.
\end{proof}
\section{Proof of Theorem \ref{thm1}}
In this section, we will prove Theorem \ref{thm1}. At first,
we derive a local estimate on the pressure $P_L(x,t)$.
\begin{lemma}\label{lem pressure estimate}
Let $(Q_L, v_L)$ be a strong solution to \eqref{RBE1}-\eqref{RBE3} in $\ensuremath{\mathbb{R}}^3\times(T_0, T_L)$. Assume that $Q\in S_\delta$ with sufficiently small $\delta$ on $\ensuremath{\mathbb{R}}^3\times(0, T_L)$ and
\begin{equation}\label{pressure est eq}
\sup_{T_0\leq t\leq T_L,x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_R(x_0)}|\nabla Q_L(x,t)|^3+|v_L(x,t)|^3dx\leq \varepsilon_0^3.
\end{equation}
Then for any $t\in(T_0,T_L)$, there exists a constant $c_L(t)\in \ensuremath{\mathbb{R}}$ such that the pressure $P_L$ satisfies the following estimate
\begin{align*}
&\sup_{x_0\in\ensuremath{\mathbb{R}}^3}\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}|P_L-c_L^\ast|^2\,dxdt
\\
\leq& C\sup_{y\in\ensuremath{\mathbb{R}}^3}\int_{T_0}^{T_L}\int_{B_{R}(y)}(|\nabla^2 Q_L|^2+|\nabla v_L|^2)+\frac{\varepsilon_0^2}{R^2}(|\nabla Q_L|^2+|v_L|^2)\,dxdt.\alabel{pressure estimate.1}
\end{align*}
\end{lemma}
\begin{proof} The proof is essentially the same as the one in \cite{FHM}. For completeness, we outline an approach here. Let $\phi$ be a cut-off function satisfying
$0\leq \phi\leq 1$, $\text{supp } \phi\subset B_{2R}(x_0)$ for some $x_0\in \ensuremath{\mathbb{R}}^3$ and $|\nabla \phi|\leq\frac{C}{R}$. Note that the pressure $P_L$ satisfies
\begin{equation*}
-\Delta P_L=\nabla_{ij}^2\([Q_L,H(Q_L,\nabla Q_L)]_{ij}-\sigma_{ij}(Q_L,\nabla Q_L) +v^i_Lv^j_L\)\quad\text{on}~~\ensuremath{\mathbb{R}}^3\times[T_0,T_L],
\end{equation*}
which implies $P_L=\mathcal{R}_i\mathcal{R}_j(F^{ij}),$ and \[|F^{ij}|=|[Q_L,H(Q_L,\nabla Q_L)]_{ij}-\sigma_{ij}(Q_L,\nabla Q_L) +v^i_Lv^j_L|\leq C(|\nabla^2 Q_L|+|\nabla Q_L|^2+|v_L|^2),\] where $\mathcal{R}_i$ is the $i$-th Riesz transform on $\ensuremath{\mathbb{R}}^3$. Then we have
\begin{equation}\label{pressure estimateP1.1}
(P_L-c_L^\ast)\phi=\mathcal{R}_i\mathcal{R}_j(F^{ij}\phi)+[\phi,\mathcal{R}_i\mathcal{R}_j](F^{ij})-c_L^\ast\phi
\end{equation}
for a cut-off function $\phi$, where the commutator $[\phi,\mathcal{R}_i\mathcal{R}_j]$ is defined by
\[[\phi,\mathcal{R}_i\mathcal{R}_j](\cdot)=\phi\mathcal{R}_i\mathcal{R}_j(\cdot)-\mathcal{R}_i\mathcal{R}_j(\cdot\,\phi).\]
By using the Riesz operator maps $L^q$ into $L^q$
spaces for any $1<q<+\infty$ and the assumption \eqref{pressure est eq}, we have
\begin{align*}
&\int_{T_0}^{T_L}\int_{\ensuremath{\mathbb{R}}^3}|\mathcal{R}_i\mathcal{R}_j(F^{ij}\phi)|^2\,dxdt\\
\leq& C \int_{T_0}^{T_L}\int_{B_{2R(x_0)}}|\nabla^2 Q_L|^2+|\nabla v_L|^2dx\,dt+\frac{C}{R^2}\int_{T_0}^{T_L}\int_{B_{2R(x_0)}}|\nabla
Q_L|^2+|v_L|^2\,dxdt.\alabel{pressure estimateP1.2}
\end{align*}
Since $\text{supp}\, \phi\subset B_{2R(x_0)}$, the commutator can be expressed as
\begin{align*}
&[\phi,\mathcal{R}_i\mathcal{R}_j](F^{ij})(x,t)-c_L^\ast(t)\phi(x)\\
=&\int_{B_{4R}(x_0)}\frac{(\phi(x)-\phi(y))(x_i-y_i)(x_j-y_j)}{|x-y|^5}F^{ij}(y,t)\,dy\\
&\quad+\phi(x)\left[\int_{\ensuremath{\mathbb{R}}^3\backslash B_{4R}(x_0)}\frac{(x_i-y_i)(x_j-y_j)}{|x-y|^5}F^{ij}(y,t)\,dy-c_L(t)\right]\\
=&:f_1(x,t)+f_2(x,t).\alabel{pressure estimateP2.1}
\end{align*}
By using the Hardy-Littlewood-Sobolev inequality (c.f. \cite{HM})
\[\left\| \int_{\ensuremath{\mathbb{R}}^n}\frac{f(y)}{|x-y|^{n-\alpha}}\,dy\right\|_{L^q(\ensuremath{\mathbb{R}}^n)}\leq C\|f\|_{L^r(\ensuremath{\mathbb{R}}^n)},\quad\frac{1}{q}=\frac{1}{r}-\frac{\alpha}{n}\] and the H\"older inequality, a standard covering argument yields
\begin{align*}
&\int_{T_0}^{T_L}\int_{\ensuremath{\mathbb{R}}^3}|f_1(x,s)|^2\,dxdt\leq CR^{-2}\int_{T_0}^{T_L}\|(F^{ij})\chi_{B_{4R(x_0)}}\|_{L^\frac{6}{5}(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
\leq& \frac C{R^2}\int_{T_0}^{T_L}\|(|\nabla Q_L|+|v_L|)\chi_{B_{4R}(x_0)}\|_{L^3(\ensuremath{\mathbb{R}}^3)}^2\|(|\nabla
Q_L|+|v_L|)\chi_{B_{4R}(x_0)}\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
&+ \frac C{R^2}\int_{T_0}^{T_L}\|\chi_{B_{4R}(x_0)}\|^2_{L^3(\ensuremath{\mathbb{R}}^3)}\|(|\nabla^2 Q_L|)\chi_{B_{4R}(x_0)}\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
\leq&\frac{C\varepsilon_0^2}{R^2}\int_{T_0}^{T_L}\int_{B_{4R}(x_0)}|\nabla Q_L|^2+|v_L|^2\,dxdt+C\int_{T_0}^{T_L}\int_{B_{4R}(x_0)}|\nabla^2
Q_L|^2\,dxdt,\alabel{pressure estimateP2.2}
\end{align*}
where $\chi_{B_{4R}(x_0)}(x)=1$ for $x\in B_{4R}(x_0)$ and $0$ for $x\in \ensuremath{\mathbb{R}}^3\backslash B_{4R}(x_0)$. Choosing \[c_L^\ast(t)=\int_{\ensuremath{\mathbb{R}}^3\backslash B_{4R}(x_0)}\frac{(x_{0i}-y_i)(x_{0j}-y_j)}{|x_0-y|^5}F^{ij}(y,t)\,dy\] and using the H\"older inequality, we estimate
\begin{align*}
&\int_{T_0}^{T_L}\int_{\ensuremath{\mathbb{R}}^3}|f_2(z,s)|^2\,dzdt
\\
\leq& CR^{5}\int_{T_0}^{T_L}\left|\sum^\infty_{k=4}\frac{C}{(kR)^4}\int_{B_{(k+1)R(x_0)}\backslash B_{kR(x_0)}}F^{ij}(x,t)\,dx\right|^2
\,dt
\\
\leq& C\sup_{y\in\ensuremath{\mathbb{R}}^3}\int_{T_0}^{T_L}\sum^\infty_{k=4}k^{-4}\int_{B_{R(y)}}|F^{ij}|^2\,dxdt
\\
\leq& C \sup_{y\in\ensuremath{\mathbb{R}}^3}\int_{T_0}^{T_L}\int_{B_{R}(y)}\frac{\varepsilon_0^2}{R^2}(|\nabla Q_L|^2+|v_L|^2)+(|\nabla^2 Q_L|^2+|\nabla v_L|^2)\,dxdt
.\alabel{pressure estimateP2.3}
\end{align*}
Combining \eqref{pressure estimateP1.2}, \eqref{pressure estimateP2.2} with \eqref{pressure estimateP2.3}, we can apply a standard covering argument to complete the proof.
\end{proof}
Using Lemma \ref{lem 1ord} and Lemma \ref{lem 2ord}, we have:
\begin{lemma}\label{L3 small}
Let $(Q_L,v_L)$ be a strong solution of $\eqref{RBE1}-\eqref{RBE3}$ in $\ensuremath{\mathbb{R}}^3 \times [T_0,T_L)$ with initial value $(Q_{L,T_0},v_{L,T_0})\in H^2_{Q_e}(\ensuremath{\mathbb{R}}^3;S_0)\times H^1(\ensuremath{\mathbb{R}}^3;\ensuremath{\mathbb{R}}^3)$ and $\div v=0$. Assume that $Q\in S_\delta$ for sufficiently small $\delta$ on $\ensuremath{\mathbb{R}}^3\times(0, T_L)$. There exist two constants $\varepsilon_0$ and $R$ that
\begin{align}\label{L3 small cond}
\sup_{T_0\leq t\leq T_L,x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(x_0)}|\nabla Q_L|^3+|v_L|^3+\frac{|Q_L-\pi(Q_L)|^3}{L^{\frac32}}\,dx\leq\varepsilon_0^3.
\end{align}
Then we have
\begin{align*}
&\sup_{T_0\leq s\leq T_L,x_0\in\ensuremath{\mathbb{R}}^3}\frac{1}{R}\int_{B_{R}(x_0)}\(|\nabla Q_L |^2+ |v_L |^2+\frac{|Q_{L}-\pi(Q_{L})|^2}{L}\)(x,s)\,dx
\\
&+\sup_{x_0\in\ensuremath{\mathbb{R}}^3}\frac{1}{R}\int^{T_L}_{T_0}\int_{B_{R}(x_0)}|\nabla^2 Q_L|^2+|\nabla v_L|^2 +|\partial_t Q_L|^2 +\frac{|\nabla(Q_L-\pi(Q_L))|^2}{L}\,dxdt
\\
\leq& \frac{C}{R}\sup_{x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(x_0)}|\nabla Q_{L,T_0}|^2+| v_{L,T_0}|^2+\frac{|Q_{L,T_0}-\pi(Q_{L,T_0})|^2}{L} \,dx
+C\varepsilon_0^2\frac{(T_L-T_0)}{R^2}
\alabel{L3 small 1st order}
\end{align*}
and
\begin{align*}
&\sup_{T_0\leq s\leq T_L,x_0\in\ensuremath{\mathbb{R}}^3}R\int_{B_{R}(x_0)}\(|\nabla^2 Q_L|^2+|\nabla v_L|^2+\frac{|\nabla(Q_L-\pi(Q_L))|^2}{L} \)(x,s)\,dx
\\
&+\sup_{x_0\in\ensuremath{\mathbb{R}}^3}R\int^{T_L}_{T_0}\int_{B_{R}(x_0)}|\nabla^3Q_L|^2+|\nabla^2v_L|^2+|\nabla\partial_t Q_L|^2 +\frac{|\nabla^2(Q_L-\pi(Q_L))|^2}{L}\,dxdt
\\
\leq& CR\sup_{x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(x_0)} |\nabla^2 Q_{L,T_0}|^2+ |\nabla v_{L,T_0}|^2+\frac{|\nabla (Q_{L,T_0}-\pi(Q_{L,T_0}))|^2}{L} \,dxdt
\\
&+\frac{C}{R}\sup_{x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(x_0)}|\nabla Q_{L,T_0}|^2+| v_{L,T_0}|^2+\frac{|Q_{L,T_0}-\pi(Q_{L,T_0})|^2}{L} \,dx
+C\varepsilon_0^2\frac{(T_L-T_0)}{R^2}
.\alabel{L3 small 2nd ord}
\end{align*}
\end{lemma}
\begin{proof} Let $\{B_R(x_i)\}^{\infty}_{i=1}$ be a standard open cover of $\ensuremath{\mathbb{R}}^3$ such that at each $x\in\ensuremath{\mathbb{R}}^3$, there are finite intersections of open balls $B_{R}(x_i)$.
Let $\phi\in C_0^\infty(B_{2R}(x_0))$ with $\phi\equiv 1$ on $B_{R}(x_0)$, $|\nabla \phi|\leq \frac{C}{R}$ and $|\nabla^2 \phi|\leq \frac{C}{R^2}$. Recall from Lemma \ref{lem 1ord} that
\begin{align*}
&\frac1R\int_{B_{R}(x_0)}\(|\nabla Q_L|^2+ |v_L|^2+\frac{|Q_L -\pi(Q_L)|^2}{L}\)(x,s) \,dx
\\
&+\frac1R\int_0^{s }\int_{B_{R}(x_0)}|\nabla^2 Q_L|^2+|\partial_t Q_L|^2+|\nabla v_L|^2+\frac{|\nabla (Q_L -\pi(Q_L ))|^2}{L}\,dxdt
\\
\leq& \frac{C}{R}\int_{B_{2R}(x_0)}|\nabla Q_{L,T_0}|^2+ |v_{L,T_0}|^2+\frac{|Q_{L,T_0}-\pi(Q_{L,T_0})|^2}{L}\,dx
\\
&+ \frac{C}{R}\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}|\nabla Q_L|^2\(|\nabla Q_L|^2+|v_L|^2+\frac{|Q_L -\pi(Q_L)|^2}{L}\)\,dxdt
\\
&+\frac{\eta}{R}\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}|P_L-c_L^*(t)|^2\,dxdt+\frac{C(\eta)}{R^3}\int_{T_0}^{s}\int_{B_{2R}(x_0)}|\nabla Q_L|^2+ |v_L|^2 \,dxdt
\alabel{1ord eq L3}
\end{align*}
for some small $\eta$ to be chosen later. Using H\"older's inequality and \eqref{L3 small cond}, we have
\begin{align*}
&\sup_{T_0\leq s\leq T_L,x_0\in\ensuremath{\mathbb{R}}^3}\frac1R\int^{s}_{T_0}\int_{B_{2R}(x_0)}\(|\nabla Q_L|^2+|v_L|^2 +\frac{|Q_L -\pi(Q_L)|^2}{L}\)(x,s)\,dxdt
\\
&\leq C\varepsilon_0^2(T_L-T_0).\alabel{L3 small L2}
\end{align*}
Then, using the Sobolev inequality, \eqref{L3 small cond} and \eqref{L3 small L2}, we find
\begin{align*}
&\sup_{ x_0\in\ensuremath{\mathbb{R}}^3}\frac1R\int^{T_L}_{T_0}\int_{B_{2R}(x_0)}|\nabla Q_L|^2|\nabla Q_L|^2\,dxdt
\\
\leq&\frac{C}{R} \sup_{ x_0\in\ensuremath{\mathbb{R}}^3}\int_{T_0}^{T_L}\sum_{i}\(\int_{B_R(x_i)}|\nabla Q_L|^3 \,dx\)^{\frac23}\(\int_{B_R(x_i)}|\nabla Q_L|^6\,dx\)^{\frac13}\,dt
\\
\leq& \frac{C\varepsilon_0^2}{R}\sup_{ y\in\ensuremath{\mathbb{R}}^3}\int^{T_L}_{T_0}\int_{B_{R}(y)}|\nabla^2 Q_L|^2\,dxdt+\frac{C}{R}\sup_{T_0\leq s\leq T_L,y\in\ensuremath{\mathbb{R}}^3}\int^{T_L}_{T_0} \int_{B_R(y)}|\nabla Q_L|^2\,dxdt
\\
\leq& \frac{C\varepsilon_0^2}{R}\sup_{ y\in\ensuremath{\mathbb{R}}^3}\int^{T_L}_{T_0}\int_{B_{R}(y)}|\nabla^2 Q_L|^2\,dxdt+C\varepsilon_0^2\frac{T_L-T_0}{R^2}
.\alabel{L3}
\end{align*}
Similarly, we obtain
\begin{align*}
&\sup_{ x_0\in\ensuremath{\mathbb{R}}^3}\frac1R\int^{T_L}_{T_0}\int_{B_{2R}(x_0)}|\nabla Q_L|^2\(|v_L|^2+\frac{|Q_L -\pi(Q_L)|^2}{L}\)\,dxdt
\\
\leq& \frac{C\varepsilon_0^2}{R}\sup_{ y\in\ensuremath{\mathbb{R}}^3}\int^{T_L}_{T_0}\int_{B_{R}(y)}|\nabla v_L|^2+\frac{|\nabla(Q_L-\pi(Q_L))|^2}{L}\,dxdt+C\varepsilon_0^2\frac{T_L-T_0}{R^2}
.\alabel{L3 small L4 part 1}
\end{align*}
Substituting \eqref{L3 small L2}-\eqref{L3 small L4 part 1} into \eqref{1ord eq L3}, using Lemma \ref{lem pressure estimate} and taking the supremum of $x_0\in \ensuremath{\mathbb{R}}^3$, we prove \eqref{L3 small 1st order} by choosing $\eta$ sufficiently small.
To show \eqref{L3 small 2nd ord}, recall from Lemma \ref{lem 2ord} that
\begin{align*}
&R\int_{B_R(x_0)}\(|\nabla^2 Q_L|^2+ |\nabla v_L |^2+\frac{|\nabla (Q_L -\pi(Q_L ))|^2}{L}\)(x,T_L)\,dx
\\
&+R\int_{T_0}^{T_L}\int_{B_R(x_0)}|\nabla^3 Q_L|^2+|\nabla\partial_t Q_L|^2+|\nabla^2 v_L|^2+\frac{|\nabla^2 (Q_L -\pi(Q_L ))|^2}{L}\,dxdt
\\
\leq& CR\int_{B_{2R}(x_0)} |\nabla^2 Q_{L,T_0}|^2+ |\nabla v_{L,T_0}|^2+\frac{|\nabla (Q_{L,T_0}-\pi(Q_{L,T_0}))|^2}{L} \,dx
\\
&+CR\int_{B_{2R}(x_0)} \frac{|Q_{L,0}-\pi(Q_{L,0})|^2}{L}|\nabla Q_{L,0}|^2+ \(\frac{|Q-\pi(Q)|^2}{L}|\nabla Q|^2\)(x,T_L)\,dx
\\
&+CR\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}e(Q_L,v_L)\(|\nabla^2 Q_L|^2+|\nabla v_L|^2+|\partial_tQ_L|^2\)\,dxdt
\\
&+CR\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}e(Q_L,v_L)\(\frac{|\nabla(Q_L-\pi(Q_L))|^2}{L}+e^2(Q_L,v_L)\)\,dxdt
\\
&+\frac{C}{R}\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}|\nabla^2 Q_L|^2+|\nabla v_L|^2+|\partial_tQ_L|^2+\frac{| \nabla (Q-\pi(Q))|^2}{L}\,dxdt
\\
&+\frac{C}{R}\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}e^2(Q_L,v_L)+|P_L-c_L^*(t)|^2\,dxdt
\\
\leq& CR\int_{B_{2R}(x_0)} |\nabla^2 Q_{L,T_0}|^2+ |\nabla v_{L,T_0}|^2+\frac{|\nabla (Q_{L,T_0}-\pi(Q_{L,T_0}))|^2}{L} \,dx
\\
&+C\varepsilon_0^2R\sup_{ y\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(y)}|\nabla^2 Q_{L,T_0}|^2 +|\nabla^2 Q_{L}(x,T_L)|^2 \,dx
\\
&+\frac{C}{R}\sup_{y\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(y)}|\nabla Q_{L,T_0}|^2+| v_{L,T_0}|^2+\frac{|Q_{L,T_0}-\pi(Q_{L,T_0})|^2}{L} \,dx
\\
&+C\varepsilon_0^2R\sup_{ y\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(y)}\int_{T_0}^{T_L}|\nabla^3 Q_L|^2+|\nabla^2 v_L|^2+|\nabla\partial_tQ_L|^2+\frac{|\nabla^2(Q_L-\pi(Q_L))|^2}{L}\,dxdt
\\
&+\varepsilon_0^2\frac{C(T_L-T_0)}{R^2}+CR\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}e^3(Q_L,v_L)\,dxdt
.\alabel{2ord eq L3}
\end{align*}
Here we used the argument in \eqref{L3}, Lemma \ref{lem pressure estimate} and substituted \eqref{L3 small 1st order}. Applying the argument in \eqref{L3} twice, we deduce the last term in \eqref{2ord eq L3} to
\begin{align*}
&R\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}e^3(Q_L,v_L)\,dxdt
\\
\leq&CR\int_{T_0}^{T_L}\int_{B_{2R}(x_0)}e(Q_L,v_L)\(|\nabla Q_L|^4+|v_L|^4+\frac{|Q_L -\pi(Q_L)|^4}{L^2}\)\,dxdt
\\
\
\leq&C\varepsilon_0^2R\sup_{ y\in\ensuremath{\mathbb{R}}^3}\int^{T_L}_{T_0}\int_{B_{R}(y)} |\nabla|\nabla Q_L|^2|^2+|\nabla|v_L|^2|^2+\frac{|\nabla|Q_L -\pi(Q_L)|^2|^2}{L^2}\,dxdt
\\
&+\frac{C\varepsilon_0^2}{R}\sup_{ y\in\ensuremath{\mathbb{R}}^3}\int^{T_L}_{T_0}\int_{B_{R}(y)} |\nabla Q_L|^4+|v_L|^4+\frac{|Q_L -\pi(Q_L)|^4}{L^2}\,dxdt
\\
\leq& C\varepsilon_0^2R\sup_{y\in\ensuremath{\mathbb{R}}^3}\int^{T_L}_{T_0}\int_{B_{R}(y)}|\nabla^3 Q_L|^2+|\nabla^2 v_L|^2+\frac{|\nabla^2 (Q_L -\pi(Q_L ))|^2}{L} \,dxdt
\\
&+\frac{C}{R}\sup_{x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(x_0)}|\nabla Q_{L,T_0}|^2+| v_{L,T_0}|^2+\frac{|Q_{L,T_0}-\pi(Q_{L,T_0})|^2}{L}\,dx+\varepsilon_0^2\frac{C(T_L-T_0)}{R^2}
.\alabel{L3 small L4 part 3}
\end{align*}
Here we also used \eqref{L3 small 1st order}. Substituting \eqref{L3 small L4 part 3} into \eqref{2ord eq L3} and then taking the supremum of $x_0\in \ensuremath{\mathbb{R}}^3$ on the resulting expression, we obtain \eqref{L3 small 2nd ord}.
\end{proof}
Using the Gagliardo-Nirenberg interpolation, we establish a uniform local existence of the strong solutions:
\begin{prop}\label{prop Extension}
Assume that $(Q_{L,T_0}, v_{L,T_0})$ satisfies
\begin{align}\label{prop eq}
\|Q_{L,T_0}\|^2_{H^1_{Q_e}(\ensuremath{\mathbb{R}}^3)}+\|v_{L,T_0}\|^2_{H^1(\ensuremath{\mathbb{R}}^3)}+
\frac {\|Q_{L,T_0}-\pi(Q_{L,T_0})\|^2_{H^1(\ensuremath{\mathbb{R}}^3)}}{L}\leq M
\end{align}
for some $M>0$. Then there are uniform constants $T_M,R_0$ in $L$ such that the system \eqref{RBE1}-\eqref{RBE3} with initial data $(Q_{L,T_0},v_{L,T_0})$ has
a unique strong solution $(Q_L,v_L)$ in $\ensuremath{\mathbb{R}}^3\times[T_0,T_M]$ satisfying
\begin{align}\alabel{prop L3 small}
\sup_{T_0\leq t\leq T_M,x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R_0}(x_0)}|\nabla Q_L|^3+|v_L|^3+\frac{|Q_L-\pi(Q_L)|^3}{L^{\frac32}}\,dx\leq\frac{\varepsilon_0^3}{2}
\end{align}
and
\begin{align*}
&\sup_{T_0\leq s\leq T_M}\left(\|\nabla Q_L(s)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\|v_L(s)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\frac 1 L\| Q_L(s)-\pi(Q_L(s))\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2\right)
\\
&+\|\partial_t Q_L\|_{L^2(T_0,T_M;H^1(\ensuremath{\mathbb{R}}^3))}^2+\|\nabla^2 Q_L\|_{L^2(T_0,T_M;H^1(\ensuremath{\mathbb{R}}^3))}^2
\\
&+\|\nabla v_L\|_{L^2(T_0,T_M;H^1(\ensuremath{\mathbb{R}}^3))}^2 +\frac 1 L\|\nabla( Q_L-\pi(Q_L))\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2\leq C\(1+\frac{\varepsilon_0^2}{R_0^2}\)M.\alabel{L3 small global}
\end{align*}
\end{prop}
\begin{proof}
It follows from the Sobolev embedding theorem that for any $0<\varepsilon_0<1$, there exists a positive constant
$R_0=:\frac{\varepsilon_0^2}{C_s^2N^2M}$ (c.f. \cite{FHM}) such that
\begin{align*}
&\sup_{x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R_0}(x_0)}|\nabla Q_{L,T_0}|^3+|v_{L,T_0}|^3+\frac{|Q_{L,T_0}-\pi(Q_{L,T_0})|^{3}}{L^\frac32} \,dx\leq \frac{\varepsilon_0^3}{N^3},\alabel{prop p1}
\end{align*}
where $N>1$ is an absolute constant independent of $L$ and $M$ to be chosen later. By using the Gagliardo–Nirenberg interpolation (c.f. \cite{FHM}) at $T_0$, we have
\begin{align*}
&\dist(Q_L(T_0);S_*)=\|\dist(Q_L(T_0);S_*)\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}
\\
\leq& C\|Q_L(T_0)-\pi(Q_L(T_0))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^{\frac14}\|\nabla^2(Q_L(T_0)-\pi(Q_L(T_0)))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^{\frac34}
\\
\leq& C(LM)^{\frac18}\Big(\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2Q_L(T_0)|^2 +|\partial_Q \pi|^2|\nabla^2Q_L(T_0)|^2+|\partial^2_{QQ} \pi|^2|\nabla Q_L(T_0)|^2\,dx\Big)^{\frac38}
\\
\leq& C_dL^{\frac18}M^{\frac14}\leq \frac{\delta}{2},
\end{align*}
where we have used the condition \eqref{prop eq} and chosen $L\leq\(\frac{\delta}{2C_dM^{\frac14}}\)^8$.
Using Theorem \ref{thm loc}, there is a unique local strong solution $(Q_L,v_L)$ such that $(Q_L,v_L)$ is continuous in $t$, which follows from the Sobolev inequality (c.f. \cite{HLX}). Then there is a time $T_L^*\in(T_0,T_L]$ such that
\begin{align}\label{prop dist 2}
\dist(Q_L;S_*)\leq \delta \text{ on }\ensuremath{\mathbb{R}}^3\times (T_0,T_L^*).
\end{align} and
\begin{align}\label{prop L3 small 2}
\sup_{T_0\leq t\leq T_L^*,x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R_0}(x_0)}|\nabla Q_L|^3+|v_L|^3+\frac{|Q_L-\pi(Q_L)|^3}{L^{\frac32}}\,dx\leq{\varepsilon_0^3}.
\end{align}
Using \eqref{prop eq}, \eqref{prop p1}, $R_0=\frac{\varepsilon_0^2}{C_s^2N^2M}$ and choosing $(T_L^*-T_0)\leq\sigma R_0^2$ for some small $\sigma$ to be chosen later, we have
\begin{align*}
&CR_0\sup_{x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R_0}(x_0)} |\nabla^2 Q_{L,T_0}|^2+ |\nabla v_{L,T_0}|^2+\frac{|\nabla (Q_{L,T_0}-\pi(Q_{L,T_0}))|^2}{L} \,dx
\\
&+\frac{C}{R_0}\sup_{x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R}(x_0)}|\nabla Q_{L,T_0}|^2+| v_{L,T_0}|^2+\frac{|Q_{L,T_0}-\pi(Q_{L,T_0})|^2}{L} \,dx+\varepsilon_0^2\frac{C(T_L-T_0)}{R_0^2}
\\
\leq& CMR_0+\frac{C|B_{R_0}|^{\frac 13}}{R_0}\frac{\varepsilon_0^2}{N^2}+C\varepsilon_0^2\sigma\leq \frac{C\varepsilon_0^2}{N^2}+C\varepsilon_0^2\sigma
.\alabel{GN}
\end{align*}
By using the Gagliardo-Nirenberg interpolation and applying \eqref{GN} to \eqref{L3 small 1st order}-\eqref{L3 small 2nd ord}, we obtain
\begin{align*}
&\sup_{T_0\leq t\leq T^*_L,x_0\in\ensuremath{\mathbb{R}}^3}\int_{B_{R_0}(x_0)} |\nabla Q_L|^3+|v_L|^3+\frac{|Q_L-\pi(Q_L)|^3}{L^{\frac32}}\,dx
\\
\leq& C\sup_{T_0\leq t\leq T^*_L,x_0\in\ensuremath{\mathbb{R}}^3}\left(\frac{1}{R_0}\int_{B_{R_0}(x_0)}|\nabla Q_L|^2+|v_L|^2+\frac{|Q_L-\pi(Q_L)|^2}{L}\,dx\right)^{3/2}
\\
&+C\sup_{T_0\leq t\leq T^*_L,x_0\in\ensuremath{\mathbb{R}}^3}\left(R_0\int_{B_{R_0}(x_0)}|\nabla^2 Q_L|^2+|\nabla v_L|^2+\frac{|\nabla(Q_L-\pi(Q_L))|^2}{L}\,dx\right)^{3/2}
\\
\leq& \left(\frac{C_1\varepsilon_0^2}{N^2}+C_2\sigma\varepsilon_0^2 \right)^{3/2}\leq\frac{\varepsilon_0^3}{2},
\end{align*}
where we choose $N\geq 2(C_1+1)^{\frac12}$ and $\sigma \leq \min\{\frac{2}{C_2},1\}$. Thus we prove \eqref{prop L3 small 2} up to the uniform time $T_M =T_0+\sigma R_0^2$.
For \eqref{L3 small global}, using a standard open cover $\{B_R(x_i)\}_{i=1}^\infty$ for $\ensuremath{\mathbb{R}}^3$ with finite intersections at each $x\in \ensuremath{\mathbb{R}}^3$, the H\"older inequality and the Sobolev inequality, we find
\begin{align*}
&\int_{T_0}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_L|^4\,dxdt \leq\int_{T_0}^{T_M}\sum_{i=1}^{\infty}\(\int_{B_{R_0}(x_i)}|\nabla Q_L|^3\,dx\)^{\frac23}\(\int_{B_{R_0}(x_i)}|\nabla Q_L|^6\,dx\)^{\frac13}dt
\\
\leq& C\varepsilon_0^2\int_{T_0}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_L|^2\,dxdt+C\varepsilon_0^2\frac{T_M-T_0}{R^2}\sup_{T_0\leq s\leq T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_L(x,s)|^2\,dx
\\
\leq& \frac12\int_{T_0}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_L|^2\,dxdt+\frac12\sup_{T_0\leq s\leq T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_L(x,s)|^2\,dx.\alabel{L3 small L4 prop}
\end{align*}
Similarly, we obtain
\begin{align*}
&\int_{T_0}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_L|^2\(|\nabla Q_L|^2+|v_L|^2+\frac{|Q_L-\pi(Q_L)|^2}L\)\,dxdt
\\
\leq& \frac12\int_{T_0}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_L|^2+|\nabla v_L|^2 +\frac{|\nabla(Q_L-\pi(Q_L))|^2}L\,dxdt
\\
&+\frac12\sup_{T_0\leq s\leq T_M}\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla Q_L|^2+|v_L|^2+\frac{|Q_L-\pi(Q_L)|^2}L\)(x,s)\,dx
.\alabel{L3 small L4 prop 2}
\end{align*}
Choosing $\phi\equiv 1$ in Lemma \ref{lem 1ord}, using \eqref{prop eq} and \eqref{L3 small L4 prop 2}, we have
\begin{align*}
&\sup_{T_0\leq s\leq T_M}\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla Q_L|^2+ |v_L|^2+\frac{|Q_L-\pi(Q_L)|^2}{L}\)(x,s)\,dx
\\
&+\int_{T_0}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_L|^2+|\nabla v_L|^2+|\partial_t Q_L|^2+\frac{|\nabla (Q_L-\pi(Q_L))|^2}{L}\,dxdt
\leq CM.\alabel{prop R3 1}
\end{align*}
Using the method in \eqref{L3 small L4 prop 2} and substituting \eqref{prop R3 1} to Lemma \ref{lem 2ord} with $\phi\equiv 1$, we find
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla^2 Q_L|^2+ |\nabla v_L|^2+\frac{|\nabla (Q_L-\pi(Q_L))|^2}{L}\)(x,s)\,dx
\\
&+\frac 12\int_{T_0}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^3 Q_L|^2+|\nabla^2 v_L|^2+|\nabla\partial_t Q_L|^2+\frac{|\nabla^2 (Q_L-\pi(Q_L))|^2}{L}\,dxdt
\\
\leq&CM+C\varepsilon_0^2\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 Q_L(x,T_M)|^2\,dx+\frac{C\varepsilon_0^2}{R_0^2}\int_{\ensuremath{\mathbb{R}}^3} |\nabla Q_L(x,T_M)|^2+|\nabla Q_{L,0}|^2\,dx
\\
&\frac{C\varepsilon_0^2}{R_0^2}\int_{T_0}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_L|^2+|\nabla v_L|^2+|\partial_t Q_L|^2+\frac{|\nabla (Q_L-\pi(Q_L))|^2}{L}\,dxdt
\\
&+\frac{C\varepsilon_0^2}{R_0^2}\int_{T_0}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_L|^4+|v_L|^4+\frac{|Q_L-\pi(Q_L)|^4}{L^2}\,dxdt
\leq C\(1+\frac{\varepsilon_0^2}{R_0^2}\)M
.\alabel{prop R3 2}
\end{align*}
Combining \eqref{prop R3 1} with \eqref{prop R3 2}, we prove \eqref{L3 small global}.
\end{proof}
\begin{proof}[\bf Proof of Theorem \ref{thm1}]
By using Proposition \ref{prop Extension} and Lemma \ref{L3 small}, there exist two uniform positive constants $T_1$
and $L_\ast$ such that for any $L\leq L_\ast$, the strong solution $(Q_L,v_L)$ to \eqref{RBE1}-\eqref{RBE3} satisfies
\begin{align*}
&\sup_{0\leq t\leq T_1}\left(\|\nabla Q_L(t)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\|v_L(t)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\frac 1L\| Q_L(t)-\pi(Q_L(t))\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2 \right)
\\
&+\|\partial_t Q_L\|_{L^2(0,T_1;H^1(\ensuremath{\mathbb{R}}^3))}^2+\|\nabla^2 Q_L\|_{L^2(0,T_1;H^1(\ensuremath{\mathbb{R}}^3))}^2\\
&+\|\nabla v_L\|_{L^2(0,T_1;H^1(\ensuremath{\mathbb{R}}^3))}^2+\frac1L \| Q_L-\pi(Q_L)\|_{H^2(\ensuremath{\mathbb{R}}^3)}^2 \leq C\(1+\frac{\varepsilon_0^2}{R^2}\)M
.\alabel{L3 small global p2}
\end{align*}
Since the pressure $P_L$ satisfies
\eqref{pressure estimateP1.1}, using \eqref{L3 small global p2}, we find
\begin{align*}
\int_{0}^{T_1}\int_{\ensuremath{\mathbb{R}}^3}|P_L|^2\,dxdt\leq \int_{0}^{T_1}\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_L|^4+|v_L|^4+|\nabla^2 Q_L|^2\,dxdt\leq C
\alabel{eq P}
\end{align*}
and
\begin{align*}
&\int_{0}^{T_1}\int_{\ensuremath{\mathbb{R}}^3}|\nabla P_L|^2\,dxdt
\\
\leq& C\int_{0}^{T_1}\int_{\ensuremath{\mathbb{R}}^3}\left(|\nabla[Q_L,\mathcal{H}(Q_L,\nabla Q_L)]|^2+|\nabla\sigma(Q_L,\nabla Q_L) |^2+|\nabla(v_L\otimes v_L)|^2\right)\,dxdt
\\
\leq& C\int_{0}^{T_1}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^3 Q_L|^2+|\nabla^2 Q_L|^2|\nabla Q_L|^2+|\nabla Q_L|^6+|\nabla v_L|^2|v_L|^2\,dxdt\leq C
.\alabel{eq P2}
\end{align*}
Multiplying \eqref{RBE1} with $(Q_L-Q_e)$, one can show that $(Q_L-Q_e)\in L^\infty(0,T_1; L^2(\ensuremath{\mathbb{R}}^3))$.
Then, letting $L\to0$ (up to a subsequence) and utilizing the Aubin-Lions Lemma, we have
\begin{align*}
Q_L\rightharpoonup& Q\text{ in } L^2(0,T_1; H^3_{Q_e}(\ensuremath{\mathbb{R}}^3))\cap H^1(0,T_1; H^2_{Q_e}(\ensuremath{\mathbb{R}}^3)),
\\
\partial_t Q_L\rightharpoonup& \partial_t Q\text{ in }L^2(0,T_1;H^1(\ensuremath{\mathbb{R}}^3)),
\\
v_L\rightharpoonup& v\text{ in } L^2(0,T_1; H^2(\ensuremath{\mathbb{R}}^3))\cap H^1(\ensuremath{\mathbb{R}}^3\times(0,T_1)),
\\
\partial_tv_L\rightharpoonup& \partial_tv\text{ in }L^2(0,T_1;L^2(\ensuremath{\mathbb{R}}^3)),
\\
P_L\rightharpoonup& P \text{ in }L^2(0,T_1;H^1(\ensuremath{\mathbb{R}}^3)),
\\
(\nabla Q_L,v_L)\rightarrow& (Q,v)\text{ in }L^2(0,T_1;H^1(B_R(0)))\cap C([0,T_1];L^2(B_R(0)))
\end{align*}
for any $R\in(0,\infty)$. We claim
\begin{align} \label{eq fB}
\lim_{L\to 0}\frac 1 L \int_{\ensuremath{\mathbb{R}}^3} f_B(Q_L(x,s)) \,dx=0
\end{align}
for all $s\in [0,T_1]$. To prove this claim, we need to estimate the $L^2$-norm of $\partial_t Q_L$. We differentiate \eqref{ROT RBE} in $t$, multiply by
$ R^T_{Q_L}\partial_tQ_LR_{Q_L}$. To estimate the right-hand side term on $g_B(Q_L)$, we apply Lemma \ref{lem f_B 1st} with $\xi=\partial_t Q_L$ and Lemma \ref{lemgb} to obtain
\begin{align*}
&\int^s_0\int_{\ensuremath{\mathbb{R}}^3}\< \partial_t\frac{g_B(\tilde Q_L)}{L} ,R^T_{Q_L}\partial_tQ_LR_{Q_L}\>\,dxdt
\\
\leq&- \frac1L \int^s_0\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{\tilde Q_{ij}\tilde Q_{kl}}f_B(\tilde Q_L)\partial_t(\tilde Q_L)_{ij}\partial_t(\tilde Q_L)_{kl}\,dxdt
\\
&+ C \int^s_0\int_{\ensuremath{\mathbb{R}}^3}\frac{|\partial_t \tilde Q_L|}{L^\frac12}|\partial_t R_{Q_L}\frac{| Q_L-\pi(Q_L)|}{L^\frac12}\,dxdt
\\
\leq& -\frac{\lambda}{4}\int^s_0\int_{\ensuremath{\mathbb{R}}^3}\frac{|\partial_t\tilde Q_L|^2}L\,dxdt+C\int_{\ensuremath{\mathbb{R}}^3}\frac{| Q_L-\pi(Q_L)|^2}{L}|\partial_t Q|^2\,dx\leq CM
.\alabel{gB tilde 3}
\end{align*}
Integrating by part and using \eqref{prop L3 small}, \eqref {L3 small global p2} and Young's inequality, we have
\begin{align*}
&\int^s_0\int_{\ensuremath{\mathbb{R}}^3}\<\partial_t( R^T_{Q_L}\mathcal{H}(Q_L,\nabla Q_L) R_{Q_L}),\, R^T_{Q_L}\partial_tQ_LR_{Q_L}\>\,dxdt\\
=&\int^s_0\int_{\ensuremath{\mathbb{R}}^3}\<\partial_t \nabla_k[\partial_{p_{ij}^k} f_E(Q_L,\nabla Q_L) - \partial_{Q_{ij}} f_E(Q_L,\nabla Q_L)] , \partial_tQ_L \>\,dxdt\\
+&\int^s_0\int_{\ensuremath{\mathbb{R}}^3}\<\partial_t R^T_{Q_L} \nabla_k[\partial_{p_{ij}^k} f_E(Q_L,\nabla Q_L) - \partial_{Q_{ij}} f_E(Q_L,\nabla Q_L)],\, \partial_tQ_LR_{Q_L} \>\,dxdt\\
+&\int^s_0\int_{\ensuremath{\mathbb{R}}^3}\< \nabla_k[\partial_{p_{ij}^k} f_E(Q_L,\nabla Q_L) - \partial_{Q_{ij}} f_E(Q_L,\nabla Q_L)] \partial_t R_{Q_L},\, R^T{Q_L} \partial_tQ_L \>\,dxdt\\
\leq&C\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\partial_t Q_L|^2 |\nabla Q_L|^2+ |\nabla\partial_t Q_L|^2\,dxdt
\leq CM
.\alabel{gB tilde 4}
\end{align*}
Similarly, for the left-hand side, we find
\begin{align*}
&\int^{s}_0\int_{\ensuremath{\mathbb{R}}^3}\<\partial_t(R^T_{Q_L}\big(\partial_t Q_L+(v\cdot \nabla Q_L) +[Q_L, \Omega_L]\big)R_{Q_L}),R^T_{Q_L}\partial_tQ_LR_{Q_L}\>\,dxdt
\\
\geq&\int^{s}_0\int_{\ensuremath{\mathbb{R}}^3}\frac12\partial_t|R^T_{Q_L}\partial_t Q_LR_{Q_L}|^2 \,dxdt -C \int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla \partial_t Q_L|^2+|\partial_tv_L|^2\,dxdt
\\
&- C \int^{s}_0\int_{\ensuremath{\mathbb{R}}^3}|\partial_t Q_L|^2\(|\nabla Q_L|^2 +|v_L|^2\) \,dxdt
\\
\geq&\frac12\int_{\ensuremath{\mathbb{R}}^3} | \partial_t Q_L(x,s) |^2 \,dx-\frac12\int_{\ensuremath{\mathbb{R}}^3} | \partial_t Q_0|^2 \,dx-CM. \alabel{gB tilde 5}
\end{align*}
Here we used $\int^s_0\int_{\ensuremath{\mathbb{R}}^3} |\partial_tv_L|^2\,dxdt\leq CM$ due to \eqref{RBE1}.
In view of \eqref{gB tilde 3}-\eqref{gB tilde 5}, we derive
\begin{align*}
\sup_{0\leq s\leq T_1} \int_{\ensuremath{\mathbb{R}}^3} | \partial_t Q_L(x,s)|^2 \,dx\leq & \int_{\ensuremath{\mathbb{R}}^3} | \partial_t Q_L(x,0)|^2 \,dx
+CM
\leq CM,
\alabel{pt Q}
\end{align*}
where we used \eqref{RBE3}, \eqref{prop L3 small}, \eqref {L3 small global p2}, \eqref{eq P2} and $g_B(Q_0)=0$ for $Q_0\in S_*$.
Using \eqref{RBE3}, \eqref{pt Q} and \eqref {L3 small global p2}, we have
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\frac{|g_B(\tilde Q_L)|^2}{L^2}\,dx=\int_{\ensuremath{\mathbb{R}}^3}\frac{|g_B( Q_L)|^2}{L^2}\,dx
\\
\leq&C \int_{\ensuremath{\mathbb{R}}^3}|\partial_t Q_L|^2+|v_L|^2|\nabla Q_L|^2+|Q_L|^2|\nabla v_L|^2+|\nabla^2 Q_L|^2+|\nabla Q_L|^4\,dx
\leq CM
.\alabel{gB tilde 2}
\end{align*}
For any $Q_L\in S_\delta$ with a sufficiently small $\delta >0$, we take the Taylor expansion and adapt the argument in \eqref{f_B p1.2} with Corollary \ref{cor fB} to obtain
\begin{align*}
0= f_B(Q^+)=& f_B(\tilde Q_L )+\nabla_{\tilde Q_{ij}}f_B(\tilde Q_L)(\tilde Q_L-Q^+ )_{ij}
\\
&+\partial^2_{\tilde Q_{ij} \tilde Q_{kl}} f_B(Q_\tau)(\tilde Q_L-Q^+ )_{ij} (\tilde Q_L-Q^+ )_{kl}\\
\geq& f_B(\tilde Q_L )-g_B(\tilde Q_L)_{ij}(\tilde Q_L-Q^+ )_{ij
},\alabel{f_B Taylor p2}
\end{align*}
where $Q_\tau$ is an intermediate point between $\tilde Q_L $ and $Q^+$. Then using \eqref{f_B Taylor p2} and \eqref{gB tilde 2}, we find
\begin{align*}
&\frac 1L \int_{\ensuremath{\mathbb{R}}^3} f_B(Q_L)\,dx= \frac 1L \int_{\ensuremath{\mathbb{R}}^3} f_B(\tilde Q_L)\,dx \leq \frac 1L \int_{\ensuremath{\mathbb{R}}^3}\<g_B(\tilde Q_L ), Q^+- \tilde Q_L\> \,dx
\\ \leq&\( \int_{\ensuremath{\mathbb{R}}^3}\frac{|g_B(\tilde Q_L)|^2}{L^2}\,dx \)^{\frac12}\( \int_{\ensuremath{\mathbb{R}}^3}|\pi(Q_L)-Q_L|^2\,dx \)^{\frac12}\leq C\( \int_{\ensuremath{\mathbb{R}}^3}|Q_L-\pi(Q_L)|^2\,dx \)^{\frac12}.
\end{align*}
Using \eqref{L3 small global p2} that $\|Q_L(s)-\pi(Q_L (s))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\leq CL$ for $s\in [0,T_1)$, we have \[\lim_{L\to 0}\|Q_L(s)-\pi(Q_L(s))\|_{L^\infty(0,T_1;L^2(\ensuremath{\mathbb{R}}^3))}=0.\]
Then, we prove the claim \eqref{eq fB}.
For $\varphi\in C^\infty_0(\ensuremath{\mathbb{R}}^3,S_0)$, we define
\begin{align*}
\Phi_{ij}(Q,\varphi)=&s^{-1}_+( Q_{jl}+\frac {s_+}3 \delta_{jl})\varphi_{il}+s^{-1}_+(Q_{il}+\frac {s_+}3 \delta_{il})\varphi_{jl}
\\
&-2s_+^{-2}(Q_{ij}+\frac {s_+}3 \delta_{ij})(Q_{lm}+\frac {s_+}3 \delta_{lm})\varphi_{lm}.
\end{align*}
As $L\to 0$, we multiply \eqref{RBE3} by $\Phi_{ij}(Q,\varphi)$ and obtain
\begin{align*}
&\<(\partial_t +v\cdot \nabla )Q+[Q, \Omega],\Phi(Q,\varphi)\>
=(\partial_t +v\cdot \nabla )(s_+u_iu_j)(2u_ju_l\varphi_{il})
\\
&+\Big(s_+(u_iu_k-\frac {1}3 \delta_{ik})\Omega_{kj}-s_+(u_ku_j-\frac {1}3 \delta_{kj})\Omega_{ik}\Big)\Phi_{ij}(Q,\varphi)
\\
=&\<(\partial_t +v\cdot \nabla )Q +[Q,\Omega],\varphi\>
,\alabel{con left}
\end{align*}
where we used that $|Q|=\sqrt{\frac32} s_+$ and $|u|=1$. Multiplying \eqref{RBE3} with $\Phi_{ij}(Q,\varphi)$
and using the condition \eqref{eq fB}, we can apply Theorem 2 in \cite{FH} that the solution $(Q_L,v_L)$ of \eqref{RBE1}-\eqref{RBE3} converges
to a solution $(Q, v)$ of \eqref{BE1}-\eqref{BE3} as $L\to0$. Taking the difference between two solutions under $L^2$ estimates, it can be shown (c.f. \cite{HLX} or \cite{FHM}) that the strong solution $(Q,v)$ is unique. The proof on uniqueness is similar to the claim 2 in the appendix, so we omit the details here. Let $T_1$ be any time with $T_1\leq T^*$. Under the criteria
\begin{align*}
\sup_{0\leq t\leq T_1,x\in\ensuremath{\mathbb{R}}^3}\int_{B_{R_0}(x_0)}(|\nabla Q|^3+|v|^3)\,dx\leq{\varepsilon_0^3},
\end{align*}
we can use the same methods in \cite{HLX} to extend the solution passing $T_1$ up to the maximal time $T^*$.
\end{proof}
\section{Smooth convergence}
In this section, we will prove Theorem \ref{thm2}. At first,
we obtain the following higher order estimate:
\begin{lemma}\label{kth} Let $(Q_L,v_L)$ be a strong solution of $\eqref{RBE1}-\eqref{RBE3}$ in $\ensuremath{\mathbb{R}}^3 \times [T_0,T_M)$ with initial value $(Q_{L,T_0},v_{L,T_0})\in H^2_{Q_e} (\ensuremath{\mathbb{R}}^3)\times H^1(\ensuremath{\mathbb{R}}^3)$ and $\div v=0$. For any $\tau>T_0$, $s\in(\tau,T_M]$ and any integer $m\geq 0$, there exists a positive constant $C_m$ independently of $Q_L$ and $L$ (but depending on $m$) such that
\begin{align*}
&\sup_{\tau\leq s\leq T_M}\int_{\ensuremath{\mathbb{R}}^3}\left(|\nabla^{m+1}Q_L|^2+|\nabla^m v_L|^2 +\frac{1}{L}|\nabla^m(Q_L-\pi(Q_L))|^2\right)(x,t)\,dx
\\
&+\int_{\tau}^{T_M}\int_{\ensuremath{\mathbb{R}}^3} |\nabla^{m+2}Q_L|^2+|\nabla^{m+1}v_L|^2+|\nabla^m\partial_t Q_L|^2\,dxdt
\\
&+\int_{\tau}^{T_M}\int_{\ensuremath{\mathbb{R}}^3} \frac{1}{L}|\nabla^{m+1}(Q_L-\pi(Q_L))|^2\,dxdt \leq C_m.
\alabel{kth estimates}
\end{align*}
\end{lemma}
\begin{proof}
We prove this lemma by induction. In view of \eqref{prop R3 1} and \eqref{prop R3 2}, one have shown \eqref{kth estimates} holds for $m=0,1$. Assume that \eqref{kth estimates} holds for $m=1,\cdots, k$ with $k\geq 1$. Then we have
\begin{align*}
&\sup_{\frac{\tau}2\leq s\leq T_M}\sum_{i=0}^{k}\int_{\ensuremath{\mathbb{R}}^3}\left(|\nabla^{i+1}Q_L|^2+|\nabla^i v_L|^2 +\frac{1}{L}|\nabla^i(Q_L-\pi(Q_L))|^2\right)(x,s)\,dx
\\
&+\sum_{i=0}^{k}\int_{\frac{\tau}2}^{T_M}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{i+2}Q_L|^2+|\nabla^{i+1}v_L|^2+|\nabla^i\partial_t Q_L|^2\,dxdt
\\
&+\sum_{i=0}^{k}\int_{\frac{\tau}2}^{T_M}\int_{\ensuremath{\mathbb{R}}^3} \frac{1}{L}|\nabla^{i+1}(Q_L-\pi(Q_L))|^2 \,dxdt \leq C_k(\tau).
\alabel{kth aspt}
\end{align*}
For $m=k$, it follows from using \eqref{kth aspt} and the mean value theorem that there exists a $\tau_{L}\in(\tau/2, \tau)$ such that
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\left(|\nabla^{k+2}Q_L|^2+|\nabla^{k+1} v_L|^2 +\frac{1}{L}|\nabla^k(Q_L-\pi(Q_L))|^2 \right)(x,\tau _L)\,dx\leq C_k(\tau).
\alabel{kth tau}
\end{align*}
Applying the Sobolev inequality to \eqref{kth aspt}, we obtain
\begin{align*}
&\sup_{\tau_L\leq s\leq T_M}\sum_{i=0}^{k-1}\|\nabla^{i}(Q_L-Q_e)(s)\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}\leq C_k(\tau).
\alabel{kth Sob1}
\end{align*}
For functions $f_1,f_2\in H^1(\ensuremath{\mathbb{R}}^3)$, by using a similar argument to one in \eqref{L3 small L4 prop}, we observe
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}|f_1|^2|f_2|^2\,dx\leq C\|f_1\|^2_{H^1(\ensuremath{\mathbb{R}}^3)}\|\nabla f_2\|^2_{L^2(\ensuremath{\mathbb{R}}^3)},\alabel{kth f1f2}
\\
&\int_{\ensuremath{\mathbb{R}}^3}|f_1|^2|f_2|^4\,dx\leq C\|\ensuremath{\nabla} f_1\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}\| \ensuremath{\nabla} f_2\|^4_{L^2(\ensuremath{\mathbb{R}}^3)}.\alabel{kth f1f2^2}
\end{align*}
Next, we show that \eqref{kth estimates} holds for $m=k+1$.
In order to derive the $L^2$-norm of $\nabla^{k+3} Q_L$, we apply $\nabla^{k}\nabla_\beta$ to \eqref{ROT RBE} and multiply by $\nabla^{k+2} (R^T_{Q_L}\nabla_\beta Q_LR_{Q_L})$ to obtain
\begin{align*}
&J_1:=\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k}\nabla_{\beta} \Big(R^T_{Q_L}\big(\partial_t Q_L+v_L\cdot \nabla Q_L +[Q_L, \Omega_L]\big)R_{Q_L}\Big), \nabla^{k+2} (R^T_{Q_L}\nabla_\beta Q_LR_{Q_L}) \> \,dx
\\
&=\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k}\nabla_{\beta}\(R^T_{Q_L}\mathcal{H}(Q_L,\nabla Q_L) R_{Q_L} \),\nabla^{k+2} (R^T_{Q_L}\nabla_\beta Q_LR_{Q_L})\> \,dx
\\
&+\frac1L\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k}\nabla_{\beta} g_B(\tilde Q_L),\nabla^{k+2} (R^T_{Q_L}\nabla_\beta Q_LR_{Q_L})\> \,dx =:J_2+J_3
.\alabel{kth Delta Q eq}
\end{align*}
For the term $J_2$, we have
\begin{align*}
J_2\geq&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k}\nabla_{\beta}\nabla_\nu\( \partial^2_{p^\nu} f_E\), \nabla^{k+2} \nabla_\beta Q_L \> \,dx-\frac{\alpha}{8}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+3} Q_L |^2\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1}\partial_{ Q} f_E|^2\,dx-\eta\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+2}(R^T_{Q_L} \nabla Q_LR_{Q_L})|^2\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2+\mu_3=k}|\nabla^{\mu_1}\mathcal{H}(Q_L,\nabla Q_L)|^2|\nabla^{\mu_2}\nabla R_{Q_L}|^2|\nabla^{\mu_3}R_{Q_L})|^2 \,dx
\alabel{kth H 1}
\end{align*}for some small $\eta$ to be chosen later. We deduce the first term on the right-hand side in \eqref{kth H 1} from \eqref{sec2 f_E} that
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\nabla^{k}\nabla_\beta\(\nabla_\nu\partial_{p^\nu_{ij} }f_E\) \nabla^{k+2}\nabla_\beta (Q_L)_{ij} \,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\nabla^{k}\nabla_\nu\(\partial^2_{p^\nu_{ij}p^\gamma_{mn} }f_E\nabla^2_{\beta \gamma}(Q_L)_{mn}\) \nabla^{k+2}\nabla_\beta (Q_L)_{ij} \,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3}\nabla^{k}\nabla_\nu\(\partial^2_{p^\nu_{ij}Q_{mn} }f_E\nabla_{\beta}(Q_L)_{mn}\) \nabla^{k+2}\nabla_\beta (Q_L)_{ij} \,dx
\\
\geq &\int_{\ensuremath{\mathbb{R}}^3} \partial^2_{p^\nu_{ij}p^\gamma_{mn} }f_E\nabla^{k+1}\nabla^2_{\beta \gamma}(Q_L)_{mn}\nabla^{k+1}\nabla^2_{\beta\nu}(Q_L)_{ij} \,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+3} Q_L|\sum_{\mu_1+\mu_2=k} |\nabla^{\mu_1}\nabla\partial^2_{pp}f_E||\nabla^{\mu_2}\nabla^2 Q_L| \,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+3} Q_L|\sum_{\mu_1+\mu_2=k+1}|\nabla^{\mu_1}\partial^2_{pQ }f_E||\nabla^{\mu_2}\nabla Q_L| \,dx
\\
\geq&\frac{3\alpha}{8}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+3} Q_L |^2\,dx-C\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2=k} |\nabla^{\mu_1}\nabla\partial^2_{pp}f_E|^2|\nabla^{\mu_2}\nabla^2 Q_L|^2 \,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2=k+1}|\nabla^{\mu_1}\partial^2_{pQ }f_E|^2|\nabla^{\mu_2}\nabla Q_L|^2 \,dx
.\alabel{kth Delta Q}
\end{align*}
To proceed further, we need to examine the right-hand side of \eqref{kth H 1}-\eqref{kth Delta Q}. Using \eqref{kth aspt}, \eqref{kth Sob1}-\eqref{kth f1f2^2}, the last term in \eqref{kth H 1} becomes
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+2}(R^T_{Q_L} \nabla Q_LR_{Q_L})|^2\,dx
\\
\leq& C\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2+\mu_3=k+2}|\nabla^{\mu_1}\nabla Q_L|^2|\nabla^{\mu_2}R_{Q_L}|^2|\nabla^{\mu_3}R_{Q_L}|^2\,dx
\\
\leq& C \int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+3} Q_L|^2+|\nabla^{k+2} Q_L|^2|\nabla Q_L|^2+|\nabla^{k+1} Q_L|^2(|\nabla^2 Q_L|^2+|\nabla Q_L|^4)\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k} Q_L|^2(|\nabla^3 Q_L|^2+|\nabla^2 Q_L|^2|\nabla Q_L|^2+|\nabla Q_L|^6) \,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}\sum_{\substack{\mu_1+\cdots+\mu_3=k+3
\\
\max\{\mu_1,\mu_2,\mu_3\}\leq k-1}} |\nabla^{\mu_1} Q_L|^2|\nabla^{\mu_2}R_{Q_L}|^2|\nabla^{\mu_3}R_{Q_L}|^2\,dx
\\
\leq& C\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+3} Q_L|^2\,dx
+C\|\nabla Q_L(x)\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}^2\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+2} Q_L|^2\,dx
\\
&+C\(\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+2} Q_L|^2+|\nabla^{k+1} Q_L|^2\,dx\)\(\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+2} Q_L|^2 \,dx+1\)
\\
&+C\(\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+2} Q_L|^2\,dx\)\(\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_L|^2\,dx\)^2\,
\\
&+C\|\nabla^{k} Q_L(x)\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}^2\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+2} Q_L|^2+|\nabla^{k+1} Q_L|^2|\nabla^k Q_L|^2\,dx
\\
&+C \|\nabla^{k} Q_L(x)\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}^2\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_L|^2\,dx\(\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_L|^2\,dx\)^2
\\
\leq&C\|\nabla^{k+3} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
+C (\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)^2
.\alabel{kth lot}
\end{align*}
Similarly, we can estimate the remaining terms in \eqref{kth Sob1}-\eqref{kth f1f2^2} as follows:
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k}\nabla_{\beta}(\partial_{ Q} f_E)|^2+\sum_{\mu_1+\mu_2+\mu_3=k}|\nabla^{\mu_1}\mathcal{H}(Q_L,\nabla Q_L)|^2|\nabla^{\mu_2}\nabla R_{Q_L}|^2|\nabla^{\mu_3}R_{Q_L})|^2 \,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3} \sum_{\mu_1+\mu_2=k}|\nabla^{\mu_1}\nabla\partial^2_{pp}f_E|^2|\nabla^{\mu_2}\nabla^2 Q_L|^2\,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2=k+1}|\nabla^{\mu_1}\partial^2_{pQ }f_E|^2|\nabla^{\mu_2}\nabla Q_L|^2 \,dx \leq C (\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)^2
.\alabel{kth lower}
\end{align*}
Substituting \eqref{kth Delta Q}-\eqref{kth lower} into \eqref{kth H 1} and choosing sufficiently small $\eta$, we have
\begin{align*}
J_2\geq&\frac{\alpha}{4}\|\nabla^{k+3} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 -C(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)^2
.\alabel{kth H}
\end{align*}
To estimate $J_3$, utilizing \eqref{kth aspt}, the Sobolev inequality and \eqref{kth f1f2^2}-\eqref{kth Sob1}, it follows from Corollary \ref{cor fB} that
\begin{align*}
&\frac1L\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{ \tilde Q_{ij} \tilde Q_{kl}} f_B(\tilde Q _L)\nabla^{k+2} (\tilde Q_L) _{ij}\nabla^{k+2}(\tilde Q_L)_{kl}\,dx
\\
\geq&\frac1L\int_{\ensuremath{\mathbb{R}}^3}\frac\lambda 2 |\nabla^{k+2} (Q_L -\pi(Q_L ))|^2\,dx - \frac CL \int_{\ensuremath{\mathbb{R}}^3} |\nabla^{k+1} (Q_L -\pi(Q_L ))|^2|\nabla Q_L|^2\,dx
\\
&- \frac CL\int_{\ensuremath{\mathbb{R}}^3} |\nabla^{k} (Q_L -\pi(Q_L ))|^2(|\nabla^2 Q_L|^2+|\nabla Q_L|^4)\,dx
\\
&-\frac CL \int_{\ensuremath{\mathbb{R}}^3} |\nabla^{k-1} (Q_L -\pi(Q_L ))|^2(|\nabla^3 Q_L|^2+|\nabla^2 Q_L|^2|\nabla Q_L|^2+ |\nabla Q_L|^6)\,dx
\\
&-\frac CL\int_{\ensuremath{\mathbb{R}}^3} \hspace{-2ex}\sum_{\substack{\mu_1\leq k-2,\\\max\{\mu_2,\cdots,\mu_{k+3}\}\leq k-1}}|\nabla^{\mu_1} (Q_L -\pi(Q_L ))|^2|\nabla^{\mu_2} Q_L|^2\cdots|\nabla^{\mu_{k+3}}Q_L|^2\,dx
\\
&-\frac CL \int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 (Q_L -\pi(Q_L ))|^2(|\nabla^{k} Q_L|^2+1)\,dx
\\
&-\frac CL\int_{\ensuremath{\mathbb{R}}^3} |\nabla (Q_L -\pi(Q_L ))|^2(|\nabla^{k+1} Q_L|^2+|\nabla^k Q_L|^2|\nabla Q_L|^2+1)\,dx
\\
&-\frac CL\int_{\ensuremath{\mathbb{R}}^3} |Q_L -\pi(Q_L )|^2(|\nabla^{k+2} Q_L|^2+|\nabla^{k+1} Q_L|^2|\nabla Q_L|^2)\,dx
\\
&-\frac CL\int_{\ensuremath{\mathbb{R}}^3} |Q_L -\pi(Q_L )|^2 |\nabla^{k} Q_L|^2(|\nabla^2 Q_L|^2+|\nabla Q_L|^4+1)\,dx
\\
&\geq\frac{\lambda }{2L} \|\nabla^{k+2} (Q_L -\pi(Q_L ))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
&
- C(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)\(\frac{1}{L}\|\nabla^{k+1} (Q_L-\pi(Q_L)) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 +1\)
.\alabel{kth Cor}
\end{align*}
Repeating the argument in \eqref{kth lot}, one has
\begin{align*}
&\frac{1}{L}\|\nabla^{k+2} \tilde Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\geq \frac{1}{2L}\|\nabla^{k+2} (Q_L -\pi(Q_L ))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
&- C(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)\(\frac{1}{L}\|\nabla^{k+1} (Q_L-\pi(Q_L)) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1\)
.\alabel{kth tilde Q +2}
\end{align*}
Applying Lemma \ref{lemgb} to $J_3$ and combining with \eqref{kth Cor}-\eqref{kth tilde Q +2} yields
\begin{align*}
J_3=&\frac{(-1)^{k+2}}L \int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{2k+2}\nabla_\beta (g_B(\tilde Q_L)), \nabla_\beta \tilde Q_L\> \,dx
\\
&+\frac{(-1)^{k+2}}L \int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{2k+2}\nabla_\beta (g_B(\tilde Q_L)), \nabla_\beta R_{Q_L}^T (\pi(Q_L)-Q_L)R_{Q_L}\> \,dx
\\
&+\frac{(-1)^{k+2}}L \int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{2k+2}\nabla_\beta (g_B(\tilde Q_L)),R_{Q_L}^T (\pi(Q_L)-Q_L)\nabla_\beta R_{Q_L}\> \,dx
\\
\geq&\frac 1L\int_{\ensuremath{\mathbb{R}}^3} \partial^2_{\tilde Q_{ij} \tilde Q_{kl}} f_B(\tilde Q_L)\nabla^{k+2} (\tilde Q_L)_{kl}\nabla^{k+2}(\tilde Q_L)_{ij} \,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}\frac{|\nabla^{k+2} \tilde Q_L|}{L^{\frac12}}\sum_{\mu_1+\mu_2=k}|\nabla^{\mu_1}\nabla \partial^2_{ \tilde Q} f_B(\tilde Q_L)|\frac{|\nabla^{\mu_2} \nabla \tilde Q_L|}{L^{\frac12}}\,dx
\\
&-C\int_{\ensuremath{\mathbb{R}}^3}\frac{|\nabla^{k+2} g_B(\tilde Q_L)|}{L^{\frac12}}\sum_{\substack{ \mu_1<\mu_2\leq\mu_3\leq k+2\\ \mu_1+\mu_2+\mu_3=k+2}}\frac{|\nabla^{\mu_1} (Q-\pi(Q_L))| }{L^{\frac12}}|\nabla^{\mu_2}R_{Q_L}^T||\nabla^{\mu_3}R_{Q_L}|\,dx
\\
\geq& \frac{\lambda}{4L} \|\nabla^{k+2} (Q_L -\pi(Q_L ))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
&-C(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)\(\frac{1}{L}\|\nabla^{k+1} (Q_L-\pi(Q_L)) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1\)
,\alabel{kth g_B}
\end{align*}
where we used that
\begin{align*}
&C\int_{\ensuremath{\mathbb{R}}^3}\frac{|\nabla^{k+2} g_B(\tilde Q_L)|^2}{L}\,dx\leq C\|\nabla^{k+2} (Q_L -\pi(Q_L ))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
&\qquad+C(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)\(\frac{1}{L}\|\nabla^{k+1} (Q_L-\pi(Q_L)) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1\).
\end{align*}
Applying \eqref{kth aspt}, \eqref{kth Sob1}-\eqref{kth f1f2^2} to the left-hand side of \eqref{kth Delta Q eq} and then substituting \eqref{kth lot}, we obtain
\begin{align*}
J_1\leq&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k}\nabla_{\beta}\partial_t Q_L, \nabla^{k+2}\nabla_\beta Q_L\> \,dx+\eta\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+2} (R^T_{Q_L}\nabla_\beta Q_LR_{Q_L})|^2\,dx
\\
&+C(\eta)\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2+\mu_3=k}|\nabla^{ \mu_1} \partial_t Q_L|^2|\nabla^{\mu_2}\nabla R_{Q_L}|^2|\nabla^{\mu_3} R_{Q_L}|^2\,dx
\\
&+C(\eta)\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2+\mu_3=k+1}|\nabla^{ \mu_1}(v_L\cdot\nabla Q_L+[Q_L,\Omega_L])|^2|\nabla^{\mu_2} R_{Q_L}|^2|\nabla^{\mu_3} R_{Q_L}|^2\,dx
\\
\leq& -\frac 12\frac{d}{dt}\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 +\frac{\alpha}8\|\nabla^{k+3} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+ C\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
&+C(\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\| \partial_tQ_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2+1)(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)
.\alabel{kth k+3 Q left}
\end{align*}
Combining $J_2,J_3$ with $J_1$ and integrating in $t$, we find
\begin{align*}
&\frac 12 \|\nabla^{k+2} Q_{L}(s)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 +\int_{\tau_L}^{s}\frac{\alpha}{8}\|\nabla^{k+3} Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+\frac{\lambda}{4L}\|\nabla^{k+2} (Q_L-\pi(Q_L)) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt
\\
\leq& C\int_{\tau_L}^{s}\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt+\int_{\tau_L}^{s}\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\frac{1}{L}\|\nabla^{k+1} (Q_L-\pi(Q_L)) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
&+C\int_{\tau_L}^{s}\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\(\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\| \partial_tQ_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\)\,dt
\\
&+C\int_{\tau_L}^{s}\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^4\,dt+C
.\alabel{kth Delta Q f}
\end{align*}
To estimate the $L^2$-norm of $\nabla^{k+1} \partial_t Q_L$, we applying $\nabla^{k+1}$ to \eqref{ROT RBE} and multiplying by $\nabla^{k+1}(R^T_{Q_L}\partial_tQ_LR_{Q_L})$ to have
\begin{align*}
&J_4:=\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k+1}\( R^T_{Q_L}(\partial_t Q_L +v_L\cdot \nabla Q_L)R^T_{Q_L}\) , \nabla^{k+1}(R^T_{Q_L}\partial_tQ_LR_{Q_L})\>\,dxdt
\\
&+\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k+1}\( R^T_{Q_L}[Q_L, \Omega_L])R_{Q_L}\), \nabla^{k+1}(R^T_{Q_L}\partial_tQ_LR_{Q_L})\>\,dxdt
\\
=&\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k+1}\(R^T_{Q_L}\mathcal{H}(Q_L,\nabla Q_L) R_{Q_L}\),\nabla^{k+1}(R^T_{Q_L}\partial_tQ_LR_{Q_L})\>\,dxdt
\\
&+\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k+1}\frac1Lg_B(\tilde Q_L),\nabla^{k+1}(R^T_{Q_L}\partial_tQ_LR_{Q_L})\>\,dxdt =:J_5+J_6
.\alabel{kth pt Q}
\end{align*}
Repeating the same argument in \eqref{kth lot}, we observe
\begin{align*}
&\|\nabla^{k+1}(R^T_{Q_L} \partial_t Q_LR_{Q_L})\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
\leq& C\|\nabla^{k+1}\partial_t Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+C\|\partial_tQ_L\|^2_{H^k(\ensuremath{\mathbb{R}}^3)}(\|\nabla^{k+2} Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+1)
\alabel{kth R pt Q}
\end{align*}
and
\begin{align*}
&\|\nabla^{k+1} \mathcal{H}(Q_L,\nabla Q_L)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\leq C\|\nabla^{k+3} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+C(\|\nabla^{k+2} Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+1)^2
.\alabel{kth R pt Q2}
\end{align*}
Using \eqref{sec2 f_E}, \eqref{kth lower}, \eqref{kth R pt Q}-\eqref{kth R pt Q2} and \eqref{kth tau} to $J_5$, we obtain
\begin{align*}
J_5 \leq& \int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k}\nabla_\beta(\nabla_\nu\partial_{p^\nu}f_E-\partial_Qf_E),\nabla^{k}\nabla_\beta\partial_t Q_L\>\,dx dt
\\
& +C\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1} \mathcal{H}(Q_L,\nabla Q_L)|\sum_{\mu_1+\mu_2+\mu_3=k}|\nabla^{\mu_1}\partial_t Q_L||\nabla^{\mu_2}\nabla R_{Q_L}||\nabla^{\mu_3} R_{Q_L}|\,dx dt
\\
&+C\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1}(R^T_{Q_L} \partial_t Q_LR_{Q_L})|\sum_{\mu_1+\mu_2+\mu_3=k}|\nabla^{\mu_1}\mathcal{H}||\nabla^{\mu_2}\nabla R_{Q_L}||\nabla^{\mu_3} R_{Q_L}|\,dx dt
\\
\leq&\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\nu_{ij} p^\gamma_{mn}}f_E \nabla^{k}\nabla^3_{\mu\gamma\nu}( Q_L)_{mn} \nabla^{k}\nabla_\beta\partial_t( Q_L)_{ij}\,dx dt
\\
&+C\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1}\partial_t Q_L|\sum_{\mu_1+\mu_2=k}|\nabla^{\mu_1}\nabla\partial^2_{pp}f_E||\nabla^{\mu_2} \nabla^2 Q_L| \,dx dt
\\
&+C\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1}\partial_t Q_L|\Big(|\nabla^{k+1}(\partial^2_{pQ}f_E\cdot\nabla Q_L)|+|\nabla^{k+1}\partial_Qf_E|\Big)\,dx dt
\\
&+\eta\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1} \mathcal{H}(Q_L,\nabla Q_L)|^2+|\nabla^{k+1}(R^T_{Q_L} \partial_t Q_LR_{Q_L})|^2\,dx dt
\\
&+C(\eta)\int_{\tau_L}^{s}(\|\nabla^{k+2}Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_tQ_L\|^2_{H^k(\ensuremath{\mathbb{R}}^3)})\|\nabla^{k+2} Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}\, dt+C
\\
\leq& -\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3} \frac12\partial_t\Big(\partial^2_{p_{ij}^\nu p_{mn}^\gamma} f_E \nabla^{k+1}\nabla_\gamma(Q_L)_{mn}\nabla^{k+1}\nabla_\nu(Q_L)_{ij}\Big)\,dxdt
\\
&+\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\frac 12\partial_t\partial^2_{p_{ij}^\nu p_{mn}^\gamma} f_E \nabla^{k+1}\nabla_\gamma(Q_L)_{mn}\nabla^{k+1}\nabla_\nu(Q_L)_{ij}\,dxdt
\\
&-\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\nabla_\nu\partial^2_{p^\nu_{ij} p^\gamma_{mn}}f_E \nabla^{k}\nabla^2_{\beta\gamma}( Q_L)_{mn} \nabla^{k}\nabla_\beta\partial_t( Q_L)_{ij}\,dx dt
\\
&+\eta\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1} \mathcal{H}(Q_L,\nabla Q_L)|^2+|\nabla^{k+1}(R^T_{Q_L} \partial_t Q_LR_{Q_L})|^2+|\nabla^{k+1}\partial_t Q_L|^2\,dx dt
\\
&+C(\eta)\int_{\tau_L}^{s}(\|\nabla^{k+2}Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_tQ_L\|^2_{H^k(\ensuremath{\mathbb{R}}^3)})\|\nabla^{k+2} Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}\, dt+C
\\
\leq&-\frac {\alpha}4\|\nabla^{k+2} Q_{L}(s) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
+\int_{\tau_L}^{s} \frac\alpha{8}\|\nabla^{k+3} Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+\frac18\|\nabla^{k+1}\partial_t Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}\, dt
\\
&+C\int_{\tau_L}^{s}(\|\nabla^{k+2}Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_tQ_L\|^2_{H^k(\ensuremath{\mathbb{R}}^3)})\|\nabla^{k+2} Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}\, dt+C
.\alabel{kth H Q_t}
\end{align*}
Utilizing Corollary \ref{cor fB}, \eqref{kth tilde Q +2}, \eqref{kth R pt Q}, integration by parts and \eqref{kth tau}, we compute $J_6$
\begin{align*}
J_6=&-\frac 1L\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3} \nabla^{k+1}\partial_t(\tilde Q_L) _{ij}\nabla^{k} \(\partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B(\tilde Q_L )\nabla (\tilde Q_L)_{kl}\)\,dxdt
\\
=&-\frac 1{2L}\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\partial_t\(\partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B(\tilde Q_L)\nabla^{k+1} (\tilde Q_L)_{kl}\nabla^{k+1}(\tilde Q_L) _{ij}\)\,dxdt
\\
&+\frac 1{2L}\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\partial_t\(\partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B(\tilde Q_L)\)\nabla^{k+1} (\tilde Q_L)_{kl}\nabla^{k+1}(\tilde Q_L) _{ij}\,dxdt
\\
&+\frac 1L\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3} |\nabla^{k}\partial_t\tilde Q_L|\sum_{\mu_1+\mu_2=k}|\nabla^{\mu_1}\nabla\partial^2_{\tilde Q \tilde Q } f_B(\tilde Q_L)||\nabla^{\mu_2}\nabla \tilde Q_L|\,dxdt
\\
\leq&-\frac 1{2L}\frac{d}{dt}\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{\tilde Q_{ij}\tilde Q_{kl}} f_B\nabla^{k+1} (\tilde Q_L)_{kl}\nabla^{k+1}(\tilde Q_L)_{ij}\,dxdt
\\
&+\eta\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k}\partial_t(R^T_{Q_L} Q_LR_{Q_L})|^2+|\nabla^2\partial_t \tilde Q|^2+\frac{1}L |\nabla^{k+2}\tilde Q_L|^2\,dxdt
\\
&+C(\eta) \int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\frac{1}{L}|\nabla^k \tilde Q_L|^2|\partial_t Q_L|^2\,dxdt
\\
&+C(\eta) \int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\frac{1}{L}|\nabla^{k-1}\tilde Q_L|^2\(| \nabla\partial_t Q_L|^2+\frac{1}{L}|\nabla^{k+1}\tilde Q_L|^2\)\,dxdt
\\
&+C(\eta) \int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3} \frac{1}{L^2} \sum_{\mu_1+\mu_2=k} |\nabla^{\mu_1}\nabla\partial^2_{\tilde Q \tilde Q } f_B(\tilde Q_L)|^2 |\nabla^{\mu_2}\nabla \tilde Q_L|^2 \,dxdt
\\
\leq&- \frac{\lambda}{4L} \|\nabla^{k+1}( Q_L-\pi(Q_L))(s)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 +C\int_{\tau_L}^{s}\frac{1}{L^2}\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^4 \,dt
\\
&+\int_{\tau_L}^{s} \frac\lambda{8L} \|\nabla^{k+2}( Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\frac18\|\nabla^{k+1}\partial_t Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}\,dt
\\
&+C\int_{\tau_L}^{s}\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\(\frac{1}L\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 +\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\)\,dt
\\
&+C\int_{\tau_L}^{s}\frac{1}L\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\,dt
.
\alabel{kth g_B Q_t p3}
\end{align*}
Using \eqref{kth R pt Q}, we deduce the left-hand side of \eqref{kth pt Q} to
\begin{align*}
J_4\geq&\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k+1}\partial_t Q_L,\nabla^{k+1} \partial_tQ_L\>\,dxdt-\eta\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1}(R^T_{Q_L}\partial_tQ_LR_{Q_L})|^2\,dxdt
\\
&+\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2+\mu_3=k}|\nabla^{\mu_1}\partial_tQ_L|^2|\nabla^{\mu_2} \nabla R_{Q_L}|^2|\nabla^{\mu_3} R_{Q_L}|^2\,dxdt
\\
&+\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2+\mu_3=k+1}|\nabla^{\mu_1}(v_L\cdot \nabla Q_L+[Q_L,\Omega_L])|^2|\nabla^{\mu_2} R_{Q_L}|^2|\nabla^{\mu_3} R_{Q_L}|^2\,dxdt
\\
\geq&\int_{\tau_L}^{s}\frac34\|\nabla^{k+1}\partial_t Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2-C\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt
\\
&-C\int_{\tau_L}^{s}(\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2+1)(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)\,dt
.\alabel{kth k+1 Q_t}
\end{align*}
Adding $J_5,J_6$ to $J_4$, we have
\begin{align*}
&\frac {\alpha}4 \|\nabla^{k+2} Q_{L}(s) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\frac{\lambda}{2L} \|\nabla^{k+1}( Q_L-\pi(Q_L))(s)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
&+\frac12\int_{\tau_L}^{s}\|\nabla^{k+1}\partial_t Q\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt
\\
\leq&C\int_{\tau_L}^{s}\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt+ \frac{\lambda}{8L}\int_{\tau_L}^{s} \|\nabla^{k+2}( Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt
\\
&
+\frac18\int_{\tau_L}^{s}\alpha\|\nabla^{k+3} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt+\frac{C}{L^2}\int_{\tau_L}^{s} \|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^4\,dt
\\
&+C\int_{\tau_L}^{s}(\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2)\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
&+\frac CL\int_{\tau_L}^{s}\(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)} ^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\) \|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt
\\
&+C\int_{\tau_L}^{s}\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^4\,dt
.\alabel{kth k+3 Q final}
\end{align*}
Combining \eqref{kth k+3 Q final} with \eqref{kth Delta Q f} yields
\begin{align*}
&\|\nabla^{k+2} Q_{L}(s) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\frac{1}{L}\|\nabla^{k+1}( Q_L-\pi(Q_L))(s)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
&+\int_{\tau_L}^{s} \|\nabla^{k+3} Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+\|\nabla^{k+1}\partial_t Q\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 +\frac{1}{L}\|\nabla^{k+2} (Q_L-\pi(Q_L)) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt
\\
\leq&C\int_{\tau_L}^{s}\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt+C\int_{\tau_L}^{s}\(\frac{1}{L}\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 +1\)^2\,dt
\\
&+C\int_{\tau_L}^{s}(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2)\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
&+C\int_{\tau_L}^{s}\(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)} ^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\)\frac{1}{L}\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt
.\alabel{kth Q}
\end{align*}
To estimate $\nabla^{k+2} v_L$ in \eqref{kth Q}, we apply $\nabla^{k+1}$ to \eqref{RBE1} and multiply by $\nabla^{k+1} v_L$ to obtain
\begin{align*}
&\frac{1}{2}\|\nabla^{k+1} v_L(s)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+ \int_{\tau_L}^{s}\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2dt
\\
=& \int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\nabla^{k+1}\(\partial_{p^j_{mn}}
f_E\nabla_i (Q_L)_{mn} -[Q_L, \mathcal{H}(Q_L,\nabla Q_L)]_{ij}\)\nabla^{k+1}\nabla_j (v_L)_i\,dxdt
\\
\leq&- \int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\nabla^{k+1}[Q_L, \mathcal{H}(Q_L,\nabla Q_L)]_{ij}\nabla^{k+1}\nabla_j (v_L)_i\,dxdt+\frac14 \int_{\tau_L}^{s}\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2dt
\\
&+C\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2=k+1}|\nabla^{\mu_1}\partial_{p}
f_E|^2|\nabla^{\mu_2}\nabla Q_L|^2\,dx.
\\
\leq&\frac14 \int_{\tau_L}^{s}\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2dt+C\int_{\tau_L}^{s}(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)^2\,dt
\\
&- \int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\nabla^{k+1}[Q_L, \mathcal{H}]_{ij}\nabla^{k+1}\nabla_j (v_L)_i\,dxdt
.\alabel{kth p2}
\end{align*}
The last step follows from the argument in \eqref{kth lower}.
In order to cancel the Lie bracket term in \eqref{kth p2}, we differentiate \eqref{ROT RBE}, multiply by $\nabla^{k+1}\(R^T_{Q_L}\big(\mathcal{H}(Q_L,\nabla Q_L) +\frac 1Lg_B(Q_L)\big)R_{Q_L}\)$ and combine with \eqref{kth H +gB}, \eqref{kth p2.3} to obtain
\begin{align*}
&\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\left|\nabla^{k+1}\(R^T_{Q_L}\big(\mathcal{H}(Q_L,\nabla Q_L) +\frac 1Lg_B(Q_L)\big)R_{Q_L}\)\right|^2\,dxdt
\\
&=\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k+1}(R^T_{Q_L}\partial_tQ_LR_{Q_L}),\nabla^{k+1}\(R^T_{Q_L}(\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q))R_{Q_L}\)\>\,dxdt
\\
&+\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k+1}(R^T_{Q_L}(v\cdot \nabla Q_L)R_{Q_L}),\nabla^{k+1}\(R^T_{Q_L}(\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q))R_{Q_L}\)\>\,dxdt
\\
&+\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\<\nabla^{k+1}\( R^T_{Q_L} [Q_L, \Omega_L]R_{Q_L}\), \nabla^{k+1} \(R^T_{Q_L}(\mathcal{H}(Q,\nabla Q)+\frac 1Lg_B(Q))R_{Q_L}\)\>\,dxdt
\\
&=:J_7+J_8+J_9
.
\alabel{kth lie}
\end{align*}
We apply Lemma \ref{Lie} to $\Delta^{k+1}\Omega_L$ with $A=Q_L, B=\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B( Q_L), F=\Delta^{k+1}\Omega_L $ and obtain
\begin{align*}
&\<[Q_L , \Delta^{k+1}\Omega_L ],\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B( Q_L)\>=\Delta^{k+1}\nabla_jv_i[Q_L,\mathcal{H}(Q_L,\nabla Q_L)]_{ij}.
\alabel{Lie k}
\end{align*}
Note from \eqref{RBE3} that
\begin{align*}
|\nabla^k(\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B(Q_L))|^2\leq&C(|\nabla^k \partial_t Q_L|^2+\sum_{\mu_1=\mu_2=k+1}|\nabla^{\mu_1} Q_L|^2|\nabla^{\mu_2} v_L|^2)
.\alabel{H gB}
\end{align*}
Then using \eqref{Lie k}, \eqref{H gB}, \eqref{kth aspt}, \eqref{kth Sob1}-\eqref{kth f1f2^2}, we observe
\begin{align*}
&J_9 \leq(-1)^{k+1}\int_{\ensuremath{\mathbb{R}}^3}\<\Delta^{k+1}[Q_L , \Omega_L ],\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B(Q_L)\> \,dx
\\
&+\eta\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1} (\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B(Q_L) )|^2+|\nabla^{k+1}(R^T_{Q_L}[Q_L , \Omega_L ]R_{Q_L})|^2\,dx
\\
&+C(\eta)\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2+\mu_3=k}|\nabla^{\mu_1}[Q_L , \Omega_L ]|^2|\nabla^{\mu_2}\nabla R_{Q_L}|^2|\nabla^{\mu_3} R_{Q_L}|^2\,dx
\\
&+C(\eta)\int_{\ensuremath{\mathbb{R}}^3}\sum_{\mu_1+\mu_2+\mu_3=k}|\nabla^{\mu_1}(\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B(Q_L))|^2|\nabla^{\mu_2}\nabla R_{Q_L}|^2|\nabla^{\mu_3}R_{Q_L}|^2 \,dx
\\
\leq& (-1)^{k+1}\int_{\ensuremath{\mathbb{R}}^3}\nabla^{2k+2}\nabla_j(v_L)_i[Q_L,(\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B(Q_L) )]_{ij}\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1} \Omega_L|\sum_{\substack{\mu_1 +\mu_2+= k}}|\nabla^{\mu_1}\nabla Q_L||\nabla^{\mu_2}(\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B(Q_L) )|\,dx
\\
&+\eta\int_{\ensuremath{\mathbb{R}}^3}|\nabla^{k+1} (\mathcal{H}(Q_L,\nabla Q_L)+\frac 1Lg_B(Q_L) )|^2+|\nabla^{k+1}(R^T_{Q_L}[Q_L , \Omega_L ]R_{Q_L})|^2\,dx
\\
&+C(\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)
\\
\leq& \int_{\ensuremath{\mathbb{R}}^3}\nabla^{k+1}\nabla_j (v_L)_i\nabla^{k+1}[Q_L, \mathcal{H}(Q_L,\nabla Q_L)]_{ij}\,dx+\frac14 \|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
&+\eta_1(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^{k+1}}^2)
\\
& +C(\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1)
.\alabel{kth p2.3}
\end{align*}
Repeating the same arguments in \eqref{kth H Q_t}-\eqref{kth g_B Q_t p3} and integrating the resulting expression in $t$, we obtain
\begin{align*}
J_7\leq&-\(\frac{\alpha}{4}\|\nabla^{k+2}Q_L(s)\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+\frac {\lambda}{2L} \|\nabla^{k+1} (Q_L-\pi(Q_L))(s)\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}\)
\\
&\hspace{-4ex}+\eta_1\int_{\tau_L}^{s}\|\nabla^{k+3} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1}\partial_t Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+ \frac{1}{L}\|\nabla^{k+2}( Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt
\\
&+C\int_{\tau_L}^{s}\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)} ^2\(\|\nabla^{k+2}Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\)\,dt
\\
&+C\int_{\tau_L}^{s}\frac{1}L\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\( \|\nabla^{k+2}Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\)\,dt
\\
&+C\int_{\tau_L}^{s}\(\frac{1}L\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1\)^2 \,dt
.\alabel{kth H +gB}
\end{align*}
We combine $J_7$ with $J_9$ and apply Young's inequality to $J_8$. Then \eqref{kth lie} reduces to
\begin{align*}
&\int_{\tau_L}^{s}\int_{\ensuremath{\mathbb{R}}^3}\left|\nabla^{k+1}\(R^T_{Q_L}\big(\mathcal{H}(Q_L,\nabla Q_L) +\frac 1Lg_B(Q_L)\big)R_{Q_L}\)\right|^2\,dxdt
\\
\leq&\int_{\tau_L}^{s} \int_{\ensuremath{\mathbb{R}}^3}\nabla^{k+1}\nabla_j (v_L)_i\nabla^{k+1}[Q_L, \mathcal{H}(Q_L,\nabla Q_L)]_{ij}\,dx+\frac14 \|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
&\hspace{-3ex}+2\eta_1\int_{\tau_L}^{s}\|\nabla^{k+3} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1}\partial_t Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+ \frac{1}L\|\nabla^{k+2}( Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
&\hspace{-3ex}+C\int_{\tau_L}^{s}\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)} ^2\(\|\nabla^{k+2}Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\)\,dt
\\
&+C\int_{\tau_L}^{s}\frac{1}L\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\( \|\nabla^{k+2}Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\)\,dt
\\
&+C\int_{\tau_L}^{s}\(\frac{1}L\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1\)^2
\\
&+\frac12\|\nabla^{k+1}(R^T_{Q_L}(\mathcal{H}+\frac 1Lg_B)R_{Q_L})\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt.
\alabel{kth p2.1}
\end{align*}
By adding \eqref{kth p2.1} to \eqref{kth p2}, we obtain
\begin{align*}
& \frac{1}{2}\|\nabla^{k+1} v_L(s)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\frac 14\int_{\tau_L}^{s}\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
\leq &2\eta_1\int_{\tau_L}^{s}\|\nabla^{k+3} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1}\partial_t Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+ \frac{1}L\|\nabla^{k+2}( Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
&\hspace{-2ex}+C\int_{\tau_L}^{s}\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)} ^2\(\|\nabla^{k+2}Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\)\,dt
\\
&+C\int_{\tau_L}^{s}\frac{1}L\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\( \|\nabla^{k+2}Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2\)\,dt
\\
&+C\int_{\tau_L}^{s}\(\frac{1}L\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+1\)^2\,dt
.\alabel{kth v all}
\end{align*}
By substituting \eqref{kth Q} into \eqref{kth v all}, choosing sufficiently small $\eta_1$ and combining with \eqref{kth Q}, we conclude
\begin{align*}
&\|\nabla^2 Q_{L}(s) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1} v_L(s)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\frac{1}{L}\|\nabla^{k+1}( Q_L-\pi(Q_L))(s)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
&+\int_{\tau_L}^{s} \|\nabla^{k+3} Q_L\|^2_{L^2(\ensuremath{\mathbb{R}}^3)}+\|\nabla^{k+2} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1}\partial_t Q\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2\,dt
\\
&+ \int_{\tau_L}^{s}\frac{1}{L}\|\nabla^{k+2} (Q_L-\pi(Q_L)) \|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \,dt
\\
\leq&C\int_{\tau_L}^{s}\(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\frac{1}{L}\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \)
\\
&\times\Big(\|\nabla^{k+2} Q_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\nabla^{k+1} v_L\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2+\|\partial_t Q_L\|_{H^k(\ensuremath{\mathbb{R}}^3)}^2
\\
&+\frac{1}{L}\|\nabla^{k+1} (Q_L-\pi(Q_L))\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2 \Big) \,dt+C
.\alabel{kth final}
\end{align*}
We apply the Gronwall inequality to \eqref{kth final} with \eqref{kth aspt} for $t\in (\tau_L,s)$ and conclude that \eqref{kth estimates} holds for $m=k+1$ on the $(\tau,s)$. Since $\tau\geq T_0$ is an arbitrary positive constant, we prove \eqref{kth estimates} for any $s\in (\tau,T_M]$ and $m=k+1$ which completes a proof of this lemma.
\end{proof}
\begin{proof}[\bf Proof of Theorem \ref{thm2}] Let $(Q,v)$ be the strong solution to \eqref{BE1}-\eqref{BE3} in $\ensuremath{\mathbb{R}}^3\times[0,T^\ast)$ with initial data $(Q_0,v_0)\in H^2_{Q_e}(\ensuremath{\mathbb{R}}^3)\times H^1(\ensuremath{\mathbb{R}}^3)$, where $T^\ast$ is its maximal existence time. Given any $T\in (0,T^\ast)$, set
\begin{equation*}
M= 2\sup_{0\leq t\leq T}\|(\nabla Q,v)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2.
\end{equation*}
It follows from using Theorem \ref{thm1}, that there exists a subsequence $(Q_{L},v_{L})$ and
\begin{equation*}
(\nabla Q_{L},v_{L})\rightarrow(\nabla Q,v),\qquad\text{in }~~L^\infty(0,T_M;L_{loc}^2(\ensuremath{\mathbb{R}}^3))\cap L^2(0,T_M;H_{loc}^1(\ensuremath{\mathbb{R}}^3)).
\end{equation*}
Suppose that $T_M<T$. By Lemma \ref{kth} with $m=2$, note that
\[ \int_{\ensuremath{\mathbb{R}}^3}| \nabla^3 Q_L (x , T_M)|^2\,dx\leq C.\]
Similarly to Lemma \ref{lem energy}, one can show the energy identity of the system \eqref{RBE1} - \eqref{RBE3}
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\(f_E(Q ,\nabla Q ) +\frac{|v |^2}{2}\)(x,s)\,dx+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\left|H(Q,\nabla Q ) \right|^2\,dxdt
+ \int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla v |^2\,dxdt
\\
&=\int_{\ensuremath{\mathbb{R}}^3}\Big(f_E(Q_{0},\nabla Q_{0}) +\frac{|v_{0}|^2}{2}\Big)\,dx.\alabel{energy Q}
\end{align*}
Then by comparing \eqref{energy Q} with \eqref{energy eq} (c.f. Lemma 4.3 \cite{FHM}) and
integrating by parts, using H\"older's inequality, we obtain
\begin{align*}
&\quad \lim\limits_{L\rightarrow 0}\|(\nabla^2 Q_L-\nabla^2 Q)(T_M)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^2
\\
\leq& \lim\limits_{L\rightarrow 0}\(\int_{\ensuremath{\mathbb{R}}^3}|(\nabla Q_L-\nabla Q)(T_M)|^2\,dx\)^{\frac12}\(\int_{\ensuremath{\mathbb{R}}^3}|(\nabla^3 Q_L-\nabla^3 Q)(T_M)|^2\,dx\)^{\frac12}
=0.
\end{align*}
Similarly, we find
\begin{equation*}
\lim\limits_{L\rightarrow 0}\|(\nabla v_L-\nabla v)(T_M)\|_{L^2}^2=0,\quad\lim\limits_{L\rightarrow 0}\frac 1L\||\nabla (Q_L-\pi(Q_L))|^2)(T_M)\|_{L^2}^2=0.
\end{equation*}
Therefore, we obtain
\begin{align*}
&\lim\limits_{L\rightarrow 0}\left(\|\nabla Q_L(T_M)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\|v_L(T_M)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\frac1L \|(Q_L-\pi(Q_L))(T_M)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2\right)\\
=&\|\nabla Q(T_M)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\|v(T_M)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2\leq\frac{M}{2}.\nonumber
\end{align*}
Hence, for sufficiently small $L$, one has
\begin{equation*}
\|\nabla Q_L(T_M)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\|v_L(T_M)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2+\frac1L \|(Q_L-\pi(Q_L))(T_M)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2\leq M.
\end{equation*}
Utilizing Proposition \ref{prop Extension} with the initial data $(Q_L(T_M),v_L(T_M))$, we extend the strong solution $(Q_L,v_L)$ to the time $T_1=:\min\{T,2T_M\}>T_M$. That is
\begin{equation}
(\nabla Q_L,v_L)\rightarrow(\nabla Q,v),\qquad\text{in }~~L^\infty(0,T_1;L^2(\ensuremath{\mathbb{R}}^3))\cap L^2(0,T_1;H^1(\ensuremath{\mathbb{R}}^3)).
\end{equation}
Repeating the same argument with $T$, we establish the convergence up to $T$ for any $T<T^\ast$ and complete the first part of Theorem \ref{thm2} as any sequence $L\to 0$ due to the uniqueness of the solution $(Q, v)$. In the view of the Lemma \ref{kth}, the argument in the proof of \eqref{con1} in Theorem \ref{thm2} leads to the statement \eqref{con2}.
\end{proof}
\section{Appendix: Local existence and proof of Theorem \ref{thm loc}}
Assume the initial data $(Q_{L,0},v_{L,0})\in H^2_{Q_e}(\ensuremath{\mathbb{R}}^3)\times H^1(\ensuremath{\mathbb{R}}^3)$ satisfies that\\ $\|Q_{L,0}\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}\leq K$, $\div v_{L,0}=0$ and
\begin{align}
\|Q_{L,0}\|^2_{H^2_{Q_e}(\ensuremath{\mathbb{R}}^3)}+\|v_{L,0}\|^2_{H^1(\ensuremath{\mathbb{R}}^3)} = M_1
.\alabel{t=0}
\end{align}
For any $f(x)\in H^1(\ensuremath{\mathbb{R}}^3)$, it follows from the Gagliardo–Nirenberg interpolation that
\[\int_{\ensuremath{\mathbb{R}}^3} |f(x)|^4\,dx\leq \(\int_{\ensuremath{\mathbb{R}}^3} |f(x)|^2\,dx\)^{\frac12}\(\int_{\ensuremath{\mathbb{R}}^3} |\nabla f(x)|^2\,dx\)^{\frac32}.\]
Then, we have
\begin{align*}
\(\int_{\ensuremath{\mathbb{R}}^3} |f(x)|^4\,dx\)^\frac12\leq \eta\int_{\ensuremath{\mathbb{R}}^3} |\nabla f(x)|^2\,dx+\frac{C}{\eta^3} \int_{\ensuremath{\mathbb{R}}^3} |f(x)|^2\,dx.\alabel{L4}
\end{align*}
With the aid of \eqref{L4}, we now prove the local existence of \eqref{RBE1}-\eqref{RBE3} with initial data $(Q_{L,0},v_{L,0})$.
\begin{proof}[\bf Proof of Theorem \ref{thm loc}.] Without loss of generality, we assume $L=1$ and omit the subscript $L$ in the proof. Define the space
\begin{align*}
\mathcal V(0,T)=\Big\{(Q,v): &\sup_{0\leq t\leq T}\big(\| Q (t)\|_{H^2_{Q_e}(\ensuremath{\mathbb{R}}^3)}^2+\|v (t)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2 \big)+\|\nabla^3 Q \|_{L^2(0,T ;L^2(\ensuremath{\mathbb{R}}^3))}^2
\\
&+\|\partial_t Q \|_{L^2(0,T ;H^1(\ensuremath{\mathbb{R}}^3))}^2 +\|\nabla^2 v \|_{L^2(0,T;L^2(\ensuremath{\mathbb{R}}^3))}^2 \leq C_1M_1,
\\
& \nabla\cdot v=0,\quad \sup_{0\leq t\leq T} \| Q (t)\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}\leq 2K \Big\}
\end{align*}
for some $T$ and $C_1$ to be chosen later.
For a given pair $(Q_m,v_m)\in \mathcal V(0,T)$, there exists a unique strong solution $(Q_{m+1},v_{m+1})$ with the initial data $(Q_0,v_0)$ of
the linearized system of \eqref{RBE1}-\eqref{RBE3}:
\begin{align*}
&(\partial_t -\Delta)v_{m+1} +\nabla P_{m+1}-\nabla\cdot[Q_{m},h(Q_m,Q_{m+1})]
\\
&=-v_m\cdot \nabla v_m -\nabla\cdot\sigma_{ij}(Q_m,\nabla Q_m),
\alabel{v +1}
\\[2ex]
&\nabla\cdot v_{m+1}=0,\alabel{div v +1}
\\[2ex]
&\partial_tQ_{m+1}+[Q_{m}, \Omega_{m+1}]-h(Q_m,Q_{m+1})
= -v_m \cdot\nabla Q_{m}+g_B(Q_m)
\alabel{Q +1}
\end{align*}
for some $T_{m+1}>0$, where
\begin{align*}
h_{ij}(Q_m,Q_{m+1}):=&\frac12\(\nabla_\beta[\partial_{p_{ij}^\beta} f_E(Q_m,\nabla Q_{m+1})]+\nabla_\beta[\partial_{p_{ji}^\beta} f_E(Q_m,\nabla Q_{m+1})]\)
\\
&- \frac12\(\partial_{Q_{ij}} f_E(Q_m,\nabla Q_m)-\partial_{Q_{ji}} f_E(Q_m,\nabla Q_m)\)
\\
&-\frac{\delta_{ij}}3\sum_{l=1}^3\(\nabla_\beta[\partial_{p_{ll}^\beta} f_E(Q_m,\nabla Q_{m+1})]-\partial_{Q_{ll}} f_E(Q_m,\nabla Q_m)\)
.\alabel{Mol m}
\end{align*}
\noindent {\bf Claim 1}: There exists a uniform $T_{M_1}$ such that
$(Q_{m+1},v_{m+1})\in \mathcal V(0,T_{M_1})$ for some $T_{M_1}\leq T_{m+1}$ for all $m\geq 1$.
To establish the $L^2$-norm of $\nabla^3 Q_{m+1}.$ we multiply \eqref{Q +1} with $\Delta^2 Q_{m+1}$ and observe
\begin{align*}
&\<\(\partial_tQ_{m+1}+[Q_m, \Omega_{m+1}]-\nabla_\beta\partial_{p^\beta}f_E(Q_m,\nabla Q_{m+1})\),\Delta^2 Q_{m+1}\>
\\
=&\< \partial_Qf_E(Q_m,\nabla Q_m)-v_m \cdot\nabla Q_{m}+g_B(Q_m),\Delta^2 Q_{m+1}\>
.\alabel{D2 Q+1}
\end{align*}
We can compute the second term in the left-hand side of \eqref{D2 Q+1}
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<[Q_m, \Omega_{m+1}] ,\Delta^2 Q_{m+1}\>\,dxdt
\\
\leq& \frac{\alpha}{8} \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^3 Q_{m+1}|^2\,dxdt+C \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |Q_m|^2|\nabla^2 v_{m+1}|^2\,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla Q_{m}|^2|\nabla v_{m+1}|^2\,dxdt
\\
\leq& \frac{\alpha}{8} \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^3 Q_{m+1}|^2\,dxdt+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v_{m+1}|^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |v_{m+1}|^2\(|\nabla Q_{m}|^4+|\nabla^2 Q_{m}|^2\)\,dxdt
\\
\leq& \frac{\alpha}{8} \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^3 Q_{m+1}|^2\,dxdt+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v_{m+1}|^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla v_{m+1}|^2\,dx \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_{m}|^2\,dxdt+\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |v_{m+1}|^2 |\nabla^2 Q_{m}|^2\,dxdt
.\alabel{Omega m}
\end{align*}
Using the Sobolev inequality and \eqref{t=0}, we have
\begin{align*}
\sup_{0\leq t\leq T_{m+1}}\(\|\nabla Q_{m+1}(t)\|^2_{H^1(\ensuremath{\mathbb{R}}^3)}+\|v_{m+1}(t)\|^2_{H^1(\ensuremath{\mathbb{R}}^3)}\)\leq CM_1
.\alabel{sup t m}
\end{align*}
We employ the inequalities \eqref{L4} and \eqref{sup t m} to show the following:
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|v_{m+1}|^2 |\nabla^2 Q_{m}|^2 \,dxdt
\leq \int_0^s\(\int_{\ensuremath{\mathbb{R}}^3}|v_{m+1}|^4\,dx\)^{\frac12}\(\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_m|^4 \,dx\)^{\frac12}dt
\\
\leq &C\int_0^s\(\int_{\ensuremath{\mathbb{R}}^3}|\nabla v_{m+1}|^2+|v_{m+1}|^2\,dx\)\(\int_{\ensuremath{\mathbb{R}}^3}\eta_1|\nabla^3 Q_m|^3 +\frac{C}{\eta^3_1}|\nabla^2 Q_m|^2\,dx\)dt
\\
\leq&CM_1^2(\eta_1 +\frac{s}{\eta_1^3})
.\alabel{key est}
\end{align*}
Then we can write \eqref{Omega m} as
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<[Q_m, \Omega_{m+1}] ,\Delta^2 Q_{m+1}\>\,dxdt
\\
\leq& \frac{\alpha}{8} \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^3 Q_{m+1}|^2\,dxdt+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v_{m+1}|^2\,dxdt
+CM_1^2(\eta_1 +s+\frac{s}{\eta_1^3})
.\alabel{Omega m2}
\end{align*}
Integrating by parts and using \eqref{sec2 f_E}, we deduce the third term in \eqref{D2 Q+1} to
\begin{align*}
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\nabla_\beta\partial_{p^\beta_{ij}}f_E(Q_m,\nabla Q_{m+1})\Delta^2( Q_{m+1})_{ij}\,dxdt
\\
=&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij}p^\nu_{kl}}f_E(Q_m,\nabla Q_{m+1})\nabla^3_{\mu\gamma\nu} (Q_{m+1})_{kl}\nabla^3_{\beta\gamma\mu} (Q_{m+1})_{ij}\,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\nabla_{\gamma}\partial^2_{p^\beta_{ij} p_{kl}^\nu }f_E(Q_m,\nabla Q_{m+1})\nabla^2_{\mu \nu} (Q_{m+1})_{kl}\nabla^3_{\beta\gamma\mu} (Q_{m+1})_{ij}\,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\nabla_{\gamma}\(\partial^2_{p^\beta_{ij} Q_{kl}}f_E(Q_m,\nabla Q_{m+1}) \nabla_{\mu}( Q_m)_{kl}\)\nabla^3_{\beta\gamma\mu} (Q_{m+1})_{ij}\,dxdt
\\
\leq&-\frac{3\alpha}8 \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^3 Q_{m+1}|^2\,dx +C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_{m+1}|^2 (|\nabla^2 Q_m|^2+|\nabla Q_m|^4)\,dxdt
\\
\leq&-\frac{3\alpha}8 \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^3 Q_{m+1}|^2\,dx+CM_1^2(\eta_1 +s+\frac{s}{\eta_1^3}),
\alabel{fE m}
\end{align*}
where we used the argument of \eqref{key est} in the last calculation.
Using the argument in \eqref{key est} again, we obtain
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla\( \partial_Qf_E(Q_m,\nabla Q_m)-v_m \cdot\nabla Q_{m}+g_B(Q_m)\)|^2\,dxdt
\\
\leq& C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_m|^2|\nabla Q_m|^4+ |\nabla Q_m|^2|\nabla^2 Q_m|^2+|\nabla v_m|^2|\nabla Q_{m}|^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|v_m|^2|\nabla^2 Q_{m}|^2+|\nabla Q_m|^2 \,dxdt \leq CM_1^2(\eta_1 +s+\frac{s}{\eta_1^3})+CM_1s
.\alabel{D2 Q+1 r}
\end{align*}
In view of \eqref{Omega m2}-\eqref{D2 Q+1 r}, we deduce \eqref{D2 Q+1} to
\begin{align*}
&\frac12 \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_{m+1}(x,s)|^2\,dx+\frac{\alpha}{4}\int_0^s \int_{\ensuremath{\mathbb{R}}^3} |\nabla^3 Q_{m+1}|^2\,dxdt
\\
\leq&\frac12 \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_0|^2\,dx+C \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v_{m+1}|^2\,dxdt +CM_1^2(\eta_1 +s+\frac{s}{\eta_1^3})+CM_1s
.\alabel{D2 Q m}
\end{align*}
In order to estimate the $L^2$-norm of $\nabla \partial_t Q_{m+1}$, we multiply \eqref{Q +1} by $\Delta \partial_t Q_{m+1}$ and compute
\begin{align*}
&\<\(\partial_tQ_{m+1}+[Q_m, \Omega_{m+1}]-\nabla_\beta\partial_{p^\beta}f_E(Q_m,\nabla Q_{m+1})\),\Delta \partial_t Q_{m+1}\>
\\
=&\<\partial_Qf_E(Q_m,\nabla Q_m)-v_m \cdot\nabla Q_{m}+g_B(Q_m),\Delta \partial_t Q_{m+1}\>
.\alabel{D pt Q m}
\end{align*}
Using a similar argument to \eqref{Omega m2}, we have
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<[Q_m, \Omega_{m+1}] ,\Delta \partial_t Q_{m+1}\>\,dxdt
\\
\leq& \frac{1}{8} \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla \partial_t Q_{m+1}|^2\,dxdt+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla [Q_m, \Omega_{m+1}] |^2\,dxdt
\\
\leq& \frac{1}{8} \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla \partial_t Q_{m+1}|^2\,dxdt+CK^2\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v_{m+1}|^2\,dxdt +CM_1^2(\eta_1 +s+\frac{s}{\eta_1^3})
.\alabel{Omega m t}
\end{align*}
Using \eqref{sec2 f_E}, we compute the third term in \eqref{D pt Q m}
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\(-\nabla_\beta\partial_{p^\beta}f_E(Q_m,\nabla Q_{m+1})\),\Delta\partial_t Q_{m+1}\>\,dxdt
\\
=&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij}p^\gamma_{kl}}f_E(Q_m,\nabla Q_{m+1})\nabla^2_{\gamma\nu}( Q_{m+1})_{kl}\partial_t\nabla^2_{\beta\nu}( Q_{m+1})_{ij}\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij}Q{kl}}f_E(Q_m,\nabla Q_{m+1})\nabla_{\nu}( Q_{m+1})_{kl}\partial_t\nabla^2_{\beta\nu}( Q_{m+1})_{ij}\,dxdt
\\
\leq&-\frac12\int_0^s\frac{d}{dt}\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij}p^\gamma_{kl}}f_E(Q_m,\nabla Q_{m+1})\nabla^2_{\gamma\nu}( Q_{m+1})_{kl} \nabla^2_{\beta\nu}( Q_{m+1})_{ij}\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_{m+1}|\(|\nabla \partial_t Q_m||\nabla^2 Q_{m+1}|+|\partial_t Q_m||\nabla^3 Q_{m+1}|\)\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_{m+1}|\(|\nabla Q_m||\partial_t Q_m||\nabla^2 Q_{m+1}|\)\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\partial_t \nabla Q_{m+1}|\(|\nabla^2 Q_{m+1}||\nabla Q_m|+|\nabla Q_{m+1}||\nabla Q_m|^2\)\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\partial_t \nabla Q_{m+1}||\nabla Q_{m+1}||\nabla^2 Q_m|\,dxdt
\\
\leq&- \frac\alpha4 \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_{m+1}(x,s)|^2\,dx+\frac{\Lambda(1+4K^2)}2\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_0|^2\,dx +CM_1^2(\eta_1 +s+\frac{s}{\eta_1^3})
\\
&+CM_1^5s+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\frac{\alpha}{8}|\nabla^3 Q_{m+1}|^2+\frac{1}{8}|\nabla \partial_t Q_{m+1}|^2\,dxdt,
\alabel{pt Q+1}
\end{align*}
where in the last step, we used the argument in \eqref{key est} and the following estimate
\begin{align*}
&C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_{m+1}|^2 |\nabla^2 Q_{m+1}|^2\,dxdt
\\
\leq& CM_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\eta|\nabla^3 Q_{m+1}|^2+\frac{1}{\eta^3}|\nabla^2 Q_{m+1}|^2\,dxdt
\leq \frac{\alpha}{16}\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^3 Q_{m+1}|^2\,dxdt+CM_1^5s.
\end{align*}
We can apply \eqref{Omega m t}, \eqref{pt Q+1} and \eqref{D2 Q+1 r} to give
\begin{align*}
&\frac\alpha4 \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_{m+1}(x,s)|^2\,dx+\frac1{2} \int^s_0\int_{\ensuremath{\mathbb{R}}^3} |\nabla \partial_t Q_{m+1}|^2\,dxdt
\\
\leq&\frac{\Lambda(1+4K^2)}2 \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_0|^2\,dx+\frac{\alpha}{4}\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla^3 Q_{m+1}|^2\,dxdt
\\
&+C\int^s_0\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v_{m+1}|^2\,dxdt+CM_1^2(\eta_1 +s+\frac{s}{\eta_1^3})+CM_1^5s.
\alabel{D t Q final m}
\end{align*}
Adding \eqref{D t Q final m} to \eqref{D2 Q m}, we obtain
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_{m+1}(x,s)|^2\,dx+\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla^3 Q_{m+1}|^2+|\nabla \partial_t Q_{m+1}|^2\,dxdt
\\
\leq&C\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_0|^2\,dx+C\int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v_{m+1}|^2\,dx +CM_1^2(\eta_1 +s+\frac{s}{\eta_1^3})+CM_1^5s+CM_1s
.\alabel{Q m final}
\end{align*}
Here $C$ only depends on the following constants $\alpha, K$ and $\Lambda$.
To estimate $\nabla^2 v_{m+1}$ in \eqref{Q m final}, we multiply \eqref{v +1} by $-\Delta v_{m+1}$ and compute
\begin{align*}
&\frac12\int_{\ensuremath{\mathbb{R}}^3} |\nabla v_{m+1}(x,s)|^2 \,dx+\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2v_{m+1}|^2\,dxdt
\\
&-\int^s_0\int_{\ensuremath{\mathbb{R}}^3}[Q_m,h(Q_m,Q_{m+1})]_{ij}\nabla_j\Delta (v_{m+1})_i\,dxdt
\\
=&\frac12\int_{\ensuremath{\mathbb{R}}^3} |\nabla v_0|^2 \,dx-\int^s_0\int_{\ensuremath{\mathbb{R}}^3}\Big((v_m)_j\nabla_j (v_m)_i+\nabla_j\sigma_{ij}(Q_m,\nabla Q_m)\Big)(\Delta v_{m+1})_i \,dxdt
\\
\leq& \frac12\int_{\ensuremath{\mathbb{R}}^3} |\nabla v_0|^2 \,dx+\frac14\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 v_{m+1}|^2\,dxdt+CM_1^2(\eta_1 +\frac{s}{\eta_1^3})
.\alabel{D v +1}
\end{align*}
To cancel the term involving $h(Q_m,Q_{m+1})$ in \eqref{D v +1}, we differentiate \eqref{Q +1} in $x$, multiply by $\nabla h(Q_m,Q_{m+1})$ and obtain
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\(\partial_t Q_{m+1}+[Q_m, \Omega_{m+1}]\),\nabla_\beta h(Q_m,Q_{m+1})\>\,dx +\int_{\ensuremath{\mathbb{R}}^3}|\nabla h(Q_m,Q_{m+1})|^2\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\(-v_m\cdot \nabla Q_m +g_B(Q_m)\), \nabla_\beta h(Q_m,Q_{m+1})\>\,dx
.\alabel{fE +1}
\end{align*}
we choose $A=Q,B=h(Q_m,Q_{m+1}), F=\Delta\nabla v$ in Lemma \ref{Lie} and obtain
\begin{align*}
\<[Q_m,\Delta\Omega_{m+1}],h(Q_m,Q_{m+1})\>=\<\Delta\nabla v_{m+1},[Q_m,h(Q_m,Q_{m+1})]\>
.\alabel{Lie h}
\end{align*}
Note that \[h(Q_m,Q_{m+1})\leq C(|\nabla^2 Q_{m+1}|+|\nabla Q_{m+1}|^2+|\nabla Q_{m+1}||\nabla Q_m|).\] Then using \eqref{Lie h}, we compute the second term in \eqref{fE +1}
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta[Q_m, \Omega_{m+1}],\nabla_\beta h(Q_m,Q_{m+1})\>\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\<[Q_m, \Delta\Omega_{m+1}], h(Q_m,Q_{m+1})\>\,dx
\\
&+\int_{\ensuremath{\mathbb{R}}^3}\<[\Delta Q_m, \Omega_{m+1}]+2[\nabla Q_m, \nabla \Omega_{m+1}], h(Q_m,Q_{m+1})\>\,dx
\\
=&\int_{\ensuremath{\mathbb{R}}^3}\<[Q_m, \Delta\Omega_{m+1}], h(Q_m,Q_{m+1})\>\,dx +\int_{\ensuremath{\mathbb{R}}^3}\<[\nabla Q_m, \nabla \Omega_{m+1}], h(Q_m,Q_{m+1})\>\,dx
\\
&-\int_{\ensuremath{\mathbb{R}}^3}\<[\nabla_\alpha Q_m, \Omega_{m+1}],\nabla_\alpha h(Q_m,Q_{m+1})\>\,dx
\\
\geq&\int_{\ensuremath{\mathbb{R}}^3}\<\Delta\nabla v_{m+1},[Q_m,h(Q_m,Q_{m+1})]\>\,dx -\frac14 \int_{\ensuremath{\mathbb{R}}^3} |\nabla^2 v_{m+1}|^2\,dx
\\
&-\frac12 \int_{\ensuremath{\mathbb{R}}^3} |\nabla h(Q_m,Q_{m+1})|^2\,dx
-\eta_1\int_{\ensuremath{\mathbb{R}}^3}|\nabla^3 Q_m|^2\,dx-CM_1^2 \frac{s}{\eta_1^3}
.\alabel{lie m}
\end{align*}
We repeat the argument in \eqref{pt Q+1} for the first term in \eqref{fE +1}, apply Young's inequality to the right-hand side of \eqref{fE +1}. Then integrate \eqref{fE +1} in $t$ and combine with \eqref{D v +1} yield
\begin{align*}
&\frac12 \int_{\ensuremath{\mathbb{R}}^3}|\nabla v_{m+1}(x,s)|^2 \,dx+\frac12\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2v_{m+1}|^2\,dxdt
\\
\leq&\frac12\int_{\ensuremath{\mathbb{R}}^3}|\nabla v_0|^2\,dx+\frac{\Lambda(1+4K^2)}2\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_0|^2\,dx+CM_1^2(\eta_1 +\frac{s}{\eta_1^3})
\\
&+ \eta_2\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla^3 Q_{m+1}|^2+|\nabla \partial_t Q_{m+1}|^2\,dxdt
\\
&+C(\eta_2)M_1^2(\eta_1 +s+\frac{s}{\eta_1^3})+C(\eta_2)M_1^5s
\alabel{D v final s1}
\end{align*}
for some small $\eta_1,\eta_2$. Substituting \eqref{Q m final} into \eqref{D v final s1} and choosing $\eta_2$ sufficiently small, we obtain the estimates for $v_{m+1}$. Combining the resulting expression with \eqref{Q m final} yields
\begin{align*}
& \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_{m+1}(x,s)|^2+|\nabla v_{m+1}(x,s)|^2 \,dx
\\
&+ \int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla^3 Q_{m+1}|^2+|\nabla \partial_t Q_{m+1}|^2+|\nabla^2v_{m+1}|^2\,dxdt
\\
\leq&CM_1+CM_1^2(\eta_1 +s+\frac{s}{\eta_1^3})+CM_1^5s+CM_1s
.\alabel{D2 final s2}
\end{align*}
Here $C$ only depends on $\alpha,K$ and $ \Lambda$.
It remains to check the $L^2$-norm of the lower order terms in $\mathcal V(0,T)$. We multiply \eqref{v +1} by $v_{m+1}$ to obtain
\begin{align*}
&\frac12\int_{\ensuremath{\mathbb{R}}^3}|v_{m+1}(x,s)|^2 \,dx + \frac12 \int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla v_{m+1}|^2\,dxdt
\\
\leq&\frac12\int_{\ensuremath{\mathbb{R}}^3}|v_0|^2 \,dx +C\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|[Q_{m},h(Q_m,Q_{m+1})]|^2+|\sigma_{ij}(Q_m,\nabla Q_m)|^2\,dxdt
\\
\leq&CM_1+C\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_{m+1}|^2+|\nabla Q_{m+1}|^4+|\nabla Q_m|^4\,dxdt
\leq CM_1+CM_1s
.\alabel{v low}
\end{align*}
By using mean value theorem with the fact that $g_B(Q_e)=0$, we find
\begin{align*}
|g_B(Q_m)|\leq C(K)|Q_m-Q_e|.
\alabel{3}
\end{align*}
Multiplying \eqref{Q +1} by $\partial_t Q_{m+1}$ and $Q_{m+1}-Q_e$ respectively then using \eqref{3} yield
\begin{align*}
&\frac12 \int_{\ensuremath{\mathbb{R}}^3}|(Q_{m+1}-Q_e)(x,s)|^2\,dx+\frac12\int^s_0\int_{\ensuremath{\mathbb{R}}^3}| \partial_t Q_{m+1}|^2\,dxdt
\\
\leq&\frac12\int_{\ensuremath{\mathbb{R}}^3}|Q_0-Q_e|^2\,dx+\frac12 \int^s_0\int_{\ensuremath{\mathbb{R}}^3}|Q_{m+1}-Q_e |^2\,dxdt
\\
&+ C\int^s_0\int_{\ensuremath{\mathbb{R}}^3}|[Q_{m}, \Omega_{m+1}]|^2+|h(Q_m,Q_{m+1})|^2+|v_m \cdot\nabla Q_{m}|^2+|g_B(Q_m)|^2 dxdt
\\
\leq& CM_1+CM_1s
.\alabel{Q m+1 low}
\end{align*}
Note that the $L^2$-norm of $\nabla Q_{m+1}$ from the Sobolev inequality and \eqref{Q m+1 low}. Now adding \eqref{v low} and \eqref{Q m+1 low} to \eqref{D2 final s2}, we have
\begin{align*}
& \big(\| Q (x,s)\|_{H^2_{Q_e}(\ensuremath{\mathbb{R}}^3)}^2+\|v (x,s)\|_{H^1(\ensuremath{\mathbb{R}}^3)}^2 \big)+\|\nabla^3 Q \|_{L^2(0,T ;L^2(\ensuremath{\mathbb{R}}^3))}^2
\\
&+\|\partial_t Q \|_{L^2(0,T ;H^1(\ensuremath{\mathbb{R}}^3))}^2 +\|\nabla^2 v \|_{L^2(0,T_0;L^2(\ensuremath{\mathbb{R}}^3))}^2
\\
\leq&\frac{C_1}4M_1+\frac{C_1}4M_1^2(\eta_1 +s+\frac{s}{\eta_1^3})+\frac{C_1}4M_1^5s+\frac{C_1}4M_1s \leq C_1M_1
\alabel{Q v m+1}
\end{align*}
for some $C_1$ depending on $\alpha, K$ and $\Lambda$. Here in the last step, we set $\eta_1=M_1^{-1}$ and $s\leq \min\left\{\frac12M_1^{-4},\frac12M_1^{-1},\frac12\right\}$.
It remains to verify that $\| Q_{m+1} (x,s)\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}\leq 2K$. Note from \eqref{Q m+1 low} that
\begin{align*}
\int_{\ensuremath{\mathbb{R}}^3}|Q_{m+1}(x,s)-Q_0|^2\,dx=& \int_{\ensuremath{\mathbb{R}}^3}\(\int_0^s \partial_t Q_{m+1}\,dt\)^2\,dx\leq s\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\partial_t Q_{m+1}|^2\,dt\,dx
\\
\leq& s \int_{\ensuremath{\mathbb{R}}^3}\int_0^s |\partial_t Q_{m+1}|^2\,dxdt\leq CM_1s(1+s)\leq CM_1s
\end{align*}for $s\leq \frac12$. By using the Gagliardo–Nirenberg interpolation (c.f. \cite{FHM}) and choosing $s\leq C_2^{-8} K^8M_1^{-4} $, we have
\begin{align*}
\|Q_{m+1}(s)-Q_0\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}\leq& C\| Q_{m+1}(s)-Q_0 \|_{L^2(\ensuremath{\mathbb{R}}^3)}^{\frac14}\|\nabla^2( Q_{m+1}(s)-Q_0)\|_{L^2(\ensuremath{\mathbb{R}}^3)}^{\frac34}
\\
\leq& C_2(M_1s )^{\frac18} M_1 ^{\frac38}\leq K,
\end{align*}
where $C_2$ is independent from $m$. Therefore, we prove Claim 1 by choosing
\[T_{M_1}:=\min\left\{ C_2^{-8} K^8M_1^{-4},\frac12M_1^{-4},\frac12M_1^{-1},\frac12\right\}.\]
\noindent{\bf Claim 2}: There exists $T>0$ such that
\begin{align*}
&\sup_{0\leq t\leq T}\big(\| Q_{m+1}- Q_{m}(t)\|_{H^1_{Q_e}(\ensuremath{\mathbb{R}}^3)}+\|v_{m+1}- v_{m} (t)\|_{L^2(\ensuremath{\mathbb{R}}^3)} \big)
\\
&+\|\nabla^2 (Q_{m+1}- Q_{m}) \|_{L^2(0,T ;L^2(\ensuremath{\mathbb{R}}^3))} +\|\nabla (v_{m+1}- v_{m}) \|_{L^2(0,T;L^2(\ensuremath{\mathbb{R}}^3))}
\\
\leq&\frac12\sup_{0\leq t\leq T}\big(\| Q_{m}- Q_{m-1}(t)\|_{H^1_{Q_e}(\ensuremath{\mathbb{R}}^3)}+\|v_{m}- v_{m-1} (t)\|_{L^2(\ensuremath{\mathbb{R}}^3)}\big)
\\
&+\frac12\|\nabla^2 (Q_{m}- Q_{m-1}) \|_{L^2(0,T ;L^2(\ensuremath{\mathbb{R}}^3))} +\frac12\|\nabla (v_{m}- v_{m-1}) \|_{L^2(0,T;L^2(\ensuremath{\mathbb{R}}^3))}
\end{align*}
For given pairs $(Q_m,v_m)$ and $(Q_{m-1},v_{m-1})\in \mathcal V$, we have
\begin{align*}
&(\partial_t -\Delta)(v_{m+1}-v_{m}) +\nabla (P_{m+1}-P_{m})
\\
=& \nabla\cdot[Q_m,h(Q_{m},Q_{m+1} )]-\nabla\cdot[Q_{m-1},h(Q_{m-1}, Q_m )]
\\
&-v_m\cdot\nabla v_m+v_{{m-1}}\cdot\nabla v_{m-1}+ \sigma(Q_m,\nabla Q_m))-\sigma(Q_{{m-1}},\nabla Q_{{m-1}})),
\alabel{v +2}
\\[2ex]
&\nabla\cdot (v_{m+1}-v_{m})=0,\alabel{div v +2}
\\[2ex]
&\partial_t(Q_{m+1}-Q_{m}) +[Q_m, \Omega_{m+1}]-[Q_{m-1}, \Omega_{m}]
\\
=&\nabla_\beta\partial_{p^\beta}f_E(Q_m,\nabla Q_{m+1})-\nabla_\beta\partial_{p^\beta}f_E(Q_{m-1},\nabla Q_{m})-v_m\cdot \nabla Q_m+v_{m-1}\cdot \nabla Q_{m-1}
\\
&-\partial_Qf_E(Q_m,\nabla Q_m)+\partial_Qf_E(Q_{m-1},\nabla Q_{m-1}) +g_B(Q_m)-g_B(Q_{m-1})
.\alabel{Q +2}
\end{align*}
Multiplying \eqref{Q +2} by $-\Delta(Q_{m+1}-Q_{m})$ yields
\begin{align*}
&\frac12\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m+1}-Q_m)(x,s)|^2\,dx
\\
=&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<[Q_m, \Omega_{m+1}]-[Q_{m-1}, \Omega_{m}],\Delta(Q_{m+1}-Q_{m})\>\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\partial_{p^\beta}f_E(Q_m,\nabla Q_{m+1})-\nabla_\beta\partial_{p^\beta}f_E(Q_{m-1},\nabla Q_{m}),\Delta(Q_{m+1}-Q_{m})\>\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<-v_m\cdot \nabla Q_m+v_{m-1}\cdot \nabla Q_{m-1},\Delta(Q_{m+1}-Q_{m})\>\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<-\partial_Qf_E(Q_m,\nabla Q_m)+\partial_Qf_E(Q_{m-1},\nabla Q_{m-1}),\Delta(Q_{m+1}-Q_{m})\>\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<g_B(Q_m)-g_B(Q_{m-1}),\Delta(Q_{m+1}-Q_{m})\>\,dxdt
.\alabel{Q d}
\end{align*}
Using Young's inequality and \eqref{L4}, we compute the first term in the right-hand side of \eqref{Q d}
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<[Q_m, \Omega_{m+1}]-[Q_{m-1}, \Omega_{m}],\Delta(Q_{m+1}-Q_{m})\>\,dxdt
\\
\leq &\eta\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2\,dxdt +C(\eta)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla (v_{m+1}-v_{m})|^2\,dxdt
\\
&+C(\eta)\int_0^s\|Q_{m}-Q_{m-1}\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}^2\int_{\ensuremath{\mathbb{R}}^3}|v_{m}|^2\,dxdt
\\
\leq &\eta\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2\,dxdt +C(\eta)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla (v_{m+1}-v_{m})|^2\,dxdt
\\
&+C(\eta)M_1\int_0^s\(\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m}-Q_{m-1})|^4\,dx\)^{\frac12}dt
\\
\leq &\eta\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2\,dxdt +C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla (v_{m+1}-v_{m})|^2\,dxdt
\\
&+\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m}-Q_{m-1})|^2\,dxdt+CM_1^4\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m}-Q_{m-1})|^2\,dxdt
,\alabel{Q d1}
\end{align*}
where $\eta$ and $\eta_1$ are some small constants to be chosen later.
Applying \eqref{L4} again to the second term in \eqref{Q d} and using \eqref{sec2 f_E} yields
\begin{align*}
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla_\beta\partial_{p^\beta}f_E(Q_m,\nabla Q_{m+1})-\nabla_\beta\partial_{p^\beta}f_E(Q_{m-1},\nabla Q_{m}),\Delta(Q_{m+1}-Q_{m})\>\,dxdt
\\
=&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} p^\nu_{kl}}f_E(Q_m,\nabla Q_{m+1})\nabla^2_{\gamma\nu}(Q_{m+1})_{kl}\nabla^2_{\beta\gamma}(Q_{m+1}-Q_{m})_{ij} \,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} p^\nu_{kl}}f_E(Q_{m-1},\nabla Q_{m})\nabla^2_{\gamma\nu}(Q_{m})_{kl}\nabla^2_{\beta\gamma}(Q_{m+1}-Q_{m})_{ij} \,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} Q_{kl}}f_E(Q_{m},\nabla Q_{m+1})\nabla_\gamma( Q_m)_{kl}\nabla^2_{\beta\gamma}(Q_{m+1}-Q_{m})_{ij}\,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} Q_{kl}}f_E(Q_{m-1},\nabla Q_{m})\nabla_\gamma( Q_{m-1})_{kl}\nabla^2_{\beta\gamma}(Q_{m+1}-Q_{m})_{ij}\,dxdt
\\
\leq&-(\frac{\alpha}2-2\eta)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 (Q_{m+1}-Q_{m})|^2 \,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} p^\nu_{kl}}f_E(Q_m,\nabla Q_{m+1})\nabla^2_{\gamma\nu}(Q_{m})_{kl}\nabla^2_{\beta\gamma}(Q_{m+1}-Q_{m})_{ij} \,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} p^\nu_{kl}}f_E(Q_{m-1},\nabla Q_{m})\nabla^2_{\gamma\nu}(Q_{m})_{kl}\nabla^2_{\beta\gamma}(Q_{m+1}-Q_{m})_{ij} \,dxdt
\\
&+C(\eta)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\partial^2_{p Q}f_E(Q_{m},\nabla Q_{m+1})\nabla Q_m-\partial^2_{p Q}f_E(Q_{m-1},\nabla Q_{m})\nabla Q_{m-1}|^2\,dxdt
\\
\leq&-(\frac{\alpha}2-2\eta)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 (Q_{m+1}-Q_{m})|^2 \,dxdt
\\
&+C(\eta)\int_0^s\(\|Q_m-Q_{m-1}\|_{L^\infty(\ensuremath{\mathbb{R}}^3)}^2\)\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 Q_{m}|^2 \,dx dt
\\
&+C(\eta) \int_0^s\int_{\ensuremath{\mathbb{R}}^3}(|\nabla Q_{m+1}-\nabla Q_{m} |^2+|\nabla Q_{m}|^2|Q_m-Q_{m-1}|^2)|\nabla Q_m|^2\,dxdt
\\
&+C(\eta) \int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla Q_{m}|^2|\nabla Q_{m}- \nabla Q_{m-1}|^2\,dxdt
\\
\leq&-(\frac{\alpha}2-2\eta)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 (Q_{m+1}-Q_{m})|^2 \,dxdt+\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2\,dxdt
\\
&+C(M_1+M_1^4)\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m}-Q_{m-1})|^2\,dxdt
.\alabel{Q d2}
\end{align*}
The remaining terms in \eqref{Q d} are
\begin{align*}
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<-v_m\cdot \nabla Q_m+v_{m-1}\cdot \nabla Q_{m-1},\Delta(Q_{m+1}-Q_{m})\>\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<-\partial_Qf_E(Q_m,\nabla Q_m)+\partial_Qf_E(Q_{m-1},\nabla Q_{m-1}),\Delta(Q_{m+1}-Q_{m})\>\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<g_B(Q_m)-g_B(Q_{m-1}),\Delta(Q_{m+1}-Q_{m})\>\,dxdt
\\
\leq&\eta\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 (Q_{m+1}-Q_{m})|^2 \,dxdt+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|v_m|^2| \nabla Q_m-\nabla Q_{m-1}|^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |v_m-v_{m-1}|^2 |\nabla Q_{m-1}|^2+|\nabla Q_m|^4|Q_m-Q_{m-1}|^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3} \(|\nabla Q_{m-1}|^2+|\nabla Q_{m}|^2\)|\nabla (Q_m-Q_{m-1})|^2+|Q_m-Q_{m-1}|^2\,dxdt
\\
\leq&\eta\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2\,dxdt +C\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|Q_m-Q_{m-1}|^2\,dxdt
\\
&+\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 (Q_{m}-Q_{m-1})|^2 +|\nabla(v_m-v_{m-1})|^2\,dxdt
\\
&+C(M_1+M_1^4)\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m}-Q_{m-1})|^2+|v_m-v_{m-1}|^2\,dxdt
.\alabel{Q d3}
\end{align*}
Substituting \eqref{Q d1}-\eqref{Q d3} into \eqref{Q d}, we find
\begin{align*}
&\frac12\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m+1}-Q_m)(x,s)|^2\,dx+\frac{\alpha}{4}\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 (Q_{m+1}-Q_{m})|^2 \,dxdt
\\
\leq&C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla (v_{m+1}-v_{m})|^2\,dxdt +C\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|Q_m-Q_{m-1}|^2\,dxdt
\\
&+C\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m}-Q_{m-1})|^2+|\nabla(v_m-v_{m-1})|^2\,dxdt
\\
&+C(M_1+M_1^4)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m}-Q_{m-1})|^2+|v_m-v_{m-1}|^2 \,dxdt
.\alabel{Q d4}
\end{align*}
Now, we compute the difference $Q_{m+1}-Q_{m}$. Multiplying $\eqref{Q +2}$ by $Q_{m+1}-Q_{m}$, one can show
\begin{align*}
&\frac12\int_{\ensuremath{\mathbb{R}}^3}|(Q_{m+1}-Q_m)(x,s)|^2\,dx
\\
\leq &CM_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|Q_{m+1}-Q_{m}|^2\,dxdt+ C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla (v_{m+1}-v_{m})|^2\,dxdt
\\
&+C\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2+|\nabla(v_m-v_{m-1})|^2\,dxdt
\\
&+C(M_1+M_1^4)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m}-Q_{m-1})|^2+|v_m-v_{m-1}|^2+|Q_m-Q_{m-1}|^2\,dxdt
.\alabel{Q d0}
\end{align*}
Combining \eqref{Q d0} with \eqref{Q d4}, we find
\begin{align*}
&\int_{\ensuremath{\mathbb{R}}^3}|(Q_{m+1}-Q_m)(x,s)|^2+|\nabla(Q_{m+1}-Q_m)(x,s)|^2\,dx
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 (Q_{m+1}-Q_{m})|^2 \,dxdt
\\
\leq&C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla (v_{m+1}-v_{m})|^2\,dxdt+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|Q_{m+1}-Q_{m}|^2\,dxdt
\\
&+C(M_1+M_1^4)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m}-Q_{m-1})|^2+|v_m-v_{m-1}|^2+|Q_m-Q_{m-1}|^2\,dxdt
\\
&+C\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2+|\nabla(v_m-v_{m-1})|^2\,dxdt
.\alabel{Q d final}
\end{align*}
Next we compute the difference involving $v_m$. Multiplying \eqref{v +2} by $(v_{m+1}-v_{m})$, we have
\begin{align*}
&\frac12\int_{\ensuremath{\mathbb{R}}^3}|(v_{m+1}-v_m)(x,s)|^2\,dx +\frac34\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(v_{m+1}-v_{m})|^2\,dxdt
\\
\leq&C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\sigma(Q_m,\nabla Q_m))-\sigma(Q_{{m-1}},\nabla Q_{{m-1}})|^2\,dxdt
\\
& +\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla\cdot[Q_m,h(Q_{m},Q_{m+1} )]-\nabla\cdot[Q_{m-1},h(Q_{m-1}, Q_m )], v_{m+1}-v_{m} \>\,dx
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<-v_m\cdot\nabla v_m+v_{{m-1}}\cdot\nabla v_{m-1},v_{m+1}-v_{m}\>\,dxdt
.\alabel{v d}
\end{align*}
Using \eqref{L4}, we find
\begin{align*}
&C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\sigma(Q_m,\nabla Q_m)-\sigma(Q_{{m-1}},\nabla Q_{{m-1}})|^2\,dxdt
\\
\leq& C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_m-Q_{m-1})|^2\(|\nabla Q_{m}|^2+|\nabla Q_{m-1}|^2\)+|Q_m-Q_{m-1}|^2|\nabla Q_{m-1}|^4\,dxdt
\\
\leq& \eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_m-Q_{m-1})|^2dxdt+C(M_1+M_1^4)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_m-Q_{m-1})|^2dxdt
.\alabel{v d1}
\end{align*}
Applying \eqref{L4} to the last term in \eqref{v d}, we have
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<-v_m\cdot\nabla v_m+v_{{m-1}}\cdot\nabla v_{m-1},v_{m+1}-v_{m}\>\,dxdt
\\
=&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\(-v_m\cdot\nabla( v_m-v_{m-1})+(v_{m-1}-v_{m})\cdot\nabla v_{m-1}\)_j(v_{m+1}-v_{m})_j\,dxdt
\\
\leq&C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|v_m||\nabla (v_m-v_{m-1})||v_{m+1}-v_{m}|\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|v_{m-1}|(|\nabla (v_m-v_{m-1})||v_{m+1}-v_{m}|+|v_m-v_{m-1}||\nabla (v_{m+1}-v_{m})|)\,dxdt
\\
\leq&\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla (v_m-v_{m-1})|^2\,dxdt+\frac14\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla (v_{m+1}-v_m)|^2\,dxdt
\\
&+CM_1^4\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|v_{m+1}-v_{m}|^2+|v_m-v_{m-1}|^2\,dxdt
.\alabel{v d2.0}
\end{align*}
Thus we can write \eqref{v d} as
\begin{align*}
&\frac12\int_{\ensuremath{\mathbb{R}}^3}|(v_{m+1}-v_m)(x,s)|^2\,dx+\frac12\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(v_{m+1}-v_{m})|^2\,dxdt
\\
\leq&3\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_m-Q_{m-1})|^2+|\nabla (v_m-v_{m-1})|^2 \,dxdt
\\
&+CM_1^4\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_m-Q_{m-1})|^2+|v_{m+1}-v_{m}|^2 +|v_m-v_{m-1}|^2\,dxdt
\\
& +\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla\cdot[Q_m,h(Q_{m},Q_{m+1} )]-\nabla\cdot[Q_{m-1},h(Q_{m-1}, Q_m )],(v_{m+1}-v_{m})\>\,dx
.\alabel{v d2}
\end{align*}
To cancel the Lie bracket term in \eqref{v d2}, it follows from Lemma \ref{Lie} with the substitution
$A=Q_m,B=h(Q_{m},Q_{m+1}), F=\Omega_{m+1}$ and the other three cases that
\begin{align*}
&\< [Q_m,\Omega_{m+1}]-[Q_{m-1},\Omega_{m}],h(Q_{m},Q_{m+1} )-h(Q_{m-1}, Q_m ))\>
\\
=&\< [Q_m,h(Q_{m},Q_{m+1} )]-[Q_{m-1},h(Q_{m-1}, Q_m )],\nabla(v_{m+1}-v_{m})\>
.\alabel{lie h-}
\end{align*}
Multiplying \eqref{Q +2} by
$h(Q_{m},Q_{m+1})-h(Q_{m-1}, Q_m)$ and using \eqref{lie h-}, we obtain
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla\cdot[Q_m,h(Q_{m},Q_{m+1} )]-\nabla\cdot[Q_{m-1},h(Q_{m-1}, Q_m )],(v_{m+1}-v_{m})\>\,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|h(Q_{m-1}, Q_m)-h(Q_{m},Q_{m+1})|^2\,dxdt
\\
=&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\partial_t(Q_{m+1}-Q_{m}),h(Q_{m},Q_{m+1})-h(Q_{m-1}, Q_m)\>\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<-v_m\cdot \nabla Q_m+v_{m-1}\cdot \nabla Q_{m-1},h(Q_{m},Q_{m+1})-h(Q_{m-1}, Q_m)\>\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<g_B(Q_m)-g_B(Q_{m-1}),h(Q_{m},Q_{m+1})-h(Q_{m-1}, Q_m)\>\,dxdt
\\
\leq&\frac12\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|h(Q_{m-1}, Q_m)-h(Q_{m},Q_{m+1})|^2\,dxdt+\eta_3\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\partial_t(Q_{m+1}-Q_{m})|^2\,dxdt
\\
&+C(\eta_3)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\partial_Qf_E(Q_m,\nabla Q_m)-\partial_Qf_E(Q_{m-1},\nabla Q_{m-1})|^2\,dxdt
\\
&+C(\eta_3)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|v_m\cdot \nabla Q_m-v_{m-1}\cdot \nabla Q_{m-1}|^2+|g_B(Q_m)-g_B(Q_{m-1})|^2\,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\partial_t(Q_{m+1}-Q_{m}),\nabla_\beta\partial_{p^\beta}f_E(Q_m,\nabla Q_{m+1})-\nabla_\beta\partial_{p^\beta}f_E(Q_{m-1}\>\,dxdt
.\alabel{h d1}
\end{align*}
In a similar calculation to \eqref{Q d2}, using \eqref{sec2 f_E}, we estimate the last term in \eqref{h d1}
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\partial_t(Q_{m+1}-Q_{m}),\nabla_\beta\(\partial_{p^\beta}f_E(Q_m,\nabla Q_{m+1})- \partial_{p^\beta}f_E(Q_{m-1},\nabla Q_{m}\)\>\,dxdt
\\
=&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} p^\nu_{kl}}f_E(Q_m,\nabla Q_{m+1})\nabla^2_{\beta\nu}(Q_{m+1})_{kl}\partial_t(Q_{m+1}-Q_{m})_{ij} \,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} p^\nu_{kl}}f_E(Q_{m-1},\nabla Q_{m})\nabla^2_{\beta\nu}(Q_{m})_{kl}\partial_t(Q_{m+1}-Q_{m})_{ij} \,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} Q_{kl}}f_E(Q_{m},\nabla Q_{m+1})\nabla_\beta( Q_m)_{kl}\partial_t(Q_{m+1}-Q_{m})_{ij}\,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} Q_{kl}}f_E(Q_{m-1},\nabla Q_{m})\nabla_\beta( Q_{m-1})_{kl}\partial_t(Q_{m+1}-Q_{m})_{ij}\,dxdt
\\
\leq&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} p^\nu_{kl}}f_E(Q_m,\nabla Q_{m+1})\nabla^2_{\beta\nu}(Q_{m+1}-Q_m)_{kl}\partial_t(Q_{m+1}-Q_{m})_{ij} \,dxdt
\\
&+\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} p^\nu_{kl}}f_E(Q_m,\nabla Q_{m+1})\nabla^2_{\beta\nu}(Q_m)_{kl}\partial_t(Q_{m+1}-Q_{m})_{ij} \,dxdt
\\
&-\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\partial^2_{p^\beta_{ij} p^\nu_{kl}}f_E(Q_{m-1},\nabla Q_{m})\nabla^2_{\beta\nu}(Q_{m})_{kl}\partial_t(Q_{m+1}-Q_{m})_{ij} \,dxdt
\\
&+\eta_3\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\partial_t(Q_{m+1}-Q_{m})|^2 \,dxdt
\\
&+C(\eta_3)\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\partial^2_{p Q}f_E(Q_{m},\nabla Q_{m+1})\nabla Q_m-\partial^2_{p Q}f_E(Q_{m-1},\nabla Q_{m})\nabla Q_{m-1}|^2\,dxdt
\\
\leq&- \frac{\alpha}4 \int_{\ensuremath{\mathbb{R}}^3}|\nabla (Q_{m+1}-Q_{m})(x,s)|^2 \,dx +2\eta_3\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\partial_t(Q_{m+1}-Q_{m})|^2 \,dxdt
\\
&+\eta_2 \int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2\,dx dt+ CM_1^2\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla (Q_{m+1}-Q_{m})|^2 \,dx dt
\\
&+\eta_1 \int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m}-Q_{m-1})|^2\,dx dt+ CM_1^4\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla (Q_{m}-Q_{m-1})|^2 \,dx dt
.\alabel{h d2}
\end{align*}
Here we used the fact from \eqref{Q +1} that
\begin{align*}
\int_{\ensuremath{\mathbb{R}}^3}|\partial_t Q_m|^2\,dx \leq& C\int_{\ensuremath{\mathbb{R}}^3}|\nabla v_{m+1}|^2+|v_m|^2|\nabla Q_m|^2+|\nabla^2 Q_{m+1}|^2\,dx
\\
&+C\int_{\ensuremath{\mathbb{R}}^3}|\nabla Q_{m+1}|^2|\nabla Q_{m}|^2+|\nabla Q_{m}|^4+|g_B(Q_m)|^2\,dx\leq CM_1.
\end{align*} For the term $\partial_t(Q_{m+1}-Q_{m})$, it follows from \eqref{Q +2} that
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\partial_t(Q_{m+1}-Q_{m})|^2 \,dxdt
\\
\leq& C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|v_m\cdot \nabla Q_m-v_{m-1}\cdot \nabla Q_{m-1}|^2+|[Q_m, \Omega_{m+1}]-[Q_{m-1}, \Omega_{m}]|^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|h(Q_m, Q_{m+1})-h(Q_{m-1}, Q_{m})|^2+|g_B(Q_m)-g_B(Q_{m-1})|^2\,dxdt
\\
\leq&C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2+|\nabla(v_{m+1}-v_{m})|^2\,dxdt
\\
&+C\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m}-Q_{m-1})|^2+|\nabla(v_{m}-v_{m-1})|^2\,dxdt
\\
&+CM_1\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m+1}-Q_{m})|^2+|\nabla(Q_{m}-Q_{m-1})|^2\,dxdt
\\
&+CM_1 \int_0^s \int_{\ensuremath{\mathbb{R}}^3}|v_m-v_{m-1}|^2+|Q_m-Q_{m-1}|^2\,dxdt
.\alabel{h d3}
\end{align*}
Substituting \eqref{h d2} into \eqref{h d1} and using with sufficiently small $\eta_3$, we find
\begin{align*}
&\int_0^s\int_{\ensuremath{\mathbb{R}}^3}\<\nabla\cdot[Q_m,h(Q_{m},Q_{m+1} )]-\nabla\cdot[Q_{m-1},h(Q_{m-1}, Q_m )],(v_{m+1}-v_{m})\>\,dxdt
\\
&+\frac12\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|h(Q_{m-1}, Q_m)-h(Q_{m},Q_{m+1})|^2\,dxdt
\\
\leq&2\eta_2 \int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2\,dx dt+\frac14\int_0^s\int_{\ensuremath{\mathbb{R}}^3} |\nabla(v_{m+1}-v_{m})|^2\,dxdt
\\
&+C\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m}-Q_{m-1})|^2+|\nabla(v_{m}-v_{m-1})|^2\,dxdt
\\
&+C(M_1+M_1^2+M_1^4)\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m+1}-Q_{m})|^2+|\nabla(Q_{m}-Q_{m-1})|^2\,dxdt
\\
&+C(M_1+M_1^4)\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|v_m-v_{m-1}|^2+|Q_m-Q_{m-1}|^2\,dxdt
.\alabel{h d f}
\end{align*}
Adding \eqref{h d f} to \eqref{v d2}, we have
\begin{align*}
&\frac12\int_{\ensuremath{\mathbb{R}}^3}|v_{m+1}-v_m|^2(x,s)\,dx+\frac14\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla(v_{m+1}-v_{m})|^2\,dxdt
\\
\leq&2\eta_2 \int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2\,dx dt
\\
&+C\eta_1\int_0^s\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m}-Q_{m-1})|^2+|\nabla(v_{m}-v_{m-1})|^2\,dxdt
\\
&+C(M_1+M_1^2+M_1^4)\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|\nabla(Q_{m+1}-Q_{m})|^2+|\nabla(Q_{m}-Q_{m-1})|^2\,dxdt
\\
&+C(M_1+M_1^4)\int_0^s \int_{\ensuremath{\mathbb{R}}^3}|v_m-v_{m-1}|^2+|Q_m-Q_{m-1}|^2\,dxdt
.\alabel{v f}
\end{align*}
Substituting \eqref{Q d final} into \eqref{v f} and choosing suitable $\eta_2$, we obtain
\begin{align*}
&\sup_{0\leq s\leq T}\int_{\ensuremath{\mathbb{R}}^3}\(|Q_{m+1}-Q_m|^2+|\nabla(Q_{m+1}-Q_m)|^2+|v_{m+1}-v_m|^2\)(x,s)\,dx
\\
&+\int_0^T\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2 (Q_{m+1}-Q_{m})|^2+|\nabla(v_{m+1}-v_{m})|^2\,dxdt
\\
\leq& C_3\eta_1\int_0^T\int_{\ensuremath{\mathbb{R}}^3}|\nabla^2(Q_{m+1}-Q_{m})|^2+ |\nabla^2(Q_{m}-Q_{m-1})|^2+|\nabla(v_{m}-v_{m-1})|^2\,dxdt
\\
&+C_3(M_1+M_1^2+M_1^4)s\sup_{0\leq s\leq T}\int_{\ensuremath{\mathbb{R}}^3}\(|\nabla(Q_{m+1}-Q_{m})|^2+|\nabla(Q_{m}-Q_{m-1})|^2\)(x,s)\,dx
\\
&+C_3(M_1+M_1^4)s\sup_{0\leq s\leq T} \int_{\ensuremath{\mathbb{R}}^3}\(|v_m-v_{m-1}|^2+|Q_m-Q_{m-1}|^2\)(x,s)\,dx,
\alabel{diff}
\end{align*}
where $C_3$ is a constant independent from $m$. Then for $m>1$, choosing $\eta_1=\frac18C_3^{-1}$, we prove the claim 2 with
\[T:=\frac1{8C_3 }\min\{M_1^{-1},M_1^{-2},M_1^{-4}\}.\]
Using the Claim 1, $(Q_{m+1},v_{m+1})$ and $(Q_m,v_m)$ have two limits. By Claim 2, $(Q_{m+1},v_{m+1})$ is a Cauchy sequence in
$L^{\infty} ([0, T]; H^1_{Q_e}\times L^2) \cap L^2([0,T]; H^2_{Q_e}\times H^1])$, so
two weak limit of $(Q_{m+1},v_{m+1})$ and $(Q_m,v_m)$ are the same. One can estimate $P_m$ using \eqref{v +1} and the argument in \eqref{eq P}-\eqref{eq P2}. As $m\to \infty$, we prove Theorem \ref{thm loc}.
\end{proof}
|
1,108,101,565,632 | arxiv | \section{Introduction}
\label{intro}
The investigation of the physical mechanisms for the generation of ocean waves by wind has a long history which starts at the beginning of the 20th century~\cite{1925RSPSA.107..189J}
and is still ongoing. The problem is highly
nonlinear~\cite{JanssenBook} and the feedback at the air-water interface between wind and water waves is difficult to study experimentally and theoretically because of turbulence in both fluids.
The problem can be simplified at first by neglecting currents in the water and by considering the so-called wind-driven wave regime which is characterised by growing seas with wave ages $c_p/u^* < 30$, where $c_p$ is the phase velocity of the water waves and $u^*$ is the friction velocity of wind over water waves~\cite{Sullivan2010}. Direct field measurements of the pressure induced by airflow on waves are rare, thus there is no agreement in the scientific community on the underlying mechanisms leading to wave amplification~(for a review see \cite[Chap.~3]{JanssenBook} and \cite{Sullivan2010}).
In the shear flow model introduced by Miles~\cite{Miles1957,Janssen1991} the rate of energy transfer from the wind to a wave propagating at phase velocity $c_p$ is
proportional to the wind profile curvature $U''(z_c)$ at the critical height $z_c$
where the wind speed
equals the phase velocity of the wave, $U(z_c) = c_p$. The Miles mechanism
has been recently confirmed in field experiments, in particular for long waves~\cite{Hristov2003}.
For a logarithmic velocity profile in the boundary layer,
the Miles growth rate $\Gamma_M$ results in~\cite{Miles1957,Banner2002,Kharif2010}
\begin{equation}
\frac{\Gamma_M}{\omega} = \frac{\Gamma_M}{2\pi f}\equiv \frac{1}{\omega E}\frac{dE}{dt} = \delta\, \alpha
\left(\frac{u^*}{c_p}\right)^2
\label{Miles}
\end{equation}
where $E$ is the wave energy, $f$ is the frequency of the carrier wave, $\delta = \rho_a/\rho_w$ is
the density ratio ($1.29\times 10^{-3}$ between air and water), and $\alpha$ is an empirical constant of the order of 32.5 in the wind-driven wave
regime~\cite{Banner2002}. The pressure
$P$ induced at the water surface then depends on the surface elevation $\eta$ as follows~\cite{Miles1957,Kharif2010}
\begin{equation}
\frac{1}{\rho_w} P(x,t) = \frac{\Gamma_M}{f} \frac{c_p^2}{2\pi}\, \eta_x(x,t)
\label{pressure}
\end{equation}
Typical values of $\Gamma_M/f$ are shown in Fig.~1 of~\cite{Banner2002} (in that figure $\Gamma_M = \gamma$) or in Fig.~1 of~\cite{Farrell2008} (where $\Gamma_M = \beta$) as a function of the wave age $c_p/u^*$.
They range from $10^{-3}$-$10^{-2}$ for fast-moving waves ($c_p/u^*> 5$) to $10^{-2}$-1 for slow-moving waves and laboratory tank experiments ($c_p/u^*\le 5$).
Thus, the growth rate can be regarded as a small parameter in the wind-driven wave regime and
generally it is assumed that $\Gamma_M/f = O(\epsilon^2)$, where $\epsilon = a k$ is the wave steepness, $a$ being the amplitude of the vertical water displacement $\eta$ and
$k$ the wavenumber of the water wave. For weak-nonlinear waves the steepness is indeed small
and in ocean waves it is smaller than 0.55, the value for which wave-breaking
occurs~\cite{Toffoli2010}.
The case $\Gamma_M/f = O(\epsilon^2)$ gives rise to the following damped/forced
nonlinear Schr\"odinger equation~\cite{Leblanc2007,Kharif2010,OnoratoProment2012}
\begin{equation}
i\frac{\partial a}{\partial t} - \frac{1}{8} \frac{\omega}{k^2}\frac{\partial^2 a}{\partial x^2} -
\frac{1}{2} \omega k^2 |a|^2 a = i \left(\frac{\Gamma_M}{2} - 2\nu k^2\right) a
\label{dampedForced}
\end{equation}
where $\nu$ is the kinematic viscosity.
Thus, the case $\Gamma_M/f = O(\epsilon^2)$ describes the quasi-equilibrium
between wind and damping effects due to viscosity.
In this Letter, we consider the case $\Gamma_M/f= O(\epsilon)$, corresponding to stronger winds, the effect of which overcomes the dissipation due to viscosity. This case turns out to be relevant
for explaining experimental results obtained in the context of dispersive focusing of waves under the action of wind~\cite{2006EJMF...25..662T,2008NPGeo..15.1023T}.
We will insert the aerodynamic pressure term, given in eq.~(\ref{pressure}),
into the Bernoulli equation evaluated at the ocean surface and we will use the method of multiple scales to obtain the
corresponding nonlinear Schr\"odinger equation in the case of fast-growing waves. Due to the
universality of the NLS equation in many other fields of physics, the considered case can in principle be of interest in other physical
situations where the multiple-scale method can be applied and the forcing term is introduced at first order in the development parameter.
\section{Governing equations and the method of multiple scales (MMS)}
We recall here the equations governing the propagation of surface gravity waves in the presence of wind and the main assumptions used in the method of multiple scales for deriving the NLS equation.
At low viscosity the water-wave problem can be set within the framework of potential flow
theory~\cite{2008Dias} and the two-dimensional flow of a
viscous, incompressible fluid is governed by the Laplace equation
\begin{equation}
\nabla^2 \phi = 0
\end{equation}
where $\phi(x,z)$ is the velocity potential.
This equation is solved together with the kinematic boundary condition at the free surface $\eta(x,t)$
\begin{equation}
\eta_t +\phi_x \eta_x - \phi_z = 2\nu \eta_{xx} \qquad{\rm{at}}~~ z=\eta(x,t)
\label{kinem}
\end{equation}
and at the bottom
\begin{equation}
\phi_z = 0 \qquad{\rm{at}}~~ z=-H
\end{equation}
The other boundary condition is given by the Bernoulli equation which at the free surface takes the form
\begin{equation}
\phi_t + \frac{1}{2}(\phi_x^2 + \phi_z^2 )+ g\eta = - \frac{P} {\rho_w} - 2 \nu \phi_{zz} \qquad{\rm{at}}~~ z=\eta(x,t)
\label{bernoulli}
\end{equation}
where $g$ is the gravity acceleration and $P$ is the excess pressure at the ocean surface in the presence of wind, given by eqs.~(\ref{pressure}) in the context of the Miles mechanism.
We use the method of multiple scales (MMS) to find the terms in the NLS equations which are related to the wind forcing with a growth rate of first order in the wave steepness,
$\Gamma_M/f= O(\epsilon)$. This method is based on the fact that temporal
and spatial scales of the
carrier wave $(1/\omega, 1/k)$ are much smaller than those of the envelope.
MMS has been used for
deriving the NLS equation under the assumption of small nonlinearity, $\epsilon = a k \ll1$, and
narrow spectral width $\Delta k/k \ll 1$~\cite{1972Hasimoto} and successfully applied for including high-order nonlinear terms~\cite{Slunyaev2005} and constant vorticity in water waves~\cite{2012PhFl...24l7102T}, or in other physical contexts. For example, in the context of the propagation of optical waves in nonlinear materials~\cite{1994PhRvA..49..574K}, this method is also known as the slowly varying envelope approximation (SVEA)~\cite{2007PhR...441...47C,2007RPPh...70.1633B}.
The velocity potential $\phi$ and the surface elevation
$\eta$ have the following representations~\cite{Slunyaev2005,2012PhFl...24l7102T}
\begin{eqnarray}
\phi & =& \sum_{j=1}^{\infty} \epsilon^j \phi_j\, , \qquad
\phi_j = \phi_{j0} + \sum_{n=1}^{j} \phi_{jn}\, {\cal E}^n + c.c.
\label{mms1} \\
\eta & =& \sum_{j=1}^{\infty} \epsilon^j \eta_j\, , \qquad
\eta_j = \eta_{j0} +\sum_{n=1}^{j} \eta_{jn}\, {\cal E}^n + c.c.
\label{mms2}
\end{eqnarray}
where the second index in the amplitudes $\phi_{jn}, \eta_{jn}$ refers to the harmonics
\begin{equation}
{\cal E}^n = \frac{1}{2} \exp[in(kx_0-\omega t_0)]
\end{equation}
The velocity potential at the free surface, $\phi(x,z=\eta,t)$, is written as a Taylor expansion around $z=0$:
\begin{eqnarray}
\phi(x,\eta,t) &=& \sum_{j=0}^{+\infty} \frac{\eta_j}{j!} \partial^j_z \phi\arrowvert_{z=0} \label{Taylor}
\end{eqnarray}
The operators for the derivatives are replaced by sums of operators
\begin{eqnarray}
\frac{\partial}{\partial t} &=& \frac{\partial}{\partial t_0} + \epsilon \frac{\partial}{\partial t_1} + \epsilon^2 \frac{\partial}{\partial t_2} + \ldots
\end{eqnarray}
corresponding to fast and slow temporal derivatives,
and analogously for $\partial/\partial x$.
We use the same notation as in Ref.~\cite{2012PhFl...24l7102T} (note however that the order of indices in the amplitudes $\phi_{jn}, \eta_{jn}$ is inverted).
\section{Wind-forced NLS equation}
\label{section:windNLS}
In this section we apply the method of multiple scales for developing the governing equations
in terms of the expansion parameter $\epsilon$.
Terms of linear order in $\epsilon$ give the dispersion relation $\omega = \sqrt{g\sigma k}$, where
$\sigma = \tanh(kH)$, and they are not affected by wind forcing. The wind forcing terms appear in the expansion at second order in the following relations:
\begin{eqnarray}
\label{eq:cg}
&&\frac{\partial A}{\partial t_1} + c_g \frac{\partial A}{\partial x_1} = \Gamma_M\frac{\sigma A}{2} \\
&&\eta_{21}
= \frac{1}{g} \left ( i \omega D + c_g \frac{\partial A}{\partial x_1} + \Gamma_M\frac{ \sigma A}{2}\right)
\label{eq:eta21new}
\end{eqnarray}
where $A= \phi_{11}|_{z=0}$ and $D=\phi_{21}|_{z=0}$.
At third order, the new terms are $\partial \eta_{21}/\partial t_1$ in the kinetic boundary
condition (\ref{kinem}), which must be evaluated using eq.~(\ref{eq:eta21new}), and $(\Gamma_M/\omega) c_p^2 (\partial \eta_{11}/\partial x_1 + \partial \eta_{21}/\partial x_0)$ in the Bernoulli equation at $z=0$, eq.~(\ref{bernoulli}). Including these terms finally
gives the wind-forced NLS equation in the limit of deep-water waves, $kH\gg1$:
\begin{equation}
i \frac{\partial a}{\partial t_2} -\beta_1 \frac{\partial^2 a}{\partial x_1^2} - \beta_2
\frac{\partial a}{\partial x_1}-\beta_3 a -M a |a|^2 = -2i\nu k^2 a
\label{NLSwind}
\end{equation}
where $a = 2iA /c_p$, $\beta_1 = -(dc_g/dk)/2 = \omega/(8k^2)$,
$\beta_2= \Gamma_M ( \omega + c_g k)/(2\omega k)= 3\Gamma_M/(4k)$,
$\beta_3 = \Gamma_M^2 / (8\omega)$ and $M = \omega k^2/2$.
Note that the equation that we obtain for $\Gamma_M/f = O(\epsilon)$ differs, as it should, from the usual equation obtained assuming $\Gamma_M/f = O(\epsilon^2)$. When
$\Gamma_M/f = O(\epsilon^2)$, eq.~(\ref{NLSwind}) reduces to eq.~(\ref{dampedForced}).
Indeed, the terms proportional
to $\Gamma_M$ in eqs.~(\ref{eq:cg})-(\ref{eq:eta21new}) become of higher order.
Moreover, in the Bernoulli equation at $z=0$, the term $(\Gamma_M/\omega)\, c_p^2 (\partial \eta_{11}/\partial x_1 + \partial \eta_{21}/\partial x_0)$ becomes
$(\Gamma_M/\omega)\, c_p^2\, \partial \eta_{11}/\partial x_0$.
This term in the forced NLS
equation reduces for $\Gamma_M/f = O(\epsilon^2)$ to the term
$- i\Gamma_M a /2$, which corresponds to the one on the right-hand side of eq.~(\ref{dampedForced}). As we will see in the next section,
the two terms in the NLS
equation~(\ref{NLSwind}) due to wind forcing correspond to a variation of the dispersion term and of the phase of the wave field.
It is interesting to calculate the energy evolution. The Miles growth rate is recovered from the relations at second-order expansion. Indeed, multiplying eq.~(\ref{eq:cg}) by the complex conjugate $a^*$, adding the obtained
equation to its complex conjugate and integrating by parts yields
\begin{equation}
\frac{dE}{dt_1} = \Gamma_M E
\label{windFirstOrder}
\end{equation}
where the wave energy is defined as
$E = \int |a|^2 dx_1$ and $\rho = |a|^2$ is the energy density. From the same procedure, but starting from the wind-forced NLS equation~(\ref{NLSwind}) at third-order expansion
we obtain
\begin{equation}
\frac{dE}{dt_2} +\int \frac{\partial j}{\partial x_1}dx_1 = \frac{\beta_2}{\beta_1} \int j\, dx_1
-4\nu k^2 E
\label{en2order}
\end{equation}
where $j$ is the energy flux
\begin{equation}
j = -i \beta_1 \left( \frac{\partial a^*}{\partial x_1} a - \frac{\partial a}{\partial x_1} a^* \right)
\end{equation}
The second term in eq.~(\ref{en2order}) disappears if we assume that there are no incoming or outgoing waves at infinity and we get
\begin{equation}
\frac{dE}{dt_2} = \frac{3 \Gamma_M}{c_g} \int j\, dx_1 -4\nu k^2 E
\label{windSecondOrder}
\end{equation}
At this third order, the wave energy is dissipated due to viscosity effects (second term on the right-hand side) and it is amplified (for $\Gamma_M>0$) under the action of wind (first term of the right-hand side).
For comparison, in the case $\Gamma_M/f = O(\epsilon^2)$ one obtains
\begin{equation}
\frac{dE}{dt_1} = 0, \qquad \frac{dE}{dt_2} = (\Gamma_M - 4\nu k^2 ) E
\end{equation}
\section{Reduction of the wind-forced NLS to the standard NLS}
\label{forcedNLS}
In the previous section we have derived the NLS equation in the case where the growth rate of the wave energy due to the wind effect is of first order in the wave steepness, while viscosity is of second order, eq.~(\ref{NLSwind}). In this section we define
a coordinate transformation to obtain the
standard NLS equation with constant coefficients in order to use its well-known solutions.
We neglect the viscosity term (since it was already discussed in~\cite{OnoratoProment2012}) and obtain the following equation
\begin{equation}
i \frac{\partial a}{\partial t} - \beta_1 \frac{\partial^2 a}{\partial x^2}
- \beta_2 \frac{\partial a}{\partial x} - \beta_3 a - M a |a|^2 = 0
\label{eq:CorrOrder1}
\end{equation}
where $\beta_1 = \omega/(8k^2)$,
$\beta_2= 3\Gamma_M/(4k)$, $\beta_3 = \Gamma_M^2 / (8\omega)$ and $M = \omega k^2/2$.
We scale the envelope amplitude as
\begin{equation}
a(x,t) = B(x,t) e^{-i\beta_3 t}
\label{scaleBeta3}
\end{equation}
by changing its phase.
Thus eq.~(\ref{eq:CorrOrder1}) becomes
\begin{equation}
i \frac{\partial B}{\partial t} - \beta_1 \frac{\partial^2 B}{\partial x^2}
- \beta_2 \frac{\partial B}{\partial x} - M B |B|^2 = 0
\label{eq:scaled}
\end{equation}
The coordinate transformation which directly reduces eq.~(\ref{eq:scaled}) to the standard NLS equation with constant coefficients is given by
\begin{equation}
y = -i\beta_2 t + x, \qquad \tau = t
\label{coordTrans}
\end{equation}
Indeed, after this transformation eq.~(\ref{eq:scaled}) results in
\begin{equation}
i \frac{\partial B}{\partial \tau} - \beta_1 \frac{\partial^2 B}{\partial y^2} -M B |B|^2 = 0
\label{standardNLS}
\end{equation}
Thus we have formally mapped the wind-forced NLS equation, eq.~(\ref{eq:CorrOrder1}),
into the standard NLS equation, which has a number of known analytical soliton solutions (Peregrine, Akhmediev and Kuznetsov-Ma solutions), which we will discuss in the next section.
Alternatively,
another coordinate transformation is useful for understanding the physical content of eq.~(\ref{eq:CorrOrder1}). We consider the coordinate transformation $\xi = - e^{-b x}/b$, where
$b = \beta_2/\beta_1 = 3\Gamma_M/c_g$. The derivatives become
\begin{eqnarray}
\frac{\partial B}{\partial x} &=& e^{-b x} \frac{\partial B}{\partial \xi} \\
\frac{\partial^2 B}{\partial x^2} &=& -b e^{-bx } \frac{\partial B}{\partial \xi} + \frac{\partial^2 B}{\partial \xi^2} e^{-2bx}
\end{eqnarray}
and eq.~(\ref{eq:scaled}) reduces to
\begin{equation}
i \frac{\partial B}{\partial t} - \beta_1 (\xi b)^2 \frac{\partial^2 B}{\partial \xi^2} -M B |B|^2 = 0
\label{dispTerm}
\end{equation}
The factor $(\xi b)^2$ modulates the dispersion term and consequently affects the focusing properties of the system.
We check that for $b\to0$, the term $\xi b \to -1$ and we recover the standard NLS equation.
\section{Soliton solutions}
\label{rw}
Eq.~(\ref{standardNLS}) being the standard NLS equation, its solutions include
the Peregrine, the Akhmediev, and the Kuzbetsov-Ma solutions. Here we discuss how the
coordinate transformation (\ref{coordTrans}) modifies these solutions.
We start from the solutions $B(y,\tau)$ of eq.~(\ref{standardNLS}) and we perform the transformation $y = -i\beta_2 t+x$,
$\tau = t$, where $\beta_2$ is the coefficient of the wind-forcing term,
$\beta_2 = 3 \Gamma_M /(4k)$.
Finally, we scale the fields as in eq.~(\ref{scaleBeta3}) to obtain
$a(x,t) = B(x,t) e^{-i\beta_3 t}$.
The analytical form of the Peregrine solution~\cite{peregrine,2013PhR...528...47O} therefore becomes
in the case of fast-growing waves
\begin{eqnarray}
&&a(x,t) = a_0 \exp[-i M a_0^2 t-i\beta_3 t]\times \nonumber \\
&& \left(\frac{4\beta_1(1-2iMa_0^2 t)}{\beta_1+\beta_1(2Ma_0^2\,t)^2 + 2Ma_0^2\, (x-i\beta_2t)^2} -1\right)
\label{Peregrine}
\end{eqnarray}
In Fig.~1 we show the Peregrine solution for $a_0=1$ and $\epsilon = 0.1$. In panels $(a)$-$(b)$
we show the unperturbed Peregrine solution,~eq.~(\ref{Peregrine}) with $\beta_2 = 0$ and $\beta_3 = 0$, while in
panels $(c)$-$(d)$ and $(e)$-$(f)$ we show the wind-forced solution with
$\Gamma_M/f = \epsilon$ and $\Gamma_M/f = 2\epsilon$, respectively.
The effect of the wind is to break the symmetry along the direction of wave propagation and to increase the temporal duration of the rogue wave event. The lifetime and the maximum amplitude of the Peregrine soliton under the effect of wind are shown in Fig.~2 (blue solid and dotted lines). The lifetime of the soliton, defined as the period of time where the rogue wave criterium $a/a_0 > 2.2$ is satisfied, increases as the growth rate increases, while its maximum amplitude remains constant for
$\Gamma_M/f < 1.72 \epsilon$ and slightly increases for $\Gamma_M/f \ge 1.72 \epsilon$. At
$\Gamma_M/f = 1.72 \epsilon$ the maximum splits into two peaks symmetrical
with respect to the $t=0$ line\footnote{It is interesting to note that the values of both amplitudes and lifetimes scale with $\epsilon$ for the Peregrine soliton, so that the results shown in Figs.~1 and 2 are valid for all $\epsilon$. This is not true for the Akhmediev soliton.}. Enhancement of both amplitudes and lifetimes of rogue waves under the effect of wind has indeed been observed in tank experiments and in numerical simulations of dispersive focusing~\cite{2006EJMF...25..662T,2008NPGeo..15.1023T} and nonlinear focusing~\cite{TouboulKharif2006}. In these papers, the parameters are chosen so that the condition
$\Gamma_M/f = O(\epsilon)$ is satisfied, ensuring that their results can be compared to the ones presented here. Note that in the case $\Gamma_M/f=O(\epsilon^2)$ only the soliton maximum amplitude increases, while its lifetime is not affected by the wind~\cite{OnoratoProment2012}. Thus, growth rates of the same order as the steepness are required to reproduce the experimental and numerical results shown in~\cite{2006EJMF...25..662T,2008NPGeo..15.1023T}.
\begin{figure}
\centering
\includegraphics[width=0.23\textwidth]{Pereg1.pdf}
\includegraphics[width=0.23\textwidth]{Pereg1c.pdf}
\includegraphics[width=0.23\textwidth]{Pereg2_small.pdf}
\includegraphics[width=0.23\textwidth]{Pereg2c_small.pdf}
\includegraphics[width=0.23\textwidth]{Pereg2_large.pdf}
\includegraphics[width=0.23\textwidth]{Pereg2c_large.pdf}
\caption{The Peregrine solution for $a_0=1$ and $\epsilon = 0.1$. Panels $(a)$-$(b)$: unperturbed solution with $\Gamma_M =0$; $(c)$-$(d)$: $\Gamma_M/f = \epsilon$; $(e)$-$(f)$:
$\Gamma_M/f = 2\epsilon$.}
\label{fig:1}
\end{figure}
The Akhmediev solution~\cite{1987TMP....72..809A}, which for large negative times corresponds to a perturbed Stokes wave, represents the nonlinear evolution of the
modulational instability. It is periodic in space and its analytical form~\cite{2013PhR...528...47O} becomes for fast-growing waves
\begin{eqnarray}
&&a(x,t) = a_0 \exp[-i M a_0^2 t-i\beta_3 t] \times \nonumber \\
&& \left(\frac{\sqrt{2}\tilde \nu^2 \cosh[\Omega t] - i\sqrt{2} \tilde \sigma \sinh[\Omega t]}{\sqrt{2} \cosh[\Omega t] - \sqrt{2-\tilde\nu^2} \cos[K(x-i\beta_2 t)]} -1\right)
\label{Akhmediev}
\end{eqnarray}
where $\tilde \nu = K\sqrt{\beta_1/M}/a_0$ ($K$ is the wavenumber of the perturbation), $\tilde \sigma = \tilde\nu\sqrt{2-\tilde\nu^2}$
and $\Omega = Ma_0^2 \tilde \sigma$.
In Fig.~3 we show the unperturbed Akhmediev solution for $a_0=1$,
$\epsilon = 0.1$ and $k/K = 5$ in panels $(a)$-$(b)$,
and the wind-perturbed solution in panels $(c)$-$(d)$ and $(e)$-$(f)$ for
$\Gamma_M/f= 0.1$ and $\Gamma_M/f= 0.17$, respectively.
The effect of the wind is similar to the case described for the Peregrine soliton,
as can be expected from the interrelation between the first-order solutions of the NLS equation described in Ref.~\cite{Akhmediev2009}, although it becomes significant at a lower wind
forcing for the same steepness.
The shape of the wave along the direction of wave propagation is distorted,
the maximum amplitude splits into two peaks symmetrical with respect to the $t=0$ line at
$\Gamma_M/f = 0.136$ (for $\epsilon=0.1$) and then it slightly increases for larger values,
and the lifetime increases as the growth rate increases,
as shown in Fig.~2 (red thick-solid and dashed lines) and Fig.~3.
The spatial periodicity is not affected.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth]{ampT.pdf}
\caption{Maximum amplitude $a_{max}/a_{max0}$ and lifetime $T/T_0$ (normalised with respect to the unperturbed values) of the Peregrine (P) soliton (blue solid and dotted lines) and the Akhmediev (A) soliton (red thick-solid and dashed lines) as a function of the growth rate $\Gamma_M/f$ for steepness $\epsilon = 0.1$ and $\Gamma_M/f = O(\epsilon)$. The position of the arrows corresponds to the value of $\Gamma_M/f$ for which the maximum splits into two peaks. }
\label{fig:2}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.23\textwidth]{Akhm1.pdf}
\includegraphics[width=0.23\textwidth]{Akhm1c.pdf}
\includegraphics[width=0.23\textwidth]{Akhm2_small.pdf}
\includegraphics[width=0.23\textwidth]{Akhm2c_small.pdf}
\includegraphics[width=0.23\textwidth]{Akhm2_gamma017.pdf}
\includegraphics[width=0.23\textwidth]{Akhm2c_gamma017.pdf}
\caption{The Akhmediev solution for $a_0=1$, $\epsilon = 0.1$ and $k/K = 5$. Panels $(a)$-$(b)$:
unperturbed solution with $\Gamma_M =0$; $(c)$-$(d)$: $\Gamma_M/f =0.1$;
$(e)$-$(f)$: $\Gamma_M/f = 0.17$.}
\label{fig:3}
\end{figure}
The Kusnetsov-Ma solution~\cite{1979StAM...60...43M} is periodic in time. While the large-time limit for the Akhmediev solution is a small perturbation of a plane wave, the Kusnetsov-Ma solution is never small and cannot grow from the modulational instability. Its analytical form~\cite{2013PhR...528...47O} becomes for fast-growing waves
\begin{eqnarray}
&&a(x,t) = a_0 \exp[-i M a_0^2 t-i\beta_3 t] \times \nonumber \\
&& \left(\frac{-\sqrt{2}\tilde \mu^2 \cos[\Omega t] + i\sqrt{2} \tilde \rho \sin[\Omega t]}{\sqrt{2} \cos[\Omega t] - \sqrt{2+\tilde\mu^2} \cosh[K(x-i\beta_2 t)]} -1\right)
\label{Ma}
\end{eqnarray}
where $\tilde \mu = K\sqrt{\beta_1/M}/a_0$, $\tilde \rho = \tilde \mu \sqrt{2+\tilde\mu^2}$ and
$\Omega = Ma_0^2\tilde\rho$.
In Fig.~4 we show the unperturbed Kusnetsov-Ma soliton for $a_0=1$, $\epsilon = 0.1$ and
$\tilde\mu =\sqrt{2}$ in panels $(a)$-$(b)$, and the wind-perturbed solution in panels
$(c)$-$(d)$ for $\Gamma_M/f = 0.0135$. Even in this case, the effect of the wind is very strong: the amplitude of the wave is modulated in time with an envelope that diverges as $t\to \pm \infty$. Therefore this solution is eliminated by imposing the boundary condition that the ocean surface is unperturbed at $t= -\infty$.
\begin{figure}
\centering
\includegraphics[width=0.23\textwidth]{Ma1.pdf}
\includegraphics[width=0.23\textwidth]{Ma1c.pdf}
\includegraphics[width=0.23\textwidth]{Ma2.pdf}
\includegraphics[width=0.23\textwidth]{Ma2c.pdf}
\caption{The Kusnetsov-Ma solution for $a_0=1$, $\epsilon = 0.1$ and $\tilde\mu =\sqrt{2}$.
Panels $(a)$-$(b)$: unperturbed solution with $\Gamma_M =0$; $(c)$-$(d)$:
$\Gamma_M/f = 0.0135$.}
\label{fig:4}
\end{figure}
\section{Conclusions}
\label{concl}
The forcing term in the physical context of water surface waves is the wind. Different mechanisms have been proposed in the literature for leading to wave amplification under the action of wind. In this Letter we have considered a weakly nonlinear model (the NLS equation) and the Miles mechanism for the wind-wave coupling. The growth rate $\Gamma_M/f$ of the wave energy has been compiled by different authors~\cite{Banner2002,Farrell2008} and its values range from $10^{-3}$-$10^{-2}$ for fast-moving waves ($c_p/u^*> 5$) to $10^{-2}$-1 for slow-moving waves and laboratory tank experiments ($c_p/u^*\le 5$).
This prompted us to investigate the case $\Gamma_M/f = O(\epsilon)$, where $\epsilon$ is the wave steepness, which both in ocean and in tank experiments is of the order of 0.1 or less.
We have used the method of multiple scales for deriving the wind-forced NLS equation with wave growth rate of first order in the wave steepness. Beside wave amplification, the effect of the wind is to modify the dispersion term. A simple coordinate transformation reduces the wind-forced NLS
equation into the standard NLS equation with constant coefficients.
We have thus shown that the
soliton solutions (Peregrine, Akhmediev and Kuznetsov-Ma solutions) are modified in the presence of wind. In particular, the lifetime of both the Peregrine and the Akhmediev solitons increases for large growth rates. We find that the maximum amplitude of these solitons slightly increases for growth rates larger than a certain value, characterised by the fact that two maxima appear at opposite positions with respect to the $t=0$ line. The enhancement of both lifetime and maximum amplitude of rogue waves under the action of wind has been observed in tank experiments and numerical simulations of dispersive focusing~\cite{2006EJMF...25..662T,2008NPGeo..15.1023T}, thus confirming the relevance in this context of the case $\Gamma_M/f=O(\epsilon)$ with respect to $\Gamma_M/f=O(\epsilon^2)$ for which the soliton lifetime does not change under the action of wind~\cite{OnoratoProment2012}.
The results presented here should be tested in wind-wave tank experiments with different ranges of growth rate and steepness to characterise the transition between the two different regimes
$\Gamma_M/f=O(\epsilon)$ and $\Gamma_M/f=O(\epsilon^2)$.
\medskip
\noindent
{\small We would like to thank Jean-Pierre Wolf and Martin Beniston for interesting discussions and the two anonymous referees for useful comments. We acknowledge financial support from the ERC advanced grant "Filatmo" and the CADMOS project.}
\bibliographystyle{elsarticle-num}
|
1,108,101,565,633 | arxiv |
\section{Case study: Toy bars data}
To understand how different training objectives impact both predictive
performance and interpretability of topic-word parameters, we consider
a version of the toy bars dataset inspired by
\cite{griffiths:2004:fst}, but changed so the optimal $\phi$ parameters are distinct for unsupervised LDA and supervised LDA objectives. Our dataset has 144 vocabulary words visualized as pixels in
a square grid in Fig.~\ref{fig:bars_results}.
To generate the observed words $x$, we use 6 true topics: 3
horizontal bars and 3 vertical bars.
However, we generate label $y_d$ using an expanded set of 10 topics, where the extra topics are \emph{combinations} of the 6 bars.
Some combinations produce positive labels, but no single bar does. We train multiple initializations of each possible training
objective and penalty weight setting, and show the best run of each
method in Fig.~\ref{fig:bars_results}. Our conclusions are listed below:
\textbf{Standard training that instantiates $\pi$ can either ignore labels or overfit.}
Fig.~\ref{fig:bars_results}'s first column shows two problematic behaviors with the optimization objective in Eq.~\eqref{eq:def_penalized_map_objective}.
First, when $w_x = 1$, the topic-word parameters are
basically identical whether labels are ignored ($w_y = 0$) or included ($w_y = 1$). Second, when the
observed data is weighted very low ($w_x = 0.01$), we see severe
overfitting, where the learned embeddings at training time are not
reproducible at test time.
\textbf{Ideal end-to-end training can be more predictive but has expensive runtime.} In contrast to the problems with standard training,
we see in the middle column of Fig.~\ref{fig:bars_results} that using
the ideal test-time embedding function $f^*$ also during training can produce much lower error rates on heldout
data. Varying the data weight $w_x$ interpolates between interpretable topic-word parameters $\phi$ and good predictions. One caveat to ideal embedding is its expensiveness: Completing 100 sweeps through this 1000 document toy dataset takes about 2.5 hours using our vectorized pure Python with \texttt{autograd}.
\textbf{Approximate end-to-end training is much cheaper and often does
as well.} We see in the far right column of
Fig.~\ref{fig:bars_results} that using our proposed approximate
embedding $f^{\lambda}$ often yields similar predictive power and
interpretable topic-word parameters when $w_x > 0$. Furthermore, it is about 3.6X faster to train due to avoiding the expensive embedding iterations at every document.
\section{Introduction}
Abundant count data---procedures, diagnoses, meds---are produced during clinical care. An important question is how such
data can assist treatment decisions. Standard pipelines usually involve some dimensionality reduction---there are over
14,000 diagnostic ICD9-CM codes alone---followed by training
on the task of interest. Topic models such as latent Dirichlet allocation (LDA) \citep{blei2012topicmodels} are
a popular tool for such dimensionality reduction
(e.g. \citet{paul2014discovering} or \citet{ghassemi2014unfolding}).
However, especially given noise and irrelevant signal in the data,
this two-stage procedure may not produce the best predictions; thus
many efforts have tried to incorporate observed labels into the dimensionality reduction model. The most natural extension is \emph{supervised LDA} \citep{blei2008sLDA}, though other attempts exist~\citep{zhu2012medlda,
lacoste2009disclda}.
Unfortunately, a recent survey by \citet{halpern2012comparison} finds
that many of these approaches have little benefit, if any, over standard LDA.
We take inspiration from
recent work \citep{chen2015bplda} to develop an optimization algorithm that prioritizes document-topic embedding functions useful for heldout data and allows a penalized balance of generative
and discriminative terms, overcoming problems with traditional maximum likelihood point estimation or more Bayesian approximate posterior estimation. We extend this work with recognition network that allows us to scale to a data set of over 800,000 patient encounters via an approximation to the ideal but expensive embedding required at each document.
\section{Methods}
We consider models for collections of $D$ documents, each drawn from
the same finite vocabulary of $V$ possible word types. Each document
consists of a supervised binary label $y_d \in \{0, 1\}$ (extensions to non-binary labels are straightforward) and $N_d$
observed word tokens $x_d = \{ x_{dn} \}_{n=1}^{N_d}$, with each word
token an indicator of a vocabulary type. We can compactly write $x_d$
as a sparse count histogram, where $x_{dv}$ indicates the \emph{count}
of how many words of type $v$ appear in document $d$.
\subsection{Supervised LDA and Its Drawbacks}
Supervised LDA \citep{blei2008sLDA} is a generative model with the following log-likelihoods:
\begin{align}
\log p(x_d | \phi, \pi_d) &=
\log \mbox{Mult}( x_d | N_d, \textstyle \sum_{k=1}^K \pi_{dk} \phi_k )
= \sum_{v=1}^V x_{dv} \log \left( \textstyle \sum_{k=1}^K \pi_{dk} \phi_{kv} \right)
\\ \notag
\log p(y_d | \pi_d, \eta) &= \log \mbox{Bern}(y_d | \sigma(\eta^T \pi_d) ) =
y_d \log \sigma(\eta^T\pi_d)
+ (1-y_d) \log (1 - \sigma(\eta^T \pi_d) )
\end{align}
where $\pi_{dk}$ is the probability of topic $k$ in document $d$, $\phi_{kv}$ is the probability of word $v$ in topic $k$, $\eta_k$ are coefficients for predicting label $y_d$ from doc-topic probabilities $\pi_d$ via logistic regression, and $\sigma(\cdot)$ is the sigmoid function. Conjugate Dirichlet priors $p(\pi_d)$ and $p(\phi_k)$ can be easily incorporated.
For many applications, we wish to either make predictions of $y_d$ or inspect the topic-word probabilities $\phi$ directly. In these cases, point estimation is a
simple and effective training goal, via the objective:
\begin{align}
\max_{\phi, \pi, \eta}~
w_y \Big( \sum_{d=1}^D \log p(y_d | \eta, \pi_d) \Big)
+ w_x \Big(\log p(\phi) + \sum_{d=1}^D \log p(x_d | \pi_d, \phi) + \log p(\pi_d) \Big)
\label{eq:def_penalized_map_objective}
\end{align}
We include penalty weights $w_x > 0, w_y >
0$ to allow adjusting the importance of the unsupervised data term and the
supervised label term. \citet{taddy2012topicmodelmapestimation} gives a coordinate ascent algorithm for the totally unsupervised objective ($w_x=1, w_y=0$), using natural parameterization to obtain simple updates. Similar algorithms exist for all valid penalty weights.
Two problems arise in practice with such training. First, the standard supervised LDA model sets $w_x = w_y = 1$. However, when $x_d$ contains many words but $y_d$ has a few binary labels, the
$\log p(x)$ term dominates the objective. We see in Fig.~\ref{fig:bars_results} that the
estimated topic word parameters $\phi$ barely change between $w_x=1, w_y=0$ and $w_x=1, w_y=1$ under this standard training.
Second, the impact of observed labels $y$ on topic-word probabilities $\phi$ can be negligible.
According to the model,
when the document-topic probabilities $\pi_d$ are represented, the variables $\phi$ are conditionally independent of $y$.
At training time the $\pi_d$ may be coerced
by direct updates using observed $y_d$ labels to make good
predictions, but such quality may \emph{not} be available at test-time, when $\pi_d$ must be updated using $\phi$ and $x_d$ alone.
Intuitively, this problem comes from the objective treating
$x_d$ and $y_d$ as ``equal'' observations when they are not. Our testing scenario always predicts labels $y_d$ from the words $x_d$. Ignoring this can lead to severe overfitting, particularly when the word weight $w_x$ is small.
\subsection{End-to-End Optimization}
Introducing weights $w_x$ and $w_y$ can help address the first concern
(and are equivalent to providing a threshold on prediction quality).
To address the second concern, we pursue gradient-based inference of a modified version of the objective in
Eq.~\eqref{eq:def_penalized_map_objective} that respects the need to
use the same embedding of observed words $x_d$ into low-dimensional
$\pi_d$ in both training and test scenarios:
\begin{align}
\max_{\phi, \eta}
w_y \Big( \sum_{d=1}^D \log p( y_d | f^*(x_d, \phi), \eta) \Big)
+ w_x \Big( \sum_{d=1}^D \log p( x_d | f^*(x_d, \phi), \phi) \Big)
\label{eq:def_penalized_map_objective_f}
\end{align}
The function $f^*$ maps the counts $x_d$ and topic-word parameters $\phi$ to the optimal unsupervised LDA proportions $\pi_d$. The question, of course, is how to define the function $f^*$.
One can estimate $\pi_d$ by solving a maximum a-posteriori (MAP)
optimization problem over the space of valid $K-$dimensional probability vectors
$\Delta^K$:
\begin{align}
\pi_d' = \max_{\pi_d \in \Delta^{K}} \ell(\pi_d),
\quad \ell(\pi_d) = \log p(x_d | \pi_d, \phi) + \log \mbox{Dir}(\pi_d | \alpha).
\label{eq:objective_for_pi_d}
\end{align}
We can compute $\pi_d'$ via the \emph{exponentiated gradient} algorithm
\citep{kivinen1997exponentiated_gradient}, as described in \citep{sontag2011complexityoflda}.
We begin with a uniform probability vector, and iteratively reweight each entry by the exponentiated gradient until convergence using fixed stepsize $\xi
> 0$:
\begin{align}
\mbox{init:~~}
\pi^{0}_{d} \gets [\frac{1}{K} \ldots \frac{1}{K}].
\quad
\mbox{until converged:~~}
\pi^{t}_{dk} \gets \frac{p^t_{dk}}{ \sum_{j=1}^{K} p^t_{dj} },
\quad
p^t_{dk} = \pi^{t-1}_{dk} \cdot e^{ \xi \nabla \ell(\pi^{t-1}_{dk})}.
\label{eq:def_f}
\end{align}
We can view the final result after $T >> 1$ iterations, $\pi'_d
\approx \pi^T_d$, as a \emph{deterministic} function $f^*(x_d, \phi)$
of the input document $x_d$ and topic-word parameters $\phi$.
\paragraph{End-to-end training with ideal embedding.}
The procedure above does not directly lead to a way to estimate $\phi$ to maximize the objective in Eq.~\eqref{eq:def_penalized_map_objective_f}.
Recently,
\citet{chen2015bplda} developed \emph{backpropagation supervised LDA}
(BP-sLDA), which optimizes
Eq.~\eqref{eq:def_penalized_map_objective_f} under the extreme
discriminative setting $w_y = 1, w_x = 0$ by pushing gradients through
the exponentiated gradient updates above.
We can further estimate $\phi$ under any valid weights with this objective.
We call this ``training with ideal embedding'', because the embedding is optimal under the unsupervised model.
\paragraph{End-to-end training with approximate embedding.}
Direct optimization of the ideal embedding function $f^*$, as done by
\cite{chen2015bplda}, has high implementation complexity and runtime cost. We find in practice that each document requires
dozens or even hundreds of the iterations in Eq.~\eqref{eq:def_f} to
converge reasonably. Performing such iterations at scale and
back-propagating through them is possible with careful C++
implementation but will still be the computational
bottleneck. Instead, we suggest an approximation: use a simpler
embedding function $f^{\lambda}(x_d, \phi)$ which has been trained to
approximate the ideal embedding. Initial experiments suggest
a simple multi-layer perceptron (MLP) recognition network architecture with one hidden layer of size $H \approx 50$ does reasonably well:
\begin{align}
f^{\lambda}_k(x_d, \phi) = \textstyle \mbox{softmax} \Big( \sum_{h=1}^H
\lambda^{\text{output}}_{hk} \sigma(\sum_{v=1}^V
\lambda^{\text{hidden}}_{hv} x_{dv} \phi_{kv}) \Big).
\end{align}
During training, we periodically pause our gradient descent over $\eta, \phi$ and update
$\lambda$ to minimize a KL-divergence loss between the approximate
embedding $f^{\lambda}$ and the ideal, expensive embedding
$f^*$.
\section{Case study: Predicting drugs to treat depression}
We study a cohort of 875080 encounters from 49322 patients drawn from two large academic medical centers
with at least one ICD9 diagnostic code for major depressive
disorder (ICD9s 296.2x or 3x or 311, or ICD10 equivalent).
Each included patient had an identified successful treatment: a prescription repeated at least 3 times in 6 months with no change.
We extracted all procedures, diagnoses, labs, and meds from the EHR
(22,000 total codewords). For each encounter, we built $x_d$ by concatenating count histograms from the last three months and all prior history.
To simplify, we reduced this to the 9621 codewords that occurred in at least 1000 distinct encounters.
The prediction goal was to identify which of 16 common anti-depressants drugs would be successful for
each patient. (Predicting all 25 primaries and 166 augments is future work).
Table~\ref{table:psychtraj_results} compares each method's area-under-the-ROC-curve (AUC) with $K=50$ topics on a held-out set of 10\% of patients. We see that our training algorithm using the ideal embedding $f^*$ improves its predictions over a baseline unsupervised LDA model as the weight $w_x$ is driven to zero. Our approximate embedding $f^{\lambda}$ is roughly 2-6X faster, allowing a full pass through all 800K encounters in about 8 hours, yet offers competitive performance on many drug tasks except for those like desipramine or imipramine for which less than 1\% of encounters have a positive label. Unfortunately, our best sLDA model is inferior to simple bag-of-words features plus a logistic regression classifier (rightmost column ``BoW''), which we guess is due to local optima.
To remedy this, future work can explore improved data-driven initializations.
|
1,108,101,565,634 | arxiv | \section{Introduction}
Continuing theoretical advances in lattice gauge theory, especially in chiral fermion formulations and fermion simulation algorithms, and increasing computational resources are making systematic continuum extrapolation
of many QCD quantities without uncontrolled systematic error a reality.
RBC and UKQCD collaborations have generated dynamical 2+1 flavor Domain Wall Fermion (DWF) ensembles with $a^{-1}\sim 1.7$Gev \cite{DWF16Ls16,DWF16Ls8,DWF24}, which has allowed extrapolation in quark mass and lattice volume.
Table~\ref{table:evol} is a list of existing 2+1 flavor DWF ensembles.
Gauge ensembles with a smaller spacing is the obvious next step in making
continuum extrapolations more systematic.
To this end, RBC and UKQCD collaborations started generating $\beta= 2.25, 32^3 \times 64 \times 16$ dynamical DWF configurations with 2 different light quark masses.
Recently, LHPC collaboration joined this effort and now part of the ensembles are being generated with the joint allocation on DOE QCDOC at Brookhaven Notional Laboratory as a result.
We are aiming at $a^{-1}\sim 2.2$Gev, $m_{PS} L >4 $, which will allow us to get the statistical and systematic errors down to a few percent level
for the lattice studies of quantities such as weak matrix elements and hadron matrix elements.
\begin{table}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|c}
\hline
$\beta$ & $L/a$ & $m_sa$ & $m_la$ & $\hat m_s/\hat m_l$ & $m_{PS} L $ & $\tau$(MD) & Accept.\\
\hline
\multirow{7}{*}{2.13} &
\multirow{3}{*}{$16^3 \times 32 \times 16$} &
\multirow{3}{*}{0.04} & 0.01 & 3.3 & 3.9 & 4000 & \Red{57\%} \\
&& & 0.02 & 1.86 & 5.2 & 4000 & \Red{56\%}\\
&& & 0.03 & 1.3 & 6.2 & 7500 & 82\% \\
\cline{2-8}
&\multirow{4}{*}{$24^3 \times 32 \times 16$} &
\multirow{4}{*}{0.04} & 0.005 & 5.4 & 4.6 & 6518+ & 73\%\\
&& & 0.01 & 3.3 & 5.9 & 4700 & 70\% \\
&& & 0.02 & 1.86 & 7.8 & 2800 & 71\% \\
&& & 0.03 & 1.3 & 9.3 & 2800 & 72\% \\
\hline
\multirow{2}{*}{2.25} &
\multirow{2}{*}{$32^3 \times 64 \times 16$} &
\multirow{2}{*}{0.03} & 0.004 &$\sim 6.6$ & $\sim 4.1$ & 1628+ & 72\% \\
&& & 0.006 & $\sim 4.6$& & 1508+ & 75\% \\
\hline
\end{tabular}
\end{center}
\caption{ (2+1) flavor dynamical DWF ensembles generated by RBC and UKQCD collaborations. $\hat m_{\{l,s\}} = m_{\{l,s\}}+m_{res}$.
The first 2 ensembles with acceptance in boldface are generated with a different variant of Rational hybrid Monte Carlo (RHMC\cite{RHMC}) (RHMC I in \cite{DWF16Ls16}). $\tau$(MD) denotes the total trajectory length in MD units and the numbers with "+" denotes ongoing productions.}
\label{table:evol}
\end{table}
A detailed description of simulation algorithm and performance is given in section~\ref{section:sim} and basic quantities and preliminary mass measurements on $m_{l}=0.004$ ensemble are presented in section~\ref{section:meas}.
\section{Simulation details}
\label{section:sim}
As described in \cite{DWF16Ls16,DWF16Ls8,DWF24,local}, we use the combination of the DWF formulation from Furman and Shamir \cite{shamir} and
Iwasaki gauge action, which is shown to suppress lattice dislocations enough to give DWF good chiral symmetry while allowing for enough topology tunneling for the range of lattice spacings we are interested in.
The simulation of 2 light and 1 strange quarks is actually done as a
combination of
(1+1+1) flavor of strange quarks, done with rational quotient approximation, and 2 flavors of light quark preconditioned by the strange quark\cite{hasenbusch}.
While the preconditioning mass does not have to be the same as the strange quark, we found the strange quark is close to be optimal as the preconditioning mass in DWF simulations on smaller volumes.
Using $ {\cal D}(m_f) = D^\dagger_{DWF}(M_5,m_f) D_{DWF}(M_5,m_f)$ where
$M_5$ is the domain wall height, fixed at 1.8, and $m_f$ is the DWF mass term,
the fermion determinant with the corresponding Pauli-Villars fields
can be written as
{
\begin{align*}
\int \left[ dU \right] \mbox{exp}\left(-\left(S_F[U]+S_{PV}[U]\right)\right) &= \mbox{det}
\left[ \frac{ {\cal D}(m_s)^{1/2}{\cal D}(m_l)} {{\cal D}(1)^{3/2}} \right]
= \mbox{det} \left[\frac{ {\cal D}(m_s)} {{\cal D}(1)} \right]^{3/2}
\mbox{det} \left[\frac{ {\cal D}(m_l)} {{\cal D}(m_s)} \right] \\
\sim
\mbox{det} \left[ {\cal R}_{\frac12} \left(\frac{ {\cal D}(m_s)} {{\cal D}(1)}\right) \right] &
\mbox{det} \left[ {\cal R}_{\frac12} \left( \frac{ {\cal D}(m_s)} {{\cal D}(1)}\right)\right]
\mbox{det} \left[ {\cal R}_{\frac12} \left(\frac{ {\cal D}(m_s)} {{\cal D}(1)}\right) \right]
\mbox{det} \left[\frac{ {\cal D}(m_l)} {{\cal D}(m_s)} \right]
\end{align*}
}
Where ${\cal R}_{a}(x)$ denotes rational approximation of $x^a$ and each determinant term is evaluated by separate pseudofermions.
Omelyan integrator\cite{omelyan} with $\lambda=0.22$ is used in each level of multiscale integrators with $N_{step} = 16$,
$\Delta t_{light} : \Delta t_{heavy} : \Delta t_{gauge} = 1/8: 1/8: 1/48$.
The suppression of force from light quarks from Hasenbusch preconditioning allows us to have the light quark have the biggest step among different terms (the nature of higher-order integrator such as Omelyan effectively makes $\Delta t_{heavy}$ half of $\Delta t_{light}$), decreasing the computational cost significantly. Also, we decided to simulate with trajectory length $\tau= 2$ to make configurations possibly decorrelate more effectively.
The combination of higher-order, multiscale integrators and (rational) quotient terms makes the evolution program a heavily nested one. One way to describe this is
{\large
\begin{align*}
\tau = 2 = 6 MInv + 1 CG + [ [ 12GF+ [ 3 MInv & + 2RF ] \times 3 ] \times 2 + 12GF + 1 CG + HF ] \nonumber \\
\times 32 +[12GF+ [3MInv + 2RF] & \times 3]\times 2 + 12GF + 6 MInv + 1CG
\end{align*}
}
Where $MInv, CG, GF, RF,$ and $ HF$ denote multimass solver for rational quotient terms, inverter for preconditioned light quarks, gauge force,
rational quotient fermion force and quotient fermion force respectively.
Expanding the expression without changing the order or terms gives all the computational routine in an MD trajectory in order.
Algorithm described above is implemented and fully optimized for QCDOC in Columbia Physics System(CPS\cite{cps}).
All of the production runs are done on 4096-node QCDOC partitions at Brookhaven National Laboratory and another 4K partition at Edinburgh Parallel Computing Center. Each partitions are running at 400MHz, which gives 800MFlops/s peak per processor.
Table~\ref{table:perf} shows performances of each routines in the $24^3$ and $32^3$ DWF evolution.
The multimass solver for rational quotient part of the action ($MInv$) is the
dominating part, especially for relatively heavy light quarks $(m_s/m_l <4)$.
While the large number of nodes in each partition and a feature of CPS which allows only even number of sites on each nodes makes it necessary to split the 5th dimension and make strictly 4 dimensional routines such as gauge force duplicate calculation along the 5th dimension in some cases, the effect is at the level of a percent of the total time.
The sum of time on individual routines are slightly less than the total time ($\sim$ 5\% of the total time for $32^3$ ensembles). The most of the descrepancy is from the eigenvalue measurement routines which are run at the time of each Metropolis step to check if the eigenvalues of DWF dirac operators are witin range of the rational quotient approximation.
While the performance of the routine is expected to be
close to that of inverters, it was not measured and we did not include the flops for the numbers included in the table.
As a result, the overall performance slightly less than 200MFlops/s per processor, 25\% of the peak.
A more detailed analysis of mass scaling of each routines can be found in \cite{nhc}.
\begin{table}
\hspace{-1.0cm}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c}
\hline
\multicolumn{7}{c} { $24^3\times 64 \times 16(m_s=0.04)$,
Local volume = $6^3\times 2 \times 8$} \\
\hline
& $m_l=$0.03 & 0.02 & &0.01 &0.005 &\\
\hline
Routines
& time(s) & time(s) & MFlops/s &
time(s) & time(s) & MFlops/s\\
\hline
MInv
& 1225 & 1213 & 221 & 1195 & 1367 & 225 \\
CG
& 173 & 223 & 273 & 370 & 634 & 258 \\
GF
& \Red{60} & \Red{60} & 257 & \Red{62} & \Red{73} & 250 \\
RF
& 218 & 218 & 36 & 232 & 274 & 34 \\
HF
& 10 & 10 & 4.5 & 10 & 12 & 4.5 \\
\hline
Total time(seconds) & 1941 & 1983 & & 2124 & 2635 & \\
\hline
Total MFlops/core& 333557 & 344555 & & 380018 & 489642 & \\
\hline
Total flops($\times 10^{12}$)& 1366 & 1411 & & 1557 & 2006 & \\
\hline
\end{tabular}
\begin{tabular}{c|c|c|c|c}
\multicolumn{5}{c} { } \\
\hline
\multicolumn{5}{c} { $32^3\times 64 \times 16 (m_s=0.03)$ } \\
\hline
$m_l$& \multicolumn{2}{c} { 0.006 } & \multicolumn{2}{|c} { 0.004} \\
\hline
Local volume& \multicolumn{2}{c} { $8^3\times 2 \times 8$ } & \multicolumn{2}{|c} { $4^3\times 8 \times 16$ } \\
\hline
Routines & time(sec) & MFlops/s & time(sec) & MFlops/s \\
\hline
MInv & 5062 & 172 & 4263 & 205 \\
CG & 1964 & 213 & 2038 & 268 \\
GF & \Red{214} & 256 & 104 & 263 \\
RF & 1130 & 25 & 939 & 28 \\
HF & 39 & 4.3 & 10 & 16 \\
\hline
Total time(seconds) & 9035 & & 7733 & \\
\hline
Total MFlops/core & 1344806 & & 1473903 &\\
\hline
Total flops ($\times 10^{12}$) & 5467 & & 6037 &\\
\hline
\end{tabular}
\end{center}
\caption{Performance of computation routines in DWF RHMC on QCDOC. Bold numbers
denotes 4-dimensional routines which are duplicated along 5th dimension when the 5th dimension is split.}
\label{table:perf}
\end{table}
\section{Basic measurements}
\label{section:meas}
Figure~\ref{fig:evol} shows the evolution of the plaquette and the chiral condensate. The time series analysis of these quantities show they have the autocorrelation time of 7-14 MD units, which is smaller than what is reported in \cite{DWF16Ls16} from meson correlators.
Measurements of meson correlators over more configurations than what is available is needed to compare how effectively the RHMC algorithm is generating decorrelated lattice configurations.
\label{section:evol}
\begin{figure}
\begin{minipage}{0.5\textwidth}
\begin{center}
\includegraphics[angle=270,width=3.3in]{figs/plaq.ps}
\end{center}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{center}
\includegraphics[angle=270,width=3.3in]{figs/pbp.ps}
\end{center}
\end{minipage}
\caption{Evolution of the average plaquette and the chiral condensate for
$\beta=2.25, 32^3\times64\times 16$ ensembles.
Average plaquette $\langle P \rangle (m_l=0.004)=0.615574(13)$ and
$\langle P \rangle (m_l=0.006)=0.615591(9)$.
$<\bar\psi \psi >$ shown here is from
$m_l=0.004$ ensemble.}
\label{fig:evol}
\end{figure}
Figures~\ref{fig:mres} and \ref{fig:vec} show the preliminary result
of the residual masses and various hadron masses measurements. Measurements were done on 30 $m_l=0.004$ lattices from MD trajectory length 300-590 with gauge fixed box sources with size 20, placed at $t=0$ and $t=32$.
This was done mostly to ensure the lattice spacing and residual masses are within estimated range and will be measured again with the sources we will use for other measurements. Chiral extrapolation is not attempted as we measurements on only one dynamical mass.
Residual mass is measured by fitting $R(t)$, a ratio of pseudoscalar and mid-point correlator defined as
\begin{displaymath}
R(t)=\frac
{\langle\sum_x J^a_{5q}(x,t)\pi^a(0)\rangle}
{\langle\sum_x J^a_{5}(x,t)\pi^a(0)\rangle}
\end{displaymath}
to a constant between $t=6$ and 32.
($J^a_{\{5,5q\}}(x,t)$ are defined in \cite{DWF16Ls8}.)
While the uncorrelated error may be an underestimate of the real error, it shows the residual mass is $\sim 6\times 10^{-4}$ in lattice units or $\sim$ 1.3Mev. Similarly, fitting meson effective masses gives $a^{-1} \sim$ 2.2Gev. A separate measurement of lattice spacing from the heavy quark potential is in progress.
\begin{figure}
\begin{minipage}{0.5\textwidth}
\begin{center}
\vspace{-5mm}
\includegraphics[angle=270,width=3.3in]{figs/mres.ps}
\end{center}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{center}
\vspace{-5mm}
\includegraphics[angle=270,width=3.3in]{figs/pseudoscalar.ps}
\end{center}
\end{minipage}
\caption{$R(t)$ for different valence quark masses
and the pseudoscalar meson effective mass
on $\beta=2.25, 32^3\times 64 \times 16, m_l=0.004$ ensemble.
Quoted error for $R(t)$ and the mass is from uncorrelated fits, necessary due to long plateaus.}
\label{fig:mres}
\end{figure}
\begin{figure}
\begin{minipage}{0.5\textwidth}
\begin{center}
\includegraphics[angle=270,width=3.3in]{figs/vector.ps}
\end{center}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{center}
\includegraphics[angle=270,width=3.3in]{figs/nucleon.ps}
\end{center}
\end{minipage}
\caption{The vector meson and nucleon effective masses on $m_l=0.004$ ensemble. Error bars are from correlated fits with $\chi^2/d.o.f \sim 1$.}
\label{fig:vec}
\end{figure}
\section{Conclusions and Discussions}
RBC, UKQCD and LHPC joint collaborations are generating dynamical DWF ensembles with a smaller lattice spacing than which are currently available. These ensembles will reduce the systematic error in continuum extrapolation of many important physics quantities. A preliminary measurements suggests $m_{res} \sim 1/500 m_s, a^{-1} \sim 2.2$Gev and the errors from residual chiral symmetry breaking are expected to be $\sim 10^{-4}$ for $B_k$ and $\sim$ 2 \% for $\epsilon' / \epsilon$, according to the estimate in \cite{sharpe}.
While recent advances in HMC algorithms made gauge configuration generation relatively inexpensive, measurements with multiple valence masses still require significant computational resources.
We are currently working to choose the source which will give optimal overlaps with hadron states we are interested in studying. Also, we are studying various deflation techniques proposed recently.\cite{defl}
\section{Acknowledgments}
The author thanks all the members of the RBC and UKQCD collaborations who contributed to the generation of gauge ensembles and the proceeding.
The computations for this work were performed on the QCDOC machines at University of Edinburgh, Columbia University and Brookhaven National Laboratory. C.J. was supported by the U.S. Dept. of Energy under contract DE-AC02-98CH10886.
|
1,108,101,565,635 | arxiv |
\section*{Acknowledgements}
The research leading to these results is supported by the SmartData@PoliTO center for data analysis and Big Data technologies.
\biboptions{sort&compress}
\small
\bibliographystyle{ieeetr}
\section{Introduction}
\label{sec:intro}
Year by year, new services are born as the Internet connection becomes faster and more reliable for end users~\cite{8976293}. The Hypertext Transfer Protocol (HTTP) protocol has been the foundation of the World Wide Web, allowing the transfer of static content through the network. In the late 1990s and early 2000s, other protocols have become popular while supporting a broader spectrum of services. Notable examples are Peer to Peer (P2P) for sharing files and Voice-over-IP (VoIP) for phone calls, made popular by Skype. In the 2010s, social networks and high-definition video streaming have become the new heavy-hitters in terms of involved users and traffic volume, making the Internet a pillar of our working and leisure activities. Such heterogeneous classes of services have different requirements from the network point of view. For example, VoIP requires low latency, while video streaming needs large bandwidth. Their popularity shows that the Internet has succeeded in providing a reliable means for these applications to run and satisfying the users' expectations.
During the last couple of years, some of the largest tech companies like Google and Sony announced new platforms for the so-called \emph{cloud gaming}. In few words, the user plays a videogame using her equipment, while the actual execution takes place on the cloud, and the multimedia content is streamed through the network from the server to the user. This paradigm implies that the user terminal no longer needs to be equipped with powerful hardware, like the Graphic Processing Units (GPUs), to play with recent demanding games but only requires fast and reliable communication with the cloud. These new platforms promise to revolutionize the videogame industry, which has steadily increased in the last decades from basically a not existing market in the 1970s to an estimated USD 159.3 billion sales in 2020 and 2.7 billion players worldwide~\cite{NEWZOO}.
The videogame sector is even estimated to be the most lucrative sector in the whole entertainment market, recently surpassing television. Given these facts, the success of cloud gaming services would be of massive importance for the companies promoting it and for the Internet network handling the resulting traffic.
Google, NVIDIA, and Sony announced three different proprietary platforms for cloud gaming, namely, {\textsf{Stadia}}\xspace, {\textsf{GeForce~Now}}\xspace, and {\textsf{PS~Now}}\xspace. When writing this paper (September 2021), these services are available to users as a complete product. Microsoft also proposed its platform named {\textsf{xCloud}}\xspace, which is currently under testing. Amazon announced {\textsf{Luna}}\xspace, currently under testing as well. The companies disclosed a limited amount of information about the infrastructure of such services and how they utilize the network in terms of employed protocols and bandwidth usage.
In this work, we target the three cloud mentioned gaming services and offer a preliminary characterization of the network workload they generate. We collect $225$ packet traces under different application settings and network conditions.
Leveraging this data, we show which protocols each application adopts for streaming multimedia content, the volume of traffic they generate and the servers and domains they contact.\footnote{For brevity, we use throughout the paper the term \emph{domain} referring to the Fully Qualified Domain Name.} We show that {\textsf{Stadia}}\xspace and {\textsf{GeForce~Now}}\xspace adopt the standard Real Time Protocol (RTP) for streaming~\cite{rfc3550}, largely adopted by Real-Time Communication applications~\cite{nistico2020comparative}, while {\textsf{PS~Now}}\xspace uses an undocumented protocol or encapsulation mechanism. {\textsf{Stadia}}\xspace and {\textsf{GeForce~Now}}\xspace transmit video up to 45 Mbit/s, while {\textsf{PS~Now}}\xspace does not exceed 13 Mbit/s. For comparison, a Netflix 4K video consumes approximately 15 Mbit/s,\footnote{\url{https://help.netflix.com/en/node/87}, accessed October 2021.} making cloud gaming potentially a new heavy hitter of the future Internet. With these characteristics, mobile networks can sustain high-definition gaming sessions only in case of good-quality $4G$ connections, as we quantify using real-world speedtest measurements. Their client applications contact cloud servers located in the autonomous systems (ASes) of the respective corporations. They are no farther than $20$ ms in terms of Round Trip Time from our testing location in northern Italy, a figure low enough for playing a videogame with no noticeable lag. To the best of our knowledge, we are the first to perform a study on these cloud gaming services, namely {\textsf{Stadia}}\xspace, {\textsf{GeForce~Now}}\xspace, and {\textsf{PS~Now}}\xspace, showing their characteristics and peculiarities in terms of network usage.
The remainder of the paper is organized as follows. Section~\ref{sec:related} presents the related work. Section~\ref{sec:datazet} describes our experimental setup, while Section~\ref{sec:results} illustrates the findings we obtain analyzing the packet traces. Finally, Section~\ref{sec:conclusion} concludes the paper. To let other researchers replicate and extend our results, we release sample packet traces available at~\cite{andrea_di_domenico_2021_5509243}.
\section{Related work}
\label{sec:related}
Several research papers have already studied cloud gaming under different perspectives~\cite{cai2016survey}. Authors in~\cite{cai2016survey} focused on the study of cloud gaming platforms and optimization techniques. At the same time, several works provided general frameworks and guidelines for deploying cloud gaming services from the technical~\cite{huang2013gaminganywhere,hong2014placing} and business~\cite{ojala2011developing} points of view.
Other studies focused on the factors affecting the subjective QoE of users~\cite{jarschel2011evaluation,lee2012all} or proposed novel techniques to improve it~\cite{hossain2015audio,cai2013cognitive}.
Some works proposed approaches for optimizing multimedia streaming~\cite{lee2015outatime,shi2011using,hong2015enabling} or GPU usage~\cite{qi2014vgris} in the specific context of cloud gaming.
Interestingly, in 2012 the authors of~\cite{choy2012brewing} concluded that the network infrastructure was unable to meet the strict latency requirements necessary for acceptable gameplay for many end-users.
Industrial pioneers of cloud gaming such as Onlive, StreamMyGame, and Gaikai have already been the object of study~\cite{claypool2012thin,shea2013cloud,chen2013quality,manzano2012empirical}. However, research papers targeting the services recently launched by leading tech companies such as Google and Sony have not yet been published. Only {\textsf{Stadia}}\xspace has been studied by Carrascosa \emph{et al.}~\cite{carrascosa2020cloud}, who investigated its network utilization; however, their work did not compare with other platforms nor explore mobile scenarios. This paper closes this gap and first characterises the generated traffic in terms of employed protocols and network workload.
\section{Measurement collection}
\label{sec:datazet}
In this section, we describe our experimental testbed and the dataset we collect. We focus on three cloud gaming services, namely {\textsf{Stadia}}\xspace, {\textsf{GeForce~Now}}\xspace, and {\textsf{PS~Now}}\xspace, on which we created standard accounts that we use for our experiments. We deploy a testbed using three gaming devices: A PC running Windows with a 4K screen, an Android smartphone, and the dedicated {\textsf{Stadia}}\xspace dongle.
All devices are connected to the Internet through a Linux gateway equipped with two 1\,Gbit/s Ethernet network interfaces and a 300 Mbit/s WiFi 802.11ac wireless card. The Windows PC is connected to the gateway on the first Ethernet card, while the Android smartphone and the {\textsf{Stadia}}\xspace dongle are connected via WiFi. The gateway uses the second Ethernet interface as an upstream link to the Internet, provided by a 1\,Gbit/s Fiber-To-The-Home subscription located in Turin, Italy. Figure~\ref{fig:testbed} sketches our testbed. {\textsf{Stadia}}\xspace runs via the Chrome browser on the Windows PC and its mobile application on the Android phone (we used version 2.13). Moreover, we perform additional experiments using the dedicated Chromecast Ultra dongle, which allows the user to play and connect it to a screen.
{\textsf{GeForce~Now}}\xspace runs from a specific application in both cases (version 1.0.9 for PC and 5.27.28247374 for Android), while {\textsf{PS~Now}}\xspace only works from the PC application (version 11.0.2 was used).
We play the three services making gaming sessions approximately 5-10 minutes long and capturing all the network traffic the devices exchange with the Internet. We seek reliable results by playing a broad spectrum of videogames on all platforms, from first-person shooters to racing and adventure -- e.g., Sniper Elite 4, Destiny, Grid, and Tomb Raider.
\begin{figure}[t]
\centering
\includegraphics[width=0.75\columnwidth]{figures/cloud_gaming_schema.pdf}
\caption{Testbed used for the experimental campaigns.}
\label{fig:testbed}
\end{figure}
With this testbed, we perform five different experimental campaigns, summarized in Table~\ref{tab:dataset}. Firstly, we run different gaming sessions for each platform using the Windows PC, the smartphone and the dongle (when possible). Secondly, we run different gaming sessions by using the Windows PC and by manually configuring the applications to stream video with different \textit{quality levels}. This option is available on {\textsf{Stadia}}\xspace and {\textsf{GeForce~Now}}\xspace. Thirdly, only for {\textsf{GeForce~Now}}\xspace, we instrument the Windows application to use one of the $14$ available data centres by tuning the \textit{server location} application setting. Next, we artificially impair the network in terms of \textit{bandwidth} and \textit{latency} and \textit{packet loss} to study the behavior of the applications under different network conditions. To this end, we run the \texttt{tc-netem} tool on the Linux gateway to progressively decrease the available bandwidth from $100$ to $5$\,Mbit/s, impose additional latency from $10$ to $300$\,ms, or $1$-$10$\% packet loss. For {\textsf{Stadia}}\xspace and {\textsf{GeForce~Now}}\xspace, we replicate all the experiments using both the PC and the smartphone.
Moreover, we perform all experiments also with the {\textsf{Stadia}}\xspace dongle. Finally, we take {\textsf{Stadia}}\xspace as a case study to understand the behaviour with different mobiles networks. To this end, we perform different gaming sessions with the PC and emulated on the Linux gateway different mobile networks using ERRANT~\cite{ERRANT}. ERRANT is a state-of-the-art network emulator which imposes realistic network conditions based on a large-scale measurement campaign under operational mobile networks. ERRANT can reproduce the variability of conditions intrinsically rooted in mobile networks due to different operators, Radio Access Technologies (RATs) (i.e., 3G or 4G), signal quality (e.g., bad quality due to weak signal). ERRANT comes with $32$ network profiles describing the typical network conditions observed in different European operators under 3G and 4G. We also use the ERRANT speedtest training dataset to study the possibility of using {\textsf{Stadia}}\xspace on different conditions under mobile networks.
\begin{table}
\begin{center}
\small
\setlength{\tabcolsep}{4.5pt}
\caption{Overview of the measurement campaign.}
\label{tab:dataset}
\begin{tabular}{|l|c|c|c|c|c|c|c|c|}
\hline Application
& PC
& \makecell{Smart-\\phone}
& \makecell{Don-\\gle}
& \makecell{Quality\\levels}
& \makecell{Server\\location}
& \makecell{Traffic\\shaping}
& \makecell{Mobile\\Networks}
& \makecell{Total\\tests}\\
\hline
{\textsf{Stadia}}\xspace & \checkmark & \checkmark & \checkmark & 3 & & \checkmark & \checkmark & 94 \\
\textsf{Geforce Now} & \checkmark & \checkmark & & 2 & \checkmark & \checkmark & & 71 \\
{\textsf{PS~Now}}\xspace & \checkmark & & & & & \checkmark & & 60 \\
\hline
\end{tabular}
\end{center}
\end{table}
In total, we collect $225$ packet traces, summing to $390$ GB of data. We share with the research community a sample of these traces from the three services at~\cite{andrea_di_domenico_2021_5509243}.
Then, we analyze the traffic traces using the Wireshark packet analyzer.\footnote{\url{https://www.wireshark.org/}.} We also use Tstat~\cite{trevisan2017traffic}, a passive meter, to obtain flow-level logs summarizing the observed TCP/UDP flows. Finally, we use the Chrome debugging console to study {\textsf{Stadia}}\xspace and the disk log files for {\textsf{GeForce~Now}}\xspace.
\section{Results}
\label{sec:results}
We now illustrate the findings we obtain from the analysis of the collected network traces. We first show which network protocols each service employs for streaming and signalling (e.g., user's commands) and analyze in detail the different approaches used for audio and video transmission. We then provide quantitative figures on the volume of traffic the services generate at different video quality levels and study the impact of mobile network scenarios. Finally, we study the contacted servers in terms of Autonomous Systems (ASs), RTT distance and discuss how the infrastructures are organized.
\subsection{Employed protocols}
In this section, we describe the protocols used by the three cloud gaming providers to stream multimedia content and transmit control data for, e.g., session setup and users' commands. Table~\ref{tab:protocols} provides an overview of the protocols we observe, as well as the employed codecs.
\vspace{2mm}
\noindent
\textit{{\textsf{Stadia}}\xspace:} The service from Google uses the most standard protocol mix as it relies on WebRTC~\cite{rfc7478}. In few words, WebRTC is a set of standard application programming interfaces (APIs) that allow real-time communication from browsers and mobile applications. It establishes sessions using the Datagram Transport Layer Security (DTLS) protocol for key exchange. The multimedia connection between client and server is set up using Interactive Connectivity Establishment (ICE), which in turn relies on the Session Traversal Utilities for Network Address Translators (STUN) and the Traversal Using Relays around NAT (TURN) protocols for NAT (Network Address Translator) traversal. We find that {\textsf{Stadia}}\xspace uses WebRTC with no substantial modifications, both from the browser and mobile application. The traffic captures using the dedicated dongle device (Chromecast) confirm that the observed traffic is consistently compatible with WebRTC. When the multimedia session begins, the client starts a normal WebRTC \emph{peer connection} towards the server, creating a UDP flow in which DTLS, STUN and RTP are multiplexed according to the RFC 7893~\cite{rfc7983}. RTP is used for multimedia streaming, while DTLS carries the user's input. We also observe packets of the Real-Time Control Protocol (RTCP)~\cite{rfc3550,rfc4585}, used to exchange out-of-band statistics between the sender and the receiver of a multimedia stream. The RTCP payload is encrypted to enhance users' privacy, preventing the in-network devices from using it for Quality of Service monitoring.
\vspace{2mm}
\noindent
\textit{{\textsf{GeForce~Now}}\xspace:} It adopts a different approach. The server is first contacted using the TLS (over TCP) protocol to set up the session. Interestingly, the Client Hello messages contain the Server Name Indication extension, which allows us to infer the server hostname (see Section~\ref{sec:location} for details).
Then, the client opens multiple UDP channels directly, without relying on the standard session establishment protocols (ICE, STUN and TURN). Only the first packet from the client contains an undocumented hello message. Each inbound flow then carries a standard RTP stream. The client sends the user commands on a dedicated UDP flow using an undocumented protocol. All flows use fixed ports on the client-side, in the range 49003-49006, while they vary on the server-side. Here, we do not observe the presence of the RTCP protocol.
\vspace{2mm}
\noindent
\textit{{\textsf{PS~Now}}\xspace:} This service adopts a completely custom approach, with no standard in-clear header. The client opens a UDP connection towards the server without relying on any documented protocol, and, as such, we can only analyze the raw payload of packets. Still, complex manual work allowed us to catch at least the high-level encapsulation schema that we briefly describe here. The first byte of the packet is used to multiplex multiple data channels. The channel $0$ is used for signalling and user's commands, while $2$ and $3$ are used for multimedia streaming from the server. This is confirmed by the plausible packet size and inter-arrival distributions and allows us to infer which kind of multimedia content is carried on each channel, as we illustrate in Section~\ref{sec:rtp}.
\textbf{Take away:}
\textit{{\textsf{Stadia}}\xspace relies on WebRTC for streaming and session setup. {\textsf{GeForce~Now}}\xspace uses RTP with no standard session establishment protocol. {\textsf{PS~Now}}\xspace employs a fully-custom approach.}
\begin{table}
\caption{Protocol usage for different gaming session components.}
\label{tab:protocols}
\centering
\begin{tabular}{|l|c|c|c|}
\hline
& {\textsf{Stadia}}\xspace & {\textsf{GeForce~Now}}\xspace & {\textsf{PS~Now}}\xspace \\
\hline
Streaming & RTP (and RTCP) & RTP & Custom (UDP) \\
Player's input & DTLS & Custom (UDP) & Custom (UDP) \\
Session setup & DTLS, STUN & TLS & Custom (UDP) \\
Network Testing & RTP & Iperf-like & Custom (UDP) \\
Video Codec & H.264, VP9 & H.264 & - \\
\hline
\end{tabular}
\end{table}
\subsection{Network testing}
All three services have built-in functionalities to probe the network between the client and (multiple) gaming server machines to determine if the conditions are sufficient for a stable gaming session. In few words, the client applications perform a speed test-like measurement to estimate the network delay and bandwidth. We notice that the network testing is not performed consistently on each session startup, but the applications tend to re-probe the network only after a variation of the client IP address.
{\textsf{Stadia}}\xspace performs a speed test based on RTP packets carried over a session established using the standard WebRTC APIs for the multimedia streams. The server (not necessarily the same used for the subsequent gaming session) sends 5-6 MB of data to the client, resulting in a UDP session $5$-$60$ seconds long, depending on the network conditions. The RTP packets are large-sized, around $1200$ bytes on average, but we cannot inspect their payload since it is encrypted.
{\textsf{GeForce~Now}}\xspace uses a schema similar to the one used in the popular tool Iperf\footnote{\url{https://iperf.fr/}, accessed October 2021.}, in which the client sets up a network test over a UDP channel on the server port $5001$ (the same port used by Iperf). The first few packets carry JSON-encoded structures to set up the test. In case the test includes a latency measure, the last flow packets indicate the measured RTT samples. In case the test is only for bandwidth, we observe a stream of large-sized UDP packets lasting 5-10 seconds. Again, the testing server is different from the one used for the subsequent gaming session. The inspection of the JSON messages allows us to understand that the client probes the latency towards multiple alternative measurement servers.
{\textsf{PS~Now}}\xspace adopts a fully-custom approach again. At the beginning of each gaming session, the client performs a few-seconds long bandwidth test using a custom or fully encrypted protocol running over UDP. We cannot infer any information from the packets, for which we only observe that they all have size $1468$ bytes. The test is performed towards a server different from the one used for the proper gaming streaming session. We note similar additional streams consisting of few packets toward a handful of other servers that we conjecture are used to probe the latency towards more endpoints.
\fakepar{Privacy concerns:}
While analyzing the {\textsf{GeForce~Now}}\xspace network testing mechanism, we notice that the client-side control packets used to set up the test expose the user to a severe privacy concern~\cite{ethics}. The user ID is sent in clear into the UDP packet, allowing an eavesdropper to uniquely identify a user even if she changes her IP address or is roaming on another network. We compared the user ID to the user account number that we obtained on the NVIDIA website profile management page, and they match, confirming that the identifier is uniquely associated with the account. Following the best practices for these cases, we signalled the issue to NVIDIA before making our paper public, which plans to resolve it on one of the following updates.
\textbf{Take away:}
\textit{{\textsf{Stadia}}\xspace uses an RTP stream for testing the network, while {\textsf{GeForce~Now}}\xspace relies on an Iperf-like mechanism. Again, {\textsf{PS~Now}}\xspace employs a fully-custom approach. We found a severe privacy leakage in the current {\textsf{GeForce~Now}}\xspace implementation that allows an eavesdropper to obtain the user identifier.}
\subsection{Multimedia streaming}
\label{sec:rtp}
We now analyze how the three services stream the multimedia content (audio and video) from the gaming server to the client. In the case of {\textsf{Stadia}}\xspace and {\textsf{GeForce~Now}}\xspace, we will provide figures extrapolated inspecting the RTP headers, while for {\textsf{PS~Now}}\xspace we can separate the different streams by looking at the first byte of the UDP payload as mentioned in the previous section. In the last row of Table~\ref{tab:protocols} we report the employed video codecs as we extract from the browser/application log files. The widespread H.264 codec is used by both {\textsf{GeForce~Now}}\xspace and {\textsf{Stadia}}\xspace, employing the newer VP9 if the device supports it. For {\textsf{PS~Now}}\xspace, we could not obtain any information about the codecs.
{\textsf{Stadia}}\xspace relies on the WebRTC APIs, and, as such, the multimedia streaming follows its general principles. A single UDP flow carries multiple RTP streams identified by different Source Stream Identifiers (SSRC). A stream is dedicated to the video, while another one to the audio track. We also find a third flow used for video retransmission, as we confirm using the Chrome debug console.\footnote{The associated RTP stream is found to have mimeType \texttt{video/rtx}.} During most of the session, it is almost inactive. At certain moments the flow becomes suddenly active, carrying large packets containing video content. Moreover, this behaviour co-occurs with packet losses and bitrate adaptations on the video stream, as we expect for a video retransmission feature.
{\textsf{GeForce~Now}}\xspace again relies on RTP for multimedia streaming, as described in the previous section. Differently from {\textsf{Stadia}}\xspace, it uses separate UDP flows for the different multimedia tracks, whose client-side port numbers can be used to distinguish the content as NVIDIA publicly declares.\footnote{\url{https://nvidia.custhelp.com/app/answers/detail/a\_id/4504/\~/how-can-i-reduce-lag-or-improve-streaming-quality-when-using-geforce-now}, accessed October 2021.} On port $49005$, a UDP flow carries a single RTP stream for the inbound video. The audio is contained in a UDP flow on port $49003$, in which we find two RTP streams active at the same time.
Regarding {\textsf{PS~Now}}\xspace, we cannot find any header belonging to publicly documented protocols. However, the inspection of several packet captures allows us to infer the encapsulation schema used by the application. A single UDP flow carries all multimedia streams. To multiplex the streams, the first byte of the UDP payload indicates the channel number, followed by a 16-bit long sequence number. Channel $2$ carries the video stream, as we can conclude by looking at packets' packet size and inter-arrival time. Channel $3$ carries the audio track as the packets are small and fixed-sized ($250$ B) and arrive at a constant pace of one every $20$\,ms. We also find channel $0$, especially at the beginning of the flow, which we conjecture is used for signalling. Finally, channel $12$ seldom becomes active, especially in correspondence of large packet losses. As such, we conjecture that it is used for video retransmission or some form of forwarding error correction (FEC), similarly to the {\textsf{Stadia}}\xspace approach.
\textbf{Take away:}
\textit{For {\textsf{Stadia}}\xspace, a single UDP flow carries separate streams for audio, video and video retransmissions. {\textsf{GeForce~Now}}\xspace uses multiple UDP flows on different client-side ports. In {\textsf{PS~Now}}\xspace, a single UDP flow appears to carry an audio, a video and a retransmission/FEC stream.}
\begin{figure}[!t]
\begin{center}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\columnwidth]{figures/stadia_example.pdf}
\caption{{\textsf{Stadia}}\xspace (1080p).}
\label{fig:bitrate_stadia}
\vspace{5mm}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\columnwidth]{figures/geforce_example.pdf}
\caption{{\textsf{GeForce~Now}}\xspace (1080p). }
\label{fig:bitrate_geforce}
\vspace{5mm}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\columnwidth]{figures/psnow_example.pdf}
\caption{{\textsf{PS~Now}}\xspace.}
\label{fig:bitrate_psnow}
\end{subfigure}
\caption{Examples of temporal evolution of bitrate for three gaming sessions. }
\label{fig:bitrate}
\end{center}
\end{figure}
\subsection{Network workload}
We now focus on the workload imposed on the network by users playing on cloud gaming services. We start our analysis with Figure~\ref{fig:bitrate}, in which we show the evolution of a gaming session of around 10 minutes for each service. We made the corresponding packet traces available to the community at~\cite{andrea_di_domenico_2021_5509243}. The picture reports the bitrate of the inbound traffic, due almost exclusively to the video multimedia stream. We first notice that {\textsf{Stadia}}\xspace (Figure~\ref{fig:bitrate_stadia}) has a constant bitrate, while for {\textsf{GeForce~Now}}\xspace and {\textsf{PS~Now}}\xspace (Figures~\ref{fig:bitrate_geforce} and Figure~\ref{fig:bitrate_psnow} respectively) it is considerably more variable. Especially for {\textsf{GeForce~Now}}\xspace, we can root this in the different video codecs, as we describe later in this section. Indeed, the application logs confirm that the variations in the bitrate are not caused by resolution adjustments or codec substitution. Looking at {\textsf{Stadia}}\xspace, the role of the video retransmission stream (green dashed line) is clear, which becomes active in correspondence of impairments in the main video stream (solid red line). We notice a very similar behaviour in {\textsf{PS~Now}}\xspace, which allows us to conjecture the presence of an analogous mechanism.
\begin{figure}[t]
\begin{center}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\columnwidth]{figures/stadia_bitrate.pdf}
\caption{{\textsf{Stadia}}\xspace.}
\label{fig:bitrate_stadia_cdf}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\columnwidth]{figures/geforce_bitrate.pdf}
\caption{{\textsf{GeForce~Now}}\xspace.}
\label{fig:bitrate_geforce_cdf}
\end{subfigure}\\
\vspace{6pt}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=\columnwidth]{figures/psnow_bitrate.pdf}
\caption{{\textsf{PS~Now}}\xspace}
\label{fig:bitrate_psnow_cdf}
\end{subfigure}
\caption{Cumulative distribution of the video bitrate, for different quality video levels.}
\label{fig:bitrate_cdf}
\end{center}
\end{figure}
We summarize the network workload showing in Figure~\ref{fig:bitrate_cdf} the empirical cumulative distribution function of the video bitrate that we observe. For {\textsf{Stadia}}\xspace and {\textsf{GeForce~Now}}\xspace, we report separate distributions for different video resolutions thanks to Chrome debug console and application logs, respectively. For {\textsf{PS~Now}}\xspace, we only report the overall distribution as we cannot extract any statistics from the client app. As mentioned before, {\textsf{Stadia}}\xspace exhibits a rather constant bitrate (Figure~\ref{fig:bitrate_stadia_cdf}). The service allows three streaming resolutions, namely 720p, 1080p and 4K (2160p), whose average bitrate is 11, 29 and 44\,Mbit/s respectively. This is consistent with what is declared in documentation.\footnote{\url{https://support.google.com/stadia/answer/9607891}, accessed October 2021.} {\textsf{Stadia}}\xspace employs both H.264 and VP9 video codecs, with 4K streaming using uniquely VP9.
Different is the picture for {\textsf{GeForce~Now}}\xspace, shown in Figure~\ref{fig:bitrate_geforce_cdf}. The service allows several video resolutions, both in 16:9 and 16:10 aspect ratios. Here, we report the bitrate of the lowest (720p) and the highest (1080p) resolutions available for the 16:9 aspect ratio. Moreover, we show the 540p resolution, which is only adopted automatically in case of bad network conditions, that we trigger imposing a bandwidth limit on the testing machine. The figure shows that the bitrate has large variability, especially for 720p and 1080p. On median, 720p (1080p) consumes 15 (20) Mbit/s, which is consistent with what is declared on the system requirements of the service.\footnote{\url{https://www.nvidia.com/it-it/geforce-now/system-reqs/}, accessed October 2021.} However, the interquartile ranges (IQRs) are in the order of 15 Mbit/s, much more than the 2-3 Mbit/s IQRs observed in {\textsf{Stadia}}\xspace. The bitrate reaches peaks of more than 30 and 40 Mbit/s for 720p and 1080p, respectively. Without access to the unencrypted raw video stream, we conjecture that {\textsf{GeForce~Now}}\xspace makes a highly-dynamic use of the H.264 compression parameters to adapt to different video characteristics (static/dynamic scenes) and network conditions. Indeed, our experiments with limited bandwidth show that, for example, {\textsf{GeForce~Now}}\xspace can sustain a 1080p video stream also with less than 15 Mbit/s available bandwidth without dropping the frame rate, likely adjusting the H.264 compression parameters.
Finally, Figure~\ref{fig:bitrate_psnow_cdf} shows the bitrate distribution for {\textsf{PS~Now}}\xspace. Given the lack of application settings or debug information, we only show the overall distribution of the bitrate. The online documentation recommends a minimum bandwidth of 5 Mbit/s and states that video streaming has a 720p resolution. However, we observe the bitrate reaching up to 13 Mbit/s, with a consistent variability. Interestingly, when we impose a 10 Mbit/s or lower bandwidth limitation on the network, the bitrate adapts consequently (see the peak in the distribution at 5 Mbit/s). However, we cannot link it with a resolution lower than 720p.
\begin{figure}[t]
\centering
\includegraphics[width=0.55\columnwidth]{figures/bitrate_multi_loss.pdf}
\caption{Bitrate distribution with different artificial packet loss.}
\label{fig:loss_multi}
\end{figure}
We now study the impact of packet losses on cloud gaming, focusing on {\textsf{Stadia}}\xspace as a case study. As described in Section~\ref{sec:datazet}, we run experiments in which we enforce artificial packet loss via the \texttt{tc-netem} tool, ranging from $1\%$ to $10\%$. We then study how the application reacts to these scenarios, focusing on the sending video bitrate and quality level. In Figure~\ref{fig:loss_multi}, we show the distribution of bitrate for experiments with 1, 5 and 10\% packet loss and report the bitrate of sessions with no packet loss as a reference (solid red line). We notice that a packet loss of 1 and 5\% (blue and green dashed lines, respectively) does not cause significant bitrate variations. Indeed, the bitrate reaches 30 Mbit/s, and we notice the game is running at 1080p most of the time, with some 720p periods. We notice a constant activity on the \emph{retransmission} stream, used to recover lost packets. Different is the case with 10\% packet loss (yellow dashed line). In this case, {\textsf{Stadia}}\xspace keeps the video resolution at 720p most of the time, and, as such, the bitrate is limited to 10 Mbit/s. We notice a high activity again on the \emph{retransmission} stream, which, however, is not deemed sufficient to go for higher video resolutions. In a few cases, 1080p is achieved, but for short 5-10 second periods. In all the cases, the games were still playable and enjoyable, as reported by the volunteers.
\textbf{Take away:}
\textit{{\textsf{Stadia}}\xspace has three video resolutions with a rather constant bitrate of (median of 11, 29 and 44 Mbit/s). Packet losses up to 5\% do not cause a reduction in resolution. {\textsf{GeForce~Now}}\xspace streams video at many different resolutions, each having a large variability in the bitrate, reaching up to 45 Mbit/s. {\textsf{PS~Now}}\xspace does not exceed 13 Mbit/s, and we cannot infer the employed quality levels. }
\subsection{Cloud gaming under mobile networks}
\label{sec:mobile-emulation}
We now investigate the case of mobile networks to understand to what extent cloud gaming is feasible and what is the reached quality level. Moreover, we are interested in understanding the impact of variable network conditions and how applications react to network impairments. Here, we focus on {\textsf{Stadia}}\xspace, which stream video content at fixed bitrates, requiring strict bandwidth constraints.
We first build on a large-scale measurement campaign run on our previous work~\cite{ERRANT}, to study to what extent current mobile networks are ready to sustain the load and offer suitable conditions for cloud gaming. The dataset we use includes more than $100$k speed test measurements from $20$ nodes/locations equipped with SIM cards of $4$ operators, located in $2$ distinct countries (Norway and Sweden). Each experiment measured latency, downlink, and uplink capacity using multiple TCP connections towards a testing server located in the same country. Moreover, the dataset indicates physical-layer properties such as Radio Access Technology (3G or 4G) and signal strength. The measurement campaign spans the entire 2018 and includes experiments on different hours of the day and days of the week. We use the dataset to understand how a {\textsf{Stadia}}\xspace cloud gaming session would have performed with the measured network characteristics. This is possible thanks to the steady network usage of {\textsf{Stadia}}\xspace, where different video resolutions utilize almost fixed bandwidth. As such, given the network conditions measured on the speedtest measurements, we consider running {\textsf{Stadia}}\xspace ~\emph{feasible} if there is at least 10\,Mbit/s available bandwidth.
In case available bandwidth is in $[10-30)$\,Mbit/s, we conclude that a user could only reach a video resolution of 720p. When bandwidth is in the range $[30-44)$\,Mbit/s, a better $1080$p video could be transmitted, while more than $44$\,Mbit/s allow the user to receive a $4K$ quality video properly.
Figure~\ref{fig:cloud_gaming_mobile} shows the reached video quality levels, offering a breakdown on 3G and 4G and different signal qualities as measured by the node's radio equipment.\footnote{The measured Received signal strength indication (RSSI) is mapped to quality levels as recommended at \url{https://wiki.teltonika.lt/view/Mobile_Signal_Strength_Recommendations.}} The figure shows how a poor 3G link does not allow using {\textsf{Stadia}}\xspace in most cases, as the available bandwidth is below 10\,Mbit/s. The picture changes with medium signal quality, where using {\textsf{Stadia}}\xspace is possible most of the time, with the lowest 720p resolution. An excellent 3G connection allows higher resolution in less than 10\% of cases. With 4G, the network generally offers higher bandwidth, and 1080p and 4K are possible, especially when good signal quality. Indeed, 4G links with high signal strength can sustain {\textsf{Stadia}}\xspace 4K streaming 40\% of the cases, and only 38\% of the times the client is limited to 720p.
\begin{figure}[t]
\centering
\includegraphics[width=0.55\columnwidth]{figures/cloud_gaming_mobile.pdf}
\caption{Estimated resolution of {\textsf{Stadia}}\xspace sessions on mobile networks.}
\label{fig:cloud_gaming_mobile}
\end{figure}
Next, we study the capability of the gaming platforms to offer a good gaming session and cope with different mobile network conditions.
To this end, we use the PC to run a gaming session and track different quality metrics about the frame rate, expressed in Frames Per Second (\textit{FPS}), the packet loss, and the video resolution quality experienced during the gaming sessions. We focus on {\textsf{Stadia}}\xspace as a case study since it is available on many devices.
To study the gaming session in a controlled mobile network environment, we emulate different mobile network conditions on the Linux gateway by using ERRANT. In detail, we perform experiments by using a 4G good network profile as, currently, it is the best quality and most widespread network, and with a 3G good network profile as it is the lowest quality profile showing enough bandwidth to support a gaming session. To reproduce the mobile network variability, we set ERRANT to resample the network condition every 10 seconds, i.e., we pick and apply new constraints for the download rate, upload rate, and latency from the selected profile.
Figure~\ref{fig:example_fps} reports an example of the frame rate (right y-axis), the packet loss (left y-axis), and the video resolution quality when a sudden change happened during a gaming session while using the 4G good profile. Interestingly, we experienced stable performance for most gaming sessions with no packet loss, 60\,FPS, and 1080p resolution. Only when ERRANT picks a download bandwidth below 10\,Mbit/s, we experience, for a short time, a reduction in frame rate and resolution down to 20\,FPS and 720p, respectively, with a seldom increase of the packet loss. Interestingly, the {\textsf{Stadia}}\xspace platform can quickly react and adapt itself by reaching a frame rate of 60\,FPS and resolution at 720p, rising again to 1080p as soon as a better bandwidth is available.
With a 3G good network profile, instead, we could not run an entire gaming session as the network variability introduced by the mobile network caused the game to stop suddenly. This shows how the promising benefits of these solutions currently have limitations that could be overcome with the increasing popularity of fast mobile network technologies -- 4G and, in particular, 5G.
\textbf{Take away:}
\textit{{\textsf{Stadia}}\xspace cannot run in most of the cases with a bad 3G network. With proper signal strength, 3G allows a resolution of 720p. With 4G, it is possible to achieve higher quality levels, and good signal strength allows 4K video resolution in 40\% of situations. Controlled experiments reveal how {\textsf{Stadia}}\xspace quickly reacts to deteriorated network conditions. }
\begin{figure}[t]
\centering
\includegraphics[width=0.7\columnwidth]{figures/example_fps.pdf}
\caption{Example of quality-level reduction in {\textsf{Stadia}}\xspace in terms of frame rate and video resolution subsequent to packet losses.}
\label{fig:example_fps}
\end{figure}
\subsection{Location of gaming machines}
\label{sec:location}
\begin{table}[t]
\begin{center}
\caption{Gaming servers characterization.}
\label{tab:destinations}
\begin{tabular}{|l|c|c|c|c|}
\hline
& Servers & Subnets & ASNs & Owner \\
\hline
{\textsf{Stadia}}\xspace & 74 & 22 & 15169 & Google \\
\hline
{\textsf{GeForce~Now}}\xspace & 37 & 23 & \makecell{11414, \\
20347, \\
50889 }
& NVIDIA \\
\hline
{\textsf{PS~Now}}\xspace & 36 & 2 & 33353 & Sony \\
\hline
\end{tabular}
\end{center}
\end{table}
This section provides a preliminary investigation of the cloud gaming infrastructure regarding the number and location of game servers and employed domains, as observed from our location. This can be useful to identify cloud gaming traffic for, e.g., traffic engineering or content filtering.
We first focus on the remote gaming machines, analyzing the server IP addresses the client applications contact, summarized in Table~\ref{tab:destinations}. Indeed, after the initial setup phase, the client exchanges traffic (almost) uniquely with a single server where the gaming is likely executed.\footnote{We cannot infer if the server IP address acts as an ingress load balancer or reverse proxy.} Considering {\textsf{Stadia}}\xspace, at each session, we contacted a different server -- i.e., we performed $74$ sessions and reached $74$ different server IPs. They lay on $22$ subnets $/24$, all belonging to the Google $15129$ AS.\footnote{We map an IP address to the corresponding AS using an updated RIB from
\url{http://www.routeviews.org/}.} Different is the case for {\textsf{GeForce~Now}}\xspace and {\textsf{PS~Now}}\xspace, for which in roughly $50$\% of the cases we contacted a server IP we had already observed. {\textsf{GeForce~Now}}\xspace servers lay on $23$ subnets belonging to three different ASes, all controlled by NVIDIA. Remind that we used all the $14$ available NVIDIA data centres in our experimental campaign by instrumenting the client application. Finally, all $36$ server IPs for {\textsf{PS~Now}}\xspace belong to only $2$ subnets $/24$ from the $33353$ Sony AS. In terms of latency, additional \texttt{ping} measures show that {\textsf{Stadia}}\xspace and {\textsf{PS~Now}}\xspace servers are 6-8\,ms away from our location while Central European {\textsf{GeForce~Now}}\xspace in the order of 15-20\,ms. However, we cannot link this to the service quality or infrastructure deployment since we perform measurements from a single vantage point. We did not qualitatively observe a significant lag between users' commands and game response without traffic shaping.
Finally, we analyze the domains that the applications contact during the gaming sessions. We extract them by inspecting the client's Domain Name System (DNS) queries before opening a new connection and extracting the Server Name Indication (SNI) field from the Transport Layer Security Security (TLS) Client Hello messages. In the case of {\textsf{Stadia}}\xspace, we only find \texttt{stadia.google.com} as domain-specific to this service. Indeed, the client application contacts a dozen of other domains, but those are shared with all Google services (e.g., \texttt{gstatic.com} and \texttt{googleapis.com}), and, as such, not specific of {\textsf{Stadia}}\xspace. Regarding {\textsf{GeForce~Now}}\xspace, the application contacts the general \texttt{nvidia.com} domain as well as the more specific \texttt{nvidiagrid.net}. Finally, {\textsf{PS~Now}}\xspace relies on the \texttt{playstation} \texttt{.com} and \texttt{.net} domains and their sub-domains, which are, thus, not specific to the {\textsf{PS~Now}}\xspace service. Interestingly, {\textsf{GeForce~Now}}\xspace is the only service that uses domains to identify gaming machines, using subdomains of \texttt{cloudmatchbeta.nvidiagrid.net}. Indeed, for the other services, the domains we find are associated uniquely to the control servers -- used for login, static resources, etc. -- while gaming machines are contacted without a prior DNS resolution.
\textbf{Take away:}
\textit{We seldom observe repeated IP addresses of the gaming machines for the three services. They are all located in the AS of the respective corporate. {\textsf{GeForce~Now}}\xspace is the only one that identifies gaming machines via DNS domain names.}
\section{Conclusions and future directions}
\label{sec:conclusion}
In this paper, we showed important aspects of the recently launched services for cloud gaming, namely {\textsf{Stadia}}\xspace, {\textsf{GeForce~Now}}\xspace, and {\textsf{PS~Now}}\xspace. Millions of users could potentially use them shortly, accounting for a large part of the Internet traffic. Indeed, {\textsf{Stadia}}\xspace and {\textsf{GeForce~Now}}\xspace stream data up to 44 Mbit/s, which is, for instance, much higher than a 4K Netflix movie. However, fast 4G mobile networks can often sustain this load, while traditional 3G connections struggle. Cloud gaming applications rely on the standard RTP protocol for multimedia streaming, except for {\textsf{PS~Now}}\xspace, which does not use any documented protocol.
The services mentioned above entered the market between 2019 and 2020, and, as such, a lot of work is still to be done for characterizing them and understanding their impact on the network. Future directions include studying their infrastructures from different points of view worldwide to find similarities and differences in the deployments. Moreover, we believe it is crucial to conduct campaigns aiming at measuring the subjective Quality of Experience (QoE) enjoyed by the users with human-in-the-loop controlled experiments and also with AI-bot players~\cite{german}. Finally, Microsoft and Amazon will release their cloud gaming platforms, namely {\textsf{xCloud}}\xspace, and {\textsf{Luna}}\xspace, which are in early deployment at the time of writing this article. Any future work on this topic must include them.
|
1,108,101,565,636 | arxiv | \section{Motivation and Examples}
If ${\mathbf{C}}$ is a category then a functor $$F : {\mathbf{C}}^{\mathrm{op}} \to {\mathbf{Set}}$$
also called a ``presheaf over ${\mathbf{C}}$'' is most naturally considered as a ``set
varying over ${\mathbf{C}}$''. Of course, one may consider also contravariant functors
on ${\mathbf{C}}$ taking their values not in ${\mathbf{Set}}$ but in some big category of
structures like {\bf Grp}, {\bf Ab}, {\bf Rng}, {\bf Sp} etc. Typically,
a presheaf $G : {\mathbf{C}}^{\mathrm{op}} \to {\mathbf{Grp}}$ of groups appears as a group
object in $\widehat{{\mathbf{C}}} = {\mathbf{Set}}^{{\mathbf{C}}^{\mathrm{op}}}$ which is a topos if the category
${\mathbf{C}}$ is small.
More generally, one may consider ``presheaves of categories''
$${\mathcal{H}} : {\mathbf{C}}^{\mathrm{op}} \to {\mathbf{Cat}}$$
which notion will soon be axiomatised and generalised to
our central notion of \emph{fibred category}. But before we consider some
examples that (hopefully) will provide some intuition and motivation.
\begin{Exa}\label{ex1}
Let ${\mathbf{C}}$ be the category of monoids and monoid homomorphisms. With every
monoid $M$ one may associate the category $${\mathcal{H}}(M) = {\mathbf{Set}}^{M^{\mathrm{op}}}$$
of right actions of $M$ on some set and with every monoid homomorphism
$h : N \to M$ one may associate the functor
$${\mathcal{H}}(h) = h^* = {\mathbf{Set}}^{h^{\mathrm{op}}} : {\mathbf{Set}}^{M^{\mathrm{op}}} \to {\mathbf{Set}}^{N^{\mathrm{op}}}$$
where $h^*(X,\alpha) : X{\times}N \to X : (x,b) \mapsto \alpha(x,h(b))$.
\hfill \mbox{\ $\lozenge$}
\end{Exa}
\begin{Exa}\label{ex2}
Of course, Example~\ref{ex1} can be generalised by taking for ${\mathbf{C}}$ some
subcategory of the category of (small) categories and instead of ${\mathbf{Set}}$ some
other big category ${\mathcal K}$ (e.g.\ ${\mathcal{K}} = {\mathbf{Ab}}$
and ${\mathbf{C}} = {\mathbf{Cat}}$). \hfill \mbox{\ $\lozenge$}
\end{Exa}
\begin{Exa}\label{ex3}
Let ${\mathbf{E}}$ be an elementary topos (see e.g.\ \cite{Joh}). Then
$${\mathbf{E}}(\_,\Omega) : {\mathbf{E}}^{\mathrm op} \to {\mathbf{Ha}}$$
is a contravariant functor from ${\mathbf{E}}$ to the category ${\mathbf{Ha}}$ of
Heyting algebras and their morphisms. \hfill \mbox{\ $\lozenge$}
\end{Exa}
\begin{Exa}\label{ex4}
Let ${\mathbf{C}}$ be the category ${\mathbf{CRng}}$ of commutative rings with $1$.
Then we may consider the functor
$${\mathcal{H}} : {\mathbf{CRng}}^{\mathrm{op}} \to {\mathbf{Cat}}$$
where ${\mathcal{H}}(R)$ is the category of $R$--modules and for a homomorphism
$h : R^\prime \to R$ the functor ${\mathcal{H}}(h)$ performs ``restriction of
scalars'', i.e.\ ${\mathcal{H}}(h)(M)$ is the $R^\prime$--module with the same
addition as $M$ and scalar multiplication given by
$r \cdot x = h(r) \cdot_M x$. \hfill \mbox{\ $\lozenge$}
\end{Exa}
\begin{Exa}\label{ex5}
Consider the following instance of Example~\ref{ex2}. Let ${\mathbf{C}} = {\mathbf{Set}}$
(where sets are considered as small discrete categories) and
${\mathcal{K}} = {\mathbf{X}}$ be some (typically not small) category. Then we have
$${\mathrm{Fam}}({\mathbf{X}}) : {\mathbf{Set}}^{\mathrm{op}} \to {\mathbf{Cat}}$$
where ${\mathrm{Fam}}({\mathbf{X}})(I) = {\mathbf{X}}^I$ and
$${\mathrm{Fam}}({\mathbf{X}})(u) = {\mathbf{X}}^u : {\mathbf{X}}^I \to {\mathbf{X}}^J$$
for $u : J \to I$ in ${\mathbf{Set}}$.\\
This example is \emph{paradigmatic} for \emph{Fibred Category Theory \`a la
B\'enabou} as it allows categories over ${\mathbf{Set}}$ to be considered as fibrations
over ${\mathbf{Set}}$. Replacing ${\mathbf{Set}}$ by more general categories ${\mathbf{B}}$ as e.g.\ toposes
or even just categories with pullbacks one may develop a fair amount of
\emph{category theory over base ${\mathbf{B}}$} !
\end{Exa}
\begin{Exa}\label{ex6}
For a category ${\mathbf{B}}$ with pullbacks we may consider
${\mathcal{H}} : {\mathbf{B}}^{\mathrm{op}} \to {\mathbf{Cat}}$ sending $I \in {\mathbf{B}}$ to
${\mathcal{H}}(I) = {\mathbf{B}}/I$ and $u : J \to I$ in ${\mathbf{B}}$ to the pullback functor
${\mathcal{H}}(u) = u^{-1} : {\mathbf{B}}/I \to {\mathbf{B}}/J$ which is right adjoint to
$\Sigma_u \equiv u \circ (-)$ (postcomposition with $u$).
Notice that this is an example only \emph{cum grano salis} as
$u^{-1} : {\mathbf{B}}/I \to {\mathbf{B}}/J$ involves some choice of pullbacks and, accordingly,
in general we do not have
${\mathcal{H}}(uv) = {\mathcal{H}}(v) \circ {\mathcal{H}}(u)$ but only
${\mathcal{H}}(uv) \cong {\mathcal{H}}(v) \circ {\mathcal{H}}(u)$ where the
components of the natural isomorphism are given by the respective mediating
arrows. Such ``functors'' preserving composition (and identity) only up
to isomorphism are usually called \emph{pseudo--functors}. \hfill \mbox{\ $\lozenge$}
\end{Exa}
We definitely do \emph{not} want to exclude the situation of Example~\ref{ex6}
as it allows one to consider the base category ${\mathbf{B}}$ as ``fibred over itself''.
Therefore, one might feel forced to accept pseudo--functors and the ensuing
bureaucratic handling of ``canonical isomorphisms''. However, as we will show
immediately one may replace pseudo--functors
${\mathcal{H}} : {\mathbf{B}}^{\mathrm{op}} \to {\mathbf{Cat}}$ by fibrations
$P:{\mathbf{X}} \to {\mathbf{B}}$ where this bureaucracy will turn out as luckily hidden from us.
To motivate the definition of a fibration let us consider a functor
${\mathcal{H}} : {\mathbf{B}}^{\mathrm{op}} \to {\mathbf{Cat}}$ from which we will
construct the ``fibration'' $P = \int {\mathcal{H}} : {\mathbf{X}}\to{\mathbf{B}}$. The objects
of ${\mathbf{X}}$ are pairs $(I,X)$ where $I \in {\mathbf{B}}$ and $X \in {\mathcal{H}}(I)$. A
morphism in ${\mathbf{X}}$ from $(J,Y)$ to $(I,X)$ is a pair $(u,\alpha)$ where
$u : J \to I$ in ${\mathbf{B}}$ and $\alpha : Y \to {\mathcal{H}}(u)(X)$ in
${\mathcal{H}}(J)$. Composition in ${\mathbf{X}}$ is defined as follows: for maps
$(v,\beta) : (K,Z) \to (J,Y)$ and $(u,\alpha) : (J,Y) \to (I,X)$
in $\int {\mathcal{H}}$ their composition $(u,\alpha) \circ (v,\beta)$ is given by
$(u \circ v, {\mathcal{H}}(u)(\alpha) \circ \beta)$. It is readily checked
that this composition is associative and identities are given by
${\mathit{id}}_{(I,X)} = ({\mathit{id}}_I,{\mathit{id}}_{X})$. Let $P = \int{\mathcal{H}} : {\mathbf{X}} \to {\mathbf{B}}$ be
the functor sending an object $(I,X)$ in ${\mathbf{X}}$ to $I$ in ${\mathbf{B}}$ and a morphism
$(u,\alpha)$ in ${\mathbf{X}}$ to $u$ in ${\mathbf{B}}$.
Similarly, the pseudo--functor from Example~\ref{ex6} may be replaced by the
functor $P_{\mathbf{B}} \equiv \partial_1 \equiv {\mathsf{cod}} : {\mathbf{B}}^\mbox{$2\hspace*{-1.2ex}1$} \to {\mathbf{B}}$ where
$\mbox{$2\hspace*{-1.2ex}1$}$ is the partial order $0 \to 1$, i.e.\ the ordinal $2$. Obviously,
$P_{\mathbf{B}}$ sends a commuting square
\begin{diagram}[small]
B & \rTo^{f} & A \\
\dTo^{b} & & \dTo_{a} \\
J & \rTo_{u} & I \\
\end{diagram}
to $u$. Just as we have written $\partial_1$ for the ``codomain'' functor
${\mathsf{cod}}$ we will write $\partial_0$ for the ``domain'' functor
${\mathsf{dom}} : {\mathbf{B}}^\mbox{$2\hspace*{-1.2ex}1$} \to {\mathbf{B}}$. As $P_{\mathbf{B}}$ allows one to consider ${\mathbf{B}}$ as
fibred over itself and this is fundamental for developing category theory over
${\mathbf{B}}$ we call $P_{\mathbf{B}}$ the \emph{fundamental fibration of ${\mathbf{B}}$}.
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a functor as described above. A morphism $\varphi$ in
${\mathbf{X}}$ is called \emph{vertical} iff $P(\varphi) = {\mathit{id}}$. We write $P(I)$ or
${\mathbf{X}}_I$ for the subcategory of ${\mathbf{X}}$ which appears as ``inverse image of $I$
under $P$'', i.e.\ which consists of objects $X$ with $P(X) = I$ and morphisms
$\varphi$ with $P(\varphi) = {\mathit{id}}_I$.
If $P = \int{\mathcal{H}}$ then $(u,\alpha)$ will be called \emph{cartesian}
iff $\alpha$ is an isomorphism and if $P = P_{\mathbf{B}}$ then a morphism in ${\mathbf{B}}^\mbox{$2\hspace*{-1.2ex}1$}$
will be called \emph{cartesian} iff the corresponding square is a pullback
in ${\mathbf{B}}$.
\newpage
\section{Basic Definitions}
From the examples in the previous section we destill the following definition
of fibred category.
\begin{Def}\label{cartdef}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a functor. A morphism $\varphi : Y \to X$ in ${\mathbf{X}}$
over $u := P(\varphi)$ is called \emph{cartesian} iff for all $v : K \to J$
in ${\mathbf{B}}$ and $\theta : Z \to X$ with $P(\theta) = u \circ v$ there is a
unique morphism $\psi : Z \to Y$ with $P(\psi) = v$ and
$\theta = \varphi \circ \psi$.
\begin{diagram}
Z & & & & \\
& \rdDotsto_{\psi} \rdTo(4,2)^{\theta} & & & \\
& & Y & \rTo_{\varphi} & X \\
K & & & & \\
& \rdTo_{v} \rdTo(4,2)^{u \circ v} & & & \\
& & J & \rTo_{u} & I \\
\end{diagram}
A morphism $\alpha : Y \to X$ is called \emph{vertical} iff $P(\alpha)$ is
an identity morphism in ${\mathbf{B}}$.
For $I \in {\mathbf{B}}$ we write ${\mathbf{X}}_I$ or $P(I)$ for the subcategory of ${\mathbf{X}}$
consisting of those morphism $\alpha$ with $P(\alpha) = {\mathit{id}}_I$. It is called
the \emph{fibre of $P$ over $I$}.
\hfill \mbox{\ $\lozenge$}
\end{Def}
It is straightforward to check that cartesian arrows are closed under
composition and that $\alpha$ is an isomorphism in ${\mathbf{X}}$ iff $\alpha$ is a
cartesian morphism over an isomorphism.
\begin{Def}\label{fibdef1}
$P : {\mathbf{X}} \to {\mathbf{B}}$ is a \emph{fibration} or \emph{category fibred over ${\mathbf{B}}$} iff
for all $u : J \to I$ in ${\mathbf{B}}$ and $X \in P(I)$ there is a cartesian arrow
$\varphi : Y \to X$ over $u$ called a \emph{cartesian lifting of $X$ along
$u$}.
\hfill \mbox{\ $\lozenge$}
\end{Def}
Obviously, the functors $\int{\mathcal{H}}$ and $P_{\mathbf{B}}$ of the previous section
are examples of fibrations and the \emph{ad hoc} notions of ``cartesian'' as
given there coincide with the official one of Definition~\ref{fibdef1}.
Notice that cartesian liftings of $X \in P(I)$ along $u : J \to I$ are unique
up to vertical isomorphism: suppose that $\varphi : Y \to X$ and
$\psi : Z \to X$ are cartesian over $u$ then there exist vertical arrows
$\alpha : Z \to Y$ and $\beta : Y \to Z$ with $\varphi \circ \alpha = \psi$
and $\psi \circ \beta = \varphi$, respectively, from which it follows by
cartesianness of $\varphi$ and $\psi$ that $\beta \circ \alpha = {\mathit{id}}_Z$ and
$\alpha \circ \beta = {\mathit{id}}_Y$ as
$\psi \circ \beta \circ \alpha = \varphi \circ \alpha = \varphi =
\varphi \circ {\mathit{id}}_Y$ and
$\varphi \circ \beta \circ \alpha = \psi \circ \alpha = \varphi =
\varphi \circ {\mathit{id}}_Y$.
\begin{Def}\label{fibdef2}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ and $Q : {\mathbf{Y}} \to {\mathbf{B}}$ be fibrations over ${\mathbf{B}}$.\\
A \emph{cartesian} or \emph{fibred} functor from $P$ to $Q$ is an ordinary
functor $F : {\mathbf{X}} \to {\mathbf{Y}}$ such that
\begin{enumerate}
\item[\rm (1)] $Q \circ F = P$ and
\item[\rm (2)] $F(\varphi)$ is cartesian w.r.t.\ $Q$ whenever $\varphi$ is
cartesian w.r.t.\ $P$.
\end{enumerate}
If $F$ and $G$ are cartesian functors from $P$ to $Q$ then a \emph{cartesian
natural transformation from $F$ to $G$} is an ordinary natural transformation
$\tau : F \Rightarrow G$ with $\tau_X$ vertical for every $X \in {\mathbf{X}}$.
The ensuing 2-category will be called ${\mathbf{Fib}}({\mathbf{B}})$. \hfill \mbox{\ $\lozenge$}
\end{Def}
Of course, if ${\mathbf{B}}$ is the terminal category then ${\mathbf{Fib}}({\mathbf{B}})$ is isomorphic
to the 2-category ${\mathbf{Cat}}$.
\bigskip\bigskip
\noindent
{\bf Remark.} What we have called ``cartesian'' in Definition~\ref{cartdef}
is usually called \emph{hypercartesian} whereas ``cartesian'' morphisms are
defined as follows: a morphism $\varphi : Y \to X$ is called \emph{cartesian}
iff for all $\psi : Z \to X$ with $P(\varphi) = P(\psi)$ there is a unique
vertical arrow $\alpha : Z \to Y$ with $\varphi \circ \alpha = \psi$.
Employing this more liberal notion of ``cartesian'' one has to strengthen
the definition of fibred category by adding the requirement that cartesian
arrows are closed under composition. It is a simple exercise to show that
this addendum ensures that every cartesian arrow (in the liberal sense) is
actually hypercartesian (i.e.\ cartesian in the more restrictive sense of
our definition) and, accordingly, both definitions of fibred category are
equivalent.
As the current notes consider only fibrations for which ``cartesian'' and
``hypercartesian'' are equivalent anyway we have adopted the somewhat
non--canonical Definition~\ref{cartdef} as in our context it will not lead
to any confusion.
Notice, however, that in more recent (unpublished) work by J.~B\'enabou
on \emph{generalised fibrations} the distinction between cartesian arrows
(in the liberal sense) and hypercartesian arrows turns out as crucial.
\newpage
\section{Split Fibrations and Fibred Yoneda Lemma}
If $P : {\mathbf{X}}\to{\mathbf{B}}$ is a fibration then using axiom of choice for classes we may
select for every $u : J \to I$ in ${\mathbf{B}}$ and $X \in P(I)$ a cartesian arrow
${\mathrm{Cart}}(u,X) : u^*X \to X$ over $u$. Such a choice of cartesian liftings is
called a \emph{cleavage} for $P$ and it induces for every map $u : J \to I$
in ${\mathbf{B}}$ a so-called \emph{reindexing functor} $u^* : P(I) \to P(J)$
in the following way
\begin{diagram}
u^*X & \rTo^{{\mathrm{Cart}}(u,X)} & X \\
\dDashto^{u^*\alpha} & & \dTo_{\alpha} \\
u^*Y & \rTo_{{\mathrm{Cart}}(u,Y)} & Y \\
\end{diagram}
where $u^*\alpha$ is the unique vertical arrow making the diagram commute.
Alas, in general for composable maps $u : J \to I$ and $v : K \to J$ in ${\mathbf{B}}$
it does not hold that
$$v^* \circ u^* = (u \circ v)^*$$
although the functors are canonically isomorphic via $c_{u,v}$
as shown in the following diagram
\begin{diagram}
v^*u^*X\; & & & & \\
& \rdTo^{{\mathrm{Cart}}(v,u^*X)}_{\mbox{cart.}} & & & \\
\dDashto^{(c_{u,v})_X}_{\cong} & & u^*X & & \\
& & & \rdTo^{{\mathrm{Cart}}(u,X)}_{\mbox{cart.}} & \\
(u \circ v)^*X & & \rTo_{{\mathrm{Cart}}(u \circ v,X)}^{\mbox{cart.}} & & X \\
\end{diagram}
where $(c_{u,v})_X$ is the unique vertical arrow making the diagram commute.
Typically, for $P_{{\mathbf{B}}} = \partial_1 : {\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}} \to {\mathbf{B}}$, the
fundamental fibration for a category ${\mathbf{B}}$ with pullbacks, we do not know how
to choose pullbacks in a functorial way. Of course, condition (1) is always
easy to achieve but the problem is condition (2) as how should one choose
canonical pullbacks in a way that they are closed under composition?
But, nevertheless, often such a functorial choice of cartesian liftings is
possible in particular situations.
\begin{Def}
A cleavage ${\mathrm{Cart}}$ of a fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ is called \emph{split}
or a \emph{splitting of $P$} iff the following two conditions are satified
\begin{itemize}
\item[\rm (1)] ${\mathrm{Cart}}({\mathit{id}},X) = {\mathit{id}}_X$
\item[\rm (2)] ${\mathrm{Cart}}(u{\circ}v,X) = {\mathrm{Cart}}(u,X) \circ {\mathrm{Cart}}(v,u^*X)$.
\end{itemize}
A \emph{split fibration} is a fibration \emph{endowed} with a split cleavage.
A \emph{split cartesian functor} between split fibrations is a cartesian
functor $F$ between split fibrations which, moreover, preserves chosen
cartesian liftings, i.e.\ satisfies
\[ F({\mathrm{Cart}}(u,X)) = {\mathrm{Cart}}(u,F(X)) \]
for all $u : J \to I$ in the base and all $X$ over $I$. We write ${\mathbf{Sp}}({\mathbf{B}})$
for the ensuing category of split fibrations over ${\mathbf{B}}$ and split cartesian
functors between them.
\hfill \mbox{\ $\lozenge$}
\end{Def}
\bigskip
\noindent
{\bf Warning.}\\
(1) There are fibrations which are not splitable. Consider for example the
groups ${\mathbf{B}} = ({\mathbb{Z}}_2,+_2)$ and ${\mathbf{X}} = ({\mathbb{Z}},+)$ (considered as categories) and
the fibration $P : {\mathbf{X}} \to {\mathbf{B}} : a \mapsto P(a) := a \, {\mathrm{mod}} \, 2$.
A splitting of $P$ would give rise to a functor $F : {\mathbf{B}} \to {\mathbf{X}}$
with $P \circ F = {\mathrm{Id}}_{\mathbf{B}}$ but that cannot exist as there is no group
homomorphism $h : ({\mathbf{Z}}_2,+_2) \to ({\mathbf{Z}},+)$ with $h(1)$ an odd number of ${\mathbf{Z}}$.\\
(2) Notice that different splittings of the same fibration may give rise to
the same presheaf of categories. Consider for example
${\mathcal{H}} : \mbox{$2\hspace*{-1.2ex}1$}^{\mathrm{op}}\to {\mathbf{Ab}}$ with
${\mathcal{H}}(1) = {\mathcal{O}}$, the zero group, and ${\mathcal{H}}(0)$ some
non--trivial abelian group $A$. Then every $g \in A$ induces a splitting
${\mathrm{Cart}}_g$ of $P \equiv \int {\mathcal H}$ by putting
\begin{enumerate}
\item[] \qquad ${\mathrm{Cart}}_g(u,\star) = (u,g)$
\qquad for $u : 0 \to 1$ in $\mbox{$2\hspace*{-1.2ex}1$}$
\end{enumerate}
but all these ${\mathrm{Cart}}_g$ induce the same functor $\mbox{$2\hspace*{-1.2ex}1$}^{\mathrm{op}} \to {\mathbf{Cat}}$,
namely ${\mathcal{H}}$ !
In the light of (2) it might appear as more appropriate to define
split fibrations over ${\mathbf{B}}$ as functors from ${\mathbf{B}}^{\mathrm{op}}$ to ${\mathbf{Cat}}$. The latter
may be considered as categories internal to $\widehat{{\mathbf{B}}} = {\mathbf{Set}}^{{\mathbf{B}}^{\mathrm{op}}}$
and organise into the (2-)category ${\mathbf{cat}}({\mathbf{B}})$ of categories and functors
internal to $\widehat{{\mathbf{B}}}$. However, as ${\mathbf{Sp}}({\mathbf{B}})$ and ${\mathbf{cat}}({\mathbf{B}})$ are strongly
equivalent as 2-categories we will not distiguish them any further in the rest
of these notes.
\bigskip
Next we will presented the \emph{Fibred Yoneda Lemma} making precise the
relation between fibred categories and split fibrations (over the same base).
\subsection*{Fibred Yoneda Lemma}
Though, as we have seen, not every fibration $P \in {\mathbf{Fib}}({\mathbf{B}})$ is isomorphic
to a splitable fibration there is always a distinguished \emph{equivalent}
split fibration as ensured by the so-called \emph{Fibred Yoneda Lemma}.
Before giving the full formulation of the Fibred Yoneda Lemma we motivate
the construction of a canonical split fibration ${\mathit{Sp}}(P)$ equivalent to a
given fibration $P \in {\mathbf{Fib}}({\mathbf{B}})$.
For an object $I \in {\mathbf{B}}$ let $\underline{I} = P_I = \partial_0 : {\mathbf{B}}/I \to {\mathbf{B}}$
be the discrete fibration corresponding to the representable presheaf
$Y_{\mathbf{B}}(I) = {\mathbf{B}}(-,I)$ and for $u : J \to I$ in ${\mathbf{B}}$ let $\underline{u} =
P_u = \Sigma_u$ be the cartesian functor from $\underline{J}$ to
$\underline{I}$ as given by postcomposition with $u$ and corresponding to
the presheaf morphism $Y_{\mathbf{B}}(u) = {\mathbf{B}}(-,u) : Y_{\mathbf{B}}(J) \to Y_{\mathbf{B}}(I)$. Then
cartesian functors from $\underline{I}$ to $P : {\mathbf{X}} \to {\mathbf{B}}$ in ${\mathbf{Fib}}({\mathbf{B}})$
correspond to choices of cartesian liftings for an object $X \in P(I)$. There
is an obvious functor $E_{P,I} : {\mathbf{Fib}}({\mathbf{B}})(\underline{I},P) \to P(I)$ sending
$F$ to $F({\mathit{id}}_I)$ and $\tau : F\to G$ to $\tau_{{\mathit{id}}_I} : F({\mathit{id}}_I)\to G({\mathit{id}}_I)$.
It is a straightforward exercise to show that $E_{P,I}$ is full and faithful
and using the axiom of choice for classes we also get that $E_{P,I}$ is
surjective on objects, i.e.\ that
$E_{P,I} : {\mathbf{Fib}}({\mathbf{B}})(\underline{I},P) \to P(I)$ is an equivalence of categories.
Now we can define ${\mathit{Sp}}(P) : {\mathbf{B}}^{\mathrm{op}} \to {\mathbf{Cat}}$ as
\[ {\mathit{Sp}}(P)(I) = {\mathbf{Fib}}({\mathbf{B}})(\underline{I},P) \]
for objects $I$ in ${\mathbf{B}}$ and
\[ {\mathit{Sp}}(P)(u) = {\mathbf{Fib}}({\mathbf{B}})(\underline{u},P) : {\mathit{Sp}}(P)(I){\to}{\mathit{Sp}}(P)(J) \]
for morphisms $u : J \to I$ in ${\mathbf{B}}$. Let us write $U({\mathit{Sp}}(P))$ for
$\int {\mathit{Sp}}(P)$, the fibration obtained from ${\mathit{Sp}}(P)$ via the
Grothendieck construction. Then the $E_{P,I}$ as described above arise as
the components of a cartesian functor $E_P : U({\mathit{Sp}}(P)) \to P$ sending
objects $(I,X)$ in $ U({\mathit{Sp}}(P)) = \int {\mathit{Sp}}(P)$ to $E_{P,I}(X)$ and
morphism $(u,\alpha) : G \to F$ in $U({\mathit{Sp}}(P)) = \int {\mathit{Sp}}(P)$
over $u : J \to I$ to the morphism
$F(u{:}u{\to}{\mathit{id}}_I) \circ \alpha_{{\mathit{id}}_J} : G({\mathit{id}}_J) \to F({\mathit{id}}_I)$ in ${\mathbf{X}}$.
As all fibres of $E_P$ are equivalences it follows\footnote{We leave it as
an exercise to show that under assumption of axiom of choice for classes a
cartesian functor is an equivalence in ${\mathbf{Fib}}({\mathbf{B}})$ iff all its fibres are
equivalence of categories.} that $E_P$ is an equivalence in the $2$-category
${\mathbf{Fib}}({\mathbf{B}})$.
Actually, the construction of ${\mathit{Sp}}(P)$ from $P$ is just the object part
of a $2$-functor ${\mathit{Sp}} : {\mathbf{Fib}}({\mathbf{B}}) \to {\mathbf{Sp}}({\mathbf{B}})$ right adjoint to the
forgetful $2$-functor from ${\mathbf{Sp}}({\mathbf{B}})$ to ${\mathbf{Fib}}({\mathbf{B}})$ as described in the
following theorem
(which, however, will not be used any further in the rest of these notes).
\begin{Thm}\label{fYl}\emph{(Fibred Yoneda Lemma)}\\
For every category ${\mathbf{B}}$ the forgetful $2$-functor $U : {\mathbf{Sp}}({\mathbf{B}}) \to {\mathbf{Fib}}({\mathbf{B}})$
has a right $2$-adjoint ${\mathit{Sp}} : {\mathbf{Fib}}({\mathbf{B}}) \to {\mathbf{Sp}}({\mathbf{B}})$, i.e.\ there is
an equivalence of categories
\[ {\mathbf{Fib}}({\mathbf{B}})(U(S),P) \simeq {\mathbf{Sp}}({\mathbf{B}})(S,{\mathit{Sp}}(P)) \]
naturally in $S \in {\mathbf{Sp}}({\mathbf{B}})$ and $P \in {\mathbf{Fib}}({\mathbf{B}})$, whose counit
$E_P : U({\mathit{Sp}}(P)) \to P$ at $P$ is an equivalence in ${\mathbf{Fib}}({\mathbf{B}})$
for all $P \in {\mathbf{Fib}}({\mathbf{B}})$.
However, in general the unit $H_S : S \to {\mathit{Sp}}(U(S))$ at $S \in {\mathbf{Sp}}({\mathbf{B}})$
is not an equivalence in ${\mathbf{Sp}}({\mathbf{B}})$ although $U(H_S)$ is always an equivalence
in ${\mathbf{Fib}}({\mathbf{B}})$.
\end{Thm}
\begin{proof}
The functor $U : {\mathbf{Sp}}({\mathbf{B}}) \to {\mathbf{Fib}}({\mathbf{B}})$ just forgets cleavages. The object
part of its right adjoint ${\mathit{Sp}}$ is as described above, namely
\[ {\mathit{Sp}}(P)(I) = {\mathbf{Fib}}({\mathbf{B}})(\underline{I},P) \qquad\quad
{\mathit{Sp}}(P)(u) = {\mathbf{Fib}}({\mathbf{B}})(\underline{u},P)\]
for $P \in {\mathbf{Fib}}({\mathbf{B}})$. For cartesian functors $F : P \to Q$ in ${\mathbf{Fib}}({\mathbf{B}})$ we
define ${\mathit{Sp}}(F) : {\mathit{Sp}}(P) \to {\mathit{Sp}}(Q)$ as
\[ {\mathit{Sp}}(F)_I = {\mathbf{Fib}}({\mathbf{B}})(\underline{I},F) \]
for objects $I$ in ${\mathbf{B}}$. Under assumption of axiom of choice for classes
the counit for $U \dashv {\mathit{Sp}}$ at $P$ is given by the equivalence
$E_P : U({\mathit{Sp}}(P)) \to P$ as described above.
The unit $H_S : S \to {\mathit{Sp}}(U(S))$ for $U \dashv {\mathit{Sp}}$ at $S \in {\mathbf{Sp}}({\mathbf{B}})$
sends $X \in P(I)$ to the cartesian functor from $\underline{I}$ to $P$ which
chooses cartesian liftings as prescribed by the underlying cleavage of $S$ and
arrows $\alpha : X \to Y$ in $P(I)$ to the cartesian natural transformation
$H_S(\alpha) : H_S(X) \to H_S(Y)$ with $H_S(\alpha)_{{\mathit{id}}_I} = \alpha$. We
leave it as a tedious, but straightforward exercise to show that these data
give rise to an equivalence
\[ {\mathbf{Fib}}({\mathbf{B}})(U(S),P) \simeq {\mathbf{Sp}}({\mathbf{B}})(S,{\mathit{Sp}}(P)) \]
naturally in $S$ and $P$.
As all components of $H_S$ are equivalences of categories it follows that
$U(H_S)$ is an equivalence in ${\mathbf{Fib}}({\mathbf{B}})$. However, it cannot be the case
that all $H_S$ are equivalences as otherwise a split cartesian functor $F$
were an equivalence in ${\mathbf{Sp}}({\mathbf{B}})$ already if $U(F)$ is an equivalence in
${\mathbf{Fib}}({\mathbf{B}})$ and this is impossible as not every epi in $\widehat{{\mathbf{B}}}$ is a
split epi.
\end{proof}
\bigskip
As $E_P : U({\mathit{Sp}}(P)) \to P$ is always an equivalence it follows
that for fibrations $P$ and $Q$
\[ {\mathit{Sp}}_{P,Q} : {\mathbf{Fib}}({\mathbf{B}})(P,Q) \to {\mathbf{Sp}}({\mathbf{B}})({\mathit{Sp}}(P),{\mathit{Sp}}(Q)) \]
is an equivalence of categories.
However, in general ${\mathit{Sp}}_{P,Q}$ is not an isomorphism of categories.
An arbitrary split cartesian functor $G : {\mathit{Sp}}(P) \to {\mathit{Sp}}(Q)$ corresponds
via the $2$-adjunction $U \dashv {\mathit{Sp}}$ to a cartesian functor
$E_Q \circ U(G) : U({\mathit{Sp}}(P)) \to Q$ which, however, need not factor as
$E_Q \circ U(G) = F \circ E_P$ for some cartesian $F : P \to Q$.\footnote{
For example, if $Q = U({{\mathit{Sp}}}(P))$ and
$E_Q \circ U(G) = {{\mathit{Id}}}_{U({{\mathit{Sp}}}(P))}$ and $E_P$ is not
one-to-one on objects which happens to be the case whenever cartesian
liftings are not unique in $P$.} One may characterise the split cartesian
functors of the form ${\mathit{Sp}}(F)$ for some cartesian $F : P \to Q$ as those
split cartesian functors $G : {\mathit{Sp}}(P) \to {\mathit{Sp}}(Q)$ satisfying
${\mathit{Sp}}(E_Q) \circ {\mathit{Sp}}(U(G)) = G \circ {\mathit{Sp}}(E_P)$. One easily sees
that this condition is necessary and if it holds then an $F$ with
$G = {\mathit{Sp}}(F)$ can be obtained as $E_Q \circ U(G) \circ E'_P$
for some $E'_P$ with $E_P \circ E'_P = {{\mathit{Id}}}_P$ because we have
${\mathit{Sp}}(F) = {\mathit{Sp}}(E_Q \circ U(G) \circ E'_P) =
{\mathit{Sp}}(E_Q) \circ {\mathit{Sp}}(U(G)) \circ {\mathit{Sp}}(E'_P) =
G \circ {\mathit{Sp}}(E_P) \circ {\mathit{Sp}}(E'_P) = G \circ {\mathit{Sp}}(E_P \circ E'_P) = G$.
Although ${\mathit{Sp}}$ is not full and faithful the adjunction $U \dashv {\mathit{Sp}}$
nevertheless is of the type ``full reflective subcategory'' albeit in the
appropriate $2$-categorical sense. This suggests that ${\mathbf{Fib}}({\mathbf{B}})$ is obtained
from ${\mathbf{Sp}}({\mathbf{B}})$ by ``freely quasi-inverting weak equivalences in ${\mathbf{Fib}}({\mathbf{B}})$''
which can be made precise as follows.
A split cartesian functor $F$ is called a \emph{weak equivalence}
iff all its fibres are equivalences of categories, i.e.\ iff $U(F)$ is an
equivalence in ${\mathbf{Fib}}({\mathbf{B}})$. Let us write $\Sigma$ for the class of weak
equivalences in ${\mathbf{Sp}}({\mathbf{B}})$. For a $2$-category ${\mathfrak{X}}$ and a $2$-functor
$\Phi : {\mathbf{Sp}}({\mathbf{B}}) \to {\mathfrak{X}}$ we say that $\Phi$ \emph{quasi-inverts} a morphism
$F$ in ${\mathbf{Sp}}({\mathbf{B}})$ iff $\Phi(F)$ is an equivalence in ${\mathfrak{X}}$. Obviously, the
$2$-functor $U : {\mathbf{Sp}}({\mathbf{B}}) \to {\mathbf{Fib}}({\mathbf{B}})$ quasi-inverts all weak equivalences.
That $U$ freely inverts the maps in $\Sigma$ can be seen as follows. Suppose
that a $2$-functor $\Phi : {\mathbf{Sp}}({\mathbf{B}}) \to {\mathfrak{X}}$ quasi-inverts all weak
equivalences. Then there exists a $2$-functor $\Psi : {\mathbf{Fib}}({\mathbf{B}}) \to {\mathfrak{X}}$
unique up to equivalence with the property that $\Psi \circ U \simeq \Phi$.
As by assumption $\Phi$ quasi-inverts weak equivalences we have
$\Phi \circ {\mathit{Sp}} \circ U \simeq \Phi$ because all $H_S$ are weak
equivalences. On the other hand if $\Psi \circ U \simeq \Phi$ then we have
$\Psi \simeq \Psi \circ U \circ {\mathit{Sp}} \simeq \Phi \circ {\mathit{Sp}}$ (because all
$E_P$ are equivalences) showing that $\Psi$ is unique up to equivalence.
\subsection*{A Left Adjoint Splitting}
The forgetful $U : {\mathbf{Sp}}({\mathbf{B}}) \to {\mathbf{Fib}}({\mathbf{B}})$ admits also a left adjoint
splitting $L : {\mathbf{Fib}}({\mathbf{B}}) \to {\mathbf{Sp}}(B)$ which like the right adjoint splitting
discussed previously was devised by J.~Giraud in the late 1960s.
This left adjoint splitting $L(P)$ of a fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ is
constructed as follows. First choose a cleavage ${\mathrm{Cart}}_P$ of $P$ which is
\emph{normalized} in the sense that ${\mathrm{Cart}}_P({\mathit{id}}_I,X) = {\mathit{id}}_X$ for all $X$
over $I$. From this cleavage one may construct a presheaf
$S(P) : {\mathbf{B}}^{\mathrm{op}} \to {\mathbf{Cat}}$ of categories giving rise to the desired split fibration
$L(P)$ over ${\mathbf{B}}$. For $I \in {\mathbf{B}}$ the objects of $S(P)(I)$ are pairs $(a,X)$ where
$X$ is an object of ${\mathbf{X}}$ and $a : I \to P(X)$. Morphisms from $(b,Y)$ to $(a,X)$
are vertical morphism $\alpha : b^*Y \to a^*X$ and composition in $S(P)(I)$ is
inherited from ${\mathbf{X}}$, i.e.\ $P(I)$. For $u : J \to I$ in ${\mathbf{B}}$ the functor $S(P)(u)
: S(P)(I) \to S(P)(J)$ is constructed as follows. For $(a,X)$ in $S(P)(I)$ let
${\mathrm{Cart}}_{L(P)}(u,(a,X)) : (au)^*X \to u^*X$ be the unique cartesian arrow $\varphi$
over $u$ with ${\mathrm{Cart}}_P(a,X) \circ \varphi = {\mathrm{Cart}}_P(au,X)$. Let $\alpha : b^*Y \to a^*X$
be a morphism from $(b,Y)$ to $(a,X)$ in $S(P)(I)$. Then we define $S(P)(u)(\alpha)$
as the unique vertical morphism making the diagram
\begin{diagram}[small]
& & Y \\
& \ruTo^{{\mathrm{Cart}}_P(bu,Y)} & \uTo_{{\mathrm{Cart}}_P(b,Y)} \\
(bu)^*Y & \rTo_{\;\;\;{\mathrm{Cart}}_{L(P)}(u,(b,Y))} & b^*Y \\
\dTo^{S(P)(u)(\alpha)} & & \dTo_\alpha \\
(au)^*X & \rTo^{\;\;\;{\mathrm{Cart}}_{L(P)}(u,(a,X))} & a^*X \\
& \rdTo_{{\mathrm{Cart}}_P(au,X)} & \dTo_{{\mathrm{Cart}}_P(a,X)} \\
& & X
\end{diagram}
commute. One readily checks that $S(P)$ is indeed a functor from ${\mathbf{B}}^{\mathrm{op}}$ to ${\mathbf{Cat}}$
since
${\mathrm{Cart}}_{L(P)}(uv,(a,X) = {\mathrm{Cart}}_{L(P)}(u,(a,X)) \circ {\mathrm{Cart}}_{L(P)}(v,(au,X))$
and ${\mathrm{Cart}}_{L(P)}({\mathit{id}}_I,(a,X)) = {\mathit{id}}_{a^*X}$ as one can see easily.
Objects of the total category of $L(P)$ are objects of $S(P)(I)$ for some $I \in {\mathbf{B}}$
and morphisms from $(b,Y)$ to $(a,X)$ are just morphisms $b^*Y \to a^*X$
whose composition is inherited from ${\mathbf{X}}$.
The functor $L(P)$ sends $(a,X)$ to the domain of $a$ and $f : b^*Y \to a^*X$ to $P(f)$.
The splitting of $L(P)$ is given by ${\mathrm{Cart}}_{L(P)}$ as defined above for specifying
the morphism part of $S(P)$.
The unit $H_P : P \to U(L(P))$ of the (2-categorical) adjunction $L \dashv U$
sends $X$ to $({\mathit{id}}_{P(X)},X)$ and $f : Y \to X$ to $f : H_P(Y) \to H_P(X)$.
Notice that the above construction of $L(P)$ is based on a choice of a cleavage
for $P$. But this may be avoided by defining morphisms from $(b,Y)$ to $(a,X)$
over $u : J \to I$ as equivalence classes of spans $(\psi,f)$ in ${\mathbf{X}}$ where $\psi$
is a cartesian morphism to $Y$ over $b$ and $f$ is a morphism to $A$ over $au$
where $(\psi,f)$ and $(\psi^\prime,f^\prime)$ get identified iff there is a vertical
isomorphism $\iota$ with $\psi \circ \iota = \psi^\prime$ and
$f \circ \iota = f^\prime$. For a given cleavage ${\mathrm{Cart}}_P$ of $P$ the equivalence
class of $(\psi,f)$ contains a unique pair whose first component is ${\mathrm{Cart}}_P(b,Y)$.
\newpage
\section{Closure Properties of Fibrations}
In this section we will give some examples of fibrations and constructions of
new fibrations from already given ones. Keeping in mind that we think of
fibrations over ${\mathbf{B}}$ as generalisations of fibrations of the form ${\mathrm{Fam}}({\mathbf{C}})$
over ${\mathbf{Set}}$ it will appear that most of these constructions are
generalisations of well-known constructions in ${\bf Cat}$.
\subsection*{Fundamental Fibrations}
For a category ${\mathbf{B}}$ the codomain functor
$$P_{\mathbf{B}} \equiv \partial_1 : {\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}} \to {\mathbf{B}}$$
is a fibration if and only if ${\mathbf{B}}$ has pullbacks. In this case $P_{\mathbf{B}}$ is
called the \emph{fundamental fibration of ${\mathbf{B}}$}.
\subsection*{Externalisations of Internal Categories}
Let $C$ be a category internal to ${\mathbf{B}}$ as given by domain and codomain
maps $d_0,d_1 : C_1 \to C_0$, the identity map $i : C_0 \to C_1$ and a
composition map $m : C_1 \times_{C_0} C_1 \to C_1$. Then one may construct the
fibration $P_C : \underline{C} \to {\mathbf{B}}$ called \emph{externalisation of $C$}.
The objects of $\underline{C}$ over $I$ are pairs $(I,a : I \to C_0)$ and a
morphism in $\underline{C}$ from $(J,b)$ to $(I,a)$ over $u : J \to I$ is
given by a morphism $f : J \to C_1$ with $d_0 \circ f = b$ and
$d_1 \circ f = a \circ u$. Composition in $C$ is defined using $m$ analogous
to ${\mathrm{Fam}}({\mathbf{C}})$. The fibration $P_C$ itself is defined as
$$P_C(I,a) = I \qquad\qquad P_C(u,f) = u$$
and the cartesian lifting of $(I,a)$ along $u : J \to I$ is given by
$i \circ a \circ u$.
In particular, every object $I \in {\mathbf{B}}$ can be considered as a \emph{discrete}
internal category of ${\mathbf{B}}$. Its externalisation is given by
$P_I = \partial_0 : {\mathbf{B}}/I \to {\mathbf{B}}$.
\subsection*{Change of Base and Glueing}
If $P \in {\mathbf{Fib}}({\mathbf{B}})$ and $F : {\mathbf{C}} \to {\mathbf{B}}$ is an ordinary functor then
$F^*P \in {\mathbf{Fib}}({\mathbf{C}})$ where
\begin{diagram}[small]
{\mathbf{Y}} \SEpbk & \rTo^{K} & {\mathbf{X}} \\
\dTo^{F^*P} & & \dTo_{P} \\
{\mathbf{C}} & \rTo_{F} & {\mathbf{B}}
\end{diagram}
is a pullback in ${\bf Cat}$. One says that fibration $F^*P$ is obtained
from $P$ \emph{by change of base along $F$}.
Notice that $(u,\varphi)$ in ${\mathbf{Y}}$ is cartesian w.r.t.\ $F^*P$ iff $\varphi$
is cartesian w.r.t.\ $P$. Accordingly, $K$ preserves cartesianness of arrows
as $K(u,\varphi) = \varphi$.
When instantiating $P$ by the fundamental fibration $P_{\mathbf{B}}$ we get the
following important particular case of change of base
\begin{diagram}[small]
{\mathbf{B}}{\downarrow}F \SEpbk & \rTo^{\partial_0\;\;\;} & {\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}} \\
\dTo^{{\mathsf{gl}}(F) \equiv \partial_1} & & \dTo_{P_{\mathbf{B}}} \\
{\mathbf{C}} & \rTo_{F\;\;\;} & {\mathbf{B}}
\end{diagram}
usually called \emph{glueing construction} or \emph{(Artin) glueing}.
Typically, in applications the functor $F$ will be the inverse image part of
a geometric morphism $F \dashv U : {\mathbf{E}} \to {\mathbf{S}}$ between toposes. In this case
${\mathsf{Gl}}(F) = {\mathbf{E}}{\downarrow}F$ is again a topos and the functor
${\mathsf{gl}}(F) : {\mathbf{E}}{\downarrow}F \to {\mathbf{S}}$ is \emph{logical}, i.e.\ preserves all topos
structure. Actually, for this to hold it suffices already that $F$ is a
pullback preserving functor between toposes. The glueing construction will be
intrinsic later on when we discuss the
\emph{Fibrational Theory of Geometric Morphisms} \`a la J.-L.~Moens.
We write ${\mathbf{Fib}}$ for the (non--full) subcategory of ${\mathbf{Cat}}^{\mbox{$2\hspace*{-1.2ex}1$}}$
whose objects are fibrations and whose morphisms are commuting squares
\begin{diagram}[small]
{\mathbf{Y}} & \rTo^{K} & {\mathbf{X}} \\
\dTo^{Q} & & \dTo_{P} \\
{\mathbf{C}} & \rTo_{F} & {\mathbf{B}}
\end{diagram}
with $K$ cartesian over $F$, i.e.\ where $K(\varphi)$ is cartesian over $F(u)$
whenever $\varphi$ is cartesian over $u$. Obviously, ${\mathbf{Fib}}$ is fibred
over ${\mathbf{Cat}}$ via the restriction of $\partial_1 : {\mathbf{Cat}}^{\mbox{$2\hspace*{-1.2ex}1$}} \to {\mathbf{Cat}}$ to
${\mathbf{Fib}}$ for which we write ${\mathbf{Fib}}/{\mathbf{Cat}} : {\mathbf{Fib}} \to {\mathbf{Cat}}$. A morphism of ${\mathbf{Fib}}$
is cartesian iff it is a pullback square in ${\mathbf{Cat}}$.
We write ${\mathbf{Fib}}({\mathbf{B}})/{\mathbf{B}}$ for the fibration obtained from ${\mathbf{Fib}}/{\mathbf{Cat}}$
by change of base along the functor $\Sigma : {\mathbf{B}} \to {\mathbf{Cat}}$
sending $I$ to ${\mathbf{B}}/I$ and $u : J \to I$ to
$\Sigma_u : {\mathbf{B}}/J \to {\mathbf{B}}/I : v \mapsto u \circ v$
\begin{diagram}[small]
{\mathbf{Fib}} {\downarrow} \Sigma \SEpbk & \rTo & {\mathbf{Fib}} \\
\dTo^{{\mathbf{Fib}}({\mathbf{B}})/{\mathbf{B}}} & & \dTo_{{\mathbf{Fib}}/{\mathbf{Cat}}} \\
{\mathbf{B}} & \rTo_{\Sigma\quad} & {\bf Cat}
\end{diagram}
We leave it as an exercise
to show that $P : {\mathbf{X}} \to {\mathbf{B}}/I$ is a fibration iff $P_I \circ P$ is
a fibration over ${\mathbf{B}}$ and $P \in {\mathbf{Fib}}({\mathbf{B}})(P_I{\circ}P,P_I)$. Accordingly,
fibrations over ${\mathbf{B}}/I$ are understood as $I$-indexed families of fibrations
over ${\mathbf{B}}$ in analogy with ordinary functors to a discrete category $I$ which
are understood $I$-indexed families of categories.
\subsection*{Composition and Product of Fibrations}
First notice that fibrations are closed under composition. Even more we have
the following
\begin{Thm}\label{fibfib}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration and $F : {\mathbf{Y}} \to {\mathbf{X}}$ be an arbitrary functor.
Then $F$ itself is a fibration over ${\mathbf{X}}$ iff
\begin{enumerate}
\item[\rm (1)] $Q \equiv P{\circ}F$ is a fibration and $F$ is a cartesian
functor from $Q$ to $P$ over ${\mathbf{B}}$ and
\item[\rm (2)] all $F_I : {\mathbf{Y}}_I \to {\mathbf{X}}_I$ are fibrations and cartesian arrows
w.r.t.\ these fibrations are stable under reindexing, i.e.\
for every commuting diagram
\begin{diagram}[small]
Y_1 & \rTo^{\varphi_1} & X_1 \\
\dTo^{\theta} & & \dTo_{\psi} \\
Y_2 & \rTo_{\varphi_2} & X_2
\end{diagram}
in ${\mathbf{Y}}$ with $\varphi_1$ and $\varphi_2$ cartesian w.r.t.\ $Q$ over the
same arrow $u : J \to I$ in ${\mathbf{B}}$ and $Q(\psi) ={\mathit{id}}_I$ and $Q(\theta) ={\mathit{id}}_J$
it holds that $\theta$ is cartesian w.r.t.\ $F_J$
whenever $\psi$ is cartesian w.r.t.\ $F_I$.
\end{enumerate}
\end{Thm}
\begin{proof} Exercise left to the reader.
\end{proof}
\medskip
The second condition means that the commuting diagram
\begin{diagram}[small]
{\mathbf{Y}}_I & \rTo^{u^*} & {\mathbf{Y}}_J \\
\dTo^{F_I} & & \dTo_{F_J} \\
{\mathbf{X}}_I & \rTo_{u^*} & {\mathbf{X}}_J
\end{diagram}
is a morphism in ${\mathbf{Fib}}$. (Notice that due to condition (1)
of Theorem~\ref{fibfib} one can choose the reindexing functor
$u^* : {\mathbf{Y}}_I \to {\mathbf{Y}}_J$ in such a way that the diagram actually commutes. For
arbitrary cartesian functors this need not be possible although for all
choices of the $u^*$ the diagram always commutes up to isomorphism.)
The relevance of Theorem~\ref{fibfib} is that it characterises
``fibred fibrations'' as those fibred functors which are themselves ordinary
fibrations. This handy characterisation cannot even be formulated in the
framework of indexed categories and, therefore, is considered as a typical
example of the superiority of the fibrational point of view.
For fibrations $P$ and $Q$ over ${\mathbf{B}}$ their product $P{\times_{{\mathbf{B}}}}Q$ in
${\mathbf{Fib}}({\mathbf{B}})$ is given by $P \circ P^*Q = Q \circ Q^*P$ as in
\begin{diagram}[small]
{\mathbf{P}} \SEpbk & \rTo^{Q^*P} & {\mathbf{Y}} \\
\dTo^{P^*Q} & & \dTo_{Q} \\
{\mathbf{X}} & \rTo_{P} & {\mathbf{B}}
\end{diagram}
and it follows from Theorem~\ref{fibfib} that $P{\times_{{\mathbf{B}}}}Q$ is a
fibration and that the projections $P^*Q$ and $Q^*P$ are cartesian functors.
\subsection*{Fibrations of Diagrams}
Let ${\mathbf{D}}$ be a category and $P : {\mathbf{X}} \to {\mathbf{B}}$ a fibration. Then the
\emph{fibration $P^{({\mathbf{D}})}$ of diagrams of shape ${\mathbf{D}}$} is given by
\begin{diagram}[small]
{\mathbf{X}}^{({\mathbf{D}})} \SEpbk & \rTo & {\mathbf{X}}^{{\mathbf{D}}} \\
\dTo^{P^{({\mathbf{D}})}} & & \dTo_{P^{\mathbf{D}}} \\
{\mathbf{B}} & \rTo_{\Delta_{\mathbf{D}}} & {\mathbf{B}}^{{\mathbf{D}}}
\end{diagram}
where the ``diagonal functor'' $\Delta_{\mathbf{D}}$ sends $I \in {\mathbf{B}}$ to the constant
functor with value $I$ and a morphism $u$ in ${\mathbf{B}}$ to the natural
transformation all whose components are $u$.
Somewhat surprisingly, as shown by A.~Kurz in spring 2019,
the functor $P^{\mathbf{D}}$ is also a fibration, however, over ${\mathbf{B}}^{\mathbf{D}}$.
\subsection*{Exponentiation of Fibrations}
For fibrations $P$ and $Q$ over ${\mathbf{B}}$ we want to construct a fibration
$[P{\to}Q]$ such that there is an equivalence
\[ {\mathbf{Fib}}({\mathbf{B}})(R,[P{\to}Q]) \simeq {\mathbf{Fib}}({\mathbf{B}})(R{\times_{\mathbf{B}}}P, Q) \]
naturally in $R \in {\mathbf{Fib}}({\mathbf{B}})$.
Analogous to the construction of exponentials in
$\widehat{{\mathbf{B}}} = {\mathbf{Set}}^{{\mathbf{B}}^{\mathrm{op}}}$ the fibred Yoneda lemma (Theorem~\ref{fYl})
suggest us to put
$$[P{\to}Q](I) = {\mathbf{Fib}}({\mathbf{B}})(\underline{I}{\times_{\mathbf{B}}}P, Q) \qquad
[P{\to}Q](u) = {\mathbf{Fib}}({\mathbf{B}})(\underline{u}{\times_{\mathbf{B}}}P, Q)$$
where $\underline{u}$ is given by
\begin{diagram}[small]
{\mathbf{B}}/J & & \rTo^{\Sigma_u} & & {\mathbf{B}}/I \\
& \rdTo_{P_J} & &\ldTo_{P_I} & \\
& & {\mathbf{B}} & &
\end{diagram}
for $u : J \to I$ in ${\mathbf{B}}$. We leave it as a tedious, but straightforward
exercise to verify that
\[ {\mathbf{Fib}}({\mathbf{B}})(R,[P{\to}Q]) \simeq {\mathbf{Fib}}({\mathbf{B}})(R{\times_{\mathbf{B}}}P, Q) \]
holds naturally in $R \in {\mathbf{Fib}}({\mathbf{B}})$.
Notice that we have
\[ {\mathbf{Fib}}({\mathbf{B}})(P_I{\times_{\mathbf{B}}}P,Q) \simeq {\mathbf{Fib}}({\mathbf{B}}/I)(P_{/I},Q_{/I}) \]
naturally in $I \in {\mathbf{B}}$ where $P_{/I} = {P_I}^*P$ and $Q_{/I} = {P_I}^*Q$
are obtained by change of base along $P_I$. Usually $P_{/I}$ is referred to
as ``localisation of $P$ to $I$''. The desired equivalence follows from the
fact that change of base along $P_I$ is right adjoint to postcomposition
with $P_I$ and the precise correspondence between
$F \in {\mathbf{Fib}}({\mathbf{B}})(\underline{I}{\times_{\mathbf{B}}}P, Q)$ and
$G \in {\mathbf{Fib}}({\mathbf{B}}/I)(P_{/I},Q_{/I})$ is indicated by the following diagram
\begin{diagram}
\cdot & & & & \\
& \rdTo~{G} \rdTo(2,4)_{P_{/I}} \rdTo(4,2)^{F} & & & \\
& & \cdot \SEpbk & \rTo & {\mathbf{Y}} \\
& & \dTo_{Q_{/I}} & & \dTo_{Q} \\
& & {\mathbf{B}}/I & \rTo_{P_I} & {\mathbf{B}} \\
\end{diagram}
\newpage
\section{The Opposite of a Fibration}
If $P : {\mathbf{X}} \to {\mathbf{B}}$ is a fibration thought of ``as of the form ${\mathrm{Fam}}({\mathbf{C}})$''
then one may want to construct the fibration $P^{\mathrm{op}}$ thought of ``of the
form ${\mathrm{Fam}}({\mathbf{C}}^{\mathrm{op}})$''. It might be tempting at first sight to apply $(-)^{\mathrm{op}}$
to the functor $P$ giving rise to the functor ${\mathbf{X}}^{\mathrm{op}} \to {\mathbf{B}}^{\mathrm{op}}$ which,
however, has the wrong base even if it were a fibration (which in general
will not be the case). If $P = \int {\mathcal{H}}$ for some ${\mathcal{H}} : {\mathbf{B}}^{\mathrm{op}} \to {\mathbf{Cat}}$
then one may consider
${\mathcal{H}}^{\mathrm{op}} = (-)^{\mathrm{op}} \circ {\mathcal{H}} : {\mathbf{B}}^{\mathrm{op}} \to {\mathbf{Cat}}$,
i.e.\ the assignment
\[ I \mapsto {\mathcal{H}}(I)^{\mathrm{op}} \qquad\quad
u : J \to I \; \mapsto \; {\mathcal{H}}(u)^{\mathrm{op}} : {\mathcal{H}}(I)^{\mathrm{op}} \to {\mathcal{H}}(I)^{\mathrm{op}}
\]
where $(-)^{\mathrm{op}}$ is applied to the fibres of ${\mathcal{H}}$ and to the reindexing
functors. Now we express $P^{\mathrm{op}} = \int {\mathcal{H}}^{\mathrm{op}}$ in terms of $P = \int {\mathcal{H}}$
directly.
The fibration $P^{\mathrm{op}} : {\mathbf{Y}} \to {\mathbf{B}}$ is constructed from the fibration
$P : {\mathbf{X}} \to {\mathbf{B}}$ in the following way. The objects of ${\mathbf{Y}}$ and ${\mathbf{X}}$ are the
same but for $X \in P(I)$, $Y \in P(J)$ and $u : J \to I$ the collection of
morphisms in ${\mathbf{Y}}$ from $Y$ to $X$ over $u$ is constructed as follows. It
consists of all spans $(\alpha,\varphi)$ with $\alpha : Z \to Y$ vertical
and $\varphi : Z \to X$ is cartesian over $u$ \emph{modulo} the equivalence
relation $\sim_{Y,u,X}$ (also denoted simply as $\sim$) where
$(\alpha,\varphi) \sim_{Y,u,X} (\alpha',\varphi')$ iff
\begin{diagram}[small]
& & Z & & \\
& \ldTo^{\alpha} & & \rdTo^{\varphi}_{\mathrm{cart}} & \\
Y & & \uTo^{\iota}_{\cong} & & X\\
& \luTo_{\alpha'} & & \ruTo_{\varphi'}^{\mathrm{cart}}& \\
& & Z' & & \\
\end{diagram}
for some (necessarily unique) vertical isomorphism $\iota : Z' \to Z$.
Composition of arrows in ${\mathbf{Y}}$ is defined as follows:
if $[(\alpha,\varphi)]_\sim : Y \to X$
over $u : J \to I$ and $[(\beta,\psi)]_\sim : Z \to Y$ over $v : K \to J$ then
$[(\alpha,\varphi)]_\sim \circ [(\beta,\psi)]_\sim :=
[(\beta \circ \widetilde{\alpha}, \varphi \circ \widetilde{\psi})]_\sim$ where
\begin{diagram}[small]
& & & & (uv)^*X & & & &\\
&&&\ldTo^{\widetilde{\alpha}}&&\rdTo^{\widetilde{\psi}}_{\mathrm{cart}} & & &\\
& & v^*Y & & \mbox{p.b.}& & u^*X & &\\
& \ldTo^{\beta} & & \rdTo_{\psi}^{\mathrm{cart}} & & \ldTo_{\alpha}& &
\rdTo^{\varphi}_{\mathrm{cart}} &\\
Z & & & & Y & & & & X\\
\end{diagram}
with $\widetilde{\alpha}$ vertical.
Actually, this definition does not depend on the choice of $\widetilde{\psi}$
as morphisms in ${\mathbf{Y}}$ are equivalence classes modulo $\sim$ which forgets about
all distinctions made by choice of cleavages. On objects $P^{\mathrm{op}}$ behaves like
$P$ and $P^{\mathrm{op}}([(\alpha,\varphi)]_\sim)$ is defined as $P(\varphi)$.
The $P^{\mathrm{op}}$-cartesian arrows are the equivalence classes
$[(\alpha,\varphi)]_\sim$ where $\alpha$ is a vertical isomorphism.
Though most constructions appear more elegant from the fibrational point of
view the construction of $P^{\mathrm{op}}$ from $P$ may appear as somewhat less
immediate though (hopefully!) not too unelegant. Notice, however, that
for small fibrations, i.e.\ externalisations of internal categories, the
construction can be performed as in the case of presheaves of categories
as we have $P_{C^{\mathrm{op}}} \simeq P_C^{\mathrm{op}}$ for internal categories $C$.
Anyway, we have generalised now enough constructions from ordinary category
theory to the fibrational level so that we can perform (analogues of) the
various constructions of (covariant and contravariant) functor categories on
the level of fibrations. In particular, for a category $C$ internal to a
category ${\mathbf{B}}$ with pullbacks we may construct the fibration
$[P_C^{\mathrm{op}}{\to}P_{\mathbf{B}}]$ which may be considered as the fibration of (families of)
${\mathbf{B}}$-valued presheaves over the internal category $C$. Moreover, for categories
$C$ and $D$ internal to ${\mathbf{B}}$ the fibration of (families of) distributors from
$C$ to $D$ is given by $[P_D^{\mathrm{op}}{\times}P_C{\to}P_{\mathbf{B}}]$.\footnote{For an
equivalent, but non-fibrational treatment of internal presheaves and
distributors see \cite{Joh}.}
\newpage
\section{Internal Sums}
Suppose that ${\mathbf{C}}$ is a category. We will identify a purely fibrational
property of the fibration ${\mathrm{Fam}}({\mathbf{C}}) \to {\mathbf{Set}}$ equivalent to the requirement
that the category ${\mathbf{C}}$ has small sums. This will provide a basis for
generalising the property of ``having small sums'' to fibrations over
arbitrary base categories with pullbacks.
Suppose that category ${\mathbf{C}}$ has small sums. Consider a family of objects
$A = (A_i)_{i \in I}$ and a map $u : I \to J$ in ${\mathbf{Set}}$. Then one may
construct the family
$B := (\coprod_{i \in u^{-1}(j)} A_i)_{j \in J}$ together with the morphism
$(u,\varphi) : (I,A) \to (J,B)$ in ${\mathrm{Fam}}({\mathbf{C}})$ where
$\varphi_i = {\mathrm{in}}_i : A_i \to B_{u(i)} =
\coprod_{k \in u^{-1}(u(i))} A_k$, i.e.\ the restriction of $\varphi$ to
$u^{-1}(j)$ is the cocone for the sum of the family $(A_i)_{i \in u^{-1}(j)}$.
One readily observes that $(u,\varphi) : A \to B$ satisfies the following
universal property: whenever $v : J \to K$ and $(v \circ u,\psi) : A \to C$
then there exists a unique $(v,\theta) : B \to C$ such that
$(v,\theta) \circ (u,\varphi) = (v \circ u,\psi)$, i.e.\
$\theta_{u(i)} \circ {\mathrm{in}}_i = \psi_i$ for all $i \in I$.
Arrows $(u,\varphi)$ satisfying this universal property are called
\emph{cocartesian} and are determined uniquely up to vertical isomorphism.
Moreover, the cocartesian arrows of ${\mathrm{Fam}}({\mathbf{C}})$ satisfy the following
so-called\footnote{Chevalley had this condition long before Beck who later
independently found it again.} \emph{Beck--Chevalley Condition} (BCC)
which says that for every pullback
\begin{diagram}[small]
K \SEpbk & \rTo^{\widetilde{u}} & L \\
\dTo^{\widetilde{h}} & \mbox{(1)} & \dTo_{h} \\
I & \rTo_{u} & J \\
\end{diagram}
in ${\mathbf{Set}}$ and cocartesian arrow $\varphi : A \to B$ over $u$ it holds that
for every commuting square
\begin{diagram}[small]
C & \rTo^{\widetilde{\varphi}} & D \\
\dTo^{\widetilde{\psi}} & & \dTo_{\psi} \\
A & \rTo_{\varphi} & B \\
\end{diagram}
over the pullback square (1) in ${\mathbf{B}}$ with $\psi$ and $\widetilde{\psi}$
cartesian the arrow $\widetilde{\varphi}$ is cocartesian, too.
Now it is a simple exercise to formulate the obvious generalisation to
fibrations over an arbitrary base category with pullbacks.
\begin{Def}\label{intsumdef}
Let ${\mathbf{B}}$ be a category with pullbacks and $P : {\mathbf{X}} \to {\mathbf{B}}$ a fibration
over ${\mathbf{B}}$.
An arrow $\varphi : X \to Y$ over $u : I \to J$ is called \emph{cocartesian}
iff for every $v : J \to K$ in ${\mathbf{B}}$ and $\psi : X \to Z$ over $v \circ u$
there is a unique arrow $\theta : Y \to Z$ over $v$ with
$\theta \circ \varphi = \psi$.\\
The fibration $P$ \emph{has internal sums} iff the following two conditions
are satisfied.
\begin{enumerate}
\item[\rm (1)] For every $X \in P(I)$ and $u : I \to J$ in ${\mathbf{B}}$ there exists
a cocartesian arrow $\varphi : X \to Y$ over $u$.
\item[\rm (2)] The \emph{Beck--Chevalley Condition (BCC)} holds, i.e.\
for every commuting square in ${\mathbf{X}}$
\begin{diagram}[small]
C & \rTo^{\widetilde{\varphi}} & D \\
\dTo^{\widetilde{\psi}} & & \dTo_{\psi} \\
A & \rTo_{\varphi} & B \\
\end{diagram}
over a pullback in the base it holds that $\widetilde{\varphi}$ is cocartesian
whenever $\varphi$ is cocartesian and $\psi$ and $\widetilde{\psi}$ are
cartesian. \hfill \mbox{\ $\lozenge$}
\end{enumerate}
\end{Def}
\bigskip \noindent
{\bf Remark.}\\
(1) One easily sees that for a fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ an arrow $\varphi :
X \to Y$ is cocartesian iff for all $\psi : X \to Z$ over $P(\varphi)$ there
exists a unique vertical arrow $\alpha : Y \to Z$ with
$\alpha \circ \varphi = \psi$.\\
(2) It is easy to see that BCC of Definition~\ref{intsumdef} is equivalent
to the requirement that for every commuting square in ${\mathbf{X}}$
\begin{diagram}[small]
C & \rTo^{\widetilde{\varphi}} & D \\
\dTo^{\widetilde{\psi}} & & \dTo_{\psi} \\
A & \rTo_{\varphi} & B \\
\end{diagram}
over a pullback in the base it holds that $\psi$ is cartesian whenever
$\widetilde{\psi}$ is cartesian and $\varphi$ and $\widetilde{\varphi}$
are cocartesian.
\bigskip
Next we give a less phenomenological explanation of the concept of internal
sums where, in particular, the Beck--Chevalley Condition arises in a less
\emph{ad hoc} way. For this purpose we first generalise the ${\mathrm{Fam}}$
construction from ordinary categories to fibrations.
\begin{Def}
Let ${\mathbf{B}}$ be a category with pullbacks and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration.
Then the \emph{family fibration ${\mathrm{Fam}}(P)$ for $P$} is defined as
$P_{\mathbf{B}} \circ \mathit{Fam}(P)$ where
\begin{diagram}[small]
P{\downarrow}{\mathbf{B}} \SEpbk & \rTo & {\mathbf{X}} \\
\dTo^{\mathit{Fam}(P)} & & \dTo_{P} \\
{\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}} & \rTo_{\partial_0}& {\mathbf{B}} \\
\end{diagram}
The cartesian functor $\mathit{Fam}(P) : {\mathrm{Fam}}(P) \to P_{\mathbf{B}}$ is called the
\emph{fibred family fibration of $P$}.
The cartesian functor $\eta_P : P \to {\mathrm{Fam}}(P)$ is defined as in the diagram
\begin{diagram}[small]
{\mathbf{X}} & & & & \\
& \rdDashto(2,2)_{\eta_P} \rdEqual(4,2) & & \\
\dTo^{P} & & P{\downarrow}{\mathbf{B}} \SEpbk & \rTo & {\mathbf{X}} \\
& & \dTo^{\mathit{Fam}(P)} & & \dTo_{P} \\
{\mathbf{B}} & \rTo_{\quad\;\;\;\Delta_{\mathbf{B}}} & {\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}} & \rTo_{\partial_0} & {\mathbf{B}} \\
& \rdEqual & \dTo_{\partial_1 = P_{\mathbf{B}}} & & \\
& & {\mathbf{B}} & & \\
\end{diagram}
where $\Delta_{\mathbf{B}}$ sends $u : I \to J$ to
\begin{diagram}[small]
I & \rTo^{u} & J \\
\dEqual & & \dEqual \\
I & \rTo_{u} & J \\
\end{diagram}
in ${\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}}$. More explicitly, $\eta_P$ sends $\varphi : X \to Y$ over
$u : I \to J$ to
\begin{diagram}[small]
X & \rTo^{\varphi} & Y \\
I & \rTo^{u} & J \\
\dEqual & & \dEqual \\
I & \rTo_{u} & J \\
\end{diagram}
in $P{\downarrow}{\mathbf{B}}$. Obviously, the functor $\eta_P$ preserves
cartesianness of arrows, i.e.\ $\eta_P$ is cartesian. \hfill \mbox{\ $\lozenge$}
\end{Def}
\noindent
{\bf Remark.}\\
(1) If $\mathit{Fam}(P)(\varphi)$ is cocartesian w.r.t.\ $P_{\mathbf{B}}$ then $\varphi$ is
cartesian w.r.t.\ $\mathit{Fam}(P)$ iff $\varphi$ is cocartesian w.r.t.\ $\mathit{Fam}(P)$.
Moreover, for every morphism
\begin{diagram}[small]
A & \rTo^v & B \\
\dTo^a & & \dTo_b \\
I & \rTo_u & J
\end{diagram}
in ${\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}}$ we have
\begin{diagram}[small]
A & \rTo^v & B & \rEqual & B & & A & \rEqual & A & \rTo^v & B \\
\dEqual & 1_v & \dEqual & \varphi_b & \dTo_b & = &
\dEqual & \varphi_a & \dTo_a & & \dTo_b \\
A & \rTo_v & B & \rTo_b & I & & A & \rTo_a & I & \rTo_u & J
\end{diagram}
where $\varphi_a$ and $\varphi_b$ are cocartesian w.r.t.\ $P_{\mathbf{B}}$.
Using these two observations one can show that for fibrations $P$ and $Q$
over ${\mathbf{B}}$ a cartesian functor $F : \mathit{Fam}(P) \to \mathit{Fam}(Q)$ is determined
uniquely up to isomorphism by its restriction along the inclusion
$\Delta_{\mathbf{B}} : {\mathbf{B}} \to {\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}}$ from which it follows that $F$ is isomorphic
to $\mathit{Fam}(\Delta_{\mathbf{B}}^*F)$. Thus, up to isomorphism all cartesian functors
from $\mathit{Fam}(P)$ to $\mathit{Fam}(Q)$ are of the form $\mathit{Fam}(F)$ for some cartesian
functor $F : P \to Q$.
\noindent
(2) Notice, however, that not every cartesian functor ${\mathrm{Fam}}(P) \to {\mathrm{Fam}}(Q)$
over ${\mathbf{B}}$ is isomorphic to one of the form ${\mathrm{Fam}}(F)$ for some cartesian functor
$F : P \to Q$. An example for this failure is the cartesian functor
$\mu_P : {\mathrm{Fam}}^2(P) \to {\mathrm{Fam}}(P)$ sending $((X,v),u)$ to $(X,uv)$ for nontrivial
${\mathbf{B}}$.\footnote{One can show that $\eta$ and $\mu$ are natural transformations
giving rise to a monad $({\mathrm{Fam}},\eta,\mu)$ on ${\mathbf{Fib}}({\mathbf{B}})$.}
\noindent
(3) If ${\mathbf{X}}$ is a category we write ${\mathrm{Fam}}({\mathbf{X}})$ for the category of families
in ${\mathbf{X}}$ and $\mathit{Fam}({\mathbf{X}}) : {\mathrm{Fam}}({\mathbf{X}}) \to {\mathbf{Set}}$ for the family fibration.
The analogon of (1) in ordinary category theory is that for categories
${\mathbf{X}}$ and ${\mathbf{Y}}$ a cartesian functor $F : \mathit{Fam}({\mathbf{X}}) \to \mathit{Fam}({\mathbf{Y}})$ is isomorphic
to $\mathit{Fam}(F_1)$ (the fibre of $F$ at $1 \in {\mathbf{Set}}$).
The analogon of (2) in ordinary category theory is that not every ordinary
functor $F : {\mathrm{Fam}}({\mathbf{X}}) \to {\mathrm{Fam}}({\mathbf{Y}})$ is isomorphic to one of the form ${\mathrm{Fam}}(G)$
for some $G : {\mathbf{X}} \to {\mathbf{Y}}$.
\bigskip
Next we characterise the property of having internal sums in terms of
the family monad ${\mathrm{Fam}}$.
\begin{Thm}
Let ${\mathbf{B}}$ be a category with pullbacks and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration.
Then $P$ has internal sums iff $\eta_P : P \to {\mathrm{Fam}}(P)$ has a fibred
left adjoint $\coprod_P : {\mathrm{Fam}}(P) \to P$ , i.e.\ $\coprod_P \dashv \eta_P$
where $\coprod_P$ is cartesian and unit and counit of the adjunction
are cartesian natural transformations.
\end{Thm}
\begin{proof}
The universal property of the unit of the adjunction $\coprod_P \dashv \eta_P$
at $(u,X)$ is explicitated in the following diagram
\begin{diagram}[small]
& & & & Z \\
X & \rTo_{\eta_{(u,X)}} & Y & \ruTo(4,1)^{\psi} \ruDashto(2,1)_{\theta} & \\
\dDots & & \dDots & & \dDots\\
I & \rTo^{u} & J & \rTo^{v}& K\\
\dTo^{u} & & \dEqual & & \dEqual \\
J & \rEqual & J & \rTo_{v} & K \\
\end{diagram}
whose left column is the unit at $(u,X)$. From this it follows that
$\eta_{(u,X)} : X \to Y$ is cocartesian over $u$.
Cartesianness of $\coprod_P$ says that the cartesian arrow $f$ as given by
\begin{diagram}[small]
X & \rTo^{\psi}_{\mathrm{cart}} & Y \\
K \SEpbk & \rTo^{q} & L \\
\dTo^{p} & & \dTo_{v} \\
I & \rTo_{u} & J \\
\end{diagram}
in $P{\downarrow}{\mathbf{B}}$ is sent by $\coprod_P$ to the cartesian arrow
$\coprod_P f$ over $u$ satisfying
\begin{diagram}[small]
X & \rTo^{\psi}_{\mathrm{cart}} & Y \\
\dTo^{\eta_{(p,X)}} & & \dTo_{\eta_{(v,Y)}} \\
A & \rTo_{\coprod_P f}^{\mathrm{cart}} & B \\
\end{diagram}
where $\eta_{(p,X)}$ and $\eta_{(v,Y)}$ are the cocartesian units of the
adjunction above $p$ and $v$, respectively.
Thus, according to the second remark after Definition~\ref{intsumdef}
cartesianness of $\coprod_P$ is just the Beck--Chevalley Condition for
internal sums.
On the other hand if $P$ has internal sums then the functor $\coprod_P$ left
adjoint to $P$ is given by sending a morphism $f$ in $P{\downarrow}{\mathbf{B}}$
as given by
\begin{diagram}[small]
X & \rTo^{\psi} & Y \\
K & \rTo^{q} & L \\
\dTo^{p} & & \dTo_{v} \\
I & \rTo_{u} & J \\
\end{diagram}
to the morphism $\coprod_P f$ over $u$ satisfying
\begin{diagram}[small]
X & \rTo^{\psi} & Y \\
\dTo^{\varphi_1} & & \dTo_{\varphi_2}\\
A & \rTo_{\coprod_P f} & B \\
\end{diagram}
where $\varphi_1$ and $\varphi_2$ are cocartesian over $p$ and $v$,
respectively. It is easy to check that $\coprod_P$ is actually left adjoint
to $\eta_P$ using for the units of the adjunction the cocartesian liftings
guaranteed for $P$. Cartesianness of $\coprod_P$ is easily seen to be
equivalent to the Beck--Chevalley condition.
\end{proof}
\newpage
\section{Internal Products}
Of course, by duality a fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ has internal products iff
the dual fibration $P^{\mathrm{op}}$ has internal sums. After some explicitation (left
to the reader) one can see that the property of having internal products can
be characterised more elementarily as follows.
\begin{Thm}
Let ${\mathbf{B}}$ be a category with pullbacks. Then a fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ has
internal products iff the following two conditions are satisfied.
\begin{itemize}
\item[\rm (i)] For every $u : I \to J$ in ${\mathbf{B}}$ and $X \in P(I)$ there is a span
$\varphi : u^*E \to E$, $\varepsilon : u^*E \to X$ with $\varphi$ cartesian
over $u$ and $\varepsilon$ vertical such that for every span
$\theta : u^*Z \to Z$, $\alpha : u^*Z \to X$ with $\theta$ cartesian over $u$
and $\alpha$ vertical there is a unique vertical arrow $\beta : Z \to E$
such that $\alpha = \varepsilon \circ u^*\beta$ where $u^*\beta$ is the
vertical arrow with $\varphi \circ u^*\beta = \beta \circ \theta$ as
illustrated in the diagram
\begin{diagram}[small]
& & u^*Z & \rTo^{\theta}_{\mathrm{cart}} & Z \\
& \ldTo^{\alpha} & \dTo_{u^*\beta }& & \dTo_{\beta}\\
X & \lTo_{\varepsilon} & u^*E & \rTo_{\varphi}^{\mathrm{cart}} & E\\
\end{diagram}
Notice that the span $(\varphi,\varepsilon)$ is determined uniquely up to
vertical isomorphism by this universal property and is called an
\emph{evaluation span for $X$ along $u$}.
\item[\rm (ii)] Whenever
\begin{diagram}[small]
L \SEpbk & \rTo^{\widetilde{v}} & I \\
\dTo^{\widetilde{u}} & \mathrm{(1)} & \dTo_{u} \\
K & \rTo_{v} & J \\
\end{diagram}
is a pullback in ${\mathbf{B}}$ and $\varphi : u^*E \to E$, $\varepsilon : u^*E \to X$
is an evaluation span for $X$ along $u$ then for every diagram
\begin{diagram}[small]
\widetilde{v}^*X & \rTo^{\psi}_{\mathrm{cart}} & X \\
\uTo^{\widetilde{\varepsilon}} & & \uTo_{\varepsilon}\\
\widetilde{u}^*\widetilde{E}&\rTo^{\widetilde{\theta}}_{\mathrm{cart}} &u^*E\\
\dTo^{\widetilde{\varphi}} & & \dTo_{\varphi}\\
\widetilde{E} & \rTo_{\theta}^{\mathrm{cart}} & E \\
\end{diagram}
where the lower square is above pullback \emph{(1)} in ${\mathbf{B}}$ and
$\widetilde{\varepsilon}$ is vertical it holds that
$(\widetilde{\varphi},\widetilde{\varepsilon})$ is an evaluation span
for $\widetilde{v}^*X$ along $\widetilde{u}$.
\end{itemize}
\end{Thm}
\begin{proof}
Tedious, but straightforward explicitation of the requirement that
$P^{\mathrm{op}}$ has internal sums.
\end{proof}
\bigskip
Condition (ii) is called Beck--Chevally Condition (BCC) for internal products
and essentially says that evaluation spans are stable under reindexing.
\bigskip\noindent
{\bf Examples.}\\
(1) ${\mathrm{Mon}}({\cal E})$ fibred over topos ${\cal E}$ has both internal
sums and internal products.\\
(2) For every category ${\mathbf{B}}$ with pullbacks the fundamental fibration
$P_{\mathbf{B}} = \partial_1 : {\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}}\to{\mathbf{B}}$ has internal sums which look as follows
\begin{diagram}[small]
A & \rEqual & A \\
\dTo^{a} & & \dTo_{\coprod_u a} \\
I & \rTo_u & J \\
\end{diagram}
The fundamental fibration $P_{\mathbf{B}}$ has internal products iff for every
$u : I \to J$ in ${\mathbf{B}}$ the pullback functor $u^{-1} : {\mathbf{B}}/J \to {\mathbf{B}}/I$ has a
right adjoint $\prod_u$. For ${\mathbf{B}} = {\mathbf{Set}}$ this right adjoint gives
\emph{dependent products} (as known from Martin-L\"of Type Theory).
\subsection*{Models of Martin--L\"of Type Theory}
A category ${\mathbf{B}}$ with finite limits such that its fundamental fibration
$P_{\mathbf{B}}$ has internal products---usually called a \emph{locally cartesian closed
category}---allows one to interpret $\Sigma$, $\Pi$ and Identity Types of
Martin--L\"of Type Theory.
Dependent sum $\Sigma$ and dependent product $\Pi$ are interpreted as
internal sums and internal products.The fibrewise diagonal $\delta_a$
\begin{diagram}[small]
A & & & & \\
& \rdTo~{\delta_a} \rdEqual(2,4) \rdEqual(4,2) & & & \\
& & A\times_I A\SEpbk & \rTo_{\pi_2} & A \\
& & \dTo_{\pi_1} & & \dTo_{a} \\
& & A & \rTo_{a} & I \\
\end{diagram}
is used for interpreting identity types: the sequent
$i{:}I,x,y{:}A \vdash {\mathrm{Id}}_A(x,y)$ is interpreted as $\delta_a$
when $i{:}I \vdash A$ is interpreted as $a$.
One may interpret W-types in ${\mathbf{B}}$ iff for $b : B \to A$ and $a : A \to I$
there is a ``least'' $w : W \to I$ such that $W \cong \coprod_a \prod_b b^*w$
mimicking on a categorical level the requirement that $W$ is the ``least''
solution of the recursive type equation $W \cong \Sigma x{:}A.W^{B(x)}$.
\newpage
\section{Fibrations of Finite Limit Categories\\ and Complete Fibrations}
Let ${\mathbf{B}}$ be a category with pullbacks remaining fixed for this section.
\begin{Lem}\label{flf1}
For a fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ we have that
\begin{enumerate}
\item[\rm (1)] a commuting square of cartesian arrows in ${\mathbf{X}}$ over
a pullback in ${\mathbf{B}}$ is always a pullback in ${\mathbf{X}}$
\item[\rm (2)] a commuting square
\begin{diagram}[small]
Y_1 & \rTo^{\varphi_1}_{\mathrm{cart}} & X_1 \\
\dTo^{\beta} & & \dTo_{\alpha} \\
Y_2 & \rTo_{\varphi_2}^{\mathrm{cart}} & X_2\\
\end{diagram}
in ${\mathbf{X}}$ is a pullback in ${\mathbf{X}}$ whenever the $\varphi_i$ are cartesian and
$\alpha$ and $\beta$ vertical.
\end{enumerate}
\end{Lem}
\begin{proof}
Straightforward exercise.
\end{proof}
\begin{Def}\label{flfdef}
$P : {\mathbf{X}} \to {\mathbf{B}}$ is a \emph{fibration of categories with pullbacks} iff every
fibre $P(I)$ has pullbacks and these are stable under reindexing along
arbitrary morphisms in the base. \hfill \mbox{\ $\lozenge$}
\end{Def}
\begin{Lem}\label{flf2}
If $P : {\mathbf{X}} \to {\mathbf{B}}$ is a fibration of categories with pullbacks then every
pullback in some fibre $P(I)$ is also a pullback in ${\mathbf{X}}$.
\end{Lem}
\begin{proof}
Suppose
\begin{diagram}[small]
Z \SEpbk& \rTo^{\beta_2} & X_2 \\
\dTo^{\beta_1} & \mbox{($\dagger$)}& \dTo_{\alpha_2} \\
X_1 & \rTo_{\alpha_1}& Y \\
\end{diagram}
is a pullback in $P(I)$ and $\theta_1$, $\theta_2$ is a cone over $\alpha_1$,
$\alpha_2$ in ${\mathbf{X}}$, i.e.\ $\alpha_1 \circ \theta_1 = \alpha_2 \circ \theta_2$.
Obviously, $\theta_1$ and $\theta_2$ are above the same arrow $u$ in ${\mathbf{B}}$.
For $i=1,2$ let $\varphi_i : u^*X_i \to X_i$ be a cartesian arrow over $u$
and $\gamma_i : V \to u^*X_i$ be a vertical arrow with
$\varphi_i \circ \gamma_i = \theta_i$. As the image of $(\dagger)$ under $u^*$
is a pullback in its fibre there is a vertical arrow $\gamma$ with $\gamma_i =
u^*\beta_i \circ \gamma$ for $i=1,2$. The situation is illustrated in the
following diagram
\begin{diagram}
V & & & & \\
& \rdTo~{\gamma} \rdTo(2,4)_{\gamma_i} \rdTo(4,2)^{\theta} & & & \\
& & u^*Z \SEpbk & \rTo^{\psi}_{\mathrm{cart}} & Z \\
& & \dTo_{u^*\beta_i} & & \dTo_{\beta_i} \\
& & u^*X_i & \rTo_{\varphi_i}^{\mathrm{cart}} & X_i \\
& & \dTo_{u^*\alpha_i} & & \dTo_{\alpha_i} \\
& & u^*Y & \rTo_{\varphi}^{\mathrm{cart}} & Y \\
\end{diagram}
where $\varphi$ and $\psi$ are cartesian over $u$. From this diagram it is
obvious that $\theta := \psi \circ \gamma$ is a mediating arrow as desired.
If $\theta'$ were another such mediating arrow then for
$\theta' = \psi \circ \gamma'$ with $\gamma'$ vertical it holds that
$\gamma' = \gamma$ as both are mediating arrows to $u^*(\dagger)$ for the
cone given by $\gamma_1$ and $\gamma_2$ and, therefore, it follows that
$\theta = \theta'$. Thus $\theta$ is the unique mediating arrow.
\end{proof}
\bigskip
Now we can give a simple characterisation of fibrations of categories
with pullbacks in terms of a preservation property.
\begin{Thm}\label{flf3}
$P : {\mathbf{X}} \to {\mathbf{B}}$ is a fibration of categories with pullbacks iff ${\mathbf{X}}$ has and
$P$ preserves pullbacks.
\end{Thm}
\begin{proof}
Suppose that $P : {\mathbf{X}} \to {\mathbf{B}}$ is a fibration of categories with pullbacks.
For $i=1,2$ let $f_i : Y_i \to X$ be arrows in ${\mathbf{X}}$ and
$f_i = \varphi_i \circ \alpha_i$ be some vertical/cartesian factorisations.
Consider the diagram
\begin{diagram}[small]
U \SEpbk & \rTo^{\beta_2} & \SEpbk & \rTo^{\varphi_1''} & Y_2 \\
\dTo^{\beta_1} & \mbox{(4)} & \dTo^{\alpha_2'} & \mbox{(3)}
& \dTo_{\alpha_2} \\
\SEpbk & \rTo^{\alpha_1'} & \SEpbk & \rTo^{\varphi_1'} & Z_2 \\
\dTo^{\varphi_2''} & \mbox{(2)}& \dTo^{\varphi_2'} & \mbox{(1)}
& \dTo_{\varphi_2} \\
Y_1 & \rTo_{\alpha_1} & Z_1 & \rTo_{\varphi_1} & X \\
\end{diagram}
where the $\varphi$'s are cartesian and the $\alpha$'s and $\beta$'s are
vertical. Square (1) is a pullback in ${\mathbf{X}}$ over a pullback in ${\mathbf{B}}$ by
Lemma~\ref{flf1}(1). Squares (2) and (3) are pullbacks in ${\mathbf{X}}$ by
Lemma~\ref{flf1}(2). Square (4) is a pullback in ${\mathbf{X}}$ by Lemma~\ref{flf2}.
Accordingly, the big square is a pullback in ${\mathbf{X}}$ over a pullback in ${\mathbf{B}}$.
Thus, ${\mathbf{X}}$ has and $P$ preserves pullbacks.
For the reverse direction assume that ${\mathbf{X}}$ has and $P$ preserves pullbacks.
Then every fibre of $P$ has pullbacks and they are preserved under reindexing
for the following reason. For every pullback
\begin{diagram}[small]
Z \SEpbk& \rTo^{\beta_2} & X_2 \\
\dTo^{\beta_1} & \mbox{($\dagger$)}& \dTo_{\alpha_2} \\
X_1 & \rTo_{\alpha_1}& Y \\
\end{diagram}
in $P(I)$ and $u: J \to I$ in ${\mathbf{B}}$ by Lemma~\ref{flf1} we have
\begin{diagram}[small]
u^*Z \SEpbk & \rTo^{\theta}_{\mathrm{cart}} & Z\\
\dTo^{u^*\beta_i} & & \dTo_{\beta_i} \\
u^*X_i \SEpbk & \rTo_{\varphi_i}^{\mathrm{cart}} & X_i \\
\dTo^{u^*\alpha_i} & & \dTo_{\alpha_i} \\
u^*Y & \rTo_{\varphi}^{\mathrm{cart}} & Y \\
\end{diagram}
and, therefore, the image of pullback $(\dagger)$ under $u^*$ is isomorphic
to the pullback of $(\dagger)$ along $\varphi$ in ${\mathbf{X}}$. As pullback functors
preserve pullbacks it follows that the reindexing of $(\dagger)$ along $u$ is
a pullback, too.
\end{proof}
\begin{Def}\label{fibtermdef}
A fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ is a \emph{fibration of categories with terminal
objects} iff every fibre $P(I)$ has a terminal object and these are stable
under reindexing.\hfill \mbox{\ $\lozenge$}
\end{Def}
One easily sees that $P$ is a fibration of categories with terminal objects
iff for every $I \in {\mathbf{B}}$ there is an object $1_I \in P(I)$ such that for every
$u : J \to I$ in ${\mathbf{B}}$ and $X \in P(J)$ there is a unique arrow $f : X \to 1_I$
in ${\mathbf{X}}$ over $u$. Such a $1_I$ is called an``$I$--indexed family of terminal
objects''. It is easy to see that this property is stable under reindexing.
\begin{Lem}\label{flf4}
Let ${\mathbf{B}}$ have a terminal object (besides having pullbacks).
Then $P : {\mathbf{X}} \to {\mathbf{B}}$ is a fibration of categories with terminal objects iff
${\mathbf{X}}$ has a terminal object $1_{\mathbf{X}}$ with $P(1_{\mathbf{X}})$ terminal in ${\mathbf{B}}$.
\end{Lem}
\begin{proof}
Simple exercise.
\end{proof}
\begin{Thm}\label{flf5}
For a category ${\mathbf{B}}$ with finite limits a fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ is a
fibration of categories with finite limits, i.e.\ all fibres of $P$ have
finite limits preserved by reindexing along arbitrary arrows in the base,
iff ${\mathbf{X}}$ has finite limits and $P$ preserves them.
\end{Thm}
\begin{proof}
Immediate from Theorem~\ref{flf3} and Lemma~\ref{flf4}.
\end{proof}
\bigskip
From ordinary category theory one knows that ${\mathbf{C}}$ has small limits iff ${\mathbf{C}}$
has finite limits and small products. Accordingly, one may define
``completeness'' of a fibration over a base category ${\mathbf{B}}$ with finite limits
by the requirements that
\begin{enumerate}
\item[(1)] $P$ is a fibration of categories with finite limits and
\item[(2)] $P$ has internal products (satisfying BCC).
\end{enumerate}
In \cite{Bor} vol.2, Ch.8 it has been shown that for a fibration $P$ complete
in the sense above it holds for all $C \in {\mathbf{cat}}({\mathbf{B}})$ that the fibred
``diagonal'' functor $\Delta_C : P \to [P_C{\to}P]$ has a fibred right
adjoint $\prod_C$ sending diagrams of shape $C$ to their limiting cone
(in the appropriate fibred sense).
Thus, requirement (2) above is necessary and sufficient for internal
completeness under the assumption of requirement (1).
\newpage
\section{Elementary Fibrations and Representability}
A fibration $P:{\mathbf{X}}\to{\mathbf{B}}$ is called \emph{discrete} iff all its fibres are
discrete categories, i.e.\ iff $P$ reflects identity morphisms. However,
already in ordinary category theory discreteness of categories is not stable
under equivalence (though, of course, it is stable under isomorphism of
categories). Notice that a category ${\mathbf{C}}$ is equivalent to a discrete one
iff it is a \emph{posetal groupoid}, i.e.\ Hom--sets contain at most one
element and all morphisms are isomorphisms. Such categories will be called
\emph{elementary}.
This looks even nicer from a fibrational point of view.
\begin{Thm}\label{elemfibchar}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration. Then we have
\begin{itemize}
\item[\rm (1)] $P$ is a fibration of groupoids iff $P$ is \emph{conservative},
i.e.\ $P$ reflects isomorphism.
\item[\rm (2)] $P$ is a fibration of posetal categories iff $P$ is faithful.
\item[\rm (3)] $P$ is a fibration of elementary categories iff
$P$ is faithful and reflects isomorphisms.
\end{itemize}
Fibrations $P : {\mathbf{X}} \to {\mathbf{B}}$ are called \emph{elementary} iff $P$ is faithful
and reflects isomorphisms.
\end{Thm}
\begin{proof}
Straightforward exercise.
\end{proof}
\bigskip
It is well known that a presheaf $A : {\mathbf{B}}^{\mathrm op} \to {\mathbf{Set}}$ is
representable iff $\int A : {\mathbf{Elts}}(A) \to {\mathbf{B}}$ has a terminal object.
This motivates the following definition.
\begin{Def}
An elementary fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ is \emph{representable} iff
${\mathbf{X}}$ has a terminal object, i.e.\ there is an object $R \in P(I)$ such that
for every $X \in {\mathbf{X}}$ there is a unique classifying morphism $u : P(X) \to I$
in ${\mathbf{B}}$ with $X \cong u^*R$, i.e.\ fibration $P$ is equivalent to
$P_I = \partial_0 : {\mathbf{B}}/I \to {\mathbf{B}}$ for some $I \in {\mathbf{B}}$, i.e.\ $P$
is equivalent to some small discrete fibration over ${\mathbf{B}}$. \hfill \mbox{\ $\lozenge$}
\end{Def}
\newpage
\section{Local Smallness}
\begin{Def}\label{locsmalldef}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration. For objects $X,Y \in P(I)$ let
${\mathbf{Hom}}_I(X,Y)$ be the category defined as follows. Its objects are spans
\begin{diagram}[small]
& & U & & \\
& \ldTo^{\varphi}_{\mathrm{cart}} & & \rdTo^{f} & \\
X & & & & Y \\
\end{diagram}
with $P(\varphi) = P(f)$ and $\varphi$ cartesian. A morphism from
$(\psi,g)$ to $(\varphi,f)$ is a morphism $\theta$ in ${\mathbf{X}}$ such that
$\varphi \circ \theta = \psi$ and $f \circ \theta = g$
\begin{diagram}[small]
& & X & & \\
& \ruTo^{\psi}_{\mathrm{cart}} & & \luTo^{\varphi}_{\mathrm{cart}} & \\
V & & \rTo_{\theta}& & U\\
& \rdTo_{g} & & \ruTo_{f}& \\
& & Y & & \\
\end{diagram} \\
Notice that $\theta$ is necessarily cartesian and fully determined by
$P(\theta)$. The category ${\mathbf{Hom}}_I(X,Y)$ is fibred over ${\mathbf{B}}/I$ by
sending $(\varphi, f)$ to $P(\varphi)$ and $\theta$ to $P(\theta)$.
The fibration $P$ is called \emph{locally small} iff for all $X,Y \in P(I)$
the elementary fibration ${\mathbf{Hom}}_I(X,Y)$ over ${\mathbf{B}}/I$ is representable,
i.e.\ ${\mathbf{Hom}}_I(X,Y)$ has a terminal object. \hfill \mbox{\ $\lozenge$}
\end{Def}
The intuition behind this definition can be seen as follows. Let
$(\varphi_0,f_0)$ be terminal in ${\mathbf{Hom}}_I(X,Y)$. Let
$d := P(\varphi_0) : {\mathrm{hom}}_I(X,Y)\to I$. Let
$f_0 = \psi_0 \circ \mu_{X,Y}$ with $\psi_0$ cartesian and $\mu_{X,Y}$
vertical. Then for every $u : J \to I$ and $\alpha : u^*X \to u^*Y$ there
exists a unique $v : J \to {\mathrm{hom}}_I(X,Y)$ with $d \circ v = u$ such that
$\alpha = v^*\mu_{X,Y}$ as illustrated in the following diagram.
\begin{diagram}[small]
& & X \\
& \ruTo^{\varphi} & \uTo_{\varphi_0} \\
u^*X & \rTo_{\theta\quad} & d^*X \\
\dTo^{\alpha = v^*\mu_{X,Y}} & & \dTo_{\mu_{X,Y}} \\
u^*Y & \rTo^{\mathrm{cart}} & d^*Y \\
& \rdTo^{\mathrm{cart}} & \dTo_{\psi_0}\\
& & Y \\
\end{diagram}
\begin{Thm}\label{lsbp}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a locally small fibration and ${\mathbf{B}}$ have binary products.
Then for all objects $X$, $Y$ in ${\mathbf{X}}$ there exist morphisms
$\varphi_0 : Z_0 \to X$ and $f_0 : Z_0 \to Y$ with $\varphi_0$ cartesian
such that for morphisms $\varphi : Z \to X$ and $f : Z \to Y$ with
$\varphi$ cartesian there exists a unique $\theta : Z \to Z_0$ making
the diagram
\goodbreak
\begin{diagram}[small]
& & X \\
& \ruTo^{\varphi} & \uTo_{\varphi_0} \\
Z & \rDashto_{\theta} & Z_0 \\
& \rdTo_{f} & \dTo_{f_0} \\
& & Y \\
\end{diagram}
commute.
\end{Thm}
\begin{proof}
Let $p : K \to I$, $q : K \to J$ be a product cone in ${\mathbf{B}}$. Then the desired
span $(\varphi_0 , f_0)$ is obtained by putting
$$\varphi_0 := \varphi_X \circ \widetilde{\varphi} \qquad
f_0 := \varphi_Y \circ \widetilde{f}$$
where $(\widetilde{\varphi}, \widetilde{f})$ is terminal in
${\mathbf{Hom}}_K(p^*X,q^*Y)$ and $\varphi_X : p^*X \to X$ and $\varphi_Y : p^*Y \to Y$
are cartesian over $p$ and $q$, respectively. We leave it as a straightforward
exercise to verify that $(\varphi_0 , f_0)$ satisfies the desired universal
property.
\end{proof}
\bigskip
Locally small categories are closed under a lot of constructions as e.g.\
finite products and functor categories. All these arguments go through for
locally small fibrations (see e.g.\ \cite{Bor} vol.~2, Ch.~8.6). There arises
the question what it means that ${\mathbf{B}}$ fibred over itself is locally small.
The answer given by the following Theorem is quite satisfactory.
\begin{Thm}\label{lslccc}
Let ${\mathbf{B}}$ be a category with pullbacks. Then the fundamental fibration
$P_{\mathbf{B}} = \partial_0 : {\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}} \to {\mathbf{B}}$ is locally small if and only if
for every $u : J \to I$ in ${\mathbf{B}}$ the pullback functor $u^{-1} : {\mathbf{B}}/I \to {\mathbf{B}}/J$
has a right adjoint $\Pi_u$ or, equivalently, iff every slice of ${\mathbf{B}}$ is
cartesian closed.
Such categories are usually called \emph{locally cartesian closed}.
\end{Thm}
\begin{proof}
Lengthy but straightforward exercise.
\end{proof}
\bigskip
Some further uses of local smallness are the following.
\begin{Obs}
Let ${\mathbf{B}}$ be a category with an initial object $0$ and $P : {\mathbf{X}} \to {\mathbf{B}}$ be
a locally small fibration. Then for $X,Y \in P(0)$ there is precisely one
vertical morphism from $X$ to $Y$.
\end{Obs}
\begin{proof}
Let $(\varphi_0,f_0)$ be terminal in ${\mathbf{Hom}}_0(X,Y)$. Then there is a
1--1--correspondence between vertical arrows $\alpha : X \to Y$ and
sections $\theta$ of $\varphi_0$
\begin{diagram}[small]
& & X \\
& \ruEqual & \uTo_{\varphi_0} \\
X & \rDashto_{\theta} & Z_0 \\
& \rdTo_{\alpha} & \dTo_{f_0} \\
& & Y \\
\end{diagram}
As there is precisely one map from $0$ to $P(Z_0)$ there is precisely one
section $\theta$ of $\varphi_0$. Accordingly, there is precisely one
vertical arrow $\alpha :X \to Y$.
\end{proof}
\begin{Obs}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a locally small fibration. Then every cartesian arrow
over an epimorphism in ${\mathbf{B}}$ is itself an epimorphism in ${\mathbf{X}}$.
\end{Obs}
\begin{proof}
Let $\varphi : Y \to X$ be cartesian with $P(\varphi)$ epic in ${\mathbf{B}}$.
For $\varphi$ being epic in ${\mathbf{X}}$ it suffices to check that $\varphi$ is epic
w.r.t.\ vertical arrows. Suppose that
$\alpha_1 \circ \varphi = \alpha_2 \circ \varphi$
for vertical $\alpha_1,\alpha_2 : X \to Z$. Due to local smallness of $P$
there is a terminal object $(\varphi_0,f_0)$ in ${\mathbf{Hom}}_{P(X)}(X,Z)$. Thus, for
$i{=}1,2$ there are unique cartesian arrows
$\psi_i$ with $\varphi_0 \circ \psi_i = {\mathit{id}}_X$ and
$f_0 \circ \psi_i = \alpha_i$. We have
\begin{enumerate}
\item[] $\varphi_0 \circ \psi_1 \circ \varphi = \varphi =
\varphi_0 \circ \psi_2 \circ \varphi$ \qquad\qquad and
\item[] $f_0 \circ \psi_1 \circ \varphi = \alpha_1 \circ \varphi =
\alpha_2 \circ \varphi = f_0 \circ \psi_2 \circ \varphi$
\end{enumerate}
from which it follows that $\psi_1 \circ \varphi = \psi_2 \circ \varphi$.
Thus, $P(\psi_1) \circ P(\varphi) = P(\psi_2) \circ P(\varphi)$ and, therefore,
as $P(\varphi)$ is epic by assumption it follows that $P(\psi_1) = P(\psi_2)$.
As $\varphi_0 \circ \psi_1 =\varphi_0 \circ \psi_2$ and $P(\psi_1) = P(\psi_2)$
it follows that $\psi_1 = \psi_2$ as $\varphi_0$ is cartesian. Thus, we finally
get $$\alpha_1 = f_0 \circ \psi_1 = f_0 \circ \psi_2 = \alpha_2$$
as desired.
\end{proof}
\bigskip\bigskip
Next we introduce the notion of generating family.
\begin{Def}\label{genfamdef}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration. A \emph{generating family} for $P$ is
an object $G \in P(I)$ such that for every parallel pair of distinct vertical
arrows $\alpha_1, \alpha_2 : X \to Y$ there exist morphisms $\varphi : Z \to G$
and $\psi : Z \to X$ with $\varphi$ cartesian and
$\alpha_1 \circ \psi \not= \alpha_2 \circ \psi$.
\end{Def}
For locally small fibrations we have the following useful characterisation of
generating families provided the base has binary products.
\begin{Thm}\label{genfamthm}
Let ${\mathbf{B}}$ have binary products and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a locally small fibration.
Then $G \in P(I)$ is a generating family for $P$ iff for every $X \in {\mathbf{X}}$
there are morphisms $\varphi_X : Z_X \to G$ and $\psi_X : Z_X \to X$ such
that $\varphi_X$ is cartesian and $\psi_X$ is \emph{collectively epic} in the
sense that vertical arrows $\alpha_1, \alpha_2 : X \to Y$ are equal
iff $\alpha_1 \circ \psi_X = \alpha_2 \circ \psi_X$.
\end{Thm}
\begin{proof}
The implication from right to left is trivial.
For the reverse direction suppose that $G \in P(I)$ is a generating family.
Let $X \in P(J)$. As ${\mathbf{B}}$ is assumed to have binary products by
Theorem~\ref{lsbp}
there exist $\varphi_0 : Z_0 \to G$ and $\psi_0 : Z_0 \to X$
with $\varphi_0$ cartesian such that for morphisms $\varphi : Z \to G$ and
$\psi : Z \to X$ with $\varphi$ cartesian there exists a unique
$\theta : Z \to Z_0$ with
\begin{diagram}[small]
& & G \\
& \ruTo^{\varphi} & \uTo_{\varphi_0} \\
Z & \rDashto_{\theta} & Z_0 \\
& \rdTo_{\psi} & \dTo_{\psi_0} \\
& & X \\
\end{diagram}
Now assume that $\alpha_1, \alpha_2 : X \to Y$ are distinct vertical arrows.
As $G$ is a generating family for $P$ there exist $\varphi : Z \to G$ and
$\psi : Z \to X$ with $\varphi$ cartesian and
$\alpha_1 \circ \psi \not= \alpha_2 \circ \psi$. But there is a
$\theta : Z \to Z_0$ with $\psi = \psi_0 \circ \theta$. Then we
have $\alpha_1 \circ \psi_0 \not= \alpha_2 \circ \psi_0$ (as otherwise
$\alpha_1 \circ \psi = \alpha_1 \circ \psi_0 \circ \theta =
\alpha_2 \circ \psi_0 \circ \theta = \alpha_2 \circ \psi$). Thus, we have shown
that $\psi_0$ is collectively epic and we may take $\varphi_0$ and $\psi_0$ as
$\varphi_X$ and $\psi_X$, respectively.
\end{proof}
\bigskip
Intuitively, this means that ``every object can be covered by a sum of
$G_i$'s'' in case the fibration has internal sums.
\newpage
\section{Well-Poweredness}
For ordinary categories well-poweredness means that for every object the
collection of its subobjects can be indexed by a set. Employing again the
notion of representability (of elementary fibrations) we can define a notion
of well--poweredness for (a wide class of) fibrations.
\begin{Def}\label{wellpowerdef}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration where vertical monos are stable under
reindexing. For $X \in P(I)$ let ${\mathrm{Sub}}_I(X)$ be the following category.
Its objects are pairs $(\varphi,m)$ where $\varphi : Y \to X$ is cartesian
and $m : S \to Y$ is a vertical mono. A morphism from $(\psi,n)$ to
$(\varphi,m)$ is a morphism $\theta$ such that $\varphi \circ \theta = \psi$
and $\theta \circ n = m \circ \widetilde{\theta}$
\begin{diagram}
T \SEpbk & \rTo^{\widetilde{\theta}}_{\mathrm{cart}} & S \\
\dEmbed^{n} & & \dEmbed_{m} \\
Z & \rTo_{\theta}^{\mathrm{cart}} & Y \\
& \rdTo_{\psi} & \dTo_{\varphi}\\
& & X \\
\end{diagram}
for a (necessarily unique) cartesian arrow $\widetilde{\theta}$. The category
${\mathrm{Sub}}_I(X)$ is fibred over ${\mathbf{B}}/I$ by sending objects $(\varphi,m)$ to
$P(\varphi)$ and morphisms $\theta$ to $P(\theta)$.
The fibration $P$ is called \emph{well--powered} iff for every $I \in {\mathbf{B}}$ and
$X \in P(I)$ the elementary fibration ${\mathrm{Sub}}_I(X)$ over ${\mathbf{B}}/I$ is representable,
i.e.\ ${\mathrm{Sub}}_I(X)$ has a terminal object. \hfill \mbox{\ $\lozenge$}
\end{Def}
If $(\varphi_X,m_X)$ is terminal in ${\mathrm{Sub}}_I(X)$ then, roughly speaking,
for every $u : J \to I$ and $m \in {\mathrm{Sub}}_{P(J)}(u^*X)$ there is a unique map
$v : u \to P(\varphi_X)$ in ${\mathbf{B}}/I$ with $v^*(m_X) \cong m$.
We write $\sigma_X : {\mathrm{S}}_I(X) \to I$ for $P(\varphi_X)$.
\medskip
Categories with finite limits whose fundamental fibration is well-powered
have the following pleasant characterisation.
\begin{Thm}
A category ${\mathbf{B}}$ with finite limits is a topos if and only if its fundamental
fibration $P_{\mathbf{B}} = \partial_1 : {\mathbf{B}}^{\mbox{$2\hspace*{-1.2ex}1$}} \to {\mathbf{B}}$ is well--powered.\\
Thus, in this particular case well--poweredness entails local smallness
as every topos is locally cartesian closed.
\end{Thm}
\begin{proof}
Lengthy, but straightforward exercise.
\end{proof}
\bigskip
One may find it amusing, reassuring or whatever that for categories ${\mathbf{B}}$
with finite limits we have
\begin{itemize}
\item[] $P_{\mathbf{B}}$ is locally small iff ${\mathbf{B}}$ is locally cartesian closed
\item[] $P_{\mathbf{B}}$ is wellpowered iff ${\mathbf{B}}$ is a topos
\end{itemize}
i.e.\ that important properties of ${\mathbf{B}}$ can be expressed by simple
conceptual properties of the corresponding fundamental fibration.
\newpage
\section{Definability}
If ${\mathbf{C}}$ is a category and $(A_i)_{i \in I}$ is a family of objects in ${\mathbf{C}}$
then for every subcategory ${\mathbf{P}}$ of ${\mathbf{C}}$ one may want to form the subset
\[ \{ i \in I \mid A_i \in {\mathbf{P}}\} \]
of $I$ consisting of all those indices $i \in I$ such that object $A_i$
belongs to the subcategory ${\mathbf{P}}$. Though intuitively ``clear'' it is somewhat
disputable from the point of view of \emph{formal axiomatic set theory} (e.g.\
ZFC or GBN) whether the set $\{ i \in I \mid A_i \in {\mathbf{P}}\}$ actually exists.
The reason is that the usual \emph{separation axiom} guarantees the existence
of (sub)sets of the form $\{ \, i \in I \mid P(i) \,\}$ only for predicates
$P(i)$ that can be expressed\footnote{i.e.\ by a first order formula using
just the binary relation symbols $=$ and $\in$} in the formal language of set
theory. Now this may appear as a purely ``foundationalist'' argument to the
working mathematician. However, we don't take any definite position w.r.t.\
this delicate foundational question but, instead, investigate the
mathematically clean concept of \emph{definability} for fibrations.
\begin{Def}\label{stabclass}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration. A class ${\mathcal{C}} \subseteq {\mathrm{Ob}}({\mathbf{X}})$ is called
\emph{$P$--stable} or simply \emph{stable} iff for $P$--cartesian arrows
$\varphi : Y \to X$ it holds that $Y \in {\mathcal{C}}$ whenever $X \in {\mathcal{C}}$, i.e.\ iff
the class ${\mathcal{C}}$ is stable under reindexing (w.r.t.\ $P$).
\hfill \mbox{\ $\lozenge$}
\end{Def}
\begin{Def}\label{dfblclass}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration. A stable class ${\mathcal{C}} \subseteq {\mathrm{Ob}}({\mathbf{X}})$
is called \emph{definable} iff for every $X \in P(I)$ there is a subobject
$m_0 : I_0 \rightarrowtail I$ such that
\begin{enumerate}
\item[\rm (1)] $m_0^*X \in {\mathcal{C}}$ and
\item[\rm (2)] $u : J \to I$ factors through $m_0$
whenever $u^*X \in {\mathcal{C}}$. \hfill \mbox{\ $\lozenge$}
\end{enumerate}
\end{Def}
Notice that $u^*X \in {\mathcal{C}}$ makes sense as stable classes
${\mathcal{C}} \subseteq {\mathrm{Ob}}({\mathbf{X}})$ are necessarily closed under (vertical) isomorphisms.
\bigskip\noindent
{\bf Remark.}
If ${\mathcal{C}} \subseteq {\mathrm{Ob}}({\mathbf{X}})$ is stable then ${\mathcal{C}}$ is definable iff for every
$X \in P(I)$ the elementary fibration $P_{{\mathcal{C}},X} : {\mathcal{C}}_X \to {\mathbf{B}}/I$ is
representable where ${\mathcal{C}}_X$ is the full subcategory of ${\mathbf{X}}/X$ on cartesian
arrows and $P_{{\mathcal{C}},X} = P_{/X}$ sends
\begin{diagram}[small]
Z & & \rTo^{\theta} & & Y \\
& \rdTo_{\psi}^{\mathrm{cart}} & & \ldTo_{\varphi}^{\mathrm{cart}} & \\
& & X & & \\
\end{diagram}
in ${\mathcal{C}}_X$ to
\begin{diagram}[small]
P(Z) & & \rTo^{P(\theta)} & & P(Y) \\
& \rdTo_{P(\psi)} & & \ldTo_{P(\varphi)} & \\
& & P(X) & & \\
\end{diagram}
in ${\mathbf{B}}/I$. Representability of the elementary fibration $P_{{\mathcal{C}},X}$ then means
that there is a cartesian arrow $\varphi_0 : X_0 \to X$ with $X_0 \in {\mathcal{C}}$
such that for every cartesian arrow $\psi : Z \to X$ with $Z \in {\mathcal{C}}$ there
exists a unique arrow $\theta : Z \to X_0$ with $\varphi_0\circ\theta = \psi$.
This $\theta$ is necessarily cartesian and, therefore, already determined by
$P(\theta)$. From uniqueness of $\theta$ it follows immediately that
$P(\varphi_0)$ is monic.
One also could describe the situation as follows. Every $X \in P(I)$ gives
rise to a subpresheaf $C_X$ of ${\mathrm{Y}}_{\mathbf{B}}(P(X))$ consisting of the arrows
$u : J \to I$ with $u^*X \in {\mathcal{C}}$. Then ${\mathcal{C}}$ is definable iff for every
$X \in {\mathbf{X}}$ the presheaf $C_X$ is representable, i.e.\
\begin{diagram}[small]
{\mathcal{C}}_X & \rEmbed & {\mathrm{Y}}_{\mathbf{B}}(P(X)) \\
\dTo_{\cong} & \ruEmbed_{{\mathrm{Y}}_{\mathbf{B}}(m_X)} & \\
{\mathrm{Y}}_{\mathbf{B}}(I_X) & & \\
\end{diagram}
where $m_X$ is monic as ${\mathrm{Y}}_{\mathbf{B}}$ reflects monos.
\hfill \mbox{\ $\lozenge$}
\medskip
Next we give an example demonstrating that \emph{definability is not vacuously
true}. Let ${\mathbf{C}} = {\mathbf{FinSet}}$ and ${\mathbf{X}} = {\mathrm{Fam}}({\mathbf{C}})$ fibred over ${\mathbf{Set}}$. Let
$ {\mathcal{C}} \subseteq {\mathrm{Fam}}({\mathbf{C}})$ consist of those families $(A_i)_{i{\in}I}$ such
that $\exists n \in {\mathbb{N}}.\, \forall i \in I. \,|A_i| \leq n$.
Obviously, the class ${\mathcal{C}}$ is stable but it is not definable as for the family
$$K_n = \{ i \in {\mathbb{N}} \mid i < n \} \qquad (n \in {\mathbb{N}})$$
there is no greatest subset $P \subseteq {\mathbb{N}}$ with
$\exists n \in {\mathbb{N}}.\, \forall i \in P. \,i < n$. Thus, the requirement of
definability is non--trivial already when the base is ${\mathbf{Set}}$.
\medskip
For a fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ one may consider the fibration
$P^{(\mbox{$2\hspace*{-1.2ex}1$})} : {\mathbf{X}}^{(\mbox{$2\hspace*{-1.2ex}1$})} \to {\mathbf{B}}$ of (vertical) arrows in ${\mathbf{X}}$. Thus, it is
clear what it means that a class ${\mathcal{M}} \subseteq {\mathrm{Ob}}({\mathbf{X}}^{(\mbox{$2\hspace*{-1.2ex}1$})})$
is ($P^{(\mbox{$2\hspace*{-1.2ex}1$})}$-)stable. Recall that ${\mathrm{Ob}}({\mathbf{X}}^{(\mbox{$2\hspace*{-1.2ex}1$})})$ is the class of
$P$--vertical arrows of ${\mathbf{X}}$. Then ${\mathcal{M}}$ is stable iff for
all $\alpha : X \to Y$ in ${\mathcal{M}}$ and cartesian arrows
$\varphi : X' \to X$ and $\psi : Y' \to Y$ over the same arrow $u$ in ${\mathbf{B}}$
the unique vertical arrow $\alpha' : X' \to Y'$ with $\psi \circ \alpha' =
\alpha \circ \varphi$ is in ${\mathcal{M}}$, too. In other words
$u^*\alpha \in {\mathcal{M}}$ whenever $\alpha \in {\mathcal{M}}$.
\begin{Def}\label{dfblclassmor}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration and ${\mathcal{M}}$ a stable class of
vertical arrows in ${\mathbf{X}}$. Then ${\mathcal{M}}$ is called \emph{definable} iff for
every $\alpha \in P(I)$ there is a subobject $m_0 : I_0 \rightarrowtail I$ such that
$m_0^*\alpha \in {\mathcal{M}}$ and $u : J \to I$ factors through $m_0$ whenever
$u^*\alpha \in {\mathcal{M}}$. \hfill \mbox{\ $\lozenge$}
\end{Def}
Next we discuss what is an appropriate notion of subfibration for a fibration
$P : {\mathbf{X}} \to {\mathbf{B}}$. Keeping in mind the analogy with ${\mathrm{Fam}}({\mathbf{C}})$ over ${\mathbf{Set}}$ we
have to generalise the situation ${\mathrm{Fam}}({\mathbf{S}}) \subseteq {\mathrm{Fam}}({\mathbf{C}})$ where ${\mathbf{S}}$ is
a subcategory of ${\mathbf{C}}$ which is \emph{replete} in the sense that
${{\mathrm{Mor}}}({\mathbf{S}})$ is stable under composition with isomorphisms in ${\mathbf{C}}$.
In this case the objects of ${\mathrm{Fam}}({\mathbf{S}})$ are stable under reindexing and so
are the vertical arrows of ${\mathrm{Fam}}({\mathbf{S}})$. This motivates the following
\begin{Def}\label{sfdef}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$. A \emph{subfibration} of $P$ is given by a subcategory
${\mathbf{Z}}$ of ${\mathbf{X}}$ such that
\begin{enumerate}
\item[\rm (1)] cartesian arrows of ${\mathbf{X}}$ are in ${\mathbf{Z}}$ whenever their codomain
is in ${\mathbf{Z}}$ (i.e.\ a cartesian arrow $\varphi : Y \to X$ is in
${\mathbf{Z}}$ whenever $X \in {\mathbf{Z}}$) and
\item[\rm (2)] for every commuting square in ${\mathbf{X}}$
\begin{diagram}[small]
X' & \rTo^{\varphi}_{\mathrm{cart}} & X \\
\dTo^{f'} & & \dTo_{f} \\
Y' & \rTo_{\psi}^{\mathrm{cart}} & Y \\
\end{diagram}
the morphism $f' \in {\mathbf{Z}}$ whenever $f \in {\mathbf{Z}}$ and
$\varphi$ and $\psi$ are cartesian.\hfill \mbox{\ $\lozenge$}
\end{enumerate}
\end{Def}
Notice that a subfibration ${\mathbf{Z}}$ of $P : {\mathbf{X}} \to {\mathbf{B}}$ is determined uniquely by
${\mathcal V} \cap {\mathbf{Z}}$ where ${\mathcal V}$ is the class of vertical arrows
of ${\mathbf{X}}$ w.r.t.\ $P$. Thus, ${\mathbf{Z}}$ gives rise to \emph{replete} subcategories
$$S(I) = {\mathbf{Z}} \cap P(I) \qquad\qquad (I \in {\mathbf{B}})$$
which are stable under reindexing in the sense that for $u : J \to I$ in ${\mathbf{B}}$
\begin{itemize}\item[]
\begin{itemize}
\item[\rm ($S_{\mathrm{obj}}$)] \quad
$u^*X \in S(J)$ whenever $X \in S(I)$ \quad and
\item[\rm ($S_{\mathrm{mor}}$)] \quad
$u^*\alpha \in S(J)$ whenever $\alpha \in S(I)$.
\end{itemize}
\end{itemize}
On the other hand for every such such system $S = (S(I) \mid I \in {\mathbf{B}})$ of
replete subcategories of the fibres of $P$ which is stable under reindexing
in the sense that the above conditions ($S_{\mathrm{obj}}$) and
($S_{\mathrm{mor}}$) are satisfied we can define a subfibration ${\mathbf{Z}}$
of $P : {\mathbf{X}} \to {\mathbf{B}}$ as follows: $f : Y \to X$ in ${\mathbf{Z}}$ iff
$X \in S(P(X))$ and $\alpha \in S(P(Y))$ where the diagram
\begin{diagram}[small]
Y & & \\
\dTo^{\alpha} & \rdTo^{f}& \\
Z & \rTo^{\mathrm{cart}}_{\varphi} & X \\
\end{diagram}
commutes and $\alpha$ is vertical and $\varphi$ is cartesian. Obviously, this
subcategory ${\mathbf{Z}}$ satisfies condition (1) of Definition~\ref{sfdef}. For
condition (2) consider the diagram
\begin{diagram}[small]
X' & \rTo^{\varphi}_{\mathrm{cart}} & X \\
\dTo^{\alpha'} & & \dTo_{\alpha} \\
Z' & \rTo_{\mathrm{cart}} & Z \\
\dTo^{\theta'} & & \dTo_{\theta} \\
Y' & \rTo_{\psi}^{\mathrm{cart}} & Y \\
\end{diagram}
where $\alpha'$ and $\alpha$ are vertical, $\theta'$ and $\theta$ are
cartesian and $f' = \theta' \circ \alpha'$ and $f = \theta \circ \alpha$
from which it is clear that $\alpha' \in S(P(X'))$ whenever
$\alpha \in S(P(X))$ and, therefore, $f' \in {\mathbf{Z}}$ whenever $f \in {\mathbf{Z}}$.
\smallskip
Now we are ready to define the notion of definability for subfibrations.
\begin{Def}\label{subfibdfbl}
A subfibration ${\mathbf{Z}}$ of a fibration $P : {\mathbf{X}} \to {\mathbf{B}}$ is \emph{definable} iff
${\mathcal{C}} = {\mathrm{Ob}}({\mathbf{Z}})$ and ${\mathcal{M}} = {\mathcal{V}} \cap {\mathbf{Z}}$ are definable classes of
objects and morphisms, respectively. \hfill \mbox{\ $\lozenge$}
\end{Def}
Without proof we mention a couple of results illustrating the strength of
definability. Proofs can be found in \cite{Bor} vol.2, Ch.8.
\begin{enumerate}
\item[(1)]
Locally small fibrations are closed under definable subfibrations.
\item[(2)]
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a locally small fibration over ${\mathbf{B}}$ with finite limits.
Then the class of vertical isomorphisms of ${\mathbf{X}}$ is a definable subclass of
objects of ${\mathbf{X}}^{(\mbox{$2\hspace*{-1.2ex}1$})}$ w.r.t.\ $P^{(\mbox{$2\hspace*{-1.2ex}1$})}$.
\item[(3)]
If, moreover, ${\mathbf{X}}$ has finite limits and $P$ preserves them then vertical
monos (w.r.t.\ their fibres) form a definable subclass of objects of
${\mathbf{X}}^{(\mbox{$2\hspace*{-1.2ex}1$})}$ w.r.t.\ $P^{(\mbox{$2\hspace*{-1.2ex}1$})}$ and fibrewise terminal objects form a
definable subclass of objects of ${\mathbf{X}}$ w.r.t.\ $P$.
\item[(4)]
Under the assumptions of (3) for every finite category ${\mathbf{D}}$ the fibrewise
limiting cones of fibrewise ${\mathbf{D}}$--diagrams from a definable class.
\end{enumerate}
A pleasant consequence of (3) is that under the assumptions of (3) the class
of pairs of the form $(\alpha,\alpha)$ for some vertical arrow $\alpha$ form
a definable subclass of the objects of ${\mathbf{X}}^{({\mathbb{G}})}$ w.r.t.\ $P^{({\mathbb{G}})}$
where ${\mathbb{G}}$ is the category with two objects $0$ and $1$ and two nontrivial
arrows from $0$ to $1$. In other words under the assumptions of (3)
\emph{equality of morphisms is definable}.
On the negative side we have to remark that for most fibrations the class
$\{ (X,X) \mid X \in {\mathrm{Ob}}(X)\}$ is not definable as a subclass of ${\mathbf{X}}^{(2)}$
(where $2 = 1+1$ is the discrete category with 2 objects) simply because this
class is not even stable (under reindexing). Actually, stability fails already
if some of the fibres contains different isomorphic objects! This observation
may be interpreted as confirming the old suspicion that equality of objects
is somewhat ``fishy'' at least for non--split fibrations.
Notice, however, that even for discrete split fibrations equality need not be
definable which can be seen as follows. Consider a presheaf
$A \in \widehat{{\mathbb{G}}}$ (where ${\mathbb{G}}$ is defined as in the previous paragraph)
which may most naturally be considered as a directed graph. Then for $A$
considered as a discrete split fibration equality of objects is definable if
and only if $A$ is subterminal, i.e.\ both $A(1)$ and $A(0)$ contain at most
one element. Thus, for interesting graphs equality of objects is not definable!
\smallskip
We conclude this section with the following positive result.
\begin{Thm}\label{descent}
Let ${\mathbf{B}}$ be a topos and $P : {\mathbf{X}} \to {\mathbf{B}}$ a fibration. If ${\mathcal{C}}$ is a definable
class of objects of ${\mathbf{X}}$ (w.r.t.\ $P$) then for every cartesian arrow
$\varphi : Y \to X$ over an epimorphism in ${\mathbf{B}}$ we have that $X \in {\mathcal{C}}$ iff
$Y \in {\mathcal{C}}$ (often refered to as ``descent property'').
\end{Thm}
\begin{proof}
The implication from left to right follows from stability of ${\mathcal{C}}$.
For the reverse direction suppose that $\varphi : Y \to X$ is cartesian over
an epi $e$ in ${\mathbf{B}}$. Then by definability of ${\mathcal{C}}$ we have $e = m \circ f$
where $m$ is a mono in ${\mathbf{B}}$ with $m^*X \in {\mathcal{C}}$. But as $e$ is epic and $m$
is monic and we are in a topos it follows that $m$ is an isomorphism and,
therefore, $Y \cong m^*X \in {\mathcal{C}}$.
\end{proof}
\bigskip
Notice that this Theorem can be generalised to regular categories ${\mathbf{B}}$
where, however, one has to require that $P(\varphi)$ is a regular epi
(as a monomorphism $m$ in
a regular category is an isomorphism if $m \circ f$ is a regular epimorphism
for some morphism $f$ in ${\mathbf{B}}$).
\newpage
\section{Preservation Properties of Change of Base}
We know already that for an arbitrary functor $F : {\mathbf{A}} \to {\mathbf{B}}$ we have that
$F^*P \in {\bf Fib}({\mathbf{A}})$ whenever $P \in {\bf Fib}({\mathbf{B}})$. The (2-)functor
$F^* : {\bf Fib}({\mathbf{B}}) \to {\bf Fib}({\mathbf{A}})$ is known as \emph{change of base
along $F$}. In this section we will give necessary and sufficient conditions
on $F$ guaranteeing that change of base along $F$ preserves
``all good properties of fibrations''.
\begin{Lem}\label{cbsm}
Let $F : {\mathbf{A}} \to {\mathbf{B}}$ be a functor. Then $F^* : {\mathbf{Fib}}({\mathbf{B}}) \to {\mathbf{Fib}}({\mathbf{A}})$
preserves smallness of fibrations if and only if $F$ has a right adjoint $U$.
\end{Lem}
\begin{proof}
Suppose that $F$ has a right adjoint $U$. If $C \in {\mathbf{cat}}({\mathbf{B}})$ then
$F^*P_C$ is isomorphic to $P_{U(C)}$ where $U(C)$ is the image of $C$ under
$U$ which preserves all existing limits as it is a right adjoint.
Suppose that $F^*$ preserves smallness of fibrations. Consider for $I \in {\mathbf{B}}$
the small fibration $P_I = \underline{I} = \partial_0 : {\mathbf{B}}/I \to {\mathbf{B}}$.
Then $F^*P_I$ is isomorphic to $\partial_0 : F{\downarrow}I \to {\mathbf{B}}$ which
is small iff there exists $U(I) \in {\mathbf{A}}$ such that $F^*P_I \cong P_{U(I)}$,
i.e.\ ${\mathbf{B}}(F(-),I) \cong {\mathbf{A}}(-,U(I))$. Thus, if $F^*$ preserves smallness
of fibrations then for all $I \in {\mathbf{B}}$ we have ${\mathbf{B}}(F(-),I) \cong {\mathbf{A}}(-,U(I))$
for some $U(I) \in {\mathbf{A}}$, i.e.\ $F$ has a right adjoint $U$.
\end{proof}
\bigskip
As a consequence of Lemma~\ref{cbsm} we get that for $u : I \to J$ in ${\mathbf{B}}$
change of base along $\Sigma_u : {\mathbf{B}}/I \to {\mathbf{B}}/J$ preserves smallness of
fibrations iff $\Sigma_u$ has a right adjoint $u^{-1} : {\mathbf{B}}/J \to {\mathbf{B}}/I$, i.e.\
pullbacks in ${\mathbf{B}}$ along $u$ do exist. Analogously, change of base along
$\Sigma_I : {\mathbf{B}}/I \to {\mathbf{B}}$ preserves smallness of fibrations iff $\Sigma_I$ has
a right adjoint $I^*$, i.e.\ for all $K \in {\mathbf{B}}$ the cartesian product of $I$
and $K$ exists. One can show that change of base along $u^{-1}$ and $I^*$
is right adjoint to change of base along $\Sigma_u$ and $\Sigma_I$,
respectively. Thus, again by Lemma~\ref{cbsm} change of base along $u^{-1}$
and $I^*$ preserves smallness of fibrations iff $u^{-1}$ and $I^*$ have right
adjoints $\Pi_u$ and $\Pi_I$, respectively.
\bigskip
From now on we make the reasonable assumption that all base categories have
pullbacks as otherwise their fundamental fibrations would not exist.
\begin{Lem}\label{Lem13.2}
Let ${\mathbf{A}}$ and ${\mathbf{B}}$ be categories with pullbacks and $F : {\mathbf{A}} \to {\mathbf{B}}$ an
arbitrary functor. Then the following conditions are equivalent
\begin{enumerate}
\item[\rm (1)] $F$ preserves pullbacks
\item[\rm (2)] $F^* : {\mathbf{Fib}}({\mathbf{B}}) \to {\mathbf{Fib}}({\mathbf{A}})$ preserves the property
of having internal sums
\item[\rm (3)] $\partial_1 : {\mathbf{B}}{\downarrow}F \to {\mathbf{A}}$ has internal sums.
\end{enumerate}
\end{Lem}
\begin{proof}
The implications (1) $\Rightarrow$ (2) and (2) $\Rightarrow$ (3) are easy.
The implication (3) $\Rightarrow$ (1) can be seen as follows. Suppose that
the bifibration $\partial_1 : {\mathbf{B}}{\downarrow}F \to {\mathbf{A}}$ satisfies BCC then
for pullbacks
\begin{diagram}[small]
L \SEpbk & \rTo^{q} & K \\
\dTo^{p} & & \dTo_{v} \\
J & \rTo_{u} & I \\
\end{diagram}
in ${\mathbf{A}}$ we have
\begin{diagram}[small]
F(L) \SEpbk & & \rTo^{F(q)} & & F(K) & & \\
& \rdDashto_{\alpha}^{\cong} & & & \dEqual & \rdEqual & \\
\dEqual & & \SEpbk & \rTo & \HonV & & F(K) \\
& & \dTo & & \dEqual & & \\
F(L) & \hLine & \VonH & \rTo^{F(q)} & F(K) & & \dTo_{F(v)} \\
& \rdTo_{F(p)} & & & & \rdTo^{F(v)} & \\
& & F(J) & & \rTo_{F(u)} & & F(I) \\
\end{diagram}
As back and front face of the cube are cartesian arrows and the right face is
a cocartesian arrow it follows from the postulated BCC for
$\partial_1 : {\mathbf{B}} {\downarrow} F \to {\mathbf{A}}$ that the left face is a cocartesian
arrow, too. Thus, the map $\alpha$ is an isomorphism from which it follows that
\begin{diagram}[small]
F(L) \SEpbk & \rTo^{F(q)} & F(K) \\
\dTo^{F(p)} & & \dTo_{F(v)} \\
F(J) & \rTo_{F(u)} & F(I) \\
\end{diagram}
is a pullback as required.
\end{proof}
\bigskip
Thus, by the previous two lemmas a functor $F : {\mathbf{A}} \to {\mathbf{B}}$ between categories
with pullbacks necessarily has to preserve pullbacks and have a right adjoint
$U$ whenever $F^*$ preserves ``all good properties of fibrations'' as being
small and having internal sums certainly are such ``good properties''.
Actually, as pointed out by B\'enabou in his 1980 Louvain-la-Neuve lectures
\cite{Ben} these requirements for $F$ are also sufficient for $F^*$ preserving
the following good properties of fibrations
\begin{itemize}
\item (co)completeness
\item smallness
\item local smallness
\item definability
\item well--poweredness.
\end{itemize}
We will not prove all these claims but instead discuss \emph{en detail}
preservation of local smallness. Already in this case the proof is
paradigmatic and the other cases can be proved analogously.
\begin{Lem}\label{lspres}
If $F : {\mathbf{A}} \to {\mathbf{B}}$ is a functor with right adjoint $U$ and ${\mathbf{A}}$ and ${\mathbf{B}}$
have pullbacks then change of base along $F$ preserves local smallness
of fibrations.
\end{Lem}
\begin{proof}
Suppose $P \in {\mathbf{Fib}}({\mathbf{B}})$ is locally small. Let $X,Y \in P(FI)$ and
\begin{diagram}[small]
& & X \\
& \ruTo^{\varphi_0}_{\mathrm{cart}} & \\
d^*X & & \\
& \rdTo_{\psi_0} & \\
& & Y
\end{diagram}
with $d = P(\varphi_0) = P(\psi_0) : \hom_{FI}(X,Y) \to FI$ be the terminal
such span. Then consider the pullback (where we write $H$ for $\hom_{FI}(X,Y)$)
\begin{diagram}[small]
\widetilde{H} \SEpbk & \rTo^{h} & UH \\
\dTo^{\widetilde{d}} & & \dTo_{Ud} \\
I & \rTo_{\eta_I} & UFI \\
\end{diagram}
in ${\mathbf{A}}$ where $\eta_I$ is the unit of $F \dashv U$ at $I$. Then there is a
natural bijection between $v : u \to \widetilde{d}$ in ${\mathbf{A}}/I$ and
$\widehat{v} : F(u) \to d$ in ${\mathbf{B}}/FI$ by sending $v$ to
$\widehat{v} = \varepsilon_H \circ F(h) \circ F(v)$
where $\varepsilon_H$ is the counit of $F \dashv U$ at $H$.
Let $\theta_1$ be a $P$-cartesian arrow over
$\varepsilon_H \circ F(h) : F\widetilde{H} \to H$ to $d^*X$. Let
$\varphi_1 := \varphi_0 \circ \theta_1$ and $\psi_1 := \psi_0 \circ \theta_1$
which are both mapped by $P$ to
\[ d \circ \varepsilon_H \circ F(h) =
\varepsilon_{FI} \circ F(Ud) \circ F(h) =
\varepsilon_{FI} \circ F\eta_I \circ F\widetilde{d} = F\widetilde{d}
\]
We show now that the span
$\bigl((\widetilde{d},\varphi_1),((\widetilde{d},\psi_1)\bigr)$ is a terminal
object in the category ${\mathbf{Hom}}_I(X,Y)$ for $F^*P$.
For that purpose suppose that $u : J \to I$ in ${\mathbf{A}}$ and $\varphi : Z \to X$,
$\psi : Z \to Y$ are arrows over $u$ w.r.t.\ $F^*P$ with $\varphi$ cartesian
w.r.t.\ $F^*P$.
There exists a unique $P$-cartesian arrow $\theta_2$ with
$\varphi = \varphi_0 \circ \theta_2$ and $\psi = \psi_0 \circ \theta_2$.
For $\widehat{v} := P(\theta_2)$ we have $d \circ \widehat{v} = F(u)$
as $P(\varphi) = F(u) = P(\psi)$. Then there exists a unique map
$v : u \to \widetilde{d}$ with
$\varepsilon_H \circ F(h) \circ F(v) = \widehat{v}$.
Now let $\theta$ be the unique $P$-cartesian arrow over $F(v)$
with $\theta_2 = \theta_1 \circ \theta$ which exists
as $P(\theta_1) = \varepsilon_H \circ F(h)$ and $P(\theta_2) = \widehat{v} =
\varepsilon_H \circ F(h) \circ F(v)$. Thus, we have $v : u \to \widetilde{d}$
and a cartesian arrow $\theta$ with
$P(\theta) = F(v)$, $\varphi_1 \circ \theta = \varphi$
and $\psi_1 \circ \theta = \psi$ as desired.
For uniqueness of $(v,\theta)$ with this property suppose that
$v' : u \to \widetilde{d}$ and $\theta'$ is a cartesian arrow with
$P(\theta') = F(v')$, $\varphi_1 \circ \theta' = \varphi$ and
$\psi_1 \circ \theta' = \psi$.
From the universal property of $(\varphi_0,\psi_0)$ it follows that
$\theta_2 = \theta_1 \circ \theta'$. Thus, we have
\[ \widehat{v} = P(\theta_2) = P(\theta_1) \circ P(\theta') =
\varepsilon_H \circ F(h) \circ P(\theta') =
\varepsilon_H \circ F(h) \circ F(v')
\]
from which it follows that $v = v'$ as by assumption we also have
$\widetilde{d} \circ v' = u$. From $\theta_2 = \theta_1 \circ \theta'$ and
$P(\theta') = F(v') = F(v)$ it follows that $\theta' = \theta$ because
$\theta_1$ is cartesian and we have $\theta_2 = \theta_1 \circ \theta$ and
$P(\theta') = F(v)$ due to the construction of $\theta$.
\end{proof}
\bigskip
Analogously one shows that under the same premisses as in Lemma~\ref{lspres}
the functor $F^*$ preserves well--poweredness of fibrations and that, for
fibrations $P : {\mathbf{X}} \to {\mathbf{B}}$ and definable classes ${\mathcal{C}} \subseteq {\mathbf{X}}$,
the class
$$F^*({\mathcal{C}}) := \{(I,X) \mid X \in P(FI) \wedge X \in {\mathcal{C}} \}$$
is definable w.r.t.\ $F^*P$.
\bigskip\noindent
{\bf Warning.} If $F \dashv U : {\mathbf{E}} \to {\mathbf{S}}$ is an unbounded geometric morphism
then $P_{\mathbf{E}} = \partial_1 : {\mathbf{E}}^\mbox{$2\hspace*{-1.2ex}1$} \to {\mathbf{E}}$ has a generic family
though $F^*P_{\mathbf{E}} \cong {\mathsf{gl}}(F)$ does not have a generic family
as otherwise by Theorem~\ref{bgmthm} (proved later on) the geometric morphism
$F \dashv U$ were bounded! Thus, the property of having a generating family is
not preserved by change of base along functors that preserve finite limits
and have a right adjoint. In this respect the property of having a small
generating family is not as ``good'' as the other properties of fibrations
mentioned before which are stable under change of base along functors
that preserve pullbacks and have a right adjoint. \hfill \mbox{\ $\lozenge$}
\bigskip
The moral of this section is that functors $F$ between categories with
pullbacks preserve ``all good'' (actually ``most good'') properties of
fibrations by change of base along $F$ if and only if $F$ preserves
pullbacks and has a right adjoint. In particular, this holds for inverse image
parts of geometric morphisms, i.e.\ finite limit preserving functors having a
right adjoint. But there are many more examples which are also important.
Let ${\mathbf{B}}$ be a category with pullbacks and $u : I \to J$ a morphism in ${\mathbf{B}}$ then
$\Sigma_u : {\mathbf{B}}/I \to {\mathbf{B}}/J$ preserves pullbacks and has a right adjoint, namely
the pullback functor $u^{-1} : {\mathbf{B}}/J \to {\mathbf{B}}/I$, but $\Sigma_u$ preserves
terminal objects if and only if $u$ is an isomorphism. Notice that for
$I \in {\mathbf{B}}$ the functor $\Sigma_I = \partial_0 : {\mathbf{B}}/I \to {\mathbf{B}}$ always preserves
pullbacks but has a right adjoint $I^*$ if and only if for all $K \in {\mathbf{B}}$ the
cartesian product of $I$ and $K$ exists. Thus, for a category ${\mathbf{B}}$ with
pullbacks the functors $\Sigma_I : {\mathbf{B}}/I \to {\mathbf{B}}$ preserve ``all good
properties'' of fibrations by change of base if and only if ${\mathbf{B}}$ has all
binary products (but not necessarily a terminal object!).
A typical such example is the full subcategory ${\mathbf{F}}$ of ${\mathbf{Set}}^{{\mathbb{N}}}$ on those
${\mathbb{N}}$-indexed families of sets which are empty for almost all indices. Notice,
however, that every slice of ${\mathbf{F}}$ actually is a (Grothendieck) topos. This
${\mathbf{F}}$ is a typical example for B\'enabou's notion of \emph{partial topos},
i.e.\ a category with binary products where every slice is a topos.
The above example can be generalised easily. Let ${\mathbf{E}}$ be some topos and
$F$ be a (downward closed) subset of ${\mathrm{Sub}}_{\mathbf{E}}(1_{\mathbf{E}})$ then ${\mathbf{E}}_{/F}$, the full
subcategory of ${\mathbf{E}}$ on those objects $A$ whose terminal projection factors
through some subterminal in $F$, is a partial topos whose subterminal objects
form a full reflective subcategory of ${\mathbf{E}}_{/F}$ and have binary infima.
\bigskip\noindent
{\bf Exercise.} Let ${\mathbf{B}}$ be an arbitrary category. Let ${{\mathrm{st}}}({\mathbf{B}})$
be the full subcategory of ${\mathbf{B}}$ on \emph{subterminal} objects, i.e.\ objects
$U$ such that for every $I \in {\mathbf{B}}$ there is at most one arrow $I \to U$
(possibly none!). We say that ${\mathbf{B}}$ \emph{has supports} iff
${{\mathrm{st}}}({\mathbf{B}})$ is a (full) reflective subcategory of ${\mathbf{B}}$.
Show that for a category ${\mathbf{B}}$ having pullbacks and supports it holds that
${\mathbf{B}}$ has binary products iff ${{\mathrm{st}}}({\mathbf{B}})$ has binary meets!
\newpage
\section{Adjoints to Change of Base}
We first show that for a functor $F : {\mathbf{A}} \to {\mathbf{B}}$ there is a left (2-)adjoint
$\coprod_F$ and a right (2-)adjoint $\prod_F$ to $F^* : {\mathbf{Fib}}({\mathbf{B}}) \to {\mathbf{Fib}}({\mathbf{A}})$,
i.e.\ change of base along $F$.
The right (2-)adjoint $\prod_F$ is easier to describe as its behaviour
is prescribed by the fibred Yoneda Lemma as
\[ \prod_F(P)(I) \simeq {\mathbf{Fib}}({\mathbf{B}})(\underline{I},\prod_F(P)) \simeq
{\mathbf{Fib}}({\mathbf{A}})(F^*\underline{I},P)
\]
for $I\in{\mathbf{B}}$. Accordingly, one verifies easily that the right adjoint
$\prod_F$ to $F^*$ is given by
$$\prod_F(P)(I) = {\mathbf{Fib}}({\mathbf{A}})(F^*\underline{I},P) \qquad\quad
\prod_F(P)(u) = {\mathbf{Fib}}({\mathbf{A}})(F^*\underline{u},P)$$
for objects $I$ and morphisms $u$ in ${\mathbf{B}}$. Obviously, as expected if ${\mathbf{B}}$ is
terminal then $\prod P=\prod_F P$ is the category of all cartesian sections
of $P$.
Notice further that in case $F$ has a right adjoint $U$ then
$F^*\underline{I} \cong \underline{UI}$ and, accordingly, we have
$\prod_F \simeq U^*$.
We now turn to the description of $\coprod_F$. We consider first the simpler
case where ${\mathbf{B}}$ is terminal. Then one easily checks that for a fibration
$P : {\mathbf{X}}\to{\mathbf{A}}$ the sum $\coprod P=\coprod_F P$ is given by ${\mathbf{X}}[{\mathrm{Cart}}(P)^{-1}]$,
i.e.\ the category obtained from ${\mathbf{X}}$ be freely inverting all cartesian arrows.
This we can extend to the case of arbitrary functors $F$ as follows.
For $I\in{\mathbf{B}}$ consider the pullback $P_{(I)}$ of $P$ along
$\partial_1 : I/F \to {\mathbf{A}}$
\begin{diagram}[small]
{\mathbf{X}}_{(I)} \SEpbk & \rTo & {\mathbf{X}} \\
\dTo^{P_{(I)}} & & \dTo_{P} \\
I/F & \rTo_{\partial_1} & {\mathbf{A}}
\end{diagram}
and for $u : J \to I$ in ${\mathbf{B}}$ let $G_u$ the mediating cartesian functor
from $P_{(I)}$ to $P_{(J)}$ over $\underline{u}/F$ (precomposition by $u$)
in the diagram
\begin{diagram}[small]
{\mathbf{X}}_{(I)} \SEpbk & \rTo^{G_u} & {\mathbf{X}}_{(J)} \SEpbk & \rTo & {\mathbf{X}} \\
\dTo^{P_{(I)}} & & \dTo^{P_{(J)}} & & \dTo_{P} \\
I/F & \rTo_{\underline{u}/F} & J/F & \rTo_{\partial_1} & {\mathbf{A}}
\end{diagram}
bearing in mind that $\partial_1 \circ \underline{u}/F = \partial_1$.
Now the reindexing functor $\coprod_F(u) : \coprod_F(I) \to \coprod_F(J)$
is the unique functor $H_u$ with
\begin{diagram}
{\mathbf{X}}_{(I)}[{\mathrm{Cart}}(P_{(I)})^{-1}] & \rTo^{H_u} & {\mathbf{X}}_{(J)}[{\mathrm{Cart}}(P_{(I)})^{-1}] \\
\uTo^{Q_I} & & \uTo_{Q_J} \\
{\mathbf{X}}_{(I)} & \rTo_{G_u} & {\mathbf{X}}_{(J)}
\end{diagram}
which exists as $Q_J \circ G_u$ inverts the cartesian arrows of $X_{(I)}$.
Notice, however, that due to the non--local character\footnote{Here we mean
that ${\mathbf{X}}_{(I)}[{\mathrm{Cart}}(P_{(I)})]$ and ${\mathrm{Cart}}(F^*\underline{I},P)$ do not depend
only on $P(I)$, the fibre of $P$ over $I$. This phenomenon already turns up
when considering reindexing of presheaves which in general for does not
preserve exponentials for example.} of the construction of $\coprod_F$
and $\prod_F$ in general the Beck--Chevalley Condition does not hold
for $\coprod$ and $\prod$.
As for adjoint functors $F \dashv U$ we have $\prod_F \simeq U^*$ it follows
that $F^* \simeq \coprod_U$.
\bigskip
Now we will consider change of base along distributors.
Recall that a distributor $\phi$ from ${\mathbf{A}}$ to ${\mathbf{B}}$ (notation
$\phi:{\mathbf{A}}\nrightarrow{\mathbf{B}}$) is a functor from ${\mathbf{B}}^{\mathrm{op}}{\times}{\mathbf{A}}$ to ${\mathbf{Set}}$, or
equivalently, a functor from ${\mathbf{A}}$ to $\widehat{{\mathbf{B}}} = {\mathbf{Set}}^{{\mathbf{B}}^{\mathrm{op}}}$.
Of course, up to isomorphism distributors from ${\mathbf{A}}$ to ${\mathbf{B}}$ are in
1--1--correspondence with cocontinuous functors from $\widehat{{\mathbf{A}}}$ to
$\widehat{{\mathbf{B}}}$ (by left Kan extension along ${\mathrm{Y}}_{\mathbf{A}}$).
Composition of distributors is defined in terms of composition of the
associated cocontinuous functors.\footnote{As the correspondence between
distributors and cocontinuous functors is only up to isomorphism composition
of distributors is defined also only up to isomorphism. That is the reason
why distributors do form only a bicategory and not an ordinary category!}
For a functor $F:{\mathbf{A}}\to{\mathbf{B}}$ one may define a distributor
$\phi_F : {\mathbf{A}}\nrightarrow{\mathbf{B}}$ as $\phi_F(B,A) = {\mathbf{B}}(B,FA)$
and a distributor $\phi^F : {\mathbf{B}}\nrightarrow{\mathbf{A}}$ in the reverse direction as
$\phi^F(A,B) = {\mathbf{B}}(FA,B)$. Notice that $\phi_F$ corresponds
to ${\mathrm{Y}}_{\mathbf{B}} \circ F$ and $\phi^F$ is right adjoint to $\phi_F$.
For a distributor $\phi : {\mathbf{A}}\to\widehat{{\mathbf{B}}}$ change of base along $\phi$
is defined as follows
(identifying presheaves over ${\mathbf{B}}$ with their corresponding discrete fibrations)
$$\phi^*(P)(I) = {\mathbf{Fib}}({\mathbf{B}})(\phi(I),P) \qquad\qquad
\phi^*(P)(u) = {\mathbf{Fib}}({\mathbf{B}})(\phi(u),P)$$
for objects $I$ and morphisms $u$ in ${\mathbf{A}}$. From this definition one easily
sees that for a functor $F : {\mathbf{A}}\to{\mathbf{B}}$ change of base along $\phi^F$ coincides
with $\prod_F$, i.e.\ we have
$$(\phi^F)^*P \cong \prod_F P$$
for all fibrations $P$ over ${\mathbf{A}}$.
This observation allows us to reduce change of base along distributors
to change of base along functors and their right adjoints. The reason is
that every distributor $\phi:{\mathbf{A}}\nrightarrow{\mathbf{B}}$ can be factorised as a
composition of the form $\phi^G\phi_F$.\footnote{Let $F$ and
$G$ be the inclusions of ${\mathbf{A}}$ and ${\mathbf{B}}$, respectively, into the \emph{display
category} ${\mathbf{D}}_\phi$ of $\phi$ which is obtained by adjoining to the disjoint
union of ${\mathbf{A}}$ and ${\mathbf{B}}$ the elements of $\phi(B,A)$ as morphisms from $B$ to
$A$ and defining $u{\circ}x{\circ}v$ as $\phi(v,u)(x)$ for $u:A\to A'$,
$x{\in}\phi(B,A)$ and $v : B'\to B$.} Thus, we obtain
$$\phi^* = (\phi^G\phi_F)^* \simeq (\phi_F)^*(\phi^G)_* \simeq F^*\prod_G$$
as $(\phi_F)^* \simeq F^*$ and $(\phi^G)^* \simeq \prod_G$ and change of base
along distributors is functorial in a contravariant way (i.e.\
$(\phi_2\phi_1)^* \simeq \phi_1^*\phi_2^*$). Thus $\phi^*$ has a left adjoint
$\coprod_\phi = G^*\coprod_F$.
One might ask whether for all distributors $\phi:{\mathbf{A}}\nrightarrow{\mathbf{B}}$ there also
exists a right adjoint $\prod_\phi$ to $\phi^*$. Of course, if $\prod_\phi$
exists then by the fibred Yoneda Lemma it must look as follows
$$(\prod_\phi P)(I) \simeq {\mathbf{Fib}}({\mathbf{B}})(\underline{I},\prod_\phi P) \simeq
{\mathbf{Fib}}({\mathbf{A}})(\phi^*\underline{I},P)$$
from which it is obvious that it restricts to an adjunction between
$\widehat{{\mathbf{A}}}$ and $\widehat{{\mathbf{B}}}$ as ${\mathbf{Fib}}({\mathbf{A}})(\phi^*\underline{I},P)$ is
discrete whenever $P$ is discrete. Thus, a necessary condition for the
existence of $\prod_\phi$ is that the functor $\phi^* : \widehat{{\mathbf{B}}} \to
\widehat{{\mathbf{A}}}$ is cocontinuous. As $\phi^* : \widehat{{\mathbf{B}}} \to \widehat{{\mathbf{A}}}$ is
right adjoint to $\widehat{\phi} : \widehat{{\mathbf{A}}} \to \widehat{{\mathbf{B}}}$, the left
Kan extension of $\phi : {\mathbf{A}}\to\widehat{{\mathbf{B}}}$ along ${\mathrm{Y}}_{\mathbf{B}}$, the distributor
$\phi$ has a right adjoint if and only if $\phi^*$ is cocontinuous.
Thus, a necessary condition for the existence of $\prod_\phi$ is the
existence of a right adjoint distributor to $\phi$. This, however, is known to
be equivalent (see e.g.\ \cite{Bor} vol.1) to the requirement that $\phi(A)$
is a retract of a representable presheaf for all objects $A$ in ${\mathbf{A}}$. In case
${\mathbf{B}}$ is Cauchy complete, i.e.\ all idempotents in ${\mathbf{B}}$ split, this means that
up to isomorphism $\phi$ is of the form $\phi_F$ for some functor $F:{\mathbf{A}}\to{\mathbf{B}}$
and then $\prod_F$ provides a right adjoint to $\phi^*$. As ${\mathbf{Fib}}({\mathbf{B}})$ is
equivalent to ${\mathbf{Fib}}({{\mathrm{IdSp}}}({\mathbf{B}}))$, where ${{\mathrm{IdSp}}}({\mathbf{B}})$ is
obtained from ${\mathbf{B}}$ by splitting all idempotents, one can show that $\phi^*$
has a right adjoint $\prod_\phi$ whenever $\phi$ has a right adjoint distributor. Thus, for a distributor $\phi$ the change of base functor $\phi^*$ has a
right adjoint $\prod_\phi$ if and only if $\phi$ has a right adjoint
distributor, i.e.\ if and only if $\phi$ is essentially a functor. An example
of a distributor $\phi$ where $\phi^*$ does not have a right adjoint can be
obtained as follows. Let ${\mathbf{A}}$ be a terminal category and ${\mathbf{B}}$ a small category
whose splitting of idempotents does not have a terminal object.
Let $\phi : {\mathbf{A}}\to\widehat{{\mathbf{B}}}$
select a terminal presheaf from $\widehat{{\mathbf{B}}}$. Then $\phi^*$ amounts to the
global sections functor on $\widehat{{\mathbf{B}}}$ which, however, does not have a
right adjoint as otherwise ${{\mathrm{IdSp}}}({\mathbf{B}})$ would have a
terminal object.
\newpage
\section{Finite Limit Preserving Functors as Fibrations}
If $F : {\mathbf{B}} \to {\mathbf{C}}$ is a finite limit preserving functor between categories
with finite limits then the fibration
$${\mathsf{gl}}(F) \equiv F^*P_{{\mathbf{B}}} = \partial_1 : {\mathbf{C}}{\downarrow}F \to {\mathbf{B}}$$
satisfies the following conditions which later will turn out as sufficient
for reconstructing the functor $F : {\mathbf{B}} \to {\mathbf{C}}$ up to equivalence.
\begin{enumerate}
\item[(1)] ${\mathsf{Gl}}(F) \equiv {\mathbf{C}}{\downarrow}F$ has finite limits and
${\mathsf{gl}}(F)$ preserves them.
\item[(2)] ${\mathsf{gl}}(F)$ has internal sums which are \emph{stable} in the sense
that cocartesian arrows are stable under pullbacks along arbitrary morphisms.
\item[(3)] The internal sums of ${\mathsf{gl}}(F)$ are \emph{disjoint} in the
sense that for every cocartesian arrow $\varphi : X \to Y$ the fibrewise
diagonal $\delta_{\varphi}$ is cocartesian, too.
\begin{diagram}
& & & & \\
& \rdTo~{\delta_{\varphi}} \rdEqual(2,4) \rdEqual(4,2) & & & \\
& & \SEpbk & \rTo_{\pi_2} & X\\
& & \dTo_{\pi_1} & & \dTo_{\varphi} \\
& & X & \rTo_{\varphi} & Y \\
\end{diagram}
\end{enumerate}
We refrain from giving the detailed verifications of properties (1)--(3).
Instead we recall some few basic facts needed intrinsically when verifying
the claims.
Notice that a morphism
\begin{diagram}[small]
A & \rTo^{f} & B \\
\dTo^{a} & & \dTo_{b} \\
FI & \rTo_{Fu} & FJ
\end{diagram}
in ${\mathsf{Gl}}(F)$ over $u : I \to J$ is cocartesian iff $f$ is an isomorphism.
Notice that pullbacks in ${\mathsf{Gl}}(F)$ are given by
\begin{diagram}[small]
D & & \rTo^{\pi_2} & & C & & \\
& \rdTo_{\pi_1} & & & \vLine_{c} & \rdTo^{g} & \\
\dTo^{d} & & B & \rTo^{f} & \HonV & & A \\
& & \dTo^{b} & & \dTo & & \\
FL & \hLine & \VonH & \rTo^{Fq} & FK & & \dTo_{a} \\
& \rdTo_{Fp} & & & & \rdTo^{Fv} & \\
& & FJ & & \rTo_{Fu} & & FI \\
\end{diagram}
where the top square is a pullback and the bottom square is the image of a
pullback under $F$. From this is is clear that $\partial_0,\partial_1 :
{\mathbf{C}}{\downarrow}F \to {\mathbf{B}}$ both preserve pullbacks. Condition (3) follows from
preservation of pullbacks by $\partial_0$ and the above characterisation of
cocartesian arrows.
Now based on work by J.-L.~Moens from his Th\'ese \cite{Moe} we will
characterise those fibrations over a category ${\mathbf{B}}$ with finite limits
which up to equivalence are of the form ${\mathsf{gl}}(F)$ for some finite limit
preserving functor $F$ from ${\mathbf{B}}$ to a category ${\mathbf{C}}$ with finite limits.
It will turn out that the three conditions above are necessary and sufficient.
In particular, we will show that the functor $F$ can be recovered from
$P = {\mathsf{gl}}(F)$ in the following way. First observe that
$\partial_0 : {\mathbf{C}}{\downarrow}F \to {\mathbf{C}}$ is isomorphic to the functor
${\boldsymbol \Delta} : {\mathbf{C}}{\downarrow}F \to {\mathbf{C}}{\downarrow}F1 \cong {\mathbf{C}}$ given by
\begin{diagram}[small]
X & \rTo^{\varphi_X}_{\mathrm{cocart}} & {\boldsymbol \Delta}(X) \\
\dTo^{f} & & \dTo_{{\boldsymbol \Delta}(f)} \\
Y & \rTo_{\varphi_Y}^{\mathrm{cocart}} & {\boldsymbol \Delta}(Y)
\end{diagram}
with ${\boldsymbol \Delta}(f)$ is vertical over the terminal object in ${\mathbf{B}}$.
Now the functor $F$ itself can be obtained up to isomorphism as
$\Delta = {\boldsymbol \Delta} \circ 1$ (where $1$ is the cartesian functor choosing
fibrewise terminal objects).
Notice that this construction makes sense for arbitrary fibrations $P$ over
${\mathbf{B}}$ with internal sums. Our goal now is to show that every fibration $P$ of
categories with finite limits over ${\mathbf{B}}$ with stable disjoint (internal) sums
is equivalent to ${\mathsf{gl}}(\Delta)$ where $\Delta$ is defined as above
and preserves finite limits.
But for this purpose we need a sequence of auxiliary lemmas.
\begin{Lem}\label{auxMoensLem}
Let ${\mathbf{B}}$ be category with finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration
of categories with finite limits and stable disjoint internal sums. Then in
\begin{diagram}[small]
U & & & & \\
& \rdTo~{\gamma} \rdEqual(2,4) \rdTo(4,2)^{\psi} & & & \\
& & W \SEpbk & \rTo & X\\
& & \dTo & & \dTo_{\varphi} \\
& & U & \rTo_{\varphi \circ \psi} & Y
\end{diagram}
the arrow $\gamma$ is cocartesian whenever $\varphi$ is cocartesian.
\end{Lem}
\begin{proof}
Consider the diagram
\begin{diagram}[small]
U \SEpbk & \rTo^{\psi} & X & & \\
\dTo^{\gamma} & & \dTo^{\delta_\varphi} & \rdEqual& \\
W \SEpbk & \rTo & Z \SEpbk & \rTo^{\pi_2} & X\\
\dTo^{\widetilde{\varphi}} & & \dTo^{\pi_1} & & \dTo_{\varphi} \\
U & \rTo_{\psi} & X & \rTo_{\varphi} & Y
\end{diagram}
with $\pi_i \delta_{\varphi} = {\mathit{id}}_X$ and
$\widetilde{\varphi} \circ \gamma = {\mathit{id}}_U$. Thus, by stability of sums
$\gamma$ is cocartesian as it appears as pullback of the cocartesian
arrow $\delta_{\varphi}$.
\end{proof}
\begin{Lem}\label{MoensLem}
Let ${\mathbf{B}}$ be category with finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration
of categories with finite limits and stable internal sums, i.e.\ $P$ is also
a cofibration whose cocartesian arrows are stable under pullbacks along
arbitrary maps in ${\mathbf{X}}$.
Then the following conditions are equivalent
\begin{enumerate}
\item[\rm (1)] The internal sums of $P$ are disjoint.
\item[\rm (2)] If $\varphi$ and $\varphi \circ \psi$ are cocartesian
then $\psi$ is cocartesian, too.
\item[\rm (3)]
If $\alpha$ is vertical and both $\varphi$ and $\varphi \circ \alpha$ are
cocartesian then $\alpha$ is an isomorphism.
\item[\rm (4)] A commuting diagram
\begin{diagram}[small]
X & \rTo^\varphi_{\mathrm{cocart}} & U \\
\dTo^{\alpha} & & \dTo_{\beta} \\
Y & \rTo_\psi^{\mathrm{cocart}} & V
\end{diagram}
is a pullback in ${\mathbf{X}}$ whenever $\varphi$, $\psi$ are cocartesian and $\alpha$,
$\beta$ are vertical.
\end{enumerate}
The equivalence of conditions (2)--(4) holds already under the weaker
assumption that cocartesian arrows are stable under pullbacks along
vertical arrows.
\end{Lem}
\begin{proof}
(1) $\Rightarrow$ (2) : Suppose that both $\varphi$ and $\varphi \circ \psi$
are cocartesian. Then for the diagram
\begin{diagram}
Z & & & & \\
& \rdTo~{\gamma} \rdEqual(2,4) \rdTo(4,2)^{\psi} & & & \\
& & \cdot \SEpbk & \rTo_{\theta} & Y \\
& & \dTo & & \dTo_{\varphi} \\
& & Z & \rTo_{\varphi \circ \psi} & X
\end{diagram}
we have that $\psi = \theta \circ \gamma$ is cocartesian as $\gamma$ is
cocartesian by Lemma~\ref{auxMoensLem} and $\theta$ is cocartesian by
stability of sums as it appears as pullback of the
cocartesian arrow $\varphi \circ \psi$.\\
(2) $\Rightarrow$ (1) : As $\pi_i$ and $\pi_i \circ \delta_{\varphi} = {\mathit{id}}$
are both cocartesian it follows from assumption (2) that $\delta_{\varphi}$
is cocartesian, too.\\
(2) $\Leftrightarrow$ (3) : Obviously, (3) is an instance of (2). For
the reverse direction assume (3) and suppose that both $\varphi$ and
$\varphi \circ \psi$ are cocartesian. Let $\psi = \alpha \circ \theta$
with $\theta$ cocartesian and $\alpha$ vertical. Then $\varphi\circ\alpha$
is cocartesian from which it follows by (3) that $\alpha$ is a vertical
isomorphism and thus $\psi = \alpha \circ \theta$ is cocartesian.\\
(3) $\Leftrightarrow$ (4) : Obviously, (4) entails (3) instantiating $\beta$
by identity as isos are stable under pullbacks. For the reverse direction
consider the diagram
\begin{diagram}
X & & & & \\
&\rdTo~{\iota} \rdTo(2,4)_{\alpha} \rdTo(4,2)^{\varphi}_{\mathrm{cocart.}}&&&\\
& & \cdot \SEpbk & \rTo^{\theta}_{\mathrm{cocart.}} & U\\
& & \dTo_{\pi} & & \dTo_{\beta} \\
& & Y & \rTo_{\psi} & V \\
\end{diagram}
with $\pi$ vertical. The morphism $\theta$ is cocartesian since it arises
as pullback of the cocartesian arrow $\psi$ along the vertical arrow $\beta$.
Moreover, the map $\iota$ is vertical since $\alpha$ and $\pi$
are vertical. Thus, by assumption (3) it follows that $\iota$ is an
isomorphism. Thus, the outer square is a pullback since it is isomorphic
to a pullback square via $\iota$.
\end{proof}
\bigskip\noindent
{\bf Remark.} Alternatively, we could have proved Lemma~\ref{MoensLem}
by showing (1) $\Rightarrow$ (4) $\Rightarrow$ (3) $\Rightarrow$ (2)
$\Rightarrow$ (1) where the last three implications have already been
established. The implication (1) $\Rightarrow$ (4) was proved in \cite{Moe}
as follows. Consider the diagram
\begin{diagram}[small]
X &&&& \\
& \rdTo(2,6)_{\alpha} \rdTo~{\gamma} \rdEqual(4,2)& & & \\
& & \SEpbk & \rTo & X \\
& & \dTo^{\theta} & & \dTo_{\varphi} \\
& & \SEpbk & \rTo & U \\
& & \dTo & & \dTo_{\beta} \\
& & Y & \rTo_{\psi} & V \\
\end{diagram}
where $\theta$ is cocartesian by stability of sums since $\theta$ appears
as pullback of the cocartesian arrow $\varphi$. From Lemma~\ref{auxMoensLem}
it follows that $\gamma$ is cocartesian as by assumption
$\beta \circ \varphi = \psi \circ \alpha$ and $\psi$ is cocartesian.
Thus, the map $\theta \circ \gamma$ is cocartesian over an isomorphism and,
therefore, an isomorphism itself. \hfill \mbox{\ $\lozenge$}
Notice that condition (3) of Lemma~\ref{MoensLem} is equivalent
to the requirement that for every map $u : I \to J$ in ${\mathbf{B}}$ the coproduct
functor $\coprod_u : {\mathbf{X}}_I \to {\mathbf{X}}_J$ reflects isomorphisms.
\medskip
As a consequence of Lemma~\ref{MoensLem} we get the following
characterisation of disjoint stable sums in terms of
extensivity.
\begin{Lem}\label{extensiveMoensLem}
Let ${\mathbf{B}}$ be category with finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration
of categories with finite limits and internal sums. Then the following
conditions are equivalent
\begin{enumerate}
\item[\rm (1)] The internal sums of $P$ are stable and disjoint.
\item[\rm (2)] The internal sums of $P$ are extensive\footnote{Recall
that a category with pullbacks and sums is
called \emph{extensive} iff for every family of squares
\begin{diagram}[small]
B_i & \rTo^{f_i} & B \\
\dTo^{a_i} & & \dTo_b \\
A_i & \rTo_{{\mathrm{in}}_i} & \coprod_{i{\in}I} A_i \\
\end{diagram}
all squares are pullbacks iff $f_i : B_i \to B$ is a
coproduct cone.},
i.e.\ for all commuting squares
\begin{diagram}[small]
X & \rTo^\varphi & U \\
\dTo^\alpha & & \dTo_\beta \\
Y & \rTo_\psi^{\mathrm{cocart}} & V \\
\end{diagram}
where $\psi$ is cocartesian and $\alpha$ and $\beta$ are vertical it holds
that $\varphi$ is cocartesian iff the square is a pullback.
\item[\rm (3)] The internal sums of $P$ are
extensive in the sense of Lawvere\footnote{Recall that
a category ${\mathbf{C}}$ is extensive in the sense of Lawvere
iff for all sets $I$ the categories
${\mathbf{C}}^I$ and ${\mathbf{C}}/\coprod_I 1$ are canonically isomorphic.},
i.e.\ for all commuting squares
\begin{diagram}[small]
X & \rTo^\varphi & U \\
\dTo^\alpha & & \dTo_\beta \\
1_I & \rTo_{\varphi_I}^{\mathrm{cocart}} & \coprod_I 1_I \\
\end{diagram}
where $\varphi_I$ is cocartesian over $!_I : I \to 1$ in ${\mathbf{B}}$,
$1_I$ is terminal in its fibre and $\alpha$ and $\beta$ are vertical
it holds that $\varphi$ is cocartesian iff the square is a pullback.
\end{enumerate}
The equivalence of (2) and (3) holds already under the weaker assumption
that cocartesian arrows are stable under pullbacks along vertical arrows.
\end{Lem}
\begin{proof}
(1)$\Leftrightarrow$(2) :
The implication from right to left in (2) is just stability of internal sums.
The implication from left to right in (2) is just condition (4) of
Lemma~\ref{MoensLem} which under assumption of stability of sums
by Lemma~\ref{MoensLem} is equivalent to the disjointness of sums.
Obviously, condition (3) is an instance of condition (2). Thus it remains
to show that (3) entails (2).
Consider the diagram
\begin{diagram}[small]
V & \rTo^{\psi_0} & V_0 \\
\dTo^\beta & ^{(*)} & \dTo_{\beta_0} \\
U \SEpbk & \rTo_{\varphi_0}^{\mathrm{cocart}} & U_0 \\
\dTo^{\gamma} & & \dTo_{\gamma_0} \\
1_I & \rTo_{\varphi_I}^{\mathrm{cocart}} & \coprod_I 1_I \\
\end{diagram}
where $\beta$, $\beta_0$, $\gamma$ and $\gamma_0$ are vertical, $\varphi_I$
is cocartesian over $!_I$ and $1_I$ is terminal in its fibre. The lower
square is a pullback due to assumption (3). If the upper square is a pullback
then the outer rectangle is a pullback and thus $\psi_0$ is cocartesian by (3).
If $\psi_0$ is cocartesian then the outer rectangle is a pullback by (3) and
thus the upper square is a pullback, too.
Thus we have shown that
\begin{enumerate}
\item[$(\dagger)$] a diagram of the form $(*)$ is a pullback
iff $\psi_0$ is cocartesian.
\end{enumerate}
Now consider a commuting diagram
\begin{diagram}[small]
Y & \rTo^\psi & V \\
\dTo^\alpha & ^{(+)} & \dTo_\beta \\
X & \rTo_\varphi^{\mathrm{cocart}} & U
\end{diagram}
with $\alpha$ and $\beta$ vertical.
We have to show that $\psi$ is cocartesian iff $(+)$ is a pullback.
Suppose $\psi$ is cocartesian. Then by $(\dagger)$ the outer rectangle and
the right square in
\begin{diagram}[small]
Y & \rTo^\psi_{\mathrm{cocart}} & V & \rTo^{\psi_0}_{\mathrm{cocart}} & V_0 \\
\dTo^\alpha & & \dTo_\beta & & \dTo_{\beta_0} \\
X & \rTo_\varphi^{\mathrm{cocart}} & U & \rTo_{\varphi_0}^{\mathrm{cocart}} & U_0
\end{diagram}
are pullbacks from which it follows that the left square, i.e.\ $(+)$,
is a pullback, too, as desired.
Suppose the square $(+)$ is a pullback. Then we have
\begin{diagram}[small]
Y \SEpbk & \rTo^\psi & V & \rTo^{\psi_0}_{\mathrm{cocart}} & V_0 \\
\dTo^\alpha & & \dTo_\beta & & \dTo_{\beta_0} \\
X & \rTo_\varphi^{\mathrm{cocart}} & U & \rTo_{\varphi_0}^{\mathrm{cocart}} & U_0
\end{diagram}
As by $(\dagger)$ the right square is a pullback it follows that
the outer rectangle is a pullback, too, from which it follows by $(\dagger)$
that $\psi_0 \psi$ is cocartesian. Now consider the diagram
\begin{diagram}[small]
Y & \rTo_\theta^{\mathrm{cocart}} & Z & \rTo_{\theta_0}^{\mathrm{cocart}} & Z_0 \\
& \rdTo_\psi & \dTo_\iota & & \dTo_{\iota_0} \\
& & V & \rTo_{\psi_0} & V_0
\end{diagram}
where $\iota$ and $\iota_0$ are vertical. Then $\iota_0$ is an isomorphism
because $\theta_0 \theta$ and $\psi_0 \psi$ start from the same source and
are both cocartesian over the same arrow in ${\mathbf{B}}$. By $(\dagger)$ the right
square is a pullback from which it follows that $\iota$ is an isomorphism
(as isomorphisms are pullback stable) and thus $\psi$ is cocartesian
as desired.
\end{proof}
\medskip
Notice that condition (3) of Lemma~\ref{extensiveMoensLem} is equivalent to
the requirements that for all $I\in{\mathbf{B}}$ the coproduct functor
$\coprod_I : {\mathbf{X}}_I \to {\mathbf{X}}_1$ reflects isomorphisms and $\beta^*\varphi_I$
is cocartesian for all vertical maps $\beta : U \to \coprod_I 1_I$.
An immediate consequence of Lemma~\ref{extensiveMoensLem} is the following
\begin{Cor}\label{corMoens1}
Let ${\mathbf{B}}$ have finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration of categories
with finite limits and stable disjoint internal sums. Then for every
$u : I \to J$ in ${\mathbf{B}}$ and $X \in P(I)$ the functor
$\coprod_u / X : {\mathbf{X}}_I / X \to {\mathbf{X}}_J / \coprod_u X$ is an equivalence.
In particular, we get that ${\mathbf{X}}_I \cong {\mathbf{X}}_I/1_I$ is equivalent to
${\mathbf{X}}_J/ \coprod_u 1_I$ via $\coprod_u / 1_I$ and that ${\mathbf{X}}_I \cong {\mathbf{X}}_I/1_I$
is equivalent to ${\mathbf{X}}_1 / \Delta(I)$ via $\coprod_I / 1_I$
where $\Delta(I) = \coprod_I 1_I$.
\end{Cor}
\begin{Cor}\label{corMoens2}
Let ${\mathbf{B}}$ have finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be fibration of categories
with finite limits and stable disjoint internal sums. Then for every
$u : I \to J$ in ${\mathbf{B}}$ the functor $\coprod_u : {\mathbf{X}}_I \to {\mathbf{X}}_J$
preserves pullbacks.
\end{Cor}
\begin{proof}
Notice that $\coprod_u = \Sigma_{{\mathbf{X}}_J/\coprod_u 1_I} \circ \coprod_u/1_I$
where we identify ${\mathbf{X}}_I$ and ${\mathbf{X}}_I/1_I$ via their canonical isomorphism.
The functor $\coprod_u/1_I$ preserves pullbacks as it is an equivalence by
Corollary~\ref{corMoens1}. The functor
$\Sigma_{{\mathbf{X}}_J/\coprod_u1_I} = \partial_0$ is known to preserve pullbacks
anyway. Thus, the functor $\coprod_u$ preserves pullbacks
as it arises as the composite of pullback preserving functors.
\end{proof}
\begin{Lem}\label{Moenslem1}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration of categories with finite limits and
stable disjoint internal sums. Then the mediating arrow $\theta$ is
cocartesian for any diagram in ${\mathbf{X}}$
\begin{diagram}[small]
U \SEpbk & & \rTo^{\phi_2} & & X_2 \\
& \rdDashto_{\theta} & & & \dTo_{\varphi_2}\\
\dTo^{\phi_1} & & V \SEpbk & \rTo^{\beta_2} & Y_2 \\
& & \dTo^{\beta_1} & & \dTo_{\alpha_2} \\
X_1 & \rTo_{\varphi_1} & Y_1 & \rTo_{\alpha_1} & Y \\
\end{diagram}
whenever the $\varphi_i$, $\phi_i$ are cocartesian, the $\alpha_i$, $\beta_i$
are vertical and the outer and the inner square are pullbacks
\end{Lem}
\begin{proof}
Consider the diagram
\begin{diagram}[small]
U \SEpbk & \rTo^{\psi_2} & U_2 \SEpbk & \rTo^{\gamma_2} & X_2 \\
\dTo^{\psi_1} & & \dTo^{\theta_2} & & \dTo_{\varphi_2} \\
U_1 \SEpbk & \rTo^{\theta_1} & V \SEpbk & \rTo^{\beta_2} & Y_2 \\
\dTo^{\gamma_1} & & \dTo^{\beta_1} & & \dTo_{\alpha_2} \\
X_1 & \rTo_{\varphi_1} & Y_1 & \rTo_{\alpha_1} & Y \\
\end{diagram}
where by stability of sums the $\psi_i$ and $\theta_i$ are cocartesian as they
arise as pullbacks of $\varphi_1$ or $\varphi_2$, respectively.
As the big outer square is a pullback we may assume that
$\phi_i = \gamma_i \circ \psi_i$ (by appropriate choice of the $\psi_i$).
Thus, $\theta = \theta_1 \circ \psi_1 = \theta_2 \circ \psi_2$ is
cocartesian as it arises as composition of cocartesian arrows.
\end{proof}
\begin{Lem}\label{Moenslem2}
Let ${\mathbf{B}}$ have finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be fibration of categories
with finite limits and stable disjoint internal sums.
Then the functor ${\boldsymbol \Delta} : {\mathbf{X}} \to {\mathbf{X}}_1$ given by
\begin{diagram}
X & \rTo^{\varphi_X}_{\mathrm{cocart}} & {\boldsymbol \Delta}(X) \\
\dTo^{f} & & \dDashto_{{\boldsymbol \Delta}(f)} \\
Y & \rTo_{\varphi_Y}^{\mathrm{cocart}} & {\boldsymbol \Delta}(Y)
\end{diagram}
with ${\boldsymbol \Delta}(f)$ over $1$ preserves finite limits.
\end{Lem}
\begin{proof}
Clearly, the functor ${\boldsymbol \Delta}$ preserves the terminal object. It remains
to show that it preserves also pullbacks. Let
\begin{diagram}[small]
U \SEpbk & \rTo^{g_2} & X_2\\
\dTo^{g_1} & & \dTo_{f_2} \\
X_1 & \rTo_{f_1} & Y\\
\end{diagram}
be a pullback in ${\mathbf{X}}$.
Then by Lemma~\ref{Moenslem1} the arrow $\theta$ is cocartesian in
\begin{diagram}[small]
U \SEpbk & & \rTo^{g_2} & & X_2 \\
& \rdDashto_{\theta} & & & \dTo_{\varphi_2}\\
\dTo^{g_1} & & V \SEpbk & \rTo^{\beta_2} & Y_2 \\
& & \dTo^{\beta_1} & & \dTo_{\alpha_2} \\
X_1 & \rTo_{\varphi_1} & Y_1 & \rTo_{\alpha_1} & Y \\
\end{diagram}
where $f_i = \alpha_i \circ \varphi_i$ with $\alpha_i$ vertical and
$\varphi_i$ cocartesian. From this we get that the square
\begin{diagram}[small]
{\boldsymbol \Delta}(U) & \rTo^{{\boldsymbol \Delta}(g_2)} & {\boldsymbol \Delta}(X_2)\\
\dTo^{{\boldsymbol \Delta}(g_1)} & & \dTo_{{\boldsymbol \Delta}(f_2)} \\
{\boldsymbol \Delta}(X_1) & \rTo_{{\boldsymbol \Delta}(f_1)} & {\boldsymbol \Delta}(Y)\\
\end{diagram}
is a pullback, too, as it is obtained by applying the pullback preserving
functor $\coprod_{P(Y)}$ to
\begin{diagram}[small]
V \SEpbk & \rTo^{\beta_2} & Y_2 \\
\dTo^{\beta_1} & & \dTo_{\alpha_2} \\
Y_1 & \rTo_{\alpha_1} & Y \\
\end{diagram}
which is a pullback in the fibre over $P(Y)$.
\end{proof}
\bigskip
Now we are ready to prove Moens's Theorem.
\begin{Thm}\label{MoensThm}
Let ${\mathbf{B}}$ have finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be fibration of categories
with finite limits and stable disjoint internal sums. Then $P$ is equivalent
to ${\mathsf{gl}}(\Delta)$ where $\Delta$ is the finite limit preserving
functor ${\boldsymbol \Delta} \circ 1$.
More explicitely, the fibred equivalence $E : P \to {\mathsf{gl}}(\Delta)$ is
given by sending $f : X \to Y$ in ${\mathbf{X}}$ over $u : I \to J$ to
\begin{diagram}[small]
{\boldsymbol \Delta}(X) & \rTo^{{\boldsymbol \Delta}(f)} & {\boldsymbol \Delta}(Y) \\
\dTo^{{\boldsymbol \Delta}(\alpha)} & E(f) & \dTo_{{\boldsymbol \Delta}(\beta)} \\
{\boldsymbol \Delta}(1_I) & \rTo_{\Delta(u)} & {\boldsymbol \Delta}(1_J) \\
\end{diagram}
where $\alpha$ and $\beta$ are terminal projections in their fibres.
\end{Thm}
\begin{proof}
As $\Delta(u) = {\boldsymbol \Delta}(1_u)$ the map $E(f)$ arises as the
image under ${\boldsymbol \Delta}$ of the square
\begin{diagram}[small]
X & \rTo^{f} & Y \\
\dTo^{\alpha} & & \dTo_{\beta} \\
1_I & \rTo_{1_u} & 1_J \\
\end{diagram}
which is a pullback if $f$ is cartesian. As by Lemma~\ref{Moenslem2}
the functor ${\boldsymbol \Delta}$ preserves pullbacks it follows that $E$ is cartesian.
Thus, the fibred functor $E$ is a fibred equivalence as by
Corollary~\ref{corMoens1} all fibres of $E$ are (ordinary) equivalences.
\end{proof}
\medskip
Thus, for categories ${\mathbf{B}}$ with finite limits we have established a
1--1--correspondence up to equivalence between fibrations of the form
${\mathsf{gl}}(F) = \partial_1 : {\mathbf{C}}{\downarrow}F \to {\mathbf{B}}$ for some finite limit preserving
$F : {\mathbf{B}} \to {\mathbf{C}}$ where ${\mathbf{C}}$ has finite limits and fibrations over ${\mathbf{B}}$ of
categories with finite limits and extensive internal sums.\footnote{
This may explain why Lawvere's notion of extensive sums is so important.
Notice, however, that Lawvere's original definition only applied to
ordinary categories ${\mathbf{C}}$ with small coproducts in the ordinary sense.
That our notion of Lawvere extensivity is slightly more general can be seen
from the discussion at the end of section 17 where we give an example (due to
Peter Johnstone) of a fibration over ${\mathbf{Set}}$ of categories with finite limits
and Lawvere extensive small sums which, however, is not of the form
${\mathrm{Fam}}({\mathbf{C}})$ for some ordinary category ${\mathbf{C}}$.}
\medskip
With little effort we get the following generalization of Moens's Theorem.
\begin{Thm}\label{GenMoensThm}
Let ${\mathbf{B}}$ be a category with finite limits. If ${\mathbf{C}}$ is a category with finite
limits and $F : {\mathbf{B}}\to{\mathbf{C}}$ preserves terminal objects then ${\mathsf{gl}}(F)$ is a fibration
of finite limit categories and a cofibration where cocartesian arrows are stable
under pullbacks along vertical arrows and one of the following equivalent
conditions holds
\begin{enumerate}
\item[\emph{(1)}] if $\varphi$ and $\varphi\circ\psi$ are cocartesian
then $\psi$ is cocartesian
\item[\emph{(2)}] if $\alpha$ is vertical and both $\varphi$ and
$\varphi\circ\alpha$ are cocartesian
then $\alpha$ is an isomorphism
\item[\emph{(3)}] a commuting square
\begin{diagram}[small]
X & \rTo^\varphi_{\mathrm{cocart}} & U \\
\dTo^{\alpha} & & \dTo_{\beta} \\
Y & \rTo_\psi^{\mathrm{cocart}} & V
\end{diagram}
is a pullback whenever $\psi,\varphi$ are cocartesian $\alpha,\beta$ are vertical.
\end{enumerate}
Bifibrations $P$ satisfying these properties are equivalent to ${\mathsf{gl}}(\Delta_P)$
where $\Delta_P$ is the functor $\Delta : {\mathbf{B}} \to {\mathbf{X}}_1$ sending $I$ to
$\Delta(I) = \coprod_I 1_I$ and $u : J \to I$ to the unique vertical arrow
$\Delta(u)$ rendering the diagram
\begin{diagram}[small]
1_J & \rTo^{\varphi_J} & \Delta(J) \\
\dTo^{1_u} & & \dTo_{\Delta(u)} \\
1_I & \rTo_{\varphi_I}^{\mathrm{cocart}} & \Delta(I)
\end{diagram}
commutive.
This correspondence is an equivalence since $\Delta_{{\mathsf{gl}}(F)}$ is isomorphic
to $F$.
\end{Thm}
\begin{proof}
We have already seen that these conditions are necessary and the equivalence of
(1)--(3) follows from Lemma~\ref{MoensLem}.
For the reverse direction first observe that the assumptions on $P$ imply
that every commuting square
\begin{diagram}[small]
X & \rTo^\varphi & U \\
\dTo^{\alpha} & & \dTo_{\beta} \\
1_I & \rTo_{\varphi_I}^{\mathrm{cocart}} & \Delta(I)
\end{diagram}
with $\alpha$ and $\beta$ vertical is a pullback iff $\varphi$ is cocartesian.
Thus pullback along the cocartesian arrow
$\varphi_I : 1_I \to \coprod_I 1_I = \Delta(I)$ induces an equivalence
between $X_I$ and ${\mathbf{X}}_1 / \coprod_I 1_I$. This extends to
an equivalence between $P$ and ${\mathsf{gl}}(\Delta : {\mathbf{B}} \to {\mathbf{X}}_1)$ since
\begin{diagram}[small]
{\mathbf{X}}_J & \lTo^{\varphi_J^*}_\simeq & {\mathbf{X}}_1 / \Delta(J) \\
\uTo^{u^*} & & \uTo_{\Delta(u)^*} \\
{\mathbf{X}}_I & \lTo_{\varphi_I^*}^\simeq & {\mathbf{X}}_1 / \Delta(I) \\
\end{diagram}
commutes up to isomorphism for all $u : J \to I$ in ${\mathbf{B}}$.
\end{proof}
\medskip
As apparent from the proof fibrations $P : {\mathbf{X}} \to {\mathbf{B}}$ over a finite limit
category ${\mathbf{B}}$ are equivalent to ${\mathsf{gl}}(F)$ for some terminal object preserving
functor $F$ to a finite limit category if and only if $P$ is a bifibration
such that ${\mathbf{X}}$ has and $P$ preserves finite limits and every commuting square
of the form
\begin{diagram}[small]
X & \rTo^\varphi & U \\
\dTo^{\alpha} & & \dTo_{\beta} \\
1_I & \rTo_{\varphi_I}^{\mathrm{cocart}} & \Delta(I)
\end{diagram}
with $\alpha$ and $\beta$ vertical is a pullback iff $\varphi$ is cocartesian.
\bigskip
Finally we discuss how the fact that finite limit preserving functors are
closed under composition is reflected on the level of their fibrations
associated via glueing. Suppose that $F : {\mathbf{B}}\to{\mathbf{C}}$ and $G:{\mathbf{C}}\to{\mathbf{D}}$ are finite
limit preserving functors between categories with finite limits.
Then ${\mathsf{gl}}(G{\circ}F) \cong 1^*F^*\mathit{Fam}({\mathsf{gl}}(G))$ as indicated in
\begin{diagram}
{\mathbf{D}}{\downarrow}G{\circ}F \SEpbk & \rInto & \cdot \SEpbk & \rTo & \cdot \SEpbk &
\rTo & {\mathbf{D}}{\downarrow}G \SEpbk & \rTo & {\mathbf{D}}^\mbox{$2\hspace*{-1.2ex}1$}\\
\dTo_{{\mathsf{gl}}(G{\circ}F)} & & \dTo_{F^*\mathit{Fam}({\mathsf{gl}}(G))} && \dTo_{\mathit{Fam}({\mathsf{gl}}(G))} &
& \dTo_{{\mathsf{gl}}(G)} & & \dTo_{\partial_1} \\
{\mathbf{B}} & \rInto_1 & {\mathbf{C}}{\downarrow}F\SEpbk &\rTo_{\partial_1^*F} & {\mathbf{C}}^\mbox{$2\hspace*{-1.2ex}1$} & \rTo_{\partial_0}&{\mathbf{C}} & \rTo_G & {\mathbf{D}}\\
& \rdEqual & \dTo_{{\mathsf{gl}}(F)} & & \dTo_{\partial_1} & & \\
& & {\mathbf{B}} & \rTo_F & {\mathbf{C}} & &
\end{diagram}
because $\partial_0 \circ \partial_1^*F \circ 1 = \partial_0 \circ 1 = F$.
The fibration $F^*\mathit{Fam}({\mathsf{Gl}}(G))$ is ${\mathsf{gl}}(G)$ shifted from ${\mathbf{C}}$ to ${\mathsf{Gl}}(F)$
via change of base along
${\boldsymbol \Delta} = \partial_0 = \partial_0 \circ \partial_1^*F : {\mathsf{Gl}}(F) \to {\mathbf{C}}$.
The fibration ${\mathsf{gl}}(G{\circ}F)$ appears as a (n in general proper) subfibration
of the composite fibration ${\mathsf{gl}}(F) \circ F^*\mathit{Fam}({\mathsf{Gl}}(G))$.
A fibration $Q : {\mathbf{Y}} \to {\mathsf{Gl}}(F)$ is isomorphic to one of the form
$F^*\mathit{Fam}({\mathsf{gl}}(G))$ iff $Q$ is a fibration of categories with finite limits
and stable disjoint internal sums such that ${\boldsymbol \Delta} : {\mathbf{Y}} \to {\mathbf{Y}}_1$
is isomorphic to one of the form $G \circ \partial_0$, i.e.\ iff ${\boldsymbol \Delta}$
inverts cocartesian arrows of ${\mathsf{Gl}}(F)$. This latter condition is equivalent
to the requirement that $1_\varphi$ is cocartesian w.r.t.\ $Q$ whenever
$\varphi$ is cocartesian w.r.t.\ ${\mathsf{gl}}(F)$.\footnote{As ${\boldsymbol \Delta}(\varphi)$ is
an isomorphism iff $1_\varphi$ is cocartesian.
This can be seen from the diagram
\begin{diagram}[small]
1_X & \rTo^{\varphi_X}_{\mathrm{cocart}} & {\boldsymbol \Delta}(X) \\
\dTo^{1_\varphi} & & \dTo_{{\boldsymbol \Delta}(\varphi)} \\
1_Y & \rTo_{\varphi_Y}^{\mathrm{cocart}} & {\boldsymbol \Delta}(Y) \\
\end{diagram}
where $\varphi_X$ and $\varphi_Y$ are cocartesian over the terminal
projections of $X$ and $Y$, respectively, and ${\boldsymbol \Delta}(\varphi)$ is vertical.
If $1_\varphi$ is cocartesian then ${\boldsymbol \Delta}(\varphi)$ is an isomorphism as it
is vertical and cocartesian. On the other hand if ${\boldsymbol \Delta}(\varphi)$ is an
isomorphism then ${\boldsymbol \Delta}(\varphi) \circ \varphi_X$ is cocartesian, too, and
thus by Lemma~\ref{MoensLem}(2) it follows that $1_\varphi$ is
cocartesian.} This fails e.g.\ for $Q \equiv {\mathsf{gl}}({\mathit{Id}}_{{\mathsf{Gl}}(F)})$ if not all
cocartesian arrows of ${\mathsf{Gl}}(F)$ are isomorphisms, i.e.\ ${\mathbf{B}}$ is not
equivalent to the trivial category $\mathbf{1}$.
\newpage
\section{Geometric Morphisms as Fibrations}
Geometric morphism are adjunctions $F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ where $F$
preserves finite limits. Though introduced originally for toposes the notion
of geometric morphism makes sense already if ${\mathbf{B}}$ and ${\mathbf{C}}$ have finite limits.
First we will characterise for functors $F$ between categories with finite
limits the property that $F$ has a right adjoint in terms of a purely
fibrational property of its associated fibration ${\mathsf{gl}}(F) = F^*P_{\mathbf{C}}$,
namely that of having \emph{small global sections}.
First we observe that the requirement $P \dashv 1 \dashv G$ is equivalent to
$P$ having small global sections since $1 \dashv G$ says that
for every $X \in P(I)$ there is an $\varepsilon_X : 1_{GX} \to X$
such that for every $\sigma : 1_J \to X$ over
$u : J \to I$ there is a unique $v : J \to GX$ with
\begin{diagram}
& & 1_I \\
& \ruTo^{1_u} & \uTo_{1_{P(\varepsilon_X)}} \\
1_J & \rDashto^{\qquad 1_v} & 1_{GX} \\
& \rdTo_{\sigma} & \dTo_{\varepsilon_X} \\
& & X \\
\end{diagram}
i.e.\ that ${\mathbf{Hom}}_I(1_I,X)$ is representable.
If $P$ is a fibration of cartesian closed categories (or even a fibred topos)
then $P$ has small global sections iff $P$ is locally small.
\begin{Thm}\label{gmthm1}
Let $F : {\mathbf{B}} \to {\mathbf{C}}$ be a functor between categories with finite limits.
Then $F$ has a right adjoint $U$ iff the fibration
${\mathsf{gl}}(F)$ has small global sections, i.e.\ ${\mathsf{gl}}(F) \dashv 1 \dashv G$.
\end{Thm}
\begin{proof}
Suppose that $F$ has a right adjoint $U$. We show that $1 \dashv G$ by
exhibiting its counit $\widetilde{\varepsilon}_a$ for an arbitrary object
$a : A \to FI$ in ${\mathsf{Gl}}(F) = {\mathbf{C}}{\downarrow}F$.
For this purpose consider the pullback
\begin{diagram}[small]
C \SEpbk & \rTo^{q} & UA \\
\dTo^{p} & & \dTo_{Ua} \\
I & \rTo_{\eta_I} & UFI \\
\end{diagram}
where $\eta_I$ is the unit of $F \dashv U$ at $I \in {\mathbf{B}}$.
Then for the transpose $\widehat{q} = \varepsilon_A \circ Fq : FC \to A$
of $q$ we have
\begin{diagram}[small]
FC & \rTo^{\widehat{q}} & A \\
& \rdTo(1,2)_{Fp} \ldTo(1,2)_{a} & \\
& FI & \\
\end{diagram}
We show that $(p,\widehat{q}) : 1_C \to a$ is the desired counit
$\widetilde{\varepsilon}_a$ of $1 \dashv G$ at $a$. Suppose that
$(u,s) : 1_J \to a$ in ${\mathsf{Gl}}(F)$, i.e.\ $u : J \to I$ and $s : FJ \to A$
with $a \circ s = Fu$ as shown in the diagram
\begin{diagram}[small]
FJ & \rTo^s & A \\
\dEqual & & \dTo_a \\
FJ & \rTo_{Fu} & FI \\
J & \rTo_u & I\\
\end{diagram}
We have to show that there is a unique $v : J \to C$ with $p \circ v = u$
and $\widehat{q} \circ Fv = s$ as shown in the diagram
\begin{diagram}[small]
FJ & & & & \\
& \rdTo_{Fv} \rdTo(4,2)^{s} & & & \\
\dEqual & & FC & \rTo_{\widehat{q}} & A \\
& & \dEqual & \widetilde{\varepsilon}_a & \dTo_{a} \\
FJ & \rTo_{Fv} & FC & \rTo_{Fp} & FI \\
\end{diagram}
But $\widehat{q} \circ Fv = s$ iff $q \circ v = Us \circ \eta_J$
due to $F \dashv U$. Thus $v$ satisfies the above requirements iff
$p \circ v = u$ and $q \circ v = Us \circ \eta_J$,
i.e.\ iff $v$ is the mediating arrow in the diagram
\begin{diagram}[small]
J & & & & \\
& \rdTo~{v} \rdTo(2,4)_{u} \rdTo(4,2)^{Us \circ \eta_J} & & & \\
& & \SEpbk & \rTo_{q} & UA \\
& & \dTo_{p} & & \dTo_{Ua} \\
& & I & \rTo_{\eta_I} & UFI
\end{diagram}
from which there follows uniqueness and existence of $v$ with the
desired properties. Thus $\widetilde{\varepsilon}_a$ actually is the counit
for $1 \dashv G$ at $a$.
\medskip
For the reverse direction assume that ${\mathsf{gl}}(F) \dashv 1 \dashv G$.
Thus, for all $X$ over $1$ we have
${\mathbf{B}}(-,GX) \cong {\mathbf{C}}{\downarrow}F(1_{(-)},X) \cong {\mathbf{C}}/F1(F_{/1}(-),X)$, i.e.\
$F_{/1} : {\mathbf{B}} \cong {\mathbf{B}}/1 \to {\mathbf{C}}/F1$ has a right adjoint (given by the
restriction of $G$ to ${\mathbf{C}}/F1$). Since $\Sigma_{F1} \dashv (F1)^* : {\mathbf{C}} \to
{\mathbf{C}}/F1$ and $F = \Sigma_{F1} \circ F_{/1} : {\mathbf{B}} \cong {\mathbf{B}}/1 \to {\mathbf{C}}$ the functor
$F$ has a right adjoint.
A slightly more abstract proof of the backwards direction goes by observing
that the inclusion $I : {\mathbf{C}}{\downarrow}F1 \hookrightarrow {\mathbf{C}}{\downarrow}F$
has a left adjoint $R$ sending $a : A \to FI$ to $F!_I \circ a : A \to F1$ and
a morphism $(u,f)$ from $b : B \to FJ$ to $a : A \to FI$ to $f : R(b) \to R(a)$
since $(u,f)$ from $a : A \to FI$ to $c : C \to F1$ is in 1-1-correspondence
with $f : R(a) \to c$ (because necessarily $u =\; !_I$). Obviously, we have
that $F_{/1} = R \circ 1$ and thus $F_{/1}$ has right adjoint $G \circ I$.
Since $F = \Sigma_{F1} \circ F_{/1}$ and $\Sigma_{F1} \dashv (F1)^*$ it follows
that $F$ has right adjoint $G \circ I \circ (F1)^*$.
\end{proof}
\medskip
Notice that the above proof goes through if ${\mathbf{C}}$ has just pullbacks and
$\Sigma_{F1}$ has a right adjoint $(F1)^*$, i.e.\ $F1 \times X$ exists
for all objects $X$ in ${\mathbf{C}}$.
Thus, we have the following lemma which has a structure analogous
to the one of Lemma~\ref{Lem13.2}.
\begin{Lem}\label{lscor}
Suppose ${\mathbf{B}}$ has finite limits and ${\mathbf{C}}$ has pullbacks and all products of
the form $F1 \times X$. Then for a functor $F : {\mathbf{B}} \to {\mathbf{C}}$
the following conditions are equivalent
\begin{enumerate}
\item[\emph{(1)}] $F$ has a right adjoint
\item[\emph{(2)}] $F^* : {\mathbf{Fib}}({\mathbf{C}}) \to {\mathbf{Fib}}({\mathbf{B}})$ preserves the property
of having small global sections
\item[\emph{(3)}] $F^*P_{\mathbf{C}} = \partial_1 : {\mathbf{C}}{\downarrow}F \to {\mathbf{B}}$
has small global sections.
\end{enumerate}
\end{Lem}
\begin{proof} The proof of (1) $\Rightarrow$ (2) is a special case of
the proof of Lemma~\ref{lspres}. Since $P_{\mathbf{C}}$ has small global sections
(3) follows from (2). Finally, claim (1) follows from (3)
by Theorem~\ref{gmthm1} and the subsequent remark on its strengthening.
\end{proof}
\bigskip
From Lemma~\ref{Lem13.2} and Lemma~\ref{lscor} it follows that for a
functor $F : {\mathbf{B}} \to {\mathbf{C}}$ between categories with finite limits the fibration
${\mathsf{gl}}(F) = F^*P_{\mathbf{C}}$ has internal sums and small global sections iff $F$
preserves pullbacks and has a right adjoint.\footnote{This was already observed
by J.~B\'enabou in \cite{montreal}.}
\emph{Thus, for categories ${\mathbf{B}}$ with finite limits we get a
1--1--correspondence (up to equivalence) between geometric morphisms to ${\mathbf{B}}$
(i.e.\ adjunctions $F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ where ${\mathbf{C}}$ has finite limits and
$F$ preserves them) and fibrations over ${\mathbf{B}}$ of categories with finite limits,
stable disjoint sums and small global sections. Such fibrations are
called {\bf geometric}.}
In Appendix~\ref{jibthm} we prove M.~Jibladze's theorem \cite{Jib} that in
fibred toposes with internal sums these are automatically stable and disjoint.
As a consequence \emph{geometric morphisms from toposes to a topos ${\mathbf{S}}$ are
(up to equivalence) in 1--1--correspondence with toposes fibred over ${\mathbf{S}}$
that are cocomplete and locally small.}
\medskip
In the rest of this section we show that in a fibred sense every geometric
morphism is of the form $\Delta \dashv \Gamma$.
First we observe that there is a fibred version of the functor
$\Delta = {\boldsymbol \Delta} \circ 1$ considered in the previous section
\begin{Def}\label{Deltafibdef}
Let ${\mathbf{B}}$ be a category with finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration
of categories with finite limits and stable disjoint internal sums. Then there
is a fibred functor $\Delta_P : P_{\mathbf{B}} \to P$ sending the morphism
\begin{diagram}[small]
I_1 & \rTo^{v} & I_2 \\
\dTo^{u_1} & & \dTo_{u_2} \\
J_1 & \rTo_{w} & J_2 \\
\end{diagram}
in $P_{\mathbf{B}}$ to the arrow $\Delta_P(w,v)$ in ${\mathbf{X}}$ over $w$ making the
following diagram commute
\goodbreak
\begin{diagram}[small]
1_{I_1} & \rTo^{1_v} & 1_{I_2} \\
\dTo^{\varphi_{u_1}} & & \dTo_{\varphi_{u_2}} \\
\Delta_P(u_1) & \rTo_{\Delta_P(w,v)} & \Delta_P(u_2) \\
\end{diagram}
where $\varphi_{u_i}$ is cocartesian over $u_i$ for $i=1,2$.
\end{Def}
Notice that $\Delta_P$ actually is cartesian as if the first square is a
pullback then $\Delta_P(w,v)$ is cartesian by BCC for internal sums as
$1_v$ is cartesian and the $\varphi_{u_i}$ are cocartesian.
Now $P$ having small global sections turns out as equivalent to $\Delta_P$
having a fibred right adjoint $\Gamma_P$.
\begin{Thm}\label{gmthm2}
Let ${\mathbf{B}}$ be a category with finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration
of categories with finite limits and stable disjoint internal sums. Then $P$
has small global sections iff $\Delta_P$ has a fibred right adjoint $\Gamma_P$.
\end{Thm}
\begin{proof}
For the implication from left to right assume that $P \dashv 1 \dashv G$.
For $X \in {\mathbf{X}}$ let $\widetilde{\varepsilon}_X$ be the unique vertical arrow
making the diagram
\begin{diagram}[small]
1_{GX} & \rTo^{\varphi}_{\mathrm{cocart}} & \Delta_P(P(\varepsilon_X))\\
& \rdTo_{\varepsilon_X} & \dTo_{\widetilde{\varepsilon}_X} \\
& & X \\
\end{diagram}
commute where $\varepsilon_X$ is the counit of $1 \dashv G$ at $X$.
Then for $u : I \to J$ and $f : \Delta_P(u) \to X$ there is a unique
morphism $(w,v) : u \to P(\varepsilon_X)$ with
$\widetilde{\varepsilon}_X \circ \Delta_P(w,v) = f$
as can be seen from the following diagram
\begin{diagram}
1_I & \rTo^{1_v} & 1_{G(X)} & & \\
\dTo^{\varphi_u} & & \dTo^{\varphi} & \rdTo(2,4)^{\varepsilon_X} & \\
\Delta_P(u) & \rTo^{\Delta_P(w,v)} & \Delta_P(P(\varepsilon_X)) & & \\
& \rdTo(4,2)_{f} & & \rdTo~{\widetilde{\varepsilon}_X} & \\
& & & & X \\
\end{diagram}
using the universal property of $\varepsilon_X$. It follows that
necessarily $w = P(f)$. Thus, for $f : \Delta_P(u) \to X$ its lower
transpose $\check{f}$ is given by $(P(f),v) : u \to P(\varepsilon_X)$
where $v : I \to G(X)$ is the unique arrow with
$\varepsilon_X \circ 1_v = f \circ \varphi_u$.
The induced right adjoint $\Gamma_P$ sends a morphism
$h : Y \to X$ in ${\mathbf{X}}$ to the morphism
\begin{diagram}
G(Y) & \rTo^{G(h)} & G(X) \\
\dTo^{P(\varepsilon_Y)} & \Gamma_P(h) & \dTo_{P(\varepsilon_X)} \\
P(Y) & \rTo_{P(h)} & P(X) \\
\end{diagram}
in ${\mathbf{B}}^\mbox{$2\hspace*{-1.2ex}1$}$ because $G(h)$ is the unique morphism $v$ with
$\varepsilon_X \circ 1_v = h \circ \varepsilon_Y =
h \circ \widetilde{\varepsilon}_Y \circ \varphi_{P(\varepsilon_Y)}$ and,
therefore, $(P(h),G(h))$ is the lower transpose
of $h \circ \widetilde{\varepsilon}_Y$ as required.
The unit $\widetilde{\eta}_u : u \to \Gamma_P(\Delta_P(u)) =
P(\varepsilon_{\Delta_P(u)})$ of $\Delta_P \dashv \Gamma_P$ at $u : I \to J$
is given by $\widetilde{\eta}_u$ making the following diagram commute
\begin{diagram}
I & & 1_I & & \\
\dDashto^{\widetilde{\eta}_u} & & \dTo^{1_{\widetilde{\eta}_u}} &
\rdTo^{\varphi_u}_{\mathrm{cocart}} & \\
G \Delta_P(u) & & 1_{G \Delta_P(u)} & \rTo_{\varepsilon_{\Delta_P(u)}} &
\Delta_P(u)\\
\end{diagram}
because $({\mathit{id}}_{P(\Delta_P(u))}, \widetilde{\eta}_u)$ is the
lower transpose of ${\mathit{id}}_{\Delta_P(u)}$. As $P_{\mathbf{B}}\circ\Gamma_P = P$ and
the components of $\widetilde{\eta}$ and $\widetilde{\varepsilon}$ are
vertical it follows\footnote{This is an instance of a general fact about
fibred adjunctions whose formulation and (easy) verification we leave as an
exercise to the reader.} that $\Gamma_P$ is cartesian and thus
$\Delta_P \dashv \Gamma_P$ is a fibred adjunction.
\smallskip
For the implication from right to left suppose that $\Delta_P$ has a fibred
right adjoint $\Gamma_P$. We write $\widetilde{\varepsilon}$ for the counit
of this adjunction. For $X \in {\mathbf{X}}$ we define $\varepsilon_X$ as
$\widetilde{\varepsilon}_X \circ \varphi$
\begin{diagram}[small]
1_{GX} & \rTo^{\varphi}_{\mbox{cocart}} & \Delta_P \Gamma_P X\\
& \rdTo_{\varepsilon_X} & \dTo_{\widetilde{\varepsilon}_X} \\
& & X \\
\end{diagram}
where $\varphi$ is cocartesian over $P(\Gamma_P(X)) : G(X) \to P(X)$.
To verify the desired universal property of $\varepsilon_X$ assume that
$\sigma : 1_I \to X$ is a morphism over $u : I \to P(X)$.
Let $\sigma = f \circ \varphi_u$ with $f$ vertical and $\varphi_u$
cocartesian. Then the existence of a unique arrow $v : I \to G(X)$
with $\varepsilon_X \circ 1_v = \sigma$ follows from considering the diagram
\begin{diagram}[small]
1_I & \rTo^{1_v} & 1_{G(X)} & & \\
\dTo^{\varphi_u} & & \dTo^{\varphi} & \rdTo(2,4)^{\varepsilon_X} & \\
\Delta_P(u) & \rTo^{\Delta_P({\mathit{id}}_{P(X)},v)} & \Delta_P \Gamma_P X & & \\
& \rdTo(4,2)_{f} & & \rdTo~{\widetilde{\varepsilon}_X} & \\
& & & & X \\
\end{diagram}
using the universal property of $\widetilde{\varepsilon}_X$.
Thus, $P$ has small global sections.
\end{proof}
\bigskip
The following explicitation of $\Delta_{{\mathsf{gl}}(F)} \dashv \Gamma_{{\mathsf{gl}}(F)}$
for finite limit preserving $F$ will be helpful later on.
\begin{Thm}\label{gmthm3}
For the geometric fibration $P = {\mathsf{gl}}(F)$ induced by a geometric morphism
$F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ the fibred adjunction $\Delta_P \dashv \Gamma_P$
can be described more concretely as follows.\\
The left adjoint $\Delta_P$ acts by application of $F$ to arrows and squares
in ${\mathbf{B}}$. The fibre of $\Gamma_P$ over $I \in {\mathbf{B}}$ is given by
$\eta_I^* \circ U_{/I}$. The unit $\widetilde{\eta}_u$ for $u : I \to J$
is given by
\begin{diagram}[small]
I & & & & \\
& \rdTo~{\widetilde{\eta}_u} \rdTo(2,4)_{u} \rdTo(4,2)^{\eta_I} & & & \\
& & K \SEpbk & \rTo_{q} & UFI \\
& & \dTo_{p} & & \dTo_{UFu} \\
& & J & \rTo_{\eta_J} & UFJ \\
\end{diagram}
and for $a : A \to FI$ the counit $\widetilde{\varepsilon}_a$ is given by
$\varepsilon_A \circ Fq : Fp \to a$ where
\begin{diagram}[small]
K \SEpbk & \rTo^{q} & UA \\
\dTo^{p} & & \dTo_{Ua} \\
I & \rTo_{\eta_I} & UFI \\
\end{diagram}
\end{Thm}
\begin{proof}
Straightforward exercise when using the description of $\varepsilon$ from the
proof of Theorem~\ref{gmthm1} and the descriptions of $\widetilde{\eta}$ and
$\widetilde{\varepsilon}$ from the proof of Theorem~\ref{gmthm2}.
\end{proof}
\newpage
\section{Fibrational Characterisation of Boundedness}
Recall (e.g.\ from \cite{Joh}) that a geometric morphism
$F \dashv U : {\mathbf{E}} \to {\mathbf{S}}$ between elementary toposes is called \emph{bounded}
iff there is an object $S \in {\mathbf{E}}$ such that for every $X \in {\mathbf{E}}$ there is an
object $I \in {\mathbf{S}}$ such that $X$ appears as a subquotient of $S{\times}FI$
\begin{diagram}[small]
C & \rEmbed & S{\times}FI \\
\dOnto & & \\
X & & \\
\end{diagram}
i.e.\ $X$ appears as quotient of some subobject $C$ of $S{\times}FI$.
Such an $S$ is called a \emph{bound} for the geometric morphism $F \dashv U$.
The importance of bounded geometric morphisms lies in the fact that they
correspond to Grothendieck toposes over ${\mathbf{S}}$ (as shown e.g.\ in \cite{Joh}).
In this section we will show that a geometric morphism
$F \dashv U : {\mathbf{E}} \to {\mathbf{S}}$ is bounded iff for its corresponding geometric
fibration ${\mathsf{gl}}(F)$ there exists a generating family.
\begin{Lem}\label{ccmon}
Let ${\mathbf{B}}$ have finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration of categories
with finite limits with stable disjoint internal sums. Then a cocartesian
arrow $\varphi : X \to Y$ is monic w.r.t.\ to vertical arrows, i.e.\
vertical arrows $\alpha_1, \alpha_2 : Z \to X$ are equal whenever
$\varphi \circ \alpha_1 = \varphi \circ \alpha_2$.
\end{Lem}
\begin{proof}
Let $\alpha_1,\alpha_2 : Z \to X$ be vertical arrows with
$\varphi \circ \alpha_1 = \varphi \circ \alpha_2$. Then there is a unique
morphism $\alpha$ with $\pi_i \circ \alpha = \alpha_i$ for $i=1,2$.
Consider the pullback
\begin{diagram}[small]
\cdot \SEpbk & \rTo^{\beta} & X \\
\dTo^{\psi} & & \dTo_{\delta_\varphi} \\
Z & \rTo_{\alpha} & \cdot \\
\end{diagram}
where $\delta_\varphi$ is the fibrewise diagonal. Notice that both $\alpha$
and $\delta_\varphi$ are above the same mono in ${\mathbf{B}}$. Thus, the map $\psi$
lies above an isomorphism in the base (as $P$ preserves pullbacks) and,
moreover, it is cocartesian as it appears as pullback of the cocartesian
arrow $\delta_\varphi$. Thus, the arrow $\psi$ is an isomorphism and we have
$\alpha = \delta_\varphi \circ \beta \circ \psi^{-1}$ from which it follows
that $\alpha_i = \beta \circ \psi^{-1}$ for $i{=}1,2$.
Thus, we have $\alpha_1 = \alpha_2$ as desired.
Alternatively, one may argue somewhat simpler as follows. For $i{=}1,2$ we have
$\alpha_i \circ \psi = \pi_i \circ \alpha \circ \psi =
\pi_i \circ \delta_\varphi \circ \beta = \beta$. Accordingly, we have
$\alpha_1 \circ \psi = \alpha_2 \circ \psi$ from which it follows
that $\alpha_1 = \alpha_2$ since $\psi$ is cocartesian and the $\alpha_i$ are
vertical.
\end{proof}
\bigskip
For formulating the next lemma we have to recall the notion of collectively
epic morphism as introduced in Theorem~\ref{genfamthm}. If
$P : {\mathbf{X}}\to{\mathbf{B}}$ is a fibration then a morphism $f : X \to Y$ in ${\mathbf{X}}$ is
called \emph{collectively epic} iff for all vertical arrows
$\alpha_1,\alpha_2 : Y \to Z$ from $\alpha_1 \circ f = \alpha_2 \circ f$
it follows that $\alpha_1 = \alpha_2$. Notice that for a collectively epic
morphism $f : X \to Y$ for maps $g_1,g_2 : Y \to Z$ with $P(g_1) = P(g_2)$
from $g_1 f = g_2 f$ it follows that $g_1 = g_2$ because
if $g_i = \varphi \alpha_i$ with $\varphi$ cartesian and $\alpha_i$ vertical
then $\alpha_1 f = \alpha_2 f$ and thus $\alpha_1 = \alpha_2$ from which it
follows that $g_1 = \varphi \alpha_1 = \varphi \alpha_2 = g_2$.
If $P$ is ${\mathrm{Fam}}({\mathbf{C}})$ for an ordinary category
${\mathbf{C}}$ then an arrow $f : X \to Y$ in the total category of ${\mathrm{Fam}}({\mathbf{C}})$
over $u : I \to J$ is collectively epic iff for all $j \in J$ the family
$(f_i : X_i \to Y_j)_{i{\in}u^{-1}(j)}$ is collectively epic in the usual
sense of ordinary category theory. Thus, it would be more precise to say
``family of collectively epic families'' but as this formulation is too
lengthy we prefer the somewhat inaccurate formulation ``collectively epic''.
Notice that for a bifibration $P : {\mathbf{X}} \to {\mathbf{B}}$ a morphism $f : X \to Y$ in
${\mathbf{X}}$ is collectively epic iff for a cocartesian/vertical factorisation
$f = \alpha \circ \varphi$ the vertical arrow $\alpha$ is epic in its fibre.
\begin{Lem}\label{bgmlem1}
Let ${\mathbf{B}}$ be a category with finite limits and $P : {\mathbf{X}} \to {\mathbf{B}}$ a geometric
fibration which is locally small and well--powered. Moreover, suppose that
collectively epic arrows in ${\mathbf{X}}$ are stable under pullbacks.
Then for $P$ there exists a generating family iff for $P$ there exists a
\emph{separator}, i.e.\ an object $S \in P(1_ {\mathbf{B}})$ such that for every
object $X \in P(1_{\mathbf{B}})$ there exist morphisms $\varphi : Y \to S$,
$m : Z \to Y$ and $\psi : Z \to X$ with $\varphi$ cartesian, $m$ a vertical
mono and $\psi$ collectively epic.
\end{Lem}
\begin{proof}
Let $P : {\mathbf{X}} \to {\mathbf{B}}$ be a fibration satisfying the conditions above.
Suppose that $G \in P(I)$ is a generating family for $P$. Let
$\psi_0: G \to S$ be a cocartesian arrow over $!_I : I \to 1$.
Let $\psi_0 = \varphi_0 \circ \eta$ with $\varphi_0$ cartesian and
$\eta$ vertical. Notice that $\eta$ is monic as by Lemma~\ref{ccmon} the
cocartesian $\psi_0$ is monic w.r.t.\ vertical arrows.
We show that $S$ is a separator for $P$. Let $X \in P(1_{\mathbf{B}})$.
As $G$ is a generating family for $P$ and ${\mathbf{B}}$ has binary products by
Theorem~\ref{genfamthm} there are morphisms $\theta : Z \to G$ and
$\psi : Z \to X$ with $\theta$ cartesian and $\psi$ collectively epic.
Then consider the diagram
\begin{diagram}[small]
X & \lTo^{\psi} & Z \SEpbk & \rTo^{\theta} & G & & \\
& & \dEmbed^{m} & & \dEmbed^{\eta} & \rdTo^{\psi_0} & \\
& & Y & \rTo_{\theta'} & I^*S & \rTo_{\varphi_0} & S \\
\end{diagram}
where $\theta'$ is cartesian over $P(\theta)$ and $m$ is vertical. Thus, the
middle square is a pullback and $m$ is a vertical mono. Furthermore,
$\varphi := \varphi_0 \circ \theta'$ is cartesian. Thus, we have constructed
morphisms $\varphi : Y \to S$, $m : Z \to Y$ and $\psi : Z \to X$ with
$\varphi$ cartesian, $m$ a vertical mono and $\psi$ collectively epic
as required.
\medskip
Suppose that $S \in P(1_{\mathbf{B}})$ is a separator for $P$. By well--poweredness
of $P$ there exists a vertical mono $m_S : G \rightarrowtail \sigma_S^*S$ classifying
families of subobjects of $S$. We show that $G$ is a generating family for $P$.
Suppose $X \in P(I)$. Let $\theta_0 : X \to X_0$ be a cocartesian arrow
over $!_I : I \to 1$ . As $S$ is a separator there exist morphisms
$\varphi_0 : Y_0 \to S$, $m_0 : Z_0 \to Y_0$ and $\psi_0 : Z_0 \to X_0$ with
$\varphi_0$ cartesian, $m_0$ a vertical mono and $\psi_0$ collectively epic.
Consider the pullback
\begin{diagram}[small]
Z \SEpbk & \rTo^{\theta} & Z_0 \\
\dTo^{\psi} & & \dTo_{\psi_0} \\
X & \rTo_{\theta_0} & X_0 \\
\end{diagram}
where $\psi$ is collectively epic and $\theta$ is cocartesian since these
classes of arrows are stable under pullbacks. Consider further the diagram
\begin{diagram}
Z & & & & \\
& \rdTo~{\eta} \rdTo(2,4)_{m} \rdTo(4,2)^{\theta} & & & \\
& & \cdot \SEpbk & \rTo_{\varphi'} & Z_0 \\
& & \dEmbed_{m'} & & \dEmbed_{m_0} \\
& & Y & \rTo_{\varphi_1} & Y_0 \\
\end{diagram}
where $\varphi_1$ and $\varphi'$ are cartesian over $P(\theta)$ and $m'$
and $\eta$ are vertical. The inner square is a pullback and thus $m'$ is
monic as it appears as pullback of the monic arrow $m_0$. The arrow $\eta$
is a vertical mono as by Lemma~\ref{ccmon} $\theta$ is monic w.r.t.\
vertical arrows. Thus $m = m' \circ \eta$ is a vertical mono, too.
Moreover, $\varphi_0 \circ \varphi_1 : Y \to S$ is cartesian. Thus, the
mono $m : Z \to Y$ is a family of subobjects of $S$ and, accordingly, we have
\begin{diagram}[small]
Z \SEpbk & \rTo^{\varphi} & G \\
\dEmbed^{m} & & \dEmbed_{m_S} \\
Y & \rTo_{\widetilde{\varphi}} & \sigma_S^*S \\
\end{diagram}
for some cartesian arrows $\varphi$ and $\widetilde{\varphi}$. Thus, we have
morphisms $\varphi : Z \to G$ and $\psi : Z \to X$ with $\varphi$ cartesian
and $\psi$ collectively epic.
Thus, by Theorem~\ref{genfamthm} it follows
that $G$ is a generating family for $P$.
\end{proof}
\bigskip
Suppose $F : {\mathbf{B}} \to {\mathbf{C}}$ is a finite limit preserving functor between
categories with finite limits. One easily checks that an arrow
\begin{diagram}[small]
B & \rTo^e & A \\
\dTo^b & f & \dTo_a \\
FJ & \rTo_{Fu} & FI \\
J & \rTo_u & I
\end{diagram}
in ${\mathsf{Gl}}(F) = {\mathbf{C}}{\downarrow}F$ is collectively epic (w.r.t.\ the fibration
${\mathsf{gl}}(F) = \partial_1 : {\mathbf{C}}{\downarrow}F \to {\mathbf{B}}$) iff the map $e$ is epic
in ${\mathbf{C}}$. Apparently, this condition is sufficient. On the other hand if $f$
is collectively epic then $e$ is epic in ${\mathbf{C}}$ which can be seen as follows:
suppose $g_1,g_2 : A \to C$ with $g_1 e = g_2 e$ then the maps
\begin{diagram}[small]
A & \rTo^{g_i} & C \\
\dTo^a & \alpha_i & \dTo \\
FI & \rTo_{F!_I} & F1 \\
I & \rTo_{!_I} & 1
\end{diagram}
are both above $I \to 1$ and satisfy $\alpha_1 f = \alpha_2 f$ from which
it follows -- since $f$ is collectively epic -- that $\alpha_1 = \alpha_2$ and
thus $g_1 = g_2$.
Thus, if in ${\mathbf{C}}$ epimorphisms are stable under pullbacks along arbitrary
morphisms then in ${\mathsf{Gl}}(F)$ collectively epic maps are stable under pullbacks
along arbitrary morphisms.
\begin{Thm}\label{bgmthm}
A geometric morphism $F \dashv U : {\mathbf{E}} \to {\mathbf{S}}$ between toposes is bounded iff
for the corresponding geometric fibration ${\mathsf{gl}}(F)$ there exists a
generating family.
\end{Thm}
\begin{proof}
Let $F \dashv U : {\mathbf{E}} \to {\mathbf{S}}$ be a geometric morphism between toposes.
Then the corresponding geometric fibration ${\mathsf{gl}}(F)$ is locally small and
well-powered.
As for ${\mathsf{gl}}(F)$ reindexing preserves the topos structure and
in toposes epis are stable under pullbacks vertical epis are stable under
pullbacks. Thus, collectively epic arrows are stable under pullbacks as
both vertical epis and cocartesian arrows are stable under pullbacks.
Alternatively, this follows from the observations immediately preceding
the current theorem and pullback stability of epimorphisms in toposes.
Thus, since the assumptions of Lemma~\ref{bgmlem1} are satisfied for ${\mathsf{gl}}(F)$
there exists a generating family for ${\mathsf{gl}}(F)$ iff there exists a separator
for ${\mathsf{gl}}(F)$ which, obviously, is equivalent to the geometric morphism
$F \dashv U$ being bounded.
\end{proof}
\bigskip
From inspection of the proof of Lemma~\ref{bgmlem1} it follows\footnote{In
more concrete terms for the fibration ${\mathsf{gl}}(F) = F^*P_{\mathbf{E}}$ this can be seen as
follows. Suppose $a : A \to F(I)$ is a map in ${\mathbf{E}}$. As $S$ is a bound
there exists $J \in {\mathbf{S}}$ and $e : C \twoheadrightarrow A$ with $n : C \rightarrowtail F(J) \times S$.
Consider the diagram
\begin{diagram}
A & \lOnto^{\;\;\;e} & C & & \\
& & \dEmbed^m & \rdEmbed^n & \\
\dTo^a & & F(I{\times}J){\times}S \SEpbk & \rTo^{F(\pi'){\times}S}
& F(J){\times}S \\
& & \dTo^\pi & & \dTo_\pi \\
F(I) & \lTo_{\;\;\;F(\pi)} & F(I{\times}J) & \rTo_{F(\pi')} & F(J) \\
\end{diagram}
(where $F(\pi)$ and $F(\pi')$ form a product cone
because $F$ preserves finite limits and $\pi$ and $\pi'$ form a product
cone) and notice that $\pi \circ m$ appears as pullback of $g_S$ along
$F(\rho)$ where $\rho : I{\times}J \to U{\mathcal{P}}(S)$ is the unique map
classifying $m$, i.e.\
$((\varepsilon_{{\mathcal{P}}(S)}{\circ}F(\rho)){\times}S)^*{\ni_S} \;\cong m$.}
in particular that if $S\in{\mathbf{E}}$ is a bound for a geometric morphism
$F\dashv U:{\mathbf{E}}\to{\mathbf{S}}$ between toposes
then $g_S = \pi \circ m_S : G_S \to F(U{\mathcal{P}}(S))$
\begin{diagram}[small]
G_S \SEpbk & \rTo & \ni_S \\
\dEmbed^{m_S} & & \dEmbed_{\ni_S} \\
FU{\mathcal{P}}(S){\times}S \SEpbk & \rTo^{\varepsilon_{{\mathcal{P}}(S)}{\times}S} &
{\mathcal{P}}(S){\times}S\\
\dTo^{\pi} & & \dTo_{\pi} \\
FU{\mathcal{P}}(S) & \rTo_{\varepsilon_{{\mathcal{P}}(S)}} & {\mathcal{P}}(S)
\end{diagram}
is a generating family for ${\mathsf{gl}}(F)$. This condition, however, also implies
that $S$ is a bound for $F \dashv U$ since if $g_S = \pi \circ m_S$ is a
generating family for ${\mathsf{gl}}(F)$ then for every $A \in {\mathbf{E}}$ there is a map
$u : I \to U{\mathcal{P}}(S)$ in ${\mathbf{S}}$ and an epi $e : u^*G_S \twoheadrightarrow A$ such that
\begin{diagram}[small]
A & \lOnto^e & u^*G_S \SEpbk & \rTo & G_S \SEpbk & \rTo & \ni_S \\
\dTo & & \dEmbed^m & & \dEmbed^{m_S} & & \dEmbed_{\ni_S} \\
& & FI{\times}S \SEpbk & \rTo^{Fu{\times}S} & FU{\mathcal{P}}(S){\times}S
\SEpbk & \rTo^{\varepsilon_{{\mathcal{P}}(S)}{\times}S} &
{\mathcal{P}}(S){\times}S\\
& & \dTo^{\pi} & & \dTo^{\pi} & & \dTo_{\pi}\\
F1 & \lTo_{F!_I} & FI & \rTo_{Fu} & FU{\mathcal{P}}(S) & \rTo_{\varepsilon_{{\mathcal{P}}(S)}}
& {\mathcal{P}}(S)
\end{diagram}
from which it follows that $A$ appears as quotient of a subobject of some
$FI{\times}S$.
Thus $S$ is a bound for a geometric morphism $F \dashv U : {\mathbf{E}}\to{\mathbf{S}}$ between
toposes iff $g_S = \pi \circ m_S : G_S \to FU{\mathcal{P}}(S)$ is a generating
family for ${\mathsf{gl}}(F)$. In case ${\mathbf{S}}$ is ${\mathbf{Set}}$ this amounts to the usual
requirement that the family of subobjects of $S$ is generating for the
topos ${\mathbf{E}}$.
\smallskip
One can characterize boundedness of geometric morphisms in terms
of preservation properties as follows.
\begin{Thm}\label{bgmthm'}
A geometric morphism $F \dashv U : {\mathbf{E}} \to {\mathbf{S}}$ between toposes is bounded
if and only if change of base along $F$ preserves existence of small
generating families for geometric fibrations of toposes.
Thus a terminal object preserving functor $F : {\mathbf{S}}\to{\mathbf{E}}$ between toposes
is the inverse image part of a bounded geometric morphism if and only if
change of base along $F$ preserves the property of being a geometric
fibration of toposes with a small generating family.
\end{Thm}
\begin{proof}
Suppose $F^*$ preserves existence of small generating families for
geometric fibrations of toposes. Then since
$P_{\mathbf{E}} = {\mathsf{gl}}({\mathit{Id}}_{\mathbf{E}}): {\mathbf{E}}^\mbox{$2\hspace*{-1.2ex}1$} \to {\mathbf{E}}$ has a small generating family
so does $F^*P_{\mathbf{E}} = {\mathsf{gl}}(F)$ and thus by Theorem~\ref{bgmthm} the
geometric morphism $F \vdash U$ is bounded.
On the other hand if a geometric morphism $F \dashv U$ is bounded
and $P$ is a geometric fibration of toposes over ${\mathbf{E}}$ with a small
generating family then by Theorem~\ref{bgmthm} $P$ is equivalent to
$G^*P_{\mathbf{F}} = {\mathsf{gl}}(G)$ for some bounded geometric morphism $G \dashv V : {\mathbf{F}} \to {\mathbf{E}}$
and thus $F^*P$ is equivalent to $F^*G^*P_{\mathbf{F}} \simeq (GF)^*P_{\mathbf{F}} = {\mathsf{gl}}(GF)$
which has a small generating family by Theorem~\ref{bgmthm}
since $GF \dashv UV$ is a bounded geometric morphism (as by \cite{Joh}
bounded geometric morphisms are closed under composition).
The second claim follows from the fact that a terminal object preserving
functor $F : {\mathbf{S}}\to{\mathbf{E}}$ between toposes is the inverse image part of a
geometric morphism iff ${\mathsf{gl}}(F) = F^*P_{\mathbf{E}}$ is a geometric fibration of toposes.
\end{proof}
\medskip
Thus we may observe that for (bounded) geometric morphisms $F \dashv U$
change of base along $F$ for geometric fibrations of toposes (with a small
generating family) corresponds to postcomposition with $F \dashv U$ for
(bounded) geometric morphisms.\footnote{A consequence of this observation
is that change of base along inverse image parts of geometric morphisms
for geometric fibrations of toposes reflects the property of having a small
generating family since as observed in \cite{Joh} for geometric morphisms
$f$ and $g$ from $fg$ bounded it follows that $g$ is bounded.}
By Theorem~\ref{bgmthm'} change of base along inverse image parts of unbounded
geometric morphisms does not preserve existence of small generating families.
From \cite{Joh} we recall the following example of an unbounded geometric
morphism. Let ${\mathbf{E}}$ be the full subcategory of $\widehat{{\mathbb{Z}}} = {\mathbf{Set}}^{{\mathbb{Z}}^{\mathrm{op}}}$
on those objects $A$ such that $\forall a{\in}A. A(n)(a) = a$ for some
$n\in{\mathbb{N}}$, i.e.\ there is a finite bound on the size of all orbits of the
action $A$. One easily sees that ${\mathbf{E}}$ is a topos and
$\Delta \dashv \Gamma : {\mathbf{E}} \to {\mathbf{Set}}$ is a geometric morphism which, however,
is not bounded (as otherwise there were an \emph{a priori} bound on the
size of all orbits of objects of $\widehat{{\mathbb{Z}}}$). Notice, however, that
${\mathbf{E}}$ admits a (countable) generating family in the sense of ordinary category
theory, namely the family $({\mathbb{Z}}_n)_{n\in{\mathbb{N}}}$ (of all finite orbits up to
isomorphism), whose sum, however, does not exist in ${\mathbf{E}}$.
Johnstone's example also demonstrates that toposes ${\mathbf{E}}$ over ${\mathbf{Set}}$ need not
be cocomplete in the sense of ordinary category theory, i.e.\ do not have all
small sums, although the associated fibration ${\mathsf{gl}}(\Delta)$ certainly has
internal sums.\footnote{Thus, the fibrations ${\mathsf{gl}}(\Delta) = \Delta^*P_{\mathbf{E}}$
and ${\mathrm{Fam}}({\mathbf{E}})$ over ${\mathbf{S}}$ are not equivalent because ${\mathsf{gl}}(\Delta)$ has internal
sums whereas ${\mathrm{Fam}}({\mathbf{E}})$ doesn't!
Consider also the following somewhat weaker counterexample. Let $\mathcal{A}$
be a \emph{partial combinatory algebra}, $\mathbf{RT}[\mathcal{A}]$ the
realizability topos over $\mathcal{A}$ (see e.g.\ \cite{realizbook})
and $\Gamma \dashv \nabla : {\mathbf{Set}} \to \mathbf{RT}[\mathcal{A}]$ the geometric
morphism where $\Gamma = \mathbf{RT}[\mathcal{A}](1,-)$ is the global elements
functor.
Then ${\mathsf{gl}}(\nabla) = \nabla^*P_{\mathbf{RT}[\mathcal{A}]}$ is a fibration
with stable and disjoint internal sums over ${\mathbf{Set}}$ although for
{\bf nontrivial} $\mathcal{A}$ in the realizability topos
$\mathbf{RT}[\mathcal{A}]$ the sum $\coprod_{|\mathcal{A}|} 1$ does not exist
for cardinality reasons.
Moreover, for nontrivial $\mathcal{A}$ internal sums w.r.t.\ the fibration
${\mathsf{gl}}(\nabla)$ in general do not coincide with the corresponding external sums
(if they exists): consider e.g.\ $\coprod_2 1$ w.r.t.\ ${\mathsf{gl}}(\nabla)$, i.e.\
$\nabla(2)$, which is not isomorphic to $1+1$ in $\mathbf{RT}(\mathcal{A})$.
Thus $\nabla(1+1) \not\cong \nabla(1) + \nabla(1)$ from which it follows
that $\nabla$ does not have a right adjoint. Accordingly, the fibration
${\mathsf{gl}}(\nabla)$ over ${\mathbf{Set}}$ does not have small global elements.}
Apparently there is a difference between \emph{internal} and
\emph{external} families of objects in ${\mathbf{E}}$ where a family $(X_i)_{i \in I}$
in ${\mathbf{E}}$ is internal if there is a map $f : Y \to \Delta(I)$ in ${\mathbf{E}}$ with
$X_i \cong {\mathrm{in}}_i^*f$ for all $i \in I$. Of course, every internal family
gives rise to an external one whereas e.g.\ $({\mathbb{Z}}_n)_{n \in {\mathbb{N}}}$ is an
external family in the topos ${\mathbf{E}}$ which is not internal.
It is an easy exercise to show that a family
$(X_i)_{i \in I}$ in a topos ${\mathbf{E}}$ over ${\mathbf{Set}}$ is internal if and only if
the family $(X_i)_{i \in I}$ is \emph{bounded} in the sense that there exists
an object $X \in {\mathbf{E}}$
such that all $X_i$ appear as subobjects of $X$.\footnote{In the example
${\mathsf{gl}}(\nabla) = \partial_1 :
\mathbf{RT}[\mathcal{A}] {\downarrow} \nabla\to{\mathbf{Set}}$ every $X \to \nabla(I)$
in $\mathbf{RT}[\mathcal{A}]$ may be understood as a family of $I$-indexed
subobjects of $X$ but the ensuing cartesian functor (over ${\mathbf{Set}}$) from
${\mathsf{gl}}(\nabla)$ to ${\mathrm{Fam}}(\mathbf{RT}[\mathcal{A}])$ is far from being an
equivalence.
Firstly, it does not reflect isos (in each fibre) since ${\mathit{id}}_{\nabla(2)}$ and
$\eta_2 : 2 \to \nabla\Gamma(2) \cong \nabla(2)$ are not isomorphic in the
slice over $\nabla(2)$ but both give rise to $(1)_{i \in 2}$
in ${\mathrm{Fam}}(\mathbf{RT}[\mathcal{A}])(2)$. Thus, different internal families
(over $2$ already) may give rise to the same external family.
Secondly, there are external ${\mathbb{N}}$-indexed families $(X_n)_{n\in{\mathbb{N}}}$ in
$\mathbf{RT}[\mathcal{A}]$ which do not arise from a morphism
$X \to \nabla({\mathbb{N}})$ because any such family would have to be isomorphic
to a family $(X^\prime_n)_{n\in{\mathbb{N}}}$ for which symmetry and transitivity
are realized by $e_1, e_2 \in \mathcal{A}$ independently from $n \in {\mathbb{N}}$.
It is left as an exercise to the reader to give a concrete counterexample.}
Notice that due to Giraud's Theorem (see \cite{Joh}) toposes bounded
over ${\mathbf{Set}}$ are precisely the Grothendieck toposes and, therefore, do have
all small sums. Actually, one may see this more directly as follows.
Suppose $S$ is a bound for the geometric morphism
$\Delta \dashv \Gamma : {\mathbf{E}} \to {\mathbf{Set}}$. Then ${\mathbf{E}}$ has all small copowers
$\coprod_{i \in I} X \cong \Delta(I) \times X$. Suppose $(X_i)_{i \in I}$ is
a family in ${\mathbf{E}}$. Then for every $i \in I$ there is a set $J_i$ such that
$X_i$ is a subquotient of $\Delta(J_i) \times S$. Thus, all $X_i$ are
subobjects of ${\mathcal{P}}(\Delta(J){\times}S)$ via some mono $m_i$
where $J = \bigcup_{i \in I} J_i$
(since $\Delta(J_i) \times S$ is a subobject of $\Delta(J) \times S$).
Let $\chi_i$ classify the subobject $m_i$ for $i \in I$ and $\chi :
\coprod_{i \in I} {\mathcal{P}}(\Delta(J){\times}S) \to \Omega$ be the source
tupling of the $\chi_i$. Then the sum $\coprod_{i \in I} X_i$ appears
as the subobject of the copower
$\coprod_{i \in I} {\mathcal{P}}(\Delta(J){\times}S) \cong
\Delta(I){\times}{\mathcal{P}}(\Delta(J){\times}S)$
classified by $\chi$ in ${\mathbf{E}}$.
But there exist toposes over ${\mathbf{Set}}$ which, in the sense of ordinary
category theory, are cocomplete but do not admit a small generating family.
A typical such example (due to Peter Freyd) is the topos ${\mathcal{G}}$ whose objects
are pairs $(A,f)$ where $A$ is a set and $f$ is a family of bijections of $A$
indexed over the class of all sets such that the class
${\mathsf{supp}}(A,f) = \{ s \mid f_s \not= {\mathit{id}}_A \}$ is a set and
whose morphisms from $(A,f)$ to $(B,g)$ are the maps $h : A \to B$
with $h(f_s(a)) = g_s(h(a))$ for all $a \in A$ and all sets $s$.
The construction of this category can be rephrased as follows.
Let ${\mathbb{G}}$ be the free group generated by the class of all sets.
Then ${\mathcal{G}}$ is isomorphic to the full subcategory of $\widehat{{\mathbb{G}}}$
on those objects $A$ where $\{s \mid A(s) \not= {\mathit{id}}_{A(*)}\}$ is a set.
The proof that ${\mathcal{G}}$ is a topos is analogous to the proof that for every
group $G$ the presheaf category ${\mathbf{Set}}^{G^{\mathrm{op}}}$ is a boolean topos. Moreover
${\mathcal{G}}$ has all small
limits and colimits (which are constructed pointwise). Suppose
$(G_i,g^{(i)})_{i \in I}$ were a small generating family for ${\mathcal{G}}$.
Let $J = \bigcup_{i \in I} {\mathsf{supp}}(G_i,g^{(i)})$ and $s_0$ be a set with
$s_0 \not\in J$. Now let $(A,f)$ be the object of ${\mathcal{G}}$ where $A = \{0,1\}$
and $f_s \not= {\mathit{id}}_A$ only for $s = s_0$. There cannot exist a morphism
$h : (G_i,g^{(i)}) \to (A,f)$ unless $G_i$ is empty as otherwise there is
a $z \in G_i$ for which we have $h(z) = h(g^{(i)}_{s_0})(z)) = f_{s_0}(h(z))$
Obviously $(A,f)$ has two different endomorphisms which, however, cannot be
distinguished by morphisms of the form $h : (G_i,g^{(i)}) \to (A,f)$. Thus,
there cannot exist a small generating family for the cocomplete boolean topos
${\mathcal{G}}$.
One easily shows that for a cocomplete topos ${\mathbf{E}}$ the functor
$\Delta : {\mathbf{E}} \to {\mathbf{Set}}$ preserves finite limits.
Thus, for a locally small topos ${\mathbf{E}}$ it holds that
$$ {\mathbf{E}} \mbox{ bounded over } {\mathbf{Set}} \Longrightarrow {\mathbf{E}} \mbox{ cocomplete }
\Longrightarrow {\mathbf{E}} \mbox{ over } {\mathbf{Set}} $$
and the above counterexamples show that none of these implications can be
reversed in general.\footnote{The category ${\mathbf{Set}}^{\mathbf{Ord}^{\mathrm{op}}}$ of ${\mathbf{Set}}$-valued presheaves over the large category $\mathbf{Ord}$ of ordinals is an example of a cocomplete topos which, however, is not locally small since there are class many subterminals and thus $\Omega$ has class many global elements.}
Freyd's counterexample shows that the first implication
cannot be reversed in general. Johnstone's counterexample shows that the
second implication cannot be reversed in general.
If ${\mathbf{E}}$ is a topos bounded over ${\mathbf{Set}}$ then for ${\mathbf{E}}$ there exists a generating
family in the sense of ordinary category theory. However, as Johnstone's
counterexample shows the reverse implication does not hold in general for
toposes over ${\mathbf{Set}}$. Freyd's counterexample shows there are toposes ${\mathbf{E}}$
over ${\mathbf{Set}}$ such that there does not even exist a generating family for ${\mathbf{E}}$
in the sense of ordinary category theory and that such toposes may even be
cocomplete.
Notice that toposes ${\mathbf{E}}$ cocomplete in the sense of ordinary category theory
are bounded over ${\mathbf{Set}}$ iff there exists a generating family for ${\mathbf{E}}$ in the
sense of ordinary category theory. The reason is that if $(G_i)_{i \in I}$
is a generating family for ${\mathbf{E}}$ in the sense of ordinary category theory then
$\coprod_{i \in I} !_{G_i} : \coprod_{i \in I} G_i \to \coprod_{i \in I} 1_{\mathbf{E}}
= \Delta(I)$ is a generating family for the fibration
$\Delta^*P_{\mathbf{E}} = {\mathsf{gl}}(\Delta)$. Thus, a topos ${\mathbf{E}}$ is bounded over ${\mathbf{Set}}$ iff
${\mathbf{E}}$ is cocomplete and there exists a generating family for ${\mathbf{E}}$ in the sense
of ordinary category theory. However, this characterisation does not generalise
to arbitrary base toposes ${\mathbf{S}}$. Formally, the fibrational characterisation
of bounded toposes over ${\mathbf{S}}$ as cocomplete locally small fibred toposes
over ${\mathbf{S}}$ with a generating family looks similar but as we have seen above
cocomplete in the sense of fibred categories is weaker than cocomplete in the
sense of ordinary category theory and generating family in the sense of fibred
categories is stronger than in the sense of ordinary category theory.
Finally we observe that a topos over ${\mathbf{Set}}$ which in the sense of ordinary
category theory is neither cocomplete nor has a small generating family can
be obtained by combining the ideas of Freyd's and Johnstone's counterexamples,
namely the full subcategory of Freyd's counterexample ${\mathcal{G}}$ on those objects
$(A,f)$ for which there exists an $n \in {\mathbb{N}}$ such that $(f_s)^n = {\mathit{id}}_A$ for
all sets $s$.
\newpage
\section{Properties of Geometric Morphisms}
In this section we will characterise some of the most common properties
of geometric morphisms $F \dashv U$ in terms of simple fibrational properties
of the corresponding geometric fibration ${\mathsf{gl}}(F)$. In the following
we simply write $\Delta \dashv \Gamma$ for the fibred adjunction
$\Delta_P \dashv \Gamma_P$ and the according unit and counit are denoted by
$\widetilde{\eta}$ and $\widetilde{\varepsilon}$, respectively.
\begin{Thm}\label{gmprop1}
Let $F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ be a geometric morphism and $P$ be the induced
geometric fibration ${\mathsf{gl}}(F)$. Then the following conditions are equivalent.
\begin{enumerate}
\item[\rm (1)] The geometric morphism $F \dashv U$ is \emph{injective},
i.e.\ $U$ is full and faithful.
\item[\rm (2)] The counit $\widetilde{\varepsilon}$ of $\Delta \dashv \Gamma$
is a natural isomorphism.
\item[\rm (3)] For the counit $\varepsilon$ of $1 \dashv G : {\mathsf{Gl}}(F) \to {\mathbf{B}}$
it holds that $\varepsilon_X$ is cocartesian for all objects $X \in {\mathsf{Gl}}(F)$.
\end{enumerate}
\end{Thm}
\begin{proof} Conditions (2) and (3) are equivalent as by Theorem~\ref{gmthm2}
we have $\varepsilon_X = \widetilde{\varepsilon}_X \circ \varphi$ with
$\varphi : 1_{GX} \to \Delta\Gamma X$ cocartesian over $P(\varepsilon_X)$.
Condition (2) says that all $\Gamma_I$ are full and faithful. In particular,
we have that $U\cong\Gamma_1$ is full and faithful. Thus (2) implies (1).
It remains to show that (1) entails (2). Condition (1) says that the counit
$\varepsilon$ of $F \dashv U$ is a natural isomorphism.
But then for every $a : A \to FI$ in ${\mathsf{Gl}}(F)$ we have
\begin{diagram}
FK \SEpbk & \rTo^{Fq} & FUA \SEpbk & \rTo^{\varepsilon_A}_{\cong} & A \\
\dTo^{Fp} & & \dTo_{FUa} & & \dTo_{a }\\
FI & \rTo_{F\eta_I} & FUFI & \rTo_{\varepsilon_{FI}}^{\cong} & FI \\
\end{diagram}
from which it follows by Theorem~\ref{gmthm3} that the map
$\widetilde{\varepsilon}_a = \varepsilon_A \circ Fq$
is an isomorphism as it appears as pullback of the identity
${\mathit{id}}_{FI} = \varepsilon_{FI} \circ F(\eta_I)$.
\end{proof}
\medskip
\begin{Thm}\label{gmprop2}
Let $F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ be a geometric morphism and $P$ be its induced
geometric fibration ${\mathsf{gl}}(F)$. Then the following conditions are
equivalent.\footnote{This holds without assuming that $F$ has a right
adjoint. It suffices that $F$ preserves finite limits.}
\begin{enumerate}
\item[\rm (1)]
The geometric morphism $F \dashv U$ is \emph{surjective},
i.e.\ $F$ reflects isomorphisms.
\item[\rm (2)]
A morphisms $u$ in ${\mathbf{B}}$ is an isomorphism whenever $1_u$ is cocartesian.
\end{enumerate}
\end{Thm}
\begin{proof}
Obviously, the functor $F$ reflects isomorphisms iff all $\Delta_I$ reflect
isomorphisms.
For a morphism $u : w \to v$ in ${\mathbf{B}}/I$ (i.e.\ $w = v \circ u$) we have
\begin{diagram}
1_K & \rTo^{\varphi_w} & \Delta_I(w) \\
\dTo^{1_u} & & \dTo_{\Delta_I(u)} \\
1_J & \rTo_{\varphi_v} & \Delta_I(v) \\
\end{diagram}
where $\varphi_w$ and $\varphi_v$ are cocartesian over $w : K \to I$ and
$v : J \to I$, respectively, and $\Delta(u)$ is vertical over $I$.
As internal sums in ${\mathsf{gl}}(F)$ are stable and disjoint it follows
from Lemma~\ref{MoensLem} that $\Delta_I(u)$ is an isomorphism iff $1_u$
is cocartesian. Thus, the functor $\Delta_I$ reflects isomorphisms iff $u$
is an isomorphism whenver $1_u$ is cocartesian.
Thus, the functor $F$ reflects isomorphisms iff it holds for all maps $u$
in ${\mathbf{B}}$ that $u$ is an isomorphism whenever $1_u$ is cocartesian.
\end{proof}
\bigskip
A geometric morphisms $F \dashv U$ between toposes is known to be surjective
iff $F$ is faithful. One easily sees that a finite limit preserving functor
$F : {\mathbf{B}} \to {\mathbf{C}}$ between categories with finite limits is faithful iff for
the associated fibration ${\mathsf{gl}}(F)$ it holds for $u,v : J \to I$ that $u = v$
whenever $\varphi_I \circ 1_u = \varphi_I \circ 1_v$ where
$\varphi_I : 1_I \to \coprod_I 1_I$ is cocartesian over $I \to 1$.
But, of course, in this general case $F$ being faithful does not imply that
$F$ reflects isos, e.g.\ if ${\mathbf{B}}$ is posetal then $F$ is always faithful but
in general does not reflect isos. However, if $F$ reflects isos then it is
also faithful since $F$ preserves equalizers.
\begin{Thm}\label{gmprop3}
Let $F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ be a geometric morphism and $P$ be its induced
geometric fibration ${\mathsf{gl}}(F)$. Then the following conditions are equivalent.
\begin{enumerate}
\item[\rm (1)]
The geometric morphism $F \dashv U$ is \emph{connected},
i.e.\ $F$ is full and faithful.
\item[\rm (2)]
The right adjoint $G$ of $1 : {\mathbf{B}} \to {\mathsf{Gl}}(F)$
sends cocartesian arrows to isomorphisms.
\item[\rm (3)] The fibred functor $\Gamma$ is cocartesian, i.e.\ preserves
cocartesian arrows.
\end{enumerate}
\end{Thm}
\begin{proof}
Obviously, the functor $F$ is full and faithful iff all $\Delta_I$ are full
and faithful, i.e.\ all $\widetilde{\eta}_u$ are isomorphisms.
Let $u : I \to J$ be a morphism in ${\mathbf{B}}$. Then we have
\begin{diagram}
1_{G(1_I)} & \rEqual^{\varepsilon_{1_I}} & 1_I \\
\dTo^{1_{\widetilde{\eta}_u}} & & \dTo_{\varphi_u} \\
1_{G(\Delta(u))} & \rTo_{\varepsilon_{1_{\Delta(u)}}} & \Delta(u) \\
\end{diagram}
where $\varphi_u : 1_I \to \Delta(u)$ is cocartesian over $u$ and, therefore,
we have $\widetilde{\eta}_u = G(\varphi_u)$.
Thus, the functor $F$ is full and faithful iff $G(\varphi)$ is an isomorphism
for cocartesian $\varphi$ whose source is terminal in its fibre. But then
$G$ sends all cocartesian arrows to isomorphisms which can be seen as follows.
Suppose $\varphi : X \to Y$ is cocartesian over $u : I \to J$.
Let $\varphi_u : 1_I \to \Delta(u)$ be cocartesian over $u$.
Then by Lemma~\ref{MoensLem} the commuting square
\begin{diagram}[small]
X \SEpbk & \rTo^{\varphi} & Y\\
\dTo^{\alpha} & & \dTo_{\beta} \\
1_I & \rTo_{\varphi_u} & \Delta(u) \\
\end{diagram}
with $\alpha$ and $\beta$ vertical over $I$ and $J$, respectively,
is a pullback. As $G$ is a right adjoint it preserves pullbacks and, therefore,
\begin{diagram}[small]
G(X) \SEpbk & \rTo^{G(\varphi)} & G(Y) \\
\dTo^{G(\alpha)} & & \dTo_{G(\beta)} \\
G(1_I) & \rTo_{G(\varphi_u)} & G(\Delta(u)) \\
\end{diagram}
is a pullback, too, from which it follows that $G(\varphi)$ is an isomorphism
as $G(\varphi_u)$ is an isomorphism by assumption.
Thus, we have shown the equivalence of conditions (1) and (2).
The equivalence of conditions (2) and (3) can be seen as follows.
From (inspection of) the proof of Theorem~\ref{gmthm1} we know that for
$\varphi : X \to Y$ its image under $\Gamma$ is given by
\begin{diagram}[small]
G(X) & \rTo^{G(\varphi)} & G(Y) \\
\dTo^{P(\varepsilon_X)} & \Gamma(\varphi) & \dTo_{P(\varepsilon_Y)} \\
P(X) & \rTo_{P(\varphi)} & P(Y) \\
\end{diagram}
Thus $\Gamma(\varphi)$ is cocartesian iff $G(\varphi)$ is an isomorphism.
Accordingly, the functor $G$ sends all cocartesian arrows to isomorphisms
iff $\Gamma$ preserves cocartesianness of arrows.
\end{proof}
\bigskip
Notice that condition (2) of Theorem~\ref{gmprop3} is equivalent to the
requirement that $G$ inverts just cocartesian arrows over terminal projections
which can be seen as follows. Suppose $\varphi : X \to Y$ is cocartesian
over $u : I \to J$. Let $\psi : Y \to \coprod_J Y$ be a cocartesian arrows
over $!_J : J \to 1$. Then $\psi \circ \varphi$ is cocartesian over
$!_I : I \to 1$. As $G(\psi \circ \varphi) = G(\psi) \circ G(\varphi)$
and by assumption $G(\psi \circ \varphi)$ and $G(\psi)$ are isomorphisms
it follows immediately that $G(\varphi)$ is an isomorphism, too. Moreover,
one easily sees that $G$ inverts cocartesian arrows over terminal projections
if and only if $G$ inverts cocartesian arrows above terminal projections whose
source is terminal in its fibre. Of course, this condition is necessary. For
the reverse direction suppose that $\varphi : X \to Y$ is cocartesian over
$!_I : I \to 1$. Then as $P$ is a geometric fibration we have
\begin{diagram}
X \SEpbk & \rTo^{\varphi}_{\mathrm{cocart}} & Y \\
\dTo^{\alpha} & & \dTo_{\beta} \\
1_I & \rTo_{\varphi_I}^{\mathrm{cocart}} & \Delta(I)
\end{diagram}
where $\varphi_I$ is cocartesian over $I \to 1$ and $\alpha$ and $\beta$ are
the unique vertical arrows making the diagram commute. As $G$ is a right
adjoint it preserves pullbacks and, therefore, we have
\begin{diagram}
G(X) \SEpbk & \rTo^{G(\varphi)} & G(Y) \\
\dTo^{G(\alpha)} & & \dTo_{G(\beta)} \\
G(1_I) & \rTo_{G(\varphi_I)} & G(\Delta(I))
\end{diagram}
from which it follows that $G(\varphi)$ is an isomorphism as $G(\varphi_I)$
is an isomorphism by assumption. Thus, a geometric fibration $P$ is connected
if and only if $G$ inverts all cocartesian arrows over terminal projections
which start from a fibrewise terminal object, i.e.\ if for all
$\sigma : 1_J \to \Delta(I)$ there exists a unique $u : J \to I$ with
$\sigma = \varphi_I \circ 1_u$ as gets immediate from the following diagram
\begin{diagram}
1_{G1_I} & \rEqual^{\;\varepsilon_{1_I}} & 1_I \\
\dTo_\cong^{1_{G\varphi_I}} & & \dTo_{\varphi_I} \\
1_{G \Delta(I)} & \rTo_{\;\;\varepsilon_{\Delta(I)}} & \Delta(I)\\
\end{diagram}
with $\varphi_I : 1_I \to \Delta(I)$ cocartesian over $I \to 1$.
Analogously, faithfulness of $F$ is equivalent to the requirement
that $u = v$ whenever $\varphi_I \circ 1_u = \varphi_I \circ 1_v$
providing an alternative characterisation of surjectivity for geometric
morphisms between toposes (as $F \dashv U$ is surjective iff $F$ is faithful).
Obviously, condition (1) of Theorem~\ref{gmprop3} is equivalent to the
requirement that $\eta : {\mathit{Id}}_{\mathbf{B}} \to UF$ is a natural isomorphism.
For the particular case of a geometric morphism $\Delta \dashv \Gamma :
{\mathbf{E}} \to {\mathbf{Set}}$ where ${\mathbf{E}}$ is a topos one easily sees that
$\eta_I : I \to \Gamma\Delta I$ (sending $i \in I$ to the injection
${\mathrm{in}}_i : 1 \to \coprod_{i \in I} 1$) is a bijection for all sets $I$ iff
the terminal object of ${\mathbf{E}}$ is \emph{indecomposable} in the sense that for all
subterminals $U$ and $V$ with $U{+}V \cong 1_{\mathbf{E}}$ either $U$ or $V$ is
isomorphic to $0_{\mathbf{E}}$.
\begin{Thm}\label{hypconn1}
Let $F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ be a geometric morphism and $P$ be the induced
geometric fibration ${\mathsf{gl}}(F)$. Then the following conditions are equivalent.
\begin{enumerate}
\item[\rm (1)]
The geometric morphism $F \dashv U$ is \emph{hyperconnected}, i.e.\
connected and all components of the counit of the adjunction are monic.
\item[\rm (2)] The right adjoint $G$ of $1 : {\mathbf{B}} \to {\mathsf{Gl}}(F)$ inverts
cocartesian arrows and $\alpha$ is monic whenever $\varepsilon_X =
\alpha \circ \varphi$ with $\alpha$ vertical and $\varphi$ cocartesian.
\item[\rm (3)]
The fibred functor $\Gamma$ preserves cocartesian arrows and all counits
$\widetilde{\varepsilon}_X : \Delta\Gamma X \to X$ are vertical monos.
\end{enumerate}
\end{Thm}
For geometric morphisms $\Delta \dashv \Gamma : {\mathbf{E}} \to {\mathbf{Set}}$ the counit
$\varepsilon_X : \Delta\Gamma X \to X$ at $X \in {\mathbf{E}}$ is monic
iff distinct global elements of $X$ are disjoint. The above Th.~\ref{hypconn1}
generalizes this condition to arbitrary bases.
But there is a different characterization of hyperconnected geometric
morphisms between elementary toposes whose fibrational analogue might
appear as more intuitive.\footnote{For the various characterizations of
hyperconnected geometric morphisms between toposes see A.4.6 of \cite{Ele}.
One of them is that $U$ preserves subobject classifiers. Another one that
$F$ is full and faithful and the image of $F$ is closed under subquotients.}
\begin{Thm}\label{hypconn2}
Let $F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ be a geometric morphism and $P$ be the induced
geometric fibration ${\mathsf{gl}}(F)$. Then the following conditions are equivalent.
\begin{enumerate}
\item[\rm (1)]
The geometric morphism $F \dashv U$ is hyperconnected, i.e.\ connected and
for all $I \in {\mathbf{B}}$ the functor $F_{/I} : {\mathbf{B}}/I \to {\mathbf{C}}/FI$
restricts to an equivalence between ${\mathrm{Sub}}_{\mathbf{B}}(I)$ and ${\mathrm{Sub}}_{\mathbf{C}}(FI)$.
\item[\rm (2)]
The right adjoint $G$ of $1 : {\mathbf{B}} \to {\mathsf{Gl}}(F)$ inverts cocartesian arrows
and every vertical subobject $m$ of $1_I$ is isomorphic to $\Delta_I(G(m))$.
\item[\rm (3)]
The fibred functor $\Gamma$ preserves cocartesian arrows and $\Delta :
P_{\mathbf{B}} \to {\mathsf{gl}}(F)$ restricts to a fibred equivalence between ${\mathrm{Sub}}_{\mathbf{B}}$
and $F^*{\mathrm{Sub}}_{\mathbf{C}}$ considered as subfibrations of $P_{\mathbf{B}}$ and ${\mathsf{gl}}(F)$,
respectively.
\end{enumerate}
\end{Thm}
The intuitive meaning of (3) in the above theorem is that the fibred
adjunction $\Delta \dashv \Gamma$ between $P_{\mathbf{B}}$ and ${\mathsf{gl}}(F)$ restricts
to a fibred equivalence between their subterminal parts.
Of course, the conditions (3) of theorems \ref{hypconn1} and \ref{hypconn2},
respectively, are in general not equivalent for geometric fibrations
where ${\mathbf{B}}$ is not a topos and ${\mathsf{gl}}(F)$ is not a fibration of toposes
(i.e.\ ${\mathbf{C}}$ is not a topos).
\begin{Thm}\label{gmprop4}
Let $F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ be a geometric morphism and $P$ be the induced
geometric fibration ${\mathsf{gl}}(F)$. Then the following conditions are equivalent.
\begin{enumerate}
\item[\rm (1)]
The geometric morphism $F \dashv U$ is \emph{local},
i.e.\ $F$ is full and faithful and $U$ has a right adjoint.
\item[\rm (2)]
The fibred functor $\Gamma$ has a fibred right adjoint $\nabla$.
\end{enumerate}
\end{Thm}
\begin{proof}
First we show that (2) implies (1). If $\Gamma$ has a fibred right adjoint
$\nabla$ then $\Gamma$ preserves cocartesian arrows as it is a fibred left
adjoint. Thus, by the previous Theorem~\ref{gmprop3} it follows that $F$
is full and faithful. As $\Gamma_1 \dashv \nabla_1$ and $U \cong \Gamma_1$
it follows that $U$ has a right adjoint.
Now we show that (1) implies (2). If $F$ is full and faithful then the
unit $\eta : {\mathit{Id}}_{\mathbf{B}} \to UF$ is an isomorphism. Therefore, the fibred functor
$\Gamma$ acts on objects and morphisms simply by applying the functor $U$
and then postcomposing with the inverse of $\eta$,
i.e.\ $\Gamma(a) = \eta_I^{-1} \circ U(a)$
for $a : A \to FI$ in ${\mathbf{C}}{\downarrow}F$. Then $\Gamma$ has a fibred right
adjoint $\nabla$ with $\nabla(v) = \zeta_J^*R(v)$ for $v : K \to J$
in ${\mathbf{C}}{\downarrow}{\mathbf{C}}$ where $\zeta_J : FJ \to RJ$ is the transpose (w.r.t.\
$U \dashv R$) of $\eta^{-1}_J : UFJ \to J$ as follows from the natural
1-1-correspondence between
\begin{diagram}[small]
UA & & \rTo^w & & K & & A & & \rTo^{\check{w}} & & RK \\
\dTo^{Ua} & & & & \dTo_v & \qquad\mbox{and}\qquad & \dTo^{a} & & & &
\dTo_{Rv} \\
UFI & \rTo_{\eta^{-1}_I} & I & \rTo_u & J & & FI & \rTo_{Fu} & FJ &
\rTo_{\zeta_J}& RJ \\
\end{diagram}
exploiting the fact that the transpose of
$u \circ \eta^{-1}_I = \eta^{-1}_J \circ UFu$ is $\zeta_J \circ Fu$.
\end{proof}
\begin{Thm}\label{gmprop5}
Let $F \dashv U : {\mathbf{C}} \to {\mathbf{B}}$ be a geometric morphism and $P$ be the induced
geometric fibration ${\mathsf{gl}}(F)$. Then the following conditions are equivalent.
\begin{enumerate}
\item[\rm (1)]
The geometric morphism $F \dashv U$ is \emph{locally connected},
i.e.\ $F$ has a left adjoint $L$ such that
\begin{diagram}[small]
A \SEpbk & \rTo^{f} & B & & & & LA \SEpbk & \rTo^{Lf} & LB \\
\dTo^{a} & & \dTo_{b} &&\mbox{implies}&& \dTo^{\hat{a}} &&\dTo_{\hat{b}}\\
FI& \rTo_{Fu} & FJ & & & & I & \rTo_{u} & J \\
\end{diagram}
where $\hat{a}$ and $\hat{b}$ are the upper transposes of $a$ and $b$,
respectively.
\item[\rm (2)]
The fibred functor $\Delta$ has a fibred left adjoint $\Pi$.
\end{enumerate}
\end{Thm}
\begin{proof}
If $L \dashv F$ then $\Delta$ has an ordinary left adjoint $\Pi_L$ sending
\begin{diagram}[small]
A & \rTo^{f} & B & && & LA & \rTo^{Lf} & LB \\
\dTo^{a} & & \dTo_{b} &&\mbox{to}&& \dTo^{\hat{a}} &&\dTo_{\hat{b}}\\
FI& \rTo_{Fu} & FJ & & & & I & \rTo_{u} & J \\
\end{diagram}
and satisfying $P_{{\mathbf{B}}} \circ \Pi_L = {\mathsf{gl}}(F)$. Obviously, this functor
$\Pi_L$ is cartesian iff $L$ satisfies the requirement of condition (1).
Thus, condition (1) entails condition (2).
On the other hand if $\Delta$ has a fibred left adjoint $\Pi$ then
$F \cong \Delta_1$ has an ordinary left adjoint $L \cong \Pi_1$ and as
$\Pi \cong \Pi_L$ in the 2-category ${\mathbf{Cat}}{\downarrow}{\mathbf{B}}$ with vertical natural
transformations as 2-cells (because both functors are left adjoints to $\Delta$
in this 2-category) it follows that $\Pi_L$ is also cartesian and, therefore,
the functor $L$ satisfies the requirement of condition (1).
Thus, condition (2) entails condition (1).
\end{proof}
\bigskip
Moreover, it follows from the fibred version of the Special Adjoint Functor
Theorem\footnote{which applies as $P_{\mathbf{S}}$ has a small generating family and,
therefore, also a small cogenerating family (as shown by R.~Par\'e and
D.~Schumacher)} that a geometric morphism $F \dashv U : {\mathbf{E}} \to {\mathbf{S}}$ between
toposes is locally connected if and only if $F$ preserves the locally
cartesian closed structure as $\Delta : P_{\mathbf{S}} \to {\mathsf{gl}}(F)$ preserves
internal limits iff $F$ preserves finite limits and dependent products.
Recall that a geometric morphism $F \dashv U : {\mathbf{E}} \to {\mathbf{S}}$ between toposes
is called \emph{atomic} iff $F : {\mathbf{S}} \to {\mathbf{E}}$ is logical. Thus, by the previous
remark atomic geometric morphisms between toposes are locally connected.
Atomic geometric morphisms can be characterised as those locally connected
geometric morphisms $F \dashv U : {\mathbf{E}}\to{\mathbf{S}}$ where all monomorphisms
$m$ in ${\mathbf{E}}$ are ${\mathbf{S}}$--definable, i.e.\ satisfy
\begin{diagram}[small]
X \SEpbk & \rTo^{\eta_X} & F L X \\
\dEmbed^{m} & & \dTo_{F L m} \\
Y & \rTo_{\eta_Y} & F L Y \\
\end{diagram}
where $\eta$ is the unit of $L \dashv F$. This can be seen as follows. Recall
that for a locally connected geometric morphism $F \dashv U : {\mathbf{E}}\to{\mathbf{S}}$
the monomorphism $F(\top_{\mathbf{S}})$ classifies ${\mathbf{S}}$--definable monomorphisms.
Now if $F$ is logical then $F(\top_{\mathbf{S}})$ is a subobject classifier in ${\mathbf{E}}$
and, therefore, all monomorphisms in ${\mathbf{E}}$ are ${\mathbf{S}}$--definable. On the other
hand, if all monomorphisms in ${\mathbf{E}}$ are ${\mathbf{S}}$--definable then $F(\top_{\mathbf{S}})$
is a subobject classifier (as it classifies all ${\mathbf{S}}$--definable monomorphisms)
and thus $F$ is logical.
\newpage
|
1,108,101,565,637 | arxiv | \section*{APPENDIX}
\bibliographystyle{./IEEEtranBST/IEEEtran}
\section{Methods}
\label{sec:methods}
Our proposed shape estimation method is a learned multi-view shape estimation from grayscale images.
First, different views of the desired object are captured by two cameras.
Then, unnecessary information in the images is removed by an image processing pipeline that transforms them into binary images.
The processed images are then fed into a convolutional neural network (CNN) trained to estimate the parameters of the shape representation.
The approach is outlined in \Cref{fig:vise_pipeline} and the following subsections detail the sub-components of our method.
\subsection{Image Preprocessing}
\label{sec:image_preprocessing}
The RGB images of the shape are preprocessed to facilitate the learning procedure.
The original images are converted to grayscale, cropped around the shape, and scaled to the size of $256\times256$ pixels.
This preprocessing preserves enough information to accurately represent the shape, while keeping the number of parameters of the CNN relatively small.
A median filter with a $7\times7$ pixel kernel size is applied to reduce noise before using adaptive thresholding to reduce the grayscale image to a binary image\cite{otsu1979threshold}.
This step removes the background variations while preserving the shape.
In addition, the adaptive nature of the thresholding operation and the resulting binary images make our trained network function in a wide range of lighting conditions without the need for retraining.
Erosion and dilation with a $7\times7$ pixel kernel are applied for three iterations each to remove the remaining artifacts.
\subsection{Network Architecture}
\label{sec:network_architecture}
\begin{figure}[!t]
\centering
\includegraphics[width=3.3in]{figures/fig5.png}
\caption{CNN Network Architecture used in ViSE. Inputs are the preprocessed images from both cameras, and output sizes depend on the selected shape representation and robot.}
\label{fig:network_arch}
\end{figure}
The CNN used in this work is adapted from the VGG architecture\cite{simonyan2014very}. Since our approach uses binary images, we can take advantage of the simplicity and versatility of VGG. To reduce the computational demand further and improve real-time performance, several convolutional layers are removed from the standard VGG network.
The final soft-max layer is also removed to perform a regression instead of a classification.
The architecture of our network is illustrated in \Cref{fig:network_arch}.
The network's main elements are convolutional layers, batch normalization, rectification (ReLU) nonlinearities, and max pooling operations.
These elements are applied in the mentioned order and repeated five times before the output is fed into two fully connected layers.
All five convolutional layers have a kernel of size three, a stride of one, and a padding of one.
The number of channel dimensions is increased from $2$ to $6$, $16$, $32$, $64$, and $128$.
Batch normalization is applied before each convolutional layer.
Every max-pooling operation reduces its input by a factor of two, reducing the initial image size of $256\times256$ pixels to $8\times8$ pixels after five operations.
Hence, the input to the first fully connected layer is of size $8'192$ ($8\times8\times128$).
A ReLU nonlinearity is applied after the first fully connected layer.
The input to the last fully connected layer has a size of $1'000$, which is reduced to the output of size of either $6$, $9$, $12$, or $18$, depending on which shape representation and data set are used.
\subsection{Shape Representations}
\label{sec:ground_truth}
We consider two different shape representations, the point, and piecewise constant curvature (PCC) model.
The point representation comprises the positions of the key points along the soft robots.
Using the motion capture software (\textit{Qualisys Track Manager}), virtual coordinate frames are placed at the center of each group of motion markers to line up with the corresponding segment's centroid.
These coordinate frames allow the tracking of not only each segment’s translation but also orientation.
We then choose to fit the PCC model\cite{webster2010design, della2020improved}, a commonly used kinematic reduction model in soft robotics, to these virtual coordinate frames. This PCC model allows the modeling of a continuous shape by approximating it with multiple constant-curvature sections of fixed length.
Each section is defined by two parameters, the curvature and an angle indicating the curving direction.
Hence, the model only requires a few parameters to represent a long, continuously deformable shape, which is useful for model-based control purposes.
\subsection{Camera Realignment}
\label{sec:camera_realignment}
No explicit camera calibration is needed and the camera configuration is implicitly learned by the CNN. This design choice limits the cameras' positions to be fixed relative to the soft robot during the data collection.
To alleviate the need for retraining, fiducial markers (\textit{AprilTags})\cite{olson2011apriltag} are attached to the robot's base.
The camera's translation and rotation relative to the base can be extracted from the image of the fiducial markers.
Our realignment utility for the camera pose compares the camera's current and previously saved positions relative to the fiducial markers. With this utility, users can set up the RGB cameras close to the configuration they were set up during data collection and reuse the trained CNN repeatedly.
While this supervised approach still requires a motion capture setup to initially collect the ground truth data for training, the cameras can be moved without requiring retraining. A user can realign their cameras to approximately match the original poses relative to the robot's base and still perform inference using the original training data. Given our realignment utility, the trained model still successfully estimates the shape of the robot without requiring a motion capture system.
\section{Real-world Experiments}
\label{sec:results}
\subsection{Experimental Setup}
\label{sec:experiment_setup}
\subsubsection{Soft Robots for Evaluation}
\label{sec:soft_robots}
\begin{figure}[th]
\centering
\includegraphics[width=\columnwidth]{figures/soft_robots.png}
\caption{Soft robots used for performance evaluation, in each panel, left shows the original RGB image, right shows the preprocessed image. A. \textit{WaxCast} Arm~\cite{katzschmann2019dynamic}. B. \textit{WaxCast} arm with visual features (black stripes)~\cite{katzschmann2019dynamic}. C. \textit{SoPrA} arm~\cite{toshimitsu2021sopra}. D. Soft fish~\cite{zhang2022creation}.}
\label{fig:soft_robots}
\end{figure}
The approach is tested on two types of soft robotic arms\cite{katzschmann2019dynamic, toshimitsu2021sopra, https://doi.org/10.1002/aisy.202200024} and a soft robotic fish\cite{zhang2021learning, zhang2022creation} (Fig.~\ref{fig:soft_robots}).
The first soft robotic arm (which we shall refer to as the \textit{WaxCast} arm) consists of three axially connected cylindrical segments, each with four separately inflatable chambers.
They are inflated using air provided through a pressure-controlled valve array (\textit{Festo SE \& Co. KG}). By inflating one side, the chambers on that side elongate and induce bending in the segment, thus, the bending direction of the arm can be chosen by selecting the corresponding combination of inflation chambers.
Each segment has a length of about $110\pm1$ \si{mm} and a diameter of $40\pm1$ \si{mm}.
The combined length of the arm is $335\pm3$ \si{mm}. The second arm, \textit{SoPrA}, is a two-segment soft robotic arm with fiber-reinforced pneumatic actuators.
Segments are made of three individually fiber-reinforced elastomer air chambers that are glued together. Combining two of these segments adds up to a total length of $268\pm2$ \si{mm}.
The robotic fish tail is similar in construction and actuation compared to the \textit{WaxCast} arm, except that it is shaped like a fishtail.
It consists of two inflatable chambers and has a total length of $115\pm1$ \si{mm}.
\subsubsection{Data Collection}
\label{sec:data_collection}
The ground truth data for learning is obtained by eight \textit{Miqus M3} motion-capture cameras from \textit{Qualisys AB} placed in the motion capture space of $1.6\times1.1\times0.8$ \si{m}.
The placement of the motion-capture markers can be seen in \Cref{fig:soft_robots}.
A group of reflective markers is placed on a rigid ring at the end of every segment of the soft arms.
Along the soft fish's tail, the markers are placed with spaces of $38\pm1$ \si{mm} between them.
Marker position data is supplied at \SI{100}{Hz} with an average accuracy of $0.1\pm0.2$ \si{mm}, while RGB image data is recorded at \SI{25}{Hz} by two depth cameras (\textit{Intel RealSense D435i})\cite{keselman2017intel}.
During the data acquisition, the robot’s workspaces are traversed as fully as possible.
The segments of the \textit{WaxCast} arm are all actuated to perform a circular motion, with periods of 100, 10, and 1 second, for segments 1, 2, and 3, respectively.
We created two data sets, one with three motion-capture marker rings on the arm and the other with six, which also contain visual features in the form of black stripes that were put on the arm (\Cref{fig:soft_robots}B).
This process was repeated for \textit{SoPrA}, but the chambers of the arm were randomly actuated.
In total, three labeled data sets are generated for two arms, each containing 12'000 poses.
The soft robotic fish is actuated to perform a tail-fin stroke with maximal deflection, resulting in a data set of 1'800 poses.
Each data set is split into 90\% training and 10\% testing sets.
\subsubsection{Network training}
\label{sec:network_training}
The network is implemented and trained using the \textit{PyTorch} framework.
\textit{AdamW} is chosen as an optimizer and used to minimize the mean absolute loss.
The network is trained using a batch size of 64 with an early stop for a maximum of 450 epochs on each data set.
The learning rate is set to $10^{-4}$ and reduced by 0.5 after each 200th epoch.
Dropout is applied with a probability of $0.5$ in the fully connected layers during training to avoid overfitting.
Training on a GPU (Nvidia GeForce RTX 3090) requires between 30 to 60 minutes to converge.
\subsection{Results}
\label{sec:results_subsec}
\begin{table}[ht]
\centering
\caption{
Estimation errors of the piecewise constant curvature (PCC) and point estimation approaches.
}
\input{tables/table-performance}
\label{tab:performance}
\end{table}
\begin{figure*}[!t]
\centering
\includegraphics[width=2\columnwidth]{figures/evaluation_result_2col.png}
\vspace{-5pt}
\caption{Estimation results of ViSE compared to ground truth positions. Exp. A, and B employ piecewise-constant curvature (PCC) model, while Exp. C-E estimate the positions of characteristic points separately. The number of sections considered in each experiment is shown in the figure. The red dots mark the ground truth positions obtained by the motion capture system and the blue dots mark the position estimated by ViSE.}
\label{fig:result}
\end{figure*}
\begin{table*}[ht!]
\centering
\caption{
Estimation errors of our approach compared to other works.
}
\input{tables/table-comparison}
\label{tab:comparison}
\end{table*}
\begin{figure*}[tb]
\vspace{5pt}
\centering
\includegraphics[width=2\columnwidth]{figures/brightness_plot_with_img.png}
\vspace{-8pt}
\caption{Tip estimation errors with varying image brightness. The brightness modification is quantified by the addition or subtraction of pixel values ($0-255$) from original grayscale images. Sample grayscale and binary images are shown at different brightness levels.}
\label{fig:lighting}
\end{figure*}
\subsubsection{Shape Representations}
\label{sec:shape_representations}
The CNN was trained using the image data and either learned to output parameters of a PCC model that was fitted to the ground truth marker data or virtual marker positions along the arm (point estimation).
We also analyze the approach’s accuracy when estimating just three PCC sections or virtual points compared to estimating six PCC sections or points.
Both PCC and point estimation approaches were tested using the data sets from our two \textit{WaxCast} arms (\Cref{fig:result}A-D).
Detailed results of the evaluation can be seen in \Cref{tab:performance} Exp. A-D, with the point estimation approach strictly outperforming the PCC approach.
The errors are normalized based on the robot’s length, which is 335 mm $\pm$ 3 mm for the soft arm.
\subsubsection{Visual Features}
\label{sec:visual_features}
To evaluate the effect of visual features on the estimation accuracy, the point estimation approach is applied to a dataset with features (\Cref{fig:result}E).
To create these features, we modified the \textit{WaxCast} arm’s appearance to have multiple black stripes perpendicular to the arm’s backbone (\Cref{fig:soft_robots}B).
In \Cref{tab:performance}, Exp. D and E show that the mean tip error for the feature-less \textit{WaxCast} arm is $3.6\pm5.0$\% and only $0.3\pm0.2$\% for the arm with features.
\subsubsection{Benchmarks}
\label{sec:benchmarks}
We compared the results of our point estimation approach with four similar works on soft robotic state estimation (see \Cref{tab:comparison})\cite{camarillo2008vision,vandini2017unified,pedari2019spatial,albeladi2021vision}.
Those works also estimated reference points along continuously deforming shapes.
Due to limited data from the benchmarks, we only compared the tip errors, which are usually the largest.
We believe that achieving low tip position reconstruction error is important for soft robotic shape estimation methods since this accuracy is critical in real-world operations involving reaching and grasping of objects.
For a fair comparison, the errors are normalized with the length of each corresponding robot.
\textit{DeepLabCut}\cite{mathis2018deeplabcut} is not included in \Cref{tab:comparison} because it estimates pixel locations instead of 3D positions.
To compare our results with \textit{DeepLabCut}, we projected the estimated and ground truth 3D positions into the input images and evaluated the pixel distance error.
Exp. C in \Cref{tab:performance} showed a root-mean-square error at the tip position of 1.13 pixels in one camera view and 1.18 pixels in the other. In comparison, \textit{DeepLabCut} achieved an accuracy similar to the human labeling error of $2.69\pm0.1$ pixels.
However, comparing pixel errors is only of limited value, since a pixel error can have a different significance depending on the image resolution and scale of the captured object.
Moreover, reprojecting the estimated pixels from multiple calibrated cameras back into 3D space may bring in additional errors due to camera calibrations.
Therefore we believe that a direct 3D position estimation is more useful and convenient for downstream applications.
\subsubsection{Maximum Estimation Errors}
\label{sec:max_estimation_errors}
The maximum estimation errors are computed as an indication of the “worst case scenarios”.
For \textit{WaxCast} arm with features, the maximum tip error is 2.6\%, for \textit{SoPrA}, it is 4.3\%, and for the soft robotic fish, it is 9.5\%.
\subsubsection{Online Estimation}
\label{sec:online_estimation}
The real-time estimation performance of ViSE was tested on a portable computer (Intel Core i7-7500U @ 2.70 GHz).
A mean estimation rate of \SI{18.4}{Hz} was achieved on this system.
Compared to human visual perception time of around \SIrange{150}{200}{\milli\second}\cite{amano2006estimation}, a single estimation update of our approach takes \SI{54}{ms} on average, of which 60\% are used for the CNN forward calculation and 38\% are used for image processing.
The remaining 2\% are used to stream the images from the RGB cameras.
\subsubsection{Different Experimental Setups}
\label{sec:diff_exp_setups}
To demonstrate the robust performance of our trained CNN in a range of lighting conditions without the need of retraining, we evaluate and present the tip estimation errors on the \textit{SoPrA} testing data set with modified brightness levels (see \Cref{fig:lighting}).
The brightness of the original images is modified by adding or subtracting pixel values from the grayscale images (pixel value range 0-255).
The experiments are conducted on the \textit{SoPrA} data set since the gray color of the \textit{SoPrA} arm is the closest to the black background.
\textit{SoPrA} provides the least contrast compared to the other soft robots and is, therefore, more sensitive to changes in brightness.
The performance of the trained network is also tested after the reassembly of the cameras.
With relative camera translations and orientations obtained from the fiducial markers, we manually realign the cameras to a configuration with \SI{1.46}{mm} difference from the one used during data collection.
The new tip estimation error after reassembly of the cameras is $2.2\pm2.4$\% for the \textit{SoPrA} test data set.
\subsection{Discussion}
\label{sec:discussion}
The results show that the estimation errors increase along the shape regardless of the data set or shape representation being used.
The reason for this increase is most likely due to the fact that the tips of these shapes typically move faster and across a larger space than the rest of their shapes. A dynamic behavior increases the estimation difficulty towards the tip.
The approach using the PCC shape representation as output produces larger estimation errors on the three tested data sets (\Cref{tab:performance}).
This error is partially because the endpoint positions are computed using forward kinematics calculation with all previous PCC sections, which accumulates the estimation errors of each section.
Another reason for the inferior performance of the PCC approach is that the constant curvature assumption is sometimes inaccurate for a real soft robotic system.
For example, the sections of the soft arm do not exactly bend with constant curvature.
The arm's weight and dynamics, the design characteristics of the inflation chambers, and the fabrication errors all introduce imperfections with regard to the constant curvature assumption.
Moreover, the arms we use in this work do not contain an inextensible backbone and therefore also extend along their center line under inflation.
This limitation could be resolved by augmenting the PCC model to allow for constant curvature sections of variable length.
The CNN would then need to be adjusted to also estimate the length of each segment.
However, the error due to non-constant curvature deformation would remain.
By estimating points separately, we could avoid both the error accumulation and the PCC model limitations.
Although the point representation does not contain any statements about connectedness or directionality, it gives a more precise estimation of the tip position.
The visual features added to the \textit{WaxCast} arm greatly improved the estimation accuracy.
This can be seen when comparing Experiments D and E in \Cref{tab:performance}.
We believe that the added features helped the CNN extract more information from the input images.
The increased information content improved the deduction of the shape parameters.
We outperform the benchmarks for both soft arms and achieve slight performance improvements for the soft fish, as shown in \Cref{tab:comparison}.
At the same time, our approach also does not require flawless shape segmentation or prior shape knowledge, suggesting the possible generalizability to different types of soft robots.
One limitation of using convolutional neural networks is that the trained network may not be reusable and needs retraining when the experimental setup changes.
We tested our trained network on input images with various levels of brightness and also after reassembly of the cameras.
The stable performance under brightness changes (\Cref{fig:lighting}) indicates that the network could work with a wide range of lighting conditions without re-training as long as there is sufficient contrast for the adaptive image preprocessing.
Although retraining would be needed for different camera configurations, we show with the aid of fiducial markers (\textit{AprilTags}), rough realignment to previous camera positions is possible and the trained network can be reused.
\section{Introduction}
\label{sec:intro}
Soft robots are experiencing a steep rise in popularity thanks to their ability to solve challenges such as compliant grasping and dexterous movement\cite{yamanaka2020development, wang2020dual, abondance2020dexterous}, tasks with which rigid robots typically struggle\cite{hawkes2021hard}.
To fully exploit the capabilities of soft robots, modern control approaches are needed, which typically rely on rich state feedback.
However, obtaining and accurately describing the state of a continuously deforming soft body or robot is challenging compared to the state of a rigid object or robot.
Encoders at the connecting joints of rigid robots readily provide precise state measurements, while soft robots mostly consist of elastomeric materials that deform with infinite degrees of freedom\cite{yasa_overview_2022}. It is, therefore, crucial to solve the challenge of soft robotic state estimation to exploit the full potential of the great variety of soft robots for manipulation and beyond.
To date, various shape estimation approaches have attempted to improve soft robotic sensing capabilities.
One type of sensing approach uses mechanical proprioception similar to classical robotic state estimation\cite{totaro2017integrated, dawood2019modelling, park2019multi, wang2020mechanoreception}.
A variety of sensor types such as resistive, capacitive, optical, and pneumatic transducers proprioceptively estimate the continuous deformations of soft robots\cite{wang2018toward, navarro2020model, hofer2021vision}.
Mechanical proprioception with built-in sensors is limited by spatial resolution and increased fabrication complexity.
This limitation has lately led to an increased popularity of exteroceptive sensing approaches that are purely vision-based\cite{bern2020soft,albeladi2021vision, 9381634}.
\begin{figure}[t]
\centering
\includegraphics[width=3in]{figures/fig1.png}
\caption{Diagram of the real-time marker-less inference pipeline for the proposed shape estimation approach ViSE. A 3D shape is captured by two RGB video cameras. Image pairs are preprocessed and run through a convolutional neural network to estimate a shape model for the 3D shape.}
\label{fig:vise_pipeline}
\end{figure}
A widely applied vision-based method for shape estimation and tracking employs motion-capture systems\cite{duriez2013control, marchese2014design,katzschmann2019dynamic, bern2020soft}.
These systems rely on placing reflective markers on an object and triangulating the marker position using multiple calibrated motion-tracking cameras.
Such marker-based approaches offer high temporal and spatial resolutions of the placed markers.
However, these tracking systems are usually costly and limited in use, hindering the application of marker-based robots in commercial settings where multiple robots are deployed simultaneously within clutter\cite{patrizi2016comparison}.
Motion-capture systems require the tedious placement of many markers on any object to be tracked, which is not a feasible option when working with high-dimensional robotic systems that interact with an environment.
Markers cannot be placed too close to each other, limiting the fidelity of shape reconstruction.
Furthermore, obstacles could occlude and eventually displace or destroy the markers.
Therefore, much research has gone into replacing marker-based approaches by developing marker-less sensing techniques\cite{hannan2005real, camarillo2008vision, croom2010visual, ceseracciu2014comparison, vandini2017unified, albeladi2021vision, hofer2021vision}.
Current marker-less approaches still require scenario-specific capturing setups and strict formatting requirements of the image data.
In this work, we choose an approach towards 3D shape estimation of continuously deformable robots that leverages data-driven deep learning.
We propose ViSE, a {V}ision-based regression approach for camera-based, 3D {S}hape {E}stimation using a convolutional neural network (CNN)~(\Cref{fig:vise_pipeline}).
The performance of our real-time shape estimation is compared against marker-based motion capture in the tasks of estimating the shape of soft robotic arms and soft robotic fish.
Specifically, our work provides the following contributions:
\begin{itemize}
\item We present a simple-to-implement approach using two cameras and applicable to various soft robots.
\item We employ an adaptive image processing pipeline that does not require specific lighting conditions or flawless background removal.
\item We demonstrate the real-time estimation capability of our system, which enables its use for downstream applications such as the closed-loop control of soft robots.
\end{itemize}
\Cref{sec:related} summarizes related work and \Cref{sec:methods} presents our methodology. In \Cref{sec:results}, we provide evaluation results to exhibit the performance of the proposed CNN-based approach and discuss the estimation accuracy under different model assumptions. Finally, \Cref{sec:conclusions} concludes the work and outlines directions for future research.
\section{Related Works}
\label{sec:related}
Previous works demonstrate vision-based continuous shape estimation approaches that either necessitate strict image requirements or work only for specific setups.
In the following, we briefly discuss the most related works, focusing on marker-less shape estimation approaches.
Hannan and Walker use basic image processing techniques, including thresholding and image segmentation, to estimate the 2D shape of a planar, cable-actuated elephant trunk manipulator from single images\cite{hannan2005real}.
However, their estimation results are only compared to another cable-based shape estimation technique but not with actual ground truth.
Camarillo et al. extend the computer-vision methods to 3D shape estimation of a thin continuum manipulator\cite{camarillo2008vision}.
If the precise positions of cameras are known, they could extract silhouettes from multiple cameras' views, project those silhouettes into a volumetric space to find their intersection, and fit a spline through the resulting 3D point cloud. This approach requires a strong contrast between the tracked shape and the background, as well as the absence of other objects in the field of view.
Strict requirements on the image data are also found in other works.
AlBeladi et al. rely on successful edge detection of their soft arm to fit a geometric strain-based model to these edges\cite{albeladi2021vision}.
Croom et al. also perform edge detection, but then fit reference points to the edges by using an unsupervised learning algorithm called the self-organizing map\cite{croom2010visual}.
All of these approaches show good estimation results but require a strong contrast between the tracked object and the background.
Vandini et al. extract and join straight lines from a monoplane fluoroscopic surgical image to estimate the shape of a concentric tube robot\cite{vandini2017unified}. By posing conditions for connecting the line features, they manage to relax image requirements and can extract curves from more unclean image data compared to the aforementioned works\cite{vandini2017robust}.
Reiter et al.\cite{reiter2012learning} take on a similar approach to ours in that they extract features from segmented binary stereo-images.
Since their feature extraction pipeline relies on the color-coded segments of their continuum robot, it does not generalize to other types of robots that do not have those features.
Mathis et al. created a deep learning framework based on transfer learning for marker-less pose estimation and tracking called DeepLabCut\cite{mathis2018deeplabcut}.
Their framework enables tracking of multiple visual features in unprocessed videos using only a small number of labeled frames for training.
They demonstrate their method by tracking body parts of mice and show that they achieve pixel tracking errors comparable to human-level labeling accuracy.
However, this framework by itself is restricted to pixel tracking in an image, and it cannot directly track 3D coordinates of features.
Our approach for 3D shape estimation of continuously deformable robots employs convolutional neural networks.
While there are many vision-based proprioceptive methods for soft robots using deep learning\cite{werner2019vision, wang2020real, she2020exoskeleton}, we focus on exteroceptive approaches that are simple to implement and do not add complexity to the manufacturing of the soft robots.
\section{Conclusions and Future Work}
\label{sec:conclusions}
ViSE is a vision-based, 3D soft robot shape estimation approach using two cameras and a CNN.
It outperforms current marker-less shape estimation approaches when evaluated on two soft robotic arms and one soft robotic fish.
While we consider the visual robustness of our approach to be an improvement over the state-of-the-art, it could be further enhanced to be calibration-free, deal with occlusions, and allow for more expressive representations.
Future work will introduce artificial occlusions in the network's training process to work with partially occluded images and also employ learning-based shape segmentation to perform robust background removal under insufficient contrast. Another future direction is to generalize the approach to the estimation of more expressive kinds of shape representations, \textit{e.g.}, mesh reconstructions, instead of being limited to the estimation of piecewise constant curvatures or characteristic points.
|
1,108,101,565,638 | arxiv | \subsubsection{The Model}
As mentioned earlier, several candidates for pairing of unequal Fermi surfaces have been proposed in the literature. The Sarma phase~\cite{Sarma} and the prior version of the BP phase~\cite{Wilczek1}, are unstable~\cite{Heron1,Heron2,Wilczek3}, and will not be considered. Regarding the LOFF state, since this phase can exist only in a very narrow window of asymmetry for the Fermi surfaces, we do not consider this realization of superfluidity in the present calculations. The investigation of the critical temperature in the MP (with fixed particle densities)~\cite{Heron1,Heron2}, and in the clarified version of the BP phase\footnote{This version takes into account the momentum structure of the interaction and large mass asymmetry.} (with fixed chemical potentials)~\cite{Wilczek3}, must be considered and will be discussed elsewhere.
We employ the Bardeen-Cooper-Schrieffer (BCS) model to derive the Ginzburg-Landau (GL) theory, following the modern formulation developed by Sakita \cite{Sakita}. For the sake of completeness, we also derive the gap equation for an asymmetrical fermionic system at finite temperature by the variational method. The derivation is shown in the appendix, and the results agree, as it should.
We consider an asymmetrical nonrelativistic dilute system of $a$ and $b$ fermion species\footnote{In (relativistic) quark matter, the study of the critical temperature from the point of view of the GL approach, has been carried out in Refs.~\cite{Iida1,Iida2}.}, having masses $m_a$ and $m_b$, with Fermi momentum $P_F^{j}=\sqrt{2m_j \mu_j}$, $j=a, b$.
Let us begin with the partition function
\begin{equation}
\label{gf1}
Z=\int D[\psi_{a,b}] D[\psi_{a,b}^\dagger] e^{[- S(\psi, \psi^\dagger)]},
\end{equation}
where $S(\psi, \psi^\dagger)=-\int d \tau \int dx L$, with $L$ being the BCS Lagrangian
\begin{equation}
\label{BCS1}
L= \sum_{i=a,b} \psi_i^\dagger(x) \left(-\partial_{\tau} + \frac{\bigtriangledown^2}{2 m_i} + \mu_i \right) \psi_i(x) +
g \psi_{a}^\dagger(x) \psi_{b}^\dagger(x) \psi_{a}(x) \psi_{b}(x),
\end{equation}
where $g>0$. Introducing auxiliary fields via the Hubbard-Stratonovich transformation, we find an effective action, in which the mean field BCS Lagrangian is expressed by
\begin{eqnarray}
\label{BCS2}
L_{MF}= \sum_{i=a,b} \psi_i^\dagger(x) \left(- \partial_{\tau} + \frac{\bigtriangledown^2}{2 m_i} + \mu_i \right) \psi_i(x)
+ \\
\nonumber
\Delta(x) \psi_{a}^\dagger(x) \psi_{b}^\dagger(x) + \Delta^*(x) \psi_{b}(x) \psi_{a}(x) + \frac{|\Delta(x)|^2}{g} .
\end{eqnarray}
Introducing a source for the $\Delta(x)$ field, we write
\begin{equation}
\label{gf3}
Z = {\cal N} \int D[\psi] D[ \psi^\dagger] D[\Delta] D[\Delta^{\dagger}]~ e^{\int d \tau \int dx \left[ L_{MF} +j^* \Delta(x) + j \Delta^*(x) \right]}
\end{equation}
The partition function is $Z=Z[j, j^*]_{j=j^*=0}$ and the generating functional for the connected Green's functions is defined as
\begin{equation}
\label{gf}
W[j, j^*]=\ln Z[j, j^*].
\end{equation}
The Legendre transformation of $W[j, j^*]$ is
\begin{equation}
\label{g}
\Gamma[\Delta, \Delta^*]=\int d^4x (j^* \Delta(x) + j \Delta^*(x))- W[j, j^*].
\end{equation}
From all one-loop diagrams contributing to $\Gamma[\Delta, \Delta^*]$, we evaluate only the one which gives a contribution for the
$\Delta^2$ term. It is shown in Fig.~(\ref{Mixed}).
\subsubsection{The one loop correction to the BCS gap parameter}
The effective action up to one-loop is
\begin{equation}
\label{ea}
\Gamma[\Delta,\Delta^*]=\alpha |\Delta|^2 +\beta |\Delta|^4 -c \Delta^* \frac{1}{8 M} {\vec{\nabla}}^2 \Delta,
\end{equation}
where $M=\frac{m_a m_b}{m_a + m_b}$ is the reduced mass, $\alpha=\frac{1}{g}-A$, and $A$ is the momentum independent contribution from $\Sigma(q_0=0,\vec{q})$, which is given by
\begin{figure}[t]
\includegraphics[height=2in]{loop2.eps}
\caption{\label{Mixed}\textit{The one-loop diagram contribution to $\Gamma[\Delta, \Delta^*]$ with two external lines. } }
\end{figure}
\begin{equation}
\label{S}
\Sigma(q_0=0,\vec{q})=\sum_{n,k}\frac{1}{(i\omega_n-\omega^a(k))(-i\omega_n-\omega^b(k'))}=A+B {\vec{q}}^2 + \ldots,
\end{equation}
where $\omega_n=(2n+1)\pi T$, $\omega^a(k)=\frac{k^2}{2m_a}-\mu_a$, $\omega^b(k')=\frac{(\vec{k}+\vec{q})^2}{2 m_b}-\mu_b$.
It is worth noting that Eq.~(\ref{ea}) is valid only in the vicinity of a second order phase transition, since it is based on the assumption that the gap parameter is small~\cite{Schafer}. While it is important that $\Delta$ be small so that higher order terms can be neglected, it is also important that the fermions have a finite gap so that they can be properly integrated out to obtain the effective potential. It is this latter condition that renders the present analysis appropriate only when the Fermi surfaces are not mismatched.
After the frequency summation, employing the imaginary time formalism of finite temperature field theory, we obtain
\begin{eqnarray}
\label{f1}
A=\int \frac{d^3 k}{(2 \pi)^3} \frac{1}{\omega + \omega'} \left[1-\frac{1}{e^{\beta \omega}+1}- \frac{1}{e^{\beta \omega'}+1} \right],
\end{eqnarray}
where, for short, we have defined $\omega = \omega^a(k)$, $\omega' = \omega^b(k)=\frac{k^2}{2m_b}-\mu_b$ and $\beta=\frac{1}{k_B T}$, where $k_B$ is Boltzmann's constant that we will set equal to one. Changing the variable of integration from $k$ to $\epsilon$, we get
\begin{eqnarray}
\label{f2}
A=\int_0^{{\cal W}_C} \rho(\epsilon) \frac{d \epsilon}{\epsilon} \left[1-\frac{1}{e^{\beta \omega(\epsilon)}+1}- \frac{1}{e^{\beta \omega'(\epsilon)}+1} \right]\\
\nonumber
\approx \rho(0)\int_0^{{\cal W}_C} \frac{d \epsilon}{\epsilon} \left[1-\frac{1}{e^{\beta \omega(\epsilon)}+1}- \frac{1}{e^{\beta \omega'(\epsilon)}+1} \right],
\end{eqnarray}
where $\rho(0)=\frac{M k_F}{2 \pi^2}$ is defined as the density of states at the Fermi level, with $k_F=\sqrt{2 M \mu}$ being the ``average'' Fermi surface and $\mu = \mu_a + \mu_b$. We have also defined ${\cal W}_C(\Lambda)=\omega(\Lambda)+\omega'(\Lambda)$, where $\Lambda$ is the cutoff in the momentum integral. The energies are given by
\begin{eqnarray}
\label{f22}
\omega(\epsilon)=\frac{M}{m_a} \epsilon+ \mu\frac{M}{m_a}-\mu_a,\\
\nonumber
\omega'(\epsilon)=\frac{M}{m_b} \epsilon+ \mu\frac{M}{m_b}-\mu_b.
\end{eqnarray}
After simple algebra, Eq.~(\ref{f2}) can be written as
\begin{equation}
\label{f3}
A=\frac{\rho(0)}{2} \int_0^{{\cal W}_C} \frac{d \epsilon}{\epsilon} \left[\tanh \left(\frac{\beta \omega(\epsilon)}{2} \right) +
\tanh \left(\frac{\beta \omega'(\epsilon)}{2} \right) \right].
\end{equation}
If we define
\begin{equation}
\label{eta}
\eta =\frac{M}{m_{a}}\mu -\mu _{a}=-\left( \frac{M}{m_{b}}\mu -\mu_{b}\right),
\end{equation}
then we can write Eq.~(\ref{f3}) with the aid of Eqs.~(\ref{f22}) and (\ref{eta}) as
\begin{eqnarray}
\label{eqmdif}
A &=&\frac{\rho (0)}{2}\left\{ \int_{0}^{{\cal W}_{C}}\frac{d\varepsilon }{%
\varepsilon }\tanh \left( \frac{\beta }{2}\frac{M}{m_{a}}\varepsilon +\frac{%
\beta }{2}\eta \right) +\int_{0}^{{\cal W}_{C}}\frac{d\varepsilon }{\varepsilon }%
\tanh \left( \frac{\beta }{2}\frac{M}{m_{b}}\varepsilon -\frac{\beta }{2}%
\eta \right) \right\} \\
\nonumber
&=&\frac{\rho (0)}{2}\left\{ \int_{0}^{\lambda _{a}}\frac{dx}{x}\tanh \left(
x+a\right) +\int_{0}^{\lambda _{b}}\frac{dx}{x}\tanh \left( x-a\right)
\right\},
\end{eqnarray}
where we have defined
\begin{equation}
\lambda _{a}=\frac{\beta }{2}\frac{M}{m_{a}}{\cal W}_{C},\,\,\,\,\lambda _{b}=\frac{%
\beta }{2}\frac{M}{m_{b}}{\cal W}_{C}\text{ and }a=\frac{\beta }{2}\eta.
\end{equation}
If we take $m_{b} \geq m_{a}$, then $\lambda _{b} \leq \lambda _{a}$ and Eq.~(\ref{eqmdif}) can be written as
\begin{equation}
\label{eqLa}
A=\frac{\rho (0)}{2}\left\{ \int_{0}^{\lambda _{a}}\frac{dx}{x}\tanh \left(
x+a\right) +\int_{0}^{\lambda _{a}}\frac{dx}{x}\tanh \left( x-a\right)
-\int_{\lambda _{b}}^{\lambda _{a}}\frac{dx}{x}\tanh \left( x-a\right) \right\}.
\end{equation}
We can solve the first two integrals in the r.h.s. of Eq.~(\ref{eqLa}) employing the residue theorem and the last one can be easily solved if we observe that $\lambda _{a}$ and $\lambda _{b}$ correspond to the regions where $\tanh (x+a)\approx \tanh (x-a)\approx 1 $. Thus,
\begin{equation}
\label{eqf}
A=\frac{\rho (0)}{2}\left\{ \ln \left( \frac{\lambda_{a}^{2}}{\pi ^{2}}\right) -{\cal F}(a) -\ln \left( \frac{\lambda _{a}}{\lambda _{b}}\right) \right\},
\end{equation}
where ${\cal F} (a)= \Psi (\frac{1}{2}+\frac{ia}{\pi })+\Psi (\frac{1}{2}-\frac{ia}{\pi })$ with $\Psi$ being the digamma function, defined as $\Psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$, where z is a complex number with a positive real component, $\Gamma$ is the gamma function, and $\Gamma'$ is the derivative of the gamma function. We also have that ${\cal F}(0)=-2\gamma -4\ln (2)$, where $\gamma$ is the Euler's constant.
The critical temperature is the solution of the equation
\begin{equation}
\label{tc0}
\alpha= \frac{1}{g}-A=0.
\end{equation}
Then we write
\begin{equation}
\label{tc1}
\frac{1}{g \rho (0)} - \ln \left(\beta \sigma \frac{\omega_D}{\pi} \right)=-\frac{1}{2} {\cal F} (a),
\end{equation}
where $\sigma \equiv \frac{M}{\sqrt{m_a m_b}}$ is a dimensionless parameter reflecting the mass asymmetry, and we have used the fact that ${\cal W}_C=2\omega_D$.
The BCS gap in the weak coupling limit, $\rho(0)g<<1$, is given by $\Delta_0=2\omega_D~ e^{-1/\rho(0)g}$. Since the gap which minimizes the free-energy of the asymmetric system retains the same size as in the symmetric case ($\Delta_0$) until a value for the asymmetry where the pairing is not afforded any more~\cite{Paulo,Sedrakian,Toki,Heron1,Heron2}, we rewrite Eq.~(\ref{tc1}) as
\begin{equation}
\label{tc2}
T_c=\frac{\sigma\Delta_0}{2 \pi} e^{-\frac{1}{2} {\cal F}(a_c)},
\end{equation}
where $a_c=\frac{\beta_c}{2} \eta = \frac{\beta_c}{2} \frac{m_b \mu_b- m_a \mu_a}{m_a + m_b}$. We evaluate Eq.~(\ref{tc2}) in the two possible configurations for the particles masses and chemical potentials constrained to $P^a_F=P^b_F$, which are the situations encountered when the fermions are fully gapped. In these cases we can obtain analytical solutions for the critical temperature, due to the simple form of the term ${\cal F} (a_c = 0)$, as showed below.
In the investigation of the phase transition when the Fermi surfaces are mismatched, one needs to compare the thermodynamic potentials of the superfluid and normal states. In fact, as showed in~\cite{Paulo,Toki,Heron1,Heron2} by the behavior of the free-energy as a function of the gap for several asymmetries in the chemical potentials, an asymmetrical fermion system stays in the superfluid phase until a maximum value for the difference in chemical potentials is reached. After this maximum value, there is a first order phase transition to the normal phase.
\subsubsection{Equal Chemical Potentials and Masses}
\label{case1}
This case configures the symmetric system, whose critical temperature is recovered for $m_a=m_b,~\mu_a=\mu_b$, resulting
${\cal F}(a_c=0)=-2 \ln(4 e^{\gamma})$, and $\sigma=\frac{1}{2}$, giving the well known BCS result
\begin{equation}
\label{tc3}
T_c=\frac{e^{\gamma}}{\pi} \Delta_0 \equiv T_c^{sym}.
\end{equation}
We notice that both the zero temperature gap parameter $\Delta_0$ and the symmetric critical temperature $T_c^{sym}$ depend on the product $\rho(0)g$, but not their ratio.
\subsubsection{Equal Fermi Surfaces with Mismatched Chemical Potentials and Masses}
\label{case2}
This situation is also characterized by $P^a_F=P^b_F$, which implies $m_a \mu_a=m_b \mu_b$, however with $m_a \neq m_b$, $\mu_a \neq \mu_b$. This is achieved by setting $a_c =0$ in Eq.~(\ref{tc2}), yielding
\begin{equation}
\label{tc4}
T_c(P^a_F=P^b_F)= 2\sigma \frac{e^{\gamma}}{\pi} \Delta_0 = 2\sigma T_c^{sym}.
\end{equation}
We note that if we set $m_a=m_b$ in the equation above (which implies $\sigma=\frac{1}{2}$), then we have the symmetric Fermi gas, since we would also have $\mu_a=\mu_b$, due to the constraint $m_a \mu_a =m_b \mu_b$. We can observe from Eq.~(\ref{tc4}) that the critical temperature for the system constrained to $m_a \mu_a =m_b \mu_b$ goes with $2 \sqrt{\frac{m_a}{m_b}}~T_c^{sym}$ for $m_b$ greater than $m_a$ and approaches zero for $m_b>>m_a$. This shows that the pair formation is disfavored for very large mass asymmetry. The same conclusion has been found for the case of fixed number of particles~\cite{Heron3}. In Fig.~(\ref{TempC1}) we show the ratio $T_c / \Delta_0$ as a function of $m_b/m_a$. As one can see, $T_c / \Delta_0$ is a smooth function of the mass asymmetry, and goes to zero for $m_b>>m_a$.
A mass ratio in the order of, or larger than, $m_b/m_a=50$, as used in Ref.~\cite{Wilczek3} would, obviously, have a smaller critical temperature than that of the symmetric gas. Although we did not consider in our calculations the momentum structure of the interaction employed in Ref.~\cite{Wilczek3}, we believe that our results would persist, at least qualitatively, after this consideration.
\begin{figure}[t]
\includegraphics[height=3in]{Figura2.eps}
\caption{\label{TempC1}\textit{$T_c / \Delta_0$ as a function of $m_b/m_a$ for a system constrained to $m_a \mu_a =m_b \mu_b$. } }
\end{figure}
\subsubsection{Conclusions}
We have calculated the critical temperature of an asymmetrical Fermi system in two configurations for the masses and chemical potentials of the two species that form Cooper pairs. Among those cases we have investigated, the one constrained to $m_a \mu_a = m_b \mu_b$, with $m_a \neq m_b$, $\mu_a \neq \mu_b$, is particularly interesting, for which we found a generalization for the expression relating the critical temperature and the zero temperature gap $\Delta_0$. Namely: $\frac{\Delta_0}{T_c}=\frac{\pi}{e^{\gamma}} \frac{1}{2\sigma} \approx 1.76 \frac{1}{2\sigma}$, where $\sigma = \frac{\sqrt{m_a m_b}}{m_a + m_b}=\frac{\sqrt{\mu_a \mu_b}}{\mu_a + \mu_b}$ and constitutes an {\it universal constant}, for given $m_a$ and $m_b$ (or $\mu_a$ and $\mu_b$), independent of $g$ and $\rho(0)$. Another remarkable feature of this result is its independence of any cutoff parameter. This is because Eq.~(\ref{tc2}) is quite insensitive to the regularization procedure.
We believe that the results achieved in this work could, in principle, be tested experimentally in, for example, experiments involving $^{6}Li$ or $^{40}K$ in atomic traps~\cite{DeMarco1,Ohara2,Truscott,DeMarco2,Ohara,Granade,Regal,Chin,Zwierlein,Kinast3,Bartenstein,Kinast4,Kinast,Kinast2}. Evidences of superfluidity in these systems were observed both microscopically, observing the pairing of fermionic atoms~\cite{Regal,Chin,Zwierlein} and macroscopically, due to anisotropic expansions~\cite{Ohara}, collective excitations~\cite{Kinast3,Bartenstein,Kinast4} and heat capacity measurements~\cite{Kinast}. In these experiments, the strength of the pairing interaction can be controlled by an applied magnetic field, for instance. Also, the density, the number of each species and the trapping potential can be altered. Thus, the species on the trap differ by theirs spin/pseudo-spin projections, and also by theirs densities, characterizing the asymmetry of the system.
Current experiments~\cite{Kinast,Kinast2,Kinast3,Kinast4} produce temperatures down to about $0.05T_{F}$, where $T_{F}$ is the Fermi temperature for a noninteracting gas with the same number of atoms and trap conditions as the experiment, typically of order of $\mu K$. However, a weakly interacting Fermi gas requires much lower $T$ to achieve superfluidity. For the conditions of these experiments, the mean field approximation with an interaction energy proportional to the scattering length is not valid. However, the mean field approximation with a unitary limit appears approximately valid, furnishing a good agreement with predictions of the collective frequencies, and a very good agreement on the transition temperature~\cite{Thomas}. Thus, even when the measurements are done in strongly interacting Fermi gases, mean field theory had qualitatively explained the behavior of these systems, and we expect that our weak coupling mean field BCS results should also be valid qualitatively. In particular, we believe that the numerical factor $1.76\frac{1}{2 \sigma}$ can be tested experimentally, provided the mass asymmetry is not so large.
When the Fermi surfaces are mismatched the phase transitions will be of first order and the present formalism is insufficient to determine the critical temperature. We will come back to this issue soon~\cite{Heron4}. For these cases $m_a \mu_a \neq m_b \mu_b$, and $\frac{\Delta_0}{T_c}$ will not be an universal number. This ``non universality'' is also manifested in the BP at finite temperature \cite{Liao} and in (dense) neutral quark matter \cite{Andreas,Igor,Kenji}, which is essentially asymmetrical \footnote{The up, down and strange quark Fermi surfaces, which are candidates to form {\it color superconductivity} are mismatched.}. Still in quark matter, in Ref. [39] was developed a systematic method of QCD expansion of the transition temperature, motivated by the non-BCS scaling of the gap parameter with coupling. In this work, the relation between the zero temperature energy gap and the critical temperature has a non BCS form too.
We find that large mass and chemical potentials asymmetries lower the critical temperatures substantially and compromise the stability of the system. This happens because any small thermal excitation breaks the (weakly bound due to the large asymmetry) pairs, and destroys superfluidity.
\begin{acknowledgments}
One of us, H.~Caldas, would like to thank P. Bedaque for valuable suggestions, Hai-Cang Ren for helpful discussions, and John Thomas for useful discussions on experimental issues related to Fermi systems. We thank the referee for pointing out to us a way to treat first order phase transitions. The authors acknowledge financial support by CNPq/Brazil.
\end{acknowledgments}
\subsubsection{Appendix: Solution by the Variational Method}
We now derive the gap equation at finite temperature for an asymmetrical fermion system, in order to determine the critical temperature. We follow the usual derivation of the textbooks, however extending the analysis for the asymmetrical systems we are investigating. Let us define $f_k$ as the probability of an $a$ particle with momentum $\bf{k}$ is excited, and similarly $g_k$ as the probability of a $b$ particle with momentum $\bf{-k}$ is excited. Then, the entropy for an asymmetrical fermion gas is found to be
\begin{equation}
\label{ap1}
S=-\sum_{k} \left\{ f_k \ln(f_k) + (1-f_k) \ln(1-f_k) + g_k \ln(g_k) + (1-g_k) \ln(1-g_k) \right\}.
\end{equation}
The free energy or thermodynamic potential is written as
\begin{equation}
\label{ap2}
F=E-TS,
\end{equation}
where $E$ is the internal energy
\begin{eqnarray}
\label{ap3}
E=\sum_{k} \left\{ \epsilon_k^a [ (1- f_k -g_k)U_k^2+ f_k] + \epsilon_k^b [ (1- f_k -g_k)U_k^2+ g_k] \right\}\\
\nonumber
-g \sum_{k,k'} U_{k'} V_{k'} U_k V_k (1-f_k - g_k)(1-f_{k'} - g_{k'}).
\end{eqnarray}
Here we have defined the particles energies relative to their Fermi surfaces in terms of our previous definitions $\epsilon_k^a \equiv \omega = \frac{k^2}{2 m_a}-\mu_a$ and $\epsilon_k^b \equiv \omega' = \frac{k^2}{2 m_b}-\mu_b$. From the minimizations
\begin{eqnarray}
\label{ap4}
\frac{\delta F}{\delta f_k}=0,\\
\nonumber
\frac{\delta F}{\delta g_k}=0,\\
\nonumber
\frac{\delta F}{\delta U_k}=0,\\
\nonumber
\end{eqnarray}
we find, respectively,
\begin{equation}
\label{ap5}
f_k=1/(e^{\beta {\cal{E}}_k^a}+1),
\end{equation}
\begin{equation}
\label{ap6}
g_k=1/(e^{\beta {\cal{E}}_k^b}+1),
\end{equation}
\begin{equation}
\label{ap7}
U_k^2=\frac{1}{2} \left(1+ \frac{\epsilon_k^{+}}{E_k} \right),
\end{equation}
where ${\cal{E}}_k^{a,b}=\pm \epsilon_k^{-} + E_k$ are the quasiparticle excitations, with $E_k^2={\epsilon_k^{+}}^2+\Delta^2(T)$ and $\epsilon_k^{\pm} \equiv \frac {\epsilon_k^a \pm \epsilon_k^b}{2}$. In the definition of ${\cal{E}}_k^{a,b}$ we have also defined
\begin{equation}
\label{ap8}
\Delta(T)=g \sum_{k} U_k V_k (1-f_k - g_k).
\end{equation}
Since $V_k^2=1-U_k^2$, then $U_k V_k=\frac{\Delta}{2 E_k}$, and the equation above can be written as
\begin{equation}
\label{ap9}
1=g \sum_{k} \frac{1}{2 E_k} \left( 1-f_k - g_k \right).
\end{equation}
At the critical temperature $\Delta=0$ in Eq.~(\ref{ap9}), and we have
\begin{equation}
\label{ap10}
1=g \sum_{k} \frac{1}{\epsilon_k^a+\epsilon_k^b} \left(1-\frac{1}{e^{\beta_c \epsilon_k^a}} - \frac{1}{e^{\beta_c \epsilon_k^b}} \right),
\end{equation}
which is equation~(\ref{tc0}). The solution for $T_c$ from (\ref{ap10}) follow as in the body of the paper.
Important remarks are now in order:\\
{\bf 1.} Although Eqs.~(\ref{ap9}) and~(\ref{ap10}) require regularization, the regulator dependence cancels from the result~(\ref{tc2}).\\
{\bf 2.} We have obtained an equivalence between two approximations to identify the zero temperature quasiparticle excitations. Introducing the temperature via the variational method, minimizing the thermodynamic potential with respect to the excitations probabilities in thermal equilibrium and then taking the zero temperature limit is equivalent to diagonalize the (zero temperature) mean field Hamiltonian (as done in Refs.~\cite{Heron1,Heron2}) and obtain the excitations energies of the fermion quasiparticles.\\
{\bf 3.} If the fermions are fully gapped, as is the case here, the gap parameter depends on the asymmetries only at finite temperature $(0<T<T_c)$. At zero temperature the excitations probabilities vanish, and the gap depends only on ``averages'', through $\epsilon_k^{+}$. When there are gapless excitations, the gap depends on the asymmetries even at $T=0$~\cite{Sarma,Wilczek2,Heron1,Heron2}.
|
1,108,101,565,639 | arxiv | \section{Introduction}
\PARstart{S}{hape} decomposition is a fundamental problem in geometry processing where an arbitrary object is regarded as an arrangement of simple primitives \cite{Kaick2015ShapeAnalysis, Zhou2015GeneralizedDecomposition} or semantic components \cite{Au2008SkeletonContraction, Berretti20093DGraphs}. Applications of shape decomposition include disciplines such as object recognition and retrieval \cite{Zuckerberger2002PolyhedralApplications, Siddiqi2008RetrievingSurfaces}, shape reconstruction \cite{Goyal2012TowardsModeling}, shape clustering \cite{Averkiou2014ShapeSynth:Synthesis}, or modeling \cite{Funkhouser2004ModelingExample}.
Our motivation for studying shape decomposition comes from biomedical image segmentation. Advanced biomedical imaging techniques, such as 3D electron microscopy, generate large image volumes whose size can range from a gigabyte to hundreds of terabytes \cite{Rubin2014CTAdvance, Abdollahzadeh2019AutomatedMatter, Zheng2018AMelanogaster}. Segmentation of such image volumes generally favors bottom-up strategies, where the image is first over-segmented into supervoxels, then supervoxels are merged subsequently \cite{Lucchi2012, Nunez-Iglesias2014Graph-basedNeuroimages, Funke2019LargeReconstruction}. This strategy is error-prone because both the over-segmentation and subsequent merge are subjected to greedy optimization as opposed to optimizing a global objective. Our idea is instead to approach the segmentation problem based on a top-down strategy, where under-segmentation is followed by subsequent split using \textit{a priori} knowledge of objects to be segmented; in our case, tubularity of neuronal processes or blood vessels. This strategy divides a large image volume into sub-domains whose geometry/topology can be analyzed based on a global objective, independently and in parallel, but necessitates the development of a fast and robust shape decomposition technique capable of processing thousands of tubular structures in a reasonable time.
This paper develops a novel decomposition algorithm called cylindrical shape decomposition (CSD), decomposing big voxel-based tubular objects in large image volumes. We demonstrate the CSD's application in segmenting tubular structures, as the split operation of a top-down strategy, and its application in decomposing general synthetic objects.
\Figure[t][width=\textwidth]{figs/outline.png}
{Outline of the CSD algorithm. (\textbf{a}) An object is a union of several tubular components. The tubular components are color-coded. (\textbf{b}) A $800 \times 400 \times 70$ voxel-based representation of the object. Intersections of the tubular components are magnified. (\textbf{c}) The curve skeleton of the synthetic object in (\textbf{b}) is the union of all skeleton branches. Skeleton branches are color-coded and denoted as $\gamma$. We define a junction-point $j$ as such a point that skeleton branches connect. Junction-points are shown as blue filled-circles. (\textbf{d}) The curve skeleton of the object is partitioned into maximal-length sub-skeletons $\psi$ over a local orientation cost. The sub-skeletons are color-coded. (\textbf{e}) On a sub-skeleton $\psi$ and in the proximity of a junction-point $j \in \psi$, we define two decomposition intervals. The boundaries of decomposition intervals are shown with red filled-circles. The object is swept along $\psi$ and towards the joint $j$ to find a critical point in each interval. At a critical point, the normalized Hausdorff distance $H_\rho$ between a cross-sectional contour and the mean of visited cross-sectional contours exceeds $\theta_H$. Sweeping directions are shown with arrows. (\textbf{f}) We cut the object at critical points to obtain object parts. (\textbf{g}) The object parts along the same sub-skeleton are assigned the same label to construct a semantic-component. The semantic-components are further reconstructed between their comprising object-parts using generalized cylinders. The synthetic object in (\textbf{a}) comprises seven object-parts, and our algorithm decomposes it into three semantic components. \label{fig:outline}}
\subsection{Related work}
We categorize shape decomposition techniques in the literature into three categories: 1) representing an object in terms of geometrically homogeneous and simple primitives, such as ellipsoids, convex components, or generalized cylinders \cite{Simari2005ExtractionData, RAAB2004VirtualModels, Mortal2004Plumber:Bodies, Asafi2013WeakLines-of-sight, Kaick2015ShapeAnalysis, Zhou2015GeneralizedDecomposition, Li2001DecomposingApplications, Goyal2012TowardsModeling}; 2) decomposing an object into its semantic components using object skeleton or Reeb graph \cite{Reniers2008ComputingMeasure, Au2008SkeletonContraction, Berretti20093DGraphs, Livesu2017ExplicitShapes}; and 3) learning-based decomposition methods \cite{Kalogerakis2010LearningLabeling, Qi2017PointNet:Segmentation, Yu2019Partnet:Segmentation, Deng2020CVXNet:Decomposition}.
Primitives are homogeneous components with a compact representation and efficient computation. Examples of primitives with a simple parametric representation include ellipsoids \cite{Simari2005ExtractionData} and straight cylinders \cite{RAAB2004VirtualModels}. This class of primitives with a simple parametric representation is typically applied in description simplification of complex geometrical models. Therefore, higher-level primitives such as tubular primitives \cite{Mortal2004Plumber:Bodies}, convex components \cite {Asafi2013WeakLines-of-sight, Kaick2015ShapeAnalysis}, generalized cylinders \cite{Zhou2015GeneralizedDecomposition}, and generalized sweep components \cite{Li2001DecomposingApplications, Goyal2012TowardsModeling} were proposed for trading-off the representation simplicity for the generality. For example, tubular primitives in Plumber \cite{Mortal2004Plumber:Bodies} are constructed applying a seeded region growing with a heuristic set of sphere positions and radii. The Plumber does not return a complete decomposition of objects but extracts ideal tubular components \cite{Mortal2004Plumber:Bodies}. Convexity-based methods are another interesting class of high-level primitive-based decomposition techniques, developed based on the human tendency to divide an object into parts around concave regions \cite{Biederman1917Recognition-by-Components:Perception}. An exact decomposition of a shape into convex components is costly and too strict for decomposition because such methods can generate many small parts. Therefore, \cite{Asafi2013WeakLines-of-sight} and \cite{Kaick2015ShapeAnalysis} apply weakly convex components, which are derived from analyzing the pairs of points in the shape visible to each other, obtaining an approximate convex decomposition (ACD). An alternative to the convex decomposition is generalized cylinder decomposition (GCD), quantifying cylindricity. \cite{Zhou2015GeneralizedDecomposition} introduces a quantitative measure for the cylindricity following the minimum description length principle \cite{Grunwald:2007:MDL:1213810} as a measure of the skeleton straightness and the variation among the profiles. In this method, the global objective for merging local generalized cylinders is to minimize the cylindricity. The approximate convexes and generalized cylinders are effective high-level primitives, where the approximate convex method generates smoother cuts between parts, and the generalized cylinders suit better the decomposition of tubular objects. The generalized cylinders method is a computationally expensive technique, e.g., \cite{Livesu2017ExplicitShapes} reported approximately 30 minutes for a single femur decomposition. Another set of high-level primitive-based decomposition techniques is based on cross-sectional sweeping. These methods are computationally less demanding as compared to the convexity and generalized cylinder methods. The sweeping algorithms analyze object cross-sections and generate homogeneous sweeping components. For example, in \cite{Li2001DecomposingApplications}, the object is swept along its curve skeleton in search of critical points, where the object geometry/topology changes substantially. This method uses the variation of the perimeter of consecutive cross-sections as the homogeneity measure, which is sensitive to the surface noise and prone to over-segmentation. The method in \cite{Goyal2012TowardsModeling} generates local prominent cross-sections from a set of initial seed points. This method is semi-automated, requiring user interactions to adjust the density of cross-sections in different object regions and avoid creating prominent cross-sections in regions with no sweep evidence.
To decompose an object into its semantic components, the object curve skeleton or Reeb graph can be used. Both concepts are object descriptors able to guide a decomposition: the curve skeleton is a 1D representation of a 3D object \cite{Cornea2007Curve-skeletonAlgorithms}, encoding its topology and geometry; the Reeb graph tracks topology changes in level sets of a scalar function \cite{Cornea2007Curve-skeletonAlgorithms}. \cite{Reniers2008ComputingMeasure} extracts object curve skeleton based on a collapse measure, i.e., a measure of importance, and subsequently provide an object decomposition by defining skeleton-to-surface mapping based on the shortest geodesics. \cite{Au2008SkeletonContraction} extracts the curve skeleton, applying an implicit Laplacian smoothing with global positional constraints, preserving the mesh connectivity and its key features. \cite{Au2008SkeletonContraction} provides an object decomposition with an approximate measure of thickness about extracted curve skeletons. The tubular decomposition in \cite{Livesu2017ExplicitShapes} aims to be as close as possible to a Voronoi partitioning, having skeleton branches as sites, while satisfying structural constraints that ensure each decomposition element is a tube-like shape. These skeleton-based decomposition methods \cite{Reniers2008ComputingMeasure, Au2008SkeletonContraction, Livesu2017ExplicitShapes} are applied to segment synthetic objects into functional parts, which is not the case for decomposing an under-segmentation error in objects acquired from biomedical imaging datasets. In \cite{Berretti20093DGraphs}, the decomposition of a 3D mesh is a two-step approach accounting for the Reeb graph construction and refinement: the Reeb graph captures the surface topology and protrusions, and the refinement step uses curvature and adjacency information on the graph critical points for fine localization of part boundaries. This approach does not provide smooth boundary cuts between parts, requiring the internal energy function to control the smoothness of boundaries. In the context of skeletonization, it is worth reviewing the L1-medial skeletonization \cite{Huang2013LCloud} and rotational symmetry axis (ROSA) \cite{Tagliasacchi2009CurveCloud} techniques. The L1-medial skeletonization employs localized L1-medians to construct a skeleton. This method uses a weighting function with a supporting radius that defines the size of the local neighborhood; gradually increasing the supporting radius yields a clean and well-connected skeleton. ROSA defines a curve skeleton as a generalized rotational symmetry axis of a shape. The position of a skeleton point in a local set of points is computed by minimizing the sum of the projected distances to the normal extensions of the data points. The L1-medial and ROSA skeletonization techniques may form cycles when two object parts are close to each other, where the skeleton is to be acyclic. These two skeletonization methods deal with incomplete point clouds, but the skeleton centeredness within the objects is not guaranteed.
Learning-based methods are an important class of shape decomposition techniques, from early statistical modeling methods \cite{Kalogerakis2010LearningLabeling} to recent deep neural network techniques \cite{Qi2017PointNet:Segmentation, Yu2019Partnet:Segmentation, Deng2020CVXNet:Decomposition}. The objective function of a learning-based method is learned from a collection of labeled training objects. Learning-based decomposition methods have demonstrated impressive results, often producing segmentation and labeling comparable to those produced by humans. However, these techniques crucially depend on large training datasets and are often impaired when the objects to be decomposed deviate substantially from the training material.
\subsection{Outline of the CSD algorithm and contributions}
The main idea of the CSD algorithm is to guide the decomposition using the object curve skeleton and cut the object by restricted translational sweeps. Fig. \ref{fig:outline}a shows an under-segmented tubular object as a union of three tubular components. CSD begins with extracting the object curve skeleton (Fig. \ref{fig:outline}c) and partitioning the skeleton into maximal-length sub-skeletons over an orientation cost function (Fig. \ref{fig:outline}d). Each sub-skeleton corresponds to a semantic tubular component. To identify intersections of the semantic components, CSD translationally sweeps the object along sub-skeletons, searching for critical points where the object cross-section changes substantially (Fig. \ref{fig:outline}e). A translational sweep is restricted in decomposition intervals; in the proximity of junction-points where sub-skeletons intersect. The object is cut at critical points to obtain object parts (Fig. \ref{fig:outline}f). A semantic component is further reconstructed at intersections, using generalized cylinders (Fig. \ref{fig:outline}g).
The CSD algorithm possesses several advantages over previous shape decomposition techniques. Unlike semi-automated methods in \cite{Goyal2012TowardsModeling} and \cite{Au2008SkeletonContraction}, CSD is a fully automatic algorithm and requires no manual interventions; the decomposition is guided using an algorithm that partitions the object skeleton curve into distinct maximal-length straight sub-skeletons. Compared to primitive-based methods in \cite{Simari2005ExtractionData, RAAB2004VirtualModels, Kaick2015ShapeAnalysis, Zhou2015GeneralizedDecomposition}, or skeleton-to-surface mapping in \cite{Reniers2008ComputingMeasure, Berretti20093DGraphs}, our method identifies the intersection of the object parts and defines smooth boundary cuts between them. Compared to \cite{Li2001DecomposingApplications}, our method is intrinsically more robust in defining critical points in the presence of noise because we measure cross-sectional changes using the mean close curve of traversed cross-sections and modified Hausdorff distance. In \cite{Zhou2015GeneralizedDecomposition}, generating a generalized cylinder requires iterative operations, yet many such primitives are required to cover the object; henceforth, these primitives must be merged to satisfy a global objective. Such methods are computationally expensive, not suitable for the decomposition of big voxel-based objects in large image volumes. Unlike \cite{Reniers2008ComputingMeasure, Au2008SkeletonContraction, Livesu2017ExplicitShapes}, and \cite{Berretti20093DGraphs}, which apply a one-to-one assignment between skeleton branches and object parts, we propose to merge skeleton branches belonging to the same semantic part. Unlike learning-based techniques \cite{Kalogerakis2010LearningLabeling, Qi2017PointNet:Segmentation, Yu2019Partnet:Segmentation, Deng2020CVXNet:Decomposition}, our proposal does not rely on training and thus generalizes to the variation of tubular objects extracted from medical images, where it attains consistent quality without the need for additional training datasets. In comparison to the L1-medial and ROSA skeletonization techniques, we use a distance-based skeletonization approach, which correctly stays at the center of the object, even when the object parts are adjacent.
In the experimental section of this paper, we demonstrate the application of CSD in the segmentation of large electron microscopy volumes of myelinated axons. We also demonstrate the CSD decomposition of vascular networks and synthetic objects. Moreover, we compare CSD to other state-of-the-art decomposition techniques (ACD \cite {Kaick2015ShapeAnalysis} and GCD \cite{Zhou2015GeneralizedDecomposition}), and our skeletonization technique to well-known skeletonization approaches (L1-medial and ROSA). We also evaluate the effect of surface noise on decomposition results and assess a methodology to reduce the CSD computation-time.
\section{Preliminaries}
This section defines the core concepts used in the paper as there are no generally accepted definitions for most of them.
\textbf{Object.} An object $\Omega \subset \mathbb{R}^3$ is a nonempty bounded open set. We assume that its boundary $\partial \Omega$ is homeomorphic to a 2-sphere. For a discrete object, which results from foreground segmentation, we define a 3D binary image as $I: X \subset \mathbb{Z} ^3 \to \{0,1\}$, and a segmented object $\Omega := \{x \in X: I(x) = 1 \}$, where $X$ is the image domain. Throughout the paper, $\Omega$, $\partial \Omega$, and $x$ are in $\mathbb{R}^3$ unless defined otherwise.
\textbf{Curve skeleton.} Given $\Omega$ and $\partial \Omega$, the curve skeleton $\Upsilon \subset \Omega$ is defined as a locus of centers of maximal inscribed balls \cite{Lieutier2004AnyAxis}. A ball $B(x,r)$ centered at $x \in \Omega$ with radius $r$ is maximally inscribed if its surface touches $\partial \Omega$ in at least two distinct points. Formally, $B$ is a maximal inscribed ball in $\Omega$ if $\forall B',\; B \subseteq B' \subseteq \Omega \Rightarrow B' = B$.
\textbf{Curve skeleton point type.} We distinguish three types of points on the curve skeleton of an object: 1) regular-points that have exactly two neighbor points on the skeleton, 2) end-points that have exactly one neighbor point on the skeleton, and 3) junction-points that have three or more neighbor points on the skeleton \cite{Cornea2007Curve-skeletonAlgorithms}. We denote the collection of junction-points as $J$ where $j \in J$ and the collection of end-points as $O$ where $o \in O$.
\textbf{Skeleton branch.} Removing junction-points $J$ from the curve skeleton $\Upsilon$ results in disconnected simple curves, known as skeleton branches. The collection of skeleton branches is denoted as $\Gamma$, and a skeleton branch is $\gamma \in \Gamma$. For $\gamma (t): [0,1] \to \mathbb{R}^3$, its arc-length is written as $ l = \int_{0}^{1} |{\dot \gamma(t)}| \, \mathrm{d}t $ with the convention $\dot \gamma(t) := \dfrac{\mathrm{d}}{\mathrm{d}t}\gamma(t)$.
\textbf{Skeleton graph.} The topology of curve-skeleton $\Upsilon$ can be represented as a connected acyclic undirected graph (i.e., a tree) $G_{\Upsilon} = (V, E, L)$. There is a one-to-one map between skeleton branches in $\Gamma$ and edges in $E$ and a one-to-one map between the union of end-points and junction-points ($O \cup J$) and vertices in $V$. This means that for each branch $\gamma \in \Gamma$ we associate exactly one edge in $e$ in $G_{\Upsilon}$. $L \subset \mathbb{R}^+$ is the set of edge lengths. The length of an edge is the arc-length of its associated skeleton branch.
\textbf{Walk, path.} A walk is a finite or infinite sequence of edges which joins a sequence of vertices. A finite walk is a sequence of edges $W = \{e_1, e_2, \ldots, e_{n'}\}$ for which there is a sequence of vertices $\{v_0, v_1, \ldots, v_{n'}\}$ such that $e_i = v_{i-1}v_{i}$ for $i = 1, \dots , n'$. The vertex sequence of the walk is $(v_0, v_1, \ldots, v_{n'})$. A path is a walk in which all vertices are distinct.
\textbf{Sub-skeleton} is a path in the curve skeleton domain. If $W = \{e_1, e_2, \ldots, e_{n'}\}$ is a path in the skeleton graph, and $\{\gamma_1, \gamma_2, \ldots, \gamma_{n'}\}$ are corresponding skeleton branches, then $\psi = \cup_i \gamma_i \subseteq \Upsilon$ is a sub-skeleton.
\textbf{Critical point.} A point on a sub-skeleton at which the cross-sectional contour of the object changes substantially. We provide a formal definition in section \ref{sec:critical_point}.
\textbf{Cut.} A closed simple curve $C \subset \partial \Omega$ is a cutting-curve if $\partial \Omega \setminus C$ is not connected. Cut means removal of a cutting-curve from the surface.
\section{Outline of the CSD algorithm}
The outline of the CSD algorithm is shown in Fig. \ref{fig:outline}, and it is as follows:
\begin{enumerate}
\item define the curve skeleton of a given object (Fig. \ref{fig:outline}c, section \ref{sec:skel});
\item partition the curve skeleton of the object into sub-skeletons (Fig. \ref{fig:outline}d, section \ref{sec:skel_partition});
\item define decomposition intervals to restrict the object sweep (Fig. \ref{fig:outline}e, section \ref{sec:dec_int});
\item sweep the object to find critical points and cut the object at critical points (Fig. \ref{fig:outline}e and Fig. \ref{fig:outline}f, section \ref{sec:critical_point});
\item reconstruct the object between parts that have the same label using generalized cylinders (Fig. \ref{fig:outline}g, section \ref{sec:obj_rec}).
\end{enumerate}
The CSD algorithm is designed for genus zeros objects.
\section{Skeleton partitioning} \label{sec:skel_part}
We use the curve skeleton of an object to drive the decomposition. For that, we partition the skeleton graph into several distinct paths union of which covers the skeleton graph. The partitioning of the skeleton graph, by extension, partitions the curve skeleton into sub-skeletons. Each sub-skeleton corresponds to exactly one semantic object component.
\subsection{Curve skeleton} \label{sec:skel}
To determine the curve skeleton of an object $\Omega$ with sub-voxel precision, we apply a method from \cite{Hassouna2005RobustSets} and \cite{VanUitert2007SubvoxelMethods}. The algorithm initiates by determining a point $x^* \in \Omega$ with the biggest distance from the object surface $\partial \Omega$ inside the object domain. This point is used to determine a skeleton branch $\gamma(t): [0,1] \to \mathbb{R}^3$, starting at $x_s$, the furthest geodesic point from $x^*$ in $\Omega$, and ending at $x^*$. A cost function $F$ is defined to enforce the path to run in the middle of $\Omega$, where $F$ should increase if the path moves away from the center. To determine $F$, we find the distance field $D(x)$ from $\partial \Omega$, and assign $F = 1 - \big(\dfrac{D(x)}{D(x^*)}\big)^2$. The distance field $D(x)$ is determined by solving an Eikonal equation on the object domain $\Omega$ using the fast marching method \cite{Sethian1996AFronts}. Starting at $x_s$, the skeleton branch $\gamma$ is traced by a back-tracking procedure on $F$ to reach $x^*$, written as
\begin{equation}
\gamma = \displaystyle \arg \min_{P} {\displaystyle \int_{x_{s}}^{x^*}}
F (P(t)) \, \mathrm{d}t,
\label{eq:skeleton_path}
\end{equation}
where $t$ traces the path $P$. We use the Euler scheme for the back-tracking procedure, which solves the ordinary differential equation with a sub-voxel accuracy. This process is repeated to determine further branches that form the curve skeleton of the object. But rather than using the single point $x^*$ as the staring point for the fast marching method, all points in the previously calculated branches are used as starting points. We propagate a new wave from the starting points with the speed $F$ to update $x_s$. The point $x_s$ is now the furthest point from the current state of the curve skeleton and the starting point of the new branch. Applying a back-tracking algorithm from the updated $x_s$ defines the new skeleton branch. Fig. \ref{fig:skeleton} shows the skeletons of two synthetic objects and a vascular network.
\Figure[t][width=0.99\columnwidth]{figs/skeleton.png}
{The curve skeleton of an object is the union of all skeleton branches. (\textbf{a}) The skeleton of the synthetic object, size: $800 \times 400 \times 70$ voxels, seven branches. The blue filled-circles show junction-points, and the red filled-circles show end-points. The skeleton graph of this object is $G_\Upsilon(V,E,L)$, where $E = \{e_1,\ldots, e_7\}$, $V=\{v_0, \ldots, v_7 \}$, and $L =\{l_1, \ldots, l_7 \} $. (\textbf{b}) The skeleton of a synthetic object, size: $128 \times 128 \times 128$ voxels, six branches. (\textbf{c}) The skeleton of a vascular network, size: $256\times256\times256$ voxels, 20 branches. Skeleton branches are color-coded. \label{fig:skeleton}}
\subsection{Skeleton graph decomposition} \label{sec:skel_partition}
Several skeleton branches are often required to represent one semantic component of an object, and therefore detecting skeleton branches is not sufficient for a semantic decomposition. An example is shown in Fig. \ref{fig:skeleton}a, where the union of three skeleton branches $\gamma_1$, $\gamma_2$, and $\gamma_3$ is required to represent one tubular component. To formalize what constitutes a semantic decomposition, we consider connectivity, length, and local orientation, to unify skeleton branches. We propose an algorithm for traversing the graph representation of the curve skeleton $G_\Upsilon (V,E,L)$, decomposing $G_\Upsilon$ into distinct paths, each corresponds to a semantic component. The algorithm starts at the root edge and explores as far as possible along edges, which provide the optimal choice at each stage.
\Figure[t][width=0.99\columnwidth]{figs/partition_skel.png}
{Partitioning the skeleton graph of the synthetic tubular object. $E$ comprises seven edges and is partitioned into three paths: $W_1 = \{ e_1, e_2, e_3 \}$, $W_2 = \{ e_4, e_5 \} $, and $W_3 = \{ e_6, e_7 \}$. We determine $W_1$ starting from the longest edge in $E$ denoted as $e^*$ towards its incident vertices. At each vertex, we traverse the edge with minimum orientation cost. Appending new edges terminates when a leaf vertex is visited, or the angle between two successive edges is smaller than $\theta_c$. We subtract $W_1$ from $E$, when $W_1$ is determined (see Algorithm \ref{alg:walk}). The blue filled-circles show vertices in $G_\Upsilon$. The edges are color-coded with full-lines. At vertices, arrows show where to traverse next when standing on $e^*$. The grey dash-lines show the previously calculated paths. \label{fig:skel_partition}}
We partition the skeleton graph of the object into several distinct paths union of which covers the set of graph edges. Formally, we partition $G_\Upsilon (V,E,L)$ into $m$ paths $W_i, i = 1, \dots ,m$ so that $\cup_i W_i = E$ and $W_i \cap W_k = \emptyset \; \forall i,k = 1, \dots, m,\; i \neq k$. To determine the paths, we require four conditions: 1) the path contains the longest edge not associated to any other path, 2) the path has the maximum number of edges, 3) the associated angle between two successive edges is bigger than $\theta_c$, and 4) the path locally minimize an orientations cost. Denoting two successive edges in a path as $e_s$ and $e_{s+1}$, the edge $e_{s+1}$ has the maximum angle compared to $e_s$ among the set of connected edges to $e_s$. The angle between two edges $e_s$ and $e_{s+1}$ is the angle between the line segments connecting endpoints of the skeleton branches associated with edge $e_s$ and $e_{s+1}$, and it lies in range $[0, \pi]$. We used Algorithm \ref{alg:walk} to determine the $m$ distinct paths on $G_\Upsilon$. Fig. \ref{fig:skel_partition} shows skeleton graph decomposition of the synthetic object $n = 7$ into three paths $m = 3$. Each path is equivalent to a sub-skeleton.
\begin{algorithm}[t]
\DontPrintSemicolon
\SetKwInOut{Input}{Input}\SetKwInOut{Output}{Output}
\Input{$G_\Upsilon=(V,E,L)$; $\theta_c$.}
\Output{Collection of distinct paths $\Lambda$.}
\SetAlgoLined
$\Lambda \leftarrow \emptyset$ \;
\While {$E \neq \emptyset$}{
$W \leftarrow \emptyset$; $e^* \leftarrow $ longest $e \in E$ \;
$V^* \leftarrow $ \{ incident vertices to $e^*$ \} \;
$W \leftarrow W \cup \{ e^* \}$ \;
\ForAll {$\upsilon \in V^*$}{
$e^{ref} \leftarrow e^*$ \;
$\upsilon^{next} \leftarrow \upsilon$ \;
\While {$ deg(\upsilon^{next}) > 1 $ {\bf and} $e^{ref} \neq \emptyset$}{
$CE \leftarrow$ \{ edges connected to $\upsilon^{next}$ \} $\setminus e^{ref}$ \;
$e^{next} \leftarrow \emptyset$ \;
\ForAll {$e^{ngb} \in CE$}{
$\theta_{max} \leftarrow \theta_c$; $\theta \leftarrow \angle (e^{ref}, e^{ngb})$ \;
\If {$\theta > \theta_{max}$}{
$\theta_{max} \leftarrow \theta$; $e^{next} \leftarrow e^{ngb}$ \;
}
}
$e^{ref} \leftarrow e^{next}$ \;
\If{$e^{ref} \neq \emptyset$}{
$\upsilon^{next} \leftarrow \upsilon_2$, where $e^{ref} = (\upsilon_2, \upsilon^{next})$ \;
$W \leftarrow W \cup \{ e^{ref} \}$ \;
}
}
}
$\Lambda \leftarrow \Lambda \cup \{W\}$ and $E \leftarrow E \setminus W$ \;
}
\caption{Decomposing the set of edges of $G_\Upsilon$ into distinct paths. A vertex and an edge are called incident if the vertex is one of the two vertices the edge connects.}
\label{alg:walk}
\end{algorithm}
\section{Cylindrical decomposition} \label{sec:critical_p}
In this section, we propose a method to decompose an object into parts and intersections by cutting the object at critical points. To determine critical points, we sweep the object along sub-skeletons in decomposition intervals to find locations where the object geometry changes substantially (see Fig. \ref{fig:cross_section}).
\subsection{Decomposition interval} \label{sec:dec_int}
We restrict the sweep of the object along each sub-skeleton to decomposition intervals in the proximity of a junction-point $j$ on sub-skeleton $\psi$, as illustrated in Fig. \ref{fig:dec_interval}. It is convenient to work with parametrized sub-skeleton $\psi (t): [0,1] \to \mathbb{R}^3$. We define two decomposition intervals $[t_s^+, t_e^+]$ and $[t_e^-, t_s^-]$ for each junction-point as in Fig. \ref{fig:dec_interval}a. To determine the boundaries of a decomposition interval, we define an upper threshold $r_s$ and a lower threshold $r_e$. We specify $r_s$ and $r_e$ based on the radius of the maximal inscribed ball at $t_j$ and two factors $\alpha_s \geq 1$ and $\alpha_e \geq 0$ where $\alpha_s \geq \alpha_e$, as $r_s = \alpha_s \times r$ and $r_e = \alpha_e \times r$.
To determine the thresholds, we use the signed arc-length from $j$. Define $t_j$ so that $j = \psi(t_j)$. Then $t_s^+$ ($t_s^-$) is such a point on the sub-skeleton that signed arc-length from $t_j$ to $t_s^+$ ($t_s^-$) equals $r_s$ ($-r_s$). And $t_e^+$ ($t_e^-$) is such a point on the sub-skeleton that signed arc-length from $t_j$ to $t_e^+$ ($t_e^-$) equals $r_e$ ($-r_e$). We have $t_s^+ < t_e^+ < t_j < t_e^- < t_s^-$.
The upper and lower thresholds may imply arc-distances outside parametrization limits of $\psi$. If the arch-length from $\psi(0)$ to $\psi(t_j)$ is smaller than $r_s$ ($r_e$) we assign $t_s^+ = 0$ ($t_e^+ = 0$). And if the arch-length from $\psi(t_j)$ to $\psi(1)$ is smaller than $r_s$ ($r_e$) we assign $t_s^- = 1$ ($t_e^- = 1$). Also, when a junction-point is at the either ends of a sub-skeleton, e.g., in a T-shape object, we define only one decomposition interval. Therefore, if $\psi(t_j) = 0$ ($\psi(t_j) = 1$) the only interval that we define is $[t_e^-, t_s^-]$ ($[t_s^+, t_e^-]$). Fig. \ref{fig:dec_interval}b shows decomposition intervals in the proximity of $j_1$ and $j_2$ on sub-skeletons $\psi_1, \psi_2$, and $\psi_3$.
\Figure[t][width=0.99\columnwidth]{figs/decomposition_int.png}
{(\textbf{a}) In the proximity of every junction-point, e.g. $j_1$ blue filled-circle, and on each sub-skeleton, e.g. $\psi_1$ green line, we define two decomposition intervals, $[t_s^+, t_e^+]$ and $[t_e^-, t_s^-]$, tracing $\psi_1$, from $t_s^+$ to $t_e^+$ and from $t_s^-$ to $t_e^-$ (red filled-circles). The lower and upper bounds of the intervals are two factors of the radius of the maximal inscribed ball at $t_j$, the green circle. (\textbf{b}) Decomposition intervals in the proximity of all junction-points $j_1$ and $j_2$ and for all sub-skeletons $\psi_1$, $\psi_2$, and $\psi_3$ are defined with the red filled-circles. Only in decomposition intervals, we are allowed to sweep the object. Arrows depict the sweeping direction to approach junction-points. \label{fig:dec_interval}}
\subsection{Critical point} \label{sec:critical_point}
A critical point on a sub-skeleton is such a point that the cross-sectional contour of the object at this point changes substantially (Fig. \ref{fig:cross_section}). We use the Hausdorff metric to compare geometrical changes between cross-sectional contours in a decomposition interval. The Hausdorff distance between two curves $C_1$ and $C_2$ is calculated as
\begin{align}
\displaystyle \mathcal{H}(C_1, C_2) &= \nonumber\\
\max &\{\sup_{p \in C_1} \; \inf_{q \in C_2} \; d(p, q),\; \sup_{q \in C_2} \; \inf_{p \in C_1} \; d(p, q)\},
\label{eq:HausD}
\end{align}
where d(.) is the Euclidean distance between two points. We sweep $\partial \Omega$ by a cross-sectional plane $\mathcal{P} \subset \mathbb{R}^3$ to extract the cross-sectional contours. A cross-sectional plane $\mathcal{P}(t)$ is a plane orthogonal to $\psi$ at every point $t$ along $\psi$. The plane normal is equal to the tangent vector to $\psi$ at point $\psi(t)$. We sweep $\partial \Omega$ by $\mathcal{P}$ along $\psi$ in $[t_s^+,t_e^+]$ interval starting at $t_s^+$ toward $t_e^+$, and in $[t_e^-,t_s^-]$ interval starting at $t_s^-$ toward $t_e^-$, as illustrated in Fig. \ref{fig:dec_interval}. Let $\mathcal{P}(t)$ intersects $\partial \Omega$ at an inquiry point $t$. Since we assumed that $\partial \Omega$ is homeomorphic to a 2-sphere, the cross-sectional contour $C(\varsigma):[0,1] \to \mathbb{R} ^2$ is a simple closed curve, where $C(0) = C(1)$. Translating $\mathcal{P}$ along $\psi(t)$ with $t$ moving in decomposition intervals, we compare the Hausdorff distance between the cross-sectional contour at $t$ denoted as $C_t$ with the average of visited cross-sectional contours $\mu$.
\Figure[t][width=0.99\columnwidth]{figs/mean_curve.png}
{(\textbf{a}) Two nearly similar curves $C_1$ and $C_2$. Turquoise arrows represent $OM(C_1,C_2)$, and pink arrows represent $OM(C_2,C_1)$. (\textbf{b}) The average curve $\mu$ obtained from the orthogonal correspondence between $C_1$, an already visited curve, and $C_2$ a new cross-sectional curve. \label{fig:mean_curve}}
To find the average curve $\mu$ between two nearly similar curves $C_1$ and $C_2$, we first need a one-to-one orthogonal mapping (OM) between $C_1$ and $C_2$. Consider that $C_1$ is parameterized by $\varsigma$. To each point $C_1(\varsigma)$ of $C_1$, the $OM(C_1,C_2)$ associates the closest point $C_2(\varsigma)$ on $C_2$ that lies on the line passing through $C_1(\varsigma)$ and having for direction the normal $N(\varsigma)$ to $C_1$ at $C_1(\varsigma)$. Having this mapping, then each point $C_2(\varsigma)$ of $C_2$ may be expressed as the normal offset $C_1(\varsigma) + d(\varsigma)N(\varsigma)$ of $C_1(\varsigma)$. We say that $C_1(\varsigma)$ is the closest normal projection of $C_2(\varsigma)$ onto $C_1$ and can express $C_2$ as a deformation of $C_1$ completely defined by the normal displacement field $d(\varsigma)$ (see Fig. \ref{fig:mean_curve}a) \cite{Chazal2005Projection-homeomorphicSurfaces}. The average curve obtained by this orthogonal correspondence is asymmetric: $OM(C_1, C_2)$ is not necessarily equal to $OM(C_2, C_1)$. Therefore, we consistently take $C_1$ as an already visited curve, $C_2$ as the new cross-sectional curve, and define the average curve over $OM(C_1,C_2)$ (see Fig. \ref{fig:mean_curve}).
We normalize the Hausdorff distance $\mathcal{H}(C_t, \mu)$ to the range $[0, 1]$ and denote it as $H_\rho(t)$. For that we first find a point interior to $C_t$ denoted as $\kappa$. We define $\kappa \in \mathbb{R}^2$ to be the intersection of $\mathcal{P}$ and $\psi$ at point $t$. Defining $d_{C_t}(\kappa) = \sup_{q \in C_t} d(\kappa, C_t)$, we write $H_\rho(t)$ as
\begin{equation}
\displaystyle H_\rho(t) = \dfrac{\mathcal{H}(C_t,\mu)}{\mathcal{H}(C_t,\mu) + d_{C_t}(\kappa)}.
\label{eq:Hous_th}
\end{equation}
We define a similarity threshold between cross-sectional contours as $\theta_H$. While sweeping $\partial \Omega$ along $\psi$ from $t_s^+$ ($t_s^-$) to $t_e^+$ ($t_e^-$), if $H_\rho(t) < \theta_H$, the inquiry continues to the next point. However, if $H_\rho(t) \geq \theta_H$ the inquiry stops at $t$ and the point is called a critical point, denoted as $t_{c_1}$ ($t_{c_2}$), as shown in Fig. \ref{fig:cross_section}. In $[t_s^+, t_e^-]$ ($[t_e^-, t_s^-]$), if at no inquiry point $H_\rho(t)$ exceeds $\theta_H$, we define the $t_{c_1}$ ($t_{c_2}$) as the point with minimum arc-distance $r$ ($-r$) to $\psi(t_j)$ at which $H_\rho$ is maximum.
\Figure[t][width=0.99\columnwidth]{figs/cross_section.png}
{Sweeping the object surface along the sub-skeleton $\psi_1$ at junction-point $j_1$ (blue filled-circle) between $[t_s^+,t_e^+]$ and $[t_s^+,t_e^+]$ (red filled-circles). Sweeping directions are shown with arrows. At any decomposition interval, if $H_\rho < \theta_H$ the inquiry continues to the next point. If $H_\rho(t) \geq \theta_H$ the inquiry stops at $t$ and the point is called a critical point. The critical point in the first interval is denoted as $t_{c_1}$ and in the second interval is denoted as $t_{c_2}$. \label{fig:cross_section}}
\Figure[t][width=0.99\columnwidth]{figs/gc.png}
{(\textbf{a}) Homotopy between two curves $C_{c_1}(\varsigma)$ and $C_{c_2}(\varsigma)$. Generalized cylinder along (\textbf{b}) a linear, (\textbf{c}) spline, and (\textbf{d}) sine interpolation between $\psi(t_{c_1})$ and $\psi(t_{c_2})$. \label{fig:gc}}
\section{Object reconstruction} \label{sec:obj_rec}
We cut the object at all critical points and decompose $\partial \Omega$ into $n$ parts, $n$ is the number of skeleton branches, and $\delta$ intersections, $\delta$ is the number of junction-points. We distinguish between an object part and an intersection such that the interior of an intersection includes a junction-point. The final decomposition step is to discard intersections and assign the same label to those object parts that are along the same sub-skeleton to obtain $m$ semantic tubular components, $m$ is the number of sub-skeletons. As we discard the intersections, we reconstruct the semantic tubular components using generalized cylinders. A generalized cylinder $\Phi(u,\varsigma) : [0,1]^2 \to \mathbb{R}^3$ represents an elongated surface on an arbitrary axis and smoothly varying cross-sections \cite{SHANI1984129}. In Cartesian coordinates $x_1,x_2,x_3$, the axis is parametrized by $u$ as $\zeta(u) = (x_1(u), x_2(u), x_3(u))$ and cross-section boundary is represented as $C_u(\varsigma) = (x_1(u, \varsigma), x_2(u, \varsigma))$. To construct $\Phi$, we apply a translational sweep along $\zeta(u)$ using closed simple curves $C_u(\varsigma)$ written as
\begin{align}
\Phi(u,\varsigma):=\{\zeta(u) \in \mathbb{R}^{3}, C_u(\varsigma) \in \mathbb{R}^{2} : u,\varsigma \in [0,1] \}.
\label{eq:genCylinder}
\end{align}
To obtain a parametric representation of generalized cylinders, it is convenient to employ a local coordinate system defined with the origin at each point of $\zeta(u)$. A convenient choice is the Frenet-Serret frame which is suitable for describing the kinematic properties of a particle moving along a continuous, differentiable curve in $\mathbb{R}^3$. The Frenet-Serret frame is an orthonormal basis composed of three unit vectors $e_T$, $e_N$, and $e_B$, where $e_T$ is the unit tangent vector, and $e_N$ and $e_B$ are the unit normal and unit binormal vectors, respectively. By defining the cross-section in the Frenet-Serret frame, we form a parametric representation of generalized cylinders \cite{BallardDanaHarryandBrown1982ComputerVision} as follows:
\begin{align}
\Phi(u,\varsigma) = \zeta(u) + x_1(u, \varsigma) e_N(u) + x_2(u, \varsigma) e_B(u)
\end{align}
To define $C_u(\varsigma)$, we use a homotopy between two curves $C_{c_1}(\varsigma)$ and $C_{c_2}(\varsigma)$, where the curves are obtained by cross-sectioning the object surface at critical points $t_{c_1}$ and $t_{c_2}$, respectively (see Fig. \ref{fig:gc}a). Let the simple closed curves $C_{c_1}(\varsigma)$ and $C_{c_2}(\varsigma)$ in $\mathbb{R} ^2$ be homotopic with a continuous map $h: [0, 1]^2 \to \mathbb{R} ^2$. So, we write:
\begin{align}
h(0,\varsigma) &= C_{c_1}(\varsigma), \; h(1,\varsigma) = C_{c_2}(\varsigma), \; \forall \varsigma \in [0,1], \\
h(u,0) &= h(u,1), \forall u \in [0,1],
\label{eq:homotopy}
\end{align}
where $h$ is called a homotopy from $C_{c_1}(\varsigma)$ to $C_{c_2}(\varsigma)$. We denote a cross-section at a point along $\zeta(u)$ as $C_u := h(u,.)$. Note that, $\mathbb{R} ^2$ is simply connected space. We use a linear homotopy between $C_{c_1}(\varsigma)$ to $C_{c_2}(\varsigma)$ defined as:
\begin{equation}
h(u,\varsigma) = (1-u) \; C_{c_1}(\varsigma) + u \; C_{c_2}(\varsigma),
\label{eq:linear_homotopy}
\end{equation}
where the computation on the right side is in $\mathbb{R} ^2$. Equation \eqref{eq:linear_homotopy} essentially indicates that we are moving from $C_{c_1}(\varsigma)$ to $C_{c_2}(\varsigma)$ along a straight line. To define the curve $\zeta(u)$, we use an interpolation between $\psi(t_{c_1})$ and $\psi(t_{c_2})$. Figs. \ref{fig:gc}b-d show $\Phi$ on different choices of $\zeta$.
\section{Experimental results}
In this section, we evaluate the effect of CSD parameters on object decomposition, present the applications and advantages of the proposed method in decomposing tubular objects, and show the performance of CSD applied on more general objects. We use the marching cubes algorithm \cite{Lorensen1987MarchingAlgorithm} to compute a triangulated mesh of the object surface, visualizing voxel-based objects.
\subsection{Parameter setting}
In all our experiments, we fix the value of $\alpha_e = 1$ (section \ref{sec:dec_int}), meaning that the distance of $t_e^+$ ($t_e^-$) to a junction-point is equal to the radius of the maximal inscribed ball at that junction-point. Here, we examine the effect of $\alpha_s$, which determines $t_s^+$ ($t_s^-$) of the decomposition intervals (section \ref{sec:dec_int}) for values equal to 10, 20, and 30. For that, we set the similarity threshold $\theta_H$ (section \ref{sec:critical_point}) equal to 0.7. We use a linear interpolation to define $\zeta$, the curve on which a generalized cylinder is defined (section \ref{sec:obj_rec}), and set the value of the angular threshold $\theta_c$ (section \ref{sec:skel_partition}) equal to $0 ^\circ$. Figs. \ref{fig:alpha_s}a-c show the decomposition of the synthetic tubular object for $\alpha_s$ equals 10, 20, and 30, respectively. The decomposition/reconstruction at $\alpha_s = 10$ is more faithful to the original object as the critical points are detected close to the junctions. Increasing the value of $\alpha_s$ enlarges the decomposition interval. Therefore, at $\alpha_s = 30$, CSD detects the critical points distant from the junctions, resulting in a bigger reconstruction error compared to small values of $\alpha_s$. It is worth noting that although setting $\alpha_s$ to small values provides more accurate decomposition/reconstruction results, it may also result in defining a critical point within an intersection. This can be the case when applying CSD to objects degraded with surface noise. The curve skeleton of an object with surface noise may not exactly lie in the center of the object, which means that the junction-point can be dislocated and the radius of the maximum inscribed ball at that junction-point be measured smaller than its true value. For $\alpha_s$ and $\alpha_e$, we suggest values in range [3, 20] and [0.5, 2], respectively.
We also examine the effect of $\theta_H$ value, which is the similarity threshold between cross-sectional contours and $\mu$. Figs. \ref{fig:theta_H}a-c show the decomposition of the tubular synthetic object at $\theta_H$ equals 0.6, 0.7, and 0.8, respectively. To better demonstrate the effect of $\theta_H$, we set $\alpha_s = 30$, which we earlier showed this could result in a substantial decomposition error. We use a linear interpolation to define $\zeta$ and set $\theta_c = 0 ^\circ$. At $\theta_H = 0.6$, CSD is sensitive to cross-sectional changes and does not tolerate the gradual increase of the tube diameter; hence critical points are detected distant from junction-points, and the reconstruction shows a low agreement with the original object. Increasing the value of $\theta_H$ to 0.7 increases the tolerance of CSD to cross-sectional changes. Therefore, despite distant starting points from junction-points, the reconstruction shows a better agreement to the original object, and at $\theta_H = 0.8$, the reconstructed object is faithful to the original object. Note that increasing $\theta_H$ elevates the tolerance of CSD to the cross-sectional changes quickly, e.g., at $\theta_H = 0.9$, the algorithm tolerates a nine times difference between a cross-section and $\mu$, and at $\theta_H = 0.95$, it tolerates a 19 times difference. We suggest $\theta_H$ to be in the range [0.7, 0.85].
\Figure[t][width=0.99\columnwidth]{figs/alpha_s.png}
{Decomposition of the synthetic tubular object at $\alpha_s = 10, 20, 30$ for fixed values of $\alpha_e = 1$ and $\theta_H = 0.7$. We use a linear interpolation to define $\zeta$ and set $\theta_c = 0 ^\circ$. At $\alpha_s = 10$, the decomposition/reconstruction is in agreement with the original object because critical points are detected close to the junctions. Increasing the value of $\alpha_s$ enlarges the decomposition intervals, which may result in inaccuracy while decomposition/reconstruction, e.g., at $\alpha_s = 30$. \label{fig:alpha_s}}
\Figure[t][width=0.99\columnwidth]{figs/theta_H.png}
{Decomposition of the synthetic tubular object at $\theta_H = 0.6, 0.7, 0.8$ for fixed values of $\alpha_s = 30$ and $\alpha_e = 1$. We use a linear interpolation to define $\zeta$ and set $\theta_c = 0 ^\circ$. Increasing the $\theta_H$ value increases the CSD tolerance in dealing with gradient cross-sectional changes of the tubes. At $\theta_H = 0.8$, CSD recognizes the critical points near to junction-points, despite distant starting points from the junctions. \label{fig:theta_H}}
In the skeleton partitioning section (section \ref{sec:skel_partition}), we showed that we merge two successive edges when the angle between them is bigger than $\theta_c$. Therefore, by setting $\theta_c$ to big values, we emphasize the straightness of a path, but then the path may not be maximal-length. Fig. \ref{fig:constraint_ang} shows how $\theta_c$ affects the number of semantic components. At $\theta_c = 0 ^\circ$, all successive edges are allowed to merge even with acute angles; therefore we obtain a minimum number of object partitions with maximal-length paths (Fig. \ref{fig:constraint_ang}b). By increasing $\theta_c$, only successive edges with a close-to-straight angle are allowed to be merged, which reduces the number of merges and increases the number of object partitions. Fig. \ref{fig:constraint_ang}c shows that decomposition for $\theta_c = 135 ^\circ$ yields four semantic components. At $\theta_c = 180 ^\circ$, no successive edges are merged, and every edge in the skeleton graph corresponds with an object part. Setting $\theta_c > 180 ^ \circ$ generates the maximum number of object parts, equal to the number of skeleton branches (Fig. \ref{fig:constraint_ang}d).
\Figure[t][width=0.99\columnwidth]{figs/theta_c.png}
{The angle between two successive edges in a path should be bigger than $\theta_c$ to be merged. (\textbf{a}) A synthetic tubular object, size: $128 \times 128 \times 128$ voxels. (\textbf{b}) Setting $\theta_c = 0 ^\circ$ produces maximal-length paths; the minimum number of object parts $m = 3$. (\textbf{c}) At $\theta_c = 135 ^\circ$, the number of object parts increases to $m = 4$. (\textbf{d}) At $\theta_c = 180 ^\circ$, every edge in the skeleton graph is a path, hence producing the maximum number of semantic components, which is equal to the number of skeleton branches, $m = 5$. \label{fig:constraint_ang}}
\Figure[t][width=0.99\columnwidth]{figs/overlapping_joints.png}
{Overlapping of decomposition intervals when junction-points are adjacent. (\textbf{a}) Two junction-points (left panel) and two intersections of parts (right panel, grey sections) when $\alpha_s$ is equal to 3. (\textbf{b}) Two junction-points (left panel) and one intersection of parts (right panel, grey section) when $\alpha_s$ is equal to 10. \label{fig:ov_junc}}
We design CSD to have the same number of part intersections as the number of junction-points. However, when two junction-points appear adjacent on a sub-skeleton, we can merge them into one, depending on the value of $\alpha_s$. Fig. \ref{fig:ov_junc}a shows a four-leg with two intersections. For $\alpha_s = 3$, the decomposition interval around $j_1$ does not cover $j_2$, and the decomposition interval around $j_2$ does not cover $j_1$; therefore, the back of the four-leg, within its body, is decomposed. Fig. \ref{fig:ov_junc}b shows that for $\alpha_s = 10$, the decomposition interval around $j_1$ includes $j_2$, and the decomposition interval around $j_2$ includes $j_1$, resulting in one object part intersection, where the four-leg body is the intersection of object parts.
\subsection{Axon segmentation in electron microscopy volumes}
\Figure[tp][width=0.95\textwidth]{figs/axon_dec.png}
{(\textbf{a}) Examples of foreground segmentation of myelinated axons with under-segmentation. (\textbf{b}) Decomposition using ACD \cite {Kaick2015ShapeAnalysis}. The point cloud representation of objects is first down-sampled to $50\,000$ points to enable the decomposition task in a reasonable time; this method over-segments the objects. (\textbf{c}) Skeleton-to-surface mapping \cite{Reniers2008ComputingMeasure} based on Voronoi partitioning of the surface using skeleton branches. (\textbf{d}) CSD decomposition provides the correct number of semantic components in under-segmented myelinated axons. The objects are reconstructed at intersections using generalized cylinders. Objects inside boxes are magnified. \label{fig:dec_axons}}
\Figure[t][width=1\textwidth]{figs/em_seg.png}
{(\textbf{a}) A large electron microscopy volume of the white matter. The size of the volume is $4\,055 \times 2\,002 \times 1\,292$ voxels in $x$, $y$, and $z$ directions, respectively. (\textbf{b}) A 3D rendering of myelinated axons (at one-third of the original resolution). CSD evaluates a preliminary segment for under-segmentation error(s), and if required, decomposes and reconstructs an under-segmented myelinated axon. (\textbf{c}) A 3D rendering of myelinated axons sampled at different locations illustrating the diversity of thickness and orientation in segmented axons. \label{fig:whole_vol}}
\Figure[t][width=1\textwidth]{figs/vessel_dec.png}
{(\textbf{a}) ACD \cite {Kaick2015ShapeAnalysis} over-segments the vascular network. (\textbf{b}) Skeleton-to-surface mapping \cite{Reniers2008ComputingMeasure} decomposes the object into 20 semantic components, and the boundaries between these components are not accurate. (\textbf{c}) CSD decomposes the object into eight semantic components and reconstructs the object at intersections. Objects inside boxes are magnified. The 3D image of the vascular network is acquired from Colin Macdonald's GitHub page. \label{fig:vessel}}
\Figure[t][width=0.97\textwidth]{figs/general_obj.png}
{A gallery of CSD decomposition of synthetic objects. \label{fig:general_objs}}
The primary purpose of developing CSD is to segment tens of thousands of myelinated axons in electron microscopy volumes of white matter, whose sizes are approximately $4000 \times 2000 \times 1300$ voxels. We generate a probability map of myelinated axons using deep convolutional neural networks (for details, we refer to \cite{Abdollahzadeh2021DeepACSONMicroscopy}). We threshold the probability map, and using connected component analysis, we obtain a preliminary foreground segmentation of myelinated axons. Fig. \ref{fig:dec_axons}a shows examples of myelinated axons after connected component analysis with an under-segmentation error(s): an axon intersects other axons or merges with the extra-axonal space. We apply CSD to evaluate every preliminary segmentation of myelinated axons for the under-segmentation error. If CSD recognizes an under-segmentation error, it decomposes the segmented component into its semantic parts. Fig. \ref{fig:dec_axons} shows the proposed decomposition of myelinated axons compared to the ACD (developed for point clouds) and skeleton-to-surface mapping approaches. To apply ACD on large objects, we first down-sample the point cloud representation of objects to $50\,000$ points, enabling the decomposition to be performed in a reasonable time (less than 10 minutes per object). Fig. \ref{fig:dec_axons}b shows that ACD over-segments myelinated axons. We perform skeleton-to-surface mapping decomposition based on the Voronoi partitioning of surfaces, using Euclidean distance to skeleton branches (Fig. \ref{fig:dec_axons}c). Because a curve skeleton captures the object geometry, skeleton-to-surface mapping decomposes an object close-to-semantic, but it does not recognize intersections of object parts and the boundary cuts are not correct. Fig. \ref{fig:dec_axons}d shows our decomposition of myelinated axons, where CSD generates the correct number of semantic parts for under-segmented myelinated axons and reconstructs axons at intersections using generalized cylinders. Fig. \ref{fig:whole_vol} shows the complete segmentation of myelinated axons in a large electron microscopy volume, where CSD scans, decomposes, and reconstructs about $30\,000$ myelinated axons.
\subsection{Decomposition of vascular networks}
We compare our method to ACD and skeleton-to-surface mapping for the decomposition of a vascular network. Fig. \ref{fig:vessel}a shows that ACD over-segments the vascular network. Fig. \ref{fig:vessel}b shows that skeleton-to-surface mapping decomposes the object into 20 semantic components based on the Euclidean distance to skeleton benches, but the method does not identify intersections, yet the boundary cuts are not correct. For example, Fig. \ref{fig:vessel}b (magnified box) shows that where the thin vessel (green partition) bends on the thick vessel (red partition), skeleton-to-surface mapping erroneously assigns a part of the thick vessel to the thin vessel, the part which is closer to the skeleton of the thin vessel. CSD decomposes the object into eight semantic components and reconstructs the object at intersections (Fig. \ref{fig:vessel}c).
\subsection{Decomposition of synthetic objects}
To demonstrate the general applicability of the CSD algorithm, we examine the proposed CSD method on synthetic voxelized objects. The synthetic objects are from the Princeton segmentation benchmark database \cite{Chen:2009:ABF}. We voxelize meshes from the Princeton database using a ray intersection method described in \cite{Patil2005Voxel-basedShapes}. The resolution of a voxelized object is determined using the bounding box of its OFF model; the bounding box values are normalized to range in $(0, 1]$ then multiplied by 128. The resolution at each dimension is proportional to the length of the bounding box at that dimension, e.g., the dimension with the maximum length is represented by 128 voxels. Fig. \ref{fig:general_objs} shows a gallery of decomposition on a mixture of objects with articulating parts, such as humans, octopuses, or pliers, and objects with moderate or small articulation, such as birds or fishes. Objects such as tables or airplanes include flat parts, which cannot be considered tubular. In Fig. \ref{fig:qual_comp}, we qualitatively compare several of the CSD results to how humans decompose an object into functional parts (the darker the seam, the more people have chosen a cut along that edge \cite{Chen:2009:ABF}). For quantitative analysis, we compare ACD, GCD, and the proposed CSD method to the human decomposition over objects acquired from the Princeton database. We select two objects per category from the Princeton database, excluding categories that have objects with the genus bigger than zero, such as cups or vases, and categories that have an ambiguous skeleton, such as busts or mechs. The objects and their corresponding human segmentations are converted from mesh to voxel-based representation. To aggregate evaluation metrics over multiple human segmentations for the same model and multiple models for the same object category, we report averages per category (averages are computed first within each model, then the results are averaged within each object category). We report Rand error \cite{Rand1971ObjectiveMethods} and boundary error \cite{Martin2001AStatistics} and propose using the variance of information (VOI), which has a better discriminative error range than Rand error \cite{Nunez-Iglesias2013MachineImages}. Table \ref{tbl:quant_syn} shows that the proposed CSD algorithm outperforms both ACD and GCD methods. We stress that these results refer to the decomposition of the voxel-based objects (as obtained in biomedical imaging experiments), and we make no claims about the superiority of CSD when, for example, mesh-based representation of the surfaces would be the natural one.
\Figure[t][width=0.99\columnwidth]{figs/qualitative_comp.png}
{Human shape decomposition compared to the CSD decomposition results. The human decomposition of meshes into functional parts are treated as probabilistic ground truth; darker lines show places where more people placed a segmentation boundary. \label{fig:qual_comp}}
\begin{table}[p]
\centering
\caption{Comparison of decomposition techniques using Rand error (RE), the variance of information (VOI), and boundary error (BE) to human shape decomposition; smaller values are better. The average decomposition time (Time) of all objects reported in this table is presented as mean $\pm$ standard deviation. These decomposition techniques are implemented using different programming languages (ACD \cite {Kaick2015ShapeAnalysis}: C++ and Matlab, GCD \cite{Zhou2015GeneralizedDecomposition}: C++, and CSD: Python) and take different object representations as input (ACD: point cloud, GCD: mesh, and CSD: voxel).}
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{p{40pt}p{40pt}p{40pt}p{40pt}p{40pt}}
\hline
Input model & Evaluation & ACD \cite{Kaick2015ShapeAnalysis} & GCD \cite{Zhou2015GeneralizedDecomposition} & Proposed \\ \hline
& RE & 0.2421 & 0.2897 & \textbf{0.2029} \\
Human & VOI & 3.0345 & \textbf{1.7049} & 1.7360 \\
& BE & 0.4002 & 0.4145 & \textbf{0.1703} \\ \hline
& RE & 0.2402 & \textbf{0.1650} & 0.2394 \\
Glasses & VOI & 1.7264 & \textbf{0.7111} & 0.8842 \\
& BE & 0.0654 & 0.0506 & \textbf{0.0088} \\ \hline
& RE & 0.2588 & \textbf{0.1450} & 0.1868 \\
Airplane & VOI & 1.9981 & \textbf{0.9944} & 1.4765 \\
& BE & 0.1066 & 0.0678 & \textbf{0.0677} \\ \hline
& RE & 0.2284 & \textbf{0.0699} & 0.0842 \\
Ant & VOI & 2.4227 & 1.0315 & \textbf{0.5978} \\
& BE & 0.4930 & 0.1573 & \textbf{0.0281} \\ \hline
& RE & 0.3135 & 0.1657 & \textbf{0.0215} \\
Octopus & VOI & 3.0222 & 0.9684 & \textbf{0.2418} \\
& BE & 0.5433 & 0.1736 & \textbf{0.0223} \\ \hline
& RE & 0.0353 & 0.0444 & \textbf{0.0109} \\
Table & VOI & 0.4656 & 0.2476 & \textbf{0.1082} \\
& BE & 0.1530 & 0.0305 & \textbf{0.0051} \\ \hline
& RE & 0.3721 & 0.3530 & \textbf{0.0621} \\
Teddy & VOI & 3.2006 & 1.1708 & \textbf{0.5542} \\
& BE & 1.5860 & 0.2210 & \textbf{0.1851} \\ \hline
& RE & 0.3625 & 0.3400 & \textbf{0.2201} \\
Hand & VOI & 2.7837 & \textbf{1.6841} & 1.1728 \\
& BE & 0.5442 & 0.2967 & \textbf{0.1739} \\ \hline
& RE & 0.2399 & 0.0830 & \textbf{0.0656} \\
Pliers & VOI & 2.7210 & 0.6070 & \textbf{0.5419} \\
& BE & 0.2736 & 0.0936 & \textbf{0.0386} \\ \hline
& RE & 0.5879 & 0.5140 & \textbf{0.2232} \\
Fish & VOI & 2.6352 & 1.7495 & \textbf{0.8839} \\
& BE & 1.1104 & 0.4107 & \textbf{0.1337} \\ \hline
& RE & 0.1764 & 0.2309 & \textbf{0.2208} \\
Bird & VOI & 1.6971 & \textbf{1.2629} & 1.3959 \\
& BE & 1.0669 & 0.1431 & \textbf{0.0901} \\ \hline
& RE & 0.2312 & 0.1897 & \textbf{0.1870} \\
Armadillo & VOI & 2.9909 & \textbf{1.3408} & 1.5887 \\
& BE & 0.8717 & 0.3583 & \textbf{0.3407} \\ \hline
& RE & 0.4082 & 0.3667 & \textbf{0.1868} \\
Four-leg & VOI & 2.9593 & 1.9207 & \textbf{1.0821} \\
& BE & 0.6796 & 0.3486 & \textbf{0.1738} \\ \hline \hline
& Time & 1287 $\pm$ 820 & 336 $\pm$ 98 & \textbf{186 $\pm$ 42} \\ \hline
\end{tabular}
\label{tbl:quant_syn}
\end{table}
\subsection{Decomposition of noisy synthetic objects}
We develop CSD to decompose voxel-based objects, e.g., objects extracted from biomedical images, where noise can degrade the object surface. To examine how noise affects decomposition techniques, as shown in Fig. \ref{fig:noise_dec_tech}a, we add impulse noise to the surface of an object for different noise density $D_n$ values; $D_n = 0, 0.1, 0.35$, and 0.6. Fig. \ref{fig:noise_dec_tech}b shows that ACD over decomposes the noise-free object to 64 parts and 82 parts when noise density equals 0.6. The GC decomposition of the noise-free object is approximately correct, but over-decomposes the noisy object at $D_n = 0.1$ (Fig. \ref{fig:noise_dec_tech}c). GCD does not decompose the object at stronger noise levels when $D_n$ equals 0.3 or 0.6. The proposed CSD method decomposes the objects at different noise levels with excellent performance, as shown in Fig. \ref{fig:noise_dec_tech}f. For the quantitative analysis, we compare decompositions against human segmentation using Rand error, VOI, and boundary error, as in Table \ref{tbl:noise_dec_tech}. The ACD and GCD decomposition errors are high on different metrics while constantly low for the proposed CSD method.
\Figure[t][width=0.99\textwidth]{figs/noisyobj_dec.png}
{Decomposition of (\textbf{a}) ACD \cite{Kaick2015ShapeAnalysis}, (\textbf{b}) GCD \cite{Zhou2015GeneralizedDecomposition}, and the proposed CSD method (\textbf{d-f}) when the object surface is degraded with the impulse noise for the noise density $D_n$ equals 0, 0.1, 0.35, and 0.6. The original Hausdorff distance as the similarity measure in the proposed CSD method is substituted with alternative metrics: (\textbf{d}) the shape context (ShCx) method \cite{Belongie2002ShapeContexts}, (\textbf{e}) original Hausdorff distance (OHD), and (\textbf{f}) modified Hausdorff distance (MHD) \cite{Dubuisson2002AMatching}. \label{fig:noise_dec_tech}}
\begin{table*}[t]
\centering
\caption{Comparison of different techniques over the voxel-based object shown in Fig. \ref{fig:noise_dec_tech} using Rand error (RE), the variance of information (VOI), and boundary error (BE). Smaller values indicate a closer decomposition to how humans decompose an object into functional parts. The average decomposition time (Time) of all objects reported in this table is presented as mean $\pm$ standard deviation. The GCD \cite{Zhou2015GeneralizedDecomposition} method does not decompose objects at strong noise levels when $D_n$ equals 0.3 or 0.6 and returns the object itself; therefore, we reported the GCD decomposition time for all experiments separately, showing the decomposition time of a failure case with $\infty$. These decomposition techniques are implemented using different programming languages (ACD \cite {Kaick2015ShapeAnalysis}: C++ and Matlab, GCD: C++, and CSD: Python) and take different object representations as input (ACD: point cloud, GCD: mesh, and CSD: voxel).}
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{p{60pt}p{60pt}p{60pt}p{60pt}p{60pt}p{60pt}p{60pt}}
\hline
Noise level & Evaluation & ACD \cite{Kaick2015ShapeAnalysis} & GCD \cite{Zhou2015GeneralizedDecomposition} & Proposed-ShCx & Proposed-OHD & Proposed-MHD \\ \hline
& RE & 0.3633 & 0.2659 & 0.0276 & \textbf{0.0211} & 0.0263 \\
Noise free & VOI & 2.9877 & 1.2150 & 0.3046 & \textbf{0.2482} & 0.2940 \\
& BE & 0.5377 & 0.1976 & 0.0213 & \textbf{0.0157} & 0.0204 \\ \hline
& RE & 0.4159 & 0.3021 & 0.0530 & \textbf{0.0207} & 0.0249 \\
$D_n$ = 0.1 & VOI & 3.5099 & 1.9760 & 0.5486 & \textbf{0.2424} & 0.2819 \\
& BE & 0.7263 & 0.4015 & 0.0795 & \textbf{0.0681} & 0.0705 \\ \hline
& RE & 0.4677 & 0.4951 & 0.1064 & 0.0432 & \textbf{0.0360} \\
$D_n$ = 0.35 & VOI & 3.7282 & 1.7636 & 0.7965 & 0.3702 & \textbf{0.3688} \\
& BE & 1.1797 & 0.7403 & 0.2295 & \textbf{0.2248} & 0.2304 \\ \hline
& RE & 0.4221 & 0.5567 & 0.2036 & 0.0999 & \textbf{0.0250} \\
$D_n$ = 0.6 & VOI & 3.9624 & 1.9533 & 1.2451 & 0.7191 & \textbf{0.2787} \\
& BE & 0.9429 & 0.4605 & 0.3615 & 0.3581 & \textbf{0.3569} \\ \hline \hline
& Time & 1302 $\pm$ 438 & [360, 482, $\infty$, $\infty$] & 1021 $\pm$ 468 & \textbf{191 $\pm$ 26} & 193 $\pm$ 25 \\ \hline
\end{tabular}
\label{tbl:noise_dec_tech}
\end{table*}
\subsection{Cross-sectional similarity metric}
To define a critical point in section \ref{sec:critical_point}, we use the Hausdorff distance as defined in \eqref{eq:HausD} to compare geometrical changes between cross-sectional contours. The Hausdorff distance can be sensitive to surface noise, showing a mismatch between cross-sectional contours that belong to the same object part. Figs. \ref{fig:noise_dec_tech}d-f show the CSD decomposition performance, substituting the original Hausdorff distance with alternative shape matching techniques: the modified Hausdorff distance \cite{Dubuisson2002AMatching} and shape context metric \cite{Belongie2002ShapeContexts}. We set $\theta_H$ equal to 0.8 for the original and modified Hausdorff distances and set a similarity threshold of 0.5 for the shape context metric. The surface of the synthetic object in Fig. \ref{fig:noise_dec_tech}a is degraded with the impulse noise for the noise density $D_n$ equals 0, 0.1, 0.35, and 0.6. In Table \ref{tbl:noise_dec_tech}, we quantitatively evaluate decompositions using different cross-sectional similarity metrics and at different noise levels to the human segmentation using Rand error, VOI, and boundary error. Results demonstrate that different CSD similarity metrics yield excellent decompositions for the noise-free object, where the original Hausdorff distance performs better than the shape context metric and modified Hausdorff distance. With increasing the noise density, however, modified Hausdorff distance performs better than the shape context and the original Hausdorff metrics. The modified Hausdorff distance produces an excellent decomposition across noise densities.
\subsection{Choice of skeletonization technique}
The skeleton partitioning step guides the CSD algorithm for semantic decomposition; therefore, the quality of the skeletonization itself is crucial for the decomposition. We compare the CSD skeletonization approach to the L1-medial skeletonization and ROSA techniques in terms of the topological correctness and centeredness of the extracted skeletons. We compare these skeletonization techniques on synthetic voxel-based objects from McGill 3D Shape Benchmark \cite{Siddiqi2008RetrievingSurfaces}. Fig. \ref{fig:skl_comp} shows that when two object parts appear very close, L1-medial (Fig. \ref{fig:skl_comp}a human hand) and ROSA (Fig. \ref{fig:skl_comp}b octopus) merge the object parts and form a cycle in genus zero objects. The CSD skeletonization, see Fig. \ref{fig:skl_comp}c, generates separate branches for different object parts. Also, flat object parts, such as tabletop, do not possess a reasonably meaningful curve skeleton, but CSD skeletonization generates a more meaningful skeleton than the other two methods. In terms of centeredness, L1-medial and ROSA do not stay within the object. The CSD distance-based skeletonization approach penalizes the skeleton where the path moves away from the center of the object. Table \ref{tbl:skl_time} shows the computation time spent on skeletonization with L1-medial, ROSA, and CSD methods. The CSD skeletonization is faster than the other two methods, rendering it suitable for large objects.
\Figure[t][width=1\textwidth]{figs/skel_comp.png}
{Skeletonization using (\textbf{a}) L1-medial \cite{Huang2013LCloud}, (\textbf{b}) ROSA \cite{Tagliasacchi2009CurveCloud}, and (\textbf{c}) CSD distance-based techniques. L1-medial and ROSA may incorrectly form cycles in the skeleton of genus zero objects when two object parts are close, while the CSD skeletonization generates separate branches for different object parts. Also, L1-medial and ROSA do not necessarily stay within the object, while the CSD distance-based skeletonization approach penalizes the skeleton where the path moves away from the center of the object. Objects inside boxes are magnified. \label{fig:skl_comp}}
\begin{table}
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.2}
\centering
\caption{The skeletonization time for objects in Fig. \ref{fig:skl_comp} using L1-medial \cite{Huang2013LCloud}, ROSA \cite{Tagliasacchi2009CurveCloud}, and the proposed CSD method. These methods are implemented using different programming languages (L1-medial: C++, ROSA: C++ and Matlab, and CSD: Python). L1-medial and ROSA take point clouds as input, while CSD takes voxel-based objects as input. Therefore, the times reported are not directly comparable but provide insight into the speed of skeletonization algorithms, which is important for our segmentation application of large objects. For L1-medial and ROSA, we report the number of points (pts). For CSD, we report the number of voxels representing objects; the bounding box of objects is $128 \times 128 \times 128$ voxels.}
\begin{tabular}{p{30pt}p{35pt}p{50pt}p{50pt}p{40pt}}
\hline
Model & Evaluation & L1-medial \cite{Huang2013LCloud} & ROSA \cite{Tagliasacchi2009CurveCloud} & Proposed \\ \hline
\multirow{2}{*}{Human} & Size & $1\,000$ pts & $13\,189$ pts & $30\,785$ vox \\
& Time (s) & 37 & 549 & \textbf{4} \\ \hline
\multirow{2}{*}{Human} & Size & $1\,000$ pts & $11\,107$ pts & $24\,897$ vox \\
& Time (s) & 31 & 403 & \textbf{4} \\ \hline
\multirow{2}{*}{Octopus} & Size & $1\,000$ pts & $6\,415$ pts & $9\,179$ vox \\
& Time (s) & 19 & 144 & \textbf{6} \\ \hline
\multirow{2}{*}{Four-leg} & Size & $1\,000$ pts & $11\,878$ pts & $30\,490$ vox \\
&Time (s)& 31 & 465 & \textbf{4} \\ \hline
\multirow{2}{*}{Table} & Size & $1\,000$ pts & $19\,499$ pts & $42\,323$ vox \\
&Time (s)& 80 & $1\,333$ & \textbf{4} \\ \hline
\end{tabular}
\label{tbl:skl_time}
\end{table}
\subsection{Computation time}
The time complexity of the sub-voxel precise skeletonization is $O(n \, N_\Omega \log N_\Omega)$, where $n$ is the number of skeleton branches, and $N_\Omega$ is the number of voxels in a discrete $\Omega$. The $N_\Omega \log N_\Omega$ factor is from the fast marching algorithm \cite{Sethian1996AFronts}. The time complexity to determine a critical point is $O(N_p)$, where $N_p$ is the number of inquiry points that we check for the cross-sectional changes in a decomposition interval. Defining the critical points is independent of $N_\Omega$. The complexity of the method is measured through the number of basic arithmetic operations performed; other factors that may also influence the execution time, such as the number of memory accesses or memory consumption, have not been considered.
A fair comparison between the computation times of different decomposition techniques by the wall clock time requires the same constraints for all techniques. Providing such constraints is challenging because different decomposition techniques use different programming languages or take different object representations as input. In spite of that, we demonstrate the average decomposition time of ACD, GCD, and the proposed CSD method in Tables \ref{tbl:quant_syn} and \ref{tbl:noise_dec_tech}. The average decomposition time of the proposed CSD is shorter than ACD and GCD in all our experiments: the average decomposition time of synthetic objects is 3 m for CSD, 21 m for ACD, and 5 m for GCD. Note that, although the computation times of the GCD and proposed CSD methods are close, we have not been able to decompose objects acquired from biomedical image datasets, e.g., axons (Fig. \ref{fig:dec_axons}) and the vascular network (Fig. \ref{fig:vessel}), using GCD within a day.
We also propose to reduce the CSD computation time by reducing the number of inquiry points $N_p$. For that, we propose to sub-sample the sub-skeletons by a sampling factor $sf$, as shown in Fig. \ref{fig:sf}. Increasing $sf$ reduces the computation time linearly while affecting decomposition results minimally. Fig. \ref{fig:sf} shows the decomposition of three objects at $sf$ equals 1, 4, 16, and 64. To evaluate the effect of sampling the sub-skeletons on decomposition results, we compared decomposition results over Rand error, VOI, and boundary error, considering decomposition at $sf = 1$ as the ground truth. Table \ref{tbl:sf} shows that decomposition by a factor of four substantially reduces the computation time, whereas the evaluation metrics worsen minimally. Reducing the computation is important when dealing with big voxel-based objects. For example, on a 2 $\times$ Intel Xeon E5 $2\,630$ CPU 2.4 GHz machine with 512 GB RAM using Python 3.6, the skeletonization of the myelinated axon shown in the first row of Fig. \ref{fig:dec_axons} ($N_\Omega = 395\,594$) consumes 117 s and defining its critical points 353 s, and sampling the sub-skeletons by $sf = 5$ reduces the decomposition time to 75 s.
\Figure[t][width=0.99\columnwidth]{figs/computation_time.png}
{Sampling object sub-skeletons to reduce the decomposition time. The decomposition of a human and a four-leg at (\textbf{a}) $sf = 1$, (\textbf{b}) $sf = 4$, (\textbf{c}) $sf = 16$, and (\textbf{d}) $sf = 64$. \label{fig:sf}}
\begin{table}
\caption{Evaluation of how sampling the sub-skeletons for $sf$ equals 1, 4, 16, and 64 affects decomposition results of objects in Fig. \ref{fig:sf} using Rand error (RE), the variance of information (VOI), and boundary error (BE), considering decomposition at $sf = 1$ as the ground truth. The decomposition results are achieved on a 4-core Intel CPU 3.41 GHz computer with 64 GB RAM using Python 3.6.}
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{p{35pt}p{35pt}p{33pt}p{33pt}p{33pt}p{33pt}}
\hline
Model & Evaluation & $sf = 1$ & $sf = 4$ & $sf = 16$ & $sf = 64$ \\ \hline
\multirow{4}{*}{Human} & RE & & \textbf{0.0140} & 0.0203 & 0.0340 \\
& VOI & & \textbf{0.2507} & 0.4082 & 0.5511 \\
& BE & & \textbf{0.0496} & 0.0915 & 0.1797 \\
& Time (s) & 108.1 & 30.5 & 9.2 & \textbf{4.5} \\ \hline
\multirow{4}{*}{Four-leg} & RE & & \textbf{0.0319} & 0.0359 & 0.0519 \\
& VOI & & \textbf{0.2149} & 0.2775 & 0.3971 \\
& BE & & \textbf{0.0600} & 0.0652 & 0.0714 \\
& Time (s) & 174.0 & 48.3 & 14.1 & \textbf{4.9} \\\hline
\end{tabular}
\label{tbl:sf}
\end{table}
\section{Conclusion}
In this paper, we proposed the application of 3D shape decomposition in image segmentation. We presented the novel CSD algorithm to decompose and reconstruct under-segmented tubular objects. The CSD method is guided by the curve skeleton decomposition, decomposing a tubular object into maximal-length, approximately straight parts. The object is cut at the intersection of parts using translational sweeps and reconstructed by generalized cylinders.
We demonstrated the application of CSD on biomedical imaging volumes and synthetic objects. In particular, we applied CSD as instance segmentation to deep learning-based semantic segmentation of myelinated axons. Hundreds of thousands of myelinated axons were automatically evaluated for under-segmentation error, and under-segmented myelinated axons were decomposed into their constituent axons, using the same parameter values for all objects in all electron microscopy datasets. We showed that CSD outperforms state-of-the-art techniques in decomposing voxel-based objects and is robust to severe surface noise. CSD is highly parallelizable, substantially reducing the computation time of the segmentation in large biomedical imaging datasets. The proposed CSD algorithm allows for including the cylindricity as a global shape-objective for a fast 3D segmentation of tubular objects in large biomedical imaging datasets.
\section*{Acknowledgmen}
The authors acknowledge CSC-IT Center for Science, Finland and Bioinformatics Center, University of Eastern Finland, Finland, for computational resources.
\section*{Competing interests}
\noindent The authors declare that they have no conflict of interest.
\section*{Availability of data and material}
\noindent The source code of the CSD algorithm is available at https://github.com/aAbdz/CylShapeDecomposition.
\bibliographystyle{IEEEtran}
|
1,108,101,565,640 | arxiv | \section{Introduction}
In a 1990 paper \cite{Knuth90} Knuth introduced the class of nested
CNF formulas and showed that their satisfiability can be decided in
polynomial time. A CNF formula is \emph{nested} if its variables can
be linearly ordered such that there is no pair of clauses that
\emph{straddle} each other; a clause $c$ straddles a clause $c'$ if
there are variables $x,y \in {\normalfont \textsf{var}}(c)$ and $z\in {\normalfont \textsf{var}}(c')$ such that
$x<z<y$ in the linear ordering under consideration. \textsc{Nested}\xspace denotes
the class of nested CNF formulas. For an example see
Figure~\ref{fig:nested}.
\begin{figure}[tbh]
\centering
\tikzset{var/.style={inner sep=.15em,circle,fill=black,draw},
clause/.style={minimum size=1mm,rectangle,fill=white,draw},
label distance=-1pt}
\centering
\longversion{
\begin{tikzpicture}
\node (1) at (1,0) [var,label=below:$\strut t$] {};
\node (2) at (2,0) [var,label=below:$\strut u$] {};
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\node (4) at (4,0) [var,label=below:$\strut w$] {};
\node (5) at (5,0) [var,label=below:$\strut x$] {};
\node (6) at (6,0) [var,label=below:$\strut y$] {};
\node (7) at (7,0) [var,label=below:$\strut z$] {};
\node (s) at (1.5,0.4) [clause,label=above:$c_1$] {};
\node (t) at (3,0.4) [clause,label=above:$c_2$] {};
\node (u) at (4.5,0.4) [clause,label=above:$c_3$] {};
\node (v) at (5.5,0.4) [clause,label=above:$c_4$] {};
\node (w) at (6.5,0.4) [clause,label=above:$c_5$] {};
\node (x) at (2,1.3) [clause,label=above:$c_6$] {};
\node (y) at (6,1.3) [clause,label=above:$c_7$] {};
\node (z) at (3,2.5) [clause,label=above:$c_8$] {};
\draw (1)--(s)--(2)--(t)--(4)--(u)--(5)--(v)--(6)--(w)--(7)
(3)--(t)
(1) ..controls +(.05,.8) ..(x)--(2)
(4) .. controls +(-.3,.8).. (x)
(5) ..controls +(.05,.8) .. (y)
(7) ..controls +(-.05,.8) .. (y)
(1) .. controls +(0,1.5) .. (z)
(5) .. controls +(0,1.5) .. (z)
(4) .. controls +(0,.8) .. (z)
;
\end{tikzpicture}
}
\shortversion{
\begin{tikzpicture}
\node (1) at (1,0) [var,label=below:$\strut t$] {};
\node (2) at (2,0) [var,label=below:$\strut u$] {};
\node (3) at (3,0) [var,label=below:$\strut v$] {};
\node (4) at (4,0) [var,label=below:$\strut w$] {};
\node (5) at (5,0) [var,label=below:$\strut x$] {};
\node (6) at (6,0) [var,label=below:$\strut y$] {};
\node (7) at (7,0) [var,label=below:$\strut z$] {};
\node (s) at (1.5,0.4) [clause,label=above:$c_1$] {};
\node (t) at (3,0.4) [clause,label=above:$c_2$] {};
\node (u) at (4.5,0.4) [clause,label=above:$c_3$] {};
\node (v) at (5.5,0.4) [clause,label=above:$c_4$] {};
\node (w) at (6.5,0.4) [clause,label=above:$c_5$] {};
\node (x) at (2,1.5) [clause,label=above:$c_6$] {};
\node (y) at (6,1.5) [clause,label=above:$c_7$] {};
\node (z) at (3,2.5) [clause,label=above:$c_8$] {};
\draw (1)--(s)--(2)--(t)--(4)--(u)--(5)--(v)--(6)--(w)--(7)
(3)--(t)
(1) ..controls +(.1,.5) ..(x)--(2)
(4) .. controls +(-.3,.5).. (x)
(5) ..controls +(.1,.5) .. (y)
(7) ..controls +(-.1,.5) .. (y)
(1) .. controls +(0,1.5) .. (z)
(5) .. controls +(0,1.5) .. (z) (4)--(z)
;
\end{tikzpicture}
}
\caption{Incidence graph of the nested formula $F=\bigwedge_{i=1}^8 c_i$
with
$c_1=t \vee \neg u$,
$c_2=u \vee v \vee w$,
$c_3=w \vee x$,
$c_4=x \vee \neg y$,
$c_5= y \vee \neg z$,
$c_6= t \vee u \vee \neg w$,
$c_7= \neg x \vee z$,
$c_8= \neg t \vee w \vee x$.}
\label{fig:nested}
\end{figure}
Since nested formulas have incidence graphs of bounded
treewidth~\cite{BiedlH04}, one can use treewidth-based
algorithms~\cite{FischerMakowskyRavve06,SamerSzeider10} to even
compute the number of satisfying truth assignments of nested formulas
in polynomial time (incidence graphs are defined in
Section~\ref{section:prelims}). Hence the problems SAT and \#SAT are
polynomial for nested formulas.
The aim of this paper is to extend the nice computational properties of
nested formulas to formulas that are not nested but are of small
distance from being nested. We measure the distance of a CNF formula $F$
from being nested as the size of a smallest set $B$ of variables, such
that for all partial truth assignments $\tau$ to $B$, the reduced
formula $F[\tau]$ is nested. Such a set $B$ is called a \emph{strong
backdoor set} with respect to the class of nested
formulas~\cite{WilliamsGomesSelman03}, or strong \textsc{Nested}\xspace-backdoor set,
for short. Once we have found such a backdoor set of size $k$, we can
decide the satisfiability of $F$ by checking the satisfiability of $2^k$
nested formulas, or for model counting, we can take the sum of the
number of models of the $2^k$ nested formulas. Thus the problems SAT and
\#SAT can be solved in time $O(2^k\Card{F}^c)$ where $\Card{F}$ denotes
the length of $F$ and $k$ denotes the size of the given strong
\textsc{Nested}\xspace-backdoor set; $c$ is a small constant. In other words, the
problems SAT and \#SAT are \emph{fixed-parameter tractable} for
parameter~$k$ (for background on fixed-parameter tractability see
Section~\ref{section:prelims}). However, in order to use the backdoor
set we must find it first. Is the detection of strong \textsc{Nested}\xspace-backdoor
sets fixed-parameter tractable as well?
Let $\mathbf{sb}_{\mathbf{N}}(F)$ denote the size of a smallest strong \textsc{Nested}\xspace-backdoor
set of a CNF formula~$F$. To find a strong backdoor set of size
$k=\mathbf{sb}_{\mathbf{N}}(F)$ one can try all possible sets of variables of size at most
$k$, and check for each set whether it is a strong backdoor
set. However, for a formula with $n$ variables we have to check
$\binom{n}{k}=\Omega(n^k)$ such sets. Thus, this brute-force approach
scales poorly in~$k$ and does not provide fixed-parameter
tractability, as the order of the polynomial increases with~$k$.
In this paper we show that one can overcome this limitation with a
more sophisticated algorithm. \emph{We show that the problems SAT and
\#SAT are fixed-parameter tractable when parameterized by $\mathbf{sb}_{\mathbf{N}}$,
the size of a smallest strong \textsc{Nested}\xspace-backdoor set, even when the
backdoor set is not provided as an input.}
Our algorithm is constructive and uses the Grid Minor Theorem of
Robertson and Seymour~\cite{RobertsonSeymour86b} to either find that
the incidence graph of the formula has bounded treewidth---a case that
is solved using model checking for monadic second order
logic~\cite{ArnborgLagergrenSeese91}---or to find many vertex-disjoint
obstructions in the incidence graph. For the latter case, new
combinatorial arguments are used to find a small strong backdoor set.
Combining both cases leads to an algorithm producing a strong backdoor
set of a given formula $F$ of size at most $2^k$ for
$k=\mathbf{sb}_{\mathbf{N}}(F)$. Solving all the $2^{2^{k}}$ resulting nested formulas
provides a solution to~$F$.
Our result provides a new parameter $\mathbf{sb}_{\mathbf{N}}$ that makes SAT and \#SAT
fixed-parameter tractable. The parameter $\mathbf{sb}_{\mathbf{N}}$ is \emph{incomparable}
with other known parameters that make SAT and \#SAT fixed-parameter
tractable. Take for instance the treewidth of the incidence graph of
a CNF formula $F$, denoted ${\mathbf{tw}}^*(F)$. As mentioned above, SAT and
\#SAT are fixed-parameter tractable for parameter
${\mathbf{tw}}^*$~\cite{FischerMakowskyRavve06,SamerSzeider10}, and
${\mathbf{tw}}^*(F)\leq 3$ holds if $\mathbf{sb}_{\mathbf{N}}(F)=0$ (i.e., if $F\in
\textsc{Nested}\xspace$)~\cite{BiedlH04}. However, by allowing only $\mathbf{sb}_{\mathbf{N}}(F)=1$ we
already get formulas with arbitrarily large ${\mathbf{tw}}^*(F)$. This can be
seen as follows. Take a CNF formula $F_n$ whose incidence graph is an
$n\times n$ square grid, with vertices $v_{i,j}$, $1\leq i,j\leq n$.
Assume the clauses correspond to vertices $v_{i,j}$ where $i+j$ is
even, and call a clause even or odd according to whether for the
corresponding vertex $v_{i,j}$, the sum $i+j$ is a multiple of $4$ or
not, respectively. It is well known that the $n\times n$ grid, $n\ge 2$, has
treewidth $n$ (folklore).
Hence we have ${\mathbf{tw}}^*(F)=n$. Now
take a new variable $x$ and add it positively to all odd clauses and
negatively to all even clauses. Let $F_n^x$ denote the new
formula. Since the incidence graph of $F_n$ is a subgraph of the
incidence graph of $F_n^x$, we have ${\mathbf{tw}}^*(F_n^x)\geq
{\mathbf{tw}}^*(F_n)=n$. However, setting~$x$ to true removes all odd clauses
and thus yields a formula whose incidence graph is a disjoint union of
paths, which is easily seen to be nested. Similarly, setting~$x$ to
false yields a nested formula as well. Hence $\{x\}$ forms a strong
\textsc{Nested}\xspace-backdoor set, and so $\mathbf{sb}_{\mathbf{N}}(F)=1$. One can also construct
formulas where $\mathbf{sb}_{\mathbf{N}}$ is large and ${\mathbf{tw}}^*$ is small, for example by taking the
variable-disjoint union $F$ of formulas $F_i = (x_i \vee y_i \vee z_i) \wedge (\neg x_i \vee y_i \vee z_i)$ with $\mathbf{sb}_{\mathbf{N}}(F_i)=1$ and
${\mathbf{tw}}^*(F_i)=2$, $1\leq i \leq n$. Then ${\mathbf{tw}}^*(F)={\mathbf{tw}}^*(F_i)=2$, but
$\mathbf{sb}_{\mathbf{N}}(F)=\sum_{i=1}^n \mathbf{sb}_{\mathbf{N}}(F_i)=n$.
One can also define \emph{deletion backdoor sets} of a CNF formula $F$
with respect to a base class of formulas by requiring that deleting
all literals $x,\neg x$ with $x\in B$ from~$F$ produces a formula that
belongs to the base class~\cite{NishimuraRagdeSzeider07}. For many
base classes it holds that every deletion backdoor set is a strong
backdoor set, but in most cases, including the base class \textsc{Nested}\xspace, the
reverse is not true. In fact, it is easy to see that if a CNF formula
$F$ has a \textsc{Nested}\xspace-deletion backdoor set of size $k$, then
${\mathbf{tw}}^*(F)\leq k+3$. In other words, the parameter ``size of a smallest
deletion \textsc{Nested}\xspace-backdoor set'' is dominated by the parameter
incidence treewidth and therefore of limited interest. We note in
passing, that one can use the algorithm from \cite{MarxSchlotter12} to
show that the detection of deletion \textsc{Nested}\xspace-backdoor sets is
fixed-parameter tractable.
\paragraph{Related Work.}
Williams \emph{et al.}\xspace~\cite{WilliamsGomesSelman03} introduced the notion of back\-door sets\xspace
to explain favorable running times and the heavy-tailed behavior of
SAT and CSP solvers on practical instances.
The parameterized complexity of finding small backdoor sets was
initiated by Nishimura \emph{et al.}\xspace~\cite{NishimuraRagdeSzeider04-informal}
who showed that with respect to the classes of Horn formulas and of
2CNF formulas, the detection of strong backdoor sets is
fixed-parameter tractable. Their algorithms exploit the fact that for
these two base classes strong and deletion backdoor sets coincide.
For other base classes,
deleting literals is a less
powerful operation than applying partial truth assignments. This is
the case for the class \textsc{Nested}\xspace but also for the class \textsc{RHorn}\xspace of
renamable Horn formulas. In fact, finding a deletion \textsc{RHorn}\xspace-backdoor
set is fixed-parameter tractable~\cite{RazgonOSullivan09}, but it is
open whether this is the case for the detection of strong
\textsc{RHorn}\xspace-backdoor sets. For clustering formulas the situation is
similar: detection of deletion backdoor sets is fixed-parameter
tractable, detection of strong backdoor sets is most probably not
\cite{NishimuraRagdeSzeider07}. Very recently, the authors of the
present paper showed that for the base class of formulas whose
incidence graph is acyclic there is a fixed-parameter approximation
algorithm for strong backdoor sets. That is, the following problem is
fixed-parameter tractable: find a strong backdoor set of size at most
$k$ or decide that there is no strong backdoor set of size at most
$2^k$~\cite{GaspersSzeider11a}. The present paper extends the ideas
from \cite{GaspersSzeider11a} to the significantly more involved case
with \textsc{Nested}\xspace as the base class.
We conclude this section by referring to a recent survey on the
parameterized complexity of backdoor sets~\cite{GaspersSzeider11festschrift}.
\section{Preliminaries}
\label{section:prelims}
\paragraph{Parameterized Complexity.}
Parameterized Complexity \cite{DowneyFellows99,FlumGrohe06,Niedermeier06} is a two-di\-men\-sio\-nal framework to classify the complexity of problems based on their
input size $n$ and some additional parameter $k$.
It distinguishes between running times of the form $f(k) n^{g(k)}$ where the degree
of the polynomial depends on $k$ and running times of the form $f(k) n^{O(1)}$ where the exponential part of the running time is independent of $n$.
A parameterized problem is
\emph{fixed-parameter tractable} (\text{\normalfont FPT}) if there exists an algorithm that
solves an input of size $n$ and parameter $k$ in time bounded by $f(k)
n^{O(1)}$. In this case we say that the \emph{parameter
dependence} of the algorithm is $f$ and we call it an
\emph{\text{\normalfont FPT}\ algorithm}.
Parameterized Complexity has a hardness theory, similar to the theory
of \text{\normalfont NP}-completeness to show that certain problems have no \text{\normalfont FPT}\
algorithm under complexity-theoretic assumptions.
\paragraph{Graphs.}
Let $G=(V,E)$ be a simple, finite graph.
Let $S \subseteq V$\longversion{ be a subset of its vertices} and $v\in V$\longversion{ be a vertex}.
We denote by $G - S$ the graph obtained from $G$ by removing all vertices in $S$ and all edges incident to vertices in $S$.
We denote by $G[S]$ the graph $G - (V\setminus S)$.
The \emph{(open) neighborhood} of $v$ is $N(v) = \set{u\in V : uv\in E}$, the \emph{(open) neighborhood} of $S$ is $N(S) = \bigcup_{u\in S}N(u)\setminus S$, and their \emph{closed
neighborhoods} are $N[v] = N(v)\cup \set{v}$ and $N[S] = N(S)\cup S$, respectively.
A $v_1$--$v_k$ \emph{path} $P$ of length $k$ in $G$ is a sequence of $k$ pairwise distinct vertices $(v_1, v_2, \cdots, v_k)$ such that $v_i v_{i+1}\in E$ for each $i\in \{1, \dots, k-1\}$.
The vertices $v_1$ and $v_k$ are the \emph{endpoints} of $P$ and all other vertices from~$P$ are \emph{internal}. An edge is \emph{internal} to
$P$ if it is incident to two internal vertices from $P$.
Two or more paths are independent if none of them contains an inner
vertex of another.
\longversion{
A \emph{tree decomposition} of $G$ is a pair
$(\{X_i : i\in I\},T)$
where $X_i \subseteq V$, $i\in I$, and $T$ is a tree with elements
of $I$ as nodes
such that:
\begin{enumerate}
\item $\bigcup_{i\in I} X_i = V$;
\item $\forall uv\in E$, $\exists i \in I$ such that $\{u,v\}
\subseteq X_i$;
\item $\forall i,j,k \in I$, if $j$ is on the path from $i$ to $k$ in $T$
then $X_i \cap X_k \subseteq X_j$.
\end{enumerate}
The \emph{width} of a tree decomposition is $\max_{i \in I} |X_i|-1$. }%
The \emph{treewidth} \cite{RobertsonSeymour86} of $G$
\longversion{is the minimum width taken over all tree decompositions
of $G$ and it }is denoted by ${\mathbf{tw}}(G)$.
A graph is \emph{planar} if it can be drawn in the plane with no crossing edges.
For other standard graph-theoretic notions not defined here, we refer to \cite{Diestel00}.
\paragraph{CNF Formulas and Satisfiability.}
We consider propositional formulas in conjunctive normal form (CNF) where no clause contains
a complementary pair of literals.
For a clause $c$, we write ${\normalfont \textsf{lit}}(c)$ and ${\normalfont \textsf{var}}(c)$ for the sets of literals and variables
occurring in $c$, respectively.
For a CNF formula $F$ we write ${\normalfont \textsf{cla}}(F)$ for its set of clauses,
${\normalfont \textsf{lit}}(F) = \bigcup_{c\in {\normalfont \textsf{cla}}(F)} {\normalfont \textsf{lit}}(c)$ for its set of literals, and
${\normalfont \textsf{var}}(F) = \bigcup_{c\in {\normalfont \textsf{cla}}(F)} {\normalfont \textsf{var}}(c)$ for its set of variables.
For a set $X\subseteq {\normalfont \textsf{var}}(F)$ we denote by $2^X$ the set of
all mappings $\tau:X\rightarrow \set{0,1}$, the \emph{truth assignments} on $X$.
A truth assignment on $X$
can be extended to
the literals over $X$
by setting $\tau(\neg x) = 1-\tau(x)$ for all $x\in X$.
Given a CNF formula $F$ and a truth assignment $\tau \in 2^X$ we define
$F[\tau]$ to be the formula obtained from $F$ by removing all clauses $c$
such that $\tau$ sets a literal of $c$ to~1, and removing the literals set to~0
from all remaining clauses.
A CNF formula $F$ is \emph{satisfiable} if there is some $\tau\in
2^{{\normalfont \textsf{var}}(F)}$ with $F[\tau]=\emptyset$.
SAT is the $\text{\normalfont NP}$-complete problem of deciding whether a given CNF formula is
satisfiable~\cite{Cook71,Levin73}. \#SAT is the \#P-complete problem of
determining the number of distinct $\tau\in 2^{{\normalfont \textsf{var}}(F)}$ with $F[\tau]=\emptyset$ \cite{Valiant79b}.
\paragraph{Nested Formulas.}
Consider a linear order $<$ of the variables of a CNF formula $F$.
A clause $c$ \emph{straddles} a clause $c'$ if there are variables $x,y \in {\normalfont \textsf{var}}(c)$ and $z\in {\normalfont \textsf{var}}(c')$
such that $x<z<y$. Two clauses \emph{overlap} if they straddle each other.
A CNF formula $F$ is \emph{nested} if there exists a linear ordering $<$ of ${\normalfont \textsf{var}}(F)$ in which no two clauses of $F$ overlap
\cite{Knuth90}.
The satisfiability of a nested CNF formula can be determined in polynomial time \cite{Knuth90}.
The \emph{incidence graph} of a CNF formula $F$ is the bipartite graph ${\normalfont \textsf{inc}}(F)=(V,E)$ with
$V = {\normalfont \textsf{var}}(F) \cup {\normalfont \textsf{cla}}(F)$ and for a variable $x \in {\normalfont \textsf{var}}(F)$ and a clause $c \in {\normalfont \textsf{cla}}(F)$
we have $x c \in E$ if $x\in {\normalfont \textsf{var}}(c)$. The \emph{sign} of the edge $x c$ is \emph{positive}
if $x\in {\normalfont \textsf{lit}}(c)$ and \emph{negative} if $\neg x \in {\normalfont \textsf{lit}}(c)$.
The graph ${\normalfont \textsf{inc+u}}(F)$ is ${\normalfont \textsf{inc}}({\normalfont \textsf{univ}}(F))$, where ${\normalfont \textsf{univ}}(F)$ is obtained from $F$ by adding a \emph{universal} clause $c^*$ containing all variables of $F$.
By a result of Kratochv\'{\i}l and K\v{r}iv\'{a}nek \cite{KratochK93}, $F$ is nested iff\xspace ${\normalfont \textsf{inc+u}}(F)$ is planar.
Since ${\normalfont \textsf{inc}}(F)$ has treewidth at most $3$ if $F$ is nested \cite{BiedlH04}, the number of satisfying assignments of
$F$ can also be counted in polynomial time \cite{FischerMakowskyRavve06,SamerSzeider10}.
\paragraph{Backdoors.}
Backdoor sets are defined with respect to a fixed class $\cC$ of CNF
formulas, the \emph{base class}.
Let $B$ be a set of propositional variables and $F$ be a CNF formula.
$B$ is a \emph{strong} \emph{$\cC$-back\-door set\xspace{}} of $F$ if $F[\tau]\in \cC$ for each $\tau \in 2^B$.
$B$ is a \emph{deletion $\cC$-back\-door set\xspace{}} of $F$ if $F - B \in \cC$, where $F - B = \set{C \setminus \set{x, \neg x : x\in B} : C \in F}$.
If we are given a strong $\cC$-back\-door set\xspace of $F$ of
size $k$, we can reduce the satisfiability of $F$ to the satisfiability
of $2^k$ formulas in $\cC$.
Thus SAT becomes \text{\normalfont FPT}\ in $k$.
If $\cC$ is clause-induced (i.e.\xspace, $F\in \cC$ implies $F'\in \cC$ for every $F'\subseteq F$),
any deletion $\cC$-back\-door set\xspace of $F$
is a strong $\cC$-back\-door set\xspace of $F$.
The interest in deletion back\-door sets\xspace is motivated for base classes where they
are easier to detect than strong back\-door sets\xspace.
The challenging problem is to find a strong
or deletion
$\cC$-back\-door set\xspace of size at most $k$ if it exists.
Denote by $\mathbf{sb}_{\mathbf{N}}(F)$ the size of a smallest strong \NBDS.
\paragraph{Minors and Grids.}
The \emph{$r$-grid} is the graph
$L_r=(V,E)$ with vertex set $V = \{(i, j) : 1 \le i \le r$, $1 \le j \le r\}$ in which two vertices
$(i,j)$ and $(i',j')$ are adjacent iff\xspace $|i-i'|+|j-j'|=1$.
We say that a vertex $(i,j)\in V$ has horizontal index $i$ and vertical index $j$.
A graph $H$ is a \emph{minor} of a graph $G$ if $H$ can be obtained from a subgraph of $G$
by contracting edges. The \emph{contraction} of an edge $uv$
makes $u$ adjacent to all vertices in $N(v)\setminus \{u\}$ and removes $v$.
If $H$ is a minor of $G$, then one can find a model of $H$ in $G$.
A \emph{model} of $H$ in $G$ is a set of vertex-disjoint connected subgraphs
of $G$, one subgraph $C_u$ for each vertex $u$ of $H$, such that if $uv$ is an edge of $H$, then
there is an edge of $G$ with one endpoint in $C_u$ and the other in $C_v$.
By Wagner's theorem \cite{Wagner37}, a graph is planar iff\xspace it has no
$K_{3,3}$ and no $K_5$ as a minor. Here, $K_5$ denotes the complete graph
on $5$ vertices and $K_{3,3}$ the complete bipartite graph with $3$ vertices in
both independent sets of the bipartition.
We will use Robertson and Seymour's grid-minor theorem.
\begin{theorem}[\cite{RobertsonSeymour86b}]
For every positive integer $r$, there exists a constant $f(r)$ such that if a graph~$G$
has treewidth at least $f(r)$, then $G$ contains an $r$-grid as a minor.
\end{theorem}
\noindent
By \cite{RobertsonSeymourThomas94}, $f(r) \le 20^{2 r^5}$.
A linear FPT algorithm (parameterized by $k$) by Bodlaender
\cite{Bodlaender96} finds a tree decomposition of width at most $k$ of a graph $G$ if ${\mathbf{tw}}(G)\le k$.
A quadratic FPT algorithm (parameterized by $r$) by Kawarabayashi \emph{et al.}\xspace~\cite{KawarabayashiKR12} finds an $r$-grid minor in a graph $G$ if $G$ contains an $r$-grid as a minor.%
\shortversion{
Proofs of statements marked with $(\star)$ can be found in the appendix.
}
\section{Detection of Strong Nested-Backdoor Sets}
Our overall approach to find strong \NBDSs resembles the approach from \cite{GaspersSzeider11a}
to find strong \textsc{Forest}-back\-door sets\xspace. Looking more closely at both algorithms,
the reader will see significant differences in how the two main cases are handled.
Let $F$ be a CNF formula and $k$ be an integer.
Our FPT algorithm will decide the satisfiability of $F$ if $F$ has a strong \NBDS of size at most $k$.
The first step of the algorithm is to find a good approximation for a smallest strong \NBDS.
Specifically, it will either determine that $F$ has no strong \NBDS of size at most~$k$, or
it will compute a strong \NBDS of size at most $2^k$. In case
$F$ has no strong \NBDS of size at most $k$, the algorithm stops, and if
it finds a strong \NBDS $B$ of size at most $2^k$, it uses Knuth's algorithm \cite{Knuth90}
to check for every assignment
$\tau \in 2^B$ whether $F[\tau]$ is satisfiable and answers \textsc{Yes} if at least one
such assignment reduced $F$ to a satisfiable formula and \textsc{No} otherwise.
Since ${\mathbf{tw}}^*(F[\tau])\le 3$ \cite{BiedlH04} for every truth assignment $\tau$ to $B$,
a tree decomposition of ${\normalfont \textsf{inc}}(F[\tau])$ can be computed in linear time \cite{Bodlaender96}, and
treewidth-based dynamic programming algorithms can be used to compute the number of satisfying assignments
of $F[\tau]$ in polynomial time \cite{FischerMakowskyRavve06,SamerSzeider10}.
We will arrive at our main theorem.
\begin{theorem}\label{thm:sat}
The problems SAT and \#SAT are fixed-pa\-ra\-me\-ter trac\-ta\-ble\xspace parameterized by $\mathbf{sb}_{\mathbf{N}}(F)$.
\end{theorem}
It only remains to find a strong \NBDS with an \text{\normalfont FPT}\ algorithm. In the
remainder of this section we present an \text{\normalfont FPT}\ algorithm that either
determines that $F$ has no strong \NBDS of size at most $k$, or computes one
of size at most $2^k$. An algorithm of that kind is called an
\emph{\text{\normalfont FPT}-approximation algorithm}~\cite{Marx08b}, as it is an \text{\normalfont FPT}\
algorithm that computes a solution that approximates the optimum with an
error bounded by a function of the parameter.
Consider the incidence graph $G=(V,E)={\normalfont \textsf{inc}}(F)$ of $F$. By \cite{RobertsonSeymourThomas94},
it either has treewidth at most $\mathsf{tw}(k)$, or it has a $\mathsf{grid}(k)$-grid as a minor.
Here,
\begin{align*}
\mathsf{tw}(k) &:= 20^{2 \mathsf{grid}(k)^5},\\
\mathsf{grid}(k) &:= 4 \cdot \sqrt{\mathsf{obs}(k)+1},\\
\mathsf{obs}(k) &:= 2^{k} \cdot \mathsf{same}(k) + k,\text{ and}\\
\mathsf{same}(k) &:= 15 \cdot 2^{2k+2}.
\end{align*}
\subsection{Large Grid Minor}
\tikzset{var/.style={inner sep=.15em,circle,fill=black,draw},
clause/.style={minimum size=1mm,rectangle,fill=white,draw},
label distance=-2pt}
\begin{figure}[tb]
\centering
\begin{tikzpicture}[xscale=1,yscale=0.8]
\node (a) at (0,0) [var,label=left:$a$] {};
\node (b) at (2.2,0) [var,label=right:$b$] {};
\draw (a)--(b) node (d) [pos=0.45,var] {};
\draw (a) .. controls +(0,1) and +(0,1) .. (b) node (e) [pos=0.3,var] {};
\node (f1) at (1.1,-0.8) [clause] {};
\draw (a)--(f1)--(b);
\node (f) at (1.7,-1.2) [var] {};
\draw (f1)--(f);
\node (c) at (1.8,1.4) [clause,label=right:$c^*$] {};
\draw (d)--(c)--(e) (c)--(f);
\end{tikzpicture}
\caption{A \Nested-ob\-struc\-tion\xspace leading to a $K_{3,3}$-minor with the universal clause $c^*$.}
\label{fig:obs}
\end{figure}
The goal of this subsection is to design an \text{\normalfont FPT}\ algorithm that, given
a $\mathsf{grid}(k)$-grid as a minor in~$G$, computes a set $S^*$ of $2^{O(k^{10})}$ variables
from ${\normalfont \textsf{var}}(F)$ such that every strong \NBDS of size at most $k$ contains a variable from $S^*$.
Suppose $G$ has a $\mathsf{grid}(k)$-grid as a minor.
\begin{definition}
An $a$--$b$ \emph{\Nested-ob\-struc\-tion\xspace\/} is a subgraph of ${\normalfont \textsf{inc}}(F)$ consisting of
\begin{itemize}
\item five distinct vertices $a,b,p_1,p_2,p_3$, such that $p_1,p_2,p_3$ are variables,
\item three independent $a$--$b$ paths $P_1,P_2,P_3$, and
\item an edge between $p_i$ and a vertex from $P_i$ for each $i\in \{1,2,3\}$.
\end{itemize}
\end{definition}
In particular, if a path $P_i$ has a variable $v$ as an interior vertex, we can take $p_i:=v$.
See Figure~\ref{fig:obs}.
\begin{lemma}\label{lem:obs}
If $F'$ is a CNF formula such that ${\normalfont \textsf{inc}}(F')$ contains a \Nested-ob\-struc\-tion\xspace, then $F' \notin \textsc{Nested}\xspace$.
\end{lemma}
\begin{proof}
Suppose ${\normalfont \textsf{inc}}(F')$ contains a \Nested-ob\-struc\-tion\xspace.
We will exhibit a $K_{3,3}$-minor in ${\normalfont \textsf{inc+u}}(F')$.
A model of a $K_{3,3}$ can be obtained by taking the subgraphs consisting of the singleton vertices $a$, $b$, and the universal clause $c^*$
for one side of the bipartition, and the three subgraphs induced by $(P_i\cup \{p_i\})\setminus \{a,b\}, 1\le i\le 3,$ for the other side.
Since, by Wagner's Theorem \cite{Wagner37}, no planar graph has a $K_{3,3}$ as a minor, and by a result of
Kratochv\'{\i}l and K\v{r}iv\'{a}nek \cite{KratochK93}, $F'$ is nested
iff\xspace ${\normalfont \textsf{inc+u}}(F')$ is planar, we conclude that
$F' \notin \textsc{Nested}\xspace$.
\qed \end{proof}
\noindent
By Lemma \ref{lem:obs}, we have that for each assignment to the variables of a
strong \NBDS, at least one variable from each \Nested-ob\-struc\-tion\xspace vanishes in the reduced formula.
Using the $r$-grid, we now find a set $\mathcal{O}$ of $\mathsf{obs}(k)$ vertex-disjoint \Nested-ob\-struc\-tions\xspace in $G$.
\begin{restatable}{lemma}{LemGrid}\label{lem:grid}\shortversion{\textup{($\star$)}}
Given a $\mathsf{grid}(k)$-grid minor of $G={\normalfont \textsf{inc}}(F)$, a set of $\mathsf{obs}(k)$ vertex-disjoint \Nested-ob\-struc\-tions\xspace can be found in polynomial time.
\end{restatable}
\longversion{\input{proofLemGrid}}
\begin{figure}[tb]
\centering
\begin{tikzpicture}[xscale=1,yscale=0.8]
\pgfmathtruncatemacro\dim{6}
\foreach \x in {1,2,...,\dim}
\foreach \y in {1,2,...,\dim} {
\node at (\x,\y) [var] {};
\ifnum\x<\dim
\draw (\x,\y) -- +(1,0);
\fi
\ifnum\y<\dim
\draw (\x,\y) -- +(0,1);
\fi
}
\node at (2,1) [var,inner sep=.2em,fill=red,draw=red,label=below:{$a=(2,1)$}] {};
\node at (2,4) [var,inner sep=.2em,fill=red,draw=red,label=above:{$b=(2,4)$}] {};
\draw[ultra thick,red] (1,1)--(1,4)--(3,4)--(3,1)--(1,1) (2,1)--(2,4);
\end{tikzpicture}
\caption{The $6$-grid and a highlighted \Nested-ob\-struc\-tion\xspace.}
\label{fig:grid}
\end{figure}
\noindent
Denote by $\mathcal{O}$ a set of $\mathsf{obs}(k)$ vertex-disjoint \Nested-ob\-struc\-tions\xspace obtained via Lemma \ref{lem:grid}.
A backdoor variable can destroy a \Nested-ob\-struc\-tion\xspace either because it participates in the \Nested-ob\-struc\-tion\xspace, or because
every setting of the variable satisfies a clause that participates in the \Nested-ob\-struc\-tion\xspace.
\begin{definition}
Let $x$ be a variable and $O$ a \Nested-ob\-struc\-tion\xspace in $G$.
We say that $x$ \emph{kills} $O$ if neither ${\normalfont \textsf{inc}}(F[x = 1} %{\mbox{\textsf{true}}])$ nor ${\normalfont \textsf{inc}}(F[x = 0} %{\mbox{\textsf{false}}])$ contains $O$ as a subgraph.
We say that $x$ kills $O$ \emph{internally} if $x\in {\normalfont \textsf{var}}(O)$, and that $x$ kills $O$ \emph{externally} if $x$ kills $O$ but does not kill it internally.
In the latter case, $O$ contains a clause $c$ containing $x$ and a clause $c'$ containing $\neg x$ and we say that
$x$ kills $O$ (externally) \emph{in} $c$ and $c'$.
\end{definition}
\noindent
Our algorithm will make a series of $O(1)$ guesses about the strong \NBDS, where each guess is made out of a number of choices that is upper bounded by a function of $k$.
At any stage of the algorithm, a \emph{valid} strong \NBDS is one that conforms to the guesses that have been made.
For a fixed series of guesses, the algorithm will compute a set $S\subseteq {\normalfont \textsf{var}}(F)$ such that every valid strong \NBDS of size at most $k$ contains a variable from $S$. The union of all
such $S$, taken over all possible series of guesses, forms a set $S^*$ and each strong \NBDS of size at most $k$ contains a variable from $S^*$.
Bounding the size of each $S$ by a function of $k$ enables us to bound $|S^*|$ by a function of $k$, and $S^*$ can then be used in a bounded search tree algorithm
(see Subsection~\ref{subsec:algo}).
For any strong \NBDS of size at most $k$, at most $k$ \Nested-ob\-struc\-tions\xspace from $\mathcal{O}$ are killed internally since they are vertex-disjoint.
The algorithm guesses $k$ \Nested-ob\-struc\-tions\xspace from $\mathcal{O}$ that may be killed internally.
Let $\mathcal{O}'$ denote the set of the remaining \Nested-ob\-struc\-tions\xspace, which need to be killed externally.
Suppose $F$ has a strong \NBDS $B$ of size $k$ killing no \Nested-ob\-struc\-tion\xspace from $\mathcal{O}'$ internally. Then, $B$ defines a partition of $\mathcal{O}'$ into
$2^k$ parts where for each part, the \Nested-ob\-struc\-tions\xspace contained in this part are killed externally by the same set of variables from $B$.
Since $|\mathcal{O}'| = \mathsf{obs}(k) - k = 2^k \cdot \mathsf{same}(k)$, at least one of these parts contains at least $\mathsf{same}(k)$ \Nested-ob\-struc\-tions\xspace from $\mathcal{O}'$.
The algorithm guesses a subset $\mathcal{O}_s \subseteq \mathcal{O}'$ of $\mathsf{same}(k)$ \Nested-ob\-struc\-tion\xspace from this part and it guesses how many variables from the
strong \NBDS kill the obstructions in this part externally.
Suppose each \Nested-ob\-struc\-tion\xspace in $\mathcal{O}_s$ is killed externally by the same set of~$\ell$ backdoor variables, and no other backdoor variable
kills any \Nested-ob\-struc\-tion\xspace from $\mathcal{O}_s$. Clearly, $1\le \ell \le k$.
Compute the set of external killers for each \Nested-ob\-struc\-tion\xspace in $\mathcal{O}_s$. Denote by $Z$ the common external killers of the \Nested-ob\-struc\-tion\xspace in $\mathcal{O}_s$.
The presumed back\-door set\xspace contains exactly $\ell$ variables from $Z$ and no other variable from the back\-door set\xspace kills any \Nested-ob\-struc\-tion\xspace from~$\mathcal{O}_s$.
We will define three rules for the construction of $S$, and the algorithm will execute the first applicable rule.
\begin{myrule}[Few Common Killers]\label{rule:fewkillers}
If $|Z|<|\mathcal{O}_s|$, then set $S:=Z$.
\end{myrule}
The correctness of this rule follows since any valid strong \NBDS contains $\ell$ variables from $Z$ and $\ell\ge 1$.
For each $O\in \mathcal{O}_s$ we define an auxiliary graph $G_O=(Z,E_O)$ whose edge set is initially empty.
As long as $G_O$ has a vertex $v$ with degree $0$ such that $v$ and some other vertex in $Z$
have a common neighbor from $O$ in $G$, select a vertex $u$ of minimum degree in $G_O$ such
that $u$ and $v$ have a common neighbor from $O$ in $G$ and add the edge $uv$ to $E_O$.
As long as $G_O$ has a vertex $v$ with degree $0$, select a vertex $u$ of minimum degree in $G_O$ such that
$v$ has a neighbor $v'\in V(O)$ in $G$ and $u$ has a neighbor $u'\in V(O)$ in $G$
and there is a $v'$--$u'$ path in $O$ in which no internal vertex is adjacent to a vertex from $Z\setminus \{v\}$;
add the edge $uv$ to $E_O$.
By the construction of $G_O$, we have the following property on the degree of all vertices.
\begin{fact}
For each $O\in \mathcal{O}_s$, the graph $G_O$ has minimum degree at least $1$.
\end{fact}
Recall that no clause contains complimentary literals.
Consider two variables $u,v\in Z$ that share an edge in $G_O$.
By the construction of $G_O$, there is a $u$--$v$ path $P$ in $G$ whose internal edges are in $O$, such
that for each variable $z\in Z$, all edges incident to $z$ and a clause from $P$ have the same sign.
Moreover, since no variable from a valid strong \NBDS kills $O$ externally, unless it is in $Z$,
for each potential backdoor variable $x\in {\normalfont \textsf{var}}(F) \setminus Z$, all edges incident to $x$ and a clause from $P$ have the same sign.
Thus, we have the following fact.
\begin{fact}\label{fact:2}
If $u,v\in Z$ share an edge in $G_O$, then for every valid strong \NBDS that does not contain $u$ and $v$, there is a truth assignment
$\tau$ to $B$ such that ${\normalfont \textsf{inc}}(F[\tau])$ contains a $u$--$v$ path whose internal edges are in $O$.
\end{fact}
Consider the multigraph $G_m(\mathcal{O}_s)= (Z, \biguplus_{O\in \mathcal{O}_s} E_O)$, i.e.\xspace, the union of all $G_O$ over all
$O\in \mathcal{O}_s$, where the multiplicity of an edge is the number of distinct sets $E_O$ where it appears, $O\in \mathcal{O}_s$.
\begin{myrule}[Multiple Edges]\label{rule:multi}
If there are two vertices $u,v\in Z$ such that $G_m(\mathcal{O}_s)$ has a $u$--$v$ edge with multiplicity at
least $2 \cdot 2^{k}+1$, then set $S:=\{u,v\}$.
\end{myrule}
Consider any valid strong \NBDS $B$ of size $k$. Then, by Fact \ref{fact:2}, for each $u$--$v$ edge
there is some truth assignment $\tau$ to $B$ such that ${\normalfont \textsf{inc}}(F[\tau])$ contains a $u$--$v$ path in $G$. Moreover, since each
$u$--$v$ edge comes from a different $O\in \mathcal{O}_s$, all these $u$--$v$ paths are independent.
Since there are $2^k$ truth assignments to $B$ but at least $2 \cdot 2^{k}+1$ $u$--$v$ edges, for at least one
truth assignment $\tau$ to $B$, there are 3 independent $u$--$v$ paths $P_1,P_2,P_3$ in ${\normalfont \textsf{inc}}(F[\tau])$.
We obtain a $u$--$v$ \Nested-ob\-struc\-tion\xspace choosing as $p_i, 1\le i\le 3$, a variable from $P_i$ or a variable neighboring a clause
from $P_i$ and belonging to the same \Nested-ob\-struc\-tion\xspace in $\mathcal{O}_s$.
Thus, any valid strong \NBDS contains $u$ or $v$.
Now, consider the graph $G(\mathcal{O}_s)$ obtained from the multigraph $G_m(\mathcal{O}_s)$ by merging multiple edges,
i.e.\xspace, we retain each edge only once.
\begin{myrule}[No Multiple Edges]\label{rule:nomulti}
Set $S$ to be the $2k$ vertices of highest degree in $G(\mathcal{O}_s)$ (ties are broken arbitrarily).
\end{myrule}
For the sake of contradiction, suppose $F$ has a valid strong \NBDS $B$ of size~$k$ with $B\cap S = \emptyset$.
First, we show a lower bound on the number of edges in $G(\mathcal{O}_s) - B$.
Since $G_m(\mathcal{O}_s)$ has at least $\frac{|Z|}{2} \mathsf{same}(k)$ edges and each edge has multiplicity at most $2^{k+1}$, the
graph $G(\mathcal{O}_s)$ has at least $\frac{|Z| \mathsf{same}(k)}{2 \cdot 2^{k+1}} = 3\cdot 5\cdot 2^k \cdot |Z|$ edges.
Let $d$ be the sum of the degrees in $G(\mathcal{O}_s)$ of the vertices in $B \cap Z$.
Now, the sum of degrees of vertices in $S$ is at least $2d$ in $G(\mathcal{O}_s)$,
and at least $d$ in $G(\mathcal{O}_s) - B$. Therefore, $G(\mathcal{O}_s) - B$ has at least $d/2$ edges.
On the other hand, the number of edges deleted to obtain $G(\mathcal{O}_s) - B$ from $G(\mathcal{O}_s)$ is at most $d$.
It follows that the number of edges in $G(\mathcal{O}_s) - B$ is at least a third the number of edges in $G(\mathcal{O}_s)$,
and thus at least $5\cdot 2^k \cdot |Z|$.
Now, we iteratively build a truth assignment $\tau$ for $B$. Set $H:=G(\mathcal{O}_s) - B$.
Order the variables of $B$ as $b_1, \dots, b_k$. For increasing~$i$, we set
$\tau(b_i)=0$ if in $G$, the vertex $v\in B$ is adjacent with a positive edge to more paths that correspond to an edge in $H$
than with a negative edge and set $\tau(b_i)=1$ otherwise; if $\tau(b_i)=0$, then remove each edge from $H$ that corresponds to a path
in $G$ that is adjacent with a negative edge to $b_i$, otherwise remove each edge from $H$ that corresponds to a path
in $G$ that is adjacent with a positive edge to $b_i$.
Observe that for a variable $v\in B$ and a path $P$ in $G$ that corresponds to an edge in $G(\mathcal{O}_s) - B$,
$v$ is not adjacent with a positive and a negative edge to $P$. If $v\in Z$ this follows by the construction of $G_O$,
and if $v\notin Z$, this follows since $v$ does not kill any \Nested-ob\-struc\-tion\xspace from $\mathcal{O}_s$.
Therefore, each of the $k$ iterations building the truth assignment $\tau$ has removed at most half the edges of $H$.
In the end, $H$ has at least $5 |Z|$ edges.
Next, we use the following theorem of Kirousis \emph{et al.}\xspace~\cite{KirousisSS93}.
\begin{theorem}[\cite{KirousisSS93}]
If a graph has $n$ vertices and $m>0$ edges, then it has an induced subgraph that is $\lceil \frac{m+n}{2n} \rceil$-vertex-connected.
\end{theorem}
\noindent
We conclude that $H$ has an induced subgraph $H'$ that is $3$-vertex-connected.
Let $x,y\in V(H')$. We use Menger's theorem \cite{Menger27}.
\begin{theorem}[\cite{Menger27}]
Let $G=(V,E)$ be a graph and $x,y\in V$. Then the size of a minimum $x,y$-vertex-cut in $G$ is
equal to the maximum number of independent $x$--$y$ paths in $G$.
\end{theorem}
\noindent
Since the minimum size of an $x,y$-vertex cut is at least $3$ in $H'$, there are 3 independent
$x$--$y$ paths in $H'$.
Replacing each edge by its corresponding path in $G$, gives rise to 3 walks from $x$ to $y$ in $G$.
Shortcutting cycles, we obtain three $x$--$y$ paths $P_1,P_2,P_3$ in~$G$.
By construction, each edge of these paths is incident to a vertex from a \Nested-ob\-struc\-tion\xspace in~$\mathcal{O}_s$.
We assume that $P_1,P_2,P_3$ are edge-disjoint.
Indeed, by the construction of the $G_O$, $O\in \mathcal{O}_s$, they can only share the first and last edges.
In case $P_1$ shares the first edge with $P_2$, replace $x$ by its neighbor on $P_1$,
remove the first edge from $P_1$ and $P_2$, and replace $P_3$ by its symmetric difference with this edge.
Act symmetrically for the other combinations of paths sharing the first or last edge.
\begin{restatable}[\cite{Gaspers12}]{lemma}{LemPaths}\label{lem:paths}\shortversion{\textup{($\star$)}}
Let $G=(V,E)$ be a graph. If there are two vertices $x,y\in V$ with 3 edge-disjoint $x$--$y$ paths in $G$,
then there are two vertices $x',y'\in V$ with 3 independent $x'$--$y'$ paths in $G$.
\end{restatable}
\longversion{\input{proofLemPaths}}
\noindent
By Lemma \ref{lem:paths} we obtain two vertices $x',y'$ in $G$ with 3 independent $x'$--$y'$ paths $P_1',P_2',P_3'$ in $G$.
Since the lemma does not presuppose any other edges in $G$ besides those from the edge-disjoint $x$--$y$ paths,
$P_1',P_2',P_3'$ use only edges from the paths $P_1,P_2,P_3$. Thus, each edge of $P_1',P_2',P_3'$ is incident to
a vertex from a \Nested-ob\-struc\-tion\xspace in $\mathcal{O}_s$.
Thus, we obtain a $x'$--$y'$ \Nested-ob\-struc\-tion\xspace with the paths $P_1',P_2',P_3'$, and for each path $P_i'$, we choose a variable from this path
or a variable from~$\mathcal{O}_s$ neighboring a clause from this path.
We arrive at a contradiction for $B$ being a valid strong \NBDS.
This proves the correctness of Rule \ref{rule:nomulti}.
\medskip
The number of possible guesses the algorithm makes is upper bounded by $\binom{\mathsf{obs}(k)}{k} \cdot \binom{\mathsf{obs}(k)-k}{\mathsf{same}(k)}\cdot k = 2^{O(k^8)}$,
and each series of guesses leads to a set $S$ of at most $\mathsf{same}(k)$ variables. Thus, the set $S^*$, the union of all such $S$,
contains at most $2^{O(k^8)}\cdot \mathsf{same}(k) = 2^{O(k^{10})}$ variables.
Finally, we have shown the following lemma in this subsection.
\begin{lemma}\label{lem:wall}
There is an \text{\normalfont FPT}\ algorithm that, given a CNF formula~$F$, a positive
integer parameter~$k$, and a $\mathsf{grid}(k)$-grid as a minor in ${\normalfont \textsf{inc}}(F)$,
computes a set $S^*\subseteq {\normalfont \textsf{var}}(F)$ of size $2^{O(k^{10})}$ such
that every strong \NBDS of size at most $k$ contains a variable from $S^*$.
\end{lemma}
\subsection{Small Treewidth}
The goal of this subsection is to design an \text{\normalfont FPT}\ algorithm that, given
a tree decomposition of $G$ of width at most $\mathsf{tw}(k)$, finds a strong \NBDS of $F$ of size $k$ or determines that $F$ has no such strong \NBDS.
Our algorithm uses Arnborg \emph{et al.}\xspace's extension \cite{ArnborgLagergrenSeese91} of Courcelle's Theorem \cite{Courcelle90}.
It gives, amongst others, an \text{\normalfont FPT}\ algorithm that takes as input a graph $\mathcal{A}$ with labeled vertices and edges and
a Monadic Second Order (MSO) sentence $\varphi(X)$, and computes a minimum-sized set of vertices $X$ such that $\varphi(X)$ is true in $\mathcal{A}$.
Here, the parameter is $|\varphi|+{\mathbf{tw}}(\mathcal{A})$.
We will define a labeled graph whose treewidth is upper bounded by a function of ${\mathbf{tw}}({\normalfont \textsf{inc}}(F))$, and an
MSO-sentence of constant length such that the graph models the MSO-sentence
whenever its argument is a strong \NBDS of $F$.
\begin{restatable}{lemma}{LemTw}\label{lem:tw}\shortversion{\textup{($\star$)}}
There is an \text{\normalfont FPT}\ algorithm that takes as input a CNF formula~$F$, a positive integer parameter $k$, and a tree decomposition of $G$ of width at most $\mathsf{tw}(k)$,
and finds a strong \NBDS of $F$ of size $k$ if one exists.
\end{restatable}
\longversion{\input{proofLemTw}}
We note that in the case where ${\mathbf{tw}}(G)$ is bounded, one could immediately solve the satisfiability problem for $F$ \cite{FischerMakowskyRavve06,SamerSzeider10}.
However, finding a strong \NBDS enables us to give an \text{\normalfont FPT}\ approximation algorithm for the backdoor detection problem.
\subsection{The FPT algorithm}
\label{subsec:algo}
Our \text{\normalfont FPT}-approximation algorithm combines the results from the previous two subsections.
In case $G$ has treewidth at most $\mathsf{tw}(k)$, Lemma \ref{lem:tw} is used to find a solution of size $k$ if one exists.
Otherwise, Lemma \ref{lem:wall} provides a set $S^*$ of $2^{O(k^{10})}$ variables such that any solution of size at most~$k$ contains
a variable from $S^*$. For each $x\in S^*$, the algorithm recurses on both formulas $F[x=0]$ and $F[x=1]$.
If both recursive calls return strong \NBDSs $B_{\neg x}$ and $B_{x}$, then $\{x\} \cup B_{x} \cup B_{\neg x}$
is a strong \NBDS of $F$, otherwise, no strong \NBDS of $F$ of size at most~$k$ contains $x$.
Since $B_{\neg x}$ could be
disjoint from $B_{x}$ in the worst case, while $F[x=0]$ and $F[x=1]$ could have strong \NBDSs $B_{\neg x}'$ and
$B_{x}'$ of size $k-1$ with $B_{\neg x}'=B_{x}'$, our approach approximates the optimum with a factor of $2^k/k$.
\begin{restatable}{theorem}{ThmStrong}\label{thm:strong}\shortversion{\textup{($\star$)}}
There is an \text{\normalfont FPT}\ algorithm, which, for a CNF formula $F$ and a positive integer parameter $k$, either concludes that $F$
has no strong \NBDS of size at most $k$ or finds a strong \NBDS of $F$ of size at most $2^k$.
\end{restatable}
\longversion{\input{proofThmStrong}}
\noindent
In particular, this proves Theorem \ref{thm:sat}.
\section{Conclusion}
We have classified the problems SAT and \#SAT as fixed-parameter
tractable when parameterized by the size of a smallest strong backdoor
set with respect to the base class of nested formulas. As argued in the
introduction, this parameter is incomparable with incidence treewidth.
The parameter dependence makes our algorithm impractical. However, we
would like to note that the class of fixed-parameter tractable problems
has proven to be quite robust: Once a problem is shown to belong to this
class, one can start to develop faster and more practical algorithms.
For many cases in the past this was successful. For instance, the
problem of recognizing graphs of genus~$k$ was originally shown to be
fixed-parameter tractable by means of non-constructive tools from graph
minor theory~\cite{FellowsLangston88}. Later a linear-time algorithm
with doubly exponential parameter dependence was found~\cite{Mohar96},
and more recently, the algorithm could be improved to a single
exponential parameter dependence~\cite{KawarabayashiMoharReed08}.
It would be interesting to see whether a similar improvement is possible for
finding or FPT-approximating strong backdoor sets with respect to nested
formulas.
\bibliographystyle{plain}
|
1,108,101,565,641 | arxiv | \section{Introduction}
\label{sec:1}
The self-gravitating collisionless fluid dynamics (SG-CFD) is the study of motion of collisionless matter under the influence of its own gravity. A typical example is the large-scale gravitational collapse of collisionless system \citep{Lukic:2007-The-halo-mass-function--High-r}. The self-organization of self-gravitating collisionless matter leads to the formation and evolution of large-scale structures due to the gravitational instability. Highly localized and virialized halos are major manifestation of nonlinear gravitational collapse \citep{Neyman:1952-A-Theory-of-the-Spatial-Distri,Cooray:2002-Halo-models-of-large-scale-str} and the building blocks of large-scale structures.
By contrast, incompressible hydrodynamics also develops instability if Reynolds number is sufficiently high, where turbulence starts to initiate and develop. The "eddies", building blocks of turbulence, are formed at different length scales and interacting with each other, as described by a famous poem :"Big whirls have little whirls, That feed on their velocity; And little whirls have lesser whirls, And so on to viscosity" \citep{Richardson:1922-Weather-Prediction-by-Numerica}. Large eddies feed smaller eddies, which feed even smaller eddies, and then lead to viscous dissipation at the smallest scale, i.e. the concept of a direct energy cascade. While direct energy cascade is the key feature of three-dimensional turbulence, two-dimensional turbulence possesses a range of scales over which kinetic energy is transferred from small to large scales , i.e. an inverse energy cascade \citep{Kraichnan:1967-Inertial-Ranges-in-2-Dimension}.
The similarity between "eddies" in turbulence and "halos" in dark matter flow (SG-CFD) allows a new poem by simply replacing "whirls" with "halos". "Little halos have big halos, That feed on their mass; And big halos have greater halos, And so on to growth". This picture describes the inverse mass cascade in dark matter flow \citep{Xu:2021-Inverse-mass-cascade-mass-function}. There exists a broad spectrum of halo size. Small halos are created, interacting, and merging with other halos. Halos pass their mass onto larger and larger halos, until halo mass growth becomes dominant over mass propagation.
While "eddy" is not a well-defined object in turbulence literature, "halos" are well-defined dynamical objects, whose abundance and internal structure have been extensively studied over several decades. The abundance of halos is described by a halo mass function, a fundamental quantity to model structure formation and evolution. The seminal Press-Schechter (PS) model \citep{Press:1974-Formation-of-Galaxies-and-Clus,Bond:1991-Excursion-Set-Mass-Functions-f} allows one to predict the shape and evolution of mass function. This model relies on a threshold value of density contrast that can be analytically derived from the nonlinear collapse of a spherical top hat over-density \citep{Tomita:1969-Formation-of-Gravitationally-B,Gunn:1972-Infall-of-Matter-into-Clusters}. Further improvement was achieved by extending the PS formalism to elliptical collapse \citep{Sheth:2001-Ellipsoidal-collapse-and-an-im,Sheth:1999-Large-scale-bias-and-the-peak-}. In addition, halo mass function can be interpreted as an intrinsic distribution to maximize system entropy during statistically steady state of dark matter flow \citep{Xu:2021-The-maximum-entropy-distributi,Xu:2021-Mass-functions-of-dark-matter-}.
The internal structure of halos is primarily described by the halo density profile, another important quantity for structure formation and evolution \citep{Del_Popolo:2009-Density-profiles-of-dark-matte}. Structure of halos can be studied both analytically and numerically with \textit{N}-body simulations \citep{Moore:1998-Resolving-the-structure-of-col,Klypin:2001-Resolving-the-structure-of-col}. The spherical collapse model relates assumed power-law density with the initial density fluctuations, which can be dependent on the effective index of the power spectrum from linear theory. This simple similarity model leads to an isothermal density profile for virialized halos. However, high-resolution \textit{N}-body simulations of structure formation have shown that the simulated halos have a density shallower than the isothermal profile at smaller radius and steeper at larger radius \citep{Navarro:1997-A-universal-density-profile-fr,Navarro:2004-The-inner-structure-of-ACDM-ha}. Many effects might contribute to this deviation. The effect of halo mass cascade (accretion) is one of the most critical effect that is absent, which renders this simple similarity model invalid. We will discuss the effect of mass cascade on halo density profile in detail (see Section \ref{sec:3.3}).
By revisiting fundamental ideas of turbulence, the inverse mass/energy cascade can be mathematically formulated and briefly reviewed here \citep{Xu:2021-Inverse-mass-cascade-mass-function,Xu:2021-Inverse-and-direct-cascade-of-}. Mass cascade is local, two-way, and asymmetric in mass space. Halos inherit/pass their mass mostly from/to halos of similar size. The net mass transfer proceeds in a "bottom-up" fashion. Two distinct ranges can be identified, i.e. a propagation range with a scale-independent rate of mass transfer $\varepsilon _{m} $ and a deposition range with cascaded mass consumed to form and grow halos. A fundamental merging frequency $f_{0} \sim m_{p}^{\left(\lambda -1\right)} a^{-\tau _{0} } $ between two single mergers of elementary mass $m_{p}$ can be identified, where \textit{a} is the scale factor, $m_{p} $ is the particle mass, $\lambda $ and $\tau _{0} $ are two key mass cascade parameters that may be dependent on the exact cosmology model. The waiting time $\tau _{g} $ (halo lifespan) for halos to pass their mass to larger halos scales as $\tau _{g} \sim m_{h}^{-\lambda } a^{\tau _{0} } $. Consequently, the everlasting inverse mass cascade with a scale-independent mass transfer rate $\varepsilon _{m} \sim a^{-\tau _{0} } $ in the propagation range is a distinct feature of the intermediate statistically steady state of dark matter flow. Entire mass cascade was also formulated as random-walk of halos in mass space \citep{Xu:2021-Inverse-mass-cascade-mass-function}. This results in a heterogeneous diffusion model with position-dependent diffusivity, where mass function can be analytically derived without relying on any specific collapse model.
In addition, the elementary step of mass cascade, i.e a two-body collapse \citep{Xu:2021-A-non-radial-two-body-collapse}, the evolution of halo mean flow, velocity dispersion \citep{Xu:2022-The-mean-flow--velocity-disper}, and halo momentum and energy \citep{Xu:2022-The-evolution-of-energy--momen} were studied in separate papers, along with the correlation-based statistical theory for correlation and structure functions in dark matter flow \citep{Xu:2022-The-statistical-theory-of-2nd,Xu:2022-The-statistical-theory-of-3rd,Xu:2022-Two-thirds-law-for-pairwise-ve}. This is an important topic with potential relevance to dark matter particle mass and properties \citep{Xu:2022-Postulating-dark-matter-partic}, MOND (modified Newtonian dynamics) theory \citep{Xu:2022-The-origin-of-MOND-acceleratio}, and baryonic-to-halo mass relation \citep{Xu:2022-The-baryonic-to-halo-mass-rela}.
This paper focus on the effects of inverse mass cascade on halo energy, momentum, size, and internal structure. Especially, it is still not clear why halos that form in SG-CFD have nearly universal profiles. We will demonstrate that the radial flow lead to an extra length scale (scale radius) for density profile where the radial flow is at its maximum. A double-power-law density is a natural result with inner density dominated by halo deformation rate and outer density controlled by halo growth. There exists a limiting halo concentration for large halos as a result of vanishing linear moment. The effects of mass cascade on velocity dispersion and surface energy are explicitly discussed and presented. Stochastic models for halo size and particle motion in halos are also discussed along with the equation of state for halos. The rest of this paper is organized as follows: Section \ref{sec:2} introduces the simulation and numerical data, followed by the effects of mass cascade on halo properties in Section \ref{sec:3}. Stochastic models for halo size and random-walk of collisionless particles in halos are presented in Section \ref{sec:4} with complete solutions provided.
\section{N-body simulations and numerical data}
\label{sec:2}
The numerical data for this work is publicly available and generated from \textit{N}-body simulations carried out by the Virgo consortium. A comprehensive description of simulation data can be found in \citep{Frenk:2000-Public-Release-of-N-body-simul,Jenkins:1998-Evolution-of-structure-in-cold}. The same set of simulation data has been widely used in a number of different studies from clustering statistics \citep{Jenkins:1998-Evolution-of-structure-in-cold} to the formation of halo clusters in large scale environments \citep{Colberg:1999-Linking-cluster-formation-to-l}, and testing models for halo abundance and mass functions \citep{Sheth:2001-Ellipsoidal-collapse-and-an-im}. More details on simulation parameters are provided in Table \ref{tab:1}.
Two relevant datasets from this N-boby simulation, i.e. halo-based and correlation-based statistics of dark matter flow, can be found at Zenodo.org \citep{Xu:2022-Dark_matter-flow-dataset-part1, Xu:2022-Dark_matter-flow-dataset-part2}, along with the accompanying presentation slides, "A comparative study of dark matter flow \& hydrodynamic turbulence and its applications" \citep{Xu:2022-Dark_matter-flow-and-hydrodynamic-turbulence-presentation}. All data files are also available on GitHub \citep{Xu:Dark_matter_flow_dataset_2022_all_files}.
\begin{table}
\caption{Numerical parameters of N-body simulation}
\begin{tabular}{p{0.25in}p{0.05in}p{0.05in}p{0.05in}p{0.05in}p{0.05in}p{0.4in}p{0.1in}p{0.4in}p{0.4in}}
\hline
Run & $\Omega_{0}$ & $\Lambda$ & $h$ & $\Gamma$ & $\sigma _{8}$ & \makecell{L\\(Mpc/h)} & $N$ & \makecell{$m_{p}$\\$M_{\odot}/h$} & \makecell{$l_{soft}$\\(Kpc/h)} \\
\hline
SCDM1 & 1.0 & 0.0 & 0.5 & 0.5 & 0.51 & \centering 239.5 & $256^{3}$ & 2.27$\times 10^{11}$ & \makecell{\centering 36} \\ \hline
\end{tabular}
\label{tab:1}
\end{table}
\section{Effects of mass cascade on halo properties}
\label{sec:3}
\subsection{Halo density profiles}
\label{sec:3.1}
The formation of halos is a complex, hierarchical, and nonlinear process. However, the radial density profile $\rho _{h} \left(r\right)$ of halos can be robustly fitted by relatively simple functions from cosmological \textit{N}-body simulations. This section briefly reviews the NFW profile \citep{Navarro:1997-A-universal-density-profile-fr}, Einasto profile \citep{Einasto:1984-Structure-of-Superclusters-and} and power-law density profile (see Appendix \ref{appendix:a}). Especially, the isothermal profile is a direct result of infinitesimal lifetime or extremely fast mass accretion with vanishing radial flow (see Fig. \ref{fig:2} and Eq. \eqref{ZEqnNum852508}).
Both NFW and Einasto profiles involve a halo concentration parameter $c={r_{h}/r_{s} }$, where $r_{h} $ and $r_{s}$ are the halo size and scale radius. Simulations have shown that the concentration $c={r_{h}/r_{s} } $ can be dependent on both halo mass and redshift. The evolution of \textit{c} depends very much on the mass accretion rate and the faster the halo grows, the slower \textit{c} increases. A constant value of \textit{c} is expected for large halos with extremely fast mass accretion and short lifespan, where $c\approx 4$ was estimated as a limiting value for large halos from \textit{N}-body simulations \citep{Zhao:2009-Accurate-Universal-Models-for-,Correa:2015-The-accretion-history-of-dark-}. The inner structures of these halos are still being dynamically adjusted due to fast mass accretion.
Figure \ref{fig:S1} plots the variation of shape parameter $\alpha $ of an Einasto profile with the concentration \textit{c} by numerically solving Eq. \eqref{ZEqnNum720481}. The shape parameter $\alpha $ decreases from 0.2 to 0.155 for concentration \textit{c} varying from 4 to 10, i.e. $\alpha $ increases with increasing halo mass that is consistent with simulations \citep{Gao:2008-The-redshift-dependence-of-the}.
\begin{figure}
\includegraphics*[width=\columnwidth]{FigS1}
\caption{The variation of shape parameter $\alpha$ of an Einasto profile with the halo concentration parameter c. Both NFW and Einasto profiles are assumed to have the same density at halo surface. This $c$-$\alpha$ relation is obtained by numerically solving Eq. \eqref{ZEqnNum720481}. Note that there is a discontinuity at c=2. Concentration parameter $\alpha$ is on the order of 0.2 and slowly decreases with increasing c for small halos.}
\label{fig:S1}
\end{figure}
\subsection{Effects of mass cascade on halo deformation and radial flow}
\label{sec:3.2}
Large halos with extremely short lifespan should have fast mass accretion rate. At the same redshift \textit{z}, these halos are dynamical objects with a constant mean density regardless of their masses. The fast halo mass accretion with short lifespan during mass cascade affects the halo density profiles by creating a non-zero radial flow. To quantitatively formulate this idea, let's first consider the time variation of mass of these halos,
\begin{equation}
\label{ZEqnNum872917}
m_{h}^{} \left(a\right)=\frac{4}{3} \pi r_{h}^{3} \Delta _{c} \bar{\rho }_{0} a^{-3} ,
\end{equation}
where $\Delta _{c} =18\pi ^{2} $ is a critical density ratio that can be obtained from spherical collapse model or a two-body collapse model \citep[see][Eq. (89)]{Xu:2021-A-non-radial-two-body-collapse}. Here $\bar{\rho }_{0} $ is the background density at the current epoch of $a=1$. The halo size $r_{h} \left(a\right)$ is defined as halo virial radius. Equation \eqref{ZEqnNum872917} implies that circular velocity at surface of halos satisfies (with $H^{2} a^{3} ={8\pi G\bar{\rho }_{0}/3} =H_{0}^{2} $ for matter dominant model),
\begin{equation}
\label{ZEqnNum837340}
v_{cir}^{2} =\frac{Gm_{h} }{r_{h} } =4\pi ^{2} \frac{r_{h}^{2} }{t^{2} } =\left(3\pi Hr_{h} \right)^{2},
\end{equation}
where $H\left(a\right)$ and $H_{0} $ are the Hubble parameter at scale factor \textit{a} and Hubble constant at the current epoch. The following relation for variation of halo size $r_{h} $ with \textit{a} can be obtained from Eq. \eqref{ZEqnNum872917},
\begin{equation}
\label{ZEqnNum728607}
\frac{\partial \ln r_{h} }{\partial \ln a} =\frac{1}{3} \frac{\partial \ln m_{h} }{\partial \ln a} +1=\frac{a}{3m_{h} } \frac{\partial m_{h} }{\partial a} +1.
\end{equation}
The time variation of typical halos of mass $m_{h} $ can be expressed in terms of time scale $\tau _{g}$ \citep[see][Eq. (6)]{Xu:2021-Inverse-mass-cascade-mass-function},
\begin{equation}
\label{ZEqnNum808457}
\frac{\partial m_{h} }{\partial a} =\frac{1}{Ha} \frac{\partial m_{h} }{\partial t} =\frac{m_{p} }{\tau _{g} Ha} , \end{equation}
where $\tau _{g} \left(m_{h} ,a\right)$ is the mean waiting time (lifespan) of a given halo for merging with a single merger of mass $m_{p} $ and passing its mass to large scale. After inserting Eq. \eqref{ZEqnNum808457} into Eq. \eqref{ZEqnNum728607},
\begin{equation}
\label{ZEqnNum587991}
\frac{\partial \ln r_{h} }{\partial \ln a} =\frac{m_{p} }{3\tau _{g} Hm_{h} } +1=\frac{1}{3\tau _{g} Hn_{p} } +1,
\end{equation}
\noindent where $n_{p} ={m_{h}/m_{p} } $ is the number of particles in that halo. Next consider the halo density at the surface of halos (as shown in Fig. \ref{fig:1}),
\begin{equation}
\label{ZEqnNum227253}
\rho _{h} \left(r=r_{h} \right)=\frac{N_{s} m_{p} }{4\pi r_{h}^{2} r_{p} } ,
\end{equation}
where $N_{s} $ is the number of elementary mass $m_{p} $ in spherical shell of thickness $r_{p} $.
Figure \ref{fig:1} illustrates how mass cascade changes the original halo size during an infinitesimal time interval $dt$. The original halo has a size $r_{h} $ at time \textit{t} (the dashed line). By the time $t+dt$, the halo size will increase from $r_{h} $ to $r_{h} -r_{p}^{'} +r_{p} $ due to mass cascade. The original halo surface (dashed line) shrinks to a smaller size of $r_{h} -r_{p}^{'} $ (solid line around green circle). First, halo mass cascade (accretion) creates a new layer of mass around the original halo with a thickness of $r_{p} -r_{p}^{'} $. Second, this layer of mass deforms the original halo (dark blue) to a new size (green) due to gravitational interaction. This deformation creates a non-zero inward radial flow of mass. For isothermal profile with vanishing radial flow, $r_{p}^{'} =0$ such that mass accretion does not deform the original halo. This is only possible for extremely fast mass accretion such that deformation is relatively much slower.
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig1}
\caption{Schematic plot of halo mass accretion and size change during an infinitesimal time interval $dt$. Original halo has a size $r_{h} $ at time \textit{t} (the dash line in the plot). By the time $t+dt$, the halo size will increase from $r_{h} $ to $r_{h} -r_{p}^{'} +r_{p} $ due to the mass accretion. The original halo at time \textit{t} deforms to the new size of $r_{h} -r_{p}^{'} $ (green). Halo mass accretion/cascade creates a new layer of mass around the original halo with a thickness of $r_{p} $. This layer of mass potentially deforms the original halo to a new size (green) due to the gravitational interaction, which creates a non-zero radial flow of mass. Special case $r_{p}^{'} =0$ (no radial flow) leads to an isothermal density profile.}
\label{fig:1}
\end{figure}
The time variation of halo radius $r_{h} $ due to mass accretion can be expressed as (with Eq. \eqref{ZEqnNum227253}),
\begin{equation}
\label{ZEqnNum258955}
\frac{\partial r_{h} }{\partial a} =\frac{1}{Ha} \frac{\partial r_{h} }{\partial t} =\frac{r_{p} -r_{p}^{'} }{N_{s} \tau _{g} Ha} =\frac{m_{p} }{\tau _{g} Ha} \frac{1}{4\pi r_{h}^{2} \rho _{h} \left(r_{h} \right)} \left(1-\frac{r_{p}^{'} }{r_{p} } \right),
\end{equation}
where $N_{s} \tau _{g} $ is the total time it takes to form the new layer of mass. Equivalently, we have
\begin{equation}
\label{ZEqnNum410355}
\frac{\partial \ln r_{h} }{\partial \ln a} =\frac{m_{p} }{4\pi r_{h}^{3} } \frac{\alpha _{h} }{\tau _{g} H\rho _{h} \left(r_{h} \right)} ,
\end{equation}
where the incremental change in halo size is $dr_{h} =r_{p} -r_{p}^{'} =\alpha _{h} r_{p} $. The halo deformation parameter $\alpha _{h} =1-{r_{p}^{'}/r_{p} } $ is introduced to reflect the effect of mass cascade on halo deformation. Mass density at halo surface can be obtained by comparing Eq. \eqref{ZEqnNum410355} with Eq. \eqref{ZEqnNum587991},
\begin{equation}
\label{ZEqnNum913739}
\rho _{h} \left(r_{h} \right)=\frac{m_{h} }{4\pi r_{h}^{3} } \frac{3\alpha _{h} }{1+3\tau _{g} Hn_{p} }. \end{equation}
From inverse mass cascade \citep[see][Eqs. (8) and (51)]{Xu:2021-Inverse-mass-cascade-mass-function}, we can estimate that on average,
\begin{equation}
\label{ZEqnNum464837}
\tau _{f} \left(m_{h}^{L} ,a\right)=\frac{1-\lambda }{1-{2\tau _{0}/3} } t\sim t,
\end{equation}
where the time scale $\tau _{f} =\tau _{g} n_{p} $ is the time it takes to form the entire halo. Therefore,
\begin{equation}
\label{ZEqnNum821124}
\tau _{g} Hn_{p} =\tau _{f} H=\frac{2\left(1-\lambda \right)}{3-2\tau _{0} } .
\end{equation}
With the help of Eq. \eqref{ZEqnNum821124}, Eqs. \eqref{ZEqnNum808457} and \eqref{ZEqnNum587991} give the time variation of halo mass and halo size,
\begin{equation}
\frac{\partial \ln m_{h} }{\partial \ln a} =\frac{3-2\tau _{0} }{2\left(1-\lambda \right)} \quad \textrm{and} \quad \frac{\partial \ln r_{h} }{\partial \ln a} =\frac{9-2\tau _{0} -6\lambda }{6\left(1-\lambda \right)},
\label{ZEqnNum952904}
\end{equation}
\noindent both of which are dependent on two mass cascade parameters $\lambda $ and $\tau _{0} $. Now, the time variation of power-law density profile can be derived with Eqs. \eqref{ZEqnNum859547} and \eqref{ZEqnNum952904},
\begin{equation}
\label{ZEqnNum808315}
\frac{\partial \ln \rho _{h} \left(r\right)}{\partial \ln a} =-3+\frac{m\left(9-2\tau _{0} -6\lambda \right)}{6\left(1-\lambda \right)} .
\end{equation}
By comparing Eq. \eqref{ZEqnNum913739} with the power-law density profile in Eq. \eqref{ZEqnNum789386}, it is found that large halos should have a density profile of $m=3-\alpha _{h} $ if parameters $\lambda ={2/3} $ and $\tau _{0} =1$. For an isothermal profile with $m=2$, it is necessary that $\alpha _{h} =1$ or $r_{p}^{'} =0$ such that mass accretion will not affect the halo internal structure. This is the limiting situation where halo mass accretion is extremely fast such that halos have no time to relax through radial deformation. The other limit is that $\alpha _{h} =0$ or $r_{p}^{'} =r_{p} $ (halo size gained from mass accretion exactly cancels the decrease in halo size due to the deformation) such that $m=3$ which is the maximum exponent for a power-law halo density profile.
We have scaling laws of $m_{h} \sim a^{{3/2} } \sim t$, $r_{h} \sim a^{{3/2} } \sim t$, $\rho _{h} \left(r=r_{h} \right)\sim r_{h}^{-2} \sim a^{-3} $, and $\rho _{h} \left(r\right)\sim a^{0} $ from Eqs. \eqref{ZEqnNum952904} and \eqref{ZEqnNum808315} for an isothermal profile. The halo density $\rho _{h} \left(r\right)$ is time-invariant as a result of $\alpha _{h} =1$ such that the halo density at any radius \textit{r} is fully determined at the moment that shell of halo is formed and will not change thereafter (Eq. \eqref{ZEqnNum808315}). This is the key feature of a power-law density profile that is different from NFW and Einasto profiles.
By comparing the density at surface of halo (Eq. \eqref{ZEqnNum913739}) with the NFW density in Eq. \eqref{ZEqnNum859547}, the halo concentration parameter $c$ can be related to deformation parameter $\alpha _{h}$,
\begin{equation}
\label{ZEqnNum412141}
\frac{\bar{\rho }_{h} \left(a\right)}{\rho _{h} \left(r_{h} ,a\right)} =3\left[1n\left(1+c\right)-\frac{c}{1+c} \right]\left(1+\frac{1}{c} \right)^{2} =\frac{1}{\alpha _{h} } \left(\frac{9-2\tau _{0} -6\lambda }{3-2\tau _{0} } \right).
\end{equation}
The concentration parameter \textit{c} is closely dependent on the halo deformation via $\alpha _{h} $ and on the mass cascade via parameters $\tau _{0} $ and $\lambda $. For large halos with $c=4$, and parameters $\lambda ={2/3} $ and $\tau _{0} =1$, the deformation parameter $\alpha _{h} \approx 0.79$. For small halos with $c=10$ and mass cascade parameter $\lambda ={2/3} $ and $\tau _{0} =1$, $\alpha _{h} \approx 0.56$. It is expected that the deformation parameter $\alpha _{h} $ increases with the halo mass (smaller halos have relatively greater deformation and smaller $\alpha _{h} $). The concentration-mass relation (the mass dependence of \textit{c}) might be related to the mass dependence of both $\alpha _{h} $ and geometry parameter $\lambda $ that should be further explored.
\subsection{Effects of radial flow on halo density distribution}
\label{sec:3.3}
The inverse mass cascade creates a new layer of mass that deforms the original halo to a new size (green in Fig. \ref{fig:1}). This creates a non-zero radial flow that can be analyzed using the continuity equation. Let's start from a general expression of mass $m_{r} \left(r,a\right)$ ,
\begin{equation}
m_{r} \left(r,a\right)=m_{h} \left(a\right)\frac{F\left(x\right)}{F\left(c\right)} \quad \textrm{and} \quad x\left(r,a\right)=\frac{r}{r_{s} \left(a\right)} =\frac{cr}{r_{h} \left(a\right)},
\label{ZEqnNum632204}
\end{equation}
\noindent where $m_{r} \left(r,a\right)$ is the mass in a sphere of radius \textit{r}, $x\left(r,a\right)$ is a reduced spatial-temporal variable that lumps the position \textit{r} and scale factor \textit{a} into a single variable. This general expression can represent both NFW (Eq. \eqref{eq:A4}) and Einasto (Eq. \eqref{ZEqnNum807556}), or any other density profiles via different functions $F\left(x\right)$. In principle, function $F\left(x\right)$ can be an arbitrary unknown function that satisfies $F\left(0\right)=0$. For example, $F\left(x\right)=x$ for an isothermal profile. Equations \eqref{ZEqnNum588557} and \eqref{ZEqnNum807556} give expressions of $F\left(x\right)$ for NFW and Einasto profiles. The halo density, potential, and velocity dispersion can all be determined in terms of unknown function $F\left(x\right)$. The halo density profile reads
\begin{equation}
\label{ZEqnNum482501}
\rho _{h} \left(r,a\right)=\frac{1}{4\pi r^{2} } \frac{\partial m_{r} \left(r,a\right)}{\partial r} =\frac{m_{h} \left(a\right)}{4\pi r_{h}^{3} } \frac{c^{3} F^{'} \left(x\right)}{x^{2} F\left(c\right)} ,
\end{equation}
and the logarithmic slope of the halo density reads
\begin{equation}
\label{ZEqnNum922312}
\frac{\partial \ln \rho _{h} }{\partial \ln x} =\frac{\partial \ln F^{'} \left(x\right)}{\partial \ln x} -2=x\frac{F^{''} \left(x\right)}{F^{'} \left(x\right)} -2.
\end{equation}
Evidently, the halo deformation parameter satisfies \\
$\alpha _{h} ={cF^{'} \left(c\right)/F\left(c\right)}$\\
by comparing the density at halo surface to Eq. \eqref{ZEqnNum913739} with $\lambda ={2/3} $ and $\tau _{0} =1$. Time variation of $\rho _{h} \left(r,a\right)$ can be obtained from Eq. \eqref{ZEqnNum482501},
\begin{equation}
\label{ZEqnNum505562}
\frac{\partial \rho _{h} \left(r,a\right)}{\partial t} =\frac{1}{4\pi r^{2} } \frac{\partial ^{2} m_{r} \left(r,a\right)}{\partial r\partial t} .
\end{equation}
The mass continuity equation for a spherical halo in spherical coordinate simply reads,
\begin{equation}
\label{ZEqnNum114919}
\frac{\partial \rho _{h} \left(r,a\right)}{\partial t} +\frac{1}{r^{2} } \frac{\partial \left[r^{2} \rho _{h} \left(r,a\right)u_{r} \left(r,a\right)\right]}{\partial r} =0,
\end{equation}
where $u_{r} \left(r,a\right)$ is the mean radial flow velocity. From Eqs. \eqref{ZEqnNum505562} and \eqref{ZEqnNum114919}, the mass $m_{r} \left(r,a\right)$ is related to the radial flow velocity as,
\begin{equation}
\label{ZEqnNum861530}
\frac{\partial m_{r} \left(r,a\right)}{\partial t} =-4\pi r^{2} u_{r} \left(r,a\right)\rho _{h} \left(r,a\right). \end{equation}
With $m_{r} \left(r,a\right)$ from Eq. \eqref{ZEqnNum632204} and $\rho _{h} \left(r,a\right)$ from Eq. \eqref{ZEqnNum482501}, the radial flow velocity reads
\begin{equation}
\label{ZEqnNum255276}
u_{r} =-\frac{1}{4\pi r^{2} } \frac{\partial \ln m_{r} }{\partial \ln t} \frac{m_{r} \left(r,a\right)}{\rho _{h} \left(r,a\right)t} =-\frac{r_{s} \left(t\right)}{t} \frac{\partial \ln m_{r} }{\partial \ln t} \frac{F\left(x\right)}{F^{'} \left(x\right)} .
\end{equation}
While from Eq. \eqref{ZEqnNum632204} for $m_{r} \left(r,a\right)$, we have
\begin{equation}
\label{ZEqnNum295393}
\frac{\partial \ln m_{r} }{\partial \ln t} =\frac{\partial \ln m_{h} }{\partial \ln t} -\frac{xF^{'} \left(x\right)}{F\left(x\right)} \frac{\partial \ln r_{s} }{\partial \ln t} -\frac{cF^{'} \left(c\right)}{F\left(c\right)} \frac{\partial \ln c}{\partial \ln t} .
\end{equation}
Substituting Eqs. \eqref{ZEqnNum295393} into \eqref{ZEqnNum255276}, the radial flow has a very simple expression,
\begin{equation}
\label{ZEqnNum853896}
u_{r} \left(r,a\right)=\left[x\frac{\partial \ln r_{s} }{\partial \ln t} +\left(\frac{\partial \ln F\left(c\right)}{\partial \ln t} -\frac{\partial \ln m_{h} }{\partial \ln t} \right)\frac{F\left(x\right)}{F^{'} \left(x\right)} \right]\frac{r_{s} }{t} ,
\end{equation}
which is a general equation for mean radial flow with a time-varying concentration \textit{c}. For small halos with a stable core and extremely slow mass accretion (${\partial r_{s}/\partial t} \approx 0$ and ${\partial m_{h}/\partial t} \approx 0$) and constant $m_{r} \left(r_{s} ,a\right)$, we shall expect that $F\left(c\right)\propto m_{h} $ is almost a constant (from Eq. \eqref{ZEqnNum632204}) and $u_{r} \left(r,a\right)=0$ (Eq. \eqref{ZEqnNum853896}) that is consistent with the stable clustering hypothesis (small halos are virialized and well bound structures).
For the other limiting situation, i.e. large halos with extremely fast mass accretion and an expanding core, the concentration \textit{c} is relatively a constant. Equation \eqref{ZEqnNum853896} reduces to
\begin{equation}
\label{ZEqnNum849591}
u_{r} \left(r,a\right)=\frac{1}{c} \left[x-\frac{\partial \ln m_{h} }{\partial \ln r_{h} } \frac{F\left(x\right)}{F^{'} \left(x\right)} \right]\frac{\partial r_{h} }{\partial t} .
\end{equation}
In principle, the non-zero halo growth rate ${\partial r_{h}/\partial t} $ should lead to a non-zero mean radial flow. A special case is the isothermal profile with $F\left(x\right)=x$ and $m_{h} \propto r_{h} $, where $u_{r} \left(r,a\right)=0$, i.e. a vanishing radial flow for isothermal profile even if ${\partial r_{h}/\partial t} \ne 0$. The radial flow $u_{r} \left(r,a\right)$ is a function of reduced position \textit{x} only and scaled by the rate of halo growth ${\partial r_{h}/ \partial t} $. We introduce a dimensionless radial flow velocity $u_{h} \left(x\right)$ (normalized by the core expanding speed ${r_{s}/t} $) as
\begin{equation}
\label{ZEqnNum278808}
\begin{split}
u_{h} \left(x\right)=\frac{cu_{r} \left(r,a\right)t}{r_{h} } &=\frac{u_{r} \left(r,a\right)}{{r_{s}/t} }\\ &=\Big[\underbrace{x}_{1}-\underbrace{\frac{\partial \ln m_{h} }{\partial \ln r_{h} } \frac{F\left(x\right)}{F^{'} \left(x\right)} }_{2}\Big]\frac{\partial \ln r_{h} }{\partial \ln t}.
\end{split}
\end{equation}
Clearly, the halo growth rate ${\partial r_{h}/\partial t} $ affects the mean radial flow in Eq. \eqref{ZEqnNum278808}. For large halos with a constant value of $c$, ${\partial r_{h}/\partial t} ={r_{h}/ t={v_{cir}/\left(2\pi \right)} } $ (Eq. \eqref{ZEqnNum837340}) does not varying with time and $u_{r} \left(r,a\right)$ is self-similar and only dependent on the reduced variable $x$. The total radial flow can be decomposed into two contributions: 1) the outward flow due to the halo growth where $u_{r} \left(r,a\right)={r/t} $ (term 1), and 2) the inward flow due to the halo deformation (term 2).
By comparing the density at surface of halos (Eqs. \eqref{ZEqnNum482501} and \eqref{ZEqnNum913739}) and using the help of Eqs. \eqref{ZEqnNum821124} and \eqref{ZEqnNum952904}, the constraints and boundary conditions for radial flow velocity are,
\begin{equation}
\begin{split}
&u_{h} \left(x=0\right)=0, \quad u_{h} \left(x=c\right)=c\left(1-\frac{1}{\alpha _{h} } \right)\frac{\partial \ln r_{h} }{\partial \ln t},\\
&\textrm{and}\\
&x_{0} =\frac{\partial \ln m_{h} }{\partial \ln r_{h} } \left. \frac{F\left(x\right)}{F^{'} \left(x\right)} \right|_{x=x_{0} },
\label{ZEqnNum871868}
\end{split}
\end{equation}
\noindent where $u_{h} \left(x_{0} \right)=0$. Figure \ref{fig:2} plots the normalized radial velocity $u_{h} \left(x\right)$ for three different density profiles. The NFW and Einasto profiles (for $c=4$ and $\alpha =0.2$) lead to a very similar radial flow velocity with out-flow ($u_{h} \left(x\right)>0$) for core region ($x<x_{0} $) and in-flow ($u_{h} \left(x\right)<0$) for outer region ($x>x_{0} $) of halos, where $u_{h} \left(x_{0} \right)=0$. The maximum radial flow is at $x=1$ or $r=r_{s} $. The total mass $m_{r} \left(x,a\right)$ decreases with time for $x<x_{0} $ and increases with time for $x>x_{0} $ (from Eq. \eqref{ZEqnNum861530}). The difference between two density profiles is that $u_{h} \left(x\to 0\right)={x/2} $ for NFW profile and $u_{h} \left(x\to 0\right)={2x/3} $ for Einasto profile.
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig2}
\caption{The normalized mean radial flow $u_{h} \left(x\right)$ (Eq. \eqref{ZEqnNum278808}) for three different density profiles. The isothermal halo density corresponds to a vanishing radial flow, i.e.halos with extremely fast mass accretion and no internal deformation. The NFW and Einasto profiles ($c=4$ and $\alpha =0.2$) lead to very similar radial flow with out-flow ($u_{h} \left(x\right)>0$) for the core region and in-flow ($u_{h} \left(x\right)<0$) for the outer region. The maximum radial flow is at $x=1$ or $r=r_{s} $. The mass $m_{r} \left(r,a\right)$ inside the radius \textit{r} decreases with time for $x<x_{0} $ and increases with time for $x>x_{0} $ (Eq. \eqref{ZEqnNum861530}). The circular velocity is at its maximum at $x=x_{0} $. The difference between two profiles is that $u_{h} \left(x\to 0\right)={x/2} $ for NFW and $u_{h} \left(x\to 0\right)={2x/3} $ for Einasto profile.}
\label{fig:2}
\end{figure}
The dimensionless peculiar radial flow can be obtained by subtracting the Hubble flow,
\begin{equation}
\label{eq:27}
u_{p} \left(x\right)=u_{h} \left(x\right)-\frac{2}{3} x=\frac{1}{3} x-\frac{F\left(x\right)}{F^{'} \left(x\right)} .
\end{equation}
Especially for an isothermal profile with $F\left(x\right)=x$, $u_{p} \left(x\right)=-{2x/3} $, which can be a good approximation of peculiar radial flow. This is consistent with the stable clustering hypothesis, i.e. the peculiar radial flow
\begin{equation}
\label{eq:28}
u_{p} \left(x\right)\frac{r_{s} }{t} =-\frac{2}{3} x\frac{r_{h} }{ct} =-\frac{2}{3} \frac{cr}{r_{h} } \cdot \frac{r_{h} }{ct} =-Hr.
\end{equation}
\subsection{The angle of incidence for mass cascade}
\label{sec:3.4}
An interesting quantity is the angle $\theta _{vr} $ (the angle of incidence) between particle peculiar velocity and its position vector from halo center of mass,
\begin{equation}
\label{ZEqnNum340811}
\begin{split}
\cot \left(\theta _{vr} \right)&=\frac{u_{p} \left(x\right)}{v_{c} \left(r,a\right)} \frac{r_{h} }{ct}\\ &=\frac{x}{2\pi c} \left(\frac{1}{3} -\frac{1}{\alpha _{h} } \cdot \frac{F^{'} \left(c\right)}{F^{'} \left(x\right)} \cdot \frac{cF\left(x\right)}{xF\left(c\right)} \right)\sqrt{\frac{xF\left(c\right)}{cF\left(x\right)} } ,
\end{split}
\end{equation}
where circular velocity (normalized) at any radius \textit{r} of the halo is
\begin{equation}
\label{ZEqnNum537303}
v_{nc}^{2} \left(r,a\right)=\frac{v_{c}^{2} \left(r,a\right)}{v_{cir}^{2} } =\frac{Gm_{r} \left(r,a\right)}{rv_{cir}^{2} } =\frac{cF\left(x\right)}{F\left(c\right)x} .
\end{equation}
Interestingly, $v_{nc}^{2} \left(x,a\right)$ is at its maximum when $x=x_{0} $ where $u_{h} \left(x_{0} \right)=0$ (using Eq. \eqref{ZEqnNum871868}). Quantity $\cot \left(\theta _{vr} \right)$ quantifies the ratio of radial motion (radial momentum) to the circular motion (angular momentum). With $\alpha _{h} ={cF^{'} \left(c\right)/F\left(c\right)} $, $\cot \left(\theta _{vr} \right)$ at halo surface ($x=c$) and halo center ($x=0$) are obtained from Eq. \eqref{ZEqnNum340811} as
\begin{equation}
\label{eq:31}
\cot \left(\theta _{vr} \left(x=c\right)\right)=\frac{1}{2\pi } \left(\frac{1}{3} -\frac{1}{\alpha _{h} } \right)
\end{equation}
and
\begin{equation}
\label{ZEqnNum184216}
\cot \left(\theta _{vr} \left(x=0\right)\right)={\mathop{\lim }\limits_{x\to 0}} \frac{1}{2\pi c} \left(\frac{1}{3} -\frac{F\left(x\right)}{xF^{'} \left(x\right)} \right)\sqrt{\frac{x^{3} F\left(c\right)}{cF\left(x\right)} } .
\end{equation}
When $x=c$ at halo surface, $\cot \left(\theta _{vr} \right)=-{1/\left(3\pi \right)} $ and $\theta _{vr} \approx 96.06^{o} $ for $\alpha _{h} =1$ (isothermal profile) and $\theta _{vr} \approx 98.44^{o} $ for $\alpha _{h} =0.79$ (NFW profile with c=4 from Eq. \eqref{ZEqnNum412141}). From Eq. \eqref{ZEqnNum184216}, angle $\theta _{vr} $ should gradually decrease to $\theta _{vr} =90^{o}$ ($\cot \left(\theta _{vr} \right)=0$ with $u_{p} \left(x\right)\to 0$) at the core region for any $F\left(x\right)\propto x^{m} $ ($m\le 3$) with $x\to 0$. This can be also demonstrated by a two-body gravitational collapse (TBCM) model \citep[see][Eq. (105)]{Xu:2021-A-non-radial-two-body-collapse}.
By taking derivative of $u_{h} \left(x\right)$ (Eq. \eqref{ZEqnNum278808}) with respect to \textit{x},
\begin{equation}
\label{ZEqnNum273026}
\frac{\partial u_{h} \left(x\right)}{\partial x} =\left[1-\frac{\partial \ln m_{h} }{\partial \ln r_{h} } +\frac{\partial \ln m_{h} }{\partial \ln r_{h} } \frac{F\left(x\right)F^{''} \left(x\right)}{F^{'2} \left(x\right)} \right]\frac{\partial \ln r_{h} }{\partial \ln t} .
\end{equation}
Particularly for a NFW profile,
\begin{equation}
\label{eq:34}
\frac{\partial u_{h} \left(x\right)}{\partial x} =\left[\ln \left(1+x\right)-\frac{x}{1+x} \right]\frac{1-x^{2} }{x^{2} } .
\end{equation}
It can be verified that for both NFW and Einasto profiles, the conditions of maximum flow ($\left. {\partial u_{h} \left(x\right)/\partial x} \right|_{x=1} =0$) and logarithmic slope of -2 ($\left. {\partial \ln \rho _{h}/\partial \ln x} \right|_{x=1} =-2$) at scale radius ($x=1$ or $r=r_{s} $) requires $\left. F^{''} \left(x\right)\right|_{x=1} =0$ (Eq. \eqref{ZEqnNum922312}). Hence, ${\partial \ln m_{h} /\partial \ln r_{h} } =1$ from Eq. \eqref{ZEqnNum273026}, which confirms $\tau _{0} ={3\lambda/2} $ for large halos (from Eq. \eqref{ZEqnNum952904}).
Existence of an extra length scale $r_{s}$ in density profile origins from mass cascade induced radial flow. The in-flow in halo outer region and the out-flow in the inner region creates a maximum mass flow rate at scale radius $r_{s} $ (or $x=1$) and introduces an extra length scale $r_{s}$ for halo density profile, which does not exist for a scale-free isothermal density profile (Fig. \ref{fig:2}).
We are especially interested in the logarithmic slope of the unknown function $F^{'} \left(x\right)$ that directly impacts the halo density profile (Eq. \eqref{ZEqnNum482501}). It can be obtained from the mean radial flow $u_{h} \left(x\right)$ using Eqs. \eqref{ZEqnNum278808} and \eqref{ZEqnNum273026},
\begin{equation}
\label{ZEqnNum696290}
\frac{\partial \ln F^{'} \left(x\right)}{\partial \ln x} =\frac{\frac{\partial u_{h} \left(x\right)}{\partial x} -\frac{\partial \ln r_{h} }{\partial \ln t} +\frac{\partial \ln m_{h} }{\partial \ln t} }{\frac{\partial \ln r_{h} }{\partial \ln t} -\frac{u_{h} \left(x\right)}{x} } .
\end{equation}
Specially, for matter dominant system, we expect,
\begin{equation}
\frac{\partial \ln r_{h} }{\partial \ln t} =\frac{\partial \ln m_{h} }{\partial \ln t} =1, \quad\frac{\partial \ln F^{'} \left(x\right)}{\partial \ln x} =\frac{xF^{''} \left(x\right)}{F^{'} \left(x\right)} =\frac{{\partial u_{h}/\partial x} }{1-{u_{h}/x} }.
\label{ZEqnNum445988}
\end{equation}
\noindent The logarithmic slope of halo density profile (from Eq. \eqref{ZEqnNum922312}) reads
\begin{equation}
\label{ZEqnNum852508}
\frac{\partial \ln \rho _{h} }{\partial \ln x} =\frac{\partial \ln F^{'} \left(x\right)}{\partial \ln x} -2=\frac{{\partial u_{h}/\partial x} }{1-{u_{h}/x} } -2.
\end{equation}
We have simple expressions for NFW and Einasto profiles, \begin{equation}
\frac{\partial \ln F^{'} \left(x\right)}{\partial \ln x} =\frac{1-x}{1+x} \quad \textrm{and} \quad \frac{\partial \ln F^{'} \left(x\right)}{\partial \ln x} =2-2x^{\alpha }.
\label{eq:38}
\end{equation}
To provide some insights into the long-standing cusp-core controversy (core/cusp problem), a double-power-law density profile can be proposed as a natural result of Eq. \eqref{ZEqnNum852508}. The inner halo density is determined by the velocity gradient (halo deformation rate) $\gamma _{h} =\left. {\partial u_{h}/\partial x} \right|_{x=0} $ such that inner halo density follows a power-law
\begin{equation}
\label{ZEqnNum973033}
\rho _{h} \left(r<r_{s} \right)\propto r^{{\left(3\gamma _{h} -2\right)/\left(1-\gamma _{h} \right)} }
\end{equation}
that is dependent on parameter $\gamma _{h} $ only. The smaller $\gamma _{h} $ (slower deformation at the halo center) leads to a steeper density profile. The baryonic feedback processes may enhance the deformation rate $\gamma _{h}$ at halo center and lead to the formation of core structure.
In addition, for a matter dominant universe, the radial flow should be exactly the Hubble flow if both gravitational and pressure forces are not present in halos. If the potential and pressure are symmetric functions of \textit{r} and regular at origin \textit{r}=0, the gravitational and pressure forces should vanish at origin such that $u_{r} \left(r\to 0\right)=Hr$. We expect the initial velocity of mass shells at the center of halo is simply the Hubble flow for halos with fast mass accretion,
\begin{equation}
\label{ZEqnNum801048}
u_{r} \left(r\to 0\right)=u_{h} \left(x\to 0\right)\frac{r_{s} }{t} =\gamma _{h} x\frac{r_{s} }{t} =\gamma _{h} \frac{r}{t} =Hr,
\end{equation}
such that $\gamma _{h} =Ht={2/3} $. This means a central core with $\rho _{h} \left(r<r_{s} \right)\propto r^{0} $ from Eq. \eqref{ZEqnNum973033} does exist for large halos with fast mass accretion, in agreement with the finding that large halos can be better fitted by an Einasto profile \citep{Klypin:2016-MultiDark-simulations--the-sto}.
For outer halo region (especially $r\gg x_{0} $ in Fig. \ref{fig:2}), we can approximate (using Eq. \eqref{ZEqnNum871868})
\begin{equation}
\frac{\partial u_{h} }{\partial x} \approx \frac{c\left(1-{1/\alpha _{h} } \right)}{c-x_{0} } \quad \textrm{and} \quad 1-\left. \frac{u_{h} }{x} \right|_{x=c} \approx \frac{1}{\alpha _{h} },
\label{eq:41}
\end{equation}
\noindent Such that from Eq. \eqref{ZEqnNum852508}
\begin{equation}
\label{ZEqnNum367971}
\rho _{h} \left(r>r_{s} \right)\propto r^{\frac{c\left(\alpha _{h} -1\right)}{c-x_{0} } -2} ,
\end{equation}
with a power-law density profile steeper than the isothermal profile of -2 for the outer halo region. Equations \eqref{ZEqnNum973033} and \eqref{ZEqnNum367971} provide a double-power-law density with inner density controlled by halo deformation rate parameter $\gamma _{h} $ and outer density controlled by the halo growth via a halo deformation parameter $\alpha _{h} $ and concentration \textit{c}.
In principle, accurate halo density profiles can be obtained only if the normalized mean flow $u_{h} \left(x\right)$ is known. Without loss of generality, the Taylor expansion of $u_{h} \left(x\right)$ around the center (up to the third order) can be given by
\begin{equation}
\label{ZEqnNum621572}
u_{h} \left(x\right)=a_{0} +\gamma _{h} x+a_{2} x^{2} +a_{3} x^{3} ,
\end{equation}
with three unknown coefficients. To satisfy the boundary conditions \eqref{ZEqnNum871868} and the constraint $u_{h} \left(0\right)=0$ and $\left. {\partial u_{h} /\partial x} \right|_{x=1} =0$, we have $a_{2} $ and $a_{3} $ expressed as
\begin{equation}
\label{ZEqnNum909055}
a_{0} =0, a_{2} =-\frac{\left(c^{2} -3\right)\gamma _{h} +3-{3/\alpha _{h} } }{2c^{2} -3c} , a_{3} =\frac{\left(c-2\right)\gamma _{h} +2-{2/\alpha _{h} } }{2c^{2} -3c} .
\end{equation}
The unknown function $F\left(x\right)$ can be analytically solved from Eq. \eqref{ZEqnNum278808} and we have the solution,
\begin{equation}
\label{ZEqnNum676651}
\begin{split}
&F\left(x\right)=\left(\frac{x^{3} }{x-u_{h} \left(x\right)} \right)^{\frac{1}{2\left(1-\gamma _{h} \right)} }\\
&\cdot \exp \left\{\frac{{a_{2}/\left(\gamma _{h} -1\right)} }{\sqrt{4a_{3} \left(\gamma _{h} -1\right)-a_{2}^{2} } }\textrm{atan}\left[\frac{a_{2} +2a_{3} x}{\sqrt{4a_{3} \left(\gamma _{h} -1\right)-a_{2}^{2} } } \right]\right\},
\end{split}
\end{equation}
with which the density profile can be obtained from Eq. \eqref{ZEqnNum482501}.
We have shown that a complete description of $u_{h} \left(x\right)$ or $F\left(x\right)$ requires at least three parameters, the deformation rate $\gamma _{h} $ at the center of halo, halo deformation parameter $\alpha _{h}$ at the surface of halo, and concentration \textit{c} for the size of halos. The location $x_{0}$ where $u_{h} \left(x_{0} \right)=0$ is estimated to be (from Eq. \eqref{ZEqnNum621572})
\begin{equation}
\label{eq:46}
x_{0} =\frac{-a_{2} -\sqrt{a_{2}^{2} -4a_{3} \gamma _{h} } }{2a_{3} } ,
\end{equation}
with limiting values
\begin{equation}
x_{0} =\frac{3}{2}\quad \textrm{for} \quad \gamma _{h} \to 0,\quad x_{0} =\frac{2c-3}{c-2}\quad \textrm{for} \quad \gamma _{h} \to \infty.
\label{eq:47}
\end{equation}
\noindent The radial flow at $x=1$ and its derivative at $x=c$ are,
\begin{equation}
\begin{split}
&u_{h} \left(x=1\right)=\frac{\left(c-1\right)^{2} \gamma _{h} -\left(1-{1/\alpha _{h} } \right)}{c\left(2c-3\right)}\\
&\textrm{and}\\
&\left.\frac{\partial u_{h} }{\partial x} \right|_{x=c} =\frac{\left(c-3\right)\gamma _{h} +6\left(1-{1/\alpha _{h} } \right)}{{\left(2c-3\right)/\left(c-1\right)} }.
\end{split}
\label{eq:48}
\end{equation}
\noindent An even simpler case is a Taylor expansion of $u_{h} \left(x\right)$ up to the second order (i.e. $a_{3} =0$) that will lead to solutions,
\begin{equation}
a_{0} =0, \quad a_{2} =-{\gamma _{h}/2}, \quad \textrm{and} \quad \gamma _{h} ={\left(1-{1/\alpha _{h} } \right)\left(1-{c/2}\right)} \label{eq:49}
\end{equation}
\noindent from Eq. \eqref{ZEqnNum909055}. We have $x_{0} =2$ from Eq. \eqref{ZEqnNum621572} for expansion of $u_{h} \left(x\right)$ up to the second order.
Alternatively, function $F\left(x\right)$ can be modelled directly with the following constraints:
\begin{equation}
\begin{split}
&F\left(0\right)=0, \quad F\left(x\to 0\right)=x^{1/\left(1-\gamma _{h} \right)},\\
&\frac{cF^{'} \left(c\right)}{F\left(c\right)} =\alpha _{h},\quad \textrm{and} \quad \left. \frac{\partial ^{2} F}{\partial x^{2} } \right|_{x=1} =0.
\end{split}
\label{ZEqnNum434585}
\end{equation}
\noindent Solutions of all relevant quantities can be easily obtained with either $u_{h} \left(x\right)$ or $F\left(x\right)$ explicitly modelled. Therefore, halo density profile can be found by the correctly modeling of either dimensionless radial flow $u_{h} \left(x\right)$ or unknown function $F\left(x\right)$.
In short, the logarithmic slope of density profile is continuously dependent on the mean radial flow $u_{h} \left(x\right)$ (Eq. \eqref{ZEqnNum852508}). An accurate model of $u_{h} \left(x\right)$ due to mass cascade can improve halo density models. Since the matter density spectrum is closely related to halo density profiles and mass functions, effects of mass cascade and mass accretion on the density spectrum can be further investigated.
\subsection{Limiting concentration c from momentum/kinetic energy}
\label{sec:3.5}
The limiting value of concentration $c\approx 4$ for large halos with fast mass accretion was estimated from \textit{N}-body simulations. It is possible to analytically derive this limiting value by requiring a vanishing radial momentum for large halos with fast mass accretion. With Eq. \eqref{ZEqnNum849591} for mean flow $u_{r} \left(r,a\right)$ and Eq. \eqref{ZEqnNum482501} for density $\rho _{h} \left(r,a\right)$, the radial linear momentum is
\begin{equation}
\label{ZEqnNum808595}
\begin{split}
L_{hr} \left(a\right)&=\int _{0}^{r_{h} }u_{r} \left(r,a\right)4\pi r^{2} \rho _{h} \left(r,a\right)dr\\
&=\frac{m_{h} v_{cir} }{2\pi cF\left(c\right)} \left(cF\left(c\right)-2\int _{0}^{c}F\left(x\right)dx \right).
\end{split}
\end{equation}
With unknown function $F\left(x\right)=\ln \left(1+x\right)-{x/\left(1+x\right)} $ (Eq. \eqref{ZEqnNum588557}) for NFW profile, the radial linear momentum reduces to
\begin{equation}
\label{eq:52}
L_{hr} \left(a\right)=\frac{m_{h} v_{cir} }{2\pi cF\left(c\right)} \left(\frac{c\left(4+3c\right)}{1+c} -\left(4+c\right)\ln \left(1+c\right)\right).
\end{equation}
The critical value of $c$ for individual halos can be identified from the condition of a vanishing linear momentum (like spherical shells at turn-around point with a zero velocity in spherical collapse model). Therefore, by requiring $L_{hr} \left(a\right)=0$, Eq. \eqref{ZEqnNum808595} becomes
\begin{equation}
\label{ZEqnNum722864}
cF\left(c\right)=2\int _{0}^{c}F\left(x\right)dx ,
\end{equation}
where it was found that $c=3.48$ for a NFW profile (Fig. \ref{fig:3}). With sufficiently fast mass accretion rate, halos keep growing with a vanishing radial momentum $L_{hr} $. For halos with $c>3.48$, this self-similar solution leads to a negative radial linear momentum $L_{hr} <0$ indicating an overall in-flow of momentum.
Next, the radial kinetic energy is given by,
\begin{equation}
\label{eq:54}
\begin{split}
&K_{hr} \left(a\right)=\frac{1}{2} \int _{0}^{r_{h} }u_{r}^{2} \left(r,a\right)4\pi r^{2} \rho _{h} \left(r,a\right)dr\\
&=\frac{m_{h} v_{cir}^{2} }{8\pi ^{2} c^{2} F\left(c\right)} \left(c^{2} F\left(c\right)-4\int _{0}^{c}xF\left(x\right)dx +\int _{0}^{c}\frac{F^{2} \left(x\right)}{F^{'} \left(x\right)} dx \right).
\end{split}
\end{equation}
Specifically, the radial kinetic energy for a NFW profile is given by the expression of,
\begin{equation}
\label{eq:55}
\begin{split}
&K_{hr} =\frac{m_{h} v_{cir}^{2} }{8\pi ^{2} c^{2} F\left(c\right)} \left[\frac{c\left(13c^{2} +7c-10\right)}{\left(1+c\right)}\right.\\
&\left.+2\ln \left(1+c\right)\left(5-10c-5c^{2}+\left(3+4c+c^{2} +2\ln \left(-c\right)\right)\ln \left(1+c\right)\right)+\right.\\
&\left.+8\ln \left(1+c\right)\textrm{Polylog}\left(2,1+c\right)-8\textrm{Polylog}\left(3,1+c\right)+8\textrm{Zeta}\left(3\right)\right],
\end{split}
\end{equation}
which involves polylogarithm and zeta functions. In general, we can express both radial linear momentum and radial kinetic energy in terms of the circular velocity $v_{cir} $ (Eq. \eqref{ZEqnNum837340}) with two coefficients that are functions of \textit{c},
\begin{equation}
L_{hr} \left(a\right)=\lambda _{Lr} \left(c\right)m_{h} v_{cir} \quad \textrm{and} \quad K_{hr} =\lambda _{Kr} \left(c\right)m_{h} v_{cir}^{2}.
\label{eq:56}
\end{equation}
\noindent With $c=3.48$ for NFW profile, we have $\lambda _{Lr} =0$ and $\lambda _{Kr} =7\times 10^{-5} $. Figure \ref{fig:3} plots the variation of two coefficients with concentration \textit{c}. Note that $\lambda _{Kr}$ is rescaled by 100 times to be plotted in the same plot as $\lambda _{Lr} $. Halos with $c>3.48$ have increasing kinetic energy with \textit{c}. Large halos with fast mass accretion have a vanishing radial linear momentum with $c=3.48$ and a (almost) minimum radial kinetic energy for all different concentration \textit{c}. Large halos with fast mass accretion tend to grow with vanishing radial momentum and minimum radial kinetic energy.
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig3}
\caption{The variation of normalized radial momentum $\lambda _{Lr} $ and kinetic energy $\lambda _{Kr} $ with the concentration parameter \textit{c} for a NFW profile. A limiting value of $c=3.48$ can be found for large halos with $\lambda _{Lr} =0$, where the linear radial momentum vanishes. The normalized radial kinetic energy is also close to its minimum at the limiting value of \textit{c}.}
\label{fig:3}
\end{figure}
\subsection{Effects of radial flow on halo velocity dispersion}
\label{sec:3.6}
The Jeans' equation coupled with inverse mass cascade can be used to study the effect of radial flow on velocity dispersion. First, the gravitational potential in terms of unknown function $F\left(x\right)$ reads
\begin{equation}
\label{ZEqnNum207666}
\phi _{h} \left(r,a\right)=-G\int _{r}^{\infty }\frac{m_{r} \left(y,a\right)}{y^{2} } dy =-v_{cir}^{2} \frac{c}{F\left(c\right)} \int _{x}^{\infty }\frac{F\left(y\right)}{y^{2} } dy .
\end{equation}
A shifted gravitation potential can be introduced to satisfy $\phi _{h}^{*} \left(r=0,a\right)=0$,
\begin{equation}
\label{ZEqnNum269276}
\begin{split}
\phi _{h}^{*} \left(r,a\right)&=\phi _{h}^{} \left(r,a\right)+v_{cir}^{2} \frac{c}{F\left(c\right)} \int _{0}^{\infty }\frac{F\left(y\right)}{y^{2} } dy\\
&=v_{cir}^{2} \frac{c}{F\left(c\right)} \int _{0}^{x}\frac{F\left(y\right)}{y^{2} } dy.
\end{split}
\end{equation}
The full dynamic Jeans' equation along radial direction is usually written as
\begin{equation}
\label{ZEqnNum535400}
\begin{split}
\underbrace{\frac{\partial u_{r} }{\partial t} +u_{r} \frac{\partial u_{r} }{\partial r} }_{1}&+\frac{1}{\rho _{h} } \frac{\partial \left(\rho _{h} \sigma _{r}^{2} \right)}{\partial r} +\frac{2}{r} \beta _{h} \sigma _{r}^{2} \left(r,a\right)\\
&=-\frac{\partial \phi _{h} \left(r,a\right)}{\partial r} =-\frac{Gm_{r} \left(r,a\right)}{r^{2} } ,
\end{split}
\end{equation}
where $\sigma _{r}^{2} \left(r,a\right)$ is the radial velocity dispersion. The anisotropy of velocity dispersion is defined through an anisotropy parameter $\beta _{h} =1-{\sigma _{t}^{2}/\sigma _{r}^{2} } $, where $\sigma _{t}^{2} \left(r,a\right)$ is the tangential velocity dispersion. For isotropic velocity dispersion, we have $\sigma _{t} =\sigma _{r} $ and $\beta _{h} =0$. The full dynamic Jeans' equation \eqref{ZEqnNum535400} relates non-zero radial flow to halo velocity dispersion $\sigma _{r}^{2}$ for a non-rotating spherical halo. The dynamics of a rotating halo with finite angular momentum is much more complicated and presented in a separate paper \citep{Xu:2022-The-mean-flow--velocity-disper}.
Term 1 from the radial flow is often neglected for small virialized halos ($u_{r} =0$) and the radial velocity dispersion can be solved by the static Jeans equation with known $m_{r} \left(r,a\right)$ or a given density profile \citep{Binney:1982-M-L-and-Velocity-Anisotropy-fr}. However, large halos with fast mass accretion are dynamic objects, where mass cascade/accretion leads to a non-zero mean radial flow $u_{r} $ that will contribute significantly to velocity dispersion (especially to the outer region of halos). Here we attempt to solve an inverse problem, i.e. solving for the velocity dispersion $\sigma _{r}^{2} $ with a known mean radial flow $u_{r} $. After substituting the expressions for $m_{r} \left(r,a\right)$ (Eq. \eqref{ZEqnNum632204}) and $u_{r} \left(r,a\right)$ (Eq. \eqref{ZEqnNum849591}) into Jeans' equation \eqref{ZEqnNum535400} with chain rule from $x={r/r_{s} \left(t\right)} $,
\begin{equation}
\frac{\partial }{\partial t} =\frac{\partial }{\partial x} \frac{\partial x}{\partial t} =-\frac{x}{t} \frac{\partial \ln r_{s} }{\partial \ln t} \frac{\partial }{\partial x} \quad \textrm{and} \quad \frac{\partial }{\partial r} =\frac{\partial }{\partial x} \frac{\partial x}{\partial r} =\frac{1}{r_{s} } \frac{\partial }{\partial x},
\label{ZEqnNum253473}
\end{equation}
\noindent The original Jeans' equation becomes
\begin{equation}
\label{eq:61}
\begin{split}
&\underbrace{\sigma _{r}^{2} \frac{\partial \ln \left(\rho _{h} \sigma _{r}^{2} \right)}{\partial \ln x} }_{1}\\
&=x\underbrace{\frac{r_{s}^{2} }{t^{2} } \left[\frac{\partial u_{h} }{\partial x} \left(x\frac{\partial \ln r_{s} }{\partial \ln t} -u_{h} \right)+u_{h} \left(1-\frac{\partial \ln r_{s} }{\partial \ln t} \right)\right]}_{2}-\underbrace{v_{c}^{2} }_{3}.
\end{split}
\end{equation}
An equivalent equation in terms of the function $F\left(x\right)$ (using Eq. \eqref{ZEqnNum482501} for density) reads
\begin{equation}
\label{ZEqnNum325109}
\frac{c^{2} }{x\rho _{h} v_{cir}^{2} } \frac{\partial \rho _{h} \sigma _{r}^{2} }{\partial x} =\frac{1}{4\pi ^{2} } \frac{F\left(x\right)^{2} F^{''} \left(x\right)}{xF^{'} \left(x\right)^{3} } -\frac{\rho _{h} \left(x\right)}{\bar{\rho }_{h} \left(a\right)} \frac{3F\left(x\right)}{xF^{'} \left(x\right)} ,
\end{equation}
where $v_{c}^{} $ is the circular velocity at radius \textit{r} (Eq. \eqref{ZEqnNum537303}) and $\bar{\rho }_{h} \left(a\right)$ is the average halo density\textit{.} Here, ${\partial \ln r_{s}/\partial \ln t} =1$ from mass cascade was used (Eq. \eqref{ZEqnNum952904}). Term 1 comes from the pressure gradient due to radial velocity dispersion, term 2 is due to the nonzero radial flow, and term 3 comes from gravity.
For small halos with a stable core, the stable clustering hypothesis is valid and $u_{h} =0$ (term 2 vanishes), where the pressure (term 1) exactly balances the gravity (term 3) everywhere. While for the other limiting situation, i.e. large halos with fast mass accretion, the Hubble flow ($u_{h} \left(x\right)={2x/3} $) at halo center leads to a central core with a finite core density $\rho _{h} \left(0\right)\equiv \rho _{h} \left(x=0\right)$ (Eq. \eqref{ZEqnNum973033}). For core region with $u_{h} \left(x\right)={2x/3} $, Eq. \eqref{ZEqnNum325109} can be transformed to
\begin{equation}
\label{ZEqnNum786564}
\frac{c^{2} }{x\rho _{h} \left(x\right)v_{cir}^{2} } \frac{\partial \left(\rho _{h} \sigma _{r}^{2} \right)}{\partial x} =\frac{1}{\Delta _{c} } -\frac{\rho _{h} \left(x\right)}{\bar{\rho }_{h} \left(a\right)} =-\delta \left(x\right)\frac{\bar{\rho }\left(a\right)}{\bar{\rho }_{h} \left(a\right)} ,
\end{equation}
where $\Delta _{c} =18\pi ^{2} $ is the critical density ratio, $\delta \left(x\right)$ is overdensity, and $\bar{\rho }\left(a\right)$ is background density. The pressure in core region can be approximated by a parabolic function of \textit{x} (from Eq. \eqref{ZEqnNum786564}),
\begin{equation}
\label{ZEqnNum253695}
p_{h} \left(x\right)=\rho _{h} \left(x\right)\sigma _{r}^{2} \left(x\right)=p_{h} \left(x=0\right)-\frac{1}{2} J_{c} x^{2} ,
\end{equation}
where constant $J_{c} $ in the unit of pressure is (from Eq. \eqref{ZEqnNum786564}),
\begin{equation}
\label{ZEqnNum594491}
J_{c} =\left(\frac{\rho _{h} \left(0\right)}{\bar{\rho }_{h} } -\frac{1}{18\pi ^{2} } \right)\frac{\rho _{h} \left(0\right)v_{cir}^{2} }{c^{2} } \approx \frac{\rho _{h}^{2} \left(0\right)v_{cir}^{2} }{\bar{\rho }_{h} c^{2} } .
\end{equation}
A core size $x_{c} $ where Hubble flow is dominant can defined by setting $p_{h} \left(x_{c} \right)=0$ in Eq. \eqref{ZEqnNum253695},
\begin{equation}
\label{ZEqnNum946518}
x_{c} =\sqrt{\frac{2\bar{\rho }_{h} \left(a\right)}{\rho _{h} \left(0\right)} } \frac{c\sigma _{r} \left(0\right)}{v_{cir} } .
\end{equation}
Next, let's work on the velocity dispersion profile. The general expression for the radial dispersion $\sigma _{r}^{2} $ reads (from Eq. \eqref{ZEqnNum325109})
\begin{equation}
\label{ZEqnNum154562}
\begin{split}
\frac{\partial }{\partial x} \left[\frac{F^{'} \left(x\right)\sigma _{r}^{2} }{x^{2} } \right]&=\frac{F^{'} \left(x\right)}{x^{2} } \frac{r_{s}^{2} }{t^{2} } \left[\frac{\partial u_{h} }{\partial x} \left(x\frac{\partial \ln r_{s} }{\partial \ln t} -u_{h} \right)\right.\\
&\left.+u_{h} \left(1-\frac{\partial \ln r_{s} }{\partial \ln t} \right)\right]-\frac{Gm_{h} }{r_{h} } \frac{cF\left(x\right)F^{'} \left(x\right)}{F\left(c\right)x^{4}}.
\end{split}
\end{equation}
If we apply the evolution of halo size $r_{h} \sim t$ or ${\partial \ln r_{h}/\partial \ln t} =1$ from mass cascade, the integration of Eq. \eqref{ZEqnNum154562} leads to an explicit expression for radial dispersion normalized by circular velocity $\sigma _{nr}^{} ={\sigma _{r}/v_{cir} =} \sigma _{r} {t/\left(2\pi r_{h} \right)} $ (with Eqs. \eqref{ZEqnNum837340} and \eqref{ZEqnNum273026}),
\begin{equation}
\label{ZEqnNum807329}
\begin{split}
\sigma _{nr}^{2}&=\underbrace{\frac{x^{2} }{4\pi ^{2} c^{2} F^{'} \left(x\right)} \int _{x}^{\infty }\frac{F\left(x\right)^{2} }{x^{2} } \left(\frac{1}{F^{'} \left(x\right)} \right)^{'} dx}_{1}\\
&\quad\quad\quad\quad\quad\quad+\underbrace{\frac{cx^{2} }{F^{'} \left(x\right)} \int _{x}^{\infty }\frac{F\left(x\right)F^{'} \left(x\right)}{F\left(c\right)x^{4} } dx}_{2}
\end{split}
\end{equation}
with two separate contributions from radial flow (term 1) and from gravitational potential (term 2), respectively. Term 1 is usually neglected for small virialized halos with $u_{r} =0$, but can be important for large halos with fast mass accretion. Here we require the pressure term $\left. \left(\rho _{h} \sigma _{r}^{2} \right)\right|_{x=\infty } =0$ at infinity when integrating Eq. \eqref{ZEqnNum154562}.
For an isothermal profile with $F\left(x\right)=x$, the normalized dispersion has a constant value of $\sigma _{nr}^{2} ={1/2} $. For NFW profile with $F\left(x\right)=\ln \left(1+x\right)-{x/\left(1+x\right)} $, two contributions can be derived explicitly from Eq. \eqref{ZEqnNum807329}. Term 1 reads
\begin{equation}
\label{ZEqnNum410484}
\begin{split}
\sigma_{nr1}^{2}=&-\frac{\left(1+x\right)^{2}}{36\pi ^{2} x^{2} c^{2}} \Big\{3x^{2} -4\pi ^{2} x^{3} +12x^{3} \textrm{Polylog} \left(2,1+x\right)\\
&-\Big[3\left(1+x\right)^{2} \left(2x-1\right)\ln\left(1+x\right)+6x-12x^{2}\\
&-12x^{3} \ln x-12i\pi x^{3}\Big]\ln\left(1+x\right)\Big\},
\end{split}
\end{equation}
with an approximation of\\
$\sigma _{nr1}^{2} \approx \left(\frac{1}{18} -\frac{1}{3\pi^{2} } \right)\frac{x}{c^{2} } $ for $x\to 0$.\\
Term 2 becomes
\begin{equation}
\label{ZEqnNum447542}
\begin{split}
&\sigma _{nr2}^{2} =\frac{c\ln \left(1+x\right)}{2xF\left(c\right)} +\frac{c}{2F\left(c\right)} \Big\{-1-9x-7x^{2}\\
&+\left[-2-8x-4x^{2} +x^{3} \right]\ln \left(1+x\right)\\+
&\Big[\pi ^{2} +6poly\log \left(2,-x\right)-\ln x+3\left(\ln \left(1+x\right)\right)^{2} \Big]x\left(1+x\right)^{2} \Big\},
\end{split}
\end{equation}
with the approximation\\
$\sigma _{nr2}^{2} \approx -\frac{c}{2F\left(c\right)} x\ln \left(x\right)$ for $x\to 0$.\\
Figure \ref{fig:4} plots the variation of total radial velocity dispersion $\sigma _{nr}^{2} \left(x\right)=\sigma _{nr1}^{2} \left(x\right)+\sigma _{nr2}^{2} \left(x\right)$ for an isothermal profile and NFW profile ($c=4$). Two separate contributions are also presented in the same plot, i.e. $\sigma _{nr1}^{2} \left(x\right)$ from the mean radial flow and $\sigma _{nr2}^{2} \left(x\right)$ from the gravitational potential, respectively. The first contribution from mean radial flow tends to enhance the radial velocity dispersion and is only significant at a large \textit{x} for the outer region of halo.
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig4}
\caption{The normalized radial velocity dispersion $\sigma _{nr}^{2} \left(x\right)$ for an isothermal profile (a constant value of 1/2) and NFW profile (varying with \textit{x} for $c=4$) with two contributions, i.e. $\sigma _{r1}^{2} \left(x\right)$ from the mean radial flow (Eq. \eqref{ZEqnNum410484}) and $\sigma _{r2}^{2} \left(x\right)$ from the gravitational potential (Eq. \eqref{ZEqnNum447542}), respectively. The radial flow tends to enhance the radial random motion and is only significant for large \textit{x} in halo outer region.}
\label{fig:4}
\end{figure}
Now we have complete solutions of radial pressure and potential for halos with a NFW profile with effect of radial flow or mass cascade included. Both are normalized by circular velocity $v_{cir} $ and read (using Eq. \eqref{ZEqnNum482501} for density and Eq. \eqref{ZEqnNum207666} for potential)
\begin{equation}
\label{ZEqnNum595258}
p_{nr} \left(x\right)=\frac{\rho _{h} \sigma _{r}^{2} }{\bar{\rho }_{h} v_{cir}^{2} } =\frac{c^{3} F^{'} \left(x\right)}{3F\left(c\right)x^{2} } \sigma _{nr}^{2} \left(x\right),
\end{equation}
\begin{equation}
\label{ZEqnNum686764}
\phi _{nh} \left(x\right)=\frac{\phi _{h} }{v_{cir}^{2} } =-\frac{c\ln \left(1+x\right)}{xF\left(c\right)}. \end{equation}
\begin{figure}
\includegraphics*[width=\columnwidth]{FigS2}
\caption{Normalized density, pressure, gravitational potential, and radial velocity dispersion for a NFW profile with a nonzero radial flow.}
\label{fig:s2}
\end{figure}
\noindent Figure \ref{fig:s2} plots the normalized pressure $p_{nr} \left(x\right)$ (Eq. \eqref{ZEqnNum595258}), gravitational potential $\phi _{nh} \left(x\right)$ (Eq. \eqref{ZEqnNum686764} and radial velocity dispersion $\sigma _{nr}^{2} \left(x\right)$ (Eq. \eqref{ZEqnNum807329}) for a NFW profile with $c=4$. The halo density $\rho _{nh} \left(x\right)$ is normalized by the halo mean density $\bar{\rho }_{h} $. The density $\rho _{nh} \left(x\to 0\right)\sim x^{-1} $ and pressure $p_{nr} \left(x\to 0\right)\sim -\log \left(x\right)$. This leads to an Equation of State $p_{nr} \left(x\to 0\right)\sim a+b\log \left(\rho _{nh} \right)$ for NFW profile. Both pressure and density fields are divergent and irregular at the center of halo for NFW profile.
A convenient formula for the logarithmic slope of radial pressure can be derived from the full Jeans' equation (Eq. \eqref{ZEqnNum325109}) for ${\partial \ln r_{h}/\partial \ln t} =1$, where
\begin{equation}
\label{eq:73}
\frac{\partial \ln p_{nr} }{\partial \ln x} =\frac{x^{2} -xu_{h} }{4\pi ^{2} c^{2} \sigma _{nr}^{2} } \frac{\partial u_{h} }{\partial x} -\frac{v_{nc}^{2} }{\sigma _{nr}^{2} } .
\end{equation}
Similarly, two contributions can be identified (from the mean radial flow $u_{h} $ and gravitational potential, respectively). At scale radius $r_{s} $, $\left. {\partial u_{h}/\partial x} \right|_{x=1} =0$ and the logarithmic slope is exactly ${-v_{nc}^{2}/\sigma _{nr}^{2} } $. The slope equals -2 everywhere for an isothermal profile.
With expressions for all relevant halo quantities explicitly derived, the scaling of these quantities in core region is summarized in Table \ref{tab:2} that is fully determined by the deformation rate parameter $\gamma _{h} $. It should be noted that circular velocity and velocity dispersion follow same scaling ($v_{c}^{2} \left(r\right)\sim \phi _{h}^{*} \left(r\right)\sim \sigma _{r}^{2} \left(r\right)$) if $\gamma _{h} <{1/2} $, regardless of the value of $\gamma _{h}$.
\begin{table}
\centering
\caption{Scaling at center of halo for different deformation rate parameter $\gamma_{h}$}
\begin{tabular}{p{0.6in}m{0.5in}m{0.45in}m{0.45in}m{0.45in}}
\hline
&{$y$} & Isothermal & NFW & Einasto\\
\hline
$\gamma _{h} \ge 0$ & & $\gamma _{h} =0$ & $\gamma _{h} ={1/2} $ & $\gamma _{h} ={2/3} $\\
\hline
$F\left(x\right)\propto x^{y} $ (Eq. \eqref{ZEqnNum434585}) & {$y=\frac{1}{1-\gamma _{h} } $} & $y=1$ & $y=2$ & $y=3$
\\ \hline
$u_{h} \left(x\right)\propto x^{y} $ (Eq. \eqref{ZEqnNum278808}) & {$y=1$} & $y=1$ & $y=1$ & $y=1$ \\ \hline
$v_{c}^{2} \left(x\right)\propto x^{y} $ (Eq. \eqref{ZEqnNum537303}) & {$y=\frac{\gamma _{h} }{1-\gamma _{h} } $} & $y=0$ & $y=1$ & $y=2$ \\ \hline
$\rho _{h} \left(x\right)\propto x^{y} $ (Eq. \eqref{ZEqnNum482501}) & {$y=\frac{3\gamma _{h} -2}{1-\gamma _{h} } $} & $y=-2$ & $y=-1$ & $y=0$ \\ \hline
$\phi _{h}^{*} \left(x\right)\propto x^{y} $ (Eq. \eqref{ZEqnNum269276}) & $y=\frac{\gamma _{h} }{1-\gamma _{h} } $ & diverge \newline at $r=0$ & $y=1$ & $y=2$ \\ \hline
$\sigma _{r}^{2} \left(x\right)\propto x^{y} $ (Eq. \eqref{ZEqnNum807329}) & \makecell{$\frac{1}{2} <\gamma _{h} <\frac{2}{3} $\\ $y=\frac{2-3\gamma _{h} }{1-\gamma _{h} } $} & N/A & N/A & $y=0$ \\ \hline
$\sigma _{r}^{2} \left(x\right)\propto x^{y} $ (Eq. \eqref{ZEqnNum807329}) & {$\gamma _{h} ={1/2} $} & N/A & $-x\ln \left(x\right)$ & N/A\\ \hline
$\sigma _{r}^{2} \left(x\right)\propto x^{y} $ (Eq. \eqref{ZEqnNum807329}) & \makecell{$\gamma _{h} <\frac{1}{2}$\\ $y=\frac{\gamma _{h} }{1-\gamma _{h}} $} & $y=0$ & N/A & N/A \\ \hline
\end{tabular}
\label{tab:2}
\end{table}
\subsection{Effects of mass cascade on halo energies and surface tension }
\label{sec:3.7}
For complete effects of mass cascade on halo properties, total energies of entire halos are studied in this section. It was found that contributions from the radial flow to velocity dispersion could be important and should not be neglected for large halos. In contrast to small halos, large halos with fast mass accretion are dynamic objects with an expanding core and size. Multiplying the continuity Eq. \eqref{ZEqnNum114919} by the mean radial flow $u_{r} $ and the Jeans' equation \eqref{ZEqnNum535400} by density $\rho _{h} $, and adding two equations together, we have the equation
\begin{equation}
\label{ZEqnNum754617}
\frac{\partial \left(\rho _{h} u_{r} \right)}{\partial t} +\frac{1}{r^{2} } \frac{\partial \left(\rho _{h} r^{2} u_{r}^{2} \right)}{\partial r} +\frac{\partial \left(\rho _{h} \sigma _{r}^{2} \right)}{\partial r} +\rho _{h} \frac{Gm_{r} \left(r,a\right)}{r^{2} } =0.
\end{equation}
Multiplying Eq. \eqref{ZEqnNum754617} by $4\pi r^{3} $ and integrating with respect to $r$ from 0 to $r_{h} $ leads to an exact energy equation for non-rotating halos ,
\begin{equation}
\label{ZEqnNum532404}
I_{h} +S_{u} +S_{\sigma } -2K_{u} -6K_{\sigma } -\Phi _{h} =0,
\end{equation}
with all terms here normalized by $m_{h} v_{cir}^{2} $. This is a generalized version of virial theorem, as the standard virial theorem does not include the contributions from a nonzero radial flow through surface energy $S_{u} $ and kinetic energy $K_{u}$. Since $r_{h} =r_{h} \left(t\right)$, the integration of the first term in Eq. \eqref{ZEqnNum754617} can be separated into two contributions using the Leibniz's rule,
\begin{equation}
\label{eq:76}
\int _{0}^{r_{h} }4\pi r^{3} \frac{\partial \left(\rho _{h} u_{r} \right)}{\partial t} dr =\frac{\partial G_{h} }{\partial t} -4\pi r_{h}^{3} \rho _{h} \left(r_{h} \right)u_{r} \left(r_{h} \right)\frac{\partial r_{h} }{\partial t} ,
\end{equation}
where a halo virial quantity (radial momentum)
\begin{equation}
\label{eq:77}
G_{h} =\int _{0}^{r_{h} }4\pi r^{3} \rho _{h} u_{r} dr =\frac{m_{h} \left(t\right)v_{cir}^{2} t}{4\pi ^{2} c^{2} F\left(c\right)} \left[c^{2} F\left(c\right)-3\int _{0}^{c}xF\left(x\right)dx \right]
\end{equation}
is defined as the first order moment of radial flow. The virial quantity for peculiar velocity that excludes the Hubble flow reads
\begin{equation}
\label{eq:78}
\begin{split}
G_{hp}^{} &=\int _{0}^{r_{h} }4\pi r^{3} \rho _{h} \left(u_{r} -Hr\right)dr\\
&=\frac{m_{h} \left(t\right)v_{cir}^{2} t}{4\pi ^{2} c^{2} F\left(c\right)} \left[\frac{1}{3} c^{2} F\left(c\right)-\frac{5}{3} \int _{0}^{c}xF\left(x\right)dx \right].
\end{split}
\end{equation}
For comparison, $L_{hr} \left(a\right)$ is the (zeroth order) linear momentum of radial flow. The (normalized) time derivative of the virial quantity ($I_{h} $) is obtained as,
\begin{equation}
\label{ZEqnNum811693}
I_{h} =\frac{1}{m_{h} v_{cir}^{2} } \frac{\partial G_{h} }{\partial t} =\left[\frac{1}{2\pi ^{2} } -\frac{3}{2\pi ^{2} c^{2} F\left(c\right)} \int _{0}^{c}xF\left(x\right)dx \right].
\end{equation}
The surface energy terms include the contribution $S_{u} $ from the surface pressure due to radial flow at halo surface,
\begin{equation}
\label{eq:80}
\begin{split}
S_{u}&=\frac{4\pi r_{h}^{3} \rho _{h} \left(r_{h} \right)}{m_{h} v_{cir}^{2} } \left[u_{r}^{2} \left(r_{h} \right)-u_{r} \left(r_{h} \right)\frac{\partial r_{h} }{\partial t} \right]\\
&=\frac{1}{4\pi ^{2} } \left[\frac{F\left(c\right)}{cF^{'} \left(c\right)} -1\right]=\frac{{1/\alpha _{h} -1} }{4\pi ^{2} }
\end{split}
\end{equation}
and the contribution $S_{\sigma }$ from the surface pressure due to velocity dispersion at halo surface. Since the radial velocity dispersion has two contributions (Eq. \eqref{ZEqnNum807329}), we have two corresponding contributions to the pressure term $S_{\sigma }$, i.e. from the radial flow ($S_{\sigma 1}$) and from the gravitational potential ($S_{\sigma 2}$), respectively,
\begin{equation}
\label{eq:81}
S_{\sigma } =\frac{4\pi r_{h}^{3} \rho _{h} \left(r_{h} \right)\sigma _{r}^{2} \left(r_{h} \right)}{\left(m_{h} v_{cir}^{2} \right)} =S_{\sigma 1} +S_{\sigma 2} ,
\end{equation}
\begin{equation}
\label{eq:82}
S_{\sigma 1} =\frac{c}{4\pi ^{2} F\left(c\right)} \int _{c}^{\infty }\frac{F^{2} \left(x\right)}{x^{2} } \left(\frac{1}{F^{'} \left(x\right)} \right) ^{'} dx
\end{equation}
or
\begin{equation}
\label{eq:83}
S_{\sigma 1} =\frac{c}{4\pi ^{2} F\left(c\right)} \left\{\left(\left. \frac{F^{2} \left(x\right)}{x^{2} F^{'} \left(x\right)} \right|_{c}^{\infty } \right)-\int _{c}^{\infty }\left[\frac{2F\left(x\right)}{x^{2} } -\frac{2F^{2} \left(x\right)}{F^{'} \left(x\right)x^{3} } \right] \right\}dx,
\end{equation}
\begin{equation}
\label{eq:84}
S_{\sigma 2} =\frac{c^{4} }{F^{2} \left(c\right)} \int _{c}^{\infty }\frac{F\left(x\right)F^{'} \left(x\right)}{x^{4} } dx.
\end{equation}
The total kinetic energy of a halo includes the contribution directly from radial flow,
\begin{equation}
\label{eq:85}
\begin{split}
K_{u}&=\lambda _{Kr} =\frac{\int _{0}^{r_{h} }4\pi r^{2} \rho _{h} u_{r}^{2} dr }{2m_{h} v_{cir}^{2} }\\
&=\frac{1}{8\pi ^{2} } -\frac{\int _{0}^{c}xF\left(x\right)dx }{2\pi ^{2} c^{2} F\left(c\right)} +\frac{1}{8\pi ^{2} c^{2} F\left(c\right)} \int _{0}^{c}\frac{F^{2} \left(x\right)}{F^{'} \left(x\right)} dx.
\end{split}
\end{equation}
Similarly, the second contribution of kinetic energy is from velocity dispersion (random motion) that again includes contributions from radial flow ($K_{\sigma 1} $) and from gravitational potential ($K_{\sigma 2} $), respectively, according to Eq. \eqref{ZEqnNum807329},
\begin{equation}
\label{eq:86}
K_{\sigma } =\frac{1}{2m_{h} v_{cir}^{2} } \int _{0}^{r_{h} }4\pi r^{2} \rho _{h} \sigma _{r}^{2} dr= K_{\sigma 1} +K_{\sigma 2} ,
\end{equation}
\begin{equation}
\label{eq:87}
K_{\sigma 1} =\frac{1}{8\pi ^{2} c^{2} F\left(c\right)} \int _{0}^{c}x^{2} \left[\int _{x}^{\infty }\frac{F\left(y\right)^{2} }{y^{2} } \left(\frac{1}{F^{'} \left(y\right)} \right) ^{'} dy\right] dx
\end{equation}
or if ${\mathop{\lim }\limits_{x\to \infty }} {F\left(x\right)^{2}/x^{2} F^{'} \left(x\right)} =0$,
\begin{equation}
\label{eq:88}
\begin{split}
K_{\sigma 1} =-\frac{1}{24\pi ^{2} c^{2} F\left(c\right)} &\left\{\int _{0}^{c}\left(\frac{F\left(x\right)}{F^{'} \left(x\right)} +2x\right)F\left(x\right)dx\right.\\
&\left.+c^{3} \int _{c}^{\infty }\left[\frac{2F\left(x\right)}{x^{2} } -\frac{2F^{2} \left(x\right)}{F^{'} \left(x\right)x^{3} }\right] dx\right\}.
\end{split}
\end{equation}
The kinetic energy of velocity dispersion due to gravitational interaction is
\begin{equation}
\label{eq:89}
K_{\sigma 2} =\frac{c^{4} }{6F{}^{2} \left(c\right)} \int _{c}^{\infty }\frac{F\left(x\right)F^{'} \left(x\right)}{x^{4} } dx +\frac{c}{6F{}^{2} \left(c\right)} \int _{0}^{c}\frac{F\left(x\right)F^{'} \left(x\right)}{x} dx.
\end{equation}
The total gravitational potential of a halo is given by,
\begin{equation}
\label{ZEqnNum765989}
\Phi _{h} =-\frac{1}{m_{h} v_{cir}^{2} } \int _{0}^{r_{h} }4\pi r^{2} \rho _{h} \frac{Gm_{r} }{r} dr=-\frac{c}{F{}^{2} \left(c\right)} \int _{0}^{c}\frac{F\left(x\right)F^{'} \left(x\right)}{x} dx.
\end{equation}
In principle, we can derive explicit expressions for all these terms for a halo with a known function of $F\left(x\right)$. For example, some of these terms for a NFW profile are presented here,
\begin{equation}
\label{eq:91}
I_{h} =\frac{\left[5c^{2} +6c-2\left(1+c\right)\left(c+3\right)\log \left(1+c\right)\right]\left(c-3\right)}{8\pi ^{2} c^{2} \left(1+c\right)\left[\log \left(1+c\right)-{c/ \left(1+c\right)} \right]} ,
\end{equation}
\begin{equation}
\label{eq:92}
S_{u} =\frac{\left[\log \left(1+c\right)-{c/ \left(1+c\right)} \right]\left(1+c\right)^{2} -c^{2} }{4\pi ^{2} c^{2} } ,
\end{equation}
\begin{equation}
\label{eq:93}
\Phi _{h} =-\frac{c\left[c\left(2+c\right)-2\left(1+c\right)\ln \left(1+c\right)\right]}{2\left[\ln \left(1+c\right)-{c/ \left(1+c\right)} \right]^{2} \left(1+c\right)^{2} } .
\end{equation}
Figure \ref{fig:5} plots the variation of these energy terms with concentration \textit{c} for halos with a NFW profile. The dynamic term $I_{h} \left(c\right)$ is positive for small \textit{c} and negative for large \textit{c } with a critical concentration around $c=3$ where $I_{h} =0$. The surface energy $S_{u}$ due to radial flow is negative for small \textit{c} and changing to be positive for large \textit{c}. The total kinetic energy $K_{u} +K_{\sigma 1}$ due to radial flow is small compared to the kinetic energy due to random motion or velocity dispersion $K_{\sigma 2}$. However, the total surface energy $S_{u} +S_{\sigma 1}$ (due to radial flow) is comparable to $S_{\sigma 2}$ (due to velocity dispersion) and should not be neglected. This can be explained by the fact that the mean radial flow is only significant in the outer region of halos.
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig5}
\caption{The log-log variation of various halo energies with halo concentration parameter \textit{c} for NFW profile. The dynamic term $I_{h} \left(c\right)$ is positive for small \textit{c} and negative for large \textit{c}, while the surface energy $S_{u} \left(c\right)$ due to radial flow is negative for small \textit{c} and changing to be positive for large \textit{c}. The kinetic energy $K_{u} +K_{\sigma 1} $ due to the radial flow is small compared to the kinetic energy purely due to the velocity dispersion $K_{\sigma 2} $. However, the surface energy $S_{u} +S_{\sigma 1} $ is comparable to $S_{\sigma 2} $. This can be explained by the fact that radial flow is only significant in the outer region of halos.}
\label{fig:5}
\end{figure}
Among these terms, $I_{h} $ due to radial flow is relatively small. The other three terms from radial flow should not be neglected and contribute to the total energy balance with $S_{\sigma 1} >K_{\sigma 1} >S_{u} $. By neglecting the dynamic term $I_{h} $ and using Eq. \eqref{ZEqnNum532404}, we write the surface energy of large halos with fast mass accretion,
\begin{equation}
\label{ZEqnNum267268}
\begin{split}
S_{eh}&=\left(S_{u} +S_{\sigma 1} +S_{\sigma 2} \right)m_{h} v_{cir}^{2}\\
&=\left[2\left(K_{u} +3K_{\sigma 2} +3K_{\sigma 1} \right)+\Phi _{h} \right]m_{h} v_{cir}^{2} ,
\end{split}
\end{equation}
which is the extra energy required to create the expanding halo surface. The halo surface energy is also the difference between the total energy of halo with and without halo surface. The equivalent halo surface tension can be introduced as the surface energy per unit area,
\begin{equation}
\label{ZEqnNum212204}
S_{th} =\frac{S_{eh} }{2A_{h} } =\frac{2\left(K_{u} +3K_{\sigma 2} +3K_{\sigma 1} \right)+\Phi _{h} }{8\pi r_{h}^{2} } m_{h} v_{cir}^{2} ,
\end{equation}
where $A_{h} =4\pi r_{h}^{2} $ is the surface area of a halo. For large halos with the limiting concentration $c=3.48$, the normalized surface tension (from Eq. \eqref{ZEqnNum212204}) is estimated to be around
\begin{equation}
\begin{split}
&S_{nth} =\frac{2A_{h} S_{th} }{m_{h} v_{cir}^{2} } =\frac{S_{eh} }{m_{h} v_{cir}^{2} } \approx 0.3 \quad \textrm{for NFW profile}\\
&\textrm{and}\\
&S_{nth} =\frac{2A_{h} S_{th} }{m_{h} v_{cir}^{2} } =\frac{S_{eh} }{m_{h} v_{cir}^{2} } =0.5 \quad \textrm{for isothermal profile.} \label{ZEqnNum336244}
\end{split}
\end{equation}
\noindent The normalized surface tension $S_{nth}$ is a constant regardless of halo mass and time, which contributes to the effective potential exponent of entire N-body system $n_e=-1.3\neq-1$ \citep[see][Fig. 1b]{Xu:2022-The-evolution-of-energy--momen} (also Eq. \eqref{ZEqnNum797157}).
An equation analog to the Young--Laplace equation can be written to relate the pressure difference across halo surface to halo radius, or equivalently halo surface curvature,
\begin{equation}
\label{eq:97}
\begin{split}
\Delta P_{h}&=\frac{2S_{th} }{r_{h} } =\frac{S_{eh} }{A_{h} r_{h} }\\ &=\frac{1}{3} \left[S_{u} +S_{\sigma 1} +S_{\sigma 2} \right]\bar{\rho }_{h} v_{cir}^{2}=\frac{1}{3} S_{nth} \bar{\rho }_{h} v_{cir}^{2} .
\end{split}
\end{equation}
The pressure difference across surface is $\Delta P_{h} \approx 0.1\bar{\rho }_{h} v_{cir}^{2} $, which is also approximately the pressure right at halo surface (pressure is zero on the outside of halo). For a non-spherical halo, pressure may be different at different location depending on the local curvature.
We can introduce surface density for a given halo as $\rho _{sur} ={N_{s} m_{p}/A_{h} } $, where $N_{s} $ is the number of particles on surface. Halo surface tension (an inherent property of halo surface) may be fully described by the surface density $\rho _{sur} $, gravitational constant \textit{G}, and halo size $r_{h} $,
\begin{equation}
\label{eq:98}
S_{th} =\alpha _{st} \left(G\right)^{\alpha _{1} } \left(\rho _{sur}^{} \right)^{\alpha _{2} } \left(r_{h} \right)^{\alpha _{3} } ,
\end{equation}
where $\alpha _{st} $ is a numerical constant. A simple dimensional analysis leads to expression
\begin{equation}
\label{ZEqnNum184683}
S_{th} =\alpha _{st} G\rho _{sur}^{2} r_{h} ,
\end{equation}
where halo surface density reads (inserting Eq. \eqref{ZEqnNum336244} into \eqref{ZEqnNum184683})
\begin{equation}
\label{ZEqnNum292775}
\rho _{sur} =\sqrt{\frac{S_{nth} }{8\pi \alpha _{st} } } \frac{m_{h} }{r_{h}^{2} } .
\end{equation}
For halos with $\lambda ={2/3} $ and $\tau _{0} =1$, we have $m_{h} \propto r_{h} $, halo surface tension $S_{th} \propto r_{h}^{-1} $, halo surface density $\rho _{sur} \propto r_{h}^{-1}$, and thickness of halo surface layer $r_{p} ={\rho _{sur}/\rho _{h} \left(r_{h} \right)} \propto r_{h}^{} $ from Eq. \eqref{ZEqnNum227253}. A complete list of dependence of these parameters on the mass cascade parameters $\lambda $ and $\tau _{0} $ is presented in Table \ref{tab:3}.
\begin{table}
\centering
\caption{The dependence of power-law exponent \textit{m} ($\mathrm{\sim}$ $a^{m} $) on mass cascade}
\begin{tabular}{p{0.4in}p{0.4in}p{0.5in}p{0.4in}p{0.35in}p{0.35in}} \hline
$m_{h} $ & $r_{h} $ & $\rho _{sur} $ & $S_{th} $\textbf{} & $v_{cir}^{2} $ & $r_{p} $ \\ \hline
$\frac{3-2\tau _{0} }{2\left(1-\lambda \right)} $ & $\frac{9-2\tau _{0} -6\lambda }{6\left(1-\lambda \right)} $ & $\frac{12\lambda -9-2\tau _{0} }{6\left(1-\lambda \right)} $ & $\frac{6\lambda -3-2\tau _{0} }{2\left(1-\lambda \right)} $ & $\frac{3\lambda -2\tau _{0} }{3\left(1-\lambda \right)} $ & $\frac{9-2\tau _{0} -6\lambda }{6\left(1-\lambda \right)} $ \\ \hline
Eq. \eqref{ZEqnNum952904} & Eq. \eqref{ZEqnNum952904} & Eq. \eqref{ZEqnNum292775} & Eq. \eqref{ZEqnNum212204} & Eq. \eqref{ZEqnNum837340} & Eq. \eqref{ZEqnNum227253} \\ \hline
\end{tabular}
\label{tab:3}
\end{table}
From Table \ref{tab:3}, the thickness of surface layer $r_{p} $ is proportional to halo size $r_{h}$, i.e. $r_{p} \propto r_{h}$, regardless of the exact values of $\lambda$ and $\tau _{0}$ that may depend on the exact cosmology. This hints a geometric Brownian process (incremental change $r_{p}$ proportional to the current value $r_{h}$) for halo size in Section \ref{sec:4.1}.
Finally, an effective exponent for gravitational interaction $n_{e} $ can be introduced based on the virial theorem for halos with fast mass accretion and expanding size,
\begin{equation}
\label{ZEqnNum797157}
n_{e} =\frac{2\left(K_{u} +3K_{\sigma 2} +3K_{\sigma 1} \right)}{\Phi _{h} } =\frac{S_{nth} }{\Phi _{h} } -1.
\end{equation}
With $c=3.48$ and $\Phi _{h} \approx -1$, the effective exponent $n_{e} \approx -1.3$. The deviation of $n_{e} \approx -1.3$ from -1 (the actual potential exponent is -1 for $V\left(r\right)\approx r^{-1} $) reflects the effects of surface energy/tension from inverse mass cascade. This can be directly confirmed by N-body simulations \citep[see][Fig. 1b]{Xu:2022-The-evolution-of-energy--momen}.
\section{Stochastic models for halo size and density profile}
\label{sec:4}
\subsection{Stochastic model for halo size evolution}
\label{sec:4.1}
The random walk of halos in mass space was applied to derive the double-$\lambda$ halo mass function \citep{Xu:2021-Inverse-mass-cascade-mass-function}. Similarly, stochastic models can be developed for halo size and particle distributions that describe the halo internal structure. The halo structure (distribution of particles) is highly dependent on the evolution of halo size, and therefore on the mass cascade.
In mass cascade, the halo waiting time $\tau _{gr} $ is a random variable and follows an exponential distribution with mean $\tau _{g} =\left\langle \tau _{gr} \right\rangle \propto m_{h}^{-\lambda }$ \citep[see][Eq. (45)]{Xu:2021-Inverse-mass-cascade-mass-function}. The evolution of halo mass $m_{h}^{} $ can be modeled by a stochastic process with perturbations due to the randomness in halo waiting time $\tau _{gr}$,
\begin{equation}
\label{ZEqnNum315498}
\frac{\partial m_{h}^{} }{\partial t} =\frac{m_{p} }{\tau _{gr} } =\frac{m_{h}}{\tau _{g} n_{p}} \cdot \frac{\tau _{g}}{\tau _{gr}}.
\end{equation}
Equation \eqref{ZEqnNum315498} becomes deterministic equation by replacing the random waiting time $\tau _{gr}$ with the mean waiting time $\tau _{g}$ (see Eq. \eqref{ZEqnNum808457}). By introducing a random variable $\xi_{gr}$ and using Eq. \eqref{ZEqnNum821124},
\begin{equation}
\xi _{gr} \left(t\right)=\frac{\tau _{g}}{\tau _{gr}} -1\quad \textrm{and} \quad \tau _{g} n_{p} =\frac{3\left(1-\lambda \right)}{3-2\tau _{0} } t,
\label{eq:103}
\end{equation}
\noindent we will have a stochastic differential equation for $m_{h}^{} $
\begin{equation}
\label{ZEqnNum513835}
\frac{\partial \ln m_{h}^{} }{\partial \ln t} =\frac{3-2\tau _{0} }{3\left(1-\lambda \right)} \left(1+\xi _{gr} \left(t\right)\right),
\end{equation}
where $\xi _{gr} $ is approximately a Gaussian random variable with a zero mean. Equation \eqref{ZEqnNum513835} reduces to Eq. \eqref{ZEqnNum952904} for $\xi _{gr} \to 0$. As shown in Eqs. \eqref{ZEqnNum808457} and \eqref{ZEqnNum410355}, the original evolution of halo mass and size can be generalized to stochastic models
\begin{equation}
\label{eq:105}
\frac{\partial \ln m_{h}^{} }{\partial \ln t} =\frac{m_{p} t}{m_{h} \tau _{gr} }
\end{equation}
and
\begin{equation}
\label{ZEqnNum173628}
\frac{\partial \ln r_{h} }{\partial \ln t} =\frac{m_{p} }{4\pi r_{h}^{3} } \frac{\alpha _{h} t}{\tau _{gr} \rho _{h} \left(r_{h} \right)} =\frac{\alpha _{h} \bar{\rho }_{h} }{3\rho _{h} \left(r_{h} \right)} \frac{m_{p} t}{m_{h}^{} \tau _{gr} } =\frac{\partial \ln m_{h}^{} }{\partial \ln t} \frac{\alpha _{h} \bar{\rho }_{h} }{3\rho _{h} \left(r_{h} \right)} ,
\end{equation}
where the average waiting time $\tau _{g} $ is simply replaced by a random waiting time $\tau _{gr} $. Finally, we have the stochastic equation for halo size $r_h$ from Eqs. \eqref{ZEqnNum513835} and \eqref{ZEqnNum173628},
\begin{equation}
\label{ZEqnNum278210}
\frac{\partial \ln r_{h}^{} }{\partial \ln t} =\frac{\partial \ln m_{h} }{\partial \ln t} \frac{\alpha _{h} \bar{\rho }_{h} }{3\rho _{h} \left(r_{h} \right)} =\frac{3-2\tau _{0} }{3\left(1-\lambda \right)} \frac{\alpha _{h} \bar{\rho }_{h} }{3\rho _{h} \left(r_{h} \right)} \left(1+\xi _{gr} \left(t\right)\right).
\end{equation}
Halos evolving with a vanishing noise term in Eq. \eqref{ZEqnNum278210} always satisfy Eq. \eqref{ZEqnNum872917} with a mean halo density $\bar{\rho }_{h} =\Delta _{c} \bar{\rho }_{0} a^{-3} $. However, the existence of noise term $\xi _{gr} $ in Eq. \eqref{ZEqnNum278210} may drive halos away from Eq. \eqref{ZEqnNum872917}. At any instant, from Eqs. \eqref{ZEqnNum173628}, we should have
\begin{equation}
\label{ZEqnNum641162}
\frac{\partial \ln m_{h} }{\partial \ln r_{h} } =\frac{3\rho _{h} \left(r_{h} \right)}{\alpha _{h} \bar{\rho }_{h} } =\frac{3\rho _{sur} }{\alpha _{h} r_{p} \bar{\rho }_{h} } =\frac{3\left(3-2\tau _{0} \right)}{9-2\tau _{0} -6\lambda }
\end{equation}
to be always valid ($\tau_0$ and $\lambda$ are constants in mass cascade), where $\bar{\rho }_{h} \left(a\right)$ is the average halo density and $\rho _{sur}=r_{p}\rho _{h}(r_h)$ is the surface density. Therefore, these stochastic models (Eqs. \eqref{ZEqnNum513835} and \eqref{ZEqnNum278210}) describe randomly evolving mass and size of halos with fast mass accretion (i.e. constant concentration $c$ and/or shape parameter $\alpha$) and satisfying condition \eqref{ZEqnNum641162} at any instant \textit{t}, i.e. $m_{h} \propto r_{h} $ for $\tau _{0} =1$ and $\lambda ={2/3} $. Finally, the halos size $r_{h} \left(t\right)$ should evolve as (from Eqs. \eqref{ZEqnNum278210} and \eqref{ZEqnNum641162}),
\begin{equation}
\frac{dr_{h} }{dt} =b_{rh} \frac{r_{h} \left(t\right)}{t} \left(1+\xi _{gr} \left(t\right)\right) \quad \textrm{with} \quad b_{rh} =\frac{9-2\tau _{0} -6\lambda }{9\left(1-\lambda \right)}
\label{ZEqnNum824858}
\end{equation}
\noindent from Eqs. \eqref{ZEqnNum278210} and \eqref{ZEqnNum641162}, which is a geometric Brownian motion with a multiplicative noise. The parameter $b_{rh} $ is from mass cascade and $b_{rh} =1$ for $\tau _{0} =1$ and $\lambda ={2/3} $.
The evolution of halo size can be also understood as a result of fluctuating halo surface with a random velocity proportional to the velocity dispersion at the halo surface, i.e. the radial velocity dispersion $\sigma _{r} \left(r=r_{h} \right)$ discussed in Section \ref{sec:3}. The stochastic differential equation with an initial halo size of $r_{h} \left(t=t_{i} \right)=r_{h0} $ reads,
\begin{equation}
\label{ZEqnNum979638}
\frac{dr_{h} \left(t\right)}{dt} =b_{rh} \frac{r_{h} \left(t\right)}{t} +\beta _{rh} \sigma _{r} \left(r_{h} \right)\bar{\xi }_{rh} \left(t\right),
\end{equation}
which is consistent with Eq. \eqref{ZEqnNum824858}. The covariance of the noise term satisfies $\left\langle \overline{\xi }_{rh} \left(t\right)\overline{\xi }_{rh} \left(t^{'} \right)\right\rangle ={\delta \left(t-t^{'} \right)/H} $.
It is shown that the velocity dispersion at halo surface $\sigma _{r} \left(r_{h} \right)=\alpha _{rh} \left(c\right)v_{cir} =3\pi \alpha _{rh} Hr_{h} $ from Eqs. \eqref{ZEqnNum837340}, \eqref{ZEqnNum410484} and \eqref{ZEqnNum447542}, where $\alpha _{rh} \left(c\right)$ is a constant. For a limiting value of $c\approx 3.48$, $\alpha _{rh} {=1/\sqrt{2} } $ for an isothermal profile. After transforming the physical time \textit{t} to scale factor \textit{a}, Eq. \eqref{ZEqnNum979638} reads
\begin{equation}
\label{ZEqnNum251003}
\frac{dr_{h} \left(t\right)}{dt} =\frac{3}{2} b_{rh} Hr_{h} \left(t\right)+Hr_{h} \left(t\right)\xi _{rh} \left(t\right),
\end{equation}
and
\begin{equation}
\label{ZEqnNum400799}
\frac{dr_{h} \left(t\right)}{d\ln a} =\frac{3}{2} b_{rh} r_{h} \left(t\right)+r_{h} \left(t\right)\xi _{rh} \left(t\right),
\end{equation}
where the multiplicative noise $r_{h} \left(t\right)\xi _{rh} \left(t\right)$ (proportional to $r_{h} $) describes the random evolution of halo size. The covariance of new noise $\xi _{rh} \left(t\right)$ satisfies
\begin{equation}
\label{eq:113}
\left\langle \xi _{rh} \left(t\right)\xi _{rh} \left(t^{'} \right)\right\rangle =2D_{rh} {\delta \left(t-t^{'} \right)/H} ,
\end{equation}
or equivalently the noise term $\xi _{gr} \left(t\right)$ in Eq. \eqref{ZEqnNum824858} satisfies
\begin{equation}
\label{eq:114}
\left\langle \xi _{gr} \left(t\right)\xi _{gr} \left(t^{'} \right)\right\rangle =4D_{rh} {t\delta \left(t-t^{'} \right)/\left(3b_{rh}^{2} \right)} ,
\end{equation}
where $D_{rh} ={\left(3\pi \alpha _{rh} \beta _{rh} \right)^{2}/2} $ is a dimensionless diffusion coefficient. The halo size $r_{h} \left(t\right)$ described by the geometric Brownian motion (Eq. \eqref{ZEqnNum400799}) has a lognormal probability distribution of ($r_{h0}$ is the initial halo size at starting time $t_i$)
\begin{equation}
\label{ZEqnNum640133}
\begin{split}
&P_{rh} \left(r_{h} ,t\right)=\frac{1}{r_{h} \sqrt{{8\pi D_{rh} \ln \left({t/t_{i} } \right)/3} } }\\
&\quad\quad \cdot \exp \left\{-\frac{\left(\ln \left({r_{h}/r_{h0} } \right)-\left(b_{rh} -{2D_{rh}/3} \right)\ln \left({t/t_{i} } \right)\right)^{2} }{{8D_{rh} \ln \left({t/t_{i} } \right)/3} } \right\},
\end{split}
\end{equation}
with the \textit{m}th order moment of
\begin{equation}
\label{eq:116}
\left\langle r_{h}^{m} \right\rangle =r_{h0}^{m} \left({t/t_{i} } \right)^{mb_{rh} +\frac{2}{3} m\left(m-1\right)D_{rh} } .
\end{equation}
The mean halo size grows linearly with time as $\left\langle r_{h} \left(t\right)\right\rangle =r_{h0} \left({t/t_{i} } \right)^{b_{rh} } \sim t^{b_{rh} } $, as expected. The mode of the halo size grows as $r_{h0} \left({t/t_{i} } \right)^{b_{rh} -2D_{rh} } $ and the median halo size grows as $r_{h0} \left({t/t_{i} } \right)^{b_{rh} -{2D_{rh} /3} } $. Finally, the root mean square of halo size scales as $\left\langle r_{h}^{2} \right\rangle ^{{1/2} } =r_{h0} \left({t/t_{i} } \right)^{b_{rh} +{2D_{rh} /3} } $. Similarly, the halo mass ($m_{h} \propto r_{h} $) will also follow a lognormal distribution.
\subsection{Stochastic model for particle distribution}
\label{sec:4.2}
To find the halo density profile, we need to derive the particle distribution function. The particle motion in halos is complicated as it is coupled to the varying halo size in previous section. This shares similarity with the derivation of diffusion equation for standard Brownian motion (see Appendix \ref{appendix:b} for a brief review).
Let's consider the motion of a collisionless particle in a halo with varying size according to Eq. \eqref{ZEqnNum251003}. The goal is to derive the particle distribution function. The random position of that particle is
\begin{equation}
\label{ZEqnNum676188}
r_{t} \left(t\right)={x_{t} \left(t\right)r_{s} \left(t\right)=x_{t} \left(t\right)r_{h} \left(t\right)/c},
\end{equation}
where $r_{h} \left(t\right)$ is a stochastic time-varying halo size, $x_{t} \left(t\right)$ is the reduced position of that particle to the center of halo. For halos with a fixed size, $x_{t} \left(t\right)$ is expected to be a smooth function of time \textit{t} due to radial flow. The time variation of particle position $r_{t} \left(t\right)$ comes from both the time variation of $x_{t} \left(t\right)$ and the variation of halo size $r_{h} \left(t\right)$, i.e. $cdr_{t} =x_{t} \left(t\right)dr_{h} +r_{h} \left(t\right)dx_{t} $. The infinitesimal change $dr_{h} $ is presented in stochastic Eq. \eqref{ZEqnNum251003}. The infinitesimal change $dx_{t} $ for a fixed $r_{h} \left(t\right)$ is determined by the mean radial flow in Eq. \eqref{ZEqnNum278808},
\begin{equation}
u_{h} \left(x_{t} \right)=b_{rh} x_{t} \underbrace{-b_{rh} \frac{F\left(x_{t} \right)}{F^{'} \left(x_{t} \right)} }_{1}\quad \textrm{with} \quad b_{rh} =\left\langle \frac{\partial \ln r_{h} }{\partial \ln t} \right\rangle.
\label{eq:118}
\end{equation}
\noindent Here, term 1 is the particle motion relative to the halo size change. Therefore, the infinitesimal change $dx_{t} $ for a fixed halo size $r_{h} \left(t\right)$ can be written as (see Appendix \ref{appendix:b} for stadnard Brownian motion)
\begin{equation}
\label{ZEqnNum803708}
r_{s} \left(t\right)\frac{dx_{tF} }{dt} =\frac{r_{s} \left(t\right)}{t} \left[u_{h} \left(x_{t} \right)-b_{rh} x_{t} +u_{h}^{*} \left(x_{t} \right)\right],
\end{equation}
and
\begin{equation}
\label{ZEqnNum517814}
r_{s} \left(t\right)\frac{dx_{tB} }{dt} =\frac{r_{s} \left(t\right)}{t} \left[u_{h} \left(x_{t} \right)-b_{rh} x_{t} -u_{h}^{*} \left(x_{t} \right)\right],
\end{equation}
for forward and backward change of $x_{t} \left(t\right)$ in time, respectively. Here $u_{h} \left(x_{t} \right)-b_{rh} x_{t} $ is the radial flow relative to the halo size change. Velocity $u_{h}^{*} \left(x_{t} \right)$ turns out to be the osmotic flow velocity acquired by particles in equilibrium to the external force. In Einstein's theory of Brownian motion, it has an origin from the osmotic pressure. Applying chain rule to Eq. \eqref{ZEqnNum676188} to obtain
\[\frac{dr_{t} }{dt} =\frac{x_{t} \left(t\right)}{c} \frac{dr_{h} }{dt} +r_{s} \left(t\right)\frac{dx_{t} }{dt} \]
and inserting Eq. \eqref{ZEqnNum251003} with $r_{s} \left(t\right)={r_{h} \left(t\right)/c} $ lead to the equation for particle position
\begin{equation}
\label{ZEqnNum620785}
\frac{dr_{t} }{dt} =\frac{r_{s} \left(t\right)}{t} \left[t\frac{dx_{t} }{dt} +b_{rh} x_{t} \right]+x_{t} \left(t\right)r_{s} \left(t\right)H\xi _{rh} \left(t\right).
\end{equation}
By inserting Eqs. \eqref{ZEqnNum803708} and \eqref{ZEqnNum517814} into Eq. \eqref{ZEqnNum620785}, the stochastic equations for $r_{t} \left(t\right)$ for forward and backward processes reads,
\begin{equation}
\label{ZEqnNum544038}
\frac{dr_{t} }{dt} =\underbrace{\frac{r_{s} \left(t\right)}{t} \left[u_{h} \left(x_{t} \right)+u_{h}^{*} \left(x_{t} \right)\right]}_{1}+\underbrace{\sigma \left(x_{t} \right)r_{s} \left(t\right)H\xi _{rh} \left(t\right)}_{2},
\end{equation}
\begin{equation}
\label{ZEqnNum855118}
\frac{dr_{t} }{dt} =\underbrace{\frac{r_{s} \left(t\right)}{t} \left[u_{h} \left(x_{t} \right)-u_{h}^{*} \left(x_{t} \right)\right]}_{3}+\sigma \left(x_{t} \right)r_{s} \left(t\right)H\xi _{rh}^{*} \left(t\right).
\end{equation}
The stochastic process $r_{t} \left(t\right)$ is not differentiable with respect to time \textit{t}, where the forward/backward velocities (the left/right side time derivatives of $r_{t} \left(t\right)$) can be different. The mean radial flow $u_{h} \left(x_{t} \right)$ (or the current velocity ) is the average of forward (term 1) and backward (term 3) velocities, while the osmotic flow $u_{h}^{*} \left(x_{t} \right)$ (or the fluctuation velocity) is the difference between forward and backward velocities that changes its sign in terms 1 and 3 (see Appendix \ref{appendix:b}). Both $u_{h} \left(x_{t} \right)$ and $u_{h}^{*} \left(x_{t} \right)$ contribute to the drift (terms 1 and 3) in Eqs. \eqref{ZEqnNum544038} and \eqref{ZEqnNum855118}, while the mean drift at given $x_{t} $ is the mean radial flow $u_{h} \left(x_{t} \right)$, as $u_{h}^{*} \left(x_{t} \right)$ is cancelled out.
Noise terms $\xi _{rh} \left(t\right)$ and $\xi _{rh}^{*} \left(t\right)$ in term 2 represent the particle random motion due to a stochastic halo size $r_{h} \left(t\right)$, where $\xi _{rh} \left(t\right)$ is independent of $r_{t} \left(s\right)$ for $s\le t$ and $\xi _{rh}^{*} \left(t\right)$ is independent of $r_{t} \left(s\right)$ for $s\ge t$. The function $\sigma \left(x_{t} \right)$ indicates that the noise is of a multiplicative nature, i.e. the noise is dependent on the process $x_{t} $ itself. Here $\sigma \left(x_{t} \right)=x_{t} $ is expected (see Eq. \eqref{ZEqnNum620785}) because the halo size follows a geometric Brownian motion (Eq. \eqref{ZEqnNum251003}).
By comparing with Eqs. \eqref{eq:B2} and \eqref{eq:B3} for regular Brownian motion in Appendix \ref{appendix:b}, Eqs. \eqref{ZEqnNum544038} and \eqref{ZEqnNum855118} describe the random motion of collisionless particles with multiplicative noise due to the random halo size. This hints the halo internal structure (density profile) is highly correlated with inverse mass cascade.
The corresponding Fokker-Planck equations (forward and backward in time) for probability $P_{r} \left(r,t\right)=P_{r} \left(x\left(t\right)\right)$ of particle position $r_{t}$ are used to describe the forward and backward processes,
\begin{equation}
\label{ZEqnNum355357}
\begin{split}
\frac{\partial P_{r} \left(r,t\right)}{\partial t} =&-\frac{r_{s} \left(t\right)}{t} \frac{\partial }{\partial r} \left[\left(u_{h} \left(x\right)+u_{h}^{*} \left(x\right)\right)P_{r} \right]\\
&+r_{s}^{2} \left(t\right)HD_{rh} \frac{\partial ^{2} }{\partial r^{2} } \left(\sigma ^{2} \left(x\right)P_{r} \right),
\end{split}
\end{equation}
\begin{equation}
\label{ZEqnNum828202}
\begin{split}
\frac{\partial P_{r} \left(r,t\right)}{\partial t} =&-\frac{r_{s} \left(t\right)}{t} \frac{\partial }{\partial r} \left[\left(u_{h} \left(x\right)-u_{h}^{*} \left(x\right)\right)P_{r} \right]\\
&-r_{s}^{2} \left(t\right)HD_{rh} \frac{\partial ^{2} }{\partial r^{2} } \left(\sigma ^{2} \left(x\right)P_{r} \right).
\end{split}
\end{equation}
Applying the chain rule from Eq. \eqref{ZEqnNum253473} and adding/subtracting Eqs. \eqref{ZEqnNum355357} to/from Eq. \eqref{ZEqnNum828202} lead to two independent equations for velocities $u_{h} \left(x\right)$ and $u_{h}^{*} \left(x\right)$,
\begin{equation}
\label{ZEqnNum533047}
b_{rh} x\frac{\partial P_{r} }{\partial x} =\frac{\partial }{\partial x} \left[u_{h} \left(x\right)P_{r} \right],
\end{equation}
\begin{equation}
\label{ZEqnNum961031}
u_{h}^{*} \left(x\right)=d_{r} \sigma ^{2} \left(x\right)\frac{\partial }{\partial x} \ln \left[\sigma ^{2} \left(x\right)P_{r} \left(x\right)\right],
\end{equation}
where $d_{r} =D_{rh} Ht={2D_{rh} /3} $. The integration of continuity Eq. \eqref{ZEqnNum533047} leads to the same expression as we have derived for $u_{h} \left(x\right)$ in Eq. \eqref{ZEqnNum278808}. For comparison, the osmotic velocity of standard Brownian motion has a dimensional form of $u_{h}^{*} =D{\nabla \rho /\rho } $ (i.e. related to the diffusion flux), where $D$ is the diffusivity and $\rho $ is the particle number density (See Appendix \ref{appendix:b} Eq. \eqref{ZEqnNum173816} for more details).
As demonstrated in Section \ref{sec:3.3} (Eqs. \eqref{ZEqnNum621572} to \eqref{ZEqnNum676651}), the radial number density function $P_{r} \left(x\right)$ can be derived if the mean radial flow $u_{h} \left(x\right)$ is known. Similarly, number density $P_{r} \left(x\right)$ can be easily found for a given osmotic flow $u_{h}^{*} \left(x\right)$ and $\sigma ^{2} \left(x\right)$ from Eq. \eqref{ZEqnNum961031},
\begin{equation}
\label{ZEqnNum790424}
P_{r} \left(x\right)=\frac{\alpha _{s} }{\sigma ^{2} \left(x\right)} \exp \left\{\frac{1}{d_{r} } \int \frac{u_{h}^{*} \left(x\right)}{\sigma ^{2} \left(x\right)} dx \right\},
\end{equation}
where $\alpha _{s} $ is a normalization constant for probability $P_{r} \left(x\right)$. For isothermal profile with constant radial number density $P_{r} \left(x\right)$, $u_{h} \left(x\right)=0$ and $u_{h}^{*} \left(x\right)=d_{r} {\partial \sigma ^{2} \left(x\right)/\partial x} $.
For a given function of $\sigma \left(x\right)$, a key relation between two velocities (current and osmotic flow) can be obtained from Eqs. \eqref{ZEqnNum533047} and \eqref{ZEqnNum961031},
\begin{equation}
\label{ZEqnNum687681}
u_{h}^{*} \left(x\right)=\frac{d_{r} \sigma ^{2} \left(x\right)}{x-u_{h} \left(x\right)} \frac{\partial u_{h} }{\partial x} +d_{r} \frac{\partial \sigma ^{2} \left(x\right)}{\partial x} .
\end{equation}
The closure problem of halo density profile is now equivalent to find an additional relation between two velocities $u_{h}^{*} \left(x\right)$ and $u_{h} \left(x\right)$. That relation combined with Eq. \eqref{ZEqnNum687681} will provide complete and consistent solutions of $u_{h} \left(x\right)$ and $u_{h}^{*} \left(x\right)$, and hence the halo density profile. Solutions for all other the relevant halo quantities can be obtained subsequently.
However, unlike the simple closure $u_{h}^{} =-u_{h}^{*} $ for Brownian motion (see Appendix \ref{appendix:b}), it is much more complicated for dark matter flow due to the nature of long range gravitational interaction. More work is required along this line to better understand the fundamental mechanism behind a universal halo structure. A simple closure is proposed and discussed in Appendix \ref{appendix:b}.
For a NFW profile, we have (from Eqs. \eqref{ZEqnNum533047} and \eqref{ZEqnNum961031})
\begin{equation}
\begin{split}
&P_{r} \left(x\right)=\frac{x}{\left(1+x\right)^{2} }, \quad u_{h} \left(x\right)=1+2x-\frac{\left(1+x\right)^{2} }{x} \ln \left(1+x\right)\\
&\textrm{and}\quad u_{h}^{*} \left(x\right)=d_{r} \frac{x\left(3+x\right)}{1+x},
\end{split}
\label{eq:130}
\end{equation}
\noindent where the integral of $P_{r} \left(x\right)$ diverges, a well-known difficulty of NFW profile. At this time, we will take a different route to model the halo internal structure by first identifying basic properties of the osmotic flow $u_{h}^{*} \left(x\right)$. A simple model of $u_{h}^{*} \left(x\right)$ is then proposed, followed by applying Eq. \eqref{ZEqnNum790424} for particle number density $P_{r} \left(x\right)$.
Without loss of generality, let's assume a general power-law of $\sigma \left(x\right)=x^{\lambda _{r} } $ (we expect $\lambda _{r} =1$ though) and from Eq. \eqref{ZEqnNum961031},
\begin{equation}
\label{eq:131}
u_{h}^{*} \left(x\right)=d_{r} x^{2\lambda _{r} -1} \left(\frac{\partial \ln P_{r} }{\partial \ln x} +2\lambda _{r} \right).
\end{equation}
The derivative of $u_{h}^{*} \left(x\right)$ is
\begin{equation}
\label{eq:132}
\frac{\partial u_{h}^{*} \left(x\right)}{\partial x} =d_{r} x^{2\lambda _{r} -2} \frac{\partial ^{2} \ln P_{r} }{\partial \left(\ln x\right)^{2} } +\left(2\lambda _{r} -1\right)\frac{u_{h}^{*} \left(x\right)}{x} .
\end{equation}
The properties of $u_{h}^{*} \left(x\right)$ can be identified from above equations,
\begin{equation}
\begin{split}
&u_{h}^{*} \left(x=0\right)=0, \quad u_{h}^{*} \left(x=x_{0}^{*} \right)=0\\
&\textrm{when}\quad \left. \frac{\partial \ln P_{r} }{\partial \ln x} \right|_{x=x_{0}^{*} } =\left. \frac{\partial \ln \rho _{h} }{\partial \ln x} \right|_{x=x_{0}^{*} } +2=-2\lambda _{r},
\end{split}
\label{ZEqnNum477779}
\end{equation}
\noindent and
\begin{equation}
\begin{split}
&u_{h}^{*} \left(x=1\right)=2\lambda _{r} d_{r}\\
&\textrm{because} \quad \left. \frac{\partial \ln P_{r} }{\partial \ln x} \right|_{x=1} =\left. \frac{\partial \ln \rho _{h} }{\partial \ln x} \right|_{x=1} +2=0,
\end{split}
\label{ZEqnNum956240}
\end{equation}
\noindent where $x=1$ is the mode of probability function $P_{r} \left(x\right)$. Specifically, for $\lambda _{r} =1$ (From Eq. \eqref{ZEqnNum687681}),
\begin{equation}
\label{ZEqnNum497738}
\left. \frac{\partial u_{h}^{*} }{\partial x} \right|_{x=0} =\gamma _{r} =d_{r} \frac{2-\gamma _{h} }{1-\gamma _{h} } , \end{equation}
where $\gamma _{h} =\left. \left({\partial u_{h} /\partial x} \right)\right|_{x=0} $ is the halo deformation parameter we defined before (Table \ref{tab:2}).
Similar to the mean radial flow $u_{h}^{} \left(x\right)$, the osmotic flow $u_{h}^{*} \left(x\right)$ initially increases as $\gamma _{r} x$ and reaches a maximum, then decreases to zero at $x=x_{0}^{*} $ where the logarithmic slope of density $\rho _{h} $ is $-2-2\lambda _{r} $ (Eq. \eqref{ZEqnNum477779}). A simple but general model of $u_{h}^{*} \left(x\right)$ (expansion around $x=0$) with three free parameters satisfying all conditions in Eqs. \eqref{ZEqnNum477779}, \eqref{ZEqnNum956240} and \eqref{ZEqnNum497738} can be written as,
\begin{equation}
\begin{split}
u_{h}^{*} \left(x\right)=\gamma _{r} x-\beta _{r} x^{1+\alpha _{r} }\quad \textrm{with} \quad \alpha _{r} >0.
\label{ZEqnNum927116}
\end{split}
\end{equation}
\noindent Obviously the condition Eq. \eqref{ZEqnNum956240} requires,
\begin{equation}
\label{eq:137}
\beta _{r} =d_{r} \left(\frac{2-\gamma _{h} }{1-\gamma _{h} } -2\lambda _{r} \right).
\end{equation}
The general solution of $P_{r}^{} \left(x\right)$ can be obtained from Eq. \eqref{ZEqnNum790424} with given $u_{h}^{*} \left(x\right)$ in Eq. \eqref{ZEqnNum927116},
\begin{equation}
P_{r} \left(x\right)=\alpha _{s} x^{-2\lambda _{r} } \exp \left\{\frac{x^{2-2\lambda _{r} } }{d_{r} } \left(\frac{\gamma _{r} }{2-2\lambda _{r} } -\frac{\beta _{r} x^{\alpha _{r} } }{2-2\lambda _{r} +\alpha _{r} } \right)\right\}
\label{ZEqnNum714625}
\end{equation}
\noindent for $\lambda _{r} \ne 1$, where $\alpha _{s}$ is a normalization constant. While for $\lambda _{r} =1$,
\begin{equation}
P_{r} \left(x\right)=\frac{\alpha _{r} x^{{\gamma _{r} /d_{r} -2} } }{\Gamma \left[{\left(\gamma _{r} -d_{r} \right)/\left(\alpha _{r} d_{r} \right)} \right]} \left(\frac{\beta _{r} }{\alpha _{r} d_{r} } \right)^{\frac{\gamma _{r} -d_{r} }{\alpha _{r} d_{r} } } \exp \left\{-\frac{\beta _{r} x^{\alpha _{r} } }{\alpha _{r} d_{r} } \right\}.
\label{ZEqnNum230459}
\end{equation}
The condition of maximum mean radial flow at $r=r_{s} $ ($\left. {\partial P_{r} /\partial x} \right|_{x=1} =0$ from Eq. \eqref{ZEqnNum273026}) requires $x=1$ is the mode of distributions $P_{r} \left(x\right)$, i.e. we will find maximum number of particles at $r=r_{s} $. For $\left. {\partial P_{r} /\partial x} \right|_{x=1} =0$ applied to Eq. \eqref{ZEqnNum714625}, we have the relation $2d_{r} \lambda _{r} =\gamma _{r} -\beta _{r} $. Specifically, for $\lambda _{r} =1$, relations between the drift and noise terms in Eqs. \eqref{ZEqnNum544038} and \eqref{ZEqnNum855118} are found as (analogy to the fluctuation-dissipation theorem)
\begin{equation}
2d_{r} =\gamma _{r} -\beta _{r} \quad \textrm{and} \quad \gamma _{r} =d_{r} \frac{2-\gamma _{h} }{1-\gamma _{h} },
\label{eq:140}
\end{equation}
\noindent such that the particle distribution function (Eq. \eqref{ZEqnNum230459}) is reduced to a two-parameter distribution
\begin{equation}
\label{ZEqnNum103079}
P_{r} \left(x\right)=\frac{b_{r} {}^{a_{r} } }{\Gamma \left(a_{r} \right)\left(a_{r} -b_{r} \right)} \exp \left(-b_{r} x^{\frac{1}{a_{r} -b_{r} } } \right)x^{\frac{b_{r} }{a_{r} -b_{r} } }
\end{equation}
in terms of $a_{r} ={\left(\gamma _{r}-d_{r} \right)/\left(\alpha _{r} d_{r} \right)} $ and $b_{r} ={\beta _{r} /\left(\alpha _{r} d_{r} \right)} $, where $\alpha _{r} ={1/\left(a_{r} -b_{r} \right)}$.
If we require $\gamma _{r} =\gamma _{h} $ for $x\to 0$ (i.e. the inward particle motion $u_{h} \left(x\right)-u_{h}^{*} \left(x\right)$ vanishes at the center of halo), we can determine that the constant $d_{r} ={1/6} $ (hence $D_{rh} ={1/4} $) with $\gamma _{h} ={2/3} $ from Eq. \eqref{ZEqnNum801048} that is required by Hubble flow at halo center. The other option is to require $\gamma _{r} =2\gamma _{h} $ for $x\to 0$ (the inward particle motion $u_{h} \left(x\right)-u_{h}^{*} \left(x\right)=-u_{h} \left(x\right)$, i.e. the inward mass flow to the halo center balances the outward mass flow). The constant $d_{r} ={1/3} $ (hence $D_{rh} ={1/2} $). The values of relevant parameters for different options are listed in Table \ref{tab:4}.
\begin{table}
\caption{Values of relevant parameters for three possible options}
\begin{tabular}{p{0.1in}p{0.1in}p{0.15in}p{0.1in}p{0.05in}p{0.1in}p{0.15in}p{0.15in}p{0.4in}p{0.4in}}
\hline
$\gamma _{r} $ & $\gamma _{h} $ & $D_{rh} $ & $d_{r} $ & $\lambda _{r} $ & $\beta _{r} $ & $a_{r} $ & $b_{r} $ & \makecell{$u_{h}$\\$(x\to 0)$} & \makecell{$u_{h}^{*}$\\$(x\to 0)$} \\ \hline
${1/2} $ & ${2/3} $ & ${3/16} $ & ${1/8} $ & $1$ & ${1/4} $ & ${3/\alpha _{r} } $ & ${2/\alpha _{r} } $ & ${2x/3} $ & ${x/2} $ \\ \hline
${2/3} $ & ${2/3} $ & ${1/4} $ & ${1/6} $ & $1$ & ${1/3} $ & ${3/\alpha _{r} } $ & ${2/\alpha _{r} } $ & ${2x/3} $ & ${2x/3} $ \\ \hline
${4/3} $ & ${2/3} $ & ${1/2} $ & ${1/3} $ & $1$ & ${2/3} $ & ${3/\alpha _{r} } $ & ${2/\alpha _{r} } $ & ${2x/3} $ & ${4x/3} $ \\ \hline
\end{tabular}
\label{tab:4}
\end{table}
The function $F_{r} \left(x\right)$ is the fraction of particles with a distance smaller than a given \textit{x}, i.e. the cumulative distribution function of probability distribution $P_{r} \left(x\right)$,
\begin{equation}
\label{ZEqnNum768710}
\begin{split}
&F_{r} \left(x=\frac{r}{r_{s} } \right)=\int _{0}^{x}P_{r} \left(y\right) dy=\frac{m_{r} }{m_{h} } \\
&=1-\frac{\Gamma \left(a_{r} ,b_{r} x^{{1/\left(a_{r} -b_{r} \right)} } \right)}{\Gamma \left(a_{r} \right)} =\frac{\gamma \left(a_{r} ,b_{r} x^{{1/\left(a_{r} -b_{r} \right)} } \right)}{\Gamma \left(a_{r} \right)},
\end{split}
\end{equation}
where $\Gamma \left(x,y\right)$ and $\gamma \left(x,y\right)$ are the upper and lower incomplete Gamma functions, respectively. The corresponding halo density profile $\rho _{h} \left(x\right)$ is given by
\begin{equation}
\label{ZEqnNum495681}
\begin{split}
\rho _{h} \left(x\right)&=\frac{m_{h} P_{r} \left(x\right)}{4\pi r_{s}^{3} x^{2} }\\
& =\frac{m_{h} }{4\pi r_{s}^{3} } \frac{b_{r} {}^{a_{r} } }{\Gamma \left(a_{r} \right)\left(a_{r} -b_{r} \right)} \exp \left(-b_{r} x^{\frac{1}{a_{r} -b_{r} } } \right)x^{\frac{3b_{r} -2a_{r} }{a_{r} -b_{r} } },
\end{split}
\end{equation}
or equivalently with $\rho _{s} =\rho _{h} \left(x=1\right)$,
\begin{equation}
\rho _{h} \left(x\right)=\rho _{s} e^{b_{r} } \exp \left(-b_{r} x^{\frac{1}{a_{r} -b_{r} } } \right)x^{\frac{3b_{r} -2a_{r} }{a_{r} -b_{r} } }.
\label{ZEqnNum239340}
\end{equation}
\noindent It can be verified that for inner profile $\rho _{h} \left(r<r_{s} \right)\propto r^{{\left(3\gamma _{h} -2\right)/\left(1-\gamma _{h} \right)} } $. The halo deformation rate parameter $\gamma _{h} $ is related to the new density profile as $\gamma _{h} ={b_{r} /a_{r} } $. The density profile in Eqs. \eqref{ZEqnNum495681} or \eqref{ZEqnNum239340} is general with four parameters ($\rho _{s}$, $r_{s}$, $a_{r}$ and $b_{r}$). For $\gamma _{h} ={b_{r} /a_{r} } ={2/3} $, Eq. \eqref{ZEqnNum239340} exactly reduces to the Einasto profile (Eq. \eqref{ZEqnNum305317}) with shape parameter $\alpha ={2/b_{r} } $.
However, the density profile we proposed is a result of the random motion of collisionless particles in a halo with stochastically varying size according to Eq. \eqref{ZEqnNum251003}. The distribution $P_{r} \left(x\right)$ in Eq. \eqref{ZEqnNum103079} can be interpreted as the probability to find a particle at any position extending to infinity or the distribution of particles in an assembled halo (composite halo) incorporating all possible halos sizes. By contrast, the Einasto profile is for individual halos of finite size. Therefore, parameters of Eq. \eqref{ZEqnNum239340} can be different from the Einasto profile for individual halos with $\alpha\approx 0.2$ (as shown in Fig. \ref{fig:7} with $\alpha\approx 0.7$ for composite halos).
The number density $P_{r} \left(x\right)$ in Eq. \eqref{ZEqnNum103079} describes the radial density of collisionless particles from all halos with different sizes evolving according to Eq. \eqref{ZEqnNum251003}. The distribution $P_{r} \left(x={r/r_{s} } \right)$ is defined for all $x\ge 0$ extending to infinity. This density describes the probability distribution of all particles in all possible halos. It is different from the usual density profiles for individual halos with a finite size.
\subsection{Density profiles from N-body simulation}
\label{sec:4.3}
In simulation, instead of working with the spherical averaged halo density profile for each individual halo with a finite size, we compute the density profile for a group of halos of same mass. The function $F_{r} \left(r\right)$ was computed for a halo group of size $n_{p} $ at a given scale factor \textit{a}, where $r$ is the distance of every particle in the halo group to the respective center of mass of the halo that it belongs to. In simulation, the cumulative function $F_{r} \left(r\right)$ is computed as the fraction of all particles in the same halo group with a distance smaller than $r$. The radial density profile $P_{r} \left(x={r/r_{s} } \right)$ (particle distribution probability) can be obtained by taking the derivative ${\partial F_{r} \left(x\right)/\partial x} $ (Eq. \eqref{ZEqnNum768710}). This procedure will significantly reduce the noise as the number of particles in the entire halo group is much greater than the number of particles in individual halos.
Figure \ref{fig:6} plots the log-log variation of the radial cumulative distribution function $F_{r} \left(r=xr_{s} \right)$ for all particles in the same halo group of size $n_{p} $ at \textit{z}=0. Symbols plot the simulation data for five different sizes of halo groups, while the solid lines plot the best fit of the simulation data for each size of halo group using the proposed model Eq. \eqref{ZEqnNum768710} with three free parameters $r_{s} $, $a_{r} $, and $b_{r} $. The fitted values of three parameters varying with group size $n_{p} $ are presented in Figs. \ref{fig:7} and \ref{fig:8}. Clearly, the scaling $F_{r} \left(x={r/r_{s} } \right)\sim x^{3} $ for small \textit{x} indicates the existence of a central core at halo center (Table \ref{tab:2}). The proposed model Eq. \eqref{ZEqnNum768710} provides very good agreement with the simulation data for a wide range of halo group size.
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig6}
\caption{The $log_{10}-log_{10}$ variation of the cumulative distribution function $F_{r} \left(r=xr_{s} \right)$ with the distance \textit{r} to the center of mass of the halo that particle belongs to. Function $F_{r} \left(r=xr_{s} \right)$ is computed based on all particles in a group of halos with same size $n_{p} $ at \textit{z}=0. Symbols of dot present the simulation data for five different sizes of halo groups. Solid lines plot the best fit of simulation data for each size of halo group using model (Eq. \eqref{ZEqnNum768710}) with three free parameters $r_{s} $, $a_{r} $, and $b_{r} $. The fitted values of these parameters are presented in Figs. \ref{fig:7} and \ref{fig:8}. The scaling $F_{r} \left(x={r/r_{s} } \right)\sim x^{3} $ for small \textit{x} clearly indicates the existence of a central core for composite halos.}
\label{fig:6}
\end{figure}
Figure \ref{fig:7} presents the variation of fitted values of $a_{r} $ and $b_{r} $ for $F_{r} \left(r=xr_{s} \right)$ with the halo group size $n_{p} $. Both values of $a_{r} $ and $b_{r} $ slowly increase with the halo size $n_{p} $. However, the ratio of ${a_{r} /b_{r} } \approx {3/2} $ is found for all halo sizes, regardless of the halo size, which is required by a finite density at halo center (Eq. \eqref{ZEqnNum239340}). Parameter $\alpha _{r} ={1/\left(a_{r} -b_{r} \right)}=\alpha$ slowly decreases with halo size from 1.2 to 0.7, which is significantly larger than shape parameter $\alpha \approx 0.2$ for Einasto profile of individual halos with finite size.
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig7}
\caption{The variation of fitted values of $a_{r} $ and $b_{r} $ for cumulative distribution function $F_{r} \left(r=xr_{s} \right)$ with halo group size $n_{p} $. Both values of $a_{r} $ and $b_{r} $ increase with the size $n_{p} $. However, the ratio of ${a_{r} /b_{r} } \approx {3/2} $ is found regardless of the halo size, as required by a finite density at halo center. The exponent parameter $\alpha _{r} ={1/\left(a_{r} -b_{r} \right)}=\alpha$ slowly decreases with the halo size from 1.2 to 0.7. This value is significantly larger than the shape parameter of $\alpha \approx 0.2$ for Einasto profile of individual halos with a finite size.}
\label{fig:7}
\end{figure}
The \textit{k}th moment of distribution $P_{r} \left(x\right)$ can be easily found as
\begin{equation}
\label{eq:145}
M_{k} =\int _{0}^{\infty }P_{r} \left(x\right) x^{k} dx=\frac{\Gamma \left(a_{r} +\left(a_{r} -b_{r} \right)k\right)}{b_{r} {}^{\left(a_{r} -b_{r} \right)k} \Gamma \left(a_{r} \right)} .
\end{equation}
The momentum generating function of the distribution $P_{r} \left(x\right)$ reads
\begin{equation}
\label{eq:146}
MGF\left(t\right)=\int _{0}^{\infty }P_{r} \left(x\right) e^{tx} dx=\sum _{k=0}^{\infty }\frac{t^{k} \Gamma \left(a_{r} +\left(a_{r} -b_{r} \right)k\right)}{k!b_{r} {}^{\left(a_{r} -b_{r} \right)k} \Gamma \left(a_{r} \right)}
\end{equation}
from which the halo mean square radius $r_{g} $ (the root mean square distance) reads
\begin{equation}
\label{eq:147}
r_{g} =r_{s} \sqrt{\frac{b_{r} {}^{-2\left(a_{r} -b_{r} \right)} \Gamma \left(3a_{r} -2b_{r} \right)}{\Gamma \left(a_{r} \right)} } .
\end{equation}
Three characteristic length scales can be identified from the simulation data for group of halos of the same mass. The scale radius (halo core size) $r_{s} $ can be found by fitting Eq. \eqref{ZEqnNum768710} to the simulation data of $F_{r} \left(r=xr_{s} \right)$ in Fig. \ref{fig:6}. The mean square radius of halo group can be computed,
\begin{equation}
\label{ZEqnNum195815}
r_{g} =\sqrt{{\sum _{m=1}^{n_{h} }\sum _{k=1}^{n_{p} }\left(r_{km}^{2} \right) /\left(n_{p} n_{h} \right)} } ,
\end{equation}
where $r_{km}$ is the distance of the \textit{k}th particle in \textit{m}th halo of the halo group to the center of that halo. Here $n_{h} $ is the number of halos in the group. The virial radius $r_{h\Delta } $ of a halo group can be found from the simulation data of $F_{r} \left(r=xr_{s} \right)$ as
\begin{equation}
\label{ZEqnNum125294}
\frac{F_{r} \left(r_{h\Delta } \right)}{r_{h\Delta }^{3} } =\frac{4\pi \Delta _{c} \bar{\rho }\left(a\right)}{3m_{h} } ,
\end{equation}
where $\bar{\rho }\left(a\right)$ is the mean background density at scale factor \textit{a}.
Figure \ref{fig:8} plots the variation of three characteristic length scales (in the unit of Mpc/h) with the halo group size $n_{p} $, i.e. the virial radius $r_{h\Delta } $ with $\Delta _{c} =178$ (Eq. \eqref{ZEqnNum125294}), the mean square radius $r_{g} $ (Eq. \eqref{ZEqnNum195815}), and the scale radius $r_{s} $ (fitted from the simulation data in Fig. \ref{fig:6}). The ratios between three halo sizes are also presented. Note that all three length scales are defined based on the statistics of the entire halos group, instead of individual halos.
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig8}
\caption{The variation of three length scales (in the unit of Mpc/h) with halo group size $n_{p} $, i.e. the virial radius $r_{h\Delta } $ with $\Delta =178$ (Eq. \eqref{ZEqnNum125294}), the mean square radius $r_{g} $ (Eq. \eqref{ZEqnNum195815}, and the scale radius $r_{s} $ (fitted from the simulation data in Fig. \ref{fig:6}).}
\label{fig:8}
\end{figure}
Finally, all other relevant halo quantities can be obtained with number density $P_{r} \left(x\right)$ from Eq. \eqref{ZEqnNum103079}. Examples are the mean radial flow $u_{h} \left(x\right)$ from Eq. \eqref{ZEqnNum533047},
\begin{equation}
\label{ZEqnNum980800}
\begin{split}
u_{h} \left(x\right)&=x-\left(a_{r} -b_{r} \right)b_{r} {}^{-a_{r} }\\
&\cdot \exp \left(b_{r} x^{\frac{1}{a_{r} -b_{r} } } \right)x^{\frac{-b_{r} }{a_{r} -b_{r} } } \gamma \left(a_{r} ,b_{r} x^{\frac{1}{a_{r} -b_{r} } } \right).
\end{split}
\end{equation}
The shifted potential $\phi _{h}^{*} \left(x,a\right)$ from Eq. \eqref{ZEqnNum269276},
\begin{equation}
\label{ZEqnNum105452}
\begin{split}
&\phi _{h}^{*} \left(x,a\right)=\frac{Gm_{h} \left(a\right)}{\Gamma \left(a_{r} \right)r_{s} \left(a\right)}\\
&\quad\quad\quad \cdot \left[b_{r} {}^{a_{r} -b_{r} } \gamma \left(b_{r} ,b_{r} x^{\frac{1}{a_{r} -b_{r} } } \right)-\frac{1}{x} \gamma \left(a_{r} ,b_{r} x^{\frac{1}{a_{r} -b_{r} } } \right)\right].
\end{split}
\end{equation}
The lower incomplete Gamma function has the properties that
\begin{equation}
\begin{split}
&\gamma \left(x,y\right)=\Gamma \left(x\right) \quad \textrm{for} \quad y\to \infty,\\
&\gamma \left(x,y\right)={y^{x}/x} \quad \textrm{for}\quad y\to 0,
\end{split}
\label{eq:152}
\end{equation}
\noindent where the shifted potential $\phi _{h}^{*} \left(x,a\right)$ simplifies to
\begin{equation}
\label{eq:153}
\phi _{h}^{*} \left(x\to \infty \right)=\frac{Gm_{h} \left(a\right)}{r_{s} \left(a\right)} \frac{b_{r}^{a_{r} -b_{r} } \Gamma \left(b_{r} \right)}{\Gamma \left(a_{r} \right)} ,
\end{equation}
\begin{equation}
\label{eq:154}
\phi _{h}^{*} \left(x\to 0\right)=\frac{Gm_{h} \left(a\right)}{r_{s} \left(a\right)} \frac{b_{r}^{a_{r} } \left(a_{r} -b_{r} \right)}{a_{r} b_{r} \Gamma \left(a_{r} \right)} x^{\frac{b_{r} }{a_{r} -b_{r} } } .
\end{equation}
\subsection{Equation of state for relative pressure and density}
\label{sec:4.4}
The velocity dispersion $\sigma _{nr}^{2} $ can be obtained with Eqs. \eqref{ZEqnNum807329} and $F_{r} \left(x\right)$ from \eqref{ZEqnNum768710}. It should be interesting to examine the equation of state (EOS) in the core region for small \textit{x}, where halo density can be well approximated (from Eq. \eqref{ZEqnNum239340} with $a_{r} ={3b_{r} /2}$) as
\begin{equation}
\begin{split}
&\rho_{h} \left(x\right)=\rho _{h} \left(0\right)\left(1-b_{r} x^{\frac{2}{b_{r} } } \right),\\
& \textrm{with} \\
&\rho _{h} \left(0\right)=\frac{m_{h} }{4\pi r_{s}^{3} } \frac{b_{r} {}^{a_{r} } }{\Gamma \left(a_{r} \right)\left(a_{r} -b_{r} \right)}.
\end{split}
\label{ZEqnNum277257}
\end{equation}
\noindent From Eqs. \eqref{ZEqnNum253695} and \eqref{ZEqnNum594491}, pressure in core region is parabolic,
\begin{equation}
\label{ZEqnNum131959}
p_{h} \left(x\right)=\rho _{h} \left(x\right)\sigma _{r}^{2} \left(x\right)=p_{h} \left(x=0\right)-\frac{1}{2} \frac{\rho _{h}^{2} \left(0\right)v_{cir}^{2} }{\bar{\rho }_{h} c^{2} } x^{2} .
\end{equation}
The equation of state in the core region for relative pressure and density can be finally written as (with Eq. \eqref{ZEqnNum277257}),
\begin{equation}
\label{ZEqnNum609386}
\left[p_{h} \left(0\right)-p_{h} \left(x\right)\right]=\frac{\left[\rho _{h}^{} \left(0\right)\right]^{2-b_{r} } v_{cir}^{2} }{2\left(b_{r} \right)^{b_{r} } \bar{\rho }_{h} c^{2} } \left[\rho _{h} \left(0\right)-\rho _{h} \left(x\right)\right]^{b_{r} }
\end{equation}
such that $\Delta p_{h} =K_{s} \left(\Delta \rho _{h} \right)^{b_{r} }$. Unlike the ideal gas with reference pressure and density being zero when molecules are infinitely far from each other, halos have their center pressure and density as a reference state where both gravitational and pressure forces vanish (gradient is zero).
The relative pressure and density $\Delta p_{h} $ and $\Delta \rho _{h} $ to the center of halo satisfy the equation of state \eqref{ZEqnNum609386}. The parameter $b_{r} $ has the physical meaning as the exponent of equation of state. The regular polytropic equation of state $p_{h} \propto \rho _{h}^{1+{1/n} } $ for absolute pressure and density may not be applicable to dark matter halos. Figure \ref{fig:7} shows that the value of $b_{r} $ slightly increases from 1.5 to 3 with increasing halo group size. The NFW profile will not lead to such equation of state because of divergent pressure/density (Eqs. \eqref{ZEqnNum410484} and \eqref{ZEqnNum447542}).
Let's assume the equilibrium center pressure and density are $p_{h} \left(0\right)$ and $p_{h} \left(0\right)$, where both gravitational and pressure forces are not present. At any location in halo, the relative pressure $\Delta p_{hn} $
\noindent and $\Delta \rho _{hn} $ (normalized and using Eqs. \eqref{ZEqnNum154562}, \eqref{ZEqnNum273026}, and \eqref{ZEqnNum482501}) can be written in terms of $F_{r} \left(x\right)$,
\begin{equation}
\label{ZEqnNum973400}
\begin{split}
&\Delta p_{hn} =\frac{p_{h} \left(0\right)-p_{h} \left(x\right)}{\left({r_{s}^{2} /t^{2} } \right)\left({m_{h} /4\pi r_{s}^{3} } \right)} \\
&=\frac{4\pi ^{2} }{F_{r} \left(1\right)} \int _{0}^{x}\frac{F_{r} \left(x\right)F_{r}^{'} \left(x\right)}{x^{4} } dx -\int _{0}^{x}\frac{F_{r}^{2} \left(x\right)F_{r}^{''} \left(x\right)}{x^{2} \left[F_{r}^{'} \left(x\right)\right]^{2} } dx
\end{split}
\end{equation}
and
\begin{equation}
\label{ZEqnNum131593}
\Delta \rho _{hn} =\frac{\rho _{h} \left(0\right)-\rho _{h} \left(x\right)}{\left({m_{h} /4\pi r_{s}^{3} } \right)} =\left[\left. \frac{F_{r}^{'} \left(x\right)}{x^{2} } \right|_{x=0} -\frac{F_{r}^{'} \left(x\right)}{x^{2} } \right].
\end{equation}
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig9}
\caption{The variation of relative pressure $\Delta p_{hn} $ with relative density $\Delta \rho _{hn} $ for different exponent $b_{r}$ using Eqs. \eqref{ZEqnNum973400} and \eqref{ZEqnNum131593}, where function $F_r(x)$ is from Eq. \eqref{ZEqnNum768710}. The proposed model yields an equation of state $\Delta p_{hn} \propto \left(\Delta \rho _{hn} \right)^{b_{r} } $ that can be clearly identified (solid lines). The equation of state \eqref{ZEqnNum805925} is plotted as dash lines for comparison. The relation (Eq. \eqref{ZEqnNum520886}) between center pressure $p_{hn} \left(0\right)$ and center density $\rho _{hn} \left(0\right)$ is presented as the red solid line for different $b_{r} $ (i.e. the maximum $\Delta p_{hn} $ vs. maximum $\Delta \rho _{hn} $). The center velocity dispersion $\sigma _{r0}^{2} $ can be identified accordingly.}
\label{fig:9}
\end{figure}
\noindent More specifically with $F_{r} \left(x\right)$ from Eq. \eqref{ZEqnNum768710} and $a_{r} ={3b_{r} /2} $, equation of state \eqref{ZEqnNum609386} reduces to
\begin{equation}
\label{ZEqnNum805925}
\Delta p_{hn} =\frac{2^{3-b_{r} } \pi ^{2} \left(b_{r} \right)^{3b_{r} -2-{3b_{r}^{2} /2} } }{3\gamma \left({3b_{r} /2} ,b_{r} \right)\Gamma \left({3b_{r} /2} \right)^{1-b_{r} } } \left(\Delta \rho _{hn} \right)^{b_{r} } ,
\end{equation}
where $\gamma \left(x,y\right)$ is the lower incomplete Gamma functions.
Next we will derive the pressure, density, and velocity dispersion at halo center. Let's assume the equation of state \eqref{ZEqnNum609386} is valid for the entire range of \textit{x} extending to infinity. By setting $p_{h} \left(\infty \right)=0$ and $\rho _{h} \left(\infty \right)=0$, a simple relation (regardless of the value of $b_{r} $) between center pressure and density can be obtained from Eq. \eqref{ZEqnNum609386},
\begin{equation}
\label{ZEqnNum520886}
p_{h} \left(0\right)=\frac{v_{cir}^{2}\rho _{h}^{2} \left(0\right)}{2\left(b_{r} \right)^{b_{r} } \bar{\rho }_{h} c^{2} } =\frac{2\pi Gr_{s}^{2} }{3\left(b_{r} \right)^{b_{r} } } \rho _{h}^{2} \left(0\right)=K_{r} \left(a\right)\rho _{h}^{2} \left(0\right),
\end{equation}
where the pre-factor $K_{r} \left(a\right)\approx 0.01aG\left(1Mpc/h\right)^{2}$ from simulation data should be independent of halo mass $m_{h} $. At the same redshift \textit{z} (or \textit{a}) and with $r_{s}^{} \propto m_{h}^{{1/3} } $ increasing with halo mass $m_{h} $, $b_{r} $ is also expected to be slightly increasing with halo mass $m_{h} $. On the other hand (from Eqs. \eqref{ZEqnNum520886} and \eqref{ZEqnNum277257}),
\begin{equation}
\label{ZEqnNum218090}
\begin{split}
p_{h} \left(0\right)&=\frac{v_{cir}^{2} }{2\left(b_{r} \right)^{b_{r} } \bar{\rho }_{h} c^{2} } \rho _{h}^{2} \left(0\right)\\
&=\frac{Gm_{h} }{3r_{s} } \frac{\left(b_{r} \right)^{{b_{r} /2} -1} }{\Gamma \left({3b_{r} /2} \right)} \rho _{h} \left(0\right)=\sigma _{r0}^{2} \rho _{h} \left(0\right),
\end{split}
\end{equation}
and
\begin{equation}
\label{163}
\sigma_{r0}^{2} \equiv \sigma _{r}^{2} \left(x=0\right)=\frac{Gm_{h} }{3r_{s} } \frac{\left(b_{r} \right)^{{b_{r} /2} -1} }{\Gamma \left({3b_{r} /2} \right)},
\end{equation}
where $\sigma _{r0}^{2}$ is the center velocity dispersion and $\sigma _{r0}^{2} \propto m_{h}^{{2/3} } a^{-1} $. For $b_{r} ={5/3} $ (exponent for adiabatic process), $\sigma _{r0}^{2} \approx 26{r_{s}^{2} /t^{2} } $ compared to $\sigma _{r0}^{2} =2\pi ^{2} c^{2} {r_{s}^{2} /t^{2} } $ for isothermal profile (Fig. \ref{fig:4}). With $p_{h} \left(0\right)$ from Eq. \eqref{ZEqnNum520886}, the core size (where Hubble flow is dominant) $x_{c} =\left(b_{r} \right)^{{-b_{r} /2} } $ (in Eq. \eqref{ZEqnNum946518}) can be easily obtained by forcing $p_{h} \left(x\right)=0$ in Eq. \eqref{ZEqnNum131959}. The core size $x_{c} =1$ for $b_{r} =0$ (core size exactly equals scale radius) and decreases with $b_{r} $. For $b_{r} ={5/3} $, $x_{c} \approx 0.65$.
Figure \ref{fig:9} summarizes the equation of state for relative pressure and density. With cumulative function $F_{r} \left(x\right)$ from Eq. \eqref{ZEqnNum768710}, the variation of relative pressure $\Delta p_{hn} $ with relative density $\Delta \rho _{hn} $ is plotted for different $b_{r} $ using Eqs. \eqref{ZEqnNum973400} and \eqref{ZEqnNum131593}. Clearly, equation of state follows a scaling law $\Delta p_{hn} \propto \left(\Delta \rho _{hn} \right)^{b_{r} } $ for most range of density, while deviation is only observed in the outer halo region with extremely low density. Analytical approximation (dash lines from Eq. \eqref{ZEqnNum805925}) is presented for comparison. The relation between center pressure and density is also plotted as red thick line.
Figure \ref{fig:10} plots the variation of center density $\rho _{h} \left(0\right)$, center pressure $p_{h} \left(0\right)$, and center dispersion $\sigma _{r0}^{2} $ with exponent $b_{r} $. For $b_{r} \to 0$, the velocity dispersion $\sigma _{r0}^{2} \approx 20{r_{s}^{2} /t^{2} } $. There exist minimum $\rho _{h} \left(0\right)$, maximum $p_{h} \left(0\right)$ and maximum $\sigma _{r0}^{2} $ at certain $b_{r}$.
\begin{figure}
\includegraphics*[width=\columnwidth]{Fig10}
\caption{The variation of normalized pressure $p_{h} \left(0\right)$ (Eq. \eqref{ZEqnNum218090}), density $\rho _{h} \left(0\right)$ (Eq. \eqref{ZEqnNum277257}), and dispersion ${\sigma_{r0}^{2}/(r_s^2/t^{2}})$ at halo center for different parameter $b_{r} ={2/\alpha } $, where $\alpha $ is the shape parameter in Einasto profile.}
\label{fig:10}
\end{figure}
\section{Conclusions}
\label{sec:5}
The gravitational collapse of dark matter is essentially a nonlinear self-gravitating collisionless fluid flow problem (SG-CFD). The inverse mass cascade is a unique feature of SG-CFD that shares many similarities with the energy cascade in turbulence. Halos are intrinsically dynamical objects that mediate the mass cascade. This paper focus on the effect of inverse mass cascade on relevant halo properties and internal structures.
The halo internal structure is highly dependent on mass cascade. The continuous mass accretion creates a new layer of mass that deforms the original halo and creates a non-zero mean radial flow (Figs. \ref{fig:1} and \ref{fig:2}, and Eq. \eqref{ZEqnNum849591}, outflow for core region and inflow for outer region). The isothermal density profile is a natural result for halos with infinitely fast mass accretion and vanishing mean radial flow (Eq. \eqref{ZEqnNum278808}). The combined in- and out-flow lead to an extra length scale (the scale radius $r_{s}$) for density profile where the radial flow is at its maximum. A double-power-law density (Eqs. \eqref{ZEqnNum973033} and \eqref{ZEqnNum367971}) is proposed with inner density dominated by the halo deformation rate $\gamma _{h}$ and outer density controlled by a halo deformation parameter $\alpha _{h}$ that is dependent on halo concentration, and mass cascade parameters $\lambda$ and $\tau _{0}$ (Eq. \eqref{ZEqnNum412141}). The cusp-core controversy is related to the deformation rate (the gradient of mean radial flow) $\gamma _{h}$ at halo center. Slower deformation leads to a steeper core density profile (Eq. \eqref{ZEqnNum973033}). For large halos with extremely fast mass accretion and an expanding core, Hubble flow is expected at halo center that hints the existence of a central halo core with $\gamma _{h} ={2/3}$ (Eq. \eqref{ZEqnNum801048}).
The momentum and energy exchange between mean flow and velocity dispersion (random motion) is studied via the Jeans' equation (Eq. \eqref{ZEqnNum535400}) for spherical non-rotating isotropic halos. The radial flow is shown to enhance the radial dispersion in outer region (Fig. \ref{fig:4}). The closure problem for halo density profile can be reduced to the correct modeling of radial flow $u_{h} \left(x\right)$. The halo density profile can be derived for a given Taylor expansion of $u_{h} \left(x\right)$ around $x=0$. The critical halo concentration $c=3.48$ is obtained as a result of vanishing linear radial momentum for large halos (Fig. \ref{fig:3} and Eq. \eqref{ZEqnNum722864}). A complete analysis of the effects of radial flow on various halo energies is also presented (Eqs. \eqref{ZEqnNum811693}-\eqref{ZEqnNum765989}). Halo surface energy and surface tension are introduced for halos with finite size because extra energy is required to create the expanding surface (Eqs. \eqref{ZEqnNum267268} and \eqref{ZEqnNum212204}). An effective exponent $n_{e}$ of gravitational interaction is discovered with an estimate value of $n_{e} =-1.3$ (deviate from the usual exponent -1.0 for gravity) to reflect the effects of mass cascade and surface energy of halos (Eq. \eqref{ZEqnNum797157}).
Halos are dynamically evolving due to inverse mass cascade. New stochastic models are formulated for the random evolution of halo size that follows a geometric Brownian motion (Eq. \eqref{ZEqnNum251003}). As a result, halo size follows a lognormal distribution (Eq. \eqref{ZEqnNum640133}). Stochastic models are also developed for the random motion of collisionless particles in halos with random size (Eqs. \eqref{ZEqnNum544038} and \eqref{ZEqnNum855118}). This model involves drift terms including both mean radial and osmotic flow ($u_{h} $ and $u_{h}^{*} $) and a multiplicative noise term due to the random halo size. The solution of that model leads to the relation between particle density distribution $P_{r} $ (the probability to find a particle at a given position) and the mean radial and osmotic flow (Eqs. \eqref{ZEqnNum533047} and \eqref{ZEqnNum961031}). It is demonstrated that the closure problem of halo density profile can be equivalently reduced to the correct modeling of either mean flow $u_{h} $, or osmotic flow $u_{h}^{*}$, or identifying an additional closure between $u_{h}$ and $u_{h}^{*}$ besides Eq. \eqref{ZEqnNum687681}. A simple closure between $u_{h} $ and $u_{h}^{*} $ is proposed for a self-consistent particle distribution function in Appendix \ref{appendix:b} (Eq. \eqref{eq:B7}).
In this work, a simple model of osmotic flow $u_{h}^{*} $ (Eq. \eqref{ZEqnNum927116}) is proposed such that the radial particle distribution function $P_{r} $ can be fully derived (Eq. \eqref{ZEqnNum103079}), as well as other relevant halo properties, including Eq. \eqref{ZEqnNum495681} for particle density distribution, Eq. \eqref{ZEqnNum980800} for radial flow, and Eq.\eqref{ZEqnNum105452} for shifted potential. The proposed model provides an excellent fit to the cumulative function of particle density $F_{r}$ that is computed for composite halos (group of halos of same sizes) from a N-body simulation. The model agrees with a very wide range of halo group sizes, where a central halo core exists with $F_{r} \left(x\right)\sim x^{3} $ due to the Hubble radial flow at halo center (Fig. \ref{fig:6}). With reference pressure and density defined at halo center where both gravitational and pressure forces are absent, equation of state for relative pressure and density is established based on this model (Fig. \ref{fig:9} and Eqs. \eqref{ZEqnNum609386} and \eqref{ZEqnNum805925}). The pressure, density, and velocity dispersion at halo center are also presented (Fig. \ref{fig:10} and Eqs. \eqref{ZEqnNum520886} and \eqref{ZEqnNum218090}).
In short, inverse mass cascade is a fundamental feature of SG-CFD. Its effects on the structure formation and evolution remain an important topic. Some examples of future work are briefly discussed here. The concentration-mass relation (the mass dependence of \textit{c}) might be related to the mass dependence of $\alpha _{h}$ and/or $\lambda$, with Eq. \eqref{ZEqnNum412141} providing a relation between concentration $c$, deformation parameter $\alpha _{h} $ and mass cascade parameter $\lambda $ and $\tau _{0} $. Further study is also desired to identify a better closure for better understanding of the origin of universal halo structures.
\section*{Data Availability}
The data underlying this article are available on Zenodo \citep{Xu:2022-Dark_matter-flow-dataset-part1,Xu:2022-Dark_matter-flow-dataset-part2,Xu:2022-Dark_matter-flow-and-hydrodynamic-turbulence-presentation}. All data files are also available on GitHub \citep{Xu:Dark_matter_flow_dataset_2022_all_files}.
\bibliographystyle{mnras}
|
1,108,101,565,642 | arxiv | \section{Introduction}
The electromagnetic signature of core-collapse supernovae has been
exploited comprehensively and thoroughly by countless observations
during the past decades, providing only indirect information about the
explosion mechanism, however. The up to now only recorded neutrino
signal of a core-collapse supernova (SN1987A) confirmed the idea that
the collapse of the core of a massive star to neutron star densities
provides the necessary energy for the explosion \citep{BaadeZwicky34}.
Because gravitational waves (GW), the only other means to probe the
supernova engine besides neutrinos, are yet to be detected, supernova
modelers are preparing for this prospective measurement by predicting
the gravitational wave signature of core-collapse supernovae with ever
increasing realism \citep[for reviews, see {\it e.g.,}\,][]{Kotake_etal06,
Ott09, FryerNew11}.
According the Einstein's theory of general relativity (GR),
gravitational waves will be generated by any mass-energy distribution
whose quadrupole (or higher) moment varies non-linearly with time. In
core-collapse supernovae this criterion is satisfied by time-dependent
rotational flattening particularly during collapse and bounce, prompt
post-shock convection, non-radial flow inside the proto-neutron star
and in the neutrino-heated hot bubble, the activity of the standing
accretion shock instability (SASI), asymmetric emission of neutrinos,
and by asymmetries associated with the effects of magnetic fields
\citep[for a recent review see, {\it e.g.,}\,][and references therein]{Ott09}.
While significant rotational deformation and dynamically relevant
magnetic fields require particular progenitors that possess some
(considerable) amount of rotational and magnetic energy or that must
rotate fast and differentially (additional differential rotation
develops during collapse) to amplify an initially weak magnetic field,
all other processes are genuinely operative in any core-collapse
supernova.
Obviously, an adequate modeling of these effects and an accurate
prediction of the GW signal ultimately requires three dimensional (3D)
hydrodynamic simulations covering the collapse, bounce, and
post-bounce evolution (at least) until a successful launch of the
explosion and including a proper treatment of neutrino transport and
the relevant microphysics. However, most studies of the past thirty
years were either concerned with the collapse and bounce signal only
\citep{Mueller82, FinnEvans90, Moenchmeyer_etal91, YamadaSato94,
ZwergerMueller97, Rampp_etal98, Dimmelmeier_etal01,
Dimmelmeier_etal02, Kotake_etal03, Shibata03, ShibataSekiguchi04,
Ott_etal04, Cerda-Duran_etal05, Saijo05, ShibataSekiguchi05,
Kotake_etal06, Dimmelmeier_etal07, Ott_etal07, Dimmelmeier_etal08},
or were restricted to axisymmetric (2D) models \citep{Mueller_etal04,
Ott_etal06, Kotake_etal07, Marek_etal09, Murphy_etal09,
Yakunin_etal10}. Several authors also investigated the influence of
magnetic fields on the GW signal during the collapse and early
post-bounce evolution assuming axisymmetry \citep{Kotake_etal04,
YamadaSawai04, Kotake_etal05, Obergaulinger_etal06a,
Obergaulinger_etal06b} and no symmetry restriction at all
\citep{Scheidegger_etal08, Scheidegger_etal10}. The GW signal caused
by aspherical neutrino emission was first studied by \citet{Epstein78}
and subsequently by \citet{BurrowsHayes96}, \citet{MuellerJanka97},
and \citet{Kotake_etal07, Kotake_etal09a, Kotake_etal09b,
Kotake_etal11}, where the investigations by \citet{ MuellerJanka97}
and \citet{Kotake_etal09b, Kotake_etal11} considered also 3D, {\it i.e.,}\,
non-axisymmetric models.
Concerning 3D post-bounce models, the topic of the study presented
here, \citet{MuellerJanka97} were the first to analyze the GW
signature of 3D non-radial flow and anisotropic neutrino emission from
prompt post-bounce convection in the outer layers of a proto-neutron
star during the first 30\,msec after supernova-shock formation.
Because of smaller convective structures and slower overturn
velocities, the GW wave amplitudes of their 3D models are more than a
factor of 10 lower than those of the corresponding 2D ones, and the
wave amplitudes associated with anisotropic neutrino emission are a
factor of 10 higher than those caused by non-radial gas flow.
\citet{Fryer04} and \citet{Fryer_etal04} presented gravitational wave
signals obtained from 3D core-collapse simulations that extend up to
150\,msec past bounce and were performed with a Newtonian smoothed
particle hydrodynamics code coupled to a three-flavor gray
flux-limited diffusion neutrino transport scheme. Gravitational wave
emission occurs owing to matter asymmetries that arise from
perturbations caused by precollapse convection, core rotation, and
low-mode convection in the explosion engine itself, and owing to
anisotropic neutrino emission. \citet{Kotake_etal09b} simulated 3D
mock-up models that mimic neutrino-driven explosions aided by the
SASI, and computed the GW signal resulting from anisotropic neutrino
emission by means of a ray-tracing method in a post-processing
step. They pointed out that the gravitational waveforms of 3D models
vary much more stochastically than those of axisymmetric ones, {\it i.e.,}\, in
3D the GW signals do not possess any template character. However,
when considering rotating models, \citet{Kotake_etal11} argue that the
GW waveforms resulting from neutrino emission exhibit a common
feature, which results from an excess of neutrino emission along the
spin axis due to the growth of spiral SASI modes.
\citet{Scheidegger_etal08} simulated the collapse of two rotating and
magnetized cores in 3D until several 10\,msec past bounce, applying a
parametrized deleptonization scheme \citep{Liebendoerfer05} that is a
good approximation until a few milliseconds past bounce.
\citet{Scheidegger_etal10} extended their study by systematically
investigating the effects of the equation of state, the initial
rotation rate, and both the initial magnetic field strength and
configuration on the GW signature. They also simulated three
representative models until $\sim\,$200\,msec of post-bounce accretion
(no development of explosions) incorporating a treatment for neutrino
transport based on a partial implementation of the isotropic diffusion
source approximation \citep{Liebendoerfer_etal09}.
In the following we present the GW analysis of a set of parametrized
3D models of neutrino-powered supernova explosions covering the
evolution from core bounce until $\sim\,$1.4\,s later, where the
high-density inner core of the proto-neutron star (PNS) is excised and
replaced by a time-dependent boundary condition and a central point
mass. The neutrino transport is treated by an approximate solver based
on a ray-by-ray treatment of the multi-dimensional effects
\citep{Scheck_etal06}. Bceause we analyze the GW signal arising from
both non-radial mass flow and anisotropic neutrino emission, we
discuss the neutrino emission of these 3D models as well, and
particularly address its multidimensional properties, some of which
have previously been investigated in 2D by \citet{JankaMoenchmeyer89a,
JankaMoenchmeyer89b}, \citet{Ott_etal08}, \citet{Kotake_etal09a},
\citet{MarekJanka09}, \citet{Marek_etal09}, and \citet{Brandt_etal11}.
Based on 2D simulations of rotational core-collapse,
\citet{JankaMoenchmeyer89a, JankaMoenchmeyer89b} pointed out that the
energy output in neutrinos seen by an observer may vary as much as a
factor of 3 with his inclination angle relative to the rotation axis,
while for the neutrino energy much smaller angular variations are to
be expected. \citet{Marek_etal09} and \cite{MarekJanka09} found that
neutrino luminosities and mean energies show quasi-periodic time
variability in their 2D simulations of non-rotating and slowly
rotating 15\,$M_\odot$ progenitors covering up to $\sim$700\,ms past
bounce. Caused by the expansion and contraction of the shock in the
course of SASI oscillations, the level of the fluctuations is
$\la$50\% for the luminosities and roughly 1\,MeV for the mean
neutrino energies. The luminosity fluctuations are somewhat bigger for
$\nu_{\rm e}$ and $\bar\nu_{\rm e}$ than for heavy-lepton neutrinos.
The neutrino quantities also depend on polar angle as a consequence of
the preference of the SASI motions along the symmetry axis of the 2D
models. Additional short-wavelength spatial variations of the average
radiated energies and of the neutrino fluxes are caused by local
downdrafts of accreted matter. This is in accordance with the findings
of \citet{MuellerJanka97}, who inferred from a post-processing
analysis of the neutrino emission anisotropy that features in the GW
signal of their 2D models of convection in the hot-bubble region are
well-correlated with structures in the neutrino signal. The features
are associated with sinking and rising lumps of matter and with
temporal variations of aspherical accretion flows toward the
proto-neutron star. \citet{Kotake_etal09a} computed neutrino
anisotropies with a ray-tracing scheme by post-processing their 2D
SASI models and derived analytical expressions for evaluating GW
signals for neutrinos in 3D models, too. A generalization of these
expressions will be presented below. \citet{Brandt_etal11} performed
2D multi-group, multi-angle neutrino transport simulations for both a
non-rotating and rapidly rotating 20\,$M_\odot$ model extending
$\sim$400\,ms beyond bounce. Their simulations predict that the
neutrino radiation fields vary much less with angle than the matter
quantities in the region of net neutrino heating because most
neutrinos are emitted from deeper radiative regions and because the
neutrino energy density combines the specific intensities as integrals
over sources at many angles and depths. The rapidly rotating model
exhibits strong, flavor-dependent asymmetries in both peak neutrino
flux and light curves, the peak flux and decline rate having
pole-equator ratios $\la$3 and $\la$2, respectively.
\citet{Brandt_etal11} also provide estimates of the detectability of
neutrino fluctuations in IceCube and Super-Kamiokande as previously
done by \citet{Lund_etal10} on the basis of the \citet{Marek_etal09}
non-rotating models.
The paper is organized as follows: in Section\,2 we discuss the
numerical methods, the input physics, and the properties of the
progenitor models and the set of 3D simulations that we analyzed.
Section\,3 contains a description of the formalism we used to extract
the observable neutrino signal of our 3D models, and a discussion of
some of its properties relevant for the corresponding GW signal. In
Section\,4 we give the formalism necessary to calculate the GW
signature of 3D non-radial flow and anisotropic neutrino emission, and
discuss the GW signature of the investigated 3D models. Finally, in
Section\,5 we summarize our results and discuss shortcomings and
possible implications of our study.
\section{Model setup}
\subsection{Code and computational grid}
\label{subsec:codegrid}
The 3D supernova models we analyzed for their neutrino and GW
signature have been simulated with the explicit finite-volume,
Eulerian, multi-fluid hydrodynamics code {\sc Prometheus}
\citep{PROMET1,PROMET2,PROMET3}. This code integrates the
multidimensional hydrodynamic equations using the dimensional
splitting method of \citet{Strang68}, the piecewise parabolic method
of \citet{PPM}, and a Riemann solver for real gases proposed by
\citet{CollelaGlaz85}. Inside grid cells with strong grid-aligned
shocks fluxes computed from the Riemann solver are replaced by the
AUSM+ fluxes of \citet{Liou96} in order to prevent odd-even decoupling
\citep{Quirk94}. Nuclear species are advected using the consistent
multi-fluid advection (CMA) scheme of \citet{CMA}.
The simulation code employs an axis-free overlapping ``Yin-Yang'' grid
\citep{YinYang} in spherical polar coordinates, which was recently
implemented into {\sc Prometheus}, for spatial discretization
\citep{Wong_etal10a}. The Yin-Yang grid relaxes the CFL-timestep
condition and avoids numerical artifacts near the polar axis.
Concerning the ray-by-ray neutrino transport no special procedure
needs to be applied for the Yin-Yang grid. The (scalar) quantities
involved in the transport algorithm are computed on both the Yin and
Yang grid and are then linearly interpolated as any other scalar
quantity.
The grid consists of $400(r) \times 47(\theta) \times 137(\phi)
\times2$ cells corresponding to an angular resolution of $2^\circ$ and
covers the full $4\pi$ solid angle. The radial grid has an
equidistant spacing of 0.3\,km from the inner grid boundary out to $r
= 80\,$km (models W15 and N20; see Table\,1) or 115\,km (model L15;
see Table\,1), respectively. Beyond this radius the radial grid is
logarithmically spaced. The outer radial grid boundary $R_\mathrm{ob}$
is at 18000\,km, which is sufficiently far out to prevent the
supernova shock from leaving the computational domain during the
simulated epoch. This radial resolution suffices the requirement that
there are always more than 15 radial zones per decade in density.
A central region, the dense inner core of the proto-neutron star (PNS)
at $\rho \ga 10^{12\ldots13}\,$gcm$^{-3}$, is excised from the
computational domain and replaced by an inner time-dependent radial
boundary condition and a point mass at the coordinate origin. The
radius of the inner radial boundary shrinks according to Eq.(13) of
\citet{Scheck_etal08}. For the W15 and N20 models the initial and
final (asymptotic) boundary radii are $R^\mathrm{i}_\mathrm{ib} =
65\,$km and $R^\mathrm{f}_\mathrm{ib} = 15\,$km, respectively. For
the L15 models the corresponding radii are 82\,km and 25\,km,
accounting for a less extreme contraction of the neutron star within
the simulation time. The timescale for the contraction is
$t_\mathrm{ib} = 1\,$sec for all models. This choice of parameters
implies $R_\mathrm{ib} \approx 19\,$km at 1.3\,s for models W15 and
N20, and $R_\mathrm{ib} \approx 30\,$km at 1.4\,s for the L15 models.
Hydrostatic equilibrium is assumed at the inner radial grid boundary
$R_\mathrm{ib}$, which is thus a Lagrangian (co-moving) position,
while a free outflow boundary condition is employed at the outer
radial grid boundary \citep[for more details, see][]{Wong11,
Wong_etal10a}.
\subsection{Input physics}
\label{subsec:physics}
Self-gravity is fully taken into account by solving Poisson's equation
in integral form using an expansion into spherical harmonics as in
\citet{MuellerSteinmetz95}. The monopole term of the potential is
corrected for general relativistic effects as described in
\citet{Scheck_etal06} and \citet{Arcones_etal07}. The cooling of the
PNS is prescribed by neutrino emission properties (luminosities and
mean neutrino energies) as boundary condition at the inner radial grid
boundary \citep[for details, see][]{Scheck_etal06}. The contraction of
the PNS is mimicked by the movement of the inner radial grid boundary
(see Sect.\,\ref{subsec:codegrid}). ``Ray-by-ray'' neutrino transport
and neutrino-matter interactions are approximated as in
\citet{Scheck_etal06} by radial integration of the one-dimensional
(spherical), gray transport equation for all angular grid directions
($\theta$,\,$\phi$) independently. This approach allows for angular
variations of the neutrino fluxes. The tabulated equation of state
(EoS) of \citet{JankaMueller96} is used to describe the stellar
fluid. It includes arbitrarily degenerate and arbitrarily relativistic
electrons and positrons, photons, and four predefined nuclear species
(n, p, $\alpha$, and a representative Fe-group nucleus) in nuclear
statistical equilibrium.
\begin{table}
\caption{Some properties of the analyzed 3D models including the mass
of the progenitor star $M_\mathrm{PS}$, the mass of the neutron star
$M_\mathrm{NS}$, the time of explosion $t_\mathrm{exp}$, and the
explosion energy $E_\mathrm{exp}$ at the time $t_\mathrm{f}$ after
bounce when we stopped the simulation. Note that $1\,\mathrm{B} =
1\,\mathrm{bethe} = 10^{51}\,$erg.}
\centering
\begin{tabular}{lccccc}
\hline\hline\multirow{2}{*}{Model}
& $M_\mathrm{PS}$ & $M_\mathrm{NS}$ & $t_\mathrm{exp}$
& $E_\mathrm{exp}$ & $t_\mathrm{f}$ \\
& [$M_\odot$] & [$M_\odot$] & [ms]
& [B] & [s]\\
\hline
W15-2 & 15 & 1.37 & 248 & 1.13 & 1.3 \\
W15-4 & 15 & 1.38 & 272 & 0.94 & 1.3 \\
L15-2 & 15 & 1.51 & 381 & 1.74 & 1.4 \\
L15-3 & 15 & 1.62 & 477 & 0.84 & 1.4 \\
N20-2 & 20 & 1.28 & 265 & 3.12 & 1.3 \\
\hline
\end{tabular}
\label{tab:models}
\end{table}
\subsection{Models}
\label{subsec:models}
We have analyzed a set of 3D simulations \citep{Wong11, Wong_etal10b,
Wong_etal11} based on two 15\,$M_\odot$ progenitor models (W15 and
L15), and a 20\,$M_\odot$ progenitor model (N20). The W15 model is
obtained from the non-rotating 15\,$M_\odot$ progenitor s15s7b2 of
\citet{WoosleyWeaver95}, the L15 model from a star evolved by
\citet{Limongi_etal00}, and the N20 model from a SN\,1987A progenitor
of \citet{ShigeyamaNomoto90}. The progenitor models were evolved
through collapse to 15\,ms after bounce with the {\sc
Prometheus-Vertex} code in one dimension (A.\,Marek and R.\,Buras,
private communication) providing the initial models for the 3D
simulations. To break the spherical symmetry of the initial models,
random seed perturbations of 0.1\% amplitude are imposed on the radial
velocity ($v_r$) field at the beginning of the 3D
simulations. Explosions are initiated by neutrino heating at a rate
that depends on suitable values of the neutrino luminosities imposed
at the lower boundary chosen such that the desired value of the
explosion energy is obtained. The evolution is followed until 1.3\.s
after bounce for the W15 and N20 progenitor models, while the L15
models are simulated until 1.4\,s post-bounce. The GW analysis
presented below comprises five models (see Table\,1), where models W15-2
and W15-4 differ only by the initial seed perturbations. The
explosion energies, $E_\mathrm{exp}$, given in Table\,1 are
instantaneous values at the end of the simulations ($t =
t_\mathrm{f}$), adding up the total energies (kinetic + internal +
gravitational) in all zones where the sum of these energies is
positive. The explosion time, $t_\mathrm{exp}$, is defined as the time
when this sum exceeds a value of $10^{48}\,$erg, roughly corresponding
to the time when the average shock radius is 400 to 500\,km (see,
however, \citet{PejchaThompson11} for an alternative definition of the
time of the onset of the explosion).
\section{Neutrino signal}
\label{sec:neutrino-signal}
The non-radial motions caused by the SASI and convection in the
neutrino-heated hot-bubble as well as by convection inside the
proto-neutron star (driven by Ledoux unstable lepton gradients) give
rise to a time-dependent, anisotropic emission of neutrinos of all
flavors, and thus to the emission of gravitational waves
\citep{Epstein78, BurrowsHayes96, MuellerJanka97, Kotake_etal07,
Kotake_etal09a, Kotake_etal09b}, as discussed in
Sect.\,\ref{sec:gw-signal}. We have analyzed this emission for the 3D
models discussed above (see Sect.\,\ref{subsec:models}), particularly
addressing its multidimensional properties.
\subsection{Formalism}
\label{subsec:nuform}
To derive \emph{observable luminosities} of an emitting source we
consider an observer located at far distance $D$ from that source
(see Fig.\,\ref{fig:applum}). According to definition the flux
measured by the observer is given by the following integral at the
position of the observer:
\begin{equation}
F(D,t) = \oint {\mathrm d} \omega\, \mu\, I(D, {\vec \omega}, t) \, ,
\label{eq:obsflux}
\end{equation}
where $\mu$ is the cosine of the angle between the direction of the
radiation and the line of sight (between the observer and the center
of the source), $\vec \omega$ denotes the radiation direction at the
observer's location (defined by a pair of angles), and ${\mathrm d}
\omega$ is the solid angle element around the radiation direction
$\vec \omega$. The intensity $I$ adopts non-zero values within the
opening angle subtended by the emitting surface (not necessarily a
sphere).
We note that here and in the following we suppress the
dependence of the intensity on the neutrino energy and assume that
energy-integrated quantities are considered (the outlined formalism,
however, is valid also for an energy-dependent treatment). The
integration over $\vec \omega$ at the observer's location can be
substituted by an integration over the emitting surface of the source,
because the radiation intensity is constant along rays, {\it i.e.,}\,
\begin{equation}
I(D, {\vec \omega}, t) = I_o({\vec R}_o, {\vec \omega}_o, t)
\label{eq:radintens}
\end{equation}
for any ray arriving at the observer from the source (and zero
otherwise), where ${\vec R}_o$ denotes the position of a surface
element of the emitting surface in the coordinate frame of the source
and ${\vec \omega}_o$ the direction of the radiation field at that
position toward the observer. Note that we ignore in
Eq.\,(\ref{eq:radintens}) the trivial effect that the time $t$ for
$I_o$ relative to the time for $I$ is subject to a retardation.
Moreover, in the following we disregard spectral and angular
corrections that may be relevant when the emitting surface is in
relative motion to the observer or sitting deep in the gravitational
potential of a compact star (in which case general relativistic (GR)
effects like redshifting and ray bending would be important).
Considering the source to be at rest is a good assumption for the
neutrinospheric region in the supernova core after bounce (the
velocities of mass motions in this layer are unlikely to be higher
than some 1000\,km/s, {\it i.e.,}\, at most around one percent of the speed of
light), while GR energy redshift is certainly of relevance on the
$\sim$10--20\% level during the proto-neutron star cooling phase ($t
\gtrsim 1$\,s after bounce), but much lower in the accretion and
shock revival phases, when the forming neutron star is still
considerably less compact than the final remnant.
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics{applum.eps}}
\caption{Sketch illustrating the various quantities involved when
deriving the observable luminosity of a radiating source, whose
visible surface is shaded in blue. When the observer is at infinity,
the emitting surface is a sphere, and $\mathbf{R}_o$ is measured
from the origin of this sphere, one obtains $\alpha = \gamma$.}
\label{fig:applum}
\end{figure}
For a distant observer $D \gg \mathrm{max} \lbrace |{\vec R}_o|
\rbrace $ holds, {\it i.e.,}\, the value of $\mu$ is very close to one for the
whole emitting surface. Denoting the solid angle subtended by a
surface element of the emitting surface by ${\rm d} \omega$, we have
${\rm d} \omega = {\rm d} A_{\perp} / D^2$, where ${\rm d} A_{\perp} =
\cos\gamma\, {\rm d} A$ is the projected area of this surface element
perpendicular to the line of sight, when $\gamma$ is the angle between
the normal unit vector ${\vec n}_A$ of the emitting surface element
${\rm d} A$ and the line of sight (see Fig.\,\ref{fig:applum}), but
specified to the case $D \gg |{\vec R}_o|$. Hence, we obtain for the
observable luminosity the expression
\begin{equation}
L_o(t) = 4\pi D^2 F(D,t) = 4\pi \int_\mathrm{vis. surf.}
{\rm d} A\, \cos\gamma\,
I_o ({\vec R}_o, {\vec \omega}_o, t) \, .
\label{eq:app-lum0}
\end{equation}
In order to evaluate the integral in Eq.\,(\ref{eq:app-lum0}), one
needs to know the intensity $I_o$ as a function of energy, emission
direction, and time at every point of the radiating surface of the
source. Determining $I_o({\vec R}_o, {\vec \omega}_o, t)$ in general
requires calculating full-scale neutrino transport. With this quantity
as the solution of the transport problem at hand,
Eq.\,(\ref{eq:app-lum0}) can be evaluated directly (in general with
energy dependence) by performing the integration over {\em any}
surface that encloses the radiation (neutrino) emitting source and
that lies outside the volume where radiation interacts with matter
({\it i.e.,}\, the intensity $I_o$ in all points on the chosen surface must be
given in the reference frame of the observer and must be preserved on
the way from the emission point to the observer).
Our ray-by-ray transport approximation, however, yields only the local
neutrino energy density $E({\vec R}_o, t)$ and the neutrino flux
density $F({\vec R}_o, t)$. To estimate the neutrinos radiated from
every point of the neutrinosphere to the observer, we therefore have to
develop an approach that yields a reasonable representation of the
direction-dependent intensity as function of the quantities delivered
by our transport approximation
\footnote{Note that the procedure described in the following does not
depend on whether the transport is performed in the gray
approximation or is energy dependent. We therefore suppress the
energy variable in all transport quantities and introduce mean
energies in our gray treatment instead of considering neutrino
energies as a function of spectral frequencies.}.
To this end, we assume that the neutrino distribution is axisymmetric
around the normal vector ${\vec n}_A$ at all points ${\vec R}_o$. This
implies that the direction dependence of the intensity $I_o$ is
described by the direction angle $\gamma$ only (see
Fig.\,\ref{fig:applum}), {\it i.e.,}\, $I_o = I_o({\vec R}_o, \gamma, t)$, and
that the flux direction is given by ${\vec n}_A$. Assuming further
that $I_o(\gamma)$ can be approximated by the lowest two terms of an
expansion in spherical harmonics (as in the diffusion approximation),
one can write
\begin{equation}
I_o ({\vec R}_o, \gamma, t) = \frac{F_o ({\vec R}_o, t)}{2\pi}
\left( 1 + \frac{3}{2} \cos\gamma \right) \, .
\label{eq:app-int}
\end{equation}
Because the radiation flux density is defined as the first angular
moment of the intensity, one can easily verify that the numerical
coefficient 3/2 of the dipole term ensures that the flux density
$F_o({\vec R}_o,t)$ (normal to the emitting surface element
$\mathrm{d}A$) is given by $\int_0^{2\pi}\mathrm{d}\varphi \int_0^1
\mathrm{d} \cos\gamma\, \cos\gamma I_o({\vec R}_o,\gamma,t)$, if the
radiating surface does not receive any incident neutrinos from outside
(i.e., $I_o({\vec R}_o, \gamma, t) = 0$ for $\cos\gamma < 0$)
\footnote{Note that the requirement $I_o\ge 0$ implies that
Eq.\,(\ref{eq:app-int}) is valid in the whole range of $\cos\gamma
\ge -\frac{2}{3}$, which includes inward going radiation for
$\cos\gamma < 0$. Extending the integration over all directions of
validity, one obtains $\frac{25}{27}F_o = \int_0^{2\pi} \mathrm{d}
\varphi \int_{-2/3}^1 \mathrm{d} \cos\gamma\, \cos\gamma I_o \approx
F_o$ and $E_o = \int_0^{2\pi} \mathrm{d} \varphi\int_{-2/3}^1
\mathrm{d} \cos\gamma\, I_o = \frac{35}{12} F_o/c \approx 3
F_o/c$. This means that we have $F_o \approx \frac{1}{3} cE_o$,
which is a reasonably good approximation of the relation between
flux density and energy density at the neutrinosphere, where one
typically obtains $f_o \equiv F_o/(E_oc) \approx \frac{1}{4} \dots
\frac{1}{3}$ for the flux factor at an optical depth between unity
and about 2/3 in detailed neutrino transport calculations in
spherical symmetry \citep{Janka91}. The ansatz of
Eq.\,(\ref{eq:app-int}) is therefore consistent with basic
properties of the neutrinospheric emission. Moreover, we note that
the expression corresponds to the limb-darkening law $I_E
(\cos\gamma) / I_E (1) = (2/5) (1 + 3/2\times \cos\gamma)$ that can
be derived on grounds of the Eddington approximation (see, {\it e.g.,}\,,
\citet{Mihalas78}, page 61, or \citet{MorseFeshbach53}, page 187).}.
Inserting Eq.\,(\ref{eq:app-int}) into Eq.\,(\ref{eq:app-lum0}), we
find for the observable neutrino luminosity the expression
\begin{equation}
L_o(t) = 2\, \int_\mathrm{vis. surf.} {\rm d} A\,
\cos\gamma\, F_o ({\vec R}_o,t)\,
\left (1 + \frac{3}{2} \cos\gamma \right) \, .
\label{eq:app-lum}
\end{equation}
We further define an \emph{observable mean neutrino energy} according
to
\begin{equation}
\langle E \rangle_o\, (t) \,=\, \frac{L_o (t)}{L_{n,o}(t)} \, ,
\label{eq:enue_mean}
\end{equation}
where
\begin{equation}
L_{n,o} (t) = 2\, \int_\mathrm{vis. surf.} {\rm d} A\, \cos\gamma\,
\frac{F_o ({\vec R}_o, t)}{\epsilon ({\vec R}_o,t)} \,
\left (1 + \frac{3}{2} \cos\gamma \right)
\label{eq:lnum}
\end{equation}
is the observable \emph{neutrino number flux} with $\epsilon$ being
the mean energy of the neutrino energy spectrum radiated from point
${\vec R}_o$.
Our 3D radiation hydrodynamics code computes the time-dependent
\emph{neutrino energy flux density}, $F_o ({\vec R}_o, t)$, and
\emph{neutrino number flux density}, $F_{n,o} ({\vec R}_o,t)$, through
a sphere of radius $R_o = |{\vec R}_o|$ in dependence of the angular
position ${\vec \Omega} \equiv (\theta, \phi)$, but actually stores
the related quantities
\begin{equation}
\Lambda ({\vec \Omega},t) \equiv 4\pi R^2_o\,
F_o (R_o, {\vec \Omega},t) \, ,
\label{eq:lambdae}
\end{equation}
and
\begin{equation}
\Lambda_n ({\vec \Omega},t)
\equiv \frac{\Lambda ({\vec \Omega},t) }{ \epsilon ({\vec \Omega},t)}
= 4\pi R^2_o\, F_{n,o} (R_o, {\vec \Omega},t) \, ,
\label{eq:lambdan}
\end{equation}
because these quantities are constant in the ray-by-ray approximation
of the free streaming region. In this approximation, both the neutrino
energy flux and the neutrino number flux are purely radial.
Using Eqs.\,(\ref{eq:lambdae}) and (\ref{eq:lambdan}), and the fact
that ${\rm d} A = R^2_o {\rm d} \Omega$ with ${\rm d}\Omega =
\sin\theta {\rm d}\theta {\rm d} \phi$ for the special case of an
emitting sphere of radius $R_o$, we can rewrite the general expression
for the observable neutrino luminosity given in
Eq.\,(\ref{eq:app-lum}) in the form
\begin{equation}
L_o(t) = \frac{1}{2\pi}\, \int_\mathrm{vis. hem.} {\rm d} \Omega\,
\cos\gamma\, \Lambda ({\vec \Omega},t)\,
\left (1 + \frac{3}{2} \cos\gamma \right)
\label{eq:app-lum_code}
\end{equation}
and that of the observable neutrino number flux given in
Eq.\,(\ref{eq:lnum}) in the form
\begin{equation}
L_{n,o} (t) = \frac{1}{2\pi}\, \int_\mathrm{vis. hem.} {\rm d} \Omega\,
\cos\gamma\, \Lambda_n ({\vec \Omega}, t)\,
\left (1 + \frac{3}{2} \cos\gamma \right) \, ,
\label{eq:lnum_code}
\end{equation}
where in both cases the integration is performed over the visible
hemisphere.
\footnote{Using of the ray-by-ray approximation has the advantage that
the evaluation of the integrals on the rhs of Eqs.\,\ref{eq:app-lum}
and \ref{eq:lnum} does not require the specification of a suitable
surface, but can be done on any sphere outside the
neutrino-decoupling region, as Eqs.\,\ref{eq:app-lum_code} and
\ref{eq:lnum_code} are independent of ${\vec R}_o$. }
For the evaluation of the gravitational wave amplitude in
Sect.\,\ref{subsec:gw-form-nu} we will also need the quantity
\begin{equation}
\frac{ {\rm d}\Lambda }{ {\rm d}\Omega} ({\vec \Omega},t)
\equiv F_o (R_0, {\vec \Omega},t)\, R^2_o
\label{eq:dlamdom}
\end{equation}
and the corresponding angle-integrated quantity
\begin{equation}
\Lambda (t) \equiv \oint_\mathrm{surf.} {\rm d}\Omega\,
\frac{ {\rm d}\Lambda }{ {\rm d}\Omega} ({\vec \Omega},t)
= \frac{1}{4\pi}
\oint_\mathrm{surf.} {\rm d} \Omega\,
\Lambda ({\vec \Omega}, t) \, .
\label{eq:lamt}
\end{equation}
For the later discussion of the results we finally define the
\emph{surface-averaged neutrino flux density}
\begin{equation}
\langle F_o \rangle\, (t)
\equiv \frac{1}{4\pi} \oint_\mathrm{surf.}
{\rm d}\Omega\, F_0 (R_0, {\vec \Omega},t)
\equiv \frac{1}{4\pi R^2_o}\, \frac{{\rm d}{\mathcal E}(t)}{{\rm d} t}
\, ,
\label{eq:flxave}
\end{equation}
where
\begin{equation}
\frac{{\rm d}{\mathcal E}(t)}{{\rm d} t} = \oint_\mathrm{surf.}
{\rm d}\Omega\, \frac{ {\rm d}\Lambda }{ {\rm d}\Omega} ({\vec \Omega},t)
= \Lambda (t)
\label{eq:eloss}
\end{equation}
is the total energy loss rate at time $t$ from the supernova core to
all directions, which (because of the flux variations over the sphere)
is no directly observable quantity.
We have also analyzed the evolution of the neutrino flux asymmetry by
calculating the angular pseudo-power spectrum of the neutrino energy
flux variation
\begin{equation}
\delta_\Lambda ({\vec \Omega},t) \equiv
\frac{ \Lambda ({\vec \Omega},t) - \Lambda(t) }{ \Lambda(t) } \, ,
\label{eq:dlam}
\end{equation}
where $\Lambda ({\vec \Omega},t)$ and $\Lambda(t)$ are defined in
Eqs.\,(\ref{eq:lambdae}) and (\ref{eq:lamt}), respectively. The
pseudo-power spectrum is given by the decomposition of $\delta_\Lambda
({\vec \Omega},t)$ in spherical harmonic coefficients
\begin{equation}
\Lambda_{lm}(t) = \oint {\rm d} \Omega\, \delta_\Lambda ({\vec \Omega},t)
Y^\ast_{lm} (\Omega) \, ,
\label{eq:lamlm}
\end{equation}
where $Y^\ast_{lm} (\Omega)$ is the respective (complex conjugate)
spherical harmonics. For our mode analysis we actually used the
pseudo-power coefficients $C_0 \equiv |\Lambda_{00}|^2$ and
\begin{equation}
C_l \equiv \frac{1}{2l+1} \left( |\Lambda_{l0}|^2 +
2 \sum_{m=1}^{m=l} |\Lambda_{lm}|^2 \right)
\label{eq:cl}
\end{equation}
for $l>0$, respectively.
\begin{figure*}[!]
\centering
\resizebox{0.47\hsize}{!}{\includegraphics*{W15-4_100.ps}}\hspace{1cm}
\resizebox{0.47\hsize}{!}{\includegraphics*{L15-3_100.ps}}\\
\resizebox{0.47\hsize}{!}{\includegraphics*{W15-4_150.ps}}\hspace{1cm}
\resizebox{0.47\hsize}{!}{\includegraphics*{L15-3_230.ps}}\\
\resizebox{0.47\hsize}{!}{\includegraphics*{W15-4_300.ps}}\hspace{1cm}
\resizebox{0.47\hsize}{!}{\includegraphics*{L15-3_500.ps}}\\
\resizebox{0.47\hsize}{!}{\includegraphics*{W15-4_900.ps}}\hspace{1cm}
\resizebox{0.47\hsize}{!}{\includegraphics*{L15-3_900.ps}}\\
\caption{Snapshots of models W15-4 (left) and L15-3 (right)
illustrating the four phases characterizing the evolution of our 3D
models (see text for details). Each snapshot shows two surfaces of
constant entropy marking the position of the shock wave (gray) and
depicting the growth of non-radial structures (greenish). The time
and linear scale are indicated for each snapshot. }
\label{fig:phases}
\end{figure*}
\begin{figure}[htp!]
\centering
\resizebox{0.95\hsize}{!}{\includegraphics*{phases.ps}}\\
\resizebox{0.95\hsize}{!}{\includegraphics*{W15-4_rsh_vs_t.ps}}\\
\resizebox{0.95\hsize}{!}{\includegraphics*{W15-4_lnue_vs_t.ps}}
\caption{Shock radius (top) and total ({\it i.e.,}\, summed over all flavors)
energy loss rate due to neutrinos (bottom) as functions of time for
model W15-4. In the upper panel, the black curve shows the
angle-averaged mean shock radius, the blue (red) curve gives the
maximum (minimum) shock radius, and the vertical dashed line marks
the time of the onset of the explosion as defined in
Sect.\,\ref{subsec:models}. In the lower panel, the blue and red
curves show the time evolution of $\Lambda_\mathrm{max}(\Omega,t)$
and $\Lambda_\mathrm{min}(\Omega,t)$, the maximum and minimum value
of $\Lambda(\Omega,t)$ (Eq.\,\ref{eq:lambdae}) on a sphere of
500\,km radius, respectively. The black line gives the corresponding
surface-averaged value $\Lambda(t)$ (Eq.\,\ref{eq:lamt}). Note that
the luminosities imposed at the inner radial grid boundary are kept
constant during the first second and later are assumed to decay like
$t^{-2/3}$.}
\label{fig:rsh+lnu_vs_t}
\end{figure}
\begin{figure}[!]
\centering
\resizebox{0.95\hsize}{!}{\includegraphics*{L15-3_rsh_vs_t.ps}}\\
\resizebox{0.95\hsize}{!}{\includegraphics*{L15-3_lnue_vs_t.ps}}\\
\resizebox{0.95\hsize}{!}{\includegraphics*{N20-2_rsh_vs_t.ps}}\\
\resizebox{0.95\hsize}{!}{\includegraphics*{N20-2_lnue_vs_t.ps}}
\caption{Same as Fig.\,\ref{fig:rsh+lnu_vs_t} but for models L15-3
(uppermost two panels) and N20-2 (lowermost two panels),
respectively.}
\label{fig:rsh+lnu_vs_t_a}
\end{figure}
\subsection{Results}
\label{subsec:nuresults}
The evolution of our models can be divided into four distinct phases
(Figs.\,\ref{fig:phases}, \ref{fig:rsh+lnu_vs_t}).
\begin{itemize}
\item[(1)] The first phase, the \emph{quasi-spherical shock-expansion
phase} (Fig.\,\ref{fig:phases}, top row), lasts from shock formation
shortly after core bounce to $80\,-\,150$\,msec, when convection
sets in. During this phase the shock rapidly propagates out to a
radius of $\sim\,$200\,km, where its expansion comes to a halt.
\item[(2)] The second phase, the hydrodynamically vigorous
\emph{pre-explosion phase}, comprises the growth of post-shock
convection and of the standing accretion shock instability, SASI
(Fig.\,\ref{fig:phases}, second row from top).
\item[(3)] The \emph{post-explosion accretion phase} begins when
energy deposition by $\nu$-heating in the post-shock layers becomes
sufficiently strong to launch the explosion, and the total energy in
the post-shock region ultimately becomes positive (see
Sect.\,\ref{subsec:models} for a definition). During this phase the
shock accelerates outward while gas is still accreted onto the PNS.
This process is commonly called "shock revival"
(Fig.\,\ref{fig:phases}, third row from top).
Non-radial instabilities during the latter two stages cause
considerable temporal and angular fluctuations of the neutrino
energy flux density as illustrated in Figs.\,\ref{fig:rsh+lnu_vs_t}
- \ref{fig:W15-4_lnue}. Besides the evolution of the shock radius,
the figures show the surface-averaged neutrino light curve
$\Lambda(t)$, {\it i.e.,}\, the total energy loss due to neutrinos versus time
(Eqs.\,\ref{eq:lamt}, \ref{eq:eloss}), together with the time
evolution of the maximum and minimum values of $\Lambda(\Omega,t)$
(Eq.\,(\ref{eq:lambdae}); the numerical evaluation is performed on
an arbitrarily chosen sphere of 500\,km radius). Distinct and
high-amplitude spikes in $\Lambda_\mathrm{max} (\Omega,t)$ are
visible for several 100\,msec and reflect violent post-shock
convection, possible SASI activity, and anisotropic accretion
fluctuations after the onset of the explosion. We have marked the
explosion time $t_\mathrm{exp}$ (see Section\,\ref{subsec:models},
and Table\,1) by a vertical dashed line in
Figs.\,\ref{fig:rsh+lnu_vs_t} and \ref{fig:rsh+lnu_vs_t_a}. The
post-explosion accretion phase lasts until $\sim\,$500\,msec (models
W15-4 and N20-2) or $\sim\,$700\,msec (model L15-3) depending on the
progenitor.
\item[(4)] During the \emph{post-accretion phase}, the fourth and
final phase characterizing the evolution of our models
(Fig.\,\ref{fig:phases}, bottom row), gas infall to the
proto-neutron star has come to an end and the newly formed neutron
star looses mass at a low rate in a nearly spherical neutrino-driven
wind. We find considerably less temporal variability and a lower
level of angular variation ($\la 10\%$) of the neutrino emission
during this fourth phase (Figs.\,\ref{fig:rsh+lnu_vs_t} -
\ref{fig:W15-4_lnue}).
While in model L15-3 the amplitudes of the neutrino emission
fluctuations decrease continuously, the other two models exhibit
growing temporal emission variations (though at a lower level than
the earlier variability) during a later stage (notice the
decrease/increase in $\Lambda_\mathrm{max} - \Lambda_\mathrm{min}$
in Figs.\,\ref{fig:rsh+lnu_vs_t} and \ref{fig:rsh+lnu_vs_t_a}),
which might be considered as a fifth evolutionary phase. This phase
is associated with growing convective activity below the
neutrinosphere. This PNS convection develops more or less strongly
in the different models depending on the location of the
convectively unstable region relative to the inner radial boundary
of our computational domain.
\end{itemize}
\begin{figure}[!]
\centering
\resizebox{0.88\hsize}{!}{\includegraphics*{W15-4_lnue_0170.eps}}\\
\resizebox{0.88\hsize}{!}{\includegraphics*{W15-4_lnue_0200.eps}}\\
\resizebox{0.88\hsize}{!}{\includegraphics*{W15-4_lnue_0342.eps}}\\
\resizebox{0.88\hsize}{!}{\includegraphics*{W15-4_lnue_0600.eps}}\\
\resizebox{0.88\hsize}{!}{\includegraphics*{W15-4_lnue_1300.eps}}
\caption{Neutrino flux asymmetry at 170\,msec, 200\,msec, 342\,msec,
600\,msec, and 1.3\,sec (from top to bottom), respectively. The
$4\pi$-maps show the relative angular variation $\Delta F_o /
\left\langle F_o \right\rangle$ of the total ({\it i.e.,}\, sum of all
neutrino flavors) neutrino energy flux density over a sphere
(normalized to its angular average) for model W15-4. The maximum
value is given in the lower right corner of each panel. Regions of
higher emission are shown in bright yellow, while orange, red,
green, and blue colors indicate successively less emission. Note
that the color scale of each panel is adjusted to the maximum and
minimum values at the corresponding time. The total energy loss
rate due to neutrinos is given in the lower left corner.}
\label{fig:W15-4_lnue}
\end{figure}
\begin{figure*}
\centering
\resizebox{0.33\hsize}{!}{\includegraphics*{W15-4_Cl.ps}}
\resizebox{0.33\hsize}{!}{\includegraphics*{L15-3_Cl.ps}}
\resizebox{0.33\hsize}{!}{\includegraphics*{N20-2_Cl.ps}}
\caption{Pseudo-power spectrogram of the electron-neutrino energy flux
density (top row) for models W15-4 (left), L15-3 (middle), and N20-2
(right), respectively. The panels in the middle row show the
corresponding maximum pseudo-power coefficient $C^\mathrm{max}_l$ as
a function of time, and the panels in the lower row give the
relative angular variation of the electron neutrino flux density
(maximum minus minimum flux density on the sphere divided by the
angle-averaged flux density in percent) with time. }
\label{fig:cl_spectrogram}
\end{figure*}
We have evaluated the time evolution of the neutrino energy flux
asymmetry by producing $4\pi$-maps that show the relative angular
variation $\Delta F_o / \left\langle F_o \right\rangle$ of the total
({\it i.e.,}\, sum of all neutrino flavors) neutrino energy flux density across
a sphere (normalized to the surface-averaged flux density;
Eq.\,\ref{eq:flxave}). Several snapshots of this evolution are shown
for model W15-4 in Fig.\,\ref{fig:W15-4_lnue}. The evolution of the
typical angular scales of the fluctuations is reflected by the
pseudo-power spectrogram of the electron neutrino energy flux
variation (Eq.\,\ref{eq:dlam}) in Fig.\,\ref{fig:cl_spectrogram}, top
panels, which give the color-coded pseudo-power coefficient
distribution normalized to the maximum value versus time. The
variation of the pseudo-power coefficients with angular mode number is
shown in Fig.\,\ref{fig:cl_vs_l} at selected times of 200\,ms (blue),
400\,ms (red), and 1000\,ms (black).
During the quasi-spherical shock expansion phase the level of angular
fluctuations of $F$ is low ($\la 10^{-2}$), while the fluctuation
amplitudes of the total neutrino energy flux density reach a level of
several 10\% during the hydrodynamically vigorous second phase and the
post-explosion accretion phase, where a few distinct regions or even
single spots with an angular size of 10$^\circ$ to 20$^\circ$ dominate
the emission (Fig.\,\ref{fig:W15-4_lnue}, panels 2 and 3). The mode
number $l$ of the dominant angular perturbation scale is of no
relevance during the first phase, as the maximum pseudo-power
coefficient $C^\mathrm{max}_l$ (see Eq.\,\ref{eq:cl}) is tiny $\la
10^{-6}$ (Fig.\,\ref{fig:cl_spectrogram}, middle panels), {\it i.e.,}\, the
dominating $l=2$ and $l=4$ modes visible in the upper panels of
Fig.\,\ref{fig:cl_spectrogram} only reflect tiny angular perturbations
imprinted presumably by the computational grid. When neutrino heating
eventually causes significant non-radial flow during the second and
third phases, $C^\mathrm{max}_l$ rises sharply to a level of $\sim
10^{-3}$ (Fig.\,\ref{fig:cl_spectrogram}, middle panels), and the
relative angular variations of the electron neutrino flux density grow
to the several ten percent level (Fig.\,\ref{fig:cl_spectrogram},
bottom panels). The latter quantity gives the maximum minus the
minimum flux density on the sphere divided by the angle-averaged flux
density in percent. Compared to the \emph{total} neutrino emission in
Figs.\,\ref{fig:rsh+lnu_vs_t} - \ref{fig:W15-4_lnue}, the temporal and
angular variations in different directions are even more pronounced
when considering the energy flux of the electron neutrinos or electron
anti-neutrinos alone (Fig.\,\ref{fig:cl_spectrogram}), where angular
variations can exceed 100\% in all models during the pre-explosion and
accretion phases, and peak values are close to 200\% during short
episodes (Fig.\,\ref{fig:cl_spectrogram}, lower panels).
During the vigorous pre-explosion phase including the post-explosion
accretion stage, electron neutrinos and antineutrinos dominate the
angular flux variations, while muon and tau neutrinos (accounting for
roughly 50\% of the total luminosity) exhibit essentially isotropic
emission in all directions. The reason of this finding is that $\nu_e$
and $\bar \nu_e$ are produced almost exclusively by efficient
charged-current reactions in the accretion region perturbed by
non-radial fluid flows. The spectrogram of the two phases is
characterized by initially very small-scale angular variations with $l
\ga 12$, which are associated with the onset of the Rayleigh-Taylor
overturn activity, and which merge to continuously larger angular
structures that correspond to $l \approx 1 \ldots 4$ modes toward the
end of the accretion period at $0.4\,-\,0.6\,$s (depending on the
model). This evolution is accompanied by a steady decrease of
$C^\mathrm{max}_l$ to a level of $\sim 10^{-5}$ and a reduction of the
electron neutrino flux density variations from values well beyond
100\% to a level of $\sim\,10\%$, only (see
Fig.\,\ref{fig:cl_spectrogram}, left panels).
When neutrino-energy deposition in the post-shock layers becomes
sufficiently strong and the explosion is eventually launched at about
250 to 500\,msec (depending on the model; Table\,1), subsequent radial
shock expansion rapidly diminishes the activity of the SASI and
freezes post-shock convection. Single, longer lasting downdrafts of
accretion flows are associated with isolated hot spots, where the
variations of the total flux density can reach peak amplitudes of up
to $\sim\,70\,$\% (Fig.\,\ref{fig:W15-4_lnue}, panel 3). When
accretion has ended, the amplitude of the angular variations of the
total neutrino energy flux reduces to a level of a few percent
(Figs.\,\ref{fig:rsh+lnu_vs_t}, \ref{fig:rsh+lnu_vs_t_a}), and the
angular pattern of the emission becomes more uniform over the sphere,
consisting of many spots with an angular size of $\sim\,$30$^\circ$
(Fig.\,\ref{fig:W15-4_lnue}, panel 4).
In the early post-accretion phase of model W15-4, $0.6\,\mathrm{s} \la
t \la 0.8\,$s, the spectrogram indicates the presence of low-amplitude
($C^\mathrm{max}_l \la 10^{-4}$), small-angular size ($l \ga 10$)
perturbations in the electron neutrino energy flux caused by some
low-amplitude turbulent flow in and below the neutrinospheric
region. When strong convection inside the PNS is encountered for $t
\ga 0.8\,$s the spectrogram drastically changes, being dominated by
angular modes with $l=4$, but still with $C^\mathrm{max}_l \sim
10^{-4}$. The electron neutrino flux density variations rise somewhat
to a level of 10\% to 20\%, and become manifest in the total energy
loss rate, too (Figs.\,\ref{fig:rsh+lnu_vs_t},
\ref{fig:rsh+lnu_vs_t_a}).
Model N20-2 exhibits quite a similar behavior as model W15-4 except
for the appearance of even larger ($l \sim 3$) angular structures
clearly recognizable in the pseudo-power spectrogram between 1.0\,s
and 1.2\,s (Figs.\,\ref{fig:cl_spectrogram}, \ref{fig:cl_vs_l}). This
differs from the behavior of model L15-3, where the amplitudes and
angular size of the energy flux density variations remain small and
even decrease in the post-explosion phase
(Figs.\,\ref{fig:rsh+lnu_vs_t_a}, \ref{fig:cl_spectrogram}).
\begin{figure}
\centering
\resizebox{0.9\hsize}{!}{\includegraphics*{Cl_vs_l.ps}}
\caption{Pseudo-power coefficients $C^\mathrm{max}_l$ of the
electron-neutrino flux density as functions of angular mode number
$l$ at 200\,ms (blue), 400\,ms (red), and 1000\,ms (black) for
models W15-4 (top), L15-3 (middle), and N20-3 (bottom),
respectively. }
\label{fig:cl_vs_l}
\end{figure}
\begin{figure}[t]
\centering
\resizebox{0.90\hsize}{!}{\includegraphics{ekin_vs_t.ps}}
\caption{Evolution of the non-radial specific kinetic energy
$(v_\theta^2 + v_\phi^2)/2$ volume averaged over the computational
domain inside the neutrinosphere for models W15-4 (solid), L15-3
(dashed), and N20-2 (dashed-dotted), respectively. }
\label{fig:ekin_vs_t}
\end{figure}
\begin{figure}
\centering \resizebox{1.0\hsize}{!}{\includegraphics*{lobs_vs_t.ps}}
\caption{Observable luminosity $L_o$ (top row), observable mean energy
$\langle E \rangle_o$ (middle row), and normalized quantity $L_o
\langle E \rangle_o^2$ (bottom row) of electron neutrinos (left
column) and electron anti-neutrinos (right column) as a function of
time for three of our models. Although we only present the results
for one particular observer direction here, the global behavior and
characteristics are very similar for all viewing directions.}
\label{fig:lobs_vs_t}
\end{figure}
The reason for the fluctuation behavior of the neutrino emission
during the vigorous pre-explosion and post-explosion accretion phases
has been discussed, but what causes the spatial and temporal
variations during the post-accretion phase? Because the explosion is
well on its way at this time, neither post-shock convection nor the
SASI nor accretion can be responsible. Hence, there only remains
non-radial gas flow in the outer layers of the proto-neutron star.
Ledoux convection in the proto-neutron star thus may become visible
eventually, {\it i.e.,}\, its presence in the inner parts of the computational
domain may become dominant in observable signals. This happens in
models W15-4 and N20-2, where the level of the non-radial specific
kinetic energy $(v_\theta^2 + v_\phi^2)/2$, volume-averaged over the
computational domain, inside the neutrinosphere shows a steep rise at
$\sim\,$0.8\,s and $\sim\,$0.9\,s, respectively
(Fig.\,\ref{fig:ekin_vs_t}). These non-radial flows that develop in
models W15-4 and N20-2 at late times also become manifest in all
discussed quantities: $\Lambda_\mathrm{max}(\Omega,t)$,
$\Lambda_\mathrm{min}(\Omega,t)$, $C^\mathrm{max}_l$, the dominant low
$l$-modes ($2 \la l \la 4$), and relative angular flux-density
variations. In contrast, no such effect is present in model L15-3
(see Fig.\,\ref{fig:ekin_vs_t}), where we find a steady decrease of
$\Lambda_\mathrm{max}(\Omega,t) - \Lambda_\mathrm{min}(\Omega,t)$,
higher $l$-modes ($l \ga 10$), smaller $C^\mathrm{max}_l$, and lower
flux-density variation amplitudes than in models W15-4 and N20-2 (see
Figs.\,\ref{fig:rsh+lnu_vs_t}, \ref{fig:rsh+lnu_vs_t_a}, and
\ref{fig:cl_spectrogram}).
Simulations with fully self-consistent treatment of the PNS interior
show the presence of convection inside the PNS, {\it i.e.,}\, below the
neutrinosphere \citep[see][]{Keil_etal96, Buras_etal06,
Dessart_etal06} more or less from the early post-bounce phase on.
With the use of our inner radial grid boundary excising the inner
parts of the PNS, and imposing neutrino luminosities at this boundary,
convective activity is triggered only when the neutrino energy (or
lepton number) inflow into the layers close to the grid boundary is
faster than neutrino transport can carry away this energy (or lepton
number). Then convectively unstable gradients develop and convective
flows begin to carry the energy and lepton-number outward. Whether
this happens or not depends on the boundary luminosities as well as on
the location of the grid boundary within the density and temperature
profiles of the PNS layers below the neutrinosphere. That location
determines the efficiency of the neutrino transport and varies with
the stellar progenitor, whose mass-infall rate decides how much mass
accumulates in the near-surface layers of the PNS outside the inner
grid boundary. The relative strength of the artificially imposed
inflow of neutrino energy and lepton number compared to the efficiency
of the neutrino transport on the grid, both sensitive to the location
and contraction of the grid boundary on the one hand and the chosen
values of the boundary luminosities on the other, therefore decides
about when, where, and how strongly convective activity develops below
the neutrinosphere.
Because the position of and the conditions imposed at the inner
boundary can thus influence the neutrino emission properties, in
particular during the post-accretion phase, our respective model
predictions must be considered with care. While they do not allow us
to make any definite statements concerning the neutrino signal of a
particular progenitor model because of the neglected treatment of the
inner parts of the proto-neutron star, the models nevertheless show
that convective flows below the neutrinosphere are likely to imprint
themselves on the neutrino emission, and hence also on the GW signal
of core-collapse supernovae. A measurement of these signals may
actually provide some insight into the conditions inside proto-neutron
stars.
Because the neutrino energy flux density varies in our models both
with latitude and longitude, the observable neutrino luminosity
$L_o(t)$ is obtained by an integral over the hemisphere visible to an
observer (Eq.\,\ref{eq:app-lum_code}). In Fig.\,\ref{fig:lobs_vs_t}
we show the observable electron neutrino and electron anti-neutrino
luminosities for one chosen viewing direction for the three models
W15-4, L15-3, and N20-2, respectively. The results for other
directions look very similar with all characteristic features being
independent of the observer position. We provide these quantities in
addition to the total neutrino energy loss rate (Eqs.\,\ref{eq:lamt}
and \ref{eq:eloss}; Figs.\,\ref{fig:rsh+lnu_vs_t} and
\ref{fig:rsh+lnu_vs_t_a}), because their temporal evolutions are the
ones expected to be measurable in the IceCube and Super-Kamiokande
detectors. These detectors (mainly for ${\bar \nu_e}$) will be
sensitive to a combination of the observable neutrino luminosity $L_o$
and the observable mean neutrino energy $\langle E \rangle_o$. Thus,
we also provide in Fig.\,\ref{fig:lobs_vs_t} the time evolution of the
observable mean neutrino energy and of the combination $L_o \langle E
\rangle_o^2$, which (roughly) enters the IceCube detection rate of
Cherenkov photons originating from the dominant inverse beta decay
reaction ${\bar \nu_e} + p \rightarrow n + e^+$ \citep{Lund_etal10}.
\footnote{Note that our transport approximation only provides
luminosities and mean energies, but not the higher moments of the
energy spectrum (see Sect.\,\ref{subsec:nuform}). }
Again one can recognize the different evolution stages, and in
particular the post-shock convection and SASI phase, during which the
quantity $L_o \langle E \rangle_o^2$ exhibits rapid low-amplitude
variations for all three models. The level of the variations is a few
percent (Fig.\,\ref{fig:lobs_vs_t}), which is considerably lower than
that of the angular fluctuation amplitudes of the flux density, which
reaches almost 100\% for the total neutrino flux density
(Fig.\,\ref{fig:W15-4_lnue}) and almost 200\% for the electron
neutrino and electron antineutrino flux densities
(Fig.\,\ref{fig:cl_spectrogram}, lower panels). However, because the
flux density variations are caused by a few individual hot spots
covering only angular areas of size $\sim (\pi/9)^2$, the observable
fluctuations (of $L_0$ and $\langle E \rangle_0$) are lower by a
factor of roughly $(\pi/9)^2 / (2\pi) \sim 1/50$. Some of this
activity is also present at late times in the two models W15-4 and
N20-2, where Ledoux convection develops in the simulated outer parts
of the proto-neutron star (see discussion above).
From the results presented above we conclude that the signals carry
clear information about the postshock hydrodynamic activity, and about
the duration and decay of the accretion period. Composition-shell
interfaces present in the progenitor star can also leave an
imprint. In model W15-4 the transition from the Fe-core to the
Si-shell manifests itself in fast drops of the luminosities of $\nu_e$
and $\bar\nu_e$ at $\sim150\,$msec, when the mass accretion rate
decreases steeply at the time the interface between the Fe-core and
the Si-shell of the 15\,$M_\odot$ progenitor falls through the shock.
\section{Gravitational wave signature}
\label{sec:gw-signal}
Non-radial mass motions caused by gravity waves in the near-surface
layers of the PNS, which are caused by the SASI and convection in the
post-shock region as well as by convective activity inside the
proto-neutron star \citep{Murphy_etal09, Marek_etal09} (driven by
Ledoux unstable lepton or entropy gradients) result in a
time-dependent, aspherical density stratification that produces
gravitational radiation. The anisotropic emission of neutrinos
associated with the non-radial mass flow (see
Sect.\,\ref{sec:neutrino-signal}) contributes to the gravitational
wave signal, too. We computed and analyzed the signature of this
gravitational radiation for the 3D models discussed in
Sect.\,\ref{subsec:models}.
\subsection{Formalism}
\label{subsec:gw-form}
\subsubsection{Non-radial mass flow}
\label{subsec:gw-form-flow}
If a source is of genuine three-dimensional nature, as it is the case
for our models, it is common to express the gravitational quadrupole
radiation tensor, $\vec h^{\rm TT}$, in the transverse traceless gauge
in the following tensorial form
\begin{equation}
\vec h^{\rm TT} (\vec X,t) = \frac{1}{R}\,
(A_+ \vec e_+ + A_{\times} \vec e_{\times})
\label{eq:htt-general}
\end{equation}
\citep[see, {\it e.g.,}\,][]{Misner_etal73}. $R$ denotes the distance between
the observer and the source, and the unit linear-polarization tensors
are given by
\begin{eqnarray}
\vec e_+ &=& \vec e_{\theta} \otimes \vec e_{\theta} -
\vec e_{\phi} \otimes \vec e_{\phi}\, ,
\label{eq:pol-tensor-p}\\
\vec e_{\times} &=& \vec e_{\theta} \otimes \vec e_{\phi} +
\vec e_{\phi} \otimes \vec e_{\theta}\, ,
\label{eq:polt-tensor-x}
\end{eqnarray}
with $\vec e_{\theta}$ and $\vec e_{\phi}$ being the unit polarization
vectors in $\theta$ and $\phi$-direction of a spherical coordinate
system, and $\otimes$ denoting the tensor product.
The wave amplitudes $A_+$ and $A_{\times}$ represent the only two
independent modes of polarization in the TT gauge
\citep{Misner_etal73}. In the slow-motion limit, they are obtained
from linear combinations of the second time derivatives (evaluated at
retarded time, and denoted by a double dot accent) of the components
of the transverse traceless mass quadrupole tensor
\citep{Misner_etal73}
\begin{eqnarray}
A_+ &=& \ddot Q_{\theta\theta} - \ddot Q_{\phi\phi} \ ,
\label{eq:aplus-tf} \\
A_{\times} &=& 2 \ddot Q_{\theta\phi} \ .
\label{eq:acros-tf}
\end{eqnarray}
We computed the latter using a post-Newtonian approach whereby the
numerically troublesome second-order time derivatives of the mass
quadrupole tensor components are transformed into much better
tractable spatial derivatives. Following \citet{NakamuraOohara89} and
\citet{Blanchet_etal90}, the second-order time derivatives read in a
Cartesian orthonormal basis (the spatial indices $i$ and $j$ run from
1 to 3)
\begin{equation}
\ddot Q_{ij} = \frac{G}{c^4} \int {\rm d}^3 x \ \rho \
\left(
2 v_i v_j - x_i \ \partial_j \Phi_{\rm eff}
- x_j \ \partial_i \Phi_{\rm eff}
\right) \ ,
\label{eq:qddot}
\end{equation}
where $G$ is Newton's gravitational constant, $c$ the speed of light
in vacuum, $\Phi_{\rm eff}$ the effective Newtonian gravitational
potential including the general relativistic ``case A" correction of
the monopole term according to \citet{Marek_etal06}, $\rho$ the
mass-density, $v_ i$ the Cartesian velocity components, and
$\partial_i$ the partial derivative with respect to the coordinate
$x^i$ of a Cartesian basis.
We note that the integrand in Eq.\,(\ref{eq:qddot}) has compact
support and is known to the (2nd order) accuracy level of the
numerical scheme employed in the hydrodynamics code. It can easily be
shown that evaluating the integral of Eq.~(\ref{eq:qddot}) by an
integration scheme (of at least 2nd order) is by one order of accuracy
superior to twice applying numerical time-differentiation methods to
quadrupole data given at discrete points of time \citep{FinnEvans90,
Moenchmeyer_etal91}.
Exploiting the coordinate transformation between the orthonormal
Cartesian basis $x^i$ and the orthonormal basis in spherical
coordinates $\hat{x}^i$ (with $\hat{x}^i \in \lbrack r, \theta, \phi
\rbrack$), the wave amplitudes $A_+$ and $A_{\times}$
(Eqs.\,(\ref{eq:aplus-tf}) \& (\ref{eq:acros-tf})) are obtained from
the following second time derivatives of the spherical components of
the mass quadrupole tensor \citep{Oohara_etal97, Scheidegger_etal08}
\begin{eqnarray}
I^{TT}_{\theta\theta} &=& \left( I^{TT}_{xx} \cos^2\phi
+ I^{TT}_{yy} \sin^2\phi
+ 2\, I^{TT}_{zz} \sin\phi \cos\phi
\right) \cos^2\theta
\nonumber\\
&\phantom{= }& + I^{TT}_{yy} \sin^2\theta
-2\, \left( I^{TT}_{xz} \cos\phi +
I^{TT}_{yz} \sin\phi \right)
\sin\theta \cos\theta \, ,
\\
I^{TT}_{\phi\phi} &=& I^{TT}_{xx} \sin^2\phi + I^{TT}_{yy} \cos^2\phi
-2\, I^{TT}_{xy} \sin\phi \cos\phi \, ,
\\
I^{TT}_{\theta \phi} &=& \left( I^{TT}_{yy} - I^{TT}_{xx} \right)
\cos\theta \sin\phi \cos\phi
+ I^{TT}_{xy} \cos\theta
\left( \cos^2\phi \right.
\nonumber\\
&\phantom{=}& \left. - \sin^2\phi \right)
+ I^{TT}_{xz} \sin\theta \sin\phi
- I^{TT}_{yz} \sin\theta \cos\phi \, ,
\end{eqnarray}
where we used the abbreviation
\begin{equation}
I^{TT}_{ij} \equiv \ddot Q^{TT}_{ij}\, .
\label{eq:ieqqddot}
\end{equation}
Choosing $\phi=0$ one obtains the polarization modes (see, {\it e.g.,}\,
\cite{Misner_etal73})
\begin{eqnarray}
A_+ &=& I^{TT}_{xx} - I^{TT}_{yy} \ , \label{eq:aplus_p} \\
A_{\times} &=& 2 I^{TT}_{xy} \ , \label{eq:acros_p}
\end{eqnarray}
for $\theta=0$, and
\begin{eqnarray}
A_+ &=& I^{TT}_{zz} - I^{TT}_{yy} \ , \label{eq:aplus_e} \\
A_{\times} &=& -2 I^{TT}_{yz} \, , \label{eq:acros_e}
\end{eqnarray}
for $\theta = \pi/2$, respectively. These expressions were already
discussed in earlier investigations concerned with the evaluation of
the gravitational wave signature of 3D core-collapse supernova models
\citep{MuellerJanka97, Fryer_etal04, Scheidegger_etal08,
Scheidegger_etal10}.
The total energy radiated in the form of gravitational waves due to
nonspherical mass flow is given in the quadrupole approximation by
\citep[see, {\it e.g.,}\,][]{Misner_etal73}
\begin{eqnarray}
E_{\rm M} &=& \frac{c^3}{5G} \int_0^\infty \sum_{ij} \left\lbrack
\frac{{\rm d}}{{\rm d}t} \left( I^{TT}_{ij} - \frac{1}{3}
\delta_{ij} \sum_l I^{TT}_{ll} \right)
\right\rbrack^2 {\rm d}t
\nonumber \\
&=& \frac{c^3}{15G} \int_0^\infty {\rm d}t \left\lbrack
(\dot{I}^{\ TT}_{xx} - \dot{I}^{\ TT}_{yy})^2 +
(\dot{I}^{\ TT}_{xx} - \dot{I}^{\ TT}_{zz})^2 \right.
\label{eq:egw} \\
&\phantom{=}& \phantom{ \frac{c^3}{15G}} \left.
+ (\dot{I}^{\ TT}_{yy} - \dot{I}^{\ TT}_{zz})^2
+ 6\, \left( (\dot{I}^{\ TT}_{xy})^2 + (\dot{I}^{\ TT}_{xz})^2 +
(\dot{I}^{\ TT}_{yz})^2 \right)
\right\rbrack \, ,
\nonumber
\end{eqnarray}
with $\dot{I}^{\ TT}_{ij} \equiv \partial I^{\ TT}_{ij}/ \partial t$,
and the corresponding GW spectral energy density is given by (where
$\nu$ denotes the frequency)
\begin{eqnarray}
\frac{{\rm d}E_{\rm M}}{{\rm d} \nu} &=& \frac{2c^3}{15G} (2\pi \nu)^2
\left\lbrack
\left|\widetilde{I}^{\ TT}_{xx} - \widetilde{I}^{\ TT}_{yy}\right|^2 +
\left|\widetilde{I}^{\ TT}_{xx} - \widetilde{I}^{\ TT}_{zz}\right|^2 \right.
\label{eq:egwnu} \\
&\phantom{=}& \phantom{ \frac{2}{15} } \left.
+ \left|\widetilde{I}^{\ TT}_{yy} - \widetilde{I}^{\ TT}_{zz}\right|^2
+ 6\, \left( \left|\widetilde{I}^{\ TT}_{xy}\right|^2 +
\left|\widetilde{I}^{\ TT}_{xz}\right|^2 +
\left|\widetilde{I}^{\ TT}_{yz}\right|^2 \right)
\right\rbrack \, ,
\nonumber
\end{eqnarray}
where
\begin{equation}
\widetilde{I}^{\ TT}_{ij}(\nu) =
\int_{-\infty}^\infty I^{TT}_{ij}(t)\, e^{-2\pi i \nu t}\, {\rm d}t
\end{equation}
is the Fourier transform of $I^{TT}_{ij} (t)$.
\subsubsection{Anisotropic neutrino emission}
\label{subsec:gw-form-nu}
To determine the gravitational wave signal associated with the
anisotropic emission of neutrinos, we follow \citet{MuellerJanka97}
and use Eq.\,(16) of \citet{Epstein78} in the limit of a distant
source, $R \to \infty$, together with the approximation that the
gravitational wave signal measured by an observer at time $t$ is
caused only by radiation emitted at time $t' = t - R/c$. Hence, we
take $t-t' = {\rm const} = R/c$, {\it i.e.,}\, we assume that only the neutrino
pulse itself causes a gravitational wave signal, whereas memory
effects, which prevail after the pulse has passed the observer, are
disregarded.
With these simplifications, the dimensionless gravitational wave
amplitudes of the two polarization modes are given in the
transverse-traceless gauge for an observer located at a distance $R$
along the $z$-axis of the observer frame by \citet{MuellerJanka97}
\begin{equation}
h_{+}(t) = \frac{2G}{c^4R} \int_0^t {\rm d}t'
\int_{4\pi} {\rm d}\Omega' \,
( 1 + \cos\theta ) \cos 2\phi \,
\frac{ {\rm d}\Lambda }{ {\rm d}\Omega'} ({\vec \Omega'},t')
\label{eq:hplus_nu}
\end{equation}
and
\begin{equation}
h_{\times}(t) = \frac{2G}{c^4R} \int_0^t {\rm d}t'
\int_{4\pi} {\rm d}\Omega' \,
( 1 + \cos\theta ) \sin 2\phi \,
\frac{ {\rm d}\Lambda }{ {\rm d}\Omega'} ({\vec \Omega'},t')\ ,
\label{eq:hcros_nu}
\end{equation}
respectively. Here ${\rm d} \Lambda (\Omega', t')/{\rm d}\Omega'$ is
given by Eq.\,(\ref{eq:dlamdom}) and denotes the total neutrino energy
radiated at time $t'$ per unit of time into a solid angle ${\rm
d}\Omega'$ in direction $(\theta', \phi')$. Except for
position-dependent factors the gravitational wave amplitudes are
simply a function of this quantity provided by the ray-by-ray
transport approximation (note that in M\"uller \& Janka (1997) we used
the symbol $L_{\nu}$ instead of $\Lambda$).
The angular integration, ${\rm d}\Omega' = -{\rm d} (\cos\theta')\,
{\rm d}\phi'$, in Eqs.\,(\ref{eq:hplus_nu}) and (\ref{eq:hcros_nu})
extends over all angles $\theta'$ and $\phi'$ in the coordinate frame
of the source $(x',y',z')$ that we identify with the (arbitrarily
chosen) spherical polar coordinate frame to which the hydrodynamic
results were mapped from the Yin-Yang grid employed in the
simulations. For the evaluation of the polarisation modes we used the
(asymptotic) values of ${\rm d} \Lambda(\Omega, t)/{\rm d}\Omega$
extracted at a radius of 500\,km from our 3D models.
\begin{figure}[!]
\centering
\resizebox{1.0\hsize}{!}{\includegraphics{obs_src_gimp.ps}}
\caption{Relation between the source coordinate system $(x^\prime,
y^\prime, z^\prime)$ and the observer coordinate system $(x, y,
z)$. Changing from the observer system to the source system involves
a rotation by an angle $\alpha$ about the $z'$-axis to an
intermediate coordinate system $(x^\ast, y^\ast, z^\ast)$, followed
by a rotation by an angle $\beta$ about the $y^\ast$-axis (which
thus is also the $y$-axis). }
\label{fig:obs_src}
\end{figure}
The angles $\theta$ and $\phi$ in Eqs.\,(\ref{eq:hplus_nu}) and
(\ref{eq:hcros_nu}) are measured in the observer frame $(x,y,z)$,
while the neutrino luminosity is measured in the source frame
$(x',y',z')$. To allow for an arbitrary orientation of the observer
relative to the source, we introduce two viewing angles $\alpha \in
[-\pi,+\pi]$ and $\beta \in [0,\pi]$ (see Fig.\,\ref{fig:obs_src}).
The coordinates measured in the observer frame are then related to the
coordinates in the source frame by the following coordinate
transformations
\begin{eqnarray}
x^\ast &=& x' \cos\alpha + y' \sin\alpha\, , \\
y^\ast &=& -x' \sin\alpha + y' \cos\alpha\, , \\
z^\ast &=& z'\, ,
\end{eqnarray}
and
\begin{eqnarray}
x &=& x^\ast \cos\beta - z^\ast \sin\beta\, , \\
y &=& y^\ast \, , \\
z &=& x^\ast \sin\beta + z^\ast \cos\beta \, .
\end{eqnarray}
With these coordinate transformations and the relations
\begin{eqnarray}
x &=& r \sin\theta \cos\phi
\label{eq:csx}\, , \\
y &=& r \sin\theta \sin\phi\, , \\
z &=& r \cos\theta\,
\label{eq:csz}
\end{eqnarray}
between Cartesian coordinates $(x,y,z)$ and spherical polar
coordinates $(r,\theta,\phi)$, we obtain
\begin{eqnarray}
\sin\theta \cos\phi &=& (\cos\phi' \cos\alpha + \sin\phi' \sin\alpha)
\sin\theta' \cos\beta\, -
\nonumber\\
& & \cos\theta' \sin\beta\, ,
\\
\sin\theta \sin\phi &=& (\sin\phi' \cos\alpha -
\cos\phi' \sin\alpha) \sin\theta'\, ,
\\
\cos\theta &=& (\cos\phi' \cos\alpha + \sin\phi' \sin\alpha)
\sin\theta' \sin\beta\, +
\nonumber\\
& & \cos\theta' \cos\beta \, .
\label{eq:angle_trans}
\end{eqnarray}
These expressions relate the angular coordinates in the observer frame
$(\theta, \phi)$ to those in the source frame $(\theta',\phi')$. For
the special case $\alpha =0$ they were already presented by
\citet{Kotake_etal09a}. Using Eq.\,(\ref{eq:angle_trans}) and the
equalities
\begin{eqnarray}
\sin 2\phi &=& \frac{2xy}{x^2+y^2}\, ,
\label{eq:sintwophi}\\
\cos 2\phi &=& \frac{x^2-y^2}{x^2+y^2}
\label{eq:costwophi}
\end{eqnarray}
derived from Eqs.\,(\ref{eq:csx}) to (\ref{eq:csz}), the two
polarization modes (Eqs.\,\ref{eq:hplus_nu} and \ref{eq:hcros_nu}) are
given by
\begin{equation}
h_{\rm S} (t,\alpha,\beta) = \frac{2G}{c^4R} \int_0^t {\rm d}t'
\Lambda(t')\, \alpha_{\rm S}(t',\alpha,\beta)\,,
\label{eq:hnue}
\end{equation}
where ${\rm S} \in (+,\times)$ and $\Lambda(t)$ is the angular
integral of the neutrino energy radiated at time $t$ per unit of time
given in Eqs.\,(\ref{eq:lamt}) and (\ref{eq:eloss}).
\begin{equation}
\alpha_{\mathrm S}(t,\alpha,\beta) = \frac{1}{\Lambda(t)}
\int_{4\pi}{\rm d}\Omega'\, W_{\rm S}(\Omega',\alpha,\beta)\,
\frac{ {\rm d}\Lambda }{ {\rm d}\Omega'} ({\vec \Omega'}, t) \, ,
\label{eq:aniso}
\end{equation}
are anisotropy parameters, which provide a quantitative measure of the
time-dependent anisotropy of the emission in both polarization
modes. Note that the evaluation of the anisotropy parameter
$\alpha(t)$ defined in Eq.\,(29) of \citet{MuellerJanka97}, which
should not be confused with the observer angle $\alpha$ introduced in
Fig.\,\ref{fig:obs_src}, does neither involve a dependence on observer
angles $(\alpha,\beta)$ nor on the polarization mode.
The angular weight functions appearing in the above expression for the
anisotropy parameters are given by
\begin{equation}
W_{\rm S} (\theta',\phi',\alpha,\beta) =
\frac{D_{\rm S} (\theta',\phi',\alpha,\beta)}{N (\theta',\phi',\alpha,\beta)} \ ,
\end{equation}
where
\begin{eqnarray}
D_+ &=& \left[ 1 + (\cos\phi' \cos\alpha + \sin\phi' \sin\alpha)
\sin\theta' \sin\beta + \cos\theta' \right.
\nonumber\\
& & \left. \cos\beta \right] \, \left\lbrace
\left[ (\cos\phi' \cos\alpha + \sin\phi' \sin\alpha)
\sin\theta' \cos\beta\, - \right. \right.
\nonumber\\
& & \left.\left. \cos\theta' \sin\beta \right]^2 -
\sin^2\theta' \left( \sin\phi' \cos\alpha -
\cos\phi' \sin\alpha \right)^2
\right\rbrace\, ,
\\
D_\times &=& \left[ 1 + ( \cos\phi' \cos\alpha + \sin\phi' \sin\alpha)
\sin\theta' \sin\beta + \cos\theta' \right.
\nonumber\\
& & \left. \cos\beta \right] \,
2 \left[ (\cos\phi' \cos\alpha + \sin\phi' \sin\alpha)
\sin\theta' \cos\beta\, - \right.
\nonumber\\
& & \left. \cos\theta' \sin\beta \right]\,
\sin\theta' \left( \sin\phi' \cos\alpha -
\cos\phi' \sin\alpha \right)\, ,
\\
N &=& \left[ (\cos\phi' \cos\alpha +
\sin\phi' \sin\alpha )
\sin\theta' \cos\beta\, - \, \cos\theta'\right.
\nonumber\\
& & \left. \sin\beta \right]^2 +
\sin^2\theta' \left( \sin\phi' \cos\alpha -
\cos\phi' \sin\alpha \right)^2 \, .
\end{eqnarray}
Choosing $\alpha = 0$ and $\beta = \pi/2$ the observer is located in
the equatorial plane of the source ({\it i.e.,}\, perpendicular to the source's
$z'$-axis) at the azimuthal position $\phi' = 0$. In that case one
obtains simpler expressions for the angular functions \citep[see
also][]{Kotake_etal09a}
\begin{eqnarray}
\left. W_+ \right|_e &=& (\cos^2\theta' - \sin^2\theta' \sin^2\phi) \,
\frac{1 + \sin\theta' \cos\phi'}{
\cos^2\theta' + \sin^2\theta'
\sin^2\phi'}\, ,
\\
\left. W_\times \right|_e &=& -2 \cos\theta' \sin\theta' \sin\phi'\,
\frac{1 + \sin\theta' \cos\phi'}{
\cos^2\theta' + \sin^2\theta'
\sin^2\phi'}\, .
\end{eqnarray}
Note that for axisymmetric sources $h_{\times} = 0$.
In general, the total energy $E_{\rm GW}(t)$ radiated to infinity by a
source in form of gravitational waves until time $t$ is given by (see,
{\it e.g.,}\, \cite{Misner_etal73}; Greek indices run from 0 to 3, and repeated
indices are summed over)
\begin{equation}
E_{\rm GW}(t) = \int_0^t {\rm d}t'
\int_{S_\infty^2} \tau_{0\nu}\ n^\nu r^2
{\rm d} \Omega \ ,
\label{eq:enue_1}
\end{equation}
where the angular integration is performed over a two-sphere at
spatial infinity $S_\infty^2$, and $n^\mu = (0,1,0,0)$ is a unit
spacelike vector in polar coordinates $\lbrace ct, r, \theta, \phi
\rbrace$ normal to $S_\infty^2$. Denoting by $\langle \dots \rangle$
an average over several wavelengths, the gravitational-wave
energy-momentum tensor $\tau_{\mu\nu}$ is given in
transverse-traceless gauge by
\begin{equation}
\tau_{\mu\nu} = \frac{c^5}{32\pi G} \left\langle
(\partial_\mu h^{\rm TT}_{\rho\sigma})\
(\partial_\nu h_{\rm TT}^{\rho\sigma}) \right\rangle \ .
\label{eq:taunue}
\end{equation}
Thus, Eq.\,(\ref{eq:enue_1}) can be rewritten as
\begin{equation}
E_{\rm GW}(t) = \frac{c^3}{32\pi G} \int_0^t {\rm d}t'
\int_{S_\infty^2} r^2 {\rm d} \Omega \left\langle
(\partial_t h^{\rm TT}_{ik})
(\partial_r h_{\rm TT}^{ik}) \right\rangle \ ,
\label{eq:enue_2}
\end{equation}
where we have used the facts that $h^{\rm TT}_{0\nu} = 0$, $h^{\rm
TT}_{i,r} = 0$, and $c\ \partial_r h^{\rm TT}_{ik} = - \partial_t
h^{\rm TT}_{ik}$ for radially outgoing gravitational
radiation. Evaluating the double sum in Eq.\,(\ref{eq:enue_2}) and
using the relations $h^{\rm TT}_{\theta\theta} = - h^{\rm
TT}_{\phi\phi} = h_+$ and $h^{\rm TT}_{\theta\phi} = h^{\rm
TT}_{\phi\theta} = h_\times$ \citep[see, {\it e.g.,}\,][]{Misner_etal73}, we
finally find
\begin{equation}
E_{\rm GW}(t) = \frac{c^3}{16\pi G} \int_0^t {\rm d}t'
\int_{S_\infty^2} r^2 {\rm d} \Omega \left\langle
(\partial_t h_+)^2 + (\partial_t h_\times)^2
\right\rangle \ .
\label{eq:hnue_alpha}
\end{equation}
Inserting the expressions for $h_+$ and $h_\times$ given in
Eq.\,(\ref{eq:hnue}) into Eq.\,(\ref{eq:hnue_alpha}), we obtain for the
energy $E_{\rm N}(t)$ radiated in form of gravitational waves until
time $t$ due to anisotropic neutrino emission
\begin{equation}
E_{\rm N}(t) = \frac{G}{4\pi c^5}
\int_0^t {\rm d}t' \int_{4\pi}{\rm d} \Omega_{\alpha\beta}
\left\lbrack l^2_+ (t',\alpha,\beta) +
l^2_\times (t',\alpha,\beta) \right\rbrack
\label{eq:enue}
\end{equation}
with ${\rm d} \Omega_{\alpha\beta} = \sin\beta\, {\rm d}\beta\, {\rm
d} \alpha$ and
\begin{equation}
l_{\rm S} (t,\alpha,\beta) = \Lambda(t)\, \alpha_{\rm S}(t,\alpha,\beta) \,.
\label{eq:lnue}
\end{equation}
The corresponding spectral energy density is given by
\begin{equation}
\frac{{\rm d} E_{\rm N}}{{\rm d} \nu} =
\frac{G}{2\pi c^5}\ \left| {\tilde l}(\nu) \right|^2 \, ,
\label{eq:de_ndnu}
\end{equation}
where ${\tilde l} (\nu)$ is the Fourier transform of
\begin{equation}
l(t) = \left\lbrace
\int_{4\pi} {\rm d} \Omega_{\alpha\beta}\
\left\lbrack l^2_+ (t,\alpha,\beta) +
l^2_\times (t,\alpha,\beta)
\right\rbrack
\right\rbrace^{1/2} \, .
\label{eq:lambda}
\end{equation}
For completeness we also provide an expression for the total energy
radiated in form of gravitational waves until time $t$, {\it i.e.,}\, due to
anisotropic mass flow {\bf and} neutrino emission. It is obtained by
inserting the total GW amplitude, {\it i.e.,}\, the sum of the amplitudes given
by Eqs.\,(\ref{eq:htt-general}) and (\ref{eq:hnue}) into
Eq.\,(\ref{eq:hnue_alpha}), which leads to
\begin{equation}
E_{\rm GW} = \frac{c^3}{16\pi G} \int_0^t {\rm d}t'
\int_{4\pi}{\rm d} \Omega
\left\lbrack
\left( \frac{2G}{c^4} l_+ + \partial_t A_+ \right)^2
+ \left( \frac{2G}{c^4} l_\times + \partial_t A_\times \right)^2
\right\rbrack \, .
\label{eq:gwetot}
\end{equation}
\subsection{Results}
\label{subsec:gw-results}
\begin{figure*}[t]
\centering
\resizebox{0.47\hsize}{!}{\includegraphics*{W15-2_c.eps}}\hspace{1cm}
\resizebox{0.47\hsize}{!}{\includegraphics*{W15-46_c.eps}}\\
\vspace{0.5cm}
\resizebox{0.47\hsize}{!}{\includegraphics*{L15-2_c.eps}}\hspace{1cm}
\resizebox{0.47\hsize}{!}{\includegraphics*{L15-3_c.eps}}\\
\caption{The four panels show the gravitational wave amplitudes (top)
and spectrograms of ${\rm d} E_{\rm M} / {\rm d} \nu$ (bottom;
normalized to the absolute maximum) arising from non-spherical mass
flow of models W15-2 (top left), W15-4 (top right), L15-2 (bottom
left), and L15-3 (bottom right), respectively. Blue curves give the
amplitude $A_+$ at the pole (solid) and the equator (dotted), while
red curves show the other independent mode of polarization
$A_\times$ from the same directions. }
\label{fig:W15-as}
\end{figure*}
\begin{figure*}
\centering
\resizebox{0.33\hsize}{!}{\includegraphics*{W15-4_htot_vs_t.ps}}
\resizebox{0.33\hsize}{!}{\includegraphics*{L15-3_htot_vs_t.ps}}
\resizebox{0.33\hsize}{!}{\includegraphics*{N20-2_htot_vs_t.ps}}
\caption{Gravitational wave amplitudes $R h^{TT}_{+}$ (blue) and
$R h^{TT}_{\times}$ (red) due to anisotropic mass flow and neutrino
emission as a function of time for models W15-4 (left), L15-3
(middle), and N20-3 (right) , respectively. The solid curves show
the amplitudes for an observer located above the north pole ($\alpha
= \beta = 0$; see Fig.\,\ref{fig:obs_src}) of the source, while the
other curves give the amplitudes at the equator ($\alpha = 0$,
$\beta = \pi/2$). }
\label{fig:htot_vs_t}
\end{figure*}
\begin{figure*}[!]
\centering
\resizebox{0.33\hsize}{!}{\includegraphics*{W15-4_aniso_vs_t.ps}}
\resizebox{0.33\hsize}{!}{\includegraphics*{L15-3_aniso_vs_t.ps}}
\resizebox{0.33\hsize}{!}{\includegraphics*{N20-2_aniso_vs_t.ps}}
\caption{Asymmetry parameter $\alpha_S$ of the neutrino emission
(Eq.\,\ref{eq:aniso}) as a function of time for models W15-4 (left),
L15-3 (middle), and N20-2 (right), respectively. The panels in the
upper row show $\alpha_+$ (blue) and $\alpha_\times$ (red) for a
particular observer direction, while the panels in the lower row
give for both parameters the maximum and minimum values in all
directions. }
\label{fig:aniso_vs_t}
\end{figure*}
Although an observer can only measure the total gravitational wave
amplitude, {\it i.e.,}\, that due to the combined effect of non-radial flow and
anisotropic neutrino emission, we will first discuss the GW signal of
non-radial mass flow only, because it reflects the various phases of
the post-bounce evolution already introduced in the discussion of the
neutrino signal above.
Until post-shock convection and the SASI are eventually mature at
around 150\,msec, the GW signal is very small
(Fig.\,\ref{fig:W15-as}). Note that our models are not able to follow
the GW emission that is caused by prompt post-shock convection because
of the excised inner region of the PNS \citep{Marek_etal09}. Later
on, sizable g-mode activity is instigated in the outer layers of the
proto-neutron star by convective overturn and the SASI during the
hydrodynamically vigorous pre-explosion phase, and by the impact of
anisotropic accretion flows during the subsequent post-explosion
accretion phase \citep{Marek_etal09}. This g-mode activity is the
cause of GW signals \citep{Marek_etal09, Murphy_etal09,
Yakunin_etal10}, whose maximum amplitudes are on the order of a few
centimeters centered around zero.
The GW frequency distribution possesses a very broad maximum in the
range of 100\,Hz to 500\,Hz, and the frequency corresponding to this
maximum slowly increases with time (Fig.\,\ref{fig:W15-as}). Partially
already during the post-explosion accretion phase, but at latest when
the shock wave starts to rapidly propagate to large radii between
$\sim\,$0.4\,sec and $\sim\,$0.7\,sec (see
Figs.\,\ref{fig:rsh+lnu_vs_t} and \ref{fig:rsh+lnu_vs_t_a}), the GW
amplitudes start to grow by about a factor of ten until approximately
asymptoting at $\sim\,$0.9\,sec in the case of the models based on the
progenitor W15, and at $\sim\,$1\,sec in the case of models based on
the progenitor L15, respectively (Fig.\,\ref{fig:W15-as}). This
growth of the amplitude is associated with the anisotropic expansion
of the shock wave, and a positive/negative wave amplitude indicates a
prolate/oblate explosion, respectively \citep{Murphy_etal09}.
While the GW amplitudes grow, the GW energy distribution $\mathrm{d}
E_\mathrm{M} / \mathrm{d} \nu$ becomes narrower and dimmer, and the
frequency at maximum power continues to increase. The latter effect
was also observed in the 2D models of \citet{Murphy_etal09}. At late
times, the GW signal of the W15 models clearly signifies the
convective activity inside the proto-neutron star through
low-amplitude, high-frequency fluctuations around the asymptotically
roughly constant mean GW amplitudes, while no such fluctuations are
present in the case of the L15 models (see discussion of the neutrino
signal above). This model discrepancy is also evident from the energy
spectrograms, which do exhibit a pronounced broad maximum (between
$\sim\,350\,$Hz and $\sim\,550\,$Hz) at $t > 0.8\,$sec in the case of
the W15 models, but none for the L15 ones. Furthermore note that until
the end of the simulations the frequency of the maximum of $\mathrm{d}
E_\mathrm{M} / \mathrm{d} \nu$ has increased from around 100\,Hz to
almost 500\,Hz for the former models (owing to the increasing speeds
of mass motions in the postshock region at times $\la 0.5\,$sec and
because of the increased compactness of the proto-neutron star at
times $\ga 0.6\,$sec, respectively).
The behavior of the total (matter plus neutrinos) GW amplitudes is
significantly different from that of the flow-only GW amplitudes for
models that exhibit PNS convection below the neutrinosphere, {\it i.e.,}\, for
the models based on the progenitors W15 and N20. Particularly at late
times, anisotropic neutrino emission causes a continuing growth of the
GW amplitudes (instead of a saturation) in these models, while this is
not the case for the L15 models (see Fig.\,\ref{fig:htot_vs_t}, and
compare with Fig.\,\ref{fig:W15-as}). The latter behavior is also
reflected in the time evolution of the asymmetry parameter
$\alpha_{\mathrm S}$ (Eq.\,\ref{eq:aniso}) of the neutrino emission
(Fig.\,\ref{fig:aniso_vs_t}). The asymmetry parameter is practically
zero in model L15-3 at late times, while it remains, after having
temporarily grown to values beyond about 0.4 - 0.5\%, at the level of
$\sim 0.3\,$\% until the end of the simulations in models W15-4 and
N20-2.
The final GW amplitudes are up to a factor of two to three higher when
taking the contribution of anisotropic neutrino emission into account.
In contrast, the amount of energy radiated in the form of GW, which is
proportional to the third time-derivative of the quadrupole moment and
hence proportional to the time derivative of the GW amplitude is only
insignificantly changed, and is practically constant for all simulated
models after the onset of the explosion (see
Fig.\,\ref{fig:etot_vs_t}). The integral value of the GW energy
radiated by neutrinos is low ($\la 1\%$) compared to that emitted by
matter through the slow variation of the GW neutrino amplitude with
time, {\it i.e.,}\, its time derivative is much smaller than that of the GW
matter amplitude. For this reason we also abstained from evaluating
the total energy radiated in form of GW (Eq.\,\ref{eq:gwetot}). It
differs little from that caused by anisotropic matter flow alone
(Eq.\,\ref{eq:egw}), because the mixed term in Eq.\,(\ref{eq:gwetot}),
resulting from the square of the sum of the matter and neutrino parts,
contributes $\la 10\%$ to the total radiated GW energy, and the pure
neutrino term $\la 1\%$. Figure \ref{fig:etot_vs_t} also shows that
the (small) contribution of anisotropic neutrino emission to the
radiated GW energy is enhanced at late times when proto-neutron star
convection occurs below the neutrinosphere, as it is the case for
models W15-4 and in particular N20-2.
\begin{figure}[!]
\centering
\resizebox{0.95\hsize}{!}{\includegraphics{etot_vs_t.ps}}
\caption{Energy emitted in form of gravitational waves due to
anisotropic mass flow (top panel) and due to anisotropic neutrino
emission (bottom panel) as a function of time for models W15-4
(solid), L15-3 (dashed), and N20-2 (dash-dotted), respectively. }
\label{fig:etot_vs_t}
\end{figure}
\begin{figure}
\centering
\resizebox{0.95\hsize}{!}{\includegraphics*{W15-4_hptot_13.eps}}\\
\resizebox{0.95\hsize}{!}{\includegraphics*{W15-4_hxtot_13.eps}}
\caption{Gravitational wave amplitudes due to anisotropic mass flow
and neutrino emission, $R h^{TT}_{+}$ (top) and $R h^{TT}_{\times}$
(bottom), as functions of the observer angles (see
Fig.\,\ref{fig:obs_src}) for model W15-4 at 1.3\,sec past
bounce. The white contours give the locations, where the amplitudes
are zero. Yellow and red areas indicate positive amplitudes, green
and blue negative ones.}
\label{fig:W15-4_hnue}
\end{figure}
The variation of the total GW amplitudes with observer angle is
illustrated in Fig.\,\ref{fig:W15-4_hnue} for model W15-4 at 1.3\,sec
(when the simulation was stopped). Both the amplitude variations and
the typical angular size of the speckled GW emission are similar for
all other simulated models. The model-independent level of the
amplitude variations is also supported by Fig.\,\ref{fig:htot_vs_t}
when comparing various amplitudes at any given (late) time.
The (normalized) amplitude spectrograms of the total gravitational
wave amplitudes (${\rm d} (A_{+, \times} + R h_{+, \times})/ \rm{d}
\nu$; Figs.\,\ref{fig:W15-4_h_pol} and \ref{fig:L15-3_h_pol})
illustrate two model-independent findings. Firstly, during the
hydrodynamically vigorous pre-explosion and post-explosion accretion
phases ($0.2 \la t \la 0.5-0.7\,$sec) the spectra of all models are
characterized by some power at low frequencies ($\la 100\,$Hz) and a
broad power maximum at frequencies $\sim 200\,$Hz and another weak one
at $\sim 800\,$Hz. The latter broad maximum at high frequency is more
pronounced in the models based on the W15 and N20 progenitors and in
the cross polarization GW mode. Secondly, during the post-accretion
phase ($t \ga 0.7\,$sec) the spectra of all models are dominated by a
low-frequency ($\la 40\,$Hz) contribution peaked toward the lower end
of the spectrogram. In the models where PNS convection occurs below
the neutrinosphere (models W15 and N20) we also find a double-peaked
high-frequency contribution decreasing/increasing from $\sim 700\,$Hz
(400\,Hz) at $t \sim 0.8\,$sec, and eventually merging into a single
power maximum at $\sim 500\,$Hz at $t \sim 1.2\,$sec. Again, this
contribution is more pronounced for the cross polarization GW mode.
The spectra of the total GW amplitudes are dominated by the
contribution from non-isotropic neutrino emission at low ($\le
100\,$Hz) frequencies (Figs.\,\ref{fig:W15-4_h_pol} and
\ref{fig:L15-3_h_pol}). At higher frequencies ($\ge 100\,$Hz) the
spectra of model W15-4 show two pronounced maxima (at 100 - 200\,Hz
and $600-800\,$Hz, respectively) at all times. These maxima are also
present in model L15-3 at times $\le 0.7\,$s, the high-frequency one
being, however, much less pronounced. The lower maximum (at
$100-200\,$Hz) results from g-mode activity in the PNS surface
instigated by non-radial flow (SASI, accretion) in the post-shock
region until $\sim 0.5 -0.7\,$sec. At later times PNS convection is
responsible for the peak between 300 and 500\,Hz. We have proposed
this explanation already for the corresponding maxima present in the
GW energy spectrograms arising from non-spherical mass flow
(Fig.\,\ref{fig:W15-as}), and discussed why the frequencies of these
maxima increase with time. The source of the high-frequency maximum
(600 - 800\,Hz) is unclear, but a further detailed analysis shows that
(i) the maximum is solely caused by non-radial gas flow, {\it i.e.,}\, it is not
connected to neutrinos, (ii) it does not result from stellar layers
below the neutrino sphere but from those close to or slightly above
it, and (iii) does not depend on the position of the observer.
Note that the high-frequency maximum present in the amplitude
spectrograms is strongly suppressed in the corresponding energy
spectrograms (Fig.\,\ref{fig:W15-as}), because the latter involve the
squared time derivatives of the amplitudes. Thus, the already high
ratio of the low- and high-frequency maxima in the amplitude
spectrograms (about two orders of magnitude) translates into an even
higher ratio for the energy spectrogram maxima, rendering the
high-frequency maximum practically invisible.
\begin{figure*}
\centering
\resizebox{0.495\hsize}{!}{\includegraphics*{W15-4_hplus_pol.eps}}
\resizebox{0.495\hsize}{!}{\includegraphics*{W15-4_hcros_pol.eps}}
\caption{Normalized (to the absolute maximum) amplitude spectrograms
of the total gravitational wave amplitudes $A_+ + Rh_+$ (left
panels) and $A_\times + Rh_\times$ (right panels) at the pole for
model W15-4. The lower panels show the spectrograms in the frequency
range 5\,Hz to 100\,Hz, and the upper ones in the frequency range
100\,Hz to 1\,kHz.}
\label{fig:W15-4_h_pol}
\end{figure*}
\begin{figure*}
\centering
\resizebox{0.495\hsize}{!}{\includegraphics*{L15-3_hplus_pol.eps}}
\resizebox{0.495\hsize}{!}{\includegraphics*{L15-3_hcros_pol.eps}}
\caption{Same as Fig.\,\ref{fig:W15-4_h_pol}, but for model L15-3.}
\label{fig:L15-3_h_pol}
\end{figure*}
\section{Discussion and conclusions}
Based on a set of three-dimensional (3D) parametrized neutrino-driven
supernova explosion models of non-rotating 15 and 20\,$M_\odot$ stars,
employing a neutrino transport description with a gray spectral
treatment and a ray-by-ray approximation of multi-dimensional effects
(the scheme is applicable in the regime outside the dense neutron star
core, {\it i.e.,}\, around and outside the neutrinosphere), we evaluated both
the time-dependent and direction-dependent neutrino and
gravitational-wave emission of these models. To this end we presented
the formalism necessary to compute both the observable neutrino and
gravitational wave signals for a three-dimensional, spherical
source. For the neutrino signal we presented formulas that allow one
to estimate the apparent luminosity when the local flux density $F$ on
a sphere is known and the ansatz of Eq.\,(4) about the angular
distribution of the intensity can be made. While in general the
location of the neutrino-decoupling surface has to be suitably defined
({\it e.g.,}\, by a criterion based on the optical depth), our ray-by-ray
transport scheme implies that $r^2 F =$const in the free-streaming
limit in every direction. Thus, it spares us such a definition of the
neutrinosphere for evaluating the angular integration for the
observable luminosity. Concerning the gravitational-wave analysis, we
extended and generalized previous studies, where the source was either
assumed to be axisymmetric or where the formulas for the signals of a
3D source were only given for special observer directions.
Our models followed the evolution from shortly after core bounce up to
more than one second into the early cooling evolution of the PNS
without imposing any symmetry restrictions and covering a full sphere.
The extension over such a relatively long evolution time in 3D was
possible through the usage of an axis-free overset grid (the Yin-Yang
grid) in spherical polar coordinates, which considerably eases the CFL
time-step restriction and avoids axis artifacts. A central region,
the dense inner core of the proto-neutron star, was excised from the
computational domain and replaced by an inner, time-dependent radial
boundary condition and a gravitating point mass at the coordinate
origin. Explosions in the models were initiated by neutrino heating
at a rate that depends on suitably chosen values of the neutrino
luminosities imposed at the inner radial boundary.
The post-bounce evolution of our models can be divided into four
distinct phases (Fig.\,\ref{fig:rsh+lnu_vs_t}). The first phase, the
\emph{quasi-spherical shock-expansion phase}, lasts from shock
formation shortly after core bounce to $80\,-\,150$\,msec, when
convection sets in. The second phase, the hydrodynamically vigorous
\emph{pre-explosion phase}, comprises the growth of post-shock
convection and of the standing accretion shock instability (SASI).
The \emph{post-explosion accretion phase} begins when energy
deposition by $\nu$-heating in the post-shock layers becomes
sufficiently strong so that the total energy in the post-shock region
ultimately becomes positive. During this phase the shock accelerates
outward while gas is still accreted onto the PNS. This process is
commonly called ``shock revival". The duration of the latter two
phases depends on the progenitor. During the \emph{post-accretion
phase}, the fourth and final phase characterizing the evolution of
our models, accretion ends and the proto-neutron star develops a
nearly spherical neutrino-driven wind.
The neutrino emission properties (fluxes and effective spectral
temperatures) of our 3D models exhibit the generic time-dependent
features already known from 2D (axisymmetric) models \citep[{\it e.g.,}\,][]{
Buras_etal06, Scheck_etal06, Marek_etal09, Brandt_etal11}, showing
fluctuations over the neutron star surface on different spatial and
temporal scales. We found that non-radial mass motions caused by the
SASI and convection in the neutrino-heated hot-bubble region as well
as by PNS convection below the neutrinosphere give rise to a
time-dependent, anisotropic emission of neutrinos, particularly of
electron neutrinos and anti-neutrinos, and thus also to the emission
of gravitational waves. Because very prominent, quasi-periodic
sloshing motions of the shock due to the standing accretion-shock
instability as visible in 2D simulations are absent and the emission
from different surface areas facing an observer adds up incoherently,
the modulation amplitudes of the measurable neutrino luminosities and
mean energies are significantly lower than predicted by 2D models
\citep[for 2D results see][]{Marek_etal09, Brandt_etal11}.
During the quasi-spherical shock expansion phase shortly after bounce
the level of temporal and angular fluctuations of the neutrino
emission is low ($\la 10^{-2}$). In contrast, the fluctuation
amplitudes reach a level of several 10\% of the average values during
the hydrodynamically vigorous pre-explosion phase and the
post-explosion accretion phase, where a few distinct, highly
time-variable regions or even short-lived single spots with an angular
size of 10$^\circ$ to 20$^\circ$ are responsible for the brightest
emission maxima. As the outward shock expansion is well on its way in
the post-explosion accretion phase, still existing accretion
downdrafts can be responsible for similar fluctuations in the neutrino
emission, though the number of corresponding hot spots decreases with
diminishing accretion. When accretion has ended and the post-accretion
phase has started, directional variations can be caused by the
occurrence of Ledoux convection in the outer layers of the
proto-neutron star, which we indeed observe in models based on two of
our three progenitors (see also the discussion of the influence of the
inner radial boundary condition below). The temporal and angular
variations of the emission in different directions are even more
pronounced when considering the energy flux of the electron neutrinos
or electron anti-neutrinos alone (instead of the emission in all
neutrino flavors). In that case the angular variations of local flux
densities can exceed 100\% in all models during the pre-explosion and
post-explosion accretion phases, and the peak values can be close to
200\% during short episodes. The total energy loss rates in neutrinos
and the observable luminosities as surface-integrated quantities,
however, are much smoother in time during all phases, showing
fluctuation amplitudes of at most several percent.
The gravitational wave emission also exhibits the generic
time-dependent features already known from 2D (axisymmetric) models,
but the 3D wave amplitudes are considerably lower (by a factor of
$2-3$) than those predicted by 2D models \citep{Mueller_etal04,
Marek_etal09, Murphy_etal09, Yakunin_etal10} owing to less coherent
mass motions and neutrino emission. Note in this respect that the GW
quadrupole amplitudes, which are usually quoted for 2D models
($A^{E2}_{20}$), have to be multiplied by a geometric factor
$\sin^2\theta\, \sqrt{15/\pi}/8$ (which is equal to $\approx 0.27$ for
$\theta = 90^o$). Violent, non-radial hydrodynamic mass motions in
the accretion layer and their interaction with the outer layers of the
proto-neutron star give rise to a GW signal with an amplitude of $\sim
2-4\,$cm in the frequency range of $\sim\,$100\,Hz to $\sim\,$400\,Hz,
while anisotropic neutrino emission is responsible for a superimposed
low-frequency evolution of the wave amplitude, which thus can grow to
maximum values of $10-20\,$cm. Variations of the mass-quadrupole
moment caused by convective activity inside the nascent neutron star
contribute a high-frequency component ($300\,-\,600\,$Hz) to the GW
signal during the post-accretion phase. The GW signals exhibit strong
variability between the two polarizations, different explosion
simulations and different observer directions, and besides common
basic features do not possess any template character.
Finally we would like to reflect on some of the deficiencies of the
presented 3D models. Because of the ray-by-ray treatment of the
$\nu$-transport, the directional variations of the neutrino emission
in response to local inhomogeneities in the star may be overestimated
\citep{Ott_etal08, Brandt_etal11}. However, when we evaluate
observable signals, these artificial effects are mostly compensated
for by integrations of the neutrino flux densities over the surface
areas visible to observers from different viewing directions (see
Eqs.\,\ref{eq:app-lum}, \ref{eq:lnum}, \ref{eq:app-lum_code},
\ref{eq:lnum_code}), or by the integration of the neutrino energy loss
in all directions (Eqs.\,\ref{eq:lamt}, \ref{eq:aniso}).
Another deficiency concerns the usage of the inner radial grid
boundary, because of which our simulations do not fully include
(either in space or time) the convective flow occurring in the PNS
interior after core bounce \citep{Keil_etal96, Buras_etal06,
Dessart_etal06}. Moreover, convective activity in the simulated
outer layers of the PNS occurs for special conditions: It is triggered
only when the artificially imposed inflow of neutrino energy and
lepton number through the inner radial boundary into the adjacent
layers is faster than the neutrinos can carry away this energy or
lepton number. Whether this is the case sensitively depends on the
employed neutrino-transport approximation, but also on the location
and contraction of the grid boundary, the chosen values of the
boundary luminosities, and on the stellar progenitor. Its mass-infall
rate decides how much mass accumulates in the near-surface layers of
the PNS outside the inner grid boundary. Because the position of and
the conditions imposed at the inner boundary can thus influence the
neutrino emission properties, in particular during the post-accretion
phase, our respective model predictions must be considered with
care. While they do not allow us to make any definite statements
concerning the detailed neutrino signal of a particular progenitor
model due to the neglected treatment of the inner parts of the
proto-neutron star, the models nevertheless show that convective flows
below the neutrinosphere are likely to imprint themselves on the
neutrino emission, and hence also on the GW signal of core-collapse
supernovae. A measurement of these signals may actually provide some
insight into the conditions inside proto-neutron stars.
\acknowledgements
This work was supported by the Deutsche Forschungsgemeinschaft through
the Transregional Collaborative Research Centers SFB/TR~27 ``Neutrinos
and Beyond'' and SFB/TR~7 ``Gravitational Wave Astronomy'' and the
Cluster of Excellence EXC~153 ``Origin and Structure of the Universe''
(http://www.universe-cluster.de).
\bibliographystyle{aa}
|
1,108,101,565,643 | arxiv | \section{\label{intr}Introduction}
Recently, a new Eddington-inspired theory of gravity with a Born-Infeld like structure \cite{Born:1934gh} has been proposed by Banados and Ferreira \cite{Banados:2010ix} (see also \cite{Deser:1998rj,Vollick:2003qp,Wohlfarth:2003ss,Nieto:2004qj,Comelli:2005tn,Vollick:2005gc,Zinoviev:2006,Ferraro:2008ey,Fiorini:2009ux,Ferraro:2009zk,Gullu:2010pc,Alishahiha:2010iq,Gullu:2010em} for other relevant studies of Born-Infeld type gravitational models and \cite{Clifton:2011jh} for a recent review on alternative theories of gravity). A key feature of the Eddington-inspired Born-Infeld (EiBI) theory of gravity introduced in \cite{Banados:2010ix} is that it is equivalent to Einstein's general relativity in vacuum. In \cite{Banados:2010ix} it was shown that, in the non-relativistic limit, the EiBI theory of gravity leads to a modified Poisson equation given by
\begin{equation}
\nabla^2 \phi = 4 \pi G \rho_m+\frac{\kappa}{4}\nabla^2 \rho_m\,,
\label{poisson}
\end{equation}
where $\phi$ is the gravitational potential and $\rho_m$ is the matter density. The gravitational constant $G$, $\kappa$ and the speed of light in vacuum $c$ are the only parameters of the theory. With these parameters it is possible to define a fundamental length, time, mass and density respectively by
\begin{equation}
R_*=\sqrt{\frac{\kappa}{G}}\,,\ \ t_*=\sqrt{\frac{\kappa}{G c^2}}\,,\ \ M_*=\sqrt{\frac{\kappa c^4}{G^3}}\,,\ \ \rho_*=\frac{c^2}{\kappa}\,,
\label{fstar}
\end{equation}
whose physical interpretation will be revealed in this letter. In \cite{Banados:2010ix} it was shown that the EiBI theory of gravity significantly changes the universe dynamics at early times, leading to a non-singular cosmological model. In \cite{Pani:2011mg} the EiBI theory has also been shown to support compact stars made of pressureless dust. Astrophysical constraints on the single extra parameter of the theory $\kappa$ have been determined considering the physics of astronomical objects such as neutron stars \cite{Pani:2011mg} and the Sun \cite{Casanellas:2011kf}.
In this letter we compute both astrophysical and cosmological constraints on the EiBI theory of gravity. In Sec. II basic results for the evolution of the Universe during the radiation era are reviewed and a constraint on the value of $\kappa$ is derived from primordial nucleosynthesis. In Sec. III the Zel'dovich approximation is generalized to account for the modification to the Poisson equation in Eq. (\ref{poisson}) which it is shown to lead to an effective fundamental effective length, in the case of pressureless cold dark matter. We use this result to compute the minimum size (allowed by the theory) of compact objects which are held together by gravity and to determine the maximum density attainable inside stable compact stars. We also obtain the critical mass above which pressureless compact stars cannot exist. We then use these results to derive stringent astrophysical constraints on the value of $\kappa$. We conclude in Sec. IV.
\section{Background evolution of the Universe}
The EiBI theory of gravity leads to modifications to the dynamics of the Universe at early times. In \cite{Banados:2010ix} it has been shown that, in the radiation era, the Friedmann equation is given by
\begin{equation}
H^2=\frac{8\pi G c^2}{3\kappa} \times f({\tilde \rho})\,,
\end{equation}
where
\begin{eqnarray}
f({\tilde \rho})&=&\left({\tilde \rho}-1+\frac{(1+{\tilde \rho})^{1/2}(3-{\tilde \rho})^{3/2}}{3^{3/2}}\right)\times \nonumber \\
&\times& \frac{(1+{\tilde \rho})(3-{\tilde \rho})^2}{(3+{\tilde \rho^2})^2}\,,
\end{eqnarray}
${\tilde \rho}=\kappa \rho /c^2$, $H={\dot a}/a$ is the Hubble parameter, $a$ is the cosmological scale factor and a dot represents a derivative with respect to the physical time $t$. Two stationary points with $H=0$ at ${\tilde \rho}=3$ and ${\tilde \rho}=-1$ were identified, for positive and negative $\kappa$, respectively. For small ${\tilde \rho}$ one recovers the usual dynamics with
\begin{equation}
H^2=8\pi G \rho/3\,,
\end{equation}
but for values of ${\tilde \rho}$ of order unity the dynamics is severely modified. In this letter we shall assume that $\kappa > 0$, so that ${\tilde \rho} > 0$, in order to avoid instabilities associated with an imaginary effective sound speed (see Sec. III).
\subsection{Primordial nucleosynthesis constraint}
Less than one second after the big bang protons and neutrons were in thermal equilibrium. However, about one second after the big bang the temperature $T$ drops below $1 \, {\rm MeV}$ and the neutron-proton inter-conversion rate ($\sim G_F^2 T^5$, where $G_F$ is the Fermi coupling constant) becomes smaller than expansion rate ($H$). This leads to the approximate freeze out of the ratio between the number of neutrons and the number of protons $t_{nuc} \sim 1 \, {\rm s}$ after the big bang, except for free neutron decay. Although the formation of $^4 He$ is delayed until the temperature of the Universe is low enough for deuterium to form (at about $T=0.1 \, {\rm MeV}$), it is crucial that the dynamics of the Universe at the start of the primordial nucleosynthesis epoch ($t_{nuc} \sim 1 \, {\rm s}$) is very close to the standard one in order that the good agreement between primordial nucleosynthesis predictions and observed light element abundances is preserved (see \cite{Nakamura} for a recent review). This implies that the following very conservative constraint must be satisfied
\begin{equation}
{\tilde \rho}_{nuc}=\frac{\kappa \rho_{nuc}}{c^2} < 3\,,
\label{nuccons1}
\end{equation}
with $ \rho_{nuc}=\rho(t_{nuc})$. Hence, taking into account that $\rho_{nuc} \sim 3H_{nuc}^2/(8\pi G)$, Eq. (\ref{nuccons1}) implies that
\begin{equation}
\kappa < 8\pi G R_{Hnuc}^2 \sim 6 \times 10^8 \, {\rm m^5 \, kg^{-1} \, s^{-2}}\,,
\label{nuccons2}
\end{equation}
where $H_{nuc}=H(t_{nuc})$, the Hubble radius is defined by $R_H=c H^{-1}$, and its value at the nucleosynthesis epoch is $R_{Hnuc} \sim R_\odot \sim 2 \, {\rm light \ seconds}$ ($R_\odot$ is the solar radius). The constraint in Eq. (\ref{nuccons2}) may be improved by almost an order of magnitude by requiring that the variation of the Hubble parameter at $t_{nuc} \sim 1 \, {\rm s}$, with respect to the standard model value, is less than $10 \, \%$. Primordial nucleosynthesis provides the strongest cosmological constraint on the EiBI theory of gravity. In the next section we shall show that the constraint in Eq. (\ref{nuccons2}) might be improved by several orders of magnitude by taking into account the mere existence of compact round astronomical objects.
\section{Linear evolution of density perturbations}
In this section we shall consider the non-relativistic regime and generalize the Zel'dovich approximation \cite{Zeld}, for the evolution of cold dark matter density perturbations in an expanding background, to account for the modification to the Poisson equation in Eq. (\ref{poisson}). Here we shall consider times much later than $t_{nuc}$ and consequently the background evolution of the Universe is that of the standard cosmological model.
The trajectory of the cold dark matter particles in a homogeneous and isotropic Friedmann-Robertson-Walker background can be written as
\begin{equation}
{\vec r}=a(t)\left[{\vec q}+{\vec \psi}({\vec q},t)\right].
\label{eq:position}
\end{equation}
where ${\vec q}$ is the unperturbed comoving position and ${\vec \psi}({\vec q},t)$ is the comoving displacement. Calculating the first derivative of Eq. (\ref{eq:position}) with respect to the physical time one obtains
\begin{equation}
{\vec v}=H{\vec r} +{\vec v}_{pec},
\label{eq:velocity}
\end{equation}
where ${\vec v}={\dot {\vec r}}$ and ${\vec v}_{pec}=a {\dot {\vec \psi}}$ is the peculiar velocity.
The gravitational acceleration felt by the cold dark matter particles is equal to
\begin{equation}
{\vec {\rm a}} \equiv {\ddot r}= -\nabla \phi.
\label{eq:accm}
\end{equation}
where $\phi$ is given by a generalized Poisson equation
\begin{equation}
\nabla^2 \phi = 4 \pi G ({\bar \rho}+3{\bar p}+\delta \rho_m)+\frac{\kappa}{4}\nabla^2 \rho_m,
\label{eq:poisson}
\end{equation}
where $\delta \rho_m \equiv \rho_m-{\bar \rho}_m$, $\rho_m$ is the matter density, ${\bar \rho}_m$ is the average matter density, ${\bar \rho}$ is the average density and ${\bar p}$ is the average pressure. Eq. (\ref{eq:poisson}) generalizes Eq. (\ref{poisson}) to account for a homogeneous cosmological background with an arbitrary equation of state. However, the two equations are equivalent if ${\bar p}=0$.
Mass conservation requires that
\begin{equation}
{\bar \rho}_m a^3 d^3 q = \rho_m d^3 r,
\label{eq:mconserv}
\end{equation}
where the infinitesimal volume elements $d^3 r$ and $d^3 q$ are related by
\begin{eqnarray}
\frac{d^3 r}{d^3 q}&=&\left|\frac{\partial {\vec r}}{\partial {\vec q}}\right|=a^3\left(1+\sum_i \psi_{i,i} + ...\right) \sim \nonumber\\
&\sim& a^3(1+a \nabla \cdot {\vec \psi}),
\label{eq:jacobian1}
\end{eqnarray}
a comma denotes a partial derivative with respect to the comoving coordinates and the approximation is valid up to first order in the comoving displacement. Hence
\begin{equation}
\delta \sim - a \nabla \cdot {\vec \psi}\,,
\label{eq:zel1}
\end{equation}
where $\delta \equiv \delta \rho_m/{\bar \rho}_m$.
Integrating Eq. (\ref{eq:poisson}) one obtains the first order solution
\begin{equation}
\nabla \phi = \frac {4 \pi G ({\bar \rho}+3{\bar p})}{3} {\vec r} -
4\pi G {\bar \rho}_m a {\vec \psi}_\parallel +\frac{\kappa {\bar \rho}_m}{4}\nabla \delta ,
\label{eq:poissonint}
\end{equation}
with ${\vec \psi}= {\vec \psi}_\parallel+ {\vec \psi}_\perp$ where ${\vec \psi}_\parallel$ and ${\vec \psi}_\perp$ are the irrotational and divergence parts of the comoving displacement, respectively ($\nabla \times {\vec \psi}_\parallel= {\vec 0}$ and $\nabla \cdot {\vec \psi}_\perp = 0$). A more general solution may be obtained by adding, to the right hand side of Eq. (\ref{eq:poisson}), the term $\nabla \varphi$ where $\varphi$ is an arbitrary scalar field satisfying the Laplace equation $\nabla^2 \varphi=0$. For simplicity, we shall ignore that extra term since it will not have any impact on our conclusions. Using the Rachaudhury equation
\begin{equation}
\frac{\ddot a}{a}=-\frac{4\pi G ({\bar \rho}+3{\bar p})}{3},
\label{eq:rach}
\end{equation}
together with Eqs. (\ref{eq:position}), (\ref{eq:accm}) and (\ref{eq:poissonint}) one may show that
\begin{equation}
{\ddot {\vec \psi}}+2H {\dot {\vec \psi}} -4\pi G {\bar \rho}_m
{\vec \psi}_\parallel = -\frac{\kappa {\bar \rho}_m}{4 a}\nabla \delta.
\label{eq:zel2}
\end{equation}
Decomposing Eq. (\ref{eq:zel2}) into its parallel and perpendicular components one finds
\begin{eqnarray}
{\ddot {\vec \psi}_\parallel}&+&2H {\dot {\vec \psi}_\parallel} -4\pi G {\bar \rho}_m {\vec \psi}_\parallel = -\frac{1}{a} \frac{\kappa}{4}\nabla \rho_m\,,
\label{eq:zelparallel} \\
{\ddot {\vec \psi}_\perp}&+&2H {\dot {\vec \psi}_\perp}
= 0\,.
\label{eq:zelperp}
\end{eqnarray}
Integrating Eq. (\ref{eq:zelperp}) one finds that ${\vec v}_{pec \perp}=a {\dot {\vec \psi}}_{pec \perp} \propto a^{-1}$, which simply expresses the conservation of angular momentum. On the other hand, using Eqs. (\ref{eq:zel1}) and (\ref{eq:zel2}) one finally obtains an equation for the evolution of the cold dark matter density perturbations
\begin{equation}
{\ddot \delta}+2H {\dot \delta} -4\pi G {\bar \rho}_m \delta =c_{seff}^2 \nabla^2 \delta\,,
\label{eq:denevol}
\end{equation}
where $c_{seff}^2=\kappa {\bar \rho}_m/4$ is the effective sound speed squared of the cold dark matter in the EiBI theory of gravity.
In Fourier space one obtains
\begin{equation}
{\ddot \delta}_{\vec k}+2H {\dot \delta}_{\vec k}
-\left(4\pi G {\bar \rho}_m -\frac{k^2 c_{seff}^2}{a^2}\right) \delta_{\vec k} = 0.
\label{eq:denevfour}
\end{equation}
where ${\vec k}$ is the comoving wavenumber and $k=|{\vec k}|$. In this letter we assume that $c_{seff}^2 > 0$, or equivalently $\kappa > 0$, in order to avoid unwanted instabilities associated with an imaginary effective sound speed.
\subsection{Astrophysical constraints}
An effective Jeans length can be defined for the cold dark matter as
\begin{equation}
\lambda_{Jeff}= \frac{2 \pi a}{k_{Jeff}}=c_{seff} {\sqrt{\frac{ \pi}{G {\bar \rho}_m}}}= {\sqrt{\frac{\pi \kappa}{4G}}} \sim R_*\,,
\label{eq:jeans}
\end{equation}
where $k_{Jeff}$ is the value of $k$ for which the term within brackets in Eq. ({\ref{eq:denevfour}) is equal to zero.
The effective Jeans length for the cold dark matter in the EiBI theory of gravity defines the critical scale bellow which the collapse of pressureless dust is no longer possible (for wavelengths $\lambda < \lambda_{Jeff} \sim R_*$ matter fields oscillate coherently). Consequently, $\lambda_{Jeff}$ determines the minimum scale of compact objects which are held together by gravity (note that including pressure increases the Jeans scale). The demonstration that $\lambda_{Jeff}$ is independent of the matter density being approximately equal to the fundamental scale of the theory $R_*$ is one of the key results of this letter.
By requiring that $\lambda_{Jeff}$ is equal to the schwarzchild radius $r_s$,
\begin{equation}
\lambda_{Jeff} = {\sqrt{\frac{\pi \kappa}{4G}}} = \frac{2GM}{c^2} =r_s\,,
\end{equation}
one obtains the critical mass above which pressureless compact stars are unstable to collapse into a black hole
\begin{equation}
M = \frac{{\sqrt{\pi}}}{4} \kappa^{1/2} c^2 G^{-3/2} \sim M_*\,,
\end{equation}
which is essentially equal to the fundamental mass of the theory $M_*$. The fundamental density, given by
\begin{equation}
\rho_*= \frac{M_*}{R_*^3}=\frac{c^2}{\kappa} \sim \frac{c^2}{G} \lambda_{eff}^{-2}\,,
\end{equation}
provides an estimate of the maximum density attainable inside stable compact stars (note that adding pressure increases the Jeans scale, leading to a decrease of the maximum density with respect to the pressureless case).
By requiring that $\lambda_{Jeff}$ is smaller than the solar radius $R_\odot$ one obtains the following conservative constraint
\begin{equation}
\kappa < \frac{4}{\pi} G R_\odot^2 \sim 3 \times 10^{7} \, {\rm m^5 \, kg^{-1} \, s^{-2}}\,, \label{astcons}
\end{equation}
which is about two orders of magnitude weaker than that derived in \cite{Casanellas:2011kf}, where a detailed model for the structure of the sun has been considered. However, much stronger constraints on $\kappa$ may be obtained by considering smaller astrophysical objects.
There are several natural satellites in the Solar System which are massive enough to relax to a rounded shape through their internal gravity. Some of them have radii of the order of $100 \, {\rm km}$ \cite{BonnieJ2003329}. Substituting $R_\odot$ by $R =100 \, {\rm km}$ in Eq. (\ref{astcons}) improves the $\kappa$ constraint by more than $7$ orders of magnitude. On the other hand, neutron stars \cite{Lattimer:2004pg} with a typical radius of about $R_{NS} \sim 12 \, {\rm km}$ (nearly 5 orders of magnitude smaller than $R_\odot$) are also held together by gravity. They have core densities which are predicted to be larger than $\rho_c \sim 10^{17} \, {\rm kg \, m^3}$. By requiring that $\rho_* > 10^{17} \, {\rm kg \, m^3}$ one obtains the constraint $\kappa < 1 \, {\rm m^5 \, kg^{-1} \, s^{-2}}$ which is similar to the one obtained in \cite{Pani:2011mg} considering a relativistic model for internal structure of the neutron star. An even tighter constraint may be obtained by requiring that the minimum scale of compact objects which are held together by gravity $\lambda_{Jeff} \sim R_*$ is smaller than $12 \, {\rm km}$. Substituting $R_\odot$ by $R =12 \, {\rm km}$ in Eq. (\ref{astcons}) one obtains the constraint
\begin{equation}
\kappa < 10^{-2} \, {\rm m^5 \, kg^{-1} \, s^{-2}}\,,\label{tcons}
\end{equation}
which is more than $9$ orders of magnitude stronger than the limit given in Eq. (\ref{astcons}) and more than $7$ orders of magnitude tighter than the solar constraints obtained in \cite{Casanellas:2011kf}. Eq. (\ref{tcons}) also strengthens the constraint given in \cite{Pani:2011mg} by two orders of magnitude.
\section{\label{conc}Conclusions}
In this letter we determined astrophysical and cosmological constraints on a recently proposed EiBI theory of gravity. We have shown, using a generalized version of the Zel'dovich approximation, that in this theory a pressureless cold dark matter fluid has a non-zero effective Jeans length. We used this result to provide a physical interpretation of the fundamental length $R_*$, mass $M_*$ and density $\rho_*$ of the theory and to obtain stringent limits on $\kappa$, the only additional parameter of theory with respect to general relativity.
The cosmological and astrophysical constraints can be summarized as $\kappa \lsim G R^2$, where $R$ is either the Hubble radius at the onset of primordial nucleosynthesis ($t_{nuc} \sim 1 \, {\rm s}$) or the scale of compact objects which are held together by gravity. The strongest astrophysical limit ($\kappa < 10^{-2} \, {\rm m^5 \, kg^{-1} \, s^{-2}}$) is about $10$ orders of magnitude stronger than big bang nucleosynthesis constraints, yielding a constraint on the fundamental mass ($M_* < 5 \, M_\odot$) and density ($\rho_* > 9 \times 10^{18} \, {\rm kg \, m^{-3}}$) of the theory. These limits imply that large changes with respect to the dynamics of the standard cosmological model in the early Universe are expected in the context of the EiBI theory of gravity but only for cosmic times $t < 10^{-5} \, {\rm s}$.
\begin{acknowledgments}
We thank Margarida Cunha for useful comments on the manuscript. This work is partially supported by FCT-Portugal through the project CERN/FP/116358/2010.
\end{acknowledgments}
|
1,108,101,565,644 | arxiv | \section{Introduction}
Frustrated magnetic systems are characterized by competing magnetic interactions, which allow for a number of different, yet energetically nearly degenerate, magnetic states.
As a result, already a subtle perturbation, e.g., a weak additional interaction or anisotropy, can completely change the magnetic ground state.
This leads to a vast variety of magnetic phases observed in such systems. \cite{lacroix2011introduction}
Prominent examples are highly-entangled disordered spin-liquid phases \cite{balents2010spin} on the one hand, and long-range-ordered spiral spin structures\cite{lacroix2011introduction} on the other hand, which besides the difference in the magnetic ordering also exhibit conceptually different excitations.
Namely, in classical long-range-ordered magnetic states, the elementary excitations are magnons (spin waves), i.e., collective spin-1 excitations of the magnetic order with precisely determined dispersion relations.\cite{des1962spin}
On the contrary, in spin liquids, typical excitations are spinons, i.e., deconfined fractional spin-1/2 excitations into which magnon fractionalize, that form a broad excitation continuum.\cite{lake2005soliton,zaliznyak2005quantum}
One of the simplest frustrated spin system is a zigzag spin-1/2 chain, where the nearest- and next-nearest-neighbor interactions compete.\cite{okunishi2003magnetic, hikihara2008vector, sudan2009emergent, hikihara2010magnetic}
The corresponding phase diagram encompasses spin-density-wave (SDW), vector-chiral (VC), as well as multipolar/spin-nematic phases, which are distinguished by specific one-dimensional correlations.
Yet, in most real systems, finite interchain interactions and anisotropies allow for spin correlations to expand in three dimensions and even lead to another magnetic ground state.\cite{furukawa2010chiral, schapers2013thermodynamic, du2016magnetic}
Such compounds display features of both one- and two-/three-dimensional spin systems.
Namely, at low temperature and low energies a particular system may behave as a three-dimensionally ordered magnet,\cite{masuda2005spin, enderle2005quantum} while at higher energies, at elevated magnetic fields, or above the ordering temperature, a one-dimensional response may prevail.\cite{enderle2010two, orlova2017nuclear}
Examples of coexistence of three-dimensional magnon and one-dimensional spinon excitations have been reported for uniform spin chains,\cite{lake2005quantum, bera2017spinon} yet for zigzag chains such reports are scarce.\cite{enderle2005quantum, enderle2010two}
\begin{figure}[b]
\centering
\includegraphics[width=\columnwidth]{exchange-network_2.pdf}
\caption{(a) The crystal structure of $\beta$-TeVO$_4$. The coordination square pyramids of vanadium are shown in blue, oxygen ions are in red, and tellurium ions are light brown. The monoclinic unit cell ($Z$\,=\,4) is presented with a thin black line. (b) Magnetic exchange network in $\beta$-TeVO$_4$. The dominant intrachain couplings are shown in red ($J_1$ and $J_2$), while the weaker interchain couplings are shown in green ($J_3$ and $J_4$).}
\label{fig-network}
\end{figure}
Here we focus on a quasi-one-dimensional frustrated zigzag chain compound $\beta$-TeVO$_4$ [space group $P2_1/c$, V(4e), $a$\,=\,4.3919(1)\,\AA, $b$\,=\,13.5155(1)\,\AA, $c$\,=\,5.4632(1)\,\AA\, and $\beta$\,=\,90.779(1)$^\circ$, at $T$\,=\,10\,K].\cite{pregelj2015spin}
Spin chains are built of distorted corner-sharing VO$_5$ pyramids with magnetic V$^{4+}$ ($S$\,=\,1/2) ions that run along the crystalographic $c$ axis [Fig.\,\ref{fig-network}(a)]. \cite{meunier1973oxyde, savina2011magnetic}
The nearest-neighbor superexchange interaction $J_1$\,$\sim$\,$-$38\,K is ferromagnetic, while the next-nearest-neighbor interaction $J_2$\,$\sim$\,$-$0.8\,$J_1$ is antiferromagnetic, \cite{pregelj2015spin} thus imposing a strong magnetic frustration.
Additional interchain interactions were estimated to be an order of magnitude weaker and also frustrated.
These are responsible for a sequence of low-temperature magnetic transitions.
At $T_{N1}$\,=\,4.65\,K the paramagnetic phase is replaced by an SDW phase with the incommensurate amplitude-modulated (ICAM) magnetic order defined by the magnetic wave vector {\bf k}\,$\approx$\,($-$0.20,\,0,\,0.41). \cite{pregelj2015spin, pregelj2016exchange}
The SDW phase is succeeded by an unconventional spin-stripe phase, which develops at $T_{N2}$\,=\,3.28\,K.
In this phase an additional weak ICAM magnetic component with {\bf k'}\,=\,({\bf k}+{\bf $\Delta$k}) and {\bf $\Delta$k}\,$\approx$\,($-$0.03,\,0,\,0.02) introduces a nanometer-sized stripe modulation of the spin structure.
Finally, at $T_{N3}$\,=\,2.28\,K, {\bf $\Delta$k} becomes zero and an elliptical-spiral magnetic order is established reflecting the dominance of the VC correlations at low temperatures.\cite{pregelj2016exchange}
Clearly, $\beta$-TeVO$_4$ can be rather well described as a frustrated zigzag spin-1/2 chain, yet the fascinating spin-stripe phase must emerge due to perturbative effects, i.e., probably due to weak interchain interactions and sizable exchange anisotropy ($\sim$20\%) of $J_1$ and $J_2$ that compete with fourth-order spin couping terms.\cite{pregelj2016exchange}
A comprehensive description thus requires a detailed knowledge about the exchange network, which calls for an in-depth investigation of the magnon and spinon excitations.
In this study we investigate the magnetic excitation spectrum in the ground state of $\beta$-TeVO$_4$ using inelastic neutron scattering.
Our results reveal a coexistence of sharp magnon excitations and a broad spinon-like excitation continuum.
The pronounced asymmetry of the continuum is in line with the fact that the $J_2/J_1$ ratio is close to $-$1.\cite{ren2012spinons}
Moreover, the modeling of the sharp dispersion by a linear spin-wave theory enabled us to estimate intrachain interactions and their anisotropies as well as to quantify the main weak interchain interactions.
\section{Experimental}
Inelastic neutron scattering experiments were performed using a sample consisting of five single crystals having a total mass of $\sim$0.8\,g.
The $a$ and $c$ axes defined the scattering plane.
The high-energy neutron scattering experiments, focused on excitations between 0.7 and 13\,meV, were performed on the thermal-neutron triple-axis spectrometer EIGER,\cite{stuhr2017thermal} at SINQ, at the Paul Scherrer Institute, Villigen, Switzerland.
The use of double focusing PG(002) monochromator and analyzer lead to the energy resolution at the elastic line of 0.7\,meV.
The final wave vector $k_f$\,=\,2.66\,\AA$^{-1}$ was filtered by a PG filter.
The low-energy experiments, focusing on excitations up to 3\,meV, were performed on the cold-neutron triple-axis spectrometer THALES, at the Institut Laue-Langevin, Grenoble, France.
In this experiment double focusing PG(002) monochromator, analyzer and a Be filter cooled to 77\,K (to remove the higher-order contamination) were used.
Here, $k_f$\,=\,1.55\,\AA$^{-1}$ and 1.3\,\AA$^{-1}$ setups lead to the resolution at the elastic line of $\sim$0.15\,meV and 0.1\,meV, respectively.
The temperature was in both cases controlled using standard ILL orange cryostats.
All measurements were performed at 1.5\,K.
In experiment performed at EIGER the sample crystals were co-aligned within $\delta\phi$\,$<$\,3$^\circ$, which is sufficient for the corresponding experimental resolution.
On the other hand, in the THALES experiment one crystal was shifted for $\sim$12$^\circ$.
To avoid the spurious signal, the dispersion relations were measured at high reciprocal space values, i.e., the dispersion along $a^*$ near reflections with large $H$ and the dispersion along $c^*$ near reflections with large $L$.
A a result, the excitation spectra measured in the same part of the Brillouin zone at the two instruments differ only in resolution and signal to noise ratio, indicating that THALES data were not effected by misalignment of the sample.
\section{Results}
\subsection{Dispersion along the spin chains}
\begin{figure}[!]
\centering
\includegraphics[width=\columnwidth]{E-cuts_l-dep_q-cuts_E-dep_v3.pdf}
\caption{(a) Energy scans at several positions in the reciprocal space measured at EIGER at 1.5\,K. Sharp features are magnon excitation modes, while broad background is due to the spinon continuum. (b) $L$ scans at ($-0.8,~0,~L$) at fixed energies. The broad feature implies the presence of the spinon continuum.}
\label{fig-Ecut-lcut}
\end{figure}
We first inspected the excitation spectrum along the spin chains (almost along $c^*$), which should be dictated mainly by the strong intrachain interactions.
We measured a series of energy scans at fixed positions in reciprocal space, crossing the magnetic reflections at ($-$0.8,~0,~$-$0.4) and ($-$0.8,~0,~0.6).
The excitation spectrum below 4\,meV exhibits several sharp features with the width comparable to the experimental resolution [Fig.\,\ref{fig-Ecut-lcut}(a)], which are therefore most likely associated with magnon excitations.\cite{des1962spin}
We note that $E$\,=\,4\,meV is comparable to the strength of $J_1$ and $J_2$, estimated to amount 3.3 and 2.6 meV, respectively. \cite{pregelj2015spin}
In addition to these sharp features we find a broad excitation continuum extending across a wide region in energy and reciprocal space.
The existence of the latter is even more apparent when measuring the neutron-scattering intensity at fixed energies along $c^*$, i.e., changing the $L$ value [Fig.\,\ref{fig-Ecut-lcut}(b)].
The combined results are summarized in a map plot in Fig.\,\ref{fig-ql}(a) that reveals the coexistence of sharp features, extending up to $\sim$4\,meV, while a broad excitation continuum becomes pronounced above 2\,meV and extends up to $\sim$12\,meV.
The latter exhibits a dome-like shape, which reaches the maximum energy at $L$\,$\sim$\,0.5 r.l.u. and drops to zero energies at $L$\,$\sim$\,0 and 1 r.l.u., and is thus most probably associated with two-spinon excitations.\cite{lake2005soliton,zaliznyak2005quantum}
\begin{figure}[!]
\centering
\includegraphics[width=\columnwidth]{map_ql_v3_b.pdf}
\caption{(a) The energy map for ($-0.8,~0,~L$) cut in the reciprocal space measured at EIGER at 1.5\,K. (b) The calculated magnon dispersion for the model presented in text added to the spinon continuum calculated in Ref.\,\onlinecite{onishi2015magnetic}. In the calculated map, experimental resolution has been already taken into account. The solid lines in (a) and (b) represent the calculated spin-wave dispersion, while the dashed lines represent the upper and lower boundaries of a two-spinon continuum calculated for a uniform antiferromagnetic spin-1/2 chain, where the magnitude of the exchange $J$ equals $J_1$.}
\label{fig-ql}
\end{figure}
In the next step, we inspected the excitation spectrum along $c^*$ at lower energies and with higher resolution using THALES spectrometer at ILL.
However, we did not detect any additional excitation branches, which implies that weak additional interactions that could induce low-energy excitations must be perpendicular to the chains, as suggested by earlier studies. \cite{pregelj2015spin, saul2014density, weickert2016magnetic}
\subsection{Dispersion perpendicular to the spin chains}
Next, we focused on the dispersion perpendicular to the chains, which should be significantly influenced by the weak interchain interactions.
In contrast to the measurements performed along $c^*$, we were unable to resolve any significant change in the excitation spectrum along $a^*$, when measuring with the high-energy setup at the EIGER spectrometer.
Hence, we performed additional high-resolution measurements at the THALES spectrometer at ILL.
This allowed us to inspect excitations with energies above $\sim$0.3\,meV, as this is the point where the intensity of the elastic contribution falls below the background level.
We performed energy scans at fixed positions in reciprocal space along $a^*$, crossing the (0.8,~0,~0.4) magnetic reflection.
The excitation spectrum exhibits several sharp features with the width comparable to the experimental resolution (Fig.\,\ref{fig-Ecut}).
In addition, we observed the broad excitation continuum above $\sim$2\,meV, which appears almost independent of $H$.
All results are summarized in the map plot in Fig.\,\ref{fig-qh}(a).
Obviously, sharp features are here better resolved than in the measurements along $c^*$, as for this orientation they appear at lower energies and are thus clearly separated from (they appear below) the continuum.
\begin{figure}[!]
\centering
\includegraphics[width=\columnwidth]{E-cuts_h-dep_v3all.pdf}
\caption{(a) Energy scans at several positions in the reciprocal space measured at THALES at 1.5\,K. Sharp features are magnon excitation modes, while broad background is due to the spinon continuum. (b) Energy scan at the magnetic peak (0.8,~0,~0.42) measured at 1.5\,K (symbols) and simulation (line) for calculated magnon dispersion and spinon continuum calculated in Ref.\,\onlinecite{onishi2015magnetic}. In the calculated spectrum, experimental resolution has been already taken into the account.}
\label{fig-Ecut}
\end{figure}
\begin{figure}[!]
\centering
\includegraphics[width=\columnwidth]{map_qh_v3_b.pdf}
\caption{(a) The energy map for ($H,~0,~0.42$) cut in the reciprocal space measured at THALES at 1.5\,K. The dashed vertical lines emphasize the scan shown in Fig.\,\ref{fig-Ecut}. (b) The calculated magnon dispersion for the model presented in text added to the spinon continuum calculated in Ref.\,\onlinecite{onishi2015magnetic}. In the calculated map, experimental resolution has been already taken into account. The solid lines in (a) and (b) represent the calculated spin-wave dispersion.}
\label{fig-qh}
\end{figure}
\section{Discussion}
\subsection{Spinon continuum}
Our measurements reveal a broad excitation continuum that is characteristic of fractional spinon excitations, which are inherent to spin-1/2-chain systems.\cite{lake2005soliton,zaliznyak2005quantum}
The main contribution to the spinon continuum is due to two-spinon excitations, which for a uniform antiferromagnetic spin-1/2 chain occur between $E_{\text{low}}$\,=\,$\pi/2$\,$J|$sin($k$)$|$ and $E_{\text{high}}$\,=\,$\pi$\,$J|$sin($k/2$)$|$, where $J$ is the antiferromagnetic exchange constant.\cite{muller1981quantum, caux2006four, lake2005soliton, zaliznyak2005quantum}
The resulting
dynamical structure factor -- the quantity measured by inelastic neutron scattering -- diverges at the lower boundary and decreases with increasing energy, $E$, as given by expression 1/($E^2$\,$-$\,$E_{\text{low}}^2(k)$)$^{1/2}$.\cite{muller1981quantum}
Assuming that the magnitude of $J$ equals $J_1$, we plot the corresponding two-spinon-continuum boundaries (Fig.\,\ref{fig-ql}).
Clearly, the measured continuum exists already below the lower theoretical boundary $E_{\text{low}}$, highlighting the discrepancy between $\beta$-TeVO$_4$ and the uniform spin-1/2 chain model.
In fact, our results are more similar to the response expected in frustrated zigzag spin-1/2 chain with ferromagnetic $J_1$, where the lower boundary is significantly lowered and becomes asymmetric with increasing next-nearest neighboring interactions.\cite{ren2012spinons, onishi2015magnetic}
Moreover, asymmetry and lowering of the spinon continuum has been observed in LiCuVO$_4$,\cite{enderle2010two} which is one of the most studied frustrated ferromagnetic zigzag spin-1/2 chain compounds.
However, in contrast to the latter, our data do not show significant scattering above the two-spinon continuum boundary $E_{\text{high}}$ (Fig.\,\ref{fig-ql}), suggesting that, if present, the contribution of four-spinon excitation in $\beta$-TeVO$_4$ is far less pronounced.
\subsection{Spin-wave modeling}
The acquired data provide detailed information about the interactions that constitute the exchange network in $\beta$-TeVO$_4$.
Previous experimental studies offered a good estimate of the main exchange interactions, $J_1$ and $J_2$, \cite{pregelj2015spin} an approximate estimation of their anisotropies, \cite{pregelj2016exchange} and a rough assessment of the interchain interactions. \cite{pregelj2015spin}
On the other hand, density-functional-theory calculations, \cite{saul2014density,weickert2016magnetic} investigating the exchange interactions up to the eight-shortest vanadium-vanadium distances, offered a quantitative estimate for all interactions included in calculations.
Since these calculations comply with previous experimental results,\cite{pregelj2015spin,pregelj2016exchange} the derived exchange parameters ensure a good foundation to describe present inelastic-neutron-scattering results as well as the modulation of the magnetic order perpendicular to spin chains.
In order to refine the exchange-network parameters we first inspect the origin of the magnetic modulation reflecting in the magnetic wave vector.
The two intrachain interactions $J_1$ and $J_2$ determine the magnetic modulation along the chains, \cite{pregelj2015spin} i.e., along $c^*$, while the modulation perpendicular to the chains (either along $a^*$ or $b^*$) depend on the weak interchain interactions.
In the reciprocal lattice units, the modulation period along $a^*$ is almost exactly half of the modulation period along $c^*$.
This can be explained already by a single exchange interaction that couples vanadium ions in the consecutive sites along the chain that are located in the neighboring chains, e.g., connecting sixth- or seventh-nearest-neighbor vanadium ions (the former being denoted as $J_4$ in Fig.\,\ref{fig-network}).
On the other hand, the modulation along $b^*$ coincides with the crystal structure, indicating that the sequence of the exchange interactions between the two equivalent magnetic ions along the $b$ axis favors parallel alignment.
Hence, such sequence must involve an even number of antiferromagnetic interactions.
To describe the observed magnetic modulation one clearly has to take into account interchain interactions.
To avoid overparametrization, our goal is to find a minimal set of exchange parameters.
Furthermore, as the magnetic wave vector in $\beta$-TeVO$_4$ is incommensurate along $a^*$ and $c^*$, we had to limit our analysis to coplanar magnetic structures with evenly sized magnetic moments to make calculations manageable.
Using the SpinW library \cite{toth2015linear} for numerically simulating magnetic structures and excitations based on the linear spin-wave theory, we thus tested different exchange combinations, involving interactions up to the twelfth-shortest distance between vanadium ions.
Surprisingly, we find that besides the intrachain interactions, $J_1$ and $J_2$, already two interchain interactions, $J_3$ and $J_4$ (Fig.\,\ref{fig-network}), corresponding to the fifth- and sixth-shortest distances between vanadium ions, \cite{saul2014density} are sufficient to reproduce the magnetic wave vector.
In particular, $J_3$ has to be antiferromagnetic and $J_4$ has to be ferromagnetic, which nicely complies with the density-functional-theory calculations. \cite{saul2014density, weickert2016magnetic}
We note that other compatible exchange combinations were dismissed, because they involved more interactions.
Next, we calculate spin-wave (magnon) dispersion relations based on the linear spin-wave theory, employing SpinW library, \cite{toth2015linear} and compare them with the inelastic neutron scattering results.
We consider exchange interactions $J_i$ ($i$\,=\,1-4), with $J_1$ and $J_2$ having a sizable exchange anisotropy along $b$, $\delta_i^b$ ($i$\,=\,1,2), i.e., imposing either easy-plane or easy-axis anisotropy that is slightly distorted by a small additional anisotropy along $c$, $\delta_i^c$ ($i$\,=\,1,2), as implied by the recent anisotropy study,\cite{pregelj2016exchange} while $J_3$ and $J_4$ were assumed to be completely isotropic.
The corresponding Hamiltonian thus has the form
\begin{align}
\begin{split}
\label{Hamiltonian}
H = J_1\sum_{n,j}({\bf S}_{n,j}\cdot{\bf S}_{n,j+1}+\delta^b_1 S^b_{n,j}S^b_{n,j+1}+\delta^c_1 S^c_{n,j}S^c_{n,j+1})\\
+ J_2\sum_{n,j}({\bf S}_{n,j}\cdot{\bf S}_{n,j+2}+\delta^b_2 S^b_{n,j}S^b_{n,j+2}+\delta^c_2 S^c_{n,j}S^c_{n,j+2})\\
+J_3\sum_{\langle n, m \rangle,j}{\bf S}_{n,j}\cdot{\bf S}_{m,j-1}
+J_4\sum_{\langle n, m' \rangle,j}{\bf S}_{n,j}\cdot{\bf S}_{m',j+1},
\end{split}
\end{align}
where $n$ labels the chains, $j$ the position of the spin along the chain, while $m$ and $m'$ denotes neighboring chains along the $b$ and $a$ axis, respectively.
To make calculations manageable, we assume that the size of all V$^{4+}$ magnetic moments is fixed to 1\,$\mu_B$, where $\mu_B$ is the Bohr magneton, and that the magnetic structure is coplanar.
Finally, we consider also the presence of the spinon continuum, which appears to dominate the excitation spectrum above 2\,meV.
In particular, we combine our spin-wave calculations with the spinon contribution calculated previously by density-matrix-renormalization-group (DMRG) method for the $J_1$\,=\,$-$$J_2$ spin-1/2 chain\cite{onishi2015magnetic} and obtain the magnetic structure factor as a sum of two contributions
\begin{align}
\begin{split}
\label{SWplusCONT}
S(E,{\bf Q}) =~ &f(E)S_{\text{sw}}(E,{\bf Q}) + \\
& [1-f(E)]S_{\text{cont}}(E,{\bf Q}),
\end{split}
\end{align}
where $S_{\text{sw/cont}}(E,{\bf Q})$ is spin-wave/spinon-continuum contribution, $f(E)$\,=\, 1/($\exp[(E\,-\,E_c)/\delta E]+1)$, $E_c$ denotes the energy of the transition between spin-wave and spinon type of excitations that has the width of $\delta E$.
The best match with the experiment was obtained by adjusting $E_c$ and $\delta E$, while the experimental resolution was approximated by convolving the calculated energy spectra with Gaussian function with the width determined from the elastic incoherent vanadium scattering.
The results showing the best agreement with the experiment are presented in Fig.\,\ref{fig-ql}(b) and Fig.\,\ref{fig-qh}(b), yielding {\bf k}$_{\text{IC}}$\,=\,($-$0.208,~0,~0.419) and magnetic moments lying in the $ac$ plane.
The corresponding parameters are $J_1$\,=\,$-$38\,K, $J_2$\,=\,38\,K, $J_3$\,=\,3\,K, $J_4$\,=\,$-$1.9\,K, $\delta_1^b$\,=\,0.106, $\delta_2^b$\,=\,$-$0.126, $\delta_1^c$\,=\,0.01, and $\delta_2^c$\,=\,0.01.
We evaluate the reliability of the derived parameters by inspecting their sensitivity to the changes of the magnetic-ordering plane, which yields the uncertainty of $\sim$5\% for $J_i$ ($i$\,=\,1-4), $\sim$20\% for $\delta_i^b$ ($i$\,=\,1,2), and $\sim$50\% for $\delta_i^c$ ($i$\,=\,1,2).
On the other hand, $E_c$ was found to roughly scale with the square root of the energy resolution, yielding $E_c$\,=\,1.3 and 2.2\,meV for THALES and EIGER instruments, respectively, whereas $\delta E$\,=\,0.5\,meV was the same.
The derived parameters are in good agreement with previous studies, \cite{pregelj2015spin, pregelj2016exchange, saul2014density,weickert2016magnetic} but also reveal a few interesting new points.
Namely, both large exchange anisotropies, $\delta^b_1$ and $\delta^b_2$, yield ferromagnetic contributions, i.e., the former increases the ferromagnetic $J_1$, while the latter reduces the antiferromagnetic $J_2$.
This suggests that they have similar origins, which could be associated with hopping of electrons between the ground states and excited states of vanadium ions in combination with the spin-orbit coupling.\cite{von2002anisotropic, eremin2006unconventional}
However, since $J_1$ and $J_2$ have different signs, $\delta^b_1$ and $\delta^b_2$ impose different types of anisotropy, i.e., the former imposes an easy-axis anisotropy, while the latter imposes an easy-plane one.
As a result, the effective ratio $(J_1/J_2)^i$, i.e., $J_1(1+\delta^i_1)/J_2(1+\delta^i_2)$, for the $b$ spin component ($i$\,=\,$b$) differs by $\sim$20\% compared to the same ratio for the $a$ and $c$ spin components ($i$\,=\,$a,c$).
This complies with the estimate derived in Ref.\,\onlinecite{pregelj2016exchange} and thus corroborates the scenario that the exchange anisotropy imposes two distinct modulations in the spin-stripe phase, i.e., for the $a$-$c$ and for the $b$ spin component.
In addition, the weak anisotropy $\delta^c$ reduces the energy difference between magnetic moments lying in the $ab$ and $bc$ planes, which is in line with the perpendicular orientations of the cycloids in the adjacent chains observed in the ground state phase.\cite{pregelj2016exchange}
Moreover, our analysis evaluates the interchain couplings that induce long-range magnetic ordering and are essential for the formation of the intriguing spin-stripe phase.
Finally, we note that the derived parameters may deviate from the exact solution of (\ref{Hamiltonian}) due to approximations undertaken in the data analysis, e.g., coplanar magnetic structure and same-sized magnetic moments.
In addition, the magnetic ground-state minimization and linear-spin-wave-theory approach used in SpinW library are optimized for large, i.e., classical, spins and three-dimensional magnetic lattices.
The fact that the agreement between the experiment and calculations is not perfect (Figs.\,\ref{fig-ql} and \ref{fig-qh}) thus suggests that there could be other contributions to the spin Hamiltonian that were not identified.
Nevertheless, the overall agreement between the experimental observations and theoretical modeling suggests that despite its limitation the proposed exchange model is not far from the actual situation and thus represents a promising starting point for more involved theoretical studies.
\section{Conclusion}
In conclusion, our low-temperature inelastic neutron scattering results indicate the coexistence of magnon and spinon excitations in $\beta$-TeVO$_4$, which has been so far rarely observed in frustrated zigzag spin-1/2 chain compounds.
Employing the linear spin-wave theory we were able to model the observed dispersion and determined the main exchange interactions and anisotropies.
Our results comply with the preceding studies of $\beta$-TeVO$_4$ and thus support the hypothesis that the exchange anisotropy and interchain interactions are most likely responsible for the establishment of the peculiar spin-stripe phase.\cite{pregelj2015spin, pregelj2016exchange}
The derived exchange parameters thus open a way for detailed theoretical studies of the corresponding spin-stripe-formation mechanism that is still puzzling.
\begin{acknowledgments}
We acknowledge the financial support of the Slovenian Research Agency (projects Z1-5443 and program No. P1-0125) and the Swiss National Science Foundation (project SCOPES IZ73Z0\_152734/1).
The neutron diffraction experiments were performed at the Swiss spallation neutron source SINQ, at the Paul Scherrer Institute, Villigen, Switzerland, and at the reactor of the Institute Laue-Langevin, Grenoble, France.
We are grateful to M. Enderle for fruitful discussion and local support at ILL.
\end{acknowledgments}
|
1,108,101,565,645 | arxiv | \section{\bf Introduction}
\label{sec:intro-jja}
The physics of Josephson junctions arrays (JJA) has been a subject of
significant interest in the last ten years \cite{mooij-schon}. A large
number of studies, both experimental
\cite{mooij_van-wees_geerligs_peters_fazio_schon,van-der-zant_geerling_mooij-92,tighe-tuominen-hergenrother-tinkham,van-der-zant-thesis,delsing-chen-haviland-harada-claeson,van_der_zant-fritschy-orlando-mooij}
and theoretical
\cite{fazio-schon-91,doniach,lozovik,zwerger,fishman-stroud,jose_1984,jacobs-jose-novotny-goldman,jacobs-jose,simkin,granato-contentino,roddick-stroud,ariosa-beck,kim-choi,zaikin,simanek,kim-choi2}
have been devoted to them. Initially, part of the
interest in JJA came from their close relation to one of the most
extensively studied theoretical spin models, i.e. the classical
2-D XY model for which JJA give a concrete experimental realization.
In non-dissipative JJA the two main contributions to the energy are the
Josephson coupling between superconducting islands due to Cooper
pair tunneling, and the electrostatic energy arising from local
deviations from charge neutrality.
In the initial experimental studies, the size of the islands was
large enough so that the charging energy contributions were very small,
thus making the arrays' behavior effectively semi-classical.
Recent advances in submicron technology have made it possible to fabricate
relatively large arrays of ultrasmall superconducting islands separated
by
insulating barriers. These islands can have areas of
the order of a few ${\mu}{\rm m}^2$,
with self capacitances $C_{\rm s}\approx 3\times 10^{-2}$ fF,
and nearest neighbors' mutual capacitance $C_{\rm m}\approx 1$ fF
\cite{van-der-zant_geerling_mooij-92}. Note
that the mutual capacitance can be at least two orders of magnitude larger
than the self capacitance. Therefore, these arrays have charging energy
contributions, $E_c$, large enough so that quantum fluctuation
effects are of paramount importance.
In the Delft \cite{van-der-zant_geerling_mooij-92} and Harvard
\cite{tighe-tuominen-hergenrother-tinkham} experiments, the island sizes
were kept constant, while varying the normal state junction resistance,
which in turn changes the Josephson coupling energy, $E_J$. This allows
one to obtain arrays with values of the quantum parameter
\begin{equation}
\label{alpha_m}
\alpha_{\rm m} = \frac{E_{C_{\rm m}}}{E_J}
\end{equation}
in the range [0.13--4.55]
\cite{van-der-zant_geerling_mooij-92}, or
values as high as 33 \cite{tighe-tuominen-hergenrother-tinkham}. In
this equation we have used the definition of charging energy,
\begin{equation}
\label{def-charging-energy}
E_{C_{\rm m}} = \frac{e^2}{2C_{\rm m}}.
\end{equation}
The experimental systems can be modeled by a quantum
generalization of the classical XY model, because the phase of the order
parameter associated with each one of the islands is canonically
conjugate to its excess Cooper pair number. The magnitude of $\alpha_{\rm m}$
determines the relevance of the quantum fluctuations. For small
$\alpha_{\rm m}$ the quantum fluctuations of the phases are small and
the system is well modeled by a renormalized classical 2-D XY model.
The nature of the phase transition in the classical 2-D XY model
is well understood, whereas in its quantum mechanical generalization
there still are unsettled issues. One of the most notorious of these
is the possibility of having a low temperature instability of the
superconducting state.
A possible reentrant transition was originally found within a mean field
theory treatment of the self-capacitive XY model
\cite{simkin,kim-choi,zaikin,simanek}. An explicit
two-dimensional study of the self-capacitative XY model, within a WKB
renormalization group (WKB-RG) analysis also found evidence of a low
temperature reentrant instability, triggered by a quantum fluctuation
induced proliferation of vortices \cite{jose_1984}.
Recently, Kim and Choi have studied the quantum induced fluctuations
in these arrays, using a variational method \cite{kim-choi2}.
They found that there is a range of values of the ratio of charging
to Josephson energy, for which there is a low temperature reentrance
from a superconducting to a normal state. Similar results had
been obtained by Simanek, also using a variational calculation,
see for example Ref. \cite{simanek}.
A non-perturbative quantum Monte Carlo study of the self-capacitive model
found a low temperature transition, but between two superconducting
states \cite{jacobs-jose-novotny-goldman}.
The fully frustrated version of this model was also
studied by quantum QMC and it yielded a larger jump discontinuity in
the superfluid density as compared to the one in zero field as well as
the critical temperature one order of magnitude
higher \cite{jacobs-jose-novotny-goldman}. A
more recent analysis of the WKB-RG analysis has shown that, to lowest
order in the quantum fluctuations, it must have the same critical temperature
for a quantum induced phase transition (QUIT) \cite{jose-rojas}.
A recent QMC study of the fully frustrated
self-capacitive model by Mikalopas et al.
\cite{mikalopas-jarrel-pinski-chung-novotny}
has suggested that the unusually large jump in the superfluid
density is dominated by metastability effects due to the particular
nature of the excitations in the frustrated model.
This result is in agreement then with the reanalysis of the RG
equations. However, this study
was carried out at relatively high temperatures and the question about the
existence of a QUIT, both in the frustrated and unfrustrated cases
remains open. We deal extensively with the later question here.
Other studies find within MFT that
to have reentrance it is necessary to include off-diagonal
capacitances \cite{fishman-stroud}, while others do not agree with
this finding
\cite{doniach,granato-contentino,roddick-stroud,ariosa-beck}.
The search for a reentrant type transition is encouraged by
some evidence of low temperature instabilities found
experimentally in arrays of Josephson junctions
\cite{mooij_van-wees_geerligs_peters_fazio_schon}, ultrathin amorphous
films \cite{belevtsev-komnik-fomin}, a multiphase high-$T_c$
system \cite{seyoum-riitano-bennett-wong}, and in granular superconductors
\cite{lin-shao-wu-hor-jin-chu-evans-bayuzick}.
Most theoretical studies have been carried out using
the self-capacitive model and different kinds of MFT
or self consistent harmonic approximations (SCHA)\cite{ariosa-beck}.
As already mentioned, these studies do not agree among each other
on some of the properties of the phase diagram, in particular about the
possible existence of a low temperature instability of the
superconducting state.
No study has been carried out that closely represents the experimental
systems where both the self and mutual capacitances
are explicitly included. The goal of this paper is to
consider a model that is expected to represent
the characteristics of the Delft experiments. In particular, we
concentrate on calculating the phase diagram using different
theoretical tools.
One of the main results of this paper is presented in Fig.
\ref{fig:fig01-phase-diagram}
which shows the $\alpha_{\rm m}$ vs. $T$ phase
diagram for an array with $C_{\rm m}> C_{\rm s}$ both for the
unfrustrated ($f=0$) and fully frustrated ($f=1/2$) cases. The left hand side
of this diagram shows the superconducting to normal phase boundary (S--N)
as data points with error bars joined by a continuous line.
These data points were calculated using a QMC
method, to be described later in the paper.
We also show (as squares) the experimental results
taken from Refs. \cite{van-der-zant_geerling_mooij-92,van-der-zant-thesis}.
For $f=0$ at small values of $\alpha_{\rm m}$, the theoretical and experimental
results agree quantitatively quite well with each other
and with the semiclassical WKB-RG approximation. On the other hand,
they only qualitatively agree for the $f=1/2$ case and
on the superconducting to insulating phase boundary. The
normal to insulating transition line is shown to the
right of the phase diagram. The latter is just a
tentative boundary since our numerical calculations were not
reliable enough to give the definitive location of this line,
as also happens in experiments.
The error bars in the calculated points used to draw the N-I line
represent a crude estimation of the region where the inverse
dielectric constant is different from zero. However, the
issue of convergence of the calculation to the path integral limit
is not resolved by these error bars. As we will
explain in the main body of the paper we found further evidence for a low
temperature instability of the superconducting state in our
numerical calculations.
We found that this instability depends strongly on the magnitude of $\alpha_{\rm m}$ and
the finite size of the imaginary time axis in the QMC
calculations. The latter finding sets strict constraints on some of the
reentrant type behavior found in previous theoretical studies.
Other studies have found reentrance very close to the
superconducting to insulating transition \cite{simanek}.
This possibility is harder to study from our Monte Carlo calculations.
The physical content of the phase diagram
is generally understood in terms of the interplay between the Josephson
and charging energies. For small $\alpha_{\rm m}$ and high temperature
the spectrum of excitations is dominated by thermally excited
vortices, which drive the superconducting to normal transition
as the temperature increases,
while the charging energy contributes with weak quantum fluctuations of the
phases. The latter produces, after averaging over the quantum fluctuations, an effective
classical action with a renormalized Josephson coupling that
lowers the critical temperature \cite{jose_1984}.
For large $\alpha_{\rm m}$ and low temperatures the charging energy
dominates. The excitations in this limit are due to the
thermally assisted Cooper pair tunneling
that produces charged polarized islands.
At low temperatures, there is not enough thermal energy to overcome the
electrostatic coulomb blockade so that the Cooper pairs are localized
and the array is insulating.
As the temperature increases, the electric dipole excitations can unbind,
driving an insulating to conducting transition (I--C). In the limit
$C_{\rm m}\gg C_{\rm s}$, it was suggested that the I-C transition could
be of the Berezinskii-Kosterlitz-Thouless (BKT) type,
for in this case the interaction between charges is essentially logarithmic
\cite{yaks-87,widom-badjou,mooij_van-wees_geerligs_peters_fazio_schon}.
However, for the experimental samples it has been shown
that rather than a true phase transition what is measured appears to be
a crossover between an insulating to conducting phase, characterized
by thermally activated processes
\cite{tighe-tuominen-hergenrother-tinkham,delsing-chen-haviland-harada-claeson}.
It is likely that the reason for the crossover is the short
screening length present in the samples ($\Lambda \approx 20$ lattice
sites). Both experimental groups
\cite{tighe-tuominen-hergenrother-tinkham,delsing-chen-haviland-harada-claeson}
find that a simple energetic argument gives an explanation for
the activation energy found in the experiments. Furthermore, the nature
of this crossover may be linked to thermal as well as to dynamical
effects. As we shall see, theoretically the I-N phase is hard to study
in detail.
For large values of $\alpha_{\rm m}$ the model can be approximated by
a 2-D lattice Coulomb gas, where a perturbative expansion can be carried out
using $E_J$ as a small parameter. This type of calculation
was performed in Ref. \cite{fazio-schon-91} in the limit
$C_{\rm s}\ll C_{\rm m}$. The analysis lead to a 2-D Coulomb gas
with a renormalized coupling constant. Here we will extend this
calculation to obtain a more accurate estimation of the
renormalized coupling constant. We do this because we are interested
in seeing if it is possible to have a QUIT instability in
the low temperature insulating phase.
This possibility is suggested by the dual symmetry of the effective action
between charges and vortices found in Ref. \cite{fazio-schon-91}, and
the fact that the $\alpha_{\rm m}$ perturbative expansion shows
a low temperature QUIT instability in the superconducting phase.
We find that the results of a first order expansion in $\alpha_{\rm m}^{-1}$
does not present this type of low temperature instability.
Among the most interesting regions of the phase
diagram is when the Josephson and charging energies are comparable.
For this nonperturbative case, using a path integral
formulation and the Villain
approximation \cite{jose-kadanoff-nelson-kirpatrick},
an effective action for logarithmically interacting
charges and vortices was derived in Ref. \cite{fazio-schon-91} in
the case where $C_{\rm s}\ll C_{\rm m}$. The action of the two
Coulomb gases shows an almost dual symmetry, so that at an
intermediate value of $\alpha_{\rm m}$, both the S--N
and the I--C transitions converge to a single
point as $T\rightarrow 0$. A similar picture was derived in Ref.
\cite{granato-contentino} using a short range electrostatic interaction
and a mean field renormalization group calculation. A nonperturbative
calculation is needed to determine the actual shape of the phase
diagram in this region. We further discuss this point in the main body of
the paper.
It has been argued that at $T=0$ the self-capacitive model is in the same
universality class as the 3-D XY model \cite{doniach,zwerger}, where
the ratio $\alpha_{\rm s}=(q^2/2C_{\rm s})/E_J$ would
play the role of temperature. This analogy would result in a transition at
some finite value of $\alpha_{\rm s}$ from a superconducting to an insulating
phase. Moreover, there should be a marked signature in the nature of
the correlation functions when crossing over from
a 2-D XY model at high temperatures to a 3-D XY model as $T\rightarrow 0$.
When $\alpha_s=0$, the correlation functions decay algebraically in the
critical region, with a temperature dependent exponent. At T=0 and
$\alpha_s \neq 0$ there is a single critical point at
$\alpha_s =\alpha_c$, so
that the correlations decay exponentially above and below $\alpha_c$, and
algebraically at $\alpha_s =\alpha_c$. The question is then, how do we go
from algebraic to exponential correlations as T changes?
This can only happen by
having a change of analyticity in the correlations, thus the possibility
of having a QUIT in the self-capacitive model.
The situation is different in the mutual-capacitance dominated limit, of
experimental interest. When $C_s=0 $ the model is equivalent to having two
interacting lattice Coulomb lattice gas models.
The general critical properties
of this case are not fully understood at present. A further complication
arises when $C_s$ is small but non-zero. The map to a higher dimensional
known model does not work in this case, and the problem has to be studied
on its own right. Because of the essential differences between the
self-capacitive and the mutual-capacitance dominated models one can not
just take results from one case and apply them to the other. It is the
goal of this paper to explicitly study the mutual capacitance dominated
model, but with nonzero $C_s$.
A brief report on some of the results of this paper has appeared
elsewhere \cite{jose-rojas}.
The outline of the rest of the paper is the following: In Section
\ref{sec:model} we define the model studied and derive the path integral
formulation of the partition function used in our calculations. In Section
\ref{sec:wkb-rg} we present a derivation of the path integral
used in the semiclassical analysis, in the limit where the Josephson
energy dominates. We carry out a WKB expansion up to
first order in $\alpha_{\rm m}$, finding an effective
classical action where the charging energy contributions are taken
into account as a renormalization
of the Josephson coupling. In section \ref{subsec:rg} we find general
renormalization group (RG) equations from which we obtain the phase
diagram for small $\alpha_{\rm m}$.
In Section \ref{sec:insulating-normal} we study the large $\alpha_{\rm m}$
limit,
in which the charging energy is dominant over the Josephson energy. There
we obtain an effective 2-D Coulomb gas model with a quantum fluctuations
renormalized coupling constant.
In Section \ref{sec:mc-operators} we discuss our QMC calculations
and define the physical quantities calculated. In Section
\ref{sec:simulation}
we present some technical details of the implementation of the
QMC simulations. In Section \ref{sec:results-for-f=0}
we give the QMC results for $f=0$ mainly, but also
for $f=1/2$. There we make a direct comparison between the semiclassical
approximation results, the QMC calculations and experiment which lead to
the phase diagram discussed above. In that section we also present an
$L_\tau$ dependent analysis of the apparent $T_{QUIT}$ for three
relatively large values of $\alpha_m
= 2.0, 2.25$ and $2.5$. The
$L_\tau\to\infty$ extrapolation of the results leads to a {\it finite}
$T_{QUIT}$ for $\alpha_m= 2.0$ and $2.25$, while for $\alpha_m=2.5$ we
get a $T_{QUIT}(L_\tau=\infty)=0$. In section \ref{sec:harmonic}
we discuss a self-consistent harmonic approximation (SCHA) analysis,
that we use
to analytically study the phase diagram, and that helps us understand
the finite size effects of the imaginary-time lattices
studied in the QMC calculations. At the end
of the paper there are two appendices where we give more technical details
of the analysis. In Section \ref{sec:conclusions} we restate the main
results of this paper.
\section{\bf The Model and the Path Integral Formalism}
\label{sec:model}
In this section we define the Josephson junction array model
considered in this paper together with the path integral formulation of
its corresponding partition function.
We assume that each superconducting island in a junction
can be characterized by a Ginzburg-Landau order parameter
$\Psi(\vec r)=|\Psi _0(\vec r)|e^{i \phi (\vec r)}$, where $\vec r$ is a
two-dimensional vector denoting the position of each island.
If the coherence length of the Cooper pairs is larger than the size of
the islands, we can assume that the phase of the
order parameter is constant in each island. Moreover,
the amplitude of the order parameter is expected to have small
fluctuations about an electrically neutral island and can then be taken as constant trough
the array. We will assume that the charge fluctuations have an effect
on the electrostatic energy but not on the Josephson contribution
to the Hamiltonian. The gauge invariant Hamiltonian studied here is
\begin{eqnarray}
\label{hamiltonian}
{\hat {\cal H}}=\hat H_C+\hat H_J
&=&
{{q^2}\over{2}}\sum_{<\vec r_1,\vec r_2>}\hat n(\vec r_1)
{\bf {\bf C}^{-1}}(\vec r_1,\vec r_2)\hat n(\vec r_2)+
E_J \sum_{<\vec r_1,\vec r_2>} \Big[1-\cos\Big(\phi(\vec r_1)-
\phi(\vec r_2) - A_{\vec r_1,\vec r_2}\Big)\Big]
,\nonumber\\
\end{eqnarray}
where $q=2e$; $\hat \phi (\vec r)$ is the quantum phase operator and
$\hat n (\vec r)$ is
its canonically conjugate number operator, which measures the excess number
of Cooper pairs in the $\vec r$ island. These operators satisfy the
commutation relations
$ [\hat n(\vec r_1),\hat\phi(\vec r_2)]=-i\delta_{\vec r_1,\vec r_2}
$ \cite{anderson}. Here $A_{\vec r_1,\vec r_2}$ is defined by the line
integral that joins the sites located at $\vec r_1$ and $\vec r_2$,
$ A_{\vec r_1,\vec r_2} = \frac{2\pi}{\Phi_0} \int_{\vec r_1}^{\vec r_2}
\vec A\cdot d\vec l$, where $\vec A$ is the vector potential
and $\Phi_0$ is the flux quantum. In Eq. (\ref{hamiltonian})
$\hat{H}_C$ is the charging energy due to the electrostatic
interaction between the excess Cooper pairs in the islands. The
${\bf C}^{-1}(\vec r_1,\vec r_2)$ matrix is the electric field propagator
and its inverse, ${\bf C}(\vec r_1,\vec r_2)$, is the geometric
capacitance matrix, which must
be calculated by solving Poisson's equation subject to the appropriate
boundary conditions. This is not easy to do in general
and typically this matrix is
approximated, both theoretically and in the experimental analysis of the
data, by diagonal plus nearest neighbor contributions
\cite{fazio-schon-91}:
\begin{equation}
\label{capacitance-matrix}
{\bf C}(\vec r_1,\vec r_2) = (C_{\rm s}+zC_{\rm m})\delta_{\vec
r_1,\vec r_2} - C_{\rm m} \sum_{\vec d} \delta_{\vec r_1,
\vec r_2+\vec d}\, .
\end{equation}
Here the vector $\vec d$ runs over nearest neighboring islands,
$z$ is the coordination number, $C_{\rm s}$ is the
self-capacitance of each island, and $C_{\rm m}$ is the mutual capacitance
between nearest neighbor islands. In the experimental arrays, typically
$C_{\rm m} \sim 10^2C_{\rm s} \sim 1 {\rm fF}$
\cite{van-der-zant_geerling_mooij-92}.
The second term in Eq.(\ref{hamiltonian}) is the Josephson energy, which
represents the probability of Cooper pair tunneling between
nearest neighboring islands. The Josephson coupling energy
$E_J=\Phi_0 i_{c}/(2\pi)$ is assumed to be temperature independent, where
$i_{c}$ is the junction critical current and $\Phi_0$ the flux quantum.
Here we are interested in calculating the thermodynamic properties of the
model defined by $\hat {\cal {H}}$. The quantity of interest is the
partition function
\begin{equation}
\label{partition_function}
Z \equiv {\rm Tr}\left\{ e^{-\beta\hat {\cal H}}\right\}\label{zeta}.
\end{equation}
The trace is taken either over the phase variables, $\hat\phi$, or
the numbers operator, $\hat{n}$.
To evaluate the partition function we will use its path integral
representation
\cite{schulman-book,kleinert-book}. To derive the path integral we use
the states
\begin{equation}
\label{charge-phase}
<n(\vec r_1)|\phi(\vec r_2)> = \delta_{\vec r_1,\vec r_2}
\frac{\exp\{ i n(\vec r_1)\phi(\vec r_1)\}}{\sqrt{2\pi}}.
\end{equation}
We will also use the fact that both $\{|n(\vec r)>\}$ and
$\{|\phi(\vec r)>\}$ form complete sets.
To start we write the partition function as a trace in the phase
representation
\begin{equation}
\label{z-in-phases}
Z = \prod_{\vec r} \int_{0}^{2\pi} d\phi(0,\vec r)
<\{\phi(0,\vec r)\}| \exp\left\{-\beta \hat {\cal H}\right\}|
\{\phi(0,\vec r)\}>.
\end{equation}
As usual we use the Trotter formula
\begin{eqnarray}
\label{trotter}
\exp\left\{-\beta(\hat H_C(\hat n)+\hat H_J(\hat \phi))\right\} =
& &\left[\exp\{-(\beta/L_{\tau}) \hat H_C(\hat n)\}
\exp\{-(\beta/L_{\tau})\hat H_J(\hat \phi)\}\right]^{L_{\tau}} \nonumber\\
& & + O\left(1/L_{\tau}^2\right).
\end{eqnarray}
Next we introduce $L_\tau-1$ complete sets $\{|\phi(\tau,\vec r)>\}$,
$\tau=1,2,\dots,L_\tau-1$ in Eq. (\ref{z-in-phases}) so that
\begin{eqnarray}
\label{z-with-trotter}
Z = \prod_{\vec r} \prod_{\tau=0}^{L_\tau-1} \int_{0}^{2\pi}
d\phi(\tau,\vec r)
& & <\{\phi(0,\vec r)\}|\exp\left\{-(\beta/L_\tau){\hat{\cal H}}
\right\}|\{\phi(1,\vec r)\}> \times \nonumber \\
& &\times <\{\phi(1,\vec r)\}|\exp\left\{-(\beta/L_\tau)
{\hat{\cal H}}\right\}|\{\phi(2,\vec r)\}> \times \nonumber\\
& & \times\cdots \times <\{\phi(L_\tau-1,\vec r)\}|\exp\left
\{-(\beta/L_\tau){\hat{\cal H}}\right\}|\{\phi(0,\vec r)\}>
\nonumber\\
& & + O\left(1/L_{\tau}^2\right).
\end{eqnarray}
At this point we need to calculate the short time propagator,
\begin{eqnarray}
\label{short-time-prop}
<\{\phi(\tau,\vec r)\}|e^{-(\beta/L_\tau){\hat{\cal H}}
}|\{\phi(\tau+1,\vec r)\}> =
\sum_{n(\tau,\vec r)= -\infty}^{\infty}
& & <\{\phi(\tau,\vec r)\}|e^{-(\beta/L_\tau)
{\hat{\cal H}} }|\{n(\tau,\vec r)\}>\times\nonumber\\
& & \times <\{n(\tau,\vec r)\}|\{\phi(\tau+1,\vec r)\}>,
\end{eqnarray}
where we used a summation over the complete set $|\{n(\tau,\vec r)\}>$.
From Eqs. (\ref{charge-phase}) and (\ref{trotter}) this propagator can
be written as
\begin{eqnarray}
\label{short-time-prop-2}
<\{\phi(\tau,\vec r)\}|e^{-(\beta/L_\tau){\hat{\cal H}}
}|\{\phi(\tau+1,\vec r)\}> = \prod_{\vec r}\frac{1}{2\pi}
\sum_{n(\tau,\vec r)=-\infty}^{\infty}
& & e^{i\ n(\tau,\vec r)[\phi(\tau+1,\vec r)-\phi(\tau,\vec r)]}
\times \nonumber\\
& & e^{-(\beta/L_\tau){\cal H}(\{n(\tau,\vec r)\},\{\phi(\tau,
\vec r)\}) } + \nonumber\\
& & + O(1/L_\tau^2).
\end{eqnarray}
Inserting this equation in Eq. (\ref{z-with-trotter}) we obtain the
following path integral representation of the partition function
\begin{eqnarray}
\label{z-path-integral}
Z = & & \
\prod_{\tau=0}^{L_{\tau}-1} \prod_{\vec r} \int_{0}^{2\pi}
\frac{d\phi(\tau,\vec r)}{2\pi} \sum_{\{ n(\tau,\vec r)\}=
-\infty}^{\infty}\exp\left[i\sum_{\tau=0}^{L_\tau-1}
n(\tau,\vec r)[\phi(\tau+1,\vec r)-\phi(\tau,\vec r)]
\right] \times \nonumber \\
& &\times \exp\left[-\frac{\beta}{L_\tau}\sum_{\tau=0}
^{L_\tau-1}\bigg[H_J(\{\phi(\tau,\vec r)\}) +
\sum_{\vec r_1,\vec r_2}\frac{q^2}{2} n(\tau,\vec r_1)
{\bf C}^{-1}(\vec r_1,\vec r_2)n(\tau,\vec r_2)\bigg]
\right] + \nonumber \\
& & + O(1/L_\tau^2).\nonumber\\
\end{eqnarray}
together with the important boundary condition
$\phi(L_\tau,\vec r)=\phi(0,\vec r)$.
These equations are our starting point
for the semiclassical approximation analysis discussed in the next section.
\section{\bf WKB and renormalization group equations}
\subsection{Semiclassical limit}
\label{sec:wkb-rg}
The semiclassical limit corresponds to taking
$q^2\rightarrow 0$, or $\alpha_{\rm m}\rightarrow 0$.
The summations over $\{n(\tau,\vec r)\}$ in Eq. (\ref{z-path-integral})
can be carried out and the result leads to
$\phi(\tau+1,\vec r)=\phi(\tau,\vec r)$, for $\tau=0,1,\dots,L_\tau-1$. In
other words, in this limit all the phase variables are constant along the
imaginary time axis, and we recover the classical 2-D XY model
\cite{berezinskii-kosterlitz-thouless,jose-kadanoff-nelson-kirpatrick}.
As the charging energy increases the value of $\phi(\tau,\vec r)$
fluctuates along the $\tau$-axis; these fluctuations suppress the XY phase
coherence in the model lowering its critical temperature.
For the self-capacitive model ($C_{\rm m}=0$), at $T=0$, one can
map the model to an anisotropic three-dimensional XY model
\cite{doniach,zwerger}.
This model should have a transition between ordered and disorder phases at a critical coupling
$(E_{C_{\rm s}}/E_J)_c$. Here $E_{C_{\rm s}}=e^2/2C_{\rm s}$, so we would
expect the phase boundary to go all the way down to $T=0$ for large
enough charging energy.
In this section we study the change of the critical temperature
as $E_{C_{\rm m}}$ increases, for small values of the ratio
$\alpha_{\rm m}=E_{C_{\rm m}}/E_J$. We start by eliminating
the $\{n's\}$ the from Eq.(\ref{z-path-integral}) using the
Poisson summation formula
\begin{equation}
\label{poisson}
\sum_{n=-\infty}^{\infty} f(n) = \sum_{m=-\infty}^{\infty}
\int_{-\infty}^{\infty} f(x) e^{2\pi imx} dx,
\end{equation}
obtaining
\begin{equation}
\label{z-with-m}
Z = \prod_{\tau=0}^{L_{\tau}-1} \sqrt{{\rm det}[ C ] }
\prod_{\vec r} \int_{0}^{2\pi}\!\!\sqrt{\frac{L_{\tau}}
{2\pi\beta q^2}} d\phi(\vec r,\tau)\!\!\!\!\sum_{\{ m(\vec r,\tau)\}
= -\infty}^{\infty}\!\!\!\!\exp\bigg[-\frac{1}{\hbar}S[\{\phi\},\{m\}]
\bigg].
\end{equation}
Here we defined the action
\begin{eqnarray}
\label{action}
\frac{1}{\hbar}S[\{\phi\},\{m\}] = & &
\sum_{\tau = 0 }^{L_{\tau}-1} \Bigg[\frac{ \beta}{L_{\tau}}
H_J(\{\phi(\tau,\vec r)\})
+\frac{L_{\tau}}{2\beta q^2}\!\!\sum_{\vec r_1,\vec r_2}
[\phi(\tau\!+\!1,\vec r_1) - \phi(\tau,\vec r_1)
+ 2\pi m(\tau,\vec r_1) ] \times \nonumber \\
& &\times{\bf C}(\vec r_1,\vec r_2)[\phi(\tau\! +\! 1,\vec r_2)
-\phi(\tau,\vec r_2) + 2\pi m(\tau,\vec r_2)]\Bigg] +
\nonumber \\
& &+ O(1/L_{\tau}^2).
\end{eqnarray}
It is convenient to write the paths in the partition function
separated into a constant part, that
corresponds to the classical model, plus a quantum fluctuating
contribution, over which we will perform the integrations to find
an effective classical action. First we eliminate the summations
over the $\{m's\}$. This is done at the same time that the integrals over the
$\{\phi's\}$ are extended from $[0,2\pi)$ to $(-\infty,\infty)$.
After a couple of standard variable changes \cite{schulman-book,kleinert-book}
we get an action where the phases and the charges are separated,
\begin{eqnarray}
\label{new-action}
\frac{1}{\hbar}\int_{0}^{\beta\hbar} & & d\tau L_E
= \frac{1}{2}
\frac{(2\pi)^2}{\beta q^2}\sum_{\vec r_1,\vec r_2} m(\vec r_1)
{\bf C}
(\vec r_1,\vec r_2) m(\vec r_2) + \frac{1}{\hbar}\int_{0}^{\beta\hbar}
d\tau \times \nonumber \\
& &\times\Bigg[ \frac{\hbar^2}{2q^2} \sum_{\vec r_1,\vec r_2}
\frac{d\psi}{d\tau}(\tau,\vec r_1){\bf C}(\vec r_1,\vec r_2)
\frac{d\psi}{d\tau}(\tau,\vec r_2) + H_J(\{\psi(\tau,\vec r)+
(2\pi/\beta\hbar)m(\vec r)\tau\})\Bigg].\nonumber\\
\end{eqnarray}
Here the variables $\psi(\beta\hbar,\vec r) = \psi(0,\vec r)$, and
the integers $m(\vec r)$ are called the winding numbers.
This equation shows that the winding numbers are the charge degrees of
freedom and that the coupling between phases and charges appears only
in the Josephson term. We can also see from this equation that in the
semi-classical limit (small charging energy) the charge fluctuations are
exponentially suppressed. This is more so for the $m's$ because they have
a discrete excitation spectrum. Therefore, to lowest order in the
semiclassical analysis we will set $m(\vec r)=0$, leaving integrals
only over the phases. Next we separate the $\psi's$ into a constant
plus a fluctuating part
\begin{equation}
\label{constant-fluctuating}
\psi(\tau,\vec r) = \overline\phi(\vec r) + \phi_f(\tau,\vec r).
\end{equation}
At this point we use the following argument
\cite{jose-kadanoff-nelson-kirpatrick,m-fisher}. First
that the Lagrangian is invariant under the transformation
$\psi(0,\vec r)\rightarrow\psi(0,\vec r)+2\pi l(\vec r)$ for all integers
$l(\vec r)$, so that we can extend the limits of
integration over $\psi(0,\vec r)$ to $(-\infty,\infty)$,
safe for an extra overall multiplicative constant.
Now, the limits of integration for $\overline\phi(\vec r) \epsilon
(-\infty,\infty)$, and because of the periodicity of $\phi_f$, we can
Fourier series expand it as
\begin{equation}
\label{fourier-series}
\phi_f(\tau,\vec r) = (\beta\hbar)^{-1/2}\sum_{k=1}^{\infty}
[\phi_k(\vec r)e^{i\omega_k\tau}+C.C.],
\end{equation}
where the $\omega_k=2\pi k/\beta\hbar$ are the Bose-Matsubara frequencies.
We have then the partition function \cite{kleinert-book}
\begin{eqnarray}
\label{z-fourier}
Z = \prod_{\vec r}\sqrt{{\rm det}{[\bf C]}}\int_{-\infty}^{\infty}
\frac{d\overline\phi(\vec r)}{(2\pi\beta q^2)^{1/2}}
\prod_{k=1}^{\infty}& &\Bigg[\frac{\omega_k^2\hbar}{\pi q^2}
{\rm det}{[\bf C]} \int_{-\infty}^{\infty}d{\rm Re}
\phi_k(\vec r)\int_{-\infty}^{\infty}d{\rm Im}
\phi_k(\vec r)\Bigg] \times \nonumber \\
& &\times \exp\left\{-\frac{1}{\hbar}S\big[\{\overline\phi\},
\{\phi_f\}\big]\right\}.
\end{eqnarray}
Next, we expand the Josephson term in the action up to
second order in $\phi_f$, for higher order terms are
suppressed in the integrations. After performing the
integrations over the Euclidean time $\tau$, and the Gaussian integrations,
the effective partition function reads
\begin{eqnarray}
\label{effective-z}
Z_{\rm eff} = \prod_{\vec r} \sqrt{{\rm det}{[\bf C]}}\int_{-\infty}
^{\infty} & &
\frac{d\overline\phi(\vec r)}{(2\pi\beta q^2)^{1/2}}
\ \ \exp\left\{-\beta H_J(\{\overline\phi\})\right\}
\times \nonumber \\
& &\times \prod_{k=1}^{\infty} \left[ {\rm det}\left\{
\delta_{\vec r_1,\vec r_2} + \frac{q^2}{\hbar^2
\omega_k^2}\sum_{\vec r}{\bf C}^{-1}(\vec r_1,\vec r)
\frac{\partial^2H_J}{\partial\phi(\vec r)\partial\phi
(\vec r_2)}\Bigg|_{\overline\phi}\right\}\right]^{-1}.
\nonumber\\
\end{eqnarray}
Here we want to expand this partition function in powers of the charging
energy. This is equivalent to expanding in powers of $q^2$, so our next
step is to expand the determinant. We use the following identities
\begin{eqnarray}
\label{det-equations}
{\rm det}[{\bf I}+{\bf D}]&=& \exp\{{\rm Tr}[{\rm ln}({\bf I}+{\bf
D})
]\}, \\
{\rm ln}({\bf I}+{\bf D}) &=& -\sum_{n=1}^{\infty} \frac{(-1)^n}{n}
{\bf D}^n,
\end{eqnarray}
where ${\bf I}$ is the identity matrix. To lowest order in $q^2$
and using the result
$\sum_{k=1}^{\infty}[q^2/(\hbar\omega_k)^2] = (q\beta)^2/24$, the
effective partition function for $\overline\phi$ is then
\begin{eqnarray}
\label{z_eff-phi}
Z_{\rm eff} = \prod_{\vec r} \sqrt{{\rm det}{[\bf C]}}
\int_{-\infty}^{\infty}
& &
\frac{d\overline\phi(\vec r)}{(2\pi\beta q^2)^{1/2}} \exp\Bigg\{
-\beta H_J(\{\overline\phi\})\nonumber\\
& &
-\frac{(q\beta)^2}
{24}\sum_{\vec r_1,\vec r_2}{\bf C}^{-1}(\vec r_1,\vec r_2)\frac
{\partial H_J}{\partial\phi(\vec r_1)\partial\phi(\vec r_2)}\Bigg|
_{\overline\phi}\Bigg\}.
\end{eqnarray}
To further advance the calculation we now use the properties of the
Josephson energy. We start by using the fact that it is a
local nearest neighbor interaction, so from Eq. (\ref{hamiltonian})
\begin{equation}
\label{josephson-energy}
H_J(\{\phi\}) = \sum_{\vec r}\sum_{\vec d} f\!\left(\phi
(\vec r+\vec d)-\phi(\vec r)\right),
\end{equation}
with the $\vec d$ running over the nearest neighbors to
$\vec r$ in the lattice. From this equation we can see that the
second derivative of $H_J(\{\phi\})$ is given by
\begin{eqnarray}
\label{secon-derivative_hj}
\frac{\partial^2 H_J}{\partial\phi(\vec r_1)\partial\phi(\vec r_2)} =
\sum_{\vec d}\Bigg[
& &
f''\!\left(\phi(\vec r_1)-\phi(\vec r_1-\vec d)
\right)\left(\delta_{\vec r_1,\vec r_2}-\delta_{\vec r_1,\vec r_2+
\vec d}\right) + \nonumber \\
& & + f''\!\left(\phi(\vec r_1+\vec d)-\phi(\vec r_1)\right)\left(
\delta_{\vec r_1,\vec r_2}-\delta_{\vec r_1+\vec d,\vec r_2}
\right)\Bigg],
\end{eqnarray}
where $f''\!(x)=d^2f(x)/dx^2$.
In this paper we consider a periodic array, which implies that
the inverse capacitance matrix is invariant under translations
and rotations, that is ${\bf C}^{-1}(\vec r_1,\vec r_2)={\bf C}^{-1}
(|\vec r_1-\vec r_2|)$. In particular, this makes ${\bf C}^{-1}(\vec r,
\vec r\pm\vec d) = {\bf C}^{-1}(|\vec d|)$, independent of the direction
of $\vec d$. Notice that here we are using $\vec d$ to denote the
vectors that connect nearest neighboring islands, therefore in a
periodic and symmetric array all of them have the same magnitude,
allowing us to take the terms ${\bf C}^{-1}(|\vec d|)$ out of the
summations. From these considerations the trace gives
\begin{equation}
\label{trace}
\sum_{\vec r_1,\vec r_2}{\bf C}^{-1}(\vec r_1,\vec r_2)\frac
{\partial^2 H_J}{\partial\phi(\vec r_1)\partial\phi(\vec r_2)}\Bigg|
_{\overline\phi} = 2\left[{\bf C}^{-1}(|\vec 0|)-{\bf C}^{-1}
(|\vec d|)\right]\sum_{\vec r}\sum_{\vec d} f''\!\left(\overline
\phi(\vec r+\vec d)-\overline\phi(\vec r)\right).
\end{equation}
Next notice that since $f''\!(x) =-f(x)$ (up to a constant), both terms in
the argument of the exponential in Eq. (\ref{z_eff-phi}) are the same
cosine function of the classical phase variables, with only different coupling
constants. Finally, the effective semi-classical partition function can
be written as
\begin{equation}
\label{final-z_eff}
Z_{\rm eff} = \prod_{\vec r} \sqrt{{\rm det}{[\bf C]}} \int_{-\infty}
^{\infty}\frac{d\overline\phi(\vec r)}{(2\pi\beta q^2)
^{1/2}}\exp\left\{-\beta_{\rm eff}
H_J(\{\overline\phi\})
\right\},
\end{equation}
where the effective temperature is explicitly given by
\begin{equation}
\label{beta_eff}
\beta_{\rm eff} = \beta - q^2 \frac{\beta^2}{12}\left[{\bf C}^{-1}
(|\vec 0|)-{\bf C}^{-1}(|\vec d|)\right].
\end{equation}
Notice that to obtain this result we have used an argument that
could be questionable, namely the extension of the $\phi(0,\vec r)$
to the $(-\infty,\infty)$ range in the path integrals. All
the other approximations are consistent with the semiclassical
approximation
and the symmetries used are exact.
However, to continue we will now restore the $[0,2\pi)$ range of the
phases to use the results known from the BKT theory.
As we will show later in the paper,
the nonperturbative QMC results do agree quantitatively with the WKB
results and experimental results to be discussed later. A similar
effective result was first obtained but for the self-capacitive model in
\cite{jose_1984}.
One of the important properties of Eqs.(\ref{final-z_eff}) and
(\ref{beta_eff}) is that up to this point we have made no
assumptions about the structure of the capacitance matrix that go
beyond translational invariance. Later on we will make specific
choices of this matrix when we make direct contact with experimental
findings \cite{van-der-zant_geerling_mooij-92}.
\subsection{Renormalization group analysis}
\label{subsec:rg}
Now that we have expressed the quantum mechanical problem as a
modified 2-D classical XY model we can directly apply the well known
results for this model
\cite{berezinskii-kosterlitz-thouless,jose-kadanoff-nelson-kirpatrick}.
The standard physical picture of the excitation spectrum in this model is
of spin-waves plus vortex pair excitations. At low temperatures the energy to
create an isolated vortex grows logarithmically with the size of the
system, therefore excitations are created as bounded vortex-antivortex
pairs. As the temperature increases, the vortex pair density increases
until they unbind at a critical dimensionless temperature
$T_{\rm BKT} = 0.894(5)$
\cite{gupta-baillie,janke-nather}. The BKT scenario is best understood
in terms of a renormalization group (RG) analysis
\cite{berezinskii-kosterlitz-thouless,jose-kadanoff-nelson-kirpatrick}.
The RG flow diagram is obtained from a perturbative expansion in powers
of the vortex pair fugacity $y$. To lowest order in $y$, the RG
equations corresponding to our problem are
\begin{eqnarray}
\label{rg-eq-1}
\frac{dK_{\rm eff}}{dl} &=& -4\pi^3 K_{\rm eff}^2 y^2, \\
\label{rg-eq-2}
\frac{dy}{dl} &=& [2-\pi K_{\rm eff}]y.
\end{eqnarray}
Here we have used the following definitions:
\begin{eqnarray}
\label{rg-definition-k_eff}
K_{\rm eff} &=& K - xK^2, \\
\label{rg-definition-x}
x &=& \frac{q^2}{12E_J}\left[{\bf C}^{-1}(|\vec 0|)-{\bf C}^{-1}
(|\vec d|)\right],\\
\label{rg-definition-k}
K &=& \beta E_J.
\end{eqnarray}
Then the equations for the coupling constants $K$ and $y$ are
\begin{eqnarray}
\label{rg-effective-1}
\frac{dK}{dl} &=& 4\pi^3 K^2 y^2 \frac{(1-xK)^2}{(2Kx-1)}, \\
\label{rg-effective-2}
\frac{dy}{dl} &=& \left[2-\pi K(1-xK)\right]y.
\end{eqnarray}
To find the critical temperature, we use the initial conditions
from the temperature and the bare vortex pair fugacity
\begin{eqnarray}
\label{K-initial}
K_{\rm eff}(l=0) &=& \beta_{\rm eff} E_J\left[1+\frac{1}
{2\beta_{\rm eff}E_J}
+O\left\{\frac{1}{(\beta_{\rm eff}E_J)^2}
\right\}\right]^{-1}, \\
\label{y-initial}
y(l=0) &=& \exp\left\{-\frac{\pi^2}{2} K_{\rm eff}(l=0)\right\}.
\end{eqnarray}
The RG equations have two nontrivial fixed points (for $x<\pi/8$). One
corresponds to the effective BKT thermal fluctuations driven
transition, and the other to a quantum fluctuations induced transition
(QUIT) \cite{jose_1984,jacobs-jose-novotny-goldman,jacobs-jose}.
One way to analyze the structure of the RG flow in the $(y,K)$
phase space is to use a conserved quantity associated with
Eqs. (\ref{rg-eq-1}) and (\ref{rg-eq-2})
\begin{equation}
\label{conserved-eff}
A = -\pi {\rm ln}K_{\rm eff}-\frac{2}{K_{\rm eff}}+2\pi^3 y^2.
\end{equation}
Using Eq. (\ref{rg-definition-x}) and expanding up to first order in
$x$ we find
\begin{equation}
\label{conserved}
A = \pi x K - \pi {\rm ln}K-\frac{2}{K}+2\pi^3 y^2.
\end{equation}
Figure \ref{fig:fig02-rg-flow} shows the RG flows obtained from
numerically solving the RG equations for different values of $A$,
where
the arrows indicate the direction of increasing $l$. We have also plotted
the set of initial conditions from Eqs.(\ref{K-initial})
and (\ref{y-initial}) as a discontinuous line. One important flow line is the
separatrix between the lines for which $y(l\!\rightarrow\!\infty)
\rightarrow\infty$ and those for $y(l\!\rightarrow\!\infty)
\rightarrow 0$. This line has $A_c=-\pi[1+{\rm ln}(2/\pi)]$, which is
determined by the condition that it must pass through the point $(y=0,
K_{\rm eff} = 2/\pi)$. The critical temperature is obtained from the
intersection of the separatrix with the initial
conditions given in Eqs. (\ref{K-initial}) and (\ref{y-initial}).
This intersection exists only if $x$ is less than the critical value
$x_c<\!\pi/8$, to be estimated below. Fortunately, we do not need to
find this intersection explicitly since we already
know the critical value of the effective coupling $K_{\rm eff}$,
which is the usual critical coupling of the classical XY model,
$K_{\rm eff}^{(c)}=K_{c}^{XY}\approx 1.1186$ \cite{gupta-baillie}.
Therefore, the values of the two critical couplings are
\begin{eqnarray}
\label{eq-critical-temps}
K(1-xK) &=& K_{c}^{XY},\\
\label{both-critical-temps}
K_{\pm} &=&\frac{1}{2x}\left[1\pm \sqrt{1-4xK_{c}^{XY}}\right],
\end{eqnarray}
which leads to $x_c=1/(4K_{c}^{XY})\approx 0.2235$.
We note from Fig. 2 that the $K^{-1}$ axis
can be divided into three different regions. If we set $K_{+}=K_{\rm QUIT}$,
and $K_{-}=K_{\rm BKT}$, then in the region
$[K_{\rm QUIT}^{-1},K_{\rm BKT}^{-1}]$, as $l$ increases,
the fugacity of the vortex-antivortex pairs decreases.
In the limit $l\!\rightarrow
\!\infty$ the energy to create a macroscopic vortex pair becomes
infinite. Therefore,
the system is superconducting for temperatures in this interval. For
temperatures $T>E_J K_{BKT}^{-1}$ the renormalized fugacity $y(l)$
increases
and the low $y$ approximation breaks down. For these temperatures, the
array is normal. For $T<E_J K_{QUIT}^{-1}$,
the vortex pair density increases due to the quantum fluctuations,
leading us to think that there may be
a low temperature transition driven by the quantum fluctuations (QUIT).
To obtain the results described above we used a high temperature
perturbative calculation. Therefore the QUIT results are in principle
outside the regimen of validity of the WKB-RG calculation.
We need, then, other calculations and approaches valid at low
temperatures to prove or
disprove the existence of the QUIT. Expanding Eq. (\ref{both-critical-temps}),
up to first order in $x$ and using Eq. (\ref{rg-definition-k}) we find
the critical temperatures,
\begin{eqnarray}
\label{T_BKT}
T_{BKT} &\approx& T_{BKT}^{(0)} - \frac{E_J}{k_{\rm B}} x +
O(x^2), \\
\label{T_QUIT}
T_{QUIT} &\approx& \frac{E_J}{k_{\rm B}} x + O(x^2).
\end{eqnarray}
Note that these equations are applicable not only in 2-D, for if
the system described by Eq. (\ref{final-z_eff}) has a transition point at
some $K_{\rm eff}^{c}$ then the equation $K_{\rm eff}^{c} = K-xK^2$ has
two solutions for $K$. In this argument we should notice that the
existence
of the second solution for $K$ depends on the higher order terms in the
$x$ expansion. Note that the change of $T_{BKT}$ for small $x$ is
correctly given by the small $x$ result. The existence of a low
temperature quantum phase is of a nonperturbative nature, however.
That is one of the reasons why we resort to using the nonperturbative
quantum QMC approach latter in the paper.
Another interesting property of Eqs. (\ref{T_BKT}) and (\ref{T_QUIT})
is that the first order correction does not depend on the specific
value of $T_{BKT}^{(0)}$. In particular, if we
add a magnetic field to the Hamiltonian in Eq. (\ref{hamiltonian}),
all the calculations leading to Eqs. (\ref{final-z_eff}) and
(\ref{beta_eff}) would be unchanged. Therefore, if $T_{c}^{(0)}(B)$ is the
superconducting to normal transition temperature for the array in
a finite magnetic field $B$ at $x=0$, then to first order in $x$ we must have
\begin{equation}
\label{tc-with-mag-field}
T_{c}(B)\approx T_{c}^{(0)}(B)-(E_J/k_{\rm B})x+O(x^2).
\end{equation}
This equation is in agreement with the results obtained in
Ref. \cite{jacobs-jose-novotny-goldman}.
Furthermore, we notice that to lowest order in $x$, the $T_{QUIT}$
must be the same with then without a magnetic field. This result
was not noted before and it can be used as a test of the QMC calculations,
in particular those of Mikalopas et al.
\cite{mikalopas-jarrel-pinski-chung-novotny}.
To make comparisons with experiment, we need to specify the
capacitance matrix. In particular, if we use Eq.
(\ref{capacitance-matrix}) and the specific geometry of the array,
we find that $x$ is given by
\begin{eqnarray}
\label{x}
x &=& \frac{q^2}{12 z E_J C_{\rm m}}\left[1-
C_{\rm s}{\bf C}^{-1}(|\vec 0|)\right],\nonumber\\
&=& \frac{q^2}{12 z E_J C_{\rm s}} g(C_{\rm m}/C_{\rm s}).
\end{eqnarray}
The function $g(w)$ can be written as an elliptic integral for a
two-dimensional square lattice \cite{kleinert-book-2}. For a general lattice
geometry, we find the following limiting behavior
\begin{equation}
\label{limits-g}
g(w) \approx \left\{\begin{array}{ll}
z-z(1+z)w, & \mbox{if $w \ll 1$,} \\
w^{-1}\left\{1-(4\pi w)^{-1}\ln w\right\},
& \mbox{if $w \gg 1$.}
\end{array}
\right.
\end{equation}
Using Eqs. (\ref{T_BKT}) and (\ref{limits-g}) we get
\begin{equation}
\label{limits-T_BKT}
\frac{k_{\rm B} T_{BKT}}{E_J} \approx \frac{k_{\rm B} T_{BKT}^{(0)}}
{E_J} - \left\{ \begin{array}{ll}
(2/3)\alpha_{\rm s} + O(\alpha_{\rm s}^2),
& \mbox{if $C_{\rm s}\gg C_{\rm m}$,} \\
& \\
(2/3z)\alpha_{\rm m} + O(\alpha_{\rm m}^2),
& \mbox{if $C_{\rm s}\ll C_{\rm m}$.}
\end{array}
\right.
\end{equation}
This result is in agreement with the Monte Carlo calculation of the
superconducting to normal transition temperature carried out in
Ref. \cite{jacobs-jose-novotny-goldman} for the self-capacitive model.
As we will show later in this paper, it is also in good agreement with
our QMC calculations for the model dominated by the mutual capacitances.
\section{\bf Insulating to Normal Cross Over.}
\label{sec:insulating-normal}
So far we have studied the normal to superconducting transition
in the limit where the Josephson energy dominates over the charging energy.
In the opposite limit, when the relevant excitations are charge
fluctuations, the transition is expected to be from a normal conducting state, where the
charges are free to move, to an insulating state where the charges are
bound into neutral dipole pairs (see Fig. \ref{fig:fig01-phase-diagram}). It
has been suggested that in the limit $(C_{s}/C_{\rm m}) \rightarrow 0$
this I-N transition would be of a BKT type
\cite{yaks-87,widom-badjou,mooij_van-wees_geerligs_peters_fazio_schon,fazio-schon-91}.
Experimental results have shown, however, that the behavior of the
fabricated samples is better explained by a crossover from a normal
to an insulating phase
\cite{tighe-tuominen-hergenrother-tinkham,van-der-zant-thesis,delsing-chen-haviland-harada-claeson}.
In finite systems, like the experimental ones, we would expect
a rounding of the transition. Furthermore, in the experiments, the
screening length is shorter than the sample size
$\Lambda\sim\sqrt{C_{\rm m}/C_{\rm s}}\approx 18$ lattice spacings
\cite{van-der-zant_geerling_mooij-92}.
Minnhagen et al. have argued that for any finite screening length, the
transition is washed out even for an infinite array
\cite{minnhagen-olsson-xu}.
This is not difficult to understand since in the BKT scenario the
superconducting to normal transition depends on the unscreened nature of
the vortex logarithmic interaction
\cite{berezinskii-kosterlitz-thouless,jose-kadanoff-nelson-kirpatrick}.
In this section we present results from a perturbative calculation
of the effect of the Josephson energy on the expected I-N crossover
temperature. We start with Eqs. (\ref{new-action})
and (\ref{constant-fluctuating}) leading to
\begin{eqnarray}
\label{z-constant-fluctuating}
Z = \sqrt{{\rm det}{[\bf C]}} \int_{0}^{2\pi}
\prod_{\vec r}
\sqrt{\frac{L_{\tau}}{2\pi \beta q^2}} d\overline\phi(\vec r)
\!\sum_{m(\vec r )=-\infty }^{\infty} \int_{-\infty}^
{\infty}
& &\prod_{\tau = 1}^{L_{\tau}-1}\!\sqrt{{\rm det}
{[\bf C]}}
\prod_{\vec r} \sqrt{\frac{L_{\tau}}{2\pi \beta q^2}}
d\phi_f(\tau,\vec r) \times \nonumber \\
& &\times\exp\bigg[-\frac{1}{\hbar}\int_{0}^{\beta\hbar}
d\tau L_E \bigg],
\end{eqnarray}
where the action is now given by
\begin{eqnarray}
\label{action-m-over_phi-phi_f}
\frac{1}{\hbar}\int_{0}^{\beta\hbar} d\tau L_E = \frac{1}{2}
\frac{(2\pi)^2}{\beta q^2}\sum_{\vec r_1,\vec r_2}
& &
m(\vec r_1){\bf C}
(\vec r_1,\vec r_2) m(\vec r_2) + \frac{1}{\hbar}\int_{0}^{\beta\hbar}
d\tau \times \nonumber \\
& & \times \Bigg[ \frac{\hbar^2}{2q^2} \sum_{\vec r_1,\vec r_2}
\frac{d\phi_f}{d\tau}(\tau,\vec r_1){\bf C}(\vec r_1,\vec r_2)
\frac{d\phi_f}{d\tau}(\tau,\vec r_2) + \nonumber \\
& & \ \ \ \ \ \ + H_J(\{\overline\phi(\vec r)+\phi_f(\tau,\vec r)+
(2\pi/\beta\hbar)m(\vec r)\tau\})\Bigg],
\nonumber\\
\end{eqnarray}
with the boundary condition
\begin{equation}
\label{boundary-for-phi_f}
\phi_f(0,\vec r) = \phi_f(\beta\hbar,\vec r) = 0.
\end{equation}
Since we are interested in the charge degrees of freedom, our task here
is to integrate out the phases. This limit has been studied before,
in particular in Ref. \cite{fazio-schon-91}. Here we are not only
interested in the crossover temperature, but we mostly want
to ascertain if there
is an equivalent QUIT in the insulating phase at low temperatures.
Since we are at the limit $E_J\ll E_C$, the Josephson energy can
be treated as perturbation. We expand the exponential
\begin{equation}
\label{h_j-expantion}
\exp\left[-\frac{1}{\hbar}\int_{0}^{\beta\hbar}d\tau H_J(\tau)\right]
\approx 1-\frac{1}{\hbar}\int_{0}^{\beta\hbar}d\tau H_J(\tau)+
\frac{1}{2\hbar^2}\int_{0}^{\beta\hbar}\int_{0}^{\beta\hbar}
d\tau d\tau' H_J(\tau)H_J(\tau')+\dots
\end{equation}
We note that Eq. (\ref{z-constant-fluctuating}) can be
written as
\begin{equation}
\label{z-with-z_eff}
Z = Z_{\phi} \prod_{\vec r} \sum_{m(\vec r)=-\infty}^{\infty}
\exp\Bigg[ -\frac{(2\pi)^2}{2\beta q^2}\sum_{\vec r_1,\vec r_2}
m(\vec r_1){\bf C}(\vec r_1,\vec r_2) m(\vec r_2)
\Bigg] Z_{\rm eff}(\{m\}),
\end{equation}
where $Z_{\phi}$ contains only phase degrees of freedom and can formally
be written as
\begin{eqnarray}
\label{z_phi}
Z_{\phi} &=& \prod_{\vec r} \int_{-\infty}^{\infty} {\cal D}
\phi_f(\vec r)
\exp\Bigg[-\frac{1}{\hbar}S_f[\phi_f]\Bigg], \\
\label{s_f}
S_f[\phi_f] &=& -\frac{\hbar^2}{2q^2}\int_{0}^{\beta\hbar}
d\tau \sum_{\vec r_1,\vec r_2} \frac{d\phi_f}
{d\tau}(\tau,\vec r_1){\bf C}(\vec r_1,\vec r_2)
\frac{d\phi_f}{d\tau}(\tau,\vec r_2).
\end{eqnarray}
Here we have used the following short hand notation for the measure
\begin{equation}
\label{path-integral-measure}
{\cal D}
\phi_f(\vec r) = \lim_{L_\tau\rightarrow\infty}
\prod_{\tau = 1}^{L_{\tau}-1}
\!\!\sqrt{{\rm det}{[\bf C]}} \prod_{\vec r}
\sqrt{\frac{L_{\tau}}{2\pi \beta q^2}} d\phi_f(\tau,\vec r),
\end{equation}
noting that strictly speaking the integrals over a finite number of
$L_\tau$'s have to be calculated before the limit $L_\tau\rightarrow\infty$
is taken \cite{feynman-stat}.
All the interactions between phases and charges are contained in
$Z_{\rm eff}(\{m\})$, the effective partition function for the
charges. The details of the explicit evaluations
of $Z_{\rm eff}$ are given in Appendix A. The result for
Eq. (\ref{z-with-z_eff}) can then be written, up to second order in
$E_J$, as
\begin{eqnarray}
\label{z-with-z_eff-2}
Z = Z_{\phi} \prod_{\vec r}
& &
\sum_{m(\vec r)=-\infty}^{\infty}
\exp\!\Bigg[ -\frac{1}{2\tilde K}\sum_{<\vec r_1,\vec r_2>}
(m(\vec r_1)-m(\vec r_2))^2 - \frac{1}{2\tilde K}\left(
\frac{C_{\rm s}}{C_{\rm m}}\right)\sum_{\vec r}
m(\vec r)^2 + \nonumber \\
& & + \frac{\tilde K^2}{2} \left(\frac{(2\pi)^2 E_J
C_{\rm m}}{q^2}\right)^2\!\!\!
\sum_{<\vec r_1,\vec r_2>}
{\cal I}\Big(m(\vec r_1)-m(\vec r_2),\tilde K
[1-C_{\rm s}{\bf C}^{-1} (|\vec 0|)]\Big)\Bigg] +
\nonumber\\
& & + O(E_j^4).
\end{eqnarray}
Here we have defined
\begin{eqnarray}
\label{def-k}
\tilde K &=& \frac{\beta q^2}{(2\pi)^2 C_{\rm m}}, \\
\label{def-i}
{\cal I}(m,\tilde K) &=& \int_{0}^{1/2}\!\! dx_1\ \cos(2\pi m x_1)\
\exp\!\Big[-\big\{2(2\pi)^2/z\big\}\tilde K \
x_1(1-x_1)\Big].
\end{eqnarray}
The function ${\cal I}(m,\tilde K)$ is an even function of $m$, so it
can be expanded in a Taylor series in $m^2$. One way to do it is to take
$\cos(x_1)\approx1+(1/2)x_1^2-(1/24)x_1^4+\dots$. This is a good
approximation if the coefficient in the exponential is large. Since
we are interested in discrete values of $m$, and considering that
values of $m$ greater than one are suppressed even near the transition
point, we can use the following approximation
\begin{equation}
\label{approx-i}
{\cal I}(m,\tilde K)={\cal I}(0,\tilde K)-\Big[{\cal I}(0,\tilde K)
-{\cal I}(1,\tilde K)\Big]m^2.
\end{equation}
With this approximation we can write Eq. (\ref{z-with-z_eff-2}) as
\begin{equation}
\label{z-with-z_eff-3}
Z = Z_{\phi} \prod_{\vec r} \sum_{m(\vec r)=-\infty}^{\infty}
\exp\!\Bigg[ -\frac{1}{2\tilde K_{\rm eff}}
\sum_{<\vec r_1,\vec r_2>}
(m(\vec r_1)-m(\vec r_2))^2 - \frac{1}{2\tilde K}\left(
\frac{C_{\rm s}}{C_{\rm m}}\right)\sum_{\vec r}
m(\vec r)^2 \Bigg] + O(E_j^4).
\end{equation}
The effective coupling constant is given by
\begin{eqnarray}
\label{k-eff}
\tilde K_{\rm eff} &=& \tilde K \Bigg[1 + \left(\frac{(2\pi)^2 E_J
C_{\rm m}} {q^2}\right)^2 h\Big(\tilde K
[1-C_{\rm s}{\bf C}^{-1}(|\vec 0|)]\Big)
\Bigg]^{-1}, \\
\label{def-h}
h(w) &=& w^3 \int_{0}^{1/2} dx [1-\cos(2\pi x)] \exp\!\Big[
-\big\{2(2\pi)^2/z\big\}w \ x(1-x)\Big].
\end{eqnarray}
The function $h(w)$ has the following limiting asymptotic behavior
\begin{equation}
\label{limits-h}
h(w) =\left\{ \begin{array}{ll}
(1/2)w^3\Big[1-w\ (1/z)(12+(2\pi)^2/3)\Big]+O(w^5),
& \mbox{if $w\ll 1$,} \\
& \\
\!\frac{(z/2)^3}{(2\pi)^4}\Big[1+\frac{12}{(2\pi)^2}
(z/2)w^{-1}-\frac{1}{(2\pi)^2}(z/2)^2\ w^{-2}\Big]
+ O(w^{-3}) ,
& \mbox{if $w\gg 1$.}
\end{array}
\right.
\end{equation}
We now use the fact that for the experimental systems
$C_{\rm s}\ll C_{\rm m}$,
so that in the limit $(C_{\rm s}/C_{\rm m})\rightarrow 0$ we can
use
\begin{equation}
\label{limit-cs_cm}
\lim_{(C_{\rm s}/C_{\rm m})\rightarrow 0} \Big[1-C_{\rm s}{\bf C}^{-1}
(|\vec 0|)\Big] = \lim_{(C_{\rm s}/C_{\rm m})\rightarrow 0} \Bigg[1+
\frac{1}{4\pi}\left(\frac{C_{\rm s}}{C_{\rm m}}\right) \ln\left(
\frac{C_{\rm s}}{C_{\rm m}}\right) \Bigg] = 1.
\end{equation}
The end result is a discrete Gaussian model with an effective coupling
constant given by Eq. (\ref{k-eff}). This effective model can
be transformed into a Villain model
\cite{janke-nather,itzykson-drouffe-book}.
The critical points of this model are given by the equation
$\tilde K_{\rm eff} = \tilde K_{c}^{\rm V}$,
where $\tilde K_c^{\rm V}$ is the critical coupling for the Villain model,
$\tilde K_{c}^{\rm V} \approx 0.752(5)$ for a square array
\cite{janke-nather}. In other words we have to solve the equations,
\begin{eqnarray}
\label{K-crit}
\tilde K_c &=& \tilde K_c^{\rm V} + \Omega\ h(\tilde K_c), \\
\label{omega}
\Omega &=& \tilde K_c^{\rm V}\ (\pi^4/4)\ (E_J/E_{C_{\rm m}})^2.
\end{eqnarray}
From these equations we find the first order correction to the crossover
temperature for a square array
\begin{equation}
\label{tc-first-correc}
\frac{T_c}{T_c^{(0)}} \approx 1 - 0.259\ \left(\frac{E_J}{E_{C_{\rm
m}}}
\right)^2 + O\Big((E_J/E_{C_{\rm }})^4
\Big).
\end{equation}
Here we have used $h(\tilde K_c^{\rm V})\approx 0.0106$.
An important property of Eq. (\ref{K-crit}) is that we can show that
it has only one solution, since the function $h(\tilde K)$ is
concave for small $\tilde K$ and it has an inflection point at
$\tilde K_{\rm infl}$,
\begin{equation}
\label{inflection}
\tilde K_{\rm infl}\approx z\ \frac{6.2}{2(2\pi)^2}.
\end{equation}
A sufficient condition for Eq. (\ref{K-crit}) to have only one solution is
$\tilde K_c^{\rm V}>\tilde K_{\rm infl}$. This condition is satisfied
for square as well as triangular arrays. This result shows that there
is no insulating QUIT phase and it is in clear
contrast to the existence of the QUIT found using the WKB-RG approximation
in the superconducting phase.
\section{\bf Quantum Monte Carlo Results}
\subsection{Definition of Physical Quantities Calculated}
\label{sec:mc-operators}
The two important physical parameters in our analysis are the temperature
and $\alpha=\frac{E_c}{E_j}$. Since in the experiments the
self-capacitance is much smaller than the mutual
capacitance, the relevant quantum parameter here is
\begin{equation}
\label{alpha-m}
\alpha_m = \frac{E_{C_{\rm m}}}{E_J} = \frac{e^2}{2C_{\rm m}E_J}.
\end{equation}
In the region where $\alpha_m$ is small, the phases dominate and we
expect a superconducting to normal transition.
The quantity we will use to characterize the coherent superconducting
phase is the helicity modulus \cite{fisher-barber-jasnow,ohta-jasnow}
defined as
\begin{equation}
\label{y-def}
\Upsilon = \frac{\partial^2 F}{\partial A_{\vec r,\vec r
+\hat x}^2}\Bigg|_{A=0}.
\end{equation}
Here $\hat x$ is the unitary vector in the $x$ direction.
The superfluid density per unit mass, $\rho_s$, is proportional $\Upsilon$,
with $\rho_s(T) = \frac{1}{V} \left(\frac{ma}{\hbar}\right)^2 \Upsilon(T)$,
where $a$ is the distance between superconducting islands,
$m$ is the mass of the Cooper pairs, and V is the volume.
From Eqs. (\ref{z-path-integral}) and (\ref{y-def}) we get
\begin{eqnarray}
\label{order-parameter-sc-nor}
\frac{1}{E_J L_x L_y}\Upsilon(T) = & &
\frac{1}{L_x L_y L_\tau}\Bigg[
\left< \sum_{\tau=0}^{L_\tau-1}\sum_{\vec r}
\cos\Big(\phi(\tau,\vec r)-\phi(\tau,\vec r+\hat x)-A_
{\vec r,\vec r+\hat x}\Big) \right> -
\nonumber \\
& & -\frac{E_J \beta}{L_\tau}\Bigg\{\left< \Bigg[
\sum_{\tau=0}^{L_\tau-1}\sum_{\vec r}
\sin\Big(\phi(\tau,\vec r)-\phi(\tau,\vec r +\hat x)-A_{\vec r,
\vec r+\hat x}\Big)\Bigg]^2\right>- \nonumber \\
& &\ \ \ \ \ \ \ \ \ \ \ -\left< \sum_{\tau=0}^{L_\tau-1}
\sum_{\tau,\vec r}\sin\Big(\phi(\vec r)-\phi
(\tau,\vec r +\hat x)-A_{\vec r,\vec r+\hat x}\Big)
\right>^2\Bigg\}\Bigg].
\nonumber\\
\end{eqnarray}
The quantity we shall use to probe the possible charge coherence in the
array is the inverse dielectric constant of the
gas of Cooper pairs, defined as \cite{minnhagen-warren,grest},
\begin{equation}
\label{ep}
\frac{1}{\varepsilon} = \lim_{\vec k\rightarrow 0} \left[
1-\frac{q^2}{k_BT}\frac{1}{{\bf C}(\vec k)} <n(\vec k)n(-\vec k)>
\right].
\end{equation}
We can obtain the Fourier transform ${\bf C}(\vec k)$ from
Eq. (\ref{capacitance-matrix}) for the capacitance matrix to get,
\begin{equation}
\label{c-k}
{\bf C}(\vec k) = C_s + 2C_m[1-\cos(k_x)]+2C_m[1-\cos(k_y)].
\end{equation}
The Fourier transform of the charge number is defined by
\begin{equation}
\label{n-k}
n(\vec k) = \frac{1}{\sqrt{L_x L_y}}\sum_{\vec r} n(\vec r)
\exp\left[i\vec k \cdot\vec r\right].
\end{equation}
Using this equation we can obtain a path integral representation for
this correlation function, given by
\begin{eqnarray}
\label{op-for-nn}
<n(\vec r_1)n(\vec r_2)> =
& &
-\frac{1}{Z}\prod_{\tau=0}^{L_\tau-1}
\sqrt{{\rm det}{[\bf C]}} \prod_{\vec r} \int_{0}^{2\pi}
\sqrt{\frac{L_\tau}{2\pi \beta q^2}} d\phi(\tau,\vec r)
\sum_{\{ m(\tau,\vec r)\}=-\infty}^{\infty} \times\nonumber\\
& & \times\frac{\partial^2}{\partial\phi(L_\tau,\vec r_1)\partial\phi
(L_\tau,\vec r_2)}\Bigg\{\exp\bigg[-\frac{1}{\hbar}
S[\{\phi\},\{m\}]\bigg]\Bigg\}\Bigg|_{\phi(L_\tau,\vec r)=
\phi(0,\vec r)}.
\end{eqnarray}
The action is given in Eq. (\ref{action}). This equation becomes
\begin{eqnarray}
\label{nxny}
<n(\vec r_1)n(\vec r_2)> =
& &\lim_{L_\tau\rightarrow\infty}\Bigg\{
\frac{L_\tau}{\beta q^2}{\bf C}(\vec r_1,\vec r_2) - \frac{1}{Z}
\prod_{\tau=0}^{L_{\tau}-1} \sqrt{{\rm det} {[\bf C]} } \prod_{\vec r}
\int_{0}^{2\pi}\sqrt{\frac{L_{\tau}} {2\pi\beta q^2} }
d\phi(\vec r,\tau) \times\nonumber\\
& &\times\sum_{\{ m(\vec r,\tau)\}= -\infty}^{\infty}
\Bigg(\frac{1}{\hbar}\frac{\partial S}{\partial
\phi(L_\tau,\vec r_1)}\ \frac{1}{\hbar}\frac{\partial S}
{\partial\phi(L_\tau,\vec r_2)} \Bigg)
\exp\bigg[-\frac{1}{\hbar}S[\{\phi\},\{m\}]\bigg]\Bigg\},
\nonumber\\
\end{eqnarray}
with
\begin{equation}
\label{ds_dphi}
\frac{1}{\hbar}\frac{\partial S}{\partial\phi(L_\tau,\vec r_1)} =
\frac{L_\tau}{\beta q^2} \sum_{\vec r}{\bf C}(\vec r_1,\vec r)
\left[\phi(L_\tau,\vec r)-\phi(L_\tau-1,\vec r)+2\pi
m(L_\tau-1,\vec r)\right].
\end{equation}
Notice, that this is not a well behaved operator since in the limit
$L_\tau\rightarrow\infty$ we would have to subtract two large numbers
and the path integral in the second term in Eq. (\ref{nxny}) would diverge.
This divergence is canceled out by the first term in Eq. (\ref{nxny}).
This can be seen explicitly by doing the calculation of $\epsilon^{-1}$
seting $E_J=0$, which leads to
\begin{equation}
\label{nxny-with-ms}
<n(\vec r_1)n(\vec r_2)> = \frac{1}{\beta q^2}\ {\bf C}(\vec r_1,
\vec r_2) + \left(\frac{2\pi}{\beta L_\tau}\right)^2
\sum_{\vec r_3,\vec r_4}{\bf C}(\vec r_1,\vec r_3)
{\bf C}(\vec r_2,\vec r_4) <m(\vec r_3)m(\vec r_4)>.
\end{equation}
This result can be put into Eq. (\ref{ep}) to obtain a finite inverse
dielectric constant,
\begin{equation}
\label{ep-2}
\frac{1}{\varepsilon} = \lim_{\vec k\rightarrow 0} \left[
\frac{(2\pi)^2}{\beta q^2}\ {\bf C}(\vec k) <|m(\vec k)|^2>
\right].
\end{equation}
Here we have used the Fourier transform defined in Eq. (\ref{n-k})
and the $m(\vec r)$ defined as
$m(\vec r) = \sum_{\tau=0}^{L_\tau-1} m(\tau,\vec r)$.
Note that in general this operator will not exactly be the
inverse dielectric constant of a gas of Cooper pairs, since it will
depend on $L_\tau$. But we expect that it does contain
most of the relevant information of the inverse dielectric constant
of our charged system. In our Monte Carlo calculations we have used
the general result Eq. (\ref{nxny}) valid for $E_J\neq 0$ and finite
$L_\tau$.
\subsection{\bf The Simulation Approach}
\label{sec:simulation}
Up to now we have seen that the partition function defined by the
Hamiltonian in Eq. (\ref{hamiltonian}) can be expressed in different
convenient representations for analytic analyses.
To carry out our QMC calculations, we have used what is, in principle,
the most straightforward representation of $Z$ given by Eqs.
(\ref{z-with-m}) and (\ref{action});
it involves the phases and the charge integer as statistical variables.
This representation is general enough to be used over all the whole
parameter range covered in the phase diagram.
In this case we have a set of angles
$\phi(\tau,\vec r)\in [0,2\pi)$, located at the nodes of a three-dimensional
lattice, with two space dimensions, $L_x$ and $L_y$, and one
imaginary time dimension, $L_\tau$. The periodic boundary condition,
comes from the trace condition in Eq.
(\ref{partition_function}), and we also have chosen to use periodic
boundary conditions in both space directions. The link variables
$m(\tau,\vec r)$ are defined in the bonds between two nodes
in the $\tau$ direction and they can take any integer value.
We have basically used the standard Metropolis algorithm to move about in
phase space \cite{metropolis}. As the phases are updated we restrict their
values to the interval $[0,2\pi)$. Moreover, the shifts along a $\tau$-column
and the individual phase moves are adjusted to keep the
acceptance rates in the range $[0.2,0.3]$.
If $\alpha_m$ is small, the system is in the semiclassical limit. In
this case the fluctuations of the phases along the imaginary time axis as
well as the fluctuations in the $m$'s are suppressed by
the second term in Eq. (\ref{action}). Attempts to change a phase variable
will have a very small success rate. Therefore we implemented
two kinds of Monte Carlo moves in the phase degrees of freedom. In one
sweep of the array we update the $L_x\times L_y$ imaginary time columns,
by shifting all the phases along a given column by the same angle. This
move does not change the second term in Eq. (\ref{action}), and thus it
probes only the Josephson energy \cite{jacobs-jose-novotny-goldman}.
To account for phase fluctuations
along the imaginary time axis, which become more likely as
$(\alpha_m/T)$ increases, we also make local updates of the phases
along the planes.
Another aspect of the implementation of the QMC algorithm is the order in
which we visit the array. This is relevant for the optimization of the
computer code in different computer architectures. In a scalar machine
we have used an algorithm that updates column by column in the array.
For a vector machine we have used the fact that for local updates, like
the ones we use, the lattice can be separated into four sublattices in
a checkerboard-like pattern. This separation
is done in such a way that each of the sublattices can be updated
using a long vector loop without problems of data dependency. Using this
last visiting scheme, the cpu time grows sublinearly with the size of the
array. One of the problems that this type of visiting scheme has in a
vector machine, like the Cray C90, is that the array's dimensions
have to be even, and this produces memory conflicts. We have not made
attempts to optimize this part of the code. We have not used parallel
machines in our calculations but the same type
of checkerboard visiting scheme would lead to a fast algorithm.
We followed Ref. \cite{jacobs-jose-novotny-goldman} and replaced the
U(1) symmetry of the problem by a discrete ${\bf Z}_N$ subgroup. We took
$N=5000$. This allows us to use integer arithmetic for the values
of the phase variables, and to store lookup tables for the Josephson
cosine part of the Boltzmann factors. This simplification can not be used for the
charging energy part of the Boltzmann factors, except in the
$C_{\rm m}=0$ case, where the $m$'s can be summed up in a virtually
exact form. In the latter case we can also store lookup
tables using the following definition of an effective potential $V_{\rm
eff}$,
\begin{equation}
\label{charging-boltzman}
\exp\left[-\left(\frac{L_\tau C_{\rm s}}{q^2\beta}\right)
V_{\rm eff}(\phi)\right] = \sum_{m=-\infty}^{\infty}
\exp\left[-\frac{1}{2}\left(\frac{L_\tau C_{\rm s}}
{q^2\beta}\right)(\phi+2\pi m)^2\right].
\end{equation}
We notice that this summation can be evaluated numerically to any
desired accuracy.
We calculated the thermodynamic averages after we had made $N$
visits to the array updating the phases and $M$ visits updating
the $m$'s. Typically, if $\alpha_{\rm m}$ is small we used $N=4$
and $M=1$. In the opposite limit we used $N=1$ and
$M=8,10,...$. This is so because our local updating algorithms
for the $m$'s have serious decorrelation time problems, due to the
long range interaction among the charges. We typically found that
in order to get reasonably small statistical errors, we needed
to perform, in most cases, about $N_{\rm meas}=2^{12}=4096$ measurements
of the thermodynamical quantities, other times we took up to
$N_{\rm meas}=2^{13}=8192$ measurements.
Once we have a long stationary string of values for the measured
operators we calculated their mean values and uncertainties.
We also have used the algorithm proposed in
Ref. \cite{flyvbjerg-petersen} for the
efficient calculation of the helicity modulus.
This method has a bias problem due to the last term
in Eq. (\ref{order-parameter-sc-nor}). However, in the zero
magnetic field case this problem is not present, since this term is
identically zero.
\subsection{\bf Results for f=0}
\label{sec:results-for-f=0}
In this subsection we present the bulk of our Monte Carlo results.
We have mostly calculated the helicity modulus in the
small $\alpha_{\rm m}$ region and the inverse dielectric
constant in the large $\alpha_{\rm m}$
regime, and both quantities in the intermediate region.
Most of the calculations we performed were for parameter values close to
or at the experimental ones. In particular, the ratio
between the self and mutual capacitances was kept fixed between the values
$C_{\rm s}/C_{\rm m}\approx 0.01$ and 0.03, with the bulk of the
calculations carried out for 0.01. We found that for the
helicity modulus both values gave essentially the same results.
Almost all of the calculations were done by lowering the temperature,
in order to reduce the possibility for the system to be trapped
in metastable states \cite{mikalopas-jarrel-pinski-chung-novotny}.
We have a clear physical understanding of the behavior of the system
in the very small $\alpha_{\rm m}$ limit, since this limit
is close to the classical 2-D XY model. Moreover, we have the
semiclassical calculation results, mentioned before,
up to first order in $\alpha_{\rm m}$,
which, as we shall see, agree very well with the Monte Carlo results.
In this limit the results are solid because the discrete
imaginary time path integral calculations converge very rapidly to the
infinite $L_\tau$ limit. Therefore in this section we will discuss our
numerical results for increasing values of $\alpha_{\rm m}$. This
will allow us to go from a well understood physical
and calculational picture to the nonperturbative region of parameter
space which is less understood. Here is where we will explore the
limits of our numerical calculational schemes. The end result will be
that a significant portion of the phase diagram can be understood. However,
some of the most interesting intermediate regimes of the phase diagram are
still very difficult to fully understand with our present calculational
techniques.
In Fig. \ref{fig:fig03-alpha-0p5} we show a typical curve for the helicity
modulus as a function of temperature, in the small $\alpha_{\rm m}$ limit.
As $\alpha_m$ increases $\Upsilon$ flattens in the superconducting region.
In order to calculate the transition
temperature we used the fact that the critical temperature and the helicity
modulus still satisfy the universal relation,
\begin{equation}
\label{tc-hel}
\Upsilon(T_c) = \frac{2}{\pi}\ T_c.
\end{equation}
Based on the first order results from the semiclassical approximation
analysis we know that this universal result is
independent of $\alpha_{\rm m}$. In other words, we can determine the
critical temperature by the intercept of $\Upsilon(T)$
with the line $(2/\pi)T$, as shown in Fig. \ref{fig:fig03-alpha-0p5}.
As can be seen from Fig. \ref{fig:fig03-alpha-0p5}, at high temperatures and
small $\alpha_{\rm m}$ the asymptotic limit
$L_\tau\rightarrow\infty$ is already reached for small
$L_\tau$. From Eq. (\ref{action}), we can see that the parameter that
determines this rate of convergence is
\begin{equation}
\label{conv-parameter}
P = \frac{L_\tau }{(\beta E_J)\alpha_{\rm m}}.
\end{equation}
The deep quantum limit is reached for a relatively large $P\gg 1$.
This progression is shown in Fig. \ref{fig:fig04-alpha-1p25}
where we plot the helicity modulus as a
function of temperature for a relatively large
$\alpha_{\rm m}=1.25$, $L_x=L_y=20$, and
three values of $L_\tau$. It can be seen that convergence is reached
for $P>5$, as found before in the self capacitive model in
Ref. \cite{jacobs-jose-novotny-goldman}.
As shown in Fig. \ref{fig5-alpha-1p25}, for a larger $\alpha_{\rm m}$
the behavior changes and the departure from the
$L_\tau\rightarrow\infty$ limit is manifested as a small dip in $\Upsilon$ at
low temperature \cite{jacobs-jose}. As the temperature is lowered
$\Upsilon$ shows an upward behavior. This is also seen in Fig.
\ref{fig:fig06-alpha-2p25}
from more extensive calculation for a still larger $\alpha_m$'s. This finite
$L_\tau$ behavior can be understood in terms of a plane decoupling along the
imaginary time direction. A way to see this is to notice that if we take
both contributions to the action given in Eq.(\ref{action}) as
independent, both of them would yield a low temperature transition.
If all the $L_\tau$ planes are considered decoupled, then the Josephson
coupling would be
$\beta/L_\tau$. Therefore in this case the N--S transition
would happen at $T_{\rm N-S}\approx 1/L_\tau$. On the other hand, if
we only consider the second term in Eq. (\ref{action}) we see that
plane decoupling would take place at
$T_{\rm decl}\approx (\alpha_{\rm m}/L_\tau)$. Now, if
$T_{\rm N-S}>T_{\rm decl}$ which implies $\alpha_{\rm m} < 1$, then only
the first transition would take place.
If however $\alpha_{\rm m}$ is large enough, we could have
$T_{\rm N-S}<T_{\rm decl}$, producing the observed dip in the helicity
modulus. This is seen in Fig. \ref{fig:fig08-Y-1_e-alpha-2p25} where the
helicity
modulus is shown together with the inverse dielectric constant.
There it can be seen that the dip starts at about the same temperature
where $\epsilon^{-1}$ becomes finite, signaling that the
nonzero winding numbers have become relevant. In this region the
fluctuations between planes have a small energy cost in the action
given by Eq. (\ref{action}).
As we increase $\alpha_{\rm m}$ further we arrive at a point where at low
temperatures, and fixed $L_\tau$, $\Upsilon$ goes to zero as shown
in Fig. \ref{fig:fig09-alpha-2p5}. To understand the nature of the low
temperature phase we have computed the equal space imaginary time
correlation function
\begin{equation}
\label{time-corr-func}
C_\tau(\tau) = \Big<\cos\left(\phi(\vec r,\tau)-
\phi(\vec r,0)\right)\Big>.
\end{equation}
We evaluated this function at three different temperatures,
for $\alpha_{\rm m}=1.75$ and $L_\tau=32$. The results are shown in
Fig. \ref{fig:fig07-corr-t-alpha-1p75} where we see that the appropriate
value for $L_\tau$ needs to be increased as T is lowered.
More extreme are the two temperature results
for $\alpha_{\rm m}=2.5$ and $L_\tau=64$, shown in Fig.
\ref{fig:fig10-corr-t-alpha-2p5}.
The upper curve corresponds to $k_BT/E_J=0.36$. As seen in Fig.
\ref{fig:fig09-alpha-2p5} at this temperature a value of $L_\tau=64$ is enough
to reliably calculate the helicity modulus, for in this case the
planes are correlated with a short decorrelation time. The lower
curve has $k_BT/E_J=0.1$. At this temperature the helicity modulus
is zero and the correlation function has a very short decorrelation
time. These results show that the low temperature discontinuity is
related to a decoupling of the planes along the imaginary time axis.
This plane decoupling does not show any dependence on $(L_x,L_y)$ for
the cases considered, and the curves shown in Fig. \ref{fig:fig09-alpha-2p5}
are reproducible within the statistical
errors for other values of $L_x$ and $L_y$.
Again we point out that upon increasing $L_\tau$ the decoupling
temperature moves closer to zero temperature. We should note that
from the WKB analysis there is a critical value for $\alpha$ above
which the superconducting state is no longer stable. So as we consider
larger values of $\alpha_{\rm m}$, the superconducting state will
become less and less stable. As we mentioned
the simulations were performed while lowering the temperature.
In contrast, in Fig.\ref{fig:fig11-alpha-2p75} we show results
from lowering the temperature for
$\alpha_m=2.75$. In this case $\Upsilon$ reaches a zero for $T\leq 0.2$.
We reversed the process increasing the temperature. Up to the last
temperature calculated the results are consistent with having zero
$\Upsilon$ for $\alpha_{\rm m}=2.75$.
The low temperature state arises from a decoupling transition between the
imaginary time planes which leads to an ensemble of decorrelated
planes. Maybe the planes could get recoupled at higher temperatures but,
as already mentioned, our local algorithm would take too long
to realign these planes so as to produce a coherent state.
As shown in Fig. \ref{fig:fig09-alpha-2p5} the temperature where $\Upsilon$
has a sharp drop changes with the size of the system. To see if in
the limit $L_\tau\rightarrow\infty$ we still have a finite low temperature
transition, we tried to extract it from the data for three
different $\alpha_m$ values by plotting them against
$1/L_\tau$ as shown in Figs. \ref{fig:fig12(a)-finite-size-alpha-2},
\ref{fig:fig12(b)-finite-size-alpha-2p25} and
\ref{fig:fig12(c)-finite-size-alpha-2p5}. From these
figures it appears that for $\alpha_{\rm m}=2.0 $ and $2.25$ there is
a nonzero transition temperature in the $L_\tau\rightarrow\infty$ limit.
We used a jackknife
calculation to estimate the infinite $L_\tau$ temperature and we found
$$T_{\rm decl}(\alpha_{\rm m}=2,L_\tau\rightarrow\infty)=(0.0183
\pm 0.009)(E_J/k_B),$$
and
$$T_{\rm decl}(\alpha_{\rm m}=
2.25, L_\tau\rightarrow\infty)=(0.0067\pm 0.0025) (E_J/k_B).$$
The same type of calculation was done for $\alpha_{\rm m}=2.5$.
The results are shown in Fig. \ref{fig:fig12(c)-finite-size-alpha-2p5}.
Here we found that
$$T_{\rm decl}(\alpha_{\rm m}=2.5, L_\tau\rightarrow\infty)=
(-0.013\pm 0.005)(E_J/k_B).$$
Therefore from these estimates we
surmise that the critical value for $\alpha_m\sim 2.5$, which is
larger than the one estimated from the WKB-RG analysis.
However, our QMC calculations, in particular at low temperatures,
are not precise enough to make a definitive determination of the
critical $\alpha_m$.
The calculations of the S--N transition
line seem to indicate that the superconductor to insulator zero temperature
transition occurs at $\alpha_{\rm m}\approx 3$, which is quantitatively
different from the T=0 estimate \cite{fazio-schon-91}.
The evaluation of the inverse dielectric constant is considerably more
complicated since the insulating region we have
$\alpha_{\rm m} > 3$, needing larger values
of $L_\tau$ at low temperatures. Moreover there are serious critical
slowing down problems due to the long range charge interactions
which worsen as the size of the system increases.
We should point out that, in comparing with the purely classical case
\cite{lee-teitel}, the quantum 2-D Coulomb gas
studied here has the extra complication of the
$n-\phi$ coupling, as seen in Eq. (\ref{z-path-integral}).
This introduces an imaginary component to the action.
Therefore we are forced to integrate the $n$'s introducing the
new variables $\{m\}$ leading to the action given in Eq. (\ref{action}).
This is the action that we used to perform the Monte Carlo
calculations.
We use the expression given in Eq. (\ref{ep-2}), which for finite $\vec k$
and $L_x=L_y$ gives $|\vec k|=2\pi/L_x$, as an upper bound for
the inverse dielectric constant \cite{olsson}.
The technical problems mentioned above made the calculation
of the dielectric constant
less reliable than that of $\Upsilon$.
The results obtained from the Monte Carlo runs were too noisy to give us
a quantitative estimate of the conductor to insulator transition
temperature if there was one. Our quoted results give
only tentative values for the
transition line.
The results are shown in Fig. \ref{fig:fig01-phase-diagram}, where we
also plotted the results of the Monte Carlo calculation of the
normal to superconductor transition temperature as well as the
experimental results from the Delft group
\cite{van-der-zant-thesis}. We have fitted a straight
line to the first seven points in this line and used a jackknife
calculation of $T_c(\alpha_{\rm m})$ for small
$\alpha_{\rm m}$. We obtained
\begin{equation}
\label{tc-first-order}
\frac{k_B T_c}{E_J} = (0.9430\pm0.0042)-(0.1800\pm0.0040)
\alpha_{\rm m}
+ O\left(\alpha_{\rm m}^2\right).
\end{equation}
The value of the slope is in good agreement with the semiclassical
approximation result given in Eq. (\ref{limits-T_BKT}).
The dashed line gives $\alpha_{\rm m}=2.8$ at T=0 and joins the
last QMC point to $T=0$. The line is only a guide
to the eye. We have not performed
detailed calculations around
$\alpha_{\rm m}=3$ since the required values of $L_\tau$ makes reliable
calculations too computationally intensive to be carried out with
current algorithms and computer capabilities.
\subsection{\bf Results for f=1/2.}
\label{sec:results-for-f=1/2}
We also have performed a few calculations of the helicity modulus for the
fully frustrated case f=1/2. The results of these calculations are shown
in Fig. \ref{fig:fig01-phase-diagram}. The experimental results of the Delft
group show a transition temperature for $\alpha_{\rm m}=0$ at
$(k_B T/E_J)\approx 0.3$ \cite{van-der-zant-thesis}, while the
classical fully frustrated case has a critical temperature close to
$(k_B T/E_J)\approx 0.5$. Taking this into account we have
rescaled the Monte Carlo results so that the calculated value
for the transition temperature for $\alpha_{\rm m}=0$, f=1/2; coincides
with the experimental result.
Taking the experimental value of the critical temperature for the $f=0$ case, we
performed a least square fit for the five smallest $\alpha_{\rm m}$'s
considered and found, using a jackknife calculation,
\begin{equation}
\label{exp-fit-for-f=0}
\left(\frac{k_B T}{E_J}\right)_{f=0} \approx (0.9787\pm 0.0070) -
(0.256\pm 0.017) \alpha_{\rm m} + O(\alpha_{\rm m}^2).
\end{equation}
In the same way, but now for the $f=1/2$ case, we found
\begin{equation}
\label{exp-fit-for-f=1/2}
\left(\frac{k_B T}{E_J}\right)_{f=1/2} \approx (0.3188\pm 0.0015) -
(0.2929\pm 0.0066) \alpha_{\rm m} + O(\alpha_{\rm m}^2).
\end{equation}
These rough calculations show that the slopes of both curves are very
close. This result can be compared
with Eq. (\ref{tc-with-mag-field}),
which confirms that the first order correction in $\alpha_{\rm m}$ to the
critical temperature should not depend on the value of the magnetic
field. A similar result for the equality of slopes was obtained
in Ref. \cite{jacobs-jose-novotny-goldman}, using a Monte Carlo
calculation for the self-capacitive model.
\section{\bf Self-consistent Harmonic Approximation}
\label{sec:harmonic}
We need an alternative analytic approach, in principle exact for fixed
and finite $L_\tau$, to further understand the QMC results at
low temperatures. This is important because of the strong $L_\tau$
dependence in the study of the QUIT temperature, and
the analytic WKB results are only strictly
valid at high temperatures and $L_\tau=\infty$.
In this section we use a variational principle to evaluate the
free energy for the JJA within a self-consistent harmonic approximation
(SCHA). This approximation gives increasingly better results
as the temperature is lowered. Previous SCHA
calculations \cite{doniach,lozovik,ariosa-beck} did not explicitly include
the charge degrees of freedom, which are of significant
importance in the analysis
presented here. As a bonus, we note that the SCHA developed here could
be used as the basis for developing an alternative QMC algorithm
to study the model at low temperatures and intermediate $\alpha$ values,
were both the WKB and the standard QMC analyses have problems.
We start with the following decomposition of the Hamiltonian given in Eq.
(\ref{hamiltonian})
\begin{equation}
\label{more-split-action}
H=H_0 + \{H_J-H_H\}.
\end{equation}
where $H_0= \left\{H+(H_H-H_J)\right\}$, and
\begin{eqnarray}
\label{def-of-H_j}
H_J &=& \frac{E_J}{L_\tau}\sum_{\tau=1}^{L_\tau-1}
\sum_{<\vec r_1,\vec r_2>} \left[1-\cos\left(\phi(\tau,\vec r_1)-
\phi(\tau,\vec r_2)\right)\right], \\
\label{def-of-H_h}
H_H &=& \frac{E_J\Gamma}{2L_\tau}\sum_{\tau=1}^{L_\tau-1}
\sum_{<\vec r_1,\vec r_2>} \left[\phi(\tau,\vec r_1)-
\phi(\tau,\vec r_2)\right]^2.
\end{eqnarray}
In other words, we replace the Josephson Hamiltonian by a spin wave
term and introduce its stiffness $\Gamma$ as the variational parameter.
$\Gamma$ is of course the helicity modulus, but to
$\underline {emphasize}$ that
it is evaluated within the SCHA we us the $\Gamma$ notation instead of
$\Upsilon$.
The spin wave approximation does not have a phase transition at any
temperature so the variational calculation is going to give a
vanishing $\Gamma$ as the signature of the JJA transition.
The variational free energy is given by
$F_V = F_H + \left< H_J - H_H \right>_H$, where
$\beta F_H = -\ln Z_H$, and $Z_H = {\rm Tr}\left\{\exp\{-\beta H_0\}\right\}$,
with the average $<,>$ defined as usual by $<A>_H =
{\rm Tr}\left\{A\exp\{-\beta H_0\}\right\}/Z_H$.
We use the change of variables
\begin{eqnarray}
\label{extra-change-of-variables}
\psi(\tau,\vec r) &=& \psi(0,\vec r) + \sqrt{\frac{2}{L_\tau}}\
\sum_{n=1}^{L_\tau-1} \chi_n(\vec r) \sin
\left(\frac{\pi n}{L_\tau} \tau\right), \\
\label{tauss}
\tau &=& 1,2,\dots,L_\tau-1,
\end{eqnarray}
and perform the integrals over the
variables $\chi_n(\vec r)$. The result is
\begin{eqnarray}
\label{z_h-mid-way}
Z_H =
& &\sqrt{{\rm det}{[\bf C]}} \prod_{\vec r}\sum_{m(\vec r)=-\infty}^
{\infty} \exp\left\{-\frac{1}{\hbar} \tilde S_m \right\}
\left[\prod_{n=1}^{L_\tau-1}\left[1+\frac{(\beta^2 q^2
E_J\Gamma/C_{\rm m})}{2L_\tau^2\left[1-
\cos\left(\frac{\pi n}{L_\tau}
\right)\right]}\right]\right]^{-L_x L_y/2} \times\nonumber\\
& &\times\prod_{\vec r}\int_{0}^{2\pi} \frac{\phi(0,\vec r)}
{\sqrt{2\pi\beta q^2}} \exp\left\{-\frac{1}{2}\sum_{\vec r_1,
\vec r_2} \phi(0,\vec r_1){\bf N}(\vec r_1,\vec r_2)
\phi(0,\vec r_2)+\sum_{\vec r} j(\vec r)
\phi(0,\vec r)\right\}.
\nonumber\\
\end{eqnarray}
where
\begin{eqnarray}
\label{def-s-tilde}
\frac{1}{\hbar}\tilde S_m &=& \frac{(2\pi)^2}{2}\left[\frac{C_{\rm m}}
{\beta q^2}+\frac{\beta E_J\Gamma}{6}\left(1-\frac{1}{L_\tau}
\right)\left(2-\frac{1}{L_\tau}\right)-(\beta E_J\Gamma)
g\!\left(\beta q\sqrt{E_J\Gamma/C_{\rm m}}\right)\right]
\times\nonumber\\
& &\times\sum_{\vec r_1,\vec r_2} m(\vec r_1){\bf O}(\vec r_1,\vec r_2)
m(\vec r_2).
\end{eqnarray}
To obtain this equation we have taken $C_{\rm s}=0$, while the general case
can also be treated as well, but since $C_{\rm s}\ll C_{\rm m}$ in the
experiment this assumption simplifies the calculations. We
also have used the following definitions to write the previous equations,
\begin{eqnarray}
\label{defs-for-o}
\sum_{<\vec r_1,\vec r_2>}\left[\phi(\vec r_1)-\phi(\vec r_2)\right]^2
&=& \sum_{\vec r_1,\vec r_2} \phi(\vec r_1) {\bf O}(\vec r_1,
\vec r_2) \phi(\vec r_2), \\
\label{def-for-c}
{\bf C} &=& C_{\rm m} {\bf O},\\
\label{def-for-n}
{\bf N} &=& (\beta E_J\Gamma)\left[1-h\left(\beta q
\sqrt{E_J\Gamma/C_{\rm m}}\right)\right]{\bf O},
\end{eqnarray}
with ${\bf O}$ the lattice Laplacian operator, and $g(*)$ and $f(*)$ are
functions defined in terms of a Matsubara sum and given in Appendix B.
The details of the variational calculation of the free energy are
presented in Appendix B. Here we discuss the main conclusions from
the calculation. The results for $\alpha_{\rm m}=0$, and 1 are shown
in Fig. \ref{fig:fig14-ys}. There we see that the $\alpha_{\rm m}=0$ case
has the right low temperature linear $T$ dependence. The result for
$\alpha_{\rm m}=1.0$ has an essentially flat low temperature
behavior for $\Gamma$, which is due to quantum phase
slips tunneling processes.
In Appendix B it is shown that the helicity modulus can be expressed as
\begin{equation}
\label{equation-for-gamma}
\Gamma = \exp\left\{ -\frac{1}{2}\left< (\Delta\phi)^2
\right>\right\}.
\end{equation}
with the fluctuations in the phase given by,
\begin{eqnarray}
\label{Dphi-for-infty}
< (\Delta \phi)^2 >_H &=& \frac{1}{2}\sqrt{\frac{2\alpha_{\rm m}}
{\Gamma}} \left(\frac{\sinh\left(\beta E_J
\sqrt{ 8\alpha_{\rm m}\Gamma}\right)}
{\cosh\left(\beta E_J \sqrt{
8\alpha_{\rm
m}\Gamma}\right)-1}\right).
\end{eqnarray}
The classical limit corresponds to setting
$\alpha_{\rm m}\rightarrow 0$, which gives
$< (\Delta\phi)^2 >_{H(cl)} = \frac{1}{2\beta E_J \Gamma}$.
Using this result and Eq. (\ref{equation-for-gamma}), at low
temperatures $\Gamma$ is given by
$\Gamma \approx 1 - {k_B T}/{4E_J} + O(T^2)$.
This is precisely the same low temperature behavior obtained from a
spin wave analysis in two dimensions \cite{ohta-jasnow}.
From $< (\Delta\phi)^2 >_{H(cl)}$ we find the
transition temperature within this
approximation, $\Gamma_c = \frac{1}{4} \left(\frac{k_B T_c}{E_J}\right)$,
$\Gamma_c = 1/e$, and ${k_B T_c^{(0)}}/{E_J} = {4}/{e}
\approx 1.472$,
which is an overestimate, (since the dimensionless 2-D critical
temperature is $T_c^{XY}\approx 0.9$).
The problem with this approximation is extending the integration intervals
from $[-\pi,\pi]$ to $(-\infty,\infty)$. As we have discussed
this is a good approximation at low temperatures but it breaks
down near the transition point.
Our conclusions from this analysis are
that for $\alpha_m=0$the classical SCHA gives
good results for low temperatures while it overestimates the critical
temperature at higher ones.
In the $\alpha_m\neq 0$ case the quantization of the
spin wave excitations leads to
a non-vanishing result for $< (\Delta\phi)^2 >_H$, given in Eq.
(\ref{Dphi-for-infty}). For $(k_B T/E_J)\ll 1$ we get
\begin{eqnarray}
\label{quantum-Dphi-for-low-T}
& & < (\Delta\phi)^2 >_H \approx \frac{1}{2}\sqrt{\frac{2\alpha_{\rm m}}
{\Gamma}} \left( 1 + 2\exp\left\{-\beta E_J
\sqrt{8\alpha_{\rm m} \Gamma}\right\} \right),\\
& & \nonumber\\
\label{quamtum-Y-for-low-T}
& & \Gamma \approx \Gamma_0
\left(1-\frac{1}{2}\sqrt{\frac{2\alpha_{\rm m}}
{\Gamma_0}} \exp\left\{-\beta E_J\sqrt{8\alpha_{\rm
m}
\Gamma_0}\right\} \right),
\end{eqnarray}
where $\Gamma_0$ is the helicity modulus at zero temperature and it
is the self-consistent solution to the equation
$\Gamma_0 = \exp\left\{-\sqrt{\frac{\alpha_{\rm m}}
{8\Gamma_0}}\right\}$,
with $\Gamma_0 \approx 1-\sqrt{\frac{\alpha_{\rm m}}{8}}$ for
$\alpha_{\rm m}\ll 1$.
The solution to this equation is shown as a function of
$\alpha_{\rm m}$ in Fig. \ref{fig:fig15-ys-T=0}, where we also show some
Monte Carlo simulation results. The $\Gamma_0$ result
also presents a transition to a zero $\Gamma$ state at
$\alpha_{\rm m}^{c}(T=0)=32/e^2\approx 4.33$, with a jump from
$\Gamma_0^{c} = 1/e^2$ to zero.
Again the result of the
SCHA overestimates the stability
of the superconducting state. Both the extension of the integration
intervals and ignoring the $m$'s in these calculations are probably
responsible for the deviations at large $\alpha_{\rm m}$.
An interesting
observation is that the result for $\Gamma_c$ is
exact up to first order in $\alpha_{\rm m}$. This is equal to the result
we obtained from the WKB-RG analysis. Also surprising is
that the first order correction to the critical temperature agrees
with Eq. (\ref{limits-T_BKT})
\begin{equation}
\label{firt-correction-tc}
\left(\frac{k_B T_c}{E_J}\right) = \left(\frac{k_B T_c^{(0)}}{E_J}\right)
- \left(\frac{2}{3z}\right) \alpha_{\rm m}
+ O(\alpha_{\rm m}^2),
\end{equation}
where $z$ is the coordination number of the lattice.
In Fig. \ref{fig:fig17-ys-lt}
we show $\Gamma$ for $\alpha_{\rm m}=1.0$ as a function of the
temperature for increasing values of $L_\tau$, which should be
compared with Fig. \ref{fig:fig04-alpha-1p25}. This result strongly suggests
that the upward tendency of
the helicity modulus at low temperatures seen in the QMC results
may be an artifact of the finite $L_\tau$ nature of the
calculations. The origin of this
increase is in the low temperature result for the finite $L_\tau$
calculation of Eq. (\ref{def-of-Dphi2})
\begin{equation}
\label{low-temp-dphi-finite-lt}
< (\Delta \phi)^2 >_H \approx \frac{L_\tau}{\beta E_J \Gamma}
\left(1-\frac{(L_\tau+2)L_\tau}{16(\beta E_J)^2\alpha_{\rm
m}\Gamma}
+ O(T^3)\right),
\end{equation}
so that for finite $L_\tau$ and at low temperatures
the helicity modulus is given by
\begin{equation}
\label{Y-low-temp-finite-lt}
\Gamma \approx 1 - \frac{L_\tau}{2}\left(\frac{k_B T}{E_J}\right) +
O(T^2).
\end{equation}
We have been able to explain the shape of the helicity modulus
curves for low temperatures, but Figs. \ref{fig:fig04-alpha-1p25} and
\ref{fig5-alpha-1p25} show
that if $\alpha_{\rm m}>\alpha_{\rm m}^{*}$, where $\alpha_{\rm m}^{*}\in
(1.25,1.75)$, then the helicity modulus has a dip before it may go to one
at low temperatures. So far we have ignored the contribution of the
Discrete Gaussian Model (DGM) to the variational free energy which
is a good approximation only for small $\alpha_{\rm m}$.
We calculated the helicity modulus for the model ignoring the DGM
and then including it as a continuous Gaussian model for finite $L_\tau$,
which is a good approximation for the effective
coupling $J_{\rm eff}\ll 1$.
For large $\alpha_{\rm m}$ the crossover point, which is when $J{\rm eff}$
becomes soft,
is seen as a finite dip in the helicity modulus.
Unfortunately, the $T^{*}$ found in the Monte Carlo calculations
is much larger than the one given by Eq. (\ref{def-of-Jeff}).
It is apparent that the effective coupling
$J_{\rm eff}$ does not contain all the contributions to the
renormalization of the DGM due to the
integration over the phases.
To illustrate the nature of this
crossover we performed several calculations of the helicity modulus
for $L_\tau=10, 20$, and 40 with $\alpha_{\rm m}=1$. The results are
shown in Fig. \ref{fig:fig19-cross-over}. There we took
$\frac{L_\tau}{\beta E_J\alpha_{\rm m}}=6$ for the crossover temperature.
These results can be compared with those of Fig. \ref{fig5-alpha-1p25}.
The Monte Carlo calculations show that this dip occurs simultaneously with the
rise in the inverse dielectric constant. This is a signal that the non-zero
effective constant for the winding numbers becomes soft, making
their contribution to the helicity modulus non vanishing. We note
that the effect of a finite lattice, necessary for the Monte Carlo
calculation, increases the softening temperature of $J_{\rm eff}$.
On the other hand, the variational calculation for $C_{\rm s}=0$ does
not depend on the size of the lattice, therefore it can not capture
these finite space size effects.
\section{\bf Conclusions}
\label{sec:conclusions}
We have presented a thorough study of the $\alpha_{\rm m}$ vs.
$T$ phase diagram for an array of ultrasmall Josephson junctions using a
series of theoretical tools.
One of our main goals was to perform these calculations for
these arrays using experimentally realistic
parameters. The model we used for the JJA is defined by a Hamiltonian
that has two contributions, a Josephson coupling and an electrostatic
interaction between the superconducting islands. The ratio of these
two contributions was defined as $\alpha_{\rm m}$
= (charging energy)/(Josephson energy). This was the important
quantum parameter in our analysis.
For convenience of calculation
we derived different path integral formulations of the quantum
partition function of the JJA. In the small $\alpha_{\rm m}$ limit
we used a WKB-RG approximation to find the first order correction in
$\alpha_{\rm m}$ to
the classical partition function. The result of this calculation was an
effective classical partition function of a 2-D XY model type, where the
coupling constant is modified by the quantum fluctuations. We used the
modified renormalization group equations for the 2-D model to find the
superconducting to normal phase boundary. We also found that up to
first order in $\alpha_{\rm m}$, the correction
to the transition temperature was $\underline{independent}$ of magnetic field.
One interesting finding from
this calculation was the possible existence of a low
temperature instability QUIT of the superconducting state.
We found evidence for the QUIT, but the evidence is at the border of validity
of the calculational approaches used. To have a definite theoretical proof
of the existence of the QUIT one needs to have better algorithms
and/or improvements in computer power. The results presented here are,
however, rather encouraging. Of course the ultimate test will be furnished
by experiment, and there too incipient indications of a low temperature
instability have also been reported in \cite{van-der-zant-thesis}.
In the large $\alpha_{\rm m}$ limit we used a perturbative expansion
in $1/\alpha_{\rm m}$ to find an effective partition function for a
quantum a 2-D Coulomb gas. This model shows a renormalized
conducting to insulating
transition as the temperature is lowered. We did not find a low
temperature QUIT instability in the large $\alpha_{\rm m}$ insulating phase.
We also performed extensive nonperturbative quantum Monte Carlo
calculations of the JJA model. We concentrated our analysis on the
helicity modulus $\Upsilon$. This quantity is directly related to
the superfluid density in the array. Using $\Upsilon$ we determined the
superconducting to normal transition boundary. We found good agreement
between the critical temperatures obtained by QMC calculation and the
semiclassical approximation. We also carried out a low temperature
$1/L_\tau$ extrapolation analysis of the $T_{QUIT}$ and found
evidence for $T_{QUIT}\neq 0$ for relatively large values of
$\alpha_m=2.0$ and $2.25$ but $T_{QUIT}= 0$ for $\alpha_m=2.5$.
These calculations have, however, a strong $L_\tau$ dependence
and our QMC algorithm is not precise enough
to completely ascertain the nature of the low temperature phase.
Nonetheless, the results found here together with the scant
emerging experimental evidence for a low temperature instability
yields further support for the possible existence of a QUIT.
We also presented some QMC calculations of the inverse dielectric
constant of the 2-D quantum Coulomb gas, in order to find the conducting to
insulating phase boundary. We found that the present
Monte Carlo path integral implementation of our model, that includes
phase and charge degrees of freedom,
does not allow us to make reliable calculations
of this quantity. Our results for this transition are only qualitative.
Further technical improvements are needed in order to make solid
quantitative statements about the $N-I$ phase boundary.
To use a QMC calculation to prove or disprove the presence of
a low temperature instability in the superconducting state is not an easy
task. However, these type of calculations give us upper temperature limits for
the instability region. As far as our calculations could determine, the
results for the superfluid density as a function of temperature are in
rather good agreement with experimental findings in JJA
\cite{van-der-zant-thesis}, except for the incipient data
on the reentrant transition in
the nonperturbative region of $\alpha_{\rm m}$.
To further understand the QMC results at low temperatures and
as a function of $L_\tau$, we have also implemented a self-consistent
harmonic approximation analysis of the model, that includes phases and
charge freedoms. We were able to make
successful qualitative, and in some instances even quantitative
comparisons between both calculational approaches.
One of the conclusions from these calculations is that the general
trend of the QMC results for $\Upsilon$ can
be traced to the discretization of the imaginary time
axis. For small $\alpha_{\rm m}$, the decrease of the helicity modulus
at low temperature is clearly due to this effect.
Among the most significant aspects of the results presented in this
paper is the quantitative agreement between our different calculational
approaches and the corresponding experimental results
in the superconducting-normal phase boundary with essentially
only the measured capacitances as adjustable parameters. The
existence of the QUIT is also a significant result of this paper,
for it had not been studied in a model including realistic capacitances
in the model.
\acknowledgments
We thank J. Houlrick and M. Novotny for contributions and helpful
discussions in the early stages of the Monte Carlo study presented
here. This work has been partially supported by
NSF grants DMR-95-21845, and PHY-94-07194, at UC Santa Barbara.
\vskip 1.0cm
|
1,108,101,565,646 | arxiv | \section{Introduction}
The ability to focus, trap, and guide the propagation of waves on subwavelength scales is of fundamental importance in physics. Systems of subwavelength resonators have, in particular, been shown to have desirable and sometimes remarkable properties thanks to their tendency to interact very strongly with waves on small length scales \cite{kaina2015negative,phononic1, pendry}. A subwavelength resonator is a cavity with material parameters that are greatly different from the background medium and which experiences resonance in response to wavelengths much greater than its size. The large material contrast is an essential prerequisite for the subwavelength resonant response.
In this review, we consider wave interaction with systems of subwavelength resonators. At particular low frequencies, known as subwavelength resonances, subwavelength resonators behave as strong wave scatterers. Using layer potential techniques and Gohberg-Sigal theory, we first derive a formula for the resonances of a system of resonators of arbitrary shapes. Then, we derive an effective medium theory for wave propagation in systems of resonators. We start with a multiple scattering formulation of the scattering problem in which an incident wave impinges on a large number of small, identical resonators in a homogeneous medium. Under certain conditions on the configuration of the resonator distribution, the point interaction approximation holds and yields an effective medium theory for the system of resonators as the number of resonators tends to infinity. As a consequence, below the resonant frequency of a single resonator, the obtained effective media may be highly refractive, making the focusing of waves at subwavelength scales achievable.
Then, we provide a mathematical theory for understanding the mechanism behind the double-negative refractive index phenomenon in systems of subwavelength resonators. The design of double-negative metamaterials generally requires the use of two different kinds of subwavelength resonators, which may limit the applicability of double-negative metamaterials. Herein we rely on media that consists of only a single type of resonant element, and show how to turn the metamaterial with a single negative effective property into a negative refractive index metamaterial, which acts as a superlens. Using dimers made of two identical resonators, we show that both effective material parameters can be negative near the anti-resonance of the two hybridized resonances for a single constituent dimer of subwavelength resonators.
Furthermore, we consider periodic structures of subwavelength resonators where subwavelength band gap opening typically occurs. This can induce rich physics on the subwavelength scale which cannot be understood by the standard homogenization theory. To demonstrate the opening of a subwavelength band gap, we exploit the strong interactions produced by subwavelength resonators among the cells in a periodic structure. We derive an approximate formula in terms of the contrast for the quasi-periodic subwavelength resonant frequencies of an arbitrarily shaped subwavelength resonator. Then, we consider the behavior of the first Bloch eigenfunction near the critical frequency where a subwavelength band
gap of the periodic structure opens. For a square lattice, we show that the critical frequency occurs at the corner of the Brillouin zone where the Bloch eigenfunctions are antiperiodic. We develop a high-frequency homogenization technique to describe the rapid variations of the Bloch eigenfunctions on the microscale (the scale of the elementary crystal cell). Compared to the effective medium theory, an effective equation can be derived only for the envelope of this first Bloch eigenfunction.
Defect modes and guided modes can be shown to exist in perturbed subwavelength resonator crystals. We use the subwavelength band gap to demonstrate cavities and waveguides of subwavelength dimensions. First, by perturbing the size of a single resonator inside the crystal, we show that this crystal has a localized eigenmode close to the defect resonator. Further, by modifying the sizes of the subwavelength resonators along a line in a crystal, we show that the line defect acts as a waveguide; waves of certain frequencies will be localized to, and guided along, the line defect.
Topological properties of periodic lattices of subwavelength resonators are also considered, and we investigate the existence of Dirac cones in honeycomb lattices and topologically protected edge modes in chains of subwavelength resonators.
We first show the existence of a Dirac dispersion cone in a honeycomb crystal comprised
of subwavelength resonators of arbitrary shape. The high-frequency homogenization technique shows that, near the Dirac points, the Bloch eigenfunction is the sum of two eigenmodes. Each eigenmode can be decomposed into two components: one which is slowly varying and satisfies a homogenized equation, while the other is periodic and highly oscillating. The slowly oscillating components of the eigenmodes satisfy a system of Dirac equations. This yields a near-zero effective refractive index near the Dirac points for the plane-wave envelopes of the Bloch eigenfunctions in a subwavelength metamaterial. The opening of a Dirac cone can create topologically protected edge modes, which are stable against geometric errors of the structure. We study a crystal which consists of a chain of subwavelength resonators arranged as dimers (often known as an SSH chain) and show that it exhibits a topologically non-trivial band gap, leading to robust localization properties at subwavelength scales.
Finally, we present a bio-inspired system of subwavelength resonators designed to mimic the cochlea. The system is inspired by the graded properties of the cochlear membranes, which are able to perform spatial frequency separation. Using layer potential techniques, the resonant modes of the system are computed and the model's ability to decompose incoming signals is explored.
This review is organized as follows. In Section~\ref{sec2}, after stating the subwavelength resonance problem and introducing some preliminaries on the layer potential techniques and Gohberg-Sigal theory, we prove the existence of subwavelength resonances for systems of resonators and show a modal decomposition for the associated eigenmodes. Then, we study in Section~\ref{sec3} the behavior of the coupled subwavelength resonant modes when the subwavelength resonators are brought close together. In Section~\ref{dilutesect} we derive an effective medium theory for dilute systems of subwavelength resonators. Section \ref{sec5} is devoted to the spectral analysis of periodic structures of subwavelength resonators. After recalling some preliminaries on the Floquet theory and quasi-periodic layer potentials, we prove the occurrence of subwavelength band gap opening in square lattices of subwavelength resonators and characterize the behavior of the first Bloch eigenfunction near the critical frequency where a subwavelength band gap of the periodic structure opens. In Section~\ref{sec6}, we consider honeycomb lattices of subwavelength resonators and prove the existence of Dirac cones. We also study a chain of subwavelength resonators which exhibit a topologically non-trivial band gap. Finally, in Section~\ref{sec7}, we present a graded array of subwavelength resonators which is designed to mimic the frequency separation proprieties of the cochlea. The review ends with some concluding remarks.
\section{Subwavelength resonances} \label{sec2}
We begin by describing the resonance problem and the main mathematical tools we will use to study a finite collection of subwavelength resonators.
\subsection{Problem setting}
We are interested in studying wave propagation in a homogeneous background medium with $N\in\mathbb{N}$ disjoint bounded inclusions, which we label as $D_1,D_2,\dots,D_N\subset\mathbb{R}^3$. We assume that the boundaries are all of class $\mathcal{C}^{1, \eta}$ with $0<\eta <1$ and write $D=D_1\cup\dots\cup D_N$.
We denote the material parameters within the bounded regions $D$ by $\rho_b$ and $\kappa_b$, respectively. The corresponding parameters for the background medium are $\rho$ and $\kappa$ and the wave speeds in $D$ and $\mathbb{R}^3\setminus \overline{D}$ are given by
$v_b=\sqrt{{\kappa_b}/{\rho_b}}$ and $v=\sqrt{{\kappa}/{\rho}}$. We define the wave numbers as
\begin{equation} \label{defkv} k = \frac{\omega}{v}, \quad k_b = \frac{\omega}{v_b}.\end{equation}
We also define the dimensionless contrast parameter
\begin{equation} \label{defdelta}
\delta=\frac{\rho_b}{\rho}.
\end{equation}
We assume that
\begin{equation} \label{defp}
\delta\ll1 \mbox{ while } v_b =\O(1) \mbox{ and } v =\O(1).\end{equation}
This high-contrast assumption will give the desired subwavelength behaviour, which we will study through an asymptotic analysis in terms of $\delta$.
For $\omega \in \mathbb{C}$, we study the scattering problem
\begin{equation} \label{eq:scattering}
\left\{
\begin{array} {ll}
\displaystyle \Delta {u}+ k^2 {u} = 0 & \text{in } \mathbb{R}^3 \setminus \overline{D}, \\[0.3em]
\displaystyle \Delta {u}+ k_b^2 {u} = 0 & \text{in } D, \\
\noalign{\smallskip}
\displaystyle {u}|_{+} -{u}|_{-} = 0 & \text{on } \partial D, \\
\noalign{\smallskip}
\displaystyle \delta \frac{\partial {u}}{\partial \nu} \bigg|_{+} - \frac{\partial {u}}{\partial \nu} \bigg|_{-} = 0 & \text{on } \partial D, \\
\noalign{\smallskip}
\displaystyle u(x) - u^{in}(x) & \text{satisfies the Sommerfeld radiation} \\ & \text{condition as } |x| \rightarrow \infty,
\end{array}
\right.
\end{equation}
where $|_+$ and $|_-$ denote the limits from the outside and inside of $D$. Here, $u^{in}$ is the incident field which we assume satisfies $\Delta u^{in} + k^2u^{in} = 0$ in $\mathbb{R}^3$ and $\nabla u^{in}\big|_D = \O(\omega)$. We restrict to frequencies such that $\Re(k) > 0$, whereby the Sommerfeld radiation condition is given by
\begin{equation}\label{eq:sommerfeld}
\lim_{|x|\rightarrow \infty}|x| \left(\frac{\partial}{\partial |x|} -\mathrm{i}\mkern1mu k\right)u = 0,
\end{equation}
which corresponds to the case where $u$ radiates energy outwards (and not inwards).
\begin{defn}[Subwavelength resonant frequency] \label{def:res^}
We define a {subwavelength resonant frequency} to be $\omega=\omega(\delta)$ such that $\Re(\omega) > 0$ and:
\begin{itemize}
\item[(i)] there exists a non-trivial solution to \eqref{eq:scattering} when $u^{in}=0$,
\item[(ii)] $\omega$ depends continuously on $\delta$ and is such that $\omega(\delta)\to0$ as $\delta\to0$.
\end{itemize}
\end{defn}
The scattering problem \eqref{eq:scattering} is a model problem for subwavelength resonators with high-contrast materials. It can be effectively studied using representations in terms of integral operators.
\subsection{Layer potential theory on bounded domains and Gohberg-Sigal theory}
\label{sec:layerpot}
The layer potential operators are the main mathematical tools used in the study of the resonance problem described above. These are operator-valued holomorphic functions, and can be studied using Gohberg-Sigal theory.
\subsubsection{Layer potential operators}
Let $\S_D^k$ be the single layer potential, defined by
\begin{equation} \label{eq:Sdef}
\S_D^k[\phi](x) := \int_{\partial D} G^k(x-y)\phi(y) \; \: \mathrm{d} \sigma(y), \quad x \in \mathbb{R}^3,
\end{equation}
where $G^k(x)$ is the outgoing Helmholtz Green's function, given by
$$
G^k(x) := -\frac{e^{\mathrm{i}\mkern1mu k|x|}}{4\pi|x|}, \quad x \in \mathbb{R}^3, \ \Re(k)\geq 0.
$$
Here, ``outgoing'' refers to the fact that $G^k$ satisfies the Sommerfeld radiation condition \eqref{eq:sommerfeld}. For $k=0$ we omit the superscript and write the fundamental solution to the Laplacian as $G$.
For the single layer potential corresponding to the Laplace equation, $\S_D^0$, we also omit the superscript and write $\S_D$. It is well known that the trace operator $\S_D: L^2(\partial D) \rightarrow H^1(\partial D)$ is invertible, where $H^1(\partial D)$ is the space of functions that are square integrable on $\partial D$ and have a weak first derivative that is also square integrable. We denote by $\L(L^2(\partial D),H^1(\partial D))$ the set of bounded linear operators from $L^2(\partial D)$ into $H^1(\partial D)$.
The Neumann-Poincar\'e operator $\mathcal{K}_D^{k,*}: L^2(\partial D) \rightarrow L^2(\partial D)$ is defined by
\begin{equation*}\label{eq:K_def}
\mathcal{K}_D^{k,*}[\phi](x) := \int_{\partial D} \frac{\partial }{\partial \nu_x}G^k(x-y) \phi(y) \; \: \mathrm{d} \sigma(y), \quad x \in \partial D,
\end{equation*}
where $\partial/\partial \nu_x$ denotes the outward normal derivative at $x\in\partial D$. For $k=0$ we omit the superscript and write $\mathcal{K}_D^{*}$.
The behaviour of $\S_D^k$ on the boundary $\partial D$ is described by the following relations, often known as \emph{jump relations},
\begin{equation}\label{eq:jump1}
\S_D^k[\phi]\big|_+ = \S_D^k[\phi]\big|_-,
\end{equation}
and
\begin{equation}\label{eq:jump2}
\frac{\partial }{\partial \nu}\S_D^k[\phi]\Big|_{\pm} = \left(\pm\frac{1}{2} I + \mathcal{K}_D^{k,*}\right) [\phi],
\end{equation}
where $|_\pm$ denote the limits from outside and inside $D$. When $k$ is small, the single layer potential satisfies
\begin{equation} \label{eq:exp_S}
\S_D^k =\S_D + k\S_{D,1} + k^2\S_{D,2} + k^3 \S_{D,3} + \O(k^4),
\end{equation}
where the error term is with respect to the operator norm $\|.\|_{\L(L^2(\partial D),H^1(\partial D))}$, and the operators $\S_{D,n}: L^2(\partial D) \rightarrow H^1(\partial D)$ for $n=1,2,3$ are given by
$$
\S_{D,n}[\phi](x) = - \frac{\mathrm{i}\mkern1mu^{n}}{4 \pi n!} \int_{\partial D}| x - y |^{n-1} \phi(y) \; \: \mathrm{d} \sigma(y) \qquad x\in \partial D.
$$
Moreover, we have
\begin{equation} \label{eq:exp_K}
\mathcal{K}_D^{k,*} = \mathcal{K}_D^{0,*} + k^2\mathcal{K}_{D,2} + k^3\mathcal{K}_{D,3} + \O(k^4),
\end{equation}
where the error term is with respect to the operator norm $\|.\|_{\L(L^2(\partial D),L^2(\partial D))}$ and where
$$\mathcal{K}_{D,2}[\phi](x) = \frac{1}{8\pi}\int_{\partial D}\frac{(x-y)\cdot \nu_x}{|x-y|}\phi(y)\: \mathrm{d} \sigma (y), \quad \mathcal{K}_{D,3}[\phi](x) = \frac{\mathrm{i}\mkern1mu}{12\pi}\int_{\partial D}(x-y)\cdot \nu_x\phi(y)\: \mathrm{d} \sigma (y),$$ for $x\in \partial D$.
We have the following lemma from \cite{ammari2017double}.
\begin{lemma} \label{lem:ints}
Let $N=2$. For any $\varphi\in L^2(\partial D)$ we have, for $i=1,2$,
\begin{equation} \label{eq:properties}
\begin{split}
\int_{\partial D_i}\left(-\frac{1}{2}I+\mathcal{K}_D^{*}\right)[\varphi]\: \mathrm{d}\sigma=0,
\qquad&\int_{\partial D_i}\left(\frac{1}{2}I+\mathcal{K}_D^{*}\right)[\varphi]\: \mathrm{d}\sigma=\int_{\partial D_i}\varphi\: \mathrm{d}\sigma,\\
\int_{\partial D_i} \mathcal{K}_{D,2}[\varphi]\: \mathrm{d}\sigma=-\int_{D_i}\S_D[\varphi]\: \mathrm{d} x, \qquad &\int_{\partial D_i} \mathcal{K}_{D,3}[\varphi]\: \mathrm{d}\sigma=\frac{\mathrm{i}\mkern1mu|D_i|}{4\pi}\int_{\partial D}\varphi\: \mathrm{d}\sigma.
\end{split}
\end{equation}
\end{lemma}
A thorough presentation of other properties of the layer potential operators and their use in wave-scattering problems can be found in \emph{e.g.} \cite{ammari2018mathematical}.
\subsubsection{Generalized argument principle and generalized Rouch\'e's theorem}
The Gohberg-Sigal theory refers to the generalization to operator-valued functions of two classical results in complex analysis, the argument principle and Rouch\'e's theorem \cite{gohsig, gohberg2009holomorphic, ammari2018mathematical}.
Let $\mathcal{B}$ and $\mathcal{B}^\prime$ be two Banach spaces and denote by $\mathcal{L }(\mathcal{B},\mathcal{B}^\prime)$ the space of bounded linear operators from $\mathcal{B}$ into $\mathcal{B}^\prime$. A point $z_{0}$ is called a {\it
characteristic value} of the operator-valued function $z \mapsto A(z) \in \mathcal{L }(\mathcal{B},\mathcal{B}^\prime)$ if $A(z)$ is holomorphic in some neighborhood of $z_0$, except possibly at
$z_0$ and there
exists a vector-valued function $\phi(z)$ with values in
$\mathcal{B}$ such that
\begin{enumerate}
\item[(i)] $\phi(z)$ is holomorphic at $z_{0}$ and $\phi(z_{0}) \not= 0$,
\item[(ii)]
$A(z)\phi(z)$ is holomorphic at $z_{0}$
and vanishes at this point.
\end{enumerate}
Let $V$ be a simply connected bounded domain with rectifiable
boundary $\partial V$. An operator-valued function
$A(z)$ is normal with respect to $\partial V$ if it is finitely
meromorphic and of Fredholm type in $V$, continuous on $\partial {V}$, and
invertible for all $z\in \partial V$.
If $A(z)$ is normal
with respect to the contour $\partial V$ and $z_{j},$ $
j=1,\ldots,\sigma$, are all its characteristic values and poles lying in $V$, the full multiplicity $\mathcal{M}(A;\partial V)$ of $A(z)$ for
$ z \in V$ is the number of characteristic
values of $A(z)$ for $ z\in V$, counted with their multiplicities, minus the
number of poles of $A(z)$ in $V$, counted with their multiplicities:
$$
\mathcal{M}(A;\partial V) := \sum_{j=1}^{\sigma} M ( A(z_{j})),
$$
with $M ( A(z_{j}))$ being the multiplicity of $z_j$; see \cite[Chap. 1]{ammari2018mathematical}.
The following results are from \cite{gohsig}.
\begin{theorem}[Generalized argument principle] \label{thmprincipal}
Suppose that $A(z)$ is an operator-valued function which is normal
with respect to $\partial V$. Let $f(z)$ be a scalar function which
is holomorphic in $ V$ and continuous in $\overline{ V}$. Then
\begin{eqnarray*}
\frac{1}{2\pi \i} \mbox{ {\rm tr} }\int_{\partial
V} f(z)A(z)^{-1}\frac{d}{dz}A(z)dz =
\sum_{j=1}^{\sigma} M ( A(z_{j}))f(z_{j}),
\end{eqnarray*}
where $z_{j},$ $j=1,\ldots,\sigma$, are all the points in $ V$
which are either poles or characteristic values of $A(z)$.
\end{theorem}
A generalization \index{generalized Rouch\'e's theorem} of
Rouch{\'e}'s theorem to operator-valued functions is stated below.
\begin{theorem}[Generalized Rouch\'e's theorem] \label{rouche}
Let $A(z)$ be an operator-valued function which is normal with
respect to $\partial V$. If an operator-valued function $S(z)$
which is finitely meromorphic in $V$ and continuous on $\partial
V$ satisfies the condition
\begin{eqnarray*}
\|A(z)^{-1}S(z)\|_{ \mathcal{L }(\mathcal{B},\mathcal{B})} < 1, \hspace{0.4cm}
z \in \partial V, \end{eqnarray*} then $A + S $ is also
normal with respect to $\partial V$ and
\begin{eqnarray*} \mathcal{M}(A;\partial V)=
\mathcal{M}(A + S;\partial V).
\end{eqnarray*}
\end{theorem}
\subsection{Capacitance matrix analysis}
\noindent The existence of subwavelength resonant frequencies is stated in the following theorem, which was proved in \cite{davies2019fully, ammari2018minnaert} using Theorem \ref{rouche}.
\begin{theorem}
A system of $N$ subwavelength resonators exhibits $N$ subwavelength resonant frequencies with positive real parts, up to multiplicity.
\end{theorem}
\begin{proof}
The solution $u$ to the scattering problem \eqref{eq:scattering} can be represented as
\begin{equation} \label{eq:layer_potential_representation}
u(x) = \begin{cases}
u^{in}+\S_{D}^{k}[\psi](x), & x\in\mathbb{R}^3\setminus \overline{D},\\
\noalign{\smallskip}
\S_{D}^{k_b}[\phi](x), & x\in D,
\end{cases}
\end{equation}
for some surface potentials $(\phi,\psi)\in L^2(\partial D)\times L^2(\partial D)$, which must be chosen so that $u$ satisfies the transmission conditions across $\partial D$. Using the jump relation between $\S_{D}^k$ and $\mathcal{K}_{D}^{k,*}$, we see that in order to satisfy the transmission conditions on $\partial D$ the densities $\phi$ and $\psi$ must satisfy, for $x\in \partial D$,
\begin{equation}
\label{repformula}
\begin{cases}
\S_{D}^{k_b}[\phi](x)-\S_{D}^{k}[\psi](x)=u^{in}(x),\\
\noalign{\smallskip}
\left(-\frac{1}{2}I+\mathcal{K}_{D}^{k_b,*}\right)[\phi](x)-\delta\left(\frac{1}{2}I+\mathcal{K}_{D}^{k,*}\right)[\psi](x)=\delta \ddp{u^{in}}{\nu}(x).
\end{cases}
\end{equation}
Therefore, $\phi$ and $\psi$ satisfy the following system of boundary integral equations:
\begin{equation} \label{eq-boundary}
\mathcal{A}(\omega, \delta)[\Psi] =F,
\end{equation}
where
\begin{equation} \label{page450}
\mathcal{A}(\omega, \delta) =
\begin{pmatrix}
\mathcal{S}_D^{k_b} & -\mathcal{S}_D^{k} \\
\noalign{\smallskip}
-\frac{1}{2}I + (\mathcal{K}_D^{k_b})^*& -\delta( \frac{1}{2}I + (\mathcal{K}_D^{k})^*)
\end{pmatrix},
\,\, \Psi=
\begin{pmatrix}
\phi\\
\psi
\end{pmatrix},
\,\,F=
\begin{pmatrix}
u^{in}\\
\delta \frac{\partial u^{in}}{\partial \nu}
\end{pmatrix}.
\end{equation}
One can show that the scattering problem (\ref{eq:scattering}) is equivalent to the system of boundary integral equations (\ref{eq-boundary}). It is clear that $\mathcal{A}(\omega, \delta)$ is a bounded linear operator from $\mathcal{H}:= L^2(\partial D) \times L^2(\partial D)$ to $\mathcal{H}_1:=H^1(\partial D) \times L^2(\partial D)$. As defined in \Cref{def:res^}, the resonant frequencies to the scattering problem (\ref{eq:scattering}) are the complex numbers $\omega$ with positive imaginary part such that there exists a nontrivial solution to the following equation:
\begin{equation} \label{eq-resonance}
\mathcal{A}(\omega, \delta)[\Psi] =0.
\end{equation}
These can be viewed as the characteristic values of the holomorphic operator-valued function (with respect to $\omega$)
$\mathcal{A}(\omega, \delta)$. The subwavelength resonant frequencies lie in the right half of a small neighborhood of the origin in the complex plane.
In what follows, we apply the Gohberg-Sigal theory to find these frequencies.
We first look at the limiting case when $\delta =\omega=0$.
It is clear that
\begin{equation} \label{eq-A_0-3d}
\mathcal{A}_0:= \mathcal{A}(0, 0) =
\begin{pmatrix}
\mathcal{S}_D & -\mathcal{S}_D \\
\noalign{\smallskip}
-\frac{1}{2}I + \mathcal{K}_D^{*}& 0
\end{pmatrix},
\end{equation}
where $\mathcal{S}_D$ and $\mathcal{K}_D^{*}$ are respectively the single-layer potential and the Neumann--Poincar\'e operator on $\partial D$ associated with the Laplacian.
Since $\mathcal{S}_D: L^2(\partial D) \rightarrow H^1(\partial D)$ is invertible in dimension three and $\mathrm{Ker } ( -\frac{1}{2}I + \mathcal{K}_D^{*})$ has dimension equal to the number of connected components of $D$, it follows that $\mathrm{Ker } (\mathcal{A}_0)$ is of dimension $N$.
This shows that $\omega=0$ is a characteristic value for the holomorphic operator-valued function $\mathcal{A}(\omega, 0)$ of full multiplicity $2N$.
By the generalized Rouch\'e's theorem, we can conclude that for any $\delta$, sufficiently small, there exist $2N$ characteristic values to the holomorphic operator-valued function $\mathcal{A}(\omega, \delta)$
such that $\omega_n(0)=0$ and
$\omega_n$ depends on $\delta$ continuously. $N$ of these characteristic values,
$\omega_n= \omega_n(\delta), n=1, \ldots, N,$ have positive real parts, and these are precisely the subwavelength resonant frequencies of the scattering problem (\ref{eq:scattering}).
\end{proof}
Our approach to approximate the subwavelength resonant frequencies is to study the \emph{(weighted) capacitance matrix}, which offers a rigorous discrete approximation to the differential problem. The eigenstates of this $N\times N$-matrix characterise, at leading order in $\delta$, the subwavelength resonant modes of the system of $N$ resonators.
In order to introduce the notion of capacitance, we define the functions $\psi_j$, for $j=1,\dots,N$, as
\begin{equation} \label{defpsi}
\psi_j:=\S_D^{-1}[\chi_{\partial D_j}],
\end{equation}
where $\chi_A:\mathbb{R}^3\to\{0,1\}$ is used to denote the characteristic function of a set $A\subset\mathbb{R}^3$. The capacitance matrix $C=(C_{ij})$ is defined, for $i,j=1,\dots,N$, as
\begin{equation} \label{defCap}
C_{ij}:=-\int_{\partial D_i} \psi_j\: \mathrm{d}\sigma.
\end{equation}
In order to capture the behaviour of an asymmetric array of resonators we need to introduce the weighted capacitance matrix $C^\mathrm{vol}=(C^\mathrm{vol}_{ij})$, given by
\begin{equation} \label{defCapw}
C^\mathrm{vol}_{ij}:=\frac{1}{|D_i|} C_{ij},
\end{equation}
which accounts for the differently sized resonators (see \emph{e.g.} \cite{ ammari2020exceptional, davies2020close, ammari2017double} for other variants in different settings).
We define the functions $S_n^\omega$, for $n=1\dots,N$, as
$$S_n^\omega(x) := \begin{cases}
\S_{D}^{k}[\psi_n](x), & x\in\mathbb{R}^3\setminus \overline{D},\\
\S_{D}^{k_b}[\psi_n](x), & x\in D.\\
\end{cases}
$$
\begin{lemma} \label{lem:modal}
The solution to the scattering problem can be written, for $x\in\mathbb{R}^3$, as
\begin{equation*}
u(x)-u^{in}(x) = \sum_{n=1}^N q_nS_n^\omega(x) - \S_D^k\left[\S_D^{-1}[u^{in}]\right](x) + \O(\omega),
\end{equation*}
for coefficients $\underline{q}=(q_1,\dots,q_N)$ which satisfy, up to an error of order $\O(\delta \omega+\omega^3)$,
\begin{equation*}\label{eq:eval_C}
\left({\omega^2}-{v_b^2\delta}\,C^\mathrm{vol}\right)\underline{q}
=
{v_b^2\delta}\begin{pmatrix} \frac{1}{|D_1|} \int_{\partial D_1}\S_D^{-1}[u^{in}]\: \mathrm{d}\sigma \\ \vdots\\
\frac{1}{|D_N|}\int_{\partial D_N}\S_D^{-1}[u^{in}]\: \mathrm{d}\sigma \end{pmatrix}.
\end{equation*}
\end{lemma}
\begin{proof}
Using the asymptotic expansions \eqref{eq:exp_S} and \eqref{eq:exp_K} for $\S_{D}^k$ and $\mathcal{K}_{D}^{k,*}$ in \eqref{repformula}, we can see that
\begin{equation*}\label{eq:psi}
\psi=\phi-\S_D^{-1}[u^{in}]+ \O(\omega),
\end{equation*}
and, further, that
\begin{multline}
\left(-\frac{1}{2}I+\mathcal{K}_D^*+\frac{\omega^2}{{v}_b^2}\mathcal{K}_{D,2}-\delta\left(\frac{1}{2}I+\mathcal{K}_D^*\right)\right)[\phi]=\\-\delta \left(\frac{1}{2}I+\mathcal{K}_D^*\right)\S_D^{-1}[u^{in}]+ \O(\delta\omega+\omega^3). \label{eq:phi}
\end{multline}
At leading order, \eqref{eq:phi} says that $\left(-\frac{1}{2}I+\mathcal{K}_D^{*}\right)[\phi]=0$ so, in light of the fact that $\{\psi_1,\dots,\psi_N\}$ forms a basis for $\mathrm{Ker } \left(-\frac{1}{2}I+\mathcal{K}_D^{*}\right)$, the solution can be written as
\begin{equation} \label{eq:psi_basis}
\phi=\sum_{n=1}^N q_n\psi_n+ \O(\omega^2+\delta),
\end{equation}
for coefficients $\underline{q}=(q_1,\dots,q_N)$.
Finally, integrating \eqref{eq:phi} over $\partial D_i$, for $1\leq i\leq N$, gives us that
\begin{equation*}
-\omega^2\int_{D_i}\S_D[\phi]\: \mathrm{d} x -{v}_b^2\delta\int_{\partial D_i}\phi\: \mathrm{d}\sigma=-{v_b^2\delta}\int_{\partial D_i}\S_D^{-1}[u^{in}]\: \mathrm{d}\sigma, \label{eq:D}
\end{equation*}
up to an error of order $\O(\delta\omega+\omega^3)$. Substituting the expression \eqref{eq:psi_basis} gives the desired result.
\end{proof}
\begin{theorem} \label{thm:res}
As $\delta \rightarrow 0$, the subwavelength resonant frequencies satisfy the asymptotic formula
$$\omega_n = \sqrt{v_b^2\lambda_n\delta} -\i\tau_n\delta+ \O(\delta^{3/2}),$$
for $n = 1,\dots,N$, where $\lambda_n$ are the eigenvalues of the weighted capacitance matrix $C^\mathrm{vol}$ and $ \tau_n$ are non-negative real numbers that depend on $C$, $v$ and $v_b$.
\end{theorem}
\begin{proof}
If $u^{in} = 0$, we find from Lemma~\ref{lem:modal} that there is a non-zero solution to the resonance problem when $\omega^2/v_b^2\delta$ is an eigenvalue of $C^\mathrm{vol}$, at leading order.
To find the imaginary part, we adopt the ansatz
\begin{equation} \label{eq:omega_ansatz}
\omega_n=\sqrt{v_b^2\lambda_n\delta} -\i\tau_n\delta+ \O(\delta^{3/2}).
\end{equation}
Using a few extra terms in the asymptotic expansions for $\S_{D}^k$ and $\mathcal{K}_{D}^{k,*}$, we have that
\begin{equation*}
\psi=\phi+\frac{k_b-k}{4\pi\i}\left(\sum_{n=1}^N\psi_n\right)\int_{\partial D}\phi\: \mathrm{d}\sigma+ \O(\omega^2),
\end{equation*}
and, hence, that
\begin{multline*}
\left(-\frac{1}{2}I+\mathcal{K}_D^*+k_b^2\mathcal{K}_{D,2}+k_b^3\mathcal{K}_{D,3}-\delta\left(\frac{1}{2}I+\mathcal{K}_D^*\right)\right)[\phi]\\-\frac{\delta(k_b-k)}{4\pi\i}\left(\sum_{n=1}^N\psi_n\right)\int_{\partial D}\phi\: \mathrm{d}\sigma= \O(\delta\omega^2+\omega^4).
\end{multline*}
We then substitute the decomposition \eqref{eq:psi_basis} and integrate over $\partial D_i$, for $i=1,\dots,N$, to find that, up to an error of order $\O(\delta\omega^2+\omega^4)$, it holds that
\begin{equation*}
\bigg(-\frac{\omega^2}{v_b^2}-\frac{\omega^3\i}{4\pi v_b^3}JC+\delta C^\mathrm{vol}+\frac{\delta\omega\i}{4\pi}\bigg(\frac{1}{v_b}-\frac{1}{v}\bigg)C^\mathrm{vol} JC \bigg)
\underline{q}=0,
\end{equation*}
where $J$ is the $N\times N$ matrix of ones (\emph{i.e.} $J_{ij}=1$ for all $i,j=1,\dots,N$).
Then, using the ansatz \eqref{eq:omega_ansatz} for $\omega_n$ we see that, if $\underline{v}_n$ is the eigenvector corresponding to $\lambda_n$, it holds that
\begin{equation}
\tau_n= \frac{v_b^2}{8\pi v} \frac{\underline{v}_n\cdot C J C\underline{v}_n}{\| \underline{v}_n\|_D^2},
\end{equation}
where we use the norm $\| x\|_D:=\big(\sum_{i=1}^N |D_i| x_i^2\big)^{1/2}$ for $x\in\mathbb{R}^N$. Since $C$ is symmetric, we can see that $\tau_n\geq0$.
\end{proof}
It is more illustrative to rephrase Lemma~\ref{lem:modal} in terms of basis functions that are associated with the resonant frequencies. Denote by $\underline{v}_n=(v_{1,n},\dots,v_{N,n})$ the eigenvector of $C^\mathrm{vol}$ with eigenvalue $\lambda_n$. Then, we have a modal decomposition with coefficients that depend on the matrix $V=(v_{i,j})$, assuming the system is such that $V$ is invertible. The following result follows from Lemma~\ref{lem:modal} by diagonalising the matrix $C^\mathrm{vol}$.
\begin{lemma} \label{lem:modal_res}
Suppose that the resonators' geometry is such that the matrix of eigenvectors $V$ is invertible. Then if $\omega=\O(\sqrt{\delta})$ the solution to the scattering problem can be written, for $x\in\mathbb{R}^3$, as
\begin{equation*}
u^s(x):= u(x)-u^{in}(x) = \sum_{n=1}^N a_n u_n(x) - \S_D\left[\S_D^{-1}[u^{in}]\right](x) + \O(\omega),
\end{equation*}
for coefficients which satisfy, up to an error of order $\O(\omega^3)$,
\begin{equation*}
a_n(\omega^2-\omega_n^2)=-A\nu_n\Re(\omega_n)^2,
\end{equation*}
where $\nu_n=\sum_{j=1}^{N} [V^{-1}]_{n,j}$, \emph{i.e.} the sum of the $n$\textsuperscript{th} row of $V^{-1}$.
\end{lemma}
\begin{remark} \label{remark12}
When $N=1$, the subwavelength resonant frequency $\omega_1$ is called the Minnaert resonance.
By writing an asymptotic expansion of
$\mathcal{A}(\omega, \delta)$ in terms of $\delta$
and applying the generalized argument principle (Theorem \ref{thmprincipal}), one can prove that
$\omega_1$ satisfies as $\delta$ goes to zero the asymptotic formula \cite{ammari2018minnaert}
\begin{equation} \label{defomegaM}
\omega_1 =
\underbrace{{\sqrt{\frac{\mathrm{Cap}_D}{|D|}} v_b {\sqrt{\delta}}}}_{:= \omega_M} - {\i} \underbrace{{\left(\frac{\mathrm{Cap}_D^2 v_b^2}{8 \pi v |D|} {\delta}\right)}}_{:= \tau_M} + \O(\delta^{\frac{3}{2}}),
\end{equation}
where \begin{equation} \label{defcap}
\mathrm{Cap}_D:= \displaystyle - \int_{\partial D} \mathcal{S}_D^{-1}[\chi_{\partial D}] \; \: \mathrm{d}\sigma \end{equation} is the capacity of $\partial D$. Moreover, the following
{monopole approximation} of the scattered field for $\omega$ near $\omega_M$ holds \cite{ammari2018minnaert}:
\begin{equation} \label{monopole}
u^s(x) = {g(\omega,\delta, D)}(1+\O(\omega)+\O(\delta)+o(1))u^{in}(0) {G^{k}(x)},
\end{equation}
with the origin $0 \in D$ and the {scattering coefficient} $g$ being given by
\begin{equation} \label{defg}
g(\omega,\delta, D) = \frac{\mathrm{Cap}_D}{1-{(\frac{\omega_M}{\omega})^2} + \i {\gamma_M}},
\end{equation}
where the damping constant $\gamma_M$ is given by
$$
\gamma_M := \frac{(v+v_b) \mathrm{Cap}_D \omega}{8 \pi v v_b} - \frac{(v-v_b)}{v} \frac{v_b \mathrm{Cap}_D^2 \delta}{8 \pi |D| \omega}.
$$
This shows the scattering enhancement near $\omega_M$.
When $N=2$, there are
{two subwavelength resonances} with positive real part for the {resonator dimer} $D$.
Assume that $D_1$ and $D_2$ are symmetric with respect to the origin $0$ and let $C_{ij}$, for $i,j=1,2$, be defined by \eqref{defCap}. Then, as $\delta \rightarrow 0$, by using Lemma \ref{lem:ints} it follows that \cite{ammari2017double}
\begin{align} \label{tau10}
\omega_1 &= \underbrace{{\sqrt{ (C_{11}+ C_{12})}} v_b {\sqrt{\delta}}}_{:= \omega_{M,1}} - \i\tau_1 {\delta} +\O(\delta^{3/2}),
\end{align}
\begin{align} \label{tau20}
\omega_2 &= \underbrace{{\sqrt{(C_{11}-C_{12})}} v_b {\sqrt{\delta}}}_{:= \omega_{M,2}} + {\delta^{3/2}} \hat\eta_1 + \i\delta^2 \hat\eta_2 + \O(\delta^{5/2}),
\end{align}
where $\hat\eta_1$ and $\hat\eta_2$ are real numbers determined by $D$, $v$, and $v_b$ and
$$
\tau_1 = \frac{v_b^2}{4\pi v}(C_{11}+C_{12})^2.
$$
The resonances $\omega_1$ and $\omega_2$ are called the {hybridized} resonances of the resonator dimmer $D$.
On the other hand, the resonator dimer can be approximated as a {point scatterer} with {resonant monopole} and {resonant dipole} modes. Assume that $D_1$ and $D_2$ are symmetric with respect to the origin. Then for $\omega=\O(\delta^{1/2})$ and $\delta \rightarrow 0$, and $|x|$ being sufficiently large, we have \cite{ammari2017double}
\begin{equation} \label{dimerh} \begin{array}{lll}
u^s(x)&=& \underbrace{g^0(\omega)u^{in}(0) {G^k(x)}}_{{monopole}} \\
\noalign{\smallskip} && + \underbrace{\nabla u^{in}(0)\cdot g^1(\omega) {\nabla G^k(x)}}_{{dipole}} +\O(\delta|x|^{-1}),
\end{array}
\end{equation}
where the
{scattering coefficients} $g^0(\omega)$ and $g^1(\omega)=(g^1_{ij}(\omega))$ are given by
\begin{align}
&g^0(\omega)=\frac{C(1,1)}{{1- \omega_1^2/\omega^2}}(1+\O(\delta^{1/2})), \quad C(1,1):= C_{11} + C_{12} + C_{21} + C_{22},\\
&~g^1_{ij}(\omega)= \int_{\partial D} \mathcal{S}_D^{-1}[x_i](y) y_j - \frac{\delta v_b^2}{\omega^2 |D|({1- \omega_2^2/\omega^2})} P^2\delta_{i1}\delta_{j1},
\end{align}
with \begin{align}
P:= \int_{\partial D} y_1(\psi_1-\psi_2)(y) \; \: \mathrm{d} \sigma(y), \label{defP}
\end{align}
$\psi_i$, for $i=1,2,$ being defined by (\ref{defpsi}), and $\delta_{i1}$ and $\delta_{j1}$ being the Kronecker delta.
As shown in \eqref{monopole}-\eqref{defg}, around $\omega_M$, a single resonator in free-space scatters waves with a greatly enhanced amplitude. If a second resonator is introduced, coupling interactions will occur giving according to \eqref{dimerh} a system that has both monopole and dipole resonant modes. This pattern continues for larger number $N$ of resonators \cite{davies2019fully}.
\end{remark}
\begin{remark}
The invertibility of $V$ is a subtle issue and depends only on the geometry of the inclusions $D=D_1\cup\dots\cup D_N$. In the case that the resonators are all identical, $V$ is clearly invertible since $C^\mathrm{vol}$ is symmetric.
\end{remark}
\begin{remark}
In many cases $\tau_n=0$ for some $n$ (see for instance formula (\ref{tau20})), meaning the imaginary parts exhibit higher-order behaviour in $\delta$. For example, the second (dipole) frequency for a pair of identical resonators is known to be $\O(\delta^{2})$ \cite{ammari2017double}. In any case, the resonant frequencies will have negative imaginary parts, due to the radiation losses.
\end{remark}
\section{Close-to-touching subwavelength resonators} \label{sec3}
In this section, we study the behaviour of the coupled subwavelength resonant modes when two subwavelength resonators are brought close together. We consider the case of a pair of spherical resonators and use bispherical coordinates to derive explicit representations for the capacitance coefficients which, as shown in Theorem \ref{thm:res}, capture the system's resonant behaviour at leading order.
We derive estimates for the rate at which the gradient of the scattered wave blows up as the resonators are brought together.
For simplicity, we study the effect of scattering by a pair of spherical inclusions, $D_1$ and $D_2$, with the same radius, which we denote by $r$, and separation distance $\epsilon$ (so that their centres are separated by $2r +\epsilon$). We refer to \cite{davies2020close} for the case of arbitrary sized spheres.
We choose the separation distance $\epsilon$ as a function of $\delta$ and will perform an asymptotic analysis in terms of $\delta$. We choose $\epsilon$ to be such that, for some $0<\beta<1$,
\begin{equation} \label{assum_epsilon}
\epsilon\sim e^{-1/\delta^{1-\beta}} \text{ as } \delta\to0.
\end{equation}
As we will see shortly, with $\epsilon$ chosen to be in this regime the subwavelength resonant frequencies are both well behaved (\emph{i.e.} $\omega=\omega(\delta)\to0$ as $\delta\to0$) and we can compute asymptotic expansions in terms of $\delta$.
From Theorem \ref{thm:res} (see also Remark \ref{remark12}),
there exist two subwavelength resonant modes, $u_1$ and $u_2$, with associated resonant frequencies $\omega_1$ and $\omega_2$ with positive real part, labelled such that $\Re(\omega_1)<\Re(\omega_2)$.
\begin{figure}
\centering
\captionsetup{width=.6\linewidth}
\includegraphics[width=.7\linewidth]{figure.pdf}
\caption{Two close-to-touching spheres, annotated with the bispherical coordinate system outlined in \Cref{sec:coordinates}.}
\label{fig:coordinates}
\end{figure}
\subsection{Coordinate system} \label{sec:coordinates}
The Helmholtz problem \eqref{eq:scattering} is invariant under translations and rotations so we are free to choose the coordinate axes. Let $R_j$ be the reflection with respect to $\partial D_j$ and let $p_1$ and $p_2$ be the unique fixed points of the combined reflections $R_1\circ R_2$ and $R_2\circ R_1$, respectively. Let $n$ be the unit vector in the direction of $p_2-p_1$. We will make use of the Cartesian coordinate system $(x_1,x_2,x_3)$ defined to be such that $p=(p_1+p_2)/2$ is the origin and the $x_3$-axis is parallel to the unit vector $n$. Then one can see that \cite{kang2019quantitative}
\begin{equation}
p_1 = (0,0,-\alpha)\quad \mbox{and} \quad p_2 = (0,0,\alpha),
\end{equation}
where
\begin{equation}
\alpha:=\frac{\sqrt{\epsilon (4r +\epsilon)}}{2}.
\end{equation}
Moreover, the sphere $D_i$ is centered at $(0,0,c_i)$ where
\begin{equation} \label{eq:centres}
c_i=(-1)^i \sqrt{r^2+\alpha^2}.
\end{equation}
We then introduce a bispherical coordinate system $(\xi,\theta,\varphi)$ which is related to the Cartesian coordinate system $(x_1,x_2,x_3)$ by
\begin{equation} \label{def:bispherical_coordinates}
x_1=\frac{\alpha\sin\theta\cos\varphi}{\cosh\xi-\cos\theta}\,,\quad
x_2=\frac{\alpha\sin\theta\sin\varphi}{\cosh\xi-\cos\theta}\,,\quad
x_3=\frac{\alpha\sinh\xi}{\cosh\xi-\cos\theta}\,,
\end{equation}
and is chosen to satisfy $-\infty<\xi<\infty$, $0\leq\theta<\pi$ and $0\leq\varphi<2\pi$. The reason for this choice of coordinate system is that $\partial D_1$ and $\partial D_2$ are given by the level sets
\begin{equation}
\partial D_1=\big\{\xi=- \sinh^{-1}\left(\frac{\alpha}{r}\right) \big\},\qquad \partial D_2= \big\{\xi= \sinh^{-1}\left(\frac{\alpha}{r}\right) \big\}.
\end{equation}
This is depicted in \Cref{fig:coordinates} (for arbitrary sized spheres). The Cartesian coordinate system is chosen so that we can define a bispherical coordinate system \eqref{def:bispherical_coordinates} such that the boundaries of the two resonators are convenient level sets.
\subsection{Resonant frequency hybridization and gradient blow-up}
Firstly, the resonant frequencies are given, in terms of the capacitance coefficients, by (see \eqref{tau10} and \eqref{tau20})
\begin{equation} \label{eq:sym_resonances}
\begin{split}
\omega_1&=\sqrt{\delta \frac{3v_b^2}{4\pi r^3}(C_{11}+C_{12})}+\O(\delta),\\
\omega_2&=\sqrt{\delta \frac{3v_b^2}{4\pi r^3}(C_{11}-C_{12})}+\O(\delta).
\end{split}
\end{equation}
Further to this, the capacitance coefficients are given by
\begin{equation} \label{eq:sym_cap}
\begin{split}
C_{11}&=C_{22}=8\pi\widetilde{\alpha}\sum_{n=0}^{\infty} \frac{e^{(2n+1)\xi_0}}{e^{2(2n+1)\xi_0}-1}, \\
C_{12}&=C_{21}=-8\pi\widetilde{\alpha}\sum_{n=0}^{\infty}\frac{1}{e^{2(2n+1)\xi_0}-1},
\end{split}
\end{equation}
where
\begin{equation*}
\widetilde{\alpha}:=\sqrt{\epsilon(r+\epsilon/4)}, \qquad
\xi_0:=\sinh^{-1}\left(\frac{\widetilde{\alpha}}{r}\right).
\end{equation*}
From \cite{lekner2011near}, we know the asymptotic behaviour of the series in \eqref{eq:sym_cap} as $\xi_0\to0$, from which we can see that as $\epsilon\to0$,
\begin{equation} \label{eq:cap_sym_asym}
\begin{split}
C_{11}&=2\pi \frac{\widetilde{\alpha}}{\xi_0}\left[\gamma+2\ln 2+\ln \left(\sqrt{r}\right)-\ln \left(\sqrt{\epsilon}\right)\right]+\O(\epsilon),\\
C_{12}&=-2\pi \frac{\widetilde{\alpha}}{\xi_0}\left[\gamma+\ln \left(\sqrt{r}\right)-\ln \left(\sqrt{\epsilon}\right)\right]+\O(\epsilon),
\end{split}
\end{equation}
where $\gamma\approx0.5772\dots$ is the Euler–Mascheroni constant.
Combining \eqref{eq:sym_resonances} and \eqref{eq:cap_sym_asym} we reach the fact that the resonant frequencies are given, as $\delta\to0$, by
\begin{equation} \label{eq:res_identical}
\begin{split}
\omega_1&=\sqrt{\delta \frac{3v_b^2\ln 2}{r^2}}+ \O\left(\delta\right),\\
\omega_2&=\sqrt{\delta \frac{3v_b^2}{2r^2} \left(\ln \left(\frac{r}{\epsilon}\right)+2\gamma+2\ln 2\right)}+ \O\left(\sqrt{\delta}\right).
\end{split}
\end{equation}
Thus, the choice of $\epsilon\sim e^{-1/\delta^{1-\beta}}$, where $0<\beta<1$, means that as $\delta\to0$ we have that $\omega_1\sim\sqrt{\delta}$ and $\omega_2\sim\delta^{\beta/2}$.
The two resonant modes, $u_1$ and $u_2$, correspond to the two resonators oscillating in phase and in antiphase with one another, respectively. Since the eigenmode $u_2$ has different signs on the two resonators, $\nabla u_2$ will blow up as the two resonators are brought together. Conversely, $u_1$ takes the same value on the two resonators so there will not be a singularity in the gradient. In particular, if we normalise the eigenmodes so that for any $x\in\partial D$
\begin{equation}
\lim_{\delta\to0} |u_1(x)|\sim1,\qquad \lim_{\delta\to0} |u_2(x)|\sim1,
\end{equation}
then the choice of $\epsilon$ to satisfy the regime $\epsilon\sim e^{-1/\delta^{1-\beta}}$ means that the maximal gradient of each eigenmode has the asymptotic behaviour, as $\delta\to0$,
\begin{equation} \label{eq:sym_mode_blowup}
\max_{x\in\mathbb{R}^3\setminus \overline{D}}|\nabla u_1(x)|\sim 1, \qquad
\max_{x\in\mathbb{R}^3\setminus \overline{D}}|\nabla u_2(x)|\sim \frac{1}{\epsilon}.
\end{equation}
By decomposing the scattered field into the two resonant modes, we can use \eqref{eq:sym_mode_blowup} to understand the singular behaviour exhibited by the acoustic pressure. The solution $u$ to the scattering problem \eqref{eq:scattering} with incoming plane wave $u^{in}$ with frequency $\omega=\O(\delta^{1/2})$ is given, for $x\in\mathbb{R}^3\setminus \overline{D}$, by
\begin{equation}
u(x)=u^{in}(x) +au_1(x)+bu_2(x),
\end{equation}
where the coefficients $a$ and $b$ satisfy, as $\delta\to0$, the equations
\begin{align*}
a(\omega^2-\omega_1^2)&= \frac{\delta v_b^2}{|D|}\int_{\partial D} \S_D^{-1}[u^{in}] \: \mathrm{d}\sigma +\O(\delta^{\hat{\beta}}),\\
b(\omega^2-\omega_2^2)&= -\frac{\delta v_b^2}{|D|}\left(\int_{\partial D_1} \S_D^{-1}[u^{in}] \: \mathrm{d} \sigma-\frac{|D_1|}{|D_2|}\int_{\partial D_2} \S_D^{-1}[u^{in}] \: \mathrm{d}\sigma \right)+\O(\delta^{\hat{\beta}}),
\end{align*}
with $\hat{\beta}:=\min(2-\beta,3/2)$ and $|D|$ being the volume of $D=D_1\cup D_2$.
\section{Effective medium theory for subwavelength resonators} \label{dilutesect}
\subsection{High refractive index effective media}
We consider a domain $\Omega$ which contains a large number of small, identical resonators. If $D_0$ is a fixed domain, then for some $r>0$ the $N$ resonators are given, for $1\leq j\leq N$, by
\begin{equation*}
D_{0,j}^{r,N}=r D_0+z_j^N,
\end{equation*}
for positions $z_j^N$. We always assume that $r$ is sufficiently small such that the resonators are not overlapping and that $D_{0}^{r,N}=\bigcup_{j=1}^N D_{0,j}^{r,N} \Subset \Omega$.
We find the effective equation in the specific case that the frequency $\omega =\O(1)$ and satisfies
\begin{equation} \label{ass:frequency}
1- (\frac{\omega_M}{\omega})^2= \beta_0 r^{\epsilon_0},
\end{equation}
for some fixed $ 0<\epsilon_0<1$ and constant $\beta_0$.
We note that there are two cases depending on whether $\omega > \omega_M$ or $\omega < \omega_M$. In the former case, $\beta_0 >0$ while in the latter case we have $\beta_0<0$.
Moreover, we assume that there exists some positive number $\Lambda$ independent of $N$ such that
\begin{equation} \label{ass:number}
r^{1-\epsilon_0}N=\Lambda \quad \mbox{and } \Lambda \mbox{ is large}.
\end{equation}
Since the resonators are small, we can use the point-scatter approximation from \eqref{monopole} to describe how they interact with incoming waves. To do so, we must make some extra assumptions on the regularity of the distribution $\{z_j^N:1\leq j\leq N\}$ so that the system is well behaved as $N\to\infty$ (under the assumption \eqref{ass:number}). In particular, we assume that there exists some constant $\eta$ such that for any $N$ it holds that
\begin{equation} \label{ass:dist}
\min_{i\neq j} |z_i^N-z_j^N| \geq \frac{\eta}{N^{1/3}},
\end{equation}
and, further, there exists some $0<\epsilon_1<1$ and constants $C_1,C_2>0$ such that for all $h\geq 2\eta N^{-1/3}$,
\begin{align}
\sum_{|x-z_j^N|\geq h} \frac{1}{|x-y_j^N|^2}\leq C_1 N |h|^{-\epsilon_1}, \qquad&\text{uniformly for all } x\in\Omega,\label{ass:reg1}\\
\sum_{2\eta N^{-1/3}\leq|x-z_j^N|\leq 3h} \frac{1}{|x-y_j^N|}\leq C_2 N |h|, \qquad&\text{uniformly for all } x\in\Omega. \label{ass:reg2}
\end{align}
Finally, we also need that
\begin{equation} \label{ass:epsilon}
\epsilon_2:=\frac{\epsilon_0}{1-\epsilon_0}-\frac{\epsilon_1}{3}>0.
\end{equation}
If we represent the field that is scattered by the collection of resonators $D_0^{r,N}=\bigcup_{j=1}^N D_{0,j}^{r,N}$ as
\begin{equation*}
u^N(x)= \begin{cases}
u^{in}(x)+\S_{D_0^{r,N}}^{k}[\psi^N](x), & x\in\mathbb{R}^3 \setminus \overline{D_0^{r,N}},\\
\S_{D_0^{r,N}}^{k_0}[\phi^N](x), & x\in D_0^{r,N},
\end{cases}
\end{equation*}
for some $\psi^N, \phi^N\in L^2(\partial D_0^{r,N})$, then we have the following lemma, which follows from \cite[Proposition~3.1]{ammari2017effective}. This justifies using a point-scatter approximation to describe the total incident field acting on the resonator $D_{0,j}^{r,N}$ and the scattered field due to $D_{0,j}^{r,N}$, defined respectively as
\begin{equation*}
u_j^{in,N}=u^{in}+\sum_{i\neq j} \S_{D_{0,i}^{r,N}}^k[\psi^N] \qquad \text{and} \qquad u_j^{s,N}=\S_{D_{0,j}^{r,N}}^k[\psi^N].
\end{equation*}
\begin{lemma} \label{lem:points}
Under the assumptions \eqref{ass:frequency}--\eqref{ass:epsilon}, it holds that the total incident field acting on the resonator $D_{0,j}^{r,N}$ is given, at $z_j^N$, by
\begin{equation*}
u_j^{in,N}(z_j^N)=u^{in}(z_j^N)+\sum_{i\neq j} \frac{r\mathrm{Cap}_{D_0}}{1-(\frac{\omega_M}{\omega})^2} G^k(z_j^N-z_i^N)u^{in}(z_j^N),
\end{equation*}
up to an error of order $\O(N^{-\epsilon_2})$. Similarly, it holds that the scattered field due to the resonator $D_{0,j}^{r,N}$ is given, at $x$ such that $|x-z_j^N|\gg r$, by
\begin{equation*}
u_j^{s,N}(x)= \frac{r\mathrm{Cap}_{D_0}}{1-(\frac{\omega_M}{\omega})^2} G^k(x-z_j^N) u_j^{in,N}(z_j^N) ,
\end{equation*}
up to an error of order $\O(N^{-\epsilon_2}+r|x-z_j^N|^{-1})$.
\end{lemma}
In order for the sums in \ref{lem:points} to be well behaved as $N\to\infty$, we make one additional assumption on the regularity of the distribution: that there exists a real-valued function $\widetilde{V}\in \mathcal{C}^1(\overline{\Omega})$ such that for any $f\in \mathcal{C}^{0,\alpha}(\Omega)$, with $0<\alpha\leq1$, there is a constant $C_3$ such that
\begin{equation} \label{ass:integral}
\max_{1\leq j\leq N} \left| \frac{1}{N}\sum_{i\neq j} G^k(z_j^N-z_i^N)f(z_i^N)-\int_\Omega G^k(z_j^N-y)\widetilde{V}(y)f(y) \: \mathrm{d} y \right| \leq C_3\frac{1}{N^{\alpha/3}}\|f\|_{\mathcal{C}^{0,\alpha}(\Omega)}.
\end{equation}
\begin{remark}
It holds that $\widetilde{V}\geq0$. If the resonators' centres $\{z_j^N:j=1,\dots,N\}$ are uniformly distributed, then $\widetilde V$ will be a positive constant, $\widetilde V = {1}/{|\Omega|}.$
\end{remark}
Under all these assumptions, we are able to derive effective equations for the system with an arbitrarily large number of small resonators. If we let $\epsilon_3\in(0,\tfrac{1}{3})$, then we will seek effective equations on the set given by
\begin{equation} \label{defomegaN}
Y_{\epsilon_3}^N:=\left\{x\in\mathbb{R}^3:|x-z_j^N|\geq N^{\epsilon_3-1} \text{ for all }1\leq j\leq N\right\},
\end{equation}
which is the set of points that are sufficiently far from the resonators, avoiding the singularities of the Green's function. The following result from \cite{ammari2017effective} holds.
\begin{theorem} \label{thm:homogenized}
Let $\omega < \omega_M$. Under the assumptions \eqref{ass:number}--\eqref{ass:integral}, the solution $u^N$ to the scattering problem \eqref{eq:scattering} with the system of resonators $D_0^{r,N}=\bigcup_{j=1}^N D_{0,j}^{r,N}$ converges to the solution of
\begin{equation*}
\begin{cases}
\left(\Delta+k^2-\frac{\Lambda \mathrm{Cap}_{D_0}}{\beta_0}\widetilde{V}(x)\right)u(x)=0, & x\in\Omega, \\
\left(\Delta+k^2\right)u(x)=0, & x\in\mathbb{R}^3\setminus \overline{\Omega},\\
u|_+ = u|_- & \mbox{on } \partial \Omega,
\end{cases}
\end{equation*}
as $N\to\infty$, together with a radiation condition governing the behaviour in the far field, which says that uniformly for all $x\in Y_{\epsilon_3}^N$ it holds that
\begin{equation*}
|u^N(x)-u(x)|\leq C N^{-\min\left\{\frac{1-\epsilon_0}{6},\epsilon_2,\epsilon_3,\frac{1-\epsilon_3}{3}\right\}}.
\end{equation*}
\end{theorem}
By our assumption, $k=\O(1)$, $\widetilde{V}=\O(1)$, and $\beta_0 <0$. When $- \Lambda \mathrm{Cap}_{D_0}/{\beta_0} \gg 1$, we see that we have an effective high refractive index medium. As a consequence, this together with \cite{ammari2015mathematical} gives a rigorous mathematical theory for the super-focusing experiment in \cite{fink}. Similarly to Theorem \ref{thm:homogenized}, if $\omega > \omega_M$, to the scattering problem \eqref{eq:scattering} with the system of resonators $D_0^{r,N}$ converges to the solution of the following dissipative equation
\begin{equation*}
\begin{cases}
\left(\Delta+k^2-\frac{\Lambda \mathrm{Cap}_{D_0}}{\beta_0}\widetilde{V}(x)\right)u(x)=0, & x\in\Omega, \\
\left(\Delta+k^2\right)u(x)=0, & x\in\mathbb{R}^3\setminus \overline{\Omega},\\
u|_+ = u|_- & \mbox{on } \partial \Omega,
\end{cases}
\end{equation*}
as $N\to\infty$, together with a radiation condition governing the behaviour in the far field, which says that uniformly for all $x\in Y_{\epsilon_3}^N$ it holds that
\begin{equation*}
|u^N(x)-u(x)|\leq C N^{-\min\left\{\frac{1-\epsilon_0}{6},\epsilon_2,\epsilon_3,\frac{1-\epsilon_3}{3}\right\}}.
\end{equation*}
\begin{remark}
At the resonant frequency $\omega= \omega_M$, the scattering coefficient $g$ defined by (\ref{defg}) is of order one.
Thus each scatterer is a point source with magnitude one. As a consequence, the addition or removal of one resonator from the medium affects the total field by a magnitude of the same order as the incident field. Therefore, we cannot expect any effective medium theory for the medium at this resonant frequency.
\end{remark}
\subsection{Double-negative metamaterials}
In this subsection, we show that, using dimers of identical subwavelength resonators, the effective material parameters of dilute system of dimers can both be negative over a non empty range of frequencies \cite{ammari2017double}.
As shown in \eqref{dimerh}, a dimer of identical subwavelength resonators can be approximated as a point scatterer with monopole and dipole modes. It features two slightly different subwavelength resonances, called the hybridized resonances. The hybridized resonances are fundamentally different modes. The first mode is, as in the case of a single resonator, a monopole mode, while the second mode is a dipole mode. The resonance associated with the dipole mode is usually referred to as the anti-resonance.
For an appropriate volume fraction, when the excitation frequency is close to the anti-resonance, a double-negative effective $\rho$ and $\kappa$ for media consisting of a large number of dimers with certain conditions on their distribution can be obtained. The dipole modes in the background medium contribute to the effective $\rho$ while the monopole modes contribute to the effective $\kappa$.
Here we consider the scattering of an incident plane wave $u^{in}$ by $N$ identical dimers with different orientations distributed in a homogeneous medium in $\mathbb{R}^3$. The $N$ identical dimers are generated by scaling the normalized dimer $D$ by a factor $r$, and then rotating the orientation and translating the center. More precisely, the dimers occupy the domain
$$
D^N:=\cup_{1\leq j \leq N}D_j^N,
$$
where $D_j^N=z_j^N + r R_{d_j^N}D$ for $1\leq j \leq N$, with $z_j^N$ being the center of the dimer $D_j^N$, $r$ being the characteristic size, and $R_{d_j^N}$ being the rotation in $\mathbb{R}^3$ which aligns the dimer $D_j^N$ in the direction $d_j^N$. Here, $d_j^N$ is a vector of unit length in $\mathbb{R}^3$. For simplicity, we suppose that $D$ is made of two identical spherical resonators.
We also assume that $0< r \ll 1$, $N\gg 1$ and that $\{z_j^N: 1\leq j \leq N\} \subset \Omega$ where $\Omega$ is a bounded domain. See Figure \ref{doublef}.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.7]
\pgfmathsetmacro{\rad}{0.07pt}
\pgfmathsetmacro{\sep}{0.1pt}
\foreach \x in {-3,...,3}
\foreach \y in {-3,...,3}
{
\begin{scope}[xshift = \x cm, yshift=\y cm, rotate=rand*360]
\draw (-\sep,0) circle (\rad);
\draw (\sep,0) circle (\rad);
\end{scope}
}
\foreach \c in {-2,...,2}
{
\begin{scope}[xshift = \c cm, yshift=4 cm, rotate=rand*360]
\draw (-\sep,0) circle (\rad);
\draw (\sep,0) circle (\rad);
\end{scope}
\begin{scope}[xshift = \c cm, yshift=-4 cm, rotate=rand*360]
\draw (-\sep,0) circle (\rad);
\draw (\sep,0) circle (\rad);
\end{scope}
\begin{scope}[xshift = 4 cm, yshift=\c cm, rotate=rand*360]
\draw (-\sep,0) circle (\rad);
\draw (\sep,0) circle (\rad);
\end{scope}
\begin{scope}[xshift = -4 cm, yshift=\c cm, rotate=rand*360]
\draw (-\sep,0) circle (\rad);
\draw (\sep,0) circle (\rad);
\end{scope}
}
\draw[name path = O] plot [smooth cycle] coordinates {(0:4.9) (60:4.8) (120:4.9) (180:4.8) (240:4.9) (300:4.8) };
\draw (3.8,-3.8) node{$\Omega$};
\end{tikzpicture}
\end{center} \caption{Illustration of the dilute system of subwavelength dimers.} \label{doublef}
\end{figure}
The scattering of waves by the dimers can be modeled by the following system of equations:
\begin{equation} \label{eq-scattering2}
\left\{
\begin{array} {ll}
&\displaystyle \nabla \cdot \frac{1}{\rho} \nabla u^N+ \frac{\omega^2}{\kappa} u^N = 0 \quad \mbox{in } \mathbb{R}^3 \backslash \overline{D^N}, \\
\noalign{\smallskip}
&\displaystyle \nabla \cdot \frac{1}{\rho_b} \nabla u^N+ \frac{\omega^2}{\kappa_b} u^N = 0 \quad \mbox{in } D^N, \\
\noalign{\smallskip}
& \displaystyle u^N_{+} -u^N_{-} =0 \quad \mbox{on } \partial D^N, \\
\noalign{\smallskip}
& \displaystyle \frac{1}{\rho} \frac{\partial u^N}{\partial \nu} \bigg|_{+} - \frac{1}{\rho_b} \frac{\partial u^N}{\partial \nu} \bigg|_{-} =0 \quad \mbox{on } \partial D^N,\\
\noalign{\smallskip}
& \displaystyle u^N- u^{in} \,\,\, \mbox{satisfies the Sommerfeld radiation condition},
\end{array}
\right.
\end{equation}
where $u^N$ is the total field.
We make the following assumptions:
\begin{itemize}
\item[(i)]
$\delta = \mu^2 r^2$ for some positive number $\mu >0$;
\item[(ii)] $\omega = \omega_{M,2} + a r^2$ for some real number $a < \mu^3
\hat \eta_1 $, where $\omega_{M,2}$ is defined in (\ref{tau20});
\item[(iii)] $r N = \Lambda $ for some positive number $\Lambda >0$;
\item[(iv)] The dimers are regularly distributed in the sense that
\[
\min_{i \neq j } |z^N_i -z^N_j| \geq \eta N^{-\frac{1}{3}},
\]
for some constant $\eta$ independent of $N$;
\item[(v)]
There exists a function $\widetilde V \in \mathcal{C}^1(\bar{\Omega})$ such that for any $f \in \mathcal{C}^{0, \alpha}(\Omega)$ with $0 < \alpha \leq 1$, \eqref{ass:integral} holds
for some constant $C$ independent of $N$;
\item[(vi)]
There exists a matrix valued function $\widetilde B \in \mathcal{C}^1(\bar{\Omega})$ such that
for $f \in (\mathcal{C}^{0, \alpha}(\Omega))^3$ with $0 < \alpha \leq 1$,
$$ \begin{array}{l} \displaystyle
\max_{1\leq j \leq N} |\frac{1}{N} \sum_{i\neq j} (f(z_i^N )\cdot d_i^N)( d_i^N \cdot \nabla G^k(z_i^N - z_j^N)) -
\int_{\Omega} f(y) \widetilde B \nabla_y G^k(y-z_j^N) \; \: \mathrm{d} y| \\
\noalign{\smallskip} \qquad \displaystyle \leq C \frac{1}{N^{\frac{\alpha}{3}}}\|f\|_{\mathcal{C}^{0, \alpha}(\Omega)}
\end{array} $$
for some constant $C$ independent of $N$;
\item[(vii)]
There exists a constant $C>0$ such that
$$
\max_{1\leq j \leq N} \frac{1}{N} \sum_{i \neq j} \frac{1}{|z_j^N-z_i^N|}
\leq C,\quad \max_{1\leq j \leq N} \frac{1}{N} \sum_{i \neq j} \frac{1}{|z_j^N-z_i^N|^2}
\leq C,
$$
for all $1\leq j \leq N$.
\end{itemize}
We introduce the two constants
$$
\widetilde g^0 = \frac{2(C_{11}+C_{12})}{1- \omega_{M,1}^2/\omega_{M,2}^2}, \quad \widetilde g^1 =
\frac{\mu^2 v_b^2 }{2|D|\omega_{M,2} (\mu^3 \hat \eta_1 -a)} P^2,
$$
where $P$ is defined by (\ref{defP}), $\omega_{M,1}$ and $\omega_{M,2}$ are the leading orders in the asymptotic expansions (\ref{tau10}) and (\ref{tau20}) of the hybridized resonant frequencies as $\delta\rightarrow 0$, and the two functions $$
M_1 = \begin{cases}
I &\quad \mbox{in } \mathbb{R}^3\setminus {\Omega},
\\
I- \Lambda \widetilde g^1 \widetilde B &\quad \mbox{in } \Omega,
\end{cases}
$$
and
$$
M_2 = \begin{cases}
k^2 &\quad \mbox{in } \mathbb{R}^3\setminus {\Omega},
\\
k^2- \Lambda\widetilde g^0 \widetilde V &\quad \mbox{in } \Omega.
\end{cases}
$$
The following result from \cite{ammari2017double} holds.
\begin{theorem} \label{lem-pointapprox1}
Suppose that there exists a unique solution $u$ to
\begin{equation} \label{eq-pde}
\nabla \cdot M_1(x) \nabla u(x) + M_2(x)u(x) =0 \quad \mbox{in } \mathbb{R}^3,
\end{equation}
such that $u-u^{in}$ satisfies the Sommerfeld radiation condition at infinity. Then, under assumptions [(i)]--[(vii)], we have $u^N(x) \rightarrow u(x)$ uniformly for $x\in\mathbb{R}^3$ such that $|x-z_j^N|\gg N^{-1} \text{ for all }1\leq j\leq N$.
\end{theorem}
Note that from (\ref{eq:sym_cap}), it follows that $\omega_{M,2} > \omega_{M,1}$.
Therefore, for large $\Lambda$, both the matrix $M_1$ and the scalar function $M_2$ are negative in $\Omega$. See Figure \ref{effectivef}.
\begin{figure}
\centering
{\includegraphics[scale=0.7]{effective}} \caption{Effective properties of the homogenized medium.} \label{effectivef}
\end{figure}
\section{Periodic structures of subwavelength resonators} \label{sec5}
In this section we investigate whether there is a possibility of subwavelength
band gap opening in subwavelength resonator crystals.
We first formulate the spectral problem for a subwavelength resonator crystal. Then we derive an asymptotic formula for the quasi-periodic resonances in terms of the contrast between the densities outside and inside the resonators. We prove the existence of a subwavelength band gap and estimate its width.
\subsection{Floquet theory} \label{sec:floquet}
Let $f(x)$ for $x \in \mathbb{R}^d, d=1,2,3,$ be a function decaying sufficiently fast. We let $l_1,..., l_d$ be linearly independent lattice vectors, and define the unit cell $Y$ and the lattice $\Lambda$ as
$$Y = \left\{\sum_{n=1}^d s_n l_n \ \Big| \ 0 < s_n < 1 \right\}, \qquad \Lambda = \left\{\sum_{n=1}^d q_n l_n \ \Big| \ q_n \in \mathbb{N} \right\}.$$
The Floquet transform of $f$ is defined as:
\begin{equation} \label{floquettransform} \mathcal{U}[f](x,\alpha) = \sum_{n
\in \Lambda} f(x-n) e^{\i \alpha \cdot n}.
\end{equation}
This transform is an analogue of the Fourier transform for the periodic case. The parameter $\alpha$ is called
the quasi-periodicity, and it is an analogue
of the dual variable in the Fourier transform. If we shift $x$ by
a period $m \in \Lambda$, we get the
Floquet condition (or quasi-periodic condition)
\begin{equation} \label{floquetcondition}
\mathcal{U}[f](x+m,\alpha) = e^{\i \alpha \cdot m}
\mathcal{U}[f](x,\alpha),\end{equation} which shows that it suffices to know
the function $\mathcal{U}[f](x,\alpha)$ on the unit cell
$Y$ in order to recover it completely as a function of
the $x$-variable. Moreover, $\mathcal{U}[f](x,\alpha)$ is periodic
with respect to $\alpha$:
\begin{equation}
\label{floquetperiodic} \mathcal{U}[f](x,\alpha + q) =
\mathcal{U}[f](x,\alpha), \quad q \in \Lambda^*.
\end{equation}
Here, $\Lambda^*$ is the dual lattice, generated by the dual lattice vectors $\alpha_1,...,\alpha_d$ defined through the relation
$$\l_i \alpha_j = 2\pi \delta_{ij}, \qquad 0 \leq i,j \leq d.$$
Therefore, $\alpha$ can be considered as an element of the torus
$\mathbb{R}^d/\Lambda^*$. Another way of saying this is that all
information about $\mathcal{U}[f](x,\alpha)$ is contained in its
values for $\alpha$ in the fundamental domain $Y^*$ of the dual
lattice $\Lambda^*$. This domain is referred to as the
Brillouin zone and is depicted in Figure \ref{bzone} for a square lattice in two dimensions.
\begin{figure}[h]
\begin{center}
\includegraphics[height=7cm]{brillouin_zone_r2}
\caption{(First) Brillouin {zone} for a square lattice in two dimensions, with the symmetry points $\Gamma, X$ and $M$. The highlighted triangle is known as the reduced Brillouin zone. \label{bzone}}
\end{center}
\end{figure}
The following result is an analogue of the Plancherel theorem when one uses the Fourier transform.
Suppose that the measures $\: \mathrm{d}\alpha$ and the Brillouin zone
$Y^*$ are normalized. The following theorem holds \cite{kuchment2}.
\begin{theorem}[Plancherel-type theorem]
The transform $$\mathcal{U} : L^2(\mathbb{R}^d) \rightarrow
L^2(Y^* , L^2(Y))$$ is isometric. Its inverse is
given by
$$\mathcal{U}^{-1}[g](x) = \int_{Y^*} g(x,\alpha)
\: \mathrm{d}\alpha,$$ where the function $g(x,\alpha) \in L^2(Y\times Y^*)$ is extended from $Y$ to all $x \in \mathbb{R}^d$
according to the Floquet condition \eqref{floquetcondition}.
\end{theorem}
Consider now a linear partial differential operator $L(x,
\partial_x)$, whose coefficients are periodic with respect to
$\Lambda$.
Due to periodicity, the operator commutes with the Floquet transform
$$ \mathcal{U}[Lf](x,\alpha) = L(x, \partial_x) \mathcal{U}[f](x,\alpha).$$
For each $\alpha$, the operator $L(x,
\partial_x)$ now acts on functions satisfying the corresponding
Floquet condition \eqref{floquetcondition}. Denoting this operator by
$L(\alpha)$, we see that the Floquet transform $\mathcal{U}$
expands the periodic partial differential operator $L$ in
$L^2(\mathbb{R}^d)$ into the direct integral of
operators \begin{equation} \label{directintegral} \int^{\oplus}_{Y^*} L(\alpha) \: \mathrm{d} \alpha.
\end{equation}
If $L$ is a self-adjoint operator, one can prove the main
spectral result: \begin{equation} \label{spectralstatement} \sigma(L) = \displaystyle
\bigcup_{\alpha \in Y^*} \sigma(L(\alpha)),\end{equation} where $\sigma$
denotes the spectrum.
If $L$ is elliptic, the operators $L(\alpha)$ have compact
resolvents and hence discrete spectra. If $L$ is bounded from
below, the spectrum of $L(\alpha)$ accumulates only at $+\infty$.
Denote by $\mu_n(\alpha)$ the $n$th eigenvalue of $L(\alpha)$
(counted in increasing order with their multiplicity). The
function $\alpha \mapsto \mu_n(\alpha)$ is continuous in $Y^*$. It
is one branch of the dispersion relations and is called a \emph{band function}. We
conclude that the spectrum $\sigma(L)$ consists of the closed
intervals (called the spectral bands)
$$\bigg[\min_\alpha \mu_n(\alpha), \max_\alpha \mu_n(\alpha)\bigg],$$ where
$\min_\alpha \mu_n(\alpha) \rightarrow +\infty$ when $n
\rightarrow +\infty$.
\subsection{Quasi-periodic layer potentials} \label{sectquasiGH}
We introduce a quasi-periodic version of the layer potentials. Again, we let $Y$ and $Y^*$ be the unit cell and dual unit cell, respectively. For $\alpha \in Y^*$, the function $G^{\alpha, k}$ is defined to satisfy
$$ (\Delta_x + k^2) G^{\alpha, k} (x,y) = \sum_{m\in \Lambda} \delta(x-y-n) e^{\i m\cdot \alpha},$$
where $\delta$ is the Dirac delta function and $G^{\alpha, k} $ is $\alpha$-quasi-periodic, \emph{i.e.}, $e^{- \i \alpha\cdot x} G^{\alpha, k}(x,y)$ is periodic in $x$ with respect to $Y$. It is known that $G^{\alpha, k} $ can be written as
$$ G^{\alpha, k}(x,y) = \sum_{q\in \Lambda^*} \frac{e^{\i (\alpha+q)\cdot (x-y)}}{k^2- |\alpha + q|^2},$$
if $k \ne |\alpha + q|$ for any $q \in Y^*$. We remark that
\begin{align}
G^{\alpha,k}(x, y)= G^{\alpha,0}(x,y) - G_l^{\alpha,\#}(x -y) := G^{\alpha,0}(x,y) - \sum_{l=1}^\infty k^{2 l}\sum_{q\in \Lambda ^*} \frac{e^{\i (\alpha+q)\cdot (x-y)}}{|\alpha+q|^{2(l+1)}} \label{eq:defGk2}
\end{align}
when $\alpha \neq 0$, and $k \rightarrow 0$.
We let $D$ be as in \Cref{sec-2} and additionally assume $D\Subset Y$. Then the quasi-periodic single layer potential $\mathcal{S}_D^{\alpha,k}$ is defined by
\begin{equation} \label{singlealpha} \mathcal{S}_D^{\alpha,k}[\phi](x) = \int_{\partial D} G^{\alpha,k} (x,y) \phi(y) \: \mathrm{d} \sigma(y),\quad x\in \mathbb{R}^3. \end{equation}
It satisfies the following jump formulas:
\begin{equation*}
\S_D^{\alpha,k}[\phi]\big|_+ = \S_D^{\alpha,k}[\phi]\big|_-,
\end{equation*}
and
$$ \frac{\partial}{\partial\nu} \mathcal{S}_D^{\alpha,k}[\phi] \Big|_{\pm} = \left( \pm \frac{1}{2} I +( \mathcal{K}_D^{-\alpha,k} )^*\right)[\phi]\quad \mbox{on}~ \partial D,$$
where $(\mathcal{K}_D^{-\alpha,k})^*$ is the operator given by
$$ (\mathcal{K}_D^{-\alpha, k} )^*[\phi](x)= \int_{\partial D} \frac{\partial}{\partial\nu_x} G^{\alpha,k}(x,y) \phi(y) \: \mathrm{d} \sigma(y).$$
We remark that $\mathcal{S}_D^{\alpha,0} : L^2(\partial D) \rightarrow H^1(\partial D)$ is invertible for $\alpha \ne 0$ \cite{ammari2018mathematical}. Moreover, the following decomposition holds for the layer potential $\mathcal{S}_D^{\alpha,k}$:
\begin{equation} \label{series-s2}
\mathcal{S}_{D}^{\alpha,k} = \mathcal{S}_D^{\alpha, 0} + k^{2}\mathcal{S}_{D,1}^{\alpha} + \O(k^4) \quad \text{with} \quad \mathcal{S}_{D,1}^\alpha[\psi] := \int_{\partial D} G_1^{\alpha,\#}(x -y) \psi(y) \: \mathrm{d} \sigma(y),
\end{equation}
where the error term is with respect to the operator norm $\|.\|_{\mathcal{L}(L^2(\partial D), H^1(\partial D))}$. Furthermore, analogously to (\ref{eq:exp_K}), we have
\begin{equation} \label{series-k2}
(\mathcal{K}_{D}^{-\alpha,k})^* = (\mathcal{K}_{D}^{-\alpha,k})^* + k^2 \mathcal{K}_{D,1}^\alpha + \O(k^3),
\end{equation}
where the error term is with respect to the operator norm $\|.\|_{\mathcal{L}(L^2(\partial D), L^2(\partial D))}$.
Finally, we introduce the $\alpha$-quasi capacity of $D$, denoted by $\mathrm{Cap}_{D,\alpha}$,
$$
\mathrm{Cap}_{D,\alpha}: = \int_{Y\setminus \overline{D}} |\nabla u|^2\; \: \mathrm{d} y,
$$
where $u$ is the $\alpha$-quasi-periodic harmonic function in $Y\setminus \overline{D}$ with $u=1$ on $\partial D$. For $\alpha\neq 0$, we have $u(x) =\mathcal{S}_D^{\alpha,0} \left(\mathcal{S}_D^{\alpha,0}\right)^{-1}[\chi_{\partial D}](x)$ for $ x \in Y \setminus \overline{D}$ and
\begin{equation} \label{alphacap}
\mathrm{Cap}_{D,\alpha} :=-\int_{\partial D} \left( \mathcal{S}_D^{\alpha,0}\right)^{-1}[\chi_{\partial D}](y)\; \: \mathrm{d} \sigma(y).
\end{equation}
Moreover, we have a variational definition of $\mathrm{Cap}_{D,\alpha}$.
Indeed, let $\mathcal{C}_{\alpha}^{\infty}(Y)$ be the set of $\mathcal{C}^{\infty}$ functions in $Y$ which can be extended to $\mathcal{C}^{\infty}$ $\alpha$-quasi-periodic functions in $\mathbb{R}^3$. Let $\mathcal{H}_{\alpha}$ be the closure of the set $\mathcal{C}_{\alpha}^{\infty}(Y)$ in $H^1(Y)$, and let
$\mathcal{V}_\alpha:= \{ v\in \mathcal{H}_{\alpha} : v=1~\mbox{on }\partial D\}$. Then we can show that
\begin{equation} \label{vcalpha}
\mathrm{Cap}_{D,\alpha}= \min_{v\in\mathcal{V}_\alpha}\int_{Y\setminus \overline{D}} |\nabla v|^2 \: \mathrm{d} y.
\end{equation}
\subsection{Square lattice subwavelength resonator crystal}
We first describe the crystal under consideration.
Assume that the resonators occupy $\cup_{n\in \mathbb{Z}^d} (D+n)$ for a bounded and simply connected domain $D \Subset Y$ with $\partial D \in \mathcal{C}^{1, \eta}$ with $0<\eta <1$. See Figure \ref{figy}.
As before, we denote by $\rho_b$ and $\kappa_b$ the material parameters inside the resonators and by $\rho$ and $\kappa$ the corresponding parameters for the background media and let $v, v_b, k,$ and $k_b$ be defined by (\ref{defkv}). We also let the dimensionless contrast parameter $\delta$ be defined by (\ref{defdelta}) and assume for simplicity that $v_b/v=1$.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1.2]
\begin{scope}[xshift=-4cm,scale=1]
\coordinate (a) at (1,0);
\coordinate (b) at (0,1);
\draw (-0.5,-0.5) -- (0.5,-0.5) -- (0.5,0.5) -- (-0.5,0.5) -- cycle;
\draw (0,0) circle(6pt);
\draw[opacity=0.2] (-0.5,0) -- (0,0)
(0.5,0) -- (0,0)
(0,-0.5) -- (0,0)
(0,0.5) -- (0,0);
\begin{scope}[shift = (a)]
\draw (0,0) circle(6pt);
\draw[opacity=0.2] (-0.5,0) -- (0,0)
(0.5,0) -- (0,0)
(0,-0.5) -- (0,0)
(0,0.5) -- (0,0);
\end{scope}
\begin{scope}[shift = (b)]
\draw (0,0) circle(6pt);
\draw[opacity=0.2] (-0.5,0) -- (0,0)
(0.5,0) -- (0,0)
(0,-0.5) -- (0,0)
(0,0.5) -- (0,0);
\end{scope}
\begin{scope}[shift = ($-1*(a)$)]
\draw (0,0) circle(6pt);
\draw[opacity=0.2] (-0.5,0) -- (0,0)
(0.5,0) -- (0,0)
(0,-0.5) -- (0,0)
(0,0.5) -- (0,0);
\end{scope}
\begin{scope}[shift = ($-1*(b)$)]
\draw (0,0) circle(6pt);
\draw[opacity=0.2] (-0.5,0) -- (0,0)
(0.5,0) -- (0,0)
(0,-0.5) -- (0,0)
(0,0.5) -- (0,0);
\end{scope}
\begin{scope}[shift = ($(a)+(b)$)]
\draw (0,0) circle(6pt);
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(0.5,0) -- (0,0)
(0,-0.5) -- (0,0)
(0,0.5) -- (0,0);
\end{scope}
\begin{scope}[shift = ($-1*(a)-(b)$)]
\draw (0,0) circle(6pt);
\draw[opacity=0.2] (-0.5,0) -- (0,0)
(0.5,0) -- (0,0)
(0,-0.5) -- (0,0)
(0,0.5) -- (0,0);
\end{scope}
\begin{scope}[shift = ($(a)-(b)$)]
\draw (0,0) circle(6pt);
\draw[opacity=0.2] (-0.5,0) -- (0,0)
(0.5,0) -- (0,0)
(0,-0.5) -- (0,0)
(0,0.5) -- (0,0);
\end{scope}
\begin{scope}[shift = ($-1*(a)+(b)$)]
\draw (0,0) circle(6pt);
\draw[opacity=0.2] (-0.5,0) -- (0,0)
(0.5,0) -- (0,0)
(0,-0.5) -- (0,0)
(0,0.5) -- (0,0);
\end{scope}
\end{scope}
\draw[dashed,opacity=0.5,->] (-3.9,0.65) .. controls(-2.9,1.8) .. (0.5,0.7);
\begin{scope}[xshift=2cm,scale=2.8]
\coordinate (a) at (1,{1/sqrt(3)});
\coordinate (b) at (1,{-1/sqrt(3)});
\coordinate (Y) at (1.8,0.45);
\coordinate (c) at (2,0);
\coordinate (x1) at ({2/3},0);
\coordinate (x0) at (1,0);
\coordinate (x2) at ({4/3},0);
\pgfmathsetmacro{\rb}{0.25pt}
\pgfmathsetmacro{\rs}{0.2pt}\
\draw[->] (-0.5,-0.5) -- (-0.5,0.5) node[left]{};
\draw[->] (-0.5,-0.5) -- (0.5,-0.5) node[below]{};
\draw (0.5,-0.5) -- (0.5,0.5) -- (-0.5,0.5);
\draw (0,0) circle(6pt);
\draw (0.3,0) node{$D$};
\draw (0.5,0.5) node[right]{$Y$};
\end{scope}
\end{tikzpicture}
\caption{Illustration of the square lattice crystal and quantities in $Y$.}\label{figy}
\end{figure}
To investigate the phononic gap of the crystal we consider the following $\alpha-$quasi-periodic equation in the unit cell $Y=[-1/2,1/2)^3$:
\begin{equation} \label{eq-scatteringB}
\left\{
\begin{array} {ll}
&\displaystyle \nabla \cdot \frac{1}{\rho} \nabla u+ \frac{\omega^2}{\kappa} u = 0 \quad \text{in} \quad Y \backslash \overline{D}, \\
\noalign{\smallskip}
&\displaystyle \nabla \cdot \frac{1}{\rho_b} \nabla u+ \frac{\omega^2}{\kappa_b} u = 0 \quad \text{in} \quad D, \\
\noalign{\smallskip}
&\displaystyle u |_{+} -u |_{-} =0 \quad \text{on} \quad \partial D, \\
\noalign{\smallskip}
& \displaystyle \frac{1}{\rho} \frac{\partial u}{\partial \nu} \bigg|_{+} - \frac{1}{\rho_b} \frac{\partial u}{\partial \nu} \bigg|_{-} =0 \quad \text{on} \quad \partial D,\\
\noalign{\smallskip}
& e^{-\i \alpha \cdot x} u \,\,\, \mbox{is periodic.}
\end{array}
\right.
\end{equation}
By choosing proper physical units, we may assume that the resonator size is of order one. We assume that the wave speeds outside and inside the resonators are comparable to each other and that condition (\ref{defp}) holds.
\subsection{Subwavelength band gaps and Bloch modes} \label{sec-2}
As described in \Cref{sec:floquet}, the problem \eqref{eq-scatteringB} has nontrivial solutions for discrete values of $\omega$ such as
$$ 0 \le \omega_1^\alpha \le \omega_2^\alpha \le \cdots$$
and we have the following band structure of propagating frequencies for the given periodic structure:
$$ [0, \max_\alpha \omega_1^\alpha] \cup [ \min_\alpha \omega_2^\alpha, \max_\alpha \omega_2^\alpha] \cup [ \min_\alpha \omega_3^\alpha, \max_\alpha \omega_3^\alpha] \cup \cdots. $$
A non-trivial solution to this problem and its corresponding frequency is called a Bloch eigenfunction and a Bloch eigenfrequency. The Bloch eigenfrequencies $\omega_i^\alpha, \ i=1,2,\ldots$ with positive real part, seen as functions of $\alpha$, are the {band functions}.
We use the quasi-periodic single-layer potential introduced in (\ref{singlealpha}) to represent the solution to the scattering problem (\ref{eq-scatteringB}) in $Y\setminus \overline{D}$.
We look for a solution $u$ of~\eqref{eq-scatteringB} of the form:
\begin{equation} \label{Helm-solutionB}
u =
\begin{cases}
\mathcal{S}_{D}^{\alpha,k} [\psi]\quad & \text{in} ~ Y \setminus \overline{D},\\
\noalign{\smallskip}
\mathcal{S}_{D}^{k_b} [\psi_b] &\text{in} ~ {D},
\end{cases}
\end{equation}
for some surface potentials $\psi, \psi_b \in L^2(\partial D)$.
Using the jump relations for the single-layer potentials, one can show that~\eqref{eq-scatteringB} is equivalent to the boundary integral equation
\begin{equation} \label{eq-boundaryB}
\mathcal{A}(\omega, \delta)[\Psi] =0,
\end{equation}
where
\[
\mathcal{A}(\omega, \delta) =
\begin{pmatrix}
\mathcal{S}_D^{k_b} & -\mathcal{S}_D^{\alpha,k} \\
\noalign{\smallskip}
-\frac{1}{2} I + \mathcal{K}_D^{k_b, *}& -\delta( \frac{1}{2} I + (\mathcal{K}_D^{ -\alpha,k})^*)
\end{pmatrix},
\,\, \Psi=
\begin{pmatrix}
\psi_b\\
\psi
\end{pmatrix}.
\]
As before, we denote by $$\mathcal{H} = L^2(\partial D) \times L^2(\partial D) \quad \mbox{ and } \quad \mathcal{H}_1 = H^1(\partial D) \times L^2(\partial D).$$ It is clear that $\mathcal{A}(\omega, \delta)$ is a bounded linear operator from $\mathcal{H}$ to $\mathcal{H}_1$, \emph{i.e.}
$\mathcal{A}(\omega, \delta) \in \mathcal{L}(\mathcal{H}, \mathcal{H}_1)$. We first look at the limiting case when $\delta =0$. The operator $\mathcal{A}(\omega, \delta)$ is a perturbation of
\begin{equation} \label{eq-A_0-3dB}
\mathcal{A}(\omega, 0) =
\begin{pmatrix}
\mathcal{S}_D^{k_b} & -\mathcal{S}_D^{\alpha,k} \\
\noalign{\smallskip}
-\frac{1}{2} I + \mathcal{K}_D^{k_b, *} & 0
\end{pmatrix}.
\end{equation}
We see that $\omega_0$ is a characteristic value of $\mathcal{A}(\omega,0)$ if and only if $(\omega_0 v^{-1}_b)^2$ is a Neumann eigenvalue of $D$ or $(\omega_0 v^{-1})^2$ is a Dirichlet eigenvalue of $Y\backslash \overline{D}$ with $\alpha$-quasi-periodicity on $\partial Y$. Since
zero is a Neumann eigenvalue of $D$,
$0$ is a characteristic value for the holomorphic operator-valued function $\mathcal{A}(\omega,0)$. {By noting that there is a positive lower bound for the other Neumann eigenvalues of $D$ and all the Dirichlet eigenvalues of $Y\backslash \overline{D}$ with $\alpha$-quasi-periodicity on $\partial Y$,} we can conclude the following result by the Gohberg-Sigal theory.
\begin{lemma}
For any $\delta$ sufficiently small, there exists {one and only one} characteristic value
$\omega_0= \omega_0(\delta)$ in a neighborhood of the origin in the complex plane to the holomorphic operator-valued function
$\mathcal{A}(\omega, \delta)$.
Moreover,
$\omega_0(0)=0$ and $\omega_0$ depends on $\delta$ continuously.
\end{lemma}
\subsubsection{Asymptotic behavior of the first Bloch eigenfrequency $\omega_1^\alpha$}
In this section we assume $\alpha \ne 0$.
We define
\begin{equation} \label{defA0}
\mathcal{A}_0 :=\mathcal{A}(0,0)=
\begin{pmatrix}
\mathcal{S}_D& -\mathcal{S}_D^{\alpha,0} \\
\noalign{\smallskip}
-\frac{1}{2} I + \mathcal{K}_D^{*}& 0
\end{pmatrix},
\end{equation}
and let $\mathcal{A}_0^* : {\mathcal{H}}_1 \to {\mathcal{H}}$ be the adjoint of $\mathcal{A}_0$. We choose an element $\psi_0\in L^2(\partial D)$ such that
$$ \big( -\frac{1}{2} I + \mathcal{K}_D^* \big)[\psi_0] =0,\quad \int_{\partial D} \psi_0 = 1.$$
We recall the definition (\ref{defcap}) of the capacity of the set $D$, $\mathrm{Cap}_D$, which is equivalent to
\begin{equation}\label{capacityB}
\mathcal{S}_D [\psi_0] = - \frac{1}{\mathrm{Cap}_D} \quad \mbox{on } {\partial D}.
\end{equation}
Then we can easily check that $\mathrm{Ker } (\mathcal{A}_0)$ and $ \mathrm{Ker } (\mathcal{A}_0^*)$ are spanned respectively by
\[
\Psi_0 = \begin{pmatrix}
\psi_0\\
\widetilde\psi_0\end{pmatrix}
\quad \text{and} \quad
\Phi_0 = \begin{pmatrix}
0\\
1 \end{pmatrix},
\]
where $\widetilde \psi_0 =( \mathcal{S}_D^{\alpha,0})^{-1} \mathcal{S}_D[\psi_0]$.
We now perturb $ \mathcal{A}_0 $ by a rank-one operator $\mathcal{P}_0$ from $\mathcal{H}$ to $\mathcal{H}_1$
given by
$
\mathcal{P}_0[\Psi]:= (\Psi, \Psi_0)\Phi_0,
$
and denote it by
$
\widetilde{\mathcal{A}_0}= \mathcal{A}_0 + \mathcal{P}_0
$.
Then the followings hold:
\begin{enumerate}
\item[(i)] $\widetilde{\mathcal{A}_0}[\Psi_0]= \| \Psi_0 \|^2 \Phi_0 $, $\widetilde{\mathcal{A}_0}^*[\Phi_0] = \| \Phi_0 \|^2\Psi_0$.
\item[(ii)] The operator $\widetilde{\mathcal{A}_0}$ and its adjoint $\widetilde{\mathcal{A}_0}^*$ are invertible in
$\mathcal{L}(\mathcal{H}, \mathcal{H}_1)$ and $\mathcal{L}(\mathcal{H}_1, \mathcal{H})$, respectively.
\end{enumerate}
Using (\ref{eq:exp_S}), (\ref{eq:exp_K}), (\ref{series-s2}), and (\ref{series-k2}), we can expand $\mathcal{A}(\omega,\delta)$ as
\begin{equation} \label{expdA}
\begin{array}{l}
\mathcal{A}(\omega, \delta):=\mathcal{A}_0 + \mathcal{B}(\omega, \delta)
= \mathcal{A}_0 + \omega \mathcal{A}_{1, 0}+ \omega^2 \mathcal{A}_{2, 0}
+ \omega^3 \mathcal{A}_{3, 0} + \delta \mathcal{A}_{0, 1}+ \delta \omega^2\mathcal{A}_{2, 1}\\
\noalign{\smallskip} \qquad \qquad \displaystyle + \O(| \omega| ^4 + |\delta \omega^3|) \end{array}
\end{equation}
where
\[
\mathcal{A}_{1,0} = \begin{pmatrix}
v_b^{-1} \mathcal{S}_{D,1} & 0 \\
\noalign{\smallskip}
0& 0
\end{pmatrix},
\,\, \mathcal{A}_{2,0}=
\begin{pmatrix}
v_b^{-2} \mathcal{S}_{D,2} & -v^{-2} \mathcal{S}_{D,1}^\alpha \\
\noalign{\smallskip}
v_b^{-2} \mathcal{K}_{D,2} & 0
\end{pmatrix},
\,\, \mathcal{A}_{3,0}=
\begin{pmatrix}
v_b^{-3} \mathcal{S}_{D,3} & 0 \\
\noalign{\smallskip}
v_b^{-3} \mathcal{K}_{D,3} & 0
\end{pmatrix},
\]
\[
\mathcal{A}_{0, 1}=
\begin{pmatrix}
0& 0\\
\noalign{\smallskip}
0 & -(\frac{1}{2}+ (\mathcal{K}_{D}^{-\alpha,0})^*)
\end{pmatrix},
\,\, \mathcal{A}_{2, 1}=
\begin{pmatrix}
0& 0\\
\noalign{\smallskip}
0 & -v^{-2} \mathcal{K}^{\alpha}_{D,1}
\end{pmatrix}.
\]
From the above expansion, it follows that
\begin{equation} \label{eq:Aomegadelta}
\begin{array}{lll}
A(\omega, \delta) &= & \displaystyle - \omega^2 \frac{ v_b^{-2} | D |}{\mathrm{Cap}_D} - \omega^3 v_b^{-3} \frac{\i c_1 | D |}{4 \pi } + c_2\delta + \omega \delta \frac{\i c_1c_2 v_b^{-1} \mathrm{Cap}_D}{4 \pi} \\
\noalign{\smallskip} &&\displaystyle + \O( | \omega |^4 + | \delta | \, |\omega|^2 + | \delta |^2),
\end{array}
\end{equation}
where \begin{equation}
c_1:=\frac{\|\psi_0\|^2}{\|\psi_0\|^2+\|\widetilde\psi_0\|^2},\end{equation}
and \begin{equation} \label{defc2B}
c_2:= \displaystyle \int_{\partial D} \widetilde\psi_0 \; \big(1/2 + \mathcal{K}_D^{-\alpha,0}\big)[\chi_{\partial D}]\; \: \mathrm{d} \sigma.\end{equation}
We now solve $A(\omega, \delta) =0$.
It is clear that $\delta = \O(\omega^2)$ and thus $\omega_0(\delta) = \O(\sqrt{\delta})$.
We write
$$
\omega_0(\delta) = a_1 \delta^{\frac{1}{2}} + a_2 \delta + \O(\delta^{\frac{3}{2}}),
$$
and get
\begin{align*}
& - \frac{ v_b^{-2} | D |}{ \mathrm{Cap}_D} \left( a_1 \delta^{\frac{1}{2}} + a_2 \delta + \O(\delta^{\frac{3}{2}}) \right)^2
- \frac{\i c_1 v_b^{-3} | D |}{4 \pi}\left( a_1 \delta^{\frac{1}{2}} + a_2 \delta + \O(\delta^{\frac{3}{2}}) \right)^3 \\
& \qquad
+c_2 \delta + \frac{\i c_1c_2 v_b^{-1} \mathrm{Cap}_D}{4 \pi } \left( a_1 \delta^{\frac{3}{2}} + a_2 \delta^2 + \O(\delta^{\frac{5}{2}}) \right) + \O(\delta^2) = 0.
\end{align*}
From the coefficients of the $\delta$ and $\delta^{\frac{3}{2}}$ terms, we obtain
\[
- a_1^2 \frac{v_b^{-2} | D |}{ \mathrm{Cap}_D} + c_2 = 0
\]
and
\[
2 a_1 a_2 \frac{- v_b^{-2} | D |}{ \mathrm{Cap}_D} - a_1^3 \frac{\i c_1 v_b^{-3} | D |}{4\pi } + a_1 \frac{\i c_1c_2 v_b^{-1} \mathrm{Cap}_D}{4 \pi } = 0,
\]
which yields
\[
a_1 = \pm \sqrt{ \frac{ c_2 \mathrm{Cap}_D}{| D |} } v_b
\quad \text{and} \quad
a_2 = 0.
\]
From the definition (\ref{alphacap}) of the $\alpha$-quasi-periodic capacity, it follows that
\begin{align*} c_2 &= \frac{\mathrm{Cap}_{D,\alpha}}{\mathrm{Cap}_D}.\end{align*}
Therefore, the following result from \cite{bandgap} holds.
\begin{theorem}\label{approx_thm} For $\alpha \ne 0$ and sufficiently small $\delta$, we have
\begin{align}
\omega_1^\alpha= \omega_M \sqrt{\frac{\mathrm{Cap}_{D,\alpha}}{\mathrm{Cap}_D}} + \O(\delta^{3/2}), \label{o_1_alpha}
\end{align}
where $\omega_M$ is defined in (\ref{defomegaM}) by $$\displaystyle \omega_M= \sqrt{ \frac{\delta \mathrm{Cap}_D}{|D|} } v_b.$$
\end{theorem}
Now from \eqref{o_1_alpha}, we can see that $$\omega_{M,\alpha}:= \omega_M \sqrt{\frac{\mathrm{Cap}_{D,\alpha}}{\mathrm{Cap}_D}} \rightarrow 0$$ as $\alpha\to 0$ because
$$ \big( (1/2) I +( \mathcal{K}_D^{-\alpha,0})^*\big)(\mathcal{S}_D^{\alpha,0})^{-1} [\chi_{\partial D}] \rightarrow 0,$$ and so $\mathrm{Cap}_{D,\alpha} \rightarrow 0$ as $\alpha\rightarrow0$. Moreover, it is clear that $\omega_{M,\alpha}$ lies in a small neighborhood of zero.
We define $\omega^1_*:= \max_{\alpha} \omega_{M,\alpha}$. Then we deduce the following
result regarding a subwavelength band gap opening.
\begin{theorem}\label{main}
For every $\epsilon>0$, there exists $\delta_0>0$ and {$\widetilde \omega> \omega^1_*+\epsilon$} such that
\begin{equation}
[ \omega^1_*+\epsilon, \widetilde\omega ] \subset [\max_\alpha \omega_1^\alpha, \min_\alpha \omega_2^\alpha]
\end{equation}
for $\delta<\delta_0$.
\end{theorem}
\begin{proof}
Using $\omega_1^0=0$ and the continuity of $\omega_1^\alpha$ in $\alpha$ and $\delta$, we get $\alpha_0$ and $\delta_1$ such that $\omega_1^\alpha < \omega^1_*$ for every $| \alpha|<\alpha_0$ and $\delta<\delta_1$. Following the derivation of \eqref{o_1_alpha}, we can check that it is valid uniformly in $\alpha$ as far as $|\alpha| \ge \alpha_0$. Thus there exists $\delta_0 < \delta_1$ such $\omega_1^\alpha \le \omega^1_* +\epsilon$ for $|\alpha| \ge \alpha_0$. We have shown that $ \max_\alpha \omega_1^\alpha \le \omega^1_*+\epsilon$ for sufficiently small $\delta$. To have $\min_\alpha \omega_2^\alpha > \omega^1_* +\epsilon$ for small $\delta$, it is enough to check that $\mathcal{A}(\omega,\delta)$ has no small characteristic value other than $\omega_1^\alpha$. For $\alpha$ away from $0$, we can see that it is true following the proof of Theorem \ref{approx_thm}. If $\alpha=0$, we have
\begin{equation}
\mathcal{A}(\omega,\delta)=\mathcal{A}(\omega,0) + \O(\delta),\end{equation}
near $\omega_2^0$ with $\delta=0$. Since $\omega_2^0\ne 0$, we have $\omega_2^0(\delta) > \omega^1_* + \epsilon$ for sufficiently small $\delta$. Finally, using the continuity of $\omega_2^\alpha$ in $\alpha$, we obtain $\min_\alpha \omega_2^\alpha > \omega^1_* +\epsilon$ for small $\delta$. This completes the proof.
\end{proof}
As shown in \cite{highfrequency}, the first Bloch eigenvalue $\omega_1^\alpha$ attains its maximum $\omega^1_*$ at $\alpha_*=(\pi,\pi,\pi)$ (\textit{i.e.} the corner $M$ of the Brillouin zone). The proof relies on the variational characterization (\ref{vcalpha}) of the quasi-periodic capacity.
\begin{theorem} \label{main2} Assume that $D$ is symmetric with respect to $\{ x_j=0\}$ for $j=1,2,3$. Then both $\mathrm{Cap}_{D,\alpha}$ and $\omega_1^\alpha$ attain their maxima at $\alpha_*=(\pi,\pi,\pi)$.
\end{theorem}
The results of Theorems \ref{main} and \ref{main2} are illustrated in Figure \ref{bloch}.
\begin{figure}[h]
\centering
\includegraphics[height=8cm]{bandgap_1}
\caption{Subwavelength band gap opening.} \label{bloch}
\end{figure}
Next, we consider the behavior of the first Bloch eigenfunction.
In \cite{highfrequency} a high-frequency homogenization approach for subwavelength resonators has been developed. An asymptotic expansion of the Bloch eigenfunction near the critical frequency has been computed. It is proved that the eigenfunction can be decomposed into two parts: one is slowly varying and satisfies a homogenized equation, while the other is periodic and varying at the microscopic scale. The microscopic oscillations explain why these structures can be used to achieve super-focusing, while the exponential decay of the slowly varying part proves the band gap opening above the critical frequency.
We need the following lemma from \cite{highfrequency}.
\begin{lemma} For $\epsilon>0$ small enough,
\begin{align*}
\mathrm{Cap}_{D, \alpha_*+\epsilon\widetilde\alpha}= \mathrm{Cap}_{D, \alpha_*} + \epsilon^2 {\Lambda_D^{\widetilde\alpha}} + \O(\epsilon^4),
\end{align*}
where $\Lambda_D^{\widetilde\alpha}$ is a {negative semi-definite quadratic function} of $\widetilde \alpha$:
$$
\frac{v_b^2}{|D|} \Lambda_D^{\widetilde\alpha} = - \sum_{1\leq i,j\leq 3} \lambda_{ij} \widetilde\alpha_i \widetilde\alpha_j
$$
with $(\lambda_{ij})$ being symmetric and positive semi-definite.
\end{lemma}
Assume that the resonators are arranged with period $r>0$ and {$\delta = \O(r^2)$}.
Then, by a scaling argument, the critical frequency $\omega_*^r = (1/r) \omega_*^1 = \O(1)$ as {$r\rightarrow 0$}.
\begin{theorem} \label{superf}
For $\omega$ near the critical frequency $\omega_*^r$:
$(\omega_*^r)^2 -\omega^2 = \O(r^2)$,
the following asymptotic of the first {Bloch eigenfunction} $u_{1,r}^{\alpha_*/r + \widetilde\alpha}$ holds:
$$
u_{1,r}^{\alpha_*/r + \widetilde\alpha}(x) = \underbrace{e^{\i \widetilde\alpha \cdot x}}_{{\mbox{macroscopic behavior}}} \underbrace{S \left(\frac{x}{r}\right)}_{{\mbox{microscpic behavior}}} + \; \O(r).
$$
The macroscopic {plane wave} $e^{\i \widetilde\alpha \cdot x}$ satisfies:
$$
{ \sum_{1\leq i, j \leq 3}\lambda_{ij} \partial_i \partial_j \widetilde{ u} (x)+ \frac{(\omega_*^r)^2 -\omega^2}{\delta}\widetilde{ u}(x)= 0}.
$$
\end{theorem}
If we write
$(\omega_*^r)^2 - \omega^2 = \beta \delta$, then
$$\sum_{1\leq i, j \leq 3}\lambda_{ij} \widetilde \alpha_i \widetilde \alpha_j = \beta + \O(r^2).$$ Moreover, for $\beta >0$, the {plane wave Bloch eigenfunction} satisfies the homogenized equation for the crystal while the microscopic field is periodic and varies on the scale of $r$. If $\beta <0$, then the Bloch eigenfunction is {exponentially growing or decaying} which is another way to see that a {band gap opening} occurs above the critical frequency.
Theorem \ref{superf} shows that the super-focusing property at subwavelength scales near the critical frequency $\omega_*^r$ holds true. Here, the mechanism is not due to effective (high-contrast below $\omega_*^r$ and negative above $\omega_*^r$) properties of the medium. The effective medium theory described in Section~\ref{dilutesect} is no longer valid in the nondilute case.
Figure \ref{figbloch} shows a one-dimensional plot along the $x_1$-axis of the real part of the {Bloch} {eigenfunction} of the {square lattice} over many unit cells.
\begin{figure}[h] \begin{center} \includegraphics[scale=0.7]{s1D.png}
\caption{Real part of the {Bloch} {eigenfunction} of the {square lattice} shown over many unit cells. } \label{figbloch} \end{center}
\end{figure}
\section{Topological metamaterials} \label{sec6}
We begin this section by studying existence and consequences of a Dirac cone singularity in a honeycomb structure. Dirac singularities are intimately connected with topologically protected edge modes, and we then study such modes in an array of subwavelength resonators.
\subsection{Dirac singularity}
The classical example of a structure with a Dirac singularity is graphene, where this singularity is responsible for many peculiar electronic properties. Graphene consists of a single layer of carbon atoms in an honeycomb lattice, and in this section we study a similar structure with subwavelength resonators.
In the homogenization theory of metamaterials, the goal is to map the metamaterial to a homogeneous material with some effective parameters. It has been demonstrated in the previous section that this approach does not apply in the case of crystals at ``high'' frequencies, \textit{i.e.}, away from the centre $\Gamma$ (corresponding to $\alpha=(0,0,0)$) of the Brillouin zone. In Theorem \ref{superf}, it is shown that around the symmetry point $M$ (corresponding to $\alpha=(\pi,\pi,\pi)$) in the Brillouin zone of a crystal with a square lattice, the Bloch eigenmodes display oscillatory behaviour on two distinct scales: small scale oscillations on the order of the size of individual
resonators, while simultaneously the plane-wave envelope oscillates at a much larger scale and satisfies a homogenized equation.
In this section we prove the near-zero effective index property in honeycomb crystal at the deep subwavelength scale. We develop a homogenization theory that captures both the macroscopic behaviour of the eigenmodes and the oscillations in the microscopic scale. The near-zero effective refractive index at the macroscale is a consequence of the existence of a Dirac dispersion cone.
We consider a two-dimensional infinite honeycomb crystal in two dimensions depicted in Figure \ref{fig:honeycomb}. Define the lattice $\Lambda$ generated by the lattice vectors
$$ l_1 = L\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right),~~l_2 = L\left( \frac{\sqrt{3}}{2}, -\frac{1}{2}\right),$$
where $L$ is the lattice constant. Denote by $Y$ a fundamental domain of the given lattice. Here, we take
$$ Y:= \left\{ s l_1+ t l_2 ~|~ 0 \le s,t \le 1 \right\}. $$
Define the three points $x_0, x_1,$ and $x_2$ as
$$x_0 = \frac{l_1 + l_2}{2}, \quad x_1 = \frac{l_1+l_2}{3}, \quad x_2 = \frac{2(l_1 + l_2)}{3} .$$
\begin{figure}[tb]
\centering
\begin{tikzpicture}
\begin{scope}[xshift=-5cm,scale=1.2]
\coordinate (a) at (1,{1/sqrt(3)});
\coordinate (b) at (1,{-1/sqrt(3)});
\pgfmathsetmacro{\rb}{0.25pt}
\pgfmathsetmacro{\rs}{0.2pt}
\draw (0,0) -- (1,{1/sqrt(3)}) -- (2,0) -- (1,{-1/sqrt(3)}) -- cycle;
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw[opacity=0.2] ({2/3},0) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})$) -- ({2/3},0)
($0.5*(1,{-1/sqrt(3)})$) -- ({2/3},0)
($(1,{1/sqrt(3)})+0.5*(1,{-1/sqrt(3)})$) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})+(1,{-1/sqrt(3)})$) -- ({4/3},0);
\begin{scope}[shift = (a)]
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw[opacity=0.2] ({2/3},0) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})$) -- ({2/3},0)
($0.5*(1,{-1/sqrt(3)})$) -- ({2/3},0)
($(1,{1/sqrt(3)})+0.5*(1,{-1/sqrt(3)})$) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})+(1,{-1/sqrt(3)})$) -- ({4/3},0);
\end{scope}
\begin{scope}[shift = (b)]
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw[opacity=0.2] ({2/3},0) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})$) -- ({2/3},0)
($0.5*(1,{-1/sqrt(3)})$) -- ({2/3},0)
($(1,{1/sqrt(3)})+0.5*(1,{-1/sqrt(3)})$) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})+(1,{-1/sqrt(3)})$) -- ({4/3},0);
\end{scope}
\begin{scope}[shift = ($-1*(a)$)]
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw[opacity=0.2] ({2/3},0) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})$) -- ({2/3},0)
($0.5*(1,{-1/sqrt(3)})$) -- ({2/3},0)
($(1,{1/sqrt(3)})+0.5*(1,{-1/sqrt(3)})$) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})+(1,{-1/sqrt(3)})$) -- ({4/3},0);
\end{scope}
\begin{scope}[shift = ($-1*(b)$)]
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw[opacity=0.2] ({2/3},0) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})$) -- ({2/3},0)
($0.5*(1,{-1/sqrt(3)})$) -- ({2/3},0)
($(1,{1/sqrt(3)})+0.5*(1,{-1/sqrt(3)})$) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})+(1,{-1/sqrt(3)})$) -- ({4/3},0);
\end{scope}
\begin{scope}[shift = ($(a)+(b)$)]
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw[opacity=0.2] ({2/3},0) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})$) -- ({2/3},0)
($0.5*(1,{-1/sqrt(3)})$) -- ({2/3},0)
($(1,{1/sqrt(3)})+0.5*(1,{-1/sqrt(3)})$) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})+(1,{-1/sqrt(3)})$) -- ({4/3},0);
\end{scope}
\begin{scope}[shift = ($-1*(a)-(b)$)]
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw[opacity=0.2] ({2/3},0) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})$) -- ({2/3},0)
($0.5*(1,{-1/sqrt(3)})$) -- ({2/3},0)
($(1,{1/sqrt(3)})+0.5*(1,{-1/sqrt(3)})$) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})+(1,{-1/sqrt(3)})$) -- ({4/3},0);
\end{scope}
\begin{scope}[shift = ($(a)-(b)$)]
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw[opacity=0.2] ({2/3},0) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})$) -- ({2/3},0)
($0.5*(1,{-1/sqrt(3)})$) -- ({2/3},0)
($(1,{1/sqrt(3)})+0.5*(1,{-1/sqrt(3)})$) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})+(1,{-1/sqrt(3)})$) -- ({4/3},0);
\end{scope}
\begin{scope}[shift = ($-1*(a)+(b)$)]
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw[opacity=0.2] ({2/3},0) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})$) -- ({2/3},0)
($0.5*(1,{-1/sqrt(3)})$) -- ({2/3},0)
($(1,{1/sqrt(3)})+0.5*(1,{-1/sqrt(3)})$) -- ({4/3},0)
($0.5*(1,{1/sqrt(3)})+(1,{-1/sqrt(3)})$) -- ({4/3},0);
\end{scope}
\end{scope}
\draw[dashed,opacity=0.5,->] (-3.9,0.65) .. controls(-1.8,1.5) .. (1,0.7);
\begin{scope}[scale=2.8]
\coordinate (a) at (1,{1/sqrt(3)});
\coordinate (b) at (1,{-1/sqrt(3)});
\coordinate (Y) at (1.8,0.45);
\coordinate (c) at (2,0);
\coordinate (x1) at ({2/3},0);
\coordinate (x0) at (1,0);
\coordinate (x2) at ({4/3},0);
\pgfmathsetmacro{\rb}{0.25pt}
\pgfmathsetmacro{\rs}{0.2pt}
\begin{scope}[xshift = 1.33333cm]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\draw (0:\rb) node[xshift=7pt] {$D_2$};
\end{scope}
\begin{scope}[xshift = 0.666667cm, rotate=60]
\draw plot [smooth cycle] coordinates {(0:\rb) (60:\rs) (120:\rb) (180:\rs) (240:\rb) (300:\rs) };
\end{scope}
\draw ({0.6666667-\rb},0) node[xshift=-7pt] {$D_1$};
\draw (Y) node{$Y$};
\draw[->] (0,0) -- (a) node[above,pos=0.7]{$l_1$};
\draw[->] (0,0) -- (b) node[below,pos=0.7]{$l_2$};
\draw (a) -- (c) -- (b);
\draw[fill] (x1) circle(0.5pt) node[xshift=6pt,yshift=-6pt]{$x_1$};
\draw[fill] (x0) circle(0.5pt) node[yshift=4pt, xshift=6pt]{$x_0$};
\draw[fill] (x2) circle(0.5pt) node[xshift=6pt,yshift=4pt]{$x_2$};
\draw[dashed] (1,0.7) node[right]{$p$} -- (1,-0.7);
\end{scope}
\end{tikzpicture}
\caption{Illustration of the honeycomb crystal and quantities in $Y$.} \label{fig:honeycomb}
\end{figure}
We will consider a general shape of the subwavelength resonators, under certain symmetry assumptions. Let $R_0$ be the rotation around $x_0$ by $\pi$, and let $R_1$ and $R_2$ be the rotations by $-\frac{2\pi}{3}$ around $x_1$ and $x_2$, respectively. These rotations can be written as
$$ R_1 x = Rx+l_1, \quad R_2 x = Rx + 2l_1, \quad R_0 x = 2x_0 - x , $$
where $R$ is the rotation by $-\frac{2\pi}{3}$ around the origin. Moreover, let $R_3$ be the reflection across the line $p = x_0 + \mathbb{R} e_2$, where $e_2$ is the second standard basis element. Assume that the unit cell contains two subwavelength resonators $D_j$, $j=1,2$, each centred at $x_j$ such that
$$R_0 D_1 = D_2, \quad R_1 D_1 = D_1,\quad R_2 D_2 = D_2, \quad R_3D_1 = D_2.$$
We denote the pair of subwavelength resonators by $D=D_1 \cup D_2$. The dual lattice of $\Lambda$, denoted $\Lambda^*$, is generated by $\alpha_1$ and $\alpha_2$ given by
$$ \alpha_1 = \frac{2\pi}{L}\left( \frac{1}{\sqrt{3}}, 1\right),~~\alpha_2 = \frac{2\pi}{L}\left(\frac{1}{\sqrt{3}}, -1 \right).$$
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=2]
\coordinate (a) at ({1/sqrt(3)},1);
\coordinate (b) at ({1/sqrt(3)},-1);
\coordinate (c) at ({2/sqrt(3)},0);
\coordinate (K1) at ({1/sqrt(3)},{1/3});
\coordinate (K2) at ({1/sqrt(3)},{-1/3});
\coordinate (K3) at (0,{-2/3});
\coordinate (K4) at ({-1/sqrt(3)},{-1/3});
\coordinate (K5) at ({-1/sqrt(3)},{1/3});
\coordinate (K6) at (0,{2/3});
\draw[->] (0,0) -- (a) node[above]{$\alpha_1$};
\draw[->] (0,0) -- (b) node[below]{$\alpha_2$};
\draw (a) -- (c) -- (b) node[pos=0.4,below right]{$Y^*$};
\draw[fill] (K1) circle(1pt) node[xshift=6pt,yshift=-4pt]{$\alpha_1^*$};
\draw[fill] (K2) circle(1pt) node[xshift=6pt,yshift=4pt]{$\alpha_2^*$};
\draw[fill] (0,0) circle(1pt) node[left]{$\Gamma$};
\draw[opacity=0.4] (K1) -- (K2) -- (K3) -- (K4) node[left]{$Y_1^*$} -- (K5) -- (K6) -- cycle;
\end{tikzpicture}
\caption{Illustration of the dual lattice and the Brillouin zone $Y^*$.}
\label{fig:bz}
\end{figure}
The Brillouin zone $Y^*:= {\mathbb{R}^2}/{\Lambda^*}$ can be represented either as
$$Y^* \simeq \left\{ s \alpha_1+ t \alpha_2 ~|~ 0 \le s,t \le 1 \right\}, $$
or as the first Brillouin zone $Y_1^*$, which is a hexagon illustrated in Figure \ref{fig:bz}.
The points $$\alpha_1^*= \frac{2\alpha_1+\alpha_2}{3}, \quad \alpha^*_2 = \frac{\alpha_1+2\alpha_2}{3},$$ in the Brillouin zone are called \emph{Dirac points}. For simplicity, we only consider the analysis around the Dirac point $\alpha_* := \alpha_1^*$, the main difference around $\alpha_2^*$ is summarized in Remark \ref{rmk:alpha2}.
Wave propagation in the honeycomb lattices of subwavelength resonators is described by the following $\alpha$-quasi-periodic Helmholtz problem in $Y$:
\begin{equation} \label{HP1}
\left\{
\begin{array} {lll}
&\displaystyle \nabla \cdot \frac{1}{\rho} \nabla u+ \frac{\omega^2}{\kappa} u = 0 \quad &\text{in} \ Y \backslash \overline{D}, \\
\noalign{\smallskip}
&\displaystyle \nabla \cdot \frac{1}{\rho_b} \nabla u+ \frac{\omega^2}{\kappa_b} u = 0 \quad &\text{in} \ D, \\
\noalign{\smallskip}
&\displaystyle u |_{+} -u |_{-} =0 \quad &\text{on} \ \partial D, \\
\noalign{\smallskip}
& \displaystyle \frac{1}{\rho} \frac{\partial u}{\partial \nu} \bigg|_{+} - \frac{1}{\rho_b} \frac{\partial u}{\partial \nu} \bigg|_{-} =0 \quad &\text{on} \ \partial D, \\
\noalign{\smallskip}
& u(x+l)= e^{\i \alpha\cdot l} u(x) \quad & \text{for all} \ l\in \Lambda.
\end{array}
\right.
\end{equation}
Let $\psi_j^{\alpha}\in L^2(\partial D)$ be given by
\begin{align} \label{psi_def}
\mathcal{S}_D^{\alpha,0}[\psi_j^{\alpha}] = \chi_{\partial D_j}\quad \mbox{on}~\partial D,\quad j=1, 2.
\end{align}
Define the capacitance matrix $C^\alpha=(C_{ij}^\alpha)$ by
\begin{equation} \label{defcapal} C_{ij}^\alpha := - \int_{\partial D_i} \psi_j^\alpha \;\: \mathrm{d} \sigma,\quad i,j=1, 2.\end{equation}
Using the symmetry of the honeycomb structure, it can be shown that the capacitance coefficients satisfy \cite{ammari2018honeycomb}
$$c_1^\alpha := C_{11}^\alpha = C_{22}^{\alpha}, \quad c_2^\alpha := C_{12}^{\alpha} = \overline{C_{21}^{\alpha}},$$
and
\begin{equation}\label{c1c2_deri}
\nabla_\alpha c_1^\alpha \Big|_{\alpha=\alpha^*} = 0,
\quad
\nabla_\alpha c_2^\alpha \Big|_{\alpha=\alpha^*} = c\begin{pmatrix} 1\\-\mathrm{i}\mkern1mu\end{pmatrix},
\end{equation}
where we denote $$c:=\frac{\partial c_2^{\alpha}}{\partial \alpha_1}\Big|_{\alpha=\alpha^*} \neq 0,$$
as proved in \cite[Lemma 3.4]{ammari2018honeycomb}.
It is shown in \cite{ammari2018honeycomb} that the first two band functions $\omega_1^\alpha$ and $\omega_2^\alpha$ form a conical dispersion relation near the Dirac point $\alpha_*$. Such a conical dispersion is referred to as a {\it Dirac cone}. More specifically, the following results which hold in the subwavelength regime are proved in \cite{ammari2018honeycomb}.
\begin{theorem}\label{thm:honeycomb}
For small $\delta$, the first two band functions $\omega_j^\alpha,j=1,2$, satisfy
\begin{equation} \label{eq:wasymp}
\omega_j^\alpha = \sqrt{\frac{\delta\lambda_j^\alpha }{|D_1|}}v_b + \O(\delta),
\end{equation}
uniformly for $\alpha$ in a neighbourhood of $\alpha_*$, where $\lambda_j^\alpha, j=1,2,$ are the two eigenvalues of $C^\alpha$ and $|D_1|$ denotes the area of one of the subwavelength resonators. Moreover, for $\alpha$ close to $\alpha_*$ and $\delta$ small enough, the first two band functions form a Dirac cone, \textit{i.e.}{},
\begin{equation} \label{eq:dirac}
\begin{matrix}
\displaystyle \omega_1^\alpha = \omega_*- \lambda|\alpha - \alpha_*| \big[ 1+ \O(|\alpha-\alpha_*|) \big], \\[0.5em]
\displaystyle \omega_2^\alpha = \omega_*+ \lambda|\alpha - \alpha_*| \big[ 1+ \O(|\alpha-\alpha_*|) \big],
\end{matrix}
\end{equation}
where $\omega_*$ and $\lambda$ are independent of $\alpha$ and satisfy
$$\omega_*= \sqrt{\frac{\delta c_1^{\alpha_*}}{|D_1|}}v_b + \O(\delta) \quad \text{and} \quad \lambda = |c|\sqrt\delta\lambda_0 + \O(\delta), \quad \lambda_0=\frac{1}{2}\sqrt{\frac{v_b^2 }{|D_1|c_1^{\alpha_*}}}$$
as $\delta \rightarrow 0$. Moreover, the error term $\O(|\alpha-\alpha_*|)$ in (\ref{eq:dirac}) is uniform in $\delta$.
\end{theorem}
The results in Theorem \ref{thm:honeycomb} are illustrated in Figure \ref{honeycombfz}. The figure shows the first three bands. Observe that the first two bands cross at the symmetry point $K$ (corresponding to $\alpha_*$) such that the dispersion relation is linear. Figure \ref{squarefz} shows the band gap structure in the subwavelength region for a rectangular array of subwavelength dimers. For such arrays, the two first bands cannot cross each other.
\begin{figure}[!h]
\begin{center}
\includegraphics[height=5.cm]{honeycomb_R_1_5.png}
\hspace{0.25cm}
\includegraphics[height=5.cm]{honeycomb_R_1_5_zoom.png}
\caption{Band gap structure upon zooming in the subwavelength region for a honeycomb lattice of subwavelength resonators.} \label{honeycombfz}
\end{center}
\end{figure}
\begin{figure}[!h]
\begin{center}
\includegraphics[height=5.cm]{rectangle_D_1R.png}
\hspace{0.25cm}
\includegraphics[height=5.cm]{rectangle_D_1R_zoom.png}
\caption{The band gap structure upon zooming in the subwavelength region for a rectangular array of subwavelength dimers.} \label{squarefz}
\end{center}
\end{figure}
Next we investigate the asymptotic behaviour of the Bloch eigenfunctions near the Dirac points. Then we show that the envelopes of the Bloch eigenfunctions satisfy a Helmholtz equation with near-zero effective refractive index and derive a two-dimensional homogenized equation of Dirac-type for the honeycomb crystal. These results are from \cite{nearzero}.
We consider the rescaled honeycomb crystal by replacing the lattice constant $L$ with $r L$ where $r>0$ is a small positive parameter.
Let $\omega_j^\alpha,j=1,2$, be the first two eigenvalues and $u_j^\alpha$ be the associated Bloch eigenfunctions for the honeycomb crystal with lattice constant $L$. Then, by a scaling argument, the honeycomb crystal with lattice constant $r L$ has the first two Bloch eigenvalues
$$
\omega_{\pm,r}^{\alpha/r} = \frac{1}{r} \omega_\pm^\alpha,
$$
and the corresponding eigenfunctions are
$$
u^{\alpha/r}_{\pm,r}(x) = u_\pm^\alpha\left( \frac{x}{r}\right).
$$
This shows that the Dirac cone is located at the point $\alpha_*/r$.
We denote the Dirac frequency by
$$
\omega^r_* = \frac{1}{r}\omega_*.
$$
We have the following result for the Bloch eigenfunctions $u_{j,r}^{\alpha/r},j=1,2,$ for $\alpha/r$ near the Dirac points $\alpha_*/r$ \cite{nearzero}.
\begin{lemma}\label{lem:Bloch_scale}
We have
$$
u_{\pm,r}^{\alpha_*/r+\tilde{\alpha}}(x) = A_\pm e^{\i \widetilde\alpha \cdot x} S_{1} \left(\frac{x}{r}\right) + B_\pm e^{\i \widetilde\alpha \cdot x} S_{2} \left(\frac{x}{r}\right) + \O(\delta + r),
$$
where
$$
S_j (x) = \mathcal{S}_D^{\alpha_*,0} [\psi_j^{\alpha_*}](x), \quad j=1,2.
$$
The functions $S_1$ and $S_2$ describe the microscopic behaviour of the Bloch eigenfunction $u_{\pm,r}^{\alpha_*/r+\widetilde{\alpha}}$ while $A_\pm e^{\i \widetilde\alpha\cdot x}$ and $B_\pm e^{\mathrm{i}\mkern1mu\tilde\alpha\cdot x}$ describe the macroscopic behaviour.
\end{lemma}
Now, we derive a homogenized equation near the Dirac frequency $\omega^r_*$. Recall that the Dirac frequency of the unscaled honeycomb crystal satisfies $\omega_* = \O(\sqrt\delta) $.
As in Theorem \ref{main2}, in order to make the order of $\omega^r_*$ fixed when $r$ tends to zero, we assume that
$\delta = \mu r^2$
for some fixed $\mu>0$.
Then we have
$$
\omega^r_* = \frac{1}{r} \omega_*= \O(1) \quad \mbox{as } r\rightarrow 0.
$$
So, in what follows, we omit the subscript $r$ in $\omega^r_*$, namely, $\omega_*:=\omega^r_*$.
Suppose the frequency $\omega$ is close to $\omega_*$, \textit{i.e.},
$$
\omega-\omega_* = \beta \sqrt\delta \quad \mbox{for some constant } \beta.
$$
We need to find the Bloch eigenfunctions or $\widetilde{\alpha}$ such that
$$
\omega = \omega_{\pm,r}^{\alpha_*/r + \widetilde\alpha}.
$$
We have that the corresponding $\widetilde{\alpha}$ satisfies
\begin{align*}
\lambda_0
\begin{bmatrix}
0 & c( \widetilde{\alpha}_1 - \i \widetilde{\alpha}_2)
\\
\overline{c}( \widetilde{\alpha}_1 + \i \widetilde{\alpha}_2) & 0
\end{bmatrix}
\begin{bmatrix}
A_\pm
\\
B_\pm
\end{bmatrix}
= \beta
\begin{bmatrix}
A_\pm
\\
B_\pm
\end{bmatrix} + \O(s).
\end{align*}
So, it is immediate to see that the macroscopic field $
[\tilde{u}_{1}, \tilde{u}_{2}]^T:=[A_\pm e^{\mathrm{i}\mkern1mu\tilde{\alpha}\cdot x}, B_\pm e^{\mathrm{i}\mkern1mu\tilde{\alpha}\cdot x}]^T$ satisfies the system of Dirac equations as follows:
$$
\lambda_0
\begin{bmatrix}
0 & (-c\mathrm{i}\mkern1mu)( \partial_1 - \mathrm{i}\mkern1mu \partial_2)
\\
(-\overline{c}\mathrm{i}\mkern1mu)( \partial_1 + \mathrm{i}\mkern1mu \partial_2) & 0
\end{bmatrix}
\begin{bmatrix}
\tilde{u}_{1}
\\
\tilde{u}_{2}
\end{bmatrix}
= \beta
\begin{bmatrix}
\tilde{u}_{1}
\\
\tilde{u}_{2}
\end{bmatrix}.
$$
Here, the superscript $T$ denotes the transpose and $\partial_i$ is the partial derivative with respect to the $i$th variable.
Note that the each component $\tilde{u}_j,j=1,2$, of the macroscopic field satisfies the Helmholtz equation
\begin{equation} \label{eq:hom}
\Delta \tilde{u}_j + \frac{\beta^2}{|c|^2\lambda_0^2} \tilde{u}_j = 0.
\end{equation}
Observe, in particular, that \eqref{eq:hom} describes a near-zero refractive index when $\beta$ is small.
The following is the main result on the homogenization theory for honeycomb lattices of subwavelength resonators \cite{nearzero}.
\begin{theorem} \label{thm:main}
For frequencies $\omega$ close to the Dirac frequency $\omega_*$, namely, $\omega-\omega_* = \beta \sqrt\delta$, the following asymptotic behaviour of the Bloch eigenfunction $u^{\alpha_*/r + \tilde{\alpha}}_r$ holds:
$$
u_{r}^{\alpha_*/r+\tilde{\alpha}}(x) = A e^{\mathrm{i}\mkern1mu\tilde\alpha \cdot x} S_{1} \left(\frac{x}{s}\right) + B e^{\mathrm{i}\mkern1mu\tilde\alpha \cdot x} S_{2} \left(\frac{x}{s}\right) + \O(s),
$$
where the macroscopic field $[\tilde{u}_{1}, \tilde{u}_{2}]^T:=[A e^{\mathrm{i}\mkern1mu\tilde{\alpha}\cdot x}, B e^{\mathrm{i}\mkern1mu\tilde{\alpha}\cdot x}]^T$ satisfies the two-dimensional Dirac equation
$$
\lambda_0
\begin{bmatrix}
0 & (-c\mathrm{i}\mkern1mu)( \partial_1 - \mathrm{i}\mkern1mu \partial_2)
\\
(-\overline{c}\mathrm{i}\mkern1mu)( \partial_1 + \mathrm{i}\mkern1mu \partial_2) & 0
\end{bmatrix}
\begin{bmatrix}
\tilde{u}_{1}
\\
\tilde{u}_{2}
\end{bmatrix}
= \frac{\omega-\omega_*}{\sqrt\delta}
\begin{bmatrix}
\tilde{u}_{1}
\\
\tilde{u}_{2}
\end{bmatrix},
$$
which can be considered as a homogenized equation for the honeycomb lattice of subwavelength resonators while the microscopic fields $S_1$ and $S_2$ vary on the scale of $r$.
\end{theorem}
Figure \ref{honeycomb} shows a one-dimensional plot along the $x_1-$axis of the real part of the {Bloch} {eigenfunction} of the honeycomb lattice shown over many unit cells.
\begin{figure}[h]
\begin{center} \includegraphics[scale=0.6]{h1D1.png}
\caption{Real part of the first Bloch eigenfunction of the honeycomb lattice shown over many unit cells. } \label{honeycomb} \end{center}
\end{figure}
\begin{remark} \label{rmk:alpha2}
Theorem \ref{thm:main} is valid around the Dirac point $\alpha_* = \alpha_1^*$. Around the other Dirac point, analogous arguments show that Theorem \ref{thm:main} is valid with all quantities instead defined using $\alpha_*=\alpha_2^*$ and the macroscopic field now satisfying
$$
\lambda_0
\begin{bmatrix}
0 & (-c\mathrm{i}\mkern1mu)( \partial_1 + \mathrm{i}\mkern1mu \partial_2)
\\
(-\overline{c}\mathrm{i}\mkern1mu)( \partial_1 - \mathrm{i}\mkern1mu \partial_2) & 0
\end{bmatrix}
\begin{bmatrix}
\tilde{u}_{1}
\\
\tilde{u}_{2}
\end{bmatrix}
= \frac{\omega-\omega_*}{\sqrt\delta}
\begin{bmatrix}
\tilde{u}_{1}
\\
\tilde{u}_{2}
\end{bmatrix}.
$$
\end{remark}
\subsection{Topologically protected edge modes}
A typical way to enable localized modes is to create a cavity inside a band gap structure. The idea is to make the frequency of the cavity mode fall within the band gap, whereby the mode will be localized to the cavity. However, localized modes created this way are highly sensitive to imperfections of the structure.
The principle that underpins the design of robust structures is that one is able to define topological invariants which capture the crystal's wave propagation properties. Then, if part of a crystalline structure is replaced with an arrangement that is associated with a different value of this invariant, not only will certain frequencies be localized to the interface but this behaviour will be stable with respect to imperfections. These eigenmodes are known as \emph{edge modes} and we say that they are \emph{topologically protected} to refer to their robustness.
\subsubsection{Sensitivity to geometric imperfections}
Subwavelength metamaterials can be used to achieve cavities of subwavelength dimensions.
The key idea is to perturb the size of a single subwavelength resonator inside the crystal, thus creating a defect mode. Observe that if we remove one resonator inside the bubbly crystal, we cannot create a defect mode. The defect created in this fashion is actually too small to support a resonant mode. In \cite{defectSIAM}, it is proved that by perturbing the radius of one resonator (see Figure \ref{fig:defect} where $D_\epsilon$ is the defect resonator) we create a detuned resonator with a resonant frequency that fall within the subwavelength band gap. Moreover, it is shown that the way to shift the frequency into the band gap depends on the crystal: in the dilute regime we have to decrease the defect resonator size while in the non-dilute regime we have to increase the size.
\begin{figure}[tb]
\centering
\begin{tikzpicture}[scale=1.6]
\centering
\draw[dashed] (0,0) circle (8pt) node{$D$};
\draw (0,0) circle (12pt) node[yshift=-20pt, xshift=15pt ]{$D_\epsilon$};
\draw (1,0) circle (8pt);
\draw (0,1) circle (8pt);
\draw (1,1) circle (8pt);
\draw (-1,0) circle (8pt);
\draw (0,-1) circle (8pt);
\draw (1,-1) circle (8pt);
\draw (-1,1) circle (8pt);
\draw (-1,-1) circle (8pt);
\draw (1.5,0) node{$\cdots$};
\draw (-1.5,0) node{$\cdots$};
\draw (0,1.5) node{$\vdots$};
\draw (0,-1.5) node{$\vdots$};
\end{tikzpicture}
\hspace{0.5cm}
{\includegraphics[width=7cm]{trapped3.png}}
\caption{Illustration of the defect crystal and the defect mode.} \label{fig:defect}
\end{figure}
In \cite{linedefect}, a waveguide is created by modifying the sizes of the resonators along a line in a dilute two-dimensional crystal, thereby creating a line defect.
It is proved that the line defect indeed acts as a waveguide; waves of certain frequencies will be localized to, and guided along, the line defect. This is depicted in Figure \ref{linedefectf}.
In wave localization due to a point defect, if the defect size is small the band structure of the defect problem will be a small perturbation of the band structure of the original problem. This way, it is possible to shift the defect band upwards, and a part of the defect band will fall into the subwavelength band gap. In \cite{linedefect} it is shown that for arbitrarily small defects, a part of the defect band will lie inside the band gap. Moreover, it is shown that for suitably large perturbation sizes, the entire defect band will fall into the band gap, and the size of the perturbation needed in order to achieve this can be explicitly quantified.
In order to have \textit{guided} waves along the line defect, the defect mode must not only be localized to the line, but also propagating along the line. In other words, we must exclude the case of standing waves in the line defect, \textit{i.e.}, modes which are localized in the direction of the line. Such modes are associated to a point spectrum of the perturbed operator which appears as a flat band in the dispersion relation. In \cite{linedefect}, it is shown that the defect band is nowhere flat, and hence does not correspond to bound modes in the direction of the line.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1.2]
\draw[dashed] (0,0) circle (10pt);
\draw (0,0) circle (6pt);
\draw[dashed] (-2,0) circle (10pt);
\draw[dashed] (-1,0) circle (10pt);
\draw[dashed] (2,0) circle (10pt);
\draw[dashed] (1,0) circle (10pt);
\draw (-2,0) circle (6pt);
\draw (-1,0) circle (6pt);
\draw (2,0) circle (6pt);
\draw (1,0) circle (6pt);
\draw (0,1) circle (10pt);
\draw (1,1) circle (10pt);
\draw (0,-1) circle (10pt);
\draw (1,-1) circle (10pt);
\draw (-1,1) circle (10pt);
\draw (-1,-1) circle (10pt);
\draw (2,1) circle (10pt);
\draw (2,-1) circle (10pt);
\draw (-2,1) circle (10pt);
\draw (-2,-1) circle (10pt);
\draw (2.65,0) node{$\cdots$};
\draw (-2.6,0) node{$\cdots$};
\draw (0,1.65) node{$\vdots$};
\draw (0,-1.5) node{$\vdots$};
\end{tikzpicture}
\hspace{0.5cm}
{\includegraphics[width=6.cm]{TwoD.png}}
\caption{Illustration of the line defect and the guided mode.} \label{linedefectf}
\end{figure}
One fundamental limitation of the above designs of subwavelength cavities and waveguides is that their properties are often very sensitive to imperfections in the crystal's structure. This is due, as illustrated in Figure \ref{defectotstable}, to the fact that the frequencies of the defect modes and guided waves are very close to the original band. In order to be able to feasibly manufacture wave-guiding devices, it is important that we are able to design subwavelength crystals that exhibit stability with respect to geometric errors.
\begin{figure}
\centering
\includegraphics[height=4.9cm]{Defect_Strong_Right} \hspace{0.25cm} \includegraphics[scale=0.38] {ndilutesmallzoom.png} \caption{Frequencies of the defect modes and guided waves.} \label{defectotstable} \end{figure}
\subsubsection{Robustness properties of one-dimensional chains of subwavelength resonators with respect to imperfections}
In the case of one-dimensional crystals such as a chain of subwavelength resonators, the natural choice of topological invariant is the Zak phase \cite{zak}. Qualitatively, a non-zero Zak phase means that the crystal has undergone \emph{band inversion}, meaning that at some point in the Brillouin zone the monopole/dipole nature of the first/second Bloch eigenmodes has swapped. In this way, the Zak phase captures the crystal's wave propagation properties. If one takes two chains of subwavelength resonators with different Zak phases and joins half of one chain to half of the other to form a new crystal, this crystal will exhibit a topologically protected edge mode at the interface, as illustrated in Figure \ref{zakf}.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}
\pgfmathsetmacro{\cubex}{4}
\pgfmathsetmacro{\cubey}{0.6}
\pgfmathsetmacro{\cubez}{0.6}
\draw [draw=blue, every edge/.append style={draw=blue, densely dashed, opacity=.5}, fill=blue!30!white]
(0,0,0) coordinate (o) -- ++(-\cubex,0,0) coordinate (a) -- ++(0,-\cubey,0) coordinate (b) edge coordinate [pos=1] (g) ++(0,0,-\cubez) -- ++(\cubex,0,0) coordinate (c) -- cycle
(o) -- ++(0,0,-\cubez) coordinate (d) -- ++(0,-\cubey,0) coordinate (e) edge (g) -- (c) -- cycle
(o) -- (a) -- ++(0,0,-\cubez) coordinate (f) edge (g) -- (d) -- cycle;
\begin{scope}[xshift=4cm]
\pgfmathsetmacro{\cubex}{4}
\pgfmathsetmacro{\cubey}{0.6}
\pgfmathsetmacro{\cubez}{0.6}
\draw [draw=red, every edge/.append style={draw=red, densely dashed, opacity=.5}, fill=red!30!white]
(0,0,0) coordinate (o) -- ++(-\cubex,0,0) coordinate (a) -- ++(0,-\cubey,0) coordinate (b) edge coordinate [pos=1] (g) ++(0,0,-\cubez) -- ++(\cubex,0,0) coordinate (c) -- cycle
(o) -- ++(0,0,-\cubez) coordinate (d) -- ++(0,-\cubey,0) coordinate (e) edge (g) -- (c) -- cycle
(o) -- (a) -- ++(0,0,-\cubez) coordinate (f) edge (g) -- (d) -- cycle;
\end{scope}
\draw[<-,out=80,in=170] (0.2,0.4) to (1,0.6);
\node at (1.65,0.6) {\scriptsize Interface};
\end{tikzpicture}
\end{center}
\caption{When two crystals with different values of the topological invariant are joined together, a protected edge mode exists at the interface.} \label{zakf}
\end{figure}
In \cite{ammari2019topological}, the bulk properties of an infinitely periodic chain of subwavelength resonator dimers are studied. Using Floquet-Bloch theory, the resonant frequencies and associated eigenmodes of this crystal are derived, and further a non-trivial band gap is proved. The analogous Zak phase takes different values for different geometries and in the \emph{dilute regime} (that is, when the distance between the resonators is an order of magnitude greater than their size) explicit expressions for its value are given. Guided by this knowledge of how the infinite (bulk) chains behave, a finite chain of resonator dimers that has a topologically protected edge mode is designed. This configuration takes inspiration from the bulk-boundary correspondence in the well-known Su-Schrieffer-Heeger (SSH) model \cite{SSH} by introducing an interface, on either side of which the resonator dimers can be associated with different Zak phases thus creating a topologically protected edge mode.
In order to present the main results obtained in \cite{ammari2019topological}, we first briefly review the topological nature of the Bloch eigenbundle. Observe that the Brillouin zone $Y^*$ has the topology of a circle. A natural question to ask, when considering the topological properties of a crystal, is whether properties are preserved after parallel transport around $Y^*$. In particular, a powerful quantity to study is the \textit{Berry-Simon connection} $A_n$, defined as
$$A_n(\alpha) := \mathrm{i}\mkern1mu \int_D u_n^\alpha \frac{\partial}{\partial \alpha} \overline{u_n^\alpha}\; \: \mathrm{d} x.$$
For any $\alpha_1,\alpha_2\in Y^*$, the parallel transport from $\alpha_1$ to $\alpha_2$ is $u_n^{\alpha_1}\mapsto e^{\mathrm{i}\mkern1mu \theta}u_n^{\alpha_2}$, where $\theta$ is given by
\begin{equation*}
\theta = \int_{\alpha_1}^{\alpha_2} A_n(\alpha) \; \: \mathrm{d} \alpha.
\end{equation*}
Thus, it is enlightening to introduce the so-called \textit{Zak phase}, $\varphi_n^z$, defined as
$$\varphi_n^{z} := \mathrm{i}\mkern1mu \int_{Y^*} \int_D u_n^\alpha \frac{\partial }{\partial \alpha} \overline{u_n^\alpha} \; \: \mathrm{d} x \, \: \mathrm{d} \alpha,$$
which corresponds to parallel transport around the whole of $Y^*$. When $\varphi_n^z$ takes a value that is not a multiple of $2\pi$, we see that the eigenmode has gained a non-zero phase after parallel transport around the circular domain $Y^*$. In this way, the Zak phase captures topological properties of the crystal. For crystals with inversion symmetry, the Zak phase is known to only attain the values $0$ or $\pi$ \cite{zak}.
Next, we study a periodic arrangement of subwavelength resonator dimers. This is an analogue of the SSH model. The goal is to derive a topological invariant which characterises the crystal's wave propagation properties and indicates when it supports topologically protected edge modes.
Assume we have a one-dimensional crystal in $\mathbb{R}^3$ with repeating unit cell $Y := [-\frac{L}{2}, \frac{L}{2}]\times \mathbb{R}^2$. Each unit cell contains a dimer surrounded by some background medium. Suppose the resonators together occupy the domain $D := D_1 \cup D_2$. We need two assumptions of symmetry for the analysis that follows. The first is that each individual resonator is symmetric in the sense that there exists some $x_1\in\mathbb{R}$ such that
\begin{equation} \label{resonator_symmetry}
R_1D_1 = D_1, \quad R_2D_2 = D_2,
\end{equation}
where $R_1$ and $R_2$ are the reflections in the planes $p_1=\{-x_1\}\times \mathbb{R}^2$ and $p_2=\{x_1\}\times \mathbb{R}^2$, respectively. We also assume that the dimer is symmetric in the sense that
\begin{equation} \label{dimer_symmetry}
D=-D.
\end{equation}
Denote the full crystal by $\mathcal{C}$, that is,
\begin{equation} \label{crystal_def}
\mathcal{C} := \bigcup_{m\in \mathbb{Z}} \left(D + (mL,0,0)\right).
\end{equation}
We denote the separation of the resonators within each unit cell, along the first coordinate axis, by $d := 2x_1$ and the separation across the boundary of the unit cell by $d' := L - d$. See Figure \ref{fig:SSH}.
\begin{figure}[tbh]
\centering
\begin{tikzpicture}[scale=2]
\begin{scope}
\draw[dashed, opacity=0.5] (-0.5,0.85) -- (-0.5,-1);
\draw[dashed, opacity=0.5] (1.8,0.85) -- (1.8,-1)node[yshift=4pt,xshift=-7pt]{};
\draw[{<[scale=1.5]}-{>[scale=1.5]}, opacity=0.5] (-0.5,-0.6) -- (1.8,-0.6) node[pos=0.5, yshift=-7pt,]{$L$};
\draw plot [smooth cycle] coordinates {(-0.1,0.1) (0.2,0) (0.5,0.1) (0.3,-0.4) (0.1,-0.4)} node[xshift=-13pt, yshift=10pt]{$D_1$};
\draw plot [smooth cycle] coordinates {(0.8,-0.1) (1.1,0) (1.4,-0.1) (1.2,0.4) (1,0.4)} node[xshift=25pt, yshift=-9pt]{$D_2$};
\draw[{<[scale=1.5]}-{>[scale=1.5]}, opacity=0.5] (0.2,0.6) -- (1.1,0.6) node[pos=0.5, yshift=-5pt,]{$d$};
\draw[dotted,opacity=0.5] (0.2,0.7) -- (0.2,-0.8) node[at end, yshift=-0.2cm]{$p_1$};
\draw[dotted,opacity=0.5] (1.1,0.7) -- (1.1,-0.8) node[at end, yshift=-0.2cm]{$p_2$};
\draw[{<[scale=1.5]}-{>[scale=1.5]}, opacity=0.5] (1.1,0.6) -- (2.5,0.6) node[pos=0.6, yshift=-5pt,]{$d'$};
\end{scope}
\begin{scope}[xshift=-2.3cm]
\draw plot [smooth cycle] coordinates {(-0.1,0.1) (0.2,0) (0.5,0.1) (0.3,-0.4) (0.1,-0.4)};
\draw plot [smooth cycle] coordinates {(0.8,-0.1) (1.1,0) (1.4,-0.1) (1.2,0.4) (1,0.4)};
\begin{scope}[xshift = 1.2cm]
\draw (-1.6,0) node{$\cdots$};
\end{scope};
\end{scope}
\begin{scope}[xshift=2.3cm]
\draw plot [smooth cycle] coordinates {(-0.1,0.1) (0.2,0) (0.5,0.1) (0.3,-0.4) (0.1,-0.4)};
\draw plot [smooth cycle] coordinates {(0.8,-0.1) (1.1,0) (1.4,-0.1) (1.2,0.4) (1,0.4)};
\draw[dotted] (0.2,0.7) -- (0.2,-0.8);
\begin{scope}[xshift = 1.1cm]
\end{scope}
\draw (1.7,0) node{$\cdots$};
\end{scope}
\begin{scope}[yshift=0.9cm]
\draw [decorate,opacity=0.5,decoration={brace,amplitude=10pt}]
(-0.5,0) -- (1.8,0) node [black,midway]{};
\node[opacity=0.5] at (0.67,0.35) {$Y$};
\end{scope}
\end{tikzpicture}
\caption{Example of a two-dimensional cross-section of a chain of subwavelength resonators satisfying the symmetry assumptions \eqref{resonator_symmetry}~and~\eqref{dimer_symmetry}. The repeating unit cell $Y$ contains the dimer $D_1 \cup D_2$.} \label{fig:SSH}
\end{figure}
Wave propagation inside the infinite periodic structure is modelled by the Helmholtz problem
\begin{equation} \label{eq:scatteringt}
\left\{
\begin{array} {ll}
\displaystyle \Delta {u}+ \omega^2 {u} = 0 & \text{in } \mathbb{R}^3 \setminus \partial \mathcal{C}, \\
\noalign{\smallskip}
\displaystyle {u}|_{+} -{u}|_{-} =0 & \text{on } \partial \mathcal{C}, \\
\noalign{\smallskip}
\displaystyle \delta \frac{\partial {u}}{\partial \nu} \bigg|_{+} - \frac{\partial {u}}{\partial \nu} \bigg|_{-} =0 & \text{on } \partial \mathcal{C}, \\
\noalign{\smallskip}
\displaystyle u(x_1,x_2,x_3) & \text{satisfies the outgoing radiation condition as } \sqrt{x_2^2+x_3^2} \rightarrow \infty.
\end{array}
\right.
\end{equation}
By applying the Floquet transform, the Bloch eigenmode $u_\alpha(x) := \mathcal{U}[u](x,\alpha)$ is the solution to the Helmholtz problem
\begin{equation} \label{eq:scattering_quasi}
\left\{
\begin{array} {ll}
\displaystyle \Delta u_\alpha+ \omega^2 {u_\alpha} = 0 &\text{in } \mathbb{R}^3 \setminus \partial \mathcal{C}, \\
\noalign{\smallskip}
\displaystyle {u_\alpha}|_{+} -{u_\alpha}|_{-} =0 & \text{on } \partial \mathcal{C}, \\
\noalign{\smallskip}
\displaystyle \delta \frac{\partial {u_\alpha}}{\partial \nu} \bigg|_{+} - \frac{\partial {u_\alpha}}{\partial \nu} \bigg|_{-} =0 & \text{on } \partial \mathcal{C}, \\
\noalign{\smallskip}
\displaystyle e^{-\mathrm{i}\mkern1mu \alpha_1 x_1} u_\alpha(x_1,x_2,x_3) \,\,\,& \mbox{is periodic in } x_1, \\
\noalign{\smallskip}
\displaystyle u_\alpha(x_1,x_2,x_3)& \text{satisfies the $\alpha$-quasi-periodic outgoing radiation condition} \\ &\hspace{0.5cm} \text{as } \sqrt{x_2^2+x_3^2} \rightarrow \infty.
\end{array}
\right.
\end{equation}
We formulate the quasi-periodic resonance problem as an integral equation.
Let $\mathcal{S}_{D}^{\alpha,\omega}$ be the single-layer potential associated to the three-dimensional Green's function which is quasi-periodic in one dimension,
$$G^{\alpha,k}(x,y) := -\sum_{m \in \mathbb{Z}} \frac{e^{\mathrm{i}\mkern1mu k|x-y-(Lm,0,0)|}}{4\pi|x-y-(Lm,0,0)|}e^{\mathrm{i}\mkern1mu \alpha Lm}.$$
The solution $u_\alpha$ of \eqref{eq:scattering_quasi} can be represented as
\begin{equation*} \label{eq:helm-solution_quasi}
u_\alpha = \mathcal{S}_{D}^{\alpha,\omega} [\Psi^\alpha],
\end{equation*}
for some density $\Psi^\alpha \in L^2(\partial D)$. Then, using the jump relations, it can be shown that~\eqref{eq:scattering_quasi} is equivalent to the boundary integral equation
\begin{equation} \label{eq:boundary_quasi}
\mathcal{A}^\alpha(\omega, \delta)[\Psi^\alpha] =0,
\end{equation}
where
\begin{equation} \label{eq:A_quasi_defn}
\mathcal{A}^\alpha(\omega, \delta) := -\lambda I + \left(\mathcal{K}_D^{ -\alpha,\omega}\right)^*, \quad \lambda := \frac{1+\delta}{2(1-\delta)}.
\end{equation}
Let $V_j^\alpha$ be the solution to
\begin{equation} \label{eq:V_quasi}
\begin{cases}
\displaystyle \Delta V_j^\alpha =0 \quad &\mbox{in } \quad Y\setminus \overline{D},\\
\displaystyle V_j^\alpha = \delta_{ij} \quad &\mbox{on } \quad \partial D_i,\\
\displaystyle V_j^\alpha(x+(mL,0,0))= e^{\mathrm{i}\mkern1mu \alpha m} V_j^\alpha(x) & \forall m \in \mathbb{Z}, \\
\displaystyle V_j^\alpha(x_1,x_2,x_3) = O\left(\tfrac{1}{\sqrt{x_2^2+x_3^2}}\right) \quad &\text{as } \sqrt{x_2^2+x_2^2}\to\infty, \text{ uniformly in } x_1,
\end{cases}
\end{equation}
where $\delta_{ij}$ is the Kronecker delta.
Analogously to \eqref{defcapal}, we then define the quasi-periodic capacitance matrix $C^\alpha=(C_{ij}^\alpha)$ by
\begin{equation} \label{eq:qp_capacitance}
C_{ij}^\alpha := \int_{Y\setminus \overline{D} }\overline{\nabla V_i^\alpha}\cdot\nabla V_j^\alpha \; \: \mathrm{d} x,\quad i,j=1, 2.
\end{equation}
Finding the eigenpairs of this matrix represents a leading order approximation to the differential problem \eqref{eq:scattering_quasi}. The following properties of $C^\alpha$ are useful.
\begin{lemma} \label{lem:quasi_matrix_form}
The matrix $C^\alpha$ is Hermitian with constant diagonal, \textit{i.e.}{},
$$C_{11}^\alpha = C_{22}^\alpha \in \mathbb{R}, \quad C_{12}^\alpha = \overline{C_{21}^\alpha} \in \mathbb{C}.$$
\end{lemma}
Since $C^\alpha$ is Hermitian, the following lemma follows directly.
\begin{lemma} \label{lem:evec}
The eigenvalues and corresponding eigenvectors of the quasi-periodic capacitance matrix are given by
\begin{align*}
\lambda_1^\alpha &= C_{11}^\alpha - \left|C_{12}^\alpha \right|, \qquad
\begin{pmatrix}
a_1 \\ b_1
\end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix}
- e^{\mathrm{i}\mkern1mu \theta_\alpha} \\ 1
\end{pmatrix}, \\
\lambda_2^\alpha &= C_{11}^\alpha + \left|C_{12}^\alpha \right|, \qquad
\begin{pmatrix}
a_2 \\ b_2
\end{pmatrix} = \frac{1}{\sqrt{2}}\begin{pmatrix}
e^{\mathrm{i}\mkern1mu \theta_\alpha} \\ 1
\end{pmatrix},
\end{align*}
where, for $\alpha$ such that $C_{12}^\alpha\neq0$, $\theta_\alpha\in[0,2\pi)$ is defined to be such that
\begin{equation}
e^{\mathrm{i}\mkern1mu \theta_\alpha} = \frac{C_{12}^\alpha}{|C_{12}^\alpha|}.
\end{equation}
\end{lemma}
In the dilute regime, we are able to compute asymptotic expansions for the band structure and topological properties. In this regime, we assume that the resonators can be obtained by rescaling fixed domains $B_1, B_2$ as follows:
\begin{equation}\label{eq:dilute}
D_1=\epsilon B_1 - \left(\frac{d}{2},0,0\right), \quad D_2=\epsilon B_2 + \left(\frac{d}{2},0,0\right),
\end{equation}
for some small parameter $\epsilon > 0$.
Let $\textrm{Cap}_{B}$ denote the capacity of $B = B_i$ for $i=1$ or $i=2$ (see \eqref{defcap} for the definition of the capacity). Due to symmetry, the capacitance is the same for the two choices $i =1, 2$. It is easy to see that, by a scaling argument,
\begin{equation}\label{eq:cap_scale}
\textrm{Cap}_{\epsilon B} = \epsilon \textrm{Cap}_B.
\end{equation}
\begin{lemma}\label{lem:cap_estim_quasi}
We assume that the resonators are in the dilute regime specified by \eqref{eq:dilute}. We also assume that $\alpha \neq 0$ is fixed. Then we have the following asymptotics of the capacitance matrix $C_{ij}^\alpha$ as $\epsilon\rightarrow 0$:
\begin{align}
C_{11}^\alpha &= \epsilon \mathrm{Cap}_B - \frac{(\epsilon \mathrm{Cap}_B)^2}{4\pi}\sum_{m \neq 0} \frac{e^{\mathrm{i}\mkern1mu m \alpha L}}{ |mL| } + \O(\epsilon^3), \label{eq:c1q}
\\
C_{12}^\alpha &= -\frac{(\epsilon \mathrm{Cap}_B)^2}{4\pi}\sum_{m =-\infty}^\infty \frac{e^{\mathrm{i}\mkern1mu m \alpha L} }{ |mL + d| } + \O(\epsilon^3). \label{eq:c2q}
\end{align}
Taking the imaginary part of \eqref{eq:c2q}, the corresponding asymptotic formula holds uniformly in $\alpha \in Y^*$.
\end{lemma}
Define normalized extensions of $V_j^\alpha$ as
$$S_j^\alpha(x) := \begin{cases} \frac{1}{\sqrt{|D_1|}}\delta_{ij} \quad &x \in D_i, \ i=1,2, \\ \noalign{\smallskip}
\frac{1}{\sqrt{|D_1|}}V_j^\alpha(x) \quad &x \in Y\setminus \overline{D}, \end{cases}$$ where $|D_1|$ is the volume of one of the resonators ($|D_1|=|D_2|$ thanks to the dimer's symmetry \eqref{dimer_symmetry}). The following two approximation results hold.
\begin{theorem} \label{thm:char_approx_infinite}
The characteristic values $\omega_j^\alpha=\omega_j^\alpha(\delta),~j=1,2$, of the operator $\mathcal{A}^{\alpha}(\omega,\delta)$, defined in \eqref{eq:A_quasi_defn}, can be approximated as
$$ \omega_j^\alpha= \sqrt{\frac{\delta \lambda_j^\alpha }{|D_1|}} + \O(\delta),$$
where $\lambda_j^\alpha,~j=1,2$, are eigenvalues of the quasi-periodic capacitance matrix $C^\alpha$.
\end{theorem}
\begin{theorem} \label{thm:mode_approx}
The Bloch eigenmodes $u_j^\alpha,~j=1,2$, corresponding to the resonances $\omega_j^\alpha$, can be approximated as
$$ u_j^\alpha(x) = a_j S^\alpha_1(x) + b_j S^\alpha_2(x) + \O(\delta),$$
where $\left(\begin{smallmatrix}a_j\\b_j\end{smallmatrix}\right),~j=1,2,$
are the eigenvectors of the quasi-periodic capacitance matrix $C^\alpha$, as given by \Cref{lem:evec}
\end{theorem}
Theorems~\ref{thm:char_approx_infinite} and \ref{thm:mode_approx} show that the capacitance matrix can be considered to be a discrete approximation of the differential problem \eqref{eq:scattering_quasi}, since its eigenpairs directly determine the resonant frequencies and the Bloch eigenmodes (at leading order in $\delta$).
\begin{figure}[tb]
\centering
\begin{tikzpicture}[scale=2]
\begin{scope}
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\draw[opacity=0.5, dashed] (0.65,1) -- (0.65,-1);
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\draw[opacity=0.5, dotted] (0.075,0.9) -- (0.075,-0.9)node[left]{$q_1$};
\draw[opacity=0.5, dotted] (1.1,0.9) node[left]{$p_2$} -- (1.1,-0.9);
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\draw[<->, opacity=0.5] (0.2,0.6) -- (1.1,0.6) node[pos=0.65, yshift=-5pt,]{$d$};
\end{scope}
\draw (2.65,0) node{$\xrightarrow[\mathcal{R}_2]{\mathcal{R}_1}$};
\begin{scope}[xshift=4cm]
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\draw[opacity=0.5] (1.8,1) -- (1.8,-1);
\draw plot [smooth cycle] coordinates {(-0.35,0.1) (-0.05,0) (0.25,0.1) (0.05,-0.4) (-0.15,-0.4)} node[xshift=25pt, yshift=10pt]{$D_1'$};
\draw plot [smooth cycle] coordinates {(1.05,-0.1) (1.35,0) (1.65,-0.1) (1.45,0.4) (1.25,0.4)} node[xshift=-15pt, yshift=-9pt]{$D_2'$};
\draw[<->, opacity=0.5] (-0.05,0.6) -- (1.35,0.6) node[pos=0.5, yshift=-5pt]{$d'$};
\draw[opacity=0.5, dotted] (1.35,0.9) node[left]{$p_2'$} -- (1.35,-0.9);
\draw[opacity=0.5, dotted] (-0.05,0.9) node[right]{$p_1'$} -- (-0.05,-0.9);
\end{scope}
\begin{scope}[yshift=1.05cm]
\draw [decorate,opacity=0.5,decoration={brace,amplitude=10pt}]
(-0.5,0) -- (0.64,0) node [black,midway]{};
\draw [decorate,opacity=0.5,decoration={brace,amplitude=10pt}]
(0.66,0) -- (1.8,0) node [black,midway]{};
\node[opacity=0.5] at (0.1,0.3) {$Y_1$};
\node[opacity=0.5] at (1.26,0.3) {$Y_2$};
\end{scope}
\begin{scope}[xshift=4cm,yshift=1.05cm]
\draw [decorate,opacity=0.5,decoration={brace,amplitude=10pt}]
(-0.5,0) -- (1.8,0) node [black,midway]{};
\node[opacity=0.5] at (0.67,0.3) {$Y'$};
\end{scope}
\end{tikzpicture}
\caption{Reflections taking $D$ to $D'$.} \label{fig:YY'}
\end{figure}
We now introduce notation which, thanks to the assumed symmetry of the resonators, will allow us to prove topological properties of the chain. Divide $Y$ into two subsets $Y=Y_1\cup Y_2$, where $Y_1 := [-\frac{L}{2},0]\times \mathbb{R}^2$ and let $Y_2 := [0,\frac{L}{2}]\times \mathbb{R}^2$, as depicted in \Cref{fig:YY'}. Define $q_1$ and $q_2$ to be the central planes of $Y_1$ and $Y_2$, that is, the planes $q_1 := \{ -\frac{L}{4}\} \times \mathbb{R}^2$ and $q_2 := \{ \frac{L}{4}\} \times \mathbb{R}^2$. Let $\mathcal{R}_1$ and $\mathcal{R}_2$ be reflections in the respective planes. Observe that, thanks to the assumed symmetry of each resonator \eqref{resonator_symmetry}, the ``complementary'' dimer $D' = D_1' \cup D_2'$, given by swapping $d$ and $d'$, satisfies $D_i' = \mathcal{R}_i D_i$ for $i=1,2$.
Define the operator $T_\alpha$ on the set of $\alpha$-quasi-periodic functions $f$ on $Y$ as
$$T_\alpha f(x) := \begin{cases} e^{-\mathrm{i}\mkern1mu \alpha L}\overline{f(\mathcal{R}_1x)}, \quad &x\in Y_1, \\ \overline{f(\mathcal{R}_2x)}, &x\in Y_2, \end{cases}$$
where the factor $e^{-\mathrm{i}\mkern1mu \alpha L}$ is chosen so that the image of a continuous ($\alpha$-quasi-periodic) function is continuous.
We now proceed to use $T_\alpha$ to analyse the different topological properties of the two dimer configurations. Define the quantity ${C_{12}^{\alpha}}'$ analogously to $C_{12}^\alpha$ but on the dimer $D'$, that is, to be the top-right element of the corresponding quasi-periodic capacitance matrix, defined in \eqref{eq:qp_capacitance}.
\begin{lemma}\label{lem:cc'}
We have
\begin{equation*}\label{eq:cc'}
{C_{12}^{\alpha}}' = e^{-\mathrm{i}\mkern1mu \alpha L} \overline{C_{12}^\alpha}.\end{equation*}
Consequently, if $d = d' = \frac{L}{2}$ then $C_{12}^{\pi/L} = 0$.
\end{lemma}
\begin{lemma}\label{lem:c=0}
We assume that $D$ is in the dilute regime specified by \eqref{eq:dilute}. Then, for $\epsilon$ small enough,
\begin{itemize}
\item[(i)] $\mathrm{Im}\ C_{12}^\alpha > 0$ for $0<\alpha<\pi/L$ and $\mathrm{Im}\ C_{12}^\alpha < 0$ for $-\pi/L<\alpha<0$. In particular, $\mathrm{Im}\ {C_{12}^{\alpha}}$ is zero if and only if $\alpha \in\{ 0, \pi/L \}$.
\item [(ii)] $C_{12}^{\alpha}$ is zero if and only if both $d = d'$ and $\alpha = \pi/L$.
\item [(iii)] $C_{12}^{\pi/L} < 0$ when $d<d'$ and $C_{12}^{\pi/L} > 0$ when $d>d'$. In both cases we have $C_{12}^{0} < 0$.
\end{itemize}
\end{lemma}
This lemma describes the crucial properties of the behaviour of the curve $\{C_{12}^\alpha:\alpha\in Y^*\}$ in the complex plane. The periodic nature of $Y^*$ means that this is a closed curve. Part (i) tells us that this curve crosses the real axis in precisely two points. Taken together with (iii), we know that this curve winds around the origin in the case $d>d'$, but not in the case $d<d'$. The following band gap result is from \cite{ammari2020robust}.
\begin{theorem} \label{thm:band_gap}
If $d\neq d'$, the first and second bands form a band gap:
$$\max_{\alpha \in Y^*} \omega_1^\alpha < \min_{\alpha \in Y^*} \omega_2^\alpha,$$
for small enough $\epsilon$ and $\delta$.
\end{theorem}
Combining the above results, we obtain the following result concerning the band inversion that takes place between the two geometric regimes $d<d'$ and $d>d'$ as illustrated in Figure \ref{bandinvf}.
\begin{proposition} \label{prop:bandinv}
For $\epsilon$ small enough, the band structure at $\alpha = \pi/L$ is inverted between the $d<d'$ and $d>d'$ regimes. In other words, the eigenfunctions associated with the first and second bands at $\alpha = \pi/L$ are given, respectively, by
\begin{equation*}
u_1^{\pi/L}(x) = S_1^{\pi/L}(x)+S_2^{\pi/L}(x)+ \O(\delta), \quad u_2^{\pi/L}(x) = S_1^{\pi/L}(x)-S_2^{\pi/L}(x)+ \O(\delta),
\end{equation*}
when $d<d'$ and by
\begin{equation*}
u_1^{\pi/L}(x) = S_1^{\pi/L}(x)-S_2^{\pi/L}(x)+ \O(\delta), \quad u_2^{\pi/L}(x) = S_1^{\pi/L}(x)+S_2^{\pi/L}(x)+ \O(\delta),
\end{equation*}
when $d>d'$.
\end{proposition}
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1.4]
\draw [decorate,decoration={brace,amplitude=10pt}]
(-0.5,0) -- (3.5,0) node [red,midway]{};
\node at (1.5,0.6) {$d<d'$};
\begin{scope}[xshift=5.3cm]
\draw [decorate,decoration={brace,amplitude=10pt}]
(-0.5,0) -- (3.5,0) node [red,midway]{};
\node at (1.5,0.6) {$d>d'$};
\end{scope}
\node at (0.1,-1.8) {\includegraphics[trim={0.65cm 0 2.6cm 0},clip,height=3.5cm]{Triv1.png}};
\node at (2.6,-1.8) {\includegraphics[trim={0.65cm 0 2.6cm 0},clip,height=3.5cm]{Triv2.png}};
\begin{scope}[xshift=5.4cm]
\node at (0,-1.8) {\includegraphics[trim={0.65cm 0 2.6cm 0},clip,height=3.5cm]{Nontriv1.png}};
\node at (2.8,-1.8) {\includegraphics[trim={0.65cm 0 0 0},clip,height=3.5cm]{Nontriv2.png}};
\end{scope}
\node at (0.9,-0.4) {\color{white}\small $u_1^{\pi/L}$};
\node at (3.4,-0.4) {\color{white}\small $u_2^{\pi/L}$};
\begin{scope}[xshift=5.3cm]
\node at (0.9,-0.4) {\color{white}\small $u_1^{\pi/L}$};
\node at (3.4,-0.4) {\color{white}\small $u_2^{\pi/L}$};
\end{scope}
\end{tikzpicture}
\caption{Band inversion: the monopole/dipole natures of the 1\textsuperscript{st} and 2\textsuperscript{nd} eigenmodes have swapped between the $d<d'$ and $d>d'$ regimes.}
\label{bandinvf}
\end{figure}
The eigenmode $S_1^{\pi/L}(x)+S_2^{\pi/L}(x)$ is constant and attains the same value on both resonators, while the eigenmode $S_1^{\pi/L}(x)-S_2^{\pi/L}(x)$ has values of opposite sign on the two resonators. They therefore correspond, respectively, to monopole and dipole modes, and \Cref{prop:bandinv} shows that the monopole/dipole nature of the first two Bloch eigenmodes are swapped between the two regimes. We will now proceed to define a topological invariant which we will use to characterise the topology of a chain and prove how its value depends on the relative sizes of $d$ and $d'$. This invariant is intimately connected with the band inversion phenomenon and is non-trivial only if $d>d'$ \cite{ammari2019topological}.
\begin{theorem} \label{thm:phase}
We assume that $D$ is in the dilute regime specified by \eqref{eq:dilute}. Then the Zak phase $\varphi_j^z, j = 1,2$, defined by
$$\varphi_j^z := \mathrm{i}\mkern1mu \int_{Y^*} \int_D u_j^\alpha \frac{\partial }{\partial \alpha} \overline{u_j^\alpha} \; \: \mathrm{d} x \, \: \mathrm{d} \alpha,$$
satisfies
$$ \varphi_j^z = \begin{cases}
0, \quad &\text{if} \ \ d < d', \\
\pi, \quad &\text{if} \ \ d > d',
\end{cases}$$
for $\epsilon$ and $\delta$ small enough.
\end{theorem}
Theorem \ref{thm:phase} shows that the Zak phase of the crystal is non-zero precisely when $d > d'$. The bulk-boundary correspondence suggests that we can create topologically protected subwavelength edge modes by joining half-space subwavelength crystals, one with $\varphi_j^z = 0$ and the other with $\varphi_j^z = \pi$.
\begin{remark}
A second approach to creating chains with robust subwavelength localized modes is to
start with a one-dimensional array of pairs of subwavelength resonators that exhibits a {subwavelength band gap}. We then introduce a defect by adding a dislocation within one of the resonator pairs. As shown in \cite{ammari2020robust}, as a result of this dislocation, mid-gap frequencies enter the band gap from either side and converge to a single frequency, within the band gap, as the dislocation becomes arbitrarily large.
Such frequency can place localized modes at any point within the band gap and corresponds to a robust {edge modes}.
\end{remark}
\section{Mimicking the cochlea with an array of graded subwavelength resonators}
\label{sec7}
In \cite{davies2019fully} an array of subwavelength resonators is used to design a to-scale artificial cochlea that mimics the first stage of human auditory processing and present a rigorous analysis of its properties. In order to replicate the spatial frequency separation of the cochlea, the array should have a size gradient, meaning each resonator is slightly larger than the previous, as depicted in Fig.~\ref{fig:geom}. The size gradient is chosen so that the resonator array mimics the spatial frequency separation performed by the cochlea. In particular, the structure can reproduce the well-known (tonotopic) relationship between incident frequency and position of maximum excitation in the cochlea. This is a consequence of the asymmetry of the eigenmodes $u_n(x)$, see \cite{davies2019fully} and \cite{davies2020hopf} for details.
\begin{figure}[h]
\centering
\includegraphics[width=11.4cm]{array_diagram.pdf}
\caption{A graded array of subwavelength resonators mimics the biomechanical properties of the cochlea in response to a sound wave. } \label{fig:geom}
\end{figure}
Such graded arrays of subwavelength resonators can mimic the biomechanical properties of the cochlea, at the same scale. In \cite{perception}, a modal time-domain expansion for the scattered pressure field due to such a structure is derived from first principles. It is proposed there that these modes should form the basis of a signal processing architecture. The properties of such an approach is investigated and it is shown that higher-order gammatone filters appear by cascading. Further, an approach for extracting meaningful global properties from the coefficients, tailored to the statistical properties of so-called natural sounds is proposed.
The subwavelength resonant frequencies of an array of $N=22$ resonators computed by using the formulation \eqref{eq-boundary}--\eqref{page450} are shown in Fig.~\ref{fig:spectrum}. This array measures 35~mm, has material parameters corresponding to air-filled resonators surrounded by water and has subwavelength resonant frequencies within the range 500~Hz~--~10~kHz. Thus, this structure has similar dimensions to the human cochlea, is made from realistic materials and experiences subwavelength resonance in response to frequencies that are audible to humans.
\begin{figure}[h]
\centering
\includegraphics[width=0.8\linewidth]{res_spec}
\caption{The resonant frequencies $\{\omega_n:n=1,\dots,N\}\subset\mathbb{C}$ lie in the right-hand complex plane, shown for an array of $N=22$ subwavelength resonators. The Helmholtz problem also has singularities in the left-hand plane, which are symmetric in the imaginary axis. The imaginary parts are all negative, due to energy losses. } \label{fig:spectrum}
\end{figure}
This analysis is useful not only for designing cochlea-like devices, but is also used in \cite{perception} as the basis for a machine hearing procedure which mimics neural processing in the auditory system. Consider the scattering of a signal, $s:[0,T]\to\mathbb{R}$, whose frequency support is wider than a single frequency and whose Fourier transform exists. Consider the Fourier transform of the incoming pressure wave, given for $\omega\in\mathbb{C}$ and $x\in\mathbb{R}^3$ by
\begin{align*}
u^{in}(x,\omega)&=\int_{-\infty}^{\infty} s(x_1/v-t) e^{\i\omega t}\: \mathrm{d} t\\
&=e^{\i\omega x_1/v}\hat{s}(\omega) = \hat{s}(\omega)+ \O(\omega),
\end{align*}
where $\hat{s}(\omega):=\int_{-\infty}^{\infty} s(-u) e^{\i\omega u}\: \mathrm{d} u$. In \Cref{def:res^}, we defined resonant frequencies as having positive real parts. However, the scattering problem \eqref{eq:scattering} is known to be symmetric in the sense that if it has a pole at $\omega\in\mathbb{C}$ then it has a pole with the same multiplicity at $-\overline{\omega}$ \cite{dyatlov2019mathematical}. As depicted in \Cref{fig:spectrum}, this means the resonant spectrum is symmetric in the imaginary axis.
Suppose that the scattered acoustic pressure field $u$ in response to the Fourier transformed signal $\hat s$ can, for $x\in\partial D$, be decomposed as
\begin{equation} \label{eq:gen_modal_decomp}
u(x,\omega
= \sum_{n=1}^N \frac{-\hat{s}(\omega)\nu_n\Re(\omega_n^+)^2}{(\omega-\omega_n)(\omega+ \overline{\omega_n})} u_n(x)
+ r(x,\omega),
\end{equation}
for some remainder $r$. We are interested in signals whose energy is mostly concentrated within the subwavelength regime. In particular, we want that
\begin{equation} \label{eq:subw_regime}
\sup_{x\in\mathbb{R}^3}\int_{-\infty}^\infty |r(x,\omega) | \: \mathrm{d}\omega = \O(\delta).
\end{equation}
Then, under the assumptions \eqref{eq:gen_modal_decomp} and \eqref{eq:subw_regime}, we can apply the inverse Fourier transform \cite{perception} to find that the scattered pressure field satisfies, for $x\in\partial D$ and $t\in\mathbb{R}$,
\begin{equation} \label{thm:timedom}
p(x,t)= \sum_{n=1}^N a_n[s](t) u_n(x) + \O(\delta),
\end{equation}
where the coefficients are given by the convolutions $a_n[s](t)=\left( s*h_n \right)(t)$ with the kernels
\begin{equation} \label{eq:hdef}
h_n(t)=
\begin{cases}
0, & t<0, \\
c_n e^{\Im(\omega_n)t} \sin(\Re(\omega_n)t), & t\geq0,
\end{cases}
\end{equation}
for $c_n=\nu_n\Re(\omega_n)$.
\begin{remark}
The assumption \eqref{eq:subw_regime} is a little difficult to interpret physically. For the purposes of informing signal processing approaches, however, it is sufficient.
\end{remark}
\begin{figure}
\centering
\includegraphics[width=0.8\linewidth]{hat_phi_n.png}
\caption{The frequency support of the band-pass filters $h_n$ created by an array of 22 subwavelength resonators. } \label{fig:bandpass}
\end{figure}
On the one hand, note that the fact that $h_n(t)=0$ for $t<0$ ensures the causality of the modal expansion in \eqref{thm:timedom}.
Moreover, as shown in Figure~\ref{fig:bandpass}, $h_n$ is a windowed oscillatory mode that acts as a band-pass filter centred at $\Re(\omega_n)$. On the other hand, the asymmetry of the spatial eigenmodes $u_n(x)$ means that the decomposition from \eqref{thm:timedom} replicates the cochlea's travelling wave behaviour. That is, in response to an impulse the position of maximum amplitude moves slowly from left to right in the array, see \cite{davies2019fully} for details. In \cite{perception}, a signal processing architecture is developed, based on the convolutional structure of \eqref{thm:timedom}. This further mimics the action of biological auditory processing by extracting global properties of behaviourally significant sounds, to which human hearing is known to be adapted.
Finally, it is worth mentioning that biological hearing is an inherently nonlinear process. In \cite{davies2020hopf} nonlinear amplification is introduced to the model in order to replicate the behaviour of the cochlear amplifier. This active structure takes the form of a fluid-coupled array of Hopf resonators. Clarifying the details of the nonlinearities that underpin cochlear function remains the largest open question in understanding biological hearing. One of the motivations for developing devices such as the one analysed here is that it will allow for the investigation of these mechanisms, which is particularly difficult to do on biological cochleas.
\section{Concluding remarks}
In this review, recent mathematical results on focusing, trapping, and guiding waves at subwavelength scales have been described in the Hermitian case. Systems of subwavelength resonators that exhibit topologically protected edge modes or that can mimic the biomechanical properties of the cochlea have been designed. A variety of mathematical tools for solving wave propagation problems at subwavelength scales have been introduced.
When sources of energy gain and loss are introduced to a wave-scattering system, the underlying mathematical formulation will be non-Hermitian. This paves the way for
new ways to control waves at subwavelength scales \cite{nh1,nh2,miri2019exceptional}.
In \cite{high-order,ammari2020exceptional}, the existence of asymptotic exceptional points, where eigenvalues coincide and eigenmodes are linearly dependent at leading order, in a parity--time-symmetric pair of subwavelength resonators is proved. Systems exhibiting exceptional points can be used for sensitivity enhancement. Moreover, a structure which exhibits asymptotic unidirectional reflectionless transmission at certain frequencies is designed. In \cite{active}, the phenomenon of topologically protected edge states in systems of subwavelength resonators with gain and loss is studied. It is demonstrated that localized edge modes appear in a periodic structure of subwavelength resonators with a defect in the gain/loss distribution, and the corresponding frequencies and decay lengths are explicitly computed. Similarly to the Hermitian case, these edge modes can be attributed to the winding of the eigenmodes. In the non-Hermitian case the topological invariants fail to be quantized, but can nevertheless predict the existence of localized edge modes.
The codes used for the numerical illustrations of the results described in this review can be downloaded at \url{http://www.sam.math.ethz.ch/~hammari/SWR.zip}.
\bibliographystyle{abbrv}
|
1,108,101,565,647 | arxiv |
\section{Centralised Sparse Representation}\label{sec:csr}
\vsp
A different approach to take into account self similarities in sparse models is
the CSR approach of \cite{dong2012nonlocally}. This approach
is easier to turn into a differentiable algorithm than the LSSC method, but we have
empirically observed that it does not perform as well. Nevertheless, we believe it to be
conceptually interesting, and we provide a brief description below.
The idea consists of regularizing each code $\alphab_i$ with the function
\begin{equation}
\label{eq:csreq}
\Psi_i(\alphab_i) = \|\alphab_i\|_1 + \gamma \|\alphab_i - \betab_i\|_1,
\end{equation}
where $\betab_i$ is obtained by a weighted average of prevous codes.
Specifically, given some codes~$\alphab_i^{(k)}$ obtained at iteration~$k$ and a similarity matrix~$\Sigmab$, we compute
\begin{equation}
\betab_i^{(k)} = \sum_{j} \frac{\Sigmab_{ij}}{\sum_{l} \Sigmab_{il}}\alphab_j^{(k)},
\end{equation}
and the weights $\betab_i^{(k)}$ are used in~(\ref{eq:csreq}) in order to compute the codes $\alphab_i^{(k+1)}$.
Note that the original CSR method of~\cite{dong2012nonlocally} uses similarities of the form
$\Sigmab_{ij} = \exp{\left(-\frac{1}{2\sigma^2} \| \Wb\alphab_i-\Wb\alphab_j\|_2^2 \right)} $, but other similarities functions may be used.
Even though~\cite{dong2012nonlocally} does not use a proximal gradient descent
method to solve the problem regularized with~(\ref{eq:csreq}), the next proposition shows
that it admits a closed form, which is a key to turn CSR into a differentiable algorithm.
To the best of our knowledge,
this expression is new; its proof is given in the~appendix.
\begin{proposition}[Proximal operator of the CSR penalty]\label{csr_prox}
Consider $\Psi_i$ defined in~(\ref{eq:csreq}). Then, for all $\ub$ in~$\Real^p$,
\begin{equation*}
\prox_{\lambda \Psi_i}[\ub] = S_\lambda \big( S_{\lambda \gamma} \left (\ub - \betab_i - \lambda \sign(\betab_i) \right ) \\ +\betab_i + \lambda\sign(\betab_i) \big ),
\end{equation*}
where $S_\lambda$ is the soft-thresholding operator, see Figure~\ref{fig:prox_csr}.
\end{proposition}
\noindent
\begin{minipage}{0.56\textwidth}
Despite the apparent complexity of the formula, it remains a continuous function of the input and is differentiable almost everywhere, hence compatible with end-to-end training.
%
Qualitatively, the shape of the proximal mapping has a simple interpretation. It pulls codes either to zero, or to the code weighted average $\betab_i$.
\end{minipage}
\hfill
\begin{minipage}{0.42\textwidth}
\centering\raisebox{\dimexpr \topskip-\height}{%
\includegraphics[width=\textwidth]{figures/fig_prox_c.png}}
\captionof{figure}{$ \prox_{\lambda \Psi_i}$ for various $\lambda, \gamma, \beta$}\label{fig:prox_csr}
\end{minipage}
At each iteration, the similarity matrix is updated along with the codes $\betab_i$. The proximal operator can then easily be plugged into our framework.
We reported performance of the CSR approach in the main paper for grayscale denoising, color denoising and demosaicking.
Performance of the CSR approach are reported in Tables \ref{colour_table_csr}, \ref{gray_table_csr}, \ref{mosaic_table_csr}.
We observe that it performs significantly
better than the baseline SC but is not as effective as GroupSC overall.
\begin{table}[htb]
\footnotesize
\centering
\caption{ \textbf{Color denoising} on CBSD68, training on CBSD400 for all methods except CSCnet (Waterloo+CBSD400).
Performance is measured in terms of average PSNR. SSIMs are reported in the appendix.}\label{colour_table_csr}
\begin{tabular}{@{}lccccccccc@{}}
\toprule
\multirow{2}{*}{Method} & \multirow{2}{*}{Trainable } & \multirow{2}{*}{Params} & \multicolumn{4}{c}{Noise level ($\sigma$)} \\
& & & 5 & 10 & 15 & 25 & 30 & 50 \\
\midrule
CBM3D \cite{dabov2007image} & \xmark & - & 40.24 & - & 33.49 & 30.68 & - & 27.36 \\
\midrule
CSCnet \cite{simon2019rethinking} & & 186k & - & - & 33.83 & 31.18 & - & 28.00 \\
CNLNet\cite{lefkimmiatis2017non} & & - & - & - & 33.69 & 30.96 & - & 27.64 \\
FFDNET \cite{zhang2018ffdnet} & & 486k & - & - & 33.87 & 31.21 & - & 27.96 \\
CDnCNN \cite{zhang2017beyond} & & 668k & 40.50 & 36.31 & 33.99 & 31.31 & - & 28.01 \\
RNAN \cite{zhang2019rnan} & & 8.96M & - & \textbf{36.60} & - & - & \textbf{30.73} & \textbf{28.35} \\
\midrule
{SC} (baseline) & & {119k} & {40.44} & - & {33.75} & 30.94 & - & 27.39 \\
{CSR} (ours) & & {119k} & {40.53} & - & {34.05} & {31.33} & - & {28.01} \\
{GroupSC} (ours) & & {119k} & \underline{40.58} & \underline{36.40} & \underline{34.11} & \underline{31.44} & \underline{30.58} & \underline{28.05} \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[htb]
\small
\centering
\caption{\textbf{Grayscale Denoising} on BSD68, training on BSD400 for all methods except CSCnet (Waterloo+BSD400).
Performance is measured in terms of average PSNR.
SSIMs are reported in the appendix.}
\label{gray_table_csr}
\begin{tabular}{@{}lcccccc@{}}
\toprule
\multirow{2}{*}{Method} & \multirow{2}{*}{Trainable} & \multirow{2}{*}{Params} & \multicolumn{4}{c}{Noise Level ($\sigma$)} \\
& & & 5 & 15 & 25 & 50 \\
\midrule
BM3D \cite{dabov2007image} & \xmark & - & 37.57 & 31.07 & 28.57 & 25.62 \\
LSSC \cite{mairal2009non} & \xmark & - & 37.70 & 31.28 & 28.71 & 25.72 \\
BM3D PCA \cite{dabov2009bm3d} & \xmark & - & 37.77 & 31.38 & 28.82 & 25.80 \\
\midrule
TNRD \cite{chen2016trainable} & & - & - & 31.42 & 28.92 & 25.97 \\
CSCnet \cite{simon2019rethinking} & & 62k & 37.84 & 31.57 & { 29.11} & 26.24 \\
CSCnet(BSD400) \cite{simon2019rethinking}\footnotemark[2] & & 62k & 37.69 & 31.40 & { 28.93} & 26.04 \\
LKSVD~\cite{scetbon2019deep} & & 45K & - & 31.54 & 29.07 & 26.13 \\
NLNet \cite{lefkimmiatis2017non} & & - & - & 31.52 & 29.03 & 26.07 \\
FFDNet \cite{zhang2018ffdnet} & & 486k & - & 31.63 & 29.19 & 26.29 \\
DnCNN \cite{zhang2017beyond} & & 556k & {37.68} & \underline{31.73} & 29.22 & 26.23 \\
N3 \cite{plotz2018neural} & & 706k & - & - & \underline{29.30} & \underline{26.39} \\
NLRN \cite{liu2018non} & & 330k & \underline{37.92} & \textbf{ 31.88} & \textbf{ 29.41} & \textbf{ 26.47} \\
\midrule
{SC} (baseline) & & {68k} & 37.84 & 31.46 & 28.90 & 25.84 \\
{CSR} (ours) & & {68k} & 37.88 & 31.64 & 29.16 & 26.08 \\
{GroupSC} (ours) & & {68k} & \textbf{37.95} & 31.71 & 29.20 & 26.17 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[htb]
\centering
\footnotesize
\caption{\textbf{Demosaicking.} Training on CBSD400 unless a larger dataset is specified between parenthesis.
Performance is measured in terms of average PSNR. SSIMs are reported in the appendix.} \label{mosaic_table_csr}
\begin{tabular}{@{}lccccc@{}}
\toprule
Method & Trainable & Params & Kodak24 & BSD68 & Urban100 \\
\midrule
LSSC & \xmark & - & 41.39 & 40.44 & 36.63 \\
\midrule
IRCNN \cite{zhang2017learning} (BSD400+Waterloo \cite{ma2016waterloo}) & & - & 40.54 & 39.9 & 36.64 \\
Kokinos \cite{kokkinos2018deep} (MIT dataset \cite{gharbi2016deep}) & & 380k & 41.5 & - & - \\
MMNet \cite{kokkinos2019iterative} (MIT dataset \cite{gharbi2016deep}) & & 380k & 42.0 & - & - \\
RNAN \cite{zhang2019rnan} & & {8.96M} & \textbf{42.86} & \underline{42.61} & - \\
\midrule
{SC} (ours) & & 119k & 42.34 & 41.88 & 37.50 \\
CSR (ours) & & 119k & 42.25 & - & - \\
{GroupSC} (ours) & & {119k} & \underline{42.71} & \textbf{42.91} & \textbf{38.21} \\
\bottomrule
\end{tabular}
\end{table}
\section{Implementation Details and Reproducibility}\label{sec:impl}
\paragraph{Training details.}
During training, we randomly extract patches $56\times56$ whose size equals the window size used for computing non-local self-similarities. We apply a mild data augmentation (random rotation by
$90^\circ$ and horizontal flips). We optimize the parameters of our models using ADAM~\cite{kingma2014adam} with a minibatch size of $32$.
All the models are trained for 300 epochs for denoising and demosaicking. The learning rate is set to $6 \times 10^{-4}$ at initialization and is sequentially lowered during training
by a factor of 0.35 every 80 training steps, in the same way for all experiments. Similar to \cite{simon2019rethinking}, we normalize the initial dictionary~$\Db_0$ by its largest singular value, which helps the LISTA
algorithm to converge faster. We initialize the matrices $\Cb$,$\Db$ and~$\Wb$ with the same value, similarly to the implementation of~\cite{simon2019rethinking} released by the authors.
\footnote{The implementation of CSCnet~\cite{simon2019rethinking} is available here \url{https://github.com/drorsimon/CSCNet/}.}
Since too large learning rates can make the model diverge (as for any neural network), we have implemented a backtracking strategy that automatically decreases the learning rate by a factor 0.8 when the loss function
increases too much on the training set, and restore a previous snapshot of the model. Divergence is monitored by computing the loss on the training set every 20 epochs.
Training the GroupSC model for color denoising takes about 2 days on a Titan RTX GPU.
\paragraph{Accelerating inference.}
In order to make the inference time of the non-local models faster, we do not update similarity maps at every step: we update patch similarities every $1/f$ steps, where $f$ is the frequency of the correlation updates.
We summarize in Table \ref{hyperparam} the set of hyperparameters that we selected for the experiments reported in the main tables.
\begin{table}[h!]
\small
\centering
\caption{Hyper-parameters chosen for every task.}
\label{hyperparam}
\begin{tabular}{lcccc}
\toprule
Experiment & Color denoising & Gray denoising & Demosaicking & Jpeg Deblocking \\
\midrule
Patch size & 7 & 9 & 7 & 9 \\
Dictionary size & 256 & 256 & 256 & 256 \\
Nr epochs & 300 & 300 & 300 & 300 \\
Batch size & 32 & 32 & 32 & 32 \\
$K$ iterations & 24 & 24 & 24 & 24 \\
Middle averaging & \cmark & \cmark & \cmark & \cmark \\
\begin{tabular}[c]{@{}l@{}}Correlation update\\ frequency $f$\end{tabular} & ${1}/{6}$ & ${1}/{6}$ & ${1}/{8}$ & ${1}/{6}$ \\
\bottomrule
\end{tabular}
\end{table}
\section{Additional Quantitative Results and Ablation Studies}\label{sec:res1}
\subsection{Results on Other Datasets and SSIM Scores}
We provide additional grayscale denoising results of our model on the datasets BSD68, Set12, and Urban100 in terms of PSNR and SSIM in Table \ref{gray_app}.
Then, we present additional results for color denoising in Table~\ref{color_app}, for demosaicking in Table~\ref{fig:supp_mosa}, and for jpeg artefact reduction in Table~\ref{tab:jpeg}.
Note that we report SSIM scores for baseline methods, either because they report SSIM in the corresponding papers, or by running
the code released by the authors.
\begin{table}[h!]
\small
\centering
\caption{\textbf{Grayscale denoising} results on different datasets. Training is performed on BSD400. Performance is measured in terms of average PSNR (left number) and SSIM (right number). }
\label{gray_app}
\begin{tabular}{lccccc}
\toprule
Dataset & Noise & BM3D & \begin{tabular}{cc}DnCNN\\556k \end{tabular} & \begin{tabular}{cc}NLRN\\330k\end{tabular} & \begin{tabular}{cc} GroupSC\\ 68k\end{tabular} \\
\midrule
\multirow{3}{*}{\textbf{Set12}}
& 15 & 32.37/0.8952 & \underline{32.86}/0.9031 & \textbf{ 33.16/0.9070} & 32.85/\underline{0.9063 } \\
& 25 & 29.97/0.8504 & 30.44/0.8622 & \textbf{30.80/0.8689} & \underline{30.44}/\underline{0.8642} \\
& 50 & 26.72/0.7676 & \underline{27.18}/\underline{0.7829} & \textbf{27.64/0.7980} & 27.14/0.7797 \\
\midrule
\multirow{3}{*}{\textbf{BSD68}}
& 15 & 31.07/0.8717 & \underline{31.73}/0.8907 & \textbf{31.88}/0.8932 & 31.70/\textbf{0.8963} \\
& 25 & 28.57/0.8013 & \underline{29.23}/0.8278 & \textbf{29.41}/0.8331 & 29.20/\textbf{0.8336} \\
& 50 & 25.62/0.6864 & \underline{26.23}/0.7189 & \textbf{26.47/0.7298} & 26.18/0.7183 \\
\midrule
\multirow{3}{*}{\textbf{Urban100}}
& 15 & 32.35/0.9220 & 32.68/0.9255 & \textbf{33.45/0.9354} & \underline{32.72}/0.9308 \\
& 25 & 29.70/0.8777 & 29.91/0.8797 & \textbf{30.94/0.9018} & \underline{30.05}/0.8912 \\
& 50 & 25.95/0.7791 & 26.28/0.7874 & \textbf{ 27.49/0.8279} & \underline{26.43}/\underline{0.8002} \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[h!]
\small
\centering
\caption{\textbf{Color denoising} results on different datasets. Training is performed on CBSD400. Performance is measured in terms of average PSNR (left number) or SSIM (right number). }
\label{color_app}
\begin{tabular}{lccc}
\toprule
Dataset & Noise & \begin{tabular}{cc}CDnCNN\\668k \end{tabular} & \begin{tabular}{cc}GroupSC\\119k \end{tabular} \\ \midrule
\multirow{4}{*}{\textbf{Kodak24}}
& 15 & \underline{34.84}/\underline{0.9233} & \textbf{35.00/0.9275 } \\
& 25 & \underline{32.34}/\underline{0.8812} & \textbf{32.51/0.8867 } \\
& 50 & \underline{29.15}/\underline{0.7985} & \textbf{29.19/0.7993 } \\
\midrule \multirow{4}{*}{\textbf{CBSD68}}
& 15 & \underline{33.98}/\underline{0.9303} & \textbf{34.11}/\textbf{0.9353} \\
& 25 & \underline{31.31}/\underline{0.8848} & \textbf{31.44}/\textbf{0.8917} \\
& 50 & \underline{28.01}/\underline{0.7925} & \textbf{28.05}/\textbf{0.7974} \\
\midrule \multirow{4}{*}{\textbf{Urban100}}
& 15 & \underline{34.11}/\underline{0.9436} & \textbf{34.14}/\textbf{0.9461} \\
& 25 & \underline{31.66}/\underline{0.9145} & \textbf{31.69}/\textbf{0.9178} \\
& 50 & \underline{28.16}/\underline{0.8410} & \textbf{28.23}/\textbf{0.8513} \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[h!]
\caption{ \textbf{Jpeg artefact reduction} on Classic5 with training on CBSD400. Performance is measured in terms of average PSNR.}
\footnotesize
\centering
\begin{tabular}{ccccc}
\toprule
\begin{tabular}[c]{@{}c@{}}Quality\\ factor\end{tabular} & AR-CNN \cite{yu2016deep} & TNRD\cite{chen2016trainable} & DnCNN-3 \cite{zhang2017beyond} & GroupSC \\
\midrule
10 & 29.04/0.7929 & 29.28/0.7992 & \underline{29.40}/\underline{0.8026} & \textbf{29.61}/ \textbf{0.8166} \\
20 & 31.16/0.8517 & 31.47/0.8576 & \underline{31.63}/\underline{0.8610} & \textbf{31.78}/ \textbf{0.8718} \\
30 & 32.52/0.8806 & 32.78/0.8837 & \underline{32.91}/\underline{0.8861} & \textbf{33.06}/ \textbf{0.8959} \\
40 & 33.34/0.8953 & - & \underline{33.75}/0.9003 & \textbf{33.91}/ \textbf{0.9093} \\
\bottomrule
\end{tabular}
\label{tab:jpeg}
\end{table}
\begin{table}[h!] \label{supp_mosa}
\centering
\footnotesize
\caption{\textbf{Demosaicking} results. Training on CBSD400 unless a larger dataset is specified between parenthesis. Performance is measured in terms of average PSNR (left) and SSIM (right).}\label{mosa_ssim}
\begin{tabular}{@{}lccccc@{}}
\toprule
Method & Params & \textbf{Kodak24} & \textbf{BSD68} & \textbf{Urban100} \\
\midrule
IRCNN (BSD400+Waterloo) & 107k & \underline{40.54}/\underline{0.9807} & \underline{39.96}/\underline{0.9850} & \underline{36.64}/\underline{0.9743} \\
{GroupSC} (CBSD400) (ours) & {118k} & \textbf{42.71/0.9901} & \textbf{42.91/0.9938} & \textbf{38.21/0.9804} \\
\bottomrule
\end{tabular}
\vspace*{-0.5cm}
\end{table}
\subsection{Inference Speed and Importance of Similarity Refinements}
In table \ref{tab:speed},
we provide a comparison of our model in terms of speed. We compare our model for demosaicking and color denoising with the methods NLRN.
This study shows how to balance the trade-off between speed and accuracy. Whereas the best model in accuracy achieves 31.71dB in PSNR with about 30s per image, a ``light'' version can
achieve 31.67dB in only 2.35s per image.
This ablation study also illustrates the need of similarity refinements during the iterations.
When they are no updates the model perfoms on average 0.15 dB lower than with 4 updates.
\begin{table}[h!]
\caption{\textbf{Inference time (s)} per image / PSNR (in dB) for gray denoising task with $\sigma=15$, computed on BSD68. Inference time is measured using a Titan RTX gpu.}\label{tab:speed}
\footnotesize
\centering
\begin{tabular}{@{}|c|c||c|c|c|c|@{}}
\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Middle \\ averaging (6) \end{tabular}} & \multirow{2}{*}{$f_{\hat{\Sigmab}}$} & \multicolumn{4}{c|}{Stride between image blocks} \\ \cline{3-6}
& & $s=56$ & $s=48$ & $s=24$ & $s=12$ \\ \hline \hline
\multirow{4}{*}{\xmark
} & $\infty$ & 1.30 / 31.29 & 1.75 / 31.57 & 6.00 / 31.58 & 22.57 / 31.59 \\
& 12 & 1.41 / 31.36 & 1.85 / 31.64 & 6.57 / 31.66 & 24.44 / 31.66 \\
& 8 & 1.51 / 31.37 & 2.90 / 31.65 & 7.06 / 31.68 & 26.05 / 31.68 \\
& 6 & 1.59 / 31.38 & 2.15 / 31.65 & 7.48 / 31.68 & 27.60 / 31.69 \\ \hline
\multirow{4}{*}{\cmark
} & $\infty $ & 1.30 / 31.29 & 1.75 / 31.57 & 6.00 / 31.58 & 22.57 / 31.59 \\
& 12 & 1.45 / 31.36 & 1.95 / 31.65 & 6.82 / 31.66 & 25.40 / 31.67 \\
& 8 & 1.63 / 31.38 & 2.17 / 31.66 & 7.61 / 31.68 & 27.92 / 31.70 \\
& 6 & 1.77 / 31.39 & 2.35 / 31.67 & 8.25 / 31.69 & 30.05 / 31.71 \\ \hline
NLRN & 330k & \multicolumn{4}{c|}{23.02 / 31.88} \\ \hline
\end{tabular}
\vspace*{-0.2cm}
\end{table}
\subsection{Influence of Patch and Dictionary Sizes}
We measure in Table \ref{ablation_table} the influence of the patch size and the dictionary size for grayscale image denoising.
For this experiment, we run a lighter version of the model groupSC in order to accelerate the training. The batch size was decreased from 25 to 16, the
frequency of the correlation updates was decreased from $1/6$ to $1/8$ and the intermediate patches are not approximated with averaging.
These changes accelerate the training but lead to slightly lower performances when compared with the model trained in the standard setting.
As can be seen in the table, better performance can be obtained by using larger dictionaries, at the cost of more computation. Note that
all other experiments conducted in the paper use a dictionary size of 256. Here as well, a trade-off between speed/number of parameters and accuracy can be chosen
by changing this default value.
\begin{table}[h!]
\small
\centering
\caption{\textbf{Influence of the dictionary size and the patch size }on the denoising performance.
Grayscale denoising on BSD68. Models are trained on BSD400. Models are trained in a light setting to accelerate training.}
\label{ablation_table}
\begin{tabular}{|c|l|c|c|c|}
\hline
Noise ($\sigma$) & Patch size & n=128 & n=256 & 512 \\
\hline
\multirow{3}{*}{$5$}
& k=7 & 37.91 & 37.92 & - \\
& k=9 & 37.90 & 37.92 & 37.96 \\
& k=11 & 37.89 & 37.89 & - \\
\hline
\multirow{3}{*}{$15$}
& k=7 & 31.60 & 31.63 & - \\
& k=9 & 31.62 & 31.67 & 31.71 \\
& k=11 & 31.63 & 31.67 & - \\
\hline
\multirow{3}{*}{$25$}
& k=7 & 29.10 & 29.11 & - \\
& k=9 & 29.12 & 29.17 & 29.20 \\
& k=11 & 29.13 & 29.18 & - \\
\hline
\end{tabular}
\end{table}
\subsection{Number of Unrolled Iterations}
We also investigated the impact of the depth of the model on the performance.
To do so, we conducted a denoising experiment using the light version of our model with a model with various number of
unrolled steps.
When changing the depth from K=12, to 36, we only measure a difference of 0.02dB.
\begin{table}[h!]
\centering
\caption{\textbf{Influence of the number of unrolled iterations}.Grayscale denoising on BSD68.
Models are trained on BSD400. Models are trained in a light setting to accelerate training.}
\begin{tabular}{|l|c|c|c|}
\hline
Model & \multicolumn{3}{l|}{Unrolled iterations } \\ \hline
SC & 28.90 & 28.91 & 28.90 \\
GroupSC (light) & 29.10 & 29.12 & 29.12 \\ \hline
\end{tabular}
\end{table}
\section{Proof of Proposition 1}\label{sec:proofs}
The proximal operator of the function $\Psi_i(\ub) = \| \ub \|_1 + \gamma \| \ub - \betab_i \|_1$ for $\ub$ in~$\Real^p$ is defined as
\begin{equation*} \label{prox_psi}
\prox_{\lambda \Psi_i} [\zb]= \argmin_{\ub \in \Real^p} \frac{1}{2}\|\zb-\ub\|^2 + \lambda \|\ub\|_1 + \lambda \gamma \| \ub - \betab_i\|_1
\end{equation*}
The optimality condition for the previous problem is
$$ 0 \in \triangledown (\frac{1}{2} || \zb - \ub ||_2^2) + \partial (\lambda ||\ub||_1) + \partial (\lambda \gamma ||\ub - \betab_i||_1) $$
$$\Leftrightarrow 0 \in \ub - \zb + \lambda \partial ||\ub||_1 + \lambda \gamma \partial ||\ub - \betab_i||_1 $$
We consider each component separately. We suppose that $\betab_i[j] \neq 0$, otherwise $\Psi_i(\ub)[j]$ boils down to the $\ell_1$ norm. And we also suppose $\lambda,\gamma >0$.
Let us examine the first case where $u[j] = 0$. The subdifferential of the $\ell_1$ norm is the interval $[-1,1]$ and the optimality condition is
\begin{align*}
0 \in \ub[j] - \zb[j] + [-\lambda,\lambda] + \lambda \gamma \sign(\ub[j]-\betab_i[j]) \\
\Leftrightarrow \zb[j] \in [-\lambda,\lambda] - \lambda \gamma \sign(\betab_i[j])
\end{align*}
Similarly if $\ub[j] = \betab_i[j]$
\begin{equation*}
\zb[j] \in \betab_i[j] + \lambda \sign(\betab_i[j]) + [-\lambda \gamma,\lambda \gamma]
\end{equation*}
Finally let us examine the case where $u[j] \neq 0$ and $u[j] \neq \betab_i[j]$: then, $\partial ||\ub||_1 = \sign(\ub[j])$ and $\partial ||\ub - \betab_i||_1 = \sign(\ub[j] - \betab_i[j])$. The minimum $u[j]^*$ is obtained as
\begin{align*}
0 = \ub[j] - \zb[j] + \lambda \sign(\ub[j]) + \lambda \gamma \sign(\ub[j]-\betab_i[j]) \\
\Leftrightarrow \ub[j]^* = \zb[j] - \lambda \sign(\ub[j]^*) - \lambda \gamma \sign(\ub[j]^*-\betab_i[j])
\end{align*}
We study separately the cases where $\ub[j]>\betab[j]$, $0<\ub[j]<\betab[j]$ and $\ub[j]<0$ when $\betab_i[j]>0$ and proceed similarly when $\betab_i<0$. With elementary operations we can derive the expression of $\zb[j]$ for each case. Putting the cases all together we obtain the formula.
\section{Additional Qualitative Results}\label{sec:res2}
We show qualitative results for jpeg artefact reduction, color denoising, grayscale denoising, and demosaicking in Figures
\ref{fig:supp_color},
\ref{fig:supp_gray},
\ref{fig:supp_mosa}, respectively.
\input{figure_jpeg}
\input{figure_color_app.tex}
\input{figure_gray_app.tex}
\input{fig_mosa.tex}
\section{Parameters visualization}\label{sec:viz}
\noindent
\begin{minipage}{0.70\textwidth}
We present in this section some visualizations of the learned parameters of our introduced model groupsc for a dnoising task.
We reported in Figure~\ref{fig:dico} learned dictionaries $\Db$ and $\Wb$ (model trained with $\Cb = \Db$).
We observe that dictionaries are coupled.
We reported in Figure~\ref{fig:lmbda} the sequence of regularization parameters $(\Lambdab_{k})_{k=0,1 \dots K-1}$ for a denoising task, and
$(\Lambda_{\sigma_0}, \dots, \Lambda_{\sigma_n} )$.
for blind denoising. Finally, we reported in Figure~\ref{fig:kappa} the learned weights $\kappab$ of the gaussian kernel for comparing patches.
\end{minipage}
\hfill
\begin{minipage}{0.25\textwidth}
\includegraphics[width=4cm]{figures/std.png}\label{fig:kappa}
\captionof{figure}{Weights $\kappab$ for comparing patches.}\label{fig:kappa}
\end{minipage}
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.3\linewidth]{figures/dict/A.png} &
\includegraphics[width=0.3\linewidth]{figures/dict/W.png} \\
$\Db$ &
$\Wb$ \\
\end{tabular}
\caption{Learned dictionnaries of groupSC for denoising.}\label{fig:dico}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.5\linewidth]{figures/lmbda.png} &
\includegraphics[width=0.5\linewidth]{figures/blind_lmbda.png} \\
\begin{tabular}{c} Sequence of regularization parameters \\
$\Lambdab_i$ of a non-blind models.\end{tabular}&
\begin{tabular}{c} Set of regularization parameters\\$(\Lambda_{\sigma_0}, \dots, \Lambda_{\sigma_n} )$ \\ of a blind model. \end{tabular}
\end{tabular}
\caption{Learned regularization parameters of groupSC for denoising and blind denoising. Models are trained on BSD400.}\label{fig:lmbda}
\end{figure}
\section{Centralised Sparse Representation}\label{sec:csr}
\vsp
A different approach to take into account self similarities in sparse models is
the CSR approach of \cite{dong2012nonlocally}. This approach
is easier to turn into a differentiable algorithm than the LSSC method, but we have
empirically observed that it does not perform as well. Nevertheless, we believe it to be
conceptually interesting, and we provide a brief description below.
The idea consists of regularizing each code $\alphab_i$ with the function
\begin{equation}
\label{eq:csreq}
\Psi_i(\alphab_i) = \|\alphab_i\|_1 + \gamma \|\alphab_i - \betab_i\|_1,
\end{equation}
where $\betab_i$ is obtained by a weighted average of prevous codes.
Specifically, given some codes~$\alphab_i^{(k)}$ obtained at iteration~$k$ and a similarity matrix~$\Sigmab$, we compute
\begin{equation}
\betab_i^{(k)} = \sum_{j} \frac{\Sigmab_{ij}}{\sum_{l} \Sigmab_{il}}\alphab_j^{(k)},
\end{equation}
and the weights $\betab_i^{(k)}$ are used in~(\ref{eq:csreq}) in order to compute the codes $\alphab_i^{(k+1)}$.
Note that the original CSR method of~\cite{dong2012nonlocally} uses similarities of the form
$\Sigmab_{ij} = \exp{\left(-\frac{1}{2\sigma^2} \| \Wb\alphab_i-\Wb\alphab_j\|_2^2 \right)} $, but other similarities functions may be used.
Even though~\cite{dong2012nonlocally} does not use a proximal gradient descent
method to solve the problem regularized with~(\ref{eq:csreq}), the next proposition shows
that it admits a closed form, which is a key to turn CSR into a differentiable algorithm.
To the best of our knowledge,
this expression is new; its proof is given in the~appendix.
\begin{proposition}[Proximal operator of the CSR penalty]\label{csr_prox}
Consider $\Psi_i$ defined in~(\ref{eq:csreq}). Then, for all $\ub$ in~$\Real^p$,
\begin{equation*}
\prox_{\lambda \Psi_i}[\ub] = S_\lambda \big( S_{\lambda \gamma} \left (\ub - \betab_i - \lambda \sign(\betab_i) \right ) \\ +\betab_i + \lambda\sign(\betab_i) \big ),
\end{equation*}
where $S_\lambda$ is the soft-thresholding operator, see Figure~\ref{fig:prox_csr}.
\end{proposition}
\noindent
\begin{minipage}{0.56\textwidth}
Despite the apparent complexity of the formula, it remains a continuous function of the input and is differentiable almost everywhere, hence compatible with end-to-end training.
%
Qualitatively, the shape of the proximal mapping has a simple interpretation. It pulls codes either to zero, or to the code weighted average $\betab_i$.
\end{minipage}
\hfill
\begin{minipage}{0.42\textwidth}
\centering\raisebox{\dimexpr \topskip-\height}{%
\includegraphics[width=\textwidth]{figures/fig_prox_c.png}}
\captionof{figure}{$ \prox_{\lambda \Psi_i}$ for various $\lambda, \gamma, \beta$}\label{fig:prox_csr}
\end{minipage}
At each iteration, the similarity matrix is updated along with the codes $\betab_i$. The proximal operator can then easily be plugged into our framework.
We reported performance of the CSR approach in the main paper for grayscale denoising, color denoising and demosaicking.
Performance of the CSR approach are reported in Tables \ref{colour_table_csr}, \ref{gray_table_csr}, \ref{mosaic_table_csr}.
We observe that it performs significantly
better than the baseline SC but is not as effective as GroupSC overall.
\begin{table}[htb]
\footnotesize
\centering
\caption{ \textbf{Color denoising} on CBSD68, training on CBSD400 for all methods except CSCnet (Waterloo+CBSD400).
Performance is measured in terms of average PSNR. SSIMs are reported in the appendix.}\label{colour_table_csr}
\begin{tabular}{@{}lccccccccc@{}}
\toprule
\multirow{2}{*}{Method} & \multirow{2}{*}{Trainable } & \multirow{2}{*}{Params} & \multicolumn{4}{c}{Noise level ($\sigma$)} \\
& & & 5 & 10 & 15 & 25 & 30 & 50 \\
\midrule
CBM3D \cite{dabov2007image} & \xmark & - & 40.24 & - & 33.49 & 30.68 & - & 27.36 \\
\midrule
CSCnet \cite{simon2019rethinking} & & 186k & - & - & 33.83 & 31.18 & - & 28.00 \\
CNLNet\cite{lefkimmiatis2017non} & & - & - & - & 33.69 & 30.96 & - & 27.64 \\
FFDNET \cite{zhang2018ffdnet} & & 486k & - & - & 33.87 & 31.21 & - & 27.96 \\
CDnCNN \cite{zhang2017beyond} & & 668k & 40.50 & 36.31 & 33.99 & 31.31 & - & 28.01 \\
RNAN \cite{zhang2019rnan} & & 8.96M & - & \textbf{36.60} & - & - & \textbf{30.73} & \textbf{28.35} \\
\midrule
{SC} (baseline) & & {119k} & {40.44} & - & {33.75} & 30.94 & - & 27.39 \\
{CSR} (ours) & & {119k} & {40.53} & - & {34.05} & {31.33} & - & {28.01} \\
{GroupSC} (ours) & & {119k} & \underline{40.58} & \underline{36.40} & \underline{34.11} & \underline{31.44} & \underline{30.58} & \underline{28.05} \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[htb]
\small
\centering
\caption{\textbf{Grayscale Denoising} on BSD68, training on BSD400 for all methods except CSCnet (Waterloo+BSD400).
Performance is measured in terms of average PSNR.
SSIMs are reported in the appendix.}
\label{gray_table_csr}
\begin{tabular}{@{}lcccccc@{}}
\toprule
\multirow{2}{*}{Method} & \multirow{2}{*}{Trainable} & \multirow{2}{*}{Params} & \multicolumn{4}{c}{Noise Level ($\sigma$)} \\
& & & 5 & 15 & 25 & 50 \\
\midrule
BM3D \cite{dabov2007image} & \xmark & - & 37.57 & 31.07 & 28.57 & 25.62 \\
LSSC \cite{mairal2009non} & \xmark & - & 37.70 & 31.28 & 28.71 & 25.72 \\
BM3D PCA \cite{dabov2009bm3d} & \xmark & - & 37.77 & 31.38 & 28.82 & 25.80 \\
\midrule
TNRD \cite{chen2016trainable} & & - & - & 31.42 & 28.92 & 25.97 \\
CSCnet \cite{simon2019rethinking} & & 62k & 37.84 & 31.57 & { 29.11} & 26.24 \\
CSCnet(BSD400) \cite{simon2019rethinking}\footnotemark[2] & & 62k & 37.69 & 31.40 & { 28.93} & 26.04 \\
LKSVD~\cite{scetbon2019deep} & & 45K & - & 31.54 & 29.07 & 26.13 \\
NLNet \cite{lefkimmiatis2017non} & & - & - & 31.52 & 29.03 & 26.07 \\
FFDNet \cite{zhang2018ffdnet} & & 486k & - & 31.63 & 29.19 & 26.29 \\
DnCNN \cite{zhang2017beyond} & & 556k & {37.68} & \underline{31.73} & 29.22 & 26.23 \\
N3 \cite{plotz2018neural} & & 706k & - & - & \underline{29.30} & \underline{26.39} \\
NLRN \cite{liu2018non} & & 330k & \underline{37.92} & \textbf{ 31.88} & \textbf{ 29.41} & \textbf{ 26.47} \\
\midrule
{SC} (baseline) & & {68k} & 37.84 & 31.46 & 28.90 & 25.84 \\
{CSR} (ours) & & {68k} & 37.88 & 31.64 & 29.16 & 26.08 \\
{GroupSC} (ours) & & {68k} & \textbf{37.95} & 31.71 & 29.20 & 26.17 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[htb]
\centering
\footnotesize
\caption{\textbf{Demosaicking.} Training on CBSD400 unless a larger dataset is specified between parenthesis.
Performance is measured in terms of average PSNR. SSIMs are reported in the appendix.} \label{mosaic_table_csr}
\begin{tabular}{@{}lccccc@{}}
\toprule
Method & Trainable & Params & Kodak24 & BSD68 & Urban100 \\
\midrule
LSSC & \xmark & - & 41.39 & 40.44 & 36.63 \\
\midrule
IRCNN \cite{zhang2017learning} (BSD400+Waterloo \cite{ma2016waterloo}) & & - & 40.54 & 39.9 & 36.64 \\
Kokinos \cite{kokkinos2018deep} (MIT dataset \cite{gharbi2016deep}) & & 380k & 41.5 & - & - \\
MMNet \cite{kokkinos2019iterative} (MIT dataset \cite{gharbi2016deep}) & & 380k & 42.0 & - & - \\
RNAN \cite{zhang2019rnan} & & {8.96M} & \textbf{42.86} & \underline{42.61} & - \\
\midrule
{SC} (ours) & & 119k & 42.34 & 41.88 & 37.50 \\
CSR (ours) & & 119k & 42.25 & - & - \\
{GroupSC} (ours) & & {119k} & \underline{42.71} & \textbf{42.91} & \textbf{38.21} \\
\bottomrule
\end{tabular}
\end{table}
\section{Implementation Details and Reproducibility}\label{sec:impl}
\paragraph{Training details.}
During training, we randomly extract patches $56\times56$ whose size equals the window size used for computing non-local self-similarities. We apply a mild data augmentation (random rotation by
$90^\circ$ and horizontal flips). We optimize the parameters of our models using ADAM~\cite{kingma2014adam} with a minibatch size of $32$.
All the models are trained for 300 epochs for denoising and demosaicking. The learning rate is set to $6 \times 10^{-4}$ at initialization and is sequentially lowered during training
by a factor of 0.35 every 80 training steps, in the same way for all experiments. Similar to \cite{simon2019rethinking}, we normalize the initial dictionary~$\Db_0$ by its largest singular value, which helps the LISTA
algorithm to converge faster. We initialize the matrices $\Cb$,$\Db$ and~$\Wb$ with the same value, similarly to the implementation of~\cite{simon2019rethinking} released by the authors.
\footnote{The implementation of CSCnet~\cite{simon2019rethinking} is available here \url{https://github.com/drorsimon/CSCNet/}.}
Since too large learning rates can make the model diverge (as for any neural network), we have implemented a backtracking strategy that automatically decreases the learning rate by a factor 0.8 when the loss function
increases too much on the training set, and restore a previous snapshot of the model. Divergence is monitored by computing the loss on the training set every 20 epochs.
Training the GroupSC model for color denoising takes about 2 days on a Titan RTX GPU.
\paragraph{Accelerating inference.}
In order to make the inference time of the non-local models faster, we do not update similarity maps at every step: we update patch similarities every $1/f$ steps, where $f$ is the frequency of the correlation updates.
We summarize in Table \ref{hyperparam} the set of hyperparameters that we selected for the experiments reported in the main tables.
\begin{table}[h!]
\small
\centering
\caption{Hyper-parameters chosen for every task.}
\label{hyperparam}
\begin{tabular}{lcccc}
\toprule
Experiment & Color denoising & Gray denoising & Demosaicking & Jpeg Deblocking \\
\midrule
Patch size & 7 & 9 & 7 & 9 \\
Dictionary size & 256 & 256 & 256 & 256 \\
Nr epochs & 300 & 300 & 300 & 300 \\
Batch size & 32 & 32 & 32 & 32 \\
$K$ iterations & 24 & 24 & 24 & 24 \\
Middle averaging & \cmark & \cmark & \cmark & \cmark \\
\begin{tabular}[c]{@{}l@{}}Correlation update\\ frequency $f$\end{tabular} & ${1}/{6}$ & ${1}/{6}$ & ${1}/{8}$ & ${1}/{6}$ \\
\bottomrule
\end{tabular}
\end{table}
\section{Additional Quantitative Results and Ablation Studies}\label{sec:res1}
\subsection{Results on Other Datasets and SSIM Scores}
We provide additional grayscale denoising results of our model on the datasets BSD68, Set12, and Urban100 in terms of PSNR and SSIM in Table \ref{gray_app}.
Then, we present additional results for color denoising in Table~\ref{color_app}, for demosaicking in Table~\ref{fig:supp_mosa}, and for jpeg artefact reduction in Table~\ref{tab:jpeg}.
Note that we report SSIM scores for baseline methods, either because they report SSIM in the corresponding papers, or by running
the code released by the authors.
\begin{table}[h!]
\small
\centering
\caption{\textbf{Grayscale denoising} results on different datasets. Training is performed on BSD400. Performance is measured in terms of average PSNR (left number) and SSIM (right number). }
\label{gray_app}
\begin{tabular}{lccccc}
\toprule
Dataset & Noise & BM3D & \begin{tabular}{cc}DnCNN\\556k \end{tabular} & \begin{tabular}{cc}NLRN\\330k\end{tabular} & \begin{tabular}{cc} GroupSC\\ 68k\end{tabular} \\
\midrule
\multirow{3}{*}{\textbf{Set12}}
& 15 & 32.37/0.8952 & \underline{32.86}/0.9031 & \textbf{ 33.16/0.9070} & 32.85/\underline{0.9063 } \\
& 25 & 29.97/0.8504 & 30.44/0.8622 & \textbf{30.80/0.8689} & \underline{30.44}/\underline{0.8642} \\
& 50 & 26.72/0.7676 & \underline{27.18}/\underline{0.7829} & \textbf{27.64/0.7980} & 27.14/0.7797 \\
\midrule
\multirow{3}{*}{\textbf{BSD68}}
& 15 & 31.07/0.8717 & \underline{31.73}/0.8907 & \textbf{31.88}/0.8932 & 31.70/\textbf{0.8963} \\
& 25 & 28.57/0.8013 & \underline{29.23}/0.8278 & \textbf{29.41}/0.8331 & 29.20/\textbf{0.8336} \\
& 50 & 25.62/0.6864 & \underline{26.23}/0.7189 & \textbf{26.47/0.7298} & 26.18/0.7183 \\
\midrule
\multirow{3}{*}{\textbf{Urban100}}
& 15 & 32.35/0.9220 & 32.68/0.9255 & \textbf{33.45/0.9354} & \underline{32.72}/0.9308 \\
& 25 & 29.70/0.8777 & 29.91/0.8797 & \textbf{30.94/0.9018} & \underline{30.05}/0.8912 \\
& 50 & 25.95/0.7791 & 26.28/0.7874 & \textbf{ 27.49/0.8279} & \underline{26.43}/\underline{0.8002} \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[h!]
\small
\centering
\caption{\textbf{Color denoising} results on different datasets. Training is performed on CBSD400. Performance is measured in terms of average PSNR (left number) or SSIM (right number). }
\label{color_app}
\begin{tabular}{lccc}
\toprule
Dataset & Noise & \begin{tabular}{cc}CDnCNN\\668k \end{tabular} & \begin{tabular}{cc}GroupSC\\119k \end{tabular} \\ \midrule
\multirow{4}{*}{\textbf{Kodak24}}
& 15 & \underline{34.84}/\underline{0.9233} & \textbf{35.00/0.9275 } \\
& 25 & \underline{32.34}/\underline{0.8812} & \textbf{32.51/0.8867 } \\
& 50 & \underline{29.15}/\underline{0.7985} & \textbf{29.19/0.7993 } \\
\midrule \multirow{4}{*}{\textbf{CBSD68}}
& 15 & \underline{33.98}/\underline{0.9303} & \textbf{34.11}/\textbf{0.9353} \\
& 25 & \underline{31.31}/\underline{0.8848} & \textbf{31.44}/\textbf{0.8917} \\
& 50 & \underline{28.01}/\underline{0.7925} & \textbf{28.05}/\textbf{0.7974} \\
\midrule \multirow{4}{*}{\textbf{Urban100}}
& 15 & \underline{34.11}/\underline{0.9436} & \textbf{34.14}/\textbf{0.9461} \\
& 25 & \underline{31.66}/\underline{0.9145} & \textbf{31.69}/\textbf{0.9178} \\
& 50 & \underline{28.16}/\underline{0.8410} & \textbf{28.23}/\textbf{0.8513} \\ \bottomrule
\end{tabular}
\end{table}
\begin{table}[h!]
\caption{ \textbf{Jpeg artefact reduction} on Classic5 with training on CBSD400. Performance is measured in terms of average PSNR.}
\footnotesize
\centering
\begin{tabular}{ccccc}
\toprule
\begin{tabular}[c]{@{}c@{}}Quality\\ factor\end{tabular} & AR-CNN \cite{yu2016deep} & TNRD\cite{chen2016trainable} & DnCNN-3 \cite{zhang2017beyond} & GroupSC \\
\midrule
10 & 29.04/0.7929 & 29.28/0.7992 & \underline{29.40}/\underline{0.8026} & \textbf{29.61}/ \textbf{0.8166} \\
20 & 31.16/0.8517 & 31.47/0.8576 & \underline{31.63}/\underline{0.8610} & \textbf{31.78}/ \textbf{0.8718} \\
30 & 32.52/0.8806 & 32.78/0.8837 & \underline{32.91}/\underline{0.8861} & \textbf{33.06}/ \textbf{0.8959} \\
40 & 33.34/0.8953 & - & \underline{33.75}/0.9003 & \textbf{33.91}/ \textbf{0.9093} \\
\bottomrule
\end{tabular}
\label{tab:jpeg}
\end{table}
\begin{table}[h!] \label{supp_mosa}
\centering
\footnotesize
\caption{\textbf{Demosaicking} results. Training on CBSD400 unless a larger dataset is specified between parenthesis. Performance is measured in terms of average PSNR (left) and SSIM (right).}\label{mosa_ssim}
\begin{tabular}{@{}lccccc@{}}
\toprule
Method & Params & \textbf{Kodak24} & \textbf{BSD68} & \textbf{Urban100} \\
\midrule
IRCNN (BSD400+Waterloo) & 107k & \underline{40.54}/\underline{0.9807} & \underline{39.96}/\underline{0.9850} & \underline{36.64}/\underline{0.9743} \\
{GroupSC} (CBSD400) (ours) & {118k} & \textbf{42.71/0.9901} & \textbf{42.91/0.9938} & \textbf{38.21/0.9804} \\
\bottomrule
\end{tabular}
\vspace*{-0.5cm}
\end{table}
\subsection{Inference Speed and Importance of Similarity Refinements}
In table \ref{tab:speed},
we provide a comparison of our model in terms of speed. We compare our model for demosaicking and color denoising with the methods NLRN.
This study shows how to balance the trade-off between speed and accuracy. Whereas the best model in accuracy achieves 31.71dB in PSNR with about 30s per image, a ``light'' version can
achieve 31.67dB in only 2.35s per image.
This ablation study also illustrates the need of similarity refinements during the iterations.
When they are no updates the model perfoms on average 0.15 dB lower than with 4 updates.
\begin{table}[h!]
\caption{\textbf{Inference time (s)} per image / PSNR (in dB) for gray denoising task with $\sigma=15$, computed on BSD68. Inference time is measured using a Titan RTX gpu.}\label{tab:speed}
\footnotesize
\centering
\begin{tabular}{@{}|c|c||c|c|c|c|@{}}
\hline
\multirow{2}{*}{\begin{tabular}[c]{@{}c@{}}Middle \\ averaging (6) \end{tabular}} & \multirow{2}{*}{$f_{\hat{\Sigmab}}$} & \multicolumn{4}{c|}{Stride between image blocks} \\ \cline{3-6}
& & $s=56$ & $s=48$ & $s=24$ & $s=12$ \\ \hline \hline
\multirow{4}{*}{\xmark
} & $\infty$ & 1.30 / 31.29 & 1.75 / 31.57 & 6.00 / 31.58 & 22.57 / 31.59 \\
& 12 & 1.41 / 31.36 & 1.85 / 31.64 & 6.57 / 31.66 & 24.44 / 31.66 \\
& 8 & 1.51 / 31.37 & 2.90 / 31.65 & 7.06 / 31.68 & 26.05 / 31.68 \\
& 6 & 1.59 / 31.38 & 2.15 / 31.65 & 7.48 / 31.68 & 27.60 / 31.69 \\ \hline
\multirow{4}{*}{\cmark
} & $\infty $ & 1.30 / 31.29 & 1.75 / 31.57 & 6.00 / 31.58 & 22.57 / 31.59 \\
& 12 & 1.45 / 31.36 & 1.95 / 31.65 & 6.82 / 31.66 & 25.40 / 31.67 \\
& 8 & 1.63 / 31.38 & 2.17 / 31.66 & 7.61 / 31.68 & 27.92 / 31.70 \\
& 6 & 1.77 / 31.39 & 2.35 / 31.67 & 8.25 / 31.69 & 30.05 / 31.71 \\ \hline
NLRN & 330k & \multicolumn{4}{c|}{23.02 / 31.88} \\ \hline
\end{tabular}
\vspace*{-0.2cm}
\end{table}
\subsection{Influence of Patch and Dictionary Sizes}
We measure in Table \ref{ablation_table} the influence of the patch size and the dictionary size for grayscale image denoising.
For this experiment, we run a lighter version of the model groupSC in order to accelerate the training. The batch size was decreased from 25 to 16, the
frequency of the correlation updates was decreased from $1/6$ to $1/8$ and the intermediate patches are not approximated with averaging.
These changes accelerate the training but lead to slightly lower performances when compared with the model trained in the standard setting.
As can be seen in the table, better performance can be obtained by using larger dictionaries, at the cost of more computation. Note that
all other experiments conducted in the paper use a dictionary size of 256. Here as well, a trade-off between speed/number of parameters and accuracy can be chosen
by changing this default value.
\begin{table}[h!]
\small
\centering
\caption{\textbf{Influence of the dictionary size and the patch size }on the denoising performance.
Grayscale denoising on BSD68. Models are trained on BSD400. Models are trained in a light setting to accelerate training.}
\label{ablation_table}
\begin{tabular}{|c|l|c|c|c|}
\hline
Noise ($\sigma$) & Patch size & n=128 & n=256 & 512 \\
\hline
\multirow{3}{*}{$5$}
& k=7 & 37.91 & 37.92 & - \\
& k=9 & 37.90 & 37.92 & 37.96 \\
& k=11 & 37.89 & 37.89 & - \\
\hline
\multirow{3}{*}{$15$}
& k=7 & 31.60 & 31.63 & - \\
& k=9 & 31.62 & 31.67 & 31.71 \\
& k=11 & 31.63 & 31.67 & - \\
\hline
\multirow{3}{*}{$25$}
& k=7 & 29.10 & 29.11 & - \\
& k=9 & 29.12 & 29.17 & 29.20 \\
& k=11 & 29.13 & 29.18 & - \\
\hline
\end{tabular}
\end{table}
\subsection{Number of Unrolled Iterations}
We also investigated the impact of the depth of the model on the performance.
To do so, we conducted a denoising experiment using the light version of our model with a model with various number of
unrolled steps.
When changing the depth from K=12, to 36, we only measure a difference of 0.02dB.
\begin{table}[h!]
\centering
\caption{\textbf{Influence of the number of unrolled iterations}.Grayscale denoising on BSD68.
Models are trained on BSD400. Models are trained in a light setting to accelerate training.}
\begin{tabular}{|l|c|c|c|}
\hline
Model & \multicolumn{3}{l|}{Unrolled iterations } \\ \hline
SC & 28.90 & 28.91 & 28.90 \\
GroupSC (light) & 29.10 & 29.12 & 29.12 \\ \hline
\end{tabular}
\end{table}
\section{Proof of Proposition 1}\label{sec:proofs}
The proximal operator of the function $\Psi_i(\ub) = \| \ub \|_1 + \gamma \| \ub - \betab_i \|_1$ for $\ub$ in~$\Real^p$ is defined as
\begin{equation*} \label{prox_psi}
\prox_{\lambda \Psi_i} [\zb]= \argmin_{\ub \in \Real^p} \frac{1}{2}\|\zb-\ub\|^2 + \lambda \|\ub\|_1 + \lambda \gamma \| \ub - \betab_i\|_1
\end{equation*}
The optimality condition for the previous problem is
$$ 0 \in \triangledown (\frac{1}{2} || \zb - \ub ||_2^2) + \partial (\lambda ||\ub||_1) + \partial (\lambda \gamma ||\ub - \betab_i||_1) $$
$$\Leftrightarrow 0 \in \ub - \zb + \lambda \partial ||\ub||_1 + \lambda \gamma \partial ||\ub - \betab_i||_1 $$
We consider each component separately. We suppose that $\betab_i[j] \neq 0$, otherwise $\Psi_i(\ub)[j]$ boils down to the $\ell_1$ norm. And we also suppose $\lambda,\gamma >0$.
Let us examine the first case where $u[j] = 0$. The subdifferential of the $\ell_1$ norm is the interval $[-1,1]$ and the optimality condition is
\begin{align*}
0 \in \ub[j] - \zb[j] + [-\lambda,\lambda] + \lambda \gamma \sign(\ub[j]-\betab_i[j]) \\
\Leftrightarrow \zb[j] \in [-\lambda,\lambda] - \lambda \gamma \sign(\betab_i[j])
\end{align*}
Similarly if $\ub[j] = \betab_i[j]$
\begin{equation*}
\zb[j] \in \betab_i[j] + \lambda \sign(\betab_i[j]) + [-\lambda \gamma,\lambda \gamma]
\end{equation*}
Finally let us examine the case where $u[j] \neq 0$ and $u[j] \neq \betab_i[j]$: then, $\partial ||\ub||_1 = \sign(\ub[j])$ and $\partial ||\ub - \betab_i||_1 = \sign(\ub[j] - \betab_i[j])$. The minimum $u[j]^*$ is obtained as
\begin{align*}
0 = \ub[j] - \zb[j] + \lambda \sign(\ub[j]) + \lambda \gamma \sign(\ub[j]-\betab_i[j]) \\
\Leftrightarrow \ub[j]^* = \zb[j] - \lambda \sign(\ub[j]^*) - \lambda \gamma \sign(\ub[j]^*-\betab_i[j])
\end{align*}
We study separately the cases where $\ub[j]>\betab[j]$, $0<\ub[j]<\betab[j]$ and $\ub[j]<0$ when $\betab_i[j]>0$ and proceed similarly when $\betab_i<0$. With elementary operations we can derive the expression of $\zb[j]$ for each case. Putting the cases all together we obtain the formula.
\section{Additional Qualitative Results}\label{sec:res2}
We show qualitative results for jpeg artefact reduction, color denoising, grayscale denoising, and demosaicking in Figures
\ref{fig:supp_color},
\ref{fig:supp_gray},
\ref{fig:supp_mosa}, respectively.
\input{figure_jpeg}
\input{figure_color_app.tex}
\input{figure_gray_app.tex}
\input{fig_mosa.tex}
\section{Parameters visualization}\label{sec:viz}
\noindent
\begin{minipage}{0.70\textwidth}
We present in this section some visualizations of the learned parameters of our introduced model groupsc for a dnoising task.
We reported in Figure~\ref{fig:dico} learned dictionaries $\Db$ and $\Wb$ (model trained with $\Cb = \Db$).
We observe that dictionaries are coupled.
We reported in Figure~\ref{fig:lmbda} the sequence of regularization parameters $(\Lambdab_{k})_{k=0,1 \dots K-1}$ for a denoising task, and
$(\Lambda_{\sigma_0}, \dots, \Lambda_{\sigma_n} )$.
for blind denoising. Finally, we reported in Figure~\ref{fig:kappa} the learned weights $\kappab$ of the gaussian kernel for comparing patches.
\end{minipage}
\hfill
\begin{minipage}{0.25\textwidth}
\includegraphics[width=4cm]{figures/std.png}\label{fig:kappa}
\captionof{figure}{Weights $\kappab$ for comparing patches.}\label{fig:kappa}
\end{minipage}
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.3\linewidth]{figures/dict/A.png} &
\includegraphics[width=0.3\linewidth]{figures/dict/W.png} \\
$\Db$ &
$\Wb$ \\
\end{tabular}
\caption{Learned dictionnaries of groupSC for denoising.}\label{fig:dico}
\end{figure}
\begin{figure}[h!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.5\linewidth]{figures/lmbda.png} &
\includegraphics[width=0.5\linewidth]{figures/blind_lmbda.png} \\
\begin{tabular}{c} Sequence of regularization parameters \\
$\Lambdab_i$ of a non-blind models.\end{tabular}&
\begin{tabular}{c} Set of regularization parameters\\$(\Lambda_{\sigma_0}, \dots, \Lambda_{\sigma_n} )$ \\ of a blind model. \end{tabular}
\end{tabular}
\caption{Learned regularization parameters of groupSC for denoising and blind denoising. Models are trained on BSD400.}\label{fig:lmbda}
\end{figure}
\subsection{Trainable Sparse Coding (without Self-Similarities)}\label{subsec:sc}
In \cite{simon2019rethinking}, the sparse coding approach (SC) is combined with the LISTA algorithm to perform
denoising tasks.\footnote{Specifically, \cite{simon2019rethinking} proposes a model based on
convolutional sparse coding (CSC).
CSC is a variant of SC, where a full image is
approximated by a linear combination of small dictionary elements. Unfortunately, CSC leads to
ill-conditioned optimization problems and has shown to perform poorly
for image denoising. For this reason,~\cite{simon2019rethinking} introduces
a hybrid approach between SC and CSC.
In our paper, we have decided to use the SC baseline and leave the investigation of CSC models for
future work.}
The only modification we introduce here is a centering step
for the patches, which empirically yields better results.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/fig1_c.pdf}
\caption{An illustration of the main inference algorithm for GroupSC.
See Figure \ref{fig:nlmodule} for an illustration of the self-similarity module.}\label{fig:diagram}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.80\linewidth]{figures/fig2_c.pdf}
\caption{An illustration of the self-similarity module used in our GroupSC algorithm.}\label{fig:nlmodule}
\end{figure}
\vsp
\paragraph{SC Model - inference with fixed parameters.}
Following the approach and notation from Section~\ref{sec:related},
the first step consists of extracting all overlapping patches $\yb_1,\ldots,\yb_n$.
Then, we perform the centering operation for every patch
\begin{equation}
\yb_i^{c} \triangleq \yb_i - \mu_i \ones_m ~~~\text{with}~~~~ \mu_i \triangleq \frac{1}{m} \ones_m^\top \yb_i.\label{eq:centering}
\end{equation}
The mean value~$\mu_i$ is recorded and added back after denoising $\yb_i^c$. Hence,
low-frequency components do not flow through the model.
The centering step is not used in~\cite{simon2019rethinking}, but we have
found it to be useful.
The next step consists of sparsely encoding each centered patch $\yb_i^c$ with
$K$ steps of the LISTA variant presented in~(\ref{eq:recurrent}), replacing $\yb_i$
by $\yb_i^c$ there, assuming the parameters $\Db, \Cb$ and $\Lambdab_{k}$ are
given.
Here, a minor change compared to~\cite{simon2019rethinking}
is the use of varying parameters $\Lambdab_{k}$ at each LISTA step.
Finally, the final image is obtained by averaging the patch estimates as in~(\ref{eq:averaging}),
after adding back $\mu_i$:
\begin{equation}
\hat{\xb} = \frac{1}{n} \sum_{i=1}^N \Rb_i (\Wb \alphab_i^{(K)} + \mu_i \ones_m), \label{eq:averaging2}
\end{equation}
but the dictionary~$\Db$ is replaced by another matrix~$\Wb$.
The reason for decoupling $\Db$ from $\Wb$ is that the $\ell_1$ penalty used by the LISTA method is known to shrink the coefficients~$\alphab_i$
too much.
For this reason, classical
denoising approaches such as~\cite{elad2006image,mairal2009non} use instead the $\ell_0$-penalty,
but we have found it ineffective for
end-to-end training. Therefore, as in~\cite{simon2019rethinking},
we have chosen
to decouple $\Wb$ from~$\Db$.
\vsp
\paragraph{Training the parameters.}
We now assume that we are given a training set of pairs of
clean/noisy images $(\xb,\yb) \sim \Pcal$, and we minimize in a supervised fashion
\begin{equation}
\min _{\Thetab} \EX_{(\xb,\yb) \sim \Pcal} \left\| \hat{\xb}(\yb) - \xb\right\|_2^2,\label{eq:cost}
\end{equation}
where $\Thetab =
\{\Cb,\Db,\Wb,(\Lambdab_{k})_{k=0,1 \dots K-1}, \kappab, \nu \}$ is the set of parameters to learn and $\hat{\xb}$ is the denoised image defined in~(\ref{eq:averaging2}).
\subsection{Differentiable Relaxation for Non-Local Sparse Priors}
\begin{algorithm}
\caption{Pseudo code for the inference model of GroupSC.}\label{alg:pseudo}
\begin{algorithmic}[1]
\State Extract patches $\Yb=[\yb_1,\ldots,\yb_n]$ and center them with~(\ref{eq:centering});
\State Initialize the codes $\alphab_i$ to $0$;
\State Initialize image estimate $\hat{\xb}$ to the noisy input $\yb$;
\State Initialize pairwise similarities~$\Sigmab$ between patches of $\hat{\xb}$;
\For {$k=1,2,\ldots K$}
\State Compute pairwise patch similarities~$\hat{\Sigmab}$ on~$\hat{\xb}$;
\State Update $\Sigmab \leftarrow (1-\nu) \Sigmab + \nu \hat{\Sigmab}$;
\For {$i=1,2,\ldots,N$ in parallel}
\State $\alphab_i \leftarrow \prox_{\Sigmab, \Lambdab_k} \left [\alphab_i+ \Cb^\top(\yb_i^c-\Db\alphab_i) \right]$;
\EndFor
\State Update the denoised image~$\hat{\xb}$ by averaging~(\ref{eq:averaging2});
\EndFor
\end{algorithmic}
\end{algorithm}
Self-similarities are modeled by replacing the $\ell_1$-norm by
structured sparsity-inducing regularization functions. In Algorithm~\ref{alg:pseudo}, we
present a generic approach to use this principle within a supervised learning approach,
based on a similarity matrix $\Sigmab$,
overcoming the difficulty of hard clustering/grouping patches together. In Figure~\ref{fig:diagram}, we also provide a diagram of one step of the inference algorithm.
At each step, the method computes pairwise patch similarities~$\Sigmab$
between patches of a current estimate~$\hat{\xb}$, using various possible
metrics that we discuss in Section~\ref{sec:similarities}.
The codes~$\alphab_i$ are updated by computing a so-called proximal operator, defined below, for a particular
penalty that depends on $\Sigmab$ and some parameters $\Lambdab_k$. Practical variants
where the pairwise similarities are only updated once in a while, are discussed in Section~\ref{sec:extensions}.
\begin{definition}[Proximal operator]
Given a convex function~$\Psi: \Real^p \!\to\! \Real$, the proximal operator of $\Psi$ is defined~as the unique solution of
\begin{equation}
\prox_{\Psi}[\zb] = \argmin_{\ub \in \Real^p} \frac{1}{2}\|\zb-\ub\|^2 + \Psi(\ub).
\end{equation}
\end{definition}
The proximal operator plays a key role in optimization and admits a closed form
for many penalties, see~\cite{mairal2014sparse}.
Indeed, given~$\Psi$, it may be shown that the iterations
$\alphab_i \leftarrow \prox_{\eta\Psi} \left [\alphab_i+ \eta \Db^\top(\yb_i^c-\Db\alphab_i) \right]$
are instances of the ISTA algorithm~\cite{beck2009fast} for minimizing
$$ \min_{\alphab_i \in \Real^p} \frac{1}{2}\|\yb_i^c - \Db\alphab_i\|^2 + \Psi(\alphab_i),$$
and the update of $\alphab_i$ in Algorithm~\ref{alg:pseudo} simply extend LISTA to deal with~$\Psi$.
Note that for the weighted $\ell_1$-norm $\Psi(\ub) = \sum_{j=1}^p \lambda_j \left | \ub[j] \right | $, the proximal operator is the soft-thresholding
operator $S_{\Lambda}$ introduced in Section~\ref{sec:related} for $\Lambdab = (\lambda_1,\ldots,\lambda_p)$ in $\Real^p$,
and we simply recover the SC algorithm from Section~\ref{subsec:sc} since $\Psi$ does not depend on the pairwise similarities~$\Sigmab$. Next, we present different structured sparsity-inducing penalties that yield more effective algorithms.
\vsp
\subsubsection{Group-SC.}\label{subsec:groupsc}
For each location~$i$, the LSSC approach~\cite{mairal2009non} defines
groups of similar patches
$S_i \triangleq \left \{ j=1, \dots, n \: \text{s.t.} \: \|\yb_i-\yb_j||_2^2 \leq \xi \right \}$ for some threshold $\xi$.
For computational reasons,
LSSC relaxes this definition in practice, and implements a clustering method
such that $S_i=S_j$ if $i$ and $j$ belong to the same group.
Then, under this clustering assumption and given a dictionary~$\Db$, LSSC minimizes
\begin{equation} \label{optim_pblm}
\min_{\Ab} \frac{1}{2} \|\Yb^c-\Db \Ab\|_{\text{F}}^2 + \sum_{i=1}^N \Psi_i(\Ab) ~~\text{with}~~ \Psi_i(\Ab)\!=\!\lambda_i\|\Ab_i\|_{1,2},
\end{equation}
where $\Ab \!=\![\alphab_1,\ldots,\alphab_N]$ in $\Real^{m \times N}$ represents all codes, $\Ab_i\!=\![\alphab_l]_{l \in S_i}$,
$\|.\|_{1,2}$ is the group sparsity regularizer defined in~(\ref{groupeq}), $\|.\|_{\text{F}}$ is the Frobenius norm, $\Yb^c=[\yb_1^c,\ldots,\yb_N^c]$, and $\lambda_i$ depends on the group size.
As explained in Section~\ref{sec:related}, the role of the Group Lasso penalty is to encourage the codes $\alphab_j$ belonging to the same cluster to share the same sparsity pattern, see Figure~\ref{fig:sparse}.
For homogeneity reasons, we also consider the normalization factor $\lambda_i={\lambda}/{\sqrt{|S_i|}}$, as in~\cite{mairal2009non}.
Minimizing~(\ref{optim_pblm}) is easy with the ISTA method since we know how to compute the proximal operator of $\Psi$, which is described below:
\begin{lemma}[Proximal operator for the Group Lasso]
Consider a matrix~$\Ub$ and call $\Zb=\prox_{\lambda\|.\|_{1,2}}[\Ub]$. Then, for all row $\Zb^j$ of $\Zb$,
\begin{equation}
\Zb^j = \max\left (1- \frac{ \lambda }{\|\Ub^j\|_2} ,0\right) \Ub^j.
\end{equation}
\end{lemma}
Unfortunately, the procedure used to design the groups~$S_i$ does not yield a
differentiable relation between the denoised image~$\hat{\xb}$ and the
parameters to learn. Therefore, we
relax the hard clustering assumption into a soft one, which is able to exploit
a similarity matrix~$\Sigmab$ representing pairwise relations between patches. Details about $\Sigmab$ are given in Section~\ref{sec:similarities}.
Yet, such a relaxation does not provide distinct groups of patches, preventing us from
using the Group Lasso penalty~(\ref{optim_pblm}).
This difficulty may be solved by introducing a joint relaxation of the Group Lasso penalty
and its proximal operator. First, we consider a similarity matrix~$\Sigmab$ that encodes the hard clustering assignment
used by LSSC---that is, $\Sigmab_{ij} = 1$ if $j$ is in~$S_i$ and~$0$ otherwise. Second, we note
that $\|\Ab_i\|_{1,2}=\|\Ab\diag(\Sigmab_i)\|_{1,2}$ where $\Sigmab_i$ is the $i$-th column of $\Sigmab$ that encodes
the $i$-th cluster membership.
Then, we adapt LISTA to problem~(\ref{optim_pblm}), with a different shrinkage parameter $\Lambdab^{(k)}_j$ per coordinate~$j$ and per iteration~$k$ as in Section~\ref{subsec:sc}, which yields
\begin{equation}
\begin{split}
\Bb & \leftarrow \Ab^{(k)} + \Cb^\top(\Yb^c-\Db\Ab^{(k)}) \\
\Ab^{(k+1)}_{ij} & \leftarrow \max\left (1- \frac{ \Lambda_j^{(k)} \sqrt{\|\Sigmab_i\|_1}}{\|(\Bb\diag(\Sigmab_i)^{\frac{1}{2}})^j\|_2} ,0\right) \Bb_{ij}, \\
\end{split}\label{eq:lista_group}
\end{equation}
where the second update is performed for all $i,j$, the superscript $^j$ denotes the $j$-th row of a matrix, as above, and $\Ab_{ij}$ is simply the $j$-th entry of $\alphab_i$.
We are now in shape to relax the hard clustering assumption by allowing any
similarity matrix $\Sigmab$ in~(\ref{eq:lista_group}), leading to a relaxation of the Group Lasso penalty in Algorithm~\ref{alg:pseudo}. The resulting model is able to encourage similar patches to share similar sparsity patterns, while being trainable by minimization of the cost~(\ref{eq:cost}).
\subsection{Similarity Metrics}\label{sec:similarities}
We have computed similarities~$\Sigmab$ in various manners,
and implemented the following practical heuristics, which improve the computional complexity.
\paragraph{Online averaging of similarity matrices.}
As shown in Algorithm~\ref{alg:pseudo}, we use a convex combination of similarity matrices (using~$\nu_k$ in $[0,1]$, also learned by backpropagation),
which provides better results than computing the similarity on the current estimate only. This is expected since the current estimate~$\hat{\xb}$ may have lost too much signal information to compute accurately similarities,
whereas online averaging allows retaining information from the original signal.
We run an ablation study of our model reported in appendix to illustrate the need of similarity refinements during the iterations.
When they are no updates the model perfoms on average 0.15 dB lower than with 4 updates.
\vsp
\paragraph{Semi-local grouping.} As in all methods that exploit non-local self
similarities in images, we restrict the search for similar patches to $\yb_i$
to a window of size $w \times w$ centered around the patch. This approach is
commonly used to reduce the size of the similarity matrix and the global memory
cost of the method. This means that we will always have $\Sigmab_{ij}=0$ if pixels~$i$ and~$j$
are too far apart.
\vsp
\paragraph{Learned distance.}
We always use a similarity function of the
form~$\Sigmab_{ij}=e^{-{d_{ij}}}$, where $d_{ij}$ is a distance
between patches~$i$ and~$j$. As in classical deep learning models using non-local
approaches~\cite{liu2018non}, we do not directly use the $\ell_2$ distance
between patches. Specifically, we consider
\begin{equation}
d_{ij} = \|\diag(\kappab)(\hat{\xb}_i - \hat{\xb}_j)\|^2,
\end{equation}
where $\hat{\xb}_i$ and $\hat{\xb}_j$ are the $i$ and $j$-th patches from the current denoised image,
and $\kappab$ in $\Real^m$ is a set of weights, which are learned by backpropagation.
\subsection{Extension to Blind Denoising and Parameter Sharing}
The regularization parameter $\lambda$ of Eq. (\ref{eq:l1_problem}) depends on the noise level.
In a blind denoising setting, it is possible to learn a shared set of dictionnaries $\{\Db,\Cb,\Wb\}$
and a set of different regularization parameters $\{\Lambda_{\sigma_0}, \dots, \Lambda_{\sigma_n} \}$ for various noise intensities.
At inference time, we use first a noise estimation algorithm from \cite{liu2013single} and then select the best regularization parameter to restore the image.
\subsection{Extension to Demosaicking}
\input{demosaicking.tex}
\subsection{Practical variants and implementation}\label{sec:extensions}
Finally, we discuss other practical variants
and implementation details.
\vsp
\paragraph{Dictionary initialization.}
A benefit of designing an architecture with a sparse coding
interpretation, is that the parameters $\Db,\Cb,\Wb$ can be initialized with a
classical dictionary learning approach, instead of using random weights, which
makes the initialization robust. To do so, we use SPAMS toolbox
\cite{mairal2010online}.
\vsp
\paragraph{Block processing and dealing with border effects.}
The size of the tensor $\Sigmab$ grows quadratically with the image size, which
requires processing sequentially image blocks.
Here, the block size is chosen to match the size~$w$ of the non local
window, which requires taking into account two important details:
(i) Pixels close to the image border belong to fewer patches than those
from the center, and thus receive less estimates in the averaging procedure.
When processing images per block, it is thus important to have a small overlap
between blocks, such that the number of estimates per pixel is consistent across the image.
(ii) We also process image blocks for training. It then is important to
take border effects into account, by rescaling the loss by the number of pixel estimates.
\subsection{Proof of propostion \ref{csr_prox}}
The proximal operator of the function $\Psi_i(\ub) = \| \ub \|_1 + \gamma \| \ub - \betab_i \|_1$ for $\ub$ in~$\Real^p$ is defined as
\begin{equation*} \label{prox_psi}
\prox_{\lambda \Psi_i} [\zb]= \argmin_{\ub \in \Real^p} \frac{1}{2}\|\zb-\ub\|^2 + \lambda \|\ub\|_1 + \lambda \gamma \| \ub - \betab_i\|_1
\end{equation*}
The optimality condition for the previous problem is
$$ 0 \in \triangledown (\frac{1}{2} || \zb - \ub ||_2^2) + \partial (\lambda ||\ub||_1) + \partial (\lambda \gamma ||\ub - \betab_i||_1) $$
$$\Leftrightarrow 0 \in \ub - \zb + \lambda \partial ||\ub||_1 + \lambda \gamma \partial ||\ub - \betab_i||_1 $$
We consider each component separately. We suppose that $\betab_i[j] \neq 0$, otherwise $\Psi_i(\ub)[j]$ boils down to the $\ell_1$ norm. And we also suppose $\lambda,\gamma >0$.
Let's examine the first case where $u[j] = 0$. The subdifferential of the $\ell_1$ norm is the interval $[-1,1]$ and the optimality condition is
\begin{align*}
0 \in \ub[j] - \zb[j] + [-\lambda,\lambda] + \lambda \gamma \sign(\ub[j]-\betab_i[j]) \\
\Leftrightarrow \zb[j] \in [-\lambda,\lambda] - \lambda \gamma \sign(\betab_i[j])
\end{align*}
Similarly if $\ub[j] = \betab_i[j]$
\begin{equation*}
\zb[j] \in \betab_i[j] + \lambda \sign(\betab_i[j]) + [-\lambda \gamma,\lambda \gamma]
\end{equation*}
Finally let's examine the case where $u[j] \neq 0$ and $u[j] \neq \betab_i[j]$: then, $\partial ||\ub||_1 = \sign(\ub[j])$ and $\partial ||\ub - \betab_i||_1 = \sign(\ub[j] - \betab_i[j])$. The minimum $u[j]^*$ is obtained as
\begin{align*}
0 = \ub[j] - \zb[j] + \lambda \sign(\ub[j]) + \lambda \gamma \sign(\ub[j]-\betab_i[j]) \\
\Leftrightarrow \ub[j]^* = \zb[j] - \lambda \sign(\ub[j]^*) - \lambda \gamma \sign(\ub[j]^*-\betab_i[j])
\end{align*}
We study separately the cases where $\ub[j]>\betab[j]$, $0<\ub[j]<\betab[j]$ and $\ub[j]<0$ when $\betab_i[j]>0$ and proceed similarly when $\betab_i<0$. With elementary operations we can derive the expression of $\zb[j]$ for each case. Putting the cases all together we obtain the formula.
\subsection{Additional experimental details}
In order to accelerate the inference time of the non-local models, we update patch similarities every $1/f$ steps. Where $f$ is the frequency of the correlation updates.
We summarize in Table \ref{hyperparam} the set of hyperparameters that we selected for the experiments reported in the main tables.
We selected the same hyper-parameters for the baselines, except that we do not compute pairwise patch similarities.
\begin{table}[h]
\small
\centering
\caption{Hyper-parameters of our experiments.}
\label{hyperparam}
\begin{tabular}{lccc}
\toprule
Experiment & Color d. & Gray d. & Demosaicking \\
\midrule
Patch size & 7 & 9 & 9 \\
Dictionnary size &256 & 256 & 256 \\
Nr epochs & 300 & 300 & 200 \\
Batch size & 25 & 25 & 16 \\
$K$ iterations & 24 & 24 & 24 \\
\begin{tabular}[c]{@{}l@{}}Correlation update\\ frequency $f$\end{tabular} & ${1}/{6}$ & ${1}/{6}$ & ${1}/{6}$ \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Influence of patch and dictionnary size}
We investigate in Table \ref{ablation_table} the influence of two hyperparameters: the patch size and the dictionnary size for grayscale image denoising. For this experiment we run a lighter version of the model groupSC, in order to accelerate the training. The batch size was decreased from 25 to 16, the frequency of the correlation updates was decreased from $1/6$ to $1/8$ and the intermediate patches are not approximated with averaging. These changes accelerate the training but lead to slightly lower performances when compared with the model trained in the standard setting. It explains the gap between the scores in Table ~\ref{ablation_table} and in Table~\ref{gray_table}.
\begin{table}[h]
\small
\centering
\caption{Influence of the dictionnary size and the patch size on the denoising performance. Grayscale denoising on BSD68. Models are trained on BSD400. Models are trained in a light setting to accelerate the training.}
\label{ablation_table}
\begin{tabular}{clccc}
\toprule
Noise ($\sigma$) & Patch size & n=128 & n=256 & 512 \\
\midrule
\multirow{3}{*}{$5$}
& k=7 & 37.91 & 37.92 & - \\
& k=9 & 37.90 & 37.92 & 37.96 \\
& k=11 & 37.89 & 37.89 & - \\
\midrule
\multirow{3}{*}{$15$}
& k=7 & 31.60 & 31.63 & - \\
& k=9 & 31.62 &31.67 & 31.71 \\
& k=11 & 31.63 & 31.67 & - \\
\midrule
\multirow{3}{*}{$25$}
& k=7 & 29.10 & 29.11 & - \\
& k=9 & 29.12 & 29.17 & 29.20 \\
& k=11 & 29.13 & 29.18 &- \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Grayscale denoising: evaluation on multiple datasets}
We provide additional grayscale denoising results on other datasets in term of PSNR of our model in Table \ref{gray_app}.
\begin{table}[h]
\small
\centering
\caption{Grayscale denoising on different datasets.Training on BSD400. Performance is measured in terms of average PSNR (in dB). }
\label{gray_app}
\begin{tabular}{cccccc}
\toprule
Dataset & Noise & BM3D & DnCnn & NLRN & {GroupSC} \\
\midrule
\multirow{4}{*}{\textbf{Set12}}
& 5 & - &-& - & 38.40 \\
& 15 & 32.37 & 32.86 & 33.16 & 32.85 \\
& 25 & 29.97 & 30.44 & 30.80 & 30.44 \\
& 50 & 26.72 & 27.18 & 27.64 & 27.14 \\
\midrule
\multirow{4}{*}{\textbf{BSD68}} & 5 & 37.57& 37.68& {37.92} & 37.95 \\
& 15 & 31.07 & 31.73 & 31.88 & 31.70 \\
& 25 & 28.57 & 29.23 & 29.41 & 29.20 \\
& 50 & 25.62 & 26.23 & 26.47 & 26.18 \\
\midrule
\multirow{4}{*}{\textbf{Urban100}}
& 5 & - & - & - & 38.51 \\
& 15 & 32.35 & 32.68 & 33.45 & 32.71 \\
& 25 & 29.70 & 29.91 & 30.94 & 30.05 \\
& 50 & 25.95 & 26.28 & 27.49 & 26.44 \\ \bottomrule
\end{tabular}
\end{table}
\subsection{Color denoising: evaluation on multiple datasets}
We provide additional color denoising results on other datasets in term of PSNR of our model in Table~\ref{color_app}.
\begin{table}[]
\small
\centering
\caption{Color denoising on different datasets.Training on CBSD400. Performance is measured in terms of average PSNR (in dB). }
\label{color_app}
\begin{tabular}{cccc}
\toprule
Dataset & Noise& CBM3D & {GroupSC} \\ \midrule
\multirow{4}{*}{\textbf{Kodak24}}
& 5 & - & 40.72 \\
& 15 & 33.25 & 34.98 \\
& 25 & 32.06 & 32.44 \\
& 50 & 28.75 & 29.16 \\
\midrule \multirow{4}{*}{\textbf{CBSD68}}
& 5 & 40.24 & 40.61 \\
& 15 & 33.49 & 34.10 \\
& 25 & 30.68 & 31.42 \\
& 50 & 27.36 & 28.03 \\
\midrule \multirow{4}{*}{\textbf{Urban100}}
& 5 & - & 39.74 \\
& 15 & 33.22 & 34.11 \\
& 25 & 30.59 & 31.63 \\
& 50 & 26.59 & 28.20 \\ \bottomrule
\end{tabular}
\end{table}
\section{Introduction}
\input{teaser}
\input{intro.tex}
\section{Preliminaries and Related Work}\label{sec:related}
\input{related}
\section{Proposed Approach}
\input{method.tex}
\section{Experiments}\label{sec:exp}
\input{figure_colore}
\input{tables_main}
\input{experiments_eccv}
\section{Conclusion}
\input{conclusion.tex}
\section*{Acknowledgements}
JM and BL were supported by the ERC grant number 714381 (SOLARIS project)
and by ANR 3IA MIAI@Grenoble Alpes (ANR-19-P3IA-0003).
JP was supported in part by the Louis Vuitton/ENS chair in artificial intelligence and the Inria/NYU collaboration.
In addition, this work was funded in part by the French government under management of Agence Nationale de la Recherche
as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute)
and was performed using HPC resources from GENCI–IDRIS (Grant 2020-AD011011252).
\bibliographystyle{splncs04}
\section{Introduction}
\input{teaser}
\input{intro.tex}
\section{Preliminaries and Related Work}\label{sec:related}
\input{related}
\section{Proposed Approach}
\input{method.tex}
\section{Experiments}\label{sec:exp}
\input{figure_colore}
\input{tables_main}
\input{experiments_eccv}
\section{Conclusion}
\input{conclusion.tex}
\section*{Acknowledgements}
JM and BL were supported by the ERC grant number 714381 (SOLARIS project)
and by ANR 3IA MIAI@Grenoble Alpes (ANR-19-P3IA-0003).
JP was supported in part by the Louis Vuitton/ENS chair in artificial intelligence and the Inria/NYU collaboration.
In addition, this work was funded in part by the French government under management of Agence Nationale de la Recherche
as part of the "Investissements d'avenir" program, reference ANR-19-P3IA-0001 (PRAIRIE 3IA Institute)
and was performed using HPC resources from GENCI–IDRIS (Grant 2020-AD011011252).
\bibliographystyle{splncs04}
\subsection{Trainable Sparse Coding (without Self-Similarities)}\label{subsec:sc}
In \cite{simon2019rethinking}, the sparse coding approach (SC) is combined with the LISTA algorithm to perform
denoising tasks.\footnote{Specifically, \cite{simon2019rethinking} proposes a model based on
convolutional sparse coding (CSC).
CSC is a variant of SC, where a full image is
approximated by a linear combination of small dictionary elements. Unfortunately, CSC leads to
ill-conditioned optimization problems and has shown to perform poorly
for image denoising. For this reason,~\cite{simon2019rethinking} introduces
a hybrid approach between SC and CSC.
In our paper, we have decided to use the SC baseline and leave the investigation of CSC models for
future work.}
The only modification we introduce here is a centering step
for the patches, which empirically yields better results.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figures/fig1_c.pdf}
\caption{An illustration of the main inference algorithm for GroupSC.
See Figure \ref{fig:nlmodule} for an illustration of the self-similarity module.}\label{fig:diagram}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.80\linewidth]{figures/fig2_c.pdf}
\caption{An illustration of the self-similarity module used in our GroupSC algorithm.}\label{fig:nlmodule}
\end{figure}
\vsp
\paragraph{SC Model - inference with fixed parameters.}
Following the approach and notation from Section~\ref{sec:related},
the first step consists of extracting all overlapping patches $\yb_1,\ldots,\yb_n$.
Then, we perform the centering operation for every patch
\begin{equation}
\yb_i^{c} \triangleq \yb_i - \mu_i \ones_m ~~~\text{with}~~~~ \mu_i \triangleq \frac{1}{m} \ones_m^\top \yb_i.\label{eq:centering}
\end{equation}
The mean value~$\mu_i$ is recorded and added back after denoising $\yb_i^c$. Hence,
low-frequency components do not flow through the model.
The centering step is not used in~\cite{simon2019rethinking}, but we have
found it to be useful.
The next step consists of sparsely encoding each centered patch $\yb_i^c$ with
$K$ steps of the LISTA variant presented in~(\ref{eq:recurrent}), replacing $\yb_i$
by $\yb_i^c$ there, assuming the parameters $\Db, \Cb$ and $\Lambdab_{k}$ are
given.
Here, a minor change compared to~\cite{simon2019rethinking}
is the use of varying parameters $\Lambdab_{k}$ at each LISTA step.
Finally, the final image is obtained by averaging the patch estimates as in~(\ref{eq:averaging}),
after adding back $\mu_i$:
\begin{equation}
\hat{\xb} = \frac{1}{n} \sum_{i=1}^N \Rb_i (\Wb \alphab_i^{(K)} + \mu_i \ones_m), \label{eq:averaging2}
\end{equation}
but the dictionary~$\Db$ is replaced by another matrix~$\Wb$.
The reason for decoupling $\Db$ from $\Wb$ is that the $\ell_1$ penalty used by the LISTA method is known to shrink the coefficients~$\alphab_i$
too much.
For this reason, classical
denoising approaches such as~\cite{elad2006image,mairal2009non} use instead the $\ell_0$-penalty,
but we have found it ineffective for
end-to-end training. Therefore, as in~\cite{simon2019rethinking},
we have chosen
to decouple $\Wb$ from~$\Db$.
\vsp
\paragraph{Training the parameters.}
We now assume that we are given a training set of pairs of
clean/noisy images $(\xb,\yb) \sim \Pcal$, and we minimize in a supervised fashion
\begin{equation}
\min _{\Thetab} \EX_{(\xb,\yb) \sim \Pcal} \left\| \hat{\xb}(\yb) - \xb\right\|_2^2,\label{eq:cost}
\end{equation}
where $\Thetab =
\{\Cb,\Db,\Wb,(\Lambdab_{k})_{k=0,1 \dots K-1}, \kappab, \nu \}$ is the set of parameters to learn and $\hat{\xb}$ is the denoised image defined in~(\ref{eq:averaging2}).
\subsection{Differentiable Relaxation for Non-Local Sparse Priors}
\begin{algorithm}
\caption{Pseudo code for the inference model of GroupSC.}\label{alg:pseudo}
\begin{algorithmic}[1]
\State Extract patches $\Yb=[\yb_1,\ldots,\yb_n]$ and center them with~(\ref{eq:centering});
\State Initialize the codes $\alphab_i$ to $0$;
\State Initialize image estimate $\hat{\xb}$ to the noisy input $\yb$;
\State Initialize pairwise similarities~$\Sigmab$ between patches of $\hat{\xb}$;
\For {$k=1,2,\ldots K$}
\State Compute pairwise patch similarities~$\hat{\Sigmab}$ on~$\hat{\xb}$;
\State Update $\Sigmab \leftarrow (1-\nu) \Sigmab + \nu \hat{\Sigmab}$;
\For {$i=1,2,\ldots,N$ in parallel}
\State $\alphab_i \leftarrow \prox_{\Sigmab, \Lambdab_k} \left [\alphab_i+ \Cb^\top(\yb_i^c-\Db\alphab_i) \right]$;
\EndFor
\State Update the denoised image~$\hat{\xb}$ by averaging~(\ref{eq:averaging2});
\EndFor
\end{algorithmic}
\end{algorithm}
Self-similarities are modeled by replacing the $\ell_1$-norm by
structured sparsity-inducing regularization functions. In Algorithm~\ref{alg:pseudo}, we
present a generic approach to use this principle within a supervised learning approach,
based on a similarity matrix $\Sigmab$,
overcoming the difficulty of hard clustering/grouping patches together. In Figure~\ref{fig:diagram}, we also provide a diagram of one step of the inference algorithm.
At each step, the method computes pairwise patch similarities~$\Sigmab$
between patches of a current estimate~$\hat{\xb}$, using various possible
metrics that we discuss in Section~\ref{sec:similarities}.
The codes~$\alphab_i$ are updated by computing a so-called proximal operator, defined below, for a particular
penalty that depends on $\Sigmab$ and some parameters $\Lambdab_k$. Practical variants
where the pairwise similarities are only updated once in a while, are discussed in Section~\ref{sec:extensions}.
\begin{definition}[Proximal operator]
Given a convex function~$\Psi: \Real^p \!\to\! \Real$, the proximal operator of $\Psi$ is defined~as the unique solution of
\begin{equation}
\prox_{\Psi}[\zb] = \argmin_{\ub \in \Real^p} \frac{1}{2}\|\zb-\ub\|^2 + \Psi(\ub).
\end{equation}
\end{definition}
The proximal operator plays a key role in optimization and admits a closed form
for many penalties, see~\cite{mairal2014sparse}.
Indeed, given~$\Psi$, it may be shown that the iterations
$\alphab_i \leftarrow \prox_{\eta\Psi} \left [\alphab_i+ \eta \Db^\top(\yb_i^c-\Db\alphab_i) \right]$
are instances of the ISTA algorithm~\cite{beck2009fast} for minimizing
$$ \min_{\alphab_i \in \Real^p} \frac{1}{2}\|\yb_i^c - \Db\alphab_i\|^2 + \Psi(\alphab_i),$$
and the update of $\alphab_i$ in Algorithm~\ref{alg:pseudo} simply extend LISTA to deal with~$\Psi$.
Note that for the weighted $\ell_1$-norm $\Psi(\ub) = \sum_{j=1}^p \lambda_j \left | \ub[j] \right | $, the proximal operator is the soft-thresholding
operator $S_{\Lambda}$ introduced in Section~\ref{sec:related} for $\Lambdab = (\lambda_1,\ldots,\lambda_p)$ in $\Real^p$,
and we simply recover the SC algorithm from Section~\ref{subsec:sc} since $\Psi$ does not depend on the pairwise similarities~$\Sigmab$. Next, we present different structured sparsity-inducing penalties that yield more effective algorithms.
\vsp
\subsubsection{Group-SC.}\label{subsec:groupsc}
For each location~$i$, the LSSC approach~\cite{mairal2009non} defines
groups of similar patches
$S_i \triangleq \left \{ j=1, \dots, n \: \text{s.t.} \: \|\yb_i-\yb_j||_2^2 \leq \xi \right \}$ for some threshold $\xi$.
For computational reasons,
LSSC relaxes this definition in practice, and implements a clustering method
such that $S_i=S_j$ if $i$ and $j$ belong to the same group.
Then, under this clustering assumption and given a dictionary~$\Db$, LSSC minimizes
\begin{equation} \label{optim_pblm}
\min_{\Ab} \frac{1}{2} \|\Yb^c-\Db \Ab\|_{\text{F}}^2 + \sum_{i=1}^N \Psi_i(\Ab) ~~\text{with}~~ \Psi_i(\Ab)\!=\!\lambda_i\|\Ab_i\|_{1,2},
\end{equation}
where $\Ab \!=\![\alphab_1,\ldots,\alphab_N]$ in $\Real^{m \times N}$ represents all codes, $\Ab_i\!=\![\alphab_l]_{l \in S_i}$,
$\|.\|_{1,2}$ is the group sparsity regularizer defined in~(\ref{groupeq}), $\|.\|_{\text{F}}$ is the Frobenius norm, $\Yb^c=[\yb_1^c,\ldots,\yb_N^c]$, and $\lambda_i$ depends on the group size.
As explained in Section~\ref{sec:related}, the role of the Group Lasso penalty is to encourage the codes $\alphab_j$ belonging to the same cluster to share the same sparsity pattern, see Figure~\ref{fig:sparse}.
For homogeneity reasons, we also consider the normalization factor $\lambda_i={\lambda}/{\sqrt{|S_i|}}$, as in~\cite{mairal2009non}.
Minimizing~(\ref{optim_pblm}) is easy with the ISTA method since we know how to compute the proximal operator of $\Psi$, which is described below:
\begin{lemma}[Proximal operator for the Group Lasso]
Consider a matrix~$\Ub$ and call $\Zb=\prox_{\lambda\|.\|_{1,2}}[\Ub]$. Then, for all row $\Zb^j$ of $\Zb$,
\begin{equation}
\Zb^j = \max\left (1- \frac{ \lambda }{\|\Ub^j\|_2} ,0\right) \Ub^j.
\end{equation}
\end{lemma}
Unfortunately, the procedure used to design the groups~$S_i$ does not yield a
differentiable relation between the denoised image~$\hat{\xb}$ and the
parameters to learn. Therefore, we
relax the hard clustering assumption into a soft one, which is able to exploit
a similarity matrix~$\Sigmab$ representing pairwise relations between patches. Details about $\Sigmab$ are given in Section~\ref{sec:similarities}.
Yet, such a relaxation does not provide distinct groups of patches, preventing us from
using the Group Lasso penalty~(\ref{optim_pblm}).
This difficulty may be solved by introducing a joint relaxation of the Group Lasso penalty
and its proximal operator. First, we consider a similarity matrix~$\Sigmab$ that encodes the hard clustering assignment
used by LSSC---that is, $\Sigmab_{ij} = 1$ if $j$ is in~$S_i$ and~$0$ otherwise. Second, we note
that $\|\Ab_i\|_{1,2}=\|\Ab\diag(\Sigmab_i)\|_{1,2}$ where $\Sigmab_i$ is the $i$-th column of $\Sigmab$ that encodes
the $i$-th cluster membership.
Then, we adapt LISTA to problem~(\ref{optim_pblm}), with a different shrinkage parameter $\Lambdab^{(k)}_j$ per coordinate~$j$ and per iteration~$k$ as in Section~\ref{subsec:sc}, which yields
\begin{equation}
\begin{split}
\Bb & \leftarrow \Ab^{(k)} + \Cb^\top(\Yb^c-\Db\Ab^{(k)}) \\
\Ab^{(k+1)}_{ij} & \leftarrow \max\left (1- \frac{ \Lambda_j^{(k)} \sqrt{\|\Sigmab_i\|_1}}{\|(\Bb\diag(\Sigmab_i)^{\frac{1}{2}})^j\|_2} ,0\right) \Bb_{ij}, \\
\end{split}\label{eq:lista_group}
\end{equation}
where the second update is performed for all $i,j$, the superscript $^j$ denotes the $j$-th row of a matrix, as above, and $\Ab_{ij}$ is simply the $j$-th entry of $\alphab_i$.
We are now in shape to relax the hard clustering assumption by allowing any
similarity matrix $\Sigmab$ in~(\ref{eq:lista_group}), leading to a relaxation of the Group Lasso penalty in Algorithm~\ref{alg:pseudo}. The resulting model is able to encourage similar patches to share similar sparsity patterns, while being trainable by minimization of the cost~(\ref{eq:cost}).
\subsection{Similarity Metrics}\label{sec:similarities}
We have computed similarities~$\Sigmab$ in various manners,
and implemented the following practical heuristics, which improve the computional complexity.
\paragraph{Online averaging of similarity matrices.}
As shown in Algorithm~\ref{alg:pseudo}, we use a convex combination of similarity matrices (using~$\nu_k$ in $[0,1]$, also learned by backpropagation),
which provides better results than computing the similarity on the current estimate only. This is expected since the current estimate~$\hat{\xb}$ may have lost too much signal information to compute accurately similarities,
whereas online averaging allows retaining information from the original signal.
We run an ablation study of our model reported in appendix to illustrate the need of similarity refinements during the iterations.
When they are no updates the model perfoms on average 0.15 dB lower than with 4 updates.
\vsp
\paragraph{Semi-local grouping.} As in all methods that exploit non-local self
similarities in images, we restrict the search for similar patches to $\yb_i$
to a window of size $w \times w$ centered around the patch. This approach is
commonly used to reduce the size of the similarity matrix and the global memory
cost of the method. This means that we will always have $\Sigmab_{ij}=0$ if pixels~$i$ and~$j$
are too far apart.
\vsp
\paragraph{Learned distance.}
We always use a similarity function of the
form~$\Sigmab_{ij}=e^{-{d_{ij}}}$, where $d_{ij}$ is a distance
between patches~$i$ and~$j$. As in classical deep learning models using non-local
approaches~\cite{liu2018non}, we do not directly use the $\ell_2$ distance
between patches. Specifically, we consider
\begin{equation}
d_{ij} = \|\diag(\kappab)(\hat{\xb}_i - \hat{\xb}_j)\|^2,
\end{equation}
where $\hat{\xb}_i$ and $\hat{\xb}_j$ are the $i$ and $j$-th patches from the current denoised image,
and $\kappab$ in $\Real^m$ is a set of weights, which are learned by backpropagation.
\subsection{Extension to Blind Denoising and Parameter Sharing}
The regularization parameter $\lambda$ of Eq. (\ref{eq:l1_problem}) depends on the noise level.
In a blind denoising setting, it is possible to learn a shared set of dictionnaries $\{\Db,\Cb,\Wb\}$
and a set of different regularization parameters $\{\Lambda_{\sigma_0}, \dots, \Lambda_{\sigma_n} \}$ for various noise intensities.
At inference time, we use first a noise estimation algorithm from \cite{liu2013single} and then select the best regularization parameter to restore the image.
\subsection{Extension to Demosaicking}
\input{demosaicking.tex}
\subsection{Practical variants and implementation}\label{sec:extensions}
Finally, we discuss other practical variants
and implementation details.
\vsp
\paragraph{Dictionary initialization.}
A benefit of designing an architecture with a sparse coding
interpretation, is that the parameters $\Db,\Cb,\Wb$ can be initialized with a
classical dictionary learning approach, instead of using random weights, which
makes the initialization robust. To do so, we use SPAMS toolbox
\cite{mairal2010online}.
\vsp
\paragraph{Block processing and dealing with border effects.}
The size of the tensor $\Sigmab$ grows quadratically with the image size, which
requires processing sequentially image blocks.
Here, the block size is chosen to match the size~$w$ of the non local
window, which requires taking into account two important details:
(i) Pixels close to the image border belong to fewer patches than those
from the center, and thus receive less estimates in the averaging procedure.
When processing images per block, it is thus important to have a small overlap
between blocks, such that the number of estimates per pixel is consistent across the image.
(ii) We also process image blocks for training. It then is important to
take border effects into account, by rescaling the loss by the number of pixel estimates.
\subsection{Proof of propostion \ref{csr_prox}}
The proximal operator of the function $\Psi_i(\ub) = \| \ub \|_1 + \gamma \| \ub - \betab_i \|_1$ for $\ub$ in~$\Real^p$ is defined as
\begin{equation*} \label{prox_psi}
\prox_{\lambda \Psi_i} [\zb]= \argmin_{\ub \in \Real^p} \frac{1}{2}\|\zb-\ub\|^2 + \lambda \|\ub\|_1 + \lambda \gamma \| \ub - \betab_i\|_1
\end{equation*}
The optimality condition for the previous problem is
$$ 0 \in \triangledown (\frac{1}{2} || \zb - \ub ||_2^2) + \partial (\lambda ||\ub||_1) + \partial (\lambda \gamma ||\ub - \betab_i||_1) $$
$$\Leftrightarrow 0 \in \ub - \zb + \lambda \partial ||\ub||_1 + \lambda \gamma \partial ||\ub - \betab_i||_1 $$
We consider each component separately. We suppose that $\betab_i[j] \neq 0$, otherwise $\Psi_i(\ub)[j]$ boils down to the $\ell_1$ norm. And we also suppose $\lambda,\gamma >0$.
Let's examine the first case where $u[j] = 0$. The subdifferential of the $\ell_1$ norm is the interval $[-1,1]$ and the optimality condition is
\begin{align*}
0 \in \ub[j] - \zb[j] + [-\lambda,\lambda] + \lambda \gamma \sign(\ub[j]-\betab_i[j]) \\
\Leftrightarrow \zb[j] \in [-\lambda,\lambda] - \lambda \gamma \sign(\betab_i[j])
\end{align*}
Similarly if $\ub[j] = \betab_i[j]$
\begin{equation*}
\zb[j] \in \betab_i[j] + \lambda \sign(\betab_i[j]) + [-\lambda \gamma,\lambda \gamma]
\end{equation*}
Finally let's examine the case where $u[j] \neq 0$ and $u[j] \neq \betab_i[j]$: then, $\partial ||\ub||_1 = \sign(\ub[j])$ and $\partial ||\ub - \betab_i||_1 = \sign(\ub[j] - \betab_i[j])$. The minimum $u[j]^*$ is obtained as
\begin{align*}
0 = \ub[j] - \zb[j] + \lambda \sign(\ub[j]) + \lambda \gamma \sign(\ub[j]-\betab_i[j]) \\
\Leftrightarrow \ub[j]^* = \zb[j] - \lambda \sign(\ub[j]^*) - \lambda \gamma \sign(\ub[j]^*-\betab_i[j])
\end{align*}
We study separately the cases where $\ub[j]>\betab[j]$, $0<\ub[j]<\betab[j]$ and $\ub[j]<0$ when $\betab_i[j]>0$ and proceed similarly when $\betab_i<0$. With elementary operations we can derive the expression of $\zb[j]$ for each case. Putting the cases all together we obtain the formula.
\subsection{Additional experimental details}
In order to accelerate the inference time of the non-local models, we update patch similarities every $1/f$ steps. Where $f$ is the frequency of the correlation updates.
We summarize in Table \ref{hyperparam} the set of hyperparameters that we selected for the experiments reported in the main tables.
We selected the same hyper-parameters for the baselines, except that we do not compute pairwise patch similarities.
\begin{table}[h]
\small
\centering
\caption{Hyper-parameters of our experiments.}
\label{hyperparam}
\begin{tabular}{lccc}
\toprule
Experiment & Color d. & Gray d. & Demosaicking \\
\midrule
Patch size & 7 & 9 & 9 \\
Dictionnary size &256 & 256 & 256 \\
Nr epochs & 300 & 300 & 200 \\
Batch size & 25 & 25 & 16 \\
$K$ iterations & 24 & 24 & 24 \\
\begin{tabular}[c]{@{}l@{}}Correlation update\\ frequency $f$\end{tabular} & ${1}/{6}$ & ${1}/{6}$ & ${1}/{6}$ \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Influence of patch and dictionnary size}
We investigate in Table \ref{ablation_table} the influence of two hyperparameters: the patch size and the dictionnary size for grayscale image denoising. For this experiment we run a lighter version of the model groupSC, in order to accelerate the training. The batch size was decreased from 25 to 16, the frequency of the correlation updates was decreased from $1/6$ to $1/8$ and the intermediate patches are not approximated with averaging. These changes accelerate the training but lead to slightly lower performances when compared with the model trained in the standard setting. It explains the gap between the scores in Table ~\ref{ablation_table} and in Table~\ref{gray_table}.
\begin{table}[h]
\small
\centering
\caption{Influence of the dictionnary size and the patch size on the denoising performance. Grayscale denoising on BSD68. Models are trained on BSD400. Models are trained in a light setting to accelerate the training.}
\label{ablation_table}
\begin{tabular}{clccc}
\toprule
Noise ($\sigma$) & Patch size & n=128 & n=256 & 512 \\
\midrule
\multirow{3}{*}{$5$}
& k=7 & 37.91 & 37.92 & - \\
& k=9 & 37.90 & 37.92 & 37.96 \\
& k=11 & 37.89 & 37.89 & - \\
\midrule
\multirow{3}{*}{$15$}
& k=7 & 31.60 & 31.63 & - \\
& k=9 & 31.62 &31.67 & 31.71 \\
& k=11 & 31.63 & 31.67 & - \\
\midrule
\multirow{3}{*}{$25$}
& k=7 & 29.10 & 29.11 & - \\
& k=9 & 29.12 & 29.17 & 29.20 \\
& k=11 & 29.13 & 29.18 &- \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Grayscale denoising: evaluation on multiple datasets}
We provide additional grayscale denoising results on other datasets in term of PSNR of our model in Table \ref{gray_app}.
\begin{table}[h]
\small
\centering
\caption{Grayscale denoising on different datasets.Training on BSD400. Performance is measured in terms of average PSNR (in dB). }
\label{gray_app}
\begin{tabular}{cccccc}
\toprule
Dataset & Noise & BM3D & DnCnn & NLRN & {GroupSC} \\
\midrule
\multirow{4}{*}{\textbf{Set12}}
& 5 & - &-& - & 38.40 \\
& 15 & 32.37 & 32.86 & 33.16 & 32.85 \\
& 25 & 29.97 & 30.44 & 30.80 & 30.44 \\
& 50 & 26.72 & 27.18 & 27.64 & 27.14 \\
\midrule
\multirow{4}{*}{\textbf{BSD68}} & 5 & 37.57& 37.68& {37.92} & 37.95 \\
& 15 & 31.07 & 31.73 & 31.88 & 31.70 \\
& 25 & 28.57 & 29.23 & 29.41 & 29.20 \\
& 50 & 25.62 & 26.23 & 26.47 & 26.18 \\
\midrule
\multirow{4}{*}{\textbf{Urban100}}
& 5 & - & - & - & 38.51 \\
& 15 & 32.35 & 32.68 & 33.45 & 32.71 \\
& 25 & 29.70 & 29.91 & 30.94 & 30.05 \\
& 50 & 25.95 & 26.28 & 27.49 & 26.44 \\ \bottomrule
\end{tabular}
\end{table}
\subsection{Color denoising: evaluation on multiple datasets}
We provide additional color denoising results on other datasets in term of PSNR of our model in Table~\ref{color_app}.
\begin{table}[]
\small
\centering
\caption{Color denoising on different datasets.Training on CBSD400. Performance is measured in terms of average PSNR (in dB). }
\label{color_app}
\begin{tabular}{cccc}
\toprule
Dataset & Noise& CBM3D & {GroupSC} \\ \midrule
\multirow{4}{*}{\textbf{Kodak24}}
& 5 & - & 40.72 \\
& 15 & 33.25 & 34.98 \\
& 25 & 32.06 & 32.44 \\
& 50 & 28.75 & 29.16 \\
\midrule \multirow{4}{*}{\textbf{CBSD68}}
& 5 & 40.24 & 40.61 \\
& 15 & 33.49 & 34.10 \\
& 25 & 30.68 & 31.42 \\
& 50 & 27.36 & 28.03 \\
\midrule \multirow{4}{*}{\textbf{Urban100}}
& 5 & - & 39.74 \\
& 15 & 33.22 & 34.11 \\
& 25 & 30.59 & 31.63 \\
& 50 & 26.59 & 28.20 \\ \bottomrule
\end{tabular}
\end{table}
|
1,108,101,565,648 | arxiv | \section*{Introduction}
\ssec{} Let $G$ be a simple complex algebraic group and $B$ its Borel
subgroup. Consider the category $\on{D}(G/B)\mod$ of left D-modules on
the flag variety $G/B$. The Lie algebra $\fg$ of $G$, and hence its
universal enveloping algebra $U(\fg)$, acts on the space
$\Gamma(G/B,{\mc F})$ of global sections of any D-module ${\mc
F}$. The center $Z(\fg)$ of $U(\fg)$ acts on $\Gamma(G/B,{\mc F})$ via
the augmentation character $\chi_0: Z(\fg) \to \C$. Let
$\fg\mod_{\chi_0}$ be the category of $\fg$-modules on which $Z(\fg)$
acts via the character $\chi_0$. Thus, we obtain a functor $$\Gamma:
\on{D}(G/B)\mod \to \fg\mod_{\chi_0}.$$ In \cite{BB} A. Beilinson and
J. Bernstein proved that this functor is an equivalence of
categories. Moreover, they generalized this equivalence to the case of
twisted D-modules, for twistings that correspond to dominant weights
$\lambda \in \fh^*$.
\medskip
Let $N$ be the unipotent radical of $B$. We can consider the
$N$--equivariant subcategories on both sides of the above
equivalence. On the D-module side this is the category
$\on{D}(G/B)\mod^N$ of $N$--equivariant D-modules on $G/B$, and on the
$\fg$-module side this is the block of the category ${\mc O}$
corresponding to the central character $\chi_0$. The resulting
equivalence of categories, which follows from \cite{BB}, and which was
proved independently by J.-L. Brylinski and M. Kashiwara \cite{BK}, is
very important in applications to representation theory of $\fg$.
\medskip
Now let $\ghat$ be the affine Kac-Moody algebra, the universal central
extension of the formal loop agebra $\fg\ppart$. Representations of
$\wh\fg$ have a parameter, an invariant bilinear form on $\fg$, which
is called the level. There is a unique inner product $\kappa_{\can}$
which is normalized so that the square length of the maximal root of
$\fg$ is equal to $2$. Any other inner product is equal to
$\kappa = k\cdot \ka_{\can}$, where $k \in \C$, and so a level
corresponds to a complex number $k$. In particular, it makes sense to
speak of {\em integral levels}. Representations, corresponding to the
bilinear form which is equal to minus one half of the Killing form
(for which $k=-h^\vee$, minus the dual Coxeter number of $\g$) are
called representations of {\em critical level}. This is really the
``middle point'' amongst all levels (and not the zero level, as one
might naively expect).
\medskip
There are several analogues of the flag variety in the affine case. In
this paper (except in the Appendix) we will consider exclusively
the {\em affine Grassmannian} $\Gr_G = G\ppart/G[[t]]$.
Another possibility is to consider the affine flag scheme $\Fl_G =
G\ppart/I$, where $I$ is the Iwahori subgroup of $G\ppart$. Most of
the results of this paper that concern the critical level have
conjectural counterparts for the affine flag variety, but they are
more difficult to formulate. In particular, one inevitably has to
consider derived categories, whereas for the affine Grassmannian
abelian categories suffice. We refer the reader to the Introduction of
our previous paper \cite{FG2} for more details.
\medskip
There is a canonical line bundle $\Ll_{\can}$ on $\Gr_G$ such that the
action of $\fg\ppart$ on $\Gr_G$ lifts to an action of
$\hg_{\kappa_{\can}}$ on $\Ll_{\can}$. For each level $\kappa$ we can
consider the category $\on{D}(\Gr_G)_\kappa\mod$ of right D-modules on
$\Gr_G$ twisted by $\Ll_{\can}^{\otimes k}$, where $\kappa=k\cdot
\kappa_{\can}$. (Recall that although the line bundle $\Ll^{\otimes
k}_{\can}$ only makes sense when $k$ is integral, the corresponding
category of twisted D-module is well-defined for an arbitrary $k$.)
Since $\Gr_G$ is an ind-scheme, the definition of these categories
requires some care (see \cite{BD} and \cite{FG1}).
Let $\hg_\kappa\mod$ be the category of (discrete) modules over
the affine Kac-Moody algebra of level $\kappa$ (see
\secref{recol}). Using the fact that the action of $\fg\ppart$ on
$\Gr_G$ lifts to an action of $\hg_{\kappa_{\can}}$ on $\Ll_{\can}$,
we obtain that for each level $\kappa$ we have a naturally defined
functor of global sections:
\begin{equation} \label{glob sections}
\Gamma: \on{D}(\Gr_G)_\kappa\mod \to \ghat_\kappa\mod.
\end{equation}
The question that we would like to address in this paper is whether
this functor is an equivalence of categories, as in the
finite-dimensional case.
\ssec{} \label{neg level}
The first results in this direction were obtained in \cite{BD,FG1}.
Namely, in {\it loc. cit.} it was shown that if $\kappa$ is such that
$\kappa=k\cdot \kappa_{\can}$ with $k+h^\vee\notin \BQ^{>0}$, then the
functor $\Gamma$ of \eqref{glob sections} is exact and faithful. (In
contrast, it is known that this functor is not exact for $k+h^\vee\in
\BQ^{>0}$.) The condition $k+h^\vee\notin \BQ^{>0}$ is analogous to
the dominant weight condition of \cite{BB}.
\medskip
Let us call $\kappa$ {\it negative} if $k+h^\vee\notin \BQ^{\geq
0}$. In this case one can show that the functor of \eqref{glob
sections} is fully faithful. In fact, in this case it makes more sense
to consider $H$-monodromic twisted D-modules on the enhanced affine
flag scheme $\wt\Fl_G = G\ppart/I^0$, rather than simply twisted
D-modules on $\Gr_G$, and the corresponding functor $\Gamma$ to
$\ghat_\kappa\mod$. The above exactness and fully-faithfulness
assertions are still valid in this context. However, the above functor
is not an equivalence of categories. Namely, the RHS of \eqref{glob
sections} has "many more" objects than the LHS.
When $\kappa$ is integral, A.~Beilinson has proposed a conjectural
intrinsic description of the image of the category
$\on{D}(\wt\Fl_G)_\kappa\mod$ inside $\ghat_\kappa\mod$ (see Remark
(ii) in the Introduction of \cite{Bei}). As far as we know, no such
description was proposed when $\kappa$ is not integral.
\medskip
It is possible, however, to establish a partial result in this
direction. Namely, let $I^0\subset I$ be the unipotent radical of the
Iwahori subgroup $I$. We can consider the category
$\on{D}(\wt\Fl_G)_\kappa\mod^{I^0}$ of $I^0$-equivariant twisted
D-modules on $\wt\Fl_G$. The corresponding functor $\Gamma$ of global
sections takes values in the affine version of category $\CO$, i.e.,
in the subcategory $\ghat_\kappa\mod^{I^0}\subset \ghat_\kappa\mod$,
whose objects are $\ghat_\kappa$-modules on which the action of the
Lie algebra $\on{Lie}(I^0) \subset \hg_\kappa$ integrates to an
algebraic action of the group $I^0$.
One can show that the functor $\Gamma$ induces an equivalence between
an appropriately defined subcategory of $H$-monodromic objects of
$\on{D}(\wt\Fl_G)_\kappa\mod^{I^0}$ and a specific block of
$\ghat_\kappa\mod^{I^0}$. This result, which is well-known to
specialists, is not available in the published literature. For the
sake of completeness, we sketch one of the possible proofs in the
Appendix of this paper.
\ssec{}
In this paper we shall concentrate on the case of the critical level,
when $k=-h^\vee$. We will see that this case is dramatically different
from the cases considered above. In \cite{FG2} we made a precise
conjecture describing the relationship between the corresponding
categories $\on{D}(\Gr_G)_\crit\mod$ and $\ghat_\crit\mod$. We shall
now review the statement of this conjecture.
\medskip
First, let us note that the image of the functor $\Gamma$ is in
a certain subcategory of $\ghat_\crit\mod$, singled out by the
condition on the action of the center.
Let $\fZ_\fg$ denote the center of the category $\ghat_\crit\mod$
(which is the same as the center of the completed enveloping algebra
of $\hg_\crit$). The fact that this center is non-trivial is what
distinguishes the critical level from all other levels. Let
$\fZ^\reg_\fg$ denote the quotient of $\fZ_\fg$, through which it acts
on the vacuum module
$\BV_\crit:=\on{Ind}^{\hg_\crit}_{\fg[[t]] \oplus \BC}(\BC)$.
Let $\ghat_\crit\mod_\reg$ be the full subcategory of
$\ghat_\crit\mod$, whose objects are $\ghat_\crit$-modules on which
the action of the center $\fZ_\fg$ factors through $\fZ^\reg_\fg$. It
is known (see \cite{FG1}) that for any $\CF \in
\on{D}(\Gr_G)_\crit\mod$, the space of global sections
$\Gamma(\Gr_G,\CF)$ is an object of $\ghat_\crit\mod_\reg$. (Here and
below we write $M \in {\mc C}$ if $M$ is an object of a category ${\mc
C}$.) Thus, $\ghat_\crit\mod_\reg$ is the category that may be viewed
as an analogue of the category $\fg\mod_{\chi_0}$ appearing on the
representation theory side of the Beilinson-Bernstein equivalence.
\medskip
However, the functor of global sections $\Gamma:
\on{D}(\Gr_G)_\crit\mod \to \ghat_\crit\mod_\reg$ is not full, and
therefore cannot possibly be an equivalence. The origin of the
non-fullness of $\Gamma$ is two-fold, with one ingredient rather
elementary, and another less so.
First, the category $\ghat_\crit\mod_\reg$ has a large center, namely,
the algebra $\fZ^\reg_\fg$ itself, while the center of the category
$\on{D}(\Gr_G)_\crit\mod$ is the group algebra of the finite group
$\pi_1(G)$ (i.e., it has a basis enumerated by the connected
components of $\Gr_G$).
\medskip
Second, the category $\on{D}(\Gr_G)_\crit\mod$
carries an additional symmetry, namely, an action of the tensor
category $\Rep(\cG)$ of the Langlands dual group $\cG$,
and this action trivializes under the functor $\Gamma$.
In more detail, let us recall that, according to \cite{FF,F:wak}, we
have a canonical isomorphism between $\Spec(\fZ^\reg_\fg)$ and the
space $\Op_\cg(\D)$ of $\cg$-opers on the formal disc $\D$ (we refer
the reader to Sect. 1 of \cite{FG2} for the definition and a detailed
review of opers). By construction, over the scheme $\Op_\cg(\D)$
there exists a canonical principal $\cG$-bundle, denoted by
$\CP_{\cG,\Op}$. Let $\CP_{\cG,\fZ}$ bethe $\cG$-bundle over
$\Spec(\fZ^\reg_\fg)$ corresponding to it under the above
isomorphism. For an object $V \in \Rep(\cG)$ let us denote by
$\CV_\fZ$ the associated vector bundle over $\Spec(\fZ^\reg_\fg)$,
i.e., $\CV_{\fZ} = \CP_{\cG,\fZ} \underset{\cG}\times V$.
Consider now the category $\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$. By
\cite{MV}, this category has a canonical tensor structure, and
as such it is equivalent to the category $\Rep(\cG)$ of algebraic
representations of $\cG$; we shall denote by
$$V\mapsto \CF_V:\Rep(\cG)\to \on{D}(\Gr_G)_\crit\mod^{G[[t]]}$$
the corresponding functor. Moreover, we have a canonical action of
$\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$ as a tensor category on
$\on{D}(\Gr_G)_\crit\mod$ by convolution functors, $\F \mapsto \F
\star \F_V$.
A. Beilinson and V. Drinfeld \cite{BD} have proved that there
are functorial isomorphisms
$$
\Gamma(\Gr_G,\F \star \F_V) \simeq \Gamma(\Gr_G,\F)
\underset{\fZ^\reg_\fg}\otimes \CV_{\fZ}, \qquad V \in
\Rep(\cG),
$$
compatible with the tensor structure. Thus, we see that there are
non-isomorphic objects of $\on{D}(\Gr_G)_\crit\mod$ that go under the
functor $\Gamma$ to isomorphic objects of $\ghat_\crit\mod_\reg$.
\ssec{}
In \cite{FG2} we showed how to modify the category
$\on{D}(\Gr_G)_\crit\mod$, by simultaneously "adding" to it
$\fZ^\reg_\fg$ as a center, and "dividing" it by the above
$\Rep(\cG)$-action, in order to obtain a category that can be
equivalent to $\ghat_\crit\mod_\reg$.
This procedure amounts to replacing $\on{D}(\Gr_G)_\crit\mod$ by the
appropriate category of {\em Hecke eigen-objects}, denoted
$\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod$.
By definition, an object of $\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod$ is
an object $\F\in \on{D}(\Gr_G)_\crit\mod$, equipped with an action of
the algebra $\fZ^\reg_\fg$ by endomorphisms and a system of
isomorphisms
$$
\al_V: \CF\star \CF_V \overset{\sim}\longrightarrow
\CV_\fZ\underset{\fZ^\reg_\fg}\otimes \CF, \qquad V \in
\Rep(\cG),
$$
compatible with the tensor structure.
\medskip
We claim that the functor $\Gamma: \on{D}(\Gr_G)_\crit\mod \to
\ghat_\crit\mod_\reg$ naturally gives rise to a functor
$\Gamma^{\Hecke_\fZ}:\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod\to
\ghat_\crit\mod_\reg$.
\medskip
This is in fact a general property. Suppose for simplicity that we
have an abelian category ${\mc C}$ which is acted upon by the tensor
category $\Rep(H)$, where $H$ is an algebraic group; we denote this
functor by $\CF \mapsto \CF \star V, V \in \Rep(H)$. Let ${\mc
C}^\Hecke$ be the category whose objects are collections
$(\CF,\{\al_V\}_{V \in \Rep(H)})$, where $\CF \in {\mc C}$ and $\{
\al_V \}$ is a compatible system of isomorphisms
$$
\al_V: \CF \star V \overset{\sim}\longrightarrow \underline{V}
\underset{\C}\otimes \CF, \qquad V \in \Rep(H),
$$
where $\underline{V}$ is the vector space underlying $V$. One may
think of ${\mc C}^\Hecke$ as the ``de-equivariantized'' category ${\mc C}$
with respect to the action of $H$. It carries a natural
action of the group $H$: for $h \in H$, we have $h \cdot
(\CF,\{\al_V\}_{V \in \Rep(H)}) = (\CF,\{ (h\otimes \on{id}_\CF) \circ
\al_V\}_{V \in \Rep(H)})$. The category ${\mc C}$ may be reconstructed
as the category of $H$-equivariant objects of ${\mc C}^\Hecke$ with
respect to this action, see \cite{Ga}.
Suppose that we have a functor $\sG: {\mc C} \to {\mc C}'$, such that we
have functorial isomorphisms
\begin{equation} \label{syst}
\sG(\CF \star V) \simeq \sG(\CF) \underset{\C}\otimes \underline{V},
\qquad V \in \Rep(H),
\end{equation}
compatible with the tensor structure. Then, according to \cite{AG},
there exists a functor ${\sG}^\Hecke: \CC^\Hecke\to \CC'$ such that
$\sG \simeq \sG^\Hecke \circ \on{Ind}$, where the functor $\on{Ind}:
{\mc C} \to {\mc C}^\Hecke$ sends $\CF$ to $\CF \star {\mc O}_{H}$,
where ${\mc O}_{H}$ is the regular representation of $H$. The functor
$\sG^\Hecke$ may be explicitly described as follows: the isomorphisms
$\al_V$ and \eqref{syst} give rise to an action of the algebra ${\mc
O}_{H}$ on $\sG(\CF)$, and ${\sG}^\Hecke(\CF)$ is obtained by taking
the fiber of $\sG(\CF)$ at $1 \in H$.
We take $\CC=\on{D}(\Gr_G)_\crit\mod$, $\CC'=\hg_\crit\mod_\reg$, and
$\sG=\Gamma$. The only difference is that now we are working over the
base $\fZ^\reg_\fg$, which we have to take into account.
\ssec{}
The conjecture suggested in \cite{FG2} states that the resulting functor
\begin{equation} \label{glob Hecke sections}
\Gamma^{\Hecke_\fZ}:\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod\to
\ghat_\crit\mod_\reg.
\end{equation}
is an equivalence. In {\it loc. cit.} we have shown that the functor
$\Gamma^{\Hecke_\fZ}$, when extended to the derived category,
is fully faithful.
\medskip
This conjecture has a number of interesting corollaries pertaining to
the structure of the category of representations at the critical level:
Let us fix a point $\chi\in \Spec(\fZ^\reg_\fg)$, and let us choose a
trivialization of the fiber $\CP_{\cG,\chi}$ of $\CP_{\cG,\fZ}$ at
$\chi$. Let $\ghat_\crit\mod_\chi$ be the subcategory of
$\hg_\crit\mod$, consisting of objects, on which the center acts
according to the character corresponding to $\chi$.
Let $\on{D}(\Gr_G)_\crit^{\Hecke}\mod$ be the category, obtained from
$\on{D}(\Gr_G)_\crit\mod$, by the procedure $\CC\mapsto \CC^\Hecke$
for $H=\cG$, described above. Our conjecture implies that we have an
equivalence
\begin{equation} \label{sect with char}
\on{D}(\Gr_G)_\crit^{\Hecke}\mod\simeq \ghat_\crit\mod_\chi.
\end{equation}
In particular, we obtain that for every two points $\chi,\chi'\in
\Spec(\fZ^\reg_\fg)$ and an isomorphism of $\cG$-torsors
$\CP_{\cG,\chi}\simeq \CP_{\cG,\chi'}$ there exists a canonical
equivalence $\ghat_\crit\mod_\chi\simeq \ghat_\crit\mod_{\chi'}$. This
may be viewed as an analogue of the translation principle that
compares the subcategories $\fg\mod_{\chi}\subset \fg\mod$ for various
central characters $\chi\in \Spec(Z(\fg))$ in the finite-dimensional
case.
By taking $\chi=\chi'$, we obtain that the group $\cG$, or, rather, its
twist with respect to $\CP_{\cG,\chi}$, acts on $\ghat_\crit\mod_\chi$.
\medskip
As we explained in the Introduction to \cite{FG2}, the conjectural
equivalence of \eqref{sect with char} fits into the general picture of
local geometric Langlands correspondence.
Namely, for a point $\chi\in \Spec(\fZ^\reg_\fg)\simeq \Op_\cg(\D)$ as
above, both sides of the equivalence \eqref{sect with char} are
natural candidates for the conjectural Langlands category associated
to the trivial $\cG$-local system on the disc ${\mc D}$. This
category, equipped with an action of the loop group $G\ppart$, should
be thought of as a "categorification" of an irreducible unramified
representation of the group $G$ over a local non-archimedian
field. Proving this conjecture would therefore be the first test of
the local geometric Langlands correspondence proposed in \cite{FG2}.
\ssec{}
Unfortunately, at the moment we are unable to prove the equivalence
\eqref{glob Hecke sections} in general. In this paper we will treat
the following particular case:
Recall that $I^0$ denotes the unipotent radical of the Iwahori
subgroup, and let us consider the corresponding $I^0$-equivariant
subcategories on both sides of \eqref{glob Hecke sections}.
On the D-module side, we obtain the category
$\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod^{I^0}$, defined in the same way
as $\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod$, but with the requirement
that the underlying D-module $\CF$ be strongly $I^0$-equivariant.
On the representation side, we obtain the category
$\ghat_\crit\mod_\reg^{I^0}$, corresponding to the condition that the
action of $\on{Lie}(I^0)\subset \hg_\crit$ integrates to an algebraic
action of $I^0$.
We shall prove that the functor $\Gamma^{\Hecke_\fZ}$ defines an
equivalence of categories
\begin{equation} \label{glob Hecke sections Iw}
\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod^{I^0}\to
\ghat_\crit\mod_\reg^{I^0}.
\end{equation}
This equivalence implies an equivalence
\begin{equation} \label{eq for chi}
\on{D}(\Gr_G)_\crit^{\Hecke}\mod^{I^0} \simeq
\ghat_\crit\mod_\chi^{I^0}
\end{equation}
for any fixed character $\chi\in \Spec(\fZ^\reg_\fg)$ and a
trivialization of $\CP_{\cG,\chi}$ as above. In particular, we obtain
the corollaries concerning the translation principle and the action of
$\cG$ on $\ghat_\crit\mod_\chi^{I^0}$.
We remark that from the point of view of the local geometric Langlands
correspondence the categories appearing in the equivalence \eqref{eq
for chi} should be viewed as "categorifications" of the space of
$I$-invariant vectors in an irreducible unramified representation of
the group $G$ over a local non-archimedian field (which is a module
over the corresponding affine Hecke algebra).
\medskip
Let us briefly describe the strategy of the proof. Due to the fact
\cite{FG2} that the functor in one direction in \eqref{glob Hecke
sections Iw} is fully-faithful at the level of the derived categories,
the statement of the theorem is essentially equivalent to the fact
that for every object $\CM\in \ghat_\crit\mod_\reg^{I^0}$ there exists
an object $\CF\in \on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod^{I^0}$ and a
non-zero map $\Gamma^{\Hecke_\fZ}(\Gr_G,\CF)\to \CM$. We explain this
in detail in \secref{main proof}.
We exhibit a collection of objects $\BM_{w,\reg}$, numbered by
elements $w\in W$, where $W$ is the Weyl group of $\fg$, which are
quotients of Verma modules over $\ghat_\crit$, such that for every
$\CM\in \ghat_\crit\mod_\reg^{I^0}$ we have
$\on{Hom}(\BM_{w,\reg},\CM)\neq 0$ for at least one $w \in W$.
We then show (see \thmref{get Wakimoto}) that each such $\BM_{w,\reg}$
is isomorphic to $\Gamma^{\Hecke_\fZ}(\Gr_G,\CF_w^\fZ)$ for some
explicit object $\CF_w^\fZ\in
\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod^{I^0}$, thereby proving the
equivalence \eqref{glob Hecke sections Iw}.
\ssec{}
It is instructive to put our results in the context of other
closely related equivalences of categories.
Using the (tautological) equivalence:
$$\on{D}(\Gr_G)\mod^{I^0}\simeq \on{D}(\wt\Fl_G)\mod^{G[[t]]}$$ (here
and below we omit the subscript $\ka$ when $\ka=0$) and the
equivalence of \thmref{KTthm}, we obtain that for every negative
integral level $\kappa=k\cdot \kappa_{\can}$ there exists an
equivalence between $\on{D}(\Gr_G)\mod^{I^0}$ and the regular block of
the category $\hg_\kappa\mod^{G[[t]]}$, studied in \cite{KL}. The
latter category is equivalent, according to {\it loc. cit.}, to the
category of modules over the quantum group $U_q^{\on{res}}(\g)$, where
$q = \exp \pi i/(k+h^\vee)$.
Using these equivalences, it was shown in \cite{AG} that the category
$\on{D}(\Gr_G)^{\Hecke}\mod^{I^0}$, defined as above, is equivalent to
the regular block $u_q(\fg)\mod_0$ of the category of modules over the
small quantum group $u_q(\g)$. The tensor product by the line bundle
$\Ll_{\can}^{-h^\vee}$ defines an equivalence
$$\on{D}(\Gr_G)^{\Hecke}\mod^{I^0}\to
\on{D}(\Gr_G)_\crit^{\Hecke}\mod^{I^0}$$ (but this equivalence does
not, of course, respect the functor of global sections). Combining
this with the equivalence of \eqref{eq for chi}, we obtain the
following diagram of equivalent categories:
\begin{equation} \label{quantum group equiv}
\ghat_\crit\mod^{I^0}_\chi \overset{\sim}\leftarrow
\on{D}(\Gr_G)_\crit^{\Hecke}\mod^{I^0} \overset{\sim}\to
u_q(\g)\mod_0.
\end{equation}
\medskip
Recall in addition that in \cite{ABBGM} it was shown that the category
$\on{D}(\Gr_G)^{\Hecke}\mod^{I^0}$ is equivalent to an appropriately
defined category $\on{D}({\mc F}l^{\frac{\infty}{2}})^{I^0}$ of
$I^0$-equivariant D-modules on the semi-infinite flag variety (it is
defined in terms of the Drinfeld compactification
$\ol{\on{Bun}}_N$). Hence, we obtain another diagram of equivalent
categories:
\begin{equation} \label{semiinf equiv}
\ghat_\crit\mod^{I^0}_\chi \overset{\sim}\leftarrow
\on{D}(\Gr_G)_\crit^{\Hecke}\mod^{I^0} \overset{\sim}\to \on{D}({\mc
F}l^{\frac{\infty}{2}})^{I^0}.
\end{equation}
In particular, we obtain a functor
$$\on{D}({\mc F}l^{\frac{\infty}{2}})^{I^0}\to
\ghat_\crit\mod^{I^0}_\chi,$$ which is, moreover, an equivalence. Its
existence had been predicted by B. Feigin and the first named author.
In fact, one would like to be able to define the category $\on{D}({\mc
F}l^{\frac{\infty}{2}})$ without imposing the $I^0$-equivariance
condition, and extend the equivalence of \cite{ABBGM} to this more
general context. Together with the equivalence of \eqref{glob Hecke
sections}, this would imply the existence of the diagram
$$\ghat_\crit\mod_\chi \overset{\sim}\leftarrow
\on{D}(\Gr_G)_\crit^{\Hecke}\mod \overset{\sim}\to \on{D}({\mc
F}l^{\frac{\infty}{2}}),$$ but we are far from being able to achieve
this goal at present.
\medskip
Finally, let us mention one more closely related category, namely, the
derived category $D\bigl(\QCoh((\cG/\cB)^{DG}\mod)\bigr)$ of complexes
of quasi-coherent sheaves over the DG-scheme
$$(\cG/\cB)^{DG}: =
\Spec\Bigl(\Sym_{\CO_{\cG/\cB}}(\Omega^1(\cG/\cB)[1])\Bigr).$$ The
above DG-scheme can be realized as the derived Cartesian product
$\wt\cg\underset{\cg}\times \on{pt}$, where $\on{pt}\to \cg$
corresponds to the point $0\in \cg$, and $\wt\cg = \{ (x,\check\bb)|x
\in \check\bb \subset \cg \}$ is Grothendieck's
alteration.
{}From the results of \cite{ABG} one can obtain an equivalence of the
derived categories
$$
D^b\bigl(\QCoh((\cG/\cB)^{DG}\mod)\bigr) \simeq
D^b\bigl(\on{D}(\Gr_G)^{\Hecke}\mod\bigr)^{I^0}.
$$
Hence we obtain an equivalence:
\begin{equation} \label{coherent equiv}
D^b\bigl(\QCoh((\cG/\cB)^{DG}\mod)\bigr)\simeq
D^b\bigl(\ghat_\crit\mod_\chi\bigr)^{I^0}.
\end{equation}
The existence of such an equivalence follows from the Main Conjecture
6.11 of \cite{FG2}. Note that, unlike the other equivalences mentioned
above, it does not preserve the t-structures, and so is inherently an
equivalence of derived categories.
\ssec{Contents}
Let us briefly describe how this paper is organized:
\medskip
In \secref{Hecke ctry}, after recalling some previous results, we
state the main result of this paper, \thmref{main}. In \secref{corol}
we review representation-theoretic corollaries of \thmref{main}. In
\secref{main proof} we show how to derive \thmref{main} from
\thmref{get Wakimoto}, and in \secref{sect get Wakimoto} we prove
\thmref{get Wakimoto}.
Finally, in the Appendix, we prove a partial localization result at
the negative level mentioned in \secref{neg level}.
\medskip
The notation in this paper follows that of \cite{FG2}.
\section{The Hecke category} \label{Hecke ctry}
In this section we recall the main definitions and state our main
result. We will rely on the concepts introduced in our previous paper
\cite{FG2}.
\ssec{Recollections} \label{recol}
Let $\fg$ be a simple finite-dimensional Lie algebra, and $G$ the
connected algebraic group of adjoint type with the Lie algebra
$\fg$. We shall fix a Borel subgroup $B\subset G$.
Let $\cG$ denote the Langlands dual group of $G$, and by
$\cg$ its Lie algebra.
\medskip
Let $\Gr_G = G\ppart/G[[t]]$ be the affine Grassmannian associated to
$G$. We denote by $\on{D}(\Gr_G)_\crit\mod$ the category of critically
twisted right D-modules on the affine Grassmannian and by
$\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$ the corresponding
$G[[t]]$-equivariant category. Recall that via the geometric Satake
equivalence (see \cite{MV}) the category
$\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$ has a natural structure of tensor
category under convolution, and as such it is equivalent to
$\Rep(\cG)$. We shall denote by $V\mapsto \CF_V$ the corresponding
tensor functor $\Rep(\cG) \to \on{D}(\Gr_G)_\crit\mod^{G[[t]]}$.
We have the convolution product functors
$$
\CF \in \on{D}(\Gr_G)_\crit\mod, \CF_V \in
\on{D}(\Gr_G)_\crit\mod^{G[[t]]} \mapsto \CF \star \CF_V \in
\on{D}(\Gr_G)_\crit\mod.
$$
These functors define an action of $\Rep(\cG)$, on the category
$\on{D}(\Gr_G)_\crit\mod$. Thus, in the terminology of \cite{Ga},
$\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$ has the structure of category
over the stack $\on{pt}/\cG$.
\medskip
Now let $\hg_\crit$ be the {\em affine Kac-Moody algebra} associated
to the critical inner product $-h^\vee \kappa_{\can}$ and
$\hg_\crit\mod$ the category of discrete $\hg_\crit$-modules (see
\cite{FG2}). Its objects are $\hg_\crit$-modules in which every vector
is annihilated by the Lie subalgebra $\fg \otimes t^n\BC[[t]]$ for
sufficiently large $n$. Let $\BV_\crit\in \hg_\crit\mod$ be the vacuum
module $\on{Ind}_{\fg[[t]]\oplus \BC}^{\hg_\crit}(\BC)$. Denote by
$\fZ_\fg$ the topological commutative algebra that is the center of
$\hg_\crit\mod$. Let $\fZ^\reg_\fg$ denote its "regular" quotient,
i.e., the quotient modulo the annihilator of $\BV_\crit$. We denote
by $\hg_\crit\mod_\reg$ the full subcategory of $\hg_\crit\mod$,
consisting of objects, on which the action of the center $\fZ_\fg$
factors through $\fZ^\reg_\fg$.
\medskip
Recall that via the Feigin-Frenkel isomorphism \cite{FF,F:wak}, the
algebra $\fZ^\reg_\fg$ is identified with the algebra of regular
functions on the scheme $\Op_{\cg}(\D)$ of $\cg$-opers on the formal
disc $\D$. In particular, $\Spec(\fZ^\reg_\fg)$ carries a canonical
$\cG$-torsor, denoted $\CP_{\cG,\fZ}$, whose fiber $\CP_{\cG,\chi}$ at
$\chi \in \Spec(\fZ^\reg_\fg) \simeq \Op_{\cg}(\D)$ is the fiber of
the $\cG$-torsor underlying the oper $\chi$ at the origin of the disc
$\D$. The $\cG$-torsor $\CP_{\cG,\fZ}$ gives rise to a morphism
$\Spec(\fZ^\reg_\fg)\to \on{pt}/\cG$. We shall denote by $$V\mapsto
\CV_\fZ$$ the resulting tensor functor from $\Rep(\cG)$ to the
category of locally free $\fZ^\reg_\fg$-modules.
\medskip
We define $\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod$ as the fiber product
category
$$\on{D}(\Gr_G)_\crit\mod\underset{\on{pt}/\cG}\times
\Spec(\fZ^\reg_\fg),$$ in the terminology of \cite{Ga}.
\medskip
Explicitly, $\on{D}(\Gr_G)_\crit^{\Hecke_\fZ}\mod$ has as objects the
data of $(\CF,\alpha_V,\,\, \forall\,\, V\in \Rep(\cG))$, where $\CF$
is an object of $\on{D}(\Gr_G)_\crit\mod$, endowed with an action of
the algebra $\fZ^\reg_\fg$ by endomorphisms, and $\alpha_V$ are
isomorphisms of D-modules
$$\CF\star \CF_V\simeq \CV_\fZ\underset{\fZ^\reg_\fg}\otimes \CF,$$
compatible with the action of $\fZ^\reg_\fg$ on both sides, and such
that the following two conditions are satisfied:
\begin{itemize}
\item
For $V$ being the trivial representations $\BC$, the morphism $\alpha_V$
is the identity map.
\item
For $V,W\in \Rep(\cG)$ and $U:=V\otimes W$, the diagram
$$
\CD
(\CF\star \CF_V)\star \CF_W @>{\sim}>> \CF\star \CF_U \\
@V{\alpha_V\star \on{id}_{\CF_W}}VV @V{\alpha_U}VV \\
(\CV_\fZ\underset{\fZ^\reg_\fg}\otimes \CF)\star \CF_W & &
\CU_\fZ\underset{\fZ^\reg_\fg}\otimes \CF \\
@V{\sim}VV @V{\sim}VV \\
\CV_\fZ\underset{\fZ^\reg_\fg}\otimes (\CF\star \CF_W)
@>{\on{id}_{\CV_\fZ}\otimes \alpha_W}>>
\CV_\fZ\underset{\fZ^\reg_\fg} \otimes
\CW_\fZ\underset{\fZ^\reg_\fg} \otimes\CF
\endCD
$$ is commutative.
\end{itemize}
Morphisms in this category between $(\CF,\alpha_V)$ and $(\CF',\alpha'_V)$
are maps of D-modules $\phi:\CF\to \CF'$ that are compatible with the actions
of $\fZ^\reg_\fg$ on both sides, and such that
$$(\on{id}_{\CV_\fZ}\otimes \phi)\circ \alpha_V=
\alpha'_V\circ (\phi\star \on{id}_{\CF_V}).$$
\ssec{Definition of the functor} \label{d of f}
Recall that according to \cite{FG1}, the functor of global sections
$$\CF\mapsto \Gamma(\Gr_G,\CF)$$ defines an exact and faithful functor
$\on{D}(\Gr_G)_\crit\mod\to \hg_\crit\mod_\reg$. Let us recall,
following \cite{FG2}, the construction of the functor
$$\Gamma^{\Hecke_\fZ}:\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\to
\hg_\crit\mod_\reg.$$
First, let us recall the following result of \cite{BD} (combined with
an observation of \cite{FG2}, Lemma 8.4.3):
\begin{thm} \label{BDisom} \hfill
\smallskip
\noindent{\em(1)}
For $\CF\in \on{D}(\Gr_G)_\crit\mod$ and $V\in \Rep(\cG)$
we have a functorial isomorphism
$$\beta_V:\Gamma(\Gr_G,\CF\star \CF_V)\simeq
\Gamma(\Gr_G,\CF)\underset{\fZ^\reg_\fg} \otimes \CV_\fZ.$$
\noindent{\em(2)}
For $\CF,V$ as above and $W\in \Rep(\cG)$, $U:=V\otimes W$ the diagram
$$
\CD
\Gamma(\Gr_G,(\CF\star \CF_V)\star \CF_W) @>{\sim}>>
\Gamma(\Gr_G,\CF\star (\CF_V\star \CF_W)) \\
@V{\beta_W}VV @V{\sim}VV \\
\Gamma(\Gr_G,(\CF\star \CF_V))
\underset{\fZ^\reg_\fg} \otimes \CW_\fZ & & \Gamma(\Gr_G,\CF\star
\CF_U) \\
@V{\beta_V}VV @V{\beta_U}VV \\
\Gamma(\Gr_G,\CF)\underset{\fZ^\reg_\fg} \otimes \CV_\fZ
\underset{\fZ^\reg_\fg} \otimes \CW_\fZ @>{\sim}>>
\Gamma(\Gr_G,\CF)\underset{\fZ^\reg_\fg} \otimes \CU_\fZ
\endCD
$$
is commutative.
\end{thm}
\medskip
Consider the scheme
$\Isom_{\fZ}:\Spec(\fZ^\reg_\fg\underset{\on{pt}/\cG}\times
\fZ^\reg_\fg)$. Let ${{\bf 1}_{\Isom_{\fZ}}}$ denote the unit section
$\Spec(\fZ^\reg_\fg)\to \Isom_{\fZ}$.
Let us denote by
$R_{\fZ}$ the direct image of the structure sheaf under
$\Spec(\fZ^\reg_\fg)\to \on{pt}/\cG$, viewed as an object of
$\Rep(\cG)$. It carries an action of $\fZ^\reg_\fg$ by endomorphisms.
Let $\CR_{\fZ}$ be the associated (infinite-dimensional) vector
bundle over $\Spec(\fZ^\reg_\fg)$; by definition, we have a
canonical isomorphism
$$\CR_\fZ\simeq \Fun(\Isom_\fZ).$$
We will think of the projection $p_r:\Isom_\fZ\to \Spec(\fZ^\reg_\fg)$ as
corresponding to the original $\fZ^\reg_\fg$-action on $R_\fZ$, and hence
on $\CR_\fZ$, by the transport of structure. We will think of the other
projection $p_l:\Isom_\fZ\to \Spec(\fZ^\reg_\fg)$, as corresponding
to the $\fZ^\reg_\fg$-module structure on $\CR_\fZ$ coming from
the fact that this is a vector bundle associated to a $\cG$-representation.
\medskip
We claim that for every object $\CF^H\in
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$, the $\hg_\crit$-module
$\Gamma(\Gr_G,\CF^H)$ carries a natural action of the algebra
$\Fun(\Isom_{\fZ})$ by endomorphisms.
First, note that
$\Gamma(\Gr_G,\CF^H)$ is a $\fZ^\reg_\fg$-bimodule:
we shall refer to the $\fZ^\reg_\fg$-action coming from its
action on any object of $\hg_\crit\mod_\reg$ as "right",
and to the one. coming from the $\fZ^\reg_\fg$-action
on $\CF^H$ as "left".
On the one hand, we have:
$$\Gamma(\Gr_G, \CF^H\star \CF_{R_{\fZ}})\overset{\beta_{R_{\fZ}}}
\simeq \Gamma(\Gr_G, \CF^H)\underset{r,\fZ^\reg_\fg,l}
\otimes\Fun(\Isom_{\fZ}),$$ and on the other hand,
$$\Gamma(\Gr_G, \CF^H\star \CF_{R_{\fZ}})\overset{\alpha_{R_{\fZ}}}
\simeq \Fun(\Isom_{\fZ})\underset{l,\fZ^\reg_\fg,l}\otimes
\Gamma(\Gr_G, \CF^H)\otimes\Fun(\Isom_{\fZ}).$$
By composing we obtain the desired action map
$$\Gamma(\Gr_G, \CF)\underset{r,\fZ^\reg_\fg,l}\otimes \Fun(\Isom_{\fZ})
\overset{\alpha_{R_\fZ}\circ \beta^{-1}_{R_\fZ}}\longrightarrow
\Fun(\Isom_{\fZ})\underset{l,\fZ^\reg_\fg,l}\otimes \Gamma(\Gr_G, \CF^H)
\overset{{\bf 1}^*_{\Isom_{\fZ}}}\longrightarrow \Gamma(\Gr_G, \CF^H).$$
The fact that it is associative follows from the second condition on
$\alpha_V$ and \thmref{BDisom}(2).
\medskip
We define the functor $\Gamma^{\Hecke_\fZ}$ by
$$\CF^H\mapsto \Gamma(\Gr_G,\CF^H) \underset{\Fun(\Isom_{\fZ}),{\bf
1}^*_{\Isom_{\fZ}}}\otimes \fZ^\reg_\fg.$$ Since the functor $\Gamma$
is exact, the functor $\Gamma^{\Hecke_\fZ}$ is evidently right-exact,
and we will denote by $\on{L}\Gamma^{\Hecke_\fZ}$ its left derived
functor $D^-(\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod)\to
D^-(\hg_\crit\mod_\reg)$
\medskip
The following was established in \cite{FG2}, Theorem 8.7.1:
\begin{thm} \label{GH fully faithful}
The functor $\on{L}\Gamma^{\Hecke_\fZ}$, restricted to
$D^b(\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod)$, is fully faithful.
\end{thm}
In \cite{FG2} we formulated the following
\begin{conj} \label{general conj}
The functor $\Gamma^{\Hecke_\fZ}$ is exact and defines an equivalence
of categories $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ and
$\hg_\crit\mod_\reg$.
\end{conj}
\ssec{The statement of the main result} \label{main result}
Recall that both categories $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ and
$\hg_\crit\mod_\reg$ carry a natural action of the group $G\ppart$
(see \cite{FG2}, Sect. 22, where this is discussed in detail). Let
$I\subset G[[t]]$ be the Iwahori subgroup, the preimage of the Borel
subgroup $B \subset G$ in $G[[t]]$ under the evaluation map $G[[t]]
\to G$. Let $I^0$ be the unipotent radical of $I$. Let us denote by
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$ and
$\hg_\crit\mod_\reg^{I^0}$ the corresponding categories if
$I$-equivariant objects. Since $I^0$ is connected, these are full
subcategories in $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ and
$\hg_\crit\mod_\reg^{I^0}$, respectively.
The functor $\Gamma^{\Hecke_\fZ}$ induces a functor
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}\to
\hg_\crit\mod_\reg^{I^0}$.
The goal of the present paper is to prove the following:
\begin{thm} \label{main} \hfill
\smallskip
\noindent{\em (1)}
For any $\CF^H\in \on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$ we have
$L^i \Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H) = 0$ for all $i>0$.
\smallskip
\noindent{\em (2)} The functor $$\Gamma^{\Hecke_\fZ}:
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}\to
\hg_\crit\mod_\reg^{I^0}$$ is an equivalence of categories.
\end{thm}
This is a special case of \conjref{general conj}.
\section{Corollaries of the main theorem} \label{corol}
We shall now discuss some applications of \thmref{main}. Note that
both sides of the equivalence stated in \thmref{main} are categories
over the algebra $\fZ^\reg_\fg$.
\ssec{Specialization to a fixed central character} \label{disc of cor}
Let us fix a point $\chi\in\Spec(\fZ^\reg_\fg)$, i.e., a character of
$\fZ^\reg_\fg$, and consider the subcategories on both sides of the
equivalence of \thmref{main}(2), corresponding to objects on which the
center acts according to this character. Let us denote the resulting
subcategory of $\hg_\crit\mod_\reg^{I^0}$ by
$\hg_\crit\mod_\chi^{I^0}$. The resulting subcategory of
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$ can be described as
follows.
\medskip
Let us denote by $\on{D}(\Gr_G)^{\Hecke}_\crit\mod$ the category,
whose objects are the data of $(\CF,\alpha_V)$, where $\CF\in
\on{D}(\Gr_G)_\crit\mod$ and $\alpha_V$ are isomorphisms of D-modules
defined for every $V\in \Rep(\cG)$
$$\CF\star\CF_V\simeq \uV\underset{\BC}\otimes \CF,$$ where $\uV$
denotes the vector space underlying the representation $V$. These
isomorphisms must be compatible with tensor products of objects of
$\Rep(\cG)$ in the same sense as in the definition of
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$.
Note that $\on{D}(\Gr_G)^{\Hecke}_\crit\mod$ carries a natural weak
action of the algebraic group $\cG$: \footnote{We refer the reader to
\cite{FG2}, Sect. 20.1, where this notion is introduced.} Given an
$S$-point $\bg$ of $\cG$ and an $S$-family of objects $(\CF,\alpha_V)$
of $\on{D}(\Gr_G)^{\Hecke}_\crit\mod$ we obtain a new $S$-family by
keeping $\CF$ the same, but replacing $\alpha_V$ by $\bg\cdot
\alpha_V$, where $\bg$ acts naturally on $\uV\otimes \CO_S$.
In addition, $\on{D}(\Gr_G)^{\Hecke}_\crit\mod$ carries a commuting
Harish-Chandra action of the group $G\ppart$; in particular, the
subcategory $\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}$ makes sense.
\medskip
Let $\CP_{\cG,\chi}$ be the fiber of the $\cG$-torsor $\CP_{\cG,\fZ}$
at $\chi$. Tautologically we have:
\begin{lem} \hfill
\smallskip
\noindent{\em (1)} For every trivialization
$\gamma:\CP_{\cG,\chi}\simeq \CP^0_\cG$ there exists a canonical
equivalence respecting the action of $G\ppart$
$$\bigl(\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod\bigr)_\chi\simeq
\on{D}(\Gr_G)^{\Hecke}_\crit\mod,$$ where the LHS denotes the fiber of
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ at $\chi$.
\smallskip
\noindent{\em (2)} If $\gamma'=\bg\cdot \gamma$ for $\bg\in \cG$, the
above equivalence is modified by the self-functor of
$\on{D}(\Gr_G)^{\Hecke}_\crit\mod$, given by the action of $\bg$.
\end{lem}
Hence, from \thmref{main} we obtain:
\begin{cor} \label{main cor}
For every trivialization $\gamma:\CP_{\cG,\chi}\simeq \CP^0_\cG$ there
exists a canonical equivalence
$$\hg_\crit\mod_\chi^{I^0}\simeq
\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}.$$
\end{cor}
\medskip
{}From \corref{main cor} we obtain:
\begin{cor} \hfill
\smallskip
\noindent{\em (1)} For any two points $\chi_1,\chi_2\in
\Spec(\fZ^\reg_\fg)$ and an isomorphism of $\cG$-torsors
$\CP_{\cG,\chi_1}\simeq \CP_{\cG,\chi_2}$ there exists a canonical
equivalence
$$\hg_\crit\mod_{\chi_1}^{I^0}\simeq \hg_\crit\mod_{\chi_2}^{I^0}.$$
\smallskip
\noindent{\em (2)} For every $\chi\in \Spec(\fZ^\reg_\fg)$, the group
of automorphisms of the $\cG$-torsor $\CP_{\cG,\chi}$ acts on the
category $\hg_\crit\mod_\chi^{I^0}$.
\end{cor}
More generally, let $S$ be an affine scheme, and let $\chi_{1,S}$ and
$\chi_{2,S}$ be two $S$-points of $\Spec(\fZ^\reg_\fg)$. Let
$\hg_\crit\mod_{S,1}^{I^0}$ and $\hg_\crit\mod_{S,2}^{I^0}$ be the
corresponding base-changed categories.
By definition, the
objects of $\hg_\crit\mod_{i,S}$ are the objects of $\hg_\crit\mod_\reg$,
endowed with an action of $\CO_S$ compatible with the initial action
of $\fZ^\reg_\fg$ on $\CM$ via the homomorphism
$\fZ^\reg_\fg\to \CO_S$, corresponding to $\chi_{i,S}$.
Morphisms in this category are $\hg_\crit$-morphisms compatible with
the action of $\CO_S$.
\medskip
We obtain:
\begin{cor} \label{main cor, families}
For every lift of the map
$$(\chi_{1,S}\times \chi_{2,S}):S\to \Spec(\fZ^\reg_\fg)\times
\Spec(\fZ^\reg_\fg)$$ to a map $S\to \Isom_\fZ$, there exists a
canonical equivalence
$$\hg_\crit\mod_{S,1}^{I^0}\simeq \hg_\crit\mod_{S,2}^{I^0}.$$
\end{cor}
\ssec{Description of irreducibles} \label{sect descr of irr}
\corref{main cor} allows to describe explicitly the set of irreducible
objects in $\hg_\crit\mod_\reg^{I^0}$. For that we will need to recall
some more notation related to the categories
$\on{D}(\Gr_G)^{\Hecke}_\crit\mod$ and
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$.
\medskip
Consider the forgetful functor $\on{D}(\Gr_G)^{\Hecke}_\crit\mod\to
\on{D}(\Gr_G)_\crit\mod$. It admits a left adjoint, denoted
$\on{Ind}^\Hecke$, which can be described as follows.
Let $R$ be the object of $\Rep(\cG)$ equal to $\CO_\cG$ under the
left regular action; let $\CF_R$ denote the corresponding
object of $\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$. Then for $\CF\in
\on{D}(\Gr_G)_\crit\mod$, the convolution $\CF\star \CF_R$ is naturally
an object of $\on{D}(\Gr_G)^{\Hecke}_\crit\mod$, and it is easy to see
that $\on{Ind}^\Hecke(\CF):=\CF\star \CF_R$ is the desired left adjoint.
\medskip
Similarly, the forgetful functor
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod\to \on{D}(\Gr_G)_\crit\mod$
admits a left adjoint functor $\on{Ind}^{\Hecke_\fZ}$ given by
$\CF\mapsto \CF\star \CF_{R_\fZ}$. The next assertion follows from the
definitions:
\begin{lem} \label{H induced} \hfill
\smallskip
\noindent{\em (1)} For $\CF\in \on{D}(\Gr_G)_\crit\mod$ there exist
canonical isomorphisms:
$$\Gamma(\Gr_G,\on{Ind}^{\Hecke_\fZ}(\CF))\simeq
\Gamma(\Gr,\CF)\underset{\fZ^\reg_\fg}\otimes \Fun(\Isom_{\fZ}),$$
where $\Fun(\Isom_{\fZ})$ is a module over $\fZ^\reg_\fg$ via one
of the projections $\Isom_{\fZ}\to \Spec(\fZ^\reg_\fg)$.
\smallskip
\noindent{\em (2)}
For $\CF$ as above,
$$\Gamma^{\Hecke_\fZ}\bigl(\Gr_G,
\on{Ind}^{\Hecke_\fZ}(\CF)\bigr)\simeq \Gamma(\Gr,\CF).$$
\end{lem}
\medskip
Let us now recall the description of irreducible objects of
$\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}$, established in \cite{ABBGM},
Corollary 1.3.10.
Recall that $I$-orbits on $\Gr_G$ are parameterized by the set
$W_{\aff}/W$, where $W_{\aff}$ denotes the extended affine Weyl
group. For an element $\wt{w}\in W_{\aff}$ let us denote by
$\IC_{\wt{w},\Gr_G}$ the corresponding irreducible object of
$\on{D}(\Gr_G)_\crit\mod^I$.
For an element $w\in W$, let $\cla_w\in W_{\aff}$ denote the unique
dominant coweight satisfying:
$$
\begin{cases}
&\langle \alpha_\imath,\cla\rangle=0 \text{ if } w(\alpha_\imath) \text{ is
positive, and} \\ &\langle \alpha_\imath,\cla\rangle=1 \text{ if }
w(\alpha_\imath) \text{ is negative,}
\end{cases}
$$
for $\imath$ running over the set of vertices of the Dynkin diagram.
It was shown in {\it loc. cit.} that the objects
$\Ind^\Hecke(\IC_{w\cdot \lambda_w})$ for $w\in W$ are the
irreducibles of $\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}$.
\medskip
Combining this with \lemref{H induced} and \corref{main cor}, we
obtain:
\begin{thm} \label{decsr of irr}
Isomorphism classes of irreducible objects of
$\hg_\crit\mod^{I^0}_\reg$ are parameterized by pairs $(\chi\in
\Spec(\fZ^\reg_\fg),w\in W)$. For each such pair the corresponding
irreducible object is given by
$$\Gamma(\Gr_G,\IC_{w\cdot \lambda_w})\underset{\fZ^\reg_\fg}\otimes
\BC_\chi.$$
\end{thm}
\ssec{The algebroid action} \label{algebroid}
Let $\isom_{\fZ}$ be the Lie algebroid of the groupoid
$\Isom_{\fZ}$. According to \cite{BD} (see also \cite{FG2}, Sect. 7.4
for a review), we have a canonical action of $\isom_{\fZ}$ on
$\wt{U}^\reg_\crit(\hg)$ by outer derivations, where
$\wt{U}^\reg_\crit(\hg)$ is the topological associative algebra
corresponding to the category $\hg_\crit\mod_\reg$ and its
tautological forgetful functor to vector spaces.
In more detail, there exists a topological associative algebra,
denoted $U^{\ren,\reg}(\hg_\crit)$, and called
the renormalized universal enveloping algebra at the critical
level. It is endowed with a natural filtration, with the $0$-th term
$U^{\ren,\reg}(\hg_\crit)_0$ being $U^\reg(\hg_\crit)$, and
$$U^{\ren,\reg}(\hg_\crit)_1/U^{\ren,\reg}(\hg_\crit)_0\simeq
U^\reg(\hg_\crit)\underset{\fZ^\reg_\fg}\hattimes \isom_\fZ.$$
The action of $\isom_{\fZ}$ on $\wt{U}^\reg_\crit(\hg)$ is given
by the adjoint action of
$\isom_\fZ$, regarded as a subset of
$\subset U^{\ren,\reg}(\hg_\crit)_1/U^{\ren,\reg}(\hg_\crit)_0$.
\medskip
Let $S$ be an affine scheme, and let $\chi_S$ be an $S$-point of
$\Spec(\fZ^\reg_\fg)$. Let $\xi_S$ be a section of
$\isom_{\fZ}|_S$. Set $S':=S\times \Spec(\BC[\epsilon]/\epsilon^2)$;
then the image of $\xi_S$ in $T(\Spec(\fZ^\reg_\fg))|_S$ gives rise to
an $S'$-point, denoted, $\chi'_S$, of $\Spec(\fZ^\reg_\fg)$.
Let $\hg_\crit\mod_S$ (resp., $\hg_\crit\mod_{S'}$) be the
corresponding base-changed category, where the latter identifies with
the category of discrete modules over
$\wt{U}^\reg_\crit(\hg)\underset{\fZ^\reg_\fg}\otimes \CO_S$ (resp.,
$\wt{U}^\reg_\crit(\hg)\underset{\fZ^\reg_\fg}\otimes \CO_{S'}$).
Then the above action of $\isom_{\fZ}$ on $\hg_\crit\mod_\reg$ gives
rise to the following construction:
\medskip
To every $\CM\in \hg_\crit\mod_S$ we can functorially attach an
extension
\begin{equation} \label{action of algebroid}
0\to \CM\to \CM'\to \CM\to 0,\,\, \qquad \CM'\in \hg_\crit\mod_{S'}.
\end{equation}
The module $\CM'$ is defined as follows. The above action of
$\isom_{\fZ}$ by outer derivations of $\wt{U}^\reg_\crit(\hg)$ allows
to lift $\xi_S$ to an isomorphism
$$A(\xi_S):\wt{U}^\reg_\crit(\hg)\underset{\fZ^\reg_\fg,\chi_{S'}}
\otimes \CO_{S'}\to
\wt{U}^\reg_\crit(\hg)\underset{\fZ^\reg_\fg,\chi_S} \otimes
\CO_S[\epsilon]/\epsilon^2.$$ We set $\CM'$ to be the
$\wt{U}^\reg_\crit(\hg)\underset{\fZ^\reg_\fg,\chi_{S'}} \otimes
\CO_{S'}$-module, corresponding via $A(\xi_S)$ to
$\CM[\epsilon]/\epsilon^2$.
The isomorphism $A(\xi_S)$ is defined up to conjugation by an element
of the form $1+\epsilon \cdot u$, $u\in
\wt{U}^\reg_\crit(\hg)\underset{\fZ^\reg_\fg,\chi_S} \otimes
\CO_S$. Since this automorphism can be canonically lifted onto
$\CM[\epsilon]/\epsilon^2$, we obtain that $\CM'$ is well-defined.
By construction, the functor $\CM\mapsto \CM'$ respects the
Harish-Chandra $G\ppart$-actions on the categories $\hg_\crit\mod_{S}$
and $\hg_\crit\mod_{S'}$, respectively.
\medskip
Let us note now that a data $(\chi_S:S\to \Spec(\fZ^\reg_\fg),
\xi_S\in \isom_{\fZ}|_S)$ as above can be regarded as a map $S'\to
\Isom_{\fZ}$, where first and second projections
$$S'\to \Isom_{\fZ}\rightrightarrows \Spec(\fZ^\reg_\fg)$$
are equal to
$$S'\to S\overset{\chi_S}\to \Spec(\fZ^\reg_\fg) \text{ and }
S'\overset{\chi'_S}\to \Spec(\fZ^\reg_\fg),$$
respectively.
Hence, \corref{main cor, families} gives rise to an equivalence
$$\hg_\crit\mod^{I^0}_S\otimes \BC[\epsilon]/\epsilon^2\simeq
\hg_\crit\mod^{I^0}_{S'},$$ and, in particular, to a functor
\begin{equation} \label{second action of algebroid}
\hg_\crit\mod^{I^0}_S\to \hg_\crit\mod^{I^0}_{S'}.
\end{equation}
\begin{prop} \label{two actions of algebroid}
The functor
$$\CM\mapsto \CM':\hg_\crit\mod_S\to \hg_\crit\mod_{S'}$$ of
\eqref{action of algebroid}, restricted to $\hg_\crit\mod^{I^0}_S$, is
canonically isomorphic to the above functor \eqref{second action of
algebroid}.
\end{prop}
\begin{proof}
The assertion follows from the fact that for $\CF\in
\on{D}(\Gr_G)_\crit\mod$, the $\hg_\crit$-action on
$\Gamma(\Gr_G,\CF)$ lifts canonically to an action of
$U^{\ren,\reg}(\hg_\crit)$ (see \cite{FG2}, Sect 7.4), so that for
$(S,\chi_S,\xi_S)$ as above we have a canonical trivialization
$$\gamma_\CF:\Gamma(\Gr_G,\CF)'\simeq
\Gamma(\Gr_G,\CF)[\epsilon]/\epsilon^2,$$ in the notation of
\eqref{action of algebroid}. Moreover, this functorial isomorphism is
compatible with that of \thmref{BDisom} in the sense that for every
$V\in \Rep(\cG)$, the diagram
$$
\CD \Gamma(\Gr_G,\CF\star \CF_V)' @>{\gamma_{\CF\star \CF_V}}>>
\Gamma(\Gr_G,\CF\star \CF_V)[\epsilon]/\epsilon^2 \\ @V{\beta_V}VV
@V{\beta_V\otimes \on{id}}VV \\
\Bigl(\Gamma(\Gr_G,\CF)\underset{\fZ^\reg_\fg}\otimes \CV\Bigr)'
@>{\gamma_\CF\otimes \xi_S}>>
\Bigl(\Gamma(\Gr_G,\CF)\underset{\fZ^\reg_\fg}\otimes
\CV\Bigr)[\epsilon]/\epsilon^2, \endCD
$$ commutes, where the bottom arrow comprises the isomorphism
$\gamma_\CF$ and the canonical action of $\xi_S$ on $\CV_\fZ$. The
latter compatibility follows assertion (b) in Theorem 8.4.2 of
\cite{FG2}.
\end{proof}
\ssec{Relation to semi-infinite cohomology}
Let us consider the functor of semi-infinite cohomology on the category
$\hg_\crit\mod_{\reg}^{I^0}$:
$$\CM \mapsto H^\semiinfb(\fn\ppart,\fn[[t]],\CM \otimes \Psi_0)$$
(see \cite{FG2}, Sect. 18 for details concerning this functor).
For an $S$-point $\chi_S$ of $\Spec(\fZ^\reg_\fg)$ and $\CM\in
\hg_\crit\mod_S$, each $H^\semiinfi(\fn\ppart,\fn[[t]],\CM\otimes
\Psi_0)$ is naturally an $\CO_S$-module.
\medskip
Let now $(\chi_{1,S},\chi_{2,S})$ be a pair of $S$-points of
$\Spec(\fZ^\reg_\fg)$, equipped with a lift $S\to \Isom_\fZ$,
and let $\CM_1\in \hg_\crit\mod_{S,1}^{I^0}$
and $\CM_2\in \hg_\crit\mod_{S,2}^{I^0}$ be two objects corresponding
to each other under the equivalence of \corref{main cor, families}.
\begin{prop} \label{behaviour of semiinf}
Under the above circumstances the $\CO_S$-modules
$$H^\semiinfi(\fn\ppart,\fn[[t]],\CM_1\otimes \Psi_0)\text{ and }
H^\semiinfi(\fn\ppart,\fn[[t]],\CM_2\otimes \Psi_0)$$
are canonically isomorphic.
\end{prop}
\begin{proof}
The assertion of the proposition can be tautologically translated as
follows:
The functor
$$\on{D}(\Gr_G)_\crit\mod \overset{\Gamma} \to \hg_\crit\mod_\reg
\overset{H^\semiinfi(\fn\ppart,\fn[[t]],?\otimes
\Psi_0)}\longrightarrow \fZ^\reg_\fg\mod$$ factors through a functor
$$H^\semiinfi_{\cG}:\on{D}(\Gr_G)_\crit\mod \to \Rep(\cG),$$ followed
by the pull-back functor, corresponding to the morphism
$\Spec(\fZ^\reg_\fg)\to \on{pt}/\cG$. Moreover, for $V\in \Rep(\cG)$
we have a functorial isomorphism
\begin{equation} \label{semiinf ident}
H^\semiinfi_{\cG}(\CF\star \CF_V)\simeq H^\semiinfi_{\cG}(\CF)\otimes V,
\end{equation}
compatible with the isomorphism of \thmref{BDisom}(1).
\medskip
The sought-after functor $H^\semiinfi_{\cG}$ has been
essentially constructed in \cite{FG2}, Sect. 18.3. Namely,
$$\Hom_{\cG}\bigl(V^\cla,H^\semiinfi_{\cG}(\CF)\bigr):=
H^i(N\ppart,\CF|_{N\ppart\cdot t^\cla}\otimes \Psi_0),$$ in the
notation of {\it loc. cit.} The isomorphisms \eqref{semiinf ident}
follow from the definitions.
\end{proof}
Finally, we would like to compare the isomorphisms of
\propref{behaviour of semiinf} and \propref{two actions of
algebroid}. Let $\CM$ be an object of $\hg_\crit\mod_\reg^{I^0}$; let
$\chi_S$ be an $S$-point of $\Spec(\fZ^\reg_\fg)$ and $\xi_S$ a
section of $\isom_\fZ|_S$.
On the one hand, in Proposition 18.3.2 of \cite{FG2} we have shown
that there exists a canonical isomorphism:
$${\bf a}_\CM:H^\semiinfi(\fn\ppart,\fn[[t]],\CM'\otimes \Psi_0)\simeq
H^\semiinfi(\fn\ppart,\fn[[t]],\CM\otimes \Psi_0)[\epsilon]/\epsilon^2,$$
valid for any $\CM\in \hg_\crit\mod_\reg$.
On the other hand, combining \propref{two actions of algebroid} and
\propref{behaviour of semiinf} we obtain another isomorphism
$${\bf b}_\CM:H^\semiinfi(\fn\ppart,\fn[[t]],\CM'\otimes \Psi_0)\simeq
H^\semiinfi(\fn\ppart,\fn[[t]],\CM\otimes \Psi_0)[\epsilon]/\epsilon^2.$$
Unraveling the two constructions, we obtain the following:
\begin{lem}
The isomorphisms ${\bf a}_\CM$ and ${\bf b}_\CM$ coincide.
\end{lem}
\section{Proof of the main theorem} \label{main proof}
In \secref{main result} we have constructed a functor
$$
\Gamma^{\Hecke_\fZ}:
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}\to
\hg_\crit\mod_\reg^{I^0}.
$$
Now we wish to show that this functor is an equivalence of
categories. This will prove \thmref{main}.
We start by constructing in \secref{Fw} certain objects $\CF^\fZ_w, w
\in W$, of the category $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$
such that $\Gamma^{\Hecke_\fZ}(\CF^\fZ_w) \simeq {\mathbb
M}_{w,\on{reg}}$, the ``standard modules'' of the category
$\hg_\crit\mod_\reg^{I^0}$. The main result of \secref{Fw},
\thmref{get Wakimoto}, will be proved in \secref{sect get
Wakimoto}. Next, in \secref{part one} we prove part (1) of
\thmref{main} that the functor $\Gamma^{\Hecke_\fZ}$ is exact. We then
outline in \secref{general} a general framework for proving that it is
an equivalence. Using this framework, we prove \thmref{main} modulo
\thmref{get Wakimoto}.
In \secref{remark on general} we explain what needs to be done in
order to prove our stronger \conjref{general conj}. Finally, in
Sects.~\ref{another}--\ref{proof of flatness} we give an alternative
proof of part (1) of \thmref{main}.
\ssec{Standard modules} \label{Fw}
For an element $w\in W$, let $\BM_w$ be the Verma module over $\ghat$,
$$
\BM_w = \on{Ind}^{\hg_\crit}_{\fg[[t]]}(M_{w(\rho)-\rho}),
$$
where for a weight $\lambda$ we denote by $M_\lambda$ the Verma module
over $\fg$ with highest weight $\lambda$.
Let $\BM_{w,\reg}$ be the maximal quotient module that belongs to
$\hg_\crit\mod_\reg$, i.e.,
$\BM_{w,\reg}=\BM_{w}\underset{\fZ_\fg}\otimes \fZ_\fg^\reg$. In fact,
it was shown in \cite{FG2}, Corollary 13.3.2, that as modules over
$\fZ_\fg$, all $\BM_w$ are supported over a quotient algebra
$\fZ^\nilp_\fg$, and are flat as $\fZ^\nilp_\fg$-modules. The
subscheme $\Spec(\fZ^\reg_\fg)\subset \Spec(\fZ_\fg)$ is contained in
$\Spec(\fZ^\nilp_\fg)$, so the definition of $\BM_{w,\reg}$ does not
neglect any lower cohomology.
The main ingredient in the remaining steps of our proof of
\thmref{main} is the following:
\begin{thm} \label{get Wakimoto}
For each $w\in W$ there exists an
object $\CF^\fZ_w\in \on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$, such that
$\Gamma^{\Hecke_\fZ}(\Gr_G, \CF_w)$ is isomorphic to $\BM_{w,\reg}$.
\end{thm}
The proof of this theorem will consist of an explicit construction of
the objects $\CF^\fZ_w$, which will be carried out in \secref{sect get
Wakimoto}.
The proof of \thmref{main} will only use a part of the assertion of
\thmref{get Wakimoto}: namely, that there exist objects $\CF^\fZ_w\in
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$, endowed with a surjection
\begin{equation} \label{surj on Wak}
\Gamma^{\Hecke_\fZ}(\Gr_G, \CF^\fZ_w)\twoheadrightarrow \BM_{w,\reg}.
\end{equation}
What we will actually use is the following corollary of this
statement:
\begin{cor} \label{F non-zero}
For every $\CM\in \hg_\crit\mod_\reg^{I^0}$ there exists
an object $\CF^H\in \on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$
and a non-zero map $\Gamma^{\Hecke_\fZ}(\Gr_G, \CF^H)\to \CM$.
\end{cor}
\begin{proof}
By \cite{FG2}, Lemma 7.8.1, for every object $\CM\in
\hg_\crit\mod_\reg^{I^0}$ there exists $w\in W$ and a non-zero map
$\BM_{w,\reg}\to \CM$.
\end{proof}
\ssec{Exactness} \label{part one}
Let us recall from \secref{sect descr of irr} the left adjoint functor
$\on{Ind}^{\Hecke_\fZ}$ to the obvious forgetful functor
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod \to \on{D}(\Gr_G)_\crit\mod$.
It is clear that every object of
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ can be covered by one of the
form $\on{Ind}^{\Hecke_\fZ}(\CF)$. From \lemref{H induced}(1) we
obtain that we can use bounded from above complexes, whose terms
consist of objects of the form $\on{Ind}^{\Hecke_\fZ}(\CF)$, in order
to compute $\on{L}\Gamma^{\Hecke_\fZ}$. Thus, we obtain:
\begin{lem} \label{L as tor}
For $\CF^H\in \on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$,
$$\on{L}^i\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H)\simeq
\on{Tor}_i^{\Fun(\Isom_{\fZ})}\bigl(\Gamma(\Gr_G,\CF^H),
\fZ^\reg_\fg\bigr).$$
\end{lem}
\medskip
We shall call an object of $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ {\it
finitely generated} if it can be obtained as a quotient of an object
of the form $\on{Ind}^{\Hecke_\fZ}(\CF)$, where $\CF$ is a finitely
generated object of $\on{D}(\Gr_G)_\crit\mod$.
It is easy to see that an object $\CF^H\in
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ is finitely generated if and
only if the functor
$\Hom_{\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod}(\CF^H,\cdot)$ commutes
with direct sums.
We shall call an object of $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ {\it
finitely presented}, if it is isomorphic to
$\on{coker}\bigl(\on{Ind}^{\Hecke_\fZ}(\CF_1)\to
\on{Ind}^{\Hecke_\fZ}(\CF_2)\bigr)$, where $\CF_1,\CF_2$ are both
finitely generated objects of $\on{D}(\Gr_G)_\crit\mod$. The following
lemma is straightforward.
\begin{lem} \label{all as ind fp} \hfill
\smallskip
\noindent{\em (1)} An object $\CF^H\in
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ is finitely presented if and
only if the functor
$\Hom_{\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod}(\CF^H,\cdot)$ commutes
with filtering direct limits.
\smallskip
\noindent{\em (2)} Every object of
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ is isomorphic to a filtering
direct limit of finitely presented ones.
\end{lem}
The proof of the following proposition will be given in \secref{sect
finite amplitude}.
\begin{prop} \label{finite amplitude}
For every finitely presented object of
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$, the corresponding object
$\on{L}\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H)\in D^-(\hg_\crit\mod_\reg)$
belongs to $D^b(\hg_\crit\mod_\reg)$.
\end{prop}
The crucial step in the proof of part (1) of \thmref{main} is the
following:
\begin{prop} \label{bounded exact}
If $\CF^H\in \on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$ is such that
$\on{L}\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H)$ belongs to
$D^b(\hg_\crit\mod_\reg)^{I^0}$, then
$$\on{L}^i\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H)\bigr)=0, \qquad i>0.$$
\end{prop}
\begin{proof}
Let $\CM$ be the lowest cohomology of
$\on{L}\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H)$, which lives, say, in degree
$-k$. By \corref{F non-zero} there exists another object $\CF_1^H\in
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$ and a non-zero map
$\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H_1)\to \CM$. Hence, we obtain a
non-zero map in $D^-(\hg_\crit\mod_\reg)$
$$\on{L}\Gamma^{\Hecke_\fZ}(\Gr_G,\CF_1^H)[k]\to
\on{L}\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H).$$
But by \thmref{GH fully faithful}, such map comes from a map
$\CF_1^H[k]\to \CF^H$, which is impossible if $k>0$.
\end{proof}
\medskip
\noindent{\em Proof of part (1) of \thmref{main}}. Combining
\propref{finite amplitude} and \propref{bounded exact}, we obtain that
$\on{L}^i\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H)=0$ for any $i>0$ and any
$\CF^H\in \on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$, which is
finitely presented.
However, by \lemref{L as tor}, the functors
$$\CF^H\mapsto \on{L}^i\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H)$$
commute with direct limits, and our assertion
follows from \lemref{all as ind fp}(2).\qed
\ssec{Proof of the equivalence} \label{general}
Consider the following general categorical framework. Let
$\sG:\CC_1\to \CC_2$ be an exact functor between abelian
categories. Assume that for $X,Y\in \CC_1$ the maps
$$\Hom_{\CC_1}(X,Y)\to \Hom_{\CC_2}(\sG(X),\sG(Y)) \text{ and }
\Ext^1_{\CC_1}(X,Y)\to \Ext^1_{\CC_2}(\sG(X),\sG(Y))$$
are isomorphisms.
\begin{lem} \label{adjoint}
If $\sG$ admits a right adjoint functor $\sF$ which is conservative,
then $\sG$ is an equivalence. \footnote{Recall that a functor $\sF$ is
called {\em conservative} if for any $X\neq 0$ we have $\sF(X)\neq
0$.}
\end{lem}
\begin{proof}
The fully faithfulness assumption on $\sG$ implies that the adjunction
map induces an isomorphism between the composition $\sF\circ \sG$ and
the identity functor on $\CC_1$. We have to show that the second
adjunction map is also an isomorphism.
For $X'\in \CC_2$ let $Y'$ and $Z'$ be the kernel and cokernel,
respectively, of the adjunction map
$$\sG\circ \sF(X')\to X'.$$
Being a right adjoint functor, $\sF$ is left-exact, hence we obtain an
exact sequence
$$0\to \sF(Y')\to \sF\circ \sG\circ \sF(X')\to \sF(X').$$ But since
$\sF(X')\to \sF\circ \sG(\sF(X'))$ is an isomorphism, we obtain that
$\sF(Y')=0$. Since $\sF$ is conservative, this implies that $Y'=0$.
\medskip
Suppose that $Z'\neq 0$. Since $\sF(Z')\neq 0$, there exists an object
$Z\in \CC_1$ with a non-zero map $\sG(Z)\to Z'$. Consider the induced
extension
$$0\to \sG\circ \sF(X')\to W'\to \sG(Z)\to 0.$$ Since $\sG$ induces a
bijection on $\Ext^1$, this extension can be obtained from an
extension
$$0\to \sF(X')\to W\to Z\to 0$$ in $\CC_1$. In other words, we obtain
a map $\sG(W)\to X'$, which does not factor through $\sG\circ
\sF(X')\subset X'$, which contradicts the $(\sG,\sF)$ adjunction.
\end{proof}
Thus, in order to prove of part (2) of \thmref{main} it remains to show
that the functor
$\Gamma^{\Hecke_\fZ}:\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}\to
\hg_\crit\mod_\reg^{I^0}$ admits a right adjoint. (The fact that it is
conservative will then follow immediately from \corref{F non-zero}.)
Recall from \cite{FG2}, Sect. 20.7, that the tautological functor
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}\hookrightarrow
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ admits a right adjoint, given by
$\on{Av}_{I^0}$. Hence, it suffices to prove the following:
\begin{prop} \label{exists right adjoint}
The functor $\Gamma^{\Hecke_\fZ}:\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod\to
\hg_\crit\mod_\reg$ admits a right adjoint.
\end{prop}
\begin{proof}
First, we will show the following:
\begin{lem} \label{kappa adjoint}
The functor $\Gamma:\on{D}(\Gr_G)_\crit\mod\to \hg_\crit\mod_\reg$
admits a right adjoint.
\end{lem}
\begin{proof}
We will prove that for any level $k$ the functor
$\Gamma:\on{D}(\Gr_G)_k\mod\to \hg_\crit\mod_k$ admits a
right adjoint (see the Introduction for the definition of these
categories). I.e., we have to prove the representability of the
functor
\begin{equation} \label{which functor}
\CF\mapsto \Hom_{\hg_k\mod}\bigl(\Gamma(\Gr_G,\CF),\CM\bigr)
\end{equation}
for every given $\CM\in \hg_k\mod$.
\medskip
Consider the following general set-up. Let $\CC$ be an abelian
category, and let $\CC^0$ be a full (but not necessarily abelian)
subcategory, such that the following holds:
\begin{itemize}
\item $\CC^0$ is equivalent to a small category.
\item The cokernel of any surjection $X''\twoheadrightarrow X'$ with
$X',X''\in \CC^0$, also belongs to $\CC^0$.
\item $\CC$ is closed under filtering direct limits.
\item For $X\in \CC^0$, the functor $\Hom_{\CC}(X,\cdot)$ commutes
with filtering direct limits.
\item Every object of $\CC$ is isomorphic to a filtering direct limit
of objects of $\CC^0$.
\end{itemize}
Then we claim that any contravariant left-exact functor $\sF\to
\Vect$, which maps direct sums to direct products (and, hence, direct
limits to inverse limits, by the previous assumption), is
representable.
Indeed, given such $\sF$, consider the category of pairs $(X,f)$,
where $X\in \CC^0$ and $f\in \sF(X)$. Morphisms between $(X,f)$ and
$(X',f')$ are maps $\phi:X\to X'$, such that $\phi^*(f')=f$. By the
first assumption on $\CC^0$, this category is small. By the second
assumption on $\CC^0$ and the left-exactness of $\sF$, this category
is filtering. It is easy to see that the object
$$\underset{\underset{(X,f)}\longrightarrow}{\lim}\, X.$$
represents the functor $\sF$.
\medskip
We apply this lemma to $\CC=\on{D}(\Gr_G)_k\mod$ with $\CC^0$
being the subcategory of finitely-generated D-modules. We set $\sF$ to
be the functor \eqref{which functor}, and the representability
assertion follows.
Note that we could have applied the above general principle to
$\CC=\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ and $\CC^0$ being the
subcategory of finitely presented objects, and obtain the assertion
of \propref{exists right adjoint} right away.
\end{proof}
Thus, for $\CM$, let $\CF$ be the object of $\on{D}(\Gr_G)_\crit\mod$
that represents the functor
$$\CF_1\mapsto
\Hom_{\hg_\crit\mod_\reg}\bigl(\Gamma(\Gr_G,\CF_1),\CM\bigr)$$ for a
given $\CM\in \hg_\crit\mod_\reg$. We claim that $\CF$ is naturally an
object of $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ and that it
represents the functor
\begin{equation} \label{which functor two}
\CF^H_1\mapsto
\Hom_{\hg_\crit\mod_\reg}\bigl(\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^H_1),\CM\bigr).
\end{equation}
First, since the algebra $\fZ^\reg_\fg$ acts on $\CM$ by
endomorphisms, the object $\CF$ carries an action of $\fZ^\reg_\fg$ by
functoriality. Let us now construct the morphisms
$\alpha_V$. Evidently, it is sufficient to do so for $V$
finite-dimensional. Let $V^*$ denote its dual.
For a test object $\CF_1\in \on{D}(\Gr_G)_\crit\mod$
we have:
\begin{align*}
&\Hom_{\on{D}(\Gr_G)_\crit\mod}(\CF_1,\CF\star \CF_V)\simeq
\Hom_{\on{D}(\Gr_G)_\crit\mod}(\CF_1\star \CF_{V^*},\CF)\simeq \\
&\simeq \Hom_{\hg_\crit\mod_\reg}\bigl(\Gamma(\Gr_G,\CF_1\star
\CF_{V^*}),\CM\bigr)\simeq \\ &\simeq
\Hom_{\hg_\crit\mod_\reg}\bigl(\Gamma(\Gr_G,\CF_1)
\underset{\fZ^\reg_\fg}\otimes \CV^*_{\fZ^\reg_\fg},\CM\bigr)\simeq
\Hom_{\hg_\crit\mod_\reg}\bigl(\Gamma(\Gr_G,\CF_1),\CV_\fZ
\underset{\fZ^\reg_\fg}\otimes\CM\bigr),
\end{align*}
where the last isomorphism takes place since $\CV_\fZ$ is locally
free. For the same reason,
$$\Hom_{\on{D}(\Gr_G)_\crit\mod}(\CF_1,\CV_\fZ
\underset{\fZ^\reg_\fg}\otimes\CF)\simeq
\Hom_{\hg_\crit\mod_\reg}\bigl(\Gamma(\Gr_G,\CF_1),\CV_\fZ
\underset{\fZ^\reg_\fg}\otimes\CM\bigr),$$
which implies that there exists a canonical isomorphism $\alpha_V$
$$\CF\star \CF_V\simeq \CV_\fZ
\underset{\fZ^\reg_\fg}\otimes\CF,$$
as required. That these isomorphisms are compatible with tensor
products of objects of $\Rep(\cG)$ follows from \thmref{BDisom}(2).
Finally, the fact that $(\CF,\alpha_V)$, thus defined, represents the
functor \eqref{which functor two}, follows from the construction. This
completes the proof of \propref{exists right adjoint}.
\end{proof}
Thus, we obtain that the functor $\Gamma^{\Hecke_\fZ}$ admits a right
adjoint functor. Moreover, this right adjoint functor is conservative
by \corref{F non-zero}. Therefore part (2) of \thmref{main} now
follows from part (1), proved in \secref{part one}, and
\lemref{adjoint}, modulo \propref{finite amplitude} and \thmref{get
Wakimoto}. It remains to prove those two statements. \propref{finite
amplitude} will be proved in the next subsection and \thmref{get
Wakimoto} will be proved in \secref{sect get Wakimoto}.
\ssec{Proof of \propref{finite amplitude}} \label{sect finite amplitude}
Recall the category $\on{D}(\Gr_G)^{\Hecke}_\crit\mod$, introduced in
\secref{sect descr of irr}. Recall also that the $\cG$-torsor
$\CP_{\cG,\fZ}$ on $\Spec(\fZ^\reg_\fg)$ is non-canonically trivial,
and let us fix such a trivialization. This choice identifies the
category $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ with
$\on{D}(\Gr_G)^{\Hecke}_\crit\mod\otimes \fZ^\reg_\fg$, i.e., with the
category of objects of $\on{D}(\Gr_G)^{\Hecke}_\crit\mod$ endowed with
an action of $\fZ^\reg_\fg$ by endomorphisms.
Under this equivalence, the functor $\CF\mapsto
\on{Ind}^{\Hecke_\fZ}(\CF)$ goes over to
$$\CF\mapsto \on{Ind}^{\Hecke}(\CF)\otimes \fZ^\reg_\fg.$$ Note also
that the trivialization of $\CP_{\cG,\fZ}$ identifies $\Isom_{\fZ}$
with $\Spec(\fZ^\reg_\fg)\times \cG\times \Spec(\fZ^\reg_\fg)$, so
that the map ${\bf 1}_{\Isom_{\fZ}}$ corresponds to
$\Delta_{\Spec(\fZ^\reg_\fg)}\times {\bf 1}_{\cG}$. For $\CF$ as
above, we have an identification
$$\Gamma\bigl(\Gr_G,\on{Ind}^{\Hecke_\fZ}(\CF)\bigr)\simeq
\Gamma(\Gr_G,\CF)\otimes \CO_\cG \otimes \fZ^\reg.$$
\medskip
Let $\CF^H$ be a finitely presented object of
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ equal to the cokernel of a map
$$\phi:\on{Ind}^{\Hecke}(\CF_1)\otimes \fZ^\reg_\fg\to
\on{Ind}^{\Hecke}(\CF_2)\otimes \fZ^\reg_\fg.$$ Recall that
$\fZ^\reg_\fg$ is isomorphic to a polynomial algebra
$\BC[x_1,...,x_n,...]$. Since $\CF_1$ was assumed finitely generated,
a map as above has the form $\phi_m\otimes
\on{id}_{\BC[x_{m+1},x_{m+2},...]}$, where $\phi_m$ is a map
$$\on{Ind}^{\Hecke}(\CF_1)\otimes \BC[x_1,...,x_m]\to
\on{Ind}^{\Hecke}(\CF_2)\otimes \BC[x_1,...,x_m]$$
defined for some $m$.
Hence, as a module over $\Fun(\Isom_{\fZ})\simeq \fZ^\reg_\fg\otimes
\CO_\cG\otimes \fZ^\reg_\fg$,
\begin{equation} \label{shape of module}
\Gamma(\Gr_G,\CF^H)\simeq \CL\otimes \BC[x_{m+1},x_{m+2},...],
\end{equation}
where $\CL$ is some module over $\fZ^\reg_\fg\otimes \CO_\cG\otimes
\BC[x_1,...,x_m]$.
\medskip
We can compute
$$\Gamma(\Gr_G,\CF^H)\underset{\Fun(\Isom_{\fZ})}{\overset{L}\otimes}
\fZ^\reg_\fg$$ in two steps, by first restricting to the preimage of
the diagonal under
$$\Spec(\fZ^\reg_\fg)\times \cG\times
\Spec(\fZ^\reg_\fg)\twoheadrightarrow
\Spec(\BC[x_{m+1},x_{m+2},...])\times
\Spec(\BC[x_{m+1},x_{m+2},...]),$$ and then by further restriction to
$\Spec(\BC[x_1,...,x_m])\times \Spec(\BC[x_{m+1},x_{m+2},...])$
sitting inside
$$\Spec(\BC[x_1,...,x_m])\times \cG\times \Spec(\BC[x_1,...,x_m])\times
\Spec(\BC[x_{m+1},x_{m+2},...]).$$
When we apply the first step to the module appearing in \eqref{shape
of module}, it is acyclic off cohomological degree $0$. The second
step has a cohomological amplitude bounded by $m+\dim(\cG)$.
Hence,
$$\on{Tor}_i^{\Fun(\Isom_{\fZ})}\bigl(\Gamma(\Gr_G,\CF^H),
\fZ^\reg_\fg\bigr)=0$$ for $i>m+\dim(\cG)$, which is what we had to
show.
This completes the proof of \propref{finite amplitude}. Therefore the
proof of \thmref{main} is now complete modulo \thmref{get Wakimoto}.
\ssec{A remark on the general case} \label{remark on general}
Let us note that the proof of \thmref{main} presented above would
enable us to prove the general \conjref{general conj} if we could
show that the functor
$$\Loc:\hg_\crit\mod_{\reg}\to \on{D}(\Gr_G)_\crit\mod,$$
right adjoint to the functor
$\Gamma:\on{D}(\Gr_G)_\crit\mod\to \hg_\crit\mod_\reg$
is conservative. In other words, in order to prove \conjref{general conj}
we need to know that for every $\CM\in \hg_\crit\mod_\reg$ there
exists a critically twisted D-module $\CF$ on $\Gr_G$ with
a non-zero map $\Gamma(\Gr_G,\CF)\to \CM$. This,
in turn, can be reformulated as follows:
Let $\on{Diff}(\Gr_G)_\crit$ be the *-sheaf of critically twisted
differential operators on $\Gr_G$. This is a pro-object of
$\on{D}(\Gr_G)_\crit\mod$, defined by the property that
$$\on{Hom}(\on{Diff}(\Gr_G)_\crit,\CF)\simeq \Gamma(\Gr_G,\CF)$$
functorially in $\CF\in \on{D}(\Gr_G)_\crit\mod$.
Explicitly, let us write $\Gr_G$ as
$\underset{\CY}{\underset{\longrightarrow}{"\lim"}}\, \CY$, where
$\CY\subset \Gr_G$ are closed sub-schemes. For each such $\CY$, let
$\on{Dist}(\CY)_\crit\in \on{D}(\Gr_G)_\crit\mod$ be the twisted
D-module of distributions on $\CY$, i.e., the object
$\on{Ind}^{\on{D}(\Gr_G)_\crit\mod}_{\QCoh(\Gr_G)}(\CO_{\CY})$, which
means by definition that
$$\on{Hom}_{\on{D}(\Gr_G)_\crit\mod}\Bigl(
\on{Ind}^{\on{D}(\Gr_G)_\crit\mod}_{\QCoh(\Gr_G)}(\CO_{\CY}),\CF\Bigr)=
\on{Hom}_{\QCoh(\Gr_G)}\Bigl(\CO_Y,\CF\Bigr).$$
Then
$$\on{Diff}(\Gr_G)_\crit: =
\underset{\CY}{\underset{\longleftarrow}{"\lim"}}\,
\on{Dist}(\CY)_\crit\in \on{Pro}(\on{D}(\Gr_G)_\crit\mod).$$
Let $\Gamma(\Gr_G,\on{Diff}(\Gr_G)_\crit)$ be the corresponding object
of $\on{Pro}(\hg_\crit\mod_\reg)$.
We obtain:
\begin{cor}
The following assertions are equivalent:
\smallskip
\noindent{\em (1)}
\conjref{general conj} holds.
\smallskip
\noindent{\em (2)}
The object $\Gamma(\Gr_G,\on{Diff}(\Gr_G)_\crit)$ is a pro-projective
generator of $\hg_\crit\mod_\reg$.
\smallskip
\noindent{\em (3)}
The functor on $\hg_\crit\mod_\reg$
$$\CM\mapsto \on{Hom}_{\hg_\crit\mod_\reg}\Bigl(
\Gr_G,\on{Diff}(\Gr_G)_\crit,\CM\Bigr)$$ is conservative.
\end{cor}
\ssec{Another proof of exactness} \label{another}
In this subsection we give shall present an alternative proof of part
(1) of \thmref{main}.
According to \lemref{L as tor}, proving the exactness property stated
in part (1) of \thmref{main} is equivalent to proving that
\begin{equation} \label{van tor}
\on{Tor}_i^{\Fun(\Isom_{\fZ})}\bigl(\Gamma(\Gr_G,\CF^H),
\fZ^\reg_\fg\bigr)=0
\end{equation}
for all $i>0$ and $\CF^H\in
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$. We will derive this from
the following weaker statement:
\begin{prop} \label{flatness}
For every $\CF\in \on{D}(\Gr_G)_\crit\mod^{I^0}$, the space of
sections $\Gamma(\Gr_G,\CF)$ is flat as a $\fZ^\reg_\fg$-module.
\end{prop}
Note that our general conjecture \eqref{general conj} predicts that
both \eqref{van tor} and the assertion of \propref{flatness} should
hold without the $I^0$-equivariance assumption. However, at the moment
we can neither prove the corresponding generalization of
\propref{flatness} nor derive \eqref{van tor} from it.
\medskip
Let us first show how \propref{flatness} implies \eqref{van tor} on
the $I^0$-equivariant category.
\begin{prop} \label{analysis of irr}
Every finitely generated object of
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$ admits a finite
filtration, whose subquotients are of the form
\begin{equation} \label{form of irr}
\on{Ind}^{\Hecke_\fZ}(\CF)\underset{\fZ^\reg_\fg}\otimes \CL,
\end{equation}
where $\CL$ is a $\fZ^\reg_\fg$-module.
\end{prop}
Let us deduce \eqref{van tor} from this proposition.
\begin{proof}
It is enough to show that \eqref{van tor} holds for finitely presented
objects of the category $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$. By
\propref{analysis of irr}, we conclude that it is enough to consider
objects of $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$ of the form
given by \eqref{form of irr}.
We have:
$$\Gamma\bigl(\Gr_G,\on{Ind}^{\Hecke_\fZ}(\CF)\underset{\fZ^\reg_\fg}
\otimes \CL\bigr) \underset{\Fun(\Isom_{\fZ})}{\overset{L}\otimes}
\fZ^\reg_\fg\simeq
\Gamma\bigl(\Gr_G,\CF)\underset{\fZ^\reg_\fg}{\overset{L}\otimes}
\CL,$$ and the assertion follows from \propref{flatness}.
\end{proof}
Let us now prove \propref{analysis of irr}.
\begin{proof}
Choosing a trivialization of $\CP_{\cG,\fZ}$ as in the previous
subsection, we can identify
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$ with
$\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}\otimes \fZ^\reg_\fg$.
\medskip
Similarly to the case of $\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$, we
shall call an object of $\on{D}(\Gr_G)^{\Hecke}_\crit\mod$ finitely
generated if it is isomorphic to a quotient of some
$\on{Ind}^\Hecke(\CF)$ for a finitely generated $\CF\in
\on{D}(\Gr_G)_\crit\mod$.
\medskip
Let us recall from \cite{ABBGM}, Corollary 1.3.10(1), that every
finitely generated object in $\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}$
has a finite length. Therefore, every finitely generated object of
$\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}\otimes \fZ^\reg_\fg$ admits a
finite filtration, whose subquotients are quotients of modules of the
form $\CF^H\otimes \fZ^\reg_\fg$ with $\CF^H\in
\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}$ being irreducible. However,
every such quotient has the form $\CF^H\otimes \CL$ for some
$\fZ^\reg_\fg$-module $\CL$.
Moreover, as was mentioned in \secref{sect descr of irr}, by
\cite{ABBGM}, Corollary 1.3.10(2), every irreducible in
$\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}$ is of the form
$\on{Ind}^\Hecke(\CF)$ for some $\CF\in
\on{D}(\Gr_G)_\crit\mod^{I^0}$. This implies the assertion of the
proposition.
\end{proof}
\ssec{Proof of \propref{flatness}} \label{proof of flatness}
We can assume that our object $\CF\in \on{D}(\Gr_G)_\crit\mod^{I^0}$
is finitely generated, which automatically implies that it has a
finite length. This reduces us to the case when $\CF$ is irreducible.
It is easy to see that any irreducible object of
$\on{D}(\Gr_G)_\crit\mod^{I^0}$ is equivariant also with respect to
$\BG_m$, which acts on $G\ppart$, and hence on $\Gr_G$, by rescalings
$t \mapsto at$. Moreover, the grading arising on its space of sections
is bounded from above. (Our conventions are such that $\BV_\crit$ is
{\it negatively} graded.)
\medskip
Recall now that the action of $\wt{U}^\reg_\crit(\hg)$ on a module of
the form $\Gamma(\Gr_G,\CF)$ for an object $\CF\in \on{D}(\Gr_G)_\crit\mod$
canonically extends to an action of the renormalized algebra
$U^{\ren,\reg}(\hg_\crit)$. Recall also that
$U^{\ren,\reg}(\hg_\crit)$ contains a $\fZ^\reg_\fg$ sub-bimodule and
a Lie subalgebra $\wt{U}^\reg_\crit(\hg)^\sharp$, which is an
extension
$$0\to \wt{U}^\reg_\crit(\hg)\to \wt{U}^\reg_\crit(\hg)^\sharp\to
\isom_{\fZ}\to 0.$$ (The resulting action of $\isom_{\fZ}$ by outer
derivations on $\wt{U}^\reg_\crit(\hg)$ is the one discussed in
\secref{algebroid}.)
We will prove the following general assertion, which implies
\propref{flatness}:
\begin{lem}
Let $\CM$ be an object of $\hg_\crit\mod_\reg$, such that the action
of $\wt{U}^\reg_\crit(\hg)$ on it extends to an action of
$U^{\ren,\reg}(\hg_\crit)$. Assume also that $\CM$ is endowed with a
grading, compatible with the one on $U^{\ren,\reg}(\hg_\crit)$, given
by rescalings $t \mapsto at$. Finally, assume that the grading on
$\CM$ is bounded from above. Then $\CM$ is flat as a
$\fZ^\reg_\fg$-module.
\end{lem}
The proof is a variation of the argument used in \cite{BD},
Sect. 6.2.2:
\begin{proof}
We can identify $\fZ^\reg_\fg$ with a polynomial algebra
$\BC[x_1,...,x_n,...]$. Moreover, we can do so in a
grading-preserving fashion, in which case each generator $x_i$ will be
homogeneous of a {\it negative} degree.
It is enough to show that $\CM$ is flat over each subalgebra
$\BC[x_1,...,x_m]\subset \fZ^\reg_\fg$. We will prove the following
assertion:
\medskip
\noindent {\it For every vector $\bv\in
\BA^m:=\Spec(\BC[x_1,...,x_m])$, the $\BC[x_1,...,x_m]$-module $\CM$
is (non-canonically) isomorphic to its translate by means of $\bv$.}
\medskip
Clearly, a module over $\BC[x_1,...,x_m]$ having this property is
flat. To prove the above claim we proceed as follows. Choose a section
$\xi$ of $\isom_{\fZ}$, which projects onto $\bv$ under
$\isom_{\fZ}\to T(\Spec(\fZ^\reg_\fg))$, where we think of $\bv$ as a
constant vector field on $\fZ^\reg_\fg\simeq
\Spec(\BC[x_1,...,x_n,...])$. Let us further lift $\xi$ to an element
$\xi'$ of $\wt{U}^\reg_\crit(\hg)^\sharp$.
Since the grading on the $x_i$'s is positive, we can choose $\xi'$
to belong to the (completion of the) sum of strictly positive graded
components of $\wt{U}^\reg_\crit(\hg)^\sharp$.
Then the assumption that the grading on $\CM$ is bounded from above,
implies that $\on{exp}(\xi')$ is a well-defined automorphism of $\CM$
as a vector space. This automorphism covers the automorphism
$\on{exp}(\bv)$ of $\BC[x_1,...,x_m]$, and the latter is the same as
the translation by $\bv$.
\end{proof}
\section{Proof of \thmref{get Wakimoto}} \label{sect get Wakimoto}
In this section we construct the objects $\CF_w^\fZ$ of the category
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod^{I^0}$ whose existence is stated
in \thmref{get Wakimoto}.
\ssec{} \label{wak without center}
We first describe the analogues of these objects in the category
$\on{D}(\Gr_G)^{\Hecke}_\crit\mod^{I^0}$. These objects, which we will
denote by $\CF_w$, were studied in \cite{ABBGM} under the name "baby
co-Verma modules".
\medskip
First, we consider the case $w=w_0$. Recall that the Langlands dual
group comes equipped with a standard Borel subgroup $\cB\subset \cG$;
we shall denote by $\cH$ the Cartan quotient of $\cB$.
Let $\cB^-\subset \cG$ be a Borel subgroup in the generic relative
position with respect to $\cB$. The latter means that $\cB\cap \cB^-$
is {\it a} Cartan subgroup; we shall identify it with $\cH$ by means
of the projection
$$\cB\cap \cB^-\hookrightarrow \cB\twoheadrightarrow \cH.$$
For $\cla\in \cLambda^+$ let $\ell^\cla$ be the line of coinvariants
$(V^\cla)_{\cN^-}$, where $V^\cla$ denotes the standard irreducible
$\cG$-representation of highest weight $\cla$ with respect to $\cB$.
The assignment $\cla\mapsto \ell^\cla$ is an $\cH$-torsor, and we obtain
a collection of maps
\begin{equation} \label{simple Plucker}
V^\cla\overset{\kappa^\cla}\twoheadrightarrow \ell^\cla,
\end{equation}
satisfying the Pl\"ucker relations, i.e., for any two dominant
coweights $\cla$ and $\cmu$, the diagram
\begin{equation} \label{Plucker rel}
\CD
V^\cla\otimes V^\cmu @>{\kappa^\cla\otimes \kappa^\cmu}>>
\ell^\cla\otimes \ell^\cmu \\ @VVV @V{\sim}VV \\ V^{\cla+\cmu}
@>{\kappa^{\cla+\cmu}}>> \ell^{\cla+\cmu} \endCD
\end{equation}
commutes.
\medskip
Let $\Fl_G = G\ppart/I$ be the affine flag variety. We have the
category $\on{D}(\Fl_G)_\crit\mod$ of right critically twisted
D-modules on $\Fl_G$ and the corresponding Iwahori equivariant
category $\on{D}(\Fl_G)_\crit\mod^I$. Given $\CF\in
\on{D}(\Gr_G)_\kappa\mod^I$ and $\CM\in \on{D}(\Fl_G)_\crit\mod^I$, we
can form their convolution, denoted by $\CM \underset{I}\star \CF$,
which is an object of $D^b(\on{D}(\Fl_G)_\crit\mod)^I$ (see \cite{FG2}
for details).
\medskip
For a dominant map $\cla$ let $j_{\cla,*}$ denote the $*$-extension of
the critically twisted D-module corresponding to the constant sheaf on
the Iwahori orbit of the point $t^\cla\in \Fl_G$. Let
$j_{\cla,\Gr_G,*}\in \on{D}(\Gr_G)_\crit\mod^I$ be
$j_{\cla,*}\underset{I}\star \delta_{1,\Gr_G}$; in other words it is
the $*$-extension of the constant D-module on the Iwahori orbit of the
point $t^\cla\in \Gr_G$. Note that for $\cmu\in \cLambda^+$ we have a
canonical map
$$j_{\cla,\Gr_G,*}\star \CF_{V^\cmu}\to j_{\cla+\cmu,\Gr_G,*},$$
obtained by identifying $\CF_{V^\cmu}$ with $\IC_{\Grb^\cmu}$.
Consider the object of $\on{D}(\Gr_G)^\Hecke_\crit\mod$ equal to the
direct sum
$$\wt{\CF}_{w_0}:=\underset{\cla\in \cLambda^+}\oplus\,
\on{Ind}^{\Hecke}\bigl(j_{\cla,\Gr_G,*}\bigr) \otimes \ell^{-\cla}.$$
\medskip
For a dominant coweight $\cmu$ we have an evident map
\begin{equation} \label{shift map}
j_{\cmu,*}\underset{I}\star \wt{\CF}_{w_0}\to \ell^\cmu \otimes
\wt{\CF}_{w_0}.
\end{equation}
We obtain two maps $\wt{\CF}_{w_0}\star \CF_{V^\cmu}\rightrightarrows
\wt{\CF}_{w_0}\otimes \ell^\cmu$ that correspond to the two circuits
of the following non-commutative diagram:
$$
\CD
\wt{\CF}_{w_0}\star \CF_{V^\cmu} @>{\alpha_V}>> \uV^\cmu\otimes
\wt{\CF}_{w_0} \\ @VVV @V{\kappa^\cmu}VV \\
j_{\cmu,*}\underset{I}\star \wt{\CF}_{w_0} @>>> \ell^\cmu \otimes
\wt{\CF}_{w_0}, \endCD
$$
where the left vertical arrow comes from the following map, defined
for each $\cla$:
$$j_{\cla,\Gr_G,*}\star \CF_{R}\star \CF_{V^\cmu}\simeq
j_{\cla,\Gr_G,*}\star \CF_{V^\cmu}\star \CF_R\to
j_{\cla+\cmu,\Gr_G,*}\star \CF_R.$$ Here we are using the object
$\CF_R$ of $\on{D}(\Gr_G)_\crit\mod^{G[[t]]}$ introduced in
\secref{sect descr of irr}, so that $\on{Ind}^\Hecke(\CF)\simeq
\CF\star \CF_R$.
We set $\CF_{w_0}$ to be the maximal quotient of $\wt{\CF}_{w_0}$,
which co-equalizes the resulting two maps
$$\ell^{-\cmu}\otimes \wt{\CF}_{w_0}\star \CF_{V^\cmu}\rightrightarrows
\wt{\CF}_{w_0}$$ for every $\cmu\in \cLambda^+$.
Note that the map \eqref{shift map} gives rise to a map
\begin{equation} \label{shift maps}
j_{\cmu,*}\underset{I}\star \CF_{w_0}\to \ell^\cmu \otimes \CF_{w_0}.
\end{equation}
By construction, $\CF_{w_0}$ has the following universal property:
\medskip
Let $\CF^H$ be an object of $\on{D}(\Gr_G)_\crit\mod^I$, endowed
with a system of morphisms
\begin{equation} \label{shift maps 2}
j_{\cmu,*}\underset{I}\star\CF^H\to \ell^\cmu \otimes \CF^H,
\end{equation}
compatible with the isomorphisms
\begin{equation} \label{j's multiply}
j_{\cmu,*}\underset{I}\star j_{\cmu',*}\simeq j_{\cmu+\cmu',*}
\end{equation}
and $\ell^\cmu\otimes \ell^{\cmu'}\simeq \ell^{\cmu+\cmu'}$.
Let $\phi: \CF_R\to \CF^H$ be a map, such that for every
$\cmu\in \cLambda$ the following diagram is commutative:
$$
\CD
\CF_R\star \CF_{V^\cmu} @>{\alpha_V}>> \uV^\cmu\otimes R
@>{\on{id}_{\uV^\cla}\otimes \phi}>> \uV^\cmu\otimes \CF^H
@>{\kappa^\cmu}>> \ell^\cmu\otimes \CF^H \\
@V{\sim}VV & & & & @AAA \\
\CF_{V^\cmu}\star \CF_R @>>> j_{\cmu,\Gr_G,*}\star \CF_R @>{\sim}>>
j_{\cmu,*}\underset{I}\star \CF_R @>{\on{id}_{j_{\cmu,*}}\star\phi}>>
j_{\cmu,*}\underset{I}\star \CF^H.
\endCD
$$
\begin{lem} \label{univ of baby Verma}
Under the above circumstances, there exists a unique map
$\CF_{w_0}\to \CF^H$, extending $\phi$, and which intertwines
the maps \eqref{shift map} and \eqref{shift maps 2}.
\end{lem}
\ssec{}
We shall now establish the equivalence between the present definition
of $\CF_{w_0}$ and the objects defined in \cite{ABBGM}.
For a weight $\cnu\in \cLambda$ consider the inductive system of
objects of $\on{D}(\Gr_G)_\crit\mod$, parameterized by pairs of elements
$\cla,\cmu\in \cLambda^+\,|\, \cla-\cmu=\cnu$, and whose terms are given
by
$$j_{\cla,\Gr_G,*}\star \CF_{(V^\cmu)^*}\otimes \ell^{-\cla+\cmu}.$$
The maps in this inductive system are defined whenever two
pairs $(\cla',\cmu')$ and $(\cla,\cmu)$ are such that
$\cla'-\cla=\cmu'-\cmu=:\ceta\in \cLambda^+$, and the corresponding
map equals the composition
\begin{align*}
& j_{\cla,\Gr_G,*}\star \CF_{(V^\cmu)^*}\otimes \ell^{-\cla+\cmu}\to
j_{\cla,\Gr_G,*}\star \CF_{V^\ceta}\star \CF_{(V^\ceta)^*}\star
\CF_{(V^\cmu)^*}\otimes \ell^{-\cla+\cmu}\to \\
&\to j_{\cla+\ceta,\Gr_G,*}\star \CF_{(V^{\cmu+\ceta})^*}\otimes
\ell^{-\cla-\ceta+(\cmu+\ceta)}.
\end{align*}
\medskip
Let $\CF'_{w_0}(\cnu)\in \on{D}(\Gr_G)_\crit\mod$ be the direct limit of
the above system. We endow $\CF'_{w_0}:=\underset{\cnu\in \cLambda}
\oplus\, \CF'_{w_0}(\cnu)$ with the structure of an object of
$\on{D}(\Gr_G)^\Hecke_\crit\mod$ as in Sect. 3.2.1 of \cite{ABBGM}.
\begin{prop} \label{two approaches}
There exists a natural isomorphism
$$\CF'_{w_0}\simeq \CF_{w_0}.$$
\end{prop}
\begin{proof}
The map $\CF_{w_0}\to \CF'_{w_0}$ is constructed using
\lemref{univ of baby Verma}, and the corresponding property
of $\CF'_{w_0}$ established in \cite{ABBGM}, Corollary 3.2.3.
To show that this map is an isomorphism, we construct
a map in the opposite direction $\CF'_{w_0}\to \CF_{w_0}$
(as mere objects of $\on{D}(\Gr_G)_\crit\mod$) as follows:
For each $\cla,\cmu\in \cLambda^+$, we let
$j_{\cla,\Gr_G,*}\star \CF_{(V^\cmu)^*}\otimes \ell^{-\cla+\cmu}$ embed into
$j_{\cla,\Gr_G,*}\star \CF_R\otimes \ell^{-\cla}$ by means of
$$\CF_{(V^\cmu)^*}\otimes \ell^{\cmu}\hookrightarrow
\CF_{(V^\cmu)^*}\otimes \uV^\cmu\hookrightarrow \CF_R,$$ where the
second arrow is given by
$$\ell^\cmu\simeq (\uV^\cmu)^{\cN}\hookrightarrow \uV^\cmu.$$
It is straightforward to check that this gives rise to a well-defined map
from the inductive system corresponding to $\CF'_{w_0}(\cnu)$,
and that the above two maps $\CF_{w_0}\leftrightarrows \CF'_{w_0}$
are mutually inverse.
\end{proof}
\begin{cor} \label{shift maps isom}
The maps \eqref{shift maps}
$j_{\cmu,*}\underset{I}\star \CF_{w_0}\to \ell^\cmu\otimes \CF_{w_0}$
are isomorphisms.
\end{cor}
\begin{proof}
The assertion follows from the fact that the maps
$$j_{\cmu,*}\underset{I}\star \CF'_{w_0}(\cnu)\to \ell^{\cmu}\otimes
\CF'_{w_0}(\cnu+\cmu)$$
are easily seen to be isomorphisms.
\end{proof}
Let us now define the objects $\CF_w$ for other elements $w\in W$.
We set
$$\CF_w:=j_{w\cdot w_0,!}\underset{I}\star \CF_{w_0}.$$
In other words, if $w_0=w'\cdot w$, then
$$\CF_{w_0}\simeq j_{w',*}\underset{I}\star \CF_w.$$
{}From \propref{two approaches} it follows that $\CF_w$ are
D-modules, i.e., that no higher cohomologies appear.
\ssec{}
Let us now define the sought-after objects $\CF_w^\fZ$ of the category
$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$.
\medskip
Consider the $\cG$-torsor $\CP_{\cG,\fZ}$ over $\Spec(\fZ^\reg_\fg)$.
Recall from \secref{recol} that we have a canonical isomorphism
$\Spec(\fZ^\reg_\fg)\simeq \Op_\cg(\D)$, under which $\CP_{\cG,\fZ}$
goes over the canonical $\cG$-torsor $\CP_{\cG,\Op}$ on the
space of opers (see \cite{FG2}, Sect. 8.3, for details). Thus, we
obtain a canonical reduction of $\CP_{\cG,\fZ}$ to $\cB$ that we will
denote by $\CP_{\cB,\fZ}$.
This $\cB$-reduction defines a $\cB^-$-reduction on
$\CP_{\cG,\fZ}$. In order to define a $\cB^-$-reduction, we need to
specify for each $\cla\in \cLambda$ a line bundle, which we will
denote by $\CL^{\cla}_{w_0}$, and for each $\cla\in \cLambda^+$ a
surjective homomorphism
$$\kappa^{\cla,\fZ}:\CV^\cla_\fZ\to \CL^\cla_{w_0}.$$ These line
bundles should be equipped with isomorphisms $\CL^{\cla+\cmu}_{w_0}
\simeq \CL^\cla_{w_0} \otimes \CL^\cmu_{w_0}$, and hence give rise to
a $\cH$-torsor on $\on{Spec}(\fZ^\reg_\fg)$, which we will denote by
$\CP_{\cH,w_0}$. In addition, the maps $\kappa^{\cla,\fZ}$ should
satisfy the Pl\"ucker relations, as in \eqref{Plucker rel}. Now
observe that our $\cB$-reduction $\CP_{\cB,\fZ}$ gives rise to a
collection of compatible line subbundles $\CL^\cla$ of
$\CV^\cla_{\fZ}$. We then define $\CL^\cla_{w_0}$ as the dual of the
line bundle $\CL^{-w_0(\cla)} \hookrightarrow \CV^{-w_0(\cla)}_{\fZ}
\simeq (\CV^\cla_{\fZ})^*$.
It follows from the definition of opers (see \cite{FG2}, Sect. 1) that
the line bundle $\CL^\cla_{w_0}$ over $\Spec(\fZ^\reg_\fg)$ is
canonically isomorphic to the trivial line bundle tensored with the
one-dimensional vector space $\omega_x^{\langle
\rho,w_0(\cla)\rangle}$, where $\omega_x$ is the fiber of
$\omega_{\D}$ at the closed point $x\in \D$.
\medskip
We define the object $\wt{\CF}^\fZ_{w_0}\in
\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod$ as a direct sum
$$\underset{\cla\in \cLambda^+}\oplus\,
\on{Ind}^{\Hecke_\fZ}\bigl(j_{\cla,\Gr_G,*}\bigr)
\underset{\fZ^\reg_\fg}\otimes \CL_{w_0}^{-\cla}.$$ We define
$\CF^\fZ_{w_0}$ to be the quotient of $\wt{\CF}^\fZ_{w_0}$ by the same
relations as those defining $\CF_{w_0}$ as a quotient of
$\wt{\CF}_{w_0}$.
If we choose a trivialization of the $\cG$-torsor $\CP_{\cG,\fZ}$ in
such a way that $\CL^\cla_{w_0}\simeq \fZ^\reg_\fg\otimes \ell^\cla$
(such a trivialization exists), then under the equivalence
$$\on{D}(\Gr_G)^{\Hecke_\fZ}_\crit\mod\simeq
\on{D}(\Gr_G)^{\Hecke}_\crit\mod\otimes \fZ^\reg_\fg,$$ the object
$\CF^\fZ_{w_0}$ corresponds to $\CF_{w_0}$.
By construction, we have a system of maps
\begin{equation} \label{shift maps 3}
j_{\cmu,*}\underset{I}\star \CF^\fZ_{w_0}\simeq \CL_{w_0}^{\cmu}
\underset{\fZ^\reg_\fg}\otimes \CF^\fZ_{w_0},
\end{equation}
which by \corref{shift maps isom} are in fact isomorphisms.
\medskip
For other elements $w\in W$ we define
$$\CF^\fZ_w:=j_{w\cdot w_0,!}\underset{I}\star \CF^\fZ_{w_0}.$$
\ssec{}
Our present goal is to define the maps
\begin{equation} \label{sought-for maps}
\phi_w:\Gamma^{\Hecke_\fZ}(\Gr_G,\CF_w^\fZ)\to \BM_{w,\reg}\otimes
\omega_x^{\langle 2\rho,\crho\rangle}.
\end{equation}
Since $\BM_{w,\reg}\simeq j_{w\cdot w_0,!}\underset{I}\star
\BM_{w_0,\reg}$, it is enough to define $\phi_w$ for $w=w_0$.
\medskip
Let $\CM$ be an object of $\hg_\crit\mod_\reg$. Assume that $\CM$ is
endowed with a system of maps
\begin{equation} \label{shift maps 4}
j_{\cmu,*}\underset{I}\star \CM\to \CL_{w_0}^{\cmu}
\underset{\fZ^\reg_\fg}\otimes \CM,
\end{equation}
defined for every $\cmu\in \cLambda^+$, compatible with the isomorphisms
\eqref{j's multiply} and $\CL_{w_0}^{\cmu} \underset{\fZ^\reg_\fg}\otimes
\CL_{w_0}^{\cmu'}\simeq \CL_{w_0}^{\cmu+\cmu'}$.
Let $\phi$ be a map $\BV_\crit\to \BM$, such that for any $\cmu\in
\cLambda^+$ the diagram
\begin{equation} \label{crucial diagram}
\CD \Gamma(\Gr_G,\CF_{V^\cmu}) @>{\beta_{V^\cmu}}>>
\CV^\cmu_\fZ\underset{\fZ^\reg_\fg} \otimes \BV_\crit
@>{\on{id}_{\CV^\cmu_\fZ}\otimes\phi}>>
\CV^\cmu_\fZ\underset{\fZ^\reg_\fg} \otimes \CM
@>{\kappa^{\cmu,\fZ}}>> \CL_{w_0}^{\cmu}
\underset{\fZ^\reg_\fg}\otimes \CM \\ @VVV & & & & @AAA \\
\Gamma(\Gr_G,j_{\cmu,\Gr_G,*}) @>{\sim}>> j_{\cmu,\Gr_G,*}\star
\BV_\crit @>{\sim}>> j_{\cmu,*}\underset{I}\star \BV_\crit
@>{\on{id}_{j_{\cmu,*}}\star \phi}>> j_{\cmu,*}\underset{I}\star \CM
\endCD
\end{equation}
is commutative.
\medskip
By the construction of $\CF_{w_0}^\fZ$, we have:
\begin{lem} \label{univ baby rep}
Under the above circumstances there exists a unique
map
$$\Gamma^{\Hecke_\fZ}(\Gr_G,\CF_{w_0}^\fZ)\to \CM,$$
which intertwines the maps \eqref{shift maps 3} and
\eqref{shift maps 4}.
\end{lem}
Thus, to construct the map as in \eqref{sought-for maps} for $w=w_0$
we need to verify that the module
$\CM:=\BM_{w_0,\reg}\otimes \omega_x^{\langle 2\rho,\crho\rangle}$
possesses the required structures.
First, the map
$$\BV_\crit\to \BM_{w_0,\reg}\otimes \omega_x^{\langle
2\rho,\crho\rangle}$$ was constructed in \cite{FG2}, Sect. 7.2.
\ssec{}
To construct the data of \eqref{shift maps 4} we need to recall some
material from \cite{FG2}, Sect. 13.4. According to {\it loc. cit.}
there exists some $\cH$-torsor $\{ \cla \mapsto \CL'{}^\cla_{w_0} \}$
on $\Spec(\fZ^\reg_\fg)$ and a system of isomorphisms
$$j_{\cmu,*}\underset{I}\star \BM_{w_0,\reg}\simeq
\CL'{}^\cla_{w_0} \underset{\fZ^\reg_\fg}\otimes \BM_{w_0,\reg}.$$
Thus, to construct the map $\phi_{w_0}$, we need to prove
the following assertion:
\begin{lem} \label{two torsors}
There exists an isomorphism of $\cH$-torsors
$$\CL^\cmu_{w_0}\simeq \CL'{}^\cmu_{w_0}$$ which makes the diagram
\eqref{crucial diagram} commutative for $\CM:=\BM_{w_0,\reg}\otimes
\omega_x^{\langle 2\rho,\crho\rangle}$.
\end{lem}
Below we will prove this assertion by a rather explicit calculation.
In a future publication, we will discuss a more
conceptual approach. The crucial step is the following statement:
\begin{lem} \label{comp non-zero}
The composition
$$\Gamma(\Gr_G,\CF_{V^\cmu})\to j_{\cmu,*}\underset{I}\star \BV_\crit
\overset{\on{id}_{j_{\cmu,*}}\star\phi}\to j_{\cmu,*}\underset{I}\star
\BM_{w_0,\reg}\otimes \omega_x^{\langle 2\rho,\crho\rangle}$$
is non-zero.
\end{lem}
This proposition will be proved in \secref{proof non-zero comp}. Let
us assume it and construct the required isomorphism
$\CL^\cmu_{w_0}\simeq \CL'{}^\cmu_{w_0}$.
\bigskip
\noindent {\em Proof of \lemref{two torsors}}. Recall from
\cite{FG2}, Corollary 13.4.2, that there exists an isomorphism,
defined up to a scalar, $\CL^\cmu_{w_0}\simeq \CL'{}^\cmu_{w_0}$,
compatible with the action of $\Aut(\D)$. \footnote{Choosing a
coordinate $t$ on $\D$, we obtain a subgroup $\BG_m\subset \Aut(\D)$
of rescalings $t \mapsto at$.} We will show that any choice of such
isomorphism makes the diagram \eqref{crucial diagram} commutative,
up to a non-zero scalar.
Thus, we are dealing with two non-zero maps
$$\CV^\cmu_\fZ\underset{\fZ^\reg_\fg}\otimes
\BV_\crit \rightrightarrows \BM_{w_0,\reg}\otimes
\omega_x^{\langle \rho,w_0(\cmu)+2\crho\rangle}.$$
Recall from \cite{FG2}, Sect. 17.2, that there exists an isomorphism
$$\fZ^\reg_\fg\simeq
\Hom_{\ghat_\crit}(\BV_\crit,\BM_{w_0,\reg}\otimes \omega_x^{\langle
\rho,2\crho\rangle}),$$ compatible with the above
$\BG_m$-action. Thus, we are reduced to showing that the space
grading-preserving maps of $\fZ^\reg_\fg$-modules
$$\CV^\cmu_\fZ\to \omega_x^{\langle \rho,w_0(\cmu)\rangle}
\otimes \fZ^\reg_\fg$$
is $1$-dimensional.
However, $\CV^\cmu_\fZ$ admits a canonical filtration, whose
subquotients are isomorphic to $\omega_x^{\langle \rho,\cmu'\rangle}
\otimes \fZ^\reg_\fg$, where $\cmu'$ runs through the set weights of
$V^\cmu$ with multiplicities. For all $\cmu'\neq w_0(\cmu)$, we have
$\langle \rho,\cmu'\rangle>\langle \rho,w_0(\cmu)\rangle$. Since
the algebra $\fZ^\reg_\fg$ is non-positively graded, the above inequality
implies that the space of grading-preserving maps
$$\omega_x^{\langle \rho,\cmu'\rangle}\otimes \fZ^\reg_\fg\to
\omega_x^{\langle \rho,w_0(\cmu)\rangle}\otimes \fZ^\reg_\fg$$
is zero for $\cmu'\neq w_0(\cmu)$, and $1$-dimensional for
$\cmu'=w_0(\cmu)$.\qed
\ssec{Proof of \lemref{comp non-zero}} \label{proof non-zero comp}
It is clear that if $\cmu=\cmu_1+\cmu_2$, with $\cmu_1,\cmu_2\in
\cLambda^+$, and the assertion of the proposition holds for $\cmu$,
then it also holds for $\cmu_1$. Hence it is sufficient to consider
the case of $\cmu$ that are regular.
\medskip
To prove the proposition we will use the semi-infinite cohomology
functor, denoted by $H^\semiinf\left(\fn\ppart,\fn[[t]], ?\otimes
\Psi_0\right)$, as in \cite{FG2}, Sect. 18. We will show that the
composition
\begin{align*}
&H^\semiinf\bigl(\fn\ppart,\fn[[t]], \Gamma(\Gr_G,\CF_{V^\cmu})
\otimes \Psi_0\bigr)\to H^\semiinf\bigl(\fn\ppart,\fn[[t]],
\Gamma(\Gr_G,j_{\cmu,\Gr_G,*}) \otimes \Psi_0\bigr)\to \\ &\to
H^\semiinf\bigl(\fn\ppart,\fn[[t]], \BM_{w_0,\reg}\otimes
\omega_x^{\langle 2\rho,\crho\rangle} \otimes \Psi_0\bigr)
\end{align*}
is non-zero (and, in fact, a surjection).
First, note that by \cite{FG2}, Sect. 18.3, the first arrow, i.e.,
$$H^\semiinf\bigl(\fn\ppart,\fn[[t]], \Gamma(\Gr_G,\CF_{V^\cmu})
\otimes \Psi_0\bigr)\to H^\semiinf\bigl(\fn\ppart,\fn[[t]],
\Gamma(\Gr_G,j_{\cmu,\Gr_G,*})\otimes \Psi_0\bigr)$$ is an
isomorphism. Hence, it remains to analyze the second arrow. By
\cite{FG2}, Proposition 18.1.1, this is equivalent to analyzing the
arrow
\begin{align*}
&H^\semiinf\bigl(\fn^-\ppart,t\fn^-[[t]],
j_{w_0\cdot\crho,*}\underset{I}\star \Gamma(\Gr_G,j_{\cmu,\Gr_G,*})
\otimes \Psi_{-\crho}\bigr)\to \\
&H^\semiinf\bigl(\fn^-\ppart,t\fn^-[[t]],
j_{w_0\cdot\crho,*}\underset{I}\star \BM_{w_0,\reg}\otimes
\omega_x^{\langle 2\rho,\crho\rangle} \otimes \Psi_{-\crho}\bigr).
\end{align*}
We claim that the corresponding map
\begin{equation} \label{surj map}
j_{w_0\cdot\crho,*}\underset{I}\star
\Gamma(\Gr_G,j_{\cmu,\Gr_G,*})\simeq
j_{w_0\cdot\crho,*}\underset{I}\star j_{\cmu,*}\underset{I}\star
\BV_\crit\to j_{w_0\cdot\crho,*}\underset{I}\star
j_{\cmu,*}\underset{I}\star \BM_{w_0,\reg}\otimes \omega_x^{\langle
2\rho,\crho\rangle}
\end{equation}
is surjective for $\cmu$ regular. This would imply our claim, since the
semi-infinite cohomology functor
$H^\semiinf\bigl(\fn^-\ppart,t\fn^-[[t]], ? \otimes \Psi_{-\crho}\bigr)$
is exact by Theorem 18.3.1 of \cite{FG2}.
\medskip
Note that $j_{w_0\cdot\crho,*}\underset{I}\star j_{\cmu,*}\simeq
j_{w_0(\cmu),*}\underset{I}\star j_{w_0\cdot\crho,*}$. Recall
from \cite{FG2}, Sect. 17.2, that we have a commutative diagram
$$ \CD j_{w_0\cdot\crho,*}\underset{I}\star \BV_\crit
@>{\on{id}_{j_{w_0\cdot\crho,*}}\star \phi}>>
j_{w_0\cdot\crho,*}\underset{I}\star \BM_{w_0,\reg}\otimes
\omega_x^{\langle 2\rho,\crho\rangle} \\ @V{\sim}VV @V{\sim}VV \\
\Gamma(\Gr_G,j_{w_0\cdot\crho,*}\underset{I}\star \delta_{1,\gr_G})
@>>> \BM_{1,\reg}\otimes \omega_x^{\langle \rho,\crho\rangle}, \endCD
$$
where the bottom arrow has the property that its cokernel, which we
denote by $\CN$, is {\it partially integrable}, i.e., it is admits
a filtration with every subquotient
integrable with respect to a sub-minimal parahoric Lie subalgebra
corresponding to some vertex $\imath$ of the Dynkin
diagram.
Thus, the map in \eqref{surj map} can be written as
$$j_{w_0(\cmu),*}\underset{I} \star
(j_{w_0\cdot\crho,*}\underset{I}\star \BV_\crit)\to
j_{w_0(\cmu),*}\underset{I} \star (\BM_{1,\reg}\otimes
\omega_x^{\langle \rho,\crho\rangle}),$$ and since the functor
$j_{w_0(\cmu),*}\underset{I} \star ?$ is right-exact, it suffices to
show that $j_{w_0(\cmu),*}\underset{I} \star \CN$ is supported in
strictly negative cohomological degrees. In fact, we claim that this
is true for any partially integrable $I$-integrable $\hg_\crit$-module
and regular dominant coweight $\cmu$.
\medskip
Indeed, by devissage we may assume that $\CN$ is integrable with
respect to a sub-minimal parahoric corresponding to some vertex
$\imath$ of the Dynkin diagram. Then $j_{s_{\imath},*}\underset{I}
\star \CN$ lives in the cohomological degree $-1$. But since $\cmu$ is
regular, $j_{w_0(\cmu),*}\underset{I}\star j_{s_\imath,!}\simeq
j_{w_0(\cmu)\cdot s_\imath,*}$, and hence,
$$j_{w_0(\cmu),*}\underset{I}\star \CN\simeq j_{w_0(\cmu)\cdot
s_\imath,*}\underset{I}\star (j_{s_{\imath},*}\underset{I} \star
\CN),$$ and our assertion follows from the fact that the functor of
convolution with $j_{w_0(\cmu)\cdot s_\imath,*}$ is right-exact.\qed
\ssec{Proof of \corref{F non-zero} and completion of the
proof of \thmref{main}}
Thus, we have proved \lemref{comp non-zero} and therefore \lemref{two
torsors}. By \lemref{univ baby rep}, this implies that we have a
canonical map
$$
\phi_{w_0}: \Gamma^{\Hecke_\fZ}(\Gr_G,\CF_{w_0}^\fZ)\to
\BM_{w,\reg}\otimes \omega_x^{\langle 2\rho,\crho\rangle}.
$$
According to the remark after formula \eqref{sought-for maps}, we then
obtain maps
$$
\phi_w:\Gamma^{\Hecke_\fZ}(\Gr_G,\CF_w^\fZ)\to \BM_{w,\reg}\otimes
\omega_x^{\langle 2\rho,\crho\rangle}
$$
for all $w \in W$ (as in formula \eqref{sought-for maps}).
\begin{prop} \label{one of the maps surj}
The map
$$\phi_1:\Gamma^{\Hecke_\fZ}(\Gr_G,\CF_1^\fZ)\to \BM_{1,\reg}\otimes
\omega_x^{\langle 2\rho,\crho\rangle}$$
is surjective.
\end{prop}
Since the functors $j_{w,*}$ are right-exact, this proposition implies
that the same surjectivity assertion holds for all $w\in W$. Hence,
\propref{one of the maps surj} implies \corref{F non-zero} and
\thmref{main}.
\bigskip
\noindent {\em Proof of \propref{one of the maps surj}}. For $\cla$,
such that $\cla-\crho$ is dominant and regular, let us consider the
map
$$j_{w_0\cdot \cla,*}\underset{I}\star \CF_{R_\fZ}\underset{\fZ^\reg_\fg}
\otimes \CL_{w_0}^{-\cla}\simeq j_{w_0,!}\underset{I}\star
j_{\cla,*}\underset{I}\star \CF_{R_\fZ}\underset{\fZ^\reg_\fg} \otimes
\CL_{w_0}^{-\cla}\to j_{w_0,!}\underset{I}\star \wt{\CF}^\fZ_{w_0}\to
j_{w_0,!}\underset{I}\star \CF^\fZ_{w_0}\simeq \CF^\fZ_1,$$ and the
resulting map
$$j_{w_0\cdot \cla,*}\underset{I}\star \BV_\crit
\underset{\fZ^\reg_\fg} \otimes \CL_{w_0}^{-\cla}\to
\Gamma^{\Hecke_\fZ}(\Gr_G,\CF_1^\fZ)\overset{\phi_1}\to
\BM_{1,\reg}\otimes \omega_x^{\langle 2\rho,\crho\rangle}.$$
By construction, this map is obtained by applying the functor
$j_{w_0\cdot \cla,*}\underset{I}\star ?$ to the map
$$\BV_\crit\to \BM_{w_0,\reg}\otimes \omega_x^{\langle
2\rho,\crho\rangle},$$ and it coincides with the map from \eqref{surj
map} for $\cmu=\cla-\crho$. Hence, it is surjective by \secref{proof
non-zero comp}.\qed
\ssec{Completion of the proof of \thmref{get Wakimoto}}
Thus, the proof of \thmref{main} is complete. Let us now finish the
proof of the fact that the morphisms $\phi_w$ are actually
isomorphisms and hence complete our proof of \thmref{get
Wakimoto}. Clearly, it is enough to do so for just one element of
$W$. We shall give two proofs.
\medskip
\noindent{\bf Proof 1.} This argument will rely on \thmref{main}. We
will analyze the map $\phi_{w_0}$. By \cite{ABBGM}, Proposition
3.2.5, the canonical map $\CF_R \to \CF_{w_0}$ identifies
$\on{Ind}^\Hecke(\delta_{1,\Gr_G})$ with the co-socle of
$\CF_{w_0}$. Hence $\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^\fZ_{w_0})$ does not
have sub-objects whose intersection with $\BV_\crit =
\Gamma^{\Hecke_\fZ}(\Gr_G,R_{\fZ})$ is zero.
Therefore, to prove the injectivity of the map $\phi_{w_0}$, it is
enough to show that the composition
$$\BV_\crit\simeq\Gamma^{\Hecke_\fZ}(\Gr_G,R_\fZ)\to
\Gamma^{\Hecke_\fZ}(\Gr_G,\CF^\fZ_{w_0})\overset{\phi_{w_0}}\to
\BM_{w_0,\reg} \otimes \omega_x^{\langle 2\rho,\crho\rangle}$$ is
injective. However, the latter map is, by construction, the map
$\BV_\crit\to \BM_{w_0,\reg}\otimes \omega_x^{\langle
2\rho,\crho\rangle}$ of \cite{FG2}, Sect. 17.2, which was injective by
definition.
\medskip
\noindent{\bf Proof 2.} This argument will be independent of
\thmref{main},(2). We will analyze the map $\phi_1$. We have a
canonical map
$$\IC_{w_0\cdot \crho,\Gr}\star \CF_{R_\fZ}\underset{\fZ^\reg_\fg}
\otimes \CL_{w_0}^{-\crho}\to
j_{w_0\cdot \crho}\underset{I}\star \CF_{R_\fZ}\underset{\fZ^\reg_\fg}
\otimes \CL_{w_0}^{-\crho}\to \CF^\fZ_1,$$
and by \cite{ABBGM}, Propositions 3.2.6 and 3.2.10, its cokernel
is partially integrable.
The composition
\begin{align*}
&\Gamma(\Gr_G, \IC_{w_0\cdot \crho,\Gr})
\underset{\fZ^\reg_\fg}\otimes \CL_{w_0}^{-\crho}\simeq
\Gamma^{\Hecke_\fZ}\bigl(\Gr_G,\IC_{w_0\cdot \crho,\Gr}
\star \CF_{R_\fZ}\underset{\fZ^\reg_\fg}\otimes \CL_{w_0}^{-\crho}\bigr)
\simeq \Gamma^{\Hecke_\fZ}\bigl(\Gr_G,\CF^\fZ_1)
\overset{\phi_1}\to \\
&\to \BM_{1,\reg}\underset{\fZ^\reg_\fg}\otimes
\omega_x^{\langle 2\rho,\crho\rangle}
\end{align*}
comes from the map
$$\Gamma(\Gr_G, \IC_{w_0\cdot \crho,\Gr})\to
\BM_{1,\reg}\underset{\fZ^\reg_\fg}\otimes \omega_x^{\langle
\rho,\crho\rangle},$$ of \cite{FG2}, Sect. 17.3, which is injective by
{\it loc.cit.}
Hence, the kernel of the map $\phi_1$ is partially integrable. But we claim
that $\Gamma^{\Hecke_\fZ}\bigl(\Gr_G,\CF^\fZ_1)$ admits no partially
integrable submodules.
Indeed, suppose that $\CN$ is a submodule of
$\Gamma^{\Hecke_\fZ}\bigl(\Gr_G,\CF^\fZ_1)$, integrable with respect
to a sub-minimal parahoric, corresponding to a vertex $\imath$ of the
Dynkin diagram. Since the functor $j_{s_\imath,*}\underset{I}\star$ is
invertible on the derived category, we would obtain a non-zero map:
$$j_{s_\imath,*}\underset{I}\star \CN\to \on{L}
\Gamma^{\Hecke_\fZ}\bigl(\Gr_G,j_{s_\imath,*}
\underset{I}\star\CF^\fZ_1).$$
But the LHS is supported in the cohomological degrees $<0$, and the
RHS is acyclic away from cohomological degree $0$. \footnote{Here we
are relying on part (1) of \thmref{main}, which was proved
independently.} This is a contradiction.
This completes the proof of \thmref{get Wakimoto}.
\section{Appendix: an equivalence at the negative level}
\ssec{}
Let $\kappa$ be a negative level, i.e., $\kappa=k\cdot \kappa_{\can}$
with $k+h^\vee\notin \BQ^{\geq 0}$.
Let $\wt\Fl_G$ be the enhanced affine flag scheme, i.e, $G\ppart/I^0$, and
let $\on{D}(\wt\Fl_G)_\kappa\mod$ be the corresponding
category of twisted D-modules.
Note that $\wt\Fl_G$ is acted on by the group $I/I^0\simeq H$ by right
multiplication. Let us denote by $\on{D}(\wt\Fl_G)_\kappa\mod^{H,w}$
the corresponding category of weakly H-equivariant objects of
$\on{D}(\wt\Fl_G)_\kappa\mod$ (see \cite{FG2}, Sect. 20.2).
For an object $\CF\in \on{D}(\wt\Fl_G)_\kappa\mod^{H,w}$, consider
$\Gamma(\wt\Fl_G,\CF)\in \hg_\kappa\mod$.
The weak $H$-equivariant structure on
$\CF$ endows $\Gamma(\wt\Fl_G,\CF)$ with a commuting action of $H$.
We let
$$\Gamma^H:\on{D}(\wt\Fl_G)_\kappa\mod^{H,w}\to \hg_\kappa\mod$$
to be the composition of $\Gamma(\wt\Fl_G,\cdot)$, followed by the
functor of $H$-invariants.
\medskip
Recall from \cite{FG2}, Sect. 20.4, that
every object of $\on{D}(\wt\Fl_G)_\kappa\mod^{H,w}$ carries
a canonical action of $\Sym(\fh)$ by endomorphism, denoted
$a^\sharp$.
For $\lambda\in \fh^*$ let
$$\on{D}(\wt\Fl_G)_\kappa\mod^{H,\lambda}\subset
\on{D}(\wt\Fl_G)_\kappa\mod^{H,w,\lambda}$$ be the full subcategories
of $\on{D}(\wt\Fl_G)_\kappa\mod^{H,w}$, corresponding to the condition
that $a^\sharp(h)=\lambda(h)$ for $h\in \fh$ in the former case, and
that $a^\sharp(h)-\lambda(h)$ acts locally nilpotently in the latter.
Since the group $H$ is connected, both of these categories are full
subcategories in $\on{D}(\wt\Fl_G)_\kappa\mod$.
\medskip
We let $D(\on{D}(\wt\Fl_G)_\kappa\mod)^{H,w,\lambda}\subset
D(\on{D}(\wt\Fl_G)_\kappa\mod)$ be the full subcategory consisting of
complexes, whose cohomologies belong to
$\on{D}(\wt\Fl_G)_\kappa\mod^{H,w,\lambda}$.
It is easy to see that the functor $\Gamma^H$, restricted to
$\on{D}(\wt\Fl_G)_\kappa\mod^{H,w,\lambda}$, extends to a functor
$$\on{R}\Gamma^H:D^+(\on{D}(\wt\Fl_G)_\kappa\mod)^{H,w,\lambda}\to
D^+(\hg_\kappa\mod).$$
\medskip
Assume now that $\lambda$ satisfies the following conditions:
$$
\begin{cases}
& \langle \lambda+\rho ,\check\alpha\rangle\notin \BZ^{\geq 0} \text{
for } \alpha \in \Delta^+ \\ &\pm \langle \lambda+\rho
,\check\alpha\rangle +2n\frac{k+h^\vee}{\kappa_{\can}(\alpha,\alpha)}
\notin \BZ^{\geq 0} \text{ for } \alpha \in \Delta^+ \text{ and } n\in
\BZ^{>0}.
\end{cases}
$$
Following \cite{BD}, Sect. 7.15, we will prove:
\begin{thm} \label{weak local, neg} \hfill
\smallskip
\noindent{\em (1)} For $\CF\in
\on{D}(\wt\Fl_G)_\kappa\mod^{H,w,\lambda}$ the higher cohomologies
$\on{R}^i\Gamma^H(\wt\Fl_G,\CF)$, $i>0$, vanish.
\smallskip
\noindent{\em (2)}
The resulting functor $\on{R}\Gamma^H:
D^b(\on{D}(\wt\Fl_G)_\kappa\mod)^{H,w,\lambda}\to
D^b(\hg_\kappa\mod)$
is fully-faithful.
\end{thm}
\ssec{}
Let $\on{D}(\wt\Fl_G)_\kappa\mod^{I^0,H,w,\lambda}\subset
\on{D}(\wt\Fl_G)_\kappa\mod)^{H,w,\lambda}$ be the full subcategory,
consisting of twisted D-modules, equivariant with respect to the
$I^0$-action on the left. Our present goal is to describe its image
under the above functor $\Gamma$.
Consider the category $\CO_{\aff}:=\hg_\kappa\mod^{I^0}$. This is a
version of the category $\CO$ for the affine Lie algebra
$\ghat_\kappa$. Its standard (resp., co-standard, irreducible) objects
are numbered by weights $\mu\in \fh^*$, and will be denoted by
$M_{\kappa,\mu}$ (resp., $M^\vee_{\kappa,\mu}$, $L_{\kappa,\mu}$).
Since $\kappa$ was assumed to be negative, every finitely generated
object of $\CO_{\aff}$ has finite length.
The extended affine Weyl group $W_{\aff}:=W\ltimes \Lambda$ acts on
$\fh^*$, with $w\in W\subset W_{\aff}$ acting as
$$w\cdot \mu=w(\mu+\rho)-\rho,$$ and $\cla\in \cLambda\subset
W_{\aff}$ by the translation by means of
$(\kappa-\kappa_{\crit})(\cla,\cdot)\in \fh^*$.
For a $W_{\aff}$-orbit $\upsilon$ in $\fh^*$ let
$(\CO_{\aff})_{\upsilon}$ be the full-subcategory of $\CO_{\aff}$,
consisting objects that admit a filtration, such that all subquotients
are isomorphic to $L_{\kappa,\lambda}$ with $\lambda\in \upsilon$.
The following assertion is known as the linkage principle (see
\cite{DGK}):
\begin{prop}
The category $\CO_{\aff}$ is the direct sum over the orbits $\upsilon$
of the subcategories $(\CO_{\aff})_{\upsilon}$.
\end{prop}
For $\lambda$ as in \thmref{weak local, neg} let $\upsilon(\lambda)$
be the $W_\aff$-orbit of $\lambda$. (Note that by assumption, the
stabilizer of $\lambda$ in $W_\aff$ is trivial.)
We shall prove the following: \footnote{This theorem is not due to the
authors of the present paper. The proof that we present is a
combination of arguments from \cite{BD}, Sect. 7.15, and \cite{KT}.}
\medskip
\begin{thm} \label{KTthm}
The functor $\Gamma^H$ defines an equivalence
$$\on{D}(\wt\Fl_G)_\kappa\mod)^{I^0,H,w,\lambda}\to
(\CO_{\aff})_{\upsilon(\lambda)}.$$
\end{thm}
\ssec{Proofs}
To prove point (1) of \thmref{weak local, neg}, it suffices to show
that $\on{R}^i\Gamma^H(\wt\Fl_G,\CF)=0$ for $\CF\in
\on{D}(\wt\Fl_G)_{\kappa}\mod^{H,\lambda}$ and $i>0$. However, this
follows immediately from \cite{BD}, Theorem 15.7.6.
\medskip
To prove point (2) of \thmref{weak local, neg} and \thmref{KTthm} we
shall rely on the following explicit computation, performed in
\cite{KT}:
For an element $\wt{w}\in W_{\aff}$ let $j_{\wt{w},*,\lambda}\in
\on{D}(\wt\Fl_G)_{\kappa}\mod^{I^0,H,\lambda}$ (resp.,
$j_{\wt{w},!,\lambda}$) be the *-extension (resp, !-extension) of the
unique $I^0$-equivariant irreducible twisted D-module on the preimage
of the corresponding $I^0$-orbit in $\Fl_G$. We have:
\begin{thm} \label{dual Verma} We have:
$$\Gamma(\Fl_G,j_{\wt{w},*,\lambda})\simeq M^\vee_{\kappa, \wt{w}\cdot
0} \text{ and } \Gamma(\Fl_G,j_{\wt{w},!,\lambda})\simeq M_{\kappa,
\wt{w}\cdot 0}.$$
\end{thm}
Let us now proceed with the proof of \thmref{weak local,
neg}(2). Clearly, it is enough to show that for two finitely generated
objects $\CF,\CF_1\in \on{D}(\wt\Fl_G)_{\kappa}\mod^{H,\lambda}$ the
map
$$\on{R}\on{Hom}_{D(\on{D}(\wt\Fl_G)_\kappa\mod)^{H,\lambda}}(\CF,
\CF_1) \to \on{R} \on{Hom}_{D(\hg_\crit\mod)}(\Gamma^H(\wt\Fl_G,
\CF),\Gamma^H(\wt\Fl_G,\CF))$$ is an isomorphism.
By adjunction (see \cite{FG2}, Sect. 22.1), the latter is equivalent
to the map
\begin{align*}
&\on{R}\on{Hom}_{D(\on{D}(\wt\Fl_G)_\kappa\mod)^{I,\lambda}}
(j_{1,!,\lambda},\CF^{op}\star \CF_1)\to \\
&\to\on{R}\on{Hom}_{D(\hg_\crit\mod)^{I,\lambda}}(
\Gamma^H(\wt\Fl_G,j_{1,!,\lambda}), \on{R}\Gamma^H(\Fl_G,\CF^{op}
\star \CF_1))
\end{align*}
being an isomorphism, where $\CF^{op}\in
\on{D}(G\ppart/K)\mod^{I,\lambda}$ is the dual D-module, where $K$ is
a sufficiently small open-compact subgroup of $G[[t]]$.
Using the stratification of $\wt\Fl_G$ by $I$-orbits, we can replace
$\CF^{op}\star \CF_1$ by its Cousin complex. In other words, it is
sufficient to show that
$$\on{R}\on{Hom}_{D(\on{D}(\wt\Fl_G)_\kappa\mod)^I}(j_{1,!,\lambda},
j_{\wt{w},\lambda,*})\to
\on{R}\on{Hom}_{D(\hg_\crit\mod)^I}(\Gamma^H(\wt\Fl_G,j_{1,!,\lambda}),
\Gamma^H(\wt\Fl_G,j_{\wt{w},\lambda,*}))$$ is an isomorphism, for all
$\wt{w}$ such that $j_{\wt{w},\lambda,*}$ is
$(I,\lambda)$-equivariant.
Note that the LHS is $0$ unless $\wt{w}=0$, and is isomorphic to $\BC$
in the latter case. Hence, taking into account \thmref{dual Verma}, it
remains to prove the following:
\begin{lem} \hfill
\smallskip
\noindent{\em (1)}
$\on{R}\on{Hom}_{D(\hg_\crit\mod)^{I,\lambda}}(M_{\kappa,\lambda},
M^\vee_{\kappa,\mu})=0$ for $\lambda\neq \mu\in \fh^*$ but such that
$M^\vee_{\kappa,\mu}\in \hg_\crit\mod^{I,\lambda}$ is
$(I,\lambda)$-equivariant.
\smallskip
\noindent{\em (2)} The map $\BC\to
\on{R}\on{Hom}_{D(\hg_\crit\mod)^{I,\lambda}}(M_{\kappa,\lambda},
M^\vee_{\kappa,\lambda})$ is an isomorphism.
\end{lem}
\begin{proof}
For any $\CM\in \hg_\crit\mod^{I,\lambda}$,
$$\on{R}\on{Hom}_{D(\hg_\crit\mod)^{I,\lambda}}(M_{\kappa,\lambda},
\CM)\simeq \on{R}\on{Hom}_{I\mod}(\BC,\CM\otimes \BC^{-\lambda}).$$
Since $M^\vee_{\kappa,\mu}$ is co-free with respect to $I^0$,
we obtain
$$\on{R}\on{Hom}_{I\mod}(\BC,M^\vee_{\kappa,\mu}\otimes
\BC^{-\lambda})\simeq \on{R}\on{Hom}_{H\mod}(\BC,\BC^\mu\otimes
\BC^{-\lambda}),$$ implying the first assertion of the lemma.
Similarly,
$$\on{R}\on{Hom}_{D(\hg_\crit\mod)^I}(M_{\kappa,\lambda},M^\vee_{\kappa,\lambda})\simeq
\on{R}\on{Hom}_{I\mod}(\BC,M^\vee_{\kappa,\lambda})\simeq
\on{R}\on{Hom}_{H\mod}(\BC,\BC)\simeq \BC,$$
implying the second assertion.
\end{proof}
Finally, let us prove \thmref{KTthm}. Taking into account \thmref{weak
local, neg}, and using Lemmas \ref{adjoint} and \ref{kappa adjoint},
it remains to show that for every $\CM\in
(\CO_{\aff})_{\upsilon(\lambda)}$ there exists an object $\CF\in
\on{D}(\wt\Fl_G)\mod^{I^0,H,w,\lambda}$ with non-zero map
$$\Gamma^H(\wt\Fl_G,\CF)\to \CM.$$
It is clear that for every $\CM\in (\CO_{\aff})_{\upsilon(\lambda)}$
there exists a Verma module $M_{\kappa,\mu}\in
(\CO_{\aff})_{\upsilon(\lambda)}$ with a non-zero map
$M_{\kappa,\mu}\to \CM$. Hence, the required property follows from
\thmref{dual Verma}.
|
1,108,101,565,649 | arxiv | \section{Introduction}
Masses of thin actinide samples, often a mix of several isotopes, are routinely inferred by counting spontaneously emitted $\alpha$-particles with silicon detectors or gas-based 2$\pi$ or 4$\pi$ counters \cite{Pomme2007}. Traditionally,
when measurements of actinide sample mass have been used to normalize fission cross-section ratio measurements, specific activity of the sample material was determined using isotopic analysis (e.g. isotopic dilution) and half-lives, while the total mass of the individual isotopes was then determined with $\alpha$ counting~\cite{Meadows1983}. The analysis of the $\alpha$-spectrum was limited to confirming the isotopic composition, rather than an independent measurement of it. In several previous publications of fission cross-section ratios that use this normalization method, many details of the exact methods employed are however omitted~\cite{Staples, Lisowski}.
In this work we provide a detailed account of how the atomic number ratio was determined. We explore several methods of combining a model $\alpha$-spectrum fit, to estimate specific activity of each isotope directly~\cite{Ihantola2011}, with mass spectrometry used to constrain the fits and determine the isotopic concentrations that are not resolved in the $\alpha$-spectrum.
The Neutron Induced Fission Fragment Tracking Experiment (NIFFTE) collaboration has constructed the fission Time Projection Chamber (fissionTPC) \cite{Heffner2014} with the aim to measure neutron-induced fission cross-section ratios for actinides with total uncertainties better than 1\%. A high precision measurement of the ratio of $^{239}$Pu and $^{235}$U atoms for the two actinide deposits, referred to hereafter as a target atom number ratio, is needed to provide an absolute $^{239}$Pu(n,f)/$^{235}$U(n,f) fission cross-section ratio \cite{Snyder2021}.
A measurement of the target atom number ratio for a target with deposits of $^{239}$Pu and $^{235}$U, made with a silicon detector, is presented in this article. This measurement was performed with a silicon detector setup, based on the prescription of Pomme~\cite{Pomme2015}, who showed that decay rate measurements with uncertainties as low as 0.1\% are achievable with such devices. Source-to-detector distances were chosen that strike a compromise between the absolute efficiency and pile-up rate for the two samples. Precise knowledge of absolute efficiency is not necessary for the purpose of measuring the target atom ratio, assuming the ratio of efficiencies for both sides of the target is unity. The sample backing was rotated, so that either the plutonium or uranium faced the detector, without disturbing the source-mount-to-detector distance, as will be described in Section~\ref{sec:silicon}.
\section{Experimental Procedure}
\subsection{Actinide Target}
The target was prepared by depositing $^{239}$Pu and $^{235}$U on a 4~cm diameter and 0.25~mm thick aluminum disk. The actinide deposits were made at the center of the disk, on opposite sides, with a diameter of 2~cm. The uranium sample was deposited using vacuum volatilization, while the plutonium sample was molecular plated~\cite{Loveland,Loveland2016}. The fissionTPC was used to image the target through tracking of the $\alpha$-particles from the decaying actinides back to their origin.
The $\alpha$-particle start vertices for each deposit are shown in Fig.~\ref{fig:vertices}. The pointing resolution of the detector for $\alpha$-particles is approximately 0.3 mm in the plane of the sample.
The capability to produce vertex images of the target made it also possible to quantify any non-uniformity in the deposit, apparent specifically on the plutonium side. This capability was essential for analyzing the impact of a non-uniform deposit on the detection efficiency of the silicon detector setup, the details of which are presented in Section~\ref{sec:acceptance}
\begin{figure}[ht]
\begin{center
\includegraphics[width=1.\linewidth]{vertexDist60deg.pdf}
\end{center}
\caption{Image of $\alpha$-particle track start vertices for plutonium (left) and uranium (right) target deposits.}
\label{fig:vertices}
\end{figure}
\subsection{Silicon detector setup}
\label{sec:silicon}
An ORTEC ULTRA ion-implanted planar silicon detector was used to measure $\alpha$-spectra. The detector had an active area of 1200~mm$^{2}$, minimum depletion depth of 100~$\mu$m and a quoted resolution (FWHM) of 37~keV~\cite{OrtecUltra}. The sensitive area of the detector was restricted by a stainless steel diaphragm, placed directly in front of the detector, with a thickness of 0.25 mm and an opening diameter of 35.6~mm. The diaphragm was positioned parallel and concentric to the sample at a distance of 127.5~mm. The fraction of solid angle subtended by the detector to the source was 0.47\%. Measurements were performed under vacuum, with a nominal absolute pressure of 20 mTorr. A rendering of the setup is shown in Fig.~\ref{fig:auxSetup}.
\begin{figure}[ht]
\begin{center}$
\begin{array}{cc}
\includegraphics[width=1.\linewidth]{isometric2.pdf}
\end{array}$
\end{center}
\caption{Rendering of the silicon detector setup. The silicon detector was mounted to an NW160 flange with standoffs. The plate holding the detector included a diaphragm with an opening diameter of 35.6~mm. Baffles with inner diameters identical to the diaphragm were mounted between the sample and the detector to block $\alpha$-particles from actinide material that might be sputtered away from the sample.}
\label{fig:auxSetup}
\end{figure}
A series of precautions were taken to minimize the effect of possible actinide contamination that might be present in the vacuum chamber. Background data were taken before and after each sample measurement. Baffles were installed to block $\alpha$-particles originating from outside the target area. Time under vacuum was minimized to prevent actinide material from leaving the sample. All background measurements yielded less than one count per four hours in the energy range that would interfere with the target atom ratio measurement.
\subsubsection{Data collection} \label{sec:data_collection}
The actinide samples were measured with the silicon detector in two separate campaigns to validate the work and ensure that the samples were not degraded or otherwise damaged.
In the first measurement campaign collection times were approximately 4 days for uranium and 10 minutes for the plutonium side. Background data was taken for at least 1 day before loading the sample each time. In the second measurement campaign uranium data was collected over 56 days, the plutonium in 20 minutes and background in 30 days. Signal processing was performed with a CAEN desktop digitizer, model DT5730, in both campaigns.
Typical energy spectra are shown in Fig.~\ref{fig:combinedEnergySpectrum}. Peaks from $^{232}$U, $^{233}$U, $^{234}$U, $^{235}$U, $^{236}$U, $^{239}$Pu background, and $\alpha$-decay daughters are evident in the uranium spectrum. For the plutonium deposit, only three strong peaks were expected from $^{238}$Pu, $^{239}$Pu, and $^{240}$Pu (not resolvable from $^{239}$Pu).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.\linewidth]{combined_plot.pdf}
\end{center}
\caption{Energy spectrum measured by a silicon detector for the uranium deposit (top panel) and plutonium deposit (bottom panel). The $^{233}$U decay chain is the dominating contributor to counts above 5 MeV on the uranium side. Peaks are labeled as follows: (a) $^{235}$U, (b) $^{233, 234}$U, (c) $^{239}$Pu, (d) $^{232}$U, (e) $^{228}$Th ($^{232}$U chain), (f) $^{225}$Ac ($^{233}$U chain), (g) $^{221}$Fr ($^{233}$U chain), (h) $^{221}$Fr ($^{233}$U chain), (i) $^{211}$Bi ($^{235}$U chain), (j) $^{216}$Ac ($^{232}$U chain), (k) $^{217}$At ($^{233}$U chain), (l) $^{215}$Po ($^{235}$U chain), (m) $^{214}$Po ($^{234}$U chain), (n) $^{213}$Po ($^{233}$U chain), and (o) $^{212}$Po ($^{232}$U chain). A peak from $^{238}$Pu (q) was well resolved from $^{239}$Pu and $^{240}$Pu (combined in p) on the plutonium side, and counts from pulse pile-up are evident at energies above 5.6 MeV.}
\label{fig:combinedEnergySpectrum}
\end{figure}
A series of peaks at energies above 5 MeV was observed for the uranium deposit. Each peak was associated with a known $\alpha$-particle decay from a daughter of $^{233}$U, $^{234}$U, or $^{235}$U, or direct decays from $^{232}$U or $^{239}$Pu. Several generations of daughters were observed, ending in the long-lived isotopes of lead. The decay daughters of $^{233}$U were the largest contributors to the spectrum at high energy because of the relatively short half life of $^{229}$Th. The total ratio of daughter and background counts to total counts was measured to be $1.61 \pm 0.02\%$. The daughter peaks are outside of the analysis region for the uranium side.
\subsubsection{Recovery time}
\label{subsec:dead_time}
The observed event rates were around 0.58 counts per second for uranium, and 3460 counts per second for plutonium. The total background rates were 0.01 counts per second. The total recovery time is on the order of microseconds, therefore it is only of consequence in the higher rate plutonium measurement.
The total recovery time is the time it takes to detect two distinct pulses, and is the result of overlapping effects: shaping time, pile-up, detector dead time, and the digitizer acquisition window. As a result of these effects the observed count rate is lower than the true count rate.
The true count rate can be estimated from the time distribution of observed events, since we expect these to follow a Poisson random process. The distribution of the time differences, $t$, between these events follows an exponential
\begin{align}
f(t) = m C \exp(-Ct) \label{eq:time_dist}
\end{align}
where $m$ is the total number of counts and $C$ is the true count rate. The estimated true count rate from the fit to the distribution of events in the plutonium measurement was 3504$\pm$2 counts per second. Therefore, the true count rate in the plutonium measurement is 1.1\% higher than the observed count rate. The recovery time was calculated from
\begin{align}
\tau = \frac{(C - N)}{C \cdot N}
\end{align}
where $N$ is the observed count rate. The total recovery time was estimated to be 3.59$\pm$0.2 $\mu s$.
\subsubsection{Absolute Efficiency Ratio}
\label{sec:acceptance}
The actinide deposits were both 2~cm in diameter and arranged back-to-back on an aluminum disk. The detector geometry was designed such that the the disk could be rotated with either deposit facing the silicon detector so that the source-to-detector distance was unchanged and therefore the absolute detector efficiency for each actinide source would be identical. The absolute efficiency of the silicon detector is defined as
\begin{align}
\epsilon= \frac{n}{N} \,\text{,}
\end{align}
where $n$ is the number of $\alpha$'s detected by the Si detector and $N$ is the total number of $\alpha$'s generated by the deposit.
The absolute detector efficiency is subject to two potential sources of systematic uncertainty. The first source of uncertainty is a result of the different distribution of target material for the $^{239}$Pu and $^{235}$U, as shown in Fig.~\ref{fig:vertices}. The second source of systematic uncertainty arises from clearance between the aluminum disk and the base flange. This clearance is 0.32 mm and could result in possible relative position offsets between the $^{239}$Pu and $^{235}$U targets, as the respective targets need not be sitting flush at the origin $(0,0,0)$.
A Geant4 simulation of the Si detector system was developed in order to perform a systematic study of the efficiency ratio. In the simulation, $\alpha$'s were considered as the primary particle with initial energies assigned from $\alpha$-decay energies of $^{239}$Pu and $^{235}$U, respectively. Starting $(x_i,y_i)$ vertex distributions of both targets were taken from the fissionTPC data of Fig.~\ref{fig:vertices}. Simulations were performed by shifting the center of the Al disk over a range of $0-0.3$ mm in the $x-y$ plane. Since the $^{239}$Pu target was non-uniform, negative displacements of the Al disk were also considered. The $z$ position was considered to be fixed for both $^{239}$Pu and $^{235}$U targets since gravity will hold the target flush with the base flange.
Calculation of the total efficiency ratio
\begin{align}
R_\epsilon= \frac{\epsilon_{Pu(x_i,y_i)}}{\epsilon_{U(x_i,y_i)}}
\end{align}
was carried out by taking the average of the permutations of the aforementioned relative efficiency ratios of $^{239}$Pu and $^{235}$U. In particular, this process involved evaluating combinatorial efficiency ratios for each assumed $i^{th}$ $^{239}$Pu start vertex $(x_i,y_i)$ over the entire suite of assumed $^{235}$U $(x_i,y_i)$ start vertices. The average of this combinatorial sum was taken to be the efficiency ratio $R_{\epsilon}=1.00006\pm5.54\times10^{-5}$, which is consistent with unity at the sub-percent level. We therefore consider the absolute detector efficiency to be equal for each actinide measurement and any contribution to the atom number ratio uncertainty to be negligible.
\section{Results}
\label{sec:results}
\subsection{Mass Spectrometry}
Mass spectrometry of surrogate samples was used to estimate isotopic concentrations in the targets, some of which were not resolvable in the $\alpha$-spectrum. Two samples were analyzed via mass spectrometry for uranium (\#1 and \#2) and two samples were analyzed for plutonium (\#3 and \#4).
The mass spectrometry results are shown in Table~\ref{table:Isotopics}.
In each case one sample (\#1 and \#3) was raw material used to make the target, and the other sample (\#2 and \#4) was a deposit prepared in a similar way to the target under study. The mass spectrometry uncertainties were determined as described in Ref.~\cite{RossUncert}. Published values for half-lives were used to calculate the decay constants and are shown in Table~\ref{table:halflivesDDEP} ~\cite{DDEP,NNDC}. The $\alpha$ energies and intensities are shown in Table~\ref{table:alpha_energies}.
\begin{table*}[ht]
\begin{center}
\caption{Uranium and plutonium target isotopic atom percentages as measured with mass spectrometry\cite{Ross}}
\label{table:Isotopics}
\begin{tabular}{ l S[table-format=3.6] S[table-format=3.6] S[table-format=3.6] S[table-format=3.6] }
\hline
\bf{Isotope} & {Sample \#1} & {Uncertainty} & {Sample \#2} & {Uncertainty} \\
\hline
$^{233}$U & .01886 & .00004 & .01893 & .00008 \\
$^{234}$U & .03448 & .00032 & .03536 & .00004 \\
$^{235}$U & 99.677 & .002 & 99.634 & .014 \\
$^{236}$U & .1701 & .00177 & .1763 & .0005 \\
$^{238}$U & .0998 & .0005 & .1355 & .0014 \\
\hline
\bf{Isotope} & {Sample \#3} & {Uncertainty} & {Sample \#4} & {Uncertainty} \\
\hline
$^{238}$Pu & \multicolumn{4}{c}{not included in analysis} \\
$^{239}$Pu & 99.1323 & 0.0024 & 99.1213 & 0.0008 \\
$^{240}$Pu & 0.8675 & 0.0023 & 0.8770 & 0.0008 \\
$^{241}$Pu & \multicolumn{2}{c}{not detected} & 0.001427 & 0.000010 \\
$^{242}$Pu & 0.000242 & 0.000042 & 0.000250 & 0.000006 \\
\hline
\end{tabular}
\end{center}
\end{table*}
\begin{table}[ht]
\begin{center}
\caption{Half-lives for $\alpha$~emitters in target, taken from DDEP~\cite{DDEP}. Note that since no decay data were available from DDEP for $^{233}$U, the half-life reported by the NNDC~\cite{NNDC} was used for that isotope.}
\label{table:halflivesDDEP}
\begin{tabular}{l S S[table-format=1.10e1]}
\hline
Isotope & {half-life (s)} & {Uncertainty (s)} \\
\hline
$^{233}$U \cite{NNDC} & \num{5.0240e12} & \num{6.3e9} \\
$^{234}$U \cite{DDEP} & \num{7.747e12} & \num{1.9e10} \\
$^{235}$U \cite{DDEP} & \num{2.2217e16} & \num{3.2e13} \\
$^{236}$U \cite{DDEP} & \num{7.394e14} & \num{1.9e12} \\
$^{238}$U \cite{DDEP} & \num{1.4100e17} & \num{1.6e14} \\
$^{238}$Pu \cite{DDEP} & \num{2.76886e9} & \num{9.5e5} \\
$^{239}$Pu \cite{DDEP} & \num{7.6054e11} & \num{3.5e8} \\
$^{240}$Pu \cite{DDEP} & \num{2.0705e11} & \num{2.2e8} \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\caption{Energies and relative intensities of $\alpha$ particles from isotopes included in the fitted alpha spectrum, taken from DDEP~\cite{DDEP}. Only intensities greater than 1\% are shown.}
\begin{center}
\label{table:alpha_energies}
\begin{tabular}{c | c | c}
\hline
{Isotope} & {Energy (keV)} & Intensity (\%) \\ \hline
\multirow{3}{*}{ $^{233}$U} & 4729.0 & 1.61 \\
& 4783.5 & 13.20 \\
& 4824.2 & 84.30 \\ \hline
\multirow{2}{*}{$ ^{234}$U} & 4722.4 & 28.42 \\
& 4774.6 & 71.37 \\ \hline
\multirow{8}{*}{$ ^{235} $U} & 4214.7 & 5.95 \\
& 4322.0 & 3.33 \\
& 4366.1 & 18.80 \\
& 4397.8 & 57.19 \\
& 4414.9 & 3.01 \\
& 4502.4 & 1.28 \\
& 4556.0 & 3.79 \\
& 4596.4 & 4.74 \\ \hline
\multirow{2}{*}{$^{236}$U} & 4445.0 & 26.10 \\
& 4494.0 & 73.80 \\ \hline
\multirow{2}{*}{$^{226}$Ra} & 4601.0 & 5.95 \\
& 4784.3 & 94.04 \\ \hline
\multirow{2}{*}{$^{238}$Pu} & 5456.3 & 28.85 \\
& 5499.0 & 71.04 \\ \hline
\multirow{3}{*}{$^{239}$Pu} & 5105.8 & 11.87 \\
& 5143.8 & 17.14 \\
& 5156.6 & 70.79 \\ \hline
\multirow{2}{*}{$^{240}$Pu} & 5123.6 & 27.16 \\
& 5168.1 & 72.74 \\ \hline
\end{tabular}
\end{center}
\end{table}
For the uranium case, the differences between the measured isotopic ratios are consistent with the addition of a small amount of natural uranium in processing from the raw material (Sample \#1) to the target deposit via vacuum volatilization (Sample \#2). This difference is demonstrated most clearly in the case of $^{238}$U, which has the largest relative discrepancy and is the majority constituent of natural uranium. The decay fraction for $^{235}$U obtained from the $\alpha$-spectrum was consistent with Sample \#2.
The $^{239}$Pu decay fraction is not highly sensitive to measured isotopic ratios for plutonium. The $^{238}$Pu was not resolvable from $^{238}$U background in the mass spectrometry measurements, but it was measured with $\alpha$-spectrometry from the silicon detector. The difference between calculated decay fractions using the two sets of ratio data is 0.03\%. The uncertainty on the decay fraction for $^{239}$Pu, 0.10\%, is dominated by uncertainties in the published half-lives, so if we assume that the variation in isotopic abundances for the target under study is within a factor of ten of the two mass spectrometry samples, this contribution to the uncertainty is negligible.
\subsection{Silicon Detector Data Analysis}
\label{subsec:analysis}
\subsubsection{Uranium Sample} \label{sec:uranium_sample}
The full $\alpha$-spectrum was modelled by assigning an exponentially modified Gaussian to each peak and summing over the decay fractions, $d$, and alpha emission intensities, $I_{\alpha}$, as
\begin{align}
S =& \sum_d \sum_{I_{\alpha}} d I_{\alpha} f_{EMG}(\mu, \sigma, \omega) \label{eq:spectrum_model} \\
\mu =& E_{\alpha} c_1 + c_0
\end{align}
where $\mu$ is the peak mean location with energy $E_{\alpha}$ and calibration constants $c_1$ and $c_0$, and $\sigma$ and $\omega$ are the standard deviation and skewness parameters. The calibration constants were determined by letting all the parameters float. Those constants were then fixed for all subsequent fits. Only alpha emission lines with relative intensities greater than 0.1\% were included. The contribution from $^{238}$U was not included because it is insignificant.
Two methods were considered when calculating the contribution of $^{235}$U to the $\alpha$-spectrum. In the first method the integrals of the fits were used directly as an estimate of the $^{235}$U contribution (Fit-Only). In the second method, a region of interest (ROI) was selected over the portion of the spectrum with majority $^{235}$U contribution. The fit results from other isotopes in the ROI were subtracted, and $^{235}$U fit results outside the ROI were added back into the estimate. The ROI method has the advantage of having lower uncertainty due to a smaller contribution of errors from the fit. The results of the ROI method are shown in Fig.~\ref{fig:u_spectrum_fit}.
In addition, we considered constraining the relative contribution of each uranium isotope from the results of mass spectrometry shown in Table~\ref{table:Isotopics}. We tested fixing all contributions (All fixed), fixing just the $^{236}$U contribution ($^{236}$U fixed), and letting all the contributions float (All floating). The last method does not require the results of mass spectrometry, which are higher fidelity but were performed on surrogate samples. We settled on using the middle approach of fixing $^{236}$U, since its contribution is unresolvable from $^{235}$U.
The last unresolved factor was the unknown contribution of $^{226}$Ra, the presence of which is indicated by the existence of resolvable uranium decay daughters in the $^{226}$Ra decay chain.
We tested the proposition of including it along with the other isotopes of uranium in the spectrum model from Eq.~\ref{eq:spectrum_model}. The results from all these approaches are shown in Table \ref{tab:uranium_fits_table}. Although the contribution of $^{226}$Ra did not have significant impact on the result with either method, we included it in our estimates for completeness since it met the criterion of $>$0.1\% relative intensity.
The Fit-Only approach resulted in a consistently lower estimate of the number of $^{235}$U atoms, except for the case with the contributions fixed by mass spectrometry results (All fixed). The Fit-Only method relies more on the ability of Eq.~\ref{eq:spectrum_model} to model the data, and by extension it is impacted by any incompleteness or inaccuracies in the accepted $\alpha$ emission intensity or any behavior of the detector that is not well modeled by an exponentially modified Gaussian.
By contrast, the ROI method is more consistent, and it produces a lower overall uncertainty in the estimate of the number of $^{235}$U atoms. Based on these results, the ROI method with $^{226}$Ra included and fixed $^{236}$U contribution was chosen as the method of determining the target atom number ratio presented in this paper and used for the fissionTPC cross-section ratio~\cite{Snyder2021}.
\begin{figure}[ht]
\begin{center}$
\begin{array}{cc}
\includegraphics[width=1.\linewidth]{uranium_spectrum_fits.pdf}
\end{array}$
\end{center}
\caption{The fits to the uranium $\alpha$-spectrum from the first measurement campaign, with $^{236}$U fixed and $^{226}$Ra included . The region of interest is shown in hashes between 3960 keV and 4600 keV. The partial integral of $^{235}$U contribution was added to the counts in the ROI, and the counts from other isotopes inside the ROI were subtracted. These adjustments are shown as solid regions under the fits.}
\label{fig:u_spectrum_fit}
\end{figure}
\begin{table*}[ht]
\caption{Summary of the various approaches to estimating the number of atoms of $^{235}$U from the $\alpha$-spectrum of the second measurement campaign. Based on these results, the ROI method with $^{226}$Ra included and fixed $^{236}$U contribution was chosen as the method of determining the target atom number ratio presented in this paper and used for the fissionTPC cross-section ratio~\cite{Snyder2021}.}
\label{tab:uranium_fits_table}
\centering
\begin{tabular}{|c|c|c|c|c|c|}
\hline
& & \multicolumn{3}{c|}{\textbf{Constraints}} & \\ \cline{3-5}
\multirow{-2}{*}{\textbf{\begin{tabular}[c]{@{}c@{}}Integration\\ Method\end{tabular}}} & \multirow{-2}{*}{$^{226}$Ra} & All floating & $^{236}$U fixed & All fixed & \multirow{-2}{*}{\textbf{$^{235}$U Atoms}} \\ \hline
& \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0}x & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0}8.61 $\pm$ 0.06 \\ \cline{2-6}
& x & x & & & 8.59 $\pm$ 0.05 \\ \cline{2-6}
& \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0}x & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0}8.63 $\pm$ 0.05 \\ \cline{2-6}
& x & & x & & 8.60 $\pm$ 0.05 \\ \cline{2-6}
\multirow{-5}{*}{\textbf{Fit-Only}} & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0}x & \cellcolor[HTML]{C0C0C0}8.69 $\pm$ 0.02 \\ \hline
& & x & & & 8.69 $\pm$ 0.03 \\ \cline{2-6}
& \cellcolor[HTML]{C0C0C0}x & \cellcolor[HTML]{C0C0C0}x & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0}8.67 $\pm$ 0.03 \\ \cline{2-6}
& & & x & & 8.70 $\pm$ 0.02 \\ \cline{2-6}
& \cellcolor[HTML]{C0C0C0}x & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0}x & \cellcolor[HTML]{C0C0C0} & \cellcolor[HTML]{C0C0C0}8.68 $\pm$ 0.02 \\ \cline{2-6}
\multirow{-5}{*}{\textbf{ROI}} & & & & x & 8.70 $\pm$ 0.02 \\ \hline
\end{tabular}
\end{table*}
\subsubsection{Plutonium Sample}
The high count rate from the $^{239}$Pu side of the sample necessitates additional analysis prior to fitting. The waveform from each event was analyzed for pulse height and pulse area, and pileup was rejected event-by-event via comparison of these extracted quantities. All events that occurred within 10 microseconds of each other were rejected and the total number of counts was determined using Eq. \ref{eq:time_dist} fit to the data. The remaining events were histogrammed and fit with a linear combination of Eq. \ref{eq:spectrum_model} comprised of every known alpha transition for $^{238}$Pu,$^{239}$Pu,$^{240}$Pu, and $^{242}$Pu. The alpha decays from $^{239}$Pu and$^{240}$Pu are not resolvable with the current resolution of our system so they were constrained in the fit as a ratio to $^{239}$Pu using the mass spectrometry results presented in Table \ref{table:decay_fractions}. The contribution from $^{238}$Pu was resolved and as such was left unconstrained in the fit. The ROI technique was employed in a similar fashion as described for the $^{235}$U in Section~\ref{sec:uranium_sample}, with a boundary located where the low-energy tail of $^{238}$Pu is of equal intensity to the high-energy tail of the remaining unresolved plutonium spectra. The results of the $^{238}$Pu fit was subtracted from the low-energy ROI and added to the high-energy ROI. The converse was performed for the remaining plutonium isotopes.
\begin{table}[h]
\begin{center}
\caption{Decay fraction estimates averaged between the two measurements.}
\label{table:decay_fractions}
\begin{tabular}{l S S }
\hline
Isotope & {Decay fraction} & {Uncertainty (\%)}\\
\hline
$^{233}$U & 0.289 & 0.60 \\
$^{234}$U & 0.349 & 0.49\\
$^{235}$U & 0.343 & 0.41 \\
$^{236}$U & 0.0184 & 0.35 \\
$^{238}$Pu & 0.0083 & 5.9 \\
$^{239}$Pu & 0.9607 & 0.19\\
$^{240}$Pu & 0.03209 & 0.22 \\
$^{242}$Pu & 1.5e-7 & 8.7 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsubsection{Target Atom Ratio}
The target atom ratio was calculated separately for both of the silicon detector measurement campaigns, introduced in Section~\ref{sec:silicon} and the results are shown in Table~\ref{table:ratioResults}. A summary of the contributing factors to the uncertainty in the total number of atoms for each isotope for the second measurement is shown in Table~\ref{table:siUncertainties}.
The two measurement campaigns were conducted independently by two different teams using the same instrument, and were in agreement. We combined the partial uncertainties from both measurements assuming they are correlated, with the exception of counting statistics which we added in quadrature. The individual measurement uncertainties shown in Table~\ref{table:ratioResults}. The average target atom ratio is 1.7343$\pm$0.0050 (0.288\%).
\begin{table}[ht]
\caption{Target atom number ratio results and uncertainties for the two measurement campaigns. }
\label{table:ratioResults}
\begin{center}
\begin{tabular}{lcc}
\hline
& Meas. 1 & Meas. 2 \\
\hline
$^{235}$U atoms & $8.647 \times 10^{17}$ & $8.682 \times 10^{17}$ \\
$^{235}$U Uncertainty & 0.257\% & 0.187\% \\
$^{239}$Pu atoms & $4.996 \times 10^{17}$ & $5.00 \times 10^{17}$ \\
$^{239}$Pu Uncertainty & 0.214\% & 0.235\% \\
Atom number ratio & 1.731 & 1.736 \\
Total Uncertainty & 0.334\% & 0.30\% \\
\hline
\end{tabular}
\end{center}
\end{table}
\iffalse
\begin{table}[ht]
\caption{Uncertainty contributions for the target atom numbers from the first measurement campaign.}
\label{table:siUncertainties_1}
\begin{tabular}{lcccc}
\hline
Source of Uncertainty & $^{235}$U & $^{239}$Pu \\
\hline
Spectrum fits error & 0.057\% & 0.007\% \\
Counting statistics & 0.204\% & 0.044\% \\
Mass spectrometry & 0.019\% & 0.10\% \\
Recovery time & n/a & 0.037\% \\
Half life & 0.144\% & 0.18\% \\
\hline
Total & 0.257\% & 0.214\% \\
\hline
\end{tabular}
\end{table}
\fi
\begin{table}[ht]
\caption{Uncertainty contributions for the target atom numbers from the second measurement campaign.}
\label{table:siUncertainties}
\begin{tabular}{lcc}
\hline
Source of Uncertainty & $^{235}$U & $^{239}$Pu \\
\hline
Spectrum fits error & 0.058\% & 0.007\% \\
Counting statistics & 0.102\% & 0.078\% \\
Mass spectrometry & 0.019\% & 0.17\% \\
Recovery time & n/a & 0.07\% \\
Half life & 0.144\% & 0.124\% \\
\hline
Total & 0.187\% & 0.235\% \\
\hline
\end{tabular}
\end{table}
\section{Conclusions}
A complementary and independent measurement of the atom number ratio for the targets used in the NIFFTE collaboration’s fissionTPC measurement of the $^{239}$Pu(n,f)/$^{235}$U(n,f) cross-section ratio have been performed using a combination of mass spectrometry and alpha spectroscopy with a silicon detector. The two silicon detector measurement campaigns were conducted independently by two different teams using the same instrument. The target atom ratio for the two measurements was in agreement within the reported uncertainties. The average target atom ratio is 1.7343$\pm$0.0050 (0.288\%). For a complete description of the results of the fissionTPC measurement of the normalized $^{239}$Pu(n,f)/$^{235}$U(n,f) cross-section ratio and comparisons to other measurements see \cite{Snyder2021}.
The methods described in this paper improve upon previous efforts to normalize fission cross-section ratio measurements by utilizing fitting of the $\alpha$-spectrum to determine the isotopic concentrations of our samples. While it was possible to use just $\alpha$-spectrum analysis to determine these quantities, we found that combining this analysis with constraints from mass spectrometry resulted in more consistent results. Uncertainty in the $\alpha$ decay data and the robustness of the $\alpha$-spectrum model ultimately limit the precision of this technique, but it provides an alternative when destructive methods like mass spectrometry are not possible.
\section{Acknowledgements}
This work was funded by the U.S. Department of Energy and operated by Los Alamos National Security, LLC, under contract DE-AC52-06NA25396. This work performed under the auspices of the U.S Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. This material is based upon work supported by the U.S. Department of Energy, National Nuclear Security Administration, Stewardship Science Academic Alliances Program, under Award Number DE-NA0002921.
LLNL-JRNL-820554
|
1,108,101,565,650 | arxiv | \section{Introduction}
\label{intro}
Formation and evolutionary processes of stellar and planetary
systems are
expected to leave their imprint on the present-day systems. One
such imprint is the stellar obliquity, the angle between the
stellar spin axis and the orbital angular momentum axis, also
referred to as the spin-orbit
angle. For star-planet systems the measurement of this angle is
a matter of intense study in recent years
\cite[e.g.,][]{triaud10, moutou11, winn11, albrecht12},
primarily for hot Jupiters ---
gas-giant planets at short-period orbits. Some of the systems
were found to be aligned, in a prograde orbit with spin-orbit
angle close to
zero, while others were found to be
misaligned, including systems in retrograde motion where the
spin-orbit angle is close to $180^\circ$
\citep[e.g.,][]{hebrard11, winn11}.
The growing sample and the wide range of spin-orbit angles
measured for hot Jupiters can be used for studying their
orbital evolutionary history. For example, \cite{winn10} have
noticed that hot stars, with an effective temperature above
6,250 K, tend to have a wide obliquity range, while cool stars
tend to have low obliquities, mostly consistent with well aligned
orbits. This was confirmed by a study of a larger sample by
\cite{albrecht12} and is consistent with the results of
\cite{schlaufman10} and \cite{hansen12} who used different
approaches. Those authors suggested that some mechanisms can
cause
the planetary orbit to attain large obliquity
\citep[e.g.,][]{fabrycky07, naoz11, batygin12}.
Then, tidal interaction with the host star
\citep[e.g.,][]{winn10} or magnetic braking
\citep[e.g.,][]{dawson14} act to realign the orbit. Since these
processes are probably inefficient for hot stars, those
systems might still retain their wide obliquity range.
So far spin-orbit alignment has been studied primarily through
the Rossiter-McLaughlin (RM) effect \citep{holt1893,
schlesinger1910, rossiter24, mclaughlin24}, originally suggested
for stellar eclipsing binaries, and observed by monitoring the
anomalous radial-velocity signal during eclipse, as the eclipsing
star moves across the disc of the eclipsed star. The RM effect is
sensitive to the sky-projected component of the spin-orbit angle,
and was successfully measured for many transiting planet systems
\cite[e.g.,][]{queloz00, winn06, triaud10}, transiting brown
dwarfs and low-mass star systems \citep{triaud13}, and stellar
binaries \citep{albrecht07, albrecht09, albrecht11, albrecht14}.
The line-of-sight component of the spin-orbit angle can be
measured using
asteroseismology \citep{gizon03, chaplin13}, or the observed
rotational broadening of spectral lines, {\it if} the
host star radius and rotation period are known with sufficient
precision \citep[][see also
\citealt{schlaufman10}]{hirano12, hirano14}.
However, these two methods require obtaining new data for each
target, using valuable resources (e.g., large telescopes or {\it Kepler}\
short-cadence data).
Other methods have been presented, based on stellar gravitational
darkening \citep{barnes09, szabo11, barnes11}, and the beaming
effect \citep[Photometric RM ---][]{shporer12, groot12}.
An interesting
approach was taken by \cite{nutzman11} and \cite{sanchis11a}, who
use the brief photometric signals during transit induced by the
transiting object moving across spots located on the surface of
the host object.
This is based on the fact that many stars show photometric
modulations resulting from the
combination of stellar rotation and non-uniform longitudinal
spots distribution \citep[e.g.,][]{irwin09, hartman11,
mcquillan14}.
When such a star displays transits by an orbiting planet, the
transiting object might
momentarily eclipse a stellar spot, inducing an increase in
observed flux, if the surface brightness of the spot-covered area
is lower than that of the non-spotted areas.
The derivation of the stellar obliquity requires identification
of such `spot-crossing' events
within a few transits, and estimate the spot and the planet
phases within their motion over the stellar disc.
The method has since
been applied to additional systems using {\it high-speed} {\it Kepler~} and
CoRoT data
\citep{sanchis11b, desert11, deming11, sanchis12, sanchis13}.
We present here another version of this approach that does not
require such high-speed photometry. Instead, we use the fact that
a spot-crossing event can induce measurable transit time
variation
\citep[TTV; e.g.,][]{sanchis11a, fabrycky12, mazeh13, szabo13,
oshagh13b},
even for data that cannot resolve the event itself. Our approach
relies on the expected correlation between the induced TTV and
the
corresponding local photometric slope immediately outside the
transit, presumably induced by the
same spot. Detected correlation or anti-correlation between the
TTVs and
their local slope can in principle differentiate between prograde
and retrograde rotation of the primary star in
stellar binaries or star-planet systems.
We present here the
basic concept and develop an analytical simplistic approximation
for the
induced TTVs and the photometric slope.
We also use the work of \citet{boisse12} and \citet{oshagh13a},
who developed a numerical tool ---
SOAP-T\footnote{http://www.astro.up.pt/resources/soap-t/}, to
simulate
a planetary transit light curve which includes a spot-crossing
event.
\citet{oshagh13b} used SOAP-T to derive detailed transit light
curves, and then fitted them with transit templates to obtain the
expected TTVs, very similar to what is performed when deriving
the TTVs from the {\it Kepler}\ actual data \citep[e.g.,][]{mazeh13}.
We show that our approximation yields TTVs with the same order of
magnitude as the results of \citet{oshagh13b}.
Using our approximation and the SOAP-T tool
we show that in some cases we can expect a negative (positive)
correlation between the
TTVs induced by spot crossing and the local photometric slopes
at the transit
timings for prograde (retrograde) motion of the planet.
We also discuss the limitations of this approach when applied to
real data, showing that it can be applied only to a limited
number of systems.
The paper is organized as follows. Section~\ref{sec:principles}
outlines the basics of our approach, while Section~\ref{sec:ttv}
presents the analytical approximation for the induced TTV for
different cases, and Section~\ref{sec:simulations} compares our
approximation with numerical simulations we performed and those
of \citet{oshagh13b}.
In Section~\ref{sec:slopes} we derive the expected derivative of
the stellar brightness at the time of transit, and in
Section~\ref{sec:correlations} display the expected correlation
between the induced TTVs and the stellar photometric slopes.
Finally,
Section~\ref{sec:discussion} discusses our results and the severe
limitations of its applicability to real data.
The present paper is the first of three studies. The next study
(Holczer et al., in preparation)
presents our analysis for the {\it Kepler~} planet
candidates \citep{batalha13}. In that paper we show that indeed a
few systems do show highly significant correlation between their
derived TTVs and the local photometric derivatives, as predicted
by this work. A forthcoming paper will
present our analysis of the {\it Kepler~} eclipsing stellar binaries
\citep{slawson11}.
\section{The principle of the approach}
\label{sec:principles}
To present our approach, we consider a transiting planet that
crosses
a stellar spot during its apparent motion over the stellar disc.
Let us assume, for the sake of simplicity, that only one spot is
present on the
stellar disc and that the stellar rotation and orbital axes are
parallel to each other. This includes both prograde (complete
alignment, with obliquity of $0^{\circ}$)
and retrograde (obliquity of $180^\circ$) configurations.
The {\it sign} of the induced time
shift depends on whether the spot-crossing event occurs in the
first (positive TTV) or second (negative TTV) half of the
transit, which is determined by
the location of the spot on the stellar disc at the time of
transit.
rotation
As depicted in Figures~\ref{fig:prograde} \&
\ref{fig:retrograde},
the location of the spot over the stellar
disc determines whether the star is becoming brighter or dimmer
at the time of transit. When the spot is
moving towards (away from) the center of the disc
the stellar intensity is decreasing (increasing), because of the
aspect effect, which changes the effective area of the spot on
the stellar visible disc ---
the projected area of the spot onto the sky plane.
Therefore, when the spot is on the disc edge
its effective area is minimal. On the other hand, the effective
area reaches its maximum when the spot is at its closest position
to the center of the visible disc, when the stellar surface is
(almost) perpendicular to our line of sight.
Another phenomenon, which also causes the star to become fainter
when the spot moves towards the center of the disc is the
limb-darkening effect,
which is ignored at this point of the discussion.
\begin{figure}
\includegraphics[width = 0.9\textwidth]{prograde.eps}
\caption{Prograde motion --- spot-crossing events during the
first (left) and second (right) halves of the transit. The top
panels display the stellar visible disc (yellow), the planet
(black, small) and the spot (gray, large). The arrows represent
the direction and speed of
the planet and spot relative to the observed stellar disc. The
middle
panels show the light curve due to the spot passage over the
stellar disc, spanning half a stellar rotation period. In the
middle panels we also see the transits, occurring at
phase 0.13 (left) and 0.37 (right) of the stellar rotation. The
bottom panels show the light curves again, now zooming on the
transits, where the small `bumps' are caused by the
spot-crossing events. We consider only a single spot, so the flux
is equal to unity when the spot is on the stellar hemisphere
hidden from the observer's view.
{\it Left (right)} --- the spot is at the first (second) half of
its crossing over the stellar disc, and therefore the local
photometric slope is negative (positive). The planet is at the
first (second) half of the transit and therefore the derived
transit timing shift (while the spot-crossing is unresolved) is
positive (negative).
}
\label{fig:prograde}
\end{figure}
\begin{figure}
\includegraphics[width = 0.9\textwidth]{retrograde.eps}
\caption{Retrograde motion --- see Figure~\ref{fig:prograde}
for details.
{\it Left (right)} --- the spot is at the first (second) half of
its crossing over the stellar disc, and therefore the local
photometric slope is negative (positive). The planet is at the
second (first) half of the transit and therefore the derived
transit timing shift (while the spot-crossing is unresolved) is
negative (positive). }
\label{fig:retrograde}
\end{figure}
Now, when the stellar rotation and planetary motion have the same
sense of rotation, the spot-crossing event in the first (second)
half of the transit should always occur when the spot is moving
towards (away from) the center of the disc.
Therefore the {\it signs} of the induced TTV and the {\it slope}
of the stellar brightness at the time of transit should be
opposite. This is depicted in Figure~\ref{fig:prograde} for
prograde motion. For retrograde
motion, positive TTV should be associated with positive
slope, as depicted in Figure~\ref{fig:retrograde}.
Therefore, we expect negative correlation between the derived
TTVs and the corresponding stellar photometric slopes for a
system with planetary prograde motion and positive correlation
for a system with retrograde motion.
In the next sections we will show that this is indeed the case
for a limited number of cases by deriving analytical
approximations for the TTVs and the photometric derivatives and
by numerical simulations for the TTVs.
\section{Analytical approximation for the TTV induced by the
spot-crossing event}
\label{sec:ttv}
\subsection{Center-of-light approximation}
To present the concept behind our method in a more quantitative
way, Figure~\ref{fig:drawing}
shows a simplified
schematic diagram of a transit light curve with a single
spot-crossing event. We neglect the transit
ingress and egress finite duration of both the transit and the
spot-crossing event.
We also assume that at the time of
each transit there is only one circular spot on the stellar disc.
In the figure we also
neglect the limb-darkening photometric modulation, and will
consider this effect later.
In the figure, $\delta_{tr}$ and $\delta_{sc}$ are the
depth of the transit and the amplitude of the photometric
increase inside the transit due to the spot-crossing event,
respectively,
$\Delta_{tr}$ and $\Delta_{sc}$ are the duration of the
transit and the spot-crossing event, respectively, and
$t_{sc}$ is the timing of the spot-crossing event relative to
mid-transit time.
\begin{figure}
\centering
\resizebox{15cm}{15cm}
{\includegraphics{drawing.eps}}
\caption{Schematic diagram of a transit light curve with
a spot-crossing event. The transit depth is $\delta_{tr}$, while
$\delta_{sc}$ is the flux increase due to the spot-crossing
event, $\Delta_{tr}$ and $\Delta_{sc}$ are the transit and
spot-crossing durations, and $t_{sc}$ is the timing of the
spot-crossing event relative to
the mid transit.
The vertical dashed black line represents the expected
mid-transit timing, without any spot-crossing event, while
the red dash-dot line represents the new mid-transit measurement,
due to the shift induced by the spot crossing.
The difference between the two lines, $TTV_{sc}$, is the induced
TTV. Approximately, $TTV_{sc}\simeq - t_{sc} \times
\delta_{sc}\Delta_{sc}/
(\delta_{tr} \Delta_{tr} - \delta_{sc} \Delta_{sc}$).
}
\label{fig:drawing}
\end{figure}
From the figure one can see, using `center-of-light'
formulation, that we expect the TTV induced by the spot-crossing
event to be:
\begin{equation}
TTV_{sc} \simeq - t_{sc} \frac{\delta_{sc} \Delta_{sc}}
{\delta_{tr} \Delta_{tr} - \delta_{sc} \Delta_{sc}} \ .
\label{eq_timing}
\end{equation}
This result is similar to Equation (3) of \citet{sanchis11a}
after neglecting $\delta _{sc} \Delta_{sc}$ in the denominator.
We will adopt this approximation below.
\newpage
\subsection{Basic Model}
To derive the analytical simplistic model we first consider a
case for which
\begin{itemize}
\item
the impact parameter of
the spot and the planet are both equal to zero, namely that they
both cross the center of the stellar disc, and
\item
there is no limb darkening.
\end{itemize}
We lessen these two assumptions below.
We denote the location of the spot on the stellar disc by the
angle $\psi$, which is the angle
between the observer and the spot, as seen from the stellar
center. {\it If} the motion of the spot is equatorial, then
$\psi$ is the longitude of the spot on the stellar visible
hemisphere:
\begin{equation}
\psi(t)=\omega_{*}t \, ,
\end{equation}
where $\omega_{*}$ is the stellar angular
velocity. When the spot is on the stellar limb
entering the visible hemisphere, $\psi$ gets the value of
$-\pi/2$, and when the location of the spot is in the middle of
its visible chord $\psi=0$.
We denote the angle corresponding to the spot crossing by
$\psi_{sc}$. The sky-projected distance of the spot from the
stellar center, as seen by the
observer, is
$d_{sc}= R_*\sin \psi_{sc}$,
where $R_*$ is the stellar radius. The timing of the
spot-crossing event,
measured relative to the middle of the transit, is therefore
\begin{equation}
t_{sc}= \frac{\Delta_{tr}}{2}\sin \psi_{sc} \, .
\label{eq_t_sc}
\end{equation}
In order to estimate the induced TTV,
let us consider two extreme cases: a small spot, for which
\begin{equation}
R_{spot}\ll R_{pl}\ll R_* \, ,
\end{equation}
and a large spot, for which
\begin{equation}
R_{pl}\ll R_{spot}\ll R_* \ ,
\end{equation}
where $R_{pl}$ and $R_{spot}$ are the radii of the planet and the
spot, respectively.
For both cases we introduce a darkness parameter, $0<\alpha<1$,
which
measures the surface brightness of the spot relative to the
surface brightness of the star immediately outside the spot. A
completely dark spot would have $\alpha=0$, while
$\alpha$ close to unity means the spot is almost as bright as the
unspotted stellar area.
For the small-spot approximation we can assume that the spot is
completely covered by the planet during the spot-crossing event
and therefore
\begin{equation}
{\rm Small \ spot\!: \ \ \ \
\ }\delta_{sc}\simeq(1-\alpha)\left(\frac{R_{spot}}{R_*}\right)^2
\cos\psi_{sc}
\ \ , \ \
\Delta_{sc}\simeq\Delta_{tr}\left(\frac{R_{pl}}{R_*}\right)\, .
\label{small_basic}
\end{equation}
As noted above, the factor $\cos\psi_{sc}$ comes from the fact
that the effective area of the spot is reduced by the aspect
ratio, which is a function
of the spot position on the visible stellar disc.
For the large-spot approximation the planet is fully contained in
the spotted area during the spot-crossing event, and therefore we
get
\begin{equation}
{\rm Large \ spot\!: \ \ \ \ \ }
\delta_{sc}\simeq(1-\alpha)\left(\frac{R_{pl}}{R_*}\right)^2
\ \ , \ \
\Delta_{sc}\simeq\Delta_{tr}\left(\frac{R_{spot}}{R_*}\right)
\cos\psi_{sc} \, .
\label{large_basic}
\end{equation}
Here the factor $\cos\psi_{sc}$ comes from the fact that the time
to
cross the spot by the relatively small planet is reduced by the
same
aspect ratio.
We now approximate the TTV to be
\begin{equation}
TTV_{sc} \simeq - t_{sc}
\frac{\delta_{sc}}{\delta_{tr}}\,
\frac{\Delta_{sc}}{\Delta_{tr}}\, ,
\end{equation}
and the transit depth $\delta_{tr}$ to be on the order of
$(R_{pl}/R_*)^2$. We therefore get for the small-spot
approximation
\begin{equation}
\begin{split}
{\rm Small \ spot\!: \ \ \ \ \ }
TTV_{sc}
&
\simeq - t_{sc} (1-\alpha)\left(\frac{R_{spot}}{R_*}\right)^2
\frac{R_{pl}}{R_*} \left(\frac{R_{pl}}{R_*}\right)^{-2}
\cos\psi_{sc}
\\&
=- t_{sc}(1-\alpha)\frac{R_{spot}^2}{R_{pl}R_*}\cos\psi_{sc}
\, ,
\end{split}
\label{eq:timing_small1}
\end{equation}
and for the large-spot approximation
\begin{equation}
{\rm Large \ spot\!: \ \ \ \ \ }
TTV_{sc}
\simeq - t_{sc} (1-\alpha) \frac{R_{spot}}{R_{*}}
\cos\psi_{sc}
\ .
\label{eq_timing_large1}
\end{equation}
Note that when $R_{spot}\rightarrow R_{pl}$,
Equation~(\ref{eq:timing_small1}) $\rightarrow$
Equation~(\ref{eq_timing_large1}). To ease the discussion we
define
$\mathcal{R}$ as:
\begin{equation}
\mathcal{R} = \left\{
\begin{array}{l l}
\frac{R_{spot}^2}{R_{pl}R_*} & $for small spot$\\
\frac{R_{spot}}{R_*} & $for large spot$ \,.
\end{array} \right.
\label{eq_factor2}
\end{equation}
Using Equation~(\ref{eq_t_sc}) we get:
\begin{equation}
TTV_{sc}
\simeq
-(1-\alpha) \mathcal{R}
\frac{\Delta_{tr}}{2}
\cos\psi_{sc}\sin\psi_{sc}
\ ,
\label{eq_TTV}
\end{equation}
which is valid both for the small- and large-spot approximations.
The maximum observed TTV induced by the spot crossing is
\begin{equation}
{\rm max}\{TTV_{sc}\}
\simeq
\frac{(1-\alpha)}{4}\mathcal{R}\,
\Delta_{tr}
\ .
\label{eq_timing_max}
\end{equation}
\subsection{Models for Limb darkening, impact parameter and
obliquity}
\label{limb_darkening}
\subsubsection{Limb darkening}
To include the stellar limb darkening effect in our model, we
consider a quadratic limb-darkening law of
$\mathcal{S}=1 - g_1(1-\cos \psi) - g_2(1-\cos \psi)^2 $, where
$\mathcal{S}$ is the scaled stellar surface brightness and $g_1$
and $g_2$
are the two limb-darkening coefficients, such that $g_1 + g_2
<1$.
The induced TTV is proportional
to $\delta_{sc}$, the increase of the stellar brightness during
the spot crossing, which depends linearly on
the stellar surface brightness $\mathcal{S}$, which is now a
function of $\psi$. Therefore we get
\begin{eqnarray}
TTV_{sc} =-(1-\alpha) \mathcal{R}\frac{\Delta_{tr}}{2}
\cos \psi(t) \sin \psi(t) \,
\big\{1 - g_1(1-\cos \psi) - g_2(1-\cos \psi)^2\big\} \nonumber
\\
=-(1-\alpha) \mathcal{R}\frac{\Delta_{tr}}{2}
\big\{(1-g_1-g_2) \sin \psi(t) + (g_1 + 2g_2) \sin \psi (t) \cos
\psi (t) \nonumber \\
- g_2 \sin \psi (t) \cos^2 \psi (t) \big\}
\cos \psi(t) \, .
\label{eq:TTV_basic_eq_LD}
\end{eqnarray}
Note that because of the limb darkening the transit light curve
does not have a rectangle shape, so our Equation (1) should be
modified. Nevertheless, as this analytical approach is aimed only
to understand the features of the TTVs as a function of the
spot-crossing phase, we neglect this effect that will affect all
phases alike.
\subsubsection{Impact parameter}
Another extension of our simplistic model accounts for a
non-zero impact parameter, $b=\cos\theta$.
Note that the stellar rotation is, as before, orthogonal to our
line of sight.
In this extension of the simplistic model, both
planet and spot still have the same impact parameter, namely both
move along the same chord on the stellar disc, a chord that does
not go through the center of the disc.
Therefore, the spot moves at a colatitude $\theta_{spot}=\theta$,
with an impact parameter $b_{spot}=\cos\theta_{spot}$.
In such a case, the angle $\psi$ fulfill the relation
\begin{equation}
\cos\psi=\sin\theta\cos\phi \, ,
\end{equation}
where now $\phi$ is the longitude of the planet, and $\phi=0$ is
when the planet crosses the projection of the stellar rotational
axis.
The range of $\psi$ is now
different: $ \pi/2-\theta\leq|\psi|\leq\pi/2$, and the timing of
the spot crossing is
\begin{equation}
t_{sc}= \frac{\Delta_{tr}^b}{2}\sin \phi_{sc}\, ,
\end{equation}
where $\Delta_{tr}^b$ is the transit duration when $b\neq0$. A
good approximation would be
$\Delta_{tr}^b= \Delta_{tr}\sin \theta$.
We now separate the discussion for the small and large spot
approximations.
For small spot, the duration of the spot-crossing event,
$\Delta_{sc}$, is still
the same as for $b=0$, but the transit duration is shorter by a
factor of $\sin\theta$. The flux increase depends on $\cos\psi$,
as for $b=0$. We can therefore write
\begin{equation}
{\rm Small \ spot\!: \ \ \ }
\delta_{sc}\simeq(1-\alpha)\left(\frac{R_{spot}}{R_*}\right)^2
\sin\theta\cos\phi_{sc}
\ , \ \
\Delta_{sc}\simeq\Delta_{tr}^b\left(\frac{R_{pl}}{R_*}\right)\frac{1}{\sin\theta}\, .
\end{equation}
Combining these expressions we get
\begin{equation}
{\rm Small \ spot\!: \ \ \ \ \ }
TTV_{sc}
\simeq
-(1-\alpha) \mathcal{R}
\frac{\Delta_{tr}^b}{2}
\cos\phi_{sc}\sin\phi_{sc}
\ ,
\label{eq_TTV_impact_small}
\end{equation}
For the large spot case, the duration of the spot-crossing event,
$\Delta_{sc}$, is now
different, as the planet is crossing a spot which forms an
ellipse on the stellar disc, whose axes are $R_{spot}$ and
$R_{spot}\cos\psi$. One can show that the length of the planet's
path on the spotted area is
$R_{spot}\sqrt{\cos^2\theta+\sin^2\theta\cos^2\phi_{sc}}$. We
therefore get
\begin{equation}
{\rm Large \ spot\!: \ \ \ \ \ }
\delta_{sc}\simeq(1-\alpha)\left(\frac{R_{pl}}{R_*}\right)^2
\ \ , \ \
\Delta_{sc}\simeq\Delta_{tr}^b\left(\frac{R_{spot}}{R_*}\right)
\sqrt{\cot^2\theta+\cos^2\phi_{sc}} \, ,
\end{equation}
and thus
\begin{equation}
{\rm Large \ spot\!: \ \ \ \ \ }
TTV_{sc}
\simeq
-(1-\alpha) \mathcal{R}
\frac{\Delta_{tr}^b}{2}
\sqrt{\cot^2\theta+\cos^2\phi_{sc}}\
\sin\phi_{sc}
\ ,
\label{eq_TTV_large}
\end{equation}
\newpage
We can see that for a non-vanishing impact parameter there is a
difference between the large and small planet cases, unlike in
the basic model.
The difference is due to the $\cot^2 \theta$ term under the
square sign in Equation~(\ref{eq_TTV_large}).
Note that the approximation of the large spot is not valid for
$|\phi|\simeq\pi/2$, where the projected area of the spot is
small. Hence, we inserted into the calculation of the large-spot
case a correction factor that turns the TTV expression to be
similar to the
small-spot one when $|\phi|\rightarrow\pi/2$.
This was done by multiplying the $\cot^2 \theta$ term with a
Fermi function that is approximately unity, except for
$|\phi|\rightarrow\pi/2$, when the correction factor goes to
zero.
\subsubsection{Limb darkening and impact parameter}
To further extend our simplistic model, we consider now a case
for non-zero impact parameter {\it and} quadratic limb darkening
together.
As before, we divide the discussion between the cases of small
and large spot.
Following Equation~(\ref{eq_TTV_impact_small}), but now
multiplying it by the limb darkening brightness factor, we get
for the small spot case:
\begin{equation}
\begin{split}
{\rm Small \ spot\!: \ \ \ \ \ }
TTV_{sc} & \simeq -(1-\alpha) \mathcal{R} \frac{\Delta_{tr}^b}{2}
\big\{ (1- g_1 -g_2) \sin\phi (t) + \\
&
+ (g_1+2g_2) \sin \theta \sin \phi (t) \cos \phi (t) - \\
&
-g_2\sin ^2\theta \sin \phi (t) \cos^2 \phi (t) \big\} \cos
\phi(t) \, ,
\end{split}
\end{equation}
while for the large spot case, following
Equation~(\ref{eq_TTV_large}), we get:
\begin{equation}
\begin{split}
{\rm Large \ spot\!: \ \ \ \ \ }
TTV_{sc}
&
\simeq -(1-\alpha) \mathcal{R} \frac{\Delta_{tr}^b}{2} \big\{ (1-
g_1 -g_2) \sin\phi (t) +
\\ &
+(g_1 +2g_2) \sin \theta \sin \phi (t) \cos \phi (t) -
\\&
- g_2\sin ^2\theta \sin \phi (t) \cos^2 \phi (t) \big\} \sqrt{
\cot ^2 \theta + \cos^2 \phi(t)} \, .
\end{split}
\label{eq:impact_parameter}
\end{equation}
\subsubsection{Stellar obliquity}
\label{sec:obliquity}
The last case we consider is when the apparent planetary chord
along the stellar disc goes through the center ($b_{pl}=0$), but
is inclined with the angle $\eta$ relative to the
stellar equator.
We nevertheless assume that in some transits spot-crossing events
happen, with spots that have different latitudes. In such cases,
$t_{sc}$ is proportional to the distance of the spot-crossing
event from the center of the disc, as in the basic model
(Equation (3)).
Similar considerations show that here also we get, as in Equation
(12):
\begin{equation}\nonumber
TTV_{sc}
\simeq
-(1-\alpha) \mathcal{R}
\frac{\Delta_{tr}}{2}
\cos\psi_{sc}\sin\psi_{sc}
\ ,
\label{eq_TTV}
\end{equation}
which is true for small and large spot cases alike. The extension
for limb darkening also holds in this case.
\subsection{Comparing the different TTV patterns}
To visualize the expected TTVs derived by our analytical
approximation for non-vanishing impact parameter cases, we
plotted in Figure \ref{fig:impact_parameter} the calculated TTVs
for different values of the impact parameter, with the large-spot
approximation, using $R_{spot}/R_*=0.15$ and $R_{pl}/R_*=0.05$
values.
We chose a typical parameters for a transiting system --- a
planet orbiting a star with solar radius in a 3 d orbit. The
duration of the transit (mid-ingress to mid-egress) is about 2.62
hours, a value on which we based our estimations.
One can see in the figure that the amplitude of the induced TTV
is about 5 min. The derived TTVs display almost linear slope as a
function of the spot-crossing position, up to a maximum at
distance of 0.6--0.85 stellar radii from the center of the
stellar disc, and then a sharp drop to zero at the edge of the
stellar disc.
\begin{figure}[p!]
\centering
\resizebox{16cm}{12cm}
{\includegraphics{AnalyticApprLarge.eps}}
\caption{The analytic approximation for the induced TTV as a
function of the spot-crossing position on the stellar disc for
different values of the impact parameter, using the large-spot
expression of Equation (\ref{eq:impact_parameter}).
The position of the spot-crossing event is measured relative to
the center of the stellar disc, in units of the stellar radius.
The graphs are for
a Jupiter-size planet that orbits a star with solar radius in a
3-d orbit. The duration of the transit (mid-ingress to
mid-egress) is about 2.62 hours, a value on which we based our
estimations. The spot and planet radii were chosen as
$R_{spot}/R_*=0.15$ and $R_{pl}/R_*=0.05$.
The limb darkening coefficients used are [g$_1$, g$_2$] =
[0.29,0.34].
}
\label{fig:impact_parameter}
\end{figure}
\section{Comparison with numerical simulations}
\label{sec:simulations}
As noted in the introduction, \citet{boisse12} and
\citet{oshagh13a} developed a numerical tool ---
SOAP-T\footnote[1]{http://www.astro.up.pt/resources/soap-t/}, to
simulate stellar photometric modulations induced by a rotating
spot, including a planetary transit light curve which includes a
spot-crossing event.
\citet{oshagh13b} used SOAP-T to derive detailed transit light
curves, and then fitted them with transit templates to obtain the
expected TTVs, very similar to what is performed when deriving
the TTVs from the {\it Kepler}\ actual data \citep[e.g.,][]{mazeh13}.
This is much more accurate derivation than that of the previous
section, where we estimated the TTVs by the center-of-light
approach.
It is therefore useful to compare the TTVs obtained by our
analytical approximation with the ones derived with the SOAP-T
numerical code and the transit fitting.
To do that we perform in this section two comparisons.
First, we used ourselves the publicly available SOAP-T tool to
produce transit light curves with spot-crossing events and fitted
them with the \citet{mazeh13} codes to produce TTVs for a few
cases and compare them with the analytical approximations.
Second, we derive with our analytical center-of-light approach
some TTVs for the cases derived by \citet{oshagh13b}, and compare
the results.
In Figure~\ref{fig:analytical_oshagh} we plotted our analytical
approximation for the same system as before --- a 3 d transiting
planet orbiting a solar-like star. We used
limb darkening of $g_1=0.29$ and $g_2=0.34$, $R_{pl}/R_*=0.1$ and
$R_{spot}/R_*=0.1$,
a dark spot, with $\alpha=0$,
and impact parameter of zero. We can see from the figure that
the maximum expected TTV based on our approximation is similar to
the one obtained when simulating the spot-crossing event. The
obvious difference is the phase dependence --- while the
analytical approximation has a smooth rise to the maximum, at
phase of 0.65, the simulated light curves yielded TTVs that are
quite small for most phases, and rise sharply towards the maximum
at phase 0.8.
\begin{figure}[t!]
\includegraphics[width =
0.9\textwidth]{Analytical_vs_fittingOshagh.eps}
\caption{Comparison of the analytic approximation for the
induced TTV with numerical simulations, as a function of the
spot-crossing phase.
The approximated TTV (red) was derived by
Equation~(\ref{eq:TTV_basic_eq_LD}), while the light curves
obtained by the SOAP-T tool (blue) were analyzed to derive the
TTV. The error bars was derived from the \citet{mazeh13} codes.
$R_{pl}/R_*=0.1$ and $R_{spot}/R_*=0.1$. The limb darkening
coefficients that were used were [g$_1$, g$_2$] = [0.29,0.34].}
\label{fig:analytical_oshagh}
\end{figure}
The reason for this difference comes from the different
approaches of obtaining the TTV. The approach that fits a model
to the simulated light curve ignores sometimes the `bump' in the
light curve caused by the spot-crossing event, yielding a small
TTV, while the center-of-light model is, in fact, integrating
over the whole transit light curve. We will see this difference
again and again. Nevertheless, this difference does not change
the result of this paper --- the negative (positive) correlation
for prograde (retrograde) motion, as will be shown below.
\citet{oshagh13b} paper includes two figures that present their
derived TTVs as a function of the orbital phase of the
spot-crossing event. We applied our analytical approximation to
all cases included in \citet{oshagh13b} figures, presented in
the next two figures.
In Figure \ref{fig:oshagh1} we plot results of our analytical
approximation that corresponding to the six cases of
\citet{oshagh13b} Figure 3, where they have considered different
spot and planet relative sizes, keeping the same limb darkening
parameters.
We chose the same $(R_{spot}/R_*)^2$ (what they call 'f')
values --- 0.01 and 0.0025,
and the same $R_{pl}/R_*$ values --- 0.05, 0.1, and 0.15.
We used the same limb darkening coefficients of [$g_1$,$g_2$] =
[0.29,0.34], and assumed a completely dark spot ($\alpha$ = 0 in
our notation).
As before, the transit duration is set to be 2.62 hours.
\begin{figure}[t!]
\centering
\resizebox{15cm}{10cm}
{\includegraphics{oshagh1.eps}}
\caption{The analytic approximation for the induced TTV as a
function of the spot-crossing phase for different spot and planet
sizes.
Expected TTV were derived by using
Equation~(\ref{eq:TTV_basic_eq_LD}).
Rp/Rs is planet to star radius ratio and f is spot to star radius
ratio squared. The limb darkening coefficients used are [g$_1$,
g$_2$] = [0.29,0.34].
}
\label{fig:oshagh1}
\end{figure}
As in the previous figure, we see here that the maximum TTV is
similar to the values obtained by
\citet{oshagh13b}, while the phase behavior of the two approaches
is different, as explained above.
Another comparison was done by constructing
Figure~\ref{fig:oshagh2} and comparing it with Figure 6 of
\citet{oshagh13b}, to study the effect of the limb darkening and
spot darkness. Here again the amplitudes of the analytical
approximation are similar to those of \citet{oshagh13b}, while
the phase dependence is different, like in our
Figure~\ref{fig:oshagh1}.
\begin{figure}[p!]
\centering
\resizebox{15cm}{10cm}
{\includegraphics{oshagh2.eps}}
\caption{Expected TTV for different limb darkening parameters,
using our analytical approximation for $R_{pl}/R_* =
R_{spot}/R_*=0.1$. The limb darkening coefficients were in
case 1 [g$_1$, g$_2$] = [0.29,0.34], in case 2 [0.38,0.37], in
case 3 [0.6,0.16], and in case 4 [0.29,0.34]. In Case 4 the spot
has half of the stellar brightness ($\alpha = 0.5$), and the spot
size was increased by 1.4, in order to get similar amplitude of
the TTVs.}
\label{fig:oshagh2}
\end{figure}
\section{Analytical approximation for the stellar photometric
slopes}
\label{sec:slopes}
We turn now to approximate the local photometric slope at the
time of the transit, assuming as before that the stellar
brightness is modulated by a single circular spot.
For no limb darkening and null impact parameter we
approximate the stellar flux, modulated by the spot as
\begin{equation}
F_*(t) \simeq 1- \mathcal{A}\cos \psi(t)\, , \ \ \
{\rm for} -\pi/2\leq\psi\leq\pi/2 \, ,
\label{eq:stellar_photo}
\end{equation}
where $\mathcal{A}$ is the observed amplitude of the
photometric
modulation. This is so because the spot area on the stellar disc
is reduced by the aspect ratio $\cos\psi$.
The {\it derivative} of the stellar photometric brightness is
therefore
\begin{equation}
\label{fdot}
\dot{F}_*(t) \simeq \omega_{*} \mathcal{A}\sin \psi (t) \ .
\end{equation}
The amplitude of the observed stellar photometric modulation is a
function of the spot
radius and darkness. To express this relation we introduce the
$0<\beta<1$ parameter,
which accounts for the possibility that the spot crossed by the
planet might not be
the only spot that contributes to the stellar modulation with the
observed
phase. Therefore, $\beta$ measures the ratio of the area of the
spot being crossed by the
planet to the total neighboring spotted area that
causes the photometric modulation {\it with the same phase}.
The total stellar modulation due to the spots, relative to the
maximum
stellar brightness, is
\begin{equation}
\mathcal{A}\simeq\frac{1-\alpha}{\beta}\left(\frac{R_{spot}}{R_*
}\right)^2 \, .
\end{equation}
In the case of limb darkening, the brightness of the spotted star
takes the form
\begin{equation}
F_*(t) \simeq 1- \mathcal{A}\cos \psi(t)
\big\{1 -g_1(1-\cos\psi(t)) - g_2(1-\cos\psi(t))^2 \big\} \, ,
\label{eq:stellar_photo_LD}
\end{equation}
as the photometry is modulated by the aspect ratio and the limb
darkening at the spot's location.
The photometric derivative is then:
\begin{equation}
\dot{F}_*(t) =\mathcal{A}\omega_*
\big\{(1-g_1-g_2)\sin \psi (t) + (2g_1+4g_2) \sin \psi (t) \cos
\psi (t) -3g_2\sin \psi (t)\cos^2 \psi (t) \big\} \, .
\end{equation}
The stellar photometry for non-vanishing impact parameter is
expressed like in Equation~(\ref{eq:stellar_photo}), but now
$\cos \psi(t)=\sin \theta \cos \omega_*t$,
and therefore the stellar photometry derivative is
\begin{equation}
\dot{F}_*(t) \simeq \omega_{*} \mathcal{A}\sin\theta\sin \phi (t)
\, ,
\label{eq:fdot_impact}
\end{equation}
where $t$ is the time since the {\it spot} was in the middle of
its trail, on the projection of the stellar spin (see below) on
the stellar disc, and $\omega_*$ is the stellar rotation rate, as
explained in Section~\ref{sec:obliquity}.
For non-vanishing impact parameter {\it and} stellar limb
darkening the stellar photometry is
\begin{equation}
F_*(t) \simeq 1- \mathcal{A} \sin \theta\cos \phi(t)\
\big\{1 -g_1(1- \sin \theta\cos\phi(t)) - g_2(1-\sin
\theta\cos\phi(t))^2 \big\} \, .
\end{equation}
and its derivative is
\begin{equation}
\begin{split}
\dot{F}_*(t) \simeq
&
\omega_{*} \mathcal{A}\big\{(1-g_1-g_2)\sin\theta\sin \phi (t) +
(2g_1 +4g_2)\sin ^2\theta \sin \phi (t) \cos \phi (t)
\\&
- 3g_2\sin ^3\theta \sin \phi (t) \cos^2 \phi (t) \big\}\, .
\end{split}
\label{eq:fdot_impact_limb}
\end{equation}
When the obliquity of the system is non-vanishing, the spot moves
on a chord orthogonal to the projection of the stellar
rotational axis, at a
colatitude $\theta_{spot}$, with $b_{spot}=cos\theta_{spot}$.
The spot chord is different from that of the planet, which we
assume goes through the center of the stellar disc.
Because of the inclination of the transit chord, at the time of
crossing
\begin{equation}
\sin\theta_{spot}\sin\omega_*t=\sin\psi_{sc}\cos\eta
\end{equation}
where $t$ is the time since the {\it spot} was in the middle of
its trail, on the projection of the stellar spin on the stellar
disc, and $\omega_*$ is the stellar rotation rate. Therefore, the
stellar photometric derivative is like
Equations~(\ref{eq:fdot_impact}) or (\ref{eq:fdot_impact_limb}),
except for a $\sin\theta_{spot}$ factor.
Note that when $\eta\rightarrow\pi/2$ then
$\dot{F}_*(t)\rightarrow0$, because the spot-crossing effect
occurs near the photometric maximum, and therefore the
correlation with the TTVs becomes difficult to detect.
\section{The correlation between $TTV_{sc}$ and the stellar
photometric slopes}
\label{sec:correlations}
We are now ready to consider the expected correlation between the
TTVs induced by the spot-crossing events and the local slope of
the stellar photometry at the time of the transit.
\subsection{TTV as a function of the photometric slope}
Figure~\ref{fig:ttvfdot} displays our analytical approximation
for the TTVs as a function of the photometric derivatives for a
few cases.
The figure shows that the slope
of the stellar brightness at the time of each transit and the
corresponding induced TTV have {\it opposite} signs for prograde
motion, and therefore we expect negative correlation between the
two.
Obviously, the slope and the induced TTV have the {\it same}
sign for retrograde motion, because of the argumentation
presented in Section~2 and plotted in Figures~\ref{fig:prograde}
and
\ref{fig:retrograde} still holds, and therefore a positive
correlation is expected in such a case.
\begin{figure}
\centering
\resizebox{15cm}{10cm}
{\includegraphics{AllRelationsLD.eps}}
\caption{
The induced TTV$_{sc}$ versus the photometric slope for {\it
prograde} motion, using arbitrary units on both axes.
The blue line is the basic model, for $b=0$ and no limb
darkening. The red line presents the limb-darkening,
$g_1=g_2=0.3$, model, the green one is for $b=0.5$ and small
spot, and the cyan line is for the same $b$ with the large spot
approximation.
}
\label{fig:ttvfdot}
\end{figure}
\subsection{Correlation as a function of noise and number of
observed transits}
Figure~\ref{fig:ttvfdot} portraits how the TTVs derived by our
analytical approximation depend on the photometric slope, but it
does not show the real expected TTV, nor includes any
observational noise, associated with every derived TTV and
photometric derivative series. To see how these two affect the
expected correlation we added normally distributed noise to both
the simulated TTVs and the photometric derivatives, the results
of which are plotted in Figure~\ref{fig:ttvvsdotanal} for our
analytically approximated TTVs, and in
Figure~\ref{fig:ttvvsdotoshagh} for TTVs derived by the SOAP-T
tool.
In both figures we used the same fiducial system, but now with
$R_{spot}/R_* = R_{pl}/R_* = 0.1$ and $[g_1,g_2] = [0.29,0.34]$.
The photometric derivative was scaled so that its maximum was
unity.
We chose at random 500 phases to be plotted in the figures, and
added randomly distributed normal noise
to both the TTVs and the slope derivatives.
The noise r.m.s.~was equal to 50\% of the maximum of the
corresponding variable.
This amounts to 150 sec error on the TTVs and $0.5$ to the scaled
slope.
\begin{figure}[pt!]
\centering
\resizebox{15cm}{10cm}
{\includegraphics{AnalyticNoise50.eps}}
\caption{
Simulation of TTV, derived by the analytical approximation,
versus the corresponding photometric slope for {\it
prograde} motion, both with added normally distributed random
noise.
The noise r.m.s.~equals to 50\% of the maximum of the
corresponding variable.
The slope is scaled such that its maximum (before adding the
noise) is unity. The simulation includes 500 phases selected at
random. Correlation is $-0.62$. See text for details.
}
\label{fig:ttvvsdotanal}
\end{figure}
\begin{figure}[pt!]
\centering
\resizebox{15cm}{10cm}
{\includegraphics{OshaghNoise50.eps}}
\caption{
Simulation of TTV, derived by analyzing the transit light curves
obtained by the SOAP-T tool, versus the corresponding photometric
slope for {\it prograde} motion, both with added normally
distributed random noise. Correlation is $-0.48$. See
Figure~\ref{fig:ttvvsdotanal} and text for details.
}
\label{fig:ttvvsdotoshagh}
\end{figure}
\newpage
The two figures show similar results --- there is a very clear
anti-correlation between the induced TTVs and the photometric
slopes at the transit timings, even when some small noise is
added. In fact, the noise covers up the fact that for some phases
the dependence of the TTVs on the slope changes its sign, as we
see in Figure~\ref{fig:ttvfdot}.
To estimate the expected effect of the noise on the measured
correlation we ran extensive simulations,
with different values of noise level and number of observed
transits.
For each choice of noise level, $\sigma_{\rm TTV}$,
$\sigma_{\rm slope}$ and number of transits, $N$, we chose $N$
random phases, derived their TTVs and photometric derivatives,
added randomly distributed noise to both the TTVs and the stellar
photometric slopes, and then derived the resulting
(anti-)correlation. We repeated this simulation for 1000 times,
with the same values of noise level and number of points. We then
derived the median and scatter of the sample of correlations
obtained, which are plotted in Figure~\ref{fig:correlation_noise}
as a function of the noise level and $N$.
We chose five values for $\sigma_{\rm TTV}$ and
$\sigma_{\rm slope}$, each scaled as a fraction of the maximum of
its corresponding variable. The five noise-to-signal ratios we
chose were [0, 0.15, 0.3, 0.5 1]. Each choice characterizes both
the noise added to the TTVs and to the photometric slopes. For
$N$ we chose values of 50, 100, 500, and 1000. For short-period
transiting planets {\it Kepler~} light curves could have on the order of
1000 transits, but 200--400 was a more typical number. All
together we had $4\times5=20$ sets of simulations, the results of
which are plotted in Figure~\ref{fig:correlation_noise}.
\begin{figure}[t!]
\centering
\resizebox{17cm}{10cm}
{\includegraphics{CorrelationNoiseN.eps}}
\caption{
The absolute value of the correlation of 1000 system samples of
simulated induced TTV with the stellar photometric slopes, for
different noise levels and different number of points. The
points are the median of each sample and the error bars are the
sample r.m.s. See text for details.
}
\label{fig:correlation_noise}
\end{figure}
The expected value of the correlation depends on the noise level.
It goes from 0.9 for no noise down to 0.3 for a SNR of unity.
The $1\sigma$ spread of the correlation depends on the noise
level and the number of points. It goes from 0.13 for $N=50$ and
SNR of unity down to 0.02 for
$N=1000$ and no noise. The figure suggests that we can easily
detect the correlation with SNR of unity, if we can measure on
the order of 500 TTVs and their corresponding photometric slopes.
\section{Discussion}
\label{sec:discussion}
We presented here a simple approach that can, in a few cases, use
the
derived TTVs of a transiting planet to distinguish between a
prograde and
a retrograde planetary motion with respect to the stellar
rotation, assuming the TTVs are induced by spot-crossing events.
Using a simplistic analytical approximation we showed that those
TTVs might
have negative (positive) correlation with the local stellar
photometric slopes at the transit
timings for prograde (retrograde) motion. We have shown that the
correlation might be detected for different stellar limb
darkening and different impact parameters. Furthermore, we
obtained similar correlated TTVs when we used the SOAP-T tool to
simulate transit light curves and derive the corresponding TTVs.
We have shown also that even if we include certain amount of
noise, the correlation is still detectable.
Can such a correlation surface above the observational noise?
The expected amplitude of the TTV can be estimated by
Equations~(\ref{eq_factor2}) and (\ref{eq_timing_max}). For
example,
a system with $R_{spot}\simeq R_{pl}\simeq 0.1 R_*$,
$(1-\alpha)/4\simeq 0.25$ and transit duration of 3 h
should show an induced TTV on the order of 5 min. So, we can
expect to observe the (anti-)correlation between the TTVs and the
photometric slopes only for systems with high enough
signal-to-noise ratio that allows timing precision of the order
of 5 min or better.
Note that for a 3 d transiting
planet in the {\it Kepler}\ field we have at hand data for up to about
400 transits,
enabling us to detect a correlation even if the noise is
comparable with the signal.
Obviously, the approximation and simulation presented here are
quite
simplistic.
First, spotted stars probably have more than one spot. The spot
eclipsed by the planet might not be the one dominating
the stellar flux modulation, and hence the local photometric
slope at the time of transit might be very different from the
expressions we developed here.
Note, however, that in our simulation we allowed an error of the
photometric slope that
can be as large as the slope itself, and showed that even in such
a case the correlation still can be detected.
Second, spots have different stellar latitudes, so some transits
might not have induced TTVs at all, contaminating the expected
correlation. To deal with this problem one might consider the
correlation of only the highly significant TTVs, which could show
the signal better.
Third, the system
obliquity can be very different from $0^{\circ}$ or
$180^{\circ}$, although most of the planets around cool stars,
with a temperature below about 6000\,K,
apparently are aligned with the stellar rotation
\citep{albrecht12,mazeh15}.
We have shown that for systems with non-vanishing obliquity and
null impact parameter the shape of the dependence of the TTV on
the photometric slope is the same, although the obliquity might
decrease or even eliminate the correlation, because many
transits might not include a spot-crossing event at all.
Note, however, that even for significant obliquity the
correlation might still exist, assuming there will be enough
induced TTVs, probably caused by spots with different latitudes.
Here again one might ignore the non-significant TTVs when
searching for a correlation.
Fourth, the observed transiting
system might have additional planets that induce dynamical TTVs,
completely shadowing the TTVs caused by spot crossing events.
Despite all these obstacles, the correlation studied here might
be solid enough
to show up for a few KOIs.
Although our method cannot give an accurate spin-orbit angle, but
can instead only indicate the sign of the orientation of the
planetary motion, the method might be useful nevertheless, as it
uses {\it Kepler~} long-cadence data that is publicly available
for all transiting planets. In the next paper (Holczer et
al., in preparation) we report on a search for
correlation between the available TTVs and the corresponding
local photometric slopes at the transit timings for all {\it
Kepler} KOIs, and indeed find five convincing cases with
significant correlations.
The approach described here can in principle be applied to any
eclipsing system, whether it is a transiting planet or a stellar
binary. For a binary system, the induced
observed minus calculated (O-C) eclipse timings can be estimated
with the small-spot approximation, for which the planetary radius
is that of the secondary. We therefore expect the TTVs to be on
the same order of magnitude as for transiting planets.
However, as eclipses in binaries are usually deeper and longer
than the planetary
transits, we expect the O-Cs in eclipsing binaries to be more
precise.
In fact, a negative correlation between the O-Cs and the local
photometric slopes was identified already for the stellar
eclipsing binary
in the Kepler-47 circumbinary planet system \citep{orosz12}.
The authors detected O-C on the order of 1 min in the timing of
the primary eclipse, and used the derived linear trend to correct
the eclipse timings. The detection of a negative
correlation for Kepler-47 is consistent with a more detailed
analysis of the spot-crossing events, also done by
\cite{orosz12}, which indicates a prograde motion.
The method presented here
can be applied in the future to a large sample of systems
monitored by
current and
future space missions, like K2 \citep{howell14},
TESS \citep{ricker14}, and PLATO \citep{rauer14}, helping
discovering, without additional observations, interesting systems
that are worth following, and possibly find what are the
conditions for alignment or misalignment of stellar rotations and
orbital
motions of planets and stellar binaries.
\acknowledgments
We are grateful to the referee for very helpful comments that
helped us substantially improve the paper. We are thankful to the
authors of the SOAP-T tool that made it publicly available.
The research leading to these results has received funding from
the European Research Council under the EU's Seventh Framework
Programme (FP7/(2007-2013)/ ERC Grant Agreement No.~291352).
T.M. also acknowledges support from
the Israel Science Foundation (grant No.\,1423/11) and the
Israeli Centers of Research Excellence (I-CORE, grant
No.\,1829/12).
T.M. is grateful to the Jesus Serra Foundation Guest Program and
to Hans Deeg and Rafaelo Rebolo, that enabled his visit to the
Instituto de Astrof\'isica de Canarias, where the last stage of this research was completed.
This work was performed in part
at the Jet Propulsion Laboratory, under contract with
the California Institute of Technology (Caltech) funded
by NASA through the Sagan Fellowship Program executed
by the NASA Exoplanet Science Institute.
|
1,108,101,565,651 | arxiv | \section{introduction}
The exploration of the connection between statistical mechanics and
quantum information has been extensive in recent years since the
work \cite{preskill}. Especially the research of entanglement in
many-body systems has contributed to the comprehensive crossover
between the two hot areas \cite{afov07}. Furthermore, the finding of
integer or fractional quantum Hall effect in two-dimensional
many-body systems imposes a challenge on the universal understanding
of phase transitions, since the traditional theory for phase
transition cannot incorporate these novel phenomena \cite{senthil}.
Recently the research of quantum entanglement in two-dimensional
many-body systems provides the clear characterization for different
quantum orders \cite{kitaev}. These facts suggest that quantum
entanglement would play a vital role in the understanding of
many-body effects.
Bipartite entanglement was first studied, and focused on the
connection to the criticality in spin-chain systems\cite{oaff02,
on02}. This interest comes from the fact that quantum phase
transition is related to the construction of the long-range
correlations in many-body systems. Hence it is a natural conjecture
that quantum entanglement, as a depiction of the non-local
correlation, could detect the appearance of long-range correlation.
Great progress has been made for the block entanglement in many-body
systems; the area law of block entanglement entropy has been
generally constructed by the conformal field theory. Furthermore the
violation of the area law has been identified as a reliable
detection of quantum phase transitions in one-dimensional
systems(see Ref. \cite{afov07} for a comprehensive review). However
the situation becomes complex for high-dimensional systems: the
violation of the area law in one-dimensional case when the system is
critical, does not seem to hold in higher dimensions\cite{afov07}.
Even for pairwise entanglement in many-body systems the results are
not satisfying . For example, the cutoff in the definition of
concurrence may lead to unphysical results when one focuses on the
connection of entanglement and phase transition in many-body systems
\cite{yang}.
The situation becomes more complex for multiparticle
entanglement(ME) because of the absence of unified measurement for
ME\cite{pv07}. However, it is a natural speculation that ME should
play a more fundamental role than a bipartite one for the
understanding of many-body effects with consideration of the
universal interaction in many-body systems. Recently the discussions
of ME in many-body systems have been given more attention because of
the availability of some special entangled states, e.g., cluster
states for one-way quantum computation\cite{br01}, $n$-party
Greenberger-Horne-Zeilinger(GHZ) state and $W$-state\cite{dvc00}.
Although great effort has been devoted to the measurement of ME, the
analytical or operational measurements exist only for some special
cases\cite{pv07}. The connection of quantum phase transition and ME
has also been discussed extensively in \cite{wei05,cffp07,
oliveira06}. However these discussions are mainly on the
one-dimensional spin-1/2 $XY$ model, and the difficulty of
calculating ME obstructs further exploration.
Recently, the global entanglement has been constructed for the
quantification of ME by Meyer and Wallach\cite{mw02}, which
possesses the virtues of the availability of analytical expression
and operability. Moreover global entanglement is measurable
experimentally since it is directly related to the mixedness of
single party\cite{brennen03}. Another important character of global
entanglement is the monotonicity under local operations and
classical communication (LOCC), if one notes that global
entanglement is intimately related to the linear
entropy\cite{horodecki07}. Consequently, Oliveira and his
collaborators improved this definition for measuring some special
entangled states, e.g. $n$-party $W$ state or GHZ
state\cite{oliveira06}. Moreover the connection of the generalized
global entanglement and quantum phase transition has also been
explored in the one-dimensional spin-1/2 $XY$ model, in which
entanglement reaches the maximum near to the transition
point\cite{oliveira06}.
It is interesting to note that the nearest neighbor coupling is
beneficial to the formation of ME in the one-dimensional spin-1/2
$XY$ model. Since the particle correlation is short-ranged in this
model\cite{on02}, one should note that the maximum of global
entanglement maybe come from the distribution of pairwise
entanglement\cite{cffp07, facchi}. Hence it is tempting to present a
discussion about ME when the correlation is long range and the
coupling is beyond the nearest neighbor case. Fortunately the
Lipkin-Meshkov-Glick(LMG) model\cite{lmg} provides us the benchmark
for exploring this point since the collective interaction in this
model. Then it is expected that ME would play a critical role.
Recently the entanglement in the LMG model has been extensively
studied, such as concurrence \cite{vidal, dv04, vmd04},
one-tangle\cite{vpa04}, entanglement entropy\cite{uv, lp05,vdb07}
and generalized entanglement\cite{somma}. The concurrence in the LMG
model displays sensitivity to the appearance of quantum phase
transition\cite{vidal, vmd04}, except for some special
cases\cite{vmd04}. A possible explanation of this discrepancy is
that the trace operation performed in the calculation of concurrence
inevitably kills some correlations between spins\cite{vmd04}. With
respect to this point, Barnum and his collaborators constructed a
subsystem-independent measure of entanglement, based on a
distinguished subspace of observables for the system\cite{bkosv04}.
The named \emph{generalized entanglement} introduced by Barnum,
\textit{et.al.} has also been discussed in the LMG model, which
displayed the ability of detecting the phase
transitions\cite{somma}. Moreover the authors show the equivalency
between generalized entanglement and the global entanglement defined
by Meyer and Wallach\cite{mw02}. However, as shown in \cite{somma},
it is indispensable for the construction of the distinguished
subspace of observables to obtain the knowledge of the ground state
in many-body systems, that in most cases is very difficult. The
research of entanglement entropy in the LMG model shows that the
entropy was divergent under thermodynamic limit near the phase
transition point, and moreover shows a discontinuity at the critical
point for the isotropic coupling case\cite{vdb07}.
The generalized global entanglement (gGE), defined by Oliveira,
\textit{et. al.}\cite{oliveira06}, is a generalization of the global
entanglement (GE). With respect to the equivalence between Barnum's
generalized entanglement and GE, gGE provides a universal
characterization of ME in many-body systems. Hence, it is
interesting to present a comprehensive research of gGE and GE in the
LMG model. Our discussion also presents detailed research for
antiferromagnetic coupling and some interesting results can be
obtained, which is rarely touched on in the previous works. I should
point out that the goal for this paper focuses on the connection
between ME, measured by gGE and GE respectively, and quantum phase
transition in the LMG model. For this purpose, the paper is
organized as following. In Sec.II the Hamiltonian is presented, and
ground states are determined analytically. The phase diagram will be
identified by introducing the proper parameter. In Sec.III the
analytical expressions for gGE and GE are presented. Based on these
formulas the multiparticle entanglements for ferro-magnetic and
antiferro-magnetic couplings are discussed respectively. The
conclusions and discussions are given in Sec. IV.
\section{Hamiltonian and ground state}
The LMG model describes a set of spin-half particles coupled to all
others with an interaction independent of the position and the
nature of the elements. The Hamiltonian can be written as
\begin{equation}\label{h}
H= - \frac{\lambda}{N}(S^2_x + \gamma S^2_y) - h_z S_z,
\end{equation}
in which $S_{\alpha}=\sum_{i=1}^{N}\sigma^i_{\alpha}/2 (\alpha=x, y,
z)$ and the $\sigma_{\alpha}$ is the Pauli operator, and $N$ is the
total particle number in this system. The prefactor $1/N$ is
essential to ensure the convergence of the free energy per spin in
the thermodynamic limit. Anti-ferromagnetic or ferromagnetic
interaction can be obtained dependent on $\lambda<0$ or
not($\lambda\neq0$). The Hamiltonian preserves the total spin and
does not couple the state having spin pointing in the direction
perpendicular to the field, i.e.
\begin{equation}\label{s}
[H, \textbf{S}^2]=0, [H, \prod_{i=1}^{N}\sigma_z^i]=0.
\end{equation}
For isotropic coupling $\gamma=1$, $[H, S_z]=0$ and the spectrum of
Eq.\eqref{h} can be determined exactly. However for $\gamma\neq 1$,
the spectrum can be determined in principle by Bethe-type
equations\cite{plo} and the analytical expressions are difficult to
obtain.
A distinguished character of Eq. \eqref{h} is the collective
interaction, which is the same for any particle and independent of
the space configuration of the system. Because of long-range
correlation between particles, the mean-field analysis is adaptive
for this model\cite{botet}. The research of phase transition in the
LMG model has shown that there is a second-order transition at
$h=h_z/|\lambda|=1$ for the ferromagnetic case and a first-order one
at $h=0$ for the antiferromagnetic case\cite{botet, vidal}.
A proper parameter for characterizing the phase diagram is the total
spin in the direction $z$ for the ground state. For ferromagnetic
coupling, one has
\begin{eqnarray}
1-2\langle S_z\rangle/N=\begin{cases}0, &h>1\\1-h, &h\in[0,
1),\end{cases}
\end{eqnarray}
which corresponds to the disorder-order transition, and obviously
the point $h=1$ is a second-order phase transition point. This phase
transition could be attributed to the disappearance of the energy
gap; at the symmetric phase $h>1$ the energy gap above the ground
state is finite, whereas at the broken phase $0\leq h<1$ the energy
gap vanishes under thermodynamic limit\cite{dv04}. For
antiferromagnetic coupling,
\begin{eqnarray} 2\langle
S_z\rangle/N=\begin{cases}1, &h>0\\-1, &h<0;\end{cases}
\end{eqnarray}
Obviously there is a first-order phase transition at the point
$h=0$. For this case the energy gap above ground state vanishes only
at the transition point $h=0$ under thermodynamic limit, and no
level crossing happens when $h\neq0$\cite{vmd04}.
The ground state for $\gamma\neq1$ can be determined analytically
with the help of Holstein-Primakoff(HP) transformation and
low-energy approximation\cite{dv04}. In Ref. \cite{cui06}, the
ground state has been obtained with the consideration of the finite
number effect. The general expression reads
\begin{eqnarray}
\label{g}
\ket{g}&=&\frac{1}{c}\sum_{n=0}^{[N/2]}(-1)^n\sqrt{\frac{(2n-1)!!}{2n!!}}\tanh^nx\ket{2n}\nonumber\\
c^2&=&\sum_{n=0}^{[N/2]}(-1)^n\frac{(2n-1)!!}{2n!!}\tanh^{2n}x
\end{eqnarray}
in which $\ket{2n}$ is the Fock state of the boson operator
introduced by Holstein-Primakoff transformation and $[N/2]$ denotes
the integer part not more than $N/2$. One should note that the
determination of the ground state Eq. \eqref{g} is based on HP
transformation, which preserves the symmetry Eq. \eqref{s}, and the
following discussion is heavily based on this ground state.
Dependent on the style of interaction, $\tanh \_ x$ has different
expressions. For ferromagnetic case $\lambda>0$, it satisfies the
relation\cite{vidal}
\begin{eqnarray}\label{ft}
\label{ferro} \tanh 2x=\begin{cases}-\frac{1-\gamma}{2h-1-\gamma},&
h>1\\-\frac{h^2-\gamma}{2-h^2-\gamma}, & 0\leq h<1\end{cases}.
\end{eqnarray}
For antiferromagnetic coupling $\lambda<0$, it is determined by
\begin{eqnarray}\label{aft}
\tanh\_2x=\frac{1-\gamma}{1+\gamma+2|h|}.
\end{eqnarray}
For isotropic case $\gamma=1$, the calculation is exact. The ground
state can be formulated generally as $\ket{g}=\ket{S=\frac{N}{2},
S_z=M}$. For ferromagnetic coupling,
\begin{eqnarray}\label{gf}
M=\begin{cases}I[h N/2],& 0\leq h<1\\ \frac{N}{2}, & h\geq1.
\end{cases}
\end{eqnarray}
in which $I[n]$ expresses the integer not more than $n$. For
antiferromagnetic coupling ,
\begin{eqnarray}\label{ga}
M=\begin{cases}\frac{N}{2},& h>0\\ -\frac{N}{2}, & h<0.
\end{cases}
\end{eqnarray}
\section{Multiparticle Entanglement}
Recently, Meyer and Wallach have constructed the global entanglement
for measuring ME in spin systems. The main procedure is to first
measure the entanglement between any party and the others, and then
calculate the average of all possible bipartition\cite{mw02}.
Although the criticism that it is not a genuine ME measure because
of the intimate connection to bipartite entanglement
\cite{horodecki07}, it has been proven that the global entanglement
is operational and more importantly, monotonic under LOCC. A
simplified expressions for global entanglement is provided by
Brennen \cite{brennen03}
\begin{equation}
Q(\ket{\phi})=2(1-\frac{1}{N}\sum_{k=0}^{N-1}\text{Tr}[\rho_k^2]).
\end{equation}
For the LMG model, one can obtain
$Q(\ket{g})=2(1-\text{Tr}\rho_1^2)$, in which $\rho_1$ stands for
the single-particle reduced density operator. Furthermore, Oliveira
and his collaborators have improved this definition in order that it
can measure some special entangle states, e.g. $\otimes_n
\ket{EPR}_n$ or $n$-party GHZ state. The main procedure is to
measure the entanglement between any two parties and the others, and
then average all possible bipartition\cite{oliveira06}. In the LMG
model, for the symmetry of particle permutation, the generalized
global entanglement can be written as
\begin{equation}
E_g=\frac{4}{3}(1- \text{Tr}[\rho^2_{2}]).
\end{equation}
in which $\rho_{2}$ denotes the reduced density operator for any two
particles. $\rho_1,\rho_{2}$ can be determined through the
correlation functions\cite{wm02}
\begin{eqnarray}\label{c}
\langle\sigma_{\alpha}\rangle&=&\frac{2}{N}\langle
S_{\alpha}\rangle,
\nonumber\\
\langle\sigma_{1\alpha}\sigma_{2\alpha}\rangle&=&\frac{4\langle
S^2_{\alpha}\rangle-N}{N(N-1)}\nonumber\\
\langle\sigma_{1\alpha}\sigma_{2\beta}\rangle&=&\frac{2\langle
[S_{\alpha}, S_{\beta}]_+\rangle-N}{N(N-1)}(\alpha\neq\beta)
\end{eqnarray}
in which $\alpha, \beta = x, y, z$.
With respect to the symmetry Eq. \eqref{s} and the ground state Eq.
\eqref{g}, one can obtain GE and gGE respectively
\begin{eqnarray}\label{me}
&Q(\ket{g})=1 - \langle\sigma_z\rangle^2\nonumber \\ &E_g=1 -
\frac{1}{3}(2\langle\sigma_z\rangle^2+\langle\sigma_{1x}\sigma_{2x}\rangle^2
+\langle\sigma_{1y}\sigma_{2y}\rangle^2+\langle\sigma_{1z}\sigma_{2z}\rangle^2).
\end{eqnarray}
Based on Eqs. \eqref{g} and \eqref{c}, ME in the LMG model can be
decided analytically, and some interesting properties can be found.
The discussion below is divided into two cases: one focuses on the
anisotropic coupling, and the other is for isotropic coupling for
which the exact results can be obtained.
\subsection{anisotropic coupling}
The analytical results can be obtained under large $N$ limit with
the hypothesis that the excitation would only happen for the low
energy states\cite{vidal}. Based on Eqs. \eqref{g} and \eqref{c},
one obtains
\begin{widetext}
\begin{eqnarray}
\langle\sigma_z\rangle&=&1-\frac{4}{Nc^2}\sum_{n=0}^{[N/2]}nc_{2n}^2\nonumber\\
\langle\sigma_{1x}\sigma_{2x}\rangle&=&\frac{2}{N(N-1)c^2}\sum_{n=0}^{[N/2]}
[\sqrt{(N-2n+2)(N-2n+1)2n(2n-1)}c_{2n-2}c_{2n}+2n(N-2n)c^2_{2n}]\nonumber\\
\langle\sigma_{1y}\sigma_{2y}\rangle&=&\frac{-
2}{N(N-1)c^2}\sum_{n=0}^{[N/2]}
[\sqrt{(N-2n+2)(N-2n+1)2n(2n-1)}c_{2n-2}c_{2n}-2n(N-2n)c^2_{2n}]\nonumber\\
\langle\sigma_{1z}\sigma_{2z}\rangle&=&1-\frac{4}{N(N-1)c^2}\sum_{n=0}^{[N/2]}2n(n-2n)c^2_{2n}
\end{eqnarray}
\end{widetext}
in which $c_{2n}=(-1)^n\sqrt{\frac{(2n-1)!!}{2n!!}}\tanh^n\_x$ and
$\tanh x$ is decided by Eqs.\eqref{ft} and \eqref{aft}. From Eq.
\eqref{me}, ME can be determined analytically.
\begin{figure}[tb]
\includegraphics{1a}
\includegraphics{1b}
\caption{\label{f1}(Color online) The multiparticle entanglement for
ferromagnetic coupling, measured by gGE (denoted by $E_g$ and solid
lines) and GE (denoted by $Q(\ket{g})$ and dashed lines) vs the
rescaled magnetic field $h$. We have chosen $\gamma=0.5$(a) and
$\gamma=0$(b) for this plot. The green and black solid lines
correspond to $N=50, 500$ respectively. }
\end{figure}
\begin{figure}[tb]
\includegraphics[bbllx=14, bblly=15, bburx=265, bbury=218, width=7cm]{2}
\caption{\label{f2} (Color online) $E_g$ (black triangles) and
$Q(\ket{g})$ (green triangles) for ferromagnetic coupling vs. the
particle number $N$ at the phase transition point $h=1$. One should
note that $\tanh 2x$ for any $\gamma\in[0, 1)$ is identical in this
case. }
\end{figure}
\emph{-Ferromagnetic case-} gGE and GE have both been plotted for
different $\gamma$ in Fig.\ref{f1}. It is obvious that ME reaches
the maximum closed to phase transition point $h=1$ and the slope of
curves tends to be infinite. In recent papers\cite{oliveira06}, the
authors have shown that the singularity of gGE is directly connected
to the degeneracy of the ground-state energy at the phase transition
point. Since the energy gap above ground state vanishes at critical
point $h=1$, the singularity of GE and gGE can be attributed to the
degeneracy of ground-state energy, and could be used as a reliable
detector for the phase transition in this case. Furthermore the
finite-size scaling at the phase transition point $h=1$ displays the
non-sensitivity both of gGE and GE to the particle number $N$, as
shown in Fig.\ref{f2}.
\begin{figure}[tb]
\includegraphics{3a}
\\[0.5cm]
\includegraphics{3b}
\caption{\label{f3}(Color online) $E_g$ (solid lines) and
$Q(\ket{g})$ (dashed lines) for antiferromagnetic coupling vs. the
rescaled magnetic field $h$. The parameter $\gamma=0.5$(a) and
$\gamma=0$(b) have been chosen for this plot. Since the system is
invariant under the changing of $h\leftrightarrow -h$, these
plottings are only for $h\ge0$ with N=50 (green lines) and N=200
(black lines) respectively.}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=8cm]{4}
\caption{\label{f4} (Color online) $E_g$ and $Q(\ket{g})$ for
antiferromagnetic coupling vs. the particle number $N$ at
first-order phase transition point $h=0$. We have choosen three
representative values of $\gamma$ for this plot.}
\end{figure}
\emph{-Antiferromagnetic case-} The situation is intricate. We have
plotted GE and gGE vs. $h$ by choosing $\gamma=0$ (a) and
$\gamma=1/2$ (b) respectively, in Fig.\ref{f3}. Since there is a
first-order quantum phase transition, the figures show that GE and
gGE both are maximum at transition point $h=0$. However a further
calculation shows two different behaviors for ME; For
$\gamma\in(0,1)$ GE and gGE show a cusp at phase transition point
$h=0$, which means that the first derivation of gGE and GE with
respect to $h$ is discontinued but finite at $h=0$. Since the
degeneracy of ground-state energy happens only at $h=0$, this
discontinuity of gGE and GE is attributed to the degeneracy of
ground-state energy, whereas for $\gamma=0$, the figure shows that
gGE and GE both have a drastic increase closed to the phase
transition point, and the first derivations of gGE and GE with $h$
tend to be divergent. Furthermore our calculation shows that with
the increasing of particle number, gGE and GE decrease for
$\gamma\in(0,1)$ at the transition point, as shown in Fig. \ref{f4}.
Moreover, GE and gGE have similar behaviors and the difference
between them is slight.
\subsection{isotropic coupling}
When $\gamma=1$, the exact results can be obtained. With respect to
Eqs. \eqref{gf}, \eqref{ga} and \eqref{c}, one has in this case
\begin{eqnarray}\label{me2}
E_g&=&1-
\frac{8M^2}{3N^2}-\frac{2(4M^2-N)^2+(4M^2-N^2)^2}{6N^2(N-1)^2}\nonumber\\
Q(\ket{g})&=&1- (\frac{2M}{N})^2.
\end{eqnarray}
\begin{figure}[tb]
\includegraphics{5}
\caption{\label{f5} $E_g$ (solid line) and $Q(\ket{g})$ (dashed
line) vs $h$ for isotropic ferromagnetic coupling with $N\rightarrow
\infty$.}
\end{figure}
\emph{-Ferromagnetic case-} With respect to Eqs. \eqref{gf} and
\eqref{me2}, both gGE and GE are zero for $h\ge1$, independent of
the particle number $N$ since the ground state is a direct-product
state of $N$ particles with the same spin orientation from Eq.
\eqref{gf}. As shown in Fig. \ref{f5}, gGE and GE both are continued
under thermodynamic limit at the phase transition $h=1$. This
behavior is different from the conclusion made in Refs. \cite{lp05,
vdb07, dv04}, in which entanglement entropy and concurrence both are
discontinued at $h=1$ under thermodynamic limit. The main reason for
this discrepancy is stated below. One notes that Eqs.\eqref{me2}
are a function of $M/N$. It is obvious from Eq. \eqref{gf} that
$M/N$ is continuous at the transition point $h=1$ under
thermodynamic limit. Hence, it is not strange that gGE and GE are
also continuous. Comparably the concurrence in \cite{dv04} is
redefined by adding a prefactor $N-1$ to keep finite under
thermodynamic limit. Similarly for the calculation of entanglement
entropy, all possible bipartition has to be considered in the
calculation to keep the entropy finite under thermodynamic
limit\cite{lp05}, which I point out plays the same function of the
prefactor $N-1$ for the calculation of concurrence. Under the
thermodynamic limit, the prefactor would play a nontrivial role.
Since the calculation for entanglement has been implemented
respectively in different regions because of Eq. \eqref{gf}, the
difference, which should disappear under thermodynamic limit, may
become finite because of this prefactor. While, since our definition
of ground state Eq.\eqref{g} has naturally considered the
finite-number effect and GE and gGE are the functions of the
correlations, one does not need this prefactor to keep the
measurements of entanglement finite under thermodynamic limit.
Together with respect that the equivalency between generalized
entanglement and GE have been proved\cite{somma}, the measure gGE
may also has the great virtue of independence on the concept of
subsystem.
\emph{-Antiferromagnetic case-} With respect to Eqs. \eqref{ga} and
\eqref{me2}, gGE and GE both vanish independently on $N$. Since the
states for $M=\pm N/2$ are degenerate at phase transition point
$h=0$, the ground state is undoubtedly the superposition of states
$\ket{N/2, N/2}$ and $\ket{N/2, -N/2}$, written on the basis of
$\{S^2, S_z\}$, with equal weight,
\begin{equation}
\ket{g}=\frac{1}{\sqrt{2}}(\ket{N/2, N/2}+\ket{N/2, -N/2}).
\end{equation}
Then in this case, a genuine maximally multiparticle entangled
state, so called n-party GHZ state\cite{dvc00}, can be obtained at
the point $h=0$ for a finite particle number, and under
thermodynamic limit, it corresponds to the celebrated (
Schr\"odinger cat ) macroscopic quantum superposition state.
Obviously, the measurement of entanglement is discontinued at $h=0$,
where a first-order phase transition happens under thermodynamic
limit\cite{footnote1}.
\section{discussions and conclusions}
Some comments and discussions should be presented. In this paper an
extensive discussion of multiparticle entanglement, measured by GE
\cite{mw02} and gGE \cite{oliveira06}, is presented in the
Lipkin-Meshkov-Glick model. Our discussion focuses on two different
situations: for ferromagnetic coupling $\lambda>0$, when the
anisotropic parameter $0<\gamma<1$ gGE and GE both reach the maximum
at the second-order phase transition point $h=1$, as shown in Figs.
\ref{f1}. Moreover they are nonsensitive to the variation of
particle number $N$, as shown in Fig.\ref{f2}. Whereas for the
isotropic case $\gamma=1$ gGE and GE are zero at the phase
transition point, shown in Fig. \ref{f5}.
Another important situation is the appearance of antiferromagnetic
coupling $\lambda<0$, for which there is a first-order phase
transition at transition point $h=0$. gGE and GE both are
calculated, as shown in Fig. \ref{f3}. It is interesting that some
different behaviors can be found in this case; one is that gGE and
GE have a cusp at phase transition point when $\gamma\in(0,1)$,
which means that the first derivation of gGE and GE with external
magnetic field is discontinued but finite at $h=0$, shown in Fig.
\ref{f3}(a). Another case happens when $\gamma=0$, in which both gGE
and GE have a drastic changing closed to $h=0$, shown in Fig.
\ref{f3}(b). Moreover our calculations show that gGE and GE for
$\gamma\in(0,1)$ decrease with the increment of $N$ as shown in
Fig.\ref{f4}, whereas for $\gamma=0$ they are non-sensitive to the
particle number $N$. It is more interesting for the isotropic case
that the entanglement is vanishing for $h\neq0$ and has a sudden
changing at $h=0$, where a genuine maximally multiparticle
entanglement state, $n$-party GHZ state for finite particle number,
can be obtained. With connection of a scheme of the realization of
the LMG model in optical cavity QED\cite{mp07}, this result provides
a powerful method to create ME experimentally.
It was naturally expected that ME should be maximum at the phase
transition point since the correlation between particles would be
long-range because of the appearance of critical quantum fluctuation
at the phase transition point. However an exceptional case appears
in our discussion, which happens for ferromagnetic and isotropic
coupling. In my own opinion, a reason for the difficulty in
constructing the connection between entanglement and quantum phase
transition is that the up-to-date measurements for entanglement are
generally a nonlinear function of correlation functions in many-body
systems. Hence the singularity of correlation functions may be
canceled\cite{footnote2}. As a concrete illustration one notes that
for antiferromagnetic coupling, $M$ has a discontinued change at
$h=0$ for $\gamma=1$, as shown in Eq.\eqref{ga}. However from
Eq.\eqref{me2} it is obvious that the discontinuity of $M$ for $h>0$
and $h<0$ has no effect on ME since gGE and GE are the functions of
$M^2$.
Regardless of this defect, some interesting information can be
obtained from the research of ME. An interesting speculation from
our discussion is that the different finite-size scales may show the
different state structures for the entanglement at the phase
transition point. As shown previously in Ref. \cite{oliveira06},
with the increment of particle number the measures for entanglement
for some states are decreasing, whereas for other states tend to be
steady values. Since the increment of particle number, or more
generally the degree of freedom, means the stronger correlation
between the particles in many-body systems, one can conclude that
some entangled states are immune to the effect imposed by the
increment of correlation between particles, whereas others are
sensitive to the changing of correlation. With respect to the
pursuit of decoherence-free space\cite{dfs}, our discussion may
provide some useful information.
With connection to the researches of the bipartite entanglement in
many-body systems, one could note that it is difficult to construct
a universal classification of phase transition based on the
entanglement and its derivation. The main obstacle, in my own
opinion, is the absence of physical definition of entanglement,
i.e., how and what to define an "entanglement operator". Since
quantum entanglement is an important physical resource and can be
measured experimentally, I believe in the existence of this
operator. Fortunately a few works have attributed to this
direction\cite{somma, cv06}. I hope our discussion will help in the
understanding of entanglement in many-body systems.
\emph{Acknowledgement} The author acknowledges the support of
Special Foundation of Theoretical Physics of NSF in China, Grant No.
10747159.
|
1,108,101,565,652 | arxiv | \section{Introduction}
The most well-known theory for describing the mechanism behind
superconductivity from microscopic perspective is the BCS theory proposed by
Bardeen, Cooper and Schrieffer. According to BCS theory, the condensation of
Cooper pairs into a boson-like state, at low temperature, is responsible for
infinite conductivity in solid state system \cite{BCS57}. However, when the
temperature increases, the Cooper pairs decouples and thus the BCS theory is
unable to explain the mechanism of superconductivity for high temperature
superconductors \cite{BCS57}. The correspondence between gravity in an Anti
de-Sitter (AdS) spacetime and a Conformal Field Theory (CFT) living on the
boundary of the spacetime provides a powerful tool for calculating
correlation functions in a strongly interacting field theory using a dual
classical gravity description \cite{Maldacena}. According to the AdS/CFT
duality proposal an $n$-dimensional conformal field theory on the boundary
is equivalent to gravity theory in $(n+1)$-dimensional AdS bulk \cite%
{Maldacena,G98,W98,H08,HR08,R10}. The dictionary of AdS/CFT duality implies
that each quantity in the bulk has a dual on the boundary. For example,
energy-momentum tensor $T_{\mu \nu }$ on the boundary corresponds to the
bulk metric $g_{\mu \nu }$\cite{G98,W98}. Based on this duality, Hartnoll et
al. proposed a model for holographic superconductor in $2008$ \cite{H08}.
Their motivation was to shed light on the problem of high temperature
superconductors. According to the holographic superconductors, we need a
hairy black hole in gravity side to describe a superconductor on its
boundary. During the past decade, the investigation on the holographic
superconductor has got a lot of attentions (see e.g. \cite%
{HR08,R10,H09,Hg09,H11,Gu09,HHH08,JCH10,SSh16,SH16,cai15,Ge10,SHsh(17),CAI11,
SHSH(16), shSh(16),Doa, Afsoon,
cai10,cai14,yao13,n3,n4,n5,n6,Gan1}).
On the other hand, BTZ (Bandos-Teitelboim-Zanelli) black holes, the
well-known solutions of general relativity in $(2+1)$-dimensional spacetime,
provide a simplified model to investigate some conceptual issues in black
hole thermodynamics, quantum gravity, string theory, gauge theory and
AdS/CFT correspondance \cite{Car1,Ash,Sar,Wit1,Car2}. It has been shown that
the quasinormal modes in this spacetime coincide with the poles of the
correlation function in the dual CFT. This gives quantitative evidence for
AdS/CFT \cite{Bir}. In addition, BTZ black holes play a crucial role for
improving our perception of gravitational interaction in low dimensional
spacetimes \cite{Wit2}. These kind solutions have been studied from
different point of views \cite{rin1,rin2,rin3,rin4}.
Holographic superconductors dual to asymptotic BTZ black holes
have been explored widely (see e.g. \cite{Wang,chaturvedi,L12,momeni,peng17,lashkari,hua,yanyan,yan,alkac,50-1,bina}%
). In order to construct the $(1+1)$-dimensional holographic superconductors
one should employ the $AdS_{3}/CFT_{2}$ correspondence. In \cite{chaturvedi}%
, the $(1+1)$-dimensional holographic superconductors were explored in the
probe limit and its distinctive features in both normal and superconducting
phases were investigated. Employing the variational method of the
Sturm-Liouville eigenvalue problem, the one-dimensional holographic
superconductors have been analytically studied in \cite{L12,momeni,peng17}.
It is also interesting to study the $(1+1)$-dimensional holographic
superconductor away from the probe limit by considering the backreaction. In
\cite{Wang}, the effects of backreaction have been studied for $s$-wave
linearly charged one-dimensional holographic superconductors.
Holographic superconductors have also been studied extensively in
the presence of nonlinear electrodynamics (see e.g.
\cite{n4,SH16,SSh16,SHsh(17),SHSH(16),shSh(16),n3,n5,n6}). The
most famous nonlinear electrodynamic is Born-Infeld
electrodynamic. This model was presented for the first time to
solve the problem of divergence of electrical field at the
position of point particle \cite{25,26,27,28,29}. It was later
showed that this model could be reproduced by string theory. In
the present work, we would like to extend the investigation on the $(1+1)$%
-dimensional holographic superconductors by taking into account
the nonlinear Born-Infeld (BI) electrodynamics, as our gauge
field. As well, we will study the effects of backreaction on our
holographic superconductors. We perform our investigation both
analytically and numerically and shall compare the result of two
methods. Our analytical approach is based on the Sturm-Liouville
variational method. In latter study, we find the relation between
critical temperature and chemical potential. Moreover, in order to
study our holographic superconductors numerically, we use the
shooting method. We show that analytical results are in good
agreement with numerical ones which implies that the
Sturm-Liouvile variation method is still powerful enough for
studying the $(1+1)$-dimensional holographic superconductor.
The structure of our paper is as follows. In section \ref{sec2}, the basic
field equations of one-dimensional holographic superconductors with
backreaction in the presence of BI nonlinear electrodynamics is introduced.
In section \ref{sec3}, we describe the procedure of analytical study of one
dimensional holographic superconductor based on Sturm-Liouvile method and
obtain the relation between critical temperature and chemical potential. In
section \ref{sec4}, the numerical approach toward the study of our
holographic superconductors will be presented. Finally, we summarize our
results in section \ref{sec5}.
\section{Basic field equations\label{sec2}}
The action of three dimensional AdS gravity in the presence of a gauge and a
scalar field is given by%
\begin{eqnarray}
S &=&\frac{1}{2\kappa ^{2}}\int d^{3}x\sqrt{-g}\left( R+\frac{2}{l^{2}}%
\right) \notag \\
&&+\int d^{3}x\sqrt{-g}\left[ \mathcal{L}(\mathcal{F})-|\nabla \psi -iqA\psi
|^{2}-m^{2}|\psi |^{2}\right] , \notag \\
&& \label{act}
\end{eqnarray}%
where $m$ and $q$ shows the mass and the charge of scalar field, $\kappa
^{2}={8\pi G_{3}}$ and $G_{3}$ is $3$-dimensional Newton gravitation
constant. Also, $g$, $R$ and $l$ are the metric determinant, Ricci scalar
and AdS radius, respectively. In (\ref{act}), $\mathcal{F}=F_{\mu \nu
}F^{\mu \nu }$ where $F_{\mu \nu }=\nabla _{\lbrack \mu }A_{\nu ]}$ is the
electrodynamics field tensor and $A_{\mu }$ is the vector potential. $%
\mathcal{L}(\mathcal{F})$ stands for the Lagrangian density of BI nonlinear
electrodynamics defined as%
\begin{equation}
\mathcal{L}(\mathcal{F})=\frac{1}{b}\left( 1-\sqrt{1+\frac{b\mathcal{F}}{2}}%
\right) ,
\end{equation}%
where $b$ is the nonlinear parameter. When $b\rightarrow 0$, $\mathcal{L}$
reduces to $-F_{\mu \nu }F^{\mu \nu }/{4}$ which is the standard Maxwell
Lagrangian \cite{H08}. Variation of the above action with respect to scalar
field $\psi $, gauge field $A_{\mu }$ and the metric $g_{\mu \nu }$ yield
the following equations of motion%
\begin{eqnarray}
0 &=&\left( \nabla _{\mu }-i{q}A_{\mu }\right) \left( \nabla ^{\mu }-i{q}%
A^{\mu }\right) \psi -m^{2}\psi \,, \label{Epsi} \\
&& \notag \\
0 &=&\nabla ^{\mu }\left( 4\mathcal{L}_{\mathcal{F}}F_{\mu \nu }\right)
\notag \\
&&-i{q}\left[ -\psi ^{\ast }(\nabla _{\nu }-i{q}A_{\nu })\psi +\psi (\nabla
_{\nu }+i{q}A_{\nu })\psi ^{\ast }\right] \,, \label{02} \\
&& \notag \\
0 &=&\frac{1}{2\kappa ^{2}}\left[ R_{\mu \nu }-g_{\mu \nu }\left( \frac{R}{2}%
+\frac{1}{l^{2}}\right) \right] +2F_{ac}F_{b}{}^{c}\mathcal{L}_{\mathcal{F}}
\notag \\
&&-\frac{g_{\mu \nu }}{2}\left[ \mathcal{L}(\mathcal{F})-m^{2}|\psi |^{2}-{%
|\nabla \psi -i{q}A\psi |^{2}}\right] \notag \\
&&-\frac{1}{2}\left[ (\nabla _{\mu }\psi -i{q}A_{\mu }\psi )(\nabla _{\nu
}\psi ^{\ast }+i{q}A_{\nu }\psi ^{\ast })+\mu \leftrightarrow \nu \right] ,
\notag \\
&& \label{Eein}
\end{eqnarray}%
where $\mathcal{L}_{\mathcal{F}}=\partial \mathcal{L}/\partial
{\mathcal{F}}$.
Since, we would like to consider the effect of the backreaction on the
holographic superconductor, we take a metric ansatz as follows \cite{Wang}%
\begin{equation}
{ds}^{2}=-f(r)e^{-\chi (r)}{dt}^{2}+\frac{{dr}^{2}}{f(r)}+\frac{r^{2}}{l^{2}}%
{dx}^{2}. \label{metric}
\end{equation}%
The Hawking temperature of the three dimensional black hole on the
outer horizon $r_{+}$ (where $r_{+}$ is the horizon radius
obtained as the largest root of $f(r_{+})=0$), may be obtained
through the use of the general definition of surface gravity
\cite{SheyKaz}
\begin{eqnarray}\label{Tem1}
T&=&\frac{\kappa_{sg}}{2
\pi}=\frac{1}{2\pi}\sqrt{-\frac{1}{2}(\nabla_{\mu}\chi _{\nu})
(\nabla^{\mu}\chi^{\nu})},
\end{eqnarray}
where $\kappa_{sg}$ is the surface gravity and
$\chi=\partial/\partial t$ is the null killing vector of the
horizon. Taking $\chi^{\nu}=(-1,0,0)$, we have
$\chi_{\nu}=(f(r_{+}) e^{-\chi (r_{+})},0,0)$ and hence on the
horizon where $f(r_{+})=0$, we find $(\nabla_{\mu}\chi _{\nu})
(\nabla^{\mu}\chi^{\nu})=-\frac{1}{2}\left[f'(r_{+}) \right]^2
e^{-\chi (r_{+})}$. Thus, the temperature is obtained as
\begin{equation}
T=\frac{e^{-\chi (r_{+})/2}f^{^{\prime }}(r_{+})}{4\pi }.
\label{temp}
\end{equation}%
We also choose the scalar and the gauge fields as \cite{H08}%
\begin{equation}
A_{\mu }=(\phi (r),0,0),\ \ \ \psi =\psi (r). \label{Aphi}
\end{equation}%
Substituting (\ref{metric}) and (\ref{Aphi}) into the field equations (\ref%
{Epsi})-(\ref{Eein}), we arrive at%
\begin{eqnarray}
0 &=&\psi ^{\prime \prime }+\psi ^{\prime }\left[ \frac{1}{r}+\frac{%
f^{\prime }}{f}-\frac{\chi ^{\prime }}{2}\right] +\psi \left[ \frac{%
q^{2}\phi ^{2}{\mathrm{e}}^{\chi }}{f^{2}}-\frac{m^{2}}{f}\right] ,
\label{psir} \\
&& \notag \\
0 &=&\phi ^{\prime \prime }+\phi ^{\prime }\left[ -\frac{be^{\chi }\phi
^{\prime 2}}{r}+\frac{\chi ^{\prime }}{2}+\frac{1}{r}\right] -\frac{%
2q^{2}\psi ^{2}\phi }{f}\left[ 1-be^{\chi }\phi ^{\prime 2}\right] ^{3/2},
\notag \\
&& \label{phir} \\
0 &=&f^{\prime }-\frac{2r}{l^{2}} \notag \\
&&+2\kappa ^{2}r\left[ \frac{q^{2}e^{\chi }\psi ^{2}\phi ^{2}}{f}+f\psi
^{\prime 2}+m^{2}\psi ^{2}-\frac{1}{b}+\frac{1}{b\sqrt{1-be^{\chi }\phi
^{\prime 2}}}\right] , \notag \\
&& \label{fr} \\
0 &=&\chi ^{\prime }+4\kappa ^{2}r\left[ \frac{q^{2}\phi ^{2}\psi ^{2}{%
\mathrm{e}}^{\chi }}{f^{2}}+\psi ^{\prime 2}\right] , \label{chir}
\end{eqnarray}%
where the prime denotes derivative with respect to $r$. Note that in the
presence of nonlinear BI electrodynamics the Eqs. (\ref{psir}) and (\ref%
{chir}) do not change compared to the linear Maxwell case. In the limiting
case where $b\rightarrow 0$ the equations of motion (\ref{phir}) and (\ref%
{fr}) turn to the corresponding equations of one dimensional holographic
superconductor with Maxwell field \cite{Wang}. The field equations (\ref%
{psir})-(\ref{chir}) enjoy the symmetries
\begin{gather}
q\rightarrow q/a,\text{ \ \ \ \ }\phi \rightarrow a\phi ,\text{ \ \ \ \ }%
\psi \rightarrow a\psi , \notag \\
\kappa \rightarrow \kappa /a,\text{ \ \ \ \ }b\rightarrow b/a^{2}, \\
\notag \\
l\rightarrow al,\text{ \ \ \ \ }r\rightarrow ar,\text{ \ \ \ \ }q\rightarrow
q/a, \notag \\
m\rightarrow m/a,\text{ \ \ \ \ }b\rightarrow a^{2}b.
\end{gather}%
Hereafter, we set $q$ and $l$ equal to unity by virtue of these symmetries.
The behavior of our model functions for large $r$ (near the boundary) read%
\begin{gather}
\chi \rightarrow 0,\ \ \ f(r)\sim r^{2}, \notag \\
\phi \sim \rho +\mu \ln (r),\ \ \ \psi \sim \psi _{-}r^{-\Delta _{-}}+\psi
_{+}r^{-\Delta _{+}}, \label{bval}
\end{gather}%
where $\mu $ and $\rho $ are the chemical potential and the charge
density of the field theory on the boundary and $\Delta _{\pm
}=1\pm \sqrt{1+m^{2}}$ which implies $m^{2}\geq -1$. Actually,
$\chi $ could be a constant near the boundary but by using the
symmetry of field equation $\mathrm{e}^{\chi }\rightarrow
a^{2}\mathrm{e}^{\chi },$ $\phi \rightarrow \phi /a$, it could be
set to zero there. From holographic superconductors point of view,
either $\psi _{+}$ or $\psi _{-}$ can be dual to the expectation
value of condensation operator (or order operator) $\left\langle
O\right\rangle $ while the other is dual to its source. We give
$\psi _{-}$ the role of
source and $\psi _{+}$ the role of expectation value of the order parameter $%
\left\langle O_{+}\right\rangle $ in this work. Since we seek for study the
effects of $b$ and $\kappa $ on our holographic superconductors and
different values of the scalar field mass do not influence this behavior
qualitatively, we consider $m^{2}=0$ in this work. With this choice, we have
$\Delta _{+}=2$, $\Delta _{-}=0$ and thus%
\begin{equation}\label{psiasy}
\psi \sim \psi _{-}+\frac{\psi _{+}}{r^{2}},
\end{equation}%
near the boundary. We set $\psi _{-}=0$ at the boundary and
consider $\psi _{+}$ as the dual of order parameter $\left\langle
O_{+}\right\rangle $. It is remarkable to note that the asymptotic
solution for $\psi$ given in Eq. (\ref{psiasy}) do not depend on
the type of electrodynamics and thus for the Maxwell case in three
dimensions the solution is the same as in Eq. (\ref{psiasy}).
While the solution depends on the spacetime dimensions. This is
due to the fact that equation for the $\psi$ given in
(\ref{psir}) is independent on the type of electrodynamics but
depends on the spacetime dimensions and the mass parameter $m$
\cite{Wang,Doa,ghor}.
The next step is to solve the coupled nonlinear field equations (\ref{psir}%
)-(\ref{chir}) simultaneously and obtain the behavior of model functions.
Then, we could figure out the behavior of different parameters of
holographic superconductor specially the order parameter $\left\langle
O_{+}\right\rangle $ and the critical temperature by using these functions.
In this work, we use both analytical and numerical methods for studying the
holographic superconductor. For analytical study, we perform Sturm-Liouville
method while for numerical study, we use shooting method.
\section{Sturm-Liouville method\label{sec3}}
\begin{table*}[t]
\caption{Analytical results of ${T_{c}}/{\protect\mu }$ for different values
of backreaction and nonlinearity parameters.}
\label{tab1}
\begin{center}
\begin{tabular}{c|c|c|c|c|c|c|}
\cline{2-3}\cline{2-7}\cline{4-7}
& \multicolumn{2}{|c|}{$b=0$} & \multicolumn{2}{|c|}{$b=0.04$} &
\multicolumn{2}{|c|}{$b=0.08$} \\ \cline{2-3}\cline{2-7}\cline{4-7}
& Analytical & Numerical & Analytical & Numerical & Analytical & Numerical
\\ \hline
\multicolumn{1}{|c|}{$\kappa ^{2}=0$} & $0.0429$ & $0.0460$ & $0.0360$ & $%
0.0410$ & $0.0275$ & $0.0362$ \\ \hline
\multicolumn{1}{|c|}{$\kappa ^{2}=0.05$} & $0.0399$ & $0.0369$ & $0.0337$ & $%
0.0326$ & $0.0260$ & $0.0286$ \\ \hline
\multicolumn{1}{|c|}{$\kappa ^{2}=0.1$} & $0.0381$ & $0.0295$ & $0.0311$ & $%
0.0260$ & $0.0218$ & $0.0227$ \\ \hline
\multicolumn{1}{|c|}{$\kappa ^{2}=0.15$} & $0.0352$ & $0.0236$ & $0.0280$ & $%
0.0207$ & $0.0174$ & $0.0180$ \\ \hline
\multicolumn{1}{|c|}{$\kappa ^{2}=0.2$} & $0.0313$ & $0.0189$ & $0.0242$ & $%
0.0165$ & $0.0136$ & $0.0143$ \\ \hline
\multicolumn{1}{|c|}{$\kappa ^{2}=0.25$} & $0.0264$ & $0.0151$ & $0.0195$ & $%
0.0131$ & $0.0089$ & $0.0114$ \\ \hline
\end{tabular}%
\end{center}
\end{table*}
In this section, we employ the Sturm-Liouville eigenvalue problem to
investigate analytically the phase transition of one dimensional $s$-wave
holographic superconductor in the presence of BI nonlinear electrodynamics.
In addition, we calculate the relation between the critical temperature $%
T_{c}$, and chemical potential $\mu $, near the horizon. Furthermore, we
study the effect of backreaction and BI nonlinear electrodynamics on the
critical temperature. For future convenience, we define a new variable $%
z=r_{+}/r$ $(\in \left[ 0,1\right] )$. With this new coordinate, the field
equations (\ref{psir})-(\ref{chir}) could be rewritten as%
\begin{eqnarray}
0 &=&\psi ^{\prime \prime }+\psi ^{\prime }\left[ \frac{f^{\prime }}{f}-%
\frac{\chi ^{\prime }}{2}+\frac{1}{z}\right] +\psi \left[ \frac{%
r_{+}^{2}e^{\chi }\phi ^{2}}{z^{4}f^{2}}-\frac{m^{2}r_{+}^{2}}{z^{4}f}\right]
, \notag \\
&& \label{psiz} \\
0 &=&\phi ^{\prime \prime }+\phi ^{\prime }\left[ \frac{bz^{3}e^{\chi }\phi
^{\prime 2}}{r_{+}^{2}}+\frac{\chi ^{\prime }}{2}+\frac{1}{z}\right] -\frac{%
2r_{+}^{2}\psi ^{2}\phi }{z^{4}f}\Upsilon ^{\frac{3}{2}}, \label{phiz} \\
&& \notag \\
0 &=&f^{\prime }+\frac{2r_{+}^{2}}{z^{3}}+\frac{2r_{+}^{2}\kappa ^{2}}{z^{3}}
\notag \\
&&\times \left[ \frac{1}{b}\left( 1-\Upsilon ^{-\frac{1}{2}}\right) -\frac{%
z^{4}f\psi ^{\prime 2}}{r_{+}^{2}}-\frac{e^{\chi }\psi ^{2}\phi ^{2}}{f}%
-m^{2}\psi ^{2}\right] , \notag \\
&& \label{fz} \\
0 &=&\chi ^{\prime }-4\kappa ^{2}\left[ \frac{r_{+}^{2}e^{\chi }\psi
^{2}\phi ^{2}}{z^{3}f^{2}}+z\psi ^{\prime 2}\right] , \label{chiz}
\end{eqnarray}%
where $\Upsilon =1-bz^{4}e^{\chi }\phi ^{\prime 2}/r_{+}^{2}$ and now the
prime indicates the derivative with respect to $z$. Since in the vicinity of
critical temperature the order parameter is small, we can consider it as an
expansion parameter%
\begin{equation*}
\epsilon \equiv \left\langle O_{i}\right\rangle ,
\end{equation*}%
where $i=+$ or $-$. We focus on solutions for small values of condensation
parameter $\epsilon $, therefore we can expand the model functions as%
\begin{gather*}
\psi \approx \epsilon \psi _{1}+\epsilon ^{3}\psi _{3}+\epsilon ^{5}\psi
_{5}+\cdots , \\
\phi \approx \phi _{0}+\epsilon ^{2}\phi _{2}+\epsilon ^{4}\phi _{4}+\cdots ,
\\
f\approx f_{0}+\epsilon ^{2}f_{2}+\epsilon ^{4}f_{4}+\cdots , \\
\chi \approx \epsilon ^{2}\chi _{2}+\epsilon ^{4}\chi _{4}+\cdots ,
\end{gather*}%
where $\epsilon \ll 1$ near the critical temperature. Moreover, by
considering $\delta \mu _{2}>0$, the chemical potential can be expressed as:%
\begin{equation*}
\mu =\mu _{0}+\epsilon ^{2}\delta \mu _{2}+...\rightarrow \epsilon
\thickapprox \Bigg(\frac{\mu -\mu _{0}}{\delta \mu _{2}}\Bigg)^{1/2}.
\end{equation*}%
During phase transition, $\mu _{c}=\mu _{0}$, thus the order parameter
vanishes. Meanwhile, the critical exponent $\beta =\frac{1}{2}$ is in a good
agreement with mean field theory result.
At zeroth order of $\epsilon $, the gauge field equation of motion (\ref%
{phiz}) reduces to%
\begin{equation}
\phi ^{\prime \prime }+\frac{\phi ^{\prime }}{z}+\frac{bz^{3}\phi ^{\prime 3}%
}{r_{+}^{2}}=0,
\end{equation}%
which could be rewritten as a first order Bernoulli differential equation by
taking $\phi ^{\prime }$ as a new function \cite{TL00}. Therefore, one
receives%
\begin{equation}
\phi ^{\prime }=\frac{\lambda r_{+}}{z\sqrt{b\lambda ^{2}z^{2}+1}},
\label{dphi}
\end{equation}%
for small values of $b$ where we define $\lambda =\mu /r_{+}$ and fix the
integration constants by looking at the behavior of $\phi $ near the
boundary given in (\ref{bval}). Integrating (\ref{dphi}) and using the fact
that $\phi (z=1)=0$\footnote{%
It is necessary so that the norm of gauge potential is finite at horizon.},
we can obtain%
\begin{eqnarray}
\phi _{0}(z) &=&\int_{1}^{z}\frac{\lambda r_{+}}{z}\left( 1-\frac{1}{2}%
b\lambda ^{2}z^{2}\right) \,dz \notag \\
&=&\lambda r_{+}\log (z)-\frac{1}{4}b\lambda ^{3}r_{+}\left( z^{2}-1\right) .
\label{phi}
\end{eqnarray}%
When $b=0$ the above equation reduces to one of \cite{L12}. Note that at the
zeroth order with respect to $\epsilon $, $\psi _{0}=\chi _{0}=0$.
Substituting (\ref{dphi}) in the (\ref{fz}), the equation for $f$ at the
zeroth order with respect to $\epsilon $ has the following form
\begin{gather}
f_{0}(z)=r_{+}^{2}g(z), \notag \\
\text{\ }g(z)=\frac{1}{z^{2}}-1+\frac{1}{8}b\kappa ^{2}\lambda ^{4}\left(
1-z^{2}\right) +\kappa ^{2}\lambda ^{2}\log (z).
\end{gather}
\begin{figure*}[t]
\centering
\subfigure[~b=0]{\includegraphics[width=0.3\textwidth]{Fig1.eps}} \qquad %
\subfigure[~b=0.04]{\includegraphics[width=0.3\textwidth]{Fig2.eps}} \qquad %
\subfigure[~b=0.08]{\includegraphics[width=0.3\textwidth]{Fig3.eps}}
\caption{The behavior of condensation parameter as a function of temperature
for different values of backreaction.}
\label{fig1}
\end{figure*}
\begin{figure*}[t]
\centering
\subfigure[~$\kappa^{2}$=0]{\includegraphics[width=0.3\textwidth]{Fig4.eps}}
\qquad %
\subfigure[~$\kappa^{2}$=0.10]{\includegraphics[width=0.3%
\textwidth]{Fig6.eps}} \qquad %
\subfigure[~$\kappa^{2}$=0.20]{\includegraphics[width=0.3%
\textwidth]{Fig8.eps}} \qquad %
\caption{The behavior of condensation parameter as a function of temperature
for different values of nonlinearity parameter $b$.}
\label{fig2}
\end{figure*}
The asymptotic behavior of the scalar field $\psi $ was given in (\ref{bval}%
). Near the boundary, we define the function $F(z)$ so that
\begin{equation}
\psi (z)=\frac{\left\langle O_{i}\right\rangle z^{\triangle _{i}}}{\sqrt{2}%
r_{+}^{\triangle _{i}}}F(z). \label{psi1F}
\end{equation}%
Inserting Eq. (\ref{psi1F}) in Eq. (\ref{psiz}) yields%
\begin{eqnarray}
&&F^{\prime \prime }(z)+F^{\prime }(z)\left[ \frac{g^{\prime }(z)}{g(z)}+%
\frac{2\Delta }{z}+\frac{1}{z}\right] \notag \\
&+&F(z)\left[ \frac{\Delta g^{\prime }(z)}{zg(z)}-\frac{m^{2}}{z^{4}g(z)}+%
\frac{\Delta ^{2}}{z^{2}}\right] \notag \\
&-&\frac{F(z)}{2z^{4}g(z)^{2}}\left[ \lambda ^{2}\log (z)\left( b\lambda
^{2}r_{+}\left( z^{2}-1\right) -2\log (z)\right) \right] =0. \notag \\
&&
\end{eqnarray}%
We can rewrite this equation in the Sturm-Liouville form as%
\begin{equation}
\left[ T(z)F^{\prime }(z)\right] ^{\prime }+P(z)T(z)F(z)+\lambda
^{2}Q(z)T(z)F(z)=0, \label{sl}
\end{equation}%
where the functions $T$, $P$, $Q$ are defined as%
\begin{equation}
T(z)=z^{2\Delta +1}\left[ \frac{1}{z^{2}}-1+\frac{1}{8}b\kappa ^{2}\lambda
^{4}\left( 1-z^{2}\right) +\kappa ^{2}\lambda ^{2}\log (z)\right] .
\end{equation}%
\begin{equation}
P(z)=\frac{\Delta }{z}\left( \frac{g^{\prime }(z)}{g(z)}+\frac{\Delta }{z}%
\right) -\frac{m^{2}}{z^{4}g(z)},
\end{equation}%
\begin{equation}
Q(z)=-\frac{\log (z)\left( b\lambda ^{2}r_{+}\left( z^{2}-1\right) -2\log
(z)\right) }{2z^{4}g(z)^{2}}.
\end{equation}%
We can consider the trial function $F(z)=1-\alpha z^{2}$ which satisfies the
required boundary conditions $F(0)=1$ and $F^{^{\prime }}(0)=0$. Then, the
eigenvalue problem could be solved for (\ref{sl}) by minimizing the
expression%
\begin{equation}
\lambda ^{2}=\frac{\int_{0}^{1}T\left( F^{\prime 2}-PF^{2}\right) dz}{%
\int_{0}^{1}TQF^{2}dz}, \label{l2}
\end{equation}%
with respect to $\alpha $. For backreacting parameter, we could use the
iteration method and define \cite{LPJW2015}
\begin{equation}
\kappa _{n}=n\Delta \kappa ,\ \ \ n=0,1,2,\cdots ,
\end{equation}%
where $\Delta \kappa =\kappa _{n+1}-\kappa _{n}$. Here, we take $\Delta
\kappa =0.05$. Since we are interested in finding the effects of
nonlinearity on backreaction up to the order $\kappa ^{2}$, we have%
\begin{equation}
\kappa ^{2}\lambda ^{2}={\kappa _{n}}^{2}\lambda ^{2}={\kappa _{n}}%
^{2}(\lambda ^{2}|_{\kappa _{n-1}})+O[(\Delta \kappa )^{4}],
\end{equation}%
where we take $\kappa _{-1}=0$ and $\lambda ^{2}|_{\kappa _{-1}}=0$. We
shall also retain the linear terms with respect to nonlinearity parameter $b$
and therefore,%
\begin{equation}
b\lambda ^{2}=b\left( \lambda ^{2}|_{b=0}\right) +\mathcal{O}(b^{2}).
\end{equation}%
Then, the minimum eigenvalue of Eq. (\ref{l2}) can be obtained. At the
critical point, temperature is defined as (see Eq. (\ref{temp}) and note
that at zeroth order with respect to $\epsilon $, $\chi $ is zero.)%
\begin{equation}
T_{c}=\frac{f^{\prime }\left( r_{+c}\right) }{4\pi }.
\end{equation}%
Using Eqs. (\ref{fr}) and (\ref{phi}), we receive
\begin{equation}
f^{\prime }\left( r_{+c}\right) =2r_{+c}+\frac{2\kappa ^{2}r_{+c}}{b}\left[
1-\frac{1}{\sqrt{1-b\phi ^{\prime }\left( r_{+c}\right) {}^{2}}}\right] ,
\label{frc}
\end{equation}%
and thus
\begin{eqnarray}
T_{c} &=&\frac{1}{4\pi }(\frac{\mu }{\lambda })[2-\kappa _{n}^{2}(\lambda
^{2}|_{\kappa _{n-1}}) \notag \\
&&-\frac{3}{4}b\kappa _{n}^{2}(\lambda ^{4}|_{\kappa _{n-1},b=0})+b\kappa
_{n}^{2}(\lambda ^{4}|_{\kappa _{n-1},b=0})].
\end{eqnarray}%
As an example, if $b=\kappa ^{2}=0$ we have%
\begin{equation*}
\lambda ^{2}=\dfrac{\frac{2}{3}\alpha ^{2}-\frac{4}{3}\alpha +1}{-\frac{%
251\alpha ^{2}}{864}+\frac{9\alpha }{16}+\frac{\alpha ^{2}\zeta (3)}{4}-%
\frac{\alpha \zeta (3)}{2}+\frac{\zeta (3)}{4}-\frac{1}{4}}.
\end{equation*}%
Inserting $\alpha =0.759$, $\lambda _{min}^{2}=13.76$ and $T_{c}=0.429\mu $.
The latter result perfectly agrees with ones in \cite{L12}.
The values of $T_{c}/\mu $ for different backreaction and nonlinearity
parameters are listed in \ref{tab1}. As it shows, the effect of increasing
the backreaction parameter $\kappa $ for a fixed value of nonlinearity
parameter $b$ follows the same trend as raising $b$ for a fixed value of $%
\kappa $. In both cases, the critical temperature $T_{c}$ diminishes by
growing the backreaction or nonlinearity parameters. It shows that the
presence of backreaction and Born-Infeld nonlinear electrodynamics make the
scalar hair harder to form. In next section, we will re-study the problem
numerically using the shooting method.
\section{Shooting method\label{sec4}}
In this section, we will study our holographic superconductor numerically.
In order to do this, we use the shooting method \cite{H09}. In this method,
the boundary values is found by setting appropriate initial conditions. So,
for doing this, we need to know the behavior of equations of motion both at
horizon and boundary. Using Taylor expansion at horizon around $z=1$, we get%
\begin{gather}
f(z)=f_{1}\left( 1-z\right) +f_{2}\left( 1-z\right) {}^{2}+\cdots , \\
\phi (z)=\phi _{1}\left( 1-z\right) +\phi _{2}\left( 1-z\right)
{}^{2}+\cdots , \\
\psi (z)=\psi _{0}+\psi _{1}\left( 1-z\right) +\psi _{2}\left( 1-z\right)
{}^{2}+\cdots , \\
\chi (z)=\chi _{0}+\chi _{1}\left( 1-z\right) +\chi _{2}\left( 1-z\right)
{}^{2}+\cdots .
\end{gather}%
Note that $\phi =0$ at horizon, otherwise it will be ill-defined there. In
our procedure, we find all coefficients in terms of $\phi _{1}$, $\psi _{0}$
and $\chi _{0}$ by using equations of motion. Varying them at the horizon,
we try to get $\psi _{-}=\chi =0$ at the boundary. So, the values of $\psi
_{+}$ and $\mu $ are achieved. In addition, we will set $r_{+}=1$ by virtue
of the equations of motion's symmetry%
\begin{equation*}
r\rightarrow ar,\text{ \ \ \ \ }f\rightarrow a^{2}f,\text{ \ \ \ \ }\phi
\rightarrow a\phi .
\end{equation*}
Performing numerical solution, we can find the values of $T_{c}/\mu $ for
different backreaction and nonlinearity parameters. In order to compare the
latter results with analytical ones, we listed both of them next to each
other in table \ref{tab1}. It is obvious that there is a reasonable
agreement between the results of both methods. Moreover, in table \ref{tab1}%
, the results of \cite{Wang} for $b=0$ has been recovered for different
values of backreaction parameter. As one could see in this table, increasing
the backreaction parameter for a fixed value of $b$, decreases the critical
temperature. This means that the larger values of backreaction parameter
makes the condensation harder to form. Similarly, for a fixed value of $%
\kappa $, increasing the nonlinearity of electrodynamic model makes scalar
hair harder to form because it diminishes the critical temperature.
Figs. \ref{fig1} and \ref{fig2} give information about the effect of
backreaction and nonlinear electrodynamic on condensation. All curves follow
a same trend. As $b\rightarrow 0$, we regain the results of Maxwell case
presented in \cite{Wang}. As figures show, the condensation gap increases by
making backreaction and nonlinearity parameters larger while the other one
is fixed. So, it can be understood that it is harder to form a
superconductor. This is in agreement with the results obtained from the
behavior of critical temperature before.
\section{Summary and discussion\label{sec5}}
In this work, by using the Sturm-Liouville eigenvalue problem, we
analytically investigated the properties of $(1+1)$-dimensional holographic
superconductor developed in BTZ black hole background in the presence of BI
nonlinear electrodynamics. We have relaxed the probe limit and further
assumed that the gauge and scalar fields do backreact on the background
metric. We determined the critical temperature for different values of
backreaction and nonlinear parameters. We have also continued our study by
using the numerical shooting method and confirmed that the analytical
results are in agreement with the numerical approach. We observed that the
formation of the scalar hair is harder in the presence of BI nonlinear
electrodynamics as well as backreaction and it becomes harder and harder to
form by increasing the strength of either the nonlinear and backreaction
parameters.
Finally, it would be of interest to extend this procedure for other
nonlinear electrodynamics like Power-Maxwell and logarithmic cases and
investigate the effects of nonlinear electrodynamics on the critical
temperature and condensation operator of one dimensional holographic
superconductors. These issues are now under investigations and the results
will be appeared elsewhere.
\begin{acknowledgments}
We thank referee for constructive comments which helped us improve
our paper significantly. We also thank Shiraz University Research
Council. The work of AS has been supported financially by Research
Institute for Astronomy and Astrophysics of Maragha (RIAAM), Iran.
MKZ would like to thank Shahid Chamran University of Ahvaz, Iran.
\end{acknowledgments}
|
1,108,101,565,653 | arxiv | \section{Introduction}
Liouville type theorems for several kinds of nonlinear elliptic equations in half space have already been extensively studied. For the semilinear elliptic equation $-\Delta u=|u|^{p-2}u$ in $\mathbf R_+^n$ with zero-Dirichlet boundary condition when $n\geq3$ and $2<p<2n/(n-2),$ Gidas and Spruck \cite{GiSp} proved that $u=0$ is the unique non-negative solution. For the real Monge-Amp\`{e}re equation, it is well known in Savin \cite{Sa} and Mooney \cite{Mo} that any convex viscosity solution of $\det\nabla^2u=1$ in $\mathbf R_+^n$ with quadratic boundary condition must be a quadratic polynomial if $u=O(|x|^2)$ as $|x|\to\infty$. For minimal surface system prescribed with an affine Dirichlet boundary condition, Ding, Jost and Xin proved in \cite{DiJoXi} that any $C^2(\mathbf R_+^n)\cap C^{1,\alpha}(\overline{\mathbf R^n_+})$ solution with small singular values and uniformly bounded gradient must be an affine function, whose one-codimensional case indicates the validity of a Liouville type theorem for minimal graph over half space $\mathbf R^n_+$. Indeed, we can establish the following Liouville type theorem:
\begin{theorem}\label{ld}
Let $n\geq 2$ be an integer and $u\in C^2(\mathbf R_+^n)\cap C(\overline{\mathbf R_+^n})$ be a solution of
\begin{align}
\mathrm{div}\(\frac{\nabla u}{\sqrt{1+|\nabla u|^2}}\)&=0\,\,\, \hbox{in $\mathbf R_+^n$,}\label{mse}\\
u&=l\,\,\, \hbox{on $\partial\mathbf R_+^n,$}\label{diri}
\end{align}
where $l:\mathbf R^n\to\mathbf R$ is an affine function.
Assume that $u:\overline{\mathbf R_+^n}\to\mathbf R$ has at most a linear growth, which means there exists a constant $K>0$ such that
\begin{equation}\label{lingro}
|u(x)|\leq K(|x|+1)\,\,\, \hbox{for any $x\in\overline{\mathbf R_+^n}$.}
\end{equation}
Then $u$ is an affine function.
\end{theorem}
\begin{remark}
With the boundary condition \eqref{diri}, we point out that $u$ is smooth up to the boundary, which follows from an approximation procedure and the $C^{1,\alpha}$-estimates for quasilinear elliptic equations. For the convenience of the reader, we provide the details in Appendix \ref{B}.
\end{remark}
For entire minimal graph, Simons \cite{Sim} proved that any minimal graph over $\mathbf R^n$ must be a hyperplane for $2\leq n\leq7$. It is of particular interest to know whether the assumption \eqref{lingro} is necessary for above theorem. In two-dimensional case, it follows from the Schwarz reflection principle and Choi-Schoen curvature estimate \cite{CiSc} for minimal surfaces in $\mathbf R^3$ that Theorem \ref{ld} is true without the linear growth condition. To the best of our knowledge, the answer is still not clear in higher dimensional cases.
With the idea of reflection, it is fairly easy to prove Theorem \ref{ld} in case $l\equiv0$. To see this, we perform a Schwarz reflection for $u$ to obtain a new function $\tilde u$. Then, $\tilde u$ is an entire solution of minimal surface equation which has at most a linear growth. Theorem \ref{ld} then follows from the Liouville theorem for entire minimal graph. When $l$ is a general affine function, the Schwarz reflection may not lead to an entire minimal graph, which appears to be a difficulty for Theorem \ref{ld}.
Different from the linear growth condition, we point out that the affine boundary value \eqref{diri} can not be removed. Otherwise, one may consider the function $$f(x)=\int_{1}^{|x|}{dt\over\sqrt{t^2-1}}$$
over the half plane
$$
P_+=\{(x_1,x_2)\in \mathbf R^2;\ x_2>2\}.
$$
Through direct calculations, it is quick to see that the function $f$ is a smooth solution of the minimal surface equation and that $|\nabla f|$ is uniformly bounded, but $f$ is not affine. From this point of view, it is interesting to know whether Liouville type theorem will be valid for Neumann boundary condition. The answer is definitely positive. In fact, we prove the following Liouville type theorem for constant Neumann boundary condition.
\begin{theorem}\label{ln}
Let $n\geq 2$ be an integer and $u\in C^2(\mathbf R_+^n)\cap C^1(\overline{\mathbf R_+^n})$ be a solution of \eqref{mse} with Neumann boundary condition
\begin{equation}
\partial_{x_n}u=\tau\ \ \ \hbox{on $\partial\mathbf R_+^n,$}\label{neu}
\end{equation}
where $\tau\in\mathbf R$ is a constant. If $u$ satisfies \eqref{lingro}, then $u$ is an affine function.
\end{theorem}
\begin{remark}
With the boundary condition \eqref{neu}, we note that $u$ is smooth up to the boundary. For more details, we refer the reader to \cite[Theorem 4.5]{Li} and \cite[Theorems 6.30-6.31]{GilTr}.
\end{remark}
As in the Dirichlet case, it is also not clear in the Neumann case whether the linear growth condition is necessary when $n\leq 7$. However, we notice $u$ is affine provided it is a solution of \eqref{mse} satisfying \eqref{diri} and \eqref{neu}, which is a direct conclusion from unique continuation property due to \cite{GaLi}.
In the following, we sketch the proof for our main theorems. For the Dirichlet case, the key ingredient is to obtain a uniform gradient estimate for the solution with affine boundary value and linear growth condition. For this purpose, we establish a boundary gradient estimate first, then the classical Bernstein technique due to \cite{Wa} yields the desired estimate. It turns out that the scaling invariance of the minimal surface equation and comparison principle make the linear growth come into play for boundary gradient estimate. To be explicit, we fix a weakly mean convex domain with some flat portion $T$ of $\mathbf R^n_+$ on its boundary. After imposing a particular smooth boundary value that coincides with the affine one on $T$, we obtain a smooth solution to the minimal surface equation as a comparison function. Compared with the rescaled solution $u_R(\cdot)=R^{-1}u(R\cdot)$, we obtain a uniformly bounded boundary gradient estimate. Then with the uniform gradient estimate derived from Bernstein method, we have a H\"older estimate for $\partial_{x_n}u$ by Krylov \cite{Kr}, which deduces that $\partial_{x_n}u$ is a constant by a scaling argument. At this stage, Theorem \ref{ld} follows easily from unique continuation \cite{GaLi} or Theorem \ref{ln}.
The proof follows a similar line for Neumann case. For gradient estimate, we apply the Bernstein method as usual but with a modified function to avoid its maximum appearing on the boundary, where the idea to push the maximum point inside is inspired from the work in Ma and Xu \cite{MaXu}. With the uniformly bounded gradient, we obtain the H\"older estimate for $\partial_{x_n}u$, which yields $\partial_{x_n}u$ to be a constant using the scaling argument. In this case, we can express $u$ to be a sum of $\tau x_n$ and an entire solution of minimal surface equation in $\mathbf R^{n-1},$ hence $u$ is affine.
The rest of this paper will be organized as follows. In section 2, we present details for gradient estimates in both Dirichlet and Neumann boundary condition. In section 3, we prove our main theorems.
\section{Gradient Estimate}
Throughout this paper, following notation will be used frequently.
\begin{itemize}
\item[(i)] For $i,j\in\{1,\ldots,n\},$ the Kronecker symbol $\delta_{ij}$ is given by
\begin{equation*}
\delta_{ij}=\begin{cases}
1&\text{if $i=j,$}\\
0&\text{if $i\neq j.$}\\
\end{cases}
\end{equation*}
\item[(ii)] We particularly distinguish the $n$-th component and write $$x=(x',x_n)\ \ \ \hbox{for any $x\in\mathbf R^{n},$}$$
where $x'\in\mathbf R^{n-1}$ and $x_n\in\mathbf R.$
\item[(iii)] For $r>0,$\ $B_r(x_0)$ is the open ball of radius $r$ and center $x_0$ in $\mathbf R^n$, and $$B_r^+(x_0)=B_r(x_0)\cap\mathbf R_+^n.$$
If $x_0=0,$ we use $B_r$ to briefly represent $B_r(x_0)$ and $$\Sigma_r=\overline{B_r}\cap\{x\in\mathbf R^n;\ x_n=0\},\ B_r^+=B_r\cap\mathbf R_+^n.$$
\item[(iv)] $C$ denotes a positive universal constant depending only on $n$ and $K$, whose meaning may be different from line to line.
\item[(v)]In section \ref{2.2}, we will use subscripts to write derivatives as$$(\cdot)_i=\partial_{x_i}(\cdot),\ (\cdot)_{ij}=\partial_{x_ix_j}(\cdot),\ (\cdot)_{ijk}=\partial_{x_ix_jx_k}(\cdot)$$
for brevity, whose meanings will be different from those subscripts of coefficients $a_{ij}.$
\end{itemize}
\subsection{Gradient Estimate for Dirichlet Problem}
In this subsection, we present the proof of global gradient estimate for Dirichlet case. We begin with the boundary gradient estimate as following.
\begin{lemma}\label{bgd}
Assume $u\in C^2(\overline{\mathbf R_+^n})$ is any solution of \eqref{mse}-\eqref{diri} satisfying \eqref{lingro}. Then
\begin{equation}\label{pd1}
\sup_{x\in\partial\mathbf R_+^n}|\nabla u(x)|<C
\end{equation}
for some universal constant $C>0$ (independent of $u$).
\end{lemma}
\begin{proof}
Without loss of generality, we may assume that $l(0)=0.$ It suffices to show that
\begin{equation}\label{pd2}
\sup_{x\in\Sigma_1}|\nabla u(x)|<C
\end{equation}
for some universal constant $C>0$.
To be explicit, we set $$u_R(x)={1\over R}u(Rx)\ \ \ \hbox{for $R>1$ and $x\in\overline{\mathbf R_+^n}$}.$$
It is clear that $u_R\in C^2(\overline{\mathbf R_+^n})$ solves \eqref{mse}-\eqref{diri}. For $R>1,$ it follows from \eqref{lingro} that $$
|u_R(x)|\leq K|x|+{K\over R}\leq K(|x|+1)\ \ \ \hbox{for every $x\in\mathbf R_+^n.$}
$$
Thanks to \eqref{pd2}, we have $$\sup_{x\in\Sigma_1}|\nabla u(Rx)|=\sup_{x\in\Sigma_1}|\nabla u_R(x)|<C\ \ \ \hbox{for every $R>1$,}$$
which implies \eqref{pd1}.
Now we turn to the proof of \eqref{pd2}. Let $\Omega\subset\mathbf R^n$ be the convex domain constructed in the Appendix \ref{A}, then the convexity of $\Omega$ implies that the boundary mean curvature $H_{\partial\Omega}$ is non-negative. Choose a smooth function $\rho:[0,+\infty)\to\mathbf R$ such that
\begin{align*}
&\rho(t)\equiv0\ \ \ \hbox{for every $t\in[0,1],$}\\
&\rho(t)\in[0,1]\ \ \ \hbox{for every $t\in[1,2],$}\\
&\rho(t)\equiv1\ \ \ \hbox{for every $t\in[2,\infty).$}
\end{align*}
For $x\in\overline\Omega$, set
\begin{equation*}
\phi(x)=6K\rho(|x|)+(1-\rho(|x|))l(x),
\end{equation*}
then $\phi\in C^3(\overline\Omega).$ By the construction of $\Omega,$ we have
\begin{equation}\label{pd5}
\phi(x)=6K\ \ \ \hbox{if $x\in\partial\Omega\cap\mathbf R_+^n;$} \,\,\,\,\,\,\,\,\phi(x)=l(x)\ \ \ \hbox{if $x\in\Sigma_1.$}
\end{equation}
Let $v\in C^2(\overline\Omega)$ be a solution to following Dirichlet problem
$$
\mathrm{div}\left({\nabla v\over\sqrt{1+|\nabla v|^2}}\right)=0\ \hbox{in $\Omega,$}\ \ \ v=\phi\ \hbox{on $\partial\Omega,$}
$$
whose existence is given by \cite{JeSe} or \cite[Theorem 16.10]{GilTr}. Notice that
\begin{equation*}
u(x)\leq K(|x|+1)\leq6K\ \ \ \hbox{for every $x\in\overline\Omega,$}
\end{equation*}
with \eqref{diri}, \eqref{pd5} and the maximum principle, we know $u\leq v$ in $\Omega.$ By (\ref{pd5}), we have $$
\partial_{x_n}u(x)\leq\partial_{x_n}v(x)\leq|\nabla v|_{L^\infty(\Omega)}\ \ \ \hbox{for every $x\in\Sigma_1$}.$$
A similar fashion gives $$\partial_{x_n}u(x)\geq-|\nabla v|_{L^\infty(\Omega)}\ \ \ \hbox{for every $x\in\Sigma_1.$}$$ Therefore, we get a uniform bound for $|\nabla u|$ on $\Sigma_1,$ which yields (\ref{pd2}).
\end{proof}
Using the classical Bernstein technique, we have
\begin{lemma}\label{ggd}
Let $n\geq2$ be an integer and $u\in C^2(\overline{\mathbf R_+^n})$ be a solution of \eqref{mse}. Assume $u$ satisfies \eqref{lingro} and \eqref{pd1}. Then $|\nabla u|\in L^\infty(\mathbf R_+^n).$
\end{lemma}
\begin{proof}
Following the calculation in \cite{Wa}, the only difficulty in our case is that the maximum point of the auxiliary function may locate on $\partial\mathbf R_+^n$. However, this can be overcome by \eqref{bgd}.
\end{proof}
\subsection{Gradient Estimate for Neumann Problem}\label{2.2}
In this subsection, we apply the Bernstein method to derive the global gradient estimate for Neumann problem.
\begin{lemma}\label{ggn}
If $u\in C^2(\overline{\mathbf R_+^n})$ satisfies \eqref{mse}, \eqref{lingro} and \eqref{neu}, then $|\nabla u|\in L^\infty(\mathbf R_+^n).$
\end{lemma}
\begin{proof}
According to the Lemma \ref{ggd}, we suffice to provide an upper bound for $|\nabla u|$ on $\partial\mathbf R_+^n$.
To this end, it is enough to show that
\begin{align}|\nabla u(0)|\leq C\label{gradu0}
\end{align} for some universal constant $C>0.$ To be explicit, for each $x_0\in\partial\mathbf R_+^n$ and $R>1+|x_0|$, we set $$
u_R(x)=\frac{1}{R}u(Rx+x_0)\ \ \ \hbox{for $x\in\overline{\mathbf R_+^n}.$}$$
Note that $u_R$ still satisfies \eqref{mse}, \eqref{lingro} and \eqref{neu}. Since $\nabla u(x_0)=\nabla u_R(0),$ we have $|\nabla u(x_0)|\leq C.$
We now prove \eqref{gradu0}. In what follows, we assume $\tau\geq0$ and $|\nabla u(0)|\geq(10+n+\tau)^{10}$. Restricting $u$ on $\overline{B_2^+(y_0)}$ for $y_0=(0,\ldots,0,1)\in\mathbf R^n,$ we may assume$$0\leq u\leq M=8K\ \hbox{in $B_2^+(y_0);$}$$ otherwise, we consider $u-\inf\limits_{B_2^+(y_0)}u$ instead. Set
\begin{align*}
&\eta(x)=\(1-\frac{|x-y_0|^2}{4}\)^2\ \ \ \hbox{for $x\in\overline{\mathbf R_+^n}$},\\
&\gamma(t)=1+\frac{t}{M}\ \ \ \hbox{for $0\leq t\leq M,$}\\
&w(x)=u(x)-\tau x_n\ \ \ \hbox{for $x\in\overline{\mathbf R_+^n}$},\\
&\Omega=\left\{x\in\overline{B_2^+(y_0)};\ |\nabla w|\geq(10+n+\tau)^{10}\right\},
\end{align*}
and we define $$\Phi(x)=\eta(x)\gamma(u(x))\log|\nabla w|^2\ \ \ \hbox{for $x\in\Omega.$}$$
Then $\Phi$ attains its maximum at some $y_1\in\Omega\setminus\partial B_2(y_0).$ The rest of the proof will be divided into three cases.
{\sl Case 1.} $y_1\in\partial\mathbf R_+^n.$ In this case,
$$(\log\Phi)_n(y_1)=\frac{4}{4-|y_1-y_0|^2}+\frac{\tau}{M+u(y_1)}>0,$$
which is a contradiction.
{\sl Case 2.} $y_1\in\partial\Omega.$ In this case, we also have $\log|\nabla w(0)|\leq C\log(10+n+\tau).$
{\sl Case 3.} $y_1$ is an interior point of $\Omega.$ Then we have $\nabla(\log\Phi)=0$ and $\nabla^2(\log\Phi)$ is negative definite at $y_1.$ In some neighborhood of $y_1$, the minimal surface equation can be written as
\begin{align}
\sum_{i,j=1}^na_{ij}(\nabla u)u_{ij}=0,\label{ndmse}
\end{align}
where each coefficient $a_{ij}$ is given by $$a_{ij}(p)=\delta_{ij}-\frac{p_ip_j}{1+|p|^2}\ \ \ \hbox{for $p=(p_1,\ldots,p_n)\in\mathbf R^n.$}$$
In order to simply our calculation, we choose a suitable coordinate such that
\begin{align}
&u_1(y_1)=|\nabla u(y_1)|>0,\,\,\, u_i(y_1)=0\ \ \ \hbox{for $2\leq i\leq n,$}\label{ui}\\
&u_{ij}(y_1)=\lambda_i\delta_{ij}\ \ \ \hbox{for $2\leq i,j\leq n,$}\label{uij}
\end{align}
then $w_1(y_1)\geq u_1(y_1)-\tau>0$ and $$\sum_{i=2}^n|w_i(y_1)|^2\leq\tau^2.$$
Under the new coordinate, $u$ still satisfies the minimal surface equation \eqref{ndmse}, it follows from \eqref{ui} and \eqref{uij} that
\begin{align}
&a_{11}(\nabla u(y_1))=\frac{1}{1+u_1^2},\ a_{ii}=1\ (i\neq1),\ a_{ij}(\nabla u(y_1))=0\ (i\neq j),\label{aij}\\
&\partial_{p_1}a_{11}(\nabla u(y_1))=-\frac{2u_1}{(1+u_1^2)^2},\ \partial_{p_i}a_{11}(\nabla u(y_1))=0\ (i\neq1),\label{pa11}\\
&\partial_{p_i}a_{1i}(\nabla u(y_1))=-\frac{u_1}{1+u_1^2}\ (i\neq1),\ \partial_{p_j}a_{1i}=0\ (j\neq1,i\neq j),\\
&\partial_{p_k}a_{ij}(\nabla u(y_1))=0\ (i\neq1,j\neq1).\label{paij}
\end{align}
In the following, we work at the point $y_1$ to evaluate $$\sum\limits_{i,j=1}^na_{ij}(\nabla u)(\log\Phi)_{ij}.$$
Through a simple differentiation, there holds
\begin{align}\label{Phi}
(\log\Phi)_i=\frac{\eta_i}{\eta}+\frac{\gamma'} {\gamma}u_i+\frac{(|\nabla w|^2)_i}{|\nabla w|^2\log|\nabla w|^2}=0.
\end{align}
Since
\begin{align*}
(\log\Phi)_{ij}=\frac{(|\nabla w|^2)_{ij}}{|\nabla w|^2\log|\nabla w|^2}-&(1+\log|\nabla w|^2)\frac{(|\nabla w|^2)_i(|\nabla w|^2)_j}{|\nabla w|^4\log^2|\nabla w|^2}\\
&+\frac{\eta_{ij}}{\eta}-\frac{\eta_i\eta_j}{\eta^2}+\left(\frac{\gamma''}{\gamma}- \frac{(\gamma')^2}{\gamma^2} \right)u_iu_j+\frac{\gamma'}{\gamma}u_{ij},
\end{align*}
by \eqref{Phi} and a direct substitution, we obtain
\begin{align}
\begin{split}
(\log\Phi)_{ij}=\frac{(|\nabla w|^2)_{ij}}{|\nabla w|^2\log|\nabla w|^2}-&(1+\log|\nabla w|^2)\left(\frac{\eta_i}{\eta}+\frac{\gamma'}{\gamma}u_i\right) \left(\frac{\eta_j}{\eta}+\frac{\gamma'}{\gamma}u_j\right)\\
&+ \frac{\eta_{ij}}{\eta}-\frac{\eta_i\eta_j}{\eta^2}+\left(\frac{\gamma''}{\gamma}- \frac{(\gamma')^2}{\gamma^2} \right)u_iu_j+\frac{\gamma'}{\gamma}u_{ij}.
\end{split}
\label{logPhi}
\end{align}
By combining \eqref{aij} and \eqref{logPhi}, there holds
\begin{align*}
\sum_{i,j=1}^na_{ij}(\nabla u)(\log\Phi)_{ij}=&\frac{1}{|\nabla w|^2\log|\nabla w|^2}\sum_{i,j=1}^na_{ij}(\nabla u)(|\nabla w|^2)_{ij}\\
&-(1+\log|\nabla w|^2)\sum_{i=1}^n\left\{\frac{1}{1+u_1^2}\left(\frac{\eta_1}{\eta}+\frac{\gamma'}{\gamma}u_1\right)^2+\sum_{i=2}^n\frac{\eta_i^2}{\eta^2}\right\}\\
&+\left\{\frac{1}{1+u_1^2}\left(\frac{\eta_{11}}{\eta}-\frac{\eta_1^2}{\eta^2}-\frac{u_1^2}{M^2\gamma^2}\right)+\sum_{i=2}^n\left(\frac{\eta_{ii}}{\eta}-\frac{\eta_i^2}{\eta^2}\right)\right\}\\
=:&I_1+I_2+I_3,
\end{align*}
then it is clear that $I_2$ is a negative term, we will use $I_1$ to bound $I_2$. We also note that $$|\nabla^2\eta|+\frac{|\nabla\eta|^2}{\eta}\leq C\ \ \ \hbox{in $B_2(y_0)$},$$
thus \begin{align}
I_3\geq-\(\frac{C}{\eta}+\frac{1}{M^2}\).&\label{I3}
\end{align}
A straightforward calculation yields that
\begin{align}
(|\nabla w|^2)_{i}&=2\sum_{k=1}^nu_{ik}w_k,\label{gw}\\
(|\nabla w|^2)_{ij}&=2\sum_{k=1}^n(u_{ik}u_{jk}+u_{ijk}w_k),\nonumber
\end{align}
thus we obtain from \eqref{uij} and \eqref{aij} that
\begin{align}
\sum_{i,j=1}^na_{ij}(\nabla u)(|\nabla w|^2)_{ij}=&2\sum_{i,k=1}^na_{ii}(\nabla u)u_{ik}^2+2\sum_{i,j,k=1}^na_{ij}(\nabla u)u_{ijk}w_k\nonumber\\
=&\frac{2}{1+u_1^2}\sum_{k=1}^nu_{1k}^2+2\sum_{i=2}^nu_{1i}^2+2\sum_{i=2}^nu_{ii}^2\nonumber\\
&+2\sum_{k=1}^n\sum_{i,j=1}^na_{ij}(\nabla u)u_{ijk}w_k.\label{uijk}
\end{align}
In order to eliminate the third derivatives of $u$ in \eqref{uijk}, we differentiate the minimal surface equation \eqref{ndmse} and get $$\sum_{i,j=1}^na_{ij}(\nabla u)u_{ijk}+\sum_{i,j,l=1}^n\partial_{p_l}a_{ij}(\nabla u)u_{ij}u_{kl}=0,$$ thus we obtain from \eqref{pa11}-\eqref{paij} that
\begin{align*}
\sum_{i,j=1}^na_{ij}(\nabla u)u_{ijk}=\frac{2u_1u_{11}u_{1k}}{(1+u_1^2)^2}+\frac{2u_1}{1+u_1^2}\sum_{j=2}^nu_{1j}u_{jk}\ \ \ \hbox{for $k=1,\ldots,n$}.\end{align*}
By a simple substition, we get
$$(|\nabla w|^2\log|\nabla w|^2)I_1=\sum_{i,j=1}^na_{ij}(\nabla u)(|\nabla w|^2)_{ij}=:J_1+J_2,$$
where
\begin{align*}
J_1&=\frac{(2+2u_1^2+4u_1w_1)u_{11}^2}{1+u_1^2}+\frac{4+2u_1^2+4u_1w_1}{1+u_1^2}\sum_{i=2}^nu_{1i}^2+2\sum_{i=2}^n\lambda_i^2,\\
J_2&=\frac{4u_1u_{11}}{(1+u_1^2)^2}\sum_{k=2}^nu_{1k}w_k+\frac{4u_1}{1+u_1^2}\sum_{i=2}^n\lambda_iu_{1i}w_i.
\end{align*}
We point out that $-|J_2|$ can be bounded by $J_1,$ to see this, we apply the Cauchy inequality to get\begin{align*}
\frac{4u_1}{(1+u_1^2)^2}\sum_{k=2}^n|u_{11}u_{1k}w_k|&\geq-\frac{4\tau u_1}{(1+u_1^2)^2}\sum_{k=2}^n|u_{11}u_{1k}|\\
&\geq-\frac{2\tau u_1}{(1+u_1^2)^2}\((n-1)u_{11}^2+\sum_{k=2}^nu_{1k}^2\),\\
\frac{4u_1}{1+u_1^2}\sum_{i=2}^n|\lambda_iu_{1i}w_i|&\geq-\frac{-4\tau u_1}{1+u_1^2}\sum_{i=2}^n|\lambda_iu_{1i}|\\
&\geq-\frac{2\tau u_1}{1+u_1^2}\(\sum_{i=2}^n\lambda_i^2+\sum_{i=2}^nu_{1i}^2\).
\end{align*}
Since $u_1\geq(10+n+\tau)^{10}$ and $$\frac{w_1}{u_1}\geq1-\frac{\tau}{u_1}\geq1-\frac{\tau}{(10+n+\tau)^{10}},$$
we have $$\frac{2+2u_1^2+4u_1w_1}{1+u_1^2}\geq\frac{11}{2},\ \hbox{and}\ \frac{2(n-1)\tau u_1}{(1+u_1^2)^2}+\frac{2\tau u_1}{1+u_1^2}\leq\frac{1}{5},$$ which imply
\begin{align}\(|\nabla w|^2\log|\nabla w|^2\)I_1\geq J_1-|J_2|\geq\frac{49}{10}\sum_{i=2}^nu_{1i}^2+\frac{49}{10}u_{11}^2+\frac{9}{5}\sum_{i=2}^n\lambda_i^2.&\label{I1}
\end{align}
Now we start to deal with the $\sum\limits_{i=2}^nu_{1i}^2$ and $u_{11}^2.$ By taking $i\geq2$ in \eqref{Phi}, we obtain from \eqref{uij} and \eqref{gw} that
\begin{align}
u_{1i}&=-\frac{\lambda_iw_i}{w_1}-\frac{\eta_i|\nabla w|^2\log|\nabla w|^2}{2\eta w_1}.\label{u1i}
\end{align}
By taking $i=1$ in \eqref{Phi} and using \eqref{u1i}, we have \begin{align*}
u_{11}&=-\sum_{j=2}^n\frac{u_{1j}w_j}{w_1}-\frac{1}{2w_1}\left(\frac{\eta_1}{\eta}+\frac{\gamma'u_1}{\gamma}\right)|\nabla w|^2\log|\nabla w|^2\nonumber\\
&=\sum_{j=2}^n\frac{\lambda_jw_j^2}{w_1^2}+\frac{1}{2w_1}\left(\sum_{j=2}^n\frac{\eta_jw_j}{\eta w_1}-\frac{\eta_1}{\eta}-\frac{\gamma'u_1}{\gamma}\right)|\nabla w|^2\log|\nabla w|^2.
\end{align*}
Hence, for $\varepsilon>w_1^{-4}$ to be determined, we have
\begin{align*}
\sum_{i=2}^nu_{1i}^2&=\sum_{i=2}^n\left(\frac{\lambda_iw_i}{w_1}+\frac{\eta_i|\nabla w|^2\log|\nabla w|^2}{2\eta w_1}\right)^2\\
&\geq\frac{1}{4w_1^2}\sum_{i=2}^n\frac{\eta_i^2|\nabla w|^4\log^2|\nabla w|^2}{\eta^2}+\sum_{i=2}^n\frac{\lambda_i\eta_iw_i|\nabla w|^2\log|\nabla w|^2}{\eta w_1^2}\\
&\geq-\varepsilon\tau^2\sum_{i=2}^n\lambda_i^2+\(\frac{1}{4w_1^2}-\frac{1}{4\varepsilon w_1^4}\)\sum_{i=2}^n\frac{\eta_i^2|\nabla w|^4\log^2|\nabla w|^2}{\eta^2},\\
u_{11}^2\geq&\(\sum_{j=2}^n\frac{\lambda_jw_j^2}{w_1^3}\)\cdot\left(\sum_{j=2}^n\frac{\eta_jw_j}{\eta w_1}-\frac{\eta_1}{\eta}-\frac{\gamma'u_1}{\gamma}\right)|\nabla w|^2\log|\nabla w|^2\\
&+\frac{1}{4w_1^2}\left(\sum_{j=2}^n\frac{\eta_jw_j}{\eta w_1}-\frac{\eta_1}{\eta}-\frac{\gamma'u_1}{\gamma}\right)^2|\nabla w|^4\log^2|\nabla w|^2\\
\geq&-\varepsilon\tau^4\sum_{j=2}^n\lambda_j^2+\(\frac{1}{4w_1^2}-\frac{1}{4\varepsilon w_1^6}\)\left(\sum_{j=2}^n\frac{\eta_jw_j}{\eta w_1}-\frac{\eta_1}{\eta}-\frac{\gamma'u_1}{\gamma}\right)^2|\nabla w|^4\log^2|\nabla w|^2\\
\geq&-\varepsilon\tau^4\sum_{j=2}^n\lambda_j^2+\(\frac{1}{4w_1^2}-\frac{1}{4\varepsilon w_1^6}\)|\nabla w|^4\log^2|\nabla w|^2\\
&\cdot\(-\varepsilon\tau^2\sum_{j=2}^n\frac{\eta_j^2}{\eta^2}+\(1-\frac{1}{\varepsilon w_1^2}\)\(\frac{\eta_1}{\eta}+\frac{\gamma'u_1}{\gamma}\)^2\).
\end{align*}
Taking $\varepsilon=4w_1^{-2}>2w_1^{-4},$ then $100\varepsilon<(1+\tau^2+\tau^4)^{-1}.$ Thus we obtain from \eqref{I1} that
\begin{align*}
I_1\geq&\frac{11}{8w_1^2}\(1-\varepsilon\tau^2-\frac{1}{\varepsilon w_1^2}\)\sum_{i=2}^n\frac{\eta_i^2|\nabla w|^2\log|\nabla w|^2}{\eta^2}\\
&+\frac{11}{8w_1^2}\(1-\frac{1}{\varepsilon w_1^6}\)\(1-\frac{1}{\varepsilon w_1^2}\)\(\frac{\eta_1}{\eta}+\frac{\gamma'u_1}{\gamma}\)^2|\nabla w|^2\log|\nabla w|^2\\
\geq&\frac{407\log|\nabla w|^2}{400}\sum_{i=2}^n\frac{\eta_i^2}{\eta^2}+\frac{99}{128}\(\frac{\eta_1}{\eta}+\frac{\gamma'u_1}{\gamma}\)^2\log|\nabla w|^2.
\end{align*}
Therefore,
\begin{equation}
\begin{split}
I_1+I_2&\geq\left(\frac{99\log|\nabla w|^2}{128}-\frac{1+\log|\nabla w|^2}{1+u_1^2}\right)\(\frac{\eta_1}{\eta}+\frac{\gamma'u_1}{\gamma}\)^2\\
&\geq\frac{\log|\nabla w|^2}{2}\(\frac{\eta_1}{\eta}+\frac{\gamma'u_1}{\gamma}\)^2.
\end{split}
\label{I1I2}
\end{equation}
We note that $$\sum_{i,j=1}^na_{ij}(\nabla u)(\log\Phi)_{ij}=I_1+I_2+I_3\leq0\ \ \ \hbox{at $y_1,$}$$ hence
\eqref{I1I2} and \eqref{I3} combined give that $$
\frac{\log|\nabla w|^2}{2}\(\frac{\eta_1}{\eta}+\frac{\gamma'u_1}{\gamma}\)^2\leq\frac{C}{\eta}+\frac{1}{M^2}.$$
To end this proof, two cases will be treated in what follows. First, if
$$
\left|\frac{\eta_1}{\eta u_1}\right|\leq \frac{\gamma'}{2\gamma},
$$
then
$$
\log |\nabla w|^2\leq C\frac{\gamma^2}{(\gamma')^2}\left(\frac{1}{\eta}+\frac{1}{M^2}\right).
$$
It follows
$$
\log|\nabla w(0)|^2\leq C\eta\log|\nabla w|^2\leq CM^2
$$
Second, if
$$
\left|\frac{\eta_1}{\eta u_1}\right|> \frac{\gamma'}{2\gamma},
$$
then
\begin{align*}
\log|\nabla w(0)|&\leq C\Phi(y_1)\leq C\eta u_1\leq\frac{2\gamma|\eta_1|}{\gamma'}\leq CM.
\end{align*}
To sum up, we have
$$
|\nabla w(0)|\leq\exp\left\{C(M^2+M)\right\}.$$
This completes the proof.
\end{proof}
\section{Proof of Main Theorems}
\begin{proof}[Proof of Theorem \ref{ln}]
For any $R>0$ and $x\in\overline{\mathbf R_+^n},$ we set $$u_R(x)=\frac{1}{R}u(Rx)\ \hbox{and}\ v_R(x)={\partial_{x_n} u_R}(x),$$ then it follows from Lemma \ref{ggn} that $|\nabla u_R|\in L^\infty(\mathbf R_+^n).$ A basic calculation shows that $v_R\in C(\overline{\mathbf R_+^n})\cap C^\infty(\mathbf R^n_+)$ satisfies following uniform elliptic equation with constant Dirichlet boundary value:
\begin{equation*}
\displaystyle\sum_{i, j=1}^n{\partial_{x_i}}\(b_{R, ij}{\partial_{x_j}v_R}\)=0\ \hbox{in $\mathbf R_+^n$,}\ \ \
v_R=\tau \ \hbox{on $\partial\mathbf R_+^n$,}
\end{equation*}
where $$b_{R,ij}={1\over\sqrt{1+|\nabla u_R|^2}}\(\delta_{ij}-{{\partial_{x_i}u_R}{\partial_{x_j}u_R}\over 1+|\nabla u_R|^2}\).$$
From \cite[Theorem 8.27, Theorem 8.29]{GilTr}, there exists $\alpha\in(0,1)$ such that
\begin{equation*}
\|v_R\|_{C^\alpha(\overline{B_1^+})}\leq C\|v_R\|_{L^2(B_2^+)}\leq C.
\end{equation*}
Therefore,
$$|v_R(x)-v_R(0)|\leq C|x|^\alpha \,\,\, \hbox{for any $x\in\overline{B_1^+} $},$$
which yields
\begin{equation}\label{hol}
|{\partial_{x_n}u}(y)-{\partial_{x_n}u}(0)|\leq C{|y|^\alpha\over R^\alpha} \,\,\, \hbox{for any $y\in\overline{B_R^+} $}.
\end{equation}
Fixing $y$ and letting $R\rightarrow \infty$ in \eqref{hol}, we know ${\partial_{x_n}u}$ is a constant in $\overline{\mathbf R_+^n}.$ Hence $u(x', x_n)=\tilde{u}(x')+\tau x_n$, where $\tilde{u}$ is a smooth function on $\mathbf R^{n-1}$. Further $\tilde{u}$ is an entire solution of the minimal surface equation in $\mathbf R^{n-1}$, which means $\tilde{u}$ is an affine function in $\mathbf R^{n-1}$ by Liouville theorem for entire minimal graph. The proof is finished.
\end{proof}
\begin{proof}[Proof of Theorem \ref{ld}]
Without loss of generality, we may assume that there exists $\beta=(\beta_1,\ldots,\beta_n)\in\mathbf R^n$ with $\beta_n=0$ such that $$l(x)=\langle\beta,x\rangle\ \ \ \hbox{for any $x\in\mathbf R_+^n.$}$$
Set $\bar u=u-l$, then $|\nabla\bar u|\in L^\infty(\mathbf R_+^n)$ and $\bar u\in C^2(\overline{\mathbf R_+^n})$ satisfies
\begin{equation*}
\sum_{i,j=1}^n\bar a_{ij}\partial_{x_ix_j}\bar u=0\ \hbox{in $\mathbf R_+^n$,}\ \ \ \bar u=0\ \hbox{on $\partial\mathbf R_+^n$,}
\end{equation*}
where $$\bar a_{ij}=\delta_{ij}-{(\partial_{x_i}\bar u+\beta_i)(\partial_{x_j}\bar u+\beta_j)\over{1+|\nabla\bar u+\beta|^2}}.$$
By the H\"{o}lder estimate for the normal derivatives of solutions on the boundary due to Krylov \cite{Kr} (see also \cite[Theorem 1.2.16]{Ha}), we get $$\partial_{x_n}u(x',0)=\partial_{x_n}\bar u(x',0)\equiv c\ \ \ \hbox{on $\partial\mathbf R_+^n,$}$$ where $c\in\mathbf R$ is a constant.
Then $u$ is an affine function by Theorem \ref{ln}.
\end{proof}
\begin{appendix}
\section{Bounded Convex Domain with $C^3$-boundary}\label{A}
In the following, we construct a bounded convex $C^3$-type domain, whose boundary contains $\Sigma_1$.
For $t\in[0,1],$ we set $$\psi(t)={64\over35}t^{1/2}-2t^2+{8\over5}t^3-{3\over7}t^4.$$ Then, $\psi:[0,1]\rightarrow[0,1]$ is continuous and concave. Straightforward calculations show that
\begin{align*}
&\psi>0,\, \psi'>0,\, \psi''<0\,\, \hbox{and}\,\, \psi'''>0\, \hbox{in $(0,1),$}\\
&\lim_{t\to0+}\psi'(t)=+\infty,\, \lim_{t\to1-}\psi'(t)=\psi'(1)=0,\\
&\lim_{t\to0+}\psi''(t)=-\infty,\, \lim_{t\to1-}\psi''(t)=\psi''(1)=0,\\
&\lim_{t\to0+}\psi'''(t)=+\infty,\, \lim_{t\to1-}\psi'''(t)=\psi'''(1)=0.\\
\end{align*}
For every $h\in[0,2],$ we define
\begin{equation*}
\tilde\psi(h)=
\begin{cases}
\psi(h) & \text{if $h\in[0,1],$}\\
\psi(2-h) & \text{if $h\in[1,2].$}\\
\end{cases}
\end{equation*}
and $$\Omega=\left\{(x',x_n)\in\mathbf R_+^n;\, |x'|<2+\tilde\psi(x_n),\, 0<x_n<2\right\},$$
we then claim that:
\medskip
\noindent{\bf Claim.} $\Omega\subset\mathbf R^n$ is a convex bounded domain with $C^3$-boundary.
\begin{proof} It is easy to see that $\Omega\subset B_5^+$ is a $C^3$-type domain. Let $(x',x_n)$ and $(y',y_n)$ be two points in $\Omega.$ We note that $\tilde\psi:[0,2]\to[0,1]$ is a concave function. For any $t\in[0,1]$, we obtain from the concavity of $\tilde\psi$ that
\begin{align*}
|tx'+(1-t)y'|&\leq t|x'|+(1-t)|y'|\\
&<t(2+\tilde\psi(x_n))+(1-t)(2+\tilde\psi(y_n))\\
&\leq2+t\tilde\psi(x_n)+(1-t)\tilde\psi(y_n)\\
&\leq2+\tilde\psi(tx_n+(1-t)y_n).
\end{align*}
Hence, we have $$(tx'+(1-t)y',tx_n+(1-t)y_n)\in\Omega,$$
which implies that $\Omega\subset\mathbf R^n$ is convex.
\end{proof}
\medskip
\section{Global Regularity for Solutions}\label{B}
In this section, we show that solutions of \eqref{mse}-\eqref{diri} are smooth up to the boundary $\partial\mathbf R_+^n,$ which is an immediate corollary of following general theorem.
\begin{theorem}
Let $\Omega\subset\mathbf R^n$ be a bounded domain with $C^3$-boundary satisfying $H_{\partial\Omega}\geq0$ on $\partial\Omega,$ where $H_{\partial\Omega}$ is the mean curvature of $\partial\Omega$ corresponding to the inner unit normal vector to $\partial\Omega.$ Suppose $T$ is a smooth portion of $\partial\Omega.$ For $\gamma\in(0,1)$ and $\varphi\in C(\partial\Omega)\cap C^{2,\gamma}(T),$ and $u\in C^2(\Omega)\cap C(\overline\Omega)$ solves the minimal surface equation \eqref{ndmse} in $\Omega$ with the Dirichlet boundary condition $u=\varphi$ on $\partial\Omega.$ Then,\ $u\in C^{2,\gamma}(\Omega\cup T).$ Furthermore, if $T$ is a smooth portion of $\partial\Omega$ and $\varphi\in C(\partial\Omega)\cap C^\infty(T),$ then $u\in C^\infty(\Omega\cup T).$
\end{theorem}
\begin{proof}
For $x_0\in T,$ put $$d={\rm dist}(x_0,\partial\Omega\setminus T)>0.$$
We suffice to show $\nabla u$ is well defined on $\overline{B_{d/16}(x_0)}\cap\overline\Omega,$ and $\nabla u\in C^{\alpha}(\overline{B_{d/16}(x_0)}\cap\overline\Omega)$ for some $\alpha\in(0,\gamma),$ which will implies $a_{ij}(\nabla u)\in C^\alpha(\overline{B_{d/16}(x_0)}\cap\overline\Omega).$ Then, by $\varphi\in C^{2,\gamma}(T)$, extension and \cite[Theorem 1.3.7]{Ha}, we know $u\in C^{2,\alpha}(\Omega\cup T),$ which implies $a_{ij}(\nabla u)\in C^{\gamma}(\Omega\cup T).$ Again we get $u\in C^{2,\gamma}(\Omega\cup T).$ As to the case that $T$ is smooth and $\varphi\in C^\infty(T),$ based on the \cite[Theorem 1.3.10]{Ha}, it will be finished by a bootstrap argument.
Let $\{\phi_k\}_{k=1}^\infty\subset C^{2,1/2}(\overline\Omega)$ be a sequence satisfying
\begin{align*}
&\phi_k\to\varphi\ \ \ \hbox{in $C(\partial\Omega),$}\\
&\phi_k\to u\ \ \ \hbox{in $C^{2,1/2}(\overline{B_{d/2}(x_0)}\cap\partial\Omega).$}
\end{align*}
We assume $$|\phi_k|_{L^\infty(\partial\Omega)}\leq M_0,\ |\phi_k|_{C^{2,1/2}(\overline{B_{d/2}(x_0)}\cap\partial\Omega)}\leq M_0$$ for some large constant $M_0>0.$ With each boundary value $\phi_k,$ we solve the minimal surface equation in $\Omega$ to obtain a solution $u_k\in C^{2,1/2}(\overline\Omega)$. It follows from comparison principle that $u_k\to u$ in $C(\overline\Omega)$ and
\begin{equation*}
|u_k|_{L^\infty(\Omega)}\leq M_0\ \ \ \hbox{for all $k\in\mathbf N_+.$}
\end{equation*}
{\sl Step 1.} We estimate the $L^\infty$-norm of $|\nabla u_k|$ near $T$. To this end, we choose a cut-off function $\rho\in C_0^\infty(B_{d/2}(x_0))$ such that
\begin{align*}
&\rho\equiv1\ \ \ \hbox{in $B_{d/4}(x_0),$}\\
&0\leq\rho\leq1\ \ \ \hbox{in $B_{d/2}(x_0),$}\\
&|\nabla\rho|\leq8/d\ \ \ \hbox{in $B_{d/2}(x_0).$}
\end{align*}
For each $k\in\mathbf N_+,$ we set $\psi_k=(1-\rho)\phi_k+\rho M_0,$ then $\phi_k\leq\psi_k$ on $\partial\Omega.$ As in the proof of Lemma \ref{bgd}, we construct the comparison function $v_k$ which is a solution of the minimal surface equation in $\Omega$ with the boundary value $\psi_k$ on $\partial\Omega.$ Hence, we obtain from Lemma \ref{bgd} that $$\partial_{x_n}u_k(x_0)\leq\partial_{x_n}v_k(x_0)\leq C(d,M_0),$$ thus we get $$|\nabla u_k|_{L^\infty(T\cap B_{d/4}(x_0))}\leq C(d,M_0),$$ where $C(d,M_0)$ are positive quantities depending only on $d$ and $M_0.$ Then, we can proceed similarly as in the proof of Lemma \ref{ggd} and apply the Bernstein technique to obtain $$|\nabla u_k|_{L^\infty(\Omega\cap B_{d/8}(x_0))}\leq C(d,M_0).$$
{\sl Step 2.} We provide an upper bound for $C^{1,\alpha}$-norms of $\{u_k\}$ near $x_0$. Since the proof is almost similar to the proof of \cite[Theorem 2.5.1]{Ha}, we sketch the procedure in the following.
{\sl Step 2.1.} Flattening the boundary $\partial\Omega$ near $x_0$, then applying \cite[Theorem 1.2.16]{Ha} to $u_k-\phi_k$ to obtain the boundary H\"{o}lder estimate for normal derivative of $u_k-\phi_k.$ Notice that the estimate is done near $x_0$ and $$|\phi_k|_{C^2(\overline\Omega\cap\overline{B_{d/8}(x_0)})}\leq C(M_0),$$ it follows from the first step of proof of \cite[Theorem 2.5.1]{Ha} that
\begin{equation}\label{B1}
[\nabla u]_{C^\beta(\overline T\cap\overline{B_{d/8}(x_0)})}\leq C(n,d,M_0,\Omega)
\end{equation}
for some $\beta=\beta(n,d,M_0,\Omega)\in(0,1).$
{\sl Step 2.2.} Using \eqref{B1} to prove that $$|\nabla u_k(x)-\nabla u_k(y)|\leq C(n,d,\beta,M_0,\Omega)|x-y|^{\beta/(1+\beta)}$$ for all $x\in\overline\Omega\cap\overline{B_{d/8}(x_0)}$ and $y\in \overline T\cap\overline{B_{d/16}(x_0)}.$
{\sl Step 2.3.} Based on the Step 2.2, directly following the Step 3 of the proof of \cite[Theorem 2.5.1]{Ha} to obtain $$[\nabla u_k]_{C^{\beta'}(\overline\Omega\cap\overline{B_{d/16}(x_0)})}\leq C(n,d,\beta,M_0,\Omega)$$ for some $\beta'\in(0,\beta).$
By the Arzel\`a-Ascoli theorem, we deduce that $\nabla u$ is well defined on $\overline\Omega\cap\overline{B_{d/16}(x_0)},$ and $\nabla u\in C^{\alpha}(\overline\Omega\cap\overline{B_{d/16}(x_0)})$ for any $\alpha\in(0,\beta').$
\end{proof}
\end{appendix}
\bigskip
\noindent
{\bf Acknowledgments.}
G. Jiang and Z. Wang would like to express their gratitude to Professor Qing Han for constant encouragements. J. Zhu is partially supported by NSFC grants No. 11671015 and 11731001. Authors would also like to thank Mr. Zhisu Li and Mr. Yongjie Liu for helpful discussions.
\bigskip
|
1,108,101,565,654 | arxiv | \section{Introduction}
Hypergeometric series with noncommutative parameters and argument,
in the special case involving square matrices, have been the subject
of recent study, see e.g.\ the papers by Duval and Ovsienko~\cite{DO},
Gr\"unbaum~\cite{G}, Tirao~\cite{T}, and some of the references
mentioned therein. Of course, this subject is also closely related to
the theory of orthogonal matrix polynomials which was initiated by
Krein~\cite{Kr} and has experienced a steady development, see
e.g.\ Dur\'an and L\'opez-Rodr\'{\i}guez~\cite{DL}.
Very recently, Tirao~\cite{T} considered a particular type
of a matrix valued hypergeometric function (which, in our terminology,
belongs to noncommutative hypergeometric series of ``type I'').
He showed, in particular, that the matrix valued hypergeometric
function satisfies the matrix valued hypergeometric differential
equation, and conversely that any solution of the latter is a
matrix valued hypergeometric function.
In \cite{S2}, the present author investigated hypergeometric and
basic hypergeometric series involving noncommutative parameters and
argument (short: {\em noncommutative hypergeometric series}, and
{\em noncommutative basic} or {\em $Q$-hypergeometric series})
over a unital ring $R$
(or, when considering nonterminating series, over a unital Banach
algebra $R$) from a different, nevertheless completely elementary,
point of view. These investigations were exclusively devoted to the
derivation of summation formulae (which quite surprisingly even exist
in the noncommutative case), aiming to build up a theory of explicit
identities analogous to the rich theory of identities for hypergeometric
and basic hypergeometric series in the classical, commutative case
(cf.\ \cite{Sl} and \cite{GR}). Two closely related types of
noncommmutative series, of ``type I'' and ``type II'', were considered
in \cite{S2}. Most of the summations obtained there concern terminating
series and were proved by induction. An exception are the
noncommutative extensions of the nonterminating $q$-binomial theorem
\cite[Th.~7.2]{S2} which were established using functional equations.
Aside from the latter and some conjectured $Q$-Gau{\ss} summations,
no other explicit summations for nonterminating noncommutative basic
hypergeometric series were given. Furthermore, noncommutative
{\em bilateral} basic hypergeometric series were not even considered.
In this paper, we define noncommutative bilateral basic hypergeometric
series of type I and type II (over an abstract unital Banach algebra $R$)
and prove, using functional equations, noncommutative extensions of
Ramanujan's ${}_1\psi_1$ summation. These generalize the noncommutative
$Q$-binomial theorem of \cite[Th.~7.2]{S2}. Our proof of the
${}_1\psi_1$ sum here is similar to Andrews and Askey's~\cite{AA} proof
in the classical commutative case. Ramanujan's ${}_1\psi_1$ summation
(displayed in (\ref{11gl})) is one of the fundamental identities in
$q$-series. It is thus just natural to look for different extensions,
including noncommutative ones.
This paper is organized as follows. In Section~\ref{secprel} we review
some standard notations for basic hypergeometric series and then explain
the notation we utilize in the noncommutative case. Section~\ref{sec11}
is devoted to the derivation of noncommutative ${}_1\psi_1$ summations.
We stress again, as in \cite{S2}, that by ``noncommutative'' we do not
mean ``$q$-com\-mutative'' or ``quasi-commutative'' (i.e., where the
variables satisfy a relation like $yx=qxy$; such series are considered
e.g.\ in \cite{K} and \cite{V}) but that the parameters in our series
are elements of some noncommutative unital ring (or unital Banach algebra).
\section{Preliminaries}\label{secprel}
\subsection{Classical (commutative) basic hypergeometric series}
For convenience, we recall some standard notations for basic
hypergeometric series (cf.\ \cite{GR}). When considering the
noncommutative extensions in Subsection~\ref{subsecnc} and in
Section~\ref{sec11}, the reader may find it useful to compare
with the classical, commutative case.
Let $q$ be a complex number such that $0<|q|<1$. Define the
{\em $q$-shifted factorial} for all integers $k$ (including
infinity) by
$$
(a;q)_k:=\prod_{j=1}^k(1-aq^j).
$$
We write
\begin{equation}\label{defhyp}
{}_r\phi_{r-1}\!\left[\!\begin{array}{c}a_1,a_2,\dots,a_r\\
b_1,b_2,\dots,b_{r-1}\end{array}\!;q,z\right]:=
\sum _{k=0}^{\infty}\frac {(a_1;q)_k(a_2;q)_k\dots (a_r;q)_k}
{(q,q)_k(b_1;q)_k\dots(b_{r-1};q)_k}\,z^k,
\end{equation}
to denote the (unilateral)
{\em basic hypergeometric ${}_r\phi_{r-1}$ series}.
Further, we write
\begin{equation}\label{defbhyp}
{}_r\psi_r\!\left[\!\begin{array}{c}a_1,a_2,\dots,a_r\\
b_1,b_2,\dots,b_r\end{array}\!;q,z\right]:=
\sum_{k=-\infty}^{\infty}\frac {(a_1;q)_k\dots (a_r;q)_k}
{(b_1;q)_k\dots(b_r;q)_k}\,z^k,
\end{equation}
to denote the {\em bilateral basic hypergeometric ${}_r\psi_r$ series}.
In (\ref{defhyp}) and (\ref{defbhyp}), $a_1,\dots,a_r$ are
called the {\em upper parameters}, $b_1,\dots,b_r$ the
{\em lower parameters}, $z$ is the {\em argument}, and $q$ the
{\em base} of the series.
The bilateral ${}_r\psi_r$ series in (\ref{defbhyp}) reduces to
a unilateral ${}_r\phi_{r-1}$ series if one of the lower parameters,
say $b_r$, equals $q$ (or more generally, an integral power of $q$).
The basic hypergeometric ${}_r\phi_{r-1}$ series
terminates if one of the upper parameters, say $a_r$, is of the
form $q^{-n}$, for a nonnegative integer $n$.
If the basic hypergeometric series does not terminate then it converges
by the ratio test when $|z|<1$.
Similarly, the bilateral basic hypergeometric series
converges when $|z|<1$ and $|b_1\dots b_r/a_1\dots a_rz|<1$.
We recall two important summations. One of them
is the (nonterminating) $q$-binomial theorem,
\begin{equation}\label{10gl}
{}_1\phi_0\!\left[\!\begin{array}{c}a\\-\end{array}\!;q,z\right]
=\frac{(az;q)_\infty}{(z;q)_\infty},
\end{equation}
where $|z|<1$ (cf.\ \cite[Appendix~(II.3)]{GR}).
It was discovered independently by several
mathematicians, including Cauchy, Gau{\ss}, and Heine.
A bilateral extension of (\ref{10gl}) was found by the legendary
Indian mathematician Ramanujan (see Hardy~\cite{Hr}),
\begin{equation}\label{11gl}
{}_1\psi_1\!\left[\!\begin{array}{c}a\\b\end{array}\!;q,z\right]
=\frac{(q;q)_\infty(b/a;q)_\infty(az;q)_\infty(q/az;q)_\infty}
{(b;q)_\infty(q/a;q)_\infty(z;q)_\infty(b/az;q)_\infty},
\end{equation}
where $|z|<1$ and $|b/az|<1$ (cf.\ \cite[Appendix~(II.29)]{GR}).
Unfortunately, Ramanujan (who very rarely gave any proofs)
did not provide a proof of the above bilateral summation.
The first proof of (\ref{11gl}) was given by
Hahn~\cite[$\kappa=0$ in Eq.~(4.7)]{H}.
Other proofs were given by Jackson~\cite{J}, Ismail~\cite{I},
Andrews and Askey~\cite{AA}, the author~\cite[Sec.~3]{S1}, and others.
Some immediate applications of Ramanujan's summation formula to
arithmetic number theory are considered in \cite[Sec.~10.6]{AAR}.
\subsection{Noncommutative basic hypergeometric series}\label{subsecnc}
Most of the following definitions are taken from \cite{S2}.
However, the definitions for noncommutative {\em bilateral}
basic hypergeometric series in (\ref{defncbhypIQ}) and
(\ref{defncbhypIIQ}) (although obvious) are new.
Let $R$ be a unital ring (i.e., a ring with a multiplicative identity).
When considering infinite series and infinite products of elements
of $R$ we shall further assume that $R$ is a Banach algebra
(with some norm $\|\cdot\|$). The elements of $R$ will be denoted by
capital letters $A,B,\dots$. In general these elements do not commute
with each other; however, we may sometimes specify certain commutation
relations explicitly. We denote the identity by $I$ and the zero element
by $O$. Whenever a multiplicative inverse element exists for any $A\in R$,
we denote it by $A^{-1}$. (Since $R$ is a unital ring, we have
$AA^{-1}=A^{-1}A=I$.) On the other hand, as we shall implicitly assume
that all the expressions which appear are well defined, whenever
we write $A^{-1}$ we assume its existence.
For instance, in (\ref{defncpochIQ}) and (\ref{defncpochIIQ})
we assume that $I-B_iQ^j$ is invertible
for all $1\le i\le r$, $0\le j<k$.
An important special case is when $R$ is the ring of $n\times n$
square matrices (our notation is certainly suggestive with respect
to this interpretation), or, more generally, one may view
$R$ as a space of some abstract operators.
Let $\mathsf Z$ be the set of integers.
For $l,m\in\mathsf Z\cup\{\pm\infty\}$ we define
the noncommutative product as follows:
\begin{equation}\label{ncprod}
\prod_{j=l}^mA_j=\left\{
\begin{array}{ll}1
&m=l-1\\
A_lA_{l+1}\dots A_m&m\ge l\\
A_{l-1}^{-1}A_{l-2}^{-1}\dots A_{m+1}^{-1}&m<l-1
\end{array}\right..
\end{equation}
Note that
\begin{equation}\label{invprod}
\prod_{j=l}^mA_j=\prod_{j=m+1}^{l-1}A_{m+l-j}^{-1},
\end{equation}
for all $l,m\in\mathsf Z\cup\{\pm\infty\}$.
Throughout this paper, $Q$ will be a parameter which commutes with
any of the other parameters appearing in the series.
(For instance, a central element such as $Q=qI$, a scalar multiple
of the unit element in $R$,
for $qI\in R$, trivially satisfies this requirement.)
Let $k\in\mathsf Z\cup\{\infty\}$.
We define the generalized {\em noncommutative $Q$-shifted factorial
of type I}\/ by
\begin{equation}\label{defncpochIQ}
\left\lceil\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rfloor_k:=
\prod_{j=1}^k\left[\left(\prod_{i=1}^r
(I-B_iQ^{k-j})^{-1}(I-A_iQ^{k-j})\right)Z\right].
\end{equation}
Similarly, we define the generalized
{\em noncommutative $Q$-shifted factorial of type II}\/ by
\begin{equation}\label{defncpochIIQ}
\left\lfloor\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rceil_k:=
\prod_{j=1}^k\left[\left(\prod_{i=1}^r
(I-B_iQ^{j-1})^{-1}(I-A_iQ^{j-1})\right)Z\right].
\end{equation}
Note the unusual usage of brackets (``floors'' and ``ceilings''
are intermixed) on the left-hand sides of (\ref{defncpochIQ}) and
(\ref{defncpochIIQ}) which is intended to suggest that the products
involve noncommuting factors in a prescribed order.
In both cases, the product,
read from left to right, starts with a denominator factor.
The brackets in the form ``$\lceil-\rfloor$'' are intended to denote
that the factors are {\em falling},
while in ``$\lfloor-\rceil$'' that they are {\em rising}.
We define the {\em noncommutative basic
hypergeometric series of type I}\/ by
\begin{equation}\label{defnchypIQ}
{}_r\phi_{r-1}\!\left\lceil\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_{r-1}\end{array}\!;Q,Z\right\rfloor:=
\sum_{k\ge 0}
\left\lceil\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_{r-1},Q\end{array}\!;Q,Z\right\rfloor_k,
\end{equation}
and the {\em noncommutative basic
hypergeometric series of type II}\/ by
\begin{equation}\label{defnchypIIQ}
{}_r\phi_{r-1}\!\left\lfloor\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_{r-1}\end{array}\!;Q,Z\right\rceil:=
\sum_{k\ge 0}
\left\lfloor\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_{r-1},Q\end{array}\!;Q,Z\right\rceil_k.
\end{equation}
Further, we define the {\em noncommutative bilateral basic
hypergeometric series of type I}\/ by
\begin{equation}\label{defncbhypIQ}
{}_r\psi_r\!\left\lceil\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rfloor:=
\sum_{k=-\infty}^\infty
\left\lceil\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rfloor_k,
\end{equation}
and the {\em noncommutative bilateral basic
hypergeometric series of type II}\/ by
\begin{equation}\label{defncbhypIIQ}
{}_r\psi_r\!\left\lfloor\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rceil:=
\sum_{k=-\infty}^\infty
\left\lfloor\!\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rceil_k.
\end{equation}
We also refer to the respective series as
{\em (noncommutative) $Q$-hypergeometric series}.
In each case (of type I and type II), the ${}_r\phi_{r-1}$ series
terminates if one of the upper parameters $A_i$ is of the form $Q^{-n}$.
If the ${}_r\phi_{r-1}$ series does not terminate, then
(implicitly assuming that $R$ is a unital Banach algebra with some
norm $\|\cdot\|$) it converges when $\|Z\|<1$.
Similarly, the ${}_r\psi_r$ series converges in $R$ when $\|Z\|<1$ and
$\|Z^{-1}\prod_{i=1}^rA_{r+1-i}^{-1}B_{r+1-i}\|<1$.
We also consider reversed (or ``transposed'') versions of
generalized noncommutative $Q$-shifted factorials and
noncommutative bilateral basic hypergeometric series of type I and II.
These are defined as follows (compare with (\ref{defncpochIQ}),
(\ref{defncpochIIQ}), (\ref{defncbhypIQ}) and (\ref{defncbhypIIQ})):
\begin{equation}\label{defncpochIr}
\phantom{xy}^{\scriptstyle\sim\atop{}}\!\!\left\lceil\!
\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rfloor_k:=
\prod_{j=1}^k\left(Z\prod_{i=1}^r(I-A_iQ^{j-1})
(I-B_iQ^{j-1})^{-1}\right),
\end{equation}
\begin{equation}\label{defncpochIIr}
\phantom{xy}^{\scriptstyle\sim\atop{}}\!\!\left\lfloor\!
\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rceil_k:=
\prod_{j=1}^k\left(Z\prod_{i=1}^r(I-A_iQ^{k-j})
(I-B_iQ^{k-j})^{-1}\right),
\end{equation}
\begin{equation}\label{defnchypIr}
{}_r\psi_r\!^{\scriptstyle\sim\atop{}}\!\!\left\lceil\!
\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rfloor:=
\sum_{k=-\infty}^\infty
{}^{\scriptstyle\sim\atop{}}\!\!\left\lceil\!
\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rfloor_k,
\end{equation}
and
\begin{equation}\label{defnchypIIr}
{}_r\psi_r\!^{\scriptstyle\sim\atop{}}\!\!\left\lfloor\!
\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rceil:=
\sum_{k=-\infty}^\infty
{}^{\scriptstyle\sim\atop{}}\!\!\left\lfloor\!
\begin{array}{c}A_1,A_2,\dots,A_r\\
B_1,B_2,\dots,B_r\end{array}\!;Q,Z\right\rceil_k.
\end{equation}
Of course, reversed versions of the unilateral noncommutative basic
hypergeometric series are defined analogously.
\section{Noncommutative ${}_1\psi_1$ summations}\label{sec11}
In \cite[Th.~7.2]{S2}, the following two noncommutative extensions of
the nonterminating $q$-binomial theorem (which generalize \cite[II.3]{GR})
were given.
\begin{proposition}\label{nc1f0Q}
Let $A$ and $Z$ be noncommutative parameters of some unital
Banach algebra, and suppose that $Q$ commutes with both $A$ and $Z$.
Further, assume that $\|Z\|<1$. Then we have the following
summation for a noncommutative basic hypergeometric series of type I.
\begin{equation}\label{nc1f0QIgl}
{}_1\phi_0\!\left\lceil\!\begin{array}{c}A\\-\end{array}\!;
Q,Z\right\rfloor=
\left\lfloor\!\begin{array}{c}AZ\\Z\end{array}\!;Q,I\right\rceil_\infty.
\end{equation}
Further, we we have the following
summation for a noncommutative basic hypergeometric series of type II.
\begin{equation}\label{nc1f0QIIgl}
{}_1\phi_0\!\left\lfloor\!\begin{array}{c}A\\-\end{array}\!;Q,Z\right\rceil=
{}^{\scriptstyle\sim\atop{}}\!\!
\left\lfloor\!\begin{array}{c}AZ\\Z\end{array}\!;Q,I\right\rceil_\infty.
\end{equation}
\end{proposition}
Here we extend Proposition~\ref{nc1f0Q} to summations
for bilateral series. Our proof is similar to that of the
classical result given in \cite{AA} (see also
\cite[p.~502, first proof of Th.~10.5.1]{AAR}), but also
similar to the proof of Proposition~\ref{nc1f0Q} given in \cite{S2}.
\begin{theorem}\label{nc11}
Let $A$, $B$ and $Z$ be noncommutative parameters of some unital
Banach algebra, suppose that $Q$ and $B$ both commute with any of the
other parameters. Further, assume that $\|Z\|<1$ and $\|BZ^{-1}A^{-1}\|<1$.
Then we have the following summation for a noncommutative
bilateral basic hypergeometric series of type I.
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{nc11Igl}
&&{}_1\psi_1\!\left\lceil\!
\begin{array}{c}A\\B\end{array}\!;Q,Z\right\rfloor=\\\nonumber&&
\left\lfloor\!\begin{array}{c}Q,BZ^{-1}A^{-1}Z\\
B,BZ^{-1}A^{-1}\end{array}\!;Q,I\right\rceil_\infty
\left\lceil\!\begin{array}{c}Z^{-1}A^{-1}Q\\
Z^{-1}A^{-1}ZQ\end{array}\!;Q,I\right\rfloor_\infty
\left\lfloor\!\begin{array}{c}AZ\\Z\end{array}\!;Q,I\right\rceil_\infty.
\end{eqnarray}}
Further, we have the following summation for a noncommutative
bilateral basic hypergeometric series of type II.
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{nc11IIgl}
&&{}_1\psi_1\!\left\lfloor\!
\begin{array}{c}A\\B\end{array}\!;
Q,Z\right\rceil=\\\nonumber
&&{}^{\scriptstyle\sim\atop{}}\!\!
\left\lfloor\!\begin{array}{c}AZ\\Z\end{array}\!;Q,I\right\rceil_\infty
\left\lfloor\!\begin{array}{c}Z^{-1}A^{-1}Q,Q\\
B,A^{-1}Q\end{array}\!;Q,I\right\rceil_\infty
{}^{\scriptstyle\sim\atop{}}\!\!\left\lfloor\!\begin{array}{c}BA^{-1}\\
BZ^{-1}A^{-1}\end{array}\!;Q,I\right\rceil_\infty.
\end{eqnarray}}
\end{theorem}
Clearly, Theorem~\ref{nc11} reduces to Proposition~\ref{nc1f0Q} when $B=Q$.
{\em Proof of Theorem~\ref{nc11}.}
We prove (\ref{nc11Igl}), leaving the proof of (\ref{nc11IIgl}),
which is similar,
to the reader.
Let $f(A,B,Z)$ denote the series on the left-hand side of
(\ref{nc11Igl}). We make use of the two simple identities
\begin{subequations}
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{fq1}
Z&=&AZQ^k+(I-AQ^k)Z,\\\label{fq2}
I&=&BQ^k+(I-BQ^k),
\end{eqnarray}}
\end{subequations}
to obtain two functional equations for $f$.
We also make use of the simple relation
\begin{equation}\label{fq}
f(A,B,Z)=f(AQ,BQ,Z)(I-B)^{-1}(I-A)Z,
\end{equation}
obtained by shifting the summation index in $f$ by one.
First, (\ref{fq1}) gives
\begin{equation}\label{fq11}
Zf(A,B,Z)=AZf(A,B,ZQ)+f(AQ,B,Z)(I-A)Z,
\end{equation}
while (\ref{fq2}) gives
\begin{equation}\label{fq21}
f(A,BQ,Z)=Bf(A,BQ,ZQ)+(I-B)f(A,B,Z).
\end{equation}
Combining (\ref{fq21}), (\ref{fq11}), and (\ref{fq}), one readily deduces
{\setlength\arraycolsep{2pt}\begin{eqnarray*}
f(A,BQ,Z)&=&(I-B)f(A,B,Z)+BZ^{-1}A^{-1}Zf(A,BQ,Z)\\&&{}
-BZ^{-1}A^{-1}f(AQ,BQ,Z)(I-A)Z\\&=&
(I-B)f(A,B,Z)+BZ^{-1}A^{-1}Zf(A,BQ,Z)\\&&{}
-BZ^{-1}A^{-1}f(A,B,Z)(I-B),
\end{eqnarray*}}
or equivalently
$$
(I-BZ^{-1}A^{-1}Z)f(A,BQ,Z)=(I-B)(I-BZ^{-1}A^{-1})f(A,B,Z),
$$
thus
\begin{equation}\label{fq0}
f(A,B,Z)=(I-BZ^{-1}A^{-1})^{-1}(I-BZ^{-1}A^{-1}Z)(I-B)^{-1}
f(A,BQ,Z).
\end{equation}
Iteration of (\ref{fq0}) gives
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{fq01}
&&f(A,B,Z)=\\\nonumber&&\prod_{j=0}^\infty\left[(I-BZ^{-1}A^{-1}Q^j)^{-1}
(I-BZ^{-1}A^{-1}ZQ^j)(I-BQ^j)^{-1}\right]
f(A,O,Z).
\end{eqnarray}}
We still need to compute $f(A,O,Z)$. It is not easy to do this directly
but we know the value of $f(A,Q,Z)$ (by Proposition~\ref{nc1f0Q}).
We set $B=Q$ in (\ref{fq01}) which gives
{\setlength\arraycolsep{2pt}\begin{eqnarray*}
&&f(A,Q,Z)=\\\nonumber&&\prod_{j=0}^\infty\left[(I-Z^{-1}A^{-1}Q^{j+1})^{-1}
(I-Z^{-1}A^{-1}ZQ^{j+1})(I-Q^{j+1})^{-1}\right]
f(A,O,Z),
\end{eqnarray*}}
thus we obtain
\begin{equation}\label{fq03}
f(A,O,Z)=\bigg[\prod_{j=0}^\infty(I-Q^{j+1})\bigg]
\left\lceil\!\begin{array}{c}Z^{-1}A^{-1}Q\\
Z^{-1}A^{-1}ZQ\end{array}\!;Q,I\right\rfloor_\infty
f(A,Q,Z).
\end{equation}
Combination of (\ref{fq01}), (\ref{fq03}) and (\ref{nc1f0QIgl})
establishes the result. \endproof
In the ${}_1\psi_1$ summations of Theorem~\ref{nc11}, the lower parameter
$B$ commutes with both $A$ and $Z$ while $A$ does not commute with $Z$.
In the next theorem the roles of $A$ and $B$ are interchanged. Here $A$
commutes with both $B$ and $Z$ while $B$ does not commute with $Z$.
\begin{theorem}\label{nc11r}
Let $A$, $B$ and $Z$ be noncommutative parameters of some Banach algebra,
suppose that $Q$ and $A$ both commute with any of the other parameters.
Further, assume that $\|Z\|<1$ and $\|BZ^{-1}A^{-1}\|<1$.
Then we have the following summation for a noncommutative
bilateral basic hypergeometric series of type I.
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{nc11rIgl}
&&{}_1\psi_1\!\left\lceil\!
\begin{array}{c}A\\B\end{array}\!;Q,Z\right\rfloor=\\\nonumber&&
Z^{-1}\left\lfloor\!\begin{array}{c}Q,ZBZ^{-1}A^{-1}\\
A^{-1}Q,Z\end{array}\!;Q,I\right\rceil_\infty
\left\lceil\!\begin{array}{c}AZ\\
ZBZ^{-1}\end{array}\!;Q,I\right\rfloor_\infty
\left\lfloor\!\begin{array}{c}Z^{-1}A^{-1}Q\\
BZ^{-1}A^{-1}\end{array}\!;Q,I\right\rceil_\infty Z.
\end{eqnarray}}
Further, we have the following summation for a noncommutative
bilateral basic hypergeometric series of type II.
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{nc11rIIgl}
&&{}_1\psi_1\!\left\lfloor\!
\begin{array}{c}A\\B\end{array}\!;
Q,Z\right\rceil=\\\nonumber
&&Z^{-1}{}^{\scriptstyle\sim\atop{}}\!\!
\left\lfloor\!\begin{array}{c}Z^{-1}A^{-1}Q\\
BZ^{-1}A^{-1}\end{array}\!;Q,I\right\rceil_\infty
\left\lfloor\!\begin{array}{c}AZ,Q\\
A^{-1}Q,B\end{array}\!;Q,I\right\rceil_\infty
{}^{\scriptstyle\sim\atop{}}\!\!\left\lfloor\!\begin{array}{c}A^{-1}B\\
Z\end{array}\!;Q,I\right\rceil_\infty Z.
\end{eqnarray}}
\end{theorem}
\begin{proof}
We indicate the derivation of (\ref{nc11rIgl}) from (\ref{nc11Igl}).
(The derivation of (\ref{nc11rIIgl}) from (\ref{nc11IIgl}) is analogous.)
The sum on the left-hand side of (\ref{nc11Igl}) remains unchanged
if the summation index, say $k$, is replaced by $-k$. Using (\ref{invprod}),
we compute
{\setlength\arraycolsep{2pt}\begin{eqnarray}
\left\lceil\!
\begin{array}{c}A\\B\end{array}\!;Q,Z\right\rfloor_{-k}&=&
\prod_{j=1}^{-k}\left[
(I-BQ^{-k-j})^{-1}(I-AQ^{-k-j})Z\right]\\\nonumber
&=&\prod_{j=1-k}^0\left[(I-BQ^{-1+j})^{-1}
(I-AQ^{-1+j})Z\right]^{-1}\\\nonumber
&=&\prod_{j=1}^k\left[Z^{-1}(I-AQ^{-1-k+j})^{-1}
(I-BQ^{-1-k+j})\right]\\\nonumber
&=&\prod_{j=1}^k\left[Z^{-1}A^{-1}(I-A^{-1}Q^{1+k-j})^{-1}
(I-B^{-1}Q^{1+k-j})B\right]\\\nonumber
&=&Z^{-1}A^{-1}\left\lceil\!
\begin{array}{c}B^{-1}Q\\
A^{-1}Q\end{array}\!;Q,BZ^{-1}A^{-1}\right\rfloor_k AZ.
\end{eqnarray}}
Thus, by performing the simultaneous replacements $A\mapsto B^{-1}Q$,
$B\mapsto A^{-1}Q$, $Z\mapsto BZ^{-1}A^{-1}$, in (\ref{nc11Igl}),
we obtain (\ref{nc11rIgl}).
\end{proof}
We complete this paper with four more ${}_1\psi_1$ summations,
immediately obtained from corresponding summations in
Theorems~\ref{nc11} and \ref{nc11r} by ``reversing all products''
(cf.\ \cite[Subsec.~8.2]{S2}) on each side of the respective identities.
\begin{theorem}\label{nc11t}
Let $A$, $B$ and $Z$ be noncommutative parameters of some unital
Banach algebra, suppose that $Q$ and $B$ both commute with any of the
other parameters. Further, assume that $\|Z\|<1$ and $\|A^{-1}Z^{-1}B\|<1$.
Then we have the following summation for a reversed noncommutative
bilateral basic hypergeometric series of type I.
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{nc11tIgl}
&&{}_1\psi_1^{\scriptstyle\sim\atop{}}\!\!\left\lceil\!
\begin{array}{c}A\\B\end{array}\!;Q,Z\right\rfloor=\\\nonumber&&
{}^{\scriptstyle\sim\atop{}}\!\!
\left\lfloor\!\begin{array}{c}ZA\\Z\end{array}\!;Q,I\right\rceil_\infty
{}^{\scriptstyle\sim\atop{}}\!\!\left\lceil\!\begin{array}{c}A^{-1}Z^{-1}Q\\
ZA^{-1}Z^{-1}Q\end{array}\!;Q,I\right\rfloor_\infty
{}^{\scriptstyle\sim\atop{}}\!\!
\left\lfloor\!\begin{array}{c}ZA^{-1}Z^{-1}B,Q\\
A^{-1}Z^{-1}B,B\end{array}\!;Q,I\right\rceil_\infty.
\end{eqnarray}}
Further, we have the following summation for a noncommutative
bilateral basic hypergeometric series of type II.
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{nc11tIIgl}
&&{}_1\psi_1^{\scriptstyle\sim\atop{}}\!\!\left\lfloor\!
\begin{array}{c}A\\B\end{array}\!;
Q,Z\right\rceil=\\\nonumber
&&\left\lfloor\!\begin{array}{c}A^{-1}B\\
A^{-1}Z^{-1}B\end{array}\!;Q,I\right\rceil_\infty
{}^{\scriptstyle\sim\atop{}}\!\!
\left\lfloor\!\begin{array}{c}Q,A^{-1}Z^{-1}Q\\
A^{-1}Q,B\end{array}\!;Q,I\right\rceil_\infty
\left\lfloor\!\begin{array}{c}ZA\\Z\end{array}\!;Q,I\right\rceil_\infty.
\end{eqnarray}}
\end{theorem}
\begin{theorem}\label{nc11tr}
Let $A$, $B$ and $Z$ be noncommutative parameters of some Banach algebra,
suppose that $Q$ and $A$ both commute with any of the other parameters.
Further, assume that $\|Z\|<1$ and $\|A^{-1}Z^{-1}B\|<1$.
Then we have the following summation for a noncommutative
bilateral basic hypergeometric series of type I.
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{nc11trIgl}
\phantom{xyz}&&{}_1\psi_1^{\scriptstyle\sim\atop{}}\!\!\left\lceil\!
\begin{array}{c}A\\B\end{array}\!;Q,Z\right\rfloor=\\\nonumber&&
Z{}^{\scriptstyle\sim\atop{}}\!\!
\left\lfloor\!\begin{array}{c}A^{-1}Z^{-1}Q\\
A^{-1}Z^{-1}B\end{array}\!;Q,I\right\rceil_\infty
{}^{\scriptstyle\sim\atop{}}\!\!\left\lceil\!\begin{array}{c}ZA\\
Z^{-1}BZ\end{array}\!;Q,I\right\rfloor_\infty
{}^{\scriptstyle\sim\atop{}}\!\!
\left\lfloor\!\begin{array}{c}A^{-1}Z^{-1}BZ,Q\\
Z,A^{-1}Q\end{array}\!;Q,I\right\rceil_\infty Z^{-1}.
\end{eqnarray}}
Further, we have the following summation for a noncommutative
bilateral basic hypergeometric series of type II.
{\setlength\arraycolsep{2pt}\begin{eqnarray}\label{nc11trIIgl}
&&{}_1\psi_1^{\scriptstyle\sim\atop{}}\!\!\left\lfloor\!
\begin{array}{c}A\\B\end{array}\!;
Q,Z\right\rceil=\\\nonumber
&&Z\left\lfloor\!\begin{array}{c}BA^{-1}\\
Z\end{array}\!;Q,I\right\rceil_\infty
{}^{\scriptstyle\sim\atop{}}\!\!\left\lfloor\!\begin{array}{c}Q,ZA\\
B,A^{-1}Q\end{array}\!;Q,I\right\rceil_\infty
\left\lfloor\!\begin{array}{c}A^{-1}Z^{-1}Q\\
A^{-1}Z^{-1}B\end{array}\!;Q,I\right\rceil_\infty
Z^{-1}.
\end{eqnarray}}
\end{theorem}
|
1,108,101,565,655 | arxiv | \section{Conclusion}
In this paper, we proposed $\mu$-cuDNN, a wrapper library for cuDNN , which divides the mini-batch to utilize high-performance convolution algorithms with limited amount of memory for workspace.
We have shown that $\mu$-cuDNN \ works well even with recent CNNs, which are composed of many convolutional layers, and can easily be integrated into existing deep learning frameworks.
The performance of $\mu$-cuDNN \ demonstrated in our work suggests that other layer types can be optimized as well, if they can be decomposed and computed by different algorithms. This is because $\mu$-cuDNN \ does not use any special properties of the convolution operator, apart from gradient accumulation.
In addition, the result of WD \ optimization (\figref{figure:caffe_time_alexnet_ws}) provides us with the insight that allocating the same workspace memory for each convolutional layer is not necessarily effective, and dynamic, adaptive assignment performs better.
This observation should be beneficial for advanced deep learning frameworks that dynamically manage GPU memory to store tensors such as neuron data, weights and their gradients, for further memory optimization.
\section{The Anatomy of Convolutional Neural Networks}
Convolution operations in Convolutional Neural Networks (CNNs) apply multiple filters to a batch of channels of two-dimensional data (Algorithm \ref{algo:conv}, \figref{fig:conv}).
In particular, input and output tensors are represented as four-dimensional tensors with dimensions ($N,C,H,W$), where $N$ is the mini-batch size, $C$ is the number of channels, and $H$ and $W$ represent image height and width, respectively.
Similarly, the filter tensor is represented as four-dimensional ($K,C,V,U$) tensor, where $K$ is the number of output channels and $V,U$ represent kernel height and width.
\begin{algorithm}[t]
\caption{Pseudo-code of two-dimensional convolution.}
\label{algo:conv}
\begin{algorithmic}[1]
\STATE \makebox[4.5cm][l]{for($n = 0$; $n < N$; $n$++)} // Mini-batch loop
\STATE \makebox[4.5cm][l]{\ for($k = 0$; $k < K$; $k$++)} // Output channel loop
\STATE \makebox[4.5cm][l]{\ \ for($h = 0$; $h < H$; $h$++)} // Height loop
\STATE \makebox[4.5cm][l]{\ \ \ for($w = 0$; $w < W$; $w$++)} // Width loop
\STATE \makebox[4.5cm][l]{\ \ \ \ for($c = 0$; $c < C$; $c$++)} // Input channel loop
\STATE \makebox[4.5cm][l]{\ \ \ \ \ for($v = 0$; $v < V$; $v$++)} // Kernel width loop
\STATE \makebox[4.5cm][l]{\ \ \ \ \ \ for($u = 0$; $u < U$; $u$++)} // Kernel height loop
\STATE \ \ \ \ \ \ \ {\small $\mathbf{Y}[n,k,h,w]$ += $\mathbf{W}[k,c,v,u] \times \mathbf{X}[n,c,h+v,w+u]$;}
\end{algorithmic}
\end{algorithm}
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{figure/conv.pdf}
\caption{Two-dimensional convolution. Each element of $\mathbf{Y}$ is set to be a sum of element-wise products between partial $C \times V \times U$ area of $\mathbf{X}$ and one filter from $\mathbf{W}$.}
\label{fig:conv}
\end{figure}
The two-dimensional convolution is composed of seven-nested loops (\algoref{algo:conv}).
The innermost three loops compute the actual convolution, where one element of the input tensor $\mathbf{X}$ is multiplied and accumulated to one element of the output tensor $\mathbf{Y}$.
The remaining loops iterate over all elements of $\mathbf{Y}$.
The key observation is that in order to solve the problem described in {Section \ref{section:introduction}}, there is no dependency inside the mini-batch loop between different iterations.
This is intuitive because in training or inference we compute parameter gradients or outputs with respect to different data samples, so this is equivalent to computing $N$ different CNNs concurrently.
This observation motivates us to apply loop tiling to the mini-batch loop, so that we can reduce the resident workspace size.
The only exception to the inter-sample independency is the computation of parameter gradients;
\begin{eqnarray}
\frac{\partial L}{\partial \mathbf{W}} = \frac{1}{N} \sum_{n=1}^{N} \frac{\partial L_n}{\partial \mathbf{Y}_n} * \mathbf{X}_n \nonumber ,
\end{eqnarray}
where $L$ and $L_n$ is the loss function with respect to a mini-batch and a sample $n$ respectively, and $*$ is the convolution operation \cite{b70224ae04784e15b91d2056c46924a6}.
The semantics of this computation is, however, not violated by the loop splitting, only if each of the iterations is performed sequentially.
In cuDNN, there are three operations related to the two-dimensional convolution;
\texttt{Forward} \ for forward computation (\figref{fig:conv}), \texttt{BackwardData} \ for computing neuron errors in back-propagation, \texttt{BackwardFilter} \ for computing parameter gradients in back-propagation.
Although \texttt{Forward} \ and \texttt{BackwardData} \ can directly be divided into several micro-batches, \texttt{BackwardFilter} \ cannot, since there are output dependencies on the accumulated parameter gradients tensor $\mathbf{dW}$.
However, we can still divide the loops by running \texttt{BackwardFilter} \ multiple times while accumulating the results, i.e., output scale $=1$ in cuDNN.
Therefore, loop splitting can be achieved by repeating cuDNN \ kernels one or more times for any convolution-related operation, regardless of the underlying method.
\section{Performance Evaluation}
We evaluate the performance of $\mu$-cuDNN \ for three different GPU architectures, NVIDIA Tesla K80 \ \cite{k80}, P100-SXM2 \ \cite{p100} and V100-SXM2 \ \cite{v100} on the TSUBAME-KFC/DL, TSUBAME 3, and DGX-1 \ supercomputers, respectively.
The specifications of these supercomputers are listed in \tabref{table:spec}.
Throughout the evaluation, we use single-precision floating point format and store tensors in the $NCHW$ storage order.
We use three different deep learning frameworks for evaluations: Caffe \ \cite{jia2014caffe}, its NVIDIA branch (NVCaffe) \cite{nvcaffe}, and TensorFlow \ \cite{tensorflow2015-whitepaper}.
Both support recent versions of cuDNN \ (6 or 7).
We use a built-in benchmarking command (Caffe's ``time'' command) or an official benchmarking script (from TensorFlow \ models repository \cite{tensorflow_models}) to measure the execution time of forward and backward passes,
and show the sum of forward and backward passes together.
In the following sections, unless explicitly mentioned,
each forward-backward passes are measured 50 times on Caffe \ and 100 times on TensorFlow.
For neural networks, we use AlexNet \cite{Krizhevsky2012}, ResNet \cite{He2016}, and DenseNet \cite{huang2017densely}.
For evaluations on Caffe, we use the AlexNet \ model defined in Caffe, ResNet-18, and ResNet-50 from NVCaffe.
We modify data prefetching size from 4 to 16 for AlexNet \ and ResNet-18 for TSUBAME 3.
For evaluations on TensorFlow, we use the definitions in an official benchmarking repository \cite{tensorflow_benchmarks}.
As for workspace limit, unless explicitly mentioned, we use 8 MiB and 64 MiB for each layer, which are the default workspace size limits of Caffe \ and Caffe2 \ \cite{caffe2} \ respectively.
In addition, we use 512 MiB of workspace per layer to investigate the case where sufficiently large workspace is provided.
To shorten the benchmarking time, we use several GPUs on the same node with the parallel evaluation function of $\mu$-cuDNN, mentioned in Section \ref{subsection:ucudnn_implementation}.
\begin{figure}[b]
\centering
\includegraphics[width=\linewidth]{figure/caffe_time_alexnet_p100_smx2_tsubame3_conv2_forward.pdf}
\caption{Benchmark results of forward convolution of AlexNet's ``conv2'' layer on P100-SXM2.
We use 64 MiB workspace size and a mini-batch size of 256.
Numbers on each rectangle represent micro-batch sizes.}
\label{figure:caffe_time_alexnet_conv2_forward}
\end{figure}
\begin{figure*}[t]
\centering
\subfloat[K80]{
\includegraphics[width=0.3\linewidth]{figure/caffe_time_alexnet_k80_tsubame_kfc.pdf}
}
\subfloat[P100-SXM2]{
\includegraphics[width=0.3\linewidth]{figure/caffe_time_alexnet_p100_smx2_tsubame3.pdf}
}
\subfloat[V100-SXM2]{
\includegraphics[width=0.4\linewidth]{figure/caffe_time_alexnet_v100_dgx1.pdf}
}
\caption{Benchmark results of AlexNet \ on three different GPUs with different workspace sizes (8, 64, 512 MiB).
The labels ``u'', ``p'' and ``a'' represent \texttt{undivided}, \texttt{powerOfTwo}, and \texttt{all}, respectively.
We use a mini-batch size of 256 on K80 \ and P100-SXM2, and 1024 on V100-SXM2.}
\label{figure:caffe_time_alexnet}
\end{figure*}
\subsection{Convolution Kernel Optimization Using WR} \label{subsection:evaluation_one_layer}
\figref{figure:caffe_time_alexnet_conv2_forward} shows the execution time of forward convolution ({\tt cudnnConvolutionForward}) of the ``conv2'' layer in AlexNet \ on P100-SXM2.
With workspace size of 64 MiB, the GEMM (GEneral Matrix-Matrix multiply)-based algorithm is the one chosen by cuDNN, requiring only \perf{caffe-time-layer-alexnet-p100-64mb-conv2-forward-gemm-ws} for workspace if the mini-batch is not divided.
On the other hand, FFT-based convolution \cite{b70224ae04784e15b91d2056c46924a6} \ is more efficient, although it requires excessive amount of workspace (\perf{caffe-time-layer-alexnet-p100-64mb-conv2-forward-fft-ws}) to store the images and filters in the frequency domain.
$\mu$-cuDNN \ with \texttt{powerOfTwo} \ option successfully enables the use of FFT within the workspace size constraints, using \perf{caffe-time-layer-alexnet-p100-64mb-conv2-forward-fft-ws-u32} over micro-batches of size 32.
The \texttt{all} \ option also enables $\mu$-cuDNN \ to use Winograd convolution \cite{Lavin2016FastAF}, an algorithm that is especially efficient for small convolution kernels, achieving \perf{caffe-time-layer-alexnet-p100-64mb-conv2-forward-speedup}x speedup over \texttt{undivided} \ in total.
\subsection{CNN Optimization Using WR}
We evaluate WR-based optimization on two different deep learning frameworks: Caffe \ and TensorFlow.
\subsubsection{Caffe}
\figref{figure:caffe_time_alexnet} shows timing breakdowns of Caffe \ on AlexNet \ with three different GPUs.
Additionally, we only highlight convolutional layers since the others (e.g., pooling) are out of the scope of this paper.
One important observation from \figref{figure:caffe_time_alexnet} is that the performance improvement of $\mu$-cuDNN \ over cuDNN \ (which is equivalent to \texttt{undivided}) is significant when the moderate amount of workspace is set by users.
For instance, if the workspace size per kernel is 64 MiB, $\mu$-cuDNN \ with the \texttt{all} \ option achieves \perf{caffe-time-alexnet-k80-64mb-speedup}x speedup with respect to the entire iteration,
and \perf{caffe-time-alexnet-k80-64mb-speedup-conv}x with respect to convolutions alone, than \texttt{undivided} \ on K80.
This is because $\mu$-cuDNN \ successfully enables cuDNN \ to use faster algorithms, as in the example from Section \ref{subsection:evaluation_one_layer}.
In addition, a similar speedup is achieved on P100-SXM2 \ (\perf{caffe-time-alexnet-p100-64mb-speedup}x for the entire iteration, and \perf{caffe-time-alexnet-p100-64mb-speedup-conv}x for convolutions alone),
and on V100-SXM2 \ (\perf{caffe-time-alexnet-v100-64mb-speedup}x for the entire iteration, and \perf{caffe-time-alexnet-v100-64mb-speedup-conv}x for convolutions alone).
In the case where workspace size is limited to 8 MiB, $\mu$-cuDNN \ cannot attain any performance improvement, because even if the mini-batch is finely divided, the specified workspace is too small to utilize.
Indeed, on P100-SXM2, only one kernel of \texttt{all} \ option seems to increase the utilization of the workspace over \texttt{undivided}.
On the other hand, when the workspace size limit is too large (512 MiB) on K80 \ and P100-SXM2 \ GPUs, performance difference between cuDNN \ and $\mu$-cuDNN \ is negligible.
This is because there is no benefit from dividing the mini-batch, as all algorithms fit into the workspace constraints.
However, this workspace limit consumes a considerable amount of workspace memory:
While the \texttt{undivided} \ option consumes \perf{caffe-time-alexnet-p100-512mb-ws-undivided} in total, \texttt{all} \ with 64 MiB limit only consumes \perf{caffe-time-alexnet-p100-64mb-ws-all},
although with \perf{caffe-time-alexnet-p100-64mb-speedup-512mb-undivided-percent} overhead caused by the choice of micro-batch algorithms.
From the viewpoint of the time to optimization, including kernel benchmarking and solving DP, \texttt{powerOfTwo} \ considerably outperforms \texttt{all}.
In particular, with 64 MiB workspace on P100-SXM2, \texttt{all} \ takes \perf{caffe-time-alexnet-p100-64mb-benchmark-all}, whereas \texttt{powerOfTwo} \ takes \perf{caffe-time-alexnet-p100-64mb-benchmark-poweroftwo}.
This result and \figref{figure:caffe_time_alexnet} imply that \texttt{powerOfTwo} \ is a reasonable choice to test the computation efficiency of new CNNs quickly.
Generally, the overhead of $\mu$-cuDNN \ is negligible with respect to the entire training time, in which the forward and backward passes are repeated hundreds of thousands of times.
\begin{figure*}[t]
\centering
\subfloat[AlexNet]{
\includegraphics[width=0.28\linewidth]{figure/tf_time_alexnet_p100_smx2_tsubame3.pdf}
}
\subfloat[ResNet-50]{
\includegraphics[width=0.28\linewidth]{figure/tf_time_resnet50_p100_smx2_tsubame3.pdf}
}
\subfloat[DenseNet-40 ($k=40$)]{
\includegraphics[width=0.42\linewidth]{figure/tf_time_densenet_p100_smx2_tsubame3.pdf}
}
\caption{Benchmark results of different CNNs on P100-SXM2 \ with different workspace sizes (8, 64, 512 MiB), using TensorFlow \ framework.
We use a mini-batch size of 256 for AlexNet \ and DenseNet, and 64 for ResNet-50.}
\label{figure:tensorflow_time}
\end{figure*}
\subsubsection{TensorFlow}
\figref{figure:tensorflow_time} presents timing breakdowns of AlexNet \ and ResNet-50, DenseNet-40 on P100-SXM2.
We set the (input width, output width) as $(224, 1000)$ for AlexNet \ and ResNet-50, or $(32, 10)$ for DenseNet-40, which are used for training ILSVRC2012 classification dataset \cite{ilsvrc} or the CIFAR dataset \cite{cifar}, respectively.
We also set $k$ of DenseNet-40, the number of feature maps of each convolutional layer, to 40 to obtain better computational efficiency.
Since TensorFlow \ 1.4.1 does not provide any workspace limits to $\mu$-cuDNN \ via cuDNN's benchmarking functions before actual convolutions,
we manually provide workspace limits of 8, 64, and 512 MiB to $\mu$-cuDNN.
$\mu$-cuDNN \ with a workspace limit of 64 MiB achieves \perf{tf-time-alexnet-p100-64mb-speedup}x speedup for AlexNet, and \perf{tf-time-resnet50-p100-64mb-speedup}x for ResNet-50.
These results prove that $\mu$-cuDNN \ has good performance portability between different deep learning frameworks that depend on cuDNN.
\subsection{Memory Consumption Using WR}
\figref{figure:forward_memory} shows the per-layer memory usage of AlexNet \ and ResNet-18 on P100-SXM2.
In \figref{figure:forward_memory}, we set a per-layer workspace limit of 512 MiB for cuDNN, and 64 MiB for $\mu$-cuDNN,
where the slowdown due to the decrease of memory limit is negligible (\perf{caffe-time-alexnet-p100-64mb-slowdown-conv-to-512mb-undivided}x).
These figures clearly show that $\mu$-cuDNN \ can cut down per-layer memory consumption by up to \perf{forward-memory-consumption-ratio-max-alexnet}x and \perf{forward-memory-consumption-ratio-max-resnet18}x on AlexNet \ and ResNet-18 respectively.
\begin{figure}[!b]
\centering
\subfloat[AlexNet \ (cuDNN)] {\includegraphics[width=0.5\linewidth]{figure/forward_memory_alexnet_p100_smx2_tsubame3_512_64_cudnn.pdf}}
\subfloat[AlexNet \ ($\mu$-cuDNN)]{\includegraphics[width=0.5\linewidth]{figure/forward_memory_alexnet_p100_smx2_tsubame3_512_64_ucudnn.pdf}}
\newline
\subfloat[ResNet-18 \ (cuDNN)] {\includegraphics[width=0.5\linewidth]{figure/forward_memory_resnet18_p100_smx2_tsubame3_512_64_cudnn.pdf}}
\subfloat[ResNet-18 \ ($\mu$-cuDNN)]{\includegraphics[width=0.5\linewidth]{figure/forward_memory_resnet18_p100_smx2_tsubame3_512_64_ucudnn.pdf}}
\caption{
Per-layer breakdowns of memory consumption of AlexNet \ and ResNet-18 on P100-SXM2.
For simplicity, we only show the memory usage of unique convolutional layers (CONV\_$n$) and fully-connected layers (fc or fc$n$) in one forward propagation.
We use a mini-batch of 256 for AlexNet \ and 128 for ResNet-18 respectively.
We set a per-layer workspace limit of 512 MiB for cuDNN, and 64 MiB for $\mu$-cuDNN.
Each bar segment of ``WS ($\mu$-cuDNN)'' represents the maximum workspace size of the layer.
}
\label{figure:forward_memory}
\end{figure}
\subsection{CNN Optimization Using WD} \label{subsection:evaluation_wdiv}
\figref{figure:caffe_time_wdiv} shows the benchmark results of using the WD \ algorithm.
The adjoined bars have the same workspace limit in total: For example, since AlexNet \ has five convolutional layers and each layer has three kernels (Forward, BackwardData, BackwardFilter), we place the result with 120 MiB WD \ workspace next to that of 8 MiB WR \ workspaces.
\begin{figure}[t]
\centering
\subfloat[AlexNet]{\includegraphics[width=\linewidth]{figure/caffe_time_alexnet_p100_smx2_tsubame3_wdiv.pdf}}
\newline
\subfloat[ResNet-50]{\includegraphics[width=\linewidth]{figure/caffe_time_resnet50_p100_smx2_tsubame3_wdiv.pdf}}
\caption{Benchmark results of AlexNet \ and ResNet-50 on P100-SXM2 \ with different workspace sizes and policies (WR \ and WD).
We use a mini-batch size of 256 for AlexNet \ and 32 for ResNet-50.
Note that the adjoined bars have the same workspace limit in total.}
\label{figure:caffe_time_wdiv}
\end{figure}
In \figref{figure:caffe_time_wdiv}, we can see that the training time decreases as the workspace constraints increase in both WR \ and WD.
At the same time, WD \ successfully manages the global memory requirements better, attaining higher performance with the same overall memory footprint (see \figref{figure:caffe_time_alexnet_ws} for breakdown).
Specifically, when 120 MiB workspace in total is provided for AlexNet, the entire execution time with WD \ optimization and \texttt{all} \ option is \perf{caffe-time-wdiv-alexnet-p100-120mb-speedup}x faster than the WR \ with \texttt{undivided} \ option for the entire iteration (or \perf{caffe-time-wdiv-alexnet-p100-120mb-speedup-conv}x for convolution).
WD \ also outperforms the baseline with 960 MiB workspace in total, which can use 8 times more memory for workspace, by \perf{caffe-time-wdiv-alexnet-p100-120mb-speedup-960mb-baseline}x in total execution time.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figure/caffe_time_alexnet_ws_alexnet_p100_smx2_tsubame3.pdf}
\caption{Assigned workspace division of AlexNet \ on P100-SXM2.
``F'', ``BF'', ``BD'' represent kernel types (Forward, BackwardFilter, BackwardData respectively).
We use a mini-batch size of 256 for AlexNet.
We set a workspace limit of 8 MiB for WR, and a total workspace limit of 120 MiB for WD.}
\label{figure:caffe_time_alexnet_ws}
\end{figure}
Furthermore, even for ResNet-50, which has 10 times more convolutional layers than AlexNet, WD \ achieves \perf{caffe-time-wdiv-resnet50-p100-2544mb-speedup}x speedup for the entire iteration (or \perf{caffe-time-wdiv-resnet50-p100-2544mb-speedup-conv}x for convolution) with 2,544 MiB of total workspace, outperforming the original version (which consumes 5,088 MiB) in terms of memory footprint as well.
In addition, the ILP for ResNet-50 is still small enough to solve in practical time.
When the workspace limit is set to 5,088 MiB, the number of 0-1 variables is 562, and the GLPK solver takes 5.46 ms to solve it.
The main reason that WD \ outperforms WR \ is that in WR, if $\mu$-cuDNN \ fails to find better algorithms and micro-batch sizes to fully utilize the assigned workspace, $\mu$-cuDNN \ must abandon that workspace slot and cannot allocate it to other kernels.
On the other hand, in WD, characteristics of different desirable workspace sizes of different kernels (\figref{figure:caffe_time_alexnet_p100_smx2_tsubame3_wdiv_desirable_set_4}) are implicitly considered in the ILP-based optimization framework.
Therefore, $\mu$-cuDNN \ can assign larger proportional workspaces to time-consuming layers, if it is expected that the kernels will be considerably faster with a larger workspace.
In \figref{figure:caffe_time_alexnet_ws}, $\mu$-cuDNN \ with the WD \ policy spares most of the workspace for ``conv2'' and ``conv3'' (\perf{caffe-time-alexnet-ws-120mb-prop-conv2-conv3}), which are the most time-consuming layers in the baseline (WR, \texttt{undivided}).
In contrast, $\mu$-cuDNN \ doesn't allocate workspace of over \perf{caffe-time-alexnet-ws-120mb-ws-max-conv4-conv5} for ``conv4'' and ``conv5'', although $\mu$-cuDNN \ lists some faster and desirable configurations than the baseline.
For instance, the fastest configuration of conv5 (forward), which uses FFT-based convolution with two micro-batches, is \perf{caffe-time-alexnet-ws-120mb-conv5-forward-best-speedup}x faster than baseline, although this configuration uses \perf{caffe-time-alexnet-ws-120mb-conv5-forward-best-ws} of workspace.
This observation implies that the WD \ does not unnecessarily allocate workspace for a specific layer but chooses the best combination, as defined by the ILP.
\section{Introduction} \label{section:introduction}
Prevalent Deep Neural Networks (DNNs) are becoming increasingly deeper and are trained with large batch sizes.
Specifically, state-of-the-art DNNs contain hundreds of layers \cite{Krizhevsky2012,He2016},
and utilize batch sizes in the order of thousands \cite{Goyal2017,2017arXiv171104325A,2017arXiv171100489S}.
Large batches are also favored by distributed data-parallel deep learning frameworks,
because they improve utilization of accelerators, as well as hiding the communication of parameter gradients in the computation efficiently.
Consequently, the batch size per accelerator (e.g., GPU) should be large to achieve better scaling.
Since the memory usage of a DNN is nearly proportional to the layer size and the batch size,
the accelerator memory tends to be used at full capacity in most real-world cases.
This ``limited memory scenario'' is also exhibited in cuDNN \ \cite{cudnn}, a deep learning kernel library for NVIDIA GPUs.
cuDNN \ provides a variety of computational primitives for deep neural networks, and is widely used in deep learning frameworks,
such as Caffe \cite{jia2014caffe} and others \cite{tensorflow2015-whitepaper,2016arXiv160502688short,chainer_learningsys2015}.
cuDNN \ provides up to eight different algorithms to perform convolutions, each of which requires different temporary storage (workspace) schemes.
To guide users to determine the best algorithm for a given maximum workspace size,
cuDNN \ provides a function \texttt{cudnnGetConvolution*Algorithm} (\texttt{*} is one of convolution types, \texttt{Forward}, \texttt{BackwardData} and \texttt{BackwardFilter}),
that benchmarks all the algorithms and chooses the best algorithm, either with respect to computation time or memory usage.
However, if the workspace size requested by a fast algorithm is one byte larger than provided, cuDNN \ will resort to a slower algorithm that requires less workspace.
In fact, the performance impact can be 4.51x in the 2nd convolutional layer of AlexNet, as shown in \figref{figure:alexnet_1byte_less_ws}.
\begin{figure*}[t]
\centering
\subfloat[Execution time of all layers.]{
\includegraphics[width=0.5\linewidth,valign=t]{figure/alexnet_1byte_less_ws.pdf}
}
\subfloat[Execution time vs. execution time of conv2. {\large $\circ$} and {\large $\diamond$} represent the ``Best'' and the ``-1 byte'' respectively.]{
\includegraphics[width=0.5\linewidth,valign=t]{figure/alexnet_1byte_less_ws_algos_conv2.pdf}
}
\caption{Execution time of cuDNN \ 7.0.1 forward convolution of single-column AlexNet \ \cite{one_wired_trick} with different workspace sizes.
The ``Best'' case always chooses the fastest algorithm regardless of workspace size, while in the ``-1 byte'' case the maximum workspace size is limited to 1 byte less than the best algorithm.}
\label{figure:alexnet_1byte_less_ws}
\end{figure*}
In this paper, we propose $\mu$-cuDNN, a transparent wrapper for cuDNN \ that attempts to mitigate the aforementioned inefficiency.
In order to utilize fast convolution algorithms with limited size of workspace,
$\mu$-cuDNN \ automatically divides layer mini-batch computation into several micro-batches and perform multiple convolutions sequentially.
$\mu$-cuDNN \ decouples the statistical efficiency (speed of accuracy/loss improvement with fixed amount of parameter updates)
from the hardware efficiency (speed of computations with fixed amount of parameter updates), improving only the latter.
Using micro-batches, $\mu$-cuDNN \ improves the utilization of the accelerators without incurring any reduction in training accuracy.
The contributions of this paper are as follows:
\begin{itemize}
\item We present a method to automatically divide mini-batch training into several ``micro-batches'', so that faster algorithms are utilized with tight workspace constraints.
\item We propose two different workspace allocation policies, which enable optimization of multiple convolutional layers with inter-dependencies.
\item We evaluate $\mu$-cuDNN \ over two different deep learning frameworks, Caffe \ and TensorFlow, showing that it can mitigate the inefficiency of cuDNN \, even with state-of-the-art Convolutional Neural Networks (CNNs), such as AlexNet \ and ResNet.
\end{itemize}
\section*{Acknowledgment}
This research was supported by the ETH Postdoctoral Fellowship (for T. B. N.), Student Summer Research Fellowship (for Y. O.), and JST CREST Grant Number JPMJCR1303, JPMJCR1687, Japan.
Part of this work is conducted as research activities of AIST - TokyoTech Real World Big-Data Computation Open Innovation Laboratory (RWBC-OIL).
\bibliographystyle{IEEEtran}
\section{Related Work}
Li et. al \cite{memeff16} propose a heuristic to automatically tune each tensor memory layout to utilize either GEMM-based or FFT-based convolution efficiently.
The proposed heuristic is, however, based on the authors' performance observation using conventional convolutional layers and specific GPU architecture,
and thus there is no guarantee that the algorithm always provides the best memory alignment for any deep neural network and GPU architecture.
On the other hand, since $\mu$-cuDNN \ uses the techniques of dynamic programming and integer linear programming,
it is mathematically guaranteed that $\mu$-cuDNN \ provides the best performance that the library can produce, provided that each convolution is independent from the others.
Rhu et al. \cite{7783721} propose a memory management technique that offloads neuron activations, parameters, and errors from the GPU memory to the CPU memory during forward-/backward-propagation,
so that larger models can be trained with the same memory constraint.
However, as \figref{figure:forward_memory} shows, even in such memory-efficient implementation or similar memory management techniques \cite{memreduction} $\mu$-cuDNN \ is expected to save the peak memory usage of each layer.
Zlateski et al. \cite{7877151} propose ZNNi, an FFT-based convolution algorithm, and mention micro-batching technique to reduce the temporal memory usage by FFT.
$\mu$-cuDNN, however, generalizes the schema so that micro-batching can be applied to any convolution algorithm, obtaining the best computational performance for the given layer configurations,
as well as maintains high portability between different existing deep learning frameworks.
\section{$\mu$-cuDNN}
$\mu$-cuDNN \ is a transparent C++ wrapper library for cuDNN, which can easily be integrated into most deep learning frameworks \cite{jia2014caffe,caffe2,tensorflow2015-whitepaper,chainer_learningsys2015}.
The key concept of $\mu$-cuDNN \ is that it automatically divides a mini-batch to several batches (referred to as ``micro-batches'' in this paper) and optimizes their sizes, to utilize faster convolution algorithms (\figref{figure:ucudnn}).
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figure/ucudnn.pdf}
\caption{The conceptual timeline of $\mu$-cuDNN.
``@256'' means that each computation is executed with batch-size of 256.
$\mu$-cuDNN \ splits one convolution operation into one or more disjoint subsets of the mini-batch.
}
\label{figure:ucudnn}
\end{figure}
\subsection{$\mu$-cuDNN \ Methodology} \label{subsection:ucudnn_methodology}
$\mu$-cuDNN \ library employs one of two workspace utilization policies to optimize micro-batches for convolution kernels (\figref{figure:wrwd}):
\begin{itemize}
\item {\bf Workspace Reuse (WR)}: WR \ allocates one workspace per layer, sharing the space between the internal micro-batches.
In this scheme, each layer is assumed to use the workspace exclusively, hence the total size of the workspaces is in proportion to the number of convolutional layers.
\item {\bf Workspace Division (WD)}: WD \ allocates one workspace per network, and assigns different segments to each convolutional layer.
WD \ enables small groups of convolution operations, as in the Inception module \cite{going_deeper_with}, to run concurrently with larger workspaces.
In WD, the actual workspace is managed by $\mu$-cuDNN \ rather than the deep learning framework.
This is because conventional frameworks allocate each workspace separately, lacking a global view of the entire network's workspace requirements.
\end{itemize}
WR \ and WD \ both rely on the parameters of one or more convolution kernel(s), the mini-batch size, and the maximum workspace size.
The output of $\mu$-cuDNN \ is a division of the mini-batch, and ``micro-configurations''; a pair of a convolution algorithm and micro-batch size for each convolution micro-batch.
In this paper, we define ``configuration'' of a segmented convolution kernel as ``a list of micro-configurations''.
For example, if a kernel with a mini-batch size of 256 is equally divided into four micro-batches and each of them uses algorithm $X$, the configuration is represented as $\{(X, 64), (X, 64), (X, 64), (X, 64)\}$.
Also we define concatenation of two lists as $+$, such as $\{a,b\} + \{c,d\} = \{a,b,c,d\}$ and $\{a\} + \emptyset = \{a\}$.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figure/wrwd.pdf}
\caption{Overview of WR \ and WD. $\mu$-cuDNN \ optimizes micro-batch sizes and internally calls cuDNN \ functions, via the cuDNN \ interfaces.}
\label{figure:wrwd}
\end{figure}
\subsection{WR \ Algorithm} \label{subsection:wr_algorithm}
The goal of the WR \ policy is to optimize $T(B)$, the total execution time with mini-batch size of $B$ using Dynamic Programming (DP), given by:
\begin{eqnarray}
T(b) = \min
\left\{
\begin{array}{l}
T_\mu(b), \\
\min_{b'=1,2,\ldots,B-1}{T(b') + T(b-b')} \\
\end{array}
\right\}, \nonumber
\end{eqnarray}
where $T_\mu(b)$ is the fastest execution time of one convolution kernel with a micro-batch size of $b$, within the workspace constraint.
If the first row of the definition of $T(B)$ is smaller than the second row, $\mu$-cuDNN \ does not have to divide the batch.
Otherwise, it is beneficial to divide the batch into two or more parts, applying the process recursively (\figref{figure:dp}).
The key point of WR \ is that the optimal micro-configuration size is deterministic and independent from other kernels.
This is because in this case, we assume that multiple kernels do not run simultaneously.
The algorithm of WR \ is three-fold, where the mini-batch size is $B$, and user-given maximum workspace size is $M$:
\begin{enumerate}
\item For $b=1, 2, \cdots, B$, WR \ benchmarks all available convolution algorithms of micro-batch size of $b$ with maximum workspace size of $M$, using cuDNN. We define the fastest micro-configuration as $c_\mu(b) = (a, b)$ (where $a$ is the fastest algorithm) and its execution time as $T_\mu(b)$. \label{algorithm:wr__benchmark_kernel}
\item For $b=1, 2, \cdots, B$, WR \ computes $T(b)$, the fastest execution time for micro-batch size of $b$, and $c(b)$, the corresponding configuration, as follows (where $T(0) = 0, c(0) = \emptyset$). $T(b)$ and $c(b)$ are memorized and reused for further iterations.
\begin{eqnarray}
\hat{b_\mu} &\leftarrow& \mathop{\rm argmin}\limits_{b_\mu = 1,2,\ldots,b}\{ T_\mu(b_\mu) + T(b-b_\mu) \} \nonumber \\
T(b) &\leftarrow& T_\mu(\hat{b_\mu}) + T(b-\hat{b_\mu}) \nonumber \\
c(b) &\leftarrow& \{c_\mu(\hat{b_\mu})\} + c(b-\hat{b_\mu}) \nonumber
\end{eqnarray}
\item Outputs the optimal configuration $c(B)$.
\end{enumerate}
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figure/dp.pdf}
\caption{DP-based optimization of WR.
Here we assume that convolution algorithm 4 with micro-batch size of 60 ($c_\mu(60) = (4, 60)$) achieves better computation efficiency, hence it is repeatedly used.}
\label{figure:dp}
\end{figure}
\subsection{WD \ Algorithm}
In the WD \ scheme, configurations for multiple convolution kernels are optimized, while at the same time the total workspace size should be less than the total workspace limit that users specify.
Therefore, WD \ is a more complex problem than WR, since the configuration of each convolution kernel is no longer independent from others, due to the total workspace size constraint.
To solve this problem, we formulate a 0-1 Integer Linear Programming (ILP)-based optimization algorithm (\figref{figure:ilp}).
Given the set of kernels $\mathcal{K}$ and sets of available configurations $C_k$ of kernel $k$, WD \ is solved by minimizing Equation \ref{eqn:wd}:
\begin{eqnarray}
{\rm min.} && T = \sum_{k \in \mathcal{K}} \sum_{c \in C_k} T_k(c) x_{k,c} \label{eqn:wd} \\
{\rm subject\ to.} && \sum_{k \in \mathcal{K}} \sum_{c \in C_k} M_k(c) x_{k,c} \leq M \label{eqn:wd_memory} \\
&& \sum_{c \in C_k} x_{k,c} = 1 \ (\forall k \in \mathcal{K}) \label{eqn:wd_config_per_kernel} \\
&& x_{k,c} \in \{0, 1\} \ (\forall k \in \mathcal{K}, \forall c \in C_k) \label{eqn:wd_end} ,
\end{eqnarray}
where $M_k(c)$ and $T_k(c)$ are the workspace size and execution time of kernel $k$ with configuration $c$, respectively.
Equation \ref{eqn:wd_memory} limits the total workspace size to the user-specified size $M$.
$\mu$-cuDNN \ uses configuration $c$ on kernel $k$ if and only if $x_{k,c} = 1$, and exactly one of them is selected for each kernel $k$, according to the constraint in Equation \ref{eqn:wd_config_per_kernel}.
\subsubsection{Desirable Configuration Selection}
The challenging problem of the above ILP-based algorithm is that if all possible configurations are evaluated (i.e., all combinations of the number of micro-batch and algorithms),
the search-space is in the order of $|\mathcal{K}| |A|^B$ (where $A$ is set of algorithms and $B$ is the mini-batch size) configurations in total, which makes the problem impractically large.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figure/ilp.pdf}
\caption{ILP-based optimization of WD.
The problem is stacking ``time $\times$ memory'' rectangles of configurations diagonally,
and obtaining the minimum total width $T$,
provided that the total height is lower than $M$.
Each configuration $u,v,\ldots,c$ is composed of one or more micro-configurations such as $c_\mu$.}
\label{figure:ilp}
\end{figure}
Here we compute a Pareto front to remove undesirable configurations from all possible configurations, without returning any sub-optimal solutions.
The resulting Pareto front $C_k$ is then input to the ILP (Equation \ref{eqn:wd}-\ref{eqn:wd_end}) to solve the entire problem.
First, we modify the DP algorithm from WR (Section \ref{subsection:wr_algorithm}) to output a set of configurations, rather than the fastest configuration, as follows:
{\small
\begin{eqnarray}
C(b)=
D \Bigg(
\bigcup_{b_\mu=1,2,\ldots,b}
\ \bigcup_{c_\mu \in C_\mu(b_\mu)}
\ \bigcup_{c \in C(b-b_\mu)}
( \{ c_\mu \} + c )
\Bigg), \nonumber
\end{eqnarray}
}where $C_\mu(b)$ is a set of available micro-configurations of micro-batch size of $b$, and $D$ is a pruning function described below.
Note that this outputs $c(B)$ of the WR \ algorithm as one of its elements;
$c(b) \in C(b)$ and $c_\mu(b) \in C_\mu(b)$ for any $b$.
Second, we define the ``desirable configuration set'' $D(C) \subset C$ as a Pareto front in the two-dimensional (execution time $\times$ workspace size) space (\figref{figure:desirable_set}):
{\small
\begin{eqnarray}
D(C)=\{c \in C | \forall c' \in C \ [ T(c) < T(c') \lor M(c) < M(c') ] \}, \nonumber
\end{eqnarray}
}where $T(c)$ and $M(c)$ is execution time and required workspace size of a configuration set $c$.
This definition implies that any $c \in D(C)$ is the fastest configuration among any of the elements of $D(C)$ using a workspace size of $M(c)$ or less.
Conversely, if an element $c \in C$ is not in $D(C)$, there is an element that is faster than $c$ and requires less workspace, hence there is no reason to choose $c$, namely ``undesirable''.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{figure/desirable_set.pdf}
\caption{The concept of desirable set. Here $c$ cannot be in $D(C)$ because a $c'$ exists for which the condition $T(c) < T(c') \lor M(c) < M(c')$ is not satisfied.}
\label{figure:desirable_set}
\end{figure}
The pruning drastically reduces the number of variables of Equation \ref{eqn:wd}, and enables solving the ILP for state-of-the-art deep CNNs in practical time.
For instance, the maximum number of desirable configurations of AlexNet's layers we examined in Section \ref{subsection:evaluation_wdiv} was 68, which is much smaller than the exponential order.
\figref{figure:caffe_time_alexnet_p100_smx2_tsubame3_wdiv_desirable_set_4} illustrates a Pareto front of one convolutional layer of AlexNet.
\begin{figure}[t]
\centering
\includegraphics[width=\linewidth]{figure/caffe_time_alexnet_p100_smx2_tsubame3_wdiv_desirable_set_4.pdf}
\caption{Desirable configurations (i.e. a Pareto front) of AlexNet's ``conv2'' layer (Forward) on P100-SXM2 \ with a maximum workspace size of 120 MiB, and a mini-batch size of 256.
Colored bars corresponding to data points represent the division of the mini-batch and the chosen micro-batch algorithms.
For example, the top-left point divides the mini-batch into two micro-batches of 128 and utilizes the {\tt FFT\_TILING} algorithm.}
\label{figure:caffe_time_alexnet_p100_smx2_tsubame3_wdiv_desirable_set_4}
\end{figure}
The validity of the pruning algorithm that the optimal solution of the ILP does not include any undesirable configurations is proved as follows:
\begin{proof}
Suppose that an optimal solution of the ILP $f: X \rightarrow \{0,1\}$, where $X$ is the set of variable symbols of the ILP,
contains an undesirable configuration $u$ of a kernel $a$ (i.e. $f(x_{a,u}) = 1$).
According to the definition of desirable sets, there is a configuration $v$ of $a$ such that
$T_{a}(v) \leq T_{a}(u)$ and $M_{a}(v) \leq M_{a}(u)$.
According to Equation \ref{eqn:wd_config_per_kernel}, $f(x_{a,c}) = 1-f(x_{a,u}) = 0$ for all $c \in C_a$.
Let $g: X \rightarrow \{0,1\}$ be defined as
\begin{eqnarray}
g(x_{k,c}) = \left\{
\begin{array}{ll}
1 & (k = a \land c = v) \\
0 & (k = a \land c \neq v) \\
f(x_{k,c}) & (\mathrm{otherwise})
\end{array}
\right.
\nonumber .
\end{eqnarray}
$g$ satisfies Equation \ref{eqn:wd_config_per_kernel} for $k = a$ as
\begin{eqnarray}
\sum_{c \in C_a} g(x_{a,c}) &=& \sum_{c \in C_a \backslash \{v\}} g(x_{a,c}) + g(x_{a,v}) = 1 \nonumber ,
\end{eqnarray}
and Equation \ref{eqn:wd_memory} as
\begin{eqnarray}
\sum_{k \in \mathcal{K}} \sum_{c \in C_k} M_k(c) g(x_{k,c})
&=& \sum_{k \in \mathcal{K} \backslash \{a\}} \sum_{c \in C_k} M_k(c) g(x_{k,c}) \nonumber \\
&& + M_a(v) g(x_{a,v}) \nonumber \\
&\leq& \sum_{k \in \mathcal{K} \backslash \{a\}} \sum_{c \in C_k} M_k(c) f(x_{k,c}) \nonumber \\
&& + M_a(u) f(x_{a,u}) \nonumber \\
&\leq& \sum_{k \in \mathcal{K}} \sum_{c \in C_k} M_k(c) f(x_{k,c}) \leq M \nonumber .
\end{eqnarray}
Similarly, by replacing $M_k$ as $T_k$ in the inequality above, the objective value of $g$ is proved to be lower than $f$,
hence $g$ is a better solution of the ILP.
Therefore it contradicts the supposition that $f$ is the optimal solution.
\end{proof}
\subsection{$\mu$-cuDNN \ Implementation} \label{subsection:ucudnn_implementation}
To enable $\mu$-cuDNN, the only modification that needs to be performed to the code is to replace the cuDNN \ handle type \texttt{cudnnHandle\_t} \ with \texttt{UcudnnHandle\_t}.
The $\mu$-cuDNN \ handle object is an opaque type that wraps the original type, such that users can call any cuDNN \ function.
When a convolution operation or benchmarking function is called with the $\mu$-cuDNN \ handle object, the $\mu$-cuDNN \ library internally computes the optimal configurations, and returns a virtual algorithm ID and zero required workspace size.
This mechanism enables users to call $\mu$-cuDNN \ with minimal modification to the original code.
For example, the number of lines to be modified to introduce $\mu$-cuDNN \ to Caffe \ (v1.0) is approximately three.
The implementation of $\mu$-cuDNN \ is based on overloading a subset of cuDNN \ functions,
where the memory of the $\mu$-cuDNN \ handle type is structured to behave to act as the cuDNN \ internal handle for the other calls.
We define a cast operator from the $\mu$-cuDNN \ handle to cuDNN \ handle so that the framework automatically adopts this method.
Using this technique, $\mu$-cuDNN \ delegates most of the functions to cuDNN, but overrides functions related to the convolutional layers.
The optimization algorithm in $\mu$-cuDNN \ is based on the methodology described in Section \ref{subsection:ucudnn_methodology}.
In practice, $\mu$-cuDNN \ provides a ``batch size policy'', which determines what micro-batch sizes are benchmarked at the step \ref{algorithm:wr__benchmark_kernel} of the WR \ algorithm, as follows:
\begin{itemize}
\item \texttt{all} \ uses all batch sizes $b \in \{1, 2, 3, \cdots, B\}$. Although this always finds the optimal solution, it takes $\mathcal{O}(B)$ time for the benchmark.
\item \texttt{powerOfTwo} \ uses only power-of-two batch sizes $b \in \{2^0, 2^1, 2^2, \cdots, B\}$. This saves a considerable amount of time since it only costs $\mathcal{O}(\log B)$ time for the benchmark.
\item \texttt{undivided} \ uses only the original mini-batch size $b \in \{B\}$. In WR, this option always selects the same configuration as cuDNN, hence this option is only useful to evaluate the overhead of $\mu$-cuDNN.
\end{itemize}
These policies can be specified via an environment variable or through a special library function in $\mu$-cuDNN.
Furthermore, $\mu$-cuDNN \ supports parallel micro-configuration evaluation via an environment variable, in which the aforementioned micro-batches are distributed to different GPUs on the same computing node and tested concurrently.
This function assumes that the node contains multiple homogeneous GPUs.
$\mu$-cuDNN \ caches the optimized configurations and the benchmark results into memory and optional file-based database respectively, to skip unnecessary recomputations.
This is especially beneficial for networks that replicate convolutional layers of the same size, such as ResNet \ \cite{He2016}.
In addition, the file-based caching enable offline benchmarking, as well as sharing the results among a homogeneous GPU cluster via network file system.
\subsection{Implementation of WD \ Optimization}
To perform WD \ optimization, $\mu$-cuDNN \ must know the number of convolutional layers and corresponding layer parameters in advance, i.e., before running any kernel.
In the current cuDNN \ API, however, the parameters are passed one layer at a time, and thus there is no way to obtain all the parameters collectively from deep learning frameworks.
To overcome this issue, we assume that the deep learning framework calls {\tt cudnnGetConvolution*Algorithm} one time for each layer prior to the computation of the entire network (e.g., training, inference).
This is the most straightforward use of the cuDNN \ interface, as memory (including workspace) is usually allocated before initiating computations.
Due to the specific implementation of Caffe, we add a $\mu$-cuDNN \ library call after network initialization, which ignores subsequent {\tt cudnnGetConvolution*Algorithm} calls.
When {\tt cudnnGetConvolution*Algorithm} is called, $\mu$-cuDNN \ pushes the kernel parameters to an internal list, and returns a dummy result.
Note that the returned results satisfy the semantics given by the cuDNN \ interface, so the framework will not raise errors and will not allocate its own workspaces.
When {\tt cudnnConvolution*} is called for the first time, $\mu$-cuDNN \ executes the optimization algorithm (namely, WD).
We use the GNU Linear Programming Kit (GLPK) \cite{glpk} as the ILP solver.
|
1,108,101,565,656 | arxiv | \section{Introduction}
\label{sec:intro}
The flavor changing processes in the ${s}-{b}$ sector are sensitive probe
of new physics (NP) beyond the standard model (SM) because they are experimentally the
least constrained.
In the minimal supersymmetric standard model (MSSM), however, the
flavor mixing in the chirality flipping
down-type squarks, $\wt{s}_{L(R)}-\wt{b}_{R(L)}$, is already strongly
constrained by the measurement of $BR(B \to X_s \gamma)$. On the other hand,
large flavor mixing in the chirality conserving
$\wt{s}_{L(R)} - \wt{b}_{L(R)}$ has been largely allowed.
Especially the large mixing scenario in the $\wt{s}_R - \wt{b}_R$ sector
has been drawing much interest
because it is well motivated by the measurement large neutrino mixing
and the idea of grand unification~\cite{Baek:GUT}.
Recently D{\O} and CDF collaborations at Fermilab Tevatron reported the results
on the measurements of $B_s - \overline{B}_s$ mass difference~\cite{D0,CDF}
\begin{eqnarray}
17 ~{\rm ps}^{-1} < \Delta m_s < 21 ~{\rm ps}^{-1} ~~(90 \% \mbox{~CL}), \nonumber\\
\Delta m_s = 17.33^{+0.42}_{-0.21}\pm 0.07 ~{\rm ps}^{-1},
\label{dms:exp}
\end{eqnarray}
respectively.
These measured values are consistent with the SM predictions~\cite{Bona:2005eu,CKMfitter}
\begin{eqnarray}
\Delta m_s^{\rm SM}({\rm UTfit}) = 21.5 \pm 2.6 ~{\rm ps}^{-1}, \quad
\Delta m_s^{\rm SM}({\rm CKMfit}) = 21.7^{+5.9}_{-4.2} ~{\rm ps}^{-1}
\label{dms:SM}
\end{eqnarray}
which are obtained from global fits,
although the experimental measurements in (\ref{dms:exp}) are slightly lower.
The implications of $\Delta m_s$ measurements have already been considered
in model independent approach~\cite{model_indep1,model_indep2,Buras:2006}, MSSM
models~\cite{MSSM,Endo:2006dm},
$Z'$-models~\cite{Zprime}, {\it etc}.
In this paper, we consider the implications of (\ref{dms:exp}) on an MSSM
scenario with large mixing in the LL and/or RR sector. We do not consider
flavor mixing in the LR(RL) sector because they are i) are already strongly
constrained by $BR(B \to X_s \gamma)$~\cite{LR} and ii) therefore relatively insensitive to
$B_s - \overline{B}_s$ mixing. We neglect mixing between the 1st and
2nd generations which are tightly constrained by $K$ meson decays and
$K - \overline{K}$ mixing, and mixing between the 1st and 3rd generations
which is also known to be small by the measurement of $B_d - \overline{B}_d$ mixing.
The paper is organized as follows. In Section~\ref{sec:BsBs}, the relevant
formulas for $B_s-\overline{B}_s$ mixing are presented. In Section~\ref{sec:num}
we perform numerical analysis and show the constraints imposed
on our scenario.
With these constraints, in Section~\ref{sec:asym}, we predict the
time-dependent CP asymmetry in $B_s \to \psi \phi$ decay
and the semileptonic asymmetry in $B_s\to \ell X$ decay.
We conclude in Section~\ref{sec:con}.
\section{$B_s-\overline{B}_s$ mixing in the MSSM scenario with large LL/RR mixing}
\label{sec:BsBs}
According to the description of our model in Section~\ref{sec:intro},
the scalar down-type mass squared matrix in the basis where down quark mass matrix
is diagonal is given by~\cite{GNK,Baek:Bs2KK}
\begin{eqnarray}
M^2_{\wt{d},LL}
=\l(\begin{array}{ccc}
\wt{m}^{d,2}_{L_{11}} & 0 & 0 \\
0 & \wt{m}^{d,2}_{L_{22}} & \wt{m}^{d,2}_{L_{23}} \\
0 & \wt{m}^{d,2}_{L_{32}} & \wt{m}^{d,2}_{L_{33}} \\
\end{array}
\r), \quad
M^2_{\wt{d},LR(RL)} \equiv 0_{3\times 3}.
\end{eqnarray}
The $M^2_{\wt{d},RR}$ can be obtained from $M^2_{\wt{d},LL}$
by exchanging $L \leftrightarrow R$.
We note that this kind of scenario is orthogonal to the one with flavor
violation controlled only by CKM matrix
(minimal flavor violation model~\cite{MFV,Buras:2006} or the effective
SUSY model considered in~\cite{Baek:1998yn}),
where large flavor violation in $s-b$ is impossible a priori.
The mass matrix $M^2_{\wt{d},LL}$ can be diagonalized by
\begin{eqnarray}
\Gamma_L M^2_{\wt{d},LL} \Gamma_L^\dagger
= {\rm diag}(m^2_{\wt{d}_L},m^2_{\wt{s}_L},m^2_{\wt{b}_L}),
\end{eqnarray}
with
\begin{eqnarray}
\Gamma_L
=\l(\begin{array}{ccc}
1 & 0 & 0 \\
0 & \cos\theta_L & \sin\theta_L\;e^{i\delta_L} \\
0 & -\sin\theta_L\;e^{-i\delta_L} & \cos\theta_L \\
\end{array}
\r).
\label{eq:Gamma_L}
\end{eqnarray}
Similarly, the exchange $L \leftrightarrow R$ in (\ref{eq:Gamma_L}) gives $\Gamma_R$.
We restrict $ -45^\circ < \theta_{L(R)} < 45^\circ$ so that the
mass eigenstate ${\wt{s}(\wt{b})}$ has more strange (beauty) flavor than beauty (strange)
flavor.
The most general effective Hamiltonian for $B_s - \overline{B}_s$ mixing
\begin{eqnarray}
H_{\rm eff} = \sum_{i=1}^5 C_i O_i + \sum_{i=1}^3 \wt{C}_i \wt{O}_i
\end{eqnarray}
has 8 independent operators
\begin{eqnarray}
O_1 &=& (\overline{s}_L \gamma_\mu b_L) \;(\overline{s}_L \gamma^\mu b_L), \nonumber\\
O_2 &=& (\overline{s}_R b_L) \;(\overline{s}_R b_L), \nonumber\\
O_3 &=& (\overline{s}_R^\alpha b_L^\beta) \;(\overline{s}_R^\beta b_L^\alpha), \nonumber\\
O_4 &=& (\overline{s}_R b_L) \;(\overline{s}_L b_R), \nonumber\\
O_5 &=& (\overline{s}_R^\alpha b_L^\beta) \;(\overline{s}_L^\beta b_R^\alpha), \nonumber\\
\wt{O}_{i=1,\cdots 3} &=& O_{i=1,\cdots 3} \l|_{L \leftrightarrow R}\r..
\end{eqnarray}
The Wilson coefficients for these
$\Delta B = \Delta S =2$ operators can be obtained
by calculating the gluino mediated box diagrams. Since the chargino and neutralino
exchanged box diagrams are suppressed by the small gauge coupling constants,
we neglect them. In the scenario we are considering, when we consider only LL (RR)
mixing, the SUSY box diagram contributes only to $C_1$ ($\wt{C}_1$).
When both LL and RR mixing exist simultaneously, there are also contributions
to $C_4$ and $C_5$. However, ${\stackrel{(\sim)}{C}}_2$ or
${\stackrel{(\sim)}{C}}_3$ are not generated at all.
Note that the induced LR (RL) mixing~\cite{Baek:1999} does not occur, either,
because we set $M^2_{\wt{d},LR(RL)} \equiv 0_{3\times 3}$. Otherwise,
the SUSY parameter space is further constrained depending
on $\tan\beta$~\cite{Baek:1999}.
The analytic formulas for the Wilson coefficients at the MSSM scale
are given by
\begin{eqnarray}
C_1^{\rm MSSM} &=& {\alpha_s^2 \over 4 m^2_{\wt{g}}} \sin^2 2\theta_L e^{2i \delta_L}
\l(
f_1(x_{\wt{b}_L,\wt{g}},x_{\wt{b}_L,\wt{g}})
-2 f_1(x_{\wt{s}_L,\wt{g}},x_{\wt{b}_L,\wt{g}})
+f_1(x_{\wt{s}_L,\wt{g}},x_{\wt{s}_L,\wt{g}})
\r),
\nonumber\\
C_{4(5)}^{\rm MSSM} &=& {\alpha_s^2 \over 4 m^2_{\wt{g}}} \sin 2\theta_L \sin 2\theta_R
e^{i (\delta_L+\delta_R)}
\l(
f_{4(5)}(x_{\wt{b}_R,\wt{g}},x_{\wt{b}_L,\wt{g}})
-f_{4(5)}(x_{\wt{b}_R,\wt{g}},x_{\wt{s}_L,\wt{g}}) \r.\nonumber\\
&&\l.-f_{4(5)}(x_{\wt{s}_R,\wt{g}},x_{\wt{b}_L,\wt{g}})
+f_{4(5)}(x_{\wt{s}_R,\wt{g}},x_{\wt{s}_L,\wt{g}})
\r), \nonumber\\
\wt{C}_1^{\rm MSSM} &=& C_1^{\rm MSSM} \l|_{L \leftrightarrow R}\r.,
\label{eq:WC}
\end{eqnarray}
where the loop functions are defined as
\begin{eqnarray}
f_1(x,y) &\equiv& {1 \over 9} j(1,x,y) + {11 \over 36} k(1,x,y), \nonumber\\
f_4(x,y) &\equiv& {7 \over 3} j(1,x,y) - {1 \over 3} k(1,x,y), \nonumber\\
f_5(x,y) &\equiv& {1 \over 9} j(1,x,y) + {5 \over 9} k(1,x,y),
\end{eqnarray}
and the $j$ and $k$ are defined in~\cite{Colangelo:1998pm}.
The RG running of the Wilson coefficients down to $m_b$ scale can be found,
for example, in~\cite{Becirevic:2001jj}.
We can calculate the $B_s - \overline{B}_s$ mixing matrix element, which is
in the form
\begin{eqnarray}
M^s_{12} = M_{12}^{s,\rm SM} (1 + R).
\label{eq:M12}
\end{eqnarray}
The mass difference of $B_s-\overline{B}_s$ system is then given by
\begin{eqnarray}
\Delta m_s &=& 2 | M^s_{12} | \nonumber\\
&=& \Delta m_s^{\rm SM} | 1 + R |.
\label{eq:dms}
\end{eqnarray}
In the SM contribution~\cite{Buras:1990fn} to the mass matrix element
\begin{eqnarray}
M_{12}^{s,\rm SM} = {G_F^2 M_W^2 \over 12 \pi^2}
M_{B_s} \l(f_{B_s} \hat{B}_{B_s}^{1/2} \r)^2
\eta_B S_0(x_t) \l(V_{tb} V_{ts}^*\r)^2,
\label{eq:M12_SM}
\end{eqnarray}
the non-perturbative parameters $f_{B_s}$ and $\hat{B}_{B_s}$
give main contribution to the theoretical uncertainty.
Using the combined lattice result~\cite{Okamoto:2005zg}
from JLQCD~\cite{Aoki:2003xb} and HPQCD~\cite{Gray:2005ad},
\begin{eqnarray}
f_{B_s} \hat{B}_{B_s}^{1/2} \Bigg|_{\rm (HP+JL)QCD} = (0.295 \pm 0.036) \;\; {\rm GeV},
\end{eqnarray}
the SM predicts
\begin{eqnarray}
\Delta m_s^{\rm SM} = (22.5 \pm 5.5) \;\; {\rm ps}^{-1},
\label{eq:dms_SM_pre}
\end{eqnarray}
which is consistent with the values in (\ref{dms:SM}) obtained from global
fits.
For the prediction in (\ref{eq:dms_SM_pre}), we used $\eta_B = 0.551$,
$\overline{m}^{\overline{MS}}_t(m_t) = 162.3$ GeV and
$V_{ts}=0.04113$~\cite{Charles:2006yw}.
Now, inserting the CDF data in (\ref{dms:exp}) and the
SM prediction in (\ref{eq:dms_SM_pre}) into (\ref{eq:dms}), we obtain
\begin{eqnarray}
|1+R| = 0.77 ^{+0.02}_{-0.01}({\rm exp}) \pm 0.19 ({\rm th}),
\label{eq:R}
\end{eqnarray}
where the experimental and theoretical errors were explicitly written.
The expression for $R$ in our scenario is given
by~\footnote{The $\hat{B}_{B_s}$ in (\ref{eq:M12_SM}) is related to
$B_1(\mu_b)$ as~\cite{Buras:1990fn}
\begin{eqnarray}
\hat{B}_{B_s} &\equiv& B_1(\mu_b) [\alpha_s^{(5)}(\mu_b)]^{-6/23}
\l[1+{\alpha_s^{(5)}(\mu_b) \over 4 \pi} J_5\r].
\end{eqnarray}
}
\begin{eqnarray}
R(\mu_b) &=& \xi_1(\mu_b) + \wt{\xi}_1(\mu_b) +
{3 \over 4} {B_4(\mu_b) \over B_1(\mu_b)}\l(M_{B_s} \over m_b(\mu_b) + m_s(\mu_b)\r)^2
\xi_4 \nonumber\\
&& + {1 \over 4}{B_5(\mu_b) \over B_1(\mu_b)}\l(M_{B_s} \over m_b(\mu_b) + m_s(\mu_b)\r)^2
\xi_5,
\end{eqnarray}
where we defined ($i=1,\cdots,5$)
\begin{eqnarray}
\xi_i(\mu_b) &\equiv& C_i^{\rm SUSY}(\mu_b)/C_1^{\rm SM}(\mu_b), \nonumber\\
\wt{\xi_i}(\mu_b) &\equiv& \wt{C}_i^{\rm SUSY}(\mu_b)/C_1^{\rm SM}(\mu_b).
\end{eqnarray}
The relevant B-parameters are given in~\cite{Becirevic:2001xt} by
\begin{eqnarray}
B_1(\mu_b) = 0.86(2)\l(^{+5}_{-4}\r), \quad
B_4(\mu_b) = 1.17(2)\l(^{+5}_{-7}\r), \quad
B_5(\mu_b) = 1.94(3)\l(^{+23}_{-7}\r).
\end{eqnarray}
Now we briefly discuss $B \to X_s \gamma$ constraint.
The SUSY parameters we consider are also directly constrained by the measured
branching ratio of inclusive radiative $B$-meson decay, $B \to X_s \gamma$.
We take this constraint
into account, although it is not expected to be so
severe as in a scenario with LR or RL mixing.
In the operator basis given in \cite{QCD_anatomy}, the
SUSY contributions to the Wilson coefficients of magnetic operators
in our scenario are
\begin{eqnarray}
C^{\rm SUSY}_{7\gamma} &=& -{4 \over 9} \; {1 \over \lambda_t}\;
{ \pi \alpha_s \sin 2\theta_L e^{i \delta_L} \over \sqrt{2} G_F m^2_{\wt{g}} }
\l[J_1(x_{b_L g}) - J_1(x_{s_L g}) \r], \nonumber\\
C^{\rm SUSY}_{8g} &=& \; {1 \over \lambda_t}\;
{ \pi \alpha_s \sin 2\theta_L e^{i \delta_L} \over \sqrt{2} G_F m^2_{\wt{g}} }
\l[\l(-{3 \over 2} I_1(x_{b_L g}) -{1 \over 6} J_1(x_{b_L
g})\r)
-(b_L \leftrightarrow s_L ) \r],
\label{eq:C7C8}
\end{eqnarray}
where $\lambda_t=V_{ts}^* V_{tb}$ and
\begin{eqnarray}
I_1(x) &=& \frac{1-6x+3x^2+2x^3-6x^2 \log x}{12(1-x)^4}, \nonumber\\
J_1(x) &=& \frac{2+3x-6x^2+x^3+6x\log x}{12(1-x)^4}.
\end{eqnarray}
There are also chirality flipped $\wt{C}_{7\gamma,8g}$ with $L$
replaced by $R$.
Therefore, we can see that in principle $\theta_{L(R)}$,$\delta_{L(R)}$ and
$m_{\wt{s}}-m_{\wt{b}}$ can be constrained.
Compared to the $LR(RL)$ mixing case where large SUSY contribution
${\cal O}(m_{\wt{g}}/m_b)$ is
possible due to the chirality flipping inside the loop,
our scenario allows only a small SUSY correction to the
SM contributions.
In addition, although LL mixing gives a linear correction
${\cal O}(C_{7\gamma,8g}^{\rm SUSY}/C_{7\gamma,8g}^{\rm SM})$
due to the interference term,
RR mixing generates only a quadratic correction
${\cal O}(|C_{7\gamma,8g}^{\rm SUSY}/C_{7\gamma,8g}^{\rm SM}|^2)$
because it is added incoherently to the SM contribution.
\section{Numerical analysis}
\label{sec:num}
In this Section, we perform numerical analysis and show the
constraints imposed by $\Delta m_s^{\rm exp}$.
We also consider the $BR(B \to X_s \gamma)$ constraint.
\begin{figure}[tbh]
\begin{center}
\psfrag{mmssLL}{$m_{ \wt{s}_L} ({\rm TeV})$}
\psfrag{ttLL}{$\theta_L$}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Fig1a.eps}
\label{fig:teL-msL-a}
}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Fig1b.eps}
\label{fig:teL-msL-b}
}
\end{center}
\caption{
Contour plots for $|1+R|$ in ($m_{\wt{s}_L}$,$\theta_L$) plane.
Sky blue region represents
2$\sigma$ allowed region ($0.39 \le |1+R| \le 1.15$),
blue 1$\sigma$ allowed region ($0.58 \le |1+R| \le 0.96$),
and white (grey) region is excluded at 95\% CL by giving too small (large)
$\Delta m_s$.
The labeled thick lines represent
the constant
$\Big(BR^{\rm tot}(B \to X_s \gamma)-BR^{\rm SM}(B \to X_s \gamma)\Big)/
BR^{\rm SM}(B \to X_s \gamma)$ contours.
Only LL mixing
is assumed to exist. The fixed parameters
are $m_{\wt{g}}=0.5$ (TeV), $m_{\wt{b}_L}=0.5$ (TeV),
(a) $\delta_L$=0, (b) $\delta_L=\pi/2$.
}
\label{fig:teL-msL}
\end{figure}
From (\ref{eq:WC}) it is obvious that the larger the mass splitting between
$\wt{s}$ and $\wt{b}$, the larger the SUSY contributions are.
Therefore we expect
that (\ref{eq:R}) constrains the mass splitting when the mixing
angle $\theta_{L(R)}$
is large. This can be seen in Figure~\ref{fig:teL-msL} where we show
filled contour plots for $|1+R|$ in ($m_{\wt{s}_L}$,$\theta_L$) plane:
sky blue region represents
2$\sigma$ allowed region ($0.39 \le |1+R| \le 1.15$),
blue 1$\sigma$ allowed region ($0.58 \le |1+R| \le 0.96$),
and white (grey) region is excluded at 95\% CL by giving too small (large)
$\Delta m_s$.
For these plots we assumed that
only LL mixing exists and fixed $m_{\wt{g}}=0.5$ TeV, $m_{\wt{b}_L} = 0.5$ TeV.
In Figure~\ref{fig:teL-msL-a}, we fixed $\delta_L = 0$. We can see that
the SUSY interferes with the SM contribution constructively
({\it i.e.} the SUSY contribution has the same sign with the SM),
and when the mixing angle
is maximal, {\it i.e.} $\theta_L = \pm\pi/4$, $m_{\wt{s}_L} - m_{\wt{b}_L}$
cannot be
greater than about 150 GeV. In Figure~\ref{fig:teL-msL-b},
we set $\delta_L = \pi/2$.
The SUSY contribution can interfere destructively
({\it i.e.} in opposite sign)
with the SM and much larger mass splitting
is allowed. Therefore we can see that the allowed parameters
are sensitive to the
CPV phase.
Also the constant
$\Big(BR^{\rm tot}(B \to X_s \gamma)-BR^{\rm SM}(B \to X_s \gamma)\Big)/
BR^{\rm SM}(B \to X_s \gamma)$
lines are shown.
For fixed $\theta_L$, larger mass splitting
$m_{\wt{s}_L}-m_{\wt{b}_L}$ gives larger deviation for the branching
ratio.
This can be understood from (\ref{eq:C7C8}).
However, for very large mass splitting the SUSY contribution
decouples and the deviation eventually
saturates.
We can see that $BR^{\rm tot}(B \to X_s \gamma)$
deviates from the SM predictions at most about 5\% in the region
allowed by $\Delta m_s$. Since
\begin{eqnarray}
BR^{\rm exp}(B \to X_s \gamma)/ BR^{\rm SM}(B \to X_s \gamma)
=1.06 \pm 0.13
\end{eqnarray}
for $E_\gamma > 1.6 $ GeV~\cite{b2sr},
it is clear that the
$BR(B \to X_s \gamma)$ constraint is completely irrelevant
in Figure~\ref{fig:teL-msL}.
The plots for the scenario with
RR mixing only are the same with Figure~\ref{fig:teL-msL} because
the expression for $B_s-\overline{B}_s$ is completely symmetric
under $L \leftrightarrow R$.
As mentioned above, the contribution to $BR(B \to X_s \gamma)$ is
much smaller than LL case.
\begin{figure}[tbh]
\begin{center}
\psfrag{ttLL}{$\theta_L$}
\psfrag{ddLL}{$\delta_L$}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Fig2a.eps}
\label{fig:teL-deL-a}
}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Fig2b.eps}
\label{fig:teL-deL-b}
}
\end{center}
\caption{Contour plots for $|1+R|$ in ($\theta_L$,$\delta_L$) plane.
(a) $m_{\wt{s}_L}=0.8$ (TeV),
(b) $m_{\wt{s}_L}=1.0$ (TeV).
The rest is the same with Figure~\ref{fig:teL-msL}.
}
\label{fig:teL-deL}
\end{figure}
In Figure~\ref{fig:teL-deL}, contour plots for
constant $|1+R|$ in ($\theta_L$,$\delta_L$) plane
are shown. For Figure~\ref{fig:teL-deL-a}(\ref{fig:teL-deL-b}),
we fixed $m_{\wt{s}_L}=0.8(1.0)$ TeV.
The other parameters used are the same with those in Figure~\ref{fig:teL-msL}.
We can again see the strong dependence on the CPV phase $\delta_L$. It can also be seen
that the parameter space with
large mixing angle $\theta_L$ can be made consistent with the experiments by
cancellation with
the SM contributions in the destructive interference region
({\it i.e.} $\delta_L \approx \pi/2$).
\begin{figure}[tbh]
\begin{center}
\psfrag{ttLL}{$\theta_L$}
\psfrag{ttRR}{$\theta_R$}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Fig3a.eps}
\label{fig:teL-teR-a}
}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Fig3b.eps}
\label{fig:teL-teR-b}
}
\end{center}
\caption{Contour plots for $|1+R|$ in ($\theta_L$,$\theta_R$) plane.
$m_{\wt{s}_L}=m_{\wt{s}_R}=0.6$ (TeV).
(a) $\delta_L = \delta_R = 0$
(b) $\delta_L = 0, \delta_R = \pi/2$.
We assume both LL and RR mixing exist.
The rest is the same with Figure~\ref{fig:teL-msL}.
}
\label{fig:teL-teR}
\end{figure}
Now we consider a scenario with both LL and RR mixing at the same time.
Then the operators $O_4$ and $O_5$ are additionally generated as
mentioned above.
They dominate $O_1$ or $\wt{O}_1$ for sizable mixing angles.
As a consequence, the constraint on the SUSY parameter space is very stringent as
can be seen in Figure~\ref{fig:teL-teR}.
In Figure~\ref{fig:teL-teR} we set $m_{\wt{g}}=0.5$ TeV,
$m_{\wt{b}_L}=m_{\wt{b}_R}=0.5$ TeV,
$m_{\wt{s}_L}=m_{\wt{s}_R}=0.6$ TeV, and
(a) $\delta_L = \delta_R = 0$ (b) $\delta_L =0, \delta_R = \pi/2$.
Even for small mass splitting most region of the parameter space is
ruled out by giving too large $\Delta m_s$.
We can see that $BR(B \to X_s \gamma)$ is almost insensitive to
the change of $\theta_R$ as mentioned before.
\section{The predictions of $S_{\psi\phi}$ and $A_{\rm SL}^s$}
\label{sec:asym}
\begin{figure}[tbh]
\begin{center}
\psfrag{mmssLL}{$m_{ \wt{s}_L} ({\rm TeV})$}
\psfrag{ddLL}{$\delta_L$}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Fig4a.eps}
\label{fig:msL-deL-a}
}
\subfigure[]{
\includegraphics[width=0.45\textwidth]{Fig4b.eps}
\label{fig:msL-deL-b}
}
\end{center}
\caption{Contour plots for $|1+R|$ in ($m_{\wt{s}_L}$,$\delta_L$) plane.
The $S_{\psi \phi}$ predictions are also shown as thick contour lines.
The thin red lines are constant $A_{SL}^s[10^{-3}]$ contours assuming
${\rm Re}(\Gamma_{12}^s / M^s_{12})^{\rm SM} = -0.0040$.
(a) Only LL mixing is assumed to exist.
We fixed $m_{\wt{g}}=m_{\wt{b}_L}=0.5$ TeV, $\delta_L = \pi/4$.
(b) Both LL and RR mixing are assumed to exist simultaneously.
We fixed $m_{\wt{g}}=2$ TeV,
$m_{\wt{b}_L}=m_{\wt{b}_R} = 1$ TeV,
$m_{\wt{s}_R} = 1.1$ TeV,
$\theta_R = \pi/4$,
$\delta_L = \pi/4$, and
$\delta_R = \pi/2$.
The rest is the same with Figure~\ref{fig:teL-msL}.
}
\label{fig:msL-deL}
\end{figure}
The CPV phase in the $B_s - \overline{B}_s$ mixing amplitude will be measured
at the LHC in the near future through the time-dependent CP
asymmetry
\begin{eqnarray}
\frac{\Gamma(\overline{B}_s(t) \to \psi\phi)-\Gamma(B_s(t) \to \psi\phi)}
{\Gamma(\overline{B}_s(t) \to \psi\phi)+\Gamma(B_s(t) \to \psi\phi)}
\equiv S_{\psi\phi} \sin(\Delta m_s t).
\end{eqnarray}
In the SM, $S_{\psi\phi}$ is predicted to be very small,
$S_{\psi\phi}^{\rm SM} = -\sin 2 \beta_s = 0.038 \pm 0.003$
($\beta_s \equiv \arg[(V_{ts}^* V_{tb})/(V_{cs}^* V_{cb})]$)~\cite{model_indep2}.
If the NP has additional CPV phases, however, the prediction
\begin{eqnarray}
S_{\psi\phi} = -\sin(2 \beta_s + \arg(1 + R))
\end{eqnarray}
can be significantly different from the SM prediction.
In Figure~\ref{fig:msL-deL}, we show $|1+R|$ constraint and the prediction
of $S_{\psi\phi}$ in $(m_{\wt{s}_L},\delta_L)$ plane.
However, the $B\to X_s \gamma$ prediction is not shown from now on
because it is irrelevant as mentioned above.
For Figure~\ref{fig:msL-deL-a}, we assumed the scenario with LL mixing only
and maximal mixing $\theta_L = \pi/4$.
We fixed $m_{\wt{g}}=0.5$ TeV, $m_{\wt{b}_L} = 0.5$ TeV.
For Figure~\ref{fig:msL-deL-b}, we allowed both LL and RR mixing simultaneously,
while fixing
$m_{\wt{g}}=2$ TeV,
$m_{\wt{b}_L}=m_{\wt{b}_R} = 1$ TeV,
$m_{\wt{s}_R} = 1.1$ TeV,
$\theta_R = \pi/4$,
$\delta_L = \pi/4$, and
$\delta_R = \pi/2$.
In both cases we can see that large $S_{\psi\phi}$ is
allowed for large mass splitting
between $m_{\wt{b}_L}$ and $m_{\wt{s}_L}$.
At the moment, $S_{\psi\phi}$
can take any value in the range $[-1,1]$ even after
imposing the current $\Delta m_s^{\rm exp}$ constraint.
\begin{figure}[tbh]
\begin{center}
\psfrag{SSpsiphi}{$S_{\psi\phi}$}
\psfrag{AASL}{$A_{SL}^s[10^{-3}]$}
\includegraphics[width=0.8\textwidth]{Fig5b.eps}
\end{center}
\caption{
The correlation between $A^s_{\rm SL}$ and $S_{\psi\phi}$.
The red line is 1-$\sigma$ upper bound.
}
\label{fig:ASL}
\end{figure}
Finally we consider the semileptonic CP asymmetry~\cite{Randall:1998te,
Baek:1998yn,model_indep2}
\begin{eqnarray}
A^s_{\rm SL} \equiv
\frac{\Gamma(\overline{B}_s\to \ell^+ X)-\Gamma(B_s\to \ell^- X)}
{\Gamma(\overline{B}_s\to \ell^+ X)+\Gamma(B_s\to \ell^- X)}
={\rm Im}\l(\Gamma_{12}^s \over M^s_{12}\r).
\end{eqnarray}
It is approximated to be~\cite{model_indep2}
\begin{eqnarray}
A^s_{\rm SL} \approx {\rm Re}\l(\Gamma_{12}^s \over M^s_{12}\r)^{\rm SM}
{\rm Im}\l(1 \over 1 + R\r),
\end{eqnarray}
where
${\rm Re}(\Gamma_{12}^s / M^s_{12})^{\rm SM}
= -0.0040 \pm 0.0016$~\cite{Beneke:2003}.
The SM prediction is
$A^s_{\rm SL}(\rm SM)=(2.1 \pm 0.4)\times 10^{-5}$~\cite{Beneke:2003,Ciuchini:2003}.
In Figure~\ref{fig:msL-deL}, the thin red lines are constant
$A_{SL}^s [10^{-3}]$ contours taking
${\rm Re}(\Gamma_{12}^s / M^s_{12})^{\rm SM} = -0.0040$.
We can readily see that the strong correlation between $S_{\psi\phi}$
and $A_{SL}^s$.
This can be seen from the relation
\begin{eqnarray}
A_{SL}^s =
- \l|{\rm Re}\l(\Gamma^s_{12} \over M^s_{12}\r)^{\rm SM}\r|
{S_{\psi\phi} \over |1+R|}.
\end{eqnarray}
For small $R$ the two observables are linearly correlated
as can be seen in Figure~\ref{fig:msL-deL}.
In Figure~\ref{fig:ASL}, we show the correlation
between $A^s_{\rm SL}$ and $S_{\psi\phi}$.
We
scanned $0.5 \le m_{\wt{g}} \le 4.0$ TeV,
$0.5 < m_{\wt{b}_{L}}, m_{\wt{s}_{L}} < 2.0$ TeV,
$-\pi/4 < \theta_L < \pi/4$
and $0 < \delta_L < 2 \pi$, while fixing $m_{\wt{g}}=m_{\wt{b}_L}=0.5$ TeV.
The $\Delta m_s$ constraint is imposed with $0.39 \le |1+R| \le 1.15$.
We have checked that in the scenario with only LL (RR) mixing,
we get the similar correlations.
The red line is experimental 1-$\sigma$ upper bound from
$A^s_{\rm SL} = -0.013 \pm 0.015$~\cite{model_indep2}.
Now several comments are in order:
i) The values for $S_{\psi\phi}$ and $A^s_{\rm SL}$
can be significantly different from the SM predictions.
ii)
The two observables are strongly correlated.
These two facts were already noted in \cite{model_indep2}.
It has been checked that in the $({\rm Re} R, {\rm Im} R)$ plane
the above scanned points can completely fill the
region allowed by $\Delta m_s$.
This explains why the correlation in Figure~\ref{fig:ASL} is
basically the same with model-independent prediction
in \cite{model_indep2}.
iii)
Although it looks like that large negative $S_{\psi\phi}$ value is
disfavored, due to large error in
${\rm Re}(\Gamma_{12}^s / M^s_{12})^{\rm SM}$
we cannot definitely rule out the region at the moment.
\section{Conclusions}
\label{sec:con}
We considered the MSSM scenario with large LL and/or RR mixing
in the down-type mass squared matrix. This scenario
is strongly constrained by the recent mesurements of $B_s - \overline{B}_s$
mass difference, $\Delta m_s$, in contrast with the MSSM
scenario where the flavor mixing is controlled only by the CKM
matrix~\cite{Baek:1998yn,Buras:2006}.
The constraint is most stringent when both LL and RR mixing exist
simultaneously. It is also shown that the allowed region is quite
sensitive to the CP violating
phase.
We also considered the time-dependent CP asymmetry, $S_{\psi\phi}$, and
the semileptonic CP asymmetry, $A_{\rm SL}^s$. It was shown that
the $S_{\psi\phi}$ and $A^s_{\rm SL}$ can take values significantly
different from the SM predictions. There is also strong
correlation between $S_{\psi\phi}$ and $A_{\rm SL}^s$.
\vskip1.3cm
\noindent {\bf Acknowledgment}\\
The author thanks P.~Ko for useful discussions and the organizers
of ``Workshop on Physics at Hadron Colliders'' at KIAS where part
of this work was motivated. The work was supported by the
Korea Research Foundation Grant funded by the Korean Government
(MOEHRD) No. KRF-2005-070-C00030.
|
1,108,101,565,657 | arxiv | \section{Introduction}
The mass metallicity relationship (MZR) is an important galaxy scaling relation that describes the coevolution of galaxies and their metal content~\citep[e.g.][]{Tremonti2004}.
Over the past several years, observational results have begun to indicate that the scatter in the MZR may be correlated with galactic star formation rates~\citep[SFR; e.g.][]{Ellison_FMR} or gas masses~\citep[e.g.][]{Bothwell2013}.
This has led to the suggestion that there is a fundamental metallicity relation~\citep[FMR;][]{LaraLopez2010, Mannucci_FMR} describing the combined relation between galactic stellar mass, SFR (or gas mass), and metallicity.
Evidence has been presented showing that this FMR holds across a wide mass range~\citep{Salim2014, Brown2017} as well as out to high redshift~\citep{Belli2013, Stott2014, Bothwell2016a} with increasingly systematic and comprehensive analyses~\citep{Sanders2017}.
Further, theoretical models have been built explaining why the FMR should naturally occur~\citep[e.g.][]{Lilly2013, Forbes2014, Zahid2014}.
However, there is no consensus about the existence of the FMR~\citep[e.g.][]{Sanchez2013, Sanchez2017}.
In particular, there are concerns that the FMR is driven by systematic uncertainties in nebular emission line metallicity diagnostics~\citep{Telford2016}
or contaminated by incomplete/non-global fibre corrections~\citep{BarreraBallesteros2017, Ellison2017, Sanchez2017}.
There is also disagreement about the strength~\citep[e.g.][]{Andrews2013} or mass dependence~\citep[e.g.][]{Yates2012} of the residual correlation as well as the persistence of the FMR in high redshift data~\citep{Yabe2015}.
In short, while the FMR provides an inviting link between the fluctuations in the metallicities and SFRs (or gas masses), it is not yet clear if the observational evidence fully supports this scenario.
Yet, in a recent paper~\citep{Torrey2017}, we demonstrated that a FMR is naturally produced in hydrodynamical simulations~\citep[see also][]{Dave2017, DeRossi2017}.
In this Letter, we explore the conditions that are required for a FMR to emerge.
In particular, the emergence of this relation relies on galactic \textit{offsets} from the MZR and star formation main sequence (SFMS) remaining anti-correlated as they evolve, which we argue requires that the SFR and metallicity share similar dominant evolution timescales.
If either the SFR or metallicity evolved much faster than its counterpart, then the anti-correlation between the two quantities would be washed out.
As we argue in this Letter, the dominant timescales for metallicity and SFR evolution are similar in our galaxy formation model.
We further argue that the evolution timescales for SFRs and metallicities are generally set by galaxy dynamics in our model, not by the adopted feedback physics.
The similarity in the SFR and metallicity evolution timescales enables the existence of the FMR.
However, as we briefly address in Section~\ref{sec:Conclusions}, we speculate that the strength of the FMR -- especially for low mass galaxies at high redshift -- may be reduced in models with particularly strong and/or globally-bursty feedback.
\vspace{-5 mm}
\section{Methods}
\label{sec:Methods}
In this Letter we analyze the time variability of galactic SFRs and metallicities using the IllustrisTNG simulation suite~\citep{Marinacci2017, Naiman2017, Nelson2017, Pillepich2017b, Springel2017}.
The IllustrisTNG simulation suite builds on the original Illustris simulation via a series of numerical and physical model improvements~\citep{Weinberger2017, Pillepich2017} over the original Illustris model~\citep{Vogelsberger2013, Torrey2014}.
In this Letter we use the TNG100 simulation which employs a simulation box of side length $L\approx100\; \mathrm{Mpc}$ and is an analog to the original Illustris simulation volume.
SFRs are determined using a subgrid model which leads to smoothly-varying, non-bursty SFR histories~\citep{SH03, vogelsberger2014a, genel2014, Sparre2015}.
Gas is converted into stars using a stochastic star formation prescription.
Each simulation stellar particle represents an unresolved full stellar population which we assume is described by a Chabrier initial mass function.
As stellar particles age, they return both mass and metals locally to the ISM resulting in time- and spatially-dependent metal enrichment.
Enrichment predictions of the IllustrisTNG simulations agree on galactic~\citep{Naiman2017, Torrey2017} and cluster scales~\citep{Vogelsberger2017} with observations.
Variability in the SFRs and metallicity values is therefore naturally driven by the variety of formation histories among the simulated galaxy population.
We always quote instantaneous (un-smoothed) SFRs and metallicity values as the SFR weighted average metallicity for all gas within a galaxy.
\begin{figure}
\centerline{\vbox{\hbox{
\includegraphics[width=0.45\textwidth]{./cbar_L75n1820TNG_mz_centrals_sfr_z10.pdf} }}}
\centerline{\vbox{\hbox{
\includegraphics[width=0.45\textwidth]{./L75n1820TNG_mz_centrals_sfr_z10.pdf} }}}
\caption{
Two-dimensional histogram of average specific SFRs for IllustrisTNG galaxies as a function of metallicity and stellar mass at $z=1$.
Solid and dashed black lines indicate the median MZR and one sigma scatter, respectively.
There is a residual correlation about the MZR where galaxies with high metallicities have low SFRs, and galaxies with low metallicities have high SFRs.
}
\label{fig:FMR}
\end{figure}
\vspace{-5 mm}
\section{Results}
\label{sec:Results}
Figure~\ref{fig:median_relations} shows the SFMS (left panel) and MZR (right panel) at several redshifts.
There is significant redshift evolution in both of these relations with higher redshift galaxies having higher SFRs~\citep{Weinberger2017b} and lower metallicities~\citep{Torrey2017}.
We use these relations to build a two-dimensional interpolation function that yields the SFR or metallicity as a function of stellar mass and redshift.
Using the median relations defined above, we track galaxies in time and identify their offsets from the scaling relations.
The insets in Figure~\ref{fig:median_relations} show tracks for the offset evolution of one randomly selected galaxy with mass $M_* \approx 10^{10} \mathrm{M}_\odot$ at $z=0$.
We highlight periods of time where this galaxy is above or below the scaling relations by shading blue or red, respectively.
There are some features in the offset evolution tracks that are repeated between the SFR and metallicity panels.
In particular, we find ``mirrored" offset evolution between the metallicity and SFR offset values
over a broad redshift range where enhancements in the SFR of this galaxy above the SFMS correspond to depressions in the metallicity below the MZR.
The anti-correlated evolution of the MZR and SFMS offsets has the observable consequence of driving residual correlations in the scatter about the MZR.
Figure~\ref{fig:FMR} shows the average specific SFR for galaxies distributed about the simulated MZR at redshift $z=1$.
There is a clear residual correlation between MZR offset and SFR for the full galaxy population, indicating that the anti-correlated offset behavior shown in the individual evolution tracks in the inset of Figure~\ref{fig:median_relations} is commonly found in the larger galaxy population.
\textcolor{black}{Although we only plot $z=1$ here for brevity, this residual trend is also present at other redshifts~\citep{Torrey2017} and we summarize the best fitting slope relating offset from the MZR with offset from the SFMS in Table~\ref{table:FMR_slopes}.
If galaxies randomly changed their SFR and metallicity values, or if the SFR and metallicity evolution had significantly different time variability, the strength of the residual correlation between SFR and MZR offset would be washed out.}
In this Letter we use a simple metric for identifying the smallest timescale with significant SFR variability by determining how well a galaxy population's offsets from the SFMS at some time, $t_0$, indicate the same galaxy population's offsets from the SFMS at some earlier time, $t_0 - \Delta t$.
We specifically use a Pearson correlation coefficient, $\rho(M_*, t_0, \Delta t)$, to describe the strength of correlation between the offsets from the SFMS for a galaxy population with average stellar mass, $M_*$, measured at the two different times.
The correlation strength is measured as a function of initial time, $t_0$, time separation, $\Delta t$, and in a series of initial stellar mass bins $M_*\pm\Delta 0.25$ dex.
For any given initial time, $t_0$, and initial stellar mass, $M_*$, we fit exponential decay curves, $\rho(M_*, t_0, \Delta t) = \mathrm{exp}(-\Delta t/\tau)$, to the measured correlation coefficients as a function of time separation, which we find provides an adequate fit.
We use the best fitting decay timescale, $\tau=\tau(M_*, t_0)$, to define the SFR evolution timescale for that particular initial time, $t_0$, and initial stellar mass, $M_*$.
We perform an identical procedure to determine metallicity evolution timescales, where we instead use offsets from the MZR as input.
There are $\sim$80 simulation snapshots over the redshift range $0\leq z \leq4$ that are used in this procedure.
\begin{table}
\begin{center}
\caption{Best fitting slopes, $\alpha$, between SFMS offset, $\Delta \mathrm{log\;SFR} $, and MZR offset, $\Delta \mathrm{log}\;Z$, in $\pm0.1$ dex stellar mass bins at several redshifts. Slopes are calculated as $\Delta \mathrm{log}\;Z = \alpha \Delta \mathrm{log\;SFR} $ via $\chi^2$ minimization. }
\label{table:FMR_slopes}
\begin{tabular}{ l c c c c }
\hline
$M_*$ & $ 10^9 \mathrm{M}_\odot$ & $ 10^{9.5} \mathrm{M}_\odot$ & $10^{10} \mathrm{M}_\odot$ & $ 10^{10.5} \mathrm{M}_\odot$ \\
\hline
\hline
$\alpha(z=0)$ & -0.19 & -0.30 & -0.30 & -0.11 \\
$\alpha(z=1)$ & -0.28 & -0.28 & -0.27 & -0.24 \\
$\alpha(z=2)$ & -0.27 & -0.29 & -0.25 & -0.29 \\
$\alpha(z=3)$ & -0.30 & -0.27 & -0.34 & -- \\
$\alpha(z=4)$ & -0.26 & -0.26 & -- & -- \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}
We note that this simple method only identifies the shortest timescale over which there is significant variation in the SFR values.
Noisy data may lead to a short derived evolution timescale, even if there is meaningful/significant longer term variability.
However, for the smoothly varying SFR and metallicity evolution histories typical of IllustrisTNG galaxies, we find this method adequately describes the physically relevant evolution/variability timescales.
\begin{figure*}
\centerline{\vbox{\hbox{
\includegraphics[width=0.45\textwidth]{./sfr_correlation_time_map_L75n1820TNG.pdf}
\includegraphics[width=0.45\textwidth]{./correlation_time_map_L75n1820TNG.pdf}
}}}
\caption{
Maps of the evolution timescale (see text for details) for the SFRs (left) and metallicities (right) in the TNG100 simulation.
In general, the SFR correlation timescales are somewhat shorter than the metallicity timescales, but both show a redshift dependence where higher redshift galaxies evolve on shorter timescales for galaxies with $M_* \lesssim 10^{10.5} \mathrm{M}_\odot$. }
\label{fig:correlation_time_map}
\end{figure*}
Figure~\ref{fig:correlation_time_map} shows the derived SFR (left) and metallicity (right) evolution timescales as a function of stellar mass and redshift for the full galaxy population.
We highlight two trends identifiable within this space.
First, the SFR and metallicity timescales associated with the highest mass galaxies (i.e. $M_* > 10^{10.5} \mathrm{M}_\odot$) are somewhat disjoint from the rest of the galaxy population owing to active galactic nuclei (AGN) feedback.
In this regime, SFRs can change rapidly as AGN feedback operates, but metallicities evolve more slowly as the low SFRs and low accretion rates make it difficult to modify the metallicity of the central gas reservoir.
Second, the timescales that describe the SFR and metallicity evolution for galaxies with stellar masses below $M_* \lesssim 10^{10.5} \mathrm{M}_\odot$ are similar.
There is a trend where higher redshift galaxies have shorter SFR and metallicity evolution timescales compared to their lower redshift companions.
While $z=0$ galaxies have evolution timescales of just over a Gyr, high redshift galaxies of the same mass have evolution timescales several times shorter than this.
There is limited change in the evolution timescale with mass at a fixed redshift for both the metallicity and SFR maps for masses $M_* \lesssim 10^{10.5} \mathrm{M}_\odot$.
\begin{figure}
\centerline{\vbox{\hbox{
\includegraphics[width=0.5\textwidth]{./correlation_time_series_L75n1820TNG.pdf}
}}}
\caption{ Metallicity (red) and SFR (blue) evolution timescales \textcolor{black}{ averaged over the mass range $10^9 < M_*/\mathrm{M}_\odot < 10^{10.5}$ } as a function of redshift for IllustrisTNG (solid) and Illustris (dot-dashed). Additionally, the halo dynamical time (black solid line) and twice the halo dynamical time (black dashed) lines are shown.
}
\label{fig:correlation_time_series}
\end{figure}
Figure~\ref{fig:correlation_time_series} shows a direct comparison of the redshift evolution of the SFR and metallicity correlation timescales.
The colored solid lines and shaded regions indicate stacked median and 10${}^{\mathrm{th}}$/90${}^{\mathrm{th}}$ percentile ranges for the evolution timescale evenly weighted over the mass range $10^9 \mathrm{M}_\odot < M_* < 10^{10.5} \mathrm{M}_\odot$.
The scatter about these evolutionary tracks is remarkably small considering the large mass range, but is consistent with the limited mass dependence shown in both panels of Figure~\ref{fig:correlation_time_map}.
Interestingly, we find that the metallicity and SFR evolution timescales are similar in magnitude and evolve in a similar sense with time.
The metallicity evolution timescales are somewhat larger (i.e. $\sim1.2-1.5$ times larger) than the SFR evolution timescales.
There is a late time drop in the SFR correlation timescale which is dominated by the highest mass bins (see Figure~\ref{fig:correlation_time_map}) where AGN feedback plays an increasing role in modulating the SFRs.
The blue and red dot-dashed lines in Figure~\ref{fig:correlation_time_series} indicate the Illustris SFR and metallicity evolution timescales.
While the Illustris SFR evolution timescales are a very good match to the IllustrisTNG trends, the Illustris metallicity evolution timescales are roughly $\sim$30 per cent larger than the IllustrisTNG metallicity evolution values.
Overall, however, the timescales of both models show a similar trend in their redshift evolution and all measured SFR and metallicity evolution timescales are the same within a factor of $\sim2$.
Although not shown here, we note that Illustris also shows a Z-SFR anti-correlation~\citep{Genel2016}.
In addition to the metallicity and SFR evolution timescales, Figure~\ref{fig:correlation_time_series} shows the halo dynamical time
\begin{equation}
\tau_{\mathrm{DM,dyn}} = \left( \frac{3 \pi}{32 G \rho_{200,\mathrm{crit} } } \right)^{1/2} \sim 0.1 \; \tau_{\mathrm{H}},
\end{equation}
where $\tau_{\mathrm{H}}$ is the Hubble time.
Without adjustment, the halo dynamical time tracks the SFR evolution timescales for both simulations reasonably well.
The metallicity evolution timescales for IllustrisTNG and Illustris also follow the halo dynamical time tracks, but are scaled up by factors of 1.5 and 2.0, respectively.
The scaling of these quantities with the halo dynamical time points to a picture where variability in the SFRs and metallicities is dominated by the evolution of the halo~\citep{Lilly2013} -- not by the specific adopted Illustris or IllustrisTNG feedback physics.
\textcolor{black}{This conclusion is supported because the halo dynamical time has no mass dependence and the measured SFR/metallicity evolution timescales have only a limited/modest mass dependence.}
The variation in the metallicity evolution timescales between Illustris and IllustrisTNG does, of course, indicate that the implemented galaxy formation physics can impact the evolution timescales at the factor of $\sim2$ level.
\textcolor{black}{However, while the FMR is not significantly impacted by factor of $\sim$2 differences in the SFR and metallicity evolution timescales~\citep{Genel2016, Torrey2017},
as we briefly discuss in the following section, we speculate that models with order-of-magnitude different/shorter evolution timescales may have a weaker FMR since
very rapid SFR variability will wash out residual correlations between SFR and metallicity offsets.}
\vspace{-8 mm}
\section{Discussion and Conclusions}
\label{sec:Conclusions}
In this Letter, we have shown that SFRs and metallicities in the Illustris and IllustrisTNG models evolve over similar timescales which are reasonably well matched to the halo dynamical time.
This similarity allows for the existence of the FMR.
The mirrored nature of the offsets from the MZR and SFMS that drive the correlation between offset from the MZR and SFR only exists because the metallicity and SFRs of galaxies evolve over \textit{similar} timescales.
If the dominant timescales for SFR and metallicity evolution were very different, then residual correlations with SFR about the MZR would be weakened.
The existence of correlated scatter as described by the FMR instead implies that the dominant timescale for variation with respect to the mean relations must be similar for galactic metallicity and SFRs.
Our models predict a continued existence of the FMR out to high redshift~\citep{Torrey2017} because the employed model gives rise to non-bursty SFR histories that evolve on timescales comparable to the halo dynamical time~\citep[see also, e.g.,][]{Dave2017, DeRossi2017}.
We do not necessarily expect that the results presented in this Letter would be recovered by simulations with galaxies dominated by globally-bursty SF histories.
For example, globally-bursty stellar feedback can strongly impact SFRs, driving significant changes to the SFR evolution without necessarily impacting metallicity in the same way.
Specifically, outflows possessing the same metallicity as the ISM can immediately change the SFR of a galaxy, while not impacting the ISM metallicity.
Globally-bursty SF histories have the ability to drive SFR evolution timescales down -- possibly by orders of magnitude to $\sim$10 or 100 Myrs~\citep[e.g.][Figure 9]{Sparre2017} -- which could drive the strength of correlation between offset from the MZR and SFR to be weaker than what our models find.
As of yet, the existence or strength of the FMR is still under debate -- both at low and high redshift -- and so it is not clear which of these models is a better match to the real Universe.
Accurately assessing the existence or strength of the FMR, especially toward higher redshift, is important because it may discriminate between bursty and non-bursty galaxy formation feedback models.
\vspace{-6 mm}
\section*{ACKNOWLEDGEMENTS}
PT thanks Alice Shapley, Ryan Sanders, and Tiantian Yuan for helpful discussion.
PT acknowledges support from NASA through Hubble Fellowship grant HST-HF2-51341.001-A awarded by STScI, which is operated under contract NAS5-26555.
RM acknowledges support from the DOE CSGF under grant number DE-FG02-97ER25308.
RW, VS and RP acknowledge support through the ERC under ERCStG grant EXAGAL-308037, and the Klaus Tschira Foundation.
RW acknowledges the IMPRS for Astronomy and Cosmic Physics at the University of Heidelberg.
VS acknowledges support from subproject EXAMAG of the Priority Programme 1648 \textit{Software for Exascale Computing} of the German Science Foundation.
MV acknowledges the support of the Alfred P. Sloan Foundation and NASA ATP grant NNX17AG29G.
JPN acknowledges support of NSF AARF award AST-1402480.
The Flatiron Institute is supported by the Simons Foundation.
The IllustrisTNG simulations and exploratory runs were run on the HazelHen Cray XC40-system (project GCS-ILLU), Stampede at TACC/XSEDE (allocation AST140063), and Hydra and Draco at the Max Planck Computing and Data Facility.
\vspace{-6 mm}
|
1,108,101,565,658 | arxiv |
\section{Introduction}
\label{sec:intro}
Rapid shifts in technology and business models have led to a mismatch between the skills needed by employers and the skills possessed by the labor force. It has been estimated that this mismatch will seriously hinder progress on climate change and reduce manufacturing output by \$2.4 trillion over ten years in the US alone \cite{giffi2018deloitte, guldimann2022green}. As such, increased attention has been placed on effective methods of ``reskilling'' workers \cite{agrawal2020beyond}. Unfortunately, reskilling will not be easy: human expertise in performing a complex task takes years of training and mastery of domain-specific knowledge \cite{bryan1899studies, ericsson1996expert}.
Of particular interest to the computer vision community is the role that Augmented Reality (AR) headsets can play in collective reskilling efforts. AR headsets are known to improve efficacy of front line workers during training and on the job, across industries as diverse as food service, manufacturing, medicine, and warehousing\cite{abraham2017augmented, clark2019educational, ruthberg2020mixed, schwerdtfeger2009pick}. AR plays a strikingly similar role across these diverse use cases: to assist the user in completing a complex task, the headset renders a sequence of visual cues on real-world objects. Note the contrast to instructional videos or even handheld AR devices; AR headsets provide hands-free, spatially-localized, step-by-step instructions \emph{in situ}. In short, an AR headset enables an instructor to scale apprenticeship across space and time.
We focus on how to aid an expert practitioner in generating the AR content necessary to fulfill this promise. Specifically, our approach focuses on extraction of key-steps of a complex task which is the most crucial component needed for automatic AR content creation.
We employ a ``learning-from-observation''-style framework \cite{michalski1983learning}, where an instructor is recorded while performing a complex task. The goal is to automatically parse the recording into {\em key steps} (KSs) that succinctly represent the complete task. This greatly streamlines the content creation process, as the trainer no longer has to manually edit the recording to find the key steps.
\input{figs/teaser}
As an example, consider the task of changing the cartridge in a printer. Using tools from the publicly available repository \cite{hl2_rm}, we captured data on a Microsoft HoloLens 2 of an ``expert'' undertaking this task. Using multiple data streams, including hand pose, head pose, eye gaze and first person video, we automatically generate the key-steps presented in Fig.~\ref{fig:teaser_figure}. We note that using multiple cues when observing experts perform procedural tasks is important when generating training materials for novices~\cite{charness2008role}.
We adopt a two-stage approach to the KS extraction problem. First, we train a task-specific model to produce a context-rich feature vector for each frame of the recording. This is followed by a cluster-and-sample approach, where these features are used to extract the frames corresponding to key-steps. As our domain is complex procedural tasks that are typically performed by an expert to train novices, we are constrained by limited availability of expert recordings and no ground truth labels. That is, KSs must be derived from at most a few unlabeled instances of the task.
To overcome paucity of data, our Multi-cue key steps (STEPs{}) approach, (Fig. \ref{fig:the_model} and \cref{sec:method}), uses multiple temporal feature sequences corresponding to different cues as input. We use {\em temporal consistency} and {\em cross-modal alignment} as proxy tasks in self-supervised training of LSTM-based encoders. We train one encoder per input stream and target a common representation space:
frames that are nearby in time should have similar representations across all modalities.
After training, the per-frame representations are clustered and KSs are sampled from the clusters.
Key step supervision is hard due to the subjective nature of what constitutes a key step. There are no large-scale datasets for real world procedural tasks of interest in the AR training domain.
This makes self-supervision a hard requirement for our setup.
Unlike recent work in self-supervised video representation learning \cite{dwibedi2019temporal,haresh2021learning, liu2021learning}, we do \emph{not} rely on alignment between multiple recordings as a proxy task. {\em Inter-video alignment} as a learning loss requires multiple recordings of the same task, thus limiting its practical applicability to the problem at hand.
Our approach is suitable for not only multi-modal data computed by first-person wearable devices such as Oculus Quest, Microsoft HoloLens and others, but also conventional procedural task videos. In this case, we employ off-the-shelf feature extraction backbones, without fine-tuning, to extract raw per-frame, multi-cue features. This enables fast LSTM encoder training; we can train a model on 60 IKEA assembly videos~\cite{ben2021ikea} for 100 epochs in under 6 minutes on a single GPU. The result is tailored to a specific setting, as is appropriate for the domain of AR content creation for specific tasks, where one needs {\em exactly} the right set of steps for a given complex task. We require multiple modalities {\em only} during training, and can work with RGB or other uni-modalities alone during key step extraction or other downstream tasks.
Assessing the efficacy of STEPs{} is challenging. Key steps are task specific and subjective, and, to our knowledge, there are no public, labeled KS datasets collected with AR headsets. As such, we provide quantitative and qualitative results on multiple video datasets in \cref{experiments}. The closest benchmark to our problem is that of key step localization. On that task, we outperform \cite{elhamifar2020self} by 1.15 (on CrossTask) and 3.5(on ProceL) F1 scores {\em without requiring task labels}. As the heart of STEPs{} is video-based representation learning, we also provide results on common procedural video tasks such as phase classification. We achieve state-of-the-art results: 6.79 percentage point (pp) improvement for phase classification accuracy on the challenging IkeaAssembly dataset and 2.76pp improvement on the PennActions dataset.
In summary, the key contributions of our work include:
\begin{enumerate}
\itemsep0em
\item An intra-video SSL-approach for per-frame video representation learning. Unlike prior approaches that train expensively large backbones, we apply SSL to pre-trained features on multiple cues.
\item Exploring the efficacy of pre-training and fine-tuning on limited videos for procedure learning.
\item Superior performance compared to state-of-the-art on several tasks: KS localization, phase classification and Kendall's Tau with a low-complexity SSL module.
\end{enumerate}
\section{Related Work}
\label{sec:related}
STEPs{} is inspired by current video understanding research including key step (KS) localization, procedure learning, self-supervised learning and video summarization.
\textbf{KS extraction \& procedure learning of complex tasks.}
To obtain key steps, prior works use a state transition model \cite{elhamifar2019unsupervised}, Mallows model ~\cite{sener2018unsupervised}, clustering and ordering of visual features ~\cite{kukleva2019unsupervised} or weak alignment between visual and linguistic cues~\cite{shen2021learning}. These approaches either rely on per-task training or additional language cues. A subset selection module is used as a teacher in \cite{elhamifar2020self} to obtain an unsupervised localization of KSs in a multi-task setting assuming access to task labels. We use multi-cue learning using features obtained from the visual stream alone. We intrinsically enable a multi-task setting since the model is agnostic to the actual task performed. Our approach also enables low-shot KS extraction. Our representation learning is not specific to KS localization. We show its efficacy for other tasks as well.
Most similar to AR applications are the videos from the Ikea-Assembly dataset \cite{ben2021ikea} that consist of unedited videos without any cuts, and do not have abrupt camera motion. Approaches that aim to learn features from complex task videos use pretext tasks to learn representations without supervision. The representations are evaluated on downstream tasks like phase classification and Kendall`s Tau. They learn features using pairs of videos from a task by finding correspondences and imposing constraints like cycle-consistency~\cite{dwibedi2019temporal}, time-warping~\cite{haresh2021learning} and optimal-transport based alignment~\cite{liu2021learning}. We use off-the-shelf features and do not require pairs of videos in training. Multiple cues in training provide significant improvement over the state-of-the-art.
\input{figs/main_model}
\noindent \textbf{SSL for video understanding.} Contrastive learning (CL) ~\cite{gutmann2010noise,hadsell2006dimensionality,oord2018representation} show impressive improvements on image-based self-supervision \cite{chen2020simple,he2020momentum,tian2020contrastive}. Videos allow for additional constraints for self-supervision like discriminating temporally transformed version of the video~\cite{jenni2020video}, predicting speediness \cite{epstein2020oops,benaim2020speednet,yao2020video}. Some works use alternate pretexts~\cite{kim2019self} while others employ temporal coherence and ordering as signals \cite{misra2016shuffle,buchler2018improving,wei2018learning,lee2017unsupervised,xu2019self,hu2021contrast}. The success of CL approaches depends on augmentations used to generate varying views. Recent works have explored new augmentations and sampling \cite{qian2021spatiotemporal,dave2021tclr,huang2021ascnet,kahatapitiya2021self} while others have explored equivariance to certain transformations~\cite{jenni2021time} or explicitly encoding augmentations~\cite{sun2021composable} to retain temporal dynamics. Most video SSL approaches focus on learning from short, trimmed videos depicting a single action. Real world task videos are rarely short or trimmed. To address this, \cite{zhukov2020learning} uses order verification to isolate actions from background, using global context and segment-based regularization~\cite{kuang2021video}, exploiting relations between clips~\cite{luo2022exploring} or devising an action boundary sensitive pretext task~\cite{xu2021boundary}.
Unlike typical SSL models trained on a large collection of videos, we work in a low-shot setting. Instead of training a feature extractor, we use off-the-shelf features and train a low complexity temporal model.
\noindent \textbf{Multiple modalities for SSL:} Multi-modal (e.g. audio and text) methods use cross-modal losses and augmentations \cite{miech2019howto100m,patrick2021compositions,morgado2021audio,recasens2021broaden,zolfaghari2021crossclr,wang2021fine} to learn features. Use of additional modalities/cues derived directly from the visual stream have been very successful for supervised learning~\cite{coskun2021domain}. Optical flow has been used by various works \cite{han2020self,xiao2021modist,xiong2021multiview,li2021motion} to learn and distill information from the motion stream while \cite{rai2021cocon} explores cooperative CL for multi-modal SSL but for training a feature encoder on short trimmed videos.
SSL relies on data augmentations which are not trivial when working with pre-extracted features. Working with multiple cues helps solve this problem by using contrastive learning similar to the ones used in approaches discussed above but without any explicit data augmentations. MM-SADA~\cite{munro2020multi} proposed a modality alignment loss assuming access to trimmed action segments and applied it for domain adaptation. Our loss function is inspired by CoCLR\cite{han2020self} but differs in that our loss objectives employ temporal sequence of features instead of one feature per video. Further, different from these prior works, we do not aim to learn a feature extractor but a temporal model.
\textbf{Unsupervised video summarization}
Video summarization without labels is another relevant thread for our work. Past works have used Reinforcement learning \cite{yoon2021interp,zhou2018deep}, subset selection~\cite{shemer2021ils}, clustering \cite{shroff2010video,turaga2009unsupervised,dhamecha2020video,jadon2020unsupervised} and GANs \cite{mahasseni2017unsupervised,yuan2019unsupervised,apostolidis2020ac,apostolidis2020unsupervised,he2019unsupervised} or SSL~\cite{gao2020unsupervised} to extract summaries from videos.
In contrast, KSs are task based and require understanding the video at an implicit semantic level. Our target use case does not have videos with abrupt changes, shot changes or extreme camera motion all of which can be cues for determining key steps. Instead we deal with videos which often have subtle changes as a task is performed. Our approach is not specifically catered to key step recovery but can be used for other high-level tasks such as phase classification. That said, our learned features can be used with existing unsupervised video summarization approaches and alternative clustering based approaches.
\section{Method}
\label{sec:method}
STEPs{} (Fig.~\ref{fig:the_model}) is designed to work with synced cues from various on-device sensors, RGB video, depth etc.
STEPs{} uses bottom-up learning of features and clustering to extract KSs.
The first stage learns multi-cue features that capture continuity and distinctiveness across the depicted task in a video.
The learned features are clustered adaptively with each cluster expected to represent one key step.
Clips and frames from each cluster are sampled to create a specific KS.
We now detail the three key processing pipelines.
\subsection{Feature Extraction}
We use an LSTM-based temporal feature extractor (Fig.~\ref{fig:the_model}) that captures the long-range, temporal dynamics of an instructional video.
We first extract features for each frame using publicly available, pre-trained backbone networks. We call these raw per-frame features (\cref{fig:the_model}). $f^{appearance}$ extracts pre-learned appearance features for the input frame, and $f^{motion}$ computes pose sequences or pre-learned motion features.
The raw appearance and motion feature sequences, $p_{t}^{appearance}$ and $p_{t}^{motion}$, are input to a bidirectional LSTM that generates a sequence of adapted per-frame features, $\tilde{q}_{t}^{appearance}$ and $\tilde{q}_{t}^{motion}$, for each modality.
Finally, an MLP extracts a projection of the input sequence and L2 normalizes it to obtain ${q}_{t}^{appearance}$, and ${q}_{t}^{motion}$ which is used to compute the losses.
Since there do not exist benchmark datasets recorded with AR devices for this task, we describe our approach with two cues : Appearance and motion which can readily be extracted from typical procedural task videos.
\subsubsection{Training procedure and losses}
The key technical innovation in this work is to use intra-video alignment from pre-extracted features as a form of self-supervision for feature learning.
We use L2 normalized per-frame \emph{sequence} features $\{q_t\}$ described above for training.
Intuitively, the LSTM learns to encode temporal dependencies in the video sequence features. This process executes for each backbone (e.g. appearance and motion) simultaneously.
\input{figs/km_qualitative}
\noindent\textbf{Per-modality Temporal loss ($\mathcal{L}_{T}$):} This loss ensures that per-modality sequence features that are close in time are close in feature space while those far away have dissimilar representations. For a fair comparison to prior work, we use CIDM~\cite{haresh2021learning} for our uni-modal temporal loss.
\\
\noindent\textbf{Cross-cue Local Loss ($\mathcal{L}_{\text{CCL}}$):}
This loss enforces that representation of different \emph{multi-modal} cues should be similar for temporally close frames.
Let us assume that we have a batch of $B$ examples during each step of the training. So we have $B$ sequences with the appearance and motion features as $\{\{q^{\text{appearance}}_t\}_i,\{q^{\text{motion}}_t\}_i\}_{i=1}^B$. We restructure this dataset as $D_B = \{q_{i,t}^m\}$ where $i \in [1,\cdots,B], t \in [1,\cdots,T] ~\text{and}~ m \in \mathcal{M} = \{\text{appearance},\text{motion}\}$.
The cross-cue local loss assumes that each feature modality is in a common latent space where it can be compared. We compute a contrastive loss enforcing nearness of positives derived from close-in-time sequence features of both modalities, while further off sequence features are considered negatives. We use an InfoNCE-based objective~\cite{chen2020simple}. Specifically, for each anchor $q_{i,t}^m$, we have the following loss:
\begin{align*}
\mathcal{L}_{i,t}^{m} &= \frac{-1}{|P_{i,t}^m|}\sum_{q_p\in P_{i,t}^m}\log{\frac{\text{exp}\left(\boldsymbol{q}_{i,t}^{m}\bigcdot\boldsymbol{q_p}/\tau\right)}{\sum\limits_{q_n\in N_{i,t}^m }\text{exp}\left(\boldsymbol{q}_{i,t}^{m}\bigcdot\boldsymbol{q_n}/\tau\right)}}, \text{where} \\
P_{i,t}^m &= \{q_{i',t'}^{m'} \in D_B |~ |t' - t| \leq \beta, m' \in \mathcal{M}\} \setminus q_{i,t}^m, \text{and} \\
N_{i,t}^m &= D_B \setminus \{q_{i,t}^m\}
\end{align*}
The loss intuitively considers frames from a $\beta$-neighborhood of the anchor in both modalities as positive and the rest as negatives. We compute the loss $\mathcal{L}_{\text{CCL}}$ averaging the loss from each anchor $q_{i,t}^m$.
Note that our approach is designed to work with temporally close features from different modalities as positive views and does not require the complex data augmentation schemes that typically appear in the SSL literature.
We train the model with the weighted loss function:
\begin{equation}
\mathcal{L} = \mathcal{L}_T + \lambda_1\mathcal{L}_{\text{CCL}}
\label{eqn:self_loss}
\end{equation}
\begin{algorithm}[!t]
\footnotesize
\caption{\label{alg:sampling_keymoments} Extracting Key steps}
\begin{algorithmic}
\STATE \textbf{Input:} Per-frame adapted features for the candidate video $\{\tilde{q}_t\}, t\in[1,\cdots,T]$, num clusters K
\STATE \textbf{Output:} Key-steps $KS(v) = \{a_k\}_{k=1}^{k=K}$,
\STATE \textcolor{gray}{\# Cluster the video}
\STATE $\mathcal{C}=\texttt{cluster}(\{\tilde{q}_t\}, K)$ where $\mathcal{C}=\{C_i\}$
\STATE \textcolor{gray}{\# Obtain assignment and distance to closest cluster}
\STATE $\{y_t,d_t\}=\texttt{predict}\_\texttt{cluster}(\{\tilde{q}_t\},\mathcal{C})$
\STATE \textcolor{gray}{\# Sampling key-steps}
\STATE \textbf{for} cluster $k = 1, \ldots, K$ \textbf{do}
\STATE $~~~~$ \textcolor{gray}{\# Obtain time indices for frames assigned to $\mathcal{C}_k$}
\STATE $~~~~$ $\mathcal{T}_k = \{t ~| y_t = k~~ \forall t \in [1,\cdots,T] \}$
\STATE $~~~~$ \textcolor{gray}{\# Remove elements farthest from the cluster center}
\STATE $~~~~$ $\mathcal{T}'_k = \texttt{background}\_\texttt{reject}(\mathcal{T}_k,\alpha)$
\STATE $~~~~$ \textcolor{gray}{\# Break into segments if adjacent indices are $\gamma$ away}
\STATE $~~~~$ $\mathcal{T}''_k = \texttt{split}\_\texttt{to}\_\texttt{segments}(\mathcal{T}'_k,\gamma)$
\STATE $~~~~$ \textcolor{gray}{\# Sample key steps from each segment}
\STATE $~~~~$ $a_k = \{t \!\sim\! \mathcal{T}''_{kj} \}$
\STATE \textbf{end for}
\STATE return $\{a_k\}_{k=1}^{k=K}$
\end{algorithmic}
\end{algorithm}
We \emph{do not} back-propagate through the backbone feature extractors, $f^\star$, when training this pipeline. As the number of parameters in the LSTM and MLP modules is substantially smaller than the number of parameters in the backbone network (5.5M for our model vs 23M for ResNet-50), precomputing the features substantially reduces the size of the gradients during training, allowing us to process very long feature sequences and large batch sizes on modest hardware. In turn, this enables us to extract rich temporal information from the videos while simultaneously making the training process efficient. In particular, the features $p_t$ are considered fixed throughout the training process and need only be computed once. They are completely determined by the choice of backbone network; we examine different choices of backbone networks in the supplementary material.
\subsection{Clustering}
We extract per-frame temporal features for a video using the LSTM temporal feature model. We use K-Means clustering to obtain partitions of the aggregated feature data for a video, each partition potentially corresponding to a sub-task or a key step. This algorithm automatically discovers groups in the data. Our approach allows for a choice of a clustering algorithm based on user constraints, including algorithms that do not require a fixed number of clusters to be specified in advance~\cite{finch}. Please refer to the Supplementary Material for additional details.
\subsection{Key Step Sampling}
To sample key steps, we first use a background frame rejection model~\cite{kukleva2019unsupervised} to reject steps which might correspond to background frames. Subsequently, each frame in the video is first assigned to one of the clusters. A step from each cluster is then chosen based on its distance to the cluster center and the key steps are temporally ordered. While sampling from a cluster, we perform an additional partitioning step based on the timestamp to ensure that we sample key steps of the same potential sub-task but happening at different time instances. Algorithm~\ref{alg:sampling_keymoments} details our approach to cluster and sample key steps. Our approach borrows the clustering and ordering approach used in ~\cite{kukleva2019unsupervised} but adapts their segmentation approach to sample \emph{key steps} given a \emph{single} video. That said, given our rich features, we could use alternative optimization-based approaches like ILS-SUMM~\cite{shemer2021ils} which we discuss in the supplementary material.
\section{Experiments and Analyses}
\label{experiments}
KS detection is a subjective and time-consuming task. To annotate a large dataset would require a large amount of AR procedural videos. But content will not exist until it is easy to detect KSs!
As such, we offer a qualitative assessment in Fig.~\ref{fig:km_qualitative} using the Ikea-Assembly Dataset~\cite{ben2021ikea} - which is the closest to the kind of videos we might deal with in the real world. We detect the key steps in a video of a person following a set of paper instructions to construct a piece of furniture.
For quantitative benchmarking, we compare our method against the state of the art in Key Step Localization (KSL). Evaluation on KSL is a good proxy since there does not exist any ground truth for key steps on procedural videos due to the subjective nature. Furthermore, to demonstrate the usefulness of our SSL approach generally, we offer a quantitative comparison on standard benchmarks used in the self-supervised video representation learning for task based videos. We also provide analyses to understand the working of our approach.
\textbf{Implementation details:} We extract off-the-shelf VGG-19~\cite{simonyan2014very} and ResNet-50~\cite{he2016deep} features to model appearance. We experiment with optical flow features extracted from RAFT~\cite{teed2020raft} encoder for motion features. For the IkeaAssembly dataset which involves furniture assembly, we make use of human pose coordinates extracted using OpenPose~\cite{cao2017realtime}. We use a light-weight LSTM to extract temporal features. While trained on multi-cue features, our approach allows the use of specific modalities at inference. During inference, we use appearance features alone for a fair comparison with competing approaches.
\input{tables/crosstask_and_procel}
\subsection{Qualitative visualizations of key steps}
In Fig.~\ref{fig:km_qualitative}, we visualize the key steps extracted on the Ikea Assembly~\cite{ben2021ikea} dataset since it closely mimics the practical AR scenario.
The dataset depicts complex furniture assembly tasks performed by multiple people across views. The tasks are composed of multiple sub-tasks like flipping the table, attaching the leg etc.
While captured in a controlled setting, the dataset consists of large foreground temporal variations and background segments.
In Fig.~\ref{fig:km_qualitative}, we visualize the extracted key steps for two randomly sampled videos for the tasks of assembling `Kallax Shelf Drawer' and `Lack Coffee Table' respectively .
We also manually map them to diagrams from the official IKEA manuals~\cite{ikea_manual}. We notice that the extracted key steps are semantically meaningful given the mapping. We did not include all steps from the manual since the curators of the dataset pre-assembled a few parts for ease of dataset collection. Note that what constitutes a key step can be very subjective and the number of steps needed to accomplish a task might depend on the application area. For example, a novice trying to learn a new task might benefit from more steps than an an experienced person trying to transfer skills from a known task. For this visualization, we simply set the number of clusters $K=10$. We can also make use of clustering algorithms like \cite{finch} which automatically determine the number of clusters. Our two-stage approach allows for this customization without having to retrain the model.
\label{aaai_fix:num_clusters}
\subsection{Quantitative results on KS Localization}
None of the procedural datasets that we work with are annotated for the task of key step recovery. Following previous works, we evaluate our method on the CrossTask~\cite{zhukov2020learning} and ProceL~\cite{elhamifar2019unsupervised} datasets on the closely related key step localization task. These are instructional video datasets sourced from YouTube; CrossTask consists of 18 primary tasks with a total of 2750 videos and 7 key steps on average, and ProceL consists of 720 videos from 12 tasks with 8 key steps on average. Although we compare our work on these videos, it should be noted that these have shot changes, abrupt camera motions etc that make detecting key steps comparatively easier. Hence, we just use these for quantitative comparisons to state-of-the-art for Key Step Localization.
We follow the same experimental setup of ~\cite{elhamifar2020self}. We first find a one-to-one matching between steps in the ground truth and clustering predictions from our method using the Hungarian algorithm. Recall is computed as the ratio of number of frames having the correct key step prediction to the ground truth number of key frames across all key steps. Precision is the ratio of the number of correctly predicted frames and number of frames predicted as key steps. F-1 score is the harmonic mean of recall and precision. Table~\ref{table:crosstask_dataset} shows our results and comparisons with other self-supervised learning approaches using appearance features. Our work leads to consistent improvements compared to previous approaches. Different from the work of \cite{elhamifar2020self}, we show an improvement without the use of spatial attention showing our improved learning capabilities. Further, we do not assume availability of task labels and hence our approach is fully self-supervised compared to the weak supervision assumed in \cite{elhamifar2020self}.
\subsection{Comparison on other tasks}
Our approach allows evaluation of learned features for other tasks involving procedural videos beyond KSL. In addition to the IkeaAssembly Dataset discussed previously, we also work with 13 actions from the PennActions dataset~\cite{zhang2013actemes}. The actions are composed of humans doing sports and exercises and are composed of 2-6 phases per action. We evaluate on two tasks. Phase classification calculates the per frame classification accuracy for fine-grained action recognition by training an SVM on the extracted per frame features of the training set. In addition, we also evaluate the Kendall's Tau (K.T) on the PennActions dataset\footnote{Since K.T assumes a strict montonic order of actions, we report it only for the PennActions dataset.}
We follow the same evaluation protocol as used in ~\cite{dwibedi2019temporal, haresh2021learningLong, liu2021learning}.
\input{tables/phase_cls_comparison_with_sota_permuted}
Table~\ref{table:ikea_penn_fair} shows results of our approach on these metrics. The improved performance shows that our approach learns generalizable features for complex videos. We see that our STEPs{} model which uses the same backbone as previous approaches (albeit without \textit{any} finetuning) outperforms the recent approach of VAVA by 6.79\% on the Ikea dataset. Note that we outperform VAVA on evaluation with and without background frames without any training time modifications. We also outperform the previous approaches on PennActions with significant improvement to Kendall's Tau owing to our improved temporal modeling.
\input{tables/finetuning_crosstask}
\subsection{Analyses and Ablations}
\noindent\textbf{How much training data do we need for pre-training?}
The curators of datasets like IkeaASM went through the arduous task of collecting their dataset. A practical system to extract key steps or phase classification on complex videos should ideally work with little training data. Also extracting unlabeled data for general easy tasks might be easy through scraping YouTube, extracting a lot of the data through AR systems is non-trivial. Previous approaches did not consider this aspect and while the Phase classification results are reported on fractions of the training set, the pre-training happens on the entire dataset. In Fig.~\ref{fig:pretraining_ikea} we plot the performance as a function of training data on the IkeaASM dataset. We see that our approach can indeed give resonable performance even with little training data. In fact, for the Ikea dataset, we observe that our performance using off-the-shelf-features with as little as 20\% of the data matches VAVA's performance on the whole dataset when finetuning the backbone.
\input{figs/pretraining_ikea}
\noindent\textbf{Transfer of trained models}
Different from the previous experiment, we explore whether we can pre-train the model on a few tasks and use that model for fine tuning on another task potentially with little training data. This is especially important for the data-scarce and resource constrained setting that motivated this work.
In Table~\ref{table:finetune_crosstask} we plot the results of transferring a model trained on all classes except the target task from the CrossTask dataset and transferring to the target task of `Make Jello Shots' with varying amount of finetuning data. Note that this setting is different than that explored in the literature where the model is trained and evaluated on all tasks (Table~\ref{table:crosstask_dataset}). It is interesting to note that the model learns good generalizable features which are useful across tasks with as little as a single video of the target task.
\noindent\textbf{Ablation analyses on various losses}
We quantitatively evaluate the effect of each our loss terms. In Table~\ref{table:loss_ablation} we show results on phase classification on the Ikea dataset (PhaseCLS 0.5). We see that each of our proposed losses brings an improvement to the final performance.
\input{tables/loss_ablation}
\noindent\textbf{Computational cost}
The use of off-the-shelf feature extraction without fine-tuning enables fast LSTM encoder training. We can train a model on 60 Ikea videos for 100 epochs in under 6 minutes on a single Nvidia 2080Ti GPU. We use a single GPU for all our experiments. Our small memory footprint (5.5M parameters for our model compared to 23M in the Res50 encoder alone) along with the use of off-the-shelf features allows training with large temporal windows unlike approaches which rely on finetuning. Precomputing of features is also quite fast. We use standard pipelines for feature extraction. Using a single GPU, it takes less than 4 minutes to extract features for the \emph{all} frames of the dataset (average of 2400 frames/video).
\noindent\textbf{Use of MM information at inference}
In Table~\ref{table:ikea_penn_fair}, following prior works we use only appearance cue during inference. In this experiment, we experiment with use of multi-modal cues during inference as well. In Table~\ref{table:mm_ikea}, we show results with Appearance+Motion cues at inference. We simply concatenate the features from both modalities and use them for the downstream task. We see that simple late fusion leads to improvements showing complementary information in the two streams. Note that while these are referred to as separate cues, the motion stream (RAFT features or Pose) is derived from the RGB stream. Further, such fusion is especially beneficial for cues in an AR device where some of these additional streams come \emph{for free} from on-device sensors.
\input{tables/phase_cls_multimodal_comparison}
\input{tables/effect_of_num_chunks}
\noindent\textbf{Importance of temporal information}
Note that many previous approaches do not focus on the temporal aspect of the problem.
We believe that temporal information is especially important for complex tasks. Since they often involve subtle and slow changes. Such changes may not be easily captured using a small temporal context used by some previous approaches. Prior works~\cite{haresh2021learning,liu2021learning} use very few sampled frames ($\sim$40) per video during training with little to no temporal context . This is sub-optimal since the videos from the IkeaASM dataset often contain 3000+ frames with sub-actions having long temporal spans (Fig. 3~\cite{ben2021ikea}). Fine-tuning a backbone while dealing with long temporal contexts is computationally expensive. Use of pre-extracted features enables us to use large temporal windows which helps in capturing more of the video and temporal contexts. In~\cref{table:temporal_chunks}, we empirically verify the effect of long temporal windows, we train our model for varying number of sampled frames. \footnote{Note that for a fair comparison, we use the \emph{same} protocol at inference following~\cite{haresh2021learning,liu2021learning}}
\section{Discussion and Conclusion}
\label{sec:conclusion}
Key step extraction in procedural task videos is a good proxy for our motivating problem of AR-enabled task guidance. For example, analysis of procedural task datasets and subsequent quantitative experiments show that any solution to key step extraction must be able to account for the long-range temporal relationships that are inherent in complex tasks. Similarly, we offer substantial evidence that using multiple cues, both in training and runtime, greatly improves key step extraction. This finding is inline with psychological research that advocates for detailed, multi-modal observation of human experts when designing training programs for novices \cite{gopher1994transfer, fadde2018training}.
We believe in the potential to use AR headsets to transform on-the-job training and guidance and hope to address several important related problems in future work. Primary among these is collecting a detailed dataset of procedural task recordings from AR headsets. Also, incorporating recent advances in meta-learning and few-shot learning as ways to potentially improve key step extraction while reducing the amount of required training data is an important line of research. In short, the present work is only the beginning.
|
1,108,101,565,659 | arxiv | \section{Introduction}\label{sec:intro}
Classification of phases of matter has been one of the most fundamental
problems in the physics of many-body systems.
Different phases of matter have been classified by their symmetries,
which led to the theory of spontaneously
symmetry breaking~\cite{landau1937theory, ginzburg1950theory,Nambu:1960tm, Nambu:1961fr}.
It is now realized that quantum phases of matter
depends also on ``topology'', and Ginzburg-Landau (GL) type
classification is not sufficient.
An important class of such states is called topological order, and
there are nontrivial long-range correlation even though no
massless excitations exist: topological degeneracy of ground states,
anyon statistics of quasiparticles, and so on~\cite{Wen:1989iv, PhysRevB.44.274, Wen200710, HANSSON2004497, sachdev2011quantum, Chen:2010gda, PhysRevLett.96.110405, PhysRevLett.96.110404}.
Low-energy effective description of topological order is given by
topological field theories, and the presence of ``deconfined'' dynamical
gauge fields plays an important role for those nontrivial
long-range phenomena with mass gap~\cite{Dijkgraaf:1990nc, Wen:1992uk}.
Certain classes of topological orders
can be understood as a consequences of spontaneous breaking of
higher-form (or generalized) global symmetries~\cite{Gaiotto:2014kfa, Wen:2018zux}.
One possible direction of further developments on quantum many-body
physics is to understand the role of topology in the presence of
gapless degrees of freedom.
Although we do not have a complete consensus about the definition of
topological order in gapless systems, let us temporarily consider it in
this paper as a quantum system which has deconfined gauge
fields in addition to local gapless excitations.
Such theoretical models have been recently studied in the context of
quantum criticality of high-$T_{\rm c}$ cuprates, using gauged GL
model~\cite{Chatterjee:2017pqv, Sachdev:2017hzd, Scheurer:2017jcp,
Sachdev:2018nbk}.
There is also an example of such description in the physics of QCD~\cite{Stephanov:2004wx, Fukushima:2010bq, Ren:2004nn, Casalbuoni:2018haw}.
At large baryon densities, QCD matter is expected to exhibit color
color superconductivity~\cite{Barrois:1977xd, Bailin:1983bm, Alford:1998mk, Schafer:1998ef, Alford:2007xm},
which has topological vortices~\cite{Balachandran:2005ev, Nakano:2007dr, Eto:2013hoa, Yamamoto:2018vgg}.
Understanding the role of topology would be important
in the discussion of the phase structure of dense nuclear matter \cite{Schafer:1998ef, Hirono:2018fjr, Cherman:2018jir} or the possible continuity of vortices
between a nuclear superfluid and a color superconductor \cite{Alford:2018mqj, Chatterjee:2018nxe}.
The low-energy theory of a color superconducting phase
is described by a topological field theory coupled with massless
Nambu-Goldstone (NG) bosons~\cite{Hirono:2018fjr, Hirono:2010gq, Cherman:2018jir}.
Motivated by these recent developments on the possibility of topological order in
gapless systems,
we study a general framework for studying superfluidity coupled to
$BF$-type topological field theory~\cite{Bergeron:1994ym, HANSSON2004497, Cho:2010rk, Putrov:2016qdo, PhysRevB.94.045113}.
Starting from a gauged GL model, we derive a dual gauge theory.
The effective theory is a generalized $BF$ theory
with a non-square $K$ matrix coupled with massless NG bosons.
The system is shown to acquire discrete and continuous 2-form/1-form symmetries.
As a consequence of the emergent symmetries,
the system is shown to exhibit fractional braiding statistics between
vortices and quasiparticles.
We examine the condition when a topological order appears,
which can be also seen as the
existence of a mixed 't~Hooft anomaly \cite{tHooft:1979rat}
between higher-form symmetries.
This paper is organized as follows.
In Sec.~\ref{sec:eft},
we introduce a low-energy effective theory of superfluidity that
can also have topological order.
In Sec.~\ref{sec:emergent},
we identify the continuous and discrete higher-form symmetries of the system
and discuss the braiding of quasiparticles and vortices.
We also discuss how to detect the topological order in this theory and
its interpretation as a mixed 't~Hooft anomaly.
In Sec.~\ref{sec:cfl}, we describe the topological properties of the color-flavor locked phase of dense QCD as an application of the framework.
This section is the follow-up of the previous paper~\cite{Hirono:2018fjr} with more detailed explanations.
In Sec.~\ref{sec:tos}, we discuss an explicit example of superfluidity with topological order.
Section~\ref{sec:summary} is devoted to a summary and outlook.
In Appendix~\ref{app:mp}, we summarize the properties of the Moore-Penrose inverse.
In Appendix~\ref{app:delta}, we provide a summary of the properties of delta-function forms.
In Appendix~\ref{app:derivation}, we give a derivation of the braiding phase using the effective theory.
Appendix~\ref{sec:smith} is devoted to a discussion about the consequence of basis changes.
\section{Effective field theory of topologically ordered superfluidity}\label{sec:eft}
We aim at describing the low-energy behavior of superfluids with topological order.
Here, let us construct a generic low-energy effective theory that consists of $2\pi$-periodic compact scalar fields and $U(1)$ gauge fields.
\subsection{Effective Lagrangian of general Abelian-Higgs models}
We consider a $3+1$-dimensional theory with multiple $U(1)$ symmetries and
some parts of them are gauged and couple to dynamical $U(1)$ 1-form gauge fields.
We are interested in the low-energy regime of the theory
and in this limit, the remaining degrees of freedom are
massless modes, that are the NG bosons associated with
the spontaneous breaking of $U(1)$ symmetries.
Thus, the system is a superfluid.
In addition, there can be topological degrees of freedom.
Let us give a derivation of an effective theory for describing such a system.
We take variables $\phi_i$,
that are (would-be) NG modes associated with
the breaking of $U(1)$ symmetries. These are $2\pi$-periodic scalar fields.
They couple to gauge fields through a covariant derivative,
\begin{equation}
\rd_a \phi_i \equiv \rd \phi_i + K_{iA} a_A ,
\end{equation}
and $a_A$ are dynamical Abelian 1-form gauge fields,
$a_A(x) = (a_A)_\mu \diff x^\mu$ ($A = 1, \cdots, |A|$),
and $K_{iA}$ is a $|i| \times |A|$\footnote{
We use the notation to represent the number of rows (or columns)
of a matrix by the absolute value of the index.
} integer-valued matrix.
Let us call $a_A(x)$ as photons.
The covariant derivative is invariant under the $0$-form gauge transformation,
\be
\phi_i\mapsto \phi_i-K_{iA}\lambda_A, \quad
a_{A}\mapsto a_A+\diff \lambda_A,
\ee
where gauge parameters $\lambda_A$ are also $2\pi$-periodic scalars.
Because of this interaction, a part of the would-be NG modes are Higgsed.
We start with an action,
\begin{equation}
S =
\frac{1}{2}
H_{ij}\int \rd_a \phi_i \wedge \star\, \rd_a \phi_j
+ \frac{1}{2}G^a_{AB}\int \rd a_A \wedge \star \, \rd a_B .
\label{eq:EFT_general_01}
\end{equation}
The positivity of the kinetic terms require that $H$ and $G^a$ are positive-definite real symmetric matrices.
We shall take an Abelian dual of this theory~\cite{Banks:2010zn}.
The action can be rewritten by introducing $\mathbb{R}$-valued $3$-form fields $h_i$ as
\begin{equation}
S = \frac{1}{8\pi^2}H^{-1}_{ij} \int h_i \wedge \star\, h_j
- \frac{\mathrm{i}}{2\pi} \int h_i \wedge \rd_a \phi_i
+ \frac{1}{2}G^a_{AB}\int \rd a_A \wedge \star \, \rd a_B ,
\label{eq:EFT_general_02}
\end{equation}
Solving the equation of motion (EOM) for $h_i$, we obtain $h_i = 2\pi \mathrm{i} H_{ij}\star\, \rd_a \phi_j$ and get the original action (\ref{eq:EFT_general_01}).
Solving the EOM for $\phi_i$, instead, we find $\rd h_i = 0$, and it can be solved as
\begin{equation}
h_i = \rd b_i ,
\label{eq:dual_gauge_field}
\end{equation}
where
$b_i(x) = \frac{1}2 (b_i)_{\mu\nu} dx^\mu \wedge
dx^\nu$ ($i = 1, \cdots, |i|$) are $2$-form $U(1)$ gauge fields\footnote{This normalization is determined by the global structure: Let us set our spacetime as $4$-torus $T^4$ of size $L$ as an example, then the $2\pi$-periodic scalar $\phi$ can be decomposed as $\phi={2\pi\over L}n_{\mu}x^{\mu}+\tilde{\phi}$, where $n_{\mu}\in\mathbb{Z}$ and $\tilde{\phi}$ is the $\mathbb{R}$-valued field. The above EOM, $\rd h=0$, comes out of the path integral over $\tilde{\phi}$. The summation over $\{n_{\mu}\}\in\mathbb{Z}^4$ further requires that $\int_{T^3}h \in 2\pi \mathbb{Z}$ for each $3$-torus and we find the correct normalization (\ref{eq:dual_gauge_field}). }.
Plugging this into $h_i$, the action is rewritten as
\begin{equation}
S = \frac{1}{2} G^b_{ij}
\int \rd b_i \wedge \star \, \rd b_j
+ \frac{1}{2}G^a_{AB} \int \rd a_A \wedge \star \rd a_B
+ \mathrm{i} \frac{K_{iA}}{2\pi} \int b_i \wedge \rd a_A,
\label{eq:EFT_general_03}
\end{equation}
where we introduce
$
G^b_{ij} \equiv H^{-1}_{ij} / 4 \pi^2 .
$
The first two terms are the usual kinetic terms, and the last term is the topological $BF$ term\footnote{One could consider further generalization by adding $a_A\wedge a_B\wedge \diff a_C$, $b_i\wedge b_j$, etc., with appropriate coefficients and modification of gauge transformations, like a twist term of Dijkgraaf-Witten theory~\cite{Dijkgraaf:1990nc, Kapustin:2014gua, Tiwari:2016zru}. In this paper, however, we do not pursue along this direction.}.
Physical observables can be calculated
by the partition function,
\begin{equation}
Z = \int \Dcal a \Dcal b \, \e^{ - S[a,b]} .
\end{equation}
In this path integral, we sum over all possible gauge fields, satisfying the canonical Dirac quantization conditions,
\begin{equation}
\int_{S} \diff a_A\in 2\pi\mathbb{Z},\; \int_{V} \diff b_i\in 2\pi\mathbb{Z},
\label{eq:normalization_gauge_01}
\end{equation}
for each closed $2$-submanifold $S$ and $3$-submanifold $V$ of the spacetime.
Physical operators that we will focus on are
Wilson loop operators and vortex operators,
\begin{equation}
W_{A}(C) = \exp \left( \ri \int_C a_A \right),
\quad
V_{i}(S) = \exp \left( \ri \int_S b_i \right),
\label{eq:Wilson_loop_surface}
\end{equation}
where $C$ is a world-line of a test particle, and $S$ is a vortex world-sheet.
\subsection{Classification of spectra}\label{sec:local_dynamics}
Since we are interested in the low-energy physics,
we only retain massless modes and topological sector.
Before studying the global nature of the theory, let us clarify its local dynamics
to identify the massless sector of $b$ and $a$.
For this purpose, let us write down the EOMs of $b_i$ and $a_A$:
\begin{eqnarray}
&&G^b_{ij} \diff \star \diff b_j-{\mathrm{i}\over 2\pi}K_{iA}\diff a_A=0,\nonumber\\
&&G^{a}_{AB}\diff \star \diff a_B+{\mathrm{i}\over 2\pi}K_{iA} \diff b_i=0.
\end{eqnarray}
Combining these EOMs, we find that $\diff b_i$ must satisfy
\begin{equation}
\left(G^b \Delta+{1\over 4\pi^2}K (G^a)^{-1}K^T\right) \diff b=0,
\end{equation}
and that $\diff a_A$ must satisfy
\begin{equation}
\left(G^a \Delta+{1\over 4\pi^2}K^T (G^b)^{-1} K\right)\diff a=0.
\end{equation}
Here, $\Delta=\diff \delta+\delta \diff$ is the form Laplacian with the codifferential $\delta=-\star \diff \star$, and $K^T$ represents the transpose of $K$ matrix.
Therefore, the mass matrix $M^b$ of $b_i$ is given by
\begin{equation}
(M^b)^2={1\over 4\pi^2}(G^b)^{-1/2} K (G^a)^{-1} K^T (G^b)^{-1/2},
\end{equation}
and the mass matrix $M^a$ of $a_A$ is given by
\begin{equation}
(M^a)^2={1\over 4\pi^2}(G^a)^{-1/2} K^T (G^b)^{-1} K (G^a)^{-1/2}.
\end{equation}
Here, we note that squared roots of $G^a$ and $G^b$ are well defined since they are positive matrix.
Let us discuss how massless degrees of freedom depend on the structure of the $K$ matrix.
When $|i| = |A|$ (i.e. $K$ is a square matrix) and $\det K \neq 0$, all the particles get nonzero mass because neither $M^b$ nor $M^a$ have zero eigenvalues,
and the $BF$ theory for superconductivity is reproduced.
We are interested in the situation where superfluidity is present.
In this case, there exists at least one massless NG modes,
which is realized when ${\rm dim} ({\rm coker}\,K) \neq 0$.
Indeed, for each vector $\bm{D}^{\bar\alpha}\in \mathrm{coker}\,\, K$, i.e.
\begin{equation}
(\bm{D}^{\bar\alpha})^T\cdot K=0,
\end{equation}
we can find the null eigenvector of the mass matrix:
\begin{equation}
(M^b)^2 \sqrt{G^b} \bm{D}^{\bar\alpha}=0.
\end{equation}
Since $K$ is in general a non-square matrix, there always exist massless NG modes if $|i|>|A|$.
There can also be remaining massless photons,
when ${\rm dim} ({\rm ker}\, K) \neq 0$.
We denote the basis of the kernel and cokernel as
$\bm C^{\alpha} \in {\rm ker} \,\, K$, and
$\bm D^{\bar\alpha} \in {\rm coker}\,\, K$, namely, they satisfy
\begin{equation}
K_{iA} C^{\alpha}_A = 0,
\quad
D^{\bar\alpha}_i K_{iA} = 0.
\end{equation}
We can identify the massless NG modes and massless photons as
\begin{equation}
b_0 \in {\rm coker}\,\, K,
\quad
a_0 \in {\rm ker}\,\, K.
\end{equation}
because $a_0$ and $b_0$ does appear in the $BF$ term.
The numbers of massless NG modes and massless photons,
$|\bar\alpha|$ and $|\alpha|$,
are given by the dimensions of the cokernel and kernel of $K$, respectively:
\begin{equation}
|\bar\alpha| = {\rm dim\,}({\rm coker}\, K),
\quad
|\alpha| = {\rm dim\,} ({\rm ker}\, K).
\end{equation}
Those massless modes can be identified by projection matrices,
\begin{equation}
(a_0)_A = P^a_{AB}\, a_B ,
\quad
(b_0)_i = P^b_{ij}\, b_j,
\end{equation}
where $P^a$ and $P^b$ are orthogonal projectors to the kernel and
cokernel of $K$.
They can be expressed using the Moore-Penrose inverse\footnote{
We summarize the properties of the
Moore-Penrose inverse in Appendix \ref{app:mp}.
} $K^+$
of the $K$ matrix, which is a generalization of a matrix inverse.
Given an arbitrary matrix, the Moore-Penrose inverse always exists and is unique.
The projectors are given by \footnote{
We use the notation where contracted matrix indices may be omitted
when there is no confusion, for example,
$[K^+ K]_{AB} = K^+_{Ai} K_{iB}$.
}
\begin{equation}
P^a_{AB} = \delta_{AB} - [K^+ K]_{AB} ,
\quad
P^b_{ij} = \delta_{ij} - [K K^+]_{ij}.
\end{equation}
Using this orthonormal projection, we denote the gauge fields as
\begin{equation}
a=a_0+a_\perp,
\quad
b=b_0+b_\perp.
\end{equation}
This decomposition diagonalizes the mass term, but it does not necessarily diagonalize the kinetic term: For example, the kinetic term of $b$ becomes
\begin{equation}
{1\over 2}\int \diff b^T\wedge G^b\star\diff b={1\over 2}\int \left(\diff b_0^T\wedge G^b\star\diff b_0+2\diff b_\perp^T\wedge G^b\star\diff b_0+\diff b_\perp^T\wedge G^b\star\diff b_\perp\right).
\end{equation}
In the low-energy limit, the last term can be neglected since it only describes the exponential decay of massive excitations, and one should retain the first and the second terms. The mixed kinetic term $\diff b_\perp^T\wedge G^b\star\diff b_0$ vanishes identically if and only if
\begin{equation}
P^b G^b (1-P^b)=0.
\end{equation}
Since $G^b$ and $P^b$ are both symmetric matrices, this condition is equivalent to
\begin{equation}
[G^b,P^b]=0.
\end{equation}
We obtain the same conclusion also for the photon fields $a$.
In the rest of this paper, we assume that
\begin{equation}
[G^a, P^a]=0,\quad
[G^b, P^b]=0, \label{eq:assumption_orthogonality}
\end{equation}
so that the mixed kinetic terms between massless and heavy modes vanish identically\footnote{
In Appendix~\ref{sec:smith}, we discuss the consequence of the mixed kinetic terms.
}.
In concrete examples, the condition (\ref{eq:assumption_orthogonality}) may be implied by a certain symmetry of the UV theory.
We expect that this condition is important to protect the topological order under the existence of gapless excitations.
As a consequence of assumption (\ref{eq:assumption_orthogonality}), we obtain the low-energy effective action as
\begin{equation}
S_{\rm eff} =
\mathrm{i} \frac{ K_{i A}}{2\pi} \int
b_{i} \wedge \rd a_{A}
+ \frac{1}{2}G^b_{ij} \int \rd (b_0)_i \wedge \star \,\rd (b_0)_j
+ \frac{1}{2}G^a_{AB} \int \rd (a_0)_A \wedge \star \,\rd (a_0)_B,
\label{eq:seff}
\end{equation}
where $b_0$ and $a_0$ are massless contributions as identified above.
We call this as the generalized $BF$ theory.
This action (\ref{eq:seff}) can describe various physical situations,
depending on the choice of $K$ matrices.
Possible physical situations are classified according to
the numbers, $|\bar\alpha|$, $|\alpha|$, $|i|$, and $|A|$.
Note that, according to Fredholm's index theorem, those numbers are
related as
\begin{equation}
|\bar \alpha| - |\alpha| = |i| -|A| .
\end{equation}
Based on this relation, possible situations of (\ref{eq:seff}) can be classified as follows:
\begin{itemize}
\item $|\bar\alpha| =|\alpha| = 0$, $|i| = |A|$: All the excitations
are massive. In this case, The $K$ matrix is square and regular,
which corresponds to $BF$ theoretical description of superconductors.
\item $|\alpha| = |A|=0$, $|\bar\alpha| = |i|$: Superfluids with no
topological order.
\item $|\bar\alpha| = |i| = 0$, $|\alpha| = |A|$: Pure Maxwell theory.
\item In other cases, superfluidity and topological order may coexist.
\end{itemize}
\subsection{Non-canonical normalization of gauge fields}
\subsubsection{Generalization to non-canonical normalization}
So far, we are working on the theory (\ref{eq:EFT_general_03}) with the canonically normalized gauge fields $b_i$ and $a_A$ as in (\ref{eq:normalization_gauge_01}).
Instead, we can work on more general normalization of these gauge fields, and let us discuss such cases in this section.
The motivation for this generalization is that gauge fields of the low-energy effective theory can be emergent and is not necessarily ensured to be canonically normalized when we derive it from the UV theory.
Therefore, it is important to establish the way to analyze such cases.
The normalization of gauge fields does not affect local dynamics, and thus the discussion in Sec.~\ref{sec:local_dynamics} is unaffected while the global nature of the theory can be changed drastically.
The normalization of gauge fields is related to the choice of physically observable Wilson loops and vortex operators.
Let us replace (\ref{eq:normalization_gauge_01}) by a generic normalization condition
(Dirac quantization condition)
for $a_A$ and $b_i$ as
\be
Q_{A B}\int_S \rd a_{B}
\in 2\pi \mathbb Z ,
\quad
\int_V \rd b_{j} R_{ji}
\in 2\pi \mathbb Z ,
\label{eq:norm}
\ee
where
$S$ and $V$ are 2D and 3D subspace without boundary,
and $Q_{AB}$ and $R_{ij}$ are invertible matrices with integer elements.
Correspondingly,
the matrices $Q$ and $R$ specify the set of generators of
gauge-invariant Wilson loops and vortex operators,
\be
W^{(Q)}_A \equiv
\exp
\left(
\mathrm{i} Q_{AB} \int_C a_B
\right),
\quad
V^{(R)}_i \equiv
\exp
\left(
\mathrm{i} \int_S b_j \, R_{ji}
\right).
\label{eq:wvqr}
\ee
The naive Wilson loop and surface operators (\ref{eq:Wilson_loop_surface}) are no longer gauge invariant in this normalization.
Let us discuss the gauge redundancy of the action.
It is invariant under $0$-form and $1$-form gauge transformations,
\begin{equation}
a_A \mapsto a_A + Q^{-1}_{AB} \, \rd \lambda_B ,
\quad
b_i \mapsto b_i + R^{-1}_{ji}\, \rd \lambda_j .
\end{equation}
where $\lm_A$ is a $2\pi$-periodic scalar and $\lm_i$ is a $U(1)$ $1$-form fields.
They satisfy
\be
\int \rd \lambda_A \in 2\pi \mathbb Z,
\quad
\int \rd \lambda_i \in 2\pi \mathbb Z,
\ee
where the integrations are over closed submanifolds of corresponding dimensions.
Introducing the matrices $Q^{-1}$ and $R^{-1}$ is necessary
for the consistency with the normalization condition (\ref{eq:norm}).
The variation of the action is
\begin{equation}
\delta_{1} S_{\rm eff} =
\frac{ K_{iA} Q^{-1}_{AB}}{2\pi}
\int \rd b_{i} \wedge \rd \lm_{B} ,
\quad
\delta_{2} S_{\rm eff} = \frac{R^{-1}_{ji} K_{iA}}{2\pi}
\int \rd \lambda_{i} \wedge \rd a_{A} .
\end{equation}
Note that the integration gives
\be
\int \rd b_{i} \wedge \rd \lm_{B} = (2\pi)^2 R^{-1}_{ji} n_j m_{B}
, \quad
\int \rd \lambda_{i} \wedge \rd a_{A}
= (2\pi)^2 Q^{-1}_{AB} m'_B n'_i
\ee
where $n_i, n_i',m_B, m'_B$ are integer vectors.
The change of the action under 1-form and 2-form gauge transformation
is now given by
\begin{equation}
\delta_{1} S_{\rm eff} =
2\pi
\,
n^T R^{-1} K Q^{-1}m
,
\, \quad
\delta_{2} S_{\rm eff} =
2\pi \,
n'^{T} R^{-1} K Q^{-1}m'
\label{eq:gauge-delta}
\end{equation}
Thus, the gauge invariance requires that the
$K$ matrix should be chosen so that the each element of
the matrix $[R^{-1} K Q^{-1}]_{iA}$ is an integer. This reproduces the fact that $K$ should be an integer-valued matrix when we take $Q=1$ and $R=1$, i.e., the canonical normalization~(\ref{eq:normalization_gauge_01}) of gauge fields.
\subsubsection{Basis change of gauge fields}\label{sec:basis_change}
As we have shown, a theory can be specified by a set of integer-valued matrices $(K,Q,R)$.
However, not every theory associated with a set $(K,Q,R)$ describes a distinct system.
This ambiguity was discussed in Ref.~\cite{Wen:1992uk} in the case of the topological $BF$ theory for Abelian anyons, and we extend it to the case of our generalized setup.
We may work on another basis of gauge fields $\widetilde{a}_A$, $\widetilde{b}_i$, which are related to the original fields as
\be
a_A = M_{AB} \, \widetilde{a}_B,
\quad
b_i = \widetilde{b}_j \, N_{ji} ,
\label{eq:basis}
\ee
with some invertible matrices $M$ and $N$.
Under the basis change, the $K$ matrix should be replaced as
\be
\widetilde{K}= N K M.
\ee
One should not forget that, together with Eq.~(\ref{eq:basis}),
we also have to transform the charge matrices,
\be
\widetilde{Q} = QM,
\quad
\widetilde{R} = N R .
\ee
Under this change of normalization, the coefficients of kinetic terms are also changed as
\be
\widetilde{G}^b=N G^b N^T,\; \widetilde{G}^a=M^T G^a M.
\ee
The theories connected by this transformation are equivalent and
give the same physical results.
For example, gauge invariance of the theory is not affected by the basis change,
since the factor $R^{-1} K Q^{-1}$ appearing in Eq.~(\ref{eq:gauge-delta}) is invariant under the basis change, Eq.~(\ref{eq:basis}):
\be
\widetilde{R}^{-1} \widetilde{K} \widetilde{Q}^{-1}=R^{-1}K Q^{-1}.
\ee
One should be cautious in changing the basis because
it in general introduce the coupling between massive and massless
gauge fields in the kinetic terms, which complicates the analysis.
One may seek a simpler expression of $K$ matrix by changing the basis\footnote{
A similar comment is true also for 3d Chern-Simons theory with a slightly different reasoning~\cite{Wen:1992uk}.};
for example, we can always obtain one of the simplest expressions for $K$ matrix by considering the Smith normal form\footnote{Authors appreciate the anonymous referee for pointing this out. }
\be
K= U^{-1} K' V^{-1},
\ee
where $U$ and $V$ are integer matrices with $\det(U)=\det(V)=\pm1$ and $K'$ takes the form
\be
K'=\begin{pmatrix}
d_1&0&0&\ldots\\
0&d_2&0&\ldots\\
0&0&\ddots&\\
\vdots & \vdots &&
\end{pmatrix}
\ee
where $d_i$ are also integers.
Since $\det(U)=\det(V)=\pm 1$, we can stay in the canonical normalization of gauge fields when the original expression is in the canonical normalization.
If (\ref{eq:assumption_orthogonality}) is maintained, this is the simplest basis to work on.
However, this is not always the case.
The theories considered in Secs.~\ref{sec:cfl} and \ref{sec:tos} are such examples.
Generically,
if the basis change matrices satisfy $N N^T = 1$ and $M^T M =1$,
the condition (\ref{eq:assumption_orthogonality}) is maintained under the basis change.
In this case, the Moore-Penrose inverses of old and new $K$ matrices
are related by\footnote{When $K$ is invertible, Eq.~(\ref{eq:k+k+}) is trivially true. }
\be
\widetilde K^+ = M^{-1} K^+ N^{-1} .
\label{eq:k+k+}
\ee
Otherwise,
the condition (\ref{eq:assumption_orthogonality}) is not kept and
Eq.~(\ref{eq:k+k+}) does not hold.
In such cases,
the change of basis introduces the coupling between massless and massive modes,
which complicates the evaluation of correlation functions and
identification of symmetry generators.
For a detailed discussion on the effects of
the coupling of massless and massive modes,
see Appendix~\ref{sec:smith}.
\section{Higher-form symmetries of generalized $BF$ theories}\label{sec:emergent}
Let us examine the global higher-form symmetry of the action
(\ref{eq:seff}).
Depending on the structure of the $K$ matrix,
there can exist discrete 1-form and 2-form symmetries.
In the presence of massless phonons and photons,
there are also continuous 1-form and 2-form symmetries.
Here let us identify those symmetries.
\subsection{Continuous higher-form symmetries}
The action (\ref{eq:seff}) has
continuous (or $U(1)$) 1-form or 2-form global symmetries
in the presence of massless photons or NG bosons, respectively.
The symmetry transformation is given by
\begin{equation}
a_A \mapsto a_A +\epsilon\, C^{\alpha}_A \, \mu_\alpha,
\quad
b_i \mapsto b_i +\epsilon\, D^{\bar\alpha}_i \, \rho_{\bar\alpha},
\label{eq:continuous_higher_form}
\end{equation}
where $\mu_\alpha$ and $\rho_{\bar\alpha}$ are flat 1-form and 2-form
fields and $\epsilon$ is an infinitesimal parameter.
One can immediately see that the effective action (\ref{eq:seff}) is invariant under the transformation
(\ref{eq:continuous_higher_form}),
using $K_{iA} C^\alpha_A = 0$ and $D_i^{\bar\alpha} K_{iA}=0$.
The number of $U(1)$ $1$-form symmetry is given by
$|\alpha|=\dim ({\rm ker}\, K)$,
and the number of $U(1)$ $2$-form symmetries is
$|\bar\alpha|= \dim ({\rm coker}\, K)$.
They are the same as the number of massless photons and massless NG modes.
Those symmetry act on Wilson loops and vortex operators as a phase rotation,
\be
W_A \mapsto
W_A \exp \left( 2\pi \mathrm{i} \epsilon \, C^{\alpha}_A \, \mu_{\alpha} \right),
\quad
V_i \mapsto
V_i \exp \left( 2\pi \mathrm{i} \epsilon \, D^{\bar\alpha}_i \rho_{\bar\alpha} \right).
\ee
\subsection{Discrete higher-form symmetries}\label{sec:discrete_higher_form}
The theory can have discrete higher-form symmetries.
Consider the following transformation of the 2-form gauge field $b_i$,
\begin{equation}
b^T_i\mapsto
b^T_i + [q^T Q K^+]_{i} \, \lm ,
\label{eq:bt}
\end{equation}
where
$\lm = \lm (x)$ is a closed 2-form connection, i.e. $\diff \lambda=0$,
with quantized holonomy $\int_S \lm \in 2 \pi \mathbb Z$
over a closed surface $S$,
and
$q_{A}$ is a charge vector.
In order to make a one-to-one correspondence between $q_A$ and this transformation, we require that the charge vector has to satisfy
\be
q^T \cdot Q\in ({\rm coker}\, K^+)^\perp=({\rm ker} \, K)^\perp .
\ee
The kinetic term for $b_0$ is invariant under (\ref{eq:bt}),
since it does not shift
massless NG modes ($\delta [P^b_{ij}b_i] = 0$
follows from the definition of the Moore-Penrose inverse).
The variation of the action is
\begin{equation}
\delta_q S_{\rm eff}
=\frac{\mathrm{i}}{2\pi} [q^T Q K^+ K]_{A B} \int \lm \wedge \rd a_B .
\end{equation}
Noting that $K^+ K$ is the projection matrix to $({\rm ker} \, K)^\perp$,
\be
(q^T Q)_A [K^+ K]_{AB} = (q^T Q)_B.
\ee
Thus,
\begin{equation}
\delta_q S_{\rm eff}
=2\pi \mathrm{i} (q^T Q)_A \int {\lm\over 2\pi} \wedge {\rd a_A\over 2\pi} \in \sum_A 2 \pi \mathrm{i} q_A \mathbb Z ,
\end{equation}
where we used the normalization condition for $a_A$.
In order for this to generate symmetry, we must require $\delta_q S_{\mathrm{eff}}\in 2\pi \mathrm{i} \mathbb{Z}$, and then $q_A$ has to be an integer-valued vector.
Another necessary requirement of symmetry is that there must exist a physical operator with nontrivial transformation.
In this case, if vortex operators in (\ref{eq:wvqr}) transforms nontrivially under (\ref{eq:bt}),
it is a $2$-form symmetry of the system.
The action of the discrete $2$-form transformation (\ref{eq:bt}) is given by
a phase rotation,
\be
V_i^{(R)}
\mapsto
V_i ^{(R)}
\exp
\left(
2\pi \mathrm{i} \, [q^T Q K^+ R]_{i}
\right).
\label{eq:vi-tr}
\ee
If there exists a charge vector $q$ with $q^T Q\in ({\mathrm{coker} \, K^+})^{\perp}$ such that the
transformation (\ref{eq:vi-tr}) is nontrivial, then
the system has a discrete $2$-form symmetry.
Similarly, the discrete 1-form symmetry transformation is given by
\begin{equation}
a_A \mapsto a_A + [K^+ R\, p]_{A}\, \omega ,
\label{eq:1form_trans}
\end{equation}
where $\omega$ is a flat connection with $\int \omega \in 2 \pi \mathbb Z$,
and $Rp \in ({\rm coker}\, K)^\perp$.
The action is varied as
\begin{equation}
\delta_p S_{\rm eff}
= \frac{\mathrm{i}}{2\pi}[K K^+ R\, p]_{i} \int \rd b_i \wedge \omega ,
\end{equation}
Noting that $K K^+R \, p = R\, p$,
we have $\delta S_{\rm eff} \in 2\pi \mathrm{i} \mathbb Z$ when $p$ is an integer vector,
and the system is invariant under the $1$-form transformation.
On the gauge-invariant Wilson loops in (\ref{eq:wvqr}), it acts as
\be
W_A^{(Q)} \mapsto
W_A^{(Q)}
\exp
\left(
2\pi \mathrm{i} \, [ Q K^+ R\, p]_A
\right).
\label{eq:wa-tr}
\ee
If there is a charge vector $p_i$ such that Eq.~(\ref{eq:wa-tr}) is a nontrivial
transformation, this is a discrete $1$-form symmetry.
\subsection{Particle-vortex statistics}\label{sec:fractional}
Let us discuss the nature of particle-vortex statistics.
Test particle of charge $q$ and vortex with charge $p$ are represented by
\begin{equation}
W_{q}^{(Q)}(C) = \exp \left( \ri \, (q^T Q)_A \int_C a_A\right),
\quad
V_{p}^{(R)}(S) = \exp \left( \ri \, \int_S b_i\, (R\, p)_i\right),
\end{equation}
respectively, and the gauge invariance requires that $q$ and $p$ are integer-valued vectors. Here, $C$ denotes the world-line of the test particle $q$, and $S$ denotes the world-sheet of the vortex $p$.
As shown in Appendix \ref{app:derivation}, the correlation function of these operators in the effective theory (\ref{eq:seff}) satisfies\footnote{
When the orthogonality condition is not satisfied and
there are mixed kinetic terms of massless and massive modes,
there will be corrections, as
explained in Appendix~\ref{sec:smith}.
Compare this equation with Eq.~(\ref{eq:app:wv}).
}
\begin{equation}
\frac{
\left< W_{q}^{(Q)}(C) V_{p}^{(R)}(S)\right>
}{ \left< W_{q}^{(Q)}(C)\right> \left< V_{p}^{(R)}(S)\right>}
= \exp \left[ - 2 \pi \ri \, (q^T QK^+R\, p) \, {\rm Lk}(S, C) \right],
\label{eq:wv}
\end{equation}
where ${\rm Lk}(S,C)$ is the linking number
of the loop $C$ and the surface $S$.
This phase $\mathrm{e}^{2\pi\mathrm{i} (q^TQ K^+R p)}$ gives the mutual statistics between test particles and vortices.
Here, we intensionally use the same symbol $q,p$ to represent the charges of $W,V$ and to parametrize the discrete higher-form transformations in Sec.~\ref{sec:discrete_higher_form}. We shall soon see that the discrete higher-form symmetries are generated by these operators when $p,q$ are chosen so that $V_p^{(R)}$, $W_q^{(Q)}$ are topological~\cite{Gaiotto:2014kfa}.
In order to identify the generators of discrete higher-form symmetries, let us discuss when the vortex operator $V_p$ becomes topological.
This can be seen by deforming the surface as
\bea
V_{p}^{(R)}(S + \delta S)
&=& V_{p}^{(R)}(S)
\exp
\left(
\mathrm{i} (R\, p)_i
\left\{\int_{S+\delta S} b_i
- \int_{S} b_i
\right\}\right)\nonumber\\
&=&
V_{p}^{(R)}(S)
\exp
\left(
\mathrm{i} (R\, p)_i
\int_V \rd b_i
\right),
\end{eqnarray}
where $V$ is the volume swept by the continuous deformation $S \rightarrow S+\delta S$, or $\delta S=\p V$. Therefore, the operator $V^{(R)}_p$ is topological, i.e. $V_{p}^{(R)}(S+\delta S) =V_{p}^{(R)}(S)$, if
\be
(R\, p)_i \diff b_i=0 ,
\label{eq:eom_pb}
\ee
by using the EOM of (\ref{eq:seff}).
This happens when the vortex decouples from the massless phonon, i.e. $R\, p\in ({\rm coker}\, K)^{\perp}$.
Indeed, the EOM of (\ref{eq:seff}) with respect to $a_A$ gives
\be
(\diff \star \diff a^T)G^a (1-K^+ K)+{\mathrm{i}\over 2\pi}\diff b^T K=0.
\ee
Multiplying $K^+ R\,p$ from the right and using $KK^+ R\, p=R\, p$ for $R\, p\in ({\rm coker} K)^{\perp}$, we get (\ref{eq:eom_pb}).
Other vortex operators with $R\, p\not \in ({\rm coker}\, K)^{\perp}$ show non-topological behaviors because of its coupling to massless NG excitations.
Let us now discuss the physical meaning of the expectation value (\ref{eq:wv}) when we consider the topological surface operator $V_p^{(R)}$ with $R\, p\in ({\rm coker}\, K)^{\perp}$.
Taking $S$ as a two sphere singly linked to $C$, we get
\be
\left< W^{(Q)}_{q}(C) V^{(R)}_{p}(S)\right>
= \mathrm{e}^{2\pi \mathrm{i} (q^T QK^+ R\, p)}\left< W_{q}(C)\right>,
\ee
because $\langle V^{(R)}_p(S)\rangle =1$ as $S$ can be shrunk to a point.
This explicitly shows that the insertion of $V_p^{(R)}$ with $R\, p\in ({\rm coker} K)^{\perp}$ is nothing but the $1$-form transformation~(\ref{eq:1form_trans}).
The same discussion applies for the Wilson loop. When $q^T Q\not\in ({\rm ker}\, K)^{\perp}$, the Wilson loop obeys Coulomb law. $W_q^{(Q)}(C)$ shows topological dependence on $C$ if and only if
\be
(q^T Q)_A \diff a_A=0,
\ee
by the EOM of (\ref{eq:seff}), which is equivalent to the decoupling condition to massless photons, i.e. $q^T Q\in ({\rm ker}\, K)^{\perp}$.
When $q^T Q\in ({\rm ker}\, K)^{\perp}$, the expectation value (\ref{eq:wv}) can be written as
\be
\left< W_{q}^{(Q)}(C) V_{p}^{(R)}(S)\right>
= \mathrm{e}^{2\pi \mathrm{i} (q^T Q K^+R\, p)}\left< V_{p}^{(R)}(S)\right>,
\ee
because $\langle W_q^{(Q)}(C)\rangle =1$ by its topological nature. This is nothing but the $2$-form transformation (\ref{eq:bt}) if the phase $\mathrm{e}^{2\pi \mathrm{i} (q^T QK^+ Rp)}$ is nontrivial for some vortex charge $p$.
Note that the braiding phase $\e^{\mathrm{i}(q^TQ K^+ R\, p)}$ is invariant under
\be
(R\,p)_i \mapsto (R\,p)_i + [K \Lambda]_i ,
\ee
where $\Lambda_A\in ({\rm ker} \,K)^{\perp}$ is an integer vector.
Two charge vectors related by this relation are
topologically equivalent.
Likewise, $\e^{\mathrm{i}(q^T Q K^+ R\,p)}$ is invariant under
\be
(q^T Q)_A \mapsto (q^T Q)_A + [(\Lambda')^T K ]_A ,
\ee
with an integer vector $\Lambda'_i \in ({\rm coker}\, K)^{\perp}$.
\subsection{Fate of the symmetries and topological order }
Let us discuss the fate of symmetries.
For notational simplicity, let us consider the case $Q=1$ and $R=1$ in this subsection.
The result for general cases can be reproduced by replacing $(q^T K^+ p)$ by $(q^T Q K^+ R p)$.
When discrete higher-form symmetries are spontaneously broken,
there appear deconfined anyons and the system acquires a topological order.
Let us diagnose the existence of topological order in the current theory.
The charged object of a discrete $2$-form symmetry is
a vortex operator with a vortex charge $p$.
Existence of a discrete 2-form symmetry indicates that
there exists a topological Wilson loop.
A Wilson loop with charge $q\in ({\rm ker}\, K)^{\perp}$ induces a $2$-form transformation on
a vortex of the form,
\be
V_{p} \mapsto V_{p} \exp [2\pi \mathrm{i} \, q^T K^+ p] .
\ee
A discrete $2$-form symmetry is spontaneously broken if
the expectation value of the vortex operator obeys the surface law
at large vortex world-surface $S$,
\be
\langle
V_{p} (S)
\rangle \simeq \exp (- \kappa \, {\rm perimeter}[S]) ).
\ee
If a vortex operator is charged under a $U(1)$ 2-form symmetry,
$\langle V_p(S)\rangle$ decays faster than the
surface law, meaning that the $2$-form symmetry is unbroken.
This is because, in $4$ spacetime dimensions,
$2$-form continuous symmetry cannot be broken
due to the generalized Coleman-Mermin-Wagner theorem for higher-form
symmetry \cite{Gaiotto:2014kfa,Lake:2018dqm}.
Physically, this corresponds to the fact that
vortices couple to massless NG bosons and feel logarithmic confining force.
In order for the system to have a topological order,
there has to be an excitation that is neutral
under $U(1)$ $1$-form or $2$-form symmetries.
A $U(1)$ $2$-form transformation (\ref{eq:continuous_higher_form}) induces a phase rotation of the form
\be
V_{p} \mapsto V_{p} \exp [2\pi \mathrm{i} (D^{\bar\alpha}_i \epsilon_{\bar\alpha}) p_i ],
\ee
where $\epsilon_{\bar\alpha}$ is a continuous parameter.
In order for a vortex to be neutral under this symmetry,
we have to take the vortex charge as
$
p \in ({\rm coker}\, K)^\perp ,
$
which is the same condition with the one that $V_p(S)$ is a topological surface operator.
If there is a generator of discrete $2$-form symmetry specified by $q$
that acts nontrivially on such vortices,
those vortices are deconfined and
they generate $1$-form symmetries.
Then the system can have a pair of broken $1$-form and $2$-form symmetries
specified by the pair $(p, q)$.
In such a case, the system is topologically ordered.
To summarize the argument above,
appearance of topological order in the theory (\ref{eq:seff}) can be detected by the following condition:
\begin{list}{$\clubsuit$}{}
\item There exists a pair of integer vectors $(p,q)\in ({\rm coker} \, K)^{\perp}\times ({\rm ker}\, K)^{\perp}$, such that $\mathrm{e}^{2\pi \mathrm{i} (q^T K^{+} p)}\not =1$
\footnote{
One might think that the former condition
$(p,q)\in ({\rm coker} \, K)^{\perp}\times ({\rm ker}\, K)^{\perp}$
is redundant, because the components in ${\rm coker}\, K$ or ${\rm ker }\, K$ are
projected out when they are contracted with $K^+$ and
we can just project $p$ or $q$ to $({\rm coker}\, K)^\perp$ and
$({\rm ker}\, K)^\perp$ to obtain topological excitations.
However, the charge vector after this projection is not necessarily an integer vector,
so we need the former condition to ensure that topological quasiparticles/vortices
indeed exist.
}.
\end{list}
The appearance of the topological order can be explained also by a 't Hooft anomaly~\cite{tHooft:1979rat, Wen:2013oza, Kapustin:2014zva, Wang:2014pma} (see also \cite{Witten:2016cio, Tachikawa:2016cha, Gaiotto:2017yup, Tanizaki:2017bam, Komargodski:2017dmc, Komargodski:2017smk, Shimizu:2017asf, Wang:2017loc,Gaiotto:2017tne, Tanizaki:2017qhf, Tanizaki:2017mtm, Guo:2017xex, Sulejmanpasic:2018upi, Tanizaki:2018xto, Cordova:2018acb, Anber:2018tcj, Anber:2018jdf, Tanizaki:2018wtg, Yonekura:2019vyz} for recent applications),
which is an obstruction in gauging a global symmetry.
To see this, we introduce background gauge fields
for a pair of discrete 1-form and 2-form symmetries.
Suppose that
$p\in ({\rm coker}\, K)^{\perp}$ generates $\mathbb Z_{N^{(1)}}$ 1-form symmetry and
$q\in ({\rm ker}\, K)^{\perp}$ generates $\mathbb Z_{N^{(2)}}$ 2-form symmetry.
The integers $N^{(1)}$ and $N^{(2)}$ are determined as the minimal integers
so that the following $\Lambda^{(1)}$ and $\Lambda^{(2)}$ are integer vectors,
\be
\Lambda^{(1)}_A = N^{(1)} [ K^+ p ]_A ,
\quad
\Lambda^{(2)}_i = N^{(2)} [ q^T K^+ ]_i .
\ee
We will couple the system to background gauge fields corresponding to those symmetries.
To do this, it is convenient to write the $BF$ coupling term of the action as
\be
S_{\rm BF} =\mathrm{i} \frac{K_{iA}}{2\pi}
\int_{M_5} \rd b_i \wedge \rd a_A ,
\ee
where $M_5$ is a 5-dimensional manifold with $\p M_5 = M_4$.
$S_{\rm BF}$ defines the four-dimensional local theory because $S_{\rm BF}\in 2\pi \mathrm{i} \mathbb{Z}$ on closed five-dimensional manifolds.
We can introduce background gauge fields as
\be
S_{\rm BF, gauged}
=
\mathrm{i} \frac{K_{iA}}{2\pi}
\int_{M_5}
\left(\rd b_i + \Lambda^{(2)}_i \mathcal B \right)
\wedge
\left(
\rd a_A + \Lambda^{(1)}_A \mathcal A
\right) ,
\ee
where $\mathcal A$
is $\mathbb Z_{N^{(1)}}$ 2-form gauge field
and
$\mathcal B$ is a
$\mathbb Z_{N^{(2)}}$
3-form gauge field,
that can be written as
\be
N^{(1)} \mathcal A = \rd \mathcal A' ,
\quad
N^{(2)} \mathcal B = \rd \mathcal B' .
\label{eq:nada-nbdb}
\ee
Since $S_{\rm BF, gauged}$ is obtained by the minimal coupling procedure, it is manifestly invariant under the $\mathbb{Z}_{N^{(1)}}$ 1-form and $\mathbb{Z}_{N^{(2)}}$ 2-form gauge transformations.
However, $S_{\rm BF, gauged}$ is no longer well-defined as a four-dimensional theory unless $(q^T K^+ p)\in \mathbb{Z}$.
Indeed, using $q^T K^+ K=q^T$ and $KK^+ p =p$, we find that
\bea
S_{\rm BF,gauged}&=&2\pi \mathrm{i}\int_{M_5}\left({\diff b^T\over 2\pi}\wedge K {\diff a\over 2\pi}+{N^{(2)}\mathcal B\over 2\pi}\wedge {q^T \diff a\over 2\pi}+{\diff b^T p\over 2\pi}\wedge {N^{(1)}\mathcal A\over 2\pi}\right)\nonumber\\
&&+{2\pi \mathrm{i}}(q^T K^+ p)\int_{M_5}{N^{(2)}\mathcal B\over 2\pi}\wedge {N^{(1)} \mathcal A\over 2\pi}.
\end{eqnarray}
The first line on the right-hand-side is well-defined as as a local functional on $M_4=\p M_5$ modulo $2\pi \mathrm{i}$, but the second line is not if $q^T K^+ p\not \in \mathbb{Z}$ and is only well-defined modulo $2\pi \mathrm{i} (q^T K^+ p)$.
This means that the partition function $Z_{M_4}[\mathcal{A},\mathcal{B}]$ of (\ref{eq:seff}) with the background gauge field $\mathcal{A}$ and $\mathcal{B}$ cannot be gauge invariant as a four-dimensional field theory,
which indicates a 't~Hooft anomaly.
To make it anomaly free, we need to regard it as a boundary theory of the five-dimensional symmetry-protected topological states,
\be
Z_{M_4}[\mathcal{A},\mathcal{B}]\exp\left[-2\pi\mathrm{i} (q^T K^+ p)\int_{M_5}{N^{(2)}\mathcal B\over 2\pi}\wedge {N^{(1)} \mathcal A\over 2\pi}\right].
\ee
This is an anomaly matching condition by Callan-Harvey's anomaly-inflow mechanism \cite{Callan:1984sa}.
The existence of a 't Hooft anomaly indicates that the ground state
cannot be a trivially gapped state.
In the current case, the anomaly matching condition is satisfied by
the appearance of topological order.
\section{Color-flavor locked phase of QCD}\label{sec:cfl}
As an application of the general framework discussed so far,
let us discuss the color superconductivity in dense QCD~\cite{Hirono:2018fjr}.
We will see that there exists the nontrivial mutual statistics between test quark and minimal winding vortices~\cite{Cherman:2018jir}, and
it indicates the emergence of a $\mathbb{Z}_3$ 2-form symmetry~\cite{Hirono:2018fjr}.
We show that any vortex operators show algebraic decay instead of the surface law, and thus this 2-form symmetry is unbroken.
This observation is important to extend the notion of quark-hadron continuity~\cite{Schafer:1998ef} to the continuity as quantum phases at
zero temperature~\cite{Hirono:2018fjr}.
We note that this section is the follow up of the previous paper~\cite{Hirono:2018fjr} by the same authors with more detailed presentations.
\subsection{Generalized $BF$ theory for CFL phase}
A color superconducting phase is realized by the condensation of the Cooper pairs of quarks.
Let us consider the $3$-color and $3$-flavor QCD with flavor-degenerate mass of fundamental quarks, then the most attractive channel between quarks near the Fermi sea is anti-symmetric in both color and flavor.
The order parameter field in the effective gauged Ginzburg-Landau description is thus the diquark condensate,
$\Phi_{\alpha i }$,
where $\alpha$ and $i$ are indices of the anti-fundamental representation for color and flavor,
respectively.
This complex scalar fields $\Phi$ has charge $2$ under $U(1)$ quark number symmetry.
In the mean-field approximation with an appropriate gauge, we get $\langle \Phi_{\alpha i}\rangle\propto \delta_{\alpha i}$ at sufficiently large quark chemical potentials, and the diagonal transformation of color and flavor is unbroken,
\be
{SU(3)_\mathrm{c}\times SU(3)_\mathrm{f}\times U(1)\over \mathbb{Z}_3\times \mathbb{Z}_3}\to {SU(3)_{\mathrm{c}+\mathrm{f}}\over \mathbb{Z}_3}\times \mathbb{Z}_2.
\ee
This phase is therefore called color-flavor locking (CFL)~\cite{ Alford:1998mk, Schafer:1998ef, Alford:2007xm}, and there is a massless NG boson associated with the
spontaneously broken $U(1)$ symmetry.
Here, let us formally generalize the flavor and color group to be $SU(N)_\mathrm{f}$ and $SU(N)_\mathrm{c}$\footnote{
A similar gauged GL model with $U(N)_{\mathrm{c}}$ gauge group is
considered in Ref.~\cite{Hidaka:2019jtv}. In this case, all the gauge fields are massive and there is no massless NG modes.
}.
The scalar field $\Phi$ is taken to be in the anti-fundamental
representation both for $SU(N)_\mathrm{c}$ color group and $SU(N)_\mathrm{f}$ flavor group, although it is no longer related to the Cooper pair of fundamental quarks\footnote{Only when $N=3$, the two-index anti-symmetric representation is the same with the anti-fundamental representation. } when $N\not =3$.
The effective Lagrangian of the CFL phase is given by
a gauged Ginzburg-Landau model,
\begin{equation}
S={1\over 2g_{\mathrm{YM}}^2}|G|^2
+{1\over 2}|(\diff + \mathrm{i} a_{SU(N)})\Phi|^2
+V_{\mathrm{eff}}(\Phi^\dagger \Phi, |\det(\Phi)|),
\end{equation}
where $a_{SU(N)}$ is the $SU(N)_\mathrm{c}$ color gauge field, $G$ is its field
strength, and the effective potential $V_{\mathrm{eff}}$ depends only on
the color-singlet order parameters, $\Phi^\dagger \Phi$ and $\det(\Phi)$.
We here choose $V_\mathrm{eff}$ so that it has the minima at
\begin{equation}
\Phi^\dagger \Phi=\Delta^2\bm{1}.
\label{eq:diquark_vev}
\end{equation}
The mean-field approximation then realizes the symmetry breaking pattern of the CFL phase,
\be
{SU(N)_\mathrm{c}\times SU(N)_\mathrm{f}\times U(1)\over \mathbb{Z}_N\times \mathbb{Z}_N}\to {SU(N)_{\mathrm{c}+\mathrm{f}}\over \mathbb{Z}_N}\times \mathbb{Z}_2.
\ee
Here, we again assign the charge $2$ to $\Phi$ under $U(1)$ symmetry as in the case of $N=3$ QCD, although this is not mandatory for $N\not =3$ because of the absence of its interpretation as Cooper pairs.
In order to apply the formulation in Sec.~\ref{sec:eft}, we take a gauge fixing so that the diquark field $\Phi$ satisfying (\ref{eq:diquark_vev}) is taken to be diagonal,
\be
\Phi = \Delta
\diag
\left(
\e^{\phi_1}, \cdots, \e^{\phi_N}
\right),
\ee
where $\phi_i$ are $2\pi$-periodic compact scalar fields to denote the phase fluctuations.
With this gauge fixing,
only Abelian part of the gauge fields remain.
As an example of the Cartan generators of $SU(N)$, let us take the Gell-Mann--type matrices,
\be
\tau_1=\diag(1,-1,0,\ldots,0),\; \tau_2=\diag(1,1,-2,0,\ldots, 0),\ldots, \tau_{N-1}=\diag(1,\ldots,1,-(N-1)).
\label{eq:Cartan_GM}
\ee
Since we can easily find that
\be
\exp\left({2\pi \mathrm{i}\over n+1}\tau_n\right)=\exp\left({2\pi\mathrm{i}\over n+1}\tau_{n+1}\right),
\ee
for $n=1,\ldots, N-2$, $\mathbb{Z}_{n+1}\subset U(1)_{\tau_n}\times U(1)_{\tau_{n+1}}$ does not generate gauge transformations, so the gauge redundancy is represented by
\be
{U(1)_{\tau_1}\times U(1)_{\tau_2} \cdots\times U(1)_{\tau_{N-1}}\over \mathbb{Z}_{2}\times \cdots\times \mathbb{Z}_{N-1}}\subset SU(N).
\label{eq:redundancy_GM}
\ee
Let us denote the $U(1)_{\tau_A}$ gauge field as $a_A$.
The scalar field $\mathrm{e}^{\mathrm{i} \phi_i}$ has a charge $(\tau_A)_{ii}$ under $U(1)_A$, and thus the charge matrix $K$ of this theory is given by
a $N \times (N-1)$ matrix,
\be
K_{iA} =
\begin{pmatrix}
1 & 1 & \ldots &1\\
-1 & 1 & \ldots &1\\
0 & -2 & \ldots &1\\
\vdots & \vdots & \ddots & \vdots \\
0 & 0 & \ldots& - (N-1) \\
\end{pmatrix}.
\label{eq:cfl_K_noncanonical}
\ee
Taking the Abelian duality, we get the effective theory of the form (\ref{eq:EFT_general_03}),
\be
S={\Delta^{-2}\over 8\pi^2}\int |\diff b_i|^2+{{\rm tr}(\tau_A\tau_B)\over 2 g_{\rm YM}^2}\int \diff a_A\wedge \star \diff a_B+{\mathrm{i}\over 2\pi}\int \diff b^T \wedge K\diff a.
\label{eq:seff_heavy}
\ee
Although 2-form gauge fields $b_i$ follow the canonical normalization, the $U(1)$ 1-form gauge fields $a_A$ obey the non-canonical normalization
\be
Q_{AB}\int \diff a_B\in 2\pi \mathbb{Z},
\label{eq:cfl_oneform_normalization}
\ee
with $Q_{AB}=K_{(i=A),B}$ for $A,B=1,\ldots,N-1$ because the gauge group is (\ref{eq:redundancy_GM}).
As a result of the diquark condensation, all the gluons become massive by Higgs mechanism and
there is no massless 1-form gauge field.
This is equivalent to ${\dim }\, ({\rm ker}\, K) = 0$, and thus we can drop the kinetic term of the gauge field in (\ref{eq:redundancy_GM}) when discussing the low-energy physics.
On the other hand,
\be
{\rm coker}\,K=\mathbb{R}\begin{pmatrix}
1\\
\vdots\\
1
\end{pmatrix},
\ee
and correspondingly there is one massless NG mode, $ \diff b_0\equiv \diff(b_1+\cdots+b_N)$,
which is associated with the spontaneous breaking of $U(1)$ baryon number symmetry.
Thanks to the permutation invariance of the kinetic term of $b_i$, which comes out of $U(N)$ flavor symmetry, the mixed kinetic term between $b_0$ and massive modes does not arise, and (\ref{eq:assumption_orthogonality}) is satisfied.
Therefore, we can obtain the effective theory of the form (\ref{eq:seff}),
\be
S_{\rm eff}[b,a]={G^b_0\over 2}\int |\diff b_0|^2+{\mathrm{i} \over 2\pi}\int b^T \wedge K\diff a,
\label{eq:seff_cfl_noncanonical}
\ee
with $G^b_0=\Delta^2/4\pi^2 N^2$.
Let us perform the basis change in Sec.~\ref{sec:basis_change} with $M=Q^{-1}$ so that we work on the canonically normalized gauge fields $\widetilde{a}_A$, i.e. $\int \diff \widetilde{a}_A\in 2\pi \mathbb{Z}$:
\be
\widetilde{a}_A=Q_{AB}a_B.
\ee
The effective action (\ref{eq:seff_cfl_noncanonical}) becomes
\be
S_{\rm eff}[b,\widetilde{a}]={G^b_0\over 2}\int |\diff b_0|^2+{\mathrm{i} \over 2\pi}\int b^T \wedge \widetilde{K}\diff \widetilde{a},
\label{eq:seff_cfl}
\ee
with the new charge matrix $\widetilde{K}$,
\be
\widetilde{K}=KQ^{-1}=
\begin{pmatrix}
1 & 0 & \ldots &0\\
0 & 1 & \ldots &0\\
\vdots & \vdots & \ddots &\vdots\\
0 & 0 & \ldots& 1 \\
-1 & -1 &\ldots & -1 \\
\end{pmatrix}.
\ee
Note that this transformation does not violate the condition (\ref{eq:assumption_orthogonality}) because it only changes the massive gauge fields $a_A$.
We can directly obtain the effective action (\ref{eq:seff_cfl}) if we use the another Cartan generator,
\be
\widetilde{\tau}_i=\diag(0,\ldots, 0, \overbrace{1}^{i\mbox{-th}},0,\ldots,0,-1),
\ee
with $i=1,\ldots,N-1$, instead of (\ref{eq:Cartan_GM}). With this Cartan basis, the Abelian subgroup of $SU(N)$ takes the simper form as
\be
U(1)_{\widetilde{\tau}_1}\times \cdots \times U(1)_{\widetilde{\tau}_{N-1}}\subset SU(N),
\ee
compared with (\ref{eq:redundancy_GM}).
We can apply the extra transformation to obtain the Smith normal form,
\be
\widetilde{K}=U^{-1}\widetilde{K}',
\ee
with
\be
U^{-1}=\begin{pmatrix}
1 & 0 & \ldots &0 &0\\
0 & 1 & \ldots &0 &0\\
\vdots & \vdots & \ddots &\vdots &\vdots\\
0 & 0 & \ldots& 1 &0\\
-1 & -1 &\ldots & -1 &1\\
\end{pmatrix},\quad
\widetilde{K}'=
\begin{pmatrix}
1 & 0 & \ldots &0\\
0 & 1 & \ldots &0\\
\vdots & \vdots & \ddots &\vdots\\
0 & 0 & \ldots& 1 \\
0 & 0 &\ldots & 0 \\
\end{pmatrix}.
\ee
The transformation violates (\ref{eq:assumption_orthogonality}), and the result in Sec.~\ref{sec:emergent} cannot be naively applied due to the mixed kinetic term between the massless and heavy degrees of freedom.
We therefore do not take the Smith normal form in this section, but put the detailed computation with the Smith normal form in Appendix~\ref{sec:smith}.
\subsection{Emergent higher-form symmetry of CFL phase}
Since the Higgs mechanism makes all the gauge fields massive, the Wilson loops
\be
W_{\widetilde{A}}(C)=\exp\left(\mathrm{i} \int_C\widetilde{a}_A\right),
\ee
obey the perimeter law. They therefore generate discrete $2$-form transformations on the vortex operators,\footnote{
The vortices with minimal energy in the CFL are called non-Abelian vortices or semi-superfluid vortices \cite{Balachandran:2005ev}.
See Ref.~\cite{Eto:2013hoa} for a review.
}
\be
V_i(S)=\exp\left(\mathrm{i} \int_S b_i\right)\mapsto
\mathrm{e}^{2\pi \mathrm{i}\widetilde{K}^+_{Ai}} V_i(S).
\ee
The Moore-Penrose inverse of $\widetilde{K}$ is given by
\be
\widetilde{K}^+_{Ai}=[QK^+]_{Ai}=\delta_{Ai}-{1\over N},
\ee
and thus this is the $\mathbb{Z}_N$ 2-form symmetry~\cite{Hirono:2018fjr}.
This shows that the test quark and the minimal winding vortex has the $\mathbb{Z}_N$ mutual statistics~\cite{Cherman:2018jir}:
\be
\langle W_{\widetilde{A}}(C) V_i(S)\rangle = \exp\left(-{2\pi\mathrm{i}\over N}{\rm Lk}(C,S)\right) \langle V_i (S)\rangle.
\ee
Note that this phase rotation is a subgroup of the $U(1)$ 2-form symmetry.
Let us show that the above $\mathbb{Z}_N$ 2-form symmetry is unbroken~\cite{Hirono:2018fjr}.
To show it, we have to see that any vortex operators $V_p(S)$ of charge $p$ decay faster than the perimeter (surface) law when the charge $p$ is nontrivial under $\mathbb{Z}_N$ 2-form symmetry.
$V_p$ is topological if and only if $p\in ({\rm coker}\, \widetilde{K})^{\perp}$, and those operators are generated by
\be
V_i(S) V_{i+1}(S)^{-1}.
\ee
Since these operators are neutral under $\mathbb{Z}_N$,
we find that this symmetry is unbroken.
This also means that the theory has no 1-form symmetry since any topological surface operators are neutral with the Wilson loops.
As a consequence, the CFL phase at the zero temperature acquires the emergent $\mathbb{Z}_N$ 2-form symmetry, but there is no topological order since it is unbroken.
\subsection{Implications for the quark-hadron continuity scenario}
Let us comment on the physical consequences of the unbroken $\mathbb Z_3$ 2-form symmetry
regarding the quark-hadron continuity scenario \cite{Schafer:1998ef, Hirono:2018fjr}.
Conventionally, the phases of matter have been classified by
the (0-form) symmetry of the system.
If there are two phases with the same symmetry, they are
considered to be in the same phase.
It means that there exists a certain deformation of the Hamiltonian
by which the two phases can be continuously connected.
When we consider quantum phases of matter,
there can be phases with the same 0-form symmetry but are
distinguished by different topological orders.
Here, we have shown here that the CFL phase does not have
a deconfined discrete gauge field and
that implies the absence of topological order,
although the appearance of fractional braiding phase has a certain similarity
to the nature of a topological ordered state.
The braiding phase is shown to be a direct consequence of a
(unbroken) $\mathbb Z_3$ 2-form symmetry.
In addition, the system does not have a 1-form symmetry, and
hence there is not mixed 't~Hooft anomaly of discrete higher-form symmetries,
which allows for a topologically trivial ground state.
So, as far as the ground state property is concerned,
the CFL can be continuously connected to a nucleon superfluid phase,
which presumably has a trivial topological structure
because of the absence of deconfined gluons.
Thus, we have extended the continuity scenario to zero temperature.
Note that this does not necessarily mean that
there is no phase transition between nucleon superfluidity and
the CFL phase as a function of baryon chemical potential $\mu_{\rm B}$.
Even if two phases have the same symmetry,
as liquid and vapor phases of water, there can be a phase transition,
depending on which path one would take in the parameter space.
The same is true for the quark-hadron continuity at finite temperatures \cite{Schafer:1998ef}.
In order to predict what would happen along a particular path
in a parameter space like ($T$, $\mu_{\rm B}$),
a more detailed approach is necessary.
One might argue that the existence of the (unbroken) $\mathbb Z_3$ 2-form symmetry
gives us a distinction of the CFL and nucleon superfluidity.
However, as we have shown, the discrete $\mathbb Z_3$ symmetry is in fact a
subgroup of the $U(1)$ 2-form symmetry,
and this symmetry is associated with the existence of a massless $U(1)$ NG mode.
Because this mode also exists in a nuclear superfluid phase,
the continuous 2-form symmetry is also present in this phase.
Therefore, nucleon superfluidity and the CFL phase have the
same 0-form and higher-form symmetries.
\section{Example of superfluidity with topological order}\label{sec:tos}
In this section, let us discuss an example of superfluidity with topological order. We consider the generalized $BF$ theory with the action (\ref{eq:seff_cfl_noncanonical}),
\be
S_{\rm eff}[b,a]={G^b_0\over 2}\int |\diff b_0|^2+{\mathrm{i} \over 2\pi}\int b^T \wedge K\diff a,
\label{eq:seff_tos}
\ee
with the $K$ matrix (\ref{eq:cfl_K_noncanonical}).
In the case of the CFL phase,
the $U(1)$ 1-form gauge fields $a_A$ obey the non-canonical normalization, (\ref{eq:cfl_oneform_normalization}), and no topological order appears.
In this section, we instead assume the canonical normalization of gauge fields, $\int \diff a_A\in 2\pi \mathbb{Z}$.
All the Wilson loops $W_A(C)=\exp\left(\mathrm{i} \int_C a_A\right)$ are topological because the theory (\ref{eq:seff_tos}) is in the Higgs phase.
The gauge-invariant correlation functions are obtained by
\be
\langle W_{A}(C) V_i(S)\rangle = \exp\left(-2\pi\mathrm{i} K^+_{Ai}{\rm Lk}(C,S)\right) \langle V_i (S)\rangle.
\ee
The Moore-Penrose inverse of (\ref{eq:cfl_K_noncanonical}) is given by
\be
K^+=
\begin{pmatrix}
\left(1-\frac{1}2\right) & -\frac{1}2 & 0 & 0 & \ldots &0\\
\left(\frac{1}{2}-\frac{1}{3}\right) & \left(\frac{1}{2}-\frac{1}{3}\right) & -\frac{1}3 & 0 & \ldots &0\\
\left(\frac{1}{3}-\frac{1}{4}\right) & \left(\frac{1}{3}-\frac{1}{4}\right) & \left(\frac{1}{3}-\frac{1}{4}\right) & -\frac{1}4 &\ldots&0\\
\vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\
\frac{1}{N(N-1)} &\frac{1}{N(N-1)} & \frac{1}{N(N-1)} &\frac{1}{N(N-1)} &\ldots & - \frac{1}{N} \\
\end{pmatrix},
\ee
and we can now find the particle-vortex statistics explicitly.
The set of topological Wilson lines generates $\mathbb{Z}_2\times \cdots \mathbb{Z}_{N-1}\times \mathbb{Z}_N$ 2-form symmetry.
To see this, let us take the basis of charge vectors $q_n$ of Wilson lines as
\be
q_1 = \begin{pmatrix}
1 \\
0\\
0\\
\vdots\\
0
\end{pmatrix},
\,\,
q_2 = \begin{pmatrix}
1 \\
1 \\
0\\
\vdots\\
0
\end{pmatrix},\, \ldots,
\,\,
q_{N-1} = \begin{pmatrix}
1 \\
1 \\
1\\
\vdots\\
1
\end{pmatrix},
\ee
that is, $W_{q_n}(C)=\exp\left(\mathrm{i}\int_C(a_1+\cdots+a_{n})\right)$.
The fractional phase is determined as
\be
q^T_n K^+=\Bigl(\overbrace{-{1\over n+1}, ,\ldots, -{1\over n+1}}^{n+1},\overbrace{0,\ldots, 0}^{N-1-n}\Bigr) \,\bmod 1.
\ee
Therefore, $W_{q_n}$ generates $\mathbb{Z}_{n+1}$ 2-form symmetry.
Similarly, we can find the $\mathbb{Z}_2\times \cdots\times \mathbb{Z}_{N-1}$ 1-form symmetry.
Topological surface operators are given by the vortex charges $p \in ({\rm coker}\, K)^\perp$, and its basis can be chosen as
\be
p_1 = \begin{pmatrix}
1 \\
-1 \\
0 \\
\vdots\\
0\\
0
\end{pmatrix},
\,\,
p_2 = \begin{pmatrix}
0 \\
1 \\
-1\\
\vdots\\
0\\
0
\end{pmatrix}, \,\ldots,\,\,
p_{N-1} = \begin{pmatrix}
0 \\
0 \\
0\\
\vdots\\
1\\
-1
\end{pmatrix},
\ee
that is, $V_{p_m}(S)=\exp\left(\mathrm{i} \int_S(b_m-b_{m+1})\right)$.
The topological surface operator $V_{p_m}$ generates $\mathbb{Z}_{m}$ 1-form symmetry for $m=2,\ldots, N-1$. Indeed, this can be found by
\be
q^T_n K^+ p_m= -{1\over n+1}\delta_{n+1,m}\, \bmod 1,
\ee
and thus $W_{q_n}$ and $V_{p_{n+1}}$ have the mutual $\mathbb{Z}_{n+1}$ statistics for $n=1,\ldots,N-2$.
This shows that $\mathbb{Z}_2\times \cdots \times \mathbb{Z}_{N-1}$ 1-form and 2-form symmetries are spontaneously broken, and thus this theory supports a topological order.
The $\mathbb{Z}_N$ 2-form symmetry generated by $W_{q_{N-1}}$ is unbroken. This fact comes out of the same discussion given in Sec.~\ref{sec:cfl}.
All the vortex operators charged under $\mathbb{Z}_N$ are coupled to NG boson, and thus those vortices are logarithmically confined.
In other words, $\mathbb{Z}_N$ 2-form symmetry is a subgroup of $U(1)$ 2-form symmetry defining the vortex winding numbers, and thus the generalized Coleman-Mermin-Wagner theorem prohibits its spontaneous breaking.
We therefore conclude that the theory describes superfluidity of one NG mode with $\mathbb{Z}_2\times \cdots \times \mathbb{Z}_{N-1}$ topological order.
\section{Summary and outlook}\label{sec:summary}
We have discussed a general effective theory for superfluids with topological order.
Starting from a gauged GL model, we have derived
a low-energy gauge theory written in terms of 2-form and 1-form gauge fields.
The theory has a structure of a $BF$ theory with a non-square $K$ matrix
that can have a nontrivial kernel/cokernel,
coupled with massless NG modes corresponding to the breaking of $U(1)$ symmetries.
Physical spectrum are classified according to the structure of the $K$ matrix.
We have discussed the symmetry of the theory.
There can be discrete 1-form and 2-form symmetries as well as
$U(1)$ 1-form and 2-form symmetries.
We have shown that the correlation of vortices and quasiparticles
is written in terms of the topological information of vortex surfaces and quasiparticle world-lines.
We have discussed how to identify the presence of topological order,
which is summarized in the condition $(\clubsuit)$.
If there is a vortex operator whose average obeys perimeter law,
the 2-form symmetry is broken, which indicates the topological order.
As an application of the framework, we have discussed the CFL phase of dense QCD matter.
We have analyzed the higher-symmetry of the phase and shown that
color Wilson loops and vortices show fractional braiding
as a consequence of $\mathbb Z_3$ 2-form symmetry.
We have shown that the nuclear superfluid phase and the CFL phase have the same symmetry including higher-form ones,
which extends the quark-hadron continuity scenario to zero temperature.
We have also discussed an example of superfluidity with topological order
in Sec.~\ref{sec:tos} and we identified the symmetry and topological order in this system.
We believe that the framework discussed in this paper would be useful
in identifying the topological structure
in systems where topological order and massless modes coexist,
which include high $T_{\rm c}$ cuprate superconductivity.
\acknowledgments
The work of Y.~T. was supported by JSPS Overseas Fellowship.
The work of Y.~H. was supported in part by the
Korean Ministry of Education, Science and Technology,
Gyeongsangbuk-do and Pohang City for Independent
Junior Research Groups at the Asia Pacific Center for
Theoretical Physics.
|
1,108,101,565,660 | arxiv | \section*{1 Introduction}
Problems associated with large order behaviour of perturbation series
coefficients were examined in many papers. It was F.Dyson \cite{D} who
argued in 1952 that perturbation series in quantum field theory diverges.
Asymptotics of coefficients at high orders which is useful, for example,
for investigation of the divergent series summation problem was found later.
This asymptotics is usually constructed by the technique used by L.Lipatov
\cite{L} in quantum field theory and by E.Br\'{e}zin, J.C.Le Guillou and
J.Zinn-Justin \cite{BLGZJ} in quantum mechanics. This method is the following.
The $k$-th order of perturbation theory for quantities like Green functions
can be represented through a functional integral which can be approximately
calculated by saddle-point technique. This approach was reusable in quantum
mechanics (see, for example, \cite{BLGZJ,BPZJ,ZJ,RS}) for study of the
high orders of perturbation theory for ground state energy and the $n$-th
excited state energy as $n$ is not large.
In this paper a new method for constructing high order asymptotics is
suggested. This technique allows us to find such asymptotics not only
for eigenvalues but also for eigenfunctions. This method is based not on the
functional integral approach but on direct analysis of recursive relations.
Consider the following dependence of the Hamiltonian on a perturbation
theory parameter $g$, momenta $p=(p_1,...,p_n)$ and coordinates
$x=(x_1,...,x_n)$:
\begin{equation}\label{11*}
{\cal H}=\frac{p^2}{2}+\frac{1}{g^2}V(gx),
\end{equation}
where $p_m=-i\partial/\partial x_m$, the potential $V$ has a local minimum
at $x=0$. These Hamiltonians were considered in \cite{BLGZJ,BPZJ,ZJ}. Without
loss of generality, one can assume that
$V(Q) \sim Q^2/2 + O(Q^3)$
as $Q \rightarrow 0$.
Contrary to the perturbation expansion in powers of $g$
for eigenvalues, perturbation expansion for eigenfunctions can be constructed
in different ways. For instance, the wave function $\Psi$ can be simply
expanded in powers of $g$ at fixed $x$,
$\Psi(x)=\sum_{k=0}^{\infty} g^k\Psi_k(x)$.
But on the other hand, one can first carry out the following change of the
wave function argument,
\begin{equation}\label{11+}
gx=Q,
\end{equation}
and expand the wave function at fixed $Q$. It appears that in this case the
expansion in powers of $g^2$ is a tunnel semiclassical expansion (see, for
example, \cite{LL,M1}). The square of perturbation theory parameter becomes
the Planck constant analog, because the equation for eigenfunction
$ {\cal H}\Psi = E_0 \Psi$
after multiplying it through by $g^2$, substitution (\ref{11+})
and renotation $g^2=\hbar$ takes the form
\begin{equation}
-\frac{\hbar^2}{2}\Delta\Psi + V(Q) \Psi
= \hbar E_0 \Psi.
\label{*3}
\end{equation}
As it is known (see, for example, \cite{LL,M1}), the tunnel asymptotics of the
ground state wave function has the form of a product of a slowly varying
pre-exponential factor by a rapidly varying exponentional function
\begin{equation}
\Phi(Q,\hbar)\exp\left(-\frac{1}{\hbar}S(Q)\right).
\label{D1*}
\end{equation}
Substitution of this formula to eq.(\ref{*3}) gives the Hamilton-Jacobi
equation for function $S$
\begin{equation}
(\nabla S(Q))^2/2=V(Q)
\label{*1}
\end{equation}
and the following equation for function $\Phi$,
\begin{equation}
-\frac{\hbar}{2}\Delta\Phi + \nabla S \nabla \Phi +
\frac{1}{2}(\Delta S - n)\Phi= (E_0-n/2)\Phi.
\label{D2*}
\end{equation}
One can apply perturbation theory to it. Both function $\Phi$ and ground
state energy can be expanded in powers of $\hbar$,
$$
\Phi(Q,\hbar)=\sum_{k=0}^{\infty} \hbar^{k} \Phi_k(Q),
E_0(\hbar)=\sum_{k=0}^{\infty} \hbar^{k} E^{(k)},
E^{(0)}=n/2.
$$
This paper deals with the asymptotical behaviour of
$\Phi_k(Q)$ as $k$ is large and $Q$ is fixed. The problem of the high order
asymptotics of the eigenfunction perturbation theory when the expansion
is considered at fixed $x$ will be discussed in the next paper.
In one-dimensional quantum mechanics of a particle in a potential shown in
fig.1 which was considered in \cite{BLGZJ,ZJ} the exponent in
(\ref{D1*}) is expressed through the action $S$ on the trajectory starting
from zero and reaching at once the point $Q$ (solid line in fig.2).
This trajectory is the solution of the euclidean equation of motion which
can be obtained from the ordinary one by changing of the real time
$t$ to the imaginary one $-i\tau$:
\begin{equation}\label{I3}
\frac{d^2}{d\tau^2} {\cal Q} = V^{'}({\cal Q}).
\end{equation}
It appears that the asymptotics of the quantity $\Phi_k$ at larges $k$
can be expressed through the action $S_s$ on the other euclidean solution
which starts from zero, reaches the turning point and finishes also
at point $Q$ ( dashed line in fig.2 ).
As well as in one-dimensional case, in multidimensional case the main
contribution to the wave function is given by the classical euclidean
solution with the least action $S$, while high order asymptotics
of the semiclassical expansion is determined by the action $S_s$ on
another euclidean solution.
It is shown in this paper that high orders of
$\Phi_k$ have the following asymptotic behaviour at larges $k$,
\begin{equation}
\Phi_k(Q) \sim \frac{(k-1)!}{(S_s(Q)-S(Q))^k},
\label{D4V}
\end{equation}
the pre-exponential factor is omited in this formula. From the expressions
obtained in this paper one can also find the asymptotic behaviour of the
ground state energy perturbation coefficients
$E^{(k)}$ and carry out a check of the formulas obtained in
\cite{BLGZJ,BPZJ,RS} for various cases by the path integral technique.
Besides the potential shown in fig.1, other examples are also considered in
this paper. First, the case of a particle moving in a radial symmetric
potential in $n$-dimensional space is also considered. If the potential
depends on the distance of the origin as it is shown in fig.1, then classical
trajectories determining the asymptotics of the eigenfunction and the high
order behaviour of semiclassical expansion are analogous with the
trajectories shown in fig.2 by solid and dashed lines for one-dimensional
case.
Second, the example of the potential with degenerate minima (fig.3) is
also discussed. Classical solution determining the asymptotics of the
eigenfunction is analogous to shown in fig.2. On the other hand, not
classical solution but ''almost classical solution'' contributes to the
large order behaviour of semiclassical expansion. This ''almost solution''
starts at $\tau \rightarrow -\infty$ from the origin, then transits to
another minimum. The ''almost solution'' resides in this minimum for a
long euclidean time and finally reaches the point $Q$.
As it was shown in \cite{BPZJ,ZJ}, if $Q=0$, then these ''almost solutions''
(instanton-anti-instanton pairs) contribute to large order
behaviour of perturbation theory for the ground state energy.
Finally, a particle on $n$-dimensional sphere in the external potential
depending only on one coordinate and having one minimum at the south pole
of the sphere which is a classical ground state (fig.4) is also considered in
this paper. One of the classical solutions shown in fig.4 by solid line
determines the behaviour of eigenfunction, another solution passing through
the north pole of the sphere determines the high order asymptotics of
semiclassical expansion. This solution is shown in fig.4 by dashed line.
An interesting feature of this asymptotics is the nullification of it at odd
$n$ which was found in ref.\cite{RS} for the ground state energy perturbation
theory. In this case high order asymptotics is determined by the ''almost
solution'' which makes a loop around the sphere, resides in the south pole for
a long time and finally reachs the point $Q$.
\section*{2 Methods of finding the large order asymptotics of semiclassical
expansion}
Asymptotics of $\Phi_k$ can be found by various methods. First, the ground
state wave function can be expressed through the path integral
(see, for example, \cite{HH})
over trajectories starting as $\tau \rightarrow -\infty$ from zero
and reaching the point $Q$ at $\tau=0$:
\begin{equation}
\int_{{\cal Q}(-\infty)=0,{\cal Q}(0)=Q} D{\cal Q} \exp(-\frac{1}{\hbar}
S[{\cal Q}])
\label{D3+}
\end{equation}
where $S$ is the euclidean action of the theory and have the form:
\begin{equation}\label{I2}
S[{\cal Q}]=\int d\tau [\dot{\cal Q}^2/2+V({\cal Q})].
\end{equation}
This integral can be evaluated as $\hbar \rightarrow 0$
by the Laplase method. As an exponential approximation this integral is
equal to
\begin{equation}
\exp\left(-\frac{1}{\hbar} \min_{{\cal Q}(-\infty)=0,{\cal Q}(0)=Q} S[{\cal
Q}]\right),
\label{exp} \end{equation}
This formula coinsides with semiclassical tunnel asymptotics.
The pre-exponential factor and the corrections can be calculated with the
help of extraction of the factor
(\ref{exp}) from the path integral (\ref{D3+}), substitution
${\cal Q}={\cal Q}_{0}+q\sqrt{\hbar}$,
where ${\cal Q}_{0}$ is the trajectory with the least action
in eq.(\ref{exp}), expansion of the integrand after this substitution
in terms of $\sqrt{\hbar}$ and calculation of integrals of the products of
some polynomial in $q$ functions
by the Gauss exponent. Coefficients of odd powers in
$\sqrt{\hbar}$ are equal to zero. Quantities $\Phi_k$
can be expressed through the integrals
\begin{equation}
\Phi_k=\int Dq \oint_C \frac{dg}{2\pi i g^{2k+1}}
\exp\left(-\frac{1}{g^2}[S({\cal Q}_{0}+gq)-S({\cal Q}_{0})]\right),
\label{***} \end{equation}
where the contour $C$ dependind in general on $q$ runs around the origin
counterclockwise. After substitution
$
{\cal Q}_{0}+gq={\cal Q}, g=\nu/\sqrt{k}
$
integrals (\ref{***}) take the form:
\begin{equation}\label{B3*}
\Phi_k=k^k \int \frac{D{\cal Q} d\nu}{2\pi i \nu^{2k+1}}
\exp\left(-k\left(
\frac{S[{\cal Q}]-S[{\cal Q}_{0}]}{\nu^2}+\ln\nu^2
\right)\right),
\end{equation}
and can be evaluated by saddle-point technique.
Consider saddle points of the exponent $({\cal Q}_{s},\nu_s)$.
It's variation in $\nu$ gives us the following condition,
$
\nu_{s}^2=S({\cal Q}_s)-S({\cal Q}_0),
$
variation in $\Phi$ leads to the condition
$
\delta S({\cal Q}_s)=0,
$
i.e. ${\cal Q}_s$ is the classical solution.
Therefore, asymptotics of the integral (\ref{B3*}) has the form (\ref{D4V})
up to a pre-exponential factor, where
$S_s(Q)=S[{\cal Q}_{s}],S(Q)=S[{\cal Q}_{0}]$.
Another technique to calculate the $\Phi_k$ high order asymptotics is based on
the direct analysis of the recursive relations for the $\Phi_k$ which can be
obtained from eq.(\ref{D2*})
\begin{equation}
-\frac{1}{2}\Delta\Phi_{k-1}+\nabla S \nabla \Phi_k + \frac{1}{2} (\Delta S
-n) \Phi_k = \sum_{p=1}^{k} E^{(p)} \Phi_{k-p} \label{D5+}
\end{equation}
Let us look for the large order asymptotics in a form
\begin{equation} \Phi_k \sim \frac{(k-1)!}{A(Q)^k}. \label{D5*}
\end{equation}
Substitution of formula (\ref{D5*}) to relations (\ref{D5+})
gives the following equation for $A$
\begin{equation} \nabla A = -2 \nabla S \label{D5V} \end{equation}
in the leading order. It has been supposed that high orders of
$\Phi_k$
are growing faster than $E^{(k)}$.
This assumption can be justified as follows. It follows from eq.(\ref{D5V})
that the value of the function $A$ in zero is more than in other
points of some vicinity of zero. Since the $E^{(k)}$ asymptotics is
distinguished from the $\Phi_k$ asymptotics only by a pre-exponential factor
and, therefore, has the form $\Gamma(k)/A(0)^k$, the right-hand side of
eq.(\ref{D5+}) can be neglected at $Q\ne 0$ because it is exponentially small.
On the other hand, when $A(0)-A(Q)\sim 1/k$, one can't ignore the
right-hand side,
so that calculations based on this neglection break down at
$Q\sim 1/\sqrt{k}$. Therefore, the asymptotics at these $Q$ and larges $k$
has another form discussed in section 5.
Section 3 contains the calculation of the pre-exponential factor in formula
(\ref{D5*}) with the help of direct analysis of the recursive relations.
This factor is defined up to a multiplier, the function $A$
is determined up to an additive constant.
In section 3 there is also a consideration of the following interesting
method for calculating the pre-exponential factor and the corrections to
asymptotic formula. Let us look for the
$\Phi_{k}$ asymptotics as $k \rightarrow \infty$ when the corrections are
taken into account in a form
\begin{equation}
\Phi_k=B(Q,e^{-\partial/\partial k}) \frac{(k-1)!}{A(Q)^k}
\label{D6*}
\end{equation}
where $B$ is a sum of power functions
of the operator $e^{-\partial/\partial k}$.
Notice that the operator $e^{-\partial/\partial k}$
playes the role of a small papameter in this case. Namely, it
transforms the sequence
$(k-1)!/A^k$ to the sequence $(k-2)!/A^{k-1}$,
the $k$-th order of the latter sequence is less than the $k$-th order of
the former one approximately in $k/A$ times.
Therefore, any asymptotics of the form
\begin{equation}
k^{\nu}\frac{(k-1)!}{A(Q)^k}(1+a_1/k+a_2/k^2+...)
\label{D6A}
\end{equation}
can be presented in a form
\begin{equation}
(b_0+b_1 e^{-\partial/\partial k}+b_2 e^{-2\partial/\partial k}+...)
e^{\nu\partial/\partial k}\frac{(k-1)!}{A(Q)^k}
\label{D6B}
\end{equation}
i.e. in the form (\ref{D6*}). Coefficients $b$ in eq.(\ref{D6B}) can be
expressed through the coefficients $a$ in eq.(\ref{D6A}).
The $k$-th order of the ground state energy perturbation theory can be
presented in the form analogous with (\ref{D6*}),too.
These expansions for $E^{(k)}$ and $\Phi_k$ can be substituted to the
relations (\ref{D5+}). Analysis of the obtained equation is analogous to
calculating of the corrections to semiclassical approximation. But in
this case the parameter of the expansion is not a number (Planck constant)
but an operator $e^{-\partial/\partial k}$.
The general theory of the semiclassical expansion when
$\hbar$ is an operator was developed in \cite{M1}.
Eq.(\ref{D5V}) determines the function $A$ up to an additive constant. For
determining it, one must analyse the behaviour of semiclassical approximation
near singular points: in one-dimensional case
it is the turning point that gives the singularity in eq.
(\ref{D4V}).
The behaviour of semiclassical expansion near the turning point which
determines the constants in formulas for the function $A$ and the
pre-exponential factor is analysed in section 4 for various types of
turning points.
It will be shown that the pre-exponential factor is divergent near the point
$Q=0$. The divergence is connected with the necessity of constructing another
asymptotics as $Q\sim 1/\sqrt{k}$, which is non-singular and allows us to
find large order asymptotics of the ground state energy perturbation
theory. Section 5 contains such derivation.
The obtained results
coincide with the obtained one in refs.\cite{BLGZJ,BPZJ,RS}.
\section*{3 Analysis of the semiclassical expansion recursive relations}
In this section the derivation of the asymptotic formulas from the recursive
relations is considered in more details. Let us examine the following form
for the asymptotics of $\Phi_k$ at larges $k$
\begin{equation}
\Phi_k(Q) \sim \frac{\Gamma(k)k^{\nu}}{A(Q)^{k+\nu}}f(Q) \label{D14*}
\end{equation}
and find conditions for the constant $\nu$ and functions $A$, $f$.
Substitute the expression (\ref{D14*}) to the left-hand side of eq.(\ref{D5+})
and consider first terms of order $\Gamma(k)k^{\nu+1}$ and then terms of order
$\Gamma(k)k^{\nu}$. To calculate the corrections to the asymptotics
(\ref{D14*}) one can consider the following terms.
Let us use the relations
\begin{equation}
\nabla \Phi_k = -
\frac{\Gamma(k)k^{\nu}(k+\nu)}{A(Q)^{k+\nu+1}}
f(Q)\nabla A +
\frac{\Gamma(k)k^{\nu}}{A(Q)^{k+\nu}}
\nabla f + ... ,
\label{D14+}
\end{equation}
$$
\Delta \Phi_{k-1} =
\frac{\Gamma(k)k^{\nu}(k+\nu)}{A(Q)^{k+\nu+1}}
(\nabla A)^2 f(Q)
$$
\begin{equation}
+ \frac{\Gamma(k)k^{\nu}}{A(Q)^{k+\nu}}
[-f(Q)\Delta A - 2 \nabla A \nabla f]+...
\label{D14-}
\end{equation}
The right-hand side of eq.(\ref{D5+}) can be omitted if
$ Q \ne 0$ because of the remark of section 2. Substitution of the relations
(\ref{D14+}) and (\ref{D14-}) to formula (\ref{D5+}) gives the equation
for $A$ (\ref{D5V}) in a leading order, so that
$$
A(Q)=A_0-2S(Q),A_0=const.
$$
The next order gives us the following equation for $f$,
\begin{equation}
\frac{1}{2} [ f\Delta A + 2 \nabla A \nabla f] + \nabla S \nabla f
+ \frac{1}{2} (\Delta S -n)f=0.
\label{*2}
\end{equation}
Separate now factor $\Phi_0$ from the function $f$ and denote
\begin{equation}
f/\Phi_0=X.
\label{D15f}
\end{equation}
It follows from the equations for $\Phi_0$ and $A$ that the function $X(Q)$
satisfies the condition,
\begin{equation}
\nabla S \nabla X = -nX
\label{D15P}
\end{equation}
and can be denote up to a multiplier. In one-dimensional case
function $X$ has the form
\begin{equation}
X=c\exp(\int_{Q}^{Q_+} dQ/\sqrt{2V(Q)} )
\label{D15X}
\end{equation}
In section 4 the vicinity of the turning point is considered in more details
and constants
$c$ and $\nu$ are found.
Other types of singular points are also considered.
Recursive relations for semiclassical expansion can be also investigated for
Hamiltonians with quantum corrections. For example, consider Hamiltonians
of the following type
\cite{RS},
\begin{equation}
H=-\frac{\hbar^2}{2}\frac{d^2}{dQ^2}
-\frac{n-1}{2Q}u(Q)\hbar^2\frac{d}{dQ}+V(Q),
\label{I12}
\end{equation}
where function $u(Q)$ is equal to 1 when $Q=0$.
When $u=Qctg Q$, this Hamiltonian describes a particle on $n$-dimensional
sphere. If $u=1$ then eq.(\ref{I12}) corresponds to the $O(n)$-symmetrical
case.
Substitution of the formula
(\ref{D1*}) to the equation $H\psi=\hbar E_0\psi$
leads to the Hamilton-Jacobi equation coinciding to
(\ref{*1}), while the equation for
$\Phi$ takes the more complicated form than (\ref{D2*}):
$$
-\frac{\hbar}{2}\Phi^{''} + S^{'}\Phi^{'} + \frac{1}{2} (S^{''}-n)\Phi
+\frac{n-1}{2Q}uS^{'}\Phi - \hbar \frac{n-1}{2Q} u \Phi^{'} =
(E_0-n/2)\Phi.
$$
Recursive relations for the perturbation series coefficients in $\hbar$
can be also obtained from the equation for $\Phi$.
Asymptotics of the $\Phi_k$ at larges $k$ can be also looked for in a form
(\ref{D14*}). Substitution of this formula to the recursive relations
gives the equation for
$f$ distinguished from (\ref{*2}). But after extracting the factor
$\Phi_0$ (eq.(\ref{D15f})) the relation for the function $X$
takes, nevertheless, the form (\ref{D15P}). Its solution has the form
$$
X=c\exp\left(n\int_{Q}^{Q_+} \frac{dQ}{\sqrt{2V(Q)}}\right).
$$
One can find the corrections to the asymptotic formula
(\ref{D14*}) by evaluating the following terms in eqs.(\ref{D14+}),
(\ref{D14-}), substituting it to the left-hand side of the eq.
(\ref{D5+}) and setting the coefficients of the corresponding orders in
$k$ equal to zero. Nevertheless, in this section another technique
to find the corrections is examined. This method illustrates the
analogy between the corrections to semiclassical approximation and
the corrections to high order asymptotics.
As it has been mentioned in section 2, let us search for the asymptotics
of the $\Phi_k$ in a form (\ref{D6*}). When $Q \ne 0$, one can present the
eq.(\ref{D5+}) in a form
$$
-\frac{1}{2}
e^{-\partial /\partial k}
\Delta \Phi_k + \nabla S \nabla \Phi_k +\frac{1}{2}(\Delta S - n)\Phi_k=
$$
\begin{equation}
=(E^{(1)} e^{-\partial /\partial k} +E^{(2)} e^{-2\partial /\partial k}
+...)\Phi_k,
\label{D16+}
\end{equation}
where the exponentially small terms associated with the ground state energy
asymptotics are omitted.
As it has been shown in section 2, the operator
$e^{-\partial /\partial k}$
plays a role of a small parameter analogous to
$\hbar$ in a case of semiclassical expansion.
Notice that the commutation rule of the derivation operator and
the function
$\Gamma(k)A(Q)^{-k}$ can be presented in a form
\begin{equation}
\frac{\partial}{\partial Q} \frac{\Gamma(k)}{A(Q)^k} = -
e^{-\partial /\partial k}
\frac{\Gamma(k)}{A(Q)^k}\frac{\partial A}{\partial Q} +
\frac{\Gamma(k)}{A(Q)^k}
\frac{\partial}{\partial Q}
\label{D16*}
\end{equation}
analogous to the commutation formula of the derivation operator with the
exponent
$$
\frac{\partial}{\partial Q}\exp(-A/\hbar)
=-\frac{1}{\hbar}\exp(-A/\hbar)\frac{\partial A}{\partial Q} +
\exp(-A/\hbar)
\frac{\partial}{\partial Q},
$$
where a number $\hbar$ is substituted by the operator
$e^{-\partial /\partial k},$
while the function of $\hbar$, $\exp(-A/\hbar)$, is changed by the set
of numbers,
$\Gamma(k)A^{-k}$. Application of eq.(\ref{D16*})
to eq.(\ref{D16+}) leads to the equation for the function
$B$ presented in formula
(\ref{D6*}),
$$
-\frac
{e^{-\partial /\partial k}}{2}
(\nabla - e^{\partial /\partial k} \nabla A)^2 B +
\nabla S
(\nabla - e^{\partial /\partial k} \nabla A) B
$$
$$
+\frac{1}{2}(\Delta S - n)B=
(e^{-\partial /\partial k}E^{(1)} +
e^{-2\partial /\partial k}E^{(2)} + ...)B
$$
which is analogous to the equation for
$\Phi$ (\ref{D2*}). Let us set the coefficients of each order in
$e^{-\partial /\partial k}$
equal to zero. We obtain the equations derived earlier and the following
formula for the corrections,
$$
B(e^{-\partial /\partial k},Q)=B_0(Q)\sum_{l=0}^{\infty} F_l(Q)
e^{-l\partial /\partial k},
$$
$$
-\frac{1}{2B_0}
\Delta(F_{l-1}B_0)+\nabla S \nabla F_l = \sum_{s=0}^{l-1} E^{(l-s)}F_s
$$
These equations define the function $B$ up to the factor which does not
depend on $Q$ but depends on
$e^{-\partial /\partial k}$.
Let us consider the singular points and find this factor.
\section*{4 The behaviour of the asymptotics near singular points}
As it has been noticed in the previous section, one must consider the
behaviour of the recursive relations near singular points in order to
find constants $c,A_0,\nu$. This problem is examined in this section. The main
idea of the consideration is the following. The potential near singular point
can be approximated by a linear, quadratic or another function depending on the
type of a singular point. All the orders of semiclassical expansion can be
evaluated exactly for this approximate potential, so one can find unknown
constants by comparing the ''exact'' results with the asymptotics discussed in
section 3.
In this section three types of one-dimensional singularities are discussed:
i) ordinary turning point (fig.1);
ii) potentials with degenerate minima (fig.3);
iii) singular quantum correction to the Hamiltonian \cite{RS}.
Consider these cases more preceisely.
\subsection*{4.1 Ordinary turning point}
In this subsection the case of one-dimensional quantum mechanics of a particle
in the potential shown in fig.1 is considered. Let us discuss the behaviour of
semiclassical expansion for the ground state wave function in the vicinity of
point $Q_+$.The potential $V$ can be approximated by a linear function,
\begin{equation}
V \sim a\xi, \xi=Q_+-Q,
\label{D19+}
\end{equation}
The function $A$ satisfying the equation (\ref{D5V}) is approximately equal to
$$
A=A_0-2S_++\frac{4}{3}\sqrt{2a}\xi^{3/2},
$$
where $S_+$ is the action
$
S=\int_{0}^{Q_+} \sqrt{2V(Q)} dQ
$
at the turning point $Q_+$. As the duration of motion from the point
$Q$ to the turning point has the form
$
\int_Q^{Q_+} \frac{dQ}{\sqrt{2V(Q)}} = \sqrt{2\xi/a},
$
eq.(\ref{D14*}) implies that the high orders of semiclassical expansion
behave as follows,
\begin{equation} \Phi_k \sim \frac{\Gamma(k)
k^{\nu}ce^{\sqrt{2\xi/a}}}
{(A_0-2S_++\frac{4}{3}\sqrt{2a}\xi^{3/2})^{k+\nu}} \Phi_0.
\label{D19*}
\end{equation}
Let us obtain the asymptotics (\ref{D19*}) in another way and find the
coefficients $c,\nu,A_0$.
First of all, notice that the semiclassical expansion for the wave function in
the case of a potential (\ref{D19+}) can be obtained as follows.
Formula (\ref{D16+}) implies that the function
$
\psi=\sum \Phi_k \hbar^k e^{-S/\hbar}
$
approximately satisfies as an asymptotic series the equation
$$
(-\frac{\hbar^2}{2}\frac{d^2}{d\xi^2} + a\xi) \psi(\xi)=0
$$
The expansion in powers of $\hbar$ of the growing at large $\xi$
solution can be obtained from the following expression,
\begin{equation}
\psi(\xi)=\int dp
\exp(-\frac{1}{\hbar}(p\xi-\frac{p^3}{6a})
\label{I10}
\end{equation}
where the integral is taken over any sufficiently small region containitg the
minimum of the exponent, $p_0=-\sqrt{2a\xi}$.
It follows from the integral presentation (\ref{I10}) that the
$\Phi_k$ asymptotics at larges $k$ has the form
$$
\Phi_k \sim \Phi_0 \frac{1}{2\pi} \frac{\Gamma(k)e^{\sqrt{2\xi/a}}}
{(\frac{4}{3}\xi \sqrt{2a \xi})^k},
$$
coinciding with eq.(\ref{D19*}) when
\begin{equation}
\nu=0,A_0=2S_+,c=1/(2\pi).
\label{I11}
\end{equation}
Substitution of these constants to the formulas obtained in section 3 gives
us the following asymptotics,
\begin{equation}
\Phi_k(Q) \sim \frac{(k-1)!\Phi_0(Q)}{2\pi (S_s(Q)-S(Q))^k}
\exp(\int_{Q}^{Q_+}\frac{dQ}{\sqrt{2V(Q)}}).
\label{D13*}
\end{equation}
\subsection*{4.2 Potential with degenerate minima }
Consider now the case of the double-well potential shown in fig.3. Namely,
let the potential $V$ have the minimum in the point $Q_+$,
besides the minimum
at zero and let $V(Q_+)$ equal to zero.In the vicinity of the point
$Q_+$ the potential can be approximated by a quadratic function,
$$
V \sim \omega^2 \xi^2/2, \xi=Q_+-Q.
$$
Action $S$ has the following form in this approximation,
$$
S=S_+-\omega \xi^2/2
$$
The duration of motion from point $Q$ to point $Q_+$ is infinite in this
case, contrary to the previous subsection. Therefore, the solution $X$ to
eq. (\ref{D15P}) has the more complicated form than (\ref{D15X}).
Namely,
\begin{equation} X=c(Q_+-Q)^{1/\omega}\exp \int_{Q}^{Q_+} dQ\left[
\frac{1}{\sqrt{2V(Q)}} - \frac{1}{\omega (Q_+ -Q)}
\right],
\label{D21*}
\end{equation}
in this formula singular contribution is substracted from the exponent in
eq.(\ref{D15X}) and the normalizing factor is redefined. One can show by
the explicit calculation that the function
(\ref{D21*}) really satisfies eq.(\ref{D15P}).
In the vicinity of the point $Q_+$ the asymptotic formula (\ref{D14*})
takes the form
\begin{equation}
\Phi_k \sim \frac{\Gamma(k) k^{\nu}c \xi^{1/\omega}}
{(A_0-2S_++\omega \xi^2)^{k+\nu}} \Phi_0
\label{D21+}
\end{equation}
On the other hand, recursive relations have the following form in the
quadratic approximation
\begin{equation}
-\frac{1}{2}\frac{d^2\Phi_{k-1}}{d\xi^2}-\omega \xi \Phi_0
\left(\frac{\Phi_k}{\Phi_0}\right)^{'} = 0
\label{D22*}
\end{equation}
and can be solved exactly,
\begin{equation}
\Phi_k=\frac{c_k}{\xi^{2k}} \Phi_0,\Phi_0=\xi^{-\frac{1+\omega}{2\omega}}
\label{D22F}
\end{equation}
where numerical coefficients $c_k$ have the form
$$
c_k=\frac{\Gamma(2k+\frac{1+\omega}{2\omega})}
{(4\omega)^k \Gamma(k+1) \Gamma(\frac{1+\omega}{2\omega})}
$$
Making use of the Stirling formula for the Gamma-function, one can obtain that
at large $k$ the $c_k$ asymptotics can be written in a form
\begin{equation}
c_k \sim \frac{(k-1)!}{\omega^k \sqrt{2\pi}} \frac{(2k)^{1/\omega}}
{\Gamma(\frac{1+\omega}{2\omega})}
\label{D22C}
\end{equation}
The constants in the $\Phi_k$ asymptotics can be found by comparing the
formulas (\ref{D22C}),(\ref{D22F}) with the formula (\ref{D21+}).
The constants are the following,
\begin{equation}
c=\frac{(2\omega)^{\frac{1}{2\omega}}}
{\sqrt{2\pi}\Gamma(\frac{1+\omega}{2\omega})},
\nu=\frac{1}{2\omega},A_0=2S_+.
\label{D22+}
\end{equation}
It follows from the formulas (\ref{D14*}),(\ref{D15f}),
(\ref{D22+}),(\ref{D21*}) that the
$k$-th order of the ground state wave function semiclassical expansion
has the following asymptotics as
$k \rightarrow \infty$ in the case of a degenerate minima potential,
$$
\Phi_k \sim \Phi_0 \frac{\Gamma(k)k^{1/2\omega}}{(2S_+-2S(Q))^{k+1/2\omega}}
\frac{(2\omega)^{\frac{1}{2\omega}}(Q_+-Q)^{1/\omega}}
{\sqrt{2\pi}\Gamma(\frac{1+\omega}{2\omega})}
$$
\begin{equation}
\times\exp \int_{Q}^{Q_+} \left[
\frac{1}{\sqrt{2V(Q)}} - \frac{1}{\omega (Q_+ -Q)}
\right]
\label{D23AS}
\end{equation}
\subsection*{4.3 Singular quantum correction to the Hamiltonian}
Consider the Hamiltonians presented as a sum of a classical (non-singular)
Hamiltonian and a quantum correction to it which is singular,
\begin{equation}
H=-\frac{\hbar^2}{2}\frac{d^2}{dQ^2} + V(\cos Q) -
\frac{\hbar^2}{2}(n-1) \frac{\cos Q}{\sin Q}\frac{d}{dQ}
\label{C12*}
\end{equation}
Evaluation of the high order asymptotics for the ground state energy
perturbation theory in this case was considered in \cite{RS}.
The $\Phi_k$ asymptotics in this case has the form
(\ref{D14*}), the function $X$ satisfies the equation (\ref{D15P})
at non-singular points ($Q \in (0,\pi) $)
and, therefore, has the form (\ref{D15X}), where $Q_+=\pi$.
When one study the vicinity of the singular point in this case, one must take
into account not only classical part of the Hamiltonian but also the
quantum correction to it because it is singular, as oppose to the previous
case. The classical Hamiltonian can be approximated by the Hamiltonian of
a free particle,
\begin{equation}
-\frac{\hbar^2}{2}\frac{d^2}{d\xi^2} + E, \xi=\pi-Q,
\label{D23A}
\end{equation}
while the quantum correction is approximately equal to the following
operator,
\begin{equation}
-\frac{\hbar^2}{2}(n-1)\frac{1}{\xi}\frac{d}{d\xi}.
\label{D23B}
\end{equation}
In the vicinity of the singular point the asymptotic formula (\ref{D14*})
takes the following approximate form,
\begin{equation}
\Phi_k \sim \frac{\Gamma(k) k^{\nu}c e^{\xi/\sqrt{2E}} }
{(A_0-2S_++2\sqrt{2E}\xi)^{k+\nu}} \Phi_0
\label{D24V}
\end{equation}
The constants can be found by comparing with this formula. The function
$
\psi=\sum \hbar^k \Phi_k e^{-S/\hbar}
$
satisfies the equation
$$
(-\frac{\hbar^2}{2}\frac{d^2}{d\xi^2} + E
-\frac{\hbar^2}{2}(n-1)\frac{1}{\xi}\frac{d}{d\xi})\psi(\xi)=0.
$$
Its solution growing at infinity is considered. It can be expressed through
the Infeld function $I_{n/2-1}$ (the Bessel function of a purely imaginary
argument),
$$
\psi(\xi)=\xi^{1-n/2} I_{n/2-1}[\xi\sqrt{2E}/\hbar].
$$
Semiclassical expansion of the fnction $\psi$ corresponds to the
Infeld function expansion at large arguments. As this asymptotic series
breaks for the Infeld function of the half-integer order, semiclassical
expansion also breaks, so in the approximation
(\ref{D23A}),(\ref{D23B}) all the orders of $\Phi_k$
begining from some $k$ are equal to zero. Therefore, singular
point
$Q_+=\pi$ does not contribute to the high order asymptotics of semiclassical
expansion at integer odd $n$. This confirms the assumption of \cite{RS}.
If $n$ is not equal to an odd integer number, the Infeld function expansion
coefficients
$$
I_{\nu}(x)=\frac{1}{\sqrt{2\pi x}} e^x (1+c_1/x+c_2/x^2+...)
$$
have the form \cite{RG}:
$$
c_k=\frac{\cos \pi\nu}{\pi} \frac{\Gamma(k-\nu+1/2)\Gamma(k+\nu+1/2)}{2^kk!}.
$$
and the following asymptotic behaviour at larges $k$
$$
c_k \sim \frac{\Gamma(k)}{2^k} \frac{\cos \pi \nu}{\pi}.
$$
Therefore, the semiclassical expansion of the function
$\psi$ behaves at high orders as follows,
$$
\Phi_k \sim \Phi_0 \frac{\Gamma(k)}{(2\xi)^k(2E)^{k/2}}
\frac{\cos\pi(n/2-1)}{\pi}
$$
Therefore, the constants $A_0,c,\nu$ in the asymptotic formula for
$\Phi_k$ have the form,
$$
A_0=2S_+,\nu=0,c=-\frac{\cos (\pi n/2)}{\pi}
$$
Thus, the following expression for the $\Phi_k$ asymptotics is obtained,
\begin{equation}
\Phi_k \sim \Phi_0 \frac{\Gamma(k)}{(2S_+-2S(Q))^{k}}
(-\frac{1}{\pi} \cos\pi n/2)
\exp \int_{Q}^{Q_+}dQ \frac{1}{\sqrt{2V(\cos Q)}}
\label{D25AS}
\end{equation}
\section*{5 The behaviour of the asymptotics in the vicinity of the minimum
and high order behaviour of the ground state energy perturbation theory}
The asymptotics obtained in the previous section for various cases have
singularities in the origin. Namely, in one-dimensional case at small
$Q$ the asymptotic formula for $\Phi_k$ is approximately equal to
\begin{equation} \Phi_k(Q) \sim
\frac{B}{|Q|}\frac{(k-1)!k^{\nu}}{(A_0-Q^2)^{k+\nu}}, \label{D26*}
\end{equation}
(the factor $B$ depends on the potential), i.e. the pre-exponential factor
diverges as $1/Q$. In the $n$-dimensional case there is a divergence as
$1/|Q|^n$.
On the other hand, in the each order of semiclassical expansion the wave
function is finite at zero. It is the behaviour of the eigenfunction
near zero that allows us to find an asymptotic behaviour of the
perturbation theory for eigenvalues.
It occurs, nevertheless, that the $\Phi_k(Q)$ asymptotics considered not
under the conditions $Q=const,k \rightarrow \infty$
but under other conditions
\begin{equation}
k \rightarrow \infty, Q\sqrt{k} \rightarrow y=const,
\label{D26+}
\end{equation}
is non-singular at zero argument.
The knowledge of this asymptotics enables us
to find high order behaviour of the eigenvalue perturbation theory.
It is convenient to change the variable $y=z\sqrt{A_0}$.
Let us search for the $\Phi_k$ asymptotics under the conditions (\ref{D26+})
in a form:
\begin{equation}
\frac{(k-1)!k^{\nu+1/2}}{A_{0}^{k+\nu+1/2}} g(z),
\label{D27+}
\end{equation}
while the $E^{(k)}$ asymptotics is seeking in a form:
\begin{equation} E^{(k)} \sim
\frac{(k-1)!k^{\nu+1/2}}{A_{0}^{k+\nu+1/2}} {\cal E} \label{D27B}
\end{equation}
Consider the substitution of the formulas (\ref{D27+}) and (\ref{D27B})
to the equation (\ref{D5+}). As the asymptotics is considered near the point
$Q=0$, one cannot omit the right-hand side of the equation
(\ref{D5+}). At larges $k$ the main contribution to the sum in the right-hand
side of the formula (\ref{D5+}) is given by the $k$-th term equal to
$E^{(k)}$ because $\Phi_0=1$ in the origin. Notice also that
$ \Delta S(0) = n$. Therefore, at larges $k$ eq.(\ref{D5+}) takes the
following form in the leading order,
\begin{equation}
-\frac{1}{2}\frac{d^2g}{dz^2}+z\frac{dg}{dz}={\cal E}.
\label{D27A}
\end{equation}
Find now the connection between the quantity $\cal E$ defining the large orders
of the ground state energy perturbation theory and the factor
$B$ obtained from the wave function high order asymptotics.
Consider first the case of a symmetric potential. Formula
(\ref{D26*}) takes then place at positive $Q$ as well as at negative $Q$.
At larges $z$ an asymptotic formula for
$\Phi_k(z\sqrt{A_0/k})$ must transform to the asymptotics (\ref{D26*}) where
the following change, $Q=z\sqrt{A_0/k}$, is made. Let us substract the factor
$A_0^{-k}$ from eq.(\ref{D26*}) and make a limit (\ref{D26+}).
Let us also take into account the relation,
$
(1-z^2/k)^{k+\nu} \rightarrow e^{-z^2}.
$
The following boundary condition for function $g$ at $+\infty$, as well as at
$-\infty$ are obtained from it:
\begin{equation}
g(z) \sim \frac{B }{|z|}e^{z^2}.
\label{D27-}
\end{equation}
On the other hand, eq.(\ref{D27A}) can be solved exactly for the derivative
$g^{'}(z)$. Comparing this solution with the asymptotics for
$g^{'}(z)$ obtained from eq.(\ref{D27-}), one can find that:
$$
{\cal E}=-\frac{2B}{\sqrt{\pi}}.
$$
The constant
$B$ and, therefore, the quantity $\cal E$ can be found for various cases
considered in the previous section. Thus, ground state energy perturbation
theory has the following large order asymptotic behaviour,
$$
E^{(k)} \simeq -\frac{k!k^{-1/2}}{\pi^{3/2}S_{B}^{k+1/2}}
Q_{+}\exp\int_{0}^{Q_{+}} dr[1/\sqrt{2V(r)}-1/r],
$$
in the case of ordinary turning point discussed in subsection 4.1, while in
the case of the potential with degenerate minima treated in subsection 4.2
the asymptotics is as follows,
$$
E^{(k)} \sim -\frac{2k!\sqrt{\omega}}{\pi A_{0}^{k+1}}
\left(\frac{2k\omega}{A_0}\right)^{\frac{1-\omega}{2\omega}}
\frac{Q_{+}^{1/\omega+1}}{\Gamma(\frac{1+\omega}{2\omega})}
$$
$$
\times \exp \int_{0}^{Q_+} dQ \left(\frac{1}{\sqrt{2V(Q)}}-
\frac{1}{\omega(Q_+-Q)}-\frac{1}{Q}\right).
$$
These formulas are in agreement with the results obtained in
\cite{BLGZJ,BPZJ}.
When the potential is not symmetric, the function $A$ in the asymptotics
(\ref{D5*}) has discontinuity at zero argument. For the definiteness, assume
that the least value of $A$ corresponds to positive $Q$.
Then the boundary condition at $+\infty$ has as in the considered case the form
(\ref{D27-}),
while the condition at $-\infty$ is the boundness of $g$.
It follows from solving eq.(\ref{D27A}) with these boundary conditions that
the quantity
$\cal E$ is in 2 times lesser than in the previous case:
$
{\cal E}=-\frac{B}{\sqrt{\pi}}.
$
This result has the following interpretation in terms of the path integral
approach. When the potential is even, there are two classical solutions
symmetric under the change $Q$ to $-Q$ which contribute to the asymptotics,
as opposed to non-even potentials.
Consider now the case of Hamiltonians
(\ref{I12}). The asymptotic formula for
$\Phi_k$ transforms at small $Q$ to the following expression,
$$
\frac{(k-1)!}{(A_0-Q^2)^k}\frac{B}{|Q|^n}.
$$
Let us seek for the asymptotics of $\Phi_k$ under the conditions (\ref{D26*})
in a form,
\begin{equation}
\frac{(k-1)!k^{n/2}}{A_{0}^{k+n/2}} g(Q\sqrt{k/A_0}),
\label{I14}
\end{equation}
analogous to one-dimensional case, where function $g(z)$ satisfies the
following boundary conditions at larges $z$,
\begin{equation}
g(z)\sim B e^{z^2}/z^n.
\label{I16}
\end{equation}
Let us look for the asymptotics of $E^{(k)}$ in a form,
\begin{equation}
E^{(k)} \sim \frac{(k-1)!k^{n/2}}{A_{0}^{k+n/2}} {\cal E}.
\label{I15}
\end{equation}
Substitution of the formulas (\ref{I14}) and (\ref{I15}) to the
recursive relations of semiclassical expansion gives us the following
equation for $g$,
$$
-\frac{1}{2}g^{''}(z) - \frac{n-1}{2z}g^{'}(z) + zg^{'}(z) = {\cal E}.
$$
Its solution which is an even function of $z$ and equal to zero at
$z=0$ have the form
$$
g(z)=\int_{0}^{z} \frac{dz}{z^{n-1}} e^{z^2}
\int_{0}^{z} [-2{\cal E}z^{'n-1}e^{-z^{'2}}]dz^{'}.
$$
Asymptotic behaviour of this formula at larges $z$ is in agreement with the
expression (\ref{I16}) when
$$
{\cal E} = -\frac{2B}{\Gamma(n/2)}.
$$
Substitution of the expression for $B$ to this formula leads to the
asymptotics (\ref{I15}) coinciding in the case of the
$O(n)$-symmetric systems with the results of
\cite{BLGZJ},
\begin{equation}
E^{(k)} \simeq -\frac{k!k^{n/2-1}}{\pi\Gamma(n/2)S_{B}^{k+n/2}}
(Q_{+}\exp\int_{0}^{Q_{+}} dr[1/\sqrt{2V(r)}-1/r])^{n}, \label{C35+}
\end{equation}
and to the asymptotics
\begin{equation} E^{(k)} \simeq \frac{k!k^{n/2-1}}{S_{SI}^{k+n/2}}\cos(\pi
n/2)
\frac{2\pi^{n-1}}{\Gamma(n/2)}\exp\left(n\int_{0}^{\pi}d\theta \left(\frac{1}{
\sqrt{2V(\cos\theta)}}-\frac{1}{\pi-\theta}\right)\right)
\label{C24*}\end{equation}
in the case of the Hamiltonians (\ref{C12*}).
The asymptotics (\ref{C24*}) is in agreement with \cite{RS}.
\section*{6 Conclusions}
In this paper large order asymptotics of the tunnel semiclassical expansion
for quantum mechanical systems is considered. As usual, the dependence of the
Hamiltonian on the semiclassical expansion parameter (Planck constant) has
the form (\ref{*3}). The results related to the high order asymptotics for the
ground state energy are in agreement with the papers
\cite{BW1,BW2,BLGZJ,BPZJ,ZJ}. Contrary to them, this paper deals with the study
of large order asymptotics not only for eigenvalues but also for
eigenfunctions.
Therefore, one can examine this problem not only by the path integral
approach but also by the direct analysis of the semiclassical expansion
recursive relations.
When one constructs the asymptotics, an important role is played by
classical euclidean solutions starting from the origin and finishing
at the point $Q$. These solutions give an exponentially small contribution
to the wave function in comparison with the contribution of another
solution satisfying the same boundary conditions. An interesting feature of
the constructed asymptotics for the semiclassical expansion large orders
is the divergence of the pre-exponential factor near zero value
of argument. This
difficulty is resolved by the investigation of the asymptotics as
$Q\sim 1/\sqrt{k}$. Another interesting feature of the obtained asymptotic
formula is the following.In one-dimensional case the most essential
growth of the asymptotics takes place near the turning point. It is the
comparison
of the asymptotics near this point with the exact coefficients of the
expansion for the linear potential that allows
us to find the unknown constants in
the asymptotic formula. Analogous tecnique is applicable to other cases of
the singular points.
Thus, the approach suggested in this paper enables us to obtain the
high order asymptotics of semiclassical expansion both for eigenvalues and
eigenfunctions. Although only the case of the ground state have been also
examined, an analogous treatment can be also applicable to the excited
states.
Probably,one can generalize the discussed method to quantum field theory and
find the asymptotic behaviour of a sum of Feynman diagramms with $N$ external
lines and $k$ loops as both $N$ and $k$ tend to infinity. The considered
technique can be also useful in the case of instantons.
The author is indebted to V.A.Rubakov for helpful discussions.
This work is supported in part by ISF, grant \# MKT000.
|
1,108,101,565,661 | arxiv | \section{Introduction}
\IEEEPARstart{I}{n} recent years, we can observe an increase in the global facial recognition market size, mainly due to the growth of importance of the surveillance industry, investment by the government and defense sector and increasing technological advancement across industry verticals~\cite{MarketsAndMarkets}.
Despite significant advances in face recognition technology with the adoption of convolutional neural networks (CNN)~\cite{lightcnn_2018, vggface2_2018}, there are still open issues.
Among them, the use of facial makeup remains a challenge for automatic face recognition systems and even for human evaluations, since it is able to change the original face appearance and cover facial flaws~\cite{Wang2016, 10.5555/3298239.3298377, 8578110, ZHANG2019, emfd_2020}.
The preliminary work that explicitly established the impact of facial makeup on automated biometric systems was proposed a few years ago.
Dantcheva et al.~\cite{ymu_2012, Chen2013} studied the impact of makeup in face verification task and showed that this simple alteration can indeed compromise the accuracy of a biometric system.
After that, Hu et al.~\cite{fam_2013} increased the accuracy in makeup scenario by measuring correlations between face images in a meta space learned using canonical correlation analysis and applying support vector machine.
Since cosmetics are generally applied to facial components as eyes and mouth, an interesting direction of investigation would be part-based representations.
As demonstrated by the cognitive science literature, there is a high probability that part-based processing may exist in human perception of the face~\cite{Sagiv2001}.
Following this inspiration from our visual system, we can see a recent attention to
this representation for face verification~\cite{Angeloni2016, 9211002}, facial gender and pose estimation~\cite{10.1145/3152125, Zavan2019}, facial age estimation~\cite{Yi2015, Angeloni2019} and facial expression~\cite{Jan2018, Happy2020}.
Coupled with this growing attention to facial parts, a set of approaches were proposed to use this idea for makeup invariant face verification.
Guo et al.~\cite{Guo2014} and Chen et al.~\cite{Chen2016} proposed two different patch scheme approaches, where a set of local features are projected onto a subspace to make the match.
Sun et al.~\cite{Sun2017} proposed a weakly supervised method pre-trained on Internet videos and fine-tuned on makeup datasets. In addition, they defined a new loss function with triplet term and two pairwise terms, and combined facial parts (and their mirroring) in the matching step using voting strategy.
Li et al.~\cite{blan2019} proposed a bi-level adversarial network (BLAN), which uses a generative adversarial network to generate non-makeup images from makeup ones preserving the person identity. To further improve the quality of their synthesized images, they used a two-path generator that consider global and local structure.
More recently, Wang et al.~\cite{emfd_2020} proposed a unified multi-branch network which can simultaneously synthesize makeup faces through face morphology network (swapping local regions of different images) and learn cosmetics-robust face representations using attention-based multi-branch learning network. This multi-branch consisted of one holistic and three local branches (two eyes and mouth) that can capture complementary and detailed information.
\begin{figure*}[!htb]
\begin{center}
\includegraphics[width=.845\linewidth]{angel1.pdf}
\end{center}
\caption{Pipeline of the proposed part-based face verification: (a) holistic approach, used as baseline; (b) proposed part-based using 4 facial parts (left periocular, right periocular, nose and mouth) and holistic; (c) proposed part-based using 3 facial thirds (upper, middle and lower) and holistic.}
\label{fig_pipeline}
\end{figure*}
In this letter, we propose a pipeline to improve the performance of face recognition with makeup by exploring part-based representations.
First, we crop specific regions of the face analyzing two strategies: four parts located around fiducial points or splitting the face in facial thirds.
Then, we extract features using publicly available state-of-the-art CNN models.
Finally, we fuse the scores of facial regions with the holistic score. We show that significant improvements can be achieved by including facial parts, even without fine-tuning the CNN models.
The major contributions of this approach are: (i) proposition and evaluation of two strategies for cropping facial parts; (ii) performance comparison on face makeup datasets of CNN features extracted from the entire face (holistic) and fusing it with facial parts; (iii) evaluation of the generalization by exploring part-based representation in a cross dataset protocol; (iv) complete reproducible experimental procedure, allowing the regeneration and extension of the obtained results.
The letter is organized as follows.
Section~\ref{Method} presents the proposed method.
Section~\ref{Experiments} reports the experimental settings and results.
Finally, conclusions are drawn in Section~\ref{Conclusion}.
\section{Method}
\label{Method}
The main goals of this letter are to propose and evaluate the adoption of facial parts in publicly available face models in order to improve the accuracy in the challenging scenario of faces with makeup.
The intuition behind the use of facial parts is that each of them suffers differently the effects of cosmetics and together they can increase the accuracy and generalization of the existing face recognition systems.
The overall pipeline of our method is illustrated in Fig.~\ref{fig_pipeline}.
In Fig.~\ref{fig_pipeline}a, the baseline of this letter is presented, where only the holistic face is used.
In Fig.~\ref{fig_pipeline}b and Fig.~\ref{fig_pipeline}c, two strategies for cropping facial parts using facial landmarks coordinates and combining them with the holistic face are presented.
A face image without makeup and another with makeup are inputs to the pipeline.
The pre-processing step starts by applying a face detector followed by a 2D facial landmarks estimator, both available in DLib~\cite{dlib2009}.
These landmarks are used to align, crop and resize the face region and facial parts.
The proposed facial parts in this letter differ from approaches that divide the face region into arbitrary blocks~\cite{7982752, YANG2020109}, as
they did not coincide or did not have their choice motivated by the fiducial regions.
Our first strategy of facial parts (Fig.~\ref{fig_pipeline}b) is composed of four components: left periocular, which includes the eye and eyebrow, right periocular, nose and mouth.
These parts were chosen due to periocular region presented complementary information to face recognition~\cite{Tiago2015, LSP2020}, and nose and mouth regions were adopted in related works~\cite{Bonnen2013, Angeloni2016, Zavan2019, Angeloni2019, Jan2018}.
We used the landmarks of each part and expand its borders until obtain a square region (expected as input to CNN models), applying a padding if it crosses the image boundaries.
In this scheme, we maintain the aspect ratio and allow overlap between facial regions.
The second strategy to crop facial parts (Fig.~\ref{fig_pipeline}c) is composed of three facial thirds, which were inspired by anthropometry studies~\cite{farkas1994anthropometry, 10.1145/280814.280823}, facial aesthetics~\cite{hashim2017ideals, harrar2018art, EGGERSTEDT2020102643} and facial beauty evaluation~ \cite{milutinovic2014evaluation, LAURENTINI2014184}.
The upper third extends from the hairline to the glabella (point between eyebrows), the middle third from the glabella to the subnasale (base of the nose), and the lower third from the subnasale to the menton (end of chin). Differently from the first approach, in this case the parts do not have any overlap between them and do not maintain the aspect ratio (due to the resize to square dimensions).
After the pre-processing, as illustrated in Fig.~\ref{fig_pipeline}, we directly extract features using the CNN models of each facial part and of the holistic face.
To extract deep features, we chose thirteen CNN models trained for face verification using four popular architectures.
The first is FaceNet~\cite{facenet_2015}, a CNN that provides a unified embedding, which maps each face image into a Euclidean space of 128 dimensions such that the distances in that space correspond to face similarity.
The second is VGGFace~\cite{vggface_2015} that trained a VGG-16 CNN architecture~\cite{Simonyan15} using a very large scale face and from which we extract the deep features (4,096 dimensions) of the fully connected layer `fc7'.
The third architecture was adopted in the VGGFace2 models~\cite{vggface2_2018}.
VGGFace2 is a very large dataset with variations in pose, age, illumination and ethnicity used to train a set of CNN models using ResNet-50 architecture~\cite{Resnet2016}, with and without Squeeze-and-Excitation (SE) blocks~\cite{SENet2018}.
In this letter, we evaluated eight models trained with VGGFace2. Four without SE blocks: ResNet-50, ResNet-50-128D (adding a 128-D embedding layer for feature representation), ResNet-50-256D (adding a 256-D embedding layer for feature representation), ResNet-50-FT (pre-trained on MS-Celeb-1M Dataset~\cite{msceleb2016}); and four with SE blocks: SE-ResNet-50, SE-ResNet-50-128D, SE-ResNet-50-256D, SE-ResNet-50-FT.
The last evaluated architecture was LightCNN~\cite{lightcnn_2018}.
Its innovation was a variation of maxout activation, called max-feature-map (MFM) that allows to obtain a 256-D face representation. In this letter, we explored three Light-CNN models, one called LightCNN-9, and the others based on LightCNN-29 architecture. It is important to mention that these models used gray-scale face images instead of RGB images as inputs, according to their authors to alleviate the influence of illumination discrepancy.
In our pipeline, we use cosine similarity among each pair as score, comparing the features extracted of the same facial part from different face images.
Finally, in order to fuse the scores computed by each facial part with the holistic score, we use the Linear Logistic Regression (LLR) available in the Bob toolbox~\cite{bob2012,bob2017}.
\section{Experiments}
\label{Experiments}
In this section, we describe the datasets, evaluation protocols and experimental results shown as an ablation study.
\begin{figure}[!htb]
\centering
{\includegraphics[width=0.15\linewidth]{angel2.01.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.02.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.03.jpg}}
\vspace{.1cm}
{\includegraphics[width=0.15\linewidth]{angel2.04.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.05.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.06.jpg}}
\vspace{.1cm}
{\includegraphics[width=0.15\linewidth]{angel2.07.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.08.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.09.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.10.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.11.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.12.jpg}}
\vspace{.1cm}
{\includegraphics[width=0.15\linewidth]{angel2.13.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.14.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.15.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.16.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.17.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.18.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.19.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.20.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.21.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.22.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.23.jpg}}
{\includegraphics[width=0.15\linewidth]{angel2.24.jpg}}
\caption{Samples of makeup datasets with three pairs of `no makeup' vs `makeup': EMFD (first row), FAM (second row), M501 (third row) and YMU (fourth row).}
\label{fig:db}
\end{figure}
We used four datasets in our experiments and analyses.
The Extended Makeup Face Dataset (EMFD)~\cite{emfd_2020} is composed of 1,102 face images.
All images were collected from the Internet, where each pair has one makeup and one non-makeup face image of the same individual.
This dataset includes male and female faces, images with different dimensions, some occlusions by glasses, large areas of acne and some variations in head pose.
The FAce Makeup (FAM) Dataset~\cite{fam_2013} contains 519 pairs of face images of celebrities available on the Internet, where each pair is composed of one image with makeup and another without makeup.
Among the 519 subjects, 222 of them are male and 297 are female.
Guo et al.~\cite{Guo2014} assembled a face dataset of 501 pairs of female individuals collected from the Internet, which we called here ``M501''.
It is composed mainly of adult women of Asian or Caucasian descent, and each pair has one makeup face image and another without makeup.
The Youtube Makeup (YMU) Dataset~\cite{ymu_2012, Chen2016} is composed of 151 female faces with images captured from Youtube makeup tutorials.
There are four shots per subject, where two of them were captured before the application of makeup and two after the application of makeup.
Some samples of these four datasets are shown in Fig.~\ref{fig:db}.
In order to verify whether adding features extracted from facial parts improves the face recognition system performance in scenarios with makeup, we adopted the following metrics~\cite{gunther2016face}: (i) Equal Error Rate (EER) is the point where false acceptance rate (FAR) and false rejection rate (FRR) intersect each other, i.e., where both rates are minimal and optimal in the evaluated system; (ii) Half Total Error Rate (HTER) is the average between FAR and FRR at a specific threshold; (iii) mean accuracy is an averaged result reported over $N$ rounds to measure the performance of algorithms.
\begin{table}
\centering
\caption{Results of holistic features vs. fusion with parts and thirds (in terms of EER (\%)). Best values marked in bold.}
\label{table}
\small
\setlength{\tabcolsep}{3pt}
\begin{tabular}{l|rrrr}
\hline
\textbf{Features} & \textbf{EMFD} & \textbf{FAM} & \textbf{M501} & \textbf{YMU} \\
\hline
FaceNet & 7.62 & 8.24 & 5.01 & 3.60 \\
VGGFace & 14.70 & 20.42 & 13.62 & 8.13 \\
VGGFace2 (ResNet-50-128D) & 5.26 & 7.71 & 3.74 & 2.94 \\
VGGFace2 (ResNet-50-256D) & 5.60 & 7.55 & 3.99 & 2.96 \\
VGGFace2 (ResNet-50-FT) & 5.26 & 8.08 & 2.99 & 2.65 \\
VGGFace2 (ResNet-50) & 5.78 & 8.82 & 3.36 & 2.67 \\
VGGFace2 (SE-ResNet-50-128D) & 4.90 & 7.71 & 3.60 & 3.19 \\
VGGFace2 (SE-ResNet-50-256D) & 4.94 & 7.68 & 3.59 & 2.98 \\
VGGFace2 (SE-ResNet-50-FT) & \underline{4.22} & 7.32 & 2.97 & 2.17 \\
VGGFace2 (SE-ResNet-50) & 5.99 & 8.85 & 3.99 & 2.63 \\
LightCNN-9 & 11.94 & 10.61 & 3.99 & 3.95 \\
LightCNN-29 & 9.29 & 7.32 & 2.40 & 2.65 \\
LightCNN-29v2 & 6.13 & \underline{5.78} & \underline{2.00} & \underline{1.48} \\
\hline
Part fusion (same features) & 3.99 & 5.37 & 1.60 & 1.49 \\
Part fusion (best combination) & \textbf{3.63} & \textbf{4.83} & \textbf{1.20} & \textbf{0.99} \\
\hline
Third fusion (same features) & 4.34 & 5.58 & 1.80 & 1.32 \\
Third fusion (best combination) & 3.81 & 5.20 & \textbf{1.20} & \textbf{0.99} \\
\hline
\end{tabular}
\label{tab_RESULTS}
\end{table}
We started our experiments by evaluating the performance of the features in the datasets separately, in terms of EER. Table~\ref{tab_RESULTS} presents the comparative results between the baseline results, i.e., holistic features (13 first rows) and the fusion results with facial parts and thirds (last 4 rows).
LightCNN-29v2 features achieved the lowest EER in the experiments using only holistic face for FAM, M501 and YMU datasets. The exception was EMFD, in which VGGFace2 (SE-ResNet-50-FT) features were the best.
The EER of facial crops individually was higher than 15\% for facial thirds and higher than 25\% for facial parts.
Although these results of facial crops are poor, when we fuse them with holistic we achieved significant improvements.
In order to understand the factors that helped in the improvements, first we extracted from the facial parts the same features that achieved the best holistic results, and fuse their scores with holistic using LLR.
Improvements were obtained in three of the four datasets, both in fusion with four facial parts and in fusion with three facial thirds.
In another experiment, we remove this restriction of to use the same features for all parts, and report the results with the best combination.
A significant improvement was achieved for all datasets, as can be seen in the rows with ``best combination'' reported in Table~\ref{tab_RESULTS}.
In the fusion results, facial parts were better or equal than facial thirds.
To check generalization, we adopted a new protocol where the threshold is defined on a dataset (EER point) and the performance is computed on all datasets using the same threshold in terms of HTER (cross dataset). Table~\ref{tab_cross} shows the results of this cross-dataset protocol. The datasets are shown in the rows, where the threshold and weights of the fusion are defined. The HTER of the four datasets are shown from the second to fourth column, and their average, standard deviation and maximum (worst result in the generalization) in the last three columns.
The features used in the experiments were the same reported as best results in Table~\ref{tab_RESULTS}.
Again, reductions in error rates are observed when fusing with the facial parts and thirds.
However, this time the best results were achieved with the fusion of holistic with facial thirds.
\begin{table}
\centering
\caption{Cross-dataset protocol (in terms of HTER (\%)).}
\label{table}
\small
\renewcommand{\arraystretch}{0.99}
\setlength{\tabcolsep}{3pt}
\begin{tabular}{l|rrrr|rr}
\hline
\textbf{Holistic} & \textbf{EMFD} & \textbf{FAM} & \textbf{M501} & \textbf{YMU} & \textbf{Avg. $\pm$ S.D.} & \textbf{Max.} \\
\hline
\textbf{EMFD} & 4.22 & 7.40 & 3.90 & 3.04 & 4.64 $\pm$ 1.91 & 7.40 \\
\textbf{FAM} & 6.18 & 5.78 & 2.17 & 2.40 & 4.13 $\pm$ 2.14 & 6.18 \\
\textbf{M501} & 6.87 & 5.46 & 2.00 & 1.98 & 4.08 $\pm$ 2.48 & 6.87 \\
\textbf{YMU} & 7.68 & 5.25 & 1.76 & 1.48 & 4.04 $\pm$ 2.97 & 7.68 \\
\hline
\hline
\textbf{Part fusion} & \multirow{2}{*}{\textbf{EMFD}} & \multirow{2}{*}{\textbf{FAM}} & \multirow{2}{*}{\textbf{M501}} & \multirow{2}{*}{\textbf{YMU}} & \multirow{2}{*}{\textbf{Avg. $\pm$ S.D.}} & \multirow{2}{*}{\textbf{Max.}} \\
\textbf{(same feat.)} & & & & & & \\
\hline
\textbf{EMFD} & 3.99 & 7.40 & 2.90 & 2.16 & 4.11 $\pm$ 2.32 & 7.40 \\
\textbf{FAM} & 6.18 & 5.37 & 3.05 & 2.45 & 4.26 $\pm$ 1.79 & 6.18 \\
\textbf{M501} & 6.75 & 5.95 & 1.60 & 1.78 & 4.02 $\pm$ 2.71 & 6.75 \\
\textbf{YMU} & 6.05 & 5.56 & 2.10 & 1.49 & 3.80 $\pm$ 2.34 & 6.11 \\
\hline
\textbf{Part fusion} & \multirow{2}{*}{\textbf{EMFD}} & \multirow{2}{*}{\textbf{FAM}} & \multirow{2}{*}{\textbf{M501}} & \multirow{2}{*}{\textbf{YMU}} & \multirow{2}{*}{\textbf{Avg. $\pm$ S.D.}} & \multirow{2}{*}{\textbf{Max.}} \\
\textbf{(best comb.)} & & & & & & \\
\hline
\textbf{EMFD} & \textbf{3.63} & 6.68 & 2.96 & 2.33 & 3.90 $\pm$ 2.23 & 6.68 \\
\textbf{FAM} & 6.11 & \textbf{4.83} & 2.82 & 2.30 & 4.02 $\pm$ 1.94 & 6.11 \\
\textbf{M501} & 7.44 & 6.60 & \textbf{1.20} & 1.52 & 4.19 $\pm$ 3.03 & 7.44 \\
\textbf{YMU} & 6.44 & 5.92 & 1.78 & \textbf{0.99} & 3.78 $\pm$ 3.01 & 6.44 \\
\hline
\hline
\textbf{Third fusion} & \multirow{2}{*}{\textbf{EMFD}} & \multirow{2}{*}{\textbf{FAM}} & \multirow{2}{*}{\textbf{M501}} & \multirow{2}{*}{\textbf{YMU}} & \multirow{2}{*}{\textbf{Avg. $\pm$ S.D.}} & \multirow{2}{*}{\textbf{Max.}} \\
\textbf{(same feat.)} & & & & & & \\
\hline
\textbf{EMFD} & 4.34 & 6.79 & 3.06 & 2.31 & 4.13 $\pm$ 1.96 & 6.79 \\
\textbf{FAM} & 5.72 & 5.58 & 3.13 & 2.13 & 4.14 $\pm$ 1.79 & 5.72 \\
\textbf{M501} & 6.32 & 6.41 & 1.80 & 1.43 & 3.99 $\pm$ 2.75 & 6.41 \\
\textbf{YMU} & 5.91 & 5.56 & 1.89 & 1.32 & 3.67 $\pm$ 2.40 & 5.91 \\
\hline
\textbf{Third fusion} & \multirow{2}{*}{\textbf{EMFD}} & \multirow{2}{*}{\textbf{FAM}} & \multirow{2}{*}{\textbf{M501}} & \multirow{2}{*}{\textbf{YMU}} & \multirow{2}{*}{\textbf{Avg. $\pm$ S.D.}} & \multirow{2}{*}{\textbf{Max.}} \\
\textbf{(best comb.)} & & & & & & \\
\hline
\textbf{EMFD} & 3.81 & 6.54 & 2.66 & 2.26 & 3.82 $\pm$ 1.93 & 6.54 \\
\textbf{FAM} & 5.64 & 5.20 & 3.10 & 2.09 & 4.01 $\pm$ 1.69 & \textbf{5.64} \\
\textbf{M501} & 6.96 & 5.82 & \textbf{1.20} & 1.37 & 3.84 $\pm$ 2.98 & 6.96 \\
\textbf{YMU} & 6.17 & 5.45 & 1.62 & \textbf{0.99} & \textbf{3.56} $\pm$ 2.63 & 6.17 \\
\hline
\end{tabular}
\label{tab_cross}
\end{table}
After verifying the gains obtained by combining the scores of holistic approaches with that based on parts, we evaluated the performance of the best combinations with results available in the literature.
To be fair, we adopted the same protocol and metrics defined in the original publications of the datasets.
The protocol of YMU dataset~\cite{ymu_2012,Chen2016} evaluates the performance of the verification for images before makeup, after makeup and matching one before with one after. The EER of the proposed approaches compared with the literature is shown in Table~\ref{tab_YMU}. As can be seen, our proposed pipeline achieved the state of the art for YMU dataset.
The protocol for other three datasets (EMFD, FAM and M501) considers only non-makeup vs makeup reported in terms of mean accuracy of 5 folds, using the same number of positive and negative trials in the test set.
The comparison of the proposed results with literature is presented in Table~\ref{tab_other}.
Our pipeline achieved competitive results in all datasets (with improvements when fusing with facial parts) even without any retraining of the CNN models with makeup face trials as the competitors.
\begin{table}
\centering
\caption{Results following the protocol of YMU dataset (in terms of EER (\%) - B vs. B: matching of before-makeup images, A vs. A: matching of after-makeup images, and A vs. B: one of the images is after-makeup, the other is before-makeup).}
\label{table}
\small
\setlength{\tabcolsep}{3pt}
\begin{tabular}{l|rrr}
\hline
\textbf{Method} & \textbf{B vs. B} & \textbf{A vs. A} & \textbf{A vs. B} \\
\hline
COTS-1~\cite{Chen2016} & 3.85 & 7.08 & 12.04 \\
COTS-2~\cite{Chen2016} & 0.69 & 1.33 & 7.69 \\
COTS-3~\cite{Chen2016} & 0.11 & 3.29 & 9.18 \\
LGGP~\cite{Chen2013_LGGP} & 5.36 & 8.01 & 19.70 \\
Patch-based ensemble~\cite{Chen2016} & 0.62 & 1.99 & 7.59 \\
\hline
LightCNN-29v2 (holistic) & 0.08 & 0.01 & 1.48 \\
Part fusion (same features) & \textbf{0.01} & \textbf{0.00} & 1.49 \\
Part fusion (best combination) & \textbf{0.01} & \textbf{0.00} & \textbf{0.99} \\
Third fusion (same features) & 0.05 & \textbf{0.00} & 1.32 \\
Third fusion (best combination) & \textbf{0.01} & 0.01 & \textbf{0.99} \\
\hline
\end{tabular}
\label{tab_YMU}
\end{table}
\begin{table}
\centering
\caption{Results following the protocol of the EMFD, FAM and M501 datasets, in terms of mean accuracy (\%) of 5-folds. Our best results marked in underline.}
\label{table}
\small
\setlength{\tabcolsep}{3pt}
\begin{tabular}{l|ccc}
\hline
\textbf{Method} & \textbf{EMFD} & \textbf{FAM} & \textbf{M501} \\
\hline
Hu et al.~\cite{fam_2013} & - & 59.60 & - \\
Guo et al.~\cite{Guo2014} & - & - & 80.50 \\
Sun et al.~\cite{Sun2017} & - & - & 82.40 \\
Li et al.~\cite{blan2019} & - & 88.10 & 94.80 \\
Wang et al.~\cite{emfd_2020} & \textbf{96.56} & \textbf{90.43} & \textbf{98.12} \\
\hline
Best holistic & 91.10 & 87.47 & 96.21 \\
Part fusion (same features) & 91.29 & 89.59 & 96.41 \\
Part fusion (best combination) & \underline{92.56} & \underline{89.79} & \underline{97.21} \\
Third fusion (same features) & 90.20 & 87.86 & 96.21 \\
Third fusion (best combination) & \underline{92.56} & 89.40 & 96.41 \\
\hline
\end{tabular}
\label{tab_other}
\end{table}
\section{Conclusion}
\label{Conclusion}
In this letter, we proposed two strategies to crop facial parts and evaluate the improvements of these adoption for the face verification with makeup task.
Experiments performed in four publicly available datasets and also in a cross-dataset protocol showed that improvements can be achieved in the evaluation metrics, even without any retraining or fine-tuning of the CNN models.
Since we provided the source code of the proposed pipeline\footnote{The source data will be available online upon this letter acceptance.}, the entire experimental procedure can be reproduced to regenerate and extend the obtained results.
Some viable future research topics include the investigation of other features to represent facial parts, other methods to fuse the scores and the evaluation of facial parts for other challenges in face verification, such face images with occlusion and noise.
\vfill\pagebreak
\balance
\bibliographystyle{ieee}
|
1,108,101,565,662 | arxiv | \section{Introduction}
The 6.4 keV Fe K$\alpha$ emission line has long been known to
be an important diagnostic of the material accreting onto supermassive black holes.
The Compton reflection hump, frequently seen in Seyfert spectra above
$\sim$7~keV and peaking near 20--30 keV (Pounds et al.\ 1990), indicates that Seyferts'
Fe K lines may have an origin in optically thick material, such as the accretion disk.
Observations with {\it ASCA} indicated that many Fe K$\alpha$ lines were broad
(FWHM velocities up to $\sim$0.3$c$) and asymmetrically skewed towards lower
energies, implying an origin in the inner accretion disk; the line profile is
sculpted by gravitational redshifting and relativistic Doppler effects (e.g.,
Tanaka et al.\ 1995, Fabian et al.\ 2002). However, {\it XMM-Newton} and {\it Chandra}
observations have been revealing a more complex picture. A narrow Fe K component
(FWHM velocities $\sim$5000 km s$^{-1}$ or less) appears to be quite
common; these lines' widths suggest emission from distant material, such as the
outer accretion disk, the optical/UV Broad Line Region (BLR) or the molecular torus.
Spectral observations in which the broad and narrow components are deconvolved
are thus a prerequisite for using the Fe K line as a tracer of the geometry of
the emitting gas.
At the same time, there is strong evidence from X-ray and UV grating observations
for the presence of ionized material in the inner regions of a large fraction of
AGN (e.g., Blustin et al.\ 2005; McKernan et al.\ 2007). High-resolution spectroscopy shows the gas is
usually outflowing from the nucleus; typical velocities are $\sim$ a few hundred
km s$^{-1}$. Absorption due to a broad range of ionic species is commonly seen;
and for many sources, there is evidence for several different photo-ionized
absorbing components, as opposed to a single absorber, along the line of sight.
In the Fe K bandpass, some Seyferts show evidence for absorption by H- or He-like
Fe, indicating a zone of highly-ionized absorbing material (e.g., NGC 3783, Reeves
et al.\ 2004).
Cold absorbing gas, with line of sight columns in excess of the Galactic value,
are routinely observed in Seyfert 2 AGN in accordance with unification schemes (Urry,
Padovani 1995), and have also been reported in some Seyfert 1 AGN. Importantly,
variations in column density on timescales from hours to years have been observed
in both Seyfert 1 AGN (e.g., in I Zw 1, Gallo et al.\ 2007;
see also Lamer et al.\ 2003, Puccetti et al.\ 2004) and Seyfert 2
AGN (Risaliti et al.\ 2002; Risaliti et al.\ 2005), suggesting that the
absorbing circumnuclear material is not homogeneous in either Seyfert type, has a
high tranverse velocity and occurs over a range of length scales.
NGC 3516 is a well-studied, nearby (z = 0.008836; Keel 1996) Seyfert 1.5 AGN that
can display strong 2--10 keV flux variability on timescales of hours to years
(e.g., Markowitz, Edelson 2004). Previous X-ray spectroscopic observations of
NGC 3516 e.g., Nandra et al.\ (1997) using {\it ASCA}, have indicated the presence
of a broad Fe K line, and this source is known to also contain complex and ionized
absorption. Numerous UV absorption lines, including N {\sc V}, C {\sc IV} and
Si {\sc IV}, were observed with the {\it International Ultraviolet Explorer}
(Ulrich, Boisson 1983); absorption line strengths vary on timescales as short
as weeks as the absorber responds to variations in the ionizing flux (e.g., Voit et al.\
1987). {\it Hubble Space Telescope} observations have revealed
that this component of absorbing gas (henceforth called the ``UV absorber'')
may consist of several distinct kinematic components (Crenshaw et al.\ 1998).
X-ray spectra of NGC 3516 can exhibit evidence for large columns
($\gtrsim$10$^{22}$ cm$^{-2}$) of absorbing gas (e.g., Kolman et al.\ 1993), and
the X-ray absorbers can display variability on timescales of years (e.g.,
Mathur et al.\ 1997). Using {\it Chandra} gratings data from observations
in April 2001 and November 2001, Turner et al.\ (2005) observed K-shell absorption lines
due to H- like Mg, Si and S, and He-like Si, evidence for a highly-ionized
absorber, likely with column density $\gtrsim$10$^{22}$ cm$^{-2}$, outflowing
at $\sim$1100 km s$^{-1}$. Simultaneous with these {\it Chandra} observations
in 2001 were two {\it XMM-Newton} observations. Turner et al.\ (2005) modeled
the continuum curvature of the two {\it XMM-Newton} EPIC spectra by including
a partial covering, mildly-ionized absorber; the column density was
$\sim$2.5 $\times$ 10$^{23}$ cm$^{-2}$, with a covering fraction of $\sim$50$\%$.
However, the formal requirement for the broad Fe line was reduced, leading to
uncertainty as to whether the broad Fe line really existed in NGC 3516.
Spectral fitting using an instrument with a wide bandpass is thus necessary
to remove such model degeneracies.
In this paper, we report on an observation of NGC 3516 made with the {\it Suzaku}
observatory in October 2005. The combination of the X-ray Imaging Spectrometer
(XIS) CCD and the Hard X-ray Detector (HXD) instruments have yielded a broadband
spectrum covering 0.3 to 76 keV, allowing us to deconvolve the various broadband
emitting and absorbing components. Furthermore, the exceptional response of the
XIS CCD and high signal-to-noise ratio of this observation have allowed us to
study narrow emission lines in great detail. $\S$2 gives a brief overview of the
{\it Suzaku} observatory, and describes the observation and data reduction. $\S$3
describes fits to the time-averaged spectrum. Variability analysis is briefly
discussed in $\S$4. Flux-resolved spectral fits are discussed
in $\S$5. In $\S$6, we describe a search for narrow red- or blue-shifted lines
in the Fe K bandpass.
The results are discussed in $\S$7, and a brief summary is given in $\S$8.
\section{Observations and Data Reduction}
The nucleus of NGC 3516 was observed by {\it Suzaku} from 2005 October 12 at
13:45 UT until October 15 at 09:07 UT. {\it Suzaku} was launched 2005 July 10
into a low-Earth orbit. It has four X-ray telescopes (XRTs; Serlemitsos et
al.\ 2007), each with a spatial resolution of 2$\arcmin$ (HPD). The XRTs focus
X-rays onto four X-ray Imaging Spectrometer (XIS; Koyama et al.\ 2007) CCDs,
which are sensitive to 0.2--12 keV X-rays on a 18$\arcmin$ by 18$\arcmin$
field of view, contain 1024 by 1024 pixel rows each, and feature an energy
resolution of $\sim$150 eV at 6 keV. Three CCDs (XIS0, 2 and 3) are
front-illuminated (FI), the fourth (XIS1) is back-illuminated (BI) and features
an enhanced soft X-ray response. The XRT/XIS combination yields effective areas
per detector of roughly 330 cm$^{2}$ (FI) or 370 cm$^{2}$ (BI) at 1.5 keV,
and 160 cm$^{2}$ (FI) or 110 cm$^{2}$ (BI) at 8 keV. Each XIS is equipped with
two $^{55}$Fe calibration sources which produce fluorescent Mn K$\alpha$ and
K$\beta$ lines and are located at the CCD corners. {\it Suzaku} also features
a non-imaging, collimated Hard X-ray Detector (HXD; Takahashi et al.\ 2007);
its two detectors, PIN and GSO, combine to yield sensitivity from $\sim$10
to $\sim$700 keV. Further details of the {\it Suzaku} observatory are given in
Mitsuda et al.\ (2007).
\subsection{XIS Reduction}
The XIS data used in this paper were version 1.2 of the screened data (Fujimoto
et al.\ 2007) provided by the Suzaku team. The screening is based on the following
criteria: grade 0, 2, 3, 4, and 6 events were used, the {\sc cleansis} script was
used to remove hot or flickering pixels, data collected within 256 s of passage
through the South Atlantic Anomaly (SAA) were discarded, and data were selected
to be 5$\arcdeg$ in elevation above the Earth rim (20$\arcdeg$ above the day-Earth
rim). The XIS-FI CCDs were in 3x3 and 5x5 editmodes, for a net exposure time after
screening of 135.0 (XIS0), 134.8 (XIS2) and 135.2 (XIS3) ks. XIS1 was also in 3x3
and 5x5 editmodes, for a net exposure of 135.4 ks. The XIS was in normal clocking mode.
The source was observed at the nominal center position of the XIS. For each XIS,
we extracted a 3$\arcmin$ radius centered on the source. The background was
extracted using four circles of radius 1.5$\arcmin$, each located $\sim$6$\arcmin$
from the source, but chosen to avoid the z = 2.1 QSO RX J110741.4+723235, located
4.5$\arcmin$ SE of NGC 3516. Spectra were binned to a minimum of 25 counts
bin$^{-1}$ to allow use of the $\chi^2$ statistic.
Response matrices and ancillary response files (ARFs) were generated for each
XIS independently using {\sc xissimrmfgen} and {\sc xissimarfgen} version
2006-10-26 (Ishisaki et al.\ 2007). The ARF generator takes into account the
level of hydrocarbon contamination on the optical blocking filter. We estimate
a carbon column density of 0.8, 1.2, 1.7, and 2.8 $\times$ 10$^{18}$ cm$^{-2}$
for XIS0, 1, 2 and 3, respectively. Finally, we co-added the three XIS-FI spectra
using {\sc mathpha}, and co-added the response files and ARFs using {\sc addrmf}
and {\sc addarf}, respectively.
To examine the accuracy of the XIS RMFs and determine residual line width due
e.g., to imperfect CTI correction, we generated spectra for the $^{55}$Fe
calibration source lines on each XIS using the above response matrices and ARFs.
We fit the calibration source spectra with three Gaussians. Two Gaussians were
for the Mn K$\alpha$ doublet (expected energies 5.899 keV and 5.888 keV), with
energy centroids fixed to be 11 eV apart, and the higher energy line flux set
to twice that of the lower energy one. The third Gaussian was used to model the
K$\beta$ line, expected at 6.490 keV. We found the average of all the calibration
line widths $\sigma$ to be $<$ 4 eV. The Mn K$\alpha$ line energy centroids for
the co-added FI spectrum were consistent with the expected energies to within
1 eV. For XIS1, the Mn K$\alpha$ line energy centroids were 3$\pm$2 eV lower
than expected. Such discrepancies are well within the accuracy ($\sim$0.2\% at
Mn-K$\alpha$) of the energy calibration of XIS. Fitting the calibration source
lines without the response file, we determined the FWHM energy resolution during
the observation to be 145 eV (average of the 4 XISes).
\subsection{HXD Reduction}
We used data from the HXD-PIN only; NGC 3516 was extremely faint in the HXD-GSO
band, and some aspects of the GSO background are still being studied, so we defer
analysis of the GSO data to a later time. The PIN source spectra were extracted
from cleaned version 1.2 (pre-1.2-r1) HXD event files provided by the HXD
instrument team. PIN background count rates are variable and strongly depend on
the time since SAA passage (Kokubun et al.\ 2007), so we selected data according
to the following criteria: at least 500 s since SAA passage, cutoff rigidity (COR) $\geq$ 8 GV,
and day- and night-Earth elevation angles each $\geq$5$\arcdeg$. Instrumental
(non-X-ray) background spectra for the PIN were provided by the HXD Team
(``Background A'' model) generated from a time-dependent model. The model utilized
the count rate of upper discriminators as the measure of cosmic ray flux that
passed through the silicon PIN diode and yielded background spectra based on a
database of non X-ray background observations with the PIN (Fukazawa et al.\ 2007).
The current accuracy of the PIN non-X-ray background (NXB) model
for a 1 day observation is about 5$\%$
(peak-to-peak residuals).
Both the source and background spectra were generated with identical good time
intervals, and the source exposure was corrected for instrument dead time
(a $\sim$5$\%$ effect). This yielded a good time exposure of 105.8 ks.
Data $<$ 12 keV were discarded due to noise contamination near the lower
threshold of the PIN diode. Data above 76 keV were also discarded: the gain
above an internal Bi K$\alpha$ calibration line at 76 keV is not well-defined,
though the photon statistics above this energy were poor anyway for this
observation. Further details of the HXD in-orbit performance are given in
Kokubun et al.\ (2007). To model the contribution to the total background from
the Cosmic X-ray Background (CXB), a spectrum of the form
9.0 $\times$ 10$^{-9}$($E$/3keV)$^{-0.29}$ exp($-E$/40keV) erg cm$^{-2}$
s$^{-1}$ sr$^{-1}$ keV$^{-1}$ (Gruber et al.\ 1999) was used. We note that some
recent works (e.g., Frontera et al.\ 2007) suggest a 10$\%$ normalization increase
compared to Gruber et al.\ (1999). In addition,
spatial fluctuations of order $\sim$5--10$\%$
over $\sim$ a square degree are
known (e.g., Barcons et al.\ 1998).
However, the effect on the net spectrum was
negligible; for instance, the change in Compton reflection component strength
was 1$\%$, typically. To simulate a CXB spectrum using {\sc xspec}, we assumed
a model of the form {\sc powerlaw*highecut}\footnote{The {\sc highecut} model is of the form $I(E) = exp( (E_{\rm c}-E)/E_{\rm f})$ for $E$ $>$ $E_{\rm c}$,
and $I(E) = 1$ for $E$ $<$ $E_{\rm c}$, where $E$ is the photon energy, $E_{\rm c}$ is the cutoff energy, and $E_{\rm f}$ is the e-folding energy.},
with photon index $\Gamma$ = 1.29, cutoff energy of 0.1 keV, and an e-folding energy of 40 keV.
The power-law normalization of 8.8 $\times$ 10$^{-4}$ ph cm$^{-2}$ s$^{-1}$ keV$^{-1}$ (at 1 keV)
was used, appropriate for a source observed in XIS-nominal mode. The 12--76
keV CXB flux was 1.4 $\times$ 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$ (using the
Gruber et al.\ 1999 normalization). The total (X-ray plus particle) background
12--76 keV flux was 4.4 $\times$ 10$^{-10}$ erg cm$^{-2}$ s$^{-1}$.
The spectrum was binned to a minimum of 400 count bin$^{-1}$. We used the
response file ae$\_$hxd$\_$pinxinom$\_$20060814.rsp. The mean 12--76 keV
net source flux and count rate were 1.1 $\times$ 10$^{-10}$ erg cm$^{-2}$ s$^{-1}$
and 0.16 ct s$^{-1}$, respectively. Figure 1 shows the net source, background, and total
(source + background) spectra. The source spectrum is always at least 15$\%$
of the total up to $\sim$40 keV.
\section{Model Fits to the Time-Averaged Spectrum}
We used 0.4--11.5 keV data in the XIS-FI spectrum and 0.3--10.0 keV data in
the XIS-BI spectrum. We ignored 1.72--1.87 keV in the co-added FI spectrum and
1.80--1.87 keV in the BI spectrum due to uncertainties in calibration associated
with the instrumental Si K edge. In all fits, we included a constant to account
for relative instrument normalizations. We left the relative XIS-BI/XIS-FI
normalization free, but best-fit values were always extremely close to 1.00.
The PIN/XIS-FI normalization was kept fixed at 1.13, a value derived using
{\it Suzaku} observations of the Crab (Ishida et al.\ 2006); the uncertainty
on the PIN/XIS-FI normalization is also discussed in $\S$3.1. All errors
on one interesting parameter correspond to $\Delta\chi^2 = 2.71$ (with the
XIS-BI/XIS-FI normalization left free).
All fits were done using {\sc xspec} v.11.3.2.
The abundances of Lodders (2003) were used. A neutral
Galactic column of 2.94 $\times$ 10$^{20}$ cm$^{-2}$ was included (Dickey, Lockman 1990).
\subsection{Preliminary Broadband Fits}
The X-ray continuum emission of Seyfert 1 and 1.5 AGN is usually dominated by a
power-law component thought to originate from Comptonization of soft
seed photons by a hot corona near the central black hole (e.g., Shapiro et al.\
1976; Sunyaev, Titarchuk 1980; Haardt et al.\ 1994). A
simple power-law (henceforth denoted the ``primary power-law'') over
0.3--76 keV yielded a very poor fit, with $\chi^2$/$dof$ (degrees of freedom) =
49708/1449. As shown in Figure 2(a), residuals strongly indicated the need
to include absorption to account for continuum curvature $\lesssim$3--4 keV. The
narrow 6.4 keV Fe K$\alpha$ line was also obvious.
We therefore added an absorbing column to the power-law, assuming systemic redshift
and initially assuming a covering fraction of unity (this component is henceforth denoted the
``primary absorber''). We used {\sc xstar} tables that assumed an underlying continuum
with a photon index of $\Gamma$ = 2.0 to model absorption in this paper.
We also added Gaussians to model Fe K$\alpha$ and K$\beta$ emission lines. The
Fe K$\beta$ energy centroid was kept fixed at 7.06 keV; the normalization was
kept fixed at 0.13 times that of the K$\alpha$ line. With these changes to
the model, $\chi^2$/$dof$ fell to 8312/1444. The best-fit ionization
parameter and column density were log($\xi$) = 2.0$\pm$0.1 erg cm s$^{-1}$ and
7.1$\pm$0.2 $\times$ 10$^{22}$ cm$^{-2}$, respectively ($\xi$ $\equiv$ 4$\pi$$F_{\rm ion}$/$n$;
$F_{\rm ion}$ is the 1--1000 Ryd ionizing continuum flux; $n$ is the density
of the reflecting material). However, as shown in Figure 2(b), this model did
not accurately describe the broadband emission.
To attempt to model the soft emission, we added a second power-law (the
``soft power-law''), with photon index $\Gamma$ tied to the primary power-law;
$\chi^2$/$dof$ fell to 3039/1443.
As shown in Figure 2(c), residuals suggested the presence of soft
X-ray emission lines, e.g., near 0.56 keV, which is likely due to O {\sc VII}.
We added 13 Gaussians to our model, widths $\sigma$ were fixed at 0.5 eV.
Energy centroids for the lower signal-to-noise ratio lines were kept fixed
at lab-frame energies. Table 1 shows the results for the lines in our best-fit
baseline model (see below). Data/model residuals for these emission lines are
shown in Figure 3. Removing lines one at a time from the final fit suggested that it was
significant at $\geq$ 99.0$\%$ confidence in an $F$-test
to include each line in the fit. However we caution that
the resulting $F$-test significance levels
were likely upper limits in a few cases, since
there could be some blending of Gaussian profiles when two lines' energy
centroids are relatively close together.
In addition, changing the order
in which lines were added might also affect
the derived significance of any one line.\footnote{For further discussion regarding justification of
including multiple lines in complex spectral fits,
we refer the reader to Pounds \& Vaughan (2006), who demonstrate
an application of a Bayesian analysis technique.}
We identify these lines as originating in H-like
C, N, O, Ne and Mg, and He-like N, O, Ne and Mg. We also report radiative
recombination continua (RRC) of O {\sc VII}, O {\sc VIII} and Ne {\sc IX} (and
possibly C {\sc VI}, blended with the 0.500 keV N {\sc VII} emission line).
The lines are likely due to photo-ionization.
Most of these lines have been reported previously; we refer the reader to
Turner et al.\ (2003) for results using the {\it XMM-Newton}-RGS. The strongest
line detected in both the RGS spectrum and the {\it Suzaku} spectrum is due to
O {\sc VII}; line intensities of N {\sc VI}, N {\sc VII}/C {\sc VI} RRC blend, O {\sc VII}, and O {\sc VIII}
as measured by {\it Suzaku} were roughly consistent with the RGS measurements.
We also included a line near 0.83 keV for Fe L {\sc XVII}, to
model any contribution from collisionally-ionized gas. Turner et al.\ (2003)
included a Mg {\sc XI} recombination edge component. However, we eschewed it in favor of
a Mg {\sc XII} line at 1.47 keV;
the higher effective area of the XIS compared to that of the RGS at this energy allowed us to
determine that a line gave a better fit than a recombination edge. Including these emission lines in the model
resulted in $\chi^2$/$dof$ falling to 2347/1424. Figure 2(d) shows the
data/model residuals after the soft X-ray lines are modeled.
Residuals in the PIN band, peaking near 20--30 keV, signaled the need to include
a Compton reflection component. We modeled this by adding a {\sc pexrav} component
(Magdziarz, Zdziarski 1995), assuming
solar abundances and an input photon index tied to that of the primary power-law.
We fixed the cut-off energy at 400 keV. This choice was somewhat arbitrary, but
the cutoff energy is not well constrained; in the baseline model below,
we found a lower limit of 120 keV. We initially fixed the inclination
at 30$\arcdeg$, as per NGC 3516's classification as a Seyfert 1.5, and
we initially assumed that the reflector is subject to the same absorption as the primary
power-law. In the best fit-model, $\chi^2$/$dof$ fell to 1895/1423, and the
value of the reflection fraction $R$ (defined as $\Omega$/2$\pi$, where $\Omega$
is the solid angle subtended by the reflector) was 2.8$^{+0.4}_{-0.2}$. The
photon index $\Gamma$ was 2.142$^{+0.020}_{-0.007}$. As shown in Figure 2(e),
the residuals in the PIN band were thus corrected.
The uncertainty on the relative PIN/XIS cross-normalization is about 3\% (Ishida et al.\ 2006); the
subsequent effect on $R$ is smaller than that associated with the uncertainty
of the PIN NXB. We modified the intensity of the PIN NXB by $\pm$2\%, which
is the 1$\sigma$ level of the current reproducibility of the PIN NXB. $R$
changed by $\pm$0.2, which is smaller than the statistical error on $R$.
Errors quoted on $R$ for the remainder of this paper are the statistical
errors only; readers should bear in mind the additional systematic uncertainty
associated with the NXB.
There remained residuals in the 5--6 keV band. We first discuss modeling these
residuals using relativistic diskline components, and later ($\S$3.2) we will discuss
if a partial covering component could provide as good a fit. We added two relativistic diskline
components for Fe K$\alpha$ and Fe K$\beta$ emission, using
a Kerr (maximally rotating) black hole line profile (Laor 1991).
Formally, the K$\beta$ diskline is not required (omission of this component does
not change $\chi^2$/$dof$ significantly), but we include it for completeness.
The normalization of the K$\beta$ diskline was fixed at 0.13 times that of the
K$\alpha$ diskline. The Fe K$\alpha$ line energy was constrained to lie within
rest-frame energies of 6.40 and 6.96 keV; the K$\beta$ line energy was fixed at
7.06 keV. All other parameters were kept tied between the K$\alpha$ and K$\beta$
components. The emissivity index $\beta$ (used when quantifying the radial
emissivity per unit area as a power-law, r$^{-\beta}$) was fixed at 3.0. Initially,
the disk inclination $i$ was fixed at 30$\arcdeg$. The outer radius $R_{\rm out}$ was
kept fixed at 400 $R_{\rm g}$ (1 $R_{\rm g}$ $\equiv$ G$M_{\rm BH}$/$c^2$). The
inner radius $R_{\rm in}$ was left free. With
the best-fit model, $\chi^2$/$dof$ fell to 1643/1420 ($\Delta\chi^2$ = --252), and the data/model
residuals near 5--6 keV were improved considerably; see Figure 2(f).
Refitting with the diskline inclination $i$ as a free parameter
yielded a significant improvement in the fit:
$\chi^2$/$dof$ fell to 1521/1419 ($\Delta\chi^2$ = --122) for $i$ $<$ 23$\arcdeg$,
significant at $>$99.99$\%$ confidence in an $F$-test.
In this and all subsequent best-fit models, we fixed the inclination
of the Compton reflection component to match that of the diskline as
opposed to leaving it fixed at 30$\arcdeg$, though in practice this usually had negligible impact
on the fit.
We refer to this model as the ``1-absorber + Compton reflection + diskline'' model.
Additional model parameters are listed in Table 2.
Visual inspection of the data/model residuals in the Fe K bandpass suggested an
additional dip near 6.9 keV, at the rest-frame energy for Fe {\sc XXVI};
see Figure 4(a). H- and He-like Fe K absorption features might be expected,
given the detection of a high-ionization absorber by Turner et al.\ (2005).
Adding a Gaussian at 6.96 keV, $\chi^2$ fell by 7.9 (for one less $dof$),
significant at 99.4$\%$ in an $F$-test.
The absolute values of the intensity and equivalent width $EW$
relative to the primary power-law were 3$\pm$2 $\times$ 10$^{-6}$ ph cm$^{-2}$ s$^{-1}$ and 9$\pm$5 eV.
Adding a narrow ($\sigma$ = 0.5 eV)
Gaussian with energy centroid fixed at 6.70 keV yielded an improvement in fit of
$\Delta\chi^2$ = --6.3, significant at 98.4$\%$ in an $F$-test.
The absolute values of the intensity and equivalent width $EW$ were
4$\pm$2 $\times$ 10$^{-6}$ ph cm$^{-2}$ s$^{-1}$ and 9$\pm$4 eV, respectively.
The {\it Chandra}-HETGS spectrum yielded narrow absorption features in the 1--3 keV band due to
highly-ionized Mg, Si, and S, but we do not significantly detect any narrow
absorption features at those energies (and we would likely not expect to,
given the XIS resolution). A small dip in the spectrum near 2.3 keV
is close to the expected energy for S {\sc XII}. However, {\sc xstar} models
demonstrate that S {\sc XV} absorption, though not significantly detected here,
is always stronger than S {\sc XII}. This feature is more likely due to
calibration uncertainty associated with an instrumental Au M edge.
To model Fe K absorption features, we added a second absorbing component
(henceforth denoted the ``high-ionization absorber''), again using an {\sc xstar} table.
Based on the results of
Turner et al.\ (2005), we assumed an outflow velocity of 1100 km s$^{-1}$.
$\chi^2$/$dof$ fell to 1485/1417 ($\Delta\chi^2$ = --36). In this model, the high-ionization absorber
had a column 4.0$^{+4.6}_{-3.1}$ $\times$ 10$^{22}$ cm$^{-2}$, similar to the value
used by Turner et al.\ (2005), and log($\xi$) = 3.7$^{+0.3}_{-0.7}$ erg cm s$^{-1}$.
We henceforth refer to this model as our ``2-absorber'' or ``baseline'' model.
Data/model residuals are shown in Figures 2(g) and 4(b); the 6.9 keV
residuals are reduced and there is some slight improvement to the $>$7 keV continuum.
The primary power-law,
with $\Gamma$ = 1.91$^{+0.04}_{-0.05}$, was absorbed by a column
5.5$\pm$0.2 $\times$ 10$^{22}$ cm$^{-2}$ and log($\xi$) = 0.3$\pm$0.1 erg cm s$^{-1}$.
Forcing the ionization parameter to a much lower value (e.g., log($\xi$) $\sim$ --0.5)
resulted in a significantly worse fit, with large residuals at 1.5--3.0 keV; this is
likely because the lower ionization does not account for absorption edges due to
higher ionization species of Si and S. We caution, however, that this result
could be influenced by residual calibration effects in the XIS near 1.8 keV and 2.3 keV
(instrumental Si K and Au M edges, respectively).
Refitting the model with the diskline emissivity as a free parameter
yielded no additional improvement to the fit. We found $\beta$ = 3.2$\pm$0.3;
we leave $\beta$ frozen at 3.0 in subsequent fits.
The values of $EW$ for the broad and narrow lines were 287$^{+49}_{-24}$ and
123$\pm$7 eV, respectively. Figure 4(c) shows the data/model
residuals when the broad lines were removed from the model. When the model was then re-fit,
there were correlated residuals in the Fe K bandpass, as illustrated in Figure 4(d).
In addition, the resulting high
value of $\chi^2$/$dof$ (1651/1421; $\Delta\chi^2$ = +166) compared to the baseline model indicated
that removing the disklines yielded a significantly worse fit ($>$99.999$\%$
in an $F$-test).\footnote{We note that when removing the PIN data, and fitting using only the XIS,
the uncertainty on the broad line flux increases by
20$\%$, and the uncertainty on $\Gamma$ increases by 80$\%$.}
In the best-fit baseline model, the strength of the Compton reflection hump was
$R$ = 1.7$^{+0.4}_{-0.5}$; a contour plot of $R$ as a function of $\Gamma$ is
shown in Figure 5. Also shown in Figure 5 is a contour plot of $R$ as a function
of the PIN/XIS-FI normalization, which had been kept frozen at 1.13; in this plot,
$\Gamma$ was a free parameter. Figure 6 shows an unfolded model spectrum. Other
model parameters are listed in Table 2 (see Table 1 for the soft X-ray emission lines).
Finally, we re-fit the model, assuming that the Compton reflection component was
not affected by the primary absorber. This yielded a goodness of fit nearly
identical to the previous fit, with $\chi^2$/$dof$ = 1479/1417.
All fit parameters were virtually identical to the previous fit; for instance,
$R$ was 1.9$\pm$0.4; we will continue to assume that the Compton reflection component
is absorbed.
We returned to focus on the soft X-ray emission lines, and attempted
to model all emission lines using an {\sc xstar} grid for photo-ionized
line emission; the grid assumed an underlying photon index of 2.0. We
also included an extra narrow Gaussian at 0.84$\pm$0.02 keV, again
likely associated with Fe L {\sc XVII} emission from collisionally-ionized
gas. Assuming a single photo-ionized emission zone with abundances for
N, O, Ne, Mg and Fe left as free parameters yielded $\chi^2$/$dof$ =
1559/1424 for values of the ionization parameter log($\xi$) =
1.3$^{+0.1}_{-0.2}$ erg cm s$^{-1}$ and column density 1.4$^{+0.3}_{-0.5}$
$\times$ 10$^{23}$ cm$^{-2}$. Best-fit abundance values relative to
solar were $Z_{\rm N}$ = 1.2$^{+1.2}_{-0.3}$, $Z_{\rm O}$ = 0.5$^{+0.3}_{-0.1}$,
$Z_{\rm Ne}$ = 1.0$^{+0.4}_{-0.2}$, $Z_{\rm Mg}$ = 2.8$\pm$0.6, and
$Z_{\rm Fe}$ = 0.3$\pm$0.3, i.e., there is a similar indication of
a high value of N/O as inferred by Turner et al.\ (2003) from
the {\it XMM-Newton} RGS spectrum. Adding a second zone of emission from
ionized gas with a significantly different ionization parameter did
not improve the fit. While the fit using the one-zone model is not
poor, for the remainder of this paper, we will continue to use the
multiple Gaussians to model the photo-ionized emission lines, as
that model yielded a lower value of $\chi^2$/$dof$.
\subsection{Additional Partial Covering Components}
We next explored the possibility of an additional, partial covering component
whose presence could potentially be manifested as curvature in the 1--5 keV continuum.
We differentiate such components from the soft power-law, which itself could
indicate ``leaked'' emission as part of a partial covering scenario (see $\S$7.1).
Starting with the baseline model, we added a partial covering component
consisting of a power-law (with photon index tied to that of the primary
power-law) absorbed by an {\sc xstar} component. We first kept the column
density $N_{\rm H, PC}$ and ionization parameter log($\xi_{\rm PC}$) tied
to those of the primary absorber. In the best-fit model (henceforth denoted
Model PC1), the added power-law had a normalization 0.32$\pm$0.03 times that
of the primary power-law. As shown in Table 3, $N_{\rm H}$ and log($\xi$) of
the primary absorber and the diskline parameters were consistent with values
of the baseline model. However, $\chi^2$/$dof$ was 1486/1416, virtually
identical to the baseline model, and so there is no formal requirement to
include the new partial covering component when the column density and
ionization states are tied to those for the primary absorber.
Next, we untied $N_{\rm H, PC}$ and log($\xi_{\rm PC}$) and refit (Model PC2).
The best-fit parameter values are listed in Table 3. Compared to Model PC1,
$\chi^2$ dropped by only 5.2 for 2 additional degrees of freedom, significant
at only $\sim$90$\%$ in an $F$-test. It is thus not highly significant to include
the new partial covering component in this case either.
Next, we addressed whether it was possible for a partial covering component
to mimic the curvature in the $\sim$4--6 keV continuum modeled above as a
relativistically-broadened Fe line. Starting with the baseline model, we
removed the disklines, and added a partial covering component consisting of
a power-law (with photon index tied to that of the primary power-law)
absorbed by gas with a relatively low value of
the ionization parameter (Model PC3). The column density needed to be
$\gtrsim$10$^{22.5}$ cm$^{-2}$ in order to have a rollover near 2--3 keV,
causing an apparent ``peak'' near $\sim$4--6 keV
(we forced it to be greater than 4 $\times$ 10$^{22}$ cm$^{-2}$).
In the best-fit model, the partial coverer had a column density of
7.8$^{+68}_{-3.8}$ $\times$ 10$^{22}$ cm$^{-2}$
(error pegged at lower limit), and
log($\xi_{\rm PC}$) = --0.4$^{+1.2}_{-2.6}$ (error pegged at lower limit).
The new partially-covered power-law had a normalization $<$0.22
times that of the primary power-law. However, $\chi^2$/$dof$ was 1677/1418 ($\Delta\chi^2$ = +192),
a much worse fit compared to the baseline model, and there were still large
data/model residuals in the Fe K bandpass; see Figure 2(h). We conclude that
a partial coverer cannot mimic the observed curvature of the diskline.
\subsection{Relativistic Reflection Fits}
The broad Fe K diskline component is a signature of reflection off a possibly-ionized disk.
{\it Suzaku}'s broad bandpass makes it an ideal instrument for attempting to
model the entire reflection spectrum (broad Fe line plus hard X-ray reflection
continuum plus soft X-ray emission) of Seyferts in a self-consistent manner.
Specifically, we used the ionized reflection models of Ross, Fabian (2005) modified by
relativistic smearing.
We first removed the disklines and Compton reflection components from
the baseline model, and modeled a blurred reflector by convolving
a Ross, Fabian (2005) reflection
model with the kernel associated with a maximally rotating Kerr black hole line profile model
(Laor 1991) using {\sc kdblur}.
The photon index of the illuminating continuum was tied to that of the primary power-law.
Free parameters here were the emissivity index $\beta$, inner disk
radius $R_{\rm in}$ and disk inclination $i$.
We also included the narrow Fe K line at 6.4 keV;
this model assumes that all of the observed Compton reflection is associated with the broadened Fe K line
and none with the narrow Fe K line. The best-fit model had $\chi^2$/$dof$ = 1486/1416, similar to the best overall fit;
the results suggested a fairly neutral or lowly-ionized reflector, with log($\xi$) $<$ 1.8 erg cm s$^{-1}$.
The best-fit value of $R$ was 3.0$\pm$0.8; the photon index $\Gamma$ was 2.06$\pm$0.08.
Parameters associated with blurring were similar to what was
obtained in the baseline model: an inclination of 26$\pm$7$\arcdeg$ and
an inner disk radius $<$ 5.5 $R_{\rm g}$ (the emissivity index was
fixed at 3.0). The absorber parameters were consistent to those obtained for the baseline model.
Next, we tested a model with both a blurred reflector and an unblurred reflector,
to test the notion that both the broad and narrow lines each be associated with contributions
to the observed Compton reflection continuum.
The blurred reflector was again modeled by convolving a Ross, Fabian (2005) reflection model
with with a relativistic diskline profile.
The unblurred reflector consisted of a narrow 6.4 keV line and a Compton reflection hump
modeled using {\sc pexrav}. The fit was very
similar to the previous one and to the baseline model: $\chi^2$/$dof$ = 1485/1415.
The strengths of the blurred and unblurred Compton reflection components
were 1.8$\pm$0.7 and 1.6$\pm$0.5, respectively. All other parameters were
consistent with those of the previous fit (e.g., the $EW$ of the narrow Fe K line
was identical to that measured in the baseline model).
\subsection{Narrow Fe K Line Properties}
In our baseline model, the best-fit energy centroid for the narrow
Fe K$\alpha$ line was 6.398$\pm$0.004 keV, consistent with neutral Fe.
The observed line width $\sigma_{\rm obs}$ was 26$^{+11}_{-13}$ eV.
The intrinsic line width $\sigma_{\rm intr}$ was found by subtracting
in quadrature the $^{55}$Fe calibration line width $\sigma$ of $<$4 eV
from the measured line width. We inferred a 99$\%$ confidence limit for
two interesting parameters of $\sigma_{\rm intr}$ $<$ 45
eV, which corresponds to a FWHM velocity width $<$4900 km s$^{-1}$.
Figure 7 shows a contour plot of line intensity versus FWHM velocity width.
This width is consistent with the results obtained by {\it Chandra}-HETGS for the
two observations in 2001, 1290$^{+1620}_{-1290}$ and 3630$^{+2350}_{-1540}$ km s$^{-1}$
(Yaqoob, Padmanabhan 2004).
Figure 8 shows a contour plot of the broad line intensity versus the
narrow line intensity, illustrating that the two lines are detected
independently at $>$4$\sigma$ confidence. Such a result is a product of
the combination of the narrow response of the XIS (yielding extremely
high signal/noise in the narrow line) and {\it Suzaku}'s broad bandpass.
Similar results have been reported e.g., for the {\it Suzaku} observation of
NGC 2992 (Yaqoob et al.\ 2007).
Finally, we discuss limits to a Fe K$\alpha$ Compton shoulder. We added
a Gaussian emission line at 6.24 keV (rest-frame), with width tied to
that of the K$\alpha$ core. We found an upper limit to the intensity of
7 $\times$ 10$^{-6}$ ph cm$^{-2}$ s$^{-1}$, or 13$\%$ of the K$\alpha$
core intensity. This limit corresponds to an $EW$ of 21 eV.
\section{Timing Analysis}
To compare the variability properties of the primary and soft power-laws,
we extracted light curves, summed over all four XISes, orbitally-binned,
and background-subtracted, for the 0.3--1.0 and 2--10 keV bands.
The 2--10 keV light curve is plotted in Figure 9, along with the 12---76 keV
background-subtracted PIN light curve, binned every three orbits.
The 0.3--1.0 keV light curve was consistent with being constant within the errors
and is not plotted.
We note that the PIN errors shown in Figure 9 are statistical only, and do not take into
account systematic uncertainty associated with subtraction of the NXB component.
For instance, Mizuno et al.\ (2006) noted that the systematic error
is roughly 6$\%$ for the 15--40 keV band over a 5760 s bin.
We calculated the fractional variability amplitude $F_{\rm var}$
(which quantifies the variability in excess of measurement noise) and
its uncertainty following Vaughan et al.\ (2003). For the 2--10 keV band,
$F_{\rm var}$ was $9.2 \pm 0.3 \%$, with a maximum/minimum flux ratio
of roughly 1.4. For the 12--76 keV band, no significant variability in excess of that
due to measurement errors was detected, with an upper limit of 4.4$\%$.
The upper limit on $F_{\rm var}$ for the 0.3--1.0 keV band was 2.5$\%$.
\section{Flux-resolved Spectral Fits}
We performed flux-resolved spectral fits to search for any physical
connection between the soft and primary power-laws. Despite the limited
flux range exhibited during this observation, we attempted to determine,
e.g., if the observed X-ray flux variability could be due to rapid variations
in column density of the primary absorber, if both power-laws vary together,
or if one power-law is constant. We split the time-averaged spectrum into
periods when the 2--10 keV flux was higher and lower than the average 2--10
keV flux of 2.31 $\times$ 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$, as illustrated
in Figure 9. Net exposure times for the high and low-flux spectra for each
XIS (and the PIN) were approximately 62.1 (46.3) and 71.3 (59.5) ks, respectively.
The average 2--10 keV fluxes were 2.47 and 2.16 $\times$ 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$,
respectively.
We applied the best-fitting baseline model from the time-average spectrum to
both spectra. All narrow Gaussian energy centroids and widths, $\beta$,
$i$ and $R_{\rm in}$ for the disklines, and log($\xi$) for the high-ionization
absorber were kept frozen at their time-averaged values. However, all freed
parameters (including $\Gamma$ and $R$) were consistent at the 90$\%$ confidence level. For instance,
we find no strong evidence that the column density of the high-ionization
absorber varies on short timescales. A broadband observation of NGC 3516 spanning a
larger flux range is thus needed to potentially distinguish determine if
the two power-laws vary in concert.
\section{The Search for Red and Blue-shifted Narrow Lines}
We searched for additional narrow absorption or emission features in the
Fe K bandpass, as seen so far in several Seyferts, including NGC 3516.
For instance, Turner et al.\ (2002) found emission lines near 5.57, 6.22,
6.41, 6.53 and 6.9 keV in the November 2001 {\it Chandra}-HETGS and
{\it XMM-Newton} EPIC spectra of NGC 3516. In the April 2001 observation,
Iwasawa et al.\ (2004) claimed the presence of a transient feature with varying
energy and with flux varying on timescales of 25 ks. One interpretation was that
such features were red- and blue-shifted Fe K lines associated
with transient ``hot spot'' emission on the inner accretion disk.
We searched for such features by adding a Gaussian component to the time-averaged
spectrum, with width frozen at 0.5 eV, sliding it over 4--9 keV in energy.
In addition to the aforementioned absorption line at 6.96 keV,
there were only two ``candidate'' feature with $\Delta\chi^2$ $<$ --5.0,
absorption lines near 6.0 and 6.7 keV. However, we performed Monte Carlo
simulations to assess the statistical significance of these features
(see $\S$3.3 of Porquet et al.\ 2004 and $\S$4.3.3 of Markowitz et al.\
2006 for a description of these simulations, also Gallo et al.\ 2005), and
we found that the lines were consistent with photon noise. Furthermore,
each candidate features was evident only in one XIS camera, and thus
was likely not real.
As an aside, we note that a 7.47 keV Ni K$\alpha$ emission line was not detected
in the time-average spectrum; adding a Gaussian at this energy,
we found an upper limit of 3 eV.
We also extracted time-resolved spectral slices by dividing the time-averaged
spectrum into five slices 48 ks in duration, with each slice having an exposure
time near 27 ks per XIS and 21 ks for the PIN. We did not investigate longer time
slices since they might miss short-lived hot-spot emission lines; shorter time
slices would have yielded poorer photon statistics. Applying a sliding narrow
Gaussian over 4--9 keV in each spectral slice revealed only 3 ``candidate''
emission or absorption lines (with $\Delta\chi^2$ $<$ --6). Again, however, the
features were seen in only one or two XIS cameras, and Monte Carlo simulations
showed that not a single candidate feature was inconsistent with photon noise
at greater than 80$\%$ confidence. Analysis of the high-and low-flux spectra similarly
yielded no significant narrow emission or absorption lines (even at 6.70 and 6.96 keV).
Typical upper limits to the equivalent width of an emission line at 5.57 keV
(one of the energies of the transient lines in Turner et al.\ 2002), were
$\lesssim$10--15 eV in the time-averaged spectrum or any of the sub-spectra.
\section{Discussion}
During the late 1990's, NGC 3516 typically displayed a 2--10 keV flux of
$\sim$4--6 $\times$ 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$ (e.g., Markowitz, Edelson 2004).
During the 2001 {\it XMM-Newton}/{\it Chandra} observations, however, the observed
2--10 keV flux was much lower: 1.6--2.3 $\times$ 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$
(Turner et al.\ 2005). Table 4 lists the inferred absorption-corrected 2--10 keV nuclear
fluxes from the {\it XMM-Newton} observations, as well as during the 2005 {\it Suzaku}
observation. In addition, Figure 10 shows the unfolded observed spectra for the {\it Suzaku} XIS and the
2001 {\it XMM-Newton} EPIC-pn data (see Turner et al.\ 2005 for details regarding the
{\it XMM-Newton} data). The {\it Suzaku} observation apparently caught the source in a similar
low level of nuclear flux as the 2001 observations. The observed 0.5--2.0 keV flux
during the {\it Suzaku} observation, however, was $\sim$2--3 times lower than during
the {\it XMM-Newton} observations, indicating that the source was still heavily
obscured, and confirming that the complex absorption in this source cannot be ignored
when fitting the broadband spectrum and modeling diskline components.
\subsection{Power-law Components}
The primary power-law observed in the hard X-rays is likely emission from a hot corona
very close to the supermassive black hole, as seen in all Seyferts.
The nature of the
soft power-law component, however, is not as clear. It could represent nuclear emission
scattered off optically-thin material, e.g., in the optical/UV Narrow Line Region (NLR).
In the baseline model, the normalization of the soft power-law relative to that
of the primary power-law was 4.2$\pm$0.4$\%$. Assuming a covering fraction of unity,
this ratio is equal to the optical depth of the scattering material,
indicating a column density of roughly 5 $\times$ 10$^{22}$ cm$^{-2}$,
consistent with this notion, though the column density is somewhat too high to
likely be associated with the NLR.
It is interesting to note that this column density is similar to that obtained
for the high-ionization absorber, suggesting
the possibility that this absorbing component could be associated with a zone of scattering.
Using {\it Chandra}-ACIS, George
et al.\ (2002) found the extended circumnuclear gas to have a 0.4--2.0 keV flux of
roughly 10$^{-14}$ erg cm$^{-2}$ s$^{-1}$. However, that emission was studied over an
annular extraction region 3$\arcsec$ to 10$\arcsec$ (0.6--1.8 kpc), and so that flux value is likely a
lower limit to the 0.4--2.0 keV flux that Suzaku would observe (given the XRTs' 2$\arcmin$ HPD). In our baseline model,
we found an unabsorbed 0.4--2.0 keV flux of 1 $\times$ 10$^{-12}$ erg cm$^{-2}$ s$^{-1}$, consistent with
the notion that the soft power-law is scattered emission.
In this case, the decrease in observed 0.5--2.0 keV flux from 2001 to 2005 could potentially
be explained by the scattered emission responding to a recent decrease in nuclear continuum flux.
However, this scenario would require the bulk of the scattered emission to lie within at most
a few light years of the black hole, and the nuclear flux would have had to decrease between 2001 and 2005
(when the source was not observed by any major X-ray mission in the 2--10 keV band) then return
to 2001 levels by the {\it Suzaku} observation.
Alternatively, the soft power-law could be unobscured, ``leaked'' nuclear
emission as part of a partial covering scenario. In this case, the primary absorber
would obscure 96$\%$ of the sky as seen from the nuclear continuum source. The lack
of significant variability in the 0.3--1.0 keV band could argue for the soft power-law to
originate in scattered emission, since we might expect to observed variability of the
same amplitude as the 2--10 keV band only if the soft power-law were leaked nuclear
emission. However, this is far from certain, as the 0.3--1.0 keV band had a low count rate
and the presence of the soft emission lines
in the XIS spectrum could contribute to dilution of intrinsic variability in the
soft power-law. A broadband observation spanning a larger observed flux range is
needed to clarify this issue. The soft power-law could of course represent a blend of
scattered emission plus leaked nuclear emission. We therefore conclude that the primary
absorber has a covering fraction between 96--100$\%$.
\subsection{Complex Absorption}
We detect two zones of absorption: in addition to the
primary absorber, which has a covering fraction of
96--100$\%$, there is the high-ionization absorber, which is
assumed here to have a covering fraction
of unity. The high-ionization absorber is potentially the same as that reported by Turner et al.\ (2005);
we find a column density $N_{\rm H}$ of 4.0$^{+4.6}_{-3.1}$ $\times$ 10$^{22}$ cm$^{-2}$,
consistent with the column density of 2 $\times$ 10$^{22}$ cm$^{-2}$ used by Turner et al.\ (2005), although we use a slightly higher
ionization parameter (see Turner et al.\ 2007).
Previous studies of NGC 3516, such as Netzer et al.\ (2002), have discussed in detail
the UV absorber, responsible for H Ly$\alpha$, C {\sc IV} and N {\sc V} absorption
features in {\it Hubble Space Telescope} spectra (Kraemer et al.\ 2002). In the X-ray band,
discrete features associated with Mg {\sc VII--IX} and Si {\sc VII--IX} are expected from
this component, but with the CCD resolution and with calibration-related artifacts
near 1.7--1.8 keV in the XIS, such features are not detected by {\it Suzaku}.
{\it Suzaku} has found the primary absorber of the hard X-ray continuum to be
lowly-ionized (log($\xi$) = 0.3$\pm$0.1 erg cm s$^{-1}$), with a column density
$N_{\rm H}$ of 5.5$\pm$0.2 $\times$ 10$^{22}$ cm$^{-2}$. It is possible that it is
the same absorber that Turner et al.\ (2005) designated as the ``heavy''
partial-covering absorber, though we use a somewhat lower ionization parameter (see Turner et al.\ 2007).
A new observation of NGC 3516 with {\it XMM-Newton} in 2006 October showed that
the source had returned to similar $>$6 keV brightness and similar
obscuration levels ($N_{\rm H}$ $\sim$ 2 $\times$ 10$^{23}$ cm$^{-2}$; covering fraction $\sim$ 45$\%$)
as during the 2001 observations (Turner et al.\ 2007).
Long-term changes in the covering fraction of the heavy absorber could explain the bulk of
the spectral variability changes between the 2001, 2005 and 2006 observations.
In this case, the column density has decreased by a
factor of 4.5, while the covering fraction has increased
from $\sim$40--60$\%$ to 96--100$\%$, from 2001 to 2005,
and subsequently returned to approximately the same levels as in 2001 within the next 12 months.
However, because we have not actually observed the entire eclipse associated with a
specific, discrete blob of absorbing gas traversing the line of sight, it is not
clear whether the covering fractions derived are associated with single, large blobs
partially blocking the line of sight to the X-ray continuum source, or if the absorber
consists of numerous, discrete blobs or has a filamentary or patchy structure.
On the other hand, given the gaps
between the 2001 and 2006 {\it XMM-Newton} and 2005 {\it Suzaku} observations, it is certainly plausible
that these observations could have caught independent, discrete blobs or filaments
with differing column densities and differing physical sizes and/or radial distances
lying on the line of sight.
To estimate the distance $r$ between the central black hole and the absorbing
gas, we can use a definition of the
ionization parameter $\xi$ = $L_{1-1000 {\rm Ryd}}$/($n$$r^2$), where $n$ is the number
density. $L_{1-1000 {\rm Ryd}}$ is the 1--1000 Ryd illuminating continuum luminosity,
and the value of the ionization parameter is taken to be the
value of 2 erg cm s$^{-1}$ measured above. We estimate the maximum possible
distance to the material by assuming that the thickness $\Delta$$r$
must be less than the distance $r$. The column density
$N_{\rm H}$ = $n$$\Delta$$r$, yielding the upper limit
$r$ $<$ $L_{1-1000 {\rm Ryd}}$/($N_{\rm H}$$\xi$). We estimate the 1-1000 Ryd
flux from the baseline model to be 9.9 $\times$ 10$^{-11}$ erg cm$^{-2}$ s$^{-1}$, which
corresponds to $L_{1-1000 {\rm Ryd}} = 1.7 \times 10^{43}$ erg s$^{-1}$ (assuming
$H_{\rm o}$ = 70 km s$^{-1}$ Mpc$^{-1}$ and $\Lambda_{\rm o}$ = 0.73).
$r$ is thus $<$2 $\times$ 10$^{20}$ cm (180 light-years), a very loose upper
limit encompassing both distances associated with the BLR ($\sim$10 light-days;
Peterson et al.\ 2004) and a possible cold molecular torus at a 1 pc radius.
Variability in the absorber properties between 2001--2005 and
2005--2006 thus imply a radial distance of at most a few light years in the case
of clouds traversing the line of sight to the nucleus.
In addition, in the case of a partial covering scenario, it is plausible that
the absorber's size could be of the same order as that of the X-ray continuum
source. The absorbing material could thus be e.g., associated with the
base of an outflow or from dense clouds associated with
magnetohydrodynamic disk turbulence (e.g., Emmering et al.\ 1992).
NGC 3516's transition from an unobscured source to a moderately-obscured
source in a 4 year span presents a challenge to standard Seyfert 1/2 unification
schemes. If the obscuration in NGC 3516 is associated with an equatorial molecular torus
usually invoked in Seyfert 1/2 unification schemes, then it is possible that
during the {\it Suzaku} observation, the inner edge of the torus
could have intersected the line of sight, but given
NGC 3516's classification as a Seyfert 1, this could only occur if the torus opening angle were
extremely small and the torus were not azimuthally symmetric.
Alternatively, variations in column density and/or covering fraction
could be due to fine structure in large-distance (tens of pc), non-equatorial
filaments that traverse the line of sight (e.g., Malkan et al.\ 1998).
\subsection{Fe K Emission Components and Compton Reflection}
We have deconvolved
the broadband emitting components, and determined that 1) the existence of the broad Fe line
is robust in that it was required in all models for an adequate fit, and 2)
a partial covering component could not mimic the curvature associated with
a relativistic broad line.
We note, for instance, that if we remove the diskline components from the baseline model and refit,
not only is the fit worse ($\chi^2$ increases by over 170), but the
value of $R$ becomes $\sim$ 3.2. This value is incompatible with the observed $EW$ of the
narrow line unless the Fe abundance is extremely sub-solar ($\lesssim$0.3; see below).
The best-fit disk inclination was typically
$\lesssim$25$\arcdeg$. The inner radius was typically $\lesssim$5$R_{\rm g}$.
The line energy was seen to be consistent with neutral to mildly-ionized Fe
(up to Fe $\sim$ {\sc XX}; Kallman et al.\ 2004).
The equivalent width with respect the primary continuum was 287$^{+49}_{-24}$ eV,
consistent with the value of 431$^{+193}_{-172}$ eV obtained by Turner et al.\ (2005) for the April 2001
{\it XMM-Newton} observation,
where the spectrum could be fit with a diskline component in addition to the
complex absorbing components.
The line energy of the narrow Fe K$\alpha$ line was also consistent with emission from neutral Fe.
The intensity of the narrow line during the {\it Suzaku} observation
is roughly 40$\%$ higher than that measured during the
2001 {\it XMM-Newton} and {\it Chandra} observations (Turner et al.\ 2002), possibly indicating
that a substantial fraction of the Fe K line photons originate in a region $\lesssim$5 lt.-yr.\ in size.
We measured a FWHM velocity line width for the narrow Fe K$\alpha$ line of $<$ 4900
km s$^{-1}$ (99$\%$ confidence level for two interesting parameters).
This velocity does not rule out an origin in the
BLR; Peterson et al.\ (2004) reported FWHM velocities
for the H$\alpha$ and H$\beta$ lines of 4770$\pm$893 and 3353$\pm$310
km s$^{-1}$, respectively. However, we also
cannot exclude a contribution from an origin in the putative molecular torus; there could
potentially be a very narrow line component with FWHM velocity $\sim$ a few hundred km s$^{-1}$,
but the XIS would be unable to separate it from the relatively broader line component.
It is possible that the same material that absorbs the hard X-rays along the line of sight is
responsible for producing the narrow Fe line.
The material producing the Fe line cannot have a column substantially less than
10$^{\sim 22}$ cm$^{-2}$ or else there would be insufficient optical depth
to produce a prominent Fe K line. The primary absorber, with its column density of
5.5 $\times$ 10$^{22}$ cm$^{-2}$ and low ionization state, is thus a plausible candidate for
the narrow line origin.
As an estimate of the Fe K$\alpha$ equivalent
width expected in this case, we can assume an origin in optically-thin gas
which completely surrounds a single X-ray continuum source and is uniform
in column density, and use the following equation:
\begin{equation}
EW_{\rm calc} = f_{\rm c} \omega f_{\rm K\alpha} A \frac{\int^{\infty}_{E_{\rm K edge}}P(E) \sigma_{\rm ph}(E) N_{\rm H} dE}{P(E_{\rm line})}
\end{equation}
Emission is assumed to be isotropic. Here, $f_{\rm c}$ is the covering
fraction, initially assumed to be 1.0. $\omega$ is the fluorescent yield,
0.34 (Kallman et al.\ 2004). $f_{\rm K\alpha}$ is the fraction of photons
that go into the K$\alpha$ line as opposed to the K$\beta$ line; this is
0.89 for Fe {\sc I}. $A$ is the number abundance relative to hydrogen.
We assumed solar abundances, using Lodders (2003). $P(E)$ is the spectrum
of the illuminating continuum at energy $E$; $E_{\rm line}$ is the
K$\alpha$ emission line energy. $\sigma_{\rm ph}(E)$ is the photo-ionization
cross section assuming absorption by K-shell electrons only
(Veigele 1973\footnote{http://www.pa.uky.edu/$\sim$verner/photo.html}).
For $N_{\rm H}$ = 5.5 $\times$ 10$^{22}$ cm$^{-2}$, $EW_{\rm calc}$ = 29 eV,
substantially lower than the observed $EW$ of 123$\pm$7 eV. We conclude that it is possible for
the primary absorber to contribute to the observed line $EW$, but there is also likely a
contribution from some other (non-continuum absorbing) material lying out of the line of sight, likely
with column densities 10$^{\sim 23}$ cm$^{-2}$ (e.g., Matt et al.\ 2002).
For instance, if the putative cold molecular torus does not intersect the line of sight, it could
contribute to the observed $EW$. The 13$\%$ upper limit to ratio of the
Compton shoulder/ narrow Fe K$\alpha$ core intensity was a 90$\%$ confidence limit only,
and does not exclude at high confidence the possibility of Compton-thick material out of the line of sight.
An additional possibility is that
the material emitting the bulk of the line photons could be responding to
a continuum flux that was higher in the past.
For instance, if the putative molecular torus is located $\sim$ a pc or so from the black hole,
the torus will yield a line $EW$ corresponding to the
continuum flux averaged over the past few years.
This situation is plausible for NGC 3516, as the 2--10 keV flux of NGC 3516
during $\sim$1998--2001 (Markowitz, Edelson 2004)
was a factor of $\sim$1.5--2 times brighter than during 2005.
We now turn our attention to properties of the Compton reflection continuum.
{\it Suzaku} has observed other Seyferts to display reduced levels of variability
in the PIN band compared to the 2--10 keV band, e.g., in MCG--6-30-15 (Miniutti et al.\
2007). This behavior is thought to be caused by the presence of the relatively non-varying
Compton reflection hump, which dilutes the observed $>$10 keV variability of the
power-law component. Gravitational light-bending in the region of strong gravity has been invoked
to explain the relative constancy of the reflection spectrum (Compton hump and Fe K diskline component) despite
large variations in the coronal power-law flux in MCG--6-30-15, for instance (Miniutti et al.\ 2007).
In the case of NGC 3516, the observed fractional variability amplitudes
for the 2--10 and 12--76 keV bands were $F_{\rm var,2-10}$ =
9.2$\pm$0.3$\%$ and $F_{\rm var,12-76}$ $<$ 4.4$\%$, respectively. These measurements allow us to
rule out the possibility that the Compton hump varies in concert with the power-law, since
the variability amplitudes would be consistent in that case.
The primary power-law and Compton hump contribute 44$\%$ and 56$\%$, respectively,
of the total 12--76 keV flux. In the case of a constant Compton hump and variable power-law,
$F_{\rm var,12-76}$ would then be equal to $F_{\rm var,2-10}$ / 2.25, or roughly 4.1$\%$.
This is roughly consistent with the observed upper limit on $F_{\rm var,12-76}$,
suggesting that the reflection component varies less strongly than the
primary power-law over the course of the observation.
To verify this, however, we would need to observe NGC 3516 over a larger
X-ray flux range than in the current {\it Suzaku} observation to potentially observe any
significant variability in the PIN band.
Finally, we discuss the origin of the material that gives rise to the observed
Compton reflection hump. The primary and high-ionization absorbers lack the necessary column
density, and are excluded. We next consider an origin in the same material that yields
either the broad or narrow Fe lines.
George, Fabian (1991) calculated that $R$ = 1 corresponds to an observed Fe K$\alpha$ line $EW$ (relative to
a primary continuum with a photon index of 1.9) of 140 eV for neutral Fe, assuming an inclination
angle of 25$\arcdeg$. However, George, Fabian (1991) used the elemental
abundances of Morrison, McCammon (1983), where the Fe number abundance relative to hydrogen
was $A_{\rm Fe}$ = 3.3 $\times$ 10$^{-5}$. More recent papers have slightly lower values of
$A_{\rm Fe}$, 2.7--3.0 $\times$ 10$^{-5}$ (Lodders 2003; Wilms et al.\ 2000).
The expected equivalent width corresponding to $R$ = 1 is thus 115--125 eV.
In our baseline model, we found a best-fit value of $R$ = 1.7, which corresponds
to an expected line $EW$ (relative to the
primary continuum) of 200--215 eV, a value in between the observed $EW$s of the broad line (287 eV
in the baseline model) and the narrow line (123 eV).
It is thus not clear from this measurement alone whether the total
Compton reflection continuum is associated with the broad line (disk), narrow line (a distant origin), or both.
That is, while is it a possibility that at least some portion
of the Compton reflection component is associated with the broad Fe K component,
we cannot exclude the possibility that the narrow line contributes as well
and that there is reflection off cold, distant material.
For example, in $\S$3.3, we demonstrated that a model wherein
there existed both blurred reflection from an ionized disk plus reflection
from cold, distant material, such as the molecular torus, provided a good fit to the data.
In addition, we demonstrated in this section that
the observed $EW$ of the narrow Fe K line means we cannot rule out a contribution to the narrow Fe line,
and to the reflection continuum as well, from Compton-thick material out of the line of sight.
\section{Summary of Main Results}
We have reported on a 150 ksec observation of NGC 3516 obtained with the {\it Suzaku}
observatory in October 2005. The good exposure times after screening were 135 ks
for each of the XIS cameras and 106 ks for the HXD-PIN.
Our best-fit broadband model included a primary power-law with photon index
$\Gamma$ = 1.904$\pm$0.025 in our baseline model, absorbed by a column of material
with $N_{\rm H}$ = 5.5$\pm$0.2 $\times$ 10$^{22}$ cm$^{-2}$ and with log($\xi$) = 0.3
erg cm s$^{-1}$. We modeled the soft band continuum emission using a power-law component
which could represent
nuclear emission off optically-thin material, unobscured ``leaked'' nuclear
emission, or a blend of both. The hard X-ray absorber could thus be
a partial coverer, with a covering fraction $>$96$\%$, or it could obscure the
X-ray continuum source completely. If this absorber is the
same ``heavy'' absorption component reported by Turner et al.\ (2005) in the 2001
{\it XMM-Newton} observations, then between 2001 and 2005 the column density
of this absorber decreased by a factor of 4.5, while the covering fraction
increased substantially, leading to an observed 0.5--2.0 keV flux a factor of
2--3 lower in 2005 than in 2001. Subsequently, by the 2006 October {\it XMM-Newton} observation,
the covering fraction returned to approximately the same
level observed in 2001.
One possibility for the variations in the properties of the obscuring material between the 2001, 2005, and 2006 observations
could be the presence of discrete clouds or filaments within a few light years of the black hole
traversing the line of sight; the equatorial molecular torus invoked in Seyfert unification schemes
is likely not directly responsible.
We also modeled a highly-ionized absorber with
a column density $N_{\rm H}$ of 4.0$^{+4.6}_{-3.1}$ $\times$ 10$^{22}$ cm$^{-2}$,
ionization parameter log($\xi$) = 3.7$^{+0.3}_{-0.7}$ erg cm s$^{-1}$, assumed
to have a covering fraction of unity.
Our baseline model also included a dozen narrow K-shell emission lines originating in
He-like N, O, Ne and Mg, H-like C, N, O, Ne and Mg and three RRC features,
consistent with an origin in photo-ionized material.
However, we cannot exclude a contribution from collisionally-ionized material, as
suggested by the presence of an Fe L-shell {\sc XVII} line near 0.83 keV.
The broad Fe K$\alpha$ line has been robustly detected:
we can distinguish between the
curvature in the observed continuum due to a partial coverer and that due to
a broad diskline; we conclude that for this observation of NGC 3516, a diskline component
is required and that neither a cold nor ionized partial coverer can mimic the continuum curvature associated with the diskline
component. The broad and narrow lines are decoupled (detected independently) at high significance,
thanks to the narrow response of the XIS and the subsequent high signal/noise ratio in the narrow line.
In our best-fit model, we find the Compton reflection strength to be
the value of $R$ = 1.7$^{+0.4}_{-0.5}$.
The narrow Fe K$\alpha$ line, meanwhile, has a FWHM velocity width of $<$4900 km s$^{-1}$ (99$\%$ confidence
level for two interesting parameters), consistent with an
origin in material with the same velocity as NGC 3516's BLR, though a contribution from
material with lower velocity widths cannot be excluded.
It is possible that the primary hard X-ray absorber may also be responsible for
emitting the narrow Fe K$\alpha$ line, though there may be a
contribution from material lying out of the line of sight, such the putative molecular torus.
\vspace{+0.5cm}
The authors gratefully acknowledge the dedication and hard
work of the {\it Suzaku} hardware teams and operations staff for making this
observation possible and for assistance with data calibration and analysis.
This research has made use of HEASARC online services, supported by NASA/GSFC.
This research has also made use of the NASA/IPAC Extragalactic Database,
operated by JPL/California Institute of Technology, under contract with NASA.
\clearpage
|
1,108,101,565,663 | arxiv | \section{Introduction}
The quest for quantum computer with intrinsic fault tolerance spurs
recent interest in searching for exotic fractional quantum Hall (FQH)
states that support non-Abelian anyons~\cite{nayak08}. While it is
easy to write down a trial wave function with highly nontrivial
statistics, the realization of it in two-dimensional electron gases
(2DEGs) is not simple. Apart from the technical difficulties of sample
preparation and low operating temperature, the lack of effective
control on the interaction between particles is a major concern. In a
realistic 2DEG, electrons interact via long-range Coulomb interaction,
which may be modified due to the presence of an adjacent gate in the
case of graphene or the effect of Landau level (LL) mixing, which
introduces effective three-body interaction (among others). In
particular, recent theoretical and experimental
studies~\cite{bishara09,xia10,wojs10,rezayi11} suggest that the
perturbative modification of the interparticle interaction can have
significant effects on the stability of FQH states. Therefore, the
question of how topological order evolves with interparticle
interaction remains an interesting question with growing experimental
capabilities of controlling microscopic parameters.
Ultracold atomic gases provide an ideal platform for simulating
quantum many-body systems~\cite{bloch}. The realizations of FQH states
in ultracold Fermi gases have been discussed in the presence of, for
example, a rapidly rotating trap~\cite{cooper,fetter} or a
laser-induced geometric gauge field~\cite{lin}. For identical
fermions, $s$-wave interactions vanish due to the Pauli exclusion
principle. Unless in the resonance regime, $p$-wave interactions in a
single-component Fermi gas are typical very small. Nevertheless,
significant interactions can still be introduced by using atoms or
molecules with strong dipole-dipole
interactions~\cite{lu,ye1,ye2}. The FQH effects in a two-dimensional
(2D) dipolar Fermi gas with isotropic dipole-dipole interaction have
been studied in Refs.~\cite{bara,Lewenstein}. The system has been
shown to undergo transitions from an integer quantum Hall (IQH) state to a
$\nu=1/3$ Laughlin state, and to a Wigner-crystal state by increasing
the rotation frequency. However, we are not aware of any work on the
FQH effects in the presence of anisotropic interaction, when the
dipoles are not oriented along the rotation axis.
In the present work, we study the FQH effects in a fast rotating
quasi-2D gas of polarized fermionic dipoles. By tilting the direction
of the dipole moments with respect to the rotation axis, we can tune
the dipole-dipole interaction to be anisotropic on the plane of
motion. Starting from a Laughlin state with isotropic dipolar
interaction, we investigate the ground state properties by varying the
tilt angle of the dipole moments. We find that for small tilt angle
the ground state can be approximately described by a FQH
state. However, as one further increases the tilt angle, the ground
state deviates from the FQH state significantly such that a
crystal-like pattern emerges in the density profile of the gas. The
ground state of the system eventually becomes an IQH state when dipole
moments are aligned in the 2D plane. For a soft confining potential,
the IQH state is noticeably anisotropic. We map out the phase diagram
in the parameter space spanned by the tilt angle and the strength of
the confining potential. The results can be explained by the
competition of the isotropic confining potential and both the
isotropic and anisotropic components of the dipole-dipole interaction.
This paper is organized as follows. In Sec.~\ref{formu}, we present
our model and calculation for relevant matrix elements of the model
Hamiltonian. Section~\ref{iso} briefly covers the FQH states with
isotropic dipolar interaction for later comparisons. In
Sec~\ref{aniso} we investigate the ground state structure in the
presence of anisotropic dipole-dipole interaction in a weak confining
potential. The full phase diagram is presented in Sec.~\ref{resu}. We
conclude our discussions in Sec.~\ref{conc}.
\section{Model}\label{formu}
We consider a system of $N$ spin polarized fermionic dipoles trapped in an axially symmetric potential
\[
U({\mathbf{r}})=\frac{1}{2}\mu(\omega ^{2}x^{2}+\omega
^{2}y^{2}+\omega _{z}^{2}z^{2}),
\]
where $\mu$ is the mass of the particle, $\omega $ and $\omega _{z}$ are the radial and axial trap frequencies, respectively. The trapping potential rotates rapidly around the $z$-axis with an angular frequency $\Omega$ ($<\omega $). We further assume that the dipole moments $d$ of all particles are polarized by an external orienting field which is at an angle $\theta $ about the $z$-axis. Since the $s$-wave collisional interaction vanishes for spin polarized fermions, particles only interact with each other via a dipole-dipole interaction. If the orienting field corotates with the trapping potential, the dipolar interaction becomes time-independent in the rotating frame, i.e.,
$$\mathcal{V}(\mathbf{r})=c_{d}V_{\theta }^{\mathrm{(3D)}}({\mathbf{r}}),$$ where $c_{d}=d^{2}/(4\pi \varepsilon _{0})$ or $\mu _{0}d^{2}/(4\pi )$ for, respectively, electric or magnetic dipoles, with $\varepsilon _{0}$ ($\mu _{0}$) being the vacuum permittivity (permeability). The spatial dependence of ${\cal V}({\mathbf r})$ can be described by
\[
V_{\theta }^{\mathrm{(3D)}}(\mathbf{r})=\frac{1}{r^{5}}\,\left[r^{2}-3(z\cos \theta +x\sin \theta )^{2}\right].
\]
Here, without loss of generality, we have assumed that the dipole moments are polarized in the $x$-$z$ plane of the rotating frame. We can tune the dipolar interaction by introducing a tilt angle $\theta $ such that $V_{\theta }^{\mathrm{(3D)}}({\mathbf{r}})$ is isotropic (anisotropic) on $x$-$y$plane for $\theta =0$ ($\theta \neq 0$). In the rotating frame, the Hamiltonian of the system becomes
\begin{eqnarray}
H_{\mathrm{3D}}=\sum_{i}\left[ \frac{{\mathbf{p}}_{i}^{2}}{2\mu}+U({\mathbf{r}}
_{i})-\Omega L_{i}^{z}\right] +c_{d}\sum_{i<j}V_{\theta }^{\mathrm{(3D)}}({
\mathbf{r}}_{i}-{\mathbf{r}}_{j}),\nonumber\\\label{hami1}
\end{eqnarray}
where $L^{z}=xp_{y}-yp_{x}$ is the $z$ component of the orbital angular momentum.
Under the condition $\omega _{z}\gg \omega $, the system can be regarded as quasi-2D. As a result, the motion of all particles along the $z$-axis is frozen to the ground state of the axial harmonic oscillator, with a wave function $\phi _{z}(z)=\pi^{-1/4}q^{-1/2}e^{-z^{2}/(2q^{2})}$, where $q=\sqrt{\hbar /(\mu\omega _{z})}$.
Integrating out the variable $z$ from Eq. (\ref{hami1}), we obtain the
Hamiltonian for the quasi-2D system as
\begin{eqnarray}
H_{\mathrm{2D}} &=&\sum_{i}\left[ \frac{({\mathbf{p}}_{i}-\mu\omega \hat{%
\mathbf{e}}_{z}\times {\boldsymbol{\rho }}_{i})^{2}}{2\mu}+\hbar (\omega
-\Omega )L_{i}^{z}\right] \nonumber \\
&&+c_{d}\sum_{i<j}V_{\theta }^{\mathrm{(2D)}}({\boldsymbol{\rho }}_{i}-{%
\boldsymbol{\rho }}_{j}). \label{hami2d}
\end{eqnarray}%
where ${\boldsymbol{\rho }}=(x,y)$, $\hat{\mathbf{e}}_{z}$ is the unit
vector along the $z$-axis and
\begin{eqnarray}
V_\theta^{\rm (2D)}({\boldsymbol{\rho}})=\frac{1}{(2\pi q^2)^{1/2}}\int dz e^{-z^2/(2q^2)} V_\theta^{(3D)}({\boldsymbol\rho},z).
\label{vdd2}
\end{eqnarray}
The first term on the righthand side of Eq.~(\ref{hami2d}) represents the single-particle Fock-Darwin Hamiltonian in the symmetric gauge~\cite{Fock,Darwin}, which can be solved exactly to yield eigenenergies~\cite{fetter}
\begin{eqnarray}
\hbar(\omega-\Omega)n_++\hbar(\omega+\Omega)n_-+\hbar\omega,
\end{eqnarray}
known as the Fock-Darwin levels, where the quantum numbers $n_+$ and $n_-$ are two non-negative integers. In the fast rotating limit $\Omega\rightarrow\omega$, the Fock-Darwin levels mimic the LLs with a level spacing $2\hbar\Omega$. Throughout this work, we assume that the interaction energy is much smaller than the LL spacing, such that particles only occupy the highly degenerate lowest Landau level (LLL).
To proceed further, it is convenient to introduce a set of dimensionless units: $\hbar$ for angular momentum, $\ell=\sqrt{\hbar/(2\mu\omega)}$ for length, and $c_{d}/\ell^{3}$ for energy. The wave function of the LLL can then be expressed as
\[
\psi _{m}(\rho,\varphi)=\frac{\rho ^{m}e^{im\varphi }e^{-\rho ^{2}/4}%
}{\sqrt{2\pi 2^{m}m!}}\quad (m\geq 0),
\]
describing a state with an angular momentum $m\hbar$. Within the LLL formalism, the Hamiltonian~(\ref{hami2d}) in the second quantization reads
\begin{equation}
H=\alpha L^{z}+\frac{1}{2}\sum_{m_{1}m_{2}m_{3}m_{4}}V_{1234}(\theta
)f_{m_{1}}^{\dag }f_{m_{2}}^{\dag }f_{m_{4}}f_{m_{3}}, \label{vx1234}
\end{equation}
where $f_{m}^{\dag }$ is the fermion creation operator that creates a particle in state $\psi _{m}$ and $L^{z}=\sum_{m}mf_{m}^{\dag }f_{m}$ the total angular momentum. The dimensionless quantity $\alpha =\hbar (\omega -\Omega )\ell ^{3}/c_{d}$ characterizes the relative strength of confining potential with respect to interaction. In the presence of anisotropy, the interaction matrix elements are
\begin{widetext}
\begin{eqnarray}
V_{1234}(\theta ) &=&\int d{\boldsymbol{\rho }}_{1}d{\boldsymbol{\rho }}%
_{2}\psi _{m_{1}}^{\ast }({\boldsymbol{\rho }}_{1})\psi _{m_{2}}^{\ast }({%
\boldsymbol{\rho }}_{2})V_{\theta }^{\mathrm{(2D)}}({\boldsymbol{\rho }}_{1}-{\boldsymbol{%
\rho }}_{2})\psi _{m_{3}}({\boldsymbol{\rho }}_{1})\psi _{m_{4}}({%
\boldsymbol{\rho }}_{2})\nonumber\\
&=&{\cal A}_{1234}\left[\frac{3\cos^2\theta-1}{2}
\left(\frac{8}{3}{\cal J}_{1234} -4{\cal
K}_{1234}\right)\delta_{m_1+m_2,m_3+m_4}
+\sin^2\theta{\cal K}_{1234}\delta_{m_1+m_2,m_3+m_4\pm2}\right],\label{v1234}
\end{eqnarray}
\end{widetext}
where
\begin{equation}
{\cal A}_{1234}=\frac{1}{2\sqrt{2\pi}q}\frac{i^{|m_3-m_1|-|m_4-m_2|}} {2^{(|m_3-m_1|+|m_4-m_2|)/2}}\sqrt{\frac{[m^<_{13}]![m^<_{24}]!}{[m^>_{13}]![m^>_{24}]!}}, \nonumber
\end{equation}
\begin{eqnarray}
{\cal J}_{1234}&=&\int d\rho \rho^{|m_3-m_1|+|m_4-m_2|+1}e^{-\rho^2}\nonumber\\
&&\!\!\!\!\times L_{m_{13}^<}^{|m_3-m_1|} \!\left(\frac{\rho^2}{2}\right)L_{m_{24}^<}^{|m_4-m_2|} \!\left(\frac{\rho^2}{2}\right),\nonumber
\end{eqnarray}
\begin{eqnarray}
{\cal K}_{1234}&=&q\sqrt{\frac{\pi}{2}}\int d\rho \rho^{|m_3-m_1|+|m_4-m_2|+2}e^{-\rho^2+q^2\rho^2/2}\nonumber\\
&&\!\!\!\!\times L_{m_{13}^<}^{|m_3-m_1|} \!\left(\frac{\rho^2}{2}\right)L_{m_{24}^<}^{|m_4-m_2|} \!\left(\frac{\rho^2}{2}\right){\rm erfc}\!\left(\frac{q\rho}{\sqrt{2}}\right),\nonumber
\end{eqnarray}
where $m_{ij}^{<}= \mathrm{min}(m_{i},m_{j})$ and $m_{ij}^{>}=\mathrm{max} (m_{i},m_{j})$. $L_{m}^{n}(\cdot)$ is the associated Laguerre polynomial and
$\mathrm{erfc}(\cdot)$ is the complementary error function. Clearly, for $\theta =0$ $V_{1234}$ is nonzero only when $m_{1}+m_{2}-m_{3}-m_{4}=0$, indicating that the total angular momentum $L_z$ is conserved in the isotropic case. However, when the interaction becomes anisotropic ($\theta \neq 0$), $V_{1234}$ are nonzero when $m_{1}+m_{2}-m_{3}-m_{4}=0$ or $\pm 2$.
Hamiltonian (\ref{vx1234}) contains three parameters: the total number
of particles $N=\sum_{m}f_{m}^{\dag }f_{m}$, the relative strength of
the confining potential $\alpha$, and the tilt angle $\theta$ of the
dipole moment. In the following sections, we will explore the quantum
states of the system in the parameter space $(N,\theta ,\alpha )$. Our
main focus is on the parameter ranges of $N\leq 10$, $0\leq \theta\leq
\frac{\pi}{2}$, and $0.01\leq\alpha \leq 0.1$. Unless otherwise
stated, the value of $q$ is chosen to be $0.01\ell$.
\section{Quantum Hall states with isotropic dipolar interaction}\label{iso}
Let us first assume that the dipolar interaction is isotropic in the
$x$-$y$ plane, which corresponds to $\theta=0$ in the
Hamiltonian~(\ref{vx1234}). In this case, the total angular momentum
is conserved. Therefore, one may numerically diagonalize the
Hamiltonian~(\ref{vx1234}) in the subspace of a given total angular
momentum $M$ to obtain
\begin{eqnarray}
H\left|\Phi^{(N)}_{M,n}\right\rangle=E^{(N)}_{M,n}\left|\Phi^{(N)}_{M,n}\right\rangle,
\end{eqnarray}
where $E^{(N)}_{M,n}$ and $\left|\Phi^{(N)}_{M,n}\right\rangle$
are eigenenergies and eigenstates, respectively. The index $n$ labels
the state in the subspace of total angular momentum $M$ with
increasing eigenenergy, i.e., $n=0$ for the lowest energy state, $n=1$
for the first excited states, etc. We emphasize that we present a
numerically exact treatment of the dipolar interaction potential for a
quasi-2D system with finite wave-function spread along the
perpendicular direction, while the ideal 2D case has been studied
previously~\cite{Lewenstein}.
\begin{widetext}
\begin{table}[h]
\caption{Magic numbers in an isotropic $N=10$ system with various
$q$s. The magic numbers obtained from the composite fermion theory
are included for comparison.}
\begin{tabular}{l|ccccccccccccccccccccc}
\hline\hline
CF&45&&55&&63&&69&&77&&83&&90&&&97&103&111&117&125&135\\
\hline
$q=0.5$&45&52&&59&&66&69&73&77&80&&85&90&93&&97&103&111&117&125&135\\
\hline
$q=0.1$&45&52&&59&&66&&73&77&80&&85&90&93&95&&103&111&117&125&135\\
\hline
$q=0.01$&45&52&&59&&66&&73&77&80&&85&90&93&95&&103&111&117&125&135\\
\hline\hline
\end{tabular}
\label{table}
\end{table}
\end{widetext}
In Fig.~\ref{speciso}, we plot the eigenenergy versus the total
angular momentum for $N=6$ and $\alpha=0$. As a guide to the eyes, we
have connected the lowest energy state in each total angular momentum
subspace by a piecewise straight line, on which a series of shoulders
are visible. The first state on the each shoulder, where a downward
cusp appears in the spectrum, represents a possible candidate for the
global ground state of the system with increasing $\alpha$. For a
given $\alpha\neq 0$, only one of these states is the global ground
state of the system; the corresponding total angular momentum is so
called a {\em magic number}. The properties of these states have been
studied extensively. Laughlin first noticed that the lowest-energy
state of the $M=3N(N-1)/2$ is closely related to the fractional
quantum Hall effect~\cite{Laughlinc}. To translate these numbers into
asymptotic filling factors in the thermodynamic limit, Girvin and Jach
proposed an explicit expression for these states in the finite
system~\cite{Girvinb},
\begin{eqnarray}
\label{eq:nu}
\nu=\frac{L_0}{M},
\end{eqnarray}
where $L_0=N(N-1)/2$. In the context of quantum dots, Jain and
Kawamura proposed an explanation of the magic numbers using the theory
of composite fermions~\cite{Jaina,Jainb}, which has been further
discussed in later works~\cite{Seki,Maksym,Landman}. We list the
magic numbers of our quasi-2D model for $N=10$ in Table~\ref{table}
for several choices of $q$. For small $q$ up to 0.1, our results are
consistent with the results in 2D rotating Fermi gases with isotropic
dipolar interaction studied earlier by Osterloh {\it et
al}~\cite{Lewenstein}. However, for larger $q$ we observe a few
discrepancies. Interestingly, the magic numbers 69 and 97, not showing
for small $q$, are consistent with the prediction of the composite
fermion theory.
\begin{figure}
\centering
\includegraphics[width=3.4in]{eval_nn6d_ll64d_alph0d.eps}
\caption{(Color online) Energy spectrum of Hamiltonian~(\ref{vx1234})
with $N=6$, $\alpha=0$, and $\theta=0$. The solid (red) line
connects the lowest energy state in each total angular momentum
subspace as a guide to the eyes. The smallest angular momentum state
on the each shoulder of the line may become the global ground state
of the system as $\alpha$ varies. } \label{speciso}
\end{figure}
\section{Quantum Hall states with anisotropic interaction}\label{aniso}
Now we turn to the study of the quantum states in systems with
anisotropic dipolar interaction. Since the total angular momentum
$L^z$ is no longer conserved, one has to numerically diagonalize the
Hamiltonian~(\ref{vx1234}) in much larger Hilbert spaces. In
practice, one may truncate the Hilbert space by introducing a cutoff,
$m_{\rm cut}$, to the angular momentum $m$, such that the
diagonalization procedures are carried out in the space constructed from
$\psi_m$s with $m\leq m_{\rm cut}$. We have chosen a sufficiently large
$m_{\rm cut}$ such that our results presented in this work are
not the choice of $m_{\rm cut}$. In numerical
diagonalization, we take advantages of the fact that dipole-dipole
interaction only couples angular momentum subspaces that differ by an
even angular momentum and diagonalize in the Hilbert space spanned by
odd and even angular momentum states separately.
For a given set of parameters $(N,\alpha,\theta)$, the ground state
wave function of the system is denoted by
$\left|\Psi^{(N)}(\alpha,\theta)\right\rangle$ with the ground state
energy $E^{(N)}(\alpha,\theta)$. Throughout this section, the strength
of the confining potential is fixed at $\alpha=0.01$, such that the
ground state is a $\nu=1/3$ Laughlin state for $\theta=0$. The
dependence of the results on $\alpha$ will be discussed in the next
section. Due to the large size of the Hilbert space involved in the
numerical diagonalization, We study systems of up to $N=8$ fermions.
\subsection{State transitions induced by varying $\theta$}\label{phatran}
\begin{figure}
\centering
\includegraphics[width=3.2in]{lzvsthe.eps}
\caption{(Color online) Mean total angular momentum $\overline M$
versus the tilt angle $\protect\theta $ for $\alpha =0.01$ in
systems with various sizes up to $N = 8$. For sufficiently large
number of particles ($N \geq 5$), plateaus corresponding to distinct
ground states develop on the curves. } \label{angular}
\end{figure}
Even though the total angular momentum $L^z$ is no longer a good
quantum number, its mean value,
\begin{eqnarray}
\overline M=\left\langle\Psi^{(N)}(\alpha,\theta)\right|L^z\left| \Psi^{(N)}(\alpha,\theta)\right\rangle,\nonumber
\end{eqnarray}
can be defined and we will see that it is sufficient to characterize
the ground state of a system. In Fig.~\ref{angular}, we plot the
$\theta$ dependence of $\overline M$ for systems with $N\leq 8$. We
find that $\overline M$ always decreases with $\theta$. As will be
shown, this monotonically decreasing behavior of $\overline M(\theta)$
is because the dipolar interaction becomes less repulsive as $\theta$
is increased, which reduces the size of the
system~\cite{Laughlinb}. Interestingly, for sufficiently large number
of particles ($N\geq 5$), plateaus develop on the $\overline
M(\theta)$ curves. Roughly speaking, for $\theta \lesssim 40^{\circ}$
the mean total angular momentum decreases abruptly from one plateau to
another as the tilt angle is increased, signaling sharp transitions
between ground states with distinct properties at various $\theta$.
For a concrete example, we examine in detail the transitions for the
system of $N=6$. The sudden drops of $\overline M$ are observed as
\begin{eqnarray}
\overline M:\quad 45\; {\stackrel{33.3^\circ}{\longrightarrow}} \;39.0 \;{\stackrel{39.9^\circ}{\longrightarrow}} \;34.89,\label{tran6}
\end{eqnarray}
where the numbers above the two arrows denote the critical tilt
angles. From the analysis presented in Sec.~\ref{iso}, the first and
second plateaus clearly correspond to the fractional quantum Hall
states with filling factors $\nu=1/3$ and $5/13$, respectively, in the
notation define in Eq.~(\ref{eq:nu}) by Girvin and
Jach~\cite{Girvinb}. The third plateau has an mean total angular
momentum of 34.89, which is 0.3\% smaller than $35$, indicating that
it mainly contains the FQH state with filling factor $\nu=3/7$. As one
further increases $\theta$, $\overline M$ decreases smoothly toward an
asymptotic value of $17$ at $\theta=90^\circ$. As shown in
Fig.~\ref{angular}, similar features also appear in the systems with
$N=5$, $7$ and $8$. For $N=3$ and $4$, however, $\overline M$ always
varies smoothly, consistent with the expectation that quantum phase
transitions happen only in thermodynamically large systems; the
absence of the sharp steps for $N<5$ is the normal finite-size
artifact.
\begin{figure}
\centering
\includegraphics[width=3.in]{spectrum.eps}
\caption{(Color online) Low-lying energy spectrum for an $N=6$ system
with $\alpha=0.01$ and a tilt angle of (a) $\theta =0^{\circ}$ and
(b) $33.3^{\circ }$. Distinct colors are used to specify different
energy-level clusters, which can be regarded as chiral excitations
of the corresponding lowest-energy state.} \label{spectrum}
\end{figure}
To gain insight into those transitions, we examine the energy spectrum
of the system. In Fig.~\ref{spectrum}, we plot the low-lying energy
levels versus the mean total angular momentum for $N=6$ for the
isotropic and the anisotropic cases. In the isotropic case
($\theta=0$), the mean total angular momentum $\overline{M}$ is simply
the total angular momentum $L^Z$, which is a good quantum number. In
both cases, energy levels group into clusters which are plotted with
different colors. For convenience, we shall refer to each energy
cluster using the angular momentum of the lowest energy level in the
cluster. Even though the boundary between two adjacent clusters are
not well defined for high-energy states, the lowest ones are clearly
well separated. In Fig.~\ref{spectrum}(a), we choose the range in
which three energy-level clusters are visible. Among them, the lowest
energy state in cluster-$45$ represents the ground state of the system
for $\theta=0$. This is the 6-particle Laughlin state as discussed in
Sec.~\ref{iso}. Apart from its mean total angular momentum, we can
also observe the feature of a Laughlin state also in the low-energy
spectrum. First of all, the low-lying excitations are chiral,
appearing only on the side of $\overline{M} > 45$. In particular, at
roughly $\overline{M} = 46$ and $47$, we find one and two states,
respectively. They can be interpreted as the edge excitations of the
ground state droplet, with their wave functions approximated by the
ground state wave function multiplied by a symmetric polynomial of the
corresponding degree. Near $\overline{M} = 48$, we expect three
low-lying states but only find two; however, there is another level
well above (near 0.7), presumably due to the influence of the cutoff
in the momentum space. The series of numbers of the low-lying states
is consistent with the chiral Luttinger liquid theory and signifies
the topological order of the corresponding ground state~\cite{7}.
As one increases the tilt angle $\theta$, all energy-level clusters
move downward to lower energies because the dipolar interaction
becomes less repulsive. Nevertheless, the counting of the low-lying
excitations remains robust, suggesting the topological order is not
destroyed by small anisotropy, as shown in Fig.~\ref{spectrum}(b). In
addition, the clusters with lower angular momentum move faster than
those with higher angular momentum, hence, for example, the lowest
energy state in cluster-$39$ becomes the ground state of the system at
$\theta=33.3^\circ$. As one further increases $\theta$, the lowest
energy state in cluster-35 will become the ground state of the
system. They have different low-energy excitation structure from that
of the $\overline{M} = 45$ ground state. In the case of the
cluster-$39$, in particular, there are two chiral excitations at
$\Delta \overline{M} \approx 1$, clearly different from the Laughlin
case.
\begin{figure}
\centering
\includegraphics[width=3.2in]{fidelity.eps}
\caption{(Color online) Fidelity ${\cal F}_{\delta\theta}$, as defined
in Eq.~(\ref{eq:fidelity}), as a function of $\theta$ for $N=6$ and
$\alpha=0.01$. The inset magnifies the fidelity in the vicinity of
the small dip around $\theta = 49.3^{\circ}$.} \label{fidelity}
\end{figure}
The phase transitions induce by tuning anisotropy can be further
confirmed by calculating the fidelity of the ground state wave
function~\cite{quan,gusj},
\begin{equation}
\label{eq:fidelity}
{\cal F}_{\delta\theta}=\left|\left\langle\Psi^{(N)}(\alpha,\theta)
\right.\left|\Psi^{(N)}(\alpha,\theta+\delta\theta)\right\rangle\right|,
\end{equation}
where $\delta\theta$ is a small quantity. The fidelity $\cal F$
measures the similarity between two adjacent states in the parameter
space~\cite{gusj}. In the bulk of a single quantum phase, two states
close in the parameter space have wave functions that are only
perturbatively different, hence ${\cal F}_{\delta\theta}$ is close to
unity for sufficiently small $\delta\theta$. Near the phase boundary,
two states that are close in the parameter space can have very
different structure in their wave functions, therefore the fidelity
can drop sharply in the quantum critical region, signaling a quantum
phase transition in the thermodynamic limit. In Fig.~\ref{fidelity},
we plot the $\theta$ dependence of ${\cal F}_{\delta\theta}$ for a
systems with $N=6$ particles. Two transition points can be clearly
identified and the critical $\theta$ values are consistent with those
obtained in Fig.~\ref{angular}. A closer look at the ${\cal
F}_{\delta\theta}(\theta)$ (inset of Fig.~\ref{fidelity}) further
reveals that there exists a third dip at $\theta\simeq
49.3^\circ$. The comparison of the fidelity ${\cal F}_{\delta\theta}$
for two different $\delta\theta$s indicates that the similarity of the
ground states decreases with the increasing distance between the
parameter $\delta\theta$ near the dip.
\subsection{Structure of the ground state wave function with anisotropic interaction}
To reveal the structure of the ground state wave function
$\left|\Psi^{(N)}(\alpha,\theta)\right\rangle$, let us calculate the
overlap integral
\begin{eqnarray}
\label{eq:overlap}
{\cal
O}_{M,n}^{(N)}(\alpha,\theta)=\left|\left\langle\Phi^{(N)}_{M,n}\right|
\left.\Psi^{(N)}(\alpha,\theta)\right\rangle\right|
\end{eqnarray}
between the isotropic and anisotropic ground states. Again, we present
the data of the system with $N=6$ particles. Figure~\ref{overlap}
shows the dependence of ${\cal O}_{M,0}^{(N)}$ on anisotropy for $M =
45$, 39, 35, and $15$. As can be seen, the $\theta$-axis is roughly
divided into four regions, the boundaries of which coincide with the
three dips of the fidelity in Fig~\ref{fidelity}.
\begin{figure}
\centering
\includegraphics[width=3.in]{overlap.eps}
\caption{(Color online) Overlap integral ${\cal O}^{(6)}_{M,0}$
[Eq.~(\ref{eq:overlap})] versus $\theta$ for $M=45$ ($\nu = 1/3$
Laughlin state), 39, 35, and 15 (IQH state) in the $N = 6$ system
with $\alpha = 0.01$. } \label{overlap}
\end{figure}
In the first region $\theta<33.2^\circ$, the overlap ${\cal
O}_{45,0}^{(6)}$ is greater than $0.974$, which indicates that the
dominate contribution to the ground state wave function comes from the
Laughlin state $\left|\Phi^{(6)}_{45,0}\right\rangle$. Nevertheless,
close to the right boundary of this region, several states in $M=47$
and $43$ manifolds are mixed into the ground state wave function, such
that ${\cal O}^{(6)}_{45,0}$ drops notably. The ground state wave
function in region $33.3^\circ<\theta<39.6^\circ$ mainly contains the
states from $M=43, 41, 39$, and $37$ manifolds. Particularly, the
state $\left|\Phi^{(6)}_{39,0}\right\rangle$ provides the largest
contribution to the ground state with $0.858 \leq {\cal
O}^{(6)}_{39,0} \leq 0.96$.
In the region $39.8^\circ<\theta<49.3^\circ$, the situation is much
more complicated compared to those in the first two regions. At the
left boundary, ${\cal O}_{35,0}^{(6)}$ is as high as $0.814$, but it
quickly drops to close to zero as the right boundary is approached. In
fact, many states from $M=21$ to $37$ in the isotropic case contribute
collectively to the ground state wave function. As a result,
complicated structures are developed in the density profile of the
system. In Fig.~\ref{crys1}, we present four typical patterns in the
density profiles
\begin{eqnarray}
\varrho(x,y)&=&\sum_{mm'}\left\langle\Psi^{(N)}(\alpha,\theta)\right|f_m^\dag f_{m'} \left|\Psi^{(N)}(\alpha,\theta)\right\rangle\nonumber\\
&&\quad\times\psi_m^*(x,y)\psi_{m'}(x,y)
\end{eqnarray}
of the system in this region. For $\theta=40^\circ$
[Fig.~\ref{crys1}(a)], besides the commonly seen ring-shaped density,
a vertical ridge appears along the $y$ axis. Five local density maxima
can be identified in the figure: 4 of them on the ring and the other
at the center of the trap. Figure~\ref{crys1}(b) shows the result for
$\theta=43.2^\circ$, on which 6 local density maxima appear on the
ring structure. Compared to the case at $\theta=40^\circ$, the density
profile is clearly stretched along the $x$ axis, which also represents
a generic trend for the density profile as $\theta$ is increased. The
reason behind this is because the interaction energy is lowered by
stretching the gas along the direction of dipole moment. As $\theta$
is further increased to $45.8^\circ$ [Fig.~\ref{crys1}(c)], the
structure with 6 local density maxima becomes more prominent, such
that each of them almost becomes an isolated island, as in a crystal
structure. In Fig.~\ref{crys1}(d), the 6 density islands start to
merge. We remark that these crystal-like structures appear as a result
of the interference between the states
$\left|\Phi_{M,n}^{(N)}\right\rangle$; the system is struggling to
maintain a balance between a large set of competing states. We note
that our calculations are based on a finite size system, this region
of competition may shrink in the thermodynamic limit, since more
plateaus are developing as system size increases as shown in
Fig.~\ref{angular}.
\begin{figure}
\centering
\includegraphics[width=3.2in]{crystal1.eps}
\caption{(Color online) Density profiles $\varrho(x,y)$ for various
$\theta$s in the $N=6$ system with $\alpha=0.01$. The brightest
points indicate the local maxima in the density of the
system. } \label{crys1}
\end{figure}
\begin{table}[h]
\caption{Largest 5 overlaps of the anisotropic state
$\left|\Psi^{(6)}(\alpha,90^\circ)\right\rangle$ with the
corresponding eigenstates in the isotropic case ($\theta = 0$). The
pair of ($M$, $n$) labels the $n$th lowest eigenstate in the total
angular momentum $M$ subspace. The system has $n = 6$ particles and
$\alpha = 0.01$. }
\begin{tabular}{c c c c c c}
\hline\hline
$(M,n)$\hspace{0.2cm}&\hspace{0.2cm}$(15,0)$\hspace{0.2cm}&\hspace{0.2cm}$(17,0)$\hspace{0.2cm}&\hspace{0.2cm}$(19,2)$\hspace{0.2cm}&\hspace{0.2cm}$(21,7)$\hspace{0.2cm}&\hspace{0.2cm}$(23,14)$\\
\hline
${\cal O}^{(N)}_{M,n}$&0.658&0.608&0.387&0.196&0.084\\
\hline\hline
\end{tabular}
\label{table2}
\end{table}
In the last region $49.3^\circ<\theta\leq 90^\circ$, the most
important contributions to the ground state wave function are provided
by the states $\left|\Phi_{15,0}^{(6)}\right\rangle$,
$\left|\Phi^{(6)}_{17,0}\right\rangle$,
$\left|\Phi^{(6)}_{19,2}\right\rangle$, and
$\left|\Phi^{(6)}_{21,7}\right\rangle$. The probability of finding the
system in either of the above 4 states is larger than $0.92$
throughout this region. To understand the properties of the ground
state in this region, we first consider the specific state at
$\theta=90^\circ$. Table~\ref{table2} lists the largest values of
${\cal O}_{M,n}^{(6)}(0.01,90^\circ)$. Clearly, the main components of
$\left|\Psi^{(6)}(0.01,90^{\circ})\right\rangle$ are the $\nu=1$
IQH state $\left|\Phi_{15,0}^{(6)}\right\rangle$ and
its edge states, which suggests that, at $\theta=90^\circ$, the ground
state is an IQH state. To confirm this, we plot the
density distribution $\varrho(x,y)$ of this state in
Fig. \ref{theta90}(a). As can be seen, the surface density for the
elliptical plateau is exactly $1/2\pi \ell ^{2}$, which is identical
to that of a $\nu=1$ IQH state. Since the filling
factor can be defined as $\nu=2\pi\ell^2n_f$ with $n_f$ being the
fermionic surface density~\cite{Lewenstein}, the filling factor for
the $\theta=90^\circ$ state is thus $1$. The homogeneous elliptical
plateau of the density profile implies that our conclusion can be
generalized to the thermodynamic limit. We plot the energy spectrum of
the system in Fig.~\ref{theta90}(b). It is well-known that the
topological properties of the $\nu=1$ IQH state is
labeled by the number of edge states, i.e., $1,1,2,3,5,\ldots$ for
$\Delta L^z=0,1,2,3,4,\ldots$, which is exactly the case shown in
Fig.~\ref{theta90}(b), although in this case we have to replace
$\Delta L^z$ by $\Delta \overline{M}$. All the above evidences mount
to the fact that the $\theta=90^\circ$ state is a $\nu=1$ integer
quantum Hall state. Since there is no phase transition observed in
this region from various criteria, we conclude that the ground state
for $\theta>49.3^\circ$ can be characterized as an anisotropic integer
quantum Hall state.
\begin{figure}
\centering
\includegraphics[width=3.2in]{theta90.eps}
\caption{(Color online) Characterization of the anisotropic IQH state
at $\theta=90^\circ$ for $N=6$ and $\alpha=0.01$. (a) The density
profiles along the $x$- and $y$-axes highlight the anisotropic
nature of the state. Near the center, the LLL is completely
filled. The inset shows the ellipsoidal density profile on the
$x$-$y$ plane. (b) The low-lying energy levels can be counted as
$1$, 1, 2, 3, 5, $\ldots$ for $\Delta \overline{M} = 0$, 1, 2, 3, 4,
$\ldots$, respectively, which is consistent with the edge theory of
an IQH state. } \label{theta90}
\end{figure}
\subsection{Understanding the anisotropic dipolar interaction}
\label{iso_aniso}
To understand how anisotropy in the dipolar interaction leads to the
emergence of the anisotropic IQH state, we decompose
the dipolar interaction into isotropic and anisotropic (on $x$-$y$
plane) components as
\begin{eqnarray}
V_\theta^{(3D)}&=&V_{\theta,\rm iso}^{(3D)}+V_{\theta,\rm{ani}}^{(3D)}\nonumber\\
&=&\eta_{\rm iso}(\theta)\frac{r^2-3z^2}{r^5}+\eta_{\rm ani}(\theta)\frac{
y^{2}-x^{2}}{r^{5}},\label{decomp}
\end{eqnarray}
where $\eta_{\rm iso}(\theta)=(3\cos^2\theta-1)/2$ and $\eta_{\rm
ani}(\theta)=3\sin^2\theta/2$ represent the strengths of the
isotropic and anisotropic components, respectively. One should note
that, to obtain Eq.~(\ref{decomp}), we have neglected the linear terms
in $z$ as they vanish after we integrate out the variable $z$ to
obtain the 2D interaction potential. The properties of the system can
be seen as determined by the competition between $V_{\theta,\rm
iso}^{(3D)}$ and $V_{\theta,\rm{ani}}^{(3D)}$. Since $\eta_{\rm
iso}$ ($\eta_{\rm ani}$) is a decreasing (increasing) function of
the tilt angle, varying $\theta$ will change the relative strength of
the isotropic and anisotropic components, which can give rise to
different quantum phases.
\begin{figure}[tbp]
\centering
\includegraphics[width=3.2in]{angular_real_fake_nn6d_ll18d.eps}
\caption{(Color online) Mean total angular momentum as a function of
$\theta$ for $N=6$ and $\alpha=0.01$ in systems of various
interactions. The solid, dashed, and dash-dotted lines correspond
to, respectively, the real system with dipolar interaction, FS-I
with the isotropic component only, and FS-II with the anisotropic
component only. The vertical dotted line indicates the position of
angle $\theta_c$, where the isotropic component becomes attractive.}
\label{fake}
\end{figure}
After introducing the decomposition Eq.~(\ref{decomp}), we explore the
contributions of the isotropic and anisotropic components of the
dipolar interaction separately. To this end, we consider two
fictitious systems, FS-I and FS-II, in which the full dipolar
interaction is replaced, respectively, by $V_{\theta,\rm iso}^{(3D)}$
and $V_{\theta,\rm ani}^{(3D)}$. The corresponding Hamiltonians take
the same form as Eq.~(\ref{vx1234}) except for that the interaction
matrix elements are replaced by those calculated using $V_{\theta,\rm
iso}^{(3D)}$ or $V_{\theta,\rm ani}^{(3D)}$.
In FS-I, the strength of the isotropic interaction, $\eta_{\rm
iso}(\theta)$ decreases with $\theta$. Therefore, increasing the
tilt angle is effectively equivalent to increasing the strength of the
confining potential $\alpha$ for the full Hamiltonian with
$\theta=0$. We plot, in Fig.~\ref{fake}, the $\theta$ dependence of
the ground state angular momentum for system-I with $N=6$ and
$\alpha=0.01$. As $\theta$ is varied, FS-I experiences the same
transitions as those studied in Sec.~\ref{iso}. For the first two
transitions, the critical values of $\theta$ roughly agree with those
obtained using full dipolar interaction potential. In particular,
$V_{\theta,{\rm iso}}^{(3D)}$ vanishes at angle $\theta_c=54.74^\circ$
and becomes attractive in the $x$-$y$ plane for
$\theta>\theta_c$. Consequently, the ground state becomes a $\nu=1$
IQH state for $\theta\geq\theta_c$. The reason that the transition to
the IQH state occurs at a tilt angle smaller than $\theta_c$ is due to
the finite $\alpha$ used in Fig.~\ref{fake}.
In FS-II, we note that the system must be in the $\nu=1$ integer
quantum state at $\theta=0$ where $V_{\theta,\rm ani}^{(3D)}$
vanishes. As one increases $\theta$, the mean total angular momentum
$\overline M$ increases smoothly from $15$ to roughly $21$
(Fig.~\ref{fake}), which suggests that the system always stays in the
IQH state independent of the tilt angle, although the quantum Hall
droplet is stretched gradually along the $x$ axis. Further study on
the ground state of FS-II can be carried out as those have been done
in the previous subsection.
\begin{figure}[tbp]
\centering
\includegraphics[width=3.2in]{phase.eps}
\caption{(Color online) Mean total angular momentum $\overline
M(\theta,\alpha)$ of the ground state for a system with $N=6$
particles. The lower left corner is the $\nu = 1/3$ Laughlin phase,
while the large (blue) region containing the upper right corner is
the IQH phase, which crosses over to an anisotropic IQH phase at the
lower right corner. }
\label{phase}
\end{figure}
From the above analysis, it becomes clear that the series of ground
state transitions induced by varying the tilt angle are mainly caused
by the isotropic component of the dipolar interaction. The anisotropic
component, on the other hand, changes the fine details of the ground
state, as it mixes in states with different total angular momentum to
the otherwise isotropic ground state, as is clearly exemplified in the
anisotropic integer quantum Hall state.
\section{Global phase diagram}\label{resu}
In Secs.~\ref{iso} and \ref{aniso}, we have shown the ground state
transitions induced by varying either the confining strength $\alpha$
or the tilt angle $\theta$. We present the complete phase diagram in
Fig.~\ref{phase}, in which we plot the mean total angular momentum
$\overline M$ as a function of $\theta$ and $\alpha$ for $N=6$. The
phase diagram is separated into regions with different value of
$\overline M$, which are well defined along the $\alpha$-axis, where
the interaction is isotropic, as discussed in Sec.~\ref{aniso}.
Within our numerical capabilities, we find the basic structure of
Fig.~\ref{phase} remains unchanged as system size $N$ varies.
We find that the Laughlin state with $\nu = 1/3$ is robust for weak
confinement and not too large tilt angle ($< 30^{\circ}$), which
assures that unintentionally introduced anisotropy in interparticle
interaction is not important. On the other hand, FQH cannot survive in
the large anisotropy of the dipolar interaction, when the isotropic
component of the interaction becomes soft, as analyzed in
Sec.~\ref{iso_aniso}.
When the confining strength becomes stronger, the mean total angular
momentum $\overline M$ becomes smaller, indicating the system of
particles becomes denser and denser. The evolution of $\overline M$ is
not smooth, but goes through a series of magic numbers, which can be
interpreted by the corresponding filling factors. In the large
confinement limit, the system develops into a maximum density droplet
with $\nu = 1$. The state crosses over to an anisotropic maximum
density droplet for large tile angles (at small confinement since the
isotropic component of the interaction changes from repulsive to
attractive interaction), as revealed in Fig.~\ref{theta90}, which
reflects the competition of the isotropic confining potential and both
the isotropic and anisotropic components of the interaction.
\section{Conclusion}\label{conc}
To summarize, we investigated the quantum Hall effects in the LLL of a
fast rotating quasi-2D Fermi gas with anisotropic dipolar interaction
through exact numerical diagonalization. With the tilt angle of the
dipole moment $\theta$, we introduced a new control knob to explore
the FQH effect in cold atomic systems. We studied in details the phase
diagram of a finite-size system and concluded that phase
transition is expected as the tilt angle $\theta$ varies in the
thermodynamic limit. When the tilt angle is small, the ground state of
the system can be described by a FQH state, whose filling fraction
depends on the strength of the confinement and hence the average
density. At large tilt angle, the system eventually becomes an
anisotropic $\nu=1$ IQH state. However, for
intermediate $\theta$ value, we find that crystal-like order develops
in the density profile of the system, suggesting the competition
between various parameters and phases. By decomposing the dipolar
interaction into isotropic and anisotropic components, we provide a
simple explanation to quantitatively understand the phase
transitions induced by anisotropy. The various competing orders and
ground states are summarized in a complete phase diagram in the
parameter space spanned by the tilt angle and the confinement.
In the presence of anisotropy in the interparticle interaction, we
lose the rotational symmetry when treating the system in a disk
geometry, hence the total angular momentum is not a good quantum
number any more. Nevertheless, we demonstrated that one can still
compute the expectation value of the total angular momentum operator
for eigenstates and use it to characterize the various ground states
emerged as the results of competitions between confinement and
anisotropy, as well as between the isotropic and anisotropic
components of the interparticle interaction. Particularly interesting
is that the resilient features of topological order, such as the
presence of hierachical FQH ground states and the low-energy
excitations pertaining to the density deformation along the edge of an
incompressible quantum Hall droplet, remain robust in the presence of
anisotropy. While we showed that the calculation of the fidelity of
the ground state can be used as a probe to detect phase transitions
between states with different topological order, we believe the mean
angular momentum treatment can be readily generalized to the
calculation of, e.g., entanglement spectrum~\cite{li08}, which also
facilitates the detection of topological order.
In this paper we demonstrated that an incompressible FQH state with a
large excitation gap can survive a fairly large amount of anisotropy.
This implies that the Laughlin state, the exact ground state produced
by the hard-core potential, is stable against the anisotropic
perturbation. The finding is not unexpected given the remarkable
stability of the Laughlin states in the presence of long-range
interaction, finite thickness of the two-dimensional electron gas, and
disorder~\cite{wan05}. It is, however, intriguing to ask the effects
of anisotropy on more exotic non-Abelian quantum Hall states. A more
challenging question would be whether it is possible, by tuning the
anisotropy, to enhance a certain FQH state, hopefully with exotic
statistics. The present paper paves a path toward these questions.
\section*{ACKNOWLEDGMENTS}
This work was supported by the NSFC (Grant Nos. 11025421, 10935010,
10874017 and 10974209), the ``Bairen" program of the Chinese Academy
of Sciences, the DOE grant No. DE-SC0002140 (Z.X.H.) and the 973
Program under Project Nos. 2009CB929100 (X.W.) and 2011CB921803 (S.P.K).
|
1,108,101,565,664 | arxiv | \section{Introduction}
In this contribution I would like to discuss some of the conceptual
and physical issues surrounding the approach to noncommutative
geometry coming out of quantum groups and braided groups, in
somewhat greater depth than I had time for in my lecture in Goslar.
The discussion is intended to be intelligible to non-specialists and
may hopefully serve as an invitation to the field. Technical
material including recent results in quantum and braided geometry may
be found in my contribution to the companion `Quantum Groups' volume
of these Proceedings. Basic material can also be found in my textbook
on quantum groups\cite{Ma:book}.
The need for some kind of quantum geometry has
been clear enough since the birth of quantum mechanics itself: how to
extend ideas of gauge theory, curvature and non-Euclidean geometry to
the the situation where coordinates are noncommuting operators. If
one ever wants to unify quantum mechanics and gravity into a single
exact theory then this is probably a prerequisite. We begin, in
Section~2, by postulating some fundamental features which any
satisfactory such quantum geometry should have. One of them is an
extension of wave-particle duality to curved
space\cite{Ma:pri}\cite{Ma:pla}.
In Section~3 we consider quantum groups as examples of quantum
geometry. Actually, this is not at all the context in which the more
famous $q$-deformed enveloping algebras $U_q(g)$ arose (which is that
of `generalised symmetries' of exactly solvable lattice models; there
$q$ is an anisotropy parameter and not directly related to Planck's
constant). However, at about the same time as these $U_q(g)$ were
being introduced in the mid 1980's, another completely different
class of quantum groups ${\Bbb C}(G_1){\blacktriangleright\!\!\!\triangleleft} U(g_0)$ was being introduced
(by the author) in another context, which was precisely the context
of
Planck scale physics and an algebraic approach to quantum-gravity.
These {\em bicrossproduct} quantum groups really do arise as quantum
algebras of observables\cite{Ma:pla}. Here $G_0G_1$ is a
Lie group factorisation and $g_0$ is the Lie algebra of $G_0$.
More recently, gauge theory etc. has been developed for quantum
geometry\cite{BrzMa:gau}\cite{Ma:diag}.
As soon as spaces become noncommutative or `quantum', their
symmetries naturally become generalised as quantum groups too. Here
the quantum groups $U_q(g)$ play a more central role. It turns out as
a new feature of quantum geometry that anything on which a quantum
group acts acquires braid statistics. In other words. not only are
algebras noncommutative but tensor products become noncommutative
too.
This means that at the Planck scale one should expect not only bosons
and fermions but complicated braid statistics as well. This gives a
systematic `braided approach'\cite{Ma:introp} to quantizing
everything
uniformly, and is the topic of Section~4.
Obviously everything we may want to say about the Planck scale here
will be speculative. However, mathematics can tell us that certain
assumptions will force us to certain conclusions on the mathematical
structure, without yet knowing realistic models.
Moreover, quantum geometry is probably necessary not only at
the Planck scale or in quantum cosmology, but also in the resolution
of those
paradoxes in quantum mechanics which are characterised by a conflict
between the macroscopic geometry of measuring apparati and quantum
mechanical evolution; it should provide the right language to
correctly formulate such questions.
Note also that most discussions of Planck scale physics, including
string theory, work within an underlying classical geometry (e.g.
inside a path integral.) This is practical
but not really justified: classical geometry should emerge from
a deeper quantum world and not vice-versa. Even Professor Fredenhagen
in his talk {\em assumed exact classical Poincar\'e symmetry} without
any real justification, except that this is a necessary assumption to
be able apply the known methods of algebraic quantum field theory.
This is still `looking for the key under the streetlight'; by
contrast the quantum geometry
programme advocated here seeks primarily to understand first what
mathematically natural quantum geometry is out there, before making
predictions. The success of General Relativity can be attributed in
part to the fact that Einstein already had a fairly complete
mathematical theory of Riemannian geometry to bring to bear.
For example, modified uncertainty relations based on modified
commutation relations $[x,p]$ are only as meaningful as the
justification that a particular operator $x$ be called position and
a particular operator $p$ be called momentum, which should generally
come from a quantum geometrical picture.
\section{Inventing quantum geometry}
Riemannian geometry as we know it arose in two stages. First, one can
work extrinsically with surfaces embedded in Euclidean space, or
submanifolds $M\subset {\Bbb R}^n$. This is a Gaussian approach. Riemann
realised, however, that one needs also a more intrinsic notion of
geometry determined by structure on $M$ itself, particularly
since actual space or (after Einstein) spacetime is what we directly
experience rather than any embedding space. This leads to our modern
form of non-Euclidean or Riemannian geometry. The situation with
quantum mechanics can be viewed analogously. The role of ${\Bbb R}^n$ is
played in a certain sense by $B(\hbox{{$\cal H$}})$, the operators on a Hilbert
space. However, the real observables in the system are usually some
$*$-subalgebra $A\subset B(\hbox{{$\cal H$}})$. We broaden the
problem a little and consider the intrinsic structure of quite
general algebras $A$ (not only normed $*$-algebras over ${\Bbb C}$). In my
opinion
we are in a similar situation to Riemann: how to develop a language
to describe the intrinsic geometric structure on an algebra. Note
that for any manifold we can consider the algebra of smooth functions
on it and formulate our usual geometrical notions in those terms; the
key difference in quantum geometry is that we do not want to be
limited to commutative algebras.
Also, let us say from the start that the real physical motivation
for quantum geometry applies directly only to phase
spaces, which lose their points on quantisation due to the
uncertainty principle; the coordinates of phase space no longer fully
commute. However, a generalised framework for phases spaces probably
entails developing a generalised framework for manifolds in general.
Apart from this zeroth assumption, we postulate also:
\begin{enumerate}
\item {\bf Richness} -- at least as much `flabbiness' in the variety
of examples and structures (gauge fields, curvature etc.) as
classically.
\item {\bf Quantization without classical assumptions} -- classical
geometry should emerge as a possible limit, not be built in from the
start by assuming a Poisson manifold.
\item {\bf Uniformity of quantisation} -- the whole geometrical
`zoo' should be quantised together, coherently, not one object at a
time.
\item {\bf Duality between geometry and quantum mechanics} -- wave
particle duality should be maintained in some form.
\end{enumerate}
Axiom~1 here is not as empty as it may seem. There are
many noncommutative algebras but most of them will be too wild to
fit recognisably into a geometric picture. In the early 1980's most
papers on noncommutative geometry focussed on the noncommutative
torus as the main noncommutative example. Quantum geometry today
contains $q$-planes, $q$-spheres,
$q$-groups $G_q$ (the coordinate rings of $U_q(g)$), $q$-monopoles,
$q$-Lie algebras, $q$-vector fields etc.
Axiom~2 involves a slight abuse of notation -- what is
quantisation if not a process starting with a Poisson manifold?
The idea is that we need deeper more intrinsically algebraic notions
for the construction of quantum algebras, from which Poisson
manifolds can sometimes (though not necessarily) be obtained by
`classicalisation' with respect to a choice of generators (cf. Lie
algebra contraction). Quantum geometry at present supplies two; one
is the idea of factorisation and the other is the idea of R-matrix or
quasitriangular structures.
Axiom~3 is a more novel issue, usually overlooked even in a Poisson
geometric setting: in classical geometry we demand commutativity {\em
uniformly} for all coordinate algebras. When we relax it for one
geometrical object (e.g. position space), should we not relax it for
another (e.g. momentum space)? And there are many different
`directions' in which one can relax commutativity (e.g. many choices
of Poisson structure or R-matrix) for each object and we need to
choose these consistently. Our quantum spheres have to be consistent
with our quantum planes etc. Section~4 explains how this can be
achieved by means of braid statistics (the braided approach to
q-deformation).
Axiom~4 probably needs the most explanation. A modern way to
think about wave-particle duality is in terms of Fourier theory. The
dual group to position space ${\Bbb R}$ (the group of irreducible
representations) is again ${\Bbb R}$, the momentum space. On the one hand,
points $x$ are fundamental `particles' while on the other hand waves
or points $p$ in momentum space are fundamental `particles'; the two
points of view are related by Fourier transform. When position space
is curved, e.g.
a simple Lie group $G$, then the irreducible representations $\hat G$
do not form a group. However, non-Abelian Fourier theory is still
possible with the right mathematical generalisation. Note that $G$
itself is `geometrical' while the `points' of $\hat G$ are more like
quantum states, so this generalised Fourier theory is an example of a
{\em quantum-geometry transformation}. We would like a similar
elucidation for some class of `group' objects and their duals in any
quantum geometry. Ideally, we might hope for a stronger form of
Axiom~4, which we call the the {\em principle of
representation-theoretic self-duality}: quantum geometry should be
general enough that the duals of `group' objects are again `group'
objects. This is the case for the kind of quantum geometry coming of
quantum groups and braided groups in Sections~3,4. Moreover, some
groups, like ${\Bbb R}^n$, are self-dual. The self-dual `groups' in quantum
geometry likewise occupy a special place as the simplest geometrical
objects.
Actually, we have argued\cite{Ma:pri} that the bifurcation into
`geometrical' and `quantum' ideas (dual to each other and related by
a generalised Fourier or quantum geometry transformation) has its
origin in our concept of physical reality itself (the dual nature of
measurement and object being measured), and is not really tied to
groups. More general manifolds with `geometrical' structure have more
complex dual notions of `representation', etc. and self-duality will
pick out geometries occupying a special place. We have postulated
this self-duality
constraint for the quantum geometry of phase space as the
philosophical origin of something like Einstein's
equation\cite{Ma:pri}.
\section{Elements of quantum geometry}
The present approach to quantum geometry is motivated particularly
from the duality Axiom~4, which leads one to formulate group objects
as Hopf algebras. A Hopf algebra is a unital algebra $H$ and algebra
homomorphisms $\Delta:H\to H\mathop{\otimes} H$ (coproduct), ${\epsilon}:H\to{\Bbb C}$
(counit) forming a counital coalgebra. The axioms of a counital
coalgebra are just those of a unital algebra with all arrows reversed
(think of the unit element as a map ${\Bbb C}\to H$). So $H^*$ is also a
unital algebra by dualising $\Delta,{\epsilon}$. There is also a kind of
`linearised inversion' $S:H\to H$ called the antipode.
In a suitable setting, if $H$ is a Hopf algebra then essentially
$H^*$ is another Hopf algebra by dualisation. The product of one
determines the coproduct of the other and vice-versa. So Axiom~4 is
satisfied in the strong form. Moreover, if $G$ is a group then its
coordinate ring ${\Bbb C}(G)$ is a Hopf algebra, at least in nice cases. In
the finite case we mean all functions $f$ on $G$ with pointwise
product and $\Delta f=f(\ \cdot\ )$, where $\cdot$ is the group
product. The blanks on the right hand side indicate a function of two
variables, i.e. an element of ${\Bbb C}(G)\mathop{\otimes}{\Bbb C}(G)$. So Hopf algebras
generalise the notion of usual groups. Dually paired to ${\Bbb C}(G)$ is
the group algebra ${\Bbb C} G$ (the linear extension of $G$ with $\Delta
g=g\mathop{\otimes} g,{\epsilon} g=1$) in the finite case or enveloping algebra $U(g)$
in the Lie group case with Lie
algebra $g$. Fourier theory is a linear isomorphism $H\to H^*$ or
${\Bbb C}(G)\to
{\Bbb C} G$
in the finite case. When $G$ is Abelian, we have ${\Bbb C} G={\Bbb C}(\hat G)$,
the coordinate ring of the dual group. But when $G$ is non-Abelian,
${\Bbb C} G$ is not commutative, so then $\hat G$ only exists as a quantum
geometry with, by definition, coordinate ring ${\Bbb C} G$. One should
consider any noncommutative Hopf algebra as, by
definition, a quantum group.
\subsection{Toy models of quantum-gravity}
Here we will see how the structural considerations in Section~2
can force one to concrete Planck scale dynamics. For our
discussion, all quantum geometries that we consider will be `group'
objects, i.e. Hopf algebras, but the ideas could ultimately be
applied more generally.
Suppose that we fix the position Hopf algebra $H_1$ and momentum
Hopf algebra $H_0$. Instead of Poisson brackets or other guesswork
about the quantum phase space (i.e. instead of guessing
position-momentum
commutation relations) let us proceed structurally and consider all
possible extensions
\eqn{ext}{H_1\to E\to H_0.}
This means a Hopf algebra $E$ and Hopf algebra maps as shown, obeying
certain conditions\cite{Ma:book}. Then theory tells us that $E$ will
be a cocycle bicrossproduct $E=H_1{\blacktriangleright\!\!\!\triangleleft}
H_0$. The simplest case is with trivial cocycles, in some
cohomological sense `close' to the tensor product $H_1\mathop{\otimes} H_0$.
Then the theorem is that $E$ is a cross product by an action of $H_0$
on $H_1$ and a cross coproduct by a coaction of $H_1$ on $H_0$.
For example, let us consider {\em all possible} extensions
\eqn{extxp}{ {\Bbb C}[x]\to E\to {\Bbb C}[p]}
of 1-dimensional position and momentum spaces ${\Bbb C}[x]$ and ${\Bbb C}[p]$.
These are classical groups and hence Hopf algebras, with $\Delta
x=x\mathop{\otimes} 1+1\mathop{\otimes} x$ and $\Delta p=p\mathop{\otimes} 1+1\mathop{\otimes} p$. We consider
all possible $E$, i.e. do not build any relations in by hand, and we
consider the Hamiltonian to be fixed in the form $p^2/2m$. Note that
this approach is
more intrinsic (in the spirit of Einstein's equivalence principle)
than keeping the commutation relations in some canonical form but
varying the Hamiltonian.
\begin{propos}\cite{Ma:pla} The possible cocycle-free extensions
(\ref{extxp}) are described by two parameters $\hbar,{\scriptscriptstyle G}$ and take
the form $E_{\hbar,{\scriptscriptstyle G}}={\Bbb C}[x]{\blacktriangleright\!\!\!\triangleleft} {\Bbb C}[p]$ with cross relations
and coproduct
\[ [x,p]=\imath\hbar(1-e^{-{x\over M{\scriptscriptstyle G}}}),\quad \Delta p=p\mathop{\otimes}
e^{-{x\over M{\scriptscriptstyle G}}}+1\mathop{\otimes} p,\]
where $M$ is a convenient fixed constant of mass dimension.
\end{propos}
The free-fall Hamiltonian $p^2/2m$ gives $\dot
x=v_\infty(1-e^{-{x\over M{\scriptscriptstyle G}}})$ to lowest order
in $\hbar$, which can be compared with infalling radial coordinates
$\dot x=-(1-(1+{x\over
2M{\scriptscriptstyle G}})^{-1})$ near a black hole of mass $M$. So the parameter
${\scriptscriptstyle G}$ plays a role in
this simple model similar to the gravitational coupling constant.
Also, the particle moves more
and more slowly as it approaches the origin and takes an infinite
time to reach it. Yet when ${\scriptscriptstyle G}$ is small, the commutation
relations are $[x,p]=\imath\hbar$ at least for states where one can
say that $x>0$, i.e. away from the origin. Further analysis of this
model gives the effective scales for these gravitational and quantum
limits as $mM>> m_P^2$ and $mM<< m_P^2$, where
$m_P=\sqrt{\hbar/{\scriptscriptstyle G}}$ is the Planck mass
\cite{Ma:pla}\cite{Ma:book}.
The same ideas work when the position and momenta are curved. Let
$G_1$ be a Lie group and $g_0$ a Lie algebra with group
$G_0$. One can consider
cocycle-free extensions
\eqn{extG}{ {\Bbb C}(G_1)\to E\to U(g_0).}
The possible extensions turn out\cite{Ma:phy} to correspond
essentially to solutions of the factorisation problem: Lie
groups factorising into $G_0G_1$. For example, the complexification
of any compact real form $G_0$ of a simple Lie group factorises as
$G_0G_1$ for a certain solvable $G_1$, and gives a
corresponding $E$. The natural Hamiltonian is the quadratic
the Casimir of $g_0$ and induces quantum dynamics on $G_1$ as
position space.
For example, the quantum group $E={\Bbb C}({\Bbb R}^2{>\!\!\!\triangleleft} {\Bbb R}){\blacktriangleright\!\!\!\triangleleft} U(so_3)$
corresponds to the Iwasawa factorisation of $SL(2,{\Bbb C})$. One can
insert two free parameters $\hbar,{\scriptscriptstyle G}$ as well. Then $E$ is
generated by the coordinates $x_i$ of ${\Bbb R}^2{>\!\!\!\triangleleft}{\Bbb R}$ and $e_i$ of
$su_2$, with\cite{Ma:book}
\ceqn{bicso3}{[e_i,e_j]=\imath\hbar {\epsilon}_{ijk}e_k,\quad
[e_i,x_j]=\imath\hbar{\epsilon}_{ijk}x_k-{\imath\hbar\over
2M{\scriptscriptstyle G}}{\epsilon}_{ij3}x\cdot
x (1+ {x_3\over M{\scriptscriptstyle G}})^{-1}\\
{}[x_i,x_j]=0,\quad\Delta x_i=x_i\mathop{\otimes} 1+ (1+{x_3\over M{\scriptscriptstyle G}})\mathop{\otimes}
x_i\\
\Delta e_i=e_i\mathop{\otimes} (1+ {x_3\over M{\scriptscriptstyle G}})^{-1}+{1\over
M{\scriptscriptstyle G}}e_3\mathop{\otimes}
x_i(1+{x_3\over M {\scriptscriptstyle G}})^{-1}+1\mathop{\otimes} e_i.}
By thinking of the $x_i$ as momenta $p_i$ (wave-particle duality
again), one can also consider this $E$ as some kind of
deformation of
$U({\Bbb R}^3{>\!\!\!\triangleleft} so_3)$, i.e. of the Poincar\'e enveloping algebra in 3
dimensions. So on the one hand, $E$ is a quantisation of particles on
orbits in ${\Bbb R}^3$ exhibiting singular dynamics, and on the other it is
a deformation of a symmetry algebra. The Minkowski spacetime version
of this model is very similar and of independent interest
\cite{MaRue:bic}.
Moreover, $E^*$ is another quantum phase space. It solves the
extension problem
\eqn{dext}{H_0^*\to E^*\to H_1^*.}
When position space is flat as in (\ref{extxp}) then essentially
${\Bbb C}[x]^*={\Bbb C}[p]$ (wave particle duality) and $E^*$ also solves
(\ref{extxp}). Hence it is of the same form as in Proposition~3.1,
i.e. these $E$ are self-dual quantum groups.
In the curved position space case, the dual quantum group is
$E^*=U(g_1){\triangleright\!\!\!\blacktriangleleft} {\Bbb C}(G_0)$. For example, the dual Hopf algebra to
(\ref{bicso3}) describes quantum particles in
$SU_2=S^3$ moving on orbits under ${\Bbb R}^2{>\!\!\!\triangleleft} {\Bbb R}$. The explicit
orbits and flows in these models are obtained by solving nonlinear
equations. Also, one can
classicalise and obtain Poisson manifolds of which these quantum
groups are quantisation, although they would not be determined
uniquely as such; see \cite{Ma:book}.
This demonstrates our algebraic approach to Planck
scale physics. It is one of the historical origins of noncommutative
(and noncocommutative) Hopf algebras or quantum groups. Recent work
on bicrossproducts is in \cite{BegMa:qua}.
\subsection{Quasitriangular structures}
It would be remiss not to mention the more famous Drinfeld-Jimbo
quantum groups $U_q(g)$ and their duals $G_q$. They have little, so
far, to do with Planck scale physics (as far as I know),
arising independently in quite a different physical context. They
can, however, be classicalised and hence viewed (if we want) as
quantisations of a certain Drinfeld-Sklyanin Poisson bracket on the
Lie group of $g$. At this level, there are connections with the
factorisation problem above\cite{Ma:phy}. Also, they again
demonstrate our Axiom~2 that
quantum geometry has its own intrinsic structure. The intrinsic
structure of $U_q(g)$ is that of a {\em quasitriangular} Hopf
algebra\cite{Dri}. It is a Hopf algebra $H$ equipped with a so-called
(by physicists) universal R-matrix $\CR\in H\mathop{\otimes} H$. Its image in
any matrix representation obeys the Yang-Baxter or braid relations.
Such generalised symmetry algebras are relevant to the next section.
The intrinsic structure of $G_q$ is therefore that of a {\em dual
quasitriangular} Hopf algebra. This is a Hopf algebra $H$ equipped
with a skew bicharacter $\CR:H\mathop{\otimes} H\to {\Bbb C}$ obeying
\eqn{dqua}{ g{}_{\scriptscriptstyle(1)} h{}_{\scriptscriptstyle(1)} \CR(h{}_{\scriptscriptstyle(2)},g{}_{\scriptscriptstyle(2)})=\CR(h{}_{\scriptscriptstyle(1)},g{}_{\scriptscriptstyle(1)})h{}_{\scriptscriptstyle(2)} g{}_{\scriptscriptstyle(2)} }
for all $h,g\in H$, where $\Delta h=h{}_{\scriptscriptstyle(1)}\mathop{\otimes} h{}_{\scriptscriptstyle(2)}$ is our notation for
the coproduct with output in $H\mathop{\otimes} H$ (summation omitted). When
$\CR$, $G_q$ are expanded in $\hbar$ with $q=e^{\hbar/2}$, one
obtains a Poisson bracket. But the quantum world is richer. For
example, there are discrete quantum groups possessing such $\CR$.
Thus the axioms for $H,\CR$ carve out a class of quantum groups
defined intrinsically and `close' to being commutative in the sense
(\ref{dqua}) rather than in the conventional sense of quantisation of
a Poisson bracket.
\section{Elements of braided geometry}
In this section we explain another approach to quantum geometry,
which has so far been applied mostly in flat space (rather than
having direct contact with the Planck scale), but which has the merit
of solving the uniformity Axiom~3. Ultimately, we would like to see
it combined with the ideas in the preceding section. This {\em
braided geometry} involves a new kind of mathematics in which
information `flows' along braids and tangles much as it flows along
the wiring in
a computer, except that under- and over-crossings of wires are now
nontrivial {\em braiding operators} $\Psi$. In usual mathematics and
computer science one wires outputs of operations into inputs of other
operations without caring about such crossings, i.e. usual
mathematics
is two-dimensional. By contrast, braided calculations, braided
Feynman diagrams etc. truly exist in a three-dimensional space where
calculations take place. Mathematically, we make use of the theory of
braided categories \cite{JoyStr:bra}. The introduction of algebras,
group theory and geometry in braided categories is due to the
author, e.g.\cite{Ma:exa}\cite{Ma:introp}.
The idea is that in quantum physics there is another kind of
noncommutativity, namely anticommutativity due
to fermionic statistics. This is a noncommutativity of $\mathop{\otimes}$
itself. Thus, when independent fermionic systems must be
exchanged during a manipulation, one uses supertransposition
$\Psi(b\mathop{\otimes} c)=(-1)^{|b||c|}c\mathop{\otimes} b$, where $|\ |$ is the degree
$0,1$. For example, the supertensor product $B\mathop{\otimes} C$ of two
superalgebras involves $\Psi$, with the result that $cb\equiv (1\mathop{\otimes}
c)(b\mathop{\otimes} 1)=\Psi(c\mathop{\otimes} b)=(-1)^{|c||b|}bc$ in $B\mathop{\otimes} C$. The idea
of braided geometry is that $\Psi$, and hence the cross relations of
$B\mathop{\otimes} C$, can be much more general than this simple $\pm1$ form.
When $\Psi^2$ is not always the identity, one says that the system
has {\em braid statistics}. Thus,
\begin{itemize}
\item quantum geometry: $\mathop{\otimes}$ usual commutative (bosonic) one,
coordinate algebras noncommutative.
\item braided geometry: $\mathop{\otimes}$ non commutative (braid statistics),
coordinate algebras may as well be `commutative' in suitably modified
sense.
\end{itemize}
Just as quantum mechanics was created with the realisation that many
construction do not require commutativity of coordinates, braided
geometry is created by a second and equally deep realisation: {\em
many constructions do not require commutativity of the notion of
independence.} In particular, we can take in place of $-1$ a
dimensionless parameter
$q$ or, more generally, an operator $\Psi$ depending on one or more
parameters $q$. Moreover, {\em classical braided geometry
$\Rightarrow$ quantum geometry} in a $q$-deformed sense because
`braided commutative' generally means noncommutative with respect to
the usual $\mathop{\otimes}$. Moreover, specifying the braid statistics
specifies such things coherently between every object and every other
object. Quantum groups still play a role:
\begin{propos}cf.\cite{Dri} All objects $B,C$ on which a quantum
group like
$U_q(g)$ (a quasitriangular Hopf algebra) acts acquire braid
statistics $\Psi(b\mathop{\otimes} c)=\CR_i.c\mathop{\otimes} \CR^i.b$, where
$\CR=\CR^i\mathop{\otimes}\CR_i$ is the universal R-matrix or quasitriangular
structure.
\end{propos}
An example is the quantum-braided plane ${\Bbb C}_q^2$ generated by a
vector of coordinates $\vecx=(x,y)$ obeying $yx=qxy$. It has braiding
and braided-coproduct\cite{Ma:poi}
\ceqn{qplane}{\Psi(x\mathop{\otimes} x)=q^2x\mathop{\otimes} x,\quad \Psi(y\mathop{\otimes}
y)=q^2y\mathop{\otimes} y,\quad \Psi(x\mathop{\otimes} y)= q y\mathop{\otimes} x\\
\Psi(y\mathop{\otimes} x)=qx\mathop{\otimes} y+(q^2-1)y\mathop{\otimes} x,\quad \Delta\vecx=\vecx\mathop{\otimes}
1+1\mathop{\otimes}\vecx.}
There are braided-plane structures for q-Euclidean and q-Minkowski
spaces. They are isomorphic to their duals
($q$-wave-particle duality). There are also braided matrices $B(R)$
generated by ${\bf u}=(u^i{}_j)$ with relations $R_{21}{\bf u}_1
R{\bf u}_2={\bf u}_2
R_{21}{\bf u}_1 R$ and\cite{Ma:exa}
\eqn{BR}{\Delta {\bf u}={\bf u}\mathop{\otimes}{\bf u},\quad \Psi({\bf u}_1\mathop{\otimes}
R{\bf u}_2)=R{\bf u}_2 R^{-1}\mathop{\otimes}{\bf u}_1 R}
in a compact notation, for any biinvertible matrix $R$ obeying the
Yang-Baxter equations. Their quotients by $q$-determinant and other
relations give braided versions $BG_q$ of the coordinate rings of
simple Lie groups. They are dual to braided versions $BU_q(g)$ of the
enveloping algebras. Here is a remarkable selfduality phenomenon:
\begin{propos}\cite{Ma:introp}\cite{Ma:lie} When $q\ne 1$ one has
essentially $BG_q{\cong} BU_q(g)$.
\end{propos}
So in the $q\ne 1$ world there is essentially only one $q$-deformed
object for each simple Lie algebra $g$, which has two limits as $q\to
1$. On the left hand side it becomes ${\Bbb C}(G)$ the commutative
coordinate ring. On the right hand side it becomes the enveloping
algebra $U(g)$. Thus these two features of classical mathematics,
conceptually dual to each other, are different scaling limits of one
object. In a similar way, one finds that $q$-Minkowski space as a
$2\times 2$ braided matrix is isomorphic to the braided enveloping
algebra of a braided Lie algebra version of $su_2\oplus
u(1)$ \cite{Ma:lie}. This is wave-particle duality in a strong form,
and is only possible when $q\ne 1$.
The $q$-Poincar\'e and $q$-conformal groups are also obtained
from braided geometry. With $q$-Minkowski space as an additive
braided group (like the quantum-braided plane above) one has a
braided adjoint action of the braided-coordinates on themselves. This
is not possible when $q=1$ since the adjoint action is then trivial.
However, when $q\ne 1$ it generates the action of $q$-special
conformal transformations\cite{Ma:geo}. The remnant of this as $q\to
1$ is
\eqn{conf}{ I\circ{\del\over\del x_i}\circ I=\lim_{q\to
1}{{\rm Ad}_{x_i}\over
q-q^{-1}}}
where $I$ is conformal inversion. This is a completely new
group-theoretical picture of conformal transformations as adjoint
action, only possible when $q\ne 1$.
The above approach to $q$-deformation has been developed over
50--60 papers by the author and collaborators since 1989. It
provides the correct meaning of $q$ as `braid statistics' (rather
than directly related to $\hbar$) and a systematic solution to the
problem of $q$-deforming everything. Moreover, we see that our
familiar $q=1$ world is merely a special limit of a deeper and more
natural $q\ne 1$ geometry.
\section*{Acknowledgments} I would like to thank the organizers,
H.-D. Doebner, P. Kramer, W. Scherer, V. Dobrev and others for an
outstanding conference.
|
1,108,101,565,665 | arxiv | \section*{Acknowledgements}
The authors wish to acknowledge the support of the grant DEC-2012/06/A/ST2/00389 from the National Science Centre Poland.
\bibliographystyle{unsrt}
|
1,108,101,565,666 | arxiv | \section{Introduction}
Figure \ref{fig:teaser} shows a high scoring detection from an object detector
with HOG features and a linear SVM classifier trained on a large database of
images. \changed{\emph{Why} does this detector think that sea water looks like a car?}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{figs/teaser_fp.png}
\vspace{-.5em}
\caption{An image from PASCAL and a high scoring car detection from
DPM \citep{felzenszwalb2010object}. Why did the detector fail?}
\label{fig:teaser}
\vspace{-.5em}
\includegraphics[width=\linewidth]{figs/teaser_fp_vis.png}
\vspace{-1em}
\caption{We show the crop for the false car detection from
Figure \ref{fig:teaser}. On the right, we show our visualization of the HOG
features for the same patch. Our visualization reveals that this false alarm
actually looks like a car in HOG space.}
\label{fig:teaser2}
\vspace{-1.5em}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{figs/false-positives.png}
\caption{We visualize some high scoring detections from the deformable parts
model \citep{felzenszwalb2010object} for person, chair,
and car. Can you guess which are false alarms? Take a minute to study
this figure, then see Figure \ref{fig:topdetsrgb} for the corresponding RGB
patches.}
\label{fig:topdets}
\vspace{-1em}
\end{figure*}
Unfortunately, computer vision researchers are often unable to explain the
failures of object detection systems. Some researchers blame the features,
others the training set, and even more the learning algorithm. Yet, if we wish
to build the next generation of object detectors, it seems crucial to
understand the failures of our current detectors.
In this paper, we introduce a tool to explain some of the failures of object
detection systems. We present algorithms to visualize the
feature spaces of object detectors. Since features are too high dimensional for
humans to directly inspect, our visualization algorithms work by inverting
features back to natural images. We found that these inversions provide an
intuitive visualization of the feature spaces used by object
detectors.
Figure \ref{fig:teaser2} shows the output from our visualization algorithm on the
features for the false car detection. This visualization reveals that,
while there are clearly no cars in the original image, there is a car hiding
in the HOG descriptor. HOG features see a slightly different visual world than
what we see, and by visualizing this space, we can gain a more intuitive
understanding of our object detectors.
Figure \ref{fig:topdets} inverts more top detections on PASCAL for a few
categories. Can you guess which are false alarms? Take a minute to study the
figure since the next sentence might ruin the surprise. Although every
visualization looks like a true positive, all of these detections are actually
false alarms. Consequently, even with a better learning
algorithm or more data, these false alarms will likely persist. In other words,
the features are responsible for these failures.
\begin{figure}
\includegraphics[width=\linewidth]{figs/teaser-multiple.pdf}
\caption{Since there are many images that map to similar features, our method recovers multiple images that are diverse in image space, but match closely in feature space.}
\label{fig:multiple}
\end{figure}
The primary contribution of this paper is a general algorithm for visualizing
features used in object detection. We present a method that inverts visual
features back to images, and show experiments for two standard features in object detection,
HOG and activations from CNNs. Since there are many images that can produce equivalent
feature descriptors, our method moreover recovers multiple images that are perceptually
different in image space, but map to similar feature vectors, illustrated in Figure \ref{fig:multiple}.
The remainder of this paper presents and analyzes our visualization algorithm.
We first review a growing body of work in feature visualization for both
handcrafted features and learned representations. We evaluate our inversions
with both automatic benchmarks and a large human study, and we found our
visualizations are perceptually more accurate at representing the content of a
HOG feature than standard methods; see Figure \ref{fig:example} for a
comparison between our visualization and HOG glyphs. We then use our
visualizations to inspect the behaviors of object detection systems and analyze
their features. Since we hope our visualizations will be useful to other
researchers, our final contribution is a public feature visualization toolbox.\footnote{Available online at \url{http://mit.edu/hoggles}}
\section{Related Work}
Our visualization algorithms are part of an actively growing body of work in
feature inversion. \cite{oliva2001modeling}, in early work, described a simple
iterative procedure to recover images given gist descriptors.
\citet{weinzaepfel2011reconstructing} were the first to reconstruct an image
given its keypoint SIFT descriptors \citep{lowe1999object}. Their approach
obtains compelling reconstructions using a nearest neighbor based approach on a
massive database. \cite{d2012beyond} then developed an algorithm to reconstruct
images given only LBP features \citep{calonder2010brief,alahi2012freak}. Their
method analytically solves for the inverse image and does not require a
dataset. \cite{kato2014image} posed feature inversion as a jigsaw puzzle
problem to invert bags of visual words.
\begin{figure}
\includegraphics[width=\linewidth]{figs/teaser2.png}
\caption{In this paper, we present algorithms to visualize features. Our visualizations are more perceptually intuitive for humans to understand.}
\label{fig:example}
\vspace{-1em}
\end{figure}
Since visual representations that are learned can be difficult to interpret,
there has been recent work to visualize and understand learned features.
\cite{zeiler2013visualizing} present a method to visualize activations from a
convolutional neural network. In related work, \cite{simonyan2013deep}
visualize class appearance models and their activations for deep networks.
\cite{girshick2013rich} proposed to visualize convolutional neural networks by
finding images that activate a specific feature. \changed{\cite{deephoggles}
describe a general method for inverting visual features from CNNs by
incorporating natural image priors.}
While these methods are good at reconstructing and visualizing images from
their respective features, our visualization algorithms have some advantages.
Firstly, while most methods are tailored for specific features, the
visualization algorithms we propose are feature independent. Since we cast
feature inversion as a machine learning problem, our algorithms can be used to
visualize any feature. In this paper, we focus on features for object
detection, and we use the same algorithm to invert both HOG and CNN features.
Secondly, our algorithms are fast: our best algorithm can invert features in
under a second on a desktop computer, enabling interactive visualization, which
we believe is important for real-time debugging of vision systems. \changed{Finally, our algorithm
explicitly optimizes for multiple inversions that are diverse in image space, yet match in feature space.}
Our method builds upon work that uses a pair of dictionaries with a coupled
representation for super resolution \citep{yang2010image,wang2012semi} and
image synthesis \citep{huangcoupled}. We extend these methods to show that
similar approaches can visualize features as well. Moreover, we incorporate
novel terms that encourage diversity in the reconstructed image in order
to recover multiple images from a single feature.
Feature visualizations have many applications in computer vision. The computer
vision community has been using these visualization largely to understand object recognition
systems so as to reveal information encoded by features
\citep{zhangspeeding,sadeghi2013fast}, interpret transformations in feature
space \citep{cheninferring}, studying diverse images with similar features
\citep{tatu2011exploring,equiv}, find security failures in machine learning systems
\citep{biggio2012poisoning,weinzaepfel2011reconstructing}, and fix problems in
convolutional neural networks
\citep{zeiler2013visualizing,simonyan2013deep,bruckner2014ml}. With many
applications, feature visualizations are an important tool for the
computer vision researcher.
Visualizations enable analysis that complement a recent line of papers that
provide tools to diagnose object recognition systems, which we briefly review
here. \cite{parikh2011human,parikh2010role} introduced a new paradigm for
human debugging of object detectors, an idea that we adopt in our experiments.
\cite{hoiem2012diagnosing} performed a large study analyzing the errors that
object detectors make. \cite{divvala2012important} analyze part-based
detectors to determine which components of object detection systems have the
most impact on performance. \cite{liu2012has} designed algorithms to highlight
which image regions contribute the most to a classifier's confidence.
\cite{zhuwe} try to determine whether we have reached Bayes risk for HOG. The
tools in this paper enable an alternative mode to analyze object detectors
through visualizations. By putting on `HOG glasses' and visualizing the world
according to the features, we are able to gain a better understanding of the
failures and behaviors of our object detection systems.
\section{Inverting Visual Features}
We now describe our feature inversion method. Let $x_0 \in
\mathbb{R}^{P}$ be a natural RGB image and $\phi = f(x_0) \in \mathbb{R}^Q$ be
its corresponding feature descriptor. \changed{Since features are many-to-one functions,} our goal is to invert the features $\phi$
by recovering a \emph{set} of images $\mathcal{X} = \{x_1, \ldots, x_N\}$ that all map to
the original feature descriptor.
We compute this inversion set $\mathcal{X}$ by solving an optimization problem. We wish
to find several $x_i$ that minimize their reconstruction error in feature space
$\left|\left| f(x_i) - \phi \right|\right|_2^2$ while simultaneously appearing
diverse in image space. We write this optimization as:
\begin{equation}
\begin{aligned}
\mathcal{X} = \argmin_{x, \xi} &\sum_{i=1}^N \left|\left| f(x_i) - \phi \right|\right|_2^2 + \gamma \sum_{j<i} \xi_{ij} \\
\textrm{s.t.} \quad &0 \le S_A(x_i, x_j) \le \xi_{ij} \; \forall_{ij}
\end{aligned}
\label{eqn:objective-multiple}
\end{equation}
The first term of this objective favors images that match in feature space and
the slack variables $\xi_{ij}$ penalize pairs of images that are too similar to
each other in image space \changed{where $S_A(x_i, x_j)$ is the} similarity cost, parametrized by $A$,
between inversions $x_i$ and $x_j$. A high similarity cost intuitively means
that $x_i$ and $x_j$ look similar and should be penalized. The hyperparameter
$\gamma \in \mathbb{R}$ controls the strength of the similarity cost. By
increasing $\gamma$, the inversions will look more different, at the expense of
matching less in feature space.
\subsection{Similarity Costs}
There are a variety of similarity costs that we could use. \changed{In this work,} we use costs of the form:
\begin{align}
S_A(x_i, x_j) = (x_i^T A x_j)^2
\label{eqn:similarity}
\end{align}
where $A \in \mathbb{R}^{P \times P}$ is an affinity matrix. Since we are
interested in images that are diverse and not negatives of each other, we
square $x_i^T A x_j$.
The identity affinity matrix, i.e.\ $A = I$, corresponds to comparing inversions
directly in the color space. However, more metrics are also possible, \changed{which we describe now.}
\emph{Edges:} We can design $A$ to favor inversions that differ in edges. Let
$A = C^T C$ where $C \in \mathbb{R}^{2P \times P}$. The first $P$ rows of $C$
correspond to the convolution with the vertical edge filters
$\left[\begin{smallmatrix}-1 & 0 & 1\end{smallmatrix}\right]$ and similarly the
second $P$ rows are for the horizontal edge filters
$\left[\begin{smallmatrix}-1 & 0 & 1\end{smallmatrix}\right]^T$.
\emph{Color:} We can also encourage the inversions to differ only in colors.
Let $A = C^T C$ where $C \in \mathbb{R}^{3 \times P}$ is a matrix that averages
each color channel such that $Cx \in \mathbb{R}^3$ is the average RGB color.
\emph{Spatial:} We can force the inversions to only differ in certain spatial
regions. Let $A = C^T C$ where $C \in \mathbb{R}^{P \times P}$ is a binary
diagonal matrix. A spatial region of $x$ will be only encouraged to be
diverse if its corresponding element on the diagonal of $C$ is $1$. Note we can
combine spatial similarity costs with both color and edge costs to encourage
color and edge diversity in only certain spatial regions as well.
\subsection{Optimization}
Unfortunately, optimizing equation \ref{eqn:objective-multiple} efficiently is challenging because
it is not convex. \changed{Instead, we will make two modifications to solve an approximation:}
\begin{figure}
\includegraphics[width=\linewidth]{figs/paired-dict-tutorial-2.png}
\caption{Inverting features using a paired dictionary. We first project the feature vector
on to a feature basis. By jointly learning a coupled
basis of features and natural images, we can transfer coefficients estimated from features
to the image basis to recover the natural image.}
\label{fig:pair-tutorial}
\end{figure}
\changed{\emph{Modification 1:} Since the first term of the objective depends on the feature function $f(\cdot)$, which is often not convex nor differentiable, efficient optimization is difficult. Consequently, we approximate an image $x_i$ and its features
$\phi = f(x_i)$ with a paired, over-complete basis to make the objective convex.
Suppose we represent an image $x_i \in \mathbb{R}^P$ and its feature $\phi \in
\mathbb{R}^Q$ in a natural image basis $U \in \mathbb{R}^{P \times K}$ and a
feature space basis $V \in \mathbb{R}^{Q \times K}$ respectively. We can
estimate $U$ and $V$ such that images and features can be encoded in their
respective bases but with shared coefficients $\alpha \in \mathbb{R}^K$:
\begin{align} x_0 = U\alpha \quad \textrm{and} \quad \phi = V\alpha \end{align}
If $U$ and $V$ have this paired representation, then we can invert features by
estimating an $\alpha$ that reconstructs the feature well. See Figure
\ref{fig:pair-tutorial} for a graphical representation of the paired
dictionaries.}
\changed{
\emph{Modification 2:} However, the objective is still not convex when there are multiple outputs. We approach solving equation \ref{eqn:objective-multiple} sub-optimally using a greedy approach.
Suppose we already computed the first $i-1$
inversions, $\{x_1, \ldots, x_{i-1}\}$. We then seek the inversion $x_i$ that
is only different from the previous inversions, but still matches $\phi$.
}
Taking these approximations into account, we solve for the inversion $x_i$ with
the optimization:
\begin{equation}
\begin{aligned}
&\alpha_i^* = \argmin_{\alpha_i, \xi} ||V\alpha_i-\phi||_2^2 + \lambda ||\alpha_i||_1 + \gamma \sum_{j=1}^{i-1} \xi_j \\
& \textrm{s.t.} \quad S_A(U\alpha_i, x_j) \le \xi_j
\end{aligned}
\label{eqn:pair-inverse}
\end{equation}
where there is a sparsity prior on $\alpha_i$ parameterized by $\lambda \in
\mathbb{R}$.\footnote{We found a sparse $\alpha_i$ improves our results. While
our method will work when regularizing with $||\alpha_i||_2$ instead, it tends to
produce more blurred images.} After estimating $\alpha_i^*$, the inversion is
$x_i = U\alpha_i^*$.
\begin{figure}
\includegraphics[width=1\linewidth]{figs/pd-pairs.jpg}
\caption{Some pairs of dictionaries for $U$ and $V$. The left of every pair is the gray scale dictionary element and the right is the positive components elements in the HOG dictionary. Notice the correlation between dictionaries.}
\label{fig:pair-basis}
\end{figure}
The similarity costs can be seen as adding a weighted Tikhonov
regularization ($\ell 2$ norm) on $\alpha_i$ because
\begin{align*}
S_A(U \alpha_i, x_j) = \alpha_i^T B \alpha_i \quad
\textrm{where} \quad
B = U^T A^T x_j^T x_j A U
\end{align*}
Since this is combined with lasso, the optimization
behaves as an elastic net \citep{zou2005regularization}.
Note that if we remove the slack variables ($\gamma =
0$), our method reduces to \citep{vondrick2013hog} and only produces one
inversion.
As the similarity costs are in the form of equation \ref{eqn:similarity}, we can
absorb $S_A(x; x_j)$ into the $\ell 2$ norm of equation \ref{eqn:pair-inverse}.
This allows us to efficiently optimize equation \ref{eqn:pair-inverse} using an
off-the-shelf sparse coding solver. We use SPAMS \citep{mairal2009online} in
our experiments. The optimization typically takes a few seconds to produce each
inversion on a desktop computer.
\subsection{Learning}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{figs/elda1.png}
\caption{We found that averaging the images of top detections from an exemplar
LDA detector provide one method to invert HOG features.}
\label{fig:elda}
\end{figure*}
The bases $U$ and $V$ can be learned such that they have paired coefficients.
We first extract millions of image patches $x_0^{(i)}$ and their corresponding
features $\phi^{(i)}$ from a large database. Then, we can solve a dictionary
learning problem similar to sparse coding, but with paired dictionaries:
\begin{equation}
\begin{aligned}
\argmin_{U, V, \alpha} \; & \sum_{i} ||x_0^{(i)} - U\alpha_i||_2^2 + ||\phi^{(i)} - V\alpha_i||_2^2 + \lambda ||\alpha_i||_1 \\
\textrm{s.t.} \quad &||U||_2^2 \le \psi_1, \; ||V||_2^2 \le \psi_2 \
\end{aligned}
\label{eqn:pairobj}
\end{equation}
for some hyperparameters $\psi_1 \in \mathbb{R}$ and $\psi_2 \in \mathbb{R}$.
We optimize the above with SPAMS \citep{mairal2009online}. Optimization
typically took a few hours, and only needs to be performed once for a fixed
feature. See Figure \ref{fig:pair-basis} for a visualization of the learned
dictionary pairs.
\section{Baseline Feature Inversion Methods}
In order to evaluate our method, we also developed several baselines that we use for
comparison. We first describe \changed{three} baselines for single feature inversion, then discuss two
baselines for multiple feature inversion.
\subsection{Exemplar LDA (ELDA)}
Consider the top detections for the exemplar object detector
\citep{hariharan2012discriminative,malisiewicz2011ensemble} for a few
images shown in Figure \ref{fig:elda}. Although all top detections are false
positives, notice that each detection captures some statistics about the query.
Even though the detections are wrong, if we squint, we can see parts of the
original object appear in each detection.
We use this observation to produce our first baseline.
Suppose we wish to invert feature $\phi$. We first train an exemplar LDA
detector \citep{hariharan2012discriminative} for this query,
$w = \Sigma^{-1}(y - \mu)$
where $\Sigma$ and $\mu$ are parameters estimated with a large dataset.
We then score $w$ against every sliding window in this database. The feature
inverse is the average of the top $K$ detections in RGB space:
$f^{-1}(\phi) = \frac{1}{K} \sum_{i=1}^K z_i$ where $z_i$ is an image of a
top detection.
This method, although simple, produces reasonable reconstructions,
even when the database does not contain the category of the feature template.
However, it is computationally expensive since it requires running an object detector across a large database.
\changed{Note that} a similar nearest neighbor method is used in brain research to
visualize what a person might be seeing \citep{nishimoto2011reconstructing}.
\subsection{Ridge Regression}
We describe a fast, parametric
inversion baseline based off ridge regression.
Let $X \in \mathbb{R}^P$ be a random variable representing a gray scale image and $\Phi \in \mathbb{R}^Q$ be a random variable of its corresponding feature. We define
these random variables to be normally distributed on a $P+Q$-variate Gaussian
$P(X, \Phi) \sim \mathcal{N}(\mu, \Sigma)$
with parameters
$\mu = \left[\begin{smallmatrix}
\mu_X & \mu_{\Phi}
\end{smallmatrix}\right]$ and $
\Sigma = \left[\begin{smallmatrix}
\Sigma_{XX} & \Sigma_{X\Phi} \\
\Sigma_{X\Phi}^T & \Sigma_{Y\Phi}
\end{smallmatrix}\right]$.
In order to invert a feature $y$, we calculate the most likely image from
the conditional Gaussian distribution $P(X | \Phi = \phi)$:
\begin{align}
f^{-1}(y) &= \argmax_{x \in \mathbb{R}^D} P(X = x | \Phi = \phi)
\end{align}
It is well known that a Gaussian distribution have a closed form conditional mode:
\begin{align}
f^{-1}(y) = \Sigma_{X\Phi} \Sigma_{\Phi\Phi}^{-1} (y - \mu_{\Phi}) + \mu_X
\end{align}
Under this inversion algorithm, any feature can be inverted by a single matrix
multiplication, allowing for inversion in under a second.
We estimate $\mu$ and $\Sigma$ on a large database. In practice, $\Sigma$ is not positive definite; we add
a small uniform prior (i.e., $\hat{\Sigma} = \Sigma + \lambda I$) so $\Sigma$
can be inverted. Since we wish to invert any feature, we assume that $P(X,
\Phi)$ is stationary \citep{hariharan2012discriminative}, allowing us to efficiently learn the covariance across
massive datasets. For features with varying spatial dimensions, we invert a feature by marginalizing
out unused dimensions.
\subsection{Direct Optimization}
We now provide a baseline that attempts to find images that, when we compute features on it, sufficiently match the original descriptor. In
order to do this efficiently, we only consider images that span a natural image basis.
Let $U \in \mathbb{R}^{D \times K}$ be the natural image basis. We found using the first $K$ eigenvectors of $\Sigma_{XX} \in \mathbb{R}^{D \times D}$ worked well for this basis. Any image $x \in \mathbb{R}^D$ can be encoded by coefficients $\rho \in \mathbb{R}^K$ in this basis: $x = U \rho$. We wish to minimize:
\begin{equation}
\begin{aligned}
&f^{-1}(y) = U\rho^* \\
&\textrm{where} \quad \rho^* = \argmin_{\rho \in \mathbb{R}^K} \left|\left| f(U\rho) - y \right|\right|_2^2
\end{aligned}
\label{eqn:highfreq-objective}
\end{equation}
Empirically we found success optimizing equation \ref{eqn:highfreq-objective} using
coordinate descent on $\rho$ with random restarts.
We use an over-complete basis corresponding to sparse Gabor-like filters for
$U$. We compute the eigenvectors of $\Sigma_{XX}$ across different scales and
translate smaller eigenvectors to form $U$.
\subsection{Nudged Dictionaries}
In order to compare our ability to recover multiple inversions, we describe two
baselines for multiple feature inversions.
Our first method modifies paired
dictionaries. Rather than incorporating similarity costs, we add noise to a feature to create a slightly different inversion by ``nudging'' it in random directions:
\begin{equation}
\begin{aligned}
&\alpha_i^* = \argmin_{\alpha_i} ||V\alpha_i-\phi + \gamma \epsilon_i||_2^2 + \lambda ||\alpha_i||_1\\
\end{aligned}
\end{equation}
where $\epsilon_i \sim \mathcal{N}(0_Q, I_Q)$ is noise from a standard normal distribution \changed{such that
$I_Q$ is the identity matrix}
and $\gamma \in \mathbb{R}$ is a hyperparameter that controls the strength of the diversity.
\subsection{Subset Dictionaries}
In addition, we compare against a second baseline that
modifies a paired dictionary by removing the basis elements that were
activated on previous iterations. Suppose the
first inversion activated the first $R$ basis elements. We
obtain a second inversion by only giving the paired dictionary the other
$K-R$ basis elements. This forces the sparse coding to use a disjoint basis
set, leading to different inversions.
\section{Evaluation of Single Inversion}
We evaluate our inversion algorithms using both qualitative and quantitative
measures. We use PASCAL VOC 2011 \citep{Everingham10} as our dataset and we
invert patches corresponding to objects. Any algorithm that required training
could only access the training set. During evaluation, only images from the
validation set are examined. The database for exemplar LDA excluded the
category of the patch we were inverting to reduce the potential effect of
dataset biases. Due to their popularity in object detection, we first focus on
evaluating HOG features.
\begin{figure}
\captionsetup[subfigure]{labelformat=empty}
\centering \subfloat[Original]{ \shortstack{
\includegraphics[width=0.18\linewidth]{figs/aeroplane_47_orig.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/sheep_199_orig.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/tvmonitor_204_orig.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/pottedplant_100_orig.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/cat_456_orig.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bottle_519_orig.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/cow_231_orig.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/bicycle_181_orig.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bird_490_orig.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bus_9_orig.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/train_311_orig.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/diningtable_181_orig.jpg}
} }
\subfloat[ELDA]{ \shortstack{
\includegraphics[width=0.18\linewidth]{figs/aeroplane_47_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/sheep_199_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/tvmonitor_204_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/pottedplant_100_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/cat_456_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bottle_519_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/cow_231_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bicycle_181_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bird_490_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bus_9_avg.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/train_311_avg.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/diningtable_181_avg.jpg}
} }
\subfloat[Ridge]{ \shortstack{
\includegraphics[width=0.18\linewidth]{figs/aeroplane_47_gmm.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/sheep_199_gmm.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/tvmonitor_204_gmm.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/pottedplant_100_gmm.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/cat_456_gmm.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bottle_519_gmm.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/cow_231_gmm.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bicycle_181_gmm.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bird_490_gmm.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bus_9_gmm.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/train_311_gmm.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/diningtable_181_gmm.jpg}
} }
\subfloat[Direct]{ \shortstack{
\includegraphics[width=0.18\linewidth]{figs/aeroplane_47_ggs.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/sheep_199_ggs.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/tvmonitor_204_ggs.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/pottedplant_100_ggs.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/cat_456_ggs.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bottle_519_ggs.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/cow_231_ggs.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/bicycle_181_ggs.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/bird_490_ggs.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bus_9_ggs.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/train_311_ggs.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/diningtable_181_ggs.jpg}
} }
\subfloat[PairDict]{ \shortstack{
\includegraphics[width=0.18\linewidth]{figs/aeroplane_47_pdl.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/sheep_199_pdl.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/tvmonitor_204_pdl.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/pottedplant_100_pdl.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/cat_456_pdl.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bottle_519_pdl.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/cow_231_pdl.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/bicycle_181_pdl.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/bird_490_pdl.jpg} \\
\includegraphics[width=0.18\linewidth]{figs/bus_9_pdl.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/train_311_pdl.jpg}\\
\includegraphics[width=0.18\linewidth]{figs/diningtable_181_pdl.jpg}
} }
\caption{We
show results for all four of our inversion algorithms on held out image
patches on similar dimensions common for object detection.}
\label{fig:results}
\end{figure}
\subsection{Qualitative Results}
We show our inversions in Figure \ref{fig:results} for a few object categories.
Exemplar LDA and ridge regression tend to produce blurred visualizations.
Direct optimization recovers high frequency details at the expense of extra
noise. Paired dictionary learning tends to produce the best visualization
for HOG descriptors. By learning a dictionary over the visual world and
the correlation between HOG and natural images, paired dictionary learning
recovered high frequencies without introducing significant noise.
\begin{figure*}
\centering
\includegraphics[width=0.45\linewidth]{figs/color/2008_002588.jpg}\hspace{2em}\includegraphics[width=0.45\linewidth]{figs/color/2011_003115.jpg}\\
\vspace{0.2em}
\includegraphics[width=0.45\linewidth]{figs/color/2011_002983.jpg}\hspace{2em}\includegraphics[width=0.45\linewidth]{figs/color/2008_002910.jpg}\\
\vspace{0.2em}
\includegraphics[width=0.45\linewidth]{figs/color/2011_001822.jpg}\hspace{2em}\includegraphics[width=0.45\linewidth]{figs/color/2011_001282.jpg}\\
\caption{We show results where our paired dictionary algorithm is trained to
recover RGB images instead of only grayscale images. The right shows the
original image and the left shows the inverse.}
\label{fig:color}
\end{figure*}
Although HOG does not explicitly encode color, we found that the paired
dictionary is able to recover color from HOG descriptors. Figure
\ref{fig:color} shows the result of training a paired dictionary to estimate
RGB images instead of grayscale images. While the paired dictionary assigns
arbitrary colors to man-made objects and indoor scenes, it frequently colors
natural objects correctly, such as grass or the sky, likely because those
categories are strongly correlated to HOG descriptors. We focus on grayscale
visualizations in this paper because we found those to be more intuitive for
humans to understand.
\changed{
We also explored whether our visualization algorithm could invert other
features besides HOG, such as deep features. Figure \ref{fig:qual-icnn} shows
how our algorithm can recover some details of the original image given only
activations from the last convolutional layer of \cite{krizhevsky2012imagenet}.
Although the visualizations are blurry, they do capture some important visual
aspects of the original images such as shapes and colors. This suggests that our visualization algorithm
may be general to the type of feature.}
\begin{figure}
\centering
\captionsetup[subfigure]{labelformat=empty}
\subfloat[Original]{
\includegraphics[width=0.32\linewidth]{figs/2009_002457-gray.jpg}
}
\subfloat[PairDict (seconds)]{
\includegraphics[width=0.32\linewidth]{figs/2009_002457-pd.jpg}
}
\subfloat[Greedy (days)]{
\includegraphics[width=0.32\linewidth]{figs/2009_002457-greedy.jpg}
}
\caption{Although our algorithms are good at inverting HOG, they are not perfect, and struggle to reconstruct high frequency detail. See text for details.}
\label{fig:notperfect}
\end{figure}
\begin{figure}
\captionsetup[subfigure]{labelformat=empty}
\centering
\subfloat[Original $x$]{
\includegraphics[width=0.32\linewidth]{figs/recursion1.jpg}
}
\subfloat[$x' = \phi^{-1}\left(\phi(x)\right)$]{
\includegraphics[width=0.32\linewidth]{figs/recursion2.jpg}
}
\subfloat[$x'' = \phi^{-1}\left(\phi(x')\right)$]{
\includegraphics[width=0.32\linewidth]{figs/recursion3.jpg}
}
\caption{We recursively compute HOG and invert it with a paired dictionary. While there is some information loss,
our visualizations still do a good job at accurately representing HOG features. $\phi(\cdot)$ is HOG, and $\phi^{-1}(\cdot)$ is the inverse.}
\label{fig:recursion}
\end{figure}
\begin{figure}
\centering
\captionsetup[subfigure]{labelformat=empty}
\subfloat[$40 \times 40$]{
\includegraphics[width=0.23\linewidth]{figs/chair-big.jpg}
}
\subfloat[$20 \times 20$]{
\includegraphics[width=0.23\linewidth]{figs/chair-medium.jpg}
}
\subfloat[$10 \times 10$]{
\includegraphics[width=0.23\linewidth]{figs/chair-small.jpg}
}
\subfloat[$5 \times 5$]{
\includegraphics[width=0.23\linewidth]{figs/chair-tiny.jpg}
}
\caption{Our inversion algorithms are sensitive to the HOG template size.
We show how performance degrades as the template becomes smaller.}
\label{fig:pyramid}
\end{figure}
While our visualizations do a good job at representing HOG features, they have
some limitations. Figure \ref{fig:notperfect} compares our best visualization
(paired dictionary) against a greedy algorithm that draws triangles of random
rotation, scale, position, and intensity, and only accepts the triangle if it
improves the reconstruction. If we allow the greedy algorithm to execute for an
extremely long time (a few days), the visualization better shows higher
frequency detail. This reveals that there exists a visualization better than
paired dictionary learning, although it may not be tractable \changed{for large scale experiments}. In a related
experiment, Figure \ref{fig:recursion} recursively computes HOG on the inverse
and inverts it again. This recursion shows that there is some loss between
iterations, although it is minor and appears to discard high frequency details.
Moreover, Figure \ref{fig:pyramid} indicates that our inversions are sensitive
to the dimensionality of the HOG template. Despite these limitations, our
visualizations are, as we will now show, still perceptually intuitive for
humans to understand.
\subsection{Quantitative Results}
We quantitatively evaluate our algorithms under two benchmarks. Firstly, we use an automatic
inversion metric that measures how well our inversions reconstruct original
images. Secondly, we conducted a large visualization challenge with human
subjects on Amazon Mechanical Turk (MTurk), which is designed to determine how
well people can infer high level semantics from our visualizations.
\emph{Pixel Level Reconstruction:}
We consider the inversion performance of our algorithm: given a HOG feature
$y$, how well does our inverse $\phi^{-1}(y)$ reconstruct the original pixels
$x$ for each algorithm? Since HOG is invariant up to a constant shift and
scale, we score each inversion against the original image with normalized cross
correlation. Our results are shown in Table \ref{tab:inversionobjective}.
Overall, exemplar LDA does the best at pixel level reconstruction.
\emph{Semantic Reconstruction:}
While the inversion benchmark evaluates how well the inversions reconstruct the
original image, it does not capture the high level content of the inverse: is
the inverse of a sheep still a sheep? To evaluate this, we conducted a study on
MTurk. We sampled 2,000 windows corresponding to objects in
PASCAL VOC 2011. We then showed participants an inversion from one of our
algorithms and asked participants to classify it into one of the 20 categories. Each
window was shown to three different users. Users were required to pass a
training course and qualification exam before participating in order to
guarantee users understood the task. Users could optionally select that they
were not confident in their answer. We also compared our algorithms against the
standard black-and-white HOG glyph popularized by \citep{dalal2005histograms}.
\begin{figure}
\centering
\includegraphics[width=12em]{figs/vis/121.jpg}\hspace{2em}
\includegraphics[width=12em]{figs/vis/13.jpg}
\includegraphics[width=12em]{figs/vis/23.jpg}\hspace{2em}
\includegraphics[width=12em]{figs/vis/34.jpg}
\includegraphics[width=12em]{figs/vis/37.jpg}\hspace{2em}
\includegraphics[width=12em]{figs/vis/54.jpg}
\includegraphics[width=12em]{figs/vis/61.jpg}\hspace{2em}
\includegraphics[width=12em]{figs/vis/74.jpg}
\caption{\changed{We show visualizations from our method to invert features
from deep convolutional networks. Although the visualizations
are blurry, they capture some key aspects of the original images, such as shapes and colors. Our visualizations are inverting
the last convolutional layer of \cite{krizhevsky2012imagenet}.}}
\label{fig:qual-icnn}
\end{figure}
\begin{table}
\centering
\begin{tabular}{l | c c c c c}
Category & ELDA & Ridge & Direct & PairDict \\
\hline
aeroplane & \textbf{0.634} & \textbf{0.633} & 0.596 & 0.609 \\
bicycle & 0.452 & \textbf{0.577} & 0.513 & 0.561 \\
bird & \textbf{0.680} & 0.650 & 0.618 & 0.638 \\
boat & \textbf{0.697} & 0.678 & 0.631 & 0.629 \\
bottle & \textbf{0.697} & 0.683 & 0.660 & 0.671 \\
bus & 0.627 & \textbf{0.632} & 0.587 & 0.585 \\
car & 0.668 & \textbf{0.677} & 0.652 & 0.639 \\
cat & \textbf{0.749} & 0.712 & 0.687 & 0.705 \\
chair & \textbf{0.660} & 0.621 & 0.604 & 0.617 \\
cow & \textbf{0.720} & 0.663 & 0.632 & 0.650 \\
table & \textbf{0.656} & 0.617 & 0.582 & 0.614 \\
dog & \textbf{0.717} & 0.676 & 0.638 & 0.667 \\
horse & \textbf{0.686} & 0.633 & 0.586 & 0.635 \\
motorbike & 0.573 & \textbf{0.617} & 0.549 & 0.592 \\
person & \textbf{0.696} & 0.667 & 0.646 & 0.646 \\
pottedplant & \textbf{0.674} & \textbf{0.679} & 0.629 & 0.649 \\
sheep & \textbf{0.743} & 0.731 & 0.692 & 0.695 \\
sofa & \textbf{0.691} & 0.657 & 0.633 & 0.657 \\
train & \textbf{0.697} & 0.684 & 0.634 & 0.645 \\
tvmonitor & \textbf{0.711} & 0.640 & 0.638 & 0.629 \\
\hline
Mean & \textbf{0.671} & 0.656 &0.620 & 0.637\\
\end{tabular}
\caption{We evaluate the performance of our inversion algorithm by comparing
the inverse to the ground truth image using the mean normalized cross
correlation. Higher is better; a score of 1 is perfect.}
\label{tab:inversionobjective}
\end{table}
\setlength{\tabcolsep}{2pt}
\begin{table}
\centering
\begin{tabular}{l | c c c c c | c}
Category & ELDA & Ridge & Direct & PairDict & Glyph & Expert \\
\hline
aeroplane & 0.433& 0.391& 0.568& \textbf{0.645}& 0.297 & 0.333\\
bicycle & 0.327& 0.127& 0.362& 0.307& \textbf{0.405} & 0.438 \\
bird & 0.364& 0.263& \textbf{0.378}& 0.372& 0.193 & 0.059\\
boat & 0.292& 0.182& 0.255& \textbf{0.329}& 0.119 & 0.352\\
bottle & 0.269& 0.282& 0.283& \textbf{0.446}& 0.312 & 0.222\\
bus & 0.473& 0.395& \textbf{0.541}& 0\textbf{.549}& 0.122 & 0.118\\
car & 0.397& 0.457& \textbf{0.617}& 0.585& 0.359 & 0.389\\
cat & 0.219& 0.178& \textbf{0.381}& 0.199& 0.139 & 0.286 \\
chair & 0.099& 0.239& 0.223& \textbf{0.386}& 0.119 & 0.167\\
cow & 0.133& 0.103& \textbf{0.230}& 0.197& 0.072 & 0.214\\
table & 0.152& 0.064& 0.162& \textbf{0.237}& 0.071 & 0.125\\
dog & 0.222& 0.316& \textbf{0.351}& 0.343& 0.107 & 0.150\\
horse & 0.260& 0.290& 0.354& \textbf{0.446}& 0.144 & 0.150\\
motorbike & 0.221& 0.232& \textbf{0.396}& 0.224& 0.298 & 0.350\\
person & 0.458& 0.546& 0.502& \textbf{0.676}& 0.301 & 0.375\\
pottedplant & 0.112& 0.109& \textbf{0.203}& 0.091& 0.080 & 0.136\\
sheep & 0.227& 0.194& \textbf{0.368}& 0.253& 0.041 & 0.000\\
sofa & 0.138& 0.100& 0.162& \textbf{0.293}& 0.104 & 0.000\\
train & 0.311& 0.244& 0.316& \textbf{0.404}& 0.173 & 0.133\\
tvmonitor & 0.537& 0.439& 0.449& \textbf{0.682}& 0.354 & 0.666\\
\hline
Mean & 0.282& 0.258& 0.355& \textbf{0.383} & 0.191 & 0.233
\end{tabular}
\caption{We evaluate visualization performance across twenty PASCAL VOC
categories by asking MTurk participants to classify our inversions. Numbers are
percent classified correctly; higher is better. Chance is $0.05$. Glyph refers
to the standard black-and-white HOG diagram popularized by
\citep{dalal2005histograms}. Paired dictionary learning provides the best
visualizations for humans. Expert refers to MIT PhD students in computer vision
performing the same visualization challenge with HOG glyphs.}
\label{tab:userstudy}
\end{table}
Our results in Table \ref{tab:userstudy} show that paired dictionary learning
and direct optimization provide the best visualization of HOG descriptors for
humans. Ridge regression and exemplar LDA perform better than the glyph, but
they suffer from blurred inversions. Human performance on the HOG glyph is
generally poor, and participants were even the slowest at completing that
study. Interestingly, the glyph does the best job at visualizing bicycles,
likely due to their unique circular gradients. Our results overall suggest that
visualizing HOG with the glyph is misleading, and richer visualizations from
our paired dictionary are useful for interpreting HOG features.
Our experiments suggest that humans can predict the performance of object detectors
by only looking at HOG visualizations. Human accuracy on
inversions and state-of-the-art object detection AP scores from
\citep{felzenszwalb2010cascade} are correlated with a Spearman's rank
correlation coefficient of 0.77.
We also asked computer vision PhD students at MIT to classify HOG glyphs in order to
compare MTurk participants with experts in HOG. Our results are summarized
in the last column of Table \ref{tab:userstudy}. HOG experts performed slightly
better than non-experts on the glyph challenge, but experts on glyphs did not
beat non-experts on other visualizations. This result suggests that our
algorithms produce more intuitive visualizations even for object detection
researchers.
\section{Evaluation of Multiple Inversions}
\changed{Since features are many-to-one functions, our visualization algorithms should be able to recover multiple inversions
for a feature descriptor.
We look at the multiple inversions from deep network features because these features appear to be robust to several invariances.}
To conduct our experiments with multiple inversions, we inverted features from the AlexNet convolutional
neural network \citep{krizhevsky2012imagenet} trained on ImageNet
\citep{deng2009imagenet,russakovsky2014imagenet}. We use the publicly
available Caffe software package \citep{Jia13caffe} to extract features. We use
features from the last convolutional layer (pool5), which has been shown to
have strong performance on recognition tasks \citep{girshick2013rich}. We
trained the dictionaries $U$ and $V$ using random windows from the PASCAL VOC
2007 training set \citep{Everingham10}. We tested on two thousand random
windows corresponding to objects in the held-out PASCAL VOC 2007 validation
set.
\begin{figure}[t]
\centering
\subfloat[Affinity = Color]{
\includegraphics[width=0.47\linewidth]{figs/color-qual.pdf}
}
\hspace{0.01\linewidth}
\subfloat[Affinity = Edge]{
\includegraphics[width=0.47\linewidth]{figs/edge-qual.pdf}
}
\vspace{1em}
\subfloat[Nudged Dict]{
\includegraphics[width=0.47\linewidth]{figs/baselineA-qual.pdf}
}
\hspace{0.01\linewidth}
\subfloat[Subset Dict]{
\includegraphics[width=0.47\linewidth]{figs/baselineB-qual.pdf}
}
\caption{We show the first three inversions for a few patches from our testing
set. Notice how the color (a) and edge (b) variants of our method tend to
produce different inversions. The baselines tend to either similar in image
space (c) or do not match well in feature space (d). Best viewed on screen. }
\label{fig:qual}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.48\linewidth]{figs/qual1/edge-matrix.jpg}
\hspace{0.02\linewidth}
\includegraphics[width=0.48\linewidth]{figs/qual1/edge-matrix-2.jpg}
\caption{The edge affinity can often result in subtle differences. Above, \changed{we show a difference matrix} between the first three inversions that highlights differences between
all pairs of a few inversions from one CNN feature. The margins show the inversions, and the inner black squares show the absolute difference. White means larger difference. Notice that our algorithm is able to recover inversions with shifts of gradients.}
\label{fig:edge}
\end{figure}
\subsection{Qualitative Results}
We first look at a few qualitative results for our multiple feature
inversions. Figure \ref{fig:qual} shows a few examples for both our method (top
rows) and the baselines (bottom rows). The 1st column shows the result of
a paired dictionary on CNN features, while the 2nd and 3rd show the
additional inversions that our method finds. While the results are blurred,
they do tend to resemble the original image in rough shape and color.
The color affinity in Figure \ref{fig:qual}a is often able to produce inversions
that vary slightly in color. Notice how the cat and the floor are changing
slightly in hue, and the grass the bird is standing on is varying slightly. The
edge affinity in Figure \ref{fig:qual}b can occasionally generate inversions with
different edges, although the differences can be subtle. To better show the differences
with the edge affinity, we visualize a difference matrix in Figure \ref{fig:edge}.
Notice how the edges of the bird and person shift between each inversion.
\begin{figure}[t]
\centering
\includegraphics[width=0.8\linewidth]{figs/hog-equiv.jpg}
\caption{\changed{The block-wise histograms of HOG allow for gradients in the
image to shift up to their bin size without affecting the feature descriptor.
By using our visualization algorithm with the edge affinity matrix, we can
recover multiple HOG inversions that differ by edges subtly shifting. Above, we
show a difference matrix between the first three inversions for a downsampled
image of a man shown in the top left corner. Notice the vertical gradient in
the background shifts between the inversions, and the man's head move
slightly.}}
\label{fig:multiple-hog}
\end{figure}
The baselines tend to either produce nearly identical inversions or inversions
that do not match well in feature space. Nudged dictionaries in Figure
\ref{fig:qual}c frequently retrieves inversions that look nearly identical.
Subset dictionaries in Figure \ref{fig:qual}d recovers different inversions,
but the inversions do not match in feature space, likely because this baseline
operates over a subset of the basis elements.
\changed{
Although HOG is not as invariant to visual transformations as deep features, we can
still recover multiple inversions from a HOG descriptor. The block-wise histograms of HOG allow for
gradients in the image to shift up to their bin size without affecting the feature descriptor. Figure \ref{fig:multiple-hog}
shows multiple inversions from a HOG descriptor of a man where the person shifts slightly between each inversion.
}
\subsection{Quantitative Results}
We wish to quantify how well our inversions
trade off matching in feature space versus having diversity in image space. To
evaluate this, we calculated Euclidean distance between the features of the
first and second inversions from each method, $||\phi(x_1) -
\phi(x_2)||_2$, and compared it to the Euclidean distance of the inversions in
Lab image space, $||L(x_1) - L(x_2)||_2$ where $L(\cdot)$ is the Lab
colorspace transformation.\footnote{We chose Lab because Euclidean distance in this
space is known to be perceptually uniform \citep{jain1989fundamentals}, which
we suspect better matches human interpretation.} We consider one inversion
algorithm to be better than another method if, for the same distance in feature space, the
image distance is larger.
We show a scatter plot of this metric in
Figure \ref{fig:eval1} for our method with different similarity costs. The thick
lines show the median image distance for a given feature distance. The overall
trend suggests that our method produces more diverse images for the same
distance in feature space. Setting the affinity matrix $A$ to perform color
averaging produces the most image variation for CNN features in order to keep
the feature space accuracy small. The baselines in general do not perform as
well, and baseline with subset dictionaries struggles to even match in feature space, causing the
green line to abruptly start in the middle of the plot. The edge affinity
produces inversions that tend to be more diverse than baselines, although this effect
is best seen qualitatively in the next section.
We consider a second evaluation metric designed
to determine how well our inversions match the original features. Since
distances in a feature space are unscaled, they can be difficult to interpret, so we use a
normalized metric. We calculate the ratio of distances that the inversions make to the original
feature: $r = \frac{||\phi(x_2) - f||_2}{||\phi(x_1) - f||_2}$ where $f$ is the original
feature and $x_1$ and $x_2$ are the first and second inversions. A value of $r = 1$ implies
the second inversion is just as close to $f$ as the first. We then compare the ratio $r$ to
the Lab distance in image space.
\begin{figure}[t]
\centering
\includegraphics[trim=15em 1em 15em 1em,clip,width=\linewidth]{figs/eval.pdf}
\caption{We evaluate the
performance of our multiple inversion algorithm. The horizontal axis is the
Euclidean distance between the first and second inversion in CNN space and the
vertical axis is the distance of the same inversions in Lab colorspace.
\changed{This plot suggests that incorporating diversity costs into the inversion are
able to produce more diverse multiple visualizations for the same reconstruction error.}
Thick lines show the median image distance for a given feature distance.}
\label{fig:eval1}
\end{figure}
\begin{figure}[t]
\centering
\subfloat[Color]{\includegraphics[width=0.32\linewidth]{figs/ratio_rgb_axis.png}}
\hspace{.01em}
\subfloat[Identity]{\includegraphics[width=0.32\linewidth]{figs/ratio_standard_axis.png}}
\hspace{.01em}
\subfloat[Edge]{\includegraphics[width=0.32\linewidth]{figs/ratio_edge_axis.png}}
\\
\subfloat[Nudged Dict]{\includegraphics[width=0.32\linewidth]{figs/ratio_baseline_axis.png}}
\hspace{.01em}
\subfloat[Subset Dict]{\includegraphics[width=0.32\linewidth]{figs/ratio_delete_axis.png}}
\caption{We show density maps that visualize image
distance versus the ratio distances in feature space: $r = \frac{||\phi(x_2)-f||_2}{||\phi(x_1)-f||_2}$.
A value of $r = 1$ means that the two inversions are the same distance from the original feature. Black means most dense
and white is zero density. Our results suggest that our method with the affinity matrix set to color averaging produces more diverse
visualizations for the same $r$ value.}
\label{fig:eval2}
\end{figure}
We show results for our second metric in
Figure \ref{fig:eval2} as a density map comparing image distance and the ratio of
distances in feature space. Black is a higher density and implies that the
method produces inversions in that region more frequently. This experiment
shows that for the same ratio $r$, our approach tends to produce more diverse
inversions when affinity is set to color averaging. Baselines frequently performed poorly,
and struggled to generate diverse images that are close in feature space.
\section{Understanding Object Detectors}
\changed{While the goal of this paper is to visualize object detection
features, in this section we will use our visualizations to inspect the
behavior of object detection systems. Due to our budget for experiments, we focus
on HOG features.}
\subsection{HOGgles}
Our visualizations reveal that the world that features see is slightly
different from the world that the human eye perceives. Figure
\ref{fig:seeintodarkA} shows a normal photograph of a man standing in a dark
room, but Figure \ref{fig:seeintodarkB} shows how HOG features see the same
man. Since HOG is invariant to illumination changes and amplifies gradients,
the background of the scene, normally invisible to the human eye, materializes
in our visualization.
In order to understand how this clutter affects object detection, we visualized
the features of some of the top false alarms from the Felzenszwalb et al.\
object detection system \citep{felzenszwalb2010object} when applied to the
PASCAL VOC 2007 test set. Figure \ref{fig:topdets} shows our visualizations of
the features of the top false alarms. Notice how the false alarms look very
similar to true positives. While there are many different types of detector
errors, this result suggests that these particular failures are due to
limitations of HOG, and consequently, even if we develop better learning
algorithms or use larger datasets, these will false alarms will likely persist.
Figure \ref{fig:topdetsrgb} shows the corresponding RGB image patches for the
false positives discussed above. Notice how when we view these detections in
image space, all of the false alarms are difficult to explain. Why do chair
detectors fire on buses, or people detectors on cherries? By
visualizing the detections in feature space, we discovered that the learning
algorithm made reasonable failures since the features are deceptively
similar to true positives.
\subsection{Human+HOG Detectors}
Although HOG features are designed for machines, how well do humans see in HOG
space? If we could quantify human vision on the HOG feature space, we could get
insights into the performance of HOG with a perfect learning algorithm
(people). Inspired by Parikh and Zitnick's methodology
\citep{parikh2011human,parikh2010role}, we conducted a large human study where
we had Amazon Mechanical Turk participants act as sliding window HOG based object
detectors.
We built an online interface for humans to look at HOG visualizations of window
patches at the same resolution as DPM. We instructed participants to either classify
a HOG visualization as a positive example or a negative example for a category.
By averaging over multiple people (we used 25 people per window), we obtain a
real value score for a HOG patch. To build our dataset, we sampled top
detections from DPM on the
PASCAL VOC 2007 dataset for a few categories. Our dataset consisted of around $5,000$
windows per category and around $20\%$ were true positives.
\begin{figure}
\centering
\subfloat[Human Vision]{
\includegraphics[height=18em]{figs/seeintodark_original.jpg}
\label{fig:seeintodarkA}
}
\subfloat[HOG Vision]{
\includegraphics[height=18em]{figs/seeintodark_inverse.jpg}
\label{fig:seeintodarkB}
}
\caption{Feature inversion reveals the world that object detectors see. The left
shows a man standing in a dark room. If we compute HOG on this
image and invert it, the previously dark scene behind the man emerges. Notice
the wall structure, the lamp post, and the chair in the bottom right hand
corner.}
\label{fig:seeintodark}
\end{figure}
Figure \ref{fig:hoggles} shows precision recall curves for the Human + HOG based
object detector. In most cases, human subjects classifying HOG visualizations
were able to rank sliding windows with either the same accuracy or better than
DPM. Humans tied DPM for recognizing cars, suggesting that performance may be
saturated for car detection on HOG. Humans were slightly superior to DPM for
chairs, although performance might be nearing saturation soon. There appears to
be the most potential for improvement for detecting cats with HOG. Subjects
performed slightly worst than DPM for detecting people, but we believe this is
the case because humans tend to be good at fabricating people in abstract
drawings.
We then repeated the same experiment as above on chairs except we instructed
users to classify the original RGB patch instead of the HOG visualization. As
expected, humans have near perfect accuracy at detecting chairs with
RGB sliding windows. The performance gap between the Human+HOG detector and Human+RGB detector
demonstrates the amount of information that HOG features discard.
\begin{figure}
\centering
\includegraphics[width=0.48\linewidth]{figs/chair-hoggles.pdf}
\includegraphics[width=0.48\linewidth]{figs/cat-hoggles.pdf}
\includegraphics[width=0.48\linewidth]{figs/car-hoggles.pdf}
\includegraphics[width=0.48\linewidth]{figs/person-hoggles.pdf}
\caption{By instructing multiple human subjects to classify the visualizations, we show
performance results with an ideal learning algorithm (i.e., humans) on the HOG
feature space. Please see text for details.}
\label{fig:hoggles}
\end{figure}
\begin{figure*}
\centering
\includegraphics[height=6.7em]{figs/models/car_4_parts.jpg}\hspace{0.5em}
\includegraphics[height=6.7em]{figs/models/person_2_parts.jpg}\hspace{0.5em}
\includegraphics[height=6.7em]{figs/models/bottle_6_parts.jpg}\hspace{0.5em}
\includegraphics[height=6.7em]{figs/models/bicycle_2_parts.jpg}\hspace{0.5em}
\includegraphics[height=6.7em]{figs/models/motorbike_2_parts.jpg}\hspace{0.5em}
\includegraphics[height=6.7em]{figs/models/pottedplant_4_parts.jpg}\hspace{0.1em}\includegraphics[height=6.7em]{figs/models/pottedplant_4_parts_hog.jpg} \\
\vspace{0.5em}
\includegraphics[height=6.7em]{figs/models/train_4_parts.jpg}\hspace{0.5em}
\includegraphics[height=6.7em]{figs/models/bus_2_parts.jpg}\hspace{0.5em}
\includegraphics[height=6.7em]{figs/models/horse_2_parts.jpg}\hspace{0.5em}
\includegraphics[height=6.7em]{figs/models/tvmonitor_4_parts.jpg}\hspace{0.5em}
\includegraphics[height=6.7em]{figs/models/chair_4_parts.jpg}\hspace{0.1em}\includegraphics[height=6.7em]{figs/models/chair_4_parts_hog.jpg}
\caption{We visualize a few deformable parts models trained with
\citep{felzenszwalb2010object}. Notice the structure that emerges with our
visualization. First row: car, person, bottle, bicycle, motorbike, potted
plant. Second row: train, bus, horse, television, chair. For the right most
visualizations, we also included the HOG glyph. Our visualizations
tend to reveal more detail than the glyph.}
\label{fig:prototypes}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{figs/false-positives-rgb.png}
\caption{We show the original RGB patches that correspond to the
visualizations from Figure \ref{fig:topdets}. We print the original patches on
a separate page to highlight how the inverses of false positives look
like true positives. We recommend comparing this figure side-by-side with
Figure \ref{fig:topdets}.} \label{fig:topdetsrgb}
\end{figure*}
Our experiments suggest that there is still some performance left to be
squeezed out of HOG. However, DPM is likely operating very close to the
performance limit of HOG. Since humans are the ideal learning agent and they
still had trouble detecting objects in HOG space, HOG may be too lossy of a
descriptor for high performance object detection. If we wish to significantly
advance the state-of-the-art in recognition, we suspect focusing effort on
building better features that capture finer details as well as higher level
information will lead to substantial performance improvements in object
detection. Indeed, recent advances in object recognition have been driven
by learning with richer features \citep{girshick2013rich}.
\subsection{Model Visualization}
We found our algorithms are also useful for visualizing the learned models of an
object detector. Figure \ref{fig:prototypes} visualizes the root templates and
the parts from \citep{felzenszwalb2010object} by inverting the
positive components of the learned weights. These visualizations provide hints on which
gradients the learning found discriminative. Notice the detailed structure
that emerges from our visualization that is not apparent in the HOG glyph. Often, one can recognize the category of the detector by only looking at the
visualizations.
\section{Conclusion}
We believe visualizations can be a powerful tool for understanding object
detection systems and advancing research in computer vision. To this end, this
paper presented and evaluated several algorithms to visualize object detection
features. We hope more intuitive visualizations will prove useful for the
community.
\emph{Acknowledgments:} We thank the CSAIL Vision Group
for many important discussions. Funding was provided by a NSF GRFP to CV, a
Facebook fellowship to AK, and a Google research award, ONR MURI N000141010933
and NSF Career Award No. 0747120 to AT.
\bibliographystyle{spbasic}
|
1,108,101,565,667 | arxiv | \section{Introduction}\label{Sec1}
We define ${\mathcal Q} _n(w)$ (where $n \ge 1$ and $-1 <w < \frac{1}{n-1}$)
as in the Abstract, and let
${\mathcal O} _n := {\mathcal Q} _n(0)$,
${\mathcal P} _n := {\mathcal Q} _n(\frac{1}{n})$.
Hadwiger~\cite{Hadw1951} showed in 1951
(see also Hertel~\cite{Hert2003})
that ${\mathcal Q} _n(w)$ is equidissectable with a cube for all $n$.
His proof is indirect and not constructive.
The simplex ${\mathcal O}_n$ is especially interesting:
it has vertices
\beql{EqH1}
000\ldots00,~
100\ldots00,~
110\ldots00,~
111\ldots00,~
\ldots,~
111\ldots10,~
111\ldots11 \,,
\end{equation}
and is an {\em orthoscheme} in Coxeter's terminology~\cite{Coxe1973}.
Because of applications to encoding
and decoding constant-weight codes~\cite{TVS2007},
we are interested in algorithms that carry out the
dissection of ${\mathcal O}_n$ in an efficient manner.
In fact our question is slightly easier than the classical
problem, because we only need to decompose
${\mathcal O}_n$ into pieces which can be
reassembled to form a rectangular parallelepiped
(or $n$-dimensional ``brick''), not necessarily a cube\footnote{For the
problems of dissecting a rectangle into a square and
a three-dimensional rectangular parallelepiped into a cube see
Boltianskii~\cite[p.~52]{Bolt1978},
Cohn~\cite{Cohn1974},
Frederickson~\cite[Page~236]{Fred1997}.}.
For the case $n=3$, Hill~\cite{Hill1895} had already shown in
1895 that the tetrahedra ${\mathcal Q}_3(w)$ are equidissectable with a cube.
It appears that that the first explicit
dissection of ${\mathcal O}_3$ into a cube was given by Sydler~\cite{Sydl1956}
in 1956.
Sydler shows that ${\mathcal O}_3$ may be cut into four pieces
which can be reassembled to form a prism with base
an isosceles right triangle.
One further cut then gives a brick.
Sydler's dissection can be seen in a number of references
(Boltianskii~\cite[p.~99]{Bolt1978},
Cromwell~\cite[p.~47]{Crom1997},
Frederickson~\cite[Fig.~20.4]{Fred1997},
Sydler~\cite{Sydl1956},
Wells~\cite[p.~251]{Well1991})
and we will not reproduce it here.
Some of these references incorrectly attribute
Sydler's dissection to Hill.
In our earlier paper~\cite{TVS2007},
we gave a dissection of ${\mathcal O}_n$ to a prism
${\mathcal O}_{n-1} \times I$ for all $n$
that requires $(n^2-n+2)/2$ pieces.
In three dimensions this uses four pieces,
the same number as Sydler's, but is somewhat simpler
than Sydler's in that all our cuts are made along
planes perpendicular to coordinate axes.
By iterating this construction we eventually
obtain a dissection of ${\mathcal O}_n$ into an $n$-dimensional brick.
The total number of pieces in the overall dissection
is large (roughly $(n!)^2/2^n$),
but the complexity of computing the coordinates
of a point in the final brick, given a initial point in ${\mathcal O}_n$, is only
$O(n^2)$.
In 1985, Sch\"obi~\cite{Scho1985}\footnote{According
to Frederickson~\cite[Page 234]{Fred1997},
this construction was independently found by Anton Hanegraaf, unpublished.} gave
a dissection of ${\mathcal Q}_3(w)$ (where $-1<w<\frac{1}{2}$)
into a prism with base an equilateral triangle that uses only three
pieces (see Figs. \ref{Fig4new}, \ref{Fig5new} below,
also Frederickson~\cite[Fig.~20.5]{Fred1997}).
There is a way to cut ${\mathcal Q}_n(w)$ for any $n$ into $n$ pieces
that is a natural generalization of Sch\"obi's dissection,
but for a long time we were convinced that
already for $n=4$ these pieces could not be reassembled to form
a prism $P \times I$ for any $(n-1)$-dimensional polytope $P$.
In fact, we were wrong, and the main goal of this paper is to
use the ``Two Tile Theorem'' (Theorem \ref{Th1}) to generalize
Sch\"obi's dissection to all dimensions.
We will show in Theorem \ref{Th2} that
${\mathcal Q}_n(w)$ can be cut into $n$ pieces
that can be reassembled to form a prism
$c {\mathcal P}_{n-1} \times I_{\ell}$,
where
$c = \sqrt{ (n-1)(w+1)/n }$ and
$\ell = \sqrt{ (1-w(n-1))/n }$.
The cross-section is always proportional to
${\mathcal P}_{n-1} = {\mathcal Q}_{n-1}(\frac{1}{n-1})$, independently of $w$.
By iterating this dissection we eventually decompose
${\mathcal Q}_n(w)$ (and in particular ${\mathcal O}_n$) into a brick.
The total number of pieces is at most $n!$
and the complexity of computing the map from ${\mathcal Q} _n(w)$
to the brick is $O(n^2)$ (Theorem \ref{Th3}).
Although this is the same order of complexity as the algorithm
given in our earlier paper \cite{TVS2007}, the present
algorithm is simpler and the number of
pieces is much smaller.
The recreational literature on dissections consists mostly
of {\em ad hoc} constructions, although there are a few
general techniques, which can be found in the books of
Lindgren~\cite{Lind1964} and Frederickson~\cite{Fred1997}, \cite{Fred2002}.
The construction we have found the most useful is
based on group theory.
We call it the ``Two Tile Theorem'',
and give our version of it in Section \ref{Sec2},
together with several examples.
In Section \ref{Sec3} we state and prove the main theorem,
and then in Section \ref{Sec4} we study the overall algorithm for
dissecting ${\mathcal O}_n$ into a brick.
Before finding the general dissection mentioned above,
we found a different generalization of Sch\"obi's dissection
which applies specifically to the $4$-dimensional case.
This is described in Section \ref{Sec5}.
It is of interest because it is partially
(and in a loose sense) a ``hinged''
dissection (cf. Frederickson~\cite{Fred2002}).
After two cuts have been made, the first two motions
each leave a two-dimensional face fixed.
We then make a third cut, giving a total of six pieces
which can reassembled to give a prism $c {\mathcal P}_3 \times I$.
This construction is also of interest because
it is symmetrical, and
it is the only {\em ad hoc} dissection we know of in four dimensions
(the dissections found by Paterson~\cite{Pate1996}
are all based on a version of the Two Tile Theorem).
A note about applications.
If we have a dissection of a polytope $P$ into a brick
$I_{\ell_1} \times I_{\ell_2} \times \cdots \times I_{\ell_n}$,
then we have a natural way to encode the points of $P$
into $n$-tuples of real numbers. This bijection provides a useful
parameterization of the points of $P$.
It may be used for source coding, if we
have a source that produces points uniformly
distributed over $P$
(for example, $P$ might be the Voronoi cell of a lattice).
Conversely, the bijection may be used in simulation,
when we wish to synthesize a uniform distribution
of points from $P$.
For the application
to constant-weight codes we refer the
reader to \cite{TVS2007}.
\vspace*{+.1in}
\noindent{\bf Notation.}
A polytope in ${\mathbb R}^n$ is a union of a finite
number of finite $n$-dimensional simplices. It need be neither
convex nor connected.
Let $P, P_1, \ldots, P_k$ be polytopes in ${\mathbb R}^n$.
By $P=P_1 + \cdots + P_k$ we mean
that the interiors of $P_1, \ldots, P_k$
are pairwise disjoint and $P=P_1 \cup \ldots \cup P_k$.
Let $\Gamma$ be a group of isometries of ${\mathbb R}^n$.
Two polytopes $P$, $Q$ in ${\mathbb R}^n$ are said to be
\emph{$\Gamma$-equidissectable}
if there are polytopes $P_1, \ldots, P_k$, $Q_1, \ldots, Q_k$
for some integer $k \ge 1$ such that
$P=P_1 + \ldots + P_k$,
$Q=Q_1 + \ldots + Q_k$, and
$P_1^{g_1} = Q_1, \ldots, P_k^{g_k} = Q_k$
for appropriate elements $g_1, \ldots, g_k \in \Gamma$.
In case $\Gamma$ is the full isometry group of ${\mathbb R}^n$
we write $P~{\thicksim}~Q$ and
say that $P$ and $Q$ are \emph{equidissectable}.
Isometries may involve reflections:
we do not insist that the dissections can be carried out
using only transformations of determinant $+1$.
$I_{\ell}$ denotes an interval of length $\ell$,
$I$ is a finite interval of unspecified length,
and $I_\infty = {\mathbb R}^1$.
For background information about dissections and Hilbert's third problem,
and any undefined terms, we refer the reader to
the excellent surveys by
Boltianskii~\cite{Bolt1978},
Dupont~\cite{Dupo2001},
Frederickson~\cite{Fred1997}, \cite{Fred2002},
Lindgren~\cite{Lind1964},
McMullen~\cite{McMu1993},
McMullen and Schneider~\cite{McSc1983},
Sah~\cite{Sah1979}
and
Yandell~\cite{Yand2002}.
\section{The ``Two Tile Theorem''}\label{Sec2}
Let $A \subset {\mathbb R}^n$ be a polytope, $\Gamma$ a group
of isometries of ${\mathbb R}^n$ and ${\Omega}$ a subset of ${\mathbb R}^n$.
If the images of $A$ under the action
of $\Gamma$ have disjoint interiors,
and ${\Omega} = \cup_{g \in \Gamma} A^g$, we
say that $A$ is a \emph{$\Gamma$-tile} for ${\Omega}$.
This implies that $\Gamma$ is discontinuous and
fixed-point-free.
Versions of the following theorem---although
not the exact version that we need---have
been given by
Aguil\'o, Fiol and Fiol~\cite[Lemma~2.2]{AFF2000},
M\"uller~\cite[Theorem~3]{Mull1988} and
Paterson~\cite{Pate1996}.
It is a more precise version of the technique of
``superposing tesselations'' used by
Macaulay \cite{Maca1914}, \cite{Maca1919},
Lindgren \cite[Chap. 2]{Lind1964}
and
Frederickson \cite[p. 29]{Fred1997}, \cite[Chap. 3]{Fred2002}.
\begin{theorem}\label{Th1}
If for some set ${\Omega} \subset {\mathbb R}^n$ and
some group $\Gamma$ of isometries of ${\mathbb R}^n$,
two $n$-dimensional polytopes $A$ and $B$ are both $\Gamma$-tiles for ${\Omega}$,
then $A$ and $B$ are $\Gamma$-equidissectable.
\end{theorem}
\vspace*{+.1in}
\noindent{\bf Proof.}
We have
$$
A = A \cap {\Omega} = A \cap \bigcup_{g \in \Gamma} B^g
= \bigcup_{g \in \Gamma} A \cap B^g \,,
$$
where only finitely many of the intersections $A \cap B^g$
are nonempty.
The set of nonempty pieces $\{ A \cap B^g \mid g \in \Gamma \}$
therefore gives a dissection of $A$,
and by symmetry
the set of nonempty pieces $\{ A^g \cap B \mid g \in \Gamma \}$
gives a dissection of $B$.
But $(A \cap B^g)^{g^{-1}} = A^{g^{-1}} \cap B$,
so the two sets of pieces are the same modulo
isometries in $\Gamma$.~~~${\vrule height .9ex width .8ex depth -.1ex }$
We give four examples; the main application will
be given in the next section.
\vspace*{+.1in}
\begin{figure}[htb]
\centerline{\includegraphics[width=5cm]{TrSq2.eps}}
\caption{Illustrating the Two Tile Theorem: $A$ is the
triangle $(0,0), (1,0), (1,1)$, $B$ (shaded) is the rectangle
$(0,0), (\frac{1}{2},0),(\frac{1}{2},1),(0,1)$, $\Omega$ is the square
$(0,0),(1,0),(1,1),(0,1)$ and $\Gamma$ is
generated by $\phi~:~(x,y) \mapsto (1-x,1-y)$.}
\label{trsq.fig}
\end{figure}
\noindent{\bf Example 1.}
Let $A={\mathcal O}_2$, the right triangle with vertices $(0,0), (1,0),
(1,1)$, let $B$ be the rectangle with vertices
$(0,0), (\frac{1}{2},0),(\frac{1}{2},1),(0,1)$
and let $\phi$ be the map $(x,y) \mapsto (1-x,1-y)$. Let
$\Gamma$ be the group of order $2$ generated by $\phi$ and let $\Omega$
be the square $(0,0),(1,0),(1,1), (0,1)$. Then $A$ and $B$ are
both $\Gamma$-tiles for $\Omega$. It follows
from Theorem \ref{Th1} that $A$ and $B$ are
equidissectable (see
Fig.~\ref{trsq.fig}).
Alternatively, we could take the origin to be at the center
of the square, and then the theorem applies with
$\phi := (x,y) \mapsto (-x,-y)$.
\begin{figure}[htb]
\centerline{\includegraphics[width=9.5cm]{TwoD.eps}}
\caption{ Another illustration of the Two Tile Theorem: $A$ is
the triangle $(0,0), (1,0), (1,1)$,
$B$ (shaded) is the square $(0,0),
(\frac{1}{2},-\frac{1}{2}),
(1,0),
(\frac{1}{2},\frac{1}{2})$
and ${\Omega}$ is the strip $x \ge y \ge x-1$. }
\label{Fig2a}
\end{figure}
\noindent{\bf Example 2.}
Again we take $A = {\mathcal O}_2$ to be the right triangle
with vertices $(0,0), (1,0), (1,1)$,
but now we take $\phi$ to be the map $(x,y) \mapsto (y+1,x)$.
Note that $\phi$ involves a reflection.
As mentioned in \S\ref{Sec1}, this is permitted by
our dissection rules.
Let $\Gamma$ be the infinite cyclic group generated by $\phi$,
and let ${\Omega}$ be the infinite strip
defined by $x \ge y \ge x-1$.
Then $A$ is a $\Gamma$-tile for ${\Omega}$ (see Fig. \ref{Fig2a}).
For $B$, the second tile, we take the square
with vertices $(0,0), (\frac{1}{2},-\frac{1}{2}),
(1,0), (\frac{1}{2},\frac{1}{2})$. This is also a $\Gamma$-tile for ${\Omega}$,
and so $A$ and $B$ are equidissectable.
The two nonempty pieces are the triangles $A \cap B$
and $A \cap B^{\phi}$. The latter is mapped
by $\phi^{-1}$ to the triangle with vertices
$(0,0), (\frac{1}{2},-\frac{1}{2}), (1,0)$.
This is a special case of the dissection given in Theorem \ref{Th2}.
Of course in this case the dissection could
also have been accomplished without using reflections.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=7.5cm]{a110312_3v.eps}
\end{center}
\caption{ Four-piece dissection of an
equilateral triangle to a square, usually attributed
to Dudeney (1902)}
\label{Fig2b}
\end{figure}
\noindent{\bf Example 3.}
One of the most elegant of all dissections is the well-known
four-piece dissection of an
equilateral triangle to a square, shown in Fig. \ref{Fig2b}.
This was published in 1902 by Dudeney, although
Frederickson~\cite[Page~136]{Fred1997} suggests that
he may not have been the original discoverer.
This dissection can be found in many references
(for example, Coffin~\cite[Chap.~1]{Coff1991},
Eves~\cite[\S5.5.1]{Eves1966},
Wells~\cite[p.~61]{Well1991}).
Gardner \cite[Chap.~3]{Gard1961} gives a proof by elementary geometry.
The usual construction of this dissection, however, is
by superimposing two strips, a technique that
Lindgren calls a $TT$-dissection
(Akiyama and Nakamura \cite{AkNa1998},
Frederickson \cite[Chaps.~11,~12]{Fred1997}, \cite[Chap.~3]{Fred2002},
Lindgren \cite[Fig.~5.2]{Lind1964}).
The literature on dissections does not appear to contain
a precise statement of conditions which guarantee that
this construction produces a dissection.
Such a theorem can be obtained as a corollary of the
Two Tile Theorem, and will be published elsewhere, together
with rigorous versions of other strip dissections.
Both Gardner and Eves mention that L.~V.~Lyons
extended Dudeney's dissection to
cut the whole plane into a ``mosaic of interlocking
squares and equilateral triangles,'' and Eves shows this ``mosaic''
in his Fig.~5.5b
(Fig.~\ref{Fig2c} below shows essentially the same figure, with the addition
of labels for certain points).
We will use this ``mosaic,'' which is really a double tiling
of the plane, to give an alternative proof that the
dissection is correct from the Two Tile Theorem.
Following Lyons, we first use the dissection to construct the
double tiling. We then ignore how the double
tiling was obtained, and apply the Two Tile Theorem
to give an immediate certificate of proof for Dudeney's dissection.
The double tiling also has some interesting properties
that are not apparent from Eves's figure,
and do not seem to have been mentioned before
in the literature.
\begin{figure}[htb]
\centerline{\includegraphics[width=14.5cm]{TriangleSquare.eps}}
\caption{
Lyons's ``mosaic,'' a double tiling
of the plane by triangles and squares.
}
\label{Fig2c}
\end{figure}
Let $\Omega = {\mathbb R}^2$, and
take the first tile to be an equilateral triangle
with edge length $1$, area $c_1 := \frac{\sqrt{3}}{4}$
and vertices
$A := (-1/4, -c_1)$, $B := -A$ and
$C := ( 3/4, -c_1)$ (see Fig. \ref{Fig2c}),
with the origin $O$ at the midpoint of $AB$.
The second tile is a square with edge length $c_2 := \sqrt{c_1}$.
The existence of the dissection imposes many constraints, such as
$|JB| = |JC| = |HI| = 1/2$,
$|OD| = |OG| = c_2/2$,
$2|LG| + |GK| = 2|JK| +|GK| = c_2$, etc.,
and after some calculation we find that the
square should have vertices
$D := ( -c_1/2, c_3/2 )$,
$E := (c_3 - c_1/2, c_3/2 + c_1)$,
$F := (c_3 + c_1/2, -c_3/2 + c_1)$ and
$G := -D$,
where $c_3 = c_2 \sqrt{1-c_1}$.
We now construct a strip of squares that replicates the square
$DEFG$ in the southwest/northeast direction,
and a strip of triangles replicating $ABC$ (with alternate triangles
inverted) in the horizontal direction.
In order to fill the plane with copies of these strips, we
must determine the offset of one strip of squares with respect to the
next strip of squares, and of one strip
of triangles with respect to the
next strip of triangles.
This implies the further constraints that
$P-O = L-H = K-E$, etc.,
and in particular that
$P$ should be the point $(1-2c_3, -2c_1)$.
Other significant points are
$H := -L := (c_3-1/2,c_1)$, $I := (c_3,c_1)$,
$J := (1/2,0)$, $K := (1-c_3-c_1/2, c_3/2-c_1)$.
The angle $CLG$ is $\arctan(c_1/c_3) = 41.15\ldots$ degrees.
We now have the desired double tiling of the plane.
Both the triangle $ABC$ and the square $DEFG$
are $\Gamma$-tiles for the whole plane,
where $\Gamma$ is the group
(of type $p2$ in the classical notation,
or type $2222$ in the orbifold notation)
generated by translation by $(1,0)$,
translation by $OP = (1-2c_3, -2c_1) = (0.009015\ldots, -0.8660\ldots)$,
and multiplication by $-1$.
We now ignore how this tiling was found, and deduce from
Theorem \ref{Th1} that Dudeney's dissection exists.
The four pieces are $OBJG$, $ODHB$, $HEI$ and $BIFJ$.
It is interesting that the horizontal strips
of triangles do not line up exactly: each strip is shifted to
the left of the one below it by
$1-2c_3 = 0.009015\ldots$.
The group is correspondingly more complicated than one might have
expected from looking at Fig. \ref{Fig2c},
since the second generator for the group is not {\em quite} translation
by $(0, -\sqrt{3}/2)$!
There is an associated lattice, generated by the vectors $OH$ and $OJ$,
and containing the points $I$ and $L$,
which is {\em nearly} rectangular, the angle between
the generators being $89.04\ldots$ degrees.
Incidentally, although Lindgren
\cite[p.~25]{Lind1964}
refers to this dissection as ``minimal'', we have never seen
a proof that a three-piece dissection of an equilateral
triangle to a square is impossible.
This appears to be an open question.
\vspace*{+.1in}
\noindent{\bf Example 4.}
It is easy to show by induction that any lattice
$\Lambda$ in ${\mathbb R}^n$ has a brick-shaped fundamental
region. The theorem then provides a dissection of the Voronoi
cell of $\Lambda$ into a brick.
For example, the Voronoi cell of the root lattice $D_n$ is described
in \cite[Chap. 21]{SPLAG}. By applying the theorem,
we obtain a dissection of the Voronoi cell into a brick that uses $2n$ pieces.
For $n=3$ this gives the well-known six-piece dissection of a rhombic
dodecahedron into a $2 \times 1 \times 1$ brick
(cf. \cite[pp.~18,~242]{Fred1997}).
\section{The main theorem}\label{Sec3}
We begin by choosing a particular realization of the
simplex ${\mathcal Q}_n(w)$.
Define the following vectors in ${\mathbb R}^n$:
\beql{EqVV}
v_1 := (a,b,b,\ldots,b), \,
v_2 := (b,a,b,\ldots,b), \,
v_3 := (b,b,a,\ldots,b), \,
\ldots, \,
v_n := (b,b,b,\ldots,a),
\end{equation}
where
\begin{eqnarray}
b & := & (\sqrt{1-w(n-1)}-\sqrt{1+w})/n \,, \nonumber \\
a & := & b+\sqrt{1+w} \,.
\label{Eqab}
\end{eqnarray}
Then $v_i \cdot v_i = 1$, $v_i \cdot v_j = -w$
for $i \ne j$, $i,j = 1, \ldots, n$.
We take the convex hull of the vectors
$0, v_1, v_1+v_2, \ldots, v_1+ \cdots + v_n$,
that is, the zero vector together with the rows of
\beql{Eqv1}
\left[ \begin{array}{ccccc}
a &b &b & \ldots &b \\
a+b &a+b &2b & \ldots &2b \\
a+2b &a+2b &a+2b & \ldots &3b \\
\ldots &\ldots &\ldots & \ldots &\ldots \\
a+(n-1)b &a+(n-1)b &a+(n-1)b & \ldots &a+(n-1)b \end{array} \right] \,,
\end{equation}
to be our standard version of ${\mathcal Q}_n(w)$.
This simplex has volume
\beql{EqVol}
(1+w)^{(n-1)/2} (1-w(n-1))^{1/2} / n! \,.
\end{equation}
Setting $w=0, a=1, b=0$
gives our standard version of ${\mathcal O}_n$, as in \eqn{EqH1},
and setting $w=1/n, a=n^{-3/2}(1+(n-1)\sqrt{n+1}),
b=n^{-3/2}(1-\sqrt{n+1})$
gives our standard version of ${\mathcal P}_n$.
Two other versions of ${\mathcal P}_n$ will also appear.
Let $p_i := 1/\sqrt{i(i+1)}$,
and construct a $n \times n$ orthogonal matrix $M_n$
as follows. For $i=1,\ldots,n-1$ the $i$th
column of $M_n$ has entries
$p_i$ ($i$ times), $-ip_i$ (once) and $0$ ($n-i-1$ times),
and the entries in the last column are all $1/\sqrt{n}$.
(The last column is in the
$(1,1,\ldots,1)$
direction and the other columns are perpendicular to it.)
For example,
\renewcommand{\arraystretch}{1.25}
$$
M_3 := \left[ \begin{array}{ccc}
\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \\
-\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{6}} & \frac{1}{\sqrt{3}} \\
0 & -\frac{2}{\sqrt{6}} & \frac{1}{\sqrt{3}} \end{array} \right] \,.
$$
\renewcommand{\arraystretch}{1}
The other two versions of ${\mathcal P}_n$ are: the convex hull
of the zero vector in ${\mathbb R}^n$ together with the rows of
\beql{Eqv2}
\sqrt{\frac{n+1}{n}} \,
\left[ \begin{array}{ccccc}
p_1 & p_2 & p_3 & \ldots & p_n \\
0 & 2p_2 & 2p_3 & \ldots & 2p_n \\
\ldots &\ldots &\ldots & \ldots &\ldots \\
0 &0 &0 & \ldots & dp_n \end{array} \right] \,,
\end{equation}
and the convex hull of
the zero vector in ${\mathbb R}^{n+1}$ together with the rows of
\renewcommand{\arraystretch}{1.25}
\beql{Eqv3}
\sqrt{\frac{n+1}{n}} \,
\left[ \begin{array}{ccccc}
\frac{n}{n+1} & -\frac{1}{n+1} & -\frac{1}{n+1} & \ldots & -\frac{1}{n+1} \\
\frac{n-1}{n+1} & \frac{n-1}{n+1} & -\frac{2}{n+1} & \ldots & -\frac{2}{n+1} \\
\ldots &\ldots &\ldots & \ldots &\ldots \\
\frac{1}{n+1} & \frac{1}{n+1} & \frac{1}{n+1} & \ldots & -\frac{n}{n+1}
\end{array} \right] \,.
\end{equation}
\renewcommand{\arraystretch}{1}
To see that both of these simplices
are congruent to the standard version of ${\mathcal P}_n$,
note that multiplying \eqn{Eqv3} on the right by
$M_{n+1}$ produces \eqn{Eqv2} supplemented by a column of zeros,
and then multiplying \eqn{Eqv2} on
the right by $M_n^{\tr}$ (where tr denotes transpose)
produces the standard version.
\noindent{\bf Remark.}
If we ignore for the moment the scale factor in front
of \eqn{Eqv3}, we see that its rows are the coset
representatives for the root lattice $A_n$
in its dual $A_n^{\ast}$ \cite[p.~109]{SPLAG}.
In other words, the rows of \eqn{Eqv3}
contain one representative of each of the classes
of vertices of the Voronoi cell for $A_n$.
${\mathcal P}_2$ is an equilateral triangle and
${\mathcal P}_3$ is a ``Scottish tetrahedron''
in the terminology of Conway and Torquato~\cite{CoTo2006}.
We can now state our main theorem.
\begin{theorem}\label{Th2}
The simplex ${\mathcal Q}_n(w)$ is equidissectable
with the prism $c {\mathcal P}_{n-1} \times I_{\ell}$,
where \\
$c := \sqrt{ (n-1)(w+1)/n }$ and
$\ell := \sqrt{ (1-w(n-1))/n }$.
\end{theorem}
\vspace*{+.1in}
\noindent{\bf Proof.}
Let ${\Omega}$ be the convex hull of the points
$\{u_i \in {\mathbb R}^n \mid i \in {\mathbb Z} \}$,
where $u_0:=(0,0,\ldots,0)$,
$u_i:=u_0^{\phi^i}$,
$\phi$ is the map
$$
\phi: (x_1,\ldots,x_n) \mapsto
(x_n+a, x_1+b, x_2+b, \ldots,x_{n-1}+b)
$$
and $a,b$ are as in \eqn{Eqab}
(see Table \ref{Tab1}).
\begin{table}[htb]
\caption{Points defining the infinite prism ${\Omega}$.
The convex hull of any
$n+1$ successive rows is a copy of ${\mathcal Q}_n(w)$.}
$$
\begin{array}{|ccccccc|}
\hline
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
u_{-1} & = & -b & -b & -b & \ldots & -a \\
u_0 & = & 0 & 0 & 0 & \ldots & 0 \\
u_{1} & = & a & b & b & \ldots & b \\
u_{2} & = & a+b & a+b & 2b & \ldots & 2b \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
u_{n} & = & a+(n-1)b & a+(n-1)b & a+(n-1)b & \ldots & a+(n-1)b \\
u_{n+1} & = & 2a+(n-1)b & a+nb & a+nb & \ldots & a+nb \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
\hline
\end{array}
$$
\label{Tab1}
\end{table}
We now argue in several easily verifiable steps.
\noindent{(i)}
For any $i \in {\mathbb Z}$, the convex hull of
$u_i, u_{i+1}, \ldots, u_{i+n}$ is a copy of
${\mathcal Q}_n(w)$, ${\mathcal Q} ^{(i)}$ (say),
with ${\mathcal Q} ^{(i)} \subset {\Omega}$ and
$({\mathcal Q} ^{(i)})^{\phi} = {\mathcal Q}^{(i+1)}$.
\noindent{(ii)}
The simplices ${\mathcal Q} ^{(i)}$ and ${\mathcal Q} ^{(i+1)}$
share a common face, the convex hull of
$u_{i+1}, \ldots, u_{i+n}$, but have disjoint interiors.
More generally, for all $i \ne j$, ${\mathcal Q} ^{(i)}$ and ${\mathcal Q} ^{(j)}$
have disjoint interiors.
\noindent{(iii)}
The points of ${\Omega}$ satisfy
\beql{EqWall}
x_1 \ge x_2 \ge \cdots \ge x_n \ge x_1 - (a-b) \,.
\end{equation}
(This is true for ${\mathcal Q} ^{(0)}$ and the property
is preserved by the action of $\phi$.)
The inequalities \eqn{EqWall} define an infinite prism
with axis in the
$(1,1,\ldots,1)$
direction.
We will show that every point in the prism belongs to
${\Omega}$, so ${\Omega}$ is in fact equal to this prism.
\noindent{(iv)}
The projection of ${\mathcal Q} ^{(0)}$ onto the hyperplane
perpendicular to the $(1,1,\ldots,1)$ direction
is congruent to $c {\mathcal P}_{n-1}$, where
$c := \sqrt{ (n-1)(w+1)/n }$.
(For multiplying \eqn{Eqv1} on the right
by $M_n$ gives a scaled copy of \eqn{Eqv2}.)
On the other hand, the intersection of the prism
defined by \eqn{EqWall} with the hyperplane $\sum_{i=1}^{n} x_i = 0$
consists of the points $(0,0,\ldots,0)$,
$ \sqrt{w+1} ((n-1)/n, -1/n, \ldots, -1/n)$,
$ \sqrt{w+1} ((n-2)/n, (n-2)/n, -2/n, \ldots, -2/n)$,
$\ldots$,
and---compare \eqn{Eqv3}---is also congruent to $c {\mathcal P}_{n-1}$.
Since the projection and the intersection have
the same volume, it follows that every point in
the prism is also in ${\Omega}$. (For consider a long
but finite segment of the prism. The total volume of
the copies of ${\mathcal Q}_n(w)$ is determined by the projection,
and the volume of the prism is determined by the cross-section,
and these coincide.) We have therefore established that ${\Omega}$ is
the infinite prism
$$c {\mathcal P}_{n-1} \times I_{\infty}$$
with walls given by \eqn{EqWall}.
Furthermore, ${\mathcal Q}_n(w)$ is a $\Gamma$-tile for ${\Omega}$,
where $\Gamma$ is the infinite cyclic group generated by $\phi$.
\noindent{(v)}
For a second tile, we take the prism
$$ B := c {\mathcal P}_{n-1} \times I_{\ell} \,,$$
where
$\ell := \sqrt{ (1-w(n-1))/n }$.
The length $\ell$ is chosen so that $B$ has the same volume as
${\mathcal Q}_n(w)$ (see \eqn{EqVol}).
We take the base of $B$ to be the particular copy of $c {\mathcal P}_{n-1}$
given by the intersection of ${\Omega}$ with
the hyperplane $\sum_{i=1}^{n} x_i = 0$, as in (iv).
The top of $B$ is found by adding $\ell/\sqrt{n}$ to
every component of the base vectors.
To show that $B$ is also a $\Gamma$-tile for ${\Omega}$,
we check that the image of the base of $B$
under $\phi$ coincides with the top of $B$. This is an easy
verification.
Since ${\mathcal Q}_n(w)$ and $B$ are both $\Gamma$-tiles for ${\Omega}$,
the desired result follows from Theorem \ref{Th1}.~~~${\vrule height .9ex width .8ex depth -.1ex }$
\noindent{\bf Remarks.}
(i) The prism $B$ consists of the portion
of the infinite prism ${\Omega}$ bounded by the
hyperplanes $\sum x_i = 0$
and $\sum x_i = \sqrt{1-w(n-1)}$.
The ``apex'' of ${\mathcal Q}_w(n)$ is the point
$(a+(n-1)b, a+(n-1)b, \ldots, a+(n-1)b)$,
which---since $a+(n-1)b = \sqrt{1-w(n-1)}$---lies on the hyperplane
$\sum x_i = n\sqrt{1-w(n-1)}$.
There are therefore $n$ pieces ${\mathcal Q}_n(w) \cap B^{{\phi}^k}$
($k= 0,1,\ldots,n-1$)
in the dissection, obtained by cutting ${\mathcal Q}_w(n)$
along the hyperplanes $\sum x_i = k\sqrt{1-w(n-1)}$
for $k=1,\ldots,n-1$.
To reassemble them to form $B$, we apply
$\phi^{-k}$ to the $k$th piece.
\begin{figure}[htb]
\begin{center}
\centerline{\includegraphics[width=9.5cm]{Tetra3Exploded2.eps}}
\end{center}
\caption{Exploded view showing three adjacent copies of ${\mathcal Q}_3(w)$ and
their intersections with the two cutting planes.}
\label{Fig4new}
\end{figure}
(ii) The case $n=2$, $w=0$ of the theorem was
illustrated in Fig. \ref{Fig2a}.
In the case $n=3$, $-1~<w<~\frac{1}{2}$,
the three pieces are exactly the same as those in
Sch\"obi's dissection \cite{Scho1985}.
However, it is interesting that we reassemble them
in a different way to form the same prism $c {\mathcal P}_2 \times I_{\ell}$,
with $c := \sqrt{2(w+1)/3}$, $\ell := \sqrt{(1-2w)/3}$.
First we describe our dissection, which is illustrated in Fig. \ref{Fig4new}.
The figure shows an exploded view of three adjacent copies of ${\mathcal Q}_3(w)$,
namely
${\mathcal Q}_3(w)^{\phi^{-1}}$ (the lower left tetrahedron),
${\mathcal Q}_3(w)$ (the upper left tetrahedron) and
${\mathcal Q}_3(w)^{\phi}$ (the tetrahedron on the right),
and their intersections with the two cutting planes.
The three pieces in the dissection can be seen in the
upper left tetrahedron ${\mathcal Q}_3(w)$. They are
$ {\tau}_{-1} = {\mathcal Q}_3(w) \cap B^{\phi^{-1}}$ (the piece $ABCG$ on the left
of this tetrahedron),
$ {\tau}_0 = {\mathcal Q}_3(w) \cap B$ (the central piece $ABCDEF$)
and $ {\tau}_1 = {\mathcal Q}_3(w) \cap B^{\phi}$ (the piece $DEFH$ on the right).
In Fig. \ref{Fig4new} we can also see an exploded view of these three pieces
reassembled to form the triangular prism:
${\tau}_1^{\phi^{-1}}$ is the right-hand piece $A'BCE$ of the lower figure and
${\tau}_{-1}^{\phi}$ is the left-hand piece $CD'EF$ of the figure on the right.
The fully assembled prism is shown in Fig. \ref{Fig5new}:
the tetrahedron $A'BCE$ is ${\tau}_1^{\phi^{-1}}$ and
the tetrahedron $CD'EF$ is ${\tau}_{-1}^{\phi}$.
\begin{figure}[htb]
\begin{center}
\centerline{\includegraphics[width=5.5cm]{SchobiFig.eps}}
\end{center}
\caption{ Three-piece dissection of the triangular prism.
Our construction and Sch\"obi's use
the same three pieces but assemble them in a different way. In
this view the points $A'$ and $B$ are at the back of the figure. }
\label{Fig5new}
\end{figure}
On the other hand, Sch\"obi
reassembles the same pieces by
rotating $ {\tau}_1$ about the edge $EF$ (which acts as a hinge),
sending $D$ to $D'$ and giving the tetrahedron $CD'EF$, and
rotating $ {\tau}_{-1}$ about the hinge $BC$,
sending $A$ to $A'$ and giving the tetrahedron $A'BCE$.
This is strictly different from our construction,
since $\phi$ has no fixed points.
The pieces are the same and the end result is the same,
but the two outer pieces ${\tau}_1$ and ${\tau}_{-1}$
have been interchanged!
(iii) By repeated application of Theorem \ref{Th2}
we can dissect ${\mathcal Q}_n(w)$ into an $n$-dimensional
brick. Each of the $n$ pieces from the first
stage is cut into at most $n-1$ pieces at the second stage, and so on,
so the total number of pieces in the final
dissection is at most $n!$.
(It could be less, if a piece from one stage is not intersected
by all of the cutting planes at the next stage.
It seems difficult to determine the exact number of pieces.)
\section{Dissecting ${\mathcal O}_n$ into a brick}\label{Sec4}
In this section we discuss in more detail the
recursive dissection of ${\mathcal Q}_n(w)$
into a rectangular parallelepiped
or ``brick'' in the case of greatest interest to
us, when we start with ${\mathcal O}_n = {\mathcal Q}_n(0)$.
From Theorem \ref{Th2} we have
\begin{eqnarray}
{\mathcal O}_n & ~{\thicksim}~ & \sqrt{\frac{n-1}{n}} \,{\mathcal P}_{n-1} \times I_{\frac{1}{\sqrt{n}}} \,, \nonumber \\
{\mathcal P}_n & ~{\thicksim}~ & \frac{\sqrt{n^2-1}}{n} {\mathcal P}_{n-1} \times I_{\frac{1}{n}} \,,
\label{Eq84}
\end{eqnarray}
and so (since ${\mathcal P}_1 = I_1$)
\beql{Eq85}
{\mathcal O}_n ~{\thicksim}~
\frac{1}{2} I_1 \times I_{\textstyle {p_2} } \times I_{\textstyle {p_3} }
\times \cdots \times
I_{\textstyle {p_{n-1}} } \times I_{\frac{1}{\sqrt{n}}} \,.
\end{equation}
The right-hand side of \eqn{Eq85} is our final brick;
we will denote it by $\Pi$. Note that
$\vol ({\mathcal O}_n) = \vol (\Pi) = 1/n!$.
Let $\Theta$ denote the map from ${\mathcal O}_n$ to $\Pi$
associated with the dissection \eqn{Eq85}.
We will show that given $x := (x_1, \ldots, x_n) \in {\mathcal O}_n$,
$(y_1, \ldots, y_n) := \Theta(x) \in \Pi$
can be computed in $O(n^2)$ steps.
The algorithm for computing $\Theta$ breaks up
naturally into two parts.
The first step involves
dissecting ${\mathcal O}_n$ into $n$ pieces
and reassembling them to form the prism
$$
B := \sqrt{\frac{n-1}{n}} \,{\mathcal P}_{n-1} \times I_{\frac{1}{\sqrt{n}}} \,.
$$
All later steps start with a point in $\lambda_k {\mathcal P}_{k}$
for $k = n-1, n-2, \ldots, 2$ and certain constants $\lambda_k$,
and produce a point in $\lambda_{k-1} {\mathcal P}_{k-1} \times I$.
For the first step we must determine which
of the pieces ${\mathcal O}_n \cap B^{\phi_1^r}$
($r= 0,1,\ldots,n-1$)
$x$ belongs to, where $\phi_1$ is the map
$(x_1,\ldots,x_n) \mapsto (x_n+1, x_1, x_2, \ldots,x_{n-1})$.
This is given by $r := \lfloor \sum_{i=1}^{n} x_i \rfloor$,
and then mapping $x$ to
$x' := x^{\phi_1^{-r}}$ corresponds to reassembling the pieces
to form $B$. However, $x'$ is expressed in terms of
the original coordinates for ${\mathcal O}_n$ and we must multiply it by $M_n$
to get coordinates perpendicular to the $(1,1,\ldots,1)$
direction, getting $x'' := (x_1'', \ldots, x_{n-1}'', y_n) = x' M_n$.
The final component of $x''$ is the projection of $x'$
in the $(1,1,\ldots,1)$ direction.
The other components of $x''$,
$(x_1'', \ldots, x_{n-1}'')$ define a point in
$\sqrt{\frac{n-1}{n}} \,{\mathcal P}_{n-1}$, but expressed in coordinates of the
form shown in \eqn{Eqv2}, and before we proceed to the next stage,
we must reexpress this in the standard coordinates for
$\sqrt{\frac{n-1}{n}} \,{\mathcal P}_{n-1}$, which we do by
multiplying it by
$M_{n-1}^{\tr}$ (see the beginning of \S\ref{Sec3}),
getting $x'''$.
The following pair of observations shorten these calculations.
First, $y_n$ can be computed
directly once we know $r$, since each application of $\phi_1^{-1}$
subtracts $1$ from the sum of the coordinates.
If $s := \sum_{i=1}^{n} x_i$, then $r := \lfloor s \rfloor$
and $y_n = (s-r)/\sqrt{n}$.
Second, the product of $M_n$-with-its-last-column-deleted
and $M_{n-1}^{\tr}$ is the $n \times (n-1)$ matrix
\renewcommand{\arraystretch}{1.25}
$$
N_n := \left[ \begin{array}{cccc}
1-p_n & -p_n & \ldots & -p_n \\
-p_n & 1-p_n & \ldots & -p_n \\
\ldots & \ldots & \ldots & \ldots \\
-p_n & -p_n & \ldots & 1-p_n \\
-\frac{1}{\sqrt{n}} & -\frac{1}{\sqrt{n}} & \ldots & -\frac{1}{\sqrt{n}}
\end{array} \right] \,.
$$
\renewcommand{\arraystretch}{1}
Multiplication by $N_n$ requires only $O(n)$ steps.
The first stage in the computation of $\Theta$
can therefore be summarized as follows:
\begin{quotation}
\noindent
Step ${\rm A}$. Given $x:=(x_1, \ldots, x_n) \in {\mathcal O}_n$.
Let $s := \sum_{i=1}^{n} x_i$, $r := \lfloor s \rfloor$. \\
Compute $x' := x^{\phi_1^{-r}}$. \\
Pass $x''' := x' N_n$ to the next stage, and output
$y_n := (\mbox{fractional~part~of~} s)/\sqrt{n}$.
\end{quotation}
In all the remaining steps we start with a point $x$
in $\lambda_k {\mathcal P}_k$ for some constant $\lambda_k$,
where $k=n-1$, $n-2, \ldots,2$. Instead of $\phi_1$ we use
the map
$\phi_2 : (x_1,\ldots,x_k) \mapsto (x_k+a, x_1+b, x_2+b, \ldots,x_{k-1}+b)$,
where
$a=k^{-3/2}(1+(k-1)\sqrt{k+1}),
b=k^{-3/2}(1-\sqrt{k+1})$.
Each application of $\phi_2^{-1}$ subtracts $1/\sqrt{k}$ from
the sum of the coordinates.
We omit the remaining details and just give the summary
of this step (for simplicity we ignore the constant $\lambda_k$):
\begin{quotation}
\noindent
Step ${\rm B}_k$. Given $x:=(x_1, \ldots, x_k) \in {\mathcal P}_k$.
Let $s := \sum_{i=1}^{k} x_i$, $r := \lfloor \sqrt{k} s \rfloor$. \\
Compute $x' := x^{\phi_2^{-r}}$. \\
Pass $x''' := x' N_k$ to the next stage, and output
$y_k := (\mbox{fractional~part~of~} \sqrt{k} s)/k$.
\end{quotation}
Since the number of computations needed at each step is linear, we conclude that:
\begin{theorem}\label{Th3}
Given $x \in {\mathcal O}_n$, $\Theta(x) \in \Pi$ can
be computed in $O(n^2)$ steps.
\end{theorem}
\noindent{\bf Remarks.}
The inverse map $\Theta ^{-1}$ is just as easy to compute,
since each of the individual steps is easily reversed.
Two details are worth mentioning.
When inverting step ${\rm B}_k$, given $x'''$ and $y_k$, we obtain $x'$ by
multiplying $x'''$ by $N_k ^{\tr}$
and adding $y_k/\sqrt{k}$ to each component.
For the computation of $r$,
it can be shown (we omit the proof) that
for inverting step ${\rm A}$, to go from $x'$ to $x$,
$r$ should be taken to be the number of strictly negative
components in $x'$. For step ${\rm B}_k$, $r$ is
the number of indices $i$ such that
$$
x_i' ~<~ b \, \sqrt{k} \, \sum_{j=1}^{k} x_j' \,.
$$
\section{An alternative dissection of ${\mathcal O}_4$}\label{Sec5}
In this section we give a six-piece dissection of ${\tau} := {\mathcal O}_4$
into a prism $\sqrt{ \frac{3}{4} } {\mathcal P}_3 \times I_{\frac{1}{2}}$.
Although it requires two more pieces than the dissection of
Theorem \ref{Th2}, it still only
uses three cuts.
It also has an appealing symmetry.
We start by subtracting $\frac{1}{2}$ from the coordinates in \eqn{EqH1},
in order to move the origin to the center of ${\tau}$.
That is, we take ${\tau}$ to be the convex hull of the points
$A:=( -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2})$,
$B:=( \frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2})$,
$C:=( \frac{1}{2}, \frac{1}{2}, -\frac{1}{2}, -\frac{1}{2})$,
$D:=( \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, -\frac{1}{2})$,
$E:=( \frac{1}{2}, \frac{1}{2}, \frac{1}{2}, \frac{1}{2})$
(see Fig. \ref{Fig5a}). We use $(w,x,y,z)$ for coordinates in ${\mathbb R}^4$.
Note that ${\tau}$ is fixed by the symmetry $(w,x,y,z) \mapsto
(-z,-y,-x,-w)$.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=7.5cm]{Tetra4.eps}
\end{center}
\caption{
${\tau} := {\mathcal O}_4$ is the convex hull
of $A,B,C,D,E$;
the first two cuts are made along the hyperplanes containing
$B,C,F,G$ and $C,D,H,I$, respectively. }
\label{Fig5a}
\end{figure}
We make two initial cuts, along the hyperplanes $w+y+z = -\frac{1}{2}$
and $w+x+z = \frac{1}{2}$. The first cut intersects the
edges of ${\tau}$ at the points $B$, $C$, $F:=(0,0,0,-\frac{1}{2})$
and $G:=( -\frac{1}{6}, -\frac{1}{6}, -\frac{1}{6}, -\frac{1}{6})$;
the second at the points $C$, $D$, $H:=(\frac{1}{2},0,0,0)$
and $I:=( \frac{1}{6}, \frac{1}{6}, \frac{1}{6}, \frac{1}{6})$.
The three pieces resulting from these cuts will
be denoted by ${\tau}_1$ (containing $A$),
${\tau}_2$ (the central piece),
and ${\tau}_3$ (containing $E$).
We apply the transformation
$\alpha := (w,x,y,z) \mapsto (-y,-x,-w,-1-z)$ to ${\tau}_1$
and
$\beta := (w,x,y,z) \mapsto (1-w,-z,-y,-x)$ to ${\tau}_3$.
$\alpha$ fixes the triangle $BCF$, although not
pointwise, and similarly $\beta$ fixes the triangle $CDH$,
and so these transformations may be regarded as hinged,
in a loose sense of that word\footnote{Since
a hinged rod in the plane has a fixed point,
and a hinged door in three dimensions
has a fixed one-dimensional subspace,
a hinged transformation in
four dimensions should, strictly speaking,
have a two-dimensional region that is {\em pointwise} fixed.}.
This is what led us to
this dissection---we were attempting
to generalize Sch\"obi's hinged three-dimensional
dissection.
After applying $\alpha$ and $\beta$,
the resulting polytope ${\tau}_4 := {\tau}_1^{\alpha}+{\tau}_2+{\tau}_3^{\beta}$
is a convex body with seven vertices and six faces. (This
and other assertions in
this section were verified with the help of the
programs Qhull~\cite{qhull}
and MATLAB~\cite{MATLAB}.)
The seven vertices are $B, C, D, G, I$,
$J := ( \frac{1}{6}, \frac{1}{6}, \frac{1}{6}, -\frac{5}{6})$
and
$K := ( \frac{5}{6}, -\frac{1}{6}, -\frac{1}{6}, -\frac{1}{6})$,
which are shown schematically in Fig. \ref{Fig5b}.
This figure is realistic in so far as it suggests
that the edges $BK$, $GI$ and $JD$ are equal
and parallel (in fact, $K-B= I-G = D-J =
( \frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{3})$).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=7.5cm]{Tetra4-1.eps}
\end{center}
\caption{
After the first two motions, we
have a polytope ${\tau}_4$ with seven vertices $B,G,J,C,K,I,D$.
The third cut is along the hyperplane containing $C,L,M,N$. }
\label{Fig5b}
\end{figure}
We now make one further cut, along the
hyperplane $w+x+y+z=0$, which
separates ${\tau}_4$ into two pieces ${\tau}_5$ (containing $B$)
and ${\tau}_6$ (containing $K$).
This hyperplane meets the edge $BK$ at the point
$L:=( \frac{3}{4}, -\frac{1}{4}, -\frac{1}{4}, -\frac{1}{4})$,
$GI$ at the point
$M:=( 0, 0, 0, 0)$,
and $JD$ at the point
$N:=( \frac{1}{4}, \frac{1}{4}, \frac{1}{4}, -\frac{3}{4})$.
The point $L$ is three-quarters of the way along $BK$,
$M$ bisects $GI$, and
$N$ is one-quarter of the way along $JD$,
The final motion is to apply
$\gamma := (w,x,y,z) \mapsto (x,y,z,w-1)$
to ${\tau}_6$, and to form ${\tau}_7 := {\tau}_5 + {\tau}_6^{\gamma}$.
The convex hull of ${\tau}_7$ involves three new points,
the images of $L$, $M$ and $N$ under $\gamma$,
namely
$P := ( -\frac{1}{4}, -\frac{1}{4}, -\frac{1}{4}, -\frac{1}{4})$,
$Q:=( 0, 0, 0, -1)$ and
$R := ( \frac{1}{4}, \frac{1}{4}, -\frac{3}{4}, -\frac{3}{4})$,
respectively.
Then ${\tau}_7$ is the convex hull of the eight points
$C, L, M, N$ and $R, B, P, Q$,
and it may be verified that the first four and the last
four of these points define copies of $\sqrt{ \frac{3}{4} } {\mathcal P}_3$,
and that ${\tau}_7$ is
indeed congruent to $\sqrt{ \frac{3}{4} } {\mathcal P}_3 \times I_{\frac{1}{2}}$,
as claimed.
We end with a question: can this construction be generalized to higher
dimensions?
\vspace*{+.1in}
\noindent{\bf Acknowledgments.}
We thank G.~N.~Frederickson for comments
on the manuscript and for drawing our attention to reference \cite{Eves1966}.
|
1,108,101,565,668 | arxiv | \section{Introduction}
This paper is devoted to a study of conservation laws and symmetries
for a class of time-dependent generalized Korteweg-de Vries equations
\begin{equation}\label{gkdv}
u_t +f(t,u)u_x + u_{xxx} =0
\end{equation}
with
\begin{equation}\label{conds}
f|_{u=0}=0,
\quad
f_{u}\neq 0 .
\end{equation}
This class is preserved under the equivalence transformations
\begin{equation}\label{equivgroup}
t\rightarrow \tilde t = t + t_0,
\quad
u\rightarrow \tilde u = u+u_0,
\quad
t_0, u_0 = \const .
\end{equation}
Many interesting evolution equations are included in this class \eqref{gkdv}:
the KdV equation ($f=u$) models the dynamics of shallow water waves;
the modified KdV equation $(f=u^2$) is a model for
acoustic waves in anharmonic lattices \cite{mkdv1}
and Alfven waves in collision-free plasmas \cite{mkdv2};
a combined KdV-mKdV equation ($f=a u+ b u^2$, with $a$ and $b$ being arbitrary nonzero constants)
arises in plasma physics and solid-state physics,
modelling wave propagation in nonlinear lattices \cite{kdv-mkdv1}
and thermal pulses in solids \cite{kdv-mkdv2,kdv-mkdv3}.
KdV-type equations having time-dependent coefficients
arise in several applications \cite{timedep-kdv1,timedep-kdv2,timedep-kdv3,timedep-kdv4}
and can be mapped into this class \eqref{gkdv} by a point transformation \cite{AncGan}.
All of these equations \eqref{gkdv} have a Hamiltonian structure,
on any fixed spatial domain $\Omega\subseteq\mathbb{R}$,
as given by
\begin{equation}\label{hamilstruc}
u_t = \Hop(\delta H/\delta u)
\end{equation}
with the local Hamiltonian functional
\begin{equation}\label{H}
H = \int_{\Omega} ( \tfrac{1}{2} u_x{}^2 - F(t,u) )\,dx,
\quad
F(t,u) = u{\textstyle\int} f(t,u)\, du - {\textstyle\int} uf(t,u)\, du,
\end{equation}
where the Hamiltonian operator \cite{Olv} is a total $x$-derivative
\begin{equation}\label{Hop}
\Hop = D_x .
\end{equation}
The Hamiltonian $H$ will be a conserved integral when and only when
the nonlinearity satisfies $f_t=0$,
which corresponds to a generalized KdV equation
$u_t + f(u)u_x + u_{xxx}=0$.
Previous work on special families of equations in the class \eqref{gkdv}
can be found in Refs.\cite{AncBlu02a,JohKhaBis,PopSer,MouKha},
In section~\ref{conslaws},
all low-order conservation laws of this class of generalized KdV equations \eqref{gkdv}--\eqref{conds}
will be classified.
In section~\ref{symms},
a corresponding classification of Hamiltonian symmetries will be derived
by using the Hamiltonian structure \eqref{hamilstruc} of the generalized KdV equations.
The physical meaning of the symmetries and the conservation laws is discussed.
Finally, some concluding remarks will be made in section~\ref{remarks}.
\section{Conservation laws}
\label{conslaws}
Conservation laws are also of basic importance in the study of evolution equations
because they provide physical, conserved quantities for all solutions $u(x,t)$,
and they can be used to check the accuracy of numerical solution methods
\cite{Olv,BluCheAnc}.
A local conservation law for a time-dependent generalized KdV equation \eqref{gkdv}
is a continuity equation
\begin{equation}\label{conslaw}
D_t T+D_x X=0
\end{equation}
holding for all solutions $u(x,t)$ of equation \eqref{gkdv},
where the conserved density $T$ and the spatial flux $X$ are
functions of $t$, $x$, $u$, and $x$-derivatives of $u$.
If $T=D_x\Theta$ and $X=-D_t\Theta$ hold for all solutions,
then the continuity equation \eqref{conslaw} becomes an identity.
Conservation laws of this form are called locally trivial,
and two conservation laws are considered to be locally equivalent if they differ
by a locally trivial conservation law.
The global form of a non-trivial conservation law is given by
\begin{equation}\label{globalconslaw}
\frac{d}{dt}\int_{\Omega} T\, dx = -X\Big|_{\partial\Omega}
\end{equation}
where $\Omega\subseteq\mathbb{R}$ is any fixed spatial domain.
Every local conservation law can be expressed in an equivalent, characteristic form
(analogous to the characteristic form for symmetries) \cite{Olv}
which is given by a divergence identity
\begin{equation}\label{chareqn}
D_t \tilde T+D_x \tilde X= (u_t +f(t,u)u_x +u_{xxx})Q
\end{equation}
holding off of the set of solutions of the evolution equation \eqref{gkdv},
where $\tilde T= T+D_x\Theta$ and $\tilde X= X-D_t\Theta$
are a conserved density and spatial flux that are locally equivalent to $T$ and $X$,
and where \cite{AncBlu02b}
\begin{equation}\label{QTrel}
Q = E_u(\tilde T)
\end{equation}
is a function of $t$, $x$, $u$, and $x$-derivatives of $u$.
This function is a called a multiplier \cite{Olv,AncBlu97,BluCheAnc}.
Here $E_u$ denotes the Euler operator with respect to $u$ \cite{Olv}.
For any evolution equation,
there is a one-to-one correspondence between non-zero multipliers
and non-trivial conservation laws up to local equivalence \cite{Olv,AncBlu02b},
and the conservation laws of basic physical interest arise from
multipliers of low order \cite{Anc16a}.
These multipliers for the evolution equation \eqref{gkdv}
take the form
\begin{equation}\label{low-order-Q}
Q(t,x,u,u_x,u_{xx})
\end{equation}
which correspond to conserved densities of the form
\begin{equation}\label{low-order-T}
T(t,x,u,u_x)
\end{equation}
modulo a trivial conserved density.
A function \eqref{low-order-Q} will be a multiplier iff
$E_u((u_t +f(t,u)u_x +u_{xxx})Q)=0$ holds identically,
since the kernel of the Euler operator consists of total divergences \cite{Olv,BluCheAnc}.
This condition splits with respects to any $x$-derivatives of $u$ that do not appear in $Q$.
The resulting overdetermined system consists of
\begin{equation}\label{adjsymm-deteqn}
0=D_tQ + f D_xQ +D_x^3 Q
\end{equation}
and
\begin{align}\label{helmholtz-deteqn}
Q_u = E_u(Q),
\
Q_{u_x} = -E_u^{(1)}(Q),
\
Q_{u_{xx}} = E_u^{(2)}(Q)
\end{align}
holding for all solutions $u(x,t)$ of the evolution equation \eqref{gkdv}.
The first equation \eqref{adjsymm-deteqn} turns out to be
the adjoint of the determining equation for symmetries
(cf.\ \eqref{symm-deteqn}).
The remaining equations \eqref{helmholtz-deteqn} constitute
the Helmholtz equations \cite{AncBlu02b,Anc16a}
which are necessary and sufficient for $Q$ to have the variational form \eqref{QTrel}.
Here $E_u^{(1)}$, and so on, denote the higher Euler operators \cite{Olv,Anc16a}.
It is straightforward to set up and solve this determining system \eqref{adjsymm-deteqn}--\eqref{helmholtz-deteqn}
subject to the classification conditions \eqref{conds}.
The computation is simplest when we separate it into two main cases:
$f_{tu}=0$, and $f_{tu}\neq 0$.
We merge the resulting subcases
after first having solved the determining system in each of these two cases
and then having used the equivalence transformations \eqref{equivgroup}.
(For solving the determining system, we use the Maple package ``rifsimp''.)
The multipliers \eqref{low-order-Q} for general $f(t,u)$
are linear combinations of
\begin{align}
&
Q_1 =1;
\label{gencase1}
\\
&
Q_2 =u .
\label{gencase2}
\end{align}
All special forms of $f(t,u)$ for which additional multipliers \eqref{low-order-Q}
are admitted consist of:
\begin{subequations}
\begin{align}
&
f(t,u) =a(u),
\quad
a(u) \text{ arbitrary} ,
\label{case1}
\\
&
Q_3 = {-}u_{xx} -\smallint a(u)\,du ;
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
&
f(t,u) = t^{-2/3} a(t^{1/3}u),
\quad
a(v) \text{ arbitrary} ,
\label{case2}
\\
&
Q_4 = {-}t u_{xx} +\tfrac{1}{3}xu -t^{1/3} \smallint a(t^{1/3}u)\,du ;
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
&
f(t,u) =a(t) u,
\quad
a(t) \text{ arbitrary} ,
\label{case3}
\\
&
Q_5 =\big(\smallint a(t)\,dt\big) u -x ;
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
&
f(u) =a(t) u,
\quad
a(t) \text{ satisfies }
a^2 a''' -13 a'' a' a +24 a'{}^3 =0 ,
\label{case4}
\\
&
\begin{aligned}
Q_6 = &
{-}2a(t)^{-3} u_{xx} -a(t)^{-2} u^2
- 2x a'(t)a(t)^{-4}u
\\&\qquad
{-}x^2(4 a'(t)^2 -a(t) a''(t))a(t)^{-6} ;
\end{aligned}
\end{align}
\end{subequations}
\begin{subequations}
\begin{align}
&
f(u) =a t^{-1/3} u + b u + c u^2,
\quad
a, b, c \text{ constant} ,
\label{case5}
\\
&
\begin{aligned}
Q_7 = &
{-}b t u_{xx} -\tfrac{1}{6} t( 2 c^2 u^3 +3cb u^2 + b^2 u )
\\&\qquad
{-}\tfrac{1}{4} a t^{2/3} (2c u^2+b u) +\tfrac{1}{6} x (2c u +b) ;
\end{aligned}
\end{align}
\end{subequations}
Note, in the case \eqref{case4},
the third-order ODE possesses two first integrals
$-2a^p a'' + (p+13)a^{p-1} a'{}^2 =c=\const$ for $p=-5$ and $p=-7$.
This yields a reduction to a first-order separable ODE
\begin{equation}
a'=a^3 \big(c_1 + c_2 a^2 \big)^{1/2} ,
\quad
c_1, c_2 \text{ constant}
\label{case4'}
\end{equation}
which has the quadrature
\begin{equation}
\begin{aligned}
& \frac{2c_1^{1/2} + \big(c_1 + c_2 a^2 \big)^{1/2}}{a} \exp\Big(-\frac{c_1^{1/2}\big(c_1 + c_2 a^2 \big)^{1/2}}{c_2a^2}\Big)
\\&
= \exp\Big( \frac{2c_1^{3/2}}{c_2} (t+c_3) \Big),
\quad
c_1, c_2, c_3 \text{ constant}
\end{aligned}
\label{case4''}
\end{equation}
For each multiplier admitted by a time-dependent generalized KdV equation \eqref{gkdv},
a corresponding conserved density and flux can be derived (up to local equivalence)
by integration of the divergence identity \eqref{chareqn}
\cite{BluCheAnc,Anc16a}.
We obtain the following results.
\begin{thm}\label{classify-conslaws}
(i) All conservation laws given by low-order conserved densities \eqref{low-order-T}
admitted by the class of time-dependent generalized KdV equations \eqref{gkdv}
for arbitrary $f(t,u)$ (satisfying conditions \eqref{conds})
are linear combinations of:
\begin{flalign}
&\label{conslaw1}
T_1= u ,
\quad
X_1= u_{xx} +\smallint f(t,u)\,du ;
&
\\
&\label{conslaw2}
T_2= \tfrac{1}{2}u^2 ,
\quad
X_2= uu_{xx}-\tfrac{1}{2}u_x^2 +\smallint uf(t,u)\,du .
&
\end{flalign}
(ii) The class of time-dependent generalized KdV equations \eqref{gkdv}
admits additional conservation laws given by low-order conserved densities \eqref{low-order-T}
only for $f(t,u)$ of the form \eqref{case1}, \eqref{case2}, \eqref{case3}, \eqref{case4}
(satisfying conditions \eqref{conds}).
The admitted conservation laws in each case are given by:
\begin{subequations}
\label{conslaw3}
\begin{flalign}
&
T_3 = \tfrac{1}{2}u_x{}^2 -\smallint A(u)\,du ,
&
\\
&
X_3 = {-}\tfrac{1}{2}u_{xx}{}^2 -A(u) u_{xx} -u_tu_x -\tfrac{1}{2} A(u)^2 ,
&
\\
& A(u) = \smallint a(u)\,du ;
\end{flalign}
\end{subequations}
\begin{subequations}
\label{conslaw4}
\begin{flalign}
&
\begin{aligned}
T_4 & = \tfrac{1}{2} t u_x{}^2 +\tfrac{1}{6} x u^2 -\smallint A(t^{1/3}u)\,du ,
\end{aligned}
&
\\
&
\begin{aligned}
X_4 & = -\tfrac{1}{2} t u_{xx}{}^2 - A(t^{1/3}u) u_{xx}
+ \tfrac{1}{6} x( 2uu_{xx} - u_x{}^2 )
\\&\qquad
+ \tfrac{1}{3} x t^{-1} \big( uA(t^{1/3}u) -\smallint A(t^{1/3}u)\,du \big)
-\tfrac{1}{2} t^{-1} A(t^{1/3}u)^2 ,
\end{aligned}
&
\\
& A(v) = \smallint a(v)\,dv ;
\end{flalign}
\end{subequations}
\begin{subequations}
\label{conslaw5}
\begin{flalign}
&
T_5 = \tfrac{1}{2} A(t) u^2 - x u ,
&
\\
&
X_5 = {-}\tfrac{1}{2} x( 2u_{xx} +a(t) u^2 ) +u_x
+\tfrac{1}{2} A(t) ( 2u u_{xx} -u_x{}^2 )
- \tfrac{1}{3} a(t) A(t) u^3 ,
&
\\
&
A(t) =\smallint a(t)\, dt ;
&
\end{flalign}
\end{subequations}
\begin{subequations}
\label{conslaw6}
\begin{flalign}
&
\begin{aligned}
T_6 & = a(t)^{-3} u_x{}^2 -\tfrac{1}{3} a(t)^{-2} u^3
-c_1 x^2 u - \big( c_2 + c_1 a(t)^{-2} \big)^{1/2} x u^2 ,
\end{aligned}
&
\\
&
\begin{aligned}
X_6 & = {-}a(t)^{-3} (u_{xx}{}^2+2 u_t u_x)
- a(t)^{-2} u^2 u_{xx}
+2\big( c_2 + c_1 a(t)^{-2} \big)^{1/2} u u_x
\\&\qquad
{-}\tfrac{1}{4} a(t)^{-1} u^4 +2c_1 (u-xu_x)
-\tfrac{1}{2}c_1 x^2( 2u_{xx} + a(t) u^2)
\\&\qquad
{-} x \big( c_2 + c_1 a(t)^{-2} \big)^{1/2} ( 2u u_{xx} - u_x{}^2 )
-\tfrac{2}{3} x \big( c_1 + c_2 a(t)^{2} \big)^{1/2} u^3 ;
\end{aligned}
&
\end{flalign}
\end{subequations}
\begin{subequations}
\label{conslaw7}
\begin{flalign}
&
\begin{aligned}
T_7 & = \tfrac{1}{2} c t u_x{}^2 -\tfrac{1}{12} t ( c u^2 + b u )^2
+\tfrac{1}{6} x( c u^2 + b u ) - \tfrac{1}{2} a t^{2/3} ( \tfrac{1}{3}c u^3 + \tfrac{1}{4} b u^2 ) ,
\end{aligned}
&
\\
&
\begin{aligned}
X_7 & = {-}\tfrac{1}{2} c t (u_{xx}{}^2 +2 u_tu_x)
-\tfrac{1}{6} t ( 2c^2 u^3 +3 bc u^2 + b^2 u )u_{xx}
+\tfrac{1}{12} t b^2 u_x{}^2
\\&\qquad
-\tfrac{1}{18} t ( c u^2 +b u )^3 +\tfrac{1}{12} x ( c u^2 + b u)^2
+\tfrac{1}{6} x ( ( 2c u + b) u_{xx} -b u_x{}^2 )
\\&\qquad
+\tfrac{1}{3}a x t^{-1/3} (\tfrac{1}{3} c u^3 + \tfrac{1}{4} b u^2)
-\tfrac{1}{12} a^2 t^{1/3} (\tfrac{3}{2}c u^4 + b u^3)
\\&\qquad
-\tfrac{1}{4} a t^{2/3} ( (2cu^2 + b u)u_{xx} - \tfrac{1}{2}b u_x{}^2
+ 2c^2 u^5 + \tfrac{5}{4} bc u^4 + \tfrac{5}{3} b^2 u^3 )
\\&\qquad
-\tfrac{1}{6} ( 2c u + b) u_x .
\end{aligned}
&
\end{flalign}
\end{subequations}
\end{thm}
Note that $u_t$ can be eliminated in the spatial flux expressions
by use of the evolution equation \eqref{gkdv}.
The physical meaning of these conservation laws \eqref{conslaw1}--\eqref{conslaw7}
can be seen by considering their global form \eqref{globalconslaw}.
For general $f(t,u)$,
the two admitted conservation laws \eqref{conslaw1} and \eqref{conslaw2}
yield the conserved integrals
\begin{align}
&\label{globalconslaw1}
C_1 = \int_{\Omega} u\, dx ,
\\
&\label{globalconslaw2}
C_2 = \int_{\Omega} \tfrac{1}{2}u^2\, dx .
\end{align}
These represent the total mass and the $L^2$-norm for solutions $u(x,t)$.
In the time-independent case \eqref{case1},
where $f(t,u) =a(u)=A'(u)$,
the conservation law \eqref{conslaw3} yields the conserved integral
\begin{equation}\label{globalconslaw3}
C_3 = \int_{\Omega} \big( \tfrac{1}{2}u_x{}^2 -\smallint A(u)\,du \big)\,dx
\end{equation}
which represents the Hamiltonian or the total energy for solutions $u(x,t)$.
In the time-dependent nonlinear case \eqref{case2},
where $f(t,u) = t^{-2/3} a(t^{1/3}u) = t^{-2/3} A'(t^{1/3}u)$,
the conservation law \eqref{conslaw4} yields the conserved integral
\begin{equation}\label{globalconslaw4}
C_4 = \int_{\Omega} \big( \tfrac{1}{2} t u_x{}^2 +\tfrac{1}{6} x u^2 - \smallint A(t^{1/3}u)\,du \big)\,dx
\end{equation}
which represents a dilational energy for solutions $u(x,t)$.
In the time-dependent linear cases \eqref{case3} and \eqref{case4},
the two conservation laws \eqref{conslaw5} and \eqref{conslaw6}
respectively yield the conserved integrals
\begin{equation}\label{globalconslaw5}
C_5 = \int_{\Omega} \big( \tfrac{1}{2} A(t) u^2 -x u \big)\,dx
\end{equation}
where $f(t,u) =a(t) u =A'(t)u$,
and
\begin{equation}\label{globalconslaw6}
C_6 = \int_{\Omega} \big( a(t)^{-3} u_x{}^2 -\tfrac{1}{3} a(t)^{-2} u^3
-\big( c_2 + c_1 a(t)^{-2} \big)^{1/2} x u^2 -c_1 x^2 u \big)\,dx
\end{equation}
where $f(t,u) =a(t) u$ with $a(t)$ given by expression \eqref{case4''}.
Since $f(t,u)$ is linear in $u$ in these two cases,
the evolution equation \eqref{gkdv} has the form
\begin{equation}\label{timedepKdV}
u_t + a(t)uu_x + u_{xxx}=0
\end{equation}
which is a KdV equation with a time-dependent coefficient,
where $a(t) u$ physically represents an advective velocity.
Then the first conserved integral \eqref{globalconslaw5}
describes a generalized Galilean momentum,
and the second conserved integral \eqref{globalconslaw6}
describes a generalized dilational energy.
In particular, when $a=\const$, these conserved integrals reduce to
the ordinary Galilean momentum
$\smallint_{\Omega} \big( \tfrac{1}{2} a t u^2 -x u \big)\,dx$
and the ordinary energy
$a^{-3} \smallint_{\Omega} \big( u_x{}^2 - a u^3 \big)\,dx$
for the KdV equation.
In the quadratic case \eqref{case5},
where $f(u) =a t^{-1/3} u + b u + c u^2$,
the conservation law \eqref{conslaw7} yields the conserved integral
\begin{equation}\label{globalconslaw7}
C_7 = \int_{\Omega} \big( \tfrac{1}{2} c t u_x{}^2 -\tfrac{1}{12} t ( c u^2 + b u )^2
+\tfrac{1}{6} x( c u^2 + b u ) - \tfrac{1}{24} a t^{2/3} ( 4c u^3 + 3 b u^2 )\big)\,dx
\end{equation}
which represents a combined Galilean energy-momentum for solutions $u(x,t)$.
In particular, when $a=b=0$,
this conserved integral reduces to the Galilean energy
$c \smallint_{\Omega} \big( \tfrac{1}{2} t u_x{}^2 -\tfrac{1}{12} c t u^4 +\tfrac{1}{6} x u^2 \big)\,dx$
for the mKdV equation,
while when $a=c=0$,
the Galilean momentum for the KdV equation is obtained.
\section{Symmetries}
\label{symms}
Symmetries are a basic structure of evolution equations
as they can be used to find invariant solutions
and yield transformations that map the set of solutions $u(x,t)$ into itself
\cite{Olv,BluCheAnc}.
An infinitesimal symmetry for a time-dependent generalized KdV equation \eqref{gkdv}
is a generator
\begin{equation}\label{generator}
\X =\xi \partial_x+\tau \partial_t+\eta \partial_u
\end{equation}
whose prolongation leaves invariant the equation \eqref{gkdv},
where $\xi$, $\tau$, and $\eta$ are functions of $t$, $x$, $u$, and $x$-derivatives of $u$.
The symmetry is trivial if it leaves invariant every solution $u(x,t)$ of the equation \eqref{gkdv}.
This occurs when (and only when) $\xi$, $\tau$, and $\eta$ satisfy the relation
\begin{equation}
\eta=u_x \xi + u_t \tau
\end{equation}
for all solutions $u(x,t)$.
The corresponding generator \eqref{generator} of a trivial symmetry
is given by
\begin{equation}\label{trivsymm}
\X_{\rm triv}= \xi \partial_x + \tau \partial_t + (\xi u_x + \tau u_t)\partial_u
\end{equation}
which has the prolongation $\pr\X_{\rm triv}=\xi D_x + \tau D_t$
Any symmetry generator is equivalent \cite{Olv,BluCheAnc} to a generator
\begin{equation}\label{symmchar}
\hat\X=\X-\X_{\rm triv} = P\partial_u,
\quad
P=\eta -\xi u_x -\tau u_t
\end{equation}
under which $u$ is infinitesimally transformed while $x$ and $t$ are invariant,
due to the relation
\begin{equation}
\pr\X-\pr\hat\X= \xi D_x + \tau D_t .
\end{equation}
This generator \eqref{symmchar} defines the \emph{characteristic form}
for the infinitesimal symmetry.
Invariance of the evolution equation \eqref{gkdv} is
then given by the condition \cite{Olv,Anc16a}
\begin{equation}\label{symm-deteqn}
0=D_tP +D_x(f P) +D_x^3 P
\end{equation}
holding for all solutions $u(x,t)$ of the equation \eqref{gkdv}.
A symmetry will generate a point transformation on $(x,t,u)$ iff
the coefficients $\xi$, $\tau$, and $\eta$ in its characteristic function $P$
depend only on $x,t,u$ \cite{Olv,BluCheAnc},
yielding a generator with the form
\begin{equation}
\X =\xi(x,t,u)\partial_x+\tau(x,t,u)\partial_t+\eta(x,t,u)\partial_u .
\end{equation}
For any Hamiltonian evolution equation,
there is a correspondence
that produces a symmetry from each admitted conservation law.
This correspondence is a Hamiltonian analog of Noether's theorem.
It can be formulated for an evolution equation \eqref{gkdv},
with the Hamiltonian structure \eqref{Hop},
through the explicit relation \cite{Olv,MaZho}
\begin{equation}\label{PQmap}
P = \Hop(\delta C/\delta u) = D_x Q
\end{equation}
involving the characteristic function $P$ of the symmetry \eqref{symmchar}
and the multiplier $Q$ associated to the conserved integral
$C = \smallint_{\Omega} T\, dx$
given by a conservation law \eqref{conslaw}.
This correspondence is one way: every conservation law yields a symmetry.
The converse holds iff the symmetry has the form \eqref{PQmap},
which requires that $E_u(P)=0$.
The classification of low-order conservation laws stated in Theorem~\ref{classify-conslaws}
for the class of time-dependent generalized KdV equations \eqref{gkdv}
yields the following symmetries
produced from the conservation law multipliers.
The multipliers \eqref{gencase1} and \eqref{gencase2}
given by the two conservation laws \eqref{conslaw1} and \eqref{conslaw2}
admitted for general $f(t,u)$
produce the symmetry characteristic functions
\begin{align}
&
P_1 =0;
\\
&
P_2 =u_x .
\label{symm1,2}
\end{align}
The additional multipliers \eqref{case1}--\eqref{case5}
given by the conservation laws \eqref{conslaw3}--\eqref{conslaw7}
which are admitted for forms of $f(t,u)$
respectively yield the symmetry characteristic functions
\begin{align}
&
P_3 = {-}u_{xxx} -a(u)u_x ;
\label{symm3}
\\
&
P_4 = {-}t u_{xxx} +\tfrac{1}{3}xu_x +\tfrac{1}{3}u -t^{2/3} a(t^{1/3}u)u_x ;
\label{symm4}
\\
&
P_5 =\big(\smallint a(t)\,dt\big) u_x -1 ;
\label{symm5}
\\
&
\begin{aligned}
P_6 = &
{-}2a(t)^{-3} u_{xxx} -2a(t)^{-2} uu_x
- 2x a'(t)a(t)^{-4}u_x
\\&\qquad
- 2 a'(t)a(t)^{-4}u
-2x(4 a'(t)^2 -a(t) a''(t))a(t)^{-6} ;
\end{aligned}
\label{symm6}
\\
&
\begin{aligned}
P_7 = &
{-}b t u_{xxx} -t( c^2 u^2 +cb u + \tfrac{1}{6} b^2 ) u_x
\\&\qquad
{-}a t^{2/3} (c u+\tfrac{1}{4} b)u_x +\tfrac{1}{3} c x u_x
+\tfrac{1}{6} (2c u +b) .
\end{aligned}
\label{symm7}
\end{align}
By evaluating these characteristic functions \eqref{symm1,2}--\eqref{symm7}
on solutions $u(x,t)$,
we can eliminate all $u_{xxx}$ terms to obtain
$P(t,x,u,u_x,u_t) =\eta(t,x,u) -\xi(t,x,u) u_x -\tau(t,x,u) u_t$
in each case.
This leads to the following symmetry classification.
\begin{thm}\label{classify-hamilsymms}
(i) The symmetries corresponding to the two low-order conserved integrals \eqref{globalconslaw1}--\eqref{globalconslaw2}
admitted by the class of time-dependent generalized KdV equations \eqref{gkdv}
for arbitrary $f(t,u)$ (satisfying conditions \eqref{conds})
are generated by:
\begin{equation}
\label{generator1,2}
\X_1 =0,
\quad
\X_2 = \partial_x .
\end{equation}
(ii) The additional symmetries corresponding to the low-order conserved integrals \eqref{globalconslaw3}--\eqref{globalconslaw7}
admitted for special forms of $f(t,u)$ (satisfying conditions \eqref{conds})
are generated in each case by:
\begin{subequations}
\label{generator3}
\begin{flalign}
&
\X_3 = \partial_t ,
&
\\
&
f(t,u) =a(u),
\quad
a(u) \text{ arbitrary} ;
\label{symmcase1}
&
\end{flalign}
\end{subequations}
\begin{subequations}
\label{generator4}
\begin{flalign}
&
\X_4 = \tfrac{1}{3}x\partial_x + t\partial_t +\tfrac{1}{3}u \partial_u ,
&
\\
&
f(t,u) = t^{-2/3} a(t^{1/3}u),
\quad
a(v) \text{ arbitrary} ;
\label{symmcase2}
&
\end{flalign}
\end{subequations}
\begin{subequations}
\label{generator5}
\begin{flalign}
&
\X_5 =a(t)\partial_x -\partial_u ,
&
\\
&
f(t,u) =a(t) u,
\quad
a(t) \text{ arbitrary} ;
\label{symmcase3}
&
\end{flalign}
\end{subequations}
\begin{subequations}
\label{generator6}
\begin{flalign}
&
\begin{aligned}
\X_6 & = - 2 a'(t)a(t)^{-4} x\partial_x +2 a(t)^{-3}\partial_t
\\&\qquad
+\big( 2x (a(t) a''(t)-4 a'(t)^2)a(t)^{-6} -2a'(t)a(t)^{-4}\big)\partial_u ,
\end{aligned}
&
\\
&
f(u) =a(t) u,
\quad
a(t) \text{ satisfies }
a^2 a''' -13 a'' a' a +24 a'{}^3 =0 ;
\label{symmcase4}
&
\end{flalign}
\end{subequations}
\begin{subequations}
\label{generator7}
\begin{flalign}
&
\begin{aligned}
\X_7 = & (\tfrac{1}{3}c x - \tfrac{1}{6}b^2 t - \tfrac{1}{4} ab t^{2/3})\partial_x
+ c t\partial_t +(\tfrac{1}{3}c u + \tfrac{1}{6}b)\partial_u ,
\end{aligned}
&
\\
&
f(u) =a t^{-1/3} u + b u + c u^2,
\quad
a, b, c \text{ constant} .
\label{symmcase5}
&
\end{flalign}
\end{subequations}
\end{thm}
Note, from the quadrature \eqref{case4''} for the ODE for $a(t)$ in case \eqref{generator6},
we can express
\begin{equation}
\X_6 = -2\big( c_2 + c_1 a(t)^{2} \big)^{1/2} x\partial_x + 2a(t)^{-3}\partial_t
-2\big(c_1 x + \big( c_2 + c_1 a(t)^{2} \big)^{1/2}\big)\partial_u .
\end{equation}
All of the symmetries \eqref{generator1,2}--\eqref{generator7} are point symmetries.
Their physical meaning will now be discussed.
For general $f(t,u)$,
the symmetry $\X_1$ obtained from the conservation law \eqref{conslaw1} is trivial.
Consequently, the conservation law \eqref{conslaw1} represents
a Casimir of the Hamiltonian structure \cite{Olv}.
The other symmetry $\X_2$ is a space translation.
In the time-independent case \eqref{symmcase1},
where the conserved integral \eqref{globalconslaw3}
represents the Hamiltonian or the total energy for solutions $u(x,t)$,
the symmetry $\X_3$ is a time translation.
In the time-dependent nonlinear case \eqref{symmcase2},
where the conserved integral \eqref{globalconslaw4}
represents a dilational energy for solutions $u(x,t)$,
the symmetry $\X_4$ is a scaling.
In the time-dependent linear cases \eqref{symmcase3} and \eqref{symmcase4},
where the two conserved integrals \eqref{globalconslaw5} and \eqref{globalconslaw6}
respectively represent a generalized Galilean momentum
and a generalized dilational energy,
the first symmetry $\X_5$ is a generalized Galilean boost
and the second symmetry $\X_6$ is a generalized dilation.
Note the evolution equation \eqref{gkdv} in these cases has the form of
a time-dependent KdV equation \eqref{timedepKdV}
in which $a(t) u$ physically represents an advective velocity.
In particular, when $a=\const$, these two symmetries reduce to
an ordinary Galilean boost and a time translation.
In the quadratic case \eqref{symmcase5},
where the conserved integral \eqref{globalconslaw7}
represents a combined Galilean energy-momentum for solutions $u(x,t)$,
the symmetry $\X_7$ is a scaling combined with a Galilean boost.
In particular, when $a=b=0$,
this symmetry reduces to the scaling symmetry for the mKdV equation,
while when $a=c=0$,
the Galilean boost symmetry for the KdV equation is obtained.
\section{Concluding remarks}
\label{remarks}
The classifications of low-order conservation laws and associated Hamiltonian symmetries
obtained in this paper can be extended to a wider class of evolution equations
\begin{equation}
u_t + f(t,u) u_x + b(t)u + c(t)u_{xxx} = s(t)
\end{equation}
by use of a mapping
\begin{equation}
x\rightarrow \tilde x = x - \zeta(t),
\quad
t\rightarrow \tilde t = \tau(t),
\quad
u\rightarrow \tilde u = \lambda(t) u+\nu(t)
\end{equation}
with $\tau'(t)\neq 0$ and $\lambda(t)\neq 0$.
This will be carried out in a subsequent paper \cite{AncGan}.
\section*{Acknowledgments}
M.R.\ and M.L.G.\ gratefully acknowledge the support of Junta de Andaluc\'ia group FQM-201.
S.C.A.\ is supported by an NSERC research grant.
|
1,108,101,565,669 | arxiv | \section{Introduction}
The rise of antiferromagnetic spintronics~\cite{Baltz2018} brought about renewed interest in 'old' materials where aspects previously disregarded have now become important.
One of the those materials, which has been known~\cite{Bronger1986} to be an antiferromagnet (AFM) since 1980's and has only recently been identified as a system with locally broken inversion symmetry~\cite{Wadley2013} --- a precondition for the observation of staggered spin-orbit torques~\cite{Zelezny2014}, a novel means of manipulation of magnetic moments in AFMs --- is CuMnAs in its tetragonal phase.
Detailed understanding of its transport properties is desirable and, with the prospect of using it under conditions of industrial applications~\cite{Jungwirth2016, Wadley2015} allowed by its relatively high N\'eel temperature $T_N\approx 490$~K~\cite{Wadley2015,Uhlirova2019}, effects of chemical and temperature-induced disorder (phonons and magnons) should be included in the model.
The alloy analogy model (AAM) has recently been implemented \cite{DW2017-IEEE, DW2017-SPIE, Drachal2018PRB-DW} within the tight-binding linear muffin-tin orbital (TB-LMTO) method with the coherent potential approximation (CPA) and used to describe FM half-Heusler NiMnSb at finite temperatures \cite{DW2019-JMMM, DW2019-PRB} and also anisotropy of hexagonal systems \cite{DW2019-hexagonal}.
Previous studies of other groups employing the AAM are based on i) the CPA and the Korringa-Kohn-Rostoker (KKR) method with the Kubo-Bastin equation \cite{Ebert, Ebert2011, Kodderitzsch_AHE} and on ii) supercells with the TB-LMTO method and the Landauer-B\"uttiker formula \cite{Glasbrenner, Starikov2018}, but they focus on transition metals.
The disordered local moment (DLM) model within the CPA was used to investigate spin disorder in NiMnSb \cite{Belashchenko2015} and relativistic generalization of the DLM approach within the KKR-CPA-AAM framework was introduced in a study of temperature dependence of magnetic anisotropy~\cite{Staunton2004, Staunton2006}.
On the experimental side, we note that apart from the phase studied in this work, an orthorhombic phase has often been investigated, see Ref.~\cite{Uhlirova2019} for a phase diagram.
This study is focused on tetragonal CuMnAs, which is stabilised by growth on suitably chosen substrates, and we aim on finding a simple yet accurate model of its magnetic disorder to describe finite-temperature electrical transport.
Recent \textit{ab initio} research on tetragonal CuMnAs~\cite{Maca2019} has dealt with transport properties in less complex situations, from the point of view of disorder.
Here, we i) investigate magnetic moments canted towards the FM state by external magnetic field or other techniques~\cite{Zelezny2017} (see Sec.\ \ref{sec_transport_canted}), ii) compare various finite temperature contributions to electrical transport properties, which may play a role in measurements (see Sec.\ \ref{sec_framework} for description of lattice vibrations and Sec.\ \ref{sec_mag_models} for spin fluctuations), and iii) show combined effect of phonons and magnons occuring under real experimental conditions (Sec.\ \ref{sec_transport_impurities}).
Because of the complexity of AFM magnetic structure, we also discuss three models for spin disorder; we present results also for the tilting model (see Sec.\ \ref{sec_mag_models}).
It should be mentioned, that similar aproaches with angular-dependent distribution of the moments are used for ferromagnets also by other authors.
For details see the Refs.\ \cite{Starikov2018} and \cite{Staunton2006}, where authors have used more complex models with a distribution of tilting angles assigned to each given temperature instead of using only one tilting angle for each temperature.
\section{Formalism, Methods, and Models}\label{sec_formalism}
\subsection{Computational framework and CuMnAs}\label{sec_framework}
The fully relativistic TB-LMTO method with the multicomponent CPA and the atomic sphere approximation \cite{IT-book} is used in this study.
For electrical transport, calculations in a framework of the Kubo linear response theory \cite{IT-relativita} with CPA-vertex corrections \cite{KC-multilayers} and a uniform mesh of at least $8\cdot 10^6$ $k-$points was used; increasing the number to $13\cdot 10^6$ resulted in corrections smaller than one percent of the resistivity value.
LSDA+U approach with nonzero Hubbard $U$ is employed for $d$-orbitals of Mn atoms, similarly to \cite{DW2019-JMMM} implemented within the scalar-relativistic TB-LMTO approach \cite{Shick2004}.
Finite-temperature atomic vibrations are approximated by frozen phonons.
Atomic displacements (root-mean-square displacements $\sqrt{\left<u^2\right>}$, later shown in the units of Bohr radius $a_{\rm{B}}$), were related to temperature using the Debye theory with zero-temperature fluctuations omitted \cite{DW2017-IEEE, DW2017-SPIE, Ebert}.
This is a good approximation unless we focus on extreme temperatures such as those occuring in Earth's core \cite{Drchal2017-DW, Drachal2019JMMM-DW}.
The \textit{spdf-}basis, necessary for inclusion of atomic displacements, was used for most of our calculations.
This study neglects an influence of the Fermi-Dirac distribution modified by finite temperatures.
Both geometry and lattice constants were taken from literature (structure ``II'' in Ref.\ \cite{Maca2019}) and the same values are used for all compositions and temperatures: lattice parameters of bulk P4/nmm CuMnAs are $a = b = 3.82$~\AA~and $c = 6.318$~\AA.
Components of the resistivity tensor $\rho_{xx}$, $\rho_{yy}$, and $\rho_{zz}$ (shown later) correspond to resistivities along $a$, $b$, and $c$, respectively.
Debye temperature of $\Theta_D = 274~\rm{K}$, measured for an orthorhombic sample \cite{Zhang2017}, was used for the lack of experimental data for tetragonal CuMnAs.
In a separate work \cite{Volny2019-CuMnAs}, various types of chemical disorder are discussed in detail while here, we investigate only prototypical and reasonable~\cite{Uhlirova2019} Cu$_{\mathrm{Mn}}$ defect with Cu impurities on Mn sublattices.
Concentrations of this
impurity are stated per formula unit.
The tetragonal structure of CuMnAs entails large empty spaces between atoms.
To remove possible errors coming from different overlap of the atomic spheres in the in-plane and out-of-plane directions, we also performed reference calculations with empty spheres placed at positions of $[0, 0, 0.5]$ and $[0.5, 0.5, 0.5]$ (with respect to the $a$, $b$, and $c$ lattice directions).
The Wigner-Seitz radius of these empty spheres was set to be smaller by 20~\% compared to other atomic spheres.
Employing eight sublattices (instead of six) increases computational expense; therefore, if not stated otherwise, presented results are obtained without the empty spheres.
Magnetic moments on the two Mn sublattices lie in the $a-b$ plane pointing in two opposite directions with respect to each other \cite{Maca2019}.
In a framework of non-collinear magnetism, we assume two modifications of the AFM ground state:
(a) Magnetic moments may be canted towards each other by an angle $\phi$ so that the moments subtend an angle of ${\pi-2\phi}$, see Fig.\ \ref{g_moments} (a).
The AFM and FM states correspond to $\phi=0$ and $\phi=\pi/2$, respectively.
This approximates a rotation of the moments towards a common direction, e.g., under effect of external magnetic field.
(b) Finite-temperature spin fluctuations are simulated by three models, described in detail in the next subsection \ref{sec_mag_models}.
We note that the canting and fluctuations may be combined in order to obtain a state influenced by both nonzero magnetic field and finite temperature (not shown here).
\subsection{Models of magnetic disorder}\label{sec_mag_models}
\begin{figure}[!htb]
\centering
\includegraphics[width=\textwidth]{{01.pdf}}
\caption{
(a) Magnetic moments on two different Mn sublattices are tilted from original AFM (blue) direction by angle $\phi$. Three models of magnetic disorder with original direction of the moments shown by blue arrows: (b) collinear uDLM, (c) tilting, and (d) tilting uDLM.
For clarity, (b) -- (d) show only one of the Mn sublattices.
}
\label{g_moments}
\end{figure}
There exist rather reliable ways to determine $M(T)$ for a given material, either experimentally or theoretically.
However, to study transport we need to know the distribution of individual spins contributing to total $M$, since it is their variation in space that leads to scattering.
One possibility would be to construct an accurate model in terms of a supercell with atomic spin directions provided for example from atomistic spin dynamics.
Spin directions could also be obtained from a mean-field theory \cite{Wysocki2007}.
However, supercell approach has high numerical demands for transport calculations especially when we need to combine it with the presence of phonons.
The spin-disorder resistivity of Fe and Ni by the supercell technique was investigated in Ref.\ \cite{Wysocki2009}.
Furthermore this construction would still have limitations, for example the incorrect treatment of the low temperature behavior in classical spin dynamics, or the limited accuracy of the mean-field model.
Another possibility is to use the CPA-DLM approach to spin disorder resistivity, which has been found to agree well with the supercell calculations for bcc-Fe, fcc-Ni, and Ni$_2$Mn based compounds \cite{DLM_Kudrnovsky}.
Here we adopt this approach, and a limited number of different spin orientations has to be selected for the alloy analogy model to provide a reasonable description of the spin direction variation that captures the most essential properties.
We thus employ two existing models here, and also develop a third one.
Their comparison leads to identification of behavior, which is independent on choice of the model.
To describe spin fluctuations, we employ a collinear uncompensated DLM (uDLM) approach and a tilting model, which were used for FM NiMnSb \cite{DW2019-JMMM} and \cite{DW2019-PRB}.
Moreover, we introduce their combination, later called ``tilting uDLM''.
For schematic illustration, see Fig.\ \ref{g_moments} (b)--(d).
The collinear uDLM model is an extension of the widely used DLM method \cite{DLM, DLM2, DLM_Drchal} with two concentrations $c_+ $ and ${c_- = 1-c_+}$ of magnetic atoms on the same sublattice but with opposite magnetic moments.
The tilting model effectively assumes four mathematically distinguishable atoms (treated within the CPA) having their moments placed on a cone (with the vertex angle $\theta$); axis of the cone corresponds the equilibrium direction \cite{DW2019-JMMM}.
Aiming at having the model as simple as possible, it is sufficient to consider four moments.
For the studied tetragonal system, eight of them (verified for selected cases) led to identical results; a lower number could result in questioning of the proper representation of the symmetry and it would make the tilting model too similar to the uDLM one (therefore, differences may not be easily investigatable).
Systems with relativistic effects have to be investigated numerically with an appropriate distribution of magnetic moments even in the maximally disordered (DLM) state \cite{DLM_Kudrnovsky}.
Such state has zero average magnetization on each sublattice, which can be within the tilting model achieved only by the configuration with $\theta=\pi/2$. However, the tilting angle $\theta=\pi/2$ applied to each magnetic atom (e.g., originally along $x$ direction) would result in moments being in the plane perpendicular to the original direction of the moments ($y-z$ plane).
Therefore, disorder in the original direction (along $x$) is suppressed, and we have introduced an artificial anisotropic distribution in the system, while the moment distribution in the maximally disordered system should be isotropic.
Because of that, we introduce the tilting uDLM model: each fluctuating moment is represented (within the CPA) by ten atoms, one heading towards the original direction, e.g., a vector $(1,0,0)$, second one to the opposite, similarly to the collinear uDLM to $(-1,0,0)$, and around each of them other eight (two times four) form two cones defined by moments tilted from the original directions towards body diagonals, i.e., they would point towards $(\pm 1, \pm 1, \pm 1)$ in a cubic system.
Our system is close to a cubic one; therefore, for a simplicity, we use this fixed vertex angle instead of directions pointing exactly along a diagonal of the tetragonal cell.
Among the three models, this one describes anisotropic material behavior in the best way.
The original direction has concentration of $1-9c_\theta$ and nine others $c_\theta$; unlike the tilting model, two opposite cones for each moments are now constructed with fixed vertex angle.
In contrast to the collinear uDLM approach, the fluctuating moments are now distributed to more directions, which may play a role, e.g., for an anisotropic electrical transport.
Moreover, in contrast with the tilting model, $c_\theta$ may be now increased to $0.1$ (the maximal spin disorder) and it does not lead to a possibility of the moments on different atomic sites being aligned.
We note, that the tilting uDLM model is similar to approach used for fully relativistic investigation of Fe within the DLM model with 26 directions of the moment \cite{DLM_Kudrnovsky}, but now with variable concentrations and applied to tetragonal structure.
In the scalar-relativistic case, the compensated collinear DLM method can be justified analytically (see Appendix of Ref.\ \cite{DLM_Kudrnovsky}) but for the fully-relativistic approach with uncompensated concentrations and non-collinear moments, the treatment of multiple magnetic moments has to be done numerically.
Magnetizations of the Mn sublattices as a function of temperature were calculated by Monte Carlo simulations \cite{Maca2019}.
We related its decrease to our CPA-averaged magnetization calculated as functions of parameters of the models.
Similarly to the FM case \cite{DW2019-JMMM}, it led to a link between temperature values (Monte Carlo simulations) and the parameters ($c_-$, $\theta$, and $c_\theta$).
Because the relation is unambiguous, experimentally well-defined temperature can be used, e.g., as an independent variable in figures.
\section{Results}
\subsection{Electronic structure}
\begin{figure}[!htb]
\centering
\includegraphics[width=\textwidth]{{02.pdf}}
\caption{
Band structures (a)--(d) and DOS (e)--(h) are calculated for CuMnAs without chemical impurities, atomic vibrations, and empty spheres on interstitial positions.
DOS for maximal spin disorder within the tilting uDLM approach is shown by red dashed lines; other data are without spin fluctuations.
See labels for employed basis and Hubbard $U$.
}
\label{g_el_structure}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=\textwidth]{{03.pdf}}
\caption{(a) Band structure calculated in GW approximation. (b) Band structure with empty LMTO spheres at positions of $[0, 0, 0.5]$ and $[0.5, 0.5, 0.5]$ with respect to the \textit{a-b-c} unit cell; other parameters are the same as for Fig.\ \ref{g_el_structure} (a).
}
\label{g_el_structure_ES}
\end{figure}
\begin{figure}[h!tpb]
\centering
\includegraphics[width=0.5\textwidth]{{04.pdf}}
\caption{Increase of DOS at $E_F$ with rising temperature caused by phonons (red squares) and magnons (blue circles) is not additive when compared to their combined effect (green crosses). Spin disorder is treated by the tilting uDLM model.
}
\label{g_dos_ef}
\end{figure}
Basis for any further theoretical study is a sound band structure of the perfect crystal.
To this end, there is a broad consensus in literature that tetragonal CuMnAs is a metal with low DOS at the Fermi level (earning it
sometimes the qualifier of a semimetal).
In literature, the most direct experimental probe into its band structure has been photoemission \cite{Veis2018} along with optical spectra obtained by ellipsometry but these integral quantities (as opposed to angular resolved ones) cannot distinguish fine details of the band structure. Such details can, on the other hand, strongly influence transport properties which are sensitive to the situation at the Fermi surface.
In general, the band structures presented below are similar to each other and when related to behavior of real disordered samples, more attention should be paid to other quantities such as electrical transport.
It has been demonstrated \cite{Veis2018} that density functional theory (DFT) calculations give better agreement with optical and photoemission spectra when extended to DFT+$U$ and Fig.\ \ref{g_el_structure} shows how the band structure and DOS depend on the value of $U$ (the two leftmost panels show, at $U=0$, that extending our basis to $spdf$ has only minor effect on the band structure).
Around the Fermi level, the largest differences among these calculations occur close to $M$ and $X$.
From our results, the $spdf$-basis and $U=0.20$~Ry band structure in Fig.\ \ref{g_el_structure}(d) is closest to Ref.\ \cite{Veis2018}.
Also, comparison to nonrelativistic quasiparticle-selfconsistent GW (QSGW) \cite{QuestaalGW} calculation in Fig.\ \ref{g_el_structure_ES}(a) is favourable; the absence of band splitting is caused by the omission of spin-orbit interaction effects.
On the other hand, adding empty spheres (see Fig.\ \ref{g_el_structure_ES} (b) for $U=0.00$~Ry), does not significantly modify the band structure, i.e., it differs only slightly from Fig.\ \ref{g_el_structure} (a).
The DOS at $E_F$ decreases to 61\,\% of its original value when $U$ is changed from 0.00~Ry to 0.20~Ry, see Fig. \ref{g_el_structure} (f)--(h).
In Figs. \ref{g_el_structure} (e)--(h), DOS for maximal spin disorder within the tilting uDLM model ($c_\theta=0.10$) is shown by dashed lines.
Sharp peaks are smeared in presence of spin fluctuations but, unlike to NiMnSb \cite{DW2019-PRB}, DOS at $E_F$ may increase.
This is caused by high DOS above $E_F$ and it may be connected to decrease of $\rho_{zz}$ later shown in Sec.\ \ref{sec_transport_fluctuations}.
With finite temperature effects included, we present a dependence of DOS at $E_F$ on temperature in Fig.\ \ref{g_dos_ef} for $U=0.00$~Ry.
Temperature value (horizontal axis) was obtained separately for atomic vibrations and spin fluctuations (tilting uDLM model).
Based on the large contribution coming from spin fluctuations, this effect is supposed to have large impact on finite-temperature electrical transport and, moreover, it is necessary to properly describe it in the whole relevant temperature range.
\subsection{Magnetic moments and total energy}
Local magnetic moments on each Mn sublattice of undistorted CuMnAs for $U=0.00$~Ry are found to be $3.72\mu_B$ and $3.71\mu_B$ for $spd-$ and $spdf-$calculations, respectively.
This value is increased by nonzero $U$, e.g., it is $4.08\mu_B$ for both bases with $U=0.10$~Ry.
When spin fluctuations are assumed, these values are almost unchanged (up to a few percent) for each direction within the CPA, but the local magnetic moment on each sublattice vanishes monotonically with increasing $\theta$, $c_-$, or $c_\theta$.
\begin{figure}[htbp]
\centering
\includegraphics[width=\textwidth]{{05.pdf}}
\caption{Mn local moments on the two magnetic sublattices (left axis; red empty squares) and energy difference from the AFM ground state (right axis; blue croses) with tilted magnetic moments present corrects symmetries with respect to $\phi=\pi/2$ and both quantities are influenced only little by increasing Cu$_{\mathrm{Mn}}$ impurity concentration from zero (a) to 5 \% (b), and 10 \% (c).
Calculated for $U=0.00$~Ry.
}
\label{g_magmom}
\end{figure}
Under strong magnetic field, originally antiparallel moments may be forced to cant towards the field direction.
To study this effect, we plot the difference between the total energy of a state with canted moments (in the $a-b$ plane) and the energy of the AFM ground state in Fig.\ \ref{g_magmom}.
Both Mn magnetic moments (Fig.\ \ref{g_magmom} -- left axis) behave equivalently and the energy differences (Fig.\ \ref{g_magmom} -- right axis) were confirmed to have a correct symmetry with respect to $\phi=0.5\pi$.
Mn local magnetic moments are practically unchanged; for stoichiometric CuMnAs, the moments are lowest for the AFM state and there is a minimum in the range of $\phi$ from $0.20\pi$ to $0.3\pi$ for Cu-rich systems.
However, this is not observed for the energy differences, which leads clearly to the AFM configuration being preferred.
Trends visible in Fig.\ \ref{g_magmom} are unchanged regardless whether empty spheres are used or not and also regardless of the basis choice ($spd$ or $spdf$).
The largest change is in the energy difference between the AFM and FM states, e.g., (for stoichiometric undistorted CuMnAs) the $spd-$basis decreases the difference to approx.\ 82~\% and empty spheres to approx.\ 44~\% of the original value.
The large difference in the energies implies, that it is difficult to experimentally observe a state with significantly canted moments by applying physically reasonable external fields.
For example, an estimation of the magnetic field required to cant the moments to $\phi=0.2\pi$ results in $B\approx 50$~T.
\subsection{Electrical transport with canted moments}\label{sec_transport_canted}
A large anisotropy of electrical transport is obtained for CuMnAs, regardless orientation of magnetic moments and other conditions.
The longitudinal in-plane resistivity $\rho_{xx}$ may be as many as seven times smaller than the out-of-plane ($\rho_{zz}$, crystallographic direction $c$) one, which is in agreement with Refs.\ \cite{Volny2019-CuMnAs} and \cite{Maca2017}.
Especially $\rho_{zz}$ can reach values of a few hundreds of $\mu\Omega \, \mathrm{cm}$, which is much more than what is usually observed for metallic systems.
These facts are compatible with the layered structure of CuMnAs as well as with its semimetallic nature.
Magnetic moments probably cannot be canted easily by external magnetic fields (see Fig.\ \ref{g_magmom} for energy differences from the AFM state)
and for the manipulation of moments by electric currents \cite{Zelezny2017}, probably, holds the same.
Together with energy analysis, we investigated electrical transport of CuMnAs with canted moments to predict changes, which could be expected.
In Fig.\ \ref{g_canting}, we show resistivities of Cu-rich CuMnAs with magnetic moments canted from the original AFM direction towards the FM orientation.
The canting dramatically reduces both $\rho_{xx}$ and $\rho_{zz}$ and the FM resistivity is much lower than the AFM one.
The in-plane resistivity is nonmonotonic with maxima slightly bellow $\phi=0.25\pi$, which coincides with minima in magnetic moments visible in Fig.\ \ref{g_magmom}.
Fig.\ \ref{g_canting} (a) is obtained without empty spheres, while data in Fig.\ \ref{g_canting} (b) are calculated with the empty spheres on interstitial positions.
The resistivity is increased by alloying but decreased with nonzero $U$.
Increasing $U$ to $0.15$ and $0.20$~Ry reduces resistivities even more (not shown in the Figure).
\begin{figure}[htpb]
\centering
\includegraphics[width=\textwidth]{{06.pdf}}
\caption{In-plane $\rho_{xx}$ (full symbols) and out-of-plane $\rho_{zz}$ (empty symbols) are calculated for CuMnAs with canted magnetic moments ($\phi=0$ is original AFM state, $\phi=\pi/2$ gives FM in-plane moments) and the $spd-$basis. Left: Cu atoms on Mn sublattice (Cu$_{\mathrm{Mn}}$) without empty LMTO spheres are assumed for 2~\% (circles), 5~\% (triangles), and 10~\% (squares) of the impurity; $U=0.00$ Ry. Right: For comparison, empty spheres are taken into account for three combinations of $U$ and Cu$_{\mathrm{Mn}}$ concentrations. All $\rho_{zz}$ values are divided by a factor of five.
}
\label{g_canting}
\end{figure}
\subsection{Electrical transport with spin fluctuations}\label{sec_transport_fluctuations}
Electrical resistivity for three models of magnetic disorder is shown in Fig.\ \ref{g_rho_mag_dis}: (a) collinear uDLM, (b) tilting, and (c) tilting uDLM.
The top horizontal axis shows temperature, which corresponds to the decrease of local Mn-sublattice magnetization \cite{Maca2019} for the given parameter of the spin disorder.
The absence of chemical impurities causes $\rho_{xx}=\rho_{zz}=0$ for $\theta = c_- = c_\theta=0$.
Nonzero Hubbard $U$ causes an increase of both $\rho_{xx}$ and $\rho_{zz}$ (except of the tilting model for $\rho_{xx}$ up to room temperature).
Differences between various values of $U$ are small for small spin fluctuations, i.e., for $\theta \lesssim 0.2\pi$, $c_-\lesssim 0.1$, and $c_\theta\lesssim 0.02$; fitting the decrease of Mn-sublattice magnetization \cite{Maca2019} with temperature $T$, these values roughly correspond to $T=$230~K, 240~K, and 220~K, respectively.
Similarity of these temperatures suggests that the three models have similar applicability to real AFM systems.
\begin{figure}[htpb]
\centering
\includegraphics[width=\textwidth]{{07.pdf}}
\caption{ Electrical resistivity of CuMnAs without chemical disorder and atomic vibrations is calculated with the $spd-$basis for three models of magnetic disorder: tilting model (a), collinear uDLM model (b), and tilting uDLM model (c). Data for $U$ of 0.00, 0.05, 0.10, and 0.15 Ry are shown by gray circles, blue triangles, green squares, and red diamonds, respectively; $\rho_{xx}$ corresponds to full symbols and $\rho_{zz}$ to empty ones. (d) Results of the three models ($U=0.15$~Ry) plotted as a function of temperature and compared with thin-film measurements \cite{Wadley2013}, see text for details.
}
\label{g_rho_mag_dis}
\end{figure}
The collinear uDLM model, Fig.\ \ref{g_rho_mag_dis} (b), assumes the moments only in the in-plane direction, which may give not completely realistic anisotropy of the transport.
Therefore, we calculated maximally disordered DLM state ($c_-=0.5$) with antiparallel moments on each sublattice along $\pm a$, $\pm b$, and $\pm c$ directions: these three cases were found to have elements of the resistivity tensors different by less than 0.1~\% (comparable with numerical errors), which leads to a conclusion, that the collinear uDLM is suitable for study of anisotropy.
Moreover, we present results for the tilting uDLM model in Fig.\ \ref{g_rho_mag_dis} (c): the model has moments more equally distributed (with cubic symmetry instead of the tetragonal one).
Results for the collinear uDLM and tilting uDLM models are almost identical, which numerically justifies the simpler collinear uDLM approach.
Temperature (obtained by fitting of the Mn-sublattice magnetization) is also useful for model comparison, because the resistivity is originally obtained as a function of three different internal parameters ($\theta$, $c_-$, and $c_\theta$).
With temperature increasing from $T=0$, both $\rho_{xx}$ and $\rho_{zz}$ increase linearly up to $T\approx 200$~K, see Fig.\ \ref{g_rho_mag_dis} (d) for $U=0.15$~Ry.
While the collinear and tilting uDLM approaches give almost the same results, they differ from the tilting model, especially at low temperatures.
Fig.\ \ref{g_rho_mag_dis} (d) also compares calculated data with measured thin-film values \cite{Wadley2013} of the planar resistivity (black solid line) and with the residual resistivity of $\rho_{xx}^0=79\,\mu\Omega \, \mathrm{cm}$ subtracted to eliminate an influence of chemical impurities independent on temperature (black dashed line).
Similar values of $\rho_{xx}(T)$ were measured for orthorhombic CuMnAs \cite{Zhang2017}.
$U=0.15$~Ry was chosen for this Figure because of the best correspondence with experiments.
In can be concluded, that the slope of the measured temperature-dependence is well reproduced by the tilting model.
It is similar to behavior of the collinear uDLM model for FM NiMnSb \cite{DW2019-PRB}; however, missing effects of impurities and phonons may be nontrivial, see the next subsection, and because of absence of experimental data, this cannot be studied in greater detail for CuMnAs.
If tetragonal bulk samples of CuMnAs are available, we suggest measuring the out-of-plane resistivity at higher temperatures in order to determine, which model of the spin fluctuations is the most
successful for the studied AFM.
Even if a more advanced model of spin fluctuations were assumed, e.g., a distribution of magnetic moments based on Heisenberg Hamiltonian, it would be difficult to estimate its correctness without comparison with experimental data for conditions of important magnetic disorder.
Therefore, we consider the presented simpler models (neglecting any magnetic short-range order and treated withing the CPA) to be sufficient in the temperature range of available experimental data.
The general trend of decreasing resistivity from room to the N\'eel temperature correlates with and can probably be attributed to large increase of DOS at the Fermi level caused by spin fluctuations, see previous Figs.\ \ref{g_el_structure} and \ref{g_dos_ef}.
Despite the fact that there is no straightforward relation between DOS at the Fermi level and electrical transport, it can be addressed, e.g., by model calculations \cite{Brouers1975}.
\subsection{Electrical transport with phonons and impurities}\label{sec_transport_impurities}
To study the decrease of resistivity due to temperature-induced increase of the DOS at $E_F$ in more details, we present a combined effect of spin fluctuations and atomic displacements in Fig.\ \ref{g_rho_mag_dis_disp}.
In contrast to Fig.\ \ref{g_rho_mag_dis}, it was obtained with the $spdf-$basis required because of the displacements; it is the reason for slightly different values for $\sqrt{\left<u^2\right>}=0.00\, a_{\rm{B}}$.
The displacements separately cause a large increase of the DOS at the Fermi level (Fig.\ \ref{g_dos_ef}) and, consequently, saturation of $\rho_{xx}$ and decrease of $\rho_{zz}$ for $\sqrt{\left<u^2\right>} \gtrsim 0.40\, a_{\rm{B}}$ compared to smaller displacements (see also Tab.\ \ref{t_rho_u}).
Similarly to the pure effect of magnons, the combined effect of the spin fluctuations and the displacements also leads to the saturation of $\rho_{xx}$ and to the decrease of $\rho_{zz}$.
Similar behavior was observed also for the tilting model having $\rho_{zz}$ lesser by approx. 15~\% in comparison to Fig.\ \ref{g_rho_mag_dis_disp}.
\begin{figure}[htpb]
\centering
\includegraphics[width=0.7\textwidth]{{08.pdf}}
\caption{
Saturation and decrease of resistivities is obtained also for the combined effect of spin fluctuations and atomic displacements (values of $\rho_{zz}$ for $\sqrt{\left<u^2\right>}$ are in $a_{\rm{B}}$, calculated with the $spdf-$basis and $U=0.00$~Ry).
}
\label{g_rho_mag_dis_disp}
\end{figure}
Nonzero values of $U$ increase resistivities when nonzero atomic vibrations are assumed, while a decrease is observed for CuMnAs with realistic \cite{Uhlirova2019} Cu$_{\mathrm{Mn}}$ impurity concentrations, see Tables \ref{t_rho_u} and \ref{t_rho_c} ($spdf-$calculations).
Both kinds of disorder explain well the large changes in resistivity and its anisotropy, which makes identification of contributions in experimental data difficult.
Ani\-so\-tro\-py $\rho_{zz}/\rho_{xx}$ is increasing with $U$, decreasing with $\sqrt{\left<u^2\right>}$ and it depends strongly especially on chemical composition.
For the atomic displacements (Tab.\ \ref{t_rho_u}), we also observe saturation of $\rho_{xx}$ and decrease of $\rho_{zz}$ for large temperatures, which can be attributed to the increasing DOS at $E_F$, similarly to the case of magnetic disorder.
The in-plane residual (4 K) resistivity in Ref.\ \cite{Wadley2013} was measured about $\rho_{xx}^0 \approx 80\,\mu\Omega \, \mathrm{cm}$ for epitaxial tetragonal CuMnAs indicating even larger chemical disorder than presented in Tab.\ \ref{t_rho_c}.
The same study reported room-temperature resistivity of $160\,\mu\Omega \, \mathrm{cm}$, which would agree with the effect of phonons and magnons, if the impurities are neglected (Fig.\ \ref{g_rho_mag_dis_disp}).
Comparing Fig.\ \ref{g_rho_mag_dis_disp} and Tab.\ \ref{t_rho_u}, combined influence of various scattering mechanisms clearly deviates from Matthiessen's rule; therefore, such room-temperature value may be realistic, but the treatment of finite-temperature effects and chemical impurities together remains a topic for further study \cite{Volny2019-CuMnAs}.
\begin{table}
\caption{ Electrical resistivities of CuMnAs without chemical disorder and spin fluctuations are given in $\mu\Omega \, \mathrm{cm}$ and calculated with the $spdf-$basis.}
\label{t_rho_u}\centering
\begin{tabular}{|c c||cc|cc|}\hline
$\sqrt{\left<u^2\right>}$ & T & \multicolumn{2}{c|}{$U=0.00$ Ry}& \multicolumn{2}{c|}{$U=0.20$ Ry}\\
$[a_{\rm{B}}]$ & [K] & $\rho_{xx}$ & $\rho_{zz}$ & $\rho_{xx}$ & $\rho_{zz}$ \\ \hline
0.1 & 50 & 9& 27& 11& 50 \\
0.2 & 110 & 40& 113& 51& 206 \\
0.3 & 185 & 133& 307& 142& 455 \\
0.4 & 290 & 198& 251& 282& 468 \\
0.5 & 415 & 203& 214& 283& 319 \\
0.6 & 570 & 209& 204& 271& 267 \\ \hline
\end{tabular}
\end{table}
\begin{table}
\caption{ Electrical resistivities of Cu-rich CuMnAs without atomic vibrations and spin fluctuations are given in $\mu\Omega \, \mathrm{cm}$ and calculated with the $spdf-$basis.}
\label{t_rho_c}\centering
\begin{tabular}{|c||cc|cc|}\hline
& \multicolumn{2}{c|}{$U=0.00$ Ry}& \multicolumn{2}{c|}{$U=0.20$ Ry}\\
Cu$_{\mathrm{Mn}}$ & $\rho_{xx}$ & $\rho_{zz}$ & $\rho_{xx}$ & $\rho_{zz}$ \\ \hline
2~\%& 10& 51& 2& 6 \\
5~\%& 23& 131& 4& 32 \\
10~\%& 40& 317& 15& 285 \\ \hline
\end{tabular}
\end{table}
\section{Conclusions}
We have presented an \textit{ab inito} investigation of electronic structure and electrical transport in tetragonal AFM CuMnAs.
For a treatment of nonzero temperatures, we have employed the CPA-AAM with frozen phonons.
Real magnetic disorder is approximated by three models, which are simple and, therefore, easy to implement to other studies, but still reasonably accurate; due to their nature, only results independent on the choice of the model can be considered as physically relevant.
Because electrical resistivity obtained for uDLM and tilting-uDLM models (which differ in directions of the moments) agree with each other and with experimental data, we consider these models to be more appropriate for the AFM CuMnAs than the tilting one.
To obtain more realistic description of the finite temperature behavior, e.g., Monte Carlo simulations and many different directions of the moments could be used.
Both phonons and magnons are treated within the CPA, there is no significant increase of numerical expenses, compared to zero-temperature calculations.
TB-LMTO band structure in LSDA$+U$ approach is compared to the GW results and the best correspondence is found for $U=0.20$ Ry.
Substantial canting of magnetic moments by external magnetic field is probably not achievable; therefore, related decrease of resistivity should not play an important role in experiments.
Saturation of $\rho_{xx}$ and decrease of $\rho_{zz}$ in CuMnAs was observed for temperatures above room temperature, which is not common for metallic systems, but it can be explained by increasing DOS at $E_F$.
For reasonable conditions of room temperature and chemical disorder featured by additional 5~\% of Cu atoms on Mn-sublattices, the largest separate contribution (among impurities, magnons, and phonons) to the resistivity is coming from spin fluctuations.
The tilting model of the spin fluctuations agrees well with the measured slope of $\rho_{xx}(T)$.
We considered also collinear and tilting uDLM approaches; they appear to overestimate the resistivity in the low temperature regime, since already for low temperatures they include moments oriented drastically differently from each other, leading to a strong scattering.
For temperatures $T \gtrsim 0.5 T_N$, the results from all three methods are rather close and we have also numerically justified the collinear uDLM, including the anisotropy of the resistivity.
Therefore, even the most simple collinear uDLM method thus provides a good description for the cases of strong spin disorder.
\section*{Acknowledgements}
Financial support from the Czech Science Foundation is acknowledged by DW, KC, and IT (Grant No.\ 18-07172S) and KV (Grant No.\ 19-28375X).
This work was also supported by The Ministry of Education, Youth and Sports from the Large Infrastructures for Research, Experimental
Development and Innovations project ``IT4Innovations National Supercomputing Center – LM2015070'' and grant No.\ LM2018110 and LNSM-LNSpin.
Access to computing and storage facilities owned by parties and projects contributing to the National Grid Infrastructure MetaCentrum provided under the programme ``Projects of Large Research, Development, and Innovations Infrastructures" (CESNET LM2015042), is greatly appreciated as well as the support of EU FET Open RIA Grant No.\ 766566.
\section*{References}
|
1,108,101,565,670 | arxiv |
\section{Method}
\label{sec:approach}
We propose a novel end-to-end framework for cross-domain label-adaptive stance detection. Our architecture (see Figure~\ref{fig:approach:moe}) is based on input representations from a pre-trained language model, adapted to source domains using Mixture-of-Experts and domain adversarial training (Section~\ref{subsec:approach:proposed_model}). We further use self-adaptive output representations obtained via label embeddings, and unsupervised alignment between seen and unseen target labels for out-of-domain datasets.
Unlike previous work, we focus on learning the relationship between datasets and their label inventories in an unsupervised fashion. Moreover, our Mixture-of-Experts model is more compact than the one proposed by \citet{wright-augenstein-2020-transformer}, as we introduce a parameter-efficient architecture with layers that are shared between the experts. Finally, we explore the capability of the model to predict from unseen user-defined target.
With this framework, we solve two main challenges: (\emph{i})~training domain-adaptive models over a large number of datasets from a variety of source domains, and (\emph{ii})~predicting an \emph{unseen} label from a disjoint set of over 50 unique labels.
\subsection{Cross-Domain Stance Detection}
\label{subsec:approach:proposed_model}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{assets/images/Domain_Adapters_v2.png}
\caption{Overview of our proposed model \textit{\OurModel\ -- Mixture-of-Experts with Label Embeddings}.}
\label{fig:approach:moe}
\end{figure}
\paragraph{Mixture-of-Experts (MoE)} is a well-known technique for multi-source domain adaptation~\citep{guo-etal-2018-multi,li-etal-2018-whats}. Recently, this framework was extended to large pre-trained Transformers~\citep{wright-augenstein-2020-transformer}.
In particular, for each domain $k \in K$, there is a domain expert $f_k$, and a shared, global model $f_g$. We define $K=4$, as we use four different domains (see Section~\ref{subsec:sources}), making this approach appealing to further encourage knowledge sharing between datasets. Further, the models produce a set of probabilities $p_k$ from each expert and $p_g$ from the global model, for all the (often shared) target labels. Then, the final output of the model is obtained by passing these predictions through a combination function, e.g.,~mean average, weighted average, or attention-based combination~\citep{wright-augenstein-2020-transformer}. We use mean average to gather the final distribution across the label space:
\begin{equation}
p_A(x, \bar{K}) = \frac{1}{|\bar{K}| + 1} \sum_{k \in \bar{K}} p_k(x) + p_g(x)
\end{equation}
\paragraph{Mixture-of-Experts with Label Embeddings (\OurModel)} We propose several changes to Mixture-of-Experts to improve the model's parameter efficiency, reduce training and inference times, and allow for different label inventories for each task.\footnote{In contrast, the model of~\citet{wright-augenstein-2020-transformer} requires that the datasets use the same labels.} First, in contrast to~\citet{wright-augenstein-2020-transformer}, we use a shared encoder, here, RoBERTa~\citep{liu2019roberta}, instead of a separate large Transformer model for each domain. Next, for each domain expert and the global shared model, we add a single Transformer~\citep{NIPS2017_7181:transformer} layer on top of the encoder block. We thereby retain the domain experts while sharing information through the encoder. This approach reduces the number of parameters by a factor of the size of the entire model divided by the size of a single layer, i.e.,~we only use four additional layers (one such encoder block per domain) instead of 48 (the number of layers in \RobertaB, not counting embedding layers). For convenience, we set the hidden sizes in the newly-added blocks to match the encoder's.
Next, each domain expert receives as input the representations from the shared encoder of all tokens in the original sequence. Finally, we obtain a domain-specific and a global representation for the input sentence from the \texttt{[CLS]} tokens. These hidden representations are denoted as $H \in \mathbb{R}^{K \times d_h}$, where $K$ is the number of domains, and $d_h$ is the model's hidden size. They are passed through a single label embedding layer to obtain the probability distributions.
\paragraph{Domain-Adversarial training} was introduced as part of the Domain-adversarial neural networks (\textbf{DANN})~\citep{pmlr-v37-ganin15}.
The aim is to learn a task classifier by confusing an auxiliary domain classifier optimised to predict a meta target, i.e.,~the domain of an example. This approach has shown promising results for many NLP applications~\citep{li-etal-2018-whats,gui-etal-2017-part,wright-augenstein-2020-transformer}. Formally, it forces the model to learn domain-invariant representations, both for the source and for the target domains. The latter is done with an adversarial loss function, where we minimise the task objective $f_g$, and maximise the confusion in the domain classifier $f_d$ for an input sample $x$ (see Eq.~\ref{eq:approach:loss_dann}). We implement this with a gradient reversal layer, which ensures that the source and the target domains are made to be similar.
\begin{equation}
\mathcal{L}_D = \max_{\theta_{D}}\min_{\theta_{G}} -d\log{f_d(f_g(x))}
\label{eq:approach:loss_dann}
\end{equation}
\subsection{Cross-Domain Label-Adaptive Prediction}
\label{subsec:approach:label_adaptive}
The second major challenge is how to obtain predictions for out-of-domain datasets. We want to emphasise that just a few of our \DatasetsCount datasets share the same set of labels (see Section~\ref{sec:datasets}); yet, many labels in different datasets are semantically related.
\paragraph{Label Embeddings (LEL)} In multi-task learning, each task typically has its own task-specific labels (in our case, dataset-specific labels), which are predicted in a joint model using separate output layers.
However, these dataset-specific labels are not entirely orthogonal to each other (see Section~\ref{sec:datasets}). Therefore, we adopt label embeddings to encourage the model to learn task relations in an unsupervised fashion using a common vector space. In particular, we add a Label Embeddings Layer, or LEL, ~\cite{augenstein-etal-2018-multi}), which learns a label compatibility function between the hidden representation of the input $h$, here the one from the \texttt{[CLS]} token, and an embedding matrix $L$:
\begin{equation}
p = \softmax(Lh)
\end{equation}
\noindent where $L \in \mathbb{R}^{(\sum_{i}{L_i}) \times h}$ is the shared label embedding matrix for all datasets, and $l$ is a hyper-parameter for the dimensionality of each vector.
We set the size of the embeddings to match the hidden size of the model, and obtain the hidden representation $h$ from the last layer of the pre-trained language model. Afterwards, we optimise a cross-entropy objective over all labels, masking the unrelated ones and keeping visible only the ones from the target datasets for a sample in the batch. We use the same masking procedure at inference time.
\paragraph{Label-Adaptive Prediction} In an unsupervised out-of-domain setting, there is no direct way to obtain a probability distribution over the set of test labels.
Label embeddings are an easy indirect option for obtaining these predictions, as they can be used to measure the similarity between source and target labels.
We investigate several alternatives.
\para{Hard Mapping}
A supervised option is to
define a set of meta-groups (\emph{hard labels}), here six, as shown in Table~\ref{tab:approach:hard_map}, then to train the model on these labels. E.g., if the out-of-domain dataset is \emph{snopes}, then its labels are replaced with meta-group labels -- \emph{agree} $\Rightarrow$ \emph{positive}, and \emph{refute} $\Rightarrow$ \emph{negative}, and thus we can directly use the predictions from the model for out-of-domain datasets.
However, this approach has several shortcomings: (\emph{i})~labels have to be grouped manually, (\emph{ii})~the meta-groups should be large enough to cover different task definitions, e.g.~the dataset's label inventory may vary in size, and, most importantly, (\emph{iii})~any change in groupings would require full model re-training.
\begin{table}[t]
\centering
\setlength{\tabcolsep}{3pt}
\resizebox{1.00\columnwidth}{!}{%
\small
\hyphenpenalty10000
\begin{tabularx}{\columnwidth}{lL{0.8\columnwidth}}
\toprule
\bf Group & \bf Task\_\_Labels Included \\
\midrule
\makecell[l]{Positive} & arc\_\_\textit{agree}, argmin\_\_\textit{argument~for}, emergent\_\_\textit{for}, fnc1\_\_\textit{agree}, iac1\_\_\textit{pro}, mtsd\_\_\textit{favor}, perspectrum\_\_\textit{support}, poldeb\_\_\textit{for}, rumor\_\_\textit{endorse}, scd\_\_\textit{for}, semeval2016t6\_\_\textit{favor}, semeval2019t7\_\_\textit{support}, snopes\_\_\textit{agree}, vast\_\_\textit{pro}, wtwt\_\_\textit{support} \\
\midrule
\makecell[l]{Negative} & arc\_\_\textit{disagree}, argmin\_\_\textit{argument~against}, emergent\_\_\textit{against}, fnc1\_\_\textit{disagree}, iac1\_\_\textit{anti}, ibmcs\_\_\textit{con}, mtsd\_\_\textit{against}, perspectrum\_\_\textit{undermine}, poldeb\_\_\textit{against}, rumor\_\_\textit{deny}, scd\_\_\textit{against}, semeval2016t6\_\_\textit{against}, semeval2019t7\_\_\textit{deny}, snopes\_\_\textit{refute}, vast\_\_\textit{con}, wtwt\_\_\textit{refute} \\
\midrule
\makecell[l]{Discuss} & arc\_\_\textit{discuss}, emergent\_\_\textit{observing}, fnc1\_\_\textit{discuss}, rumor\_\_\textit{question}, semeval2019t7\_\_\textit{query}, wtwt\_\_\textit{comment} \\
\midrule
\makecell[l]{Other} & arc\_\_\textit{unrelated}, fnc1\_\_\textit{unrelated}, iac1\_\_\textit{other}, mtsd\_\_\textit{none}, rumor\_\_\textit{unrelated}, semeval2019t7\_\_\textit{comment}, wtwt\_\_\textit{unrelated} \\
\midrule
\makecell[l]{Neutral} & rumor\_\_\textit{neutral}, vast\_\_\textit{neutral} \\
\bottomrule
\end{tabularx}
}
\caption{Hard mapping of labels to categories.}
\label{tab:approach:hard_map}
\end{table}
\para{Soft Mapping} To overcome these limitations, we propose a simple, yet effective, entirely unsupervised procedure involving only the label names. More precisely, we measure the similarity between the names of the labels across datasets. This is an intuitive approach for finding a matching label without further context, e.g.,~\textit{for} is probably close to \textit{agree}, and \textit{refute} is close to \textit{against}.
In particular, given a set of out-of-domain target labels $Y^{\tau} \in \{y^{\tau}_1, \dots, y^{\tau}_k\}$, and a set of predictions from in-domain labels $P_{\delta} \in \{p^{\delta}_1, \dots, p^{\delta}_m\}, p^{\delta}_i \in \{y^{\delta}_1$, $\dots, y^{\delta}_j\}$, we select the label from $Y_{\tau}$ with the highest cosine similarity to the predicted label $p^{\delta}_i$:
\begin{equation}
p^{\tau}_i = \argmax_{y^{\tau} \in Y^{\tau}} cos(y^{\tau}, p^{\delta}_i)
\end{equation}
where $k$ is the number of out-of-domain labels, $m$ the number of out-of-domain examples, and $j$ the number of in-domain labels. The procedure can generalise to any labels, without the need for additional supervision. To illustrate this, the embedding spaces of pre-trained embedding models for our \DatasetsCount datasets are visualised in Appendix~\ref{sec:appendix:lel_spaces}.
\para{Weak Mapping} Nevertheless, as proposed, this procedure only takes label names into account, in contrast to the hard labels that rely on human expertise. This makes combining the labels in a weakly supervised manner an appealing alternative. For this, we measure label similarities as proposed, but incorporate some supervision for defining the embeddings.
We first group the labels into six separate categories to define their nearest neighbours (see Table~\ref{tab:approach:hard_map}). Then, we choose the most similar label for the target domain from these neighbours.
The list of neighbours is defined by the group of the predicted label. However, there is no guarantee that there will be a match for the target domain within the same group, and thus we further define group-level neighbourhoods (see Table~\ref{tab:hand_neightbours}), as it is not feasible to define the neighbours for all (more than 50) labels individually.
One drawback is that each new label/group must define a neighbourhood with similar labels -- and vice-versa, it should be assigned a position in the neighbourhoods of the existing labels.
\begin{table}[t]
\centering
\small
\begin{tabular}{l|l}
\toprule
\bf Group & \bf Closest Neighbours \\
\midrule
Positive & Other, Neutral, Discuss, Negative \\
Other & Neutral, Discuss, Positive, Negative \\
Neutral & Discuss, Other, Positive, Negative \\
Discuss & Neutral, Other, Negative, Positive \\
Negative & Discuss, Neutral, Other, Positive \\
\bottomrule
\end{tabular}
\caption{Label grouping and the closest neighbours of each, sorted from closest to most distant.}
\label{tab:hand_neightbours}
\end{table}
\subsection{Training}
We train the model using the following loss:
\begin{gather}
\mathcal{L}_s = \frac{1}{N}\sum_i{y_i\log{p_{X}(x, S')}} \\
\mathcal{L}_t = \frac{1}{N}\sum_i{y_i\log{p_{t}(x)}} \\
\mathcal{L} = \lambda\mathcal{L}_s + (1-\lambda)\mathcal{L}_t + \gamma\mathcal{L}_D
\end{gather}
First, we sum the source-domain loss ($\mathcal{L}_s$) with the meta-target loss from the domain expert sub-network ($\mathcal{L}_t$), where the contribution of each is balanced by a single hyper-parameter $\lambda$, set to $0.5$. Next, we add the domain adversarial loss ($\mathcal{L}_D$), and we multiply it by a weighting factor $\gamma$, which is set to a small positive number to prevent this regulariser from dominating the overall loss. We set $\gamma$ to $0.01$. Furthermore, since our dataset is quite diverse even in the four source domains that we outlined, we optimise the domain-adaptive loss towards a meta-class for each dataset, instead of the domain.
\section*{Acknowledgments}
We thank the anonymous reviewers for their helpful questions and comments, which have helped us improve the quality of the paper.
We also would like to thank Guillaume Bouchard for the useful feedback. Finally, we thank the authors of the stance datasets for open-sourcing and providing us with their data.
\section{Discussion}
\label{sec:dicussions}
\begin{figure}[t!]
\centering
\includegraphics[width=0.87\columnwidth]{assets/images/pcorrelation.png}
\caption{Pearson correlation between the out-of-domain results for our model (\OurModel w/ \emph{weak mappings}) in terms of \fmacro \ score and dataset characteristics. The (type) of the features is shown in parenthesis.}
\label{fig:correlations}
\end{figure}
We further study the correlation between the scores for the best model in the out-of-domain setup \emph{\OurModel w/ weak mappings} and a rich set of quantitative and stance-related characteristics of the datasets (these are further discussed in Section~\ref{sec:datasets} and in Appendix~\ref{sec:appendix:dataset}). In particular, we represent each dataset as a set of features, e.g.,~\emph{fnc1} would have \emph{target -- News}, training set size of \emph{42,476}, etc., and then we measure the Pearson correlation between these features and the model's F\textsubscript{1} scores per dataset.
Figure~\ref{fig:correlations} shows the most important factors for out-of-domain performance.\footnote{Some of the factors in the analysis are not independent, e.g., Social Media as a domain, and Tweet as a context.}
We see positive correlations of F\textsubscript{1} with the training, and the development set sizes, and a negative one with the testing set size, which suggests that large datasets are indeed harder for the model. Interestingly, if there is an overlap in the targets between the testing and the training sets, the model's \fmacro\ is worse; however, this is not true for context overlap. Unsurprisingly, the size of the vocabulary is a factor that negatively impacts F\textsubscript{1}, and its moderate negative correlation with the model's scores confirms that.
The domain, the target and the context types are also important facets: the News domain has a sizable positive correlation with \fmacro, which is also true for the related features Headline target and Article body. Another positive correlation is for having a Claim as the context. On the contrary, a key factor that hinders model performance is Social Media text, i.e.,~having a tweet as a context.
\section{Conclusion and Future Work}
\label{sec:conclusion}
We have proposed a novel end-to-end unsupervised framework for out-of-domain stance detection with respect to unseen labels. In particular, we combined domain adaptation techniques such as Mixture-of-Experts and domain-adversarial training with label embeddings, which yielded sizable performance gains on \DatasetsCount datasets over strong baselines: both in-domain, i.e.,~for seen targets, and out-of-domain, i.e., for unseen targets. Moreover, we performed an exhaustive analysis of the cross-domain results, and we highlighted the most important factors influencing the model performance.
In future work, we plan to experiment with more datasets, including non-English ones, as well as with
other formulations of the stance detection task, e.g., stance of a person \cite{ICWSM2020:Unsupervised:Stance:Twitter} or of a news medium \cite{stefanov-etal-2020-user-stance} with respect to a claim or a topic.
\section*{Ethics and Broader Impact}
\paragraph{Dataset Collection}
We use publicly available datasets and we have no control over the way they were collected. For datasets that distributed their data as Twitter IDs, we used the Twitter API\footnote{\url{http://developer.twitter.com/en/docs}} to obtain the full text of the tweets, which is in accordance with the terms of use outlined by Twitter.\footnote{\url{http://developer.twitter.com/en/developer-terms/agreement-and-policy}} Note that we only downloaded public tweets.
\paragraph{Biases}
We note that some of the annotations are subjective. Thus, it is inevitable that there would be certain biases in the datasets. These biases, in turn, will likely be exacerbated by the supervised models trained on them~\citep{waseem2021disembodied}. This is beyond our control, as are the potential biases in pre-trained large-scale transformers such as BERT and RoBERTa, which we use in our experiments.
\paragraph{Intended Use and Misuse Potential}
Our models can enable analysis of text and social media content, which could be of interest to business, to fact-checkers, journalists, social media platforms, and policymakers.
However, they could also be misused by malicious actors, especially as most of the datasets we consider in this paper are obtained from social media. Most datasets compiled from social media present some risk of misuse. We, therefore, ask researchers to exercise caution.
\paragraph{Environmental Impact}
We would also like to note that the use of large-scale Transformers requires a lot of computations and the use of GPUs/TPUs for training, which contributes to global warming \cite{strubell-etal-2019-energy}. This is a bit less of an issue in our case, as we do not train such models from scratch; rather, we fine-tune them on relatively small datasets. Moreover, running on a CPU for inference, once the model has been fine-tuned, is perfectly feasible, and CPUs contribute much less to global warming.
\section{Stance Detection Datasets}
\label{sec:datasets}
In this section, we provide a brief overview of the \DatasetsCount stance datasets included in our study, and we show their key characteristics in Table~\ref{tab:dataset:domains}. More details are given in Section~\ref{subsec:datasets} and in the Appendix (Section~\ref{sec:appendix:data_splits}). We further motivate the source groupings used in our experiments and analysis (Section~\ref{subsec:sources}).
\subsection{Datasets}
\label{subsec:datasets}
\para{arc} The Argument Reasoning Comprehension dataset has posts from the New York Times debate section on \textit{immigration} and \textit{international affairs}.
\para{argmin} The Argument Mining corpus presents arguments relevant to a particular topic from heterogenous texts. Topics include controversial keywords like \textit{death penalty} and \textit{gun control}.
\para{emergent} The Emergent\footnote{\label{emergent}\url{http://www.emergent.info/}} dataset is a collection of articles from rumour sites annotated by journalists.
\para{fnc1} The Fake News Challenge dataset consists of news articles whose stance towards headlines is provided. It spans 300 topics from Emergent.\footnotemark[2]
\para{iac1} The Internet Argument Corpus V1 consists of Quote--Response pairs from a debating forum on topics related to US politics.
\para{ibmcs} This dataset expands the IBM argumentative structure dataset \citep{aharoni-etal-2014-ibm-debater} to 55 topics and provides topic--claim pairs (from IBM Project Debater\footnote{IBM
Project Debater \url{http://www.research.ibm.com/artificial-intelligence/project-debater/}}) along with their stance annotations.
\para{mtsd} The Multi-Target Stance Detection dataset includes tweets related to the 2016 US Presidential election with a specific focus on multiple targets of interest expressed in each tweet.
\para{perspectrum} The Perspectrum dataset provides several perspectives towards a given claim gathered from a number of debating websites.
\para{poldeb} The Ideological On-Line Debates corpus provides opinion--target pairs from several debating platforms encapsulating different domains.
\para{rumor} The Rumor Has It dataset presents tweets for the task of Belief Classification, where users believe, question, or refute curated rumours.
\para{vast} The Varied Stance Topics dataset consists of topic--comment pairs from the The New York Times \textit{Room for Debate} section. The dataset covers a large variety of topics in order to facilitate zero-shot learning on new unseen topics.
\para{wtwt} Will-They-Won't-They presents a large number of annotated tweets from the financial domain relating to five merger and acquisition operations.
\para{scd} The Stance Classification dataset provides debate posts from four domains including \textit{Obama} and \textit{Gay Rights}. As highlighted in Table \ref{tab:dataset:stats}, while the posts are gathered from defined domains, they are not part of the training set and need to be inferred.\footnotetext[4]{\url{http://www.snopes.com}}
\para{semeval2016t6} The SemEval-2016 Task 6 dataset provides tweet--target pairs for 5 targets including \textit{Atheism}, \textit{Feminist Movement}, and \textit{Climate Change}.
\para{semeval2019t7} The SemEval-2019 Task 9 dataset aims to model authors' stance towards a particular rumour. It provides annotated tweets supporting, denying, querying, or commenting on the rumour.
\para{snopes} The Snopes dataset provides several controversial claims and their corresponding evidence texts from the US-based fact-checking website Snopes,\footnotemark{} annotated for the text in support of, refuting, or having no stance towards a claim.
\subsection{Dataset Characteristics}
\label{subsec:dataset_characteristics}
\begin{table}[t]
\centering
\resizebox{1.00\columnwidth}{!}{%
\begin{tabular}{l|rrrr}
\toprule
\bf Dataset & \bf Train & \bf Dev & \bf Test & \bf Total \\
\midrule
arc & 12,382 & 1,851 & 3,559 & 17,792 \\
argmin & 6,845 & 1,568 & 2,726 & 11,139 \\
emergent & 1,770 & 301 & 524 & 2,595 \\
fnc1 & 42,476 & 7,496 & 25,413 & 75,385 \\
iac1 & 4,227 & 454 & 924 & 5,605 \\
ibmcs & 935 & 104 & 1,355 & 2,394 \\
mtsd* & 3,718 & 520 & 1,092 & 5,330 (8,910) \\
perspectrum & 6,978 & 2,071 & 2,773 & 11,822 \\
poldeb & 4,753 & 1,151 & 1,230 & 7,134 \\
rumor* & 6,093 & 471 & 505 & 7,276 (10,237) \\
scd & 3,251 & 624 & 964 & 4,839 \\
semeval2016t6 & 2,497 & 417 & 1,249 & 4,163 \\
semeval2019t7 & 5,217 & 1,485 & 1,827 & 8,529 \\
snopes & 14,416 & 1,868 & 3,154 & 19,438 \\
vast & 13,477 & 2,062 & 3,006 & 18,545 \\
wtwt & 25,193 & 7,897 & 18,194 & 51,284 \\
\midrule
Total & 154,228 & 30,547 & 68,495 & 253,270 \\
\bottomrule
\end{tabular}
}
\caption{Number of examples per data split. For datasets marked with \textsuperscript{*}, not all tweets could be downloaded; the original number of tweets is in parentheses.}
\label{tab:dataset:stats}
\end{table}
\begin{figure}[t]
\centering
\includegraphics[width=1.0\columnwidth]{assets/images/roberta-base-no-fine-tuning-v3.png}
\caption{tSNE plot of \textit{[CLS]} representations of each dataset; highlighted points denote cluster centroids.}
\label{fig:dataset:tsne_cls}
\end{figure}
As is readily apparent from Table~\ref{tab:dataset:domains}, the datasets differ based on the nature of the \emph{target} and the \emph{context}, as well as the stance \emph{labels}.
The \textit{Target} is the object of the stance. It can be a
\textit{Claim}, e.g.,~``\emph{Corporal punishment be used in K-12 schools.}'',
a \textit{Headline}, e.g.,~``\emph{A meteorite landed in Nicaragua}'',
a \textit{Person},
a \textit{Topic}, e.g.,~\emph{abortion}, \emph{healthcare},
or \textit{None} (i.e.,~an implicit target).
Respectively, the \textit{Context}, which is where the stance is expressed, can be
an \textit{Article},
a \textit{Claim},
a \textit{Post}, e.g.,~in a debate,
a \textit{Thread}, i.e.,~a chain of forum posts,
a \textit{Sentence},
or a \textit{Tweet}.
More examples from each dataset can be found in Table~\ref{tab:examples} in Appendix~\ref{sec:appendix:dataset}.
Moreover, the diversity of the datasets is also reflected in their label names, ranging from different variants of \textit{positive}, \textit{negative}, and \textit{neutral} to labels such as \emph{query} or \emph{comment}. The mapping between them is one of the core challenges we address, and it is discussed in more detail in Section~\ref{subsec:approach:label_adaptive}.
Finally, the datasets differ in their size (see Table~\ref{tab:dataset:stats}), varying from 800 to 75K examples. A complementary analysis of their quantitative characteristics, such as how the splits were chosen, the similarity between their training and testing parts, and their vocabularies, can be found in Appendix~\ref{sec:appendix:dataset}.
\subsection{Source Groups}
\label{subsec:sources}
Defining source groups/domains is an important part of this study, as they allow for better understanding of the relationship between datasets, which we leverage through domain-adaptive modelling (Section~\ref{sec:approach}). Moreover, we use them to outline phenomena in the results that similar datasets share (Section~\ref{sec:experiments}). Table~\ref{tab:dataset:domains} shows these groupings.
Based on the aforementioned definitions of targets and context, we define the following groups: (\emph{i})~Debates, (\emph{ii})~News, (\emph{iii})~Social Media, and (\emph{iv})~Various.
We combine \textit{argmin} (Web searches), \textit{ibmcs} (Encyclopedia), and \textit{vast} into \emph{Various}, since they do not fit into any other group.
To demonstrate the feasibility of our groupings, we plot the \DatasetsCount datasets in a latent vector space. We proportionally sample 25K examples, and we pass them through a \RobertaB~\citep{liu2019roberta} cased model without any training. The input has the following form: \texttt{[CLS]} context \texttt{[SEP]} target. Next, we take the \texttt{[CLS]} token representations, and we plot them in Figure~\ref{fig:dataset:tsne_cls} using tSNE~\citep{JMLR:v9:vandermaaten08a:tSNE}. We can see that Social Media datasets are grouped top-right, Debates are in the middle, and News are on the left (except for Snopes). The Various datasets, \textit{ibmcs} and \textit{argmin},
are placed in between the aforementioned groups (i.e.,~Debates and News), and \textit{argmin} is scattered into small clusters, confirming that they do not fit well into other source categories. Moreover, the figure reflects the strong connections between \textit{vast} and \textit{arc}, as well as between \textit{fnc1} and \textit{emergent}, as the former is derived from the latter. Finally, the clusters are well-separated and do not overlap, which highlights the rich diversity of the datasets, each of which has its own definition of stance.
\section{Introduction}
\label{sec:introduction}
There are many different scenarios in which it is useful to study the attitude expressed in texts, e.g.,~of politicians with respect to newly proposed legislation~\citep{somasundaran-wiebe-2010-poldeb}, of customers regarding new products~\citep{somasundaran-wiebe-2009-recognizing}, or of the general public towards public health measures, e.g., aiming to reduce the spread of COVID-19~\citep{hossain-etal-2020-covidlies,glandt-etal-2021-stance}. This task, commonly referred to as \emph{stance detection}, has been studied in many different forms: not just for different domains, but with more substantial differences in the settings, e.g., stance (\emph{i})~expressed in tweets~\citep{qazvinian-etal-2011-rumor,mohammad-etal-2016-semeval,conforti-etal-2020-will-they}
vs.~long news articles~\citep{pomerleau-2017-FNC,ferreira-vlachos-2016-emergent} vs. news outlets \cite{stefanov-etal-2020-user-stance} vs. people \cite{ICWSM2020:Unsupervised:Stance:Twitter},
(\emph{ii})~with respect to a claim~\citep{chen-etal-2019-perspectrum} vs. a topic, either explicit~\citep{qazvinian-etal-2011-rumor,walker-etal-2012-iac-corpus} or implicit~\citep{hasan-ng-2013-stance,gorrell-etal-2019-semeval}.
Moreover, there is substantial variation in (\emph{iii})~the label inventory, in the exact label definition, in the data collection, in the annotation setup, in the domain, etc.
The most crucial of these, which has not been investigated, currently preventing cross-domain studies, is that the label inventories differ between the settings, as shown in Table~\ref{tab:dataset:domains}. Labels include not only variants of \emph{agree}, \emph{disagree}, and \emph{unrelated}, but also difficult to cross-map ones, such as \emph{discuss} and \emph{question}.
Our goal in this paper is to design a common stance detection framework to facilitate future work on the problem is a cross-domain setting. To this end, we make the following contributions:
\begin{itemize}
\item We present the largest holistic study of stance detection to date, covering \DatasetsCount datasets.
\item We propose a novel framework (\OurModel) that combines domain-adaptation and label embeddings for learning heterogeneous target labels.
\item We further adapt the framework for out-of-domain predictions from a set of unseen targets, based on the label name similarity.
\item Our proposed approach outperforms strong baselines both in-domain and out-of-domain.
\item We perform an exhaustive analysis of cross-domain results, and find that the source domain, the vocabulary size, and the number of unique target labels are the most important factors for successful knowledge transfer.
\end{itemize}
Finally, we release our code, models, and data.\footnote{The datasets and code are available for research purposes:\\\dataurl}
\section{Experiments}
\label{sec:experiments}
\begin{table*}[t]
\centering
\setlength{\tabcolsep}{3pt}
\resizebox{1.00\textwidth}{!}{%
\begin{tabular}{lc|ccccc|ccc|ccccc|ccc}
\toprule
{} & \fmacro avg. & arc & iac1 & perspectrum & poldeb & scd & emergent & fnc1 & snopes & mtsd & rumor & semeval16 & semeval19 & wtwt & argmin & ibmcs & vast \\
\midrule
Majority class baseline & 27.60 & 21.45 & 21.27 & 34.66 & 39.38 & 35.30 & 21.30 & 20.96 & 43.98 & 19.49 & 25.15 & 24.27 & 22.34 & 15.91 & 33.83 & 34.06 & 17.19 \\
Random baseline & 35.19 & 18.50 & 30.66 & 50.06 & 48.67 & 50.08 & 31.83 & 18.64 & 45.49 & 33.15 & 20.43 & 31.11 & 17.02 & 20.01 & 49.94 & 50.08 & 33.25 \\
Logistic Regression & 41.35 & 21.43 & 28.68 & 61.33 & 72.30 & 44.63 & 61.30 & 24.02 & 59.32 & 44.29 & 19.31 & 48.92 & 22.34 & 32.32 & 51.06 & 37.13 & 33.31\\
\midrule
MTL w/ BERT\textsubscript{Base} & 63.11 & 63.19 & \bf{45.30} & 78.62 & 50.76 & 64.03 & \bf{86.23} & 74.48 & 71.55 & 56.36 & 60.26 & 68.28 & \bf{61.03} & 63.59 & 59.05 & 68.55 & 38.42 \\
MTL w/ \RobertaB & 65.12 & 64.52 & 35.73 & 82.38 & 53.83 & 59.43 & 83.91 & 75.29 & 74.95 & \bf{65.87} & \bf{71.23} & 70.46 & {59.42} & 67.64 & 61.79 & 77.27 & 38.21 \\
\midrule
\OurModel (Our Model) & \bf{65.55} & 63.17 & {38.50} & \bf{85.27} & 50.76 & \bf{65.91} & 83.74 & \bf{75.82} & \bf{75.07} & 65.08 & 67.24 & 70.05 & 57.78 & 68.37 & \bf{63.73} & \bf{79.38} & \bf{38.92} \\
$-$ DANN & 65.40 & 64.28 & 37.20 & 83.93 & \bf{53.99} & 62.79 & 83.44 & 75.47 & 74.77 & 65.44 & 70.41 & \bf{72.08} & 54.68 & 68.90 & 62.29 & 78.42 & 38.24 \\
$-$ MoE & 64.68 & \bf{65.18} & 38.41 & 81.46 & 51.34 & 64.57 & {84.60} & 75.79 & 74.05 & 65.69 & 61.07 & 69.99 & 56.67 & \bf{69.03} & 62.25 & 76.87 & 37.91 \\
\bottomrule
\end{tabular}
}
\caption{\textbf{In-domain experiments.} Results are shown in terms of \fmacro. In the rows below \OurModel, we remove (\emph{--}) the components sequentially from it.}
\label{tab:experiments:in_domain}
\end{table*}
\begin{table*}[t]
\centering
\setlength{\tabcolsep}{3pt}
\resizebox{1.00\textwidth}{!}{%
\begin{tabular}{lc|ccccc|ccc|ccccc|ccc}
\toprule
{} & \fmacro avg. & arc & iac1 & perspectrum & poldeb & scd & emergent & fnc1 & snopes & mtsd & rumor & semeval16 & semeval19 & wtwt & argmin & ibmcs & vast \\
\midrule
\OurModel w/ Hard Mapping & 32.78 & 25.29 & 35.15 & 29.55 & 22.80 & 16.13 & 58.49 & 47.05 & 29.28 & 23.34 & 32.93 & \bf{37.01} & 21.85 & 16.10 & 34.16 & 72.93 & 22.89 \\
\OurModel w/ Weak Mapping & 49.20 & \bf{51.81} & \bf{38.97} & 58.48 & 47.23 & \bf{53.96} & \bf{82.07} & 51.57 & \bf{56.97} & \bf{40.13} & \bf{51.29} & 36.31 & \bf{31.75} & 22.75 & 50.71 & \bf{75.69} & \bf{37.15} \\
\OurModel w/ Soft Mapping \\
w/ fasttext & \bf{42.67} & \bf{48.31} & 13.23 & \bf{62.73} & \bf{54.19} & 49.58 & \bf{46.86} & 53.46 & \bf{53.58} & \bf{37.88} & 44.38 & \bf{36.77} & 24.40 & 21.53 & 56.48 & 59.26 & 19.67 \\
w/ glove & 39.00 & 46.54 & 9.32 & 48.87 & 52.20 & \bf{51.97} & 40.32 & 48.36 & 49.32 & 34.38 & \bf{44.46} & 24.07 & 7.68 & \bf{28.97} & \bf{57.78} & 59.14 & 19.80 \\
w/ roberta-base & 32.22 & 44.88 & \bf{32.12} & 36.14 & 39.38 & 31.24 & 23.02 & 33.07 & 49.60 & 33.84 & 12.10 & 17.76 & 6.97 & 25.51 & 33.90 & 65.32 & \bf{30.96} \\
w/ roberta-sentiment & 37.06 & 44.81 & 26.67 & 35.18 & 50.69 & 50.65 & 19.55 & 42.75 & 45.94 & 28.65 & 15.66 & 23.25 & \bf{28.92} & 24.64 & 55.90 & \bf{72.11} & 28.05 \\
w/ sswe & 37.10 & 45.11 & 23.80 & 36.14 & 45.73 & 51.23 & 38.30 & \bf{57.31} & 43.93 & 28.97 & 18.94 & 34.02 & 6.38 & 21.18 & 57.26 & 60.03 & 24.31 \\
\bottomrule
\end{tabular}
}
\caption{\textbf{Out-of-domain experiments.} Results are shown in terms of \fmacro.}
\label{tab:experiments:out_domain}
\end{table*}
We consider three evaluation setups: (\emph{i})~\emph{no training}, random and majority class baselines; (\emph{ii})~\textit{in-domain}, training then testing on all datasets; and (\emph{iii})~\textit{out-of-domain}, i.e.,~leave-one-dataset-out training for all datasets. The reported per-dataset scores are
macro-averaged \fmacro,
which are additionally averaged to obtain per-experiment scores.
\subsection{Baselines}
\label{subsec:baselines}
\paragraph{Majority class baseline}
calculated from the distributions of the labels in each test set.
\paragraph{Random baseline}
Each test instance is assigned a target label at random with equal probability.
\paragraph{Logistic Regression} A logistic regression trained using TF.IDF word unigrams. The input is a concatenation of the target and context vectors.
\paragraph{Multi-task learning (MTL)} A single projection layer for each dataset is added on top of a pre-trained language model (BERT~\citep{devlin2019bert} or RoBERTa~\citep{liu2019roberta}). We then pass the \texttt{[CLS]} token representations through the dataset-specific layer. Finally, we propagate the errors only through that layer (and the base model), without updating parameters for other datasets.
\subsection{Evaluation Results}
\label{subsec:evaluation_results}
\paragraph{In-Domain Experiments}
We train and test on all datasets;
the results are in Table~\ref{tab:experiments:in_domain}. First, to find the best base model and set a baseline for \OurModel, we evaluate two strong models: BERT\textsubscript{Base}
uncased \citep{devlin2019bert}, and \RobertaB
cased\footnote{We choose the uncased version of BERT due to its wide use in similar tasks; RoBERTa is cased by nature. We use the base versions of the models for computational efficiency.} \citep{liu2019roberta}. On our \DatasetsCount datasets, RoBERTa outperforms BERT by 2 F\textsubscript{1} points absolute on average.
In the following rows of Table~\ref{tab:experiments:in_domain}, we show results for our model (\OurModel), i.e.,~Mixture of Experts with Label Embeddings and Domain-Adversarial Training (see Section~\ref{sec:approach}).
Its full version scores the highest in terms of \fmacro\ -- 65.55, which is 0.43 absolute points better than the MTL (\RobertaB) baseline. In particular, it outperforms this strong baseline on nine of the \DatasetsCount datasets.
Nevertheless, neither \OurModel, nor any of its variations improves the results for \emph{mtsd}, \emph{rumor}, and \emph{semeval2019t6} over the MTL (\RobertaB) model. We attribute this to their specifics: \emph{mtsd} is the only dataset where the target is a \emph{Person}, \emph{rumor} and \emph{semeval2019t6} both focus on stance towards rumors, but the data for \emph{rumor} is from 2009--2011, and \emph{semeval2019t6} has an implicit target.
Next, we present ablations -- we sequentially remove a prominent component from the proposed model (\OurModel). First, we optimise the model without the domain-adversarial loss. Removing the DANN leads to worse results on ten of the datasets, and a drop in the average \fmacro{} compared to \OurModel. However, this model does better in terms of points absolute on \emph{arc} (1\%), \emph{poldeb} (3\%), \emph{rumor} (3\%), and \emph{semeval2016t7} (+2\%). We attribute that to the more specialised domain representations being helpful, as some of the other datasets we trained on are very similar to those, e.g.,~\emph{vast} is derived from \emph{arc}.
Moreover, removing domain adversarial training has a negative impact on the datasets with source \emph{Various} (i.e.,~\emph{argmin, ibmcs, vast}).
Clearly, forcing similar representations aids knowledge sharing among domain experts, as they score between 0.7 and 1.5 \fmacro{} lower compared to \OurModel, the same behaviour as observed in other ablations.
The last row of Table~\ref{tab:experiments:in_domain} (\emph{$-$ MoE}) shows results for \RobertaB with Label Embeddings. It performs the worst of all RoBERTa-based models, scoring 0.5 points lower than MTL overall.
Note that it is not possible to present results for a MoE-based model without Label Embeddings, due to the discrepancy in the label inventories, both between and within domains,
which means a standard voting procedure cannot be applied (see Section~\ref{subsec:approach:label_adaptive}).
\para{Out-of-Domain Experiments}
In the out-of-domain setup, we leave one dataset out for testing, and we train on the rest.
We present results with the best model (\textbf{\OurModel}) on the in-domain setup as it outperforms other strong alternatives (see Table~\ref{tab:experiments:in_domain}).
In Table~\ref{tab:experiments:out_domain}, each column denotes when that dataset is held-out for training and instead evaluated on.
We further evaluate all mapping procedures proposed in Section~\ref{subsec:approach:label_adaptive} for out-of-domain prediction: (\emph{i})~\emph{hard}
(\emph{ii})~\emph{weak},
and (\emph{iii})~\emph{soft mapping}.
The \emph{hard mapping} approach outperforms the majority class baseline, but it falls almost 3 points absolute short compared to the random baseline, while failing to do better than random on more than half of the datasets. The two main factors for this are that (\emph{i})~the predictions are dominated by the meta-targets with the most examples, i.e.,~\emph{discuss}, (\emph{ii})~the model struggles to converge on the training set, due to diversity in the datasets and their labels.
The \emph{weak} and the \emph{soft embeddings} share the same set of predictions, as their training procedure is the same -- the only difference between them are the embeddings used to align the prediction to the set of unseen targets. The \emph{weak mappings} achieve the highest average \fmacro{} among the out-of-domain models. For context, note that it is still 16\% behind the best in-domain model.
Furthermore, in this setup, we see that \emph{emergent} scores 82\%, just few points below the in-domain result -- we suspect that this is due to the good alignment of labels with \emph{fnc1}, as the two datasets are closely related.
For the \emph{soft mappings}, we evaluate five well-established embedding models, i.e.,~\emph{fastText}~\citep{joulin-etal-2017-bag}, \emph{GloVe}~\citep{pennington-etal-2014-glove}, \emph{\RobertaB}, and two sentiment-enriched ones, i.e.~\emph{sentiment-specific word embedding} (sswe,~\citet{tang-etal-2014-learning}), and \emph{RoBERTa Twitter Sentiment} (roberta-sentiment,~\citet{barbieri-etal-2020-tweeteval}). Our motivation for including the latter is that the names of the stance labels are often sentiment-related, and thus encoding that information into the latent space might yield better groupings (see Appendix~\ref{sec:appendix:lel_spaces}).
We examine the performance of \emph{soft mapping w/ fastText} in more detail as they score the highest among other strong alternatives. Interestingly, the soft mappings benefit from splitting the predictions for the labels in the same group, such as \emph{wtwt\_\_comment} and all \emph{discuss-related}, which leads to the better performance on \emph{perspectrum, poldeb, fnc1, argmin} in comparison to the \emph{weak mappings}. Nevertheless, this also introduces some errors. An illustrative example are short words -- \emph{anti, pro, con}, which are distant from all other label names in our pool (see Figure~\ref{fig:lel_spaces:embeddings} in Appendix~\ref{sec:appendix:lel_spaces} for an illustration).
The neighbourhoods are sometimes hard to interpret, e.g.,~\emph{con} is not the closest word for any predicted labels in \emph{vast}, and is aligned only with \emph{undermine, unrelated} in \emph{ibmcs}.
\section{Related Work}
\label{sec:relatedwork}
\paragraph{Stance Detection}
Prior work on stance explored its connection to argument mining~\citep{Boltuzic2014Back-ur-Stance,sobhani2015-from-am-to-sc}, opinion mining~\citep{Wang2019-survey-opinion-mining}, and sentiment analysis~\citep{mohammad2017-stance-sentiment,aldayel-2019-stance-infer}.
Debating platforms were used as data source for stance~\citep{somasundaran-wiebe-2010-poldeb,Hasan2014WhyAY,aharoni-etal-2014-ibm-debater}, and more recently it was Twitter~\citep{mohammad-etal-2016-semeval,gorrell-etal-2019-semeval}.
With time, the definition of stance has become more nuanced
\citep{kucuk-2020-stance-survey}, as well as its applications~\cite{zubiaga2018survey,hardalov2021survey}. Settings vary with respect to implicit~\citep{hasan-ng-2013-stance,gorrell-etal-2019-semeval} or explicit topics~\citep{augenstein-etal-2016-stance,stab-etal-2018-argmin,allaway-mckeown-2020-vast},
claims~\citep{baly-etal-2018-integrating,chen-etal-2019-perspectrum, hanselowski-etal-2019-snopes,conforti-etal-2020-stander,conforti-etal-2020-will-they}
or headlines~\citep{ferreira-vlachos-2016-emergent,habernal-etal-2018-arc,mohtarami-etal-2018-automatic}.
The focus, however, has been on homogeneous text, as opposed to cross-platform or cross-domain.
Exceptions are \citet{stab-etal-2018-argmin}, who worked on heterogeneous text, but limited to eight topics, and \citet{schiller2021stance}, who combined datasets from different domains, but used in-domain multi-task learning, and \citet{mohtarami-etal-2019-contrastive} and \citet{hardalov2021fewshot}, who used a cross-lingual setup.
In contrast, we focus on cross-domain learning on \DatasetsCount datasets, and out-of-domain evaluation.
\paragraph{Domain Adaptation}
Domain adaptation was studied in \emph{supervised} settings, where in addition to the source-domain data, a (small) amount of labeled data in the target domain is also available~\citep{daume-iii-2007-frustratingly,finkel-manning-2009-hierarchical,donahue2013semi,yao2015semi,mou-etal-2016-transferable,lin-lu-2018-neural}, and in \emph{unsupervised} settings, without labeled target-domain data~\citep{blitzer-etal-2006-domain,lipton2018detecting,shah-etal-2018-adversarial,mohtarami-etal-2019-contrastive,Bjerva19-Future,wright-augenstein-2020-transformer}.
Recently, domain adaptation was applied to pre-trained Transformers \citep{lin-etal-2020-clinicalDA}.
One direction therein are architectural changes (method-centric): \citet{ma-etal-2019-domain} proposed curriculum learning with domain-discriminative data selection, \citet{wright-augenstein-2020-transformer} investigated an unsupervised multi-source approach with Mixture of Experts and domain adversarial training~\citep{ganin-etal-2016-dann}.
Another direction is data-centric adaptation: \citet{han-eisenstein-2019-unsupervised,rietzler-etal-2020-adapt} used MLM fine-tuning on target-domain data.
\citet{gururangan-etal-2020-dont} showed alternate domain-adaptive (in-domain data) and task-adaptive (out-of-domain unlabelled data) pre-training.
\paragraph{Label Embeddings} Label embeddings can capture, in an unsupervised fashion, the complex relations between target labels for multiple datasets or tasks. They can boost the end-task performance for various deep learning architectures, e.g.,~CNNs~\citep{zhang-etal-2018-multi,doi:10.1162/pappas_labelembed}, RNNs~\citep{augenstein-etal-2018-multi,augenstein-etal-2019-multifc}, and Transformers~\citep{chang2020taming}. Recent work has proposed different perspectives for learning label embeddings: \citet{beryozkin-etal-2019-joint} trained a named entity recogniser from heterogeneous tag sets,
\citet{chai2020description} used label descriptions for text classification,
\citet{rethmeier2021dataefficient} explored contrastive label embeddings for long-tail learning.
In our work, we propose an end-to-end framework to learn from heterogeneous labels based on unsupervised domain adaptation and label embeddings, and an unsupervised approach to obtain predictions for an unseen set of user-defined targets, using the similarity between label names.
\section{Fine-Tuning and Hyper-Parameters}
\begin{itemize}
\item All models are developed in Python using PyTorch~\citep{NEURIPS2019_9015} and the Transformers library~\citep{wolf-etal-2020-transformers}.
\item All models use Adam~\citep{DBLP:journals/corr/KingmaB14} with weight decay 1e-8, $\beta_1$ 0.9, $\beta_2$ 0.999, $\epsilon$ 1e-08, and are trained for five epochs with batch size 64, and maximum length of 100 tokens.\footnote{When needed, we truncated the sequences token by token, starting from the longest sequence in the pair.}
\item RoBERTa~\citep{liu2019roberta} is trained w/ LR 1e-05, warmup 0.06, BERT~\citep{devlin2019bert} is trained w/ LR 3e-05, warmup 0.1.
\item The values of the hyper-parameters were selected on the development set.
\item We chose the best model checkpoint based on the performance on the development set.
\item For the MTL/MoE models, we sampled each batch from a single randomly selected dataset/domain.
\item We used the same seed for all experiments.
\item Each experiment took around 1h 15m on a single NVIDIA V100 GPU using half precision.
\item For logistic regression, we converted the text to lowercase, removed the stop words, and limited the dictionary in the TF.IDF to 15,000 unigrams. We built the vocabulary using the concatenated target and context. The target and the context were transformed separately and concatenated to form the input vector.
\end{itemize}
\section{Dataset Analysis}
\label{sec:appendix:dataset}
\subsection{Data Splits}
\label{sec:appendix:data_splits}
We could not reconstruct some of the Social Media datasets in full (marked with a \textsuperscript{*} symbol in Table~\ref{tab:dataset:stats}), as with only tweet IDs, we could not obtain the actual tweet text in some cases. This is a known phenomenon in Twitter: with time, older tweets become unavailable for various reasons, such as tweets/accounts being deleted or accounts being made private~\citep{zubiaga2018longitudinal}. The missing tweets were evenly distributed among the splits of the datasets except for \emph{rumor}, where we chose a topic for the test set for which all example texts were available.
Here, we provide more detail about the splits we used for the datasets, in cases where there is a deviation from the original. For the datasets in common, we used the splitting by \citet{schiller2021stance}. We further tried to enforce a larger domain diversity between the training, the development, and the testing sets; hereby, we put (whenever possible) all examples from a particular topic (domain) strictly into a single split.
\paragraph{argmin} We removed all non-arguments. The training, the development, and the test data splits consist of five, one, and two topics, respectively.
\paragraph{iac1} Split with no intersection of topics between the training, the development, and the testing sets.
\paragraph{ibmcs} Pre-defined training and testing splits. We further reserved 10\% of the training data for development set.
\paragraph{mtsd} We used the pre-defined splits, but we created two pairs for each example: a positive and a negative one with respect to the target.
\paragraph{poledb} We used the domains \textit{Healthcare, Guns, Gay Rights} and \textit{God} for training, \textit{Abortion} for development, and \textit{Creation} for testing.
\paragraph{rumor} We used the \textit{airfrance} rumour for our test set, and we split the remaining data in ratio 9:1 for training and development, respectively.
\paragraph{wtwt} We used \textit{DIS\_FOXA} operation for testing, \textit{AET\_HUM} for development, and the rest for training. To standardize the targets, we rewrote them as sentences, i.e.,~\emph{company X} acquires \emph{company Y}.
\paragraph{scd} We used a split with \textit{Marijuana} for development, \textit{Obama} for testing, and the rest for training.
\paragraph{semeval2016t6} We split it to increase the size of the development set.
\paragraph{snopes} We adjusted the splits for compatibility with the stance setup. We further extracted and converted the rumours and their evidence into target--context pairs.
\begin{table}[t]
\centering
\resizebox{1.00\columnwidth}{!}{%
\begin{tabular}{l|rrrrrr}
\toprule
{} & \multicolumn{3}{c}{\bf Dev} & \multicolumn{3}{c}{\bf Test} \\
{\% of split in Train} & F & T & C & F & T & C \\
\midrule
arc & 1.9 & 100.0 & 93.7 & 1.5 & 100.0 & 93.8 \\
iac1 & 0.0 & 0.0 & 0.2 & 0.0 & 0.0 & 0.1 \\
perspectrum & 1.5 & 1.5 & 37.2 & 0.0 & 0.0 & 26.4 \\
poldeb & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\
scd & --- & --- & 0.0 & --- & --- & 0.0 \\
emergent & 0.0 & 0.0 & 3.0 & 0.0 & 0.0 & 1.7 \\
fnc1 & 1.7 & 100.0 & 99.8 & 0.0 & 0.6 & 0.9 \\
snopes & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.3 \\
mtsd & 0.8 & 100.0 & 1.5 & 0.3 & 100.0 & 0.5 \\
rumor & 17.6 & 100.0 & 17.6 & 0.0 & 0.0 & 0.0 \\
semeval2016t6 & 0.0 & 100.0 & 0.0 & 0.0 & 100.0 & 0.0 \\
semeval2019t7 & --- & --- & 1.4 & --- & --- & 4.8 \\
wtwt & 0.0 & 0.0 & 11.4 & 0.0 & 0.0 & 0.0 \\
argmin & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 & 0.0 \\
ibmcs & 0.0 & 100.0 & 1.0 & 0.0 & 0.0 & 0.2 \\
vast & 0.0 & 43.6 & 0.0 & 0.0 & 49.5 & 0.0 \\
\bottomrule
\end{tabular}
}
\caption{Percentage of overlap between development/testing and training data. The \textit{(T)arget} and the \textit{(C)ontext} columns show the overlap in the respective individual fields; \textit{(F)ull} shows the overall overlap.}
\label{tab:dataset:overlaps}
\end{table}
\begin{table}[t]
\centering
\resizebox{1.00\columnwidth}{!}{%
\begin{tabular}{l|rrrrr}
\toprule
{\bf Tokenization} & \multicolumn{1}{c}{\bf Words} & \multicolumn{1}{c}{\bf Tokens (Words)} & \multicolumn{3}{c}{\bf Tokens} \\
{} & \multicolumn{1}{c}{\bf $|$Unique$|$} & \multicolumn{1}{c}{\bf Mean} & \bf 25\% & \bf Median & \bf Max \\
\midrule
arc & 27,835 & 126.0 (118.3) & 84 & 116 & 286 \\
argmin & 17,990 & 30.6 (28.6) & 20 & 28 & 208 \\
emergent & 6,940 & 27.6 (23.1) & 22 & 26 & 111 \\
fnc1 & 40,738 & 503.2 (432.2) & 279 & 413 & 6,182 \\
iac1 & 88,478 & 1,554.7 (1,347.9) & 132 & 390 & 104,034 \\
ibmcs & 5,007 & 23.4 (21.9) & 18 & 23 & 55 \\
mtsd & 9,799 & 36.3 (25.9) & 33 & 36 & 65 \\
perspectrum & 9,999 & 22.4 (20.4) & 17 & 21 & 75 \\
poldeb & 40,422 & 178.0 (160.6) & 55 & 112 & 2,144 \\
rumor & 15,801 & 38.4 (24.4) & 32 & 39 & 78 \\
scd & 23,592 & 151.4 (134.8) & 39 & 78 & 6,358 \\
semeval2016t6 & 15,016 & 32.8 (22.1) & 28 & 33 & 68 \\
semeval2019t7 & 20,789 & 33.5 (22.0) & 17 & 27 & 1,466 \\
snopes & 33,896 & 53.4 (45.5) & 40 & 51 & 327 \\
vast & 24,644 & 123.2 (115.7) & 80 & 114 & 271 \\
wtwt & 102,672 & 45.7 (23.1) & 39 & 46 & 193 \\
\bottomrule
\end{tabular}
}
\caption{Statistics about the sub-word tokens for each dataset (using the \RobertaB tokeniser). The numbers in parenthesis show the word counts after the NLTK tokeniser was used.}
\label{tab:dataset:vocab}
\end{table}
\subsection{Overlap Statistics}
\label{subsec:overlaps}
Next, in Table~\ref{tab:dataset:overlaps}, we examine the proportion of contexts and targets from the development and the testing datasets that are also present in the training split. We did not change the original data in any way, and we used the splits as described in Section~\ref{sec:appendix:data_splits}.
Table~\ref{tab:dataset:overlaps} further shows statistics about the datasets in terms of the number of words and sub-words they contain (see Table~\ref{tab:dataset:vocab}). The first column in the table shows the number of unique tokens (word types) in each dataset after tokenisation using NLTK's casual tokeniser~\citep{loper-2002-nltk}, which retains the casing of the words; thus word types of different casing are counted separately. We observe that the datasets with the largest vocabularies are those (\emph{i})~with higher numbers of examples (\emph{fnc1} and \emph{wtwt}), (\emph{ii})~whose contexts are threads rather than single posts (\emph{iac1} has over 1,300 words on average), and (\emph{iii})~that cover diverse topics such as
\textit{poldeb} with six unrelated ones. In contrast, small or narrow datasets such as \textit{ibmcs} have the smallest vocabularies (fewer than 5,000 words).
In the subsequent columns, we report statistics in terms of number of sub-words (i.e.,~SentencePieces~\citep{kudo-richardson-2018-sentencepiece} from \RobertaB's tokeniser). With that, we want to present the expected coverage in terms of tokens for a pre-trained model. On average, most of the datasets are well under 100 tokens in length, which is commonly observed for tweets,\footnote{Tweets have a strict character limit. Depending on the time period, this limit can vary.} but some datasets have a higher average number of tokens, e.g.,~debate-related datasets such as \textit{arc}, \textit{poldeb}, \textit{scd}, \textit{vast} fit in 200 tokens on average, which is also the case for datasets containing large news articles or use social media threads as context (\textit{fnc1}, \textit{iac1}), where the average length is over 500.
\begin{figure*}
\centering
\includegraphics[width=0.7\textwidth]{assets/images/vocab_overlap.png}
\caption{Proportion of word types in the dataset in row $i$ that are also present in the dataset in column $j$ (stop-words removed, and case preserved).}
\label{fig:dataset:word_overlap}
\end{figure*}
Finally, Figure~\ref{fig:dataset:word_overlap} shows the relative word overlap between datasets. The numbers in each cell shows how much of the word types in dataset $i$ (row) are contained in the dataset $j$ (column). For example, in the first column in the last row (\textit{vast} $\Rightarrow$ \textit{arc}), we see that 97\% of the words in \textit{vast} are also present in \textit{arc}. Similarly, in the first row and the last column, we see that 86\% of the words in \textit{arc} are also in \textit{vast}. Note that we sort the columns and the rows by their sources (see Table~\ref{tab:dataset:domains}). We can see that datasets with the largest vocabularies (\textit{iac1} and \emph{wtwt}) have low overlap with other datasets, including with each other, up to 28\% only (row-wise).
When looking at how many words in other datasets are contained in them (column-wise), we see that \textit{iac1} has 50\% or more vocabulary overlap with the other datasets, even with ones from different sources. Then, \textit{wtwt}'s overlaps are 30--70\%, which is expected as its texts are from social media and cover a single topic (company acquisitions). For datasets that are either small or cover few topics (\emph{emergant, ibmcs, perspectrum}), we see that moderate to large part of their vocabularies is contained in other datasets; yet, the opposite in not true. Moreover, social media datasets are orthogonal to each other, with cross-overlaps of up to 50\% (both row-wise and column-wise), except for \emph{wtwt}. This is also seen when measuring how much of other datasets' vocabulary they contain (column-wise).
\begin{figure*}[th!]
\centering
\subfloat[FastText \label{subfig:embeddings:fasttext}]{
\includegraphics[width=0.30\textwidth]{assets/images/embeddings/fasttext.png}
}
\subfloat[GloVe \label{subfig:embeddings:glove}]{%
\includegraphics[width=0.30\textwidth]{assets/images/embeddings/glove.png}
}
\subfloat[RoBERTa-base \label{subfig:embeddings:roberta}]{
\includegraphics[width=0.30\textwidth]{assets/images/embeddings/roberta-base.png}
} \\
\subfloat[RoBERTa-sentiment \label{subfig:embeddings:roberta_senti}]{
\includegraphics[width=0.30\textwidth]{assets/images/embeddings/cardiffnlp_twitter-roberta-base-sentiment.png}
}
\noindent\subfloat[SSWE \label{subfig:embeddings:sswe}]{%
\includegraphics[width=0.30\textwidth]{assets/images/embeddings/sswe.png}
}
\caption{Embedding spaces of the label names' representations (PCA) from different embedding models.}
\label{fig:lel_spaces:embeddings}
\end{figure*}
\section{Embedding Label Spaces}
\label{sec:appendix:lel_spaces}
Here, we present the embeddings based on the labels' names. We explore five sets of embeddings: (\emph{i})~well-established ones such as \emph{fastText}~\citep{joulin-etal-2017-bag}, \emph{GloVe}~\citep{pennington-etal-2014-glove}, and \emph{RoBERTa}~\citep{liu2019roberta}, and (\emph{ii})~sentiment-enriched ones like \emph{Sentiment-specific word embedding} (SSWE)~\citep{tang-etal-2014-learning} and \emph{RoBERTa Twitter Sentiment} (roberta-sentiment)~\citep{barbieri-etal-2020-tweeteval}. Our motivation for including the latter is that the names of the stance labels are often sentiment-related, and thus encoding that information into the latent space could yield better grouping. We encode each label with the corresponding word from the embedding's directory for non-contextualized embeddings, if the name contains multiple words, e.g.,~\emph{argument for}, then we split on white space, and we take the average of each word's embeddings. For RoBERTa-based models, we take the representation from the \texttt{[CLS]} token. Finally, we project the obtained vectors into two dimensions using PCA. The resulting plots are shown in Figure~\ref{fig:lel_spaces:embeddings}. |
1,108,101,565,671 | arxiv | \section{Introduction}
\label{sec:intro}
The present 3\% experimental precision on $\mathcal{B}(\pi^0_\text{D})$\footnote{%
We use the shorthand notation $\pi_f^0\coloneqq\pi^0\to f$, with $\text{D}\coloneqq e^+e^-\gamma$ and $\text{DD}\coloneqq e^+e^-e^+e^-$.
}~\cite{Tanabashi:2018oca} represents a limitation for rare-$\pi^0$-decay measurements, which commonly use the Dalitz decay $\pi^0_\text{D}$ for normalization, and is also becoming a limiting factor for rare-kaon-decay measurements.
An example is the $K^+\to\pi^+e^+e^-$ decay~\cite{Batley:2009aa}: accurate knowledge of $\mathcal{B}(\pi^0_\text{D})$ would improve the precision on the rate measurement by 30\,\%, and the precision on the low-energy parameter $a_+$~\cite{DAmbrosio:1998gur} by 10\,\%.
The uncertainty on $\mathcal{B}(\pi^0_\text{D})$ also dominates the precision on the $K^\pm\to\pi^\pm\pi^0e^+e^-$ rate measurement~\cite{Batley:2018hxd}, and is among the principal contributions to the uncertainties on the measured $K_\text{L,S}\to\pi^+\pi^-e^+e^-$ rates~\cite{Lai:2003ad}.
In these circumstances, considering the improving precision on rare-decay measurements, and the recent progress on the $\pi^0$-form-factor measurement~\cite{TheNA62:2016fhr} and radiative corrections for the $\pi^0_\text{D}$ decay~\cite{Husek:2015sma}, both a precision measurement of $\mathcal{B}(\pi^0_\text{D})$ and an updated theoretical evaluation of this quantity are becoming more important.
Branching ratios can serve to translate lifetimes into decay widths and vice versa.
There are several methods to determine the $\pi^0$ lifetime: a direct average-distance measurement of the highly-relativistic pion, the conserved-vector-current hypothesis connecting the vector form factor (i.e.\ charged pions) to the $\pi^0$ lifetime~\cite{Bychkov:2008ws}, and the Primakoff effect~\cite{Pirmakoff:1951pj}.
Since 2012 its Particle Data Group (PDG) value settled to $\tau_{\pi^0}^\text{PDG}=8.52(18)\times10^{-17}$\,s~\cite{Beringer:1900zz}.
Presently, the most precise $\pi^0$-lifetime measurements are given by two different methods: $\tau_{\pi^0}^\text{PrimEx}=8.32(23)\times10^{-17}$\,s~\cite{Larin:2010kq} (Primakoff effect, PrimEx experiment at JLab) and $\tau_{\pi^0}^\text{CERN}=8.97(28)\times10^{-17}$\,s~\cite{Atherton:1985av} (direct measurement, CERN).
It is clear that the situation is unsatisfactory and a new independent measurement is desirable.
For the Primakoff-effect-type $\pi^0$-lifetime measurements, $\mathcal{B}(\pi^0_{2\gamma})$ constitutes an essential input.
In this work we discuss the theoretical determination of the following ratio of decay widths:
\begin{equation}
R
\equiv\frac{\Gamma(\pi^0\to e^+e^-\gamma)}{\Gamma(\pi^0\to\gamma\gamma)}
=\frac{\mathcal{B}(\pi^0_\text{D})}{\mathcal{B}(\pi^0_{2\gamma})}\,.
\label{eq:R}
\end{equation}
The current PDG value $R|_\text{PDG}=11.88(35)\times10^{-3}$ is an average of experimental results, the most recent of which comes from 2008 and is based on archived ALEPH data~\cite{Beddall:2008zza}.
Other measurements with competitive uncertainties date back to 1981~\cite{Schardt:1980qd} and 1961~\cite{Samios:1961zz}.
The branching ratios $\mathcal{B}(\pi^0_{2\gamma})|_\text{PDG}=98.823(34)\,\%$ and $\mathcal{B}(\pi^0_\text{D})|_\text{PDG}=1.174(35)\,\%$~\cite{Tanabashi:2018oca} are subsequently obtained from a constrained fit.
Besides the direct extraction of $R$ from experiment, the shape of the singly-virtual $\pi^0$ transition form factor (TFF) can be measured.
This can be expanded in the transferred momentum squared, with the linear coefficient called the (TFF) slope $a_\pi$.
Since the slope embodies the most relevant input to the ratio $R$ regarding the (non-perturbative) low-energy QCD sector (the peculiar~\cite{Kampf:2009tk} TFF normalization $\mathcal{F}(0)$ conveniently drops out), its knowledge from experiment is crucial to obtain a model-independent prediction of $R$.
Recently, it was measured in the NA62 experiment, which analyzed 1.1$\times10^6$ fully reconstructed $\pi_\text{D}^0$ decays with the result $a_\pi^\text{NA62}=3.68(57)\,\%$~\cite{TheNA62:2016fhr}, taking into account the complete set of next-to-leading-order (NLO) radiative corrections in the QED sector~\cite{Husek:2015sma}.
The current PDG value is dominated by two inputs: the above NA62 result and the value provided by the CELLO Collaboration ($a_\pi^\text{CELLO}=3.26(37)\,\%$)~\cite{Behrend:1990sr} by (model-dependent) extrapolation from the space-like region.
Our calculation combines a wide range of theoretical models and available experimental results on the TFF shape and the well-established QED calculation including the complete set of NLO corrections, taking into account higher orders in the QED expansion in a conservative uncertainty estimate.
As such, it represents a precise and reliable improvement (by two orders of magnitude) to the current PDG-based value of $R$, which might be further used in various theoretical predictions and experimental analyses.
Moreover, for the first time, the slope corrections were not neglected in the bremsstrahlung contribution.
Finally, we present $R$ for the full as well as partial kinematic regions.
Measurements of the TFF shape or the ratio $R$ require significant theoretical input and depend crucially on the proper incorporation of radiative corrections.
Consequently, a statement that experiment itself provides a more relevant value of $R$ than our theoretical prediction is by its nature imprecise.
However, the computation would not be possible without the experimental evidence that the TFF slope lies within a certain range of values.
An example of how theoretical inputs influence the experimental values in this sector is the well-known discrepancy in the rare decay $\pi^0_{e^+e^-}$ driven most probably by the approximate radiative corrections~\cite{Bergstrom:1982wk} which do not agree with the exact calculation~\cite{Vasko:2011pi}; for details and discussion see Refs.~\cite{Husek:2014tna,Husek:2015wta}.
\section{Theoretical framework}
\label{sec:framework}
Considering the QED expansion, the leading-order (LO) $\pi^0$-Dalitz-decay differential width reads~\cite{Mikaelian:1972yg,Kampf:2005tz,Husek:2015sma}
\begin{equation}
\begin{split}
&\frac{\diff^2\Gamma^\text{LO}(x,y)}{\diff x\diff y}\\
&=\frac{\alpha}{\pi}\,\Gamma(\pi^0\to\gamma\gamma)\bigg|\frac{\mathcal{F}(M_\pi^2x)}{\mathcal{F}(0)}\bigg|^2\frac{(1-x)^3}{4x}\left[1+y^2+\frac{4m_e^2}{M_\pi^2x}\right],
\end{split}
\label{eq:dLOxy}
\end{equation}
where the two-photon decay width is parametrized as
\begin{equation}
\Gamma(\pi^0\to\gamma\gamma)
\equiv\frac{e^4M_\pi^3}{64\pi}|\mathcal{F}(0)|^2\,.
\label{eq:G2g}
\end{equation}
Above, $M_\pi$ and $m_e$ are the neutral-pion and electron masses, respectively.
The definition (\ref{eq:G2g}) holds to all orders in the QED and Chiral Perturbation Theory expansions~\cite{Kampf:2009tk} and covers also possible physics from beyond the Standard Model, simply putting these nontrivial dynamical effects into the TFF normalization $\mathcal{F}(0)$.
As usual, kinematical variables $x$ and $y$ are defined as
\begin{equation}
x=\frac{(p_{e^-}+p_{e^+})^2}{M_\pi^2}\,,\quad y=-\frac{2}{M_\pi^2}\frac{p_{\pi^0}\cdot(p_{e^-}-p_{e^+})}{(1-x)}\,,
\label{eq:defxy}
\end{equation}
with $p$ denoting four-momenta of respective particles.
\begin{table}[t]
\begin{ruledtabular}
{\scriptsize
\begin{tabular}{c | c c c c c c c}
source & VMD & LMD & THS & dispers. & Pad\'e aps.\ & NA62 & PDG\\
\hline
$a_\pi\,[\%]$ & 3.00 & 2.45 & 2.92(4) & 3.15(9) & 3.21(19) & 3.68(57) & 3.35(31)\\
$b_\pi\,[10^{-3}]$ & 0.90 & 0.74 & 0.87(2) & 1.14(4) & 1.04(22) & $\times$ & $\times$\\
\end{tabular}}
\end{ruledtabular}
\caption{
The slope and curvature of the singly-virtual pion TFF in various approaches.
The values given by the VMD, LMD and THS models are compared with the results of the recent dispersive calculation~\cite{Hoferichter:2018dmo,Hoferichter:2018kwz} incorporating inputs from both the space- and time-like regions (and updating Ref.~\cite{Hoferichter:2014vra}), the method of Pad\'e approximants~\cite{Masjuan:2017tvw} mainly based on the extrapolation of the space-like data (as was the previous work~\cite{Masjuan:2012wy}) and supported by the low-energy time-like data, the recent measurement performed by the NA62 experiment~\cite{TheNA62:2016fhr} or the PDG average~\cite{Tanabashi:2018oca}.
Inherent model uncertainties (due to large-$N_\text{c}$ and chiral limits) are not fully included in the THS value~\cite{Husek:2015wta}.
Additionally, a recent time-like-region measurement by the A2 Collaboration reads $a_\pi^\text{A2}=3.0(1.0)\,\%$~\cite{Adlarson:2016ykr}.
}
\label{tab:slope}
\end{table}
The slope $a_\pi$ and curvature $b_\pi$ of the singly-virtual pion TFF are defined in terms of the Taylor expansion in the invariant mass of the vector current~\cite{Berman:1960zz,Landsberg:1986fd}:
\begin{equation}
\bigg|\frac{\mathcal{F}(M_\pi^2x)}{\mathcal{F}(0)}\bigg|
\equiv f(x)
=1+a_\pi x+b_\pi x^2+\mathcal{O}(x^3)\,.
\label{eq:fx}
\end{equation}
This parametrization is sufficient in the whole (small) region relevant to the $\pi^0_\text{D}$ decay.
Having particular theoretical models at hand one can immediately explore the properties of $\mathcal{F}(q^2)$ and calculate $a_\pi$ and $b_\pi$.
As examples we briefly mention the vector-meson-dominance (VMD) ansatz~\cite{sakuraiVMD,Landsberg:1986fd} together with the lowest-meson-dominance (LMD)~\cite{Peris:1998nj,Knecht:1999gb} and two-hadron-saturation (THS)~\cite{Husek:2015wta} models; see Ref.~\cite{FFmodels} for more details.
These belong to a family of large-$N_\text{c}$ motivated analytic resonance-saturation models and as such they can be straightforwardly used in the calculation of radiative corrections.
By means of the first and second derivatives it is easy to find the analytic expressions for $a_\pi$ and $b_\pi$ within these models; for details see Section 5 of Ref.~\cite{Husek:2015wta}.
Numerical results are shown in Table~\ref{tab:slope}, together with other theoretical approaches and experimental results.
From Eq.~(\ref{eq:fx}) it follows that
\begin{equation}
f^2(x)
=1+2a_\pi x+(a_\pi^2+2b_\pi)x^2+\mathcal{O}(x^3)\,.
\label{eq:f2x}
\end{equation}
We can use the expansion (\ref{eq:f2x}) to obtain a simple formula for the LO width.
Inserting Eq.~(\ref{eq:f2x}) into Eq.~(\ref{eq:dLOxy}) and taking into account that $x\in(4m_e^2/M_\pi^2,1)$ and $y\in\left(-\beta(M_\pi^2x),\beta(M_\pi^2x)\right)$ with $\beta(s)\equiv\sqrt{1-{4m_e^2}/s}$, we get
\begin{equation}
\begin{split}
R^\text{LO}
=\frac{\alpha}{\pi}
\bigg[
&\frac43\ln\frac{M_\pi}{m_e}
-\frac13(7-a_\pi)
+\frac1{30}(a_\pi^2+2b_\pi)\\
&+(12-8a_\pi)\frac{m_e^2}{M_\pi^2}
+\mathcal{O}\bigg(\frac{m_e^4}{M_\pi^4}\bigg)
\bigg]\,.
\end{split}
\label{eq:RLO}
\end{equation}
This expression is a very good approximation with the precision $\eta(R^\text{LO})\simeq10^{-9}$ (evaluated for parameters close to the physical ones).
As is a common practice, e.g.\ in Ref.~\cite{TheNA62:2016fhr}, further on the prescription $f^2(x)\big|_\text{lin.}\equiv(1+a_\pi x)^2$ is used, which is accurate to first order in $x$.
The effect of the missing $2b_\pi x^2$ term is negligible, since the region where such a difference arises (i.e.\ when $x\simeq1$) is suppressed; cf.\ Eq.~(\ref{eq:dLOxy}) and also Table~\ref{tab:xcut} later on.
If we consider $b_\pi\simeq a_\pi^2$, as suggested by the models (cf.\ Table~\ref{tab:slope}), we introduce an error of $\sigma(a_\pi)\simeq a_\pi^2/5$, i.e.\ a relative error $\eta(a_\pi)\lesssim1$\,\%, on the estimate of $a_\pi$ being well under the current experimental precision.
The previous discussion also implies that the effect of the $a_\pi x$ term on the Dalitz-decay rate is limited, letting us provide a very precise determination of $R$ while allowing for 20\,\% uncertainty on $a_\pi$.
Finally, let us note that dropping $b_\pi$ out of Eq.~(\ref{eq:RLO}) decreases its precision down to $\eta(R^\text{LO}|_{b_\pi=0})\simeq10^{-5}$, being still a good approximation for our purpose in view of the above discussion.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{diagrams.eps}
\caption{
\label{fig:diagrams}
NLO QED radiative corrections for $\pi^0_\text{D}$: (a)~vacuum-polarization insertion; (b),(c)~one-loop 1$\gamma$IR contribution; (d)~vertex correction; (e)~bremsstrahlung.
}
\end{figure}
In the rest of the section we address the NLO QED-sector radiative corrections; see Fig.~\ref{fig:diagrams} for Feynman diagrams.
It is convenient to introduce the NLO correction $\delta$ to the LO differential width (and thus to write schematically $\text{d}\Gamma=(1+\delta+\ldots)\,\text{d}\Gamma^\text{LO}$), which can be in general (for the two- and one-fold differential case, respectively) defined as
\begin{equation}
\delta(x,y)
=\frac{\diff^2\Gamma^\text{NLO}}{\diff x\diff y}\bigg/\frac{\diff^2\Gamma^\text{LO}}{\diff x\diff y}\,,\quad
\delta(x)
=\frac{\diff\Gamma^\text{NLO}}{\diff x}\bigg/\frac{\diff\Gamma^\text{LO}}{\diff x}\,.
\label{eq:dxy}
\end{equation}
One can obtain $\delta(x)$ from $\delta(x,y)$ using the following prescription:
\begin{equation}
\begin{split}
\delta(x)
&=\frac38\frac{1}{\beta(M_\pi^2x)}\left(1+\frac{2m_e^2}{M_\pi^2x}\right)^{-1}\\
&\times\int_{-\beta(M_\pi^2x)}^{\beta(M_\pi^2x)}\delta(x,y)\left[1+y^2+\frac{4m_e^2}{M_\pi^2x}\right]\diff y\,.
\end{split}
\label{eq:dx}
\end{equation}
To calculate the NLO radiative corrections we use the approach documented in Refs.~\cite{Husek:2015sma,Husek:2017vmo}, which reviewed and extended the classical work of Mikaelian and Smith~\cite{Mikaelian:1972yg}.
Hence, together with the bremsstrahlung (BS) beyond the soft-photon approximation, we take into account in the following calculations the one-photon-irreducible (1$\gamma$IR) contribution; see Figs.~\ref{fig:diagrams}(b) and \ref{fig:diagrams}(c).
For historical reasons~\cite{Mikaelian:1972yg,Tupper:1983uw,Lambin:1985sb,Tupper:1986yk,Kampf:2005tz}, let us discuss the case when the 1$\gamma$IR contribution to the NLO radiative corrections is not considered in the analysis to extract the TFF slope from the data.
If we start with the equation among the measured spectral shapes (one-fold differential widths) and eliminate ${\diff\Gamma^\text{LO}}/{\diff x}|_{f(x)=1}$ from both sides, take the expansion (\ref{eq:f2x}) to the linear order, and neglect the corrections of order $\alpha a_\pi$, we find
\begin{equation}
\Delta a_\pi x
\equiv\left(a_\pi-a_\pi^{1\gamma\text{IR}}\right)x
\simeq-\frac12\delta_{1\gamma\text{IR}}^\text{NLO}(x)\,,\quad x\ll1\,.
\end{equation}
Numerically, $\Delta a_\pi\simeq0.5\,\%$~\cite{Kampf:2005tz}.
This is the value to be {\em added} to the experimental value $a_\pi^{1\gamma\text{IR}}$ (extracted neglecting the 1$\gamma$IR contribution) in order to find an estimate of the pure-low-energy-QCD-sector parameter $a_\pi$ with all the QED radiative corrections subtracted.
Above, $\delta_{1\gamma\text{IR}}^\text{NLO}(x)$ is calculated from $\delta^{1\gamma\text{IR}}(x,y)$ stated in Section IV of Ref.~\cite{Husek:2015sma} using the prescription (\ref{eq:dx}).
Note that the 1$\gamma$IR contribution was already taken into account in the NA62 analysis~\cite{TheNA62:2016fhr} and $a_\pi^\text{NA62}$ does not need to be corrected by $\Delta a_\pi$.
Finally, taking the prescription $f^2(x)\big|_\text{lin.}$ and the NLO QED radiative corrections to approximate the exact Dalitz-decay differential width beyond LO (and consequently $\Delta R\equiv R-R^\text{LO}$) we arrive at
\begin{equation}
\begin{split}
\Delta R
\simeq
R^\text{NLO}
&\equiv\frac{\alpha}{\pi}
\iint
\,(1+a_\pi x)^2
\delta(x,y)\\
&\times\frac{(1-x)^3}{4x}\left[1+y^2+\frac{4m_e^2}{M_\pi^2x}\right]\diff x\diff y\,.
\label{eq:RatNLO}
\end{split}
\end{equation}
\section{Calculation and uncertainty estimation}
\label{sec:calculation}
Our aim now is to precisely and reliably (using conservative error estimates) determine $R$.
In the following we choose $a_\pi^\text{univ}\equiv3.55(70)\,\%$ by stretching the uncertainty band over the whole interval of values suggested by different approaches; cf.\ Table~\ref{tab:slope}.
From Eq.~(\ref{eq:RLO}) we get
\begin{equation}
R^\text{LO}
=11.879(5)\times10^{-3}
\label{eq:RLOnum}
\end{equation}
and based on (\ref{eq:RatNLO}) we arrive at
\begin{equation}
\Delta R
=0.099(3)\times10^{-3}\,.
\label{eq:dR}
\end{equation}
During the estimation of the above uncertainty, higher orders in the QED expansion were considered, surpassing in size the uncertainty stemming from the TFF dependence.
In this regard, we can take the absolute value of the {\em dominant} correction ($R_\text{BS,div}^\text{NLO}$; see Table~\ref{tab:corr}) as the typical expected maximum value appearing in NLO and anticipate that the NNLO correction is suppressed compared to the NLO one in the similar manner as NLO is suppressed with respect to LO: circa on the level of 3\,\%.
\begin{table}[t]
\begin{ruledtabular}
\begin{tabular}{c c c c | c}
$R_\text{virt}^\text{NLO}$ & $R_\text{BS,conv}^\text{NLO}$ & $R_\text{BS,div}^\text{NLO}$ & $R_{1\gamma\text{IR}}^\text{NLO}$ & $R^\text{NLO}$ \\
\hline
$-$0.0750(2) & $-$0.15759(2) & 0.3363(3) & $-$0.00466(2) & 0.09911(7)\\
\end{tabular}
\end{ruledtabular}
\caption{
Individual contributions of the NLO radiative corrections for $R$ in [$10^{-3}$].
The `virt' label stands for virtual corrections (Figs.~\ref{fig:diagrams}(a) and \ref{fig:diagrams}(d)) and `div' (`conv') label the divergent (convergent) parts of the bremsstrahlung contribution (Fig.~\ref{fig:diagrams}(e))~\cite{details}.
The listed uncertainties stem from the uncertainty of $a_\pi^\text{univ}$.
In the case of the 1$\gamma$IR correction (Figs.~\ref{fig:diagrams}(b) and \ref{fig:diagrams}(c)), a particular model for the doubly-virtual TFF (LMD, etc.) is necessary to introduce.
The resulting model dependence is suppressed~\cite{Husek:2015sma} and related uncertainty included.
}
\label{tab:corr}
\end{table}
This uncertainty is already conservative: the total NLO correction is on the level of 1\,\%.
Summing Eqs.~(\ref{eq:RLOnum}) and (\ref{eq:dR}) we finally obtain
\begin{equation}
R
=11.978(5)(3)\times10^{-3}\,.
\label{eq:Rall}
\end{equation}
This is one of the main results of the presented work.
The former uncertainty stands for the TFF effects and the latter for neglecting the higher-order corrections;%
\footnote{
Relaxing the requirement of providing a conservative value, one can significantly reduce the former uncertainty (stemming from the TFF effects) taking into account a particular result from Table~\ref{tab:slope}: e.g.\ with the most precise entry --- the dispersion-theoretical result~\cite{Hoferichter:2018dmo,Hoferichter:2018kwz} --- by factor of 8.
Higher-order QED corrections would need to be computed to achieve an additional gain of precision.
}
$m_e$, $M_\pi$ and $\alpha$ are known very precisely.
This calculation also includes all contributions from the decays where additional photon(s) with arbitrarily high (kinematically allowed) energies are radiated.
Indeed, the bremsstrahlung correction at NLO (calculated {\em \`a la} Refs.~\cite{Mikaelian:1972yg,Husek:2015sma}) takes into account an additional final-state photon and integrates over its energy and emission angle without any additional cuts.
The results are thus meant to be used for the {\em inclusive} process.
However, quantities for {\em exclusive} processes can be obtained in a similar way while introducing some specific cut-off in the bremsstrahlung correction $\delta^\text{BS}(x,y)$.
A combined approach was used in the analysis of the recent NA62 measurement~\cite{TheNA62:2016fhr}, when an additional photon was simulated above the cut-off given by the detector sensitivity.
To conclude, for each experimental setup the specific approach for including radiative corrections must be used.
When it applies, we explicitly state (as in the abstract) that the results include an additional final-state photon, denoting it as ($\gamma$).
We also take this tacitly into account in the results for $R$, e.g.\ in Eq.~(\ref{eq:Rall}).
In experiments, specific kinematic regions might be considered.
The sample values for
\begin{equation}
R(x_\text{cut})
\equiv\frac{\mathcal{B}(\pi^0\to e^+e^-\gamma(\gamma),x>x_\text{cut})}{\mathcal{B}(\pi^0_{2\gamma})}
\end{equation}
are listed in Table~\ref{tab:xcut}, using which one can also obtain values for any intermediate region.
\begin{table*}[bt]
\begin{ruledtabular}
\begin{tabular}{c r c | c r c | c r c | c r}
$x_\text{cut}$ & $R(x_\text{cut})\,[10^{-5}]$ && $x_\text{cut}$ & $R(x_\text{cut})\,[10^{-5}]$ && $x_\text{cut}$ & $R(x_\text{cut})\,[10^{-6}]$ && $x_\text{cut}$ & $R(x_\text{cut})\,[10^{-8}]$ \\
\hline
0.05 & 203.72(45) && 0.30 & 22.13(13)~\, && 0.55 & 24.20(22)~\,& & 0.80 & 67.28(85)~~\; \\
0.10 & 117.03(36) && 0.35 & 14.787(97) && 0.60 & 14.07(14)~\, && 0.85 & 19.81(27)~~\; \\
0.15 & 74.38(29) && 0.40 & 9.756(71) && 0.65 & 7.703(80) && 0.90 & 3.594(54)~\, \\
0.20 & 49.11(23) && 0.45 & 6.313(50) && 0.70 & 3.890(43) && 0.95 & 0.1967(35) \\
0.25 & 32.92(17) && 0.50 & 3.978(34) && 0.75 & 1.757(21) && 1.00 & 0\hspace{1.32cm} \\
\end{tabular}
\end{ruledtabular}
\caption{
The values of $R(x_\text{cut})$ for chosen sample values of $x_\text{cut}$.
To be suitable for interpolation, higher precision is used.
The quoted uncertainties are dominated by the TFF-slope knowledge (for its value we assume $a_\pi^\text{univ}$); the additional 3\% uncertainty covering the higher-order corrections is also included.
Note different additional multiplicative factors depending on $x_\text{cut}$.
}
\label{tab:xcut}
\end{table*}
As an example, in the $\pi^0$-rare-decay measurement performed by KTeV~\cite{Abouzaid:2006kk} the region $x>x_\text{cut}=0.232$ was used for the Dalitz decay, which served as the normalization channel in this search.
The direct calculation based on this work leads to $R(0.232)=0.380(2)\times10^{-3}$ and the interpolation based on Table~\ref{tab:xcut} gives $R(0.232)|_\text{intpol.}=0.379(2)\times10^{-3}$, which is compatible within uncertainties.
In Ref.~\cite{Niclasen:2006tz} the value $[R(0.2319)/R]|_\text{KTeV}=0.0319$ was used, which is compatible with our calculation: $R(0.232)/R=0.0317(2)$.
Based on $R$, we can predict $\mathcal{B}(\pi^0_{2\gamma})$ and $\mathcal{B}(\pi^0_\text{D})$.
Considering the uncertainty of $R$, we can write
\begin{equation}
\begin{split}
&1-\mathcal{B}(\pi^0_\text{DD})
\simeq
\mathcal{B}(\pi^0_{2\gamma})+\mathcal{B}(\pi^0_{\text{D}(\gamma)})\,,
\end{split}
\label{eq:1eqB}
\end{equation}
since the branching ratios of other decay modes are smaller than 10$^{-6}$.
Using $\mathcal{B}(\pi^0_\text{DD})=3.3(2)\times10^{-5}$~\cite{Tanabashi:2018oca,Kampf:2018wau} (double-Dalitz decay), we find
\begin{equation}
\mathcal{B}(\pi^0_{2\gamma})
\simeq
\frac{1-\mathcal{B}(\pi^0_\text{DD})}{1+R}
=98.8131(6)\,\%\,.
\label{eq:Bgg}
\end{equation}
Note that taking tacitly into account inclusive Dalitz decays in Eq.~(\ref{eq:1eqB}) is justified and contributes to the relevant decay modes.
Finally, the Dalitz-decay branching ratio reads
\begin{equation}
\begin{split}
&\mathcal{B}(\pi^0_{\text{D}(\gamma)})
\simeq
\frac{R}{1+R}\,[1-\mathcal{B}(\pi^0_\text{DD})]
=1.1836(6)\,\%\,.
\end{split}
\label{eq:BDalitz}
\end{equation}
The above results are compatible with the PDG averages, exhibiting much higher precision.
Let us see how the new result on the Dalitz-decay branching ratio (\ref{eq:BDalitz}) influences a completely different family of processes on a simple example of the $K^+\to\pi^+ e^+e^-$ decay measurements.
The low-energy parameters $a_+$ and $b_+$ were measured by the NA48/2 Collaboration to be $a_+=-0.578(16)$ and $b_+=-0.779(66)$, leading to the model-dependent branching ratio $\mathcal{B}(K^+\to\pi^+ e^+e^-)=3.11(12)\times10^{-7}$, using the 2008 PDG average $\mathcal{B}(\pi^0_\text{D})=1.198(32)\,\%$~\cite{Amsler:2008zzb} for normalization~\cite{Batley:2009aa}.
The central value of our result (\ref{eq:BDalitz}) is 1.2\,\% lower than the quoted PDG average and has a negligible error.
The remaining external uncertainty on the measurement~\cite{Batley:2009aa} related to the normalization comes from $\mathcal{B}(K^+\to\pi^+\pi^0)$ known to 0.4\% precision.
The corrected values are $a_+=-0.575(14)$, $b_+=-0.771(64)$ and $\mathcal{B}(K^+\to\pi^+ e^+e^-)=3.07(10)\times10^{-7}$.
Note that considering the external errors on $a_+$ and $b_+$ quoted in Ref.~\cite{Batley:2009aa}, further experimental progress on $K^+\to\pi^+ e^+e^-$ measurement would be impossible without improvement on $\mathcal{B}(\pi^0_\text{D})$.
\section{Comparison and conclusion}
\label{sec:comparison}
Radiative corrections for the integral decay width were first addressed by Joseph~\cite{Joseph:1960zz}, who numerically arrived to $\Delta R|_\text{Jph.}=0.105\times10^{-3}$ neglecting, among others, the pion TFF slope.
A simple analytical prescription in the limit of the vanishing electron mass was later found by Lautrup and Smith~\cite{Lautrup:1971ew}:
\begin{equation}
\begin{split}
\Delta R\big|_\text{L\&S}
&=\left(\frac{\alpha}{\pi}\right)^2
\left[
\frac89\ln^2
\frac{M_\pi}{m_e}
-\frac19\left(19-4a_\pi\right)\ln
\frac{M_\pi}{m_e}
\right.\\
&+\left.2\zeta(3)
-\frac2{27}\pi^2
+\frac{137}{81}
-\frac{63}{108}a_\pi
+\mathcal{O}\bigg(\frac{m_e}{M_\pi}\bigg)
\right].
\end{split}
\end{equation}
Numerically, $\Delta R\big|_\text{L\&S}^{a_\pi=0}=0.10378\times10^{-3}$ and $\Delta R\big|_\text{L\&S}^{a_\pi^\text{univ}}=0.10414(7)\times10^{-3}$.
The two approaches are compatible and should be compared with our result (\ref{eq:dR}).
However, the 1$\gamma$IR contribution was, due to inappropriate assumptions and arguments based on Low's theorem~\cite{Low:1958sn,Adler:1966gc,Pestleau:1967snm}, considered negligible
and left out; see also Refs.~\cite{Mikaelian:1972yg,Lambin:1985sb}.
The exact calculation shows its significance~\cite{Tupper:1983uw,Tupper:1986yk,Kampf:2005tz,Husek:2015sma} and it thus embodies the main source of the difference between our result and the previous works.
Moreover, the symmetrization with respect to the two photons in the bremsstrahlung contribution was neglected in Refs.~\cite{Joseph:1960zz,Lautrup:1971ew}.
This interference of the diagrams from Fig.~\ref{fig:diagrams}(e) is indeed negligible and corresponds (for $a_\pi=0$) to $\Delta R_\text{interf}^\text{BS}=0.000360\times10^{-3}$.
Let us stress again that the prediction (\ref{eq:Rall}) is based on the complete calculation which includes the entire bremsstrahlung and 1$\gamma$IR contributions.
Here, TFF effects were taken into account also in the bremsstrahlung correction~\cite{details} and the mass of the final-state leptons was {\em not} neglected.
Our main result (\ref{eq:Rall}) together with the value (\ref{eq:Bgg}) should be considered as an alternative to the current PDG averages which opens the way to a new level of precision for a whole class of other processes, for instance for the already mentioned kaon decays.
Similarly, the current situation, when the precision on $\mathcal{B}(\pi^0_\text{D})|_\text{PDG}$ dominates the uncertainty on $\mathcal{B}(\pi^0_\text{DD})$~\cite{Abouzaid:2008cd} and is the largest source of uncertainty on $\mathcal{B}(\pi^0_{e^+e^-})$~\cite{Abouzaid:2006kk}, is improved.
\begin{acknowledgments}
We thank G.\ D'Ambrosio, M.\ Hoferichter and A.\ Portelli for initial suggestions, P. Sanchez-Puertas for helpful discussions and J. Portol\'es for comments on the manuscript.
This work has been supported in part by
the Agencia Estatal de Investigaci\'on (AEI, ES) and the European Regional Development Fund (ERDF, EU) [Grants No.\ FPA2014-53631-C2-1-P, FPA2017-84445-P and SEV-2014-0398],
by Generalitat Valenciana [Grant No.\ PROMETEO/2017/053],
by the Czech Science Foundation grant GA\v{C}R 18-17224S and by the ERC starting grant 336581 ``KaonLepton''.
\end{acknowledgments}
|
1,108,101,565,672 | arxiv | \section{introduction}
In this work, we consider the problem of Quantitative Group Testing (QGT). Consider a set of $N$ items among which $K$ items are defective. The QGT problem is to identify (all or a sufficiently large fraction of) the defective items, where the result of a test reveals the number of defective items in the tested group. The key difference between the QGT problem and the original group testing problem is that, unlike the former, in the latter the result of each test is either $1$ or $0$ depending on whether the tested group contains any defective items or not. The objective of QGT is to design a test plan with minimum number of tests that identifies (all or a sufficiently large fraction of) the defective items.
There are two general categories of test strategies: \emph{non-adaptive} and \emph{adaptive}. In an adaptive scheme, each test depends on the outcomes of the previous tests. On the other hand, in a non-adaptive scheme, all tests are planned in advance. In other words, the result of one test does not affect the design of another test. Although, in general, adaptive algorithms require fewer tests, in most practical applications non-adaptive algorithms are preferred since they allow one to perform all tests at once in parallel.
Let $S$ be the index set of the defective items and $\hat{S}$ be an estimation of $S$. Depending on the application at hand, there can be different requirements for the \emph{closeness} of $\hat{S}$ to $S$ \cite{7953326,DBLP:journals/corr/LeePR15}. The strongest condition for closeness is \emph{exact recovery} when it is required that $\hat{S}=S$. Two weaker conditions are \emph{partial recovery without false detections} when it is required that $\hat{S} \subseteq S$ and $|\hat{S}|\geq (1-\epsilon)|S|$, and \emph{partial recovery without missed detections} when it is required that $S \subseteq \hat{S}$ and $|\hat{S}|\leq (1+\epsilon)|S|$. There are also different types of the \emph{recovery guarantees}~\cite{DBLP:journals/corr/LeePR15}. The strongest guarantee is \emph{perfect recovery guarantee} when the exact or partial recovery needs to be achieved with probability $1$ (over the space of all problem instances). A slightly weaker guarantee is \emph{probabilistic recovery guarantee} when it suffices to achieve the exact or partial recovery with high probability only (and not necessarily with probability $1$). In this work, we are interested in the exact recovery of all defective items with the probabilistic recovery guarantee.
\subsection{Related Work and Applications}
The QGT problem has been extensively studied for a wide range of applications, e.g., multi-access communication, spectrum sensing, and network tomography, see, e.g.,~\cite{WZC:17,heidarzadehfast,heidarzadehuser}, and references therein. This problem was first introduced by Shapiro in~\cite{S:60}.
Several non-adaptive and adaptive QGT strategies have been previously proposed, see, e.g.,~\cite{B:09,WZC:17,8437774}.
It was shown in~\cite{L:75} that any non-adaptive algorithm must perform at least $(2K\log_2 (N/K))/\log_2 K$ tests.
Various order optimal or near-optimal non-adaptive strategies were previously proposed, see, e.g.,~\cite{L:75,8437774,B:09}. The best known polynomial-time non-adaptive algorithms require $K\log N$ tests~\cite{lindstrom1972b2,L:75}.
Recently, a semi-quantitative group testing scheme based on sparse graph codes was proposed in~\cite{8335478}, where the result of each test is an integer in the set $\{0,1,2,\dots,L\}$. This strategy identifies a $(1-\epsilon)$ fraction of defective items using $c(\epsilon,L)K\log_2 N$ tests with high probability, where $c(\epsilon,L)$ depends only on $\epsilon$ and $L$.
\subsection{Connection with Compressed Sensing}
A closely related problem to QGT is the problem of compressed sensing (CS) in which the goal is to recover a sparse signal from a set of (linear) measurements. Given an $N$-dimensional sparse signal with a support size up to $K$, the objective is to identify the indices and the values of non-zero elements of the signal with minimum number of measurements. The main differences between the CS problem and the QGT problem are in the signal model and the constraints on the measurement matrix. Most of the existing works on the CS problem consider real-valued signals and measurement matrices. The QGT problem, however, deals with binary signals and requires the measurement matrix to be binary-valued.
There are a number of CS algorithms in the literature that use binary-valued measurement matrices, see, e.g.~\cite{8368314,IWEN20141} and references therein. However, these strategies either use techniques which are not applicable to binary signals, or provide different types of closeness and guarantee than those required in this work. There are also several CS algorithms for the support recovery where the objective is to determine the indices of the non-zero elements of the signal but not their values~\cite{DBLP:journals/corr/LiPR14,pmlr-v51-scarlett16,5766202}. The support recovery problem is indeed equivalent to the QGT problem. Notwithstanding, the existing schemes for support recovery rely on non-binary measurement matrices, and hence are not suitable for the QGT problem.
Last but not least, to the best of our knowledge, the majority of works on the CS problem focus mainly on the order optimality of the number of measurements, whereas in this work for the QGT problem we are also interested in minimizing the constant factor hidden in the order.
\subsection{Main Contributions}
In this work, we propose a non-adaptive quantitative group testing strategy for the sub-linear regime where $\frac{K}{N}$ vanishes as $K,N\rightarrow\infty$. We utilize sparse graph codes over bi-regular bipartite graphs with left-degree $\ell$ and right-degree $r$ and binary $t$-error-correcting BCH codes for the design of the proposed strategy. Leveraging powerful density evolution techniques for the analysis enables us not only to determine the exact value of constants in the number of tests needed but also to provide provable performance guarantees. We show that the proposed scheme provides exact recovery with probabilistic guarantee, i.e. recovers all the defective items with high probability. In particular, for the sub-linear regime, the proposed algorithm requires at most ${m=c(t)K\left(t\log_2\left(\frac{\ell N}{c(t)K}+1\right)+1\right)+1}$ tests to recover all defective items with probability approaching one as ${K,N\rightarrow\infty}$, where $c(t)$ depends only on $t$.
The results of our theoretical analysis reveal that the minimum number of required tests for the proposed algorithm is achieved by $t=2$. Moreover, for any $t\leq 4$, the encoding and decoding of the proposed algorithm have the computational complexity of $\mathcal{O}(K\log^2 \frac{N}{K})$ and $\mathcal{O}(K\log \frac{N}{K})$, respectively.
\section{Problem Setup and Notation}\label{sec:SN}
Throughout the paper, we use bold-face small and capital letters to denote vectors and matrices, respectively.
In this work, we consider the problem of quantitative group testing (QGT) with exact recovery and probabilistic guarantee, defined as follows. Consider a set of $N$ items among which $K$ items are defective. We focus on the sub-linear regime where the ratio $\frac{K}{N}$ vanishes as $K,N\rightarrow\infty$. The problem is to identify all the defective items with high probability while using minimum number of tests on subsets (groups) of the items, where the result of each test shows the number of defective items in the tested group.
Let the vector $\mathbf{x}\in \{0,1\}^N$ represent the set of $N$ items in which the coordinates with value $1$ correspond to the defective items. A non-adaptive group testing problem consisting of $m$ tests can be represented by a measurement matrix ${\textbf{A}\in \{0,1\}^{m\times N}}$, where the $i$-th row of the matrix corresponds to the $i$-th test. That is, the coordinates with value $1$ in the $i$-th row correspond to the items in the $i$-th test.
The results of the $m$ tests are expressed in the test vector $\mathbf{y}\in\{ 0,1,\dots\}^m$, i.e.,
\begin{equation}\label{eq:gtresult}
\mathbf{y}=[y_{1},\cdots,y_{m}]^{\mathsf{T}}=\mathbf{A}\mathbf{x}.
\end{equation}
The goal is to design a testing matrix $\mathbf{A}$ that has a small number of rows (tests), $m$, and can identify with high probability all the defective items given the test vector $\mathbf{y}$.
\section{Proposed Algorithm \label{sec:main results}}
\subsection{Binary $t$-error-correcting codes and $t$-separable matrices}
\begin{definition}\label{def:d-separable}($t$-separable matrix)
A binary matrix ${\mathbf{D}\in\{0,1\}^{m\times n}}$ (for $n>t$) is $t$-separable over field $\mathbb{F}$ if the sum (over field $\mathbb{F}$) of any set of $t$ columns is distinct.
\end{definition}
\begin{example}
Consider the following matrix,
\begin{align*}
\mathbf{D}= \begin{bmatrix}
0 & 1 & 0 & 1 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 \\
\end{bmatrix}.
\end{align*}
The matrix $\mathbf{D}$ is $2$-separable over real field $\mathbb{R}$, but it is not $2$-separable over $\mathbb{F}_2$ since, for instance, the sum of the first and second columns over $\mathbb{F}_2$ is the same as the sum of the third and fourth columns over $\mathbb{F}_2$.
\begin{align*}
\begin{bmatrix}
0 \\
0 \\
0
\end{bmatrix} \oplus \begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}=\begin{bmatrix}
0 \\
1 \\
1
\end{bmatrix} \oplus \begin{bmatrix}
1 \\
0 \\
1
\end{bmatrix}=\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}.
\end{align*}
\end{example}
From the definition, it can be easily seen that if a matrix $\mathbf{D}$ (with $n$ columns) is $t$-separable over a field $\mathbb{F}$, then $\mathbf{D}$ is also $s$-separable over $\mathbb{F}$ for any $1\leq s < t < n$.
The vector of test results, $\mbf{y}$, is the sum of the columns in the testing matrix corresponding to the coordinates of the defective items. When a $t$-separable matrix over $\mathbb{R}$ is used as the testing matrix, the vector $\mbf{y}$ will be distinct for any set of $t$ defective items. Thus, a $t$-separable matrix over $\mathbb{R}$ can be used as the testing matrix for identifying $t$ defective items. However, the construction of {$t$-separable} matrices for arbitrary $t$ with minimum number of rows is an open problem. Instead, we can leverage the idea that the parity-check matrix of any binary $t$-error-correcting code is a $t$-separable matrix over $\mathbb{F}_2$. Note that $t$-separability over $\mathbb{F}_2$ results in $t$-separability over $\mathbb{R}$. Hence, a possible choice for designing a $t$-separable matrix over $\mathbb{R}$ is utilizing the parity-check matrix of a binary $t$-error-correcting code.
In this work, we use binary BCH codes for this purpose. The key feature of the BCH codes which make them suitable for designing $t$-separable matrices is that it is possible to design binary BCH codes, capable of correcting any combination of $t$ or fewer errors.
\begin{definition}\label{def:bch}\cite{lin2001error} (Binary BCH codes)
For any positive integers $m\geq 3$ and $t < 2^{m-1}$, there exists a binary $t$-error-correcting BCH code with the following parameters:
\[
\begin{cases}
\text{$n=2^m-1$} &\quad\text{block length}\\
\text{$n-k \leq mt$} &\quad \text{number of parity-check digits}\\
\text{$d_{\min}\geq 2t+1$} &\quad\text{minimum Hamming distance}\\
\end{cases}
\]
The $t\times n$ parity-check matrix of such a code is given by
\[
\mathbf{H}_t= \begin{bmatrix}
1 & \alpha & \alpha^2 & \dots & \alpha^{n-1}\\
1 & (\alpha^3) & (\alpha^3)^2 & \dots & (\alpha^3)^{n-1}\\
1 & (\alpha^5) & (\alpha^5)^2 & \dots & (\alpha^5)^{n-1}\\
\vdots& \vdots& \vdots & \ddots & \vdots\\
1 & (\alpha^{2t-1}) & (\alpha^{2t-1})^2 & \dots & (\alpha^{2t-1})^{n-1}\\
\end{bmatrix},
\]
where $\alpha$ is a primitive element in $\mathbb{F}_{2^m}$.
\end{definition}
Since each entry of $\mathbf{H}_t$ is an element in $\mathbb{F}_{2^m}$, it can be represented by an $m$-tuple over $\mathbb{F}_2$. Thus, the number of rows in the binary representation of $\mathbf{H}_t$ is given by
\begin{equation}\label{eq:rows}
R=tm=t\log_2 ({n+1}).
\end{equation}
\subsection{Encoding algorithm}
The design of the measurement matrix $\mathbf{A}$ in our scheme is based on an architectural philosophy that was proposed in~\cite{DBLP:journals/corr/LeePR15} and~\cite{DBLP:journals/corr/VemJN17}.
The key idea is to design $\mathbf{A}$ using a sparse bi-regular bipartite graph and to apply a peeling-based iterative algorithm for recovering the defective items given $\mathbf{y}$.
Let $G_{\ell,r} (N,M)$ be a randomly generated bipartite graph where each of the $N$ left nodes is connected to $\ell$ right nodes uniformly at random, and each of the $M$ right nodes is connected to $r$ left nodes uniformly at random. Note that there are $N\ell$ edge connections from the left side and $Mr$ edge connections from the right side,
\begin{equation}\label{eq:connections}
N\ell=Mr
\end{equation}
Let $\mathbf{T}_{G} \in \{0,1\}^{M\times N}$ be the adjacency matrix of the graph $G_{\ell,r} (N,M)$, where each column in $\mathbf{T}_{\mathcal{G}}$ corresponds to a left node and has exactly $\ell$ ones, and each row corresponds to a right node and has exactly $r$ ones. Let $\mathbf{t}_i \in \{0,1\}^{N}$ denote the $i$-th row of $\mathbf{T}_{G}$, i.e., $\mathbf{T}_{G}=[\mathbf{t}_{1}^{\mathsf{T}},\mathbf{t}_{2}^{\mathsf{T}},\cdots,\mathbf{t}_{M}^{\mathsf{T}}]^{\mathsf{T}}$. We assign $s$ tests to each right node based on a signature matrix $\mathbf{U}\in \{0,1\}^{s\times r}$. The signature matrix is chosen as ${\mathbf{U}=[\mathbf{1}_{1\times r}^{\mathsf{T}} ,\mathbf{H}_t^{\mathsf{T}}]^{\mathsf{T}}}$, where $\mathbf{1}_{1\times r} $ is an all-ones row of length $r$, and ${\mathbf{H}_t \in \{0,1\}^{t\log_2(r+1)\times r}}$ is the parity-check matrix of a binary $t$-error-correcting BCH code of length $r$. From~\eqref{eq:rows}, it can be easily seen that ${s=R+1=t\log_2 (r+1)+1}$.
The measurement matrix is given by ${\mathbf{A}=[\mathbf{A}_{1}^{\mathsf{T}},\cdots,\mathbf{A}_{M}^{\mathsf{T}}]^{\mathsf{T}}}$ where ${\mathbf{A}_i\in \{0,1\}^{s\times N}}$ is a matrix that defines the $s$ tests at the $i$-th right node. There are exactly $r$ ones in each row $\mathbf{t}_i$ of $\mathbf{T}_{G}$, and the signature matrix $\mathbf{U}=[\mathbf{u}_1,\mathbf{u}_2,\cdots,\mathbf{u}_r]$ has $r$ columns. Note that $\mbf{u}_i=[1,\mbf{h}_i^{\mathsf{T}}]^{\mathsf{T}}$ is the $i$-th column of $\mbf{U}$, where $\mbf{h}_i$ is the $i$-th column of $\mbf{H}_t$. $\mathbf{A}_i$ is obtained by placing the $r$ columns of $\mathbf{U}$ at the coordinates of the $r$ ones of the row vector $\mathbf{t}_i$, and replacing zeros by all-zero columns,
\begin{align}\label{eq:measureblock}
\mbf{A}_i=[\mbf{0},\ldots,\mbf{0},\mbf{u}_1, \mbf{0},\ldots, \mbf{u}_2,\mbf{0}, \ldots, \mbf{u}_{r}]
\end{align} where $\mbf{t}_i =[0,\ldots,0,\hspace{0.6ex}1,\hspace{0.9ex} 0, \ldots,\hspace{0.6ex}1,\hspace{0.9ex}0, \ldots, \hspace{0.9ex}1]$.
The number of rows in the measurement matrix $\mathbf{A}$, ${m=M\times s}$ where $s=t\log_2 (r+1)+1$, represents the total number of tests in the proposed scheme.
\begin{example}\label{ex:example1}
Let $N=14$ be the total number of items. Let $G$ be a randomly generated left-and-right-regular graph with $N$ left nodes of degree $\ell=2$ and $M=4$ right nodes of degree $r=7$. For this example, suppose that the adjacency matrix $\mathbf{T}_{G}$ of the graph $G$ is given by
\setcounter{MaxMatrixCols}{14}
\[
\mathbf{T}_{\mathcal{G}}= \begin{bmatrix}
\textcolor{blue}{1} & 0 & \textcolor{lime}{1} & 0 & \textcolor{orange}{1} & 0 & \textcolor{green}{1} & 0 & \textcolor{red}{1} & 0 & \textcolor{brown}{1} & 0 & 0 & \textcolor{yellow}{1} \\
0 & \textcolor{blue}{1} & \textcolor{lime}{1} & 0 & 0 & \textcolor{orange}{1} & 0 & \textcolor{green}{1} & 0 & \textcolor{red}{1} & 0 & \textcolor{brown}{1} & 0 & \textcolor{yellow}{1} \\
0 & \textcolor{blue}{1} & 0 & \textcolor{lime}{1} & 0 & \textcolor{orange}{1} & \textcolor{green}{1} & 0 & 0 & \textcolor{red}{1} & \textcolor{brown}{1} & 0 & \textcolor{yellow}{1} & 0 \\
\textcolor{blue}{1} & 0 & 0 & \textcolor{lime}{1} & \textcolor{orange}{1} & 0 & 0 & \textcolor{green}{1} & \textcolor{red}{1} & 0 & 0 & \textcolor{brown}{1} & \textcolor{yellow}{1} & 0
\end{bmatrix}.
\] Consider the parity-check matrix $\mathbf{H}_1$ of a binary $t=1$-error-correcting BCH code of length $r=7$ given by
\[
\mbf{H}_1 =\begin{bmatrix}
1 & \alpha & \cdots & \alpha^6
\end{bmatrix}=
\begin{bmatrix}
0 & 0 & 1 & 0 & 1 & 1 & 1\\
0 & 1 & 0 & 1 & 1 & 1 & 0\\
1 & 0 & 0 & 1 & 0 & 1 & 1
\end{bmatrix},
\] where $\alpha \in \mathbb{F}_{2^3}$ is a root of the primitive polynomial ${\alpha^3+\alpha+1=0}$. The signature matrix ${\mathbf{U}=[\mathbf{1}_{1\times 7}^{\mathsf{T}},\mathbf{H}_1^{\mathsf{T}}]^{\mathsf{T}}}$ is then given by
\[
\mbf{U} = \begin{bmatrix}
\textcolor{blue}{1} & \textcolor{lime}{1} & \textcolor{orange}{1} & \textcolor{green}{1} & \textcolor{red}{1} & \textcolor{brown}{1} & \textcolor{yellow}{1}\\
\textcolor{blue}{0} & \textcolor{lime}{0} & \textcolor{orange}{1} & \textcolor{green}{0} & \textcolor{red}{1} & \textcolor{brown}{1} & \textcolor{yellow}{1}\\
\textcolor{blue}{0} & \textcolor{lime}{1} & \textcolor{orange}{0} & \textcolor{green}{1} & \textcolor{red}{1} & \textcolor{brown}{1} & \textcolor{yellow}{0}\\
\textcolor{blue}{1} & \textcolor{lime}{0} & \textcolor{orange}{0} & \textcolor{green}{1} & \textcolor{red}{0} & \textcolor{brown}{1} & \textcolor{yellow}{1}
\end{bmatrix}.
\] Following the construction procedure explained earlier, the testing matrix $\mathbf{A}$ is then given by
\[
\mathbf{A}= \begin{bmatrix}
\textcolor{blue}{1} & 0 & \textcolor{lime}{1} & 0 & \textcolor{orange}{1} & 0 & \textcolor{green}{1} & 0 & \textcolor{red}{1} & 0 & \textcolor{brown}{1} & 0 & 0 & \textcolor{yellow}{1} \\
\textcolor{blue}{0} & 0 & \textcolor{lime}{0} & 0 & \textcolor{orange}{1} & 0 & \textcolor{green}{0} & 0 & \textcolor{red}{1} & 0 & \textcolor{brown}{1} & 0 & 0 & \textcolor{yellow}{1}\\
\textcolor{blue}{0} & 0 & \textcolor{lime}{1} & 0 & \textcolor{orange}{0} & 0 & \textcolor{green}{1} & 0 & \textcolor{red}{1} & 0 & \textcolor{brown}{1} & 0 & 0 & \textcolor{yellow}{0}\\
\textcolor{blue}{1} & 0 & \textcolor{lime}{0} & 0 & \textcolor{orange}{0} & 0 & \textcolor{green}{1} & 0 & \textcolor{red}{0} & 0 & \textcolor{brown}{1} & 0 & 0 & \textcolor{yellow}{1}\\ \hline
0 & \textcolor{blue}{1} & \textcolor{lime}{1} & 0 & 0 & \textcolor{orange}{1} & 0 & \textcolor{green}{1} & 0 & \textcolor{red}{1} & 0 & \textcolor{brown}{1} & 0 & \textcolor{yellow}{1}\\
0 & \textcolor{blue}{0} & \textcolor{lime}{0} & 0 & 0 & \textcolor{orange}{1} & 0 & \textcolor{green}{0} & 0 & \textcolor{red}{1} & 0 & \textcolor{brown}{1} & 0 & \textcolor{yellow}{1} \\
0 & \textcolor{blue}{0} & \textcolor{lime}{1} & 0 & 0 & \textcolor{orange}{0} & 0 & \textcolor{green}{1} & 0 & \textcolor{red}{1} & 0 & \textcolor{brown}{1} & 0 & \textcolor{yellow}{0}\\
0 & \textcolor{blue}{1} & \textcolor{lime}{0} & 0 & 0 & \textcolor{orange}{0} & 0 & \textcolor{green}{1} & 0 & \textcolor{red}{0} & 0 & \textcolor{brown}{1} & 0 & \textcolor{yellow}{1}\\ \hline
0 & \textcolor{blue}{1} & 0 & \textcolor{lime}{1} & 0 & \textcolor{orange}{1} & \textcolor{green}{1} & 0 & 0 & \textcolor{red}{1} & \textcolor{brown}{1} & 0 & \textcolor{yellow}{1} & 0\\
0 & \textcolor{blue}{0} & 0 & \textcolor{lime}{0} & 0 & \textcolor{orange}{1} & \textcolor{green}{0} & 0 & 0 & \textcolor{red}{1} & \textcolor{brown}{1} & 0 & \textcolor{yellow}{1} & 0 \\
0 & \textcolor{blue}{0} & 0 & \textcolor{lime}{1} & 0 & \textcolor{orange}{0} & \textcolor{green}{1} & 0 & 0 & \textcolor{red}{1} & \textcolor{brown}{1} & 0 & \textcolor{yellow}{0} & 0\\
0 & \textcolor{blue}{1} & 0 & \textcolor{lime}{0} & 0 & \textcolor{orange}{0} & \textcolor{green}{1} & 0 & 0 & \textcolor{red}{0} & \textcolor{brown}{1} & 0 & \textcolor{yellow}{1} & 0\\\hline
\textcolor{blue}{1} & 0 & 0 & \textcolor{lime}{1} & \textcolor{orange}{1} & 0 & 0 & \textcolor{green}{1} & \textcolor{red}{1} & 0 & 0 & \textcolor{brown}{1} & \textcolor{yellow}{1} & 0\\
\textcolor{blue}{0} & 0 & 0 & \textcolor{lime}{0} & \textcolor{orange}{1} & 0 & 0 & \textcolor{green}{0} & \textcolor{red}{1} & 0 & 0 & \textcolor{brown}{1} & \textcolor{yellow}{1} & 0 \\
\textcolor{blue}{0} & 0 & 0 & \textcolor{lime}{1} & \textcolor{orange}{0} & 0 & 0 & \textcolor{green}{1} & \textcolor{red}{1} & 0 & 0 & \textcolor{brown}{1} & \textcolor{yellow}{0} & 0\\
\textcolor{blue}{1} & 0 & 0 & \textcolor{lime}{0} & \textcolor{orange}{0} & 0 & 0 & \textcolor{green}{1} & \textcolor{red}{0} & 0 & 0 & \textcolor{brown}{1} & \textcolor{yellow}{1} & 0\\
\end{bmatrix}.
\]
\end{example}
\subsection{Decoding algorithm}\label{decoding}
Let the observation vector corresponding to the $i$-th right node be defined as
\begin{equation}\label{eq:obsvec}
\mathbf{z}_i=[z_{i,1},z_{i,2},\cdots,z_{i,s}]^{\mathsf{T}}=\mathbf{A}_i\mathbf{x},\ \forall i\in\{1,\cdots,M\}.
\end{equation}
Note that
${\mbf{z}_i=[y_{(i-1)s+1},\cdots,y_{is}]^{\mathsf{T}}}$.
\begin{definition}($t$-\emph{resolvable} right node)
A right node is called $t$-\emph{resolvable} if it is connected to $t$ or fewer defective items.
\end{definition}
The following lemma is useful for resolving the right nodes. (The proofs of all lemmas can be found in the appendix.)
\begin{lemma}\label{lem:complexity}
The proposed algorithm detects and resolves all the $t$-resolvable right nodes
\end{lemma}
The decoding algorithm performs in rounds as follows. In each round, the decoding algorithm first iterates through all the right node observation vectors $\{\mathbf{z}_i\}_{i=1}^M$, and resolves all $t$-resolvable right nodes (by BCH decoding, as discussed in the proof of Lemma~\ref{lem:complexity}). Then, given the identities of the recovered left nodes, the edges connected to these defective items are peeled off the graph. That is, the contributions of the recovered defective items will be removed from the unresolved right nodes so that new right nodes may become $t$-resolvable for the next round. The decoding algorithm terminates when there is no more $t$-resolvable right nodes.
\begin{example}
Consider the group testing problem in the Example~\ref{ex:example1}. Let the number of defective items be $K=3$ and let $\mathbf{x}=[1,0,0,1,0,0,0,0,0,1,0,0,0,0]^T$, i.e., item $1$, item $4$, and item $10$ are defective items. We show how the proposed scheme can identify the defective items. The result of the tests can be expressed as follows,
\begin{align*}
\mathbf{y}= \begin{bmatrix}
\mathbf{z}_1 \\
\mathbf{z}_2\\
\mathbf{z}_3 \\
\mathbf{z}_4\\
\end{bmatrix}=\mathbf{A}\mathbf{x}=
\begin{bmatrix}
\mathbf{u}_1 \\
\mathbf{u}_{5} \\
\mathbf{u}_{2}+\mathbf{u}_{5} \\
\mathbf{u}_{1}+\mathbf{u}_{2} \\
\end{bmatrix}
\end{align*}
Then, the right-node observation vectors are given by
\[ \mathbf{z}_1=\mathbf{u}_1=[1,0,0,1]^{\mathsf{T}}\]
\[ \mathbf{z}_2=\mathbf{u}_{5}=[1,1,1,0]^{\mathsf{T}}\]
\[ \mathbf{z}_3=\mathbf{u}_{2}+\mathbf{u}_{5}=[2,1,2,0]^{\mathsf{T}}\]
\[ \mathbf{z}_4=\mathbf{u}_{1}+\mathbf{u}_{2}=[2,0,1,1]^{\mathsf{T}}\]
Because the signature matrix is built using a {$1$-separable} matrix, each right node can be resolved if it is connected to at most one defective item.
Iteration $1$: we first find the {$1$-resolvable} right nodes. The first and second right nodes are $1$-resolvable because $z_{1,1}=z_{2,1}=1$. Using a BCH decoding algorithm, one can find that the defective items connected to the first and second right nodes are item $1$ and item $10$, respectively. Next, we remove the contributions of the items $1$ and $10$ from the unresolved right nodes. The new observation vectors will be as follows,
\[ \mathbf{z}_3=\mathbf{u}_{2}=[1,0,1,0]^{\mathsf{T}}\]
\[ \mathbf{z}_4=\mathbf{u}_{2}=[1,0,1,0]^{\mathsf{T}}\]
Iteration $2$: it can be easily observed that the third and forth right nodes are {$1$-resolvable} since $z_{3,1}=z_{4,1}=1$. Using a BCH decoding algorithm, it follows that the item $4$ is the defective item connected to both right nodes $3$ and $4$. Since all the $K=3$ defective items are identified, the decoding algorithm terminates.
\end{example}
\section{Main Results}\label{sec:main}
In this section, we present our main results. Theorem~\ref{theo:main1} characterizes the required number of tests that guarantees the identification of all defective items with probability approaching one as $K,N\rightarrow\infty$. Theorem~\ref{thm:main2} presents the computational complexity of the proposed algorithm. The proofs of Theorems~\ref{theo:main1} and~\ref{thm:main2} are given in Section~\ref{sec:proofs}.
\begin{theorem}\label{theo:main1}
For the sub-linear regime, the proposed scheme recovers all defective items with probability approaching one (as $K,N\rightarrow\infty$) with at most ${m=c(t)K\left(t\log_2\left(\frac{\ell N}{c(t)K}+1\right)+1\right)+1}$ tests, where $c(t)$ depends only on $t$. Table~\ref{Table:constant} shows the values of $c(t)$ for $t\leq 8$.
\begin{table}[h]
\centering
\scriptsize{
\begin{tabular}{| c | c | c | c |c | c | c | c | c |}
\hline
$t$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8\\ \hline
$c(t)$ & 1.222 & 0.597 & 0.388 & 0.294 & 0.239 & 0.202 & 0.176 & 0.156\\ \hline
$\ell^{\star}$ & 3 & 2 & 2 & 2 & 2 & 2 & 2 & 2 \\ \hline
\end{tabular}}
\caption{\small{The function $c(t)$ and the optimal left degree $\ell^{\star}$.}}
\label{Table:constant}
\end{table}
\end{theorem}
\begin{theorem}\label{thm:main2}
The encoding and decoding of the proposed algorithm for any $t\leq 4$ have the computational complexity of $\mathcal{O}(K\log^2 \frac{N}{K})$ and $\mathcal{O}(K\log \frac{N}{K})$, respectively.
\end{theorem}
\section{Proofs of Main Theorems}\label{sec:proofs}
\subsection{Proof of Theorem~\ref{theo:main1}}
Let $N$ be the total number of items, out of which $K$ items are defective. Note that in the QGT problem, performing one initial test (on all items) would suffice to obtain the number of defective items.
As mentioned in Section~\ref{decoding}, our scheme employs an iterative decoding algorithm. In each iteration, the algorithm finds and resolves all the $t$-\emph{resolvable} right nodes. At the end of each iteration, the decoder subtracts the contribution of the identified defective items from the unresolved right nodes. This process is repeated until there is no $t$-\emph{resolvable} right nodes left in the graph. The fraction of defective items that remain unidentified when the decoding algorithm terminates can be analyzed using density evolution as follows.
Assuming that the exact number of the defective items, $K$, is known and the values assigned to the defective and non-defective items are one and zero, respectively, the left-and-right-regular bipartite graph can be pruned. All the zero left nodes and their respective edges are removed from the graph. The number of left nodes in the pruned graph is $K$, but the degree of these nodes remains unchanged. On the other hand, the number of right nodes remains unchanged, but the resulting graph is not right-regular any longer.
Let $\lambda$ be the average right degree, i.e., $\lambda=\frac{K\ell}{M}$. Let $\rho(x)\triangleq \sum_{i=1}^{\min(K,r)}\rho_ix^{i-1}$ be the right edge degree distribution, where $\rho_i$ is the probability that a randomly picked edge in the pruned graph is connected to a right node of degree $i$, and $\min(K,r)$ is the maximum degree of a right node. As shown in~\cite{DBLP:journals/corr/VemJN17}, as $K,N\rightarrow\infty$, we have $\rho_i=e^{-\lambda}\frac{\lambda^{i-1}}{(i-1)!}$.
The following lemma is useful for computing the fraction of unidentified defective items at each iteration $j$ of the decoding algorithm.
\begin{lemma}\label{lem:DenEvol}
Let $p_j$ be the probability that a randomly chosen defective item is not recovered at iteration $j$ of the decoding algorithm; and let $q_j$ be the probability that a randomly picked right node is resolved at iteration $j$ of the decoding algorithm. The relation between $p_j$ and $p_{j+1}$ is determined by the following density evolution equations:
\begin{equation}\label{eq:det-denevol1}
q_j=\sum_{i=1}^{t}\rho_i+
\sum_{i=t+1}^{\min(K,r)}\rho_i\sum_{k=0}^{t-1}{i-1 \choose k}p_j^k(1-p_j)^{i-k-1},
\end{equation}
\begin{equation}\label{eq:det-denevol2}
p_{j+1}=(1-q_j)^{\ell-1},
\end{equation}
where $t$ is the level of separability, and $\rho_i$ is the probability that a randomly picked edge in the pruned graph is connected to a right node of degree $i$.
\end{lemma}
\begin{figure}\hspace{0.5cm}
\begin{tikzpicture}
\coordinate (O1) at (0,-0.7);
\coordinate (O2) at (0,-3.5);
\coordinate (O3) at (4,-2.15);
\draw (O1) circle (0.4);
\draw (O2) circle (0.4);
\draw (O3) circle (0.4);
\draw[black] (2.2,-1.9) ++(-.65*0.5, -7/6*0.5) rectangle ++(0.7,0.7);
\draw[black] (6.5,-0.45) ++(-.65*0.5, -7/6*0.5) rectangle ++(0.7,0.7);
\draw[black] (6.5,-3.35) ++(-.65*0.5, -7/6*0.5) rectangle ++(0.7,0.7);
\draw [dotted] (0,-1.2) -- (0,-3);
\draw [dotted] (6.5,-1.1) -- (6.5,-3.2);
\path[black] (-90:0.7) node []{$v_1$};
\path[black] (-90:3.5) node []{$v_{i-1}$};
\path[black] (-44:3.05) node []{$c$};
\path[black] (-28:4.5) node []{$v$};
\path[black] (-5.75:6.6) node []{$c_1$};
\path[black] (-29:7.45) node []{$c_{\ell-1}$};
\draw (0.33,-0.92) -- (1.87,-2.15);
\draw (0.33,-3.27) -- (1.87,-2.15);
\draw (0,-1.71) -- (1.87,-2.15);
\draw (0,-2.05) -- (1.87,-2.15);
\draw (0,-2.4) -- (1.87,-2.15);
\draw (4.4,-2.15) -- (6.5,-2.13);
\draw (4.4,-2.15) -- (6.5,-2.45);
\draw (4.4,-2.15) -- (6.5,-1.78);
\draw (2.6,-2.15) -- (3.6,-2.15);
\draw (4.283,-1.867) -- (6.17,-0.65);
\draw (4.283,-2.433) -- (6.17,-3.6);
\end{tikzpicture}
\caption{Tree-like representation of neighborhood of the edge between a left node $v$ and a right node $c$ in the pruned graph.}\label{fig:denevol}
\end{figure}
Note that $p_j$ is only a function of the variables $t$, $\ell$, and $\lambda$ when $\min(K,r)\rightarrow \infty$. Recall that the goal is to minimize the total number of tests, i.e., ${M\times s}$, where $M$ is the number of right nodes, and $s$ is the number of rows in the signature matrix. The number of rows, $s$, in the signature matrix depends only on the level of separability, $t$. For a given $t$, we can minimize the number of right nodes ${M=\frac{\ell}{\lambda}K}$ subject to the constraint ${\lim_{j\rightarrow \infty }p_j(\ell,\lambda)=0}$, so as to minimize the total number of the tests. The constraint $ \lim_{j\rightarrow \infty }p_j(\ell,\lambda)=0$ guarantees that running the decoding algorithm for sufficiently large number of iterations, the probability that a randomly chosen defective item remains unidentified approaches zero. For any ${\ell \geq 2}$, let ${\lambda_T(\ell) \triangleq \sup\{\lambda: \lim_{j\rightarrow \infty }p_j(\ell,\lambda)=0\}}$. Then, for any ${\ell \geq 2}$ and $\lambda < \lambda_T(\ell)$, we have ${\lim_{j\rightarrow \infty }p_j(\ell,\lambda)=0}$. Accordingly, for any ${\ell \geq 2}$ and $M=\frac{\ell}{\lambda}K > \frac{\ell}{\lambda_T(\ell)}K$, it follows that $\lim_{j\rightarrow \infty }p_j(\ell,\lambda)=0$. Our goal is then to compute
\begin{equation}\label{eq:opt}
\min_{\ell \in \{2,3,\dots\}}\frac{\ell}{\lambda_T(\ell)}K.
\end{equation}
We can solve this problem numerically and attain the optimal value of $\ell$, i.e., $\ell^{\star}$. Let ${c(t)\triangleq \frac{\ell^{\star}}{\lambda_T(\ell^{\star})}}$. The number of right nodes can then be chosen as ${M=c(t)K\beta}$ for any $\beta> 1$ to guarantee that ${M > c(t)K=\frac{\ell^{\star}}{\lambda_T(\ell^{\star})}K}$. Substituting ${M=c(t)K\beta}$ in \eqref{eq:connections} results in ${r=\frac{\ell N}{c(t)K\beta}}$. Therefore, the total number of tests will become ${M\times s=c(t)K\beta\left(t\log_2\left(\frac{\ell N}{c(t)K\beta}+1\right)+1\right)}$.
\begin{lemma}\label{lem:lem3}
There exist some $\beta>1$ such that
\begin{multline*}
c(t)K\left(t\log_2\left(\frac{\ell N}{c(t)K}+1\right)+1\right)+1 \geq \\ c(t)K\beta\left(t\log_2\left(\frac{\ell N}{c(t)K\beta}+1\right)+1\right).
\end{multline*}
\end{lemma}
By combining the result of Lemma~\ref{lem:lem3} and the preceding arguments, it follows that with probability approaching one as $K,N\rightarrow\infty$, $m=c(t)K\left(t\log_2\left(\frac{\ell N}{c(t)K}+1\right)+1\right)+1$ tests would suffice for the proposed algorithm to recover all defective items. This completes the proof.
\subsection{Proof of Theorem~\ref{thm:main2}}
\begin{lemma}\label{lem:complexity2}
For any $t\leq 4$, the computational complexity of resolving each $t$-resolvable right node is $\mathcal{O}(\log r)$.
\end{lemma}
The total number of right nodes, $M$, is $\mathcal{O}(K)$. From Lemma~\ref{lem:complexity2}, it then follows that the complexity of the decoding algorithm is $\mathcal{O}(K\log r)$. Using \eqref{eq:connections}, it is easy to see that for any $t\leq 4$ the decoding algorithm has complexity $\mathcal{O}(K\log \frac{N}{K})$. The total number of measurements is $m$ and for each measurement $r$ summations are performed. Hence, the complexity of the encoding algorithm is $\mathcal{O}(mr)$, which becomes equivalent to $\mathcal{O}(K\log^2 \frac{N}{K})$ for any $t\leq 4$.
\section{Evaluation of $c(t)$}
In this section, we present the complete analysis for the case of $t=1$, and show how one can evaluate $c(t)$ at $t=1$, i.e., $c(1)$. The same procedure can be used for evaluating $c(t)$ at any $t>1$.
To compute ${c(1)= \frac{\ell^{\star}}{\lambda_T(\ell^{\star})}}$, we compute the ratio $\frac{\ell}{\lambda_T(\ell)}$ for each $\ell\geq 2$ and its corresponding $\lambda_T(\ell)$. The optimal $\ell$, i.e., $\ell^{\star}$, is the one that yields the minimum value for $\frac{\ell}{\lambda_T(\ell)}$.
For the case of $t=1$, the density evolution equations~\eqref{eq:det-denevol1} and~\eqref{eq:det-denevol2} can be combined as \begin{equation}\label{eq:pj1} p_{j+1}=\left(1-\sum_{i=1}^{\min(K,r)}\rho_i(1-p_j)^{i-1}\right)^{\ell-1}.
\end{equation} Obviously, $p_1=1$. Substituting ${\rho_i=e^{-\lambda}\frac{\lambda^{i-1}}{(i-1)!}}$, we can rewrite~\eqref{eq:pj1} as
\begin{equation}\label{eq:pj12}
p_{j+1}=\left(1-e^{-\lambda}\sum_{i=1}^{\min(K,r)}\frac{\lambda^{i-1}}{(i-1)!}(1-p_j)^{i-1}\right)^{\ell-1}.
\end{equation} For the sub-linear regime, $\frac{K}{N}\rightarrow 0$ (by definition) as $K,N\rightarrow \infty$, and hence, $r\rightarrow \infty$ (by~\eqref{eq:connections}). Thus, in the asymptotic regime of our interest, $\min(K,r)\rightarrow \infty$. Letting $\min(K,r)\rightarrow \infty$, the equation~\eqref{eq:pj12} reduces to
\begin{equation}\label{eq:t=1}
p_{j+1}=\left(1-e^{-\lambda p_j}\right)^{\ell-1}.
\end{equation}
Using~\eqref{eq:t=1}, we can write
\[
{\lambda= \left( \frac{\ln \left(1-p_{j+1}^{\frac{1}{\ell-1}}\right)}{-p_j}\right)}.
\]
The following two lemmas are useful for computing $\lambda_T(\ell) = \sup\{\lambda: \lim_{j\rightarrow \infty }p_j(\ell,\lambda)=0\}$ for each $\ell \geq 2$.
\begin{lemma}\label{lem:convergence}
For any $\ell \geq 2$ and any $\lambda > 0$, the infinite sequence $\{p_1,p_2,\cdots\}$ converges.
\end{lemma}
\begin{lemma}\label{lem:limit}
Let $p^{*}$ be the limit of the sequence $\{p_1,p_2,\cdots\}$, and let \[{\lambda_T(\ell)\triangleq \displaystyle \inf_{0<x<1} \left( \frac{\ln(1-x^{\frac{1}{\ell-1}})}{-x}\right)}.\] Then, for any $\ell \geq 2$, we have
\[
\begin{cases}
p^{*}=0, & \quad 0 < \lambda < \lambda_T(\ell),\\
p^{*}>0, &\quad \lambda \geq \lambda_T(\ell).\\
\end{cases}
\]
\end{lemma}
By the result of Lemma~\ref{lem:limit}, for any ${\ell\geq 2}$ the value of $\lambda_T(\ell)$ can be computed numerically. One can then obtain the optimal value of $\ell$, i.e., $\ell^{\star}$, which minimizes the ratio of $\frac{\ell}{\lambda_T(\ell)}$, and accordingly ${c(1)= \frac{\ell^{\star}}{\lambda_T(\ell^{\star})}}$ can be computed.
\begin{figure}
\includegraphics[width=0.52\textwidth]{analysis.eps}\vspace{-0.2cm}
\caption{The number of required tests ($m$) to identify all $K$ defective items (for different values of $K$) among $N=2^{16}$ items for different values of ${t}$ obtained via analysis.
}
\label{fig:best}
\end{figure}
\section{Comparison Results}
In this section we will evaluate the performance of the proposed algorithm based on our theoretical analysis and the Monte Carlo simulations.
Based on the results in Theorem~\ref{theo:main1} and Table~\ref{Table:constant}, Fig.~\ref{fig:best} depicts the total number of tests ($m$) required to identify all the defective items for different values of $t$. The number of items is assumed to be $N=2^{16}$. As it can be seen, when $t\in \{1,2,3\}$ the required number of tests for identifying all the defective items is less than that for larger values of $t$.
Using the Monte Carlo simulation, we also compare the performance of the proposed scheme for $t\in \{1,2,3\}$ with the performance of the Multi-Level Group Testing (MLGT) algorithm from \cite{8335478}. The MLGT scheme is a semi-quantitative group testing scheme where the result of each test is an integer in the set $\{0,1,2,\cdots,L\}$. Letting ${L\rightarrow \infty}$, the MLGT scheme becomes a QGT scheme. Based on the optimization that we have performed, the optimal left degree for the MLGT scheme is $\ell^{\star}=3$ when $L\rightarrow \infty$. For $K=100$ defective items among a population of $N=2^{16}$ items, the average fraction of unidentified defective items for the MLGT scheme and the proposed scheme are shown in Fig.~\ref{fig:sim} for different values of $m/K$. As it can be observed, the proposed scheme for all the three tested values of $t$ outperforms the MLGT scheme significantly. For instance, when the fraction of unidentified defective items is $2\times 10^{-4}$, the required number of tests for the MLGT scheme (for $\ell=3$) is $3$ times, $5$ times, and $7$ times more than that of the proposed scheme for $t=1$, $t=2$, and $t=3$, respectively.
\begin{figure}
\includegraphics[width=0.52\textwidth]{res_comp.eps}\vspace{-0.2cm}
\caption{{The average fraction of unidentified defective items obtained via Monte Carlo simulations for $N=2^{16}$ items among which $K=100$ items are defective.}}\label{fig:sim}
\end{figure}
\bibliographystyle{IEEEtran}
|
1,108,101,565,673 | arxiv | \section{Introduction}
Magnetism and superconductivity appear nearby in typical phase diagrams of transition-metal and heavy-fermion compounds.
Magnetism is related to the Mott insulating state and heavy-fermion formation, which can be described in terms of local correlations.
On the other hand, unconventional superconductivity requires spatial correlations to be taken into account.
For a comprehensive understanding, therefore, one needs a unified treatment of local correlations and spatial fluctuations, which has been a theoretical challenge in the field of strongly correlated electron systems.
A long-standing problem, which may be related to magnetism and superconductivity, is the pseudo-gap state in the low-doped regime of cuprates~\cite{Timusk99}. One of the candidates for its origin is a hidden order, i.e. the staggered flux state or the d-density wave (d-DW)~\cite{Chakravarty01}.
There are some experiments which indicate broken time-reversal symmetry in the pseudo-gap regime~\cite{Fauque06, Shekhter13}.
Theoretically, the mean-field approximation based on the slave-boson representation yields a d-DW in the $t$-$J$ model~\cite{Kotliar88,Ubbens92,Wen96}.
However, no clear evidence for the transition has been found in the Hubbard model~\cite{Honerkamp02, Macridin04, Lu12, Yokoyama-commun}.
Another feature, which possibly emerges near the Mott insulator, is a uniform charge instability, i.e., phase separation between two states with different electron density. It was pointed out for the $t$-$J$ model on the basis of energy arguments~\cite{Emery90}, and was indeed demonstrated numerically in the one-dimensional system~\cite{Ogata91} and in infinite dimensions~\cite{Otsuki-Vollhardt}.
In contrast to the d-DW, the phase separation has been observed also in the Hubbard model by means of various numerical methods~\cite{Zitzler02,Kotliar02,Capone06,Macridin06,Werner07,Eckstein07,Aichhorn07, Yokoyama13, Misawa-arXiv},
while quantum Monte Carlo investigations reported no evidence of phase separation~\cite{Moreo91,Becca00}.
The unresolved problems described above motivate us to investigate the two-dimensional Hubbard model as a prototypical model of strongly correlated electron systems, and to develop new theories which could clarify these issues.
The dynamical mean-field theory (DMFT) provides a description of the Mott transition~\cite{Georges96} and its cluster extensions provide a route to the d-wave superconductivity (d-SC) in the doped regime~\cite{Maier-RMP, Potthoff03}.
The d-SC has indeed been obtained in several numerical calculations~\cite{Lichtenstein00, Maier05, Gull13, Capone06}.
We note that cluster DMFT particularly accounts for short-range correlations in addition to the local ones.
A different kind of extension of single-site DMFT has been worked on, which,
in contrast to cluster extensions, aims at incorporating long-range correlations~\cite{Kusunose06, Toschi07, Held08, Katanin09, Slezak09, Taranto-arXiv, Kitatani}. The common idea of these approaches is to introduce an additional step of solving the lattice problem in a certain way after the DMFT equations are solved.
The various formulations differ (i) physically, in the sets of diagrams which are summed beyond DMFT and (ii) technically, how double counting of correlation effects is avoided, that may arise when two different methods are combined.
Rubtsov \textit{et al.} introduced an auxiliary fermion
which mediates itinerancy of electrons~\cite{Rubtsov08, Rubtsov09}.
With this dual fermion, a perturbation expansion around the DMFT has been made possible
without the double-counting problem; the zeroth-order approximation in this theory corresponds to DMFT, and spatial correlations are systematically incorporated by summing up a series of diagrams.
In particular, ladder diagrams similar to those in the fluctuation exchange approximation (FLEX)~\cite{FLEX1, FLEX2, Takimoto02, Yanase03} yield descriptions of collective modes (long-range fluctuations).
Indeed, it has been shown that inclusion of the ladder diagrams in the dual-fermion approach leads to paramagnon excitations that exhibit antiferromagnetic (AFM) fluctuations in the paramagnetic state~\cite{Hafermann09, Hafermann-book}.
At the same time, the ladder approximation yields suppression of the AFM phase transition in two dimensions~\cite{Hafermann09, Hafermann-book} and the expected critical exponents in case that phase transitions are found~\cite{Antipov14},
demonstrating that long-range fluctuations essential for the critical behavior are appropriately included.
Therefore, the dual-fermion approach with ladder-type diagrams provides a combined description of strong local correlations and long-range correlations.
Although first results of the ladder approximation have been presented in 2009~\cite{Hafermann09,Hafermann09-2}, its exemplary results for doped Mott insulators have been limited because of some technical difficulties arising from strong AFM fluctuations. In this paper, we overcome these limitations and present systematic results for the doped regime of the two-dimensional Hubbard model. We address possible phase transitions of the d-DW and the phase separation in the doped Mott insulator as well as the d-SC. Our results reveal further characteristics of the ladder approximation.
The rest of this paper is organized as follows.
In the next section, we first present phase diagrams obtained in this investigation to give an overview of our results.
Afterwards, the dual-fermion formalism and the self-energy equation are presented in Section~\ref{sec:dualfermion}.
Succeeding Sections~\ref{sec:AFM}--\ref{sec:DW} present detailed numerical results and related formulas for the AFM susceptibility, superconductivity, phase separation, and unconventional density waves.
The paper is closed with discussions in Section~\ref{sec:summary}.
\section{Overview}
\label{sec:overview}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\linewidth]{phase-n1.eps}
\end{center}
\caption{(Color online) A phase diagram at half filling, $\delta=0$.}
\label{fig:phase-n1}
\end{figure}
Prior to presenting formalism and detailed numerical results, we first give an overview of our results obtained in this paper.
We investigate the two-dimensional Hubbard model:
\begin{align}
H = \sum_{\bm{k}\sigma} \epsilon_{\bm{k}} c_{\bm{k}\sigma}^{\dag} c_{\bm{k}\sigma}
+ U \sum_{\bm{r}} n_{\bm{r}\uparrow} n_{\bm{r}\downarrow},
\end{align}
with $\epsilon_{\bm{k}}=-2t (\cos k_x + \cos k_y)$.
The number operator $n_{\bm{r}\sigma}$ is defined by
$n_{\bm{r}\sigma}=N^{-1} \sum_{\bm{k}\bm{q}} c_{\bm{k}\sigma}^{\dag} c_{\bm{k}+\bm{q}\sigma} e^{i\bm{q}\cdot \bm{r}}$,
where $N$ denotes the number of lattice sites. We take $t=1$ as the unit of energy.
In two-dimensional systems, the AFM transition is forbidden at $T>0$ by the Mermin-Wagner theorem~\cite{Mermin-Wagner}. This leads to the critical behavior $\chi \sim e^{c \beta}$ of the susceptibility at low temperatures~\cite{Chakravarty88, Hasenfratz91}.
Our approximation indeed shows no AFM transition within calculated temperatures.
To quantify the AFM fluctuations, we define a ``phase boundary" by the points where the fluctuations exceed a certain criterion (see Section~\ref{sec:AFM} for details). We may regard this line as a phase boundary in quasi-two dimensions. The phase diagram at half filling obtained in this way is shown in Fig.~\ref{fig:phase-n1}.
We plot three phase boundaries corresponding to different criteria. In DMFT, there exists a real phase transition, which is plotted for comparison.
According to a cluster DMFT calculation with a paramagnetic bath~\cite{Park08}, the Mott transition takes place at $U\simeq 6$ and below $T \simeq 0.1$~\cite{footnote-Mott}.
We could not reach this regime due to the critical AFM fluctuations, which renders the self-energy calculation unstable.
We note, however, that cluster DMFT does not take into account critical fluctuations characteristic of two dimensions, meaning that the AFM transition takes place at a higher temperature than the Mott transition. Hence the latter is actually hidden by the AFM phase in cluster DMFT.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\linewidth]{phase-U8.eps}
\end{center}
\caption{(Color online) Phase diagrams under doping $\delta=1-n$ for $U=8$.}
\label{fig:phase-dope}
\end{figure}
Figure~\ref{fig:phase-dope} shows the phase diagram of temperature against doping $\delta=1-n$ for $U=8$.
The d-SC is obtained in the region $T\lesssim 0.05$ and $\delta \lesssim 0.18$.
The superconducting transition temperature $T_{\rm c}$ monotonically increases approaching half filling ($\delta=0$). This behavior is reminiscent of the FLEX~\cite{Takimoto02, Yanase03} and differs from that in cluster DMFT, where the d-SC phase exhibits a maximum at finite doping~\cite{Lichtenstein00, Gull13}.
We consider that the monotonic behavior of $T_{\rm c}$ in our results is due to insufficient treatment of short-range spin fluctuations, which will be discussed in Sec.~\ref{sec:summary}.
In the low-doping regime above $T_{\rm c}$, we found a phase separation. The line $T_{\rm PS}$ in Fig.~\ref{fig:phase-dope} shows the spinodal line, where the uniform charge susceptibility diverges.
The phase separation extends up to $\delta \simeq 0.15$.
At $T<T_{\rm PS}$, the solution is thermodynamically unstable because of $\partial n/\partial \mu < 0$.
Thermodynamic stability is acquired by inhomogeneous coexistence of regions with different doping levels: Mott insulating regions with $\delta=0$ and metallic regions with larger doping $\delta \neq 0$.
The phase boundary for the d-SC has been computed with the homogeneous solution, which, in fact, is thermodynamically unstable in the region $\delta \lesssim 0.15$.
Therefore, pure d-SC only realizes in the region $0.15 \lesssim \delta \lesssim 0.18$, while it may not occur for $\delta \lesssim 0.15$. We have also examined the possibility of a d-DW. We find that the d-DW dominates over DWs with other symmetries, but the corresponding susceptibility shows no divergence.
\section{Dual-ladder approximation}
\label{sec:dualfermion}
\subsection{Dual action}
In the dual-fermion approach, the lattice model is solved in two steps. First, an effective impurity model, which is the same as in DMFT, is solved with the aid of some numerical methods. The local correlations, which are essential for formation of the Mott gap, are fully taken into account at this stage.
In the next step, an interacting lattice model is constructed by quantities evaluated in the first step, and is solved by a diagrammatic perturbation theory.
This way, spatial fluctuations are incorporated in addition to the local correlations in DMFT.
In the following, we first give a brief summary of the dual-fermion approach~\cite{Rubtsov08, Rubtsov09, Hafermann-book}.
It is convenient to work in the path-integral representation.
The partition function $Z$ is written in terms of Grassmann variables
$c_{\bm{r}}(\tau)$ and $c_{\bm{r}}^*(\tau)$:
$Z = \int \prod_{\bm{r}} {\cal D}[c_{\bm{r}}^* c_{\bm{r}}] e^{-{\cal S}[c^*, c]}.$
The action ${\cal S}$ is given by
\begin{align}
{\cal S}[c^*, c]
&= \sum_{\bm{r}} {\cal S}_{\rm imp}[c_{\bm{r}}^*, c_{\bm{r}}]
+ \sum_{\omega \bm{k} \sigma} (\epsilon_{\bm{k}} -\Delta_{\omega})
c_{\omega \bm{k} \sigma}^* c_{\omega \bm{k} \sigma},
\label{eq:S_lat}
\end{align}
where the Fourier transform of $c(\tau)$ is defined by
$c_{\omega}=\beta^{-1/2} \int_0^{\beta} c(\tau) e^{i\omega\tau}$
with $\omega$ being the fermionic Matsubara frequency.
The first term ${\cal S}_{\rm imp}$ describes the effective impurity model of DMFT~\cite{Georges96}:
\begin{align}
{\cal S}_{\rm imp}[c_{\bm{r}}^*, c_{\bm{r}}]
= &- \sum_{\omega \sigma} (i\omega + \mu - \Delta_{\omega})
c^*_{\omega \bm{r} \sigma} c_{\omega \bm{r} \sigma}
\nonumber \\
&+ U \int d\tau n_{\bm{r}\uparrow}(\tau) n_{\bm{r}\downarrow}(\tau).
\label{eq:S_imp}
\end{align}
The hybridization function $\Delta_{\omega}$ is actually canceled out in Eq.~(\ref{eq:S_lat}),
but an approximate solution may depend on $\Delta_{\omega}$.
A condition for determining $\Delta_{\omega}$ will be discussed later.
In order to construct a lattice model for which the solution of ${\cal S}_{\rm imp}$ is the starting point, Rubtsov \textit{et al.} introduced an auxiliary fermion which ``decouples'' the kinetic-energy term~\cite{Rubtsov08,Rubtsov09} [the second term in Eq.~(\ref{eq:S_lat})].
This fermion is termed dual fermion and represented by $f$.
The dual fermions locally hybridize with the electrons and mediate the electron itinerancy.
The point is that the transformed action written with $c$ and $f$ variables has only local terms concerning $c$ variables.
Therefore, one can integrate out $c$ variables at each site independently. This process corresponds to solving the effective impurity problem expressed by ${\cal S}_{\rm imp}$.
The local hybridization between $c$ and $f$ introduces effective interaction terms among the $f$ variables, which are local in space but non-local in the time domain.
The resulting partition function thus consists only of the dual variables $f$.
Hence, our task now is to solve the dual system described by
$\tilde{Z} =
\int \prod_{\bm{r}} {\cal D}[f_{\bm{r}}^* f_{\bm{r}}]
e^{-\tilde{\cal S}[f^*, f]}$.
The action $\tilde{\cal S}$ is given by
\begin{align}
\tilde{\cal S}[f^*, f]
&= -\sum_{\omega \bm{k} \sigma}
(\tilde{G}^0_{\omega \bm{k}})^{-1} f_{\omega \bm{k} \sigma}^* f_{\omega \bm{k} \sigma}
+ \tilde{V}[f^*, f],
\label{eq:S-dual}
\end{align}
with the bare dual Green's function $\tilde{G}^0_{\omega\bm{k}}$ defined by
\begin{align}
\tilde{G}^0_{\omega \bm{k}}
= (g_{\omega}^{-1} + \Delta_{\omega} - \epsilon_{\bm{k}})^{-1}
- g_{\omega}.
\label{eq:G0_dual}
\end{align}
Here $g_{\omega}=-\langle c_{\omega \bm{r} \sigma} c_{\omega \bm{r} \sigma}^* \rangle_{\rm imp}$ is the impurity Green's function, with $\langle \cdots \rangle_{\rm imp}$ being a thermal average with respect to the action ${\cal S}_{\rm imp}$.
The first term in Eq.~(\ref{eq:G0_dual}) corresponds to the lattice Green's function in DMFT. Subtracting the second term excludes double counting of local correlations.
The term $\tilde{V}$ denotes local interactions, which include many-body interactions as well as a two-body term.
The point of the transformed action $\tilde{\cal S}$ is that the bare propagator $\tilde{G}^0$ and the interaction $\tilde{V}$ fully include local correlations. Hence, the (undressed) dual fermions $f$ may be regarded as particles which involve all the local interaction processes. Residual interactions between the dressed particles are described by $\tilde{V}$.
Once the dual Green's function $\tilde{G}_{\omega\bm{k}}=-\langle f_{\omega\bm{k}\sigma} f_{\omega\bm{k}\sigma}^* \rangle_{\tilde{S}}$ is evaluated,
it is readily transformed to the Green's function $G_{\omega \bm{k}}$ of the original electrons
by means of the exact relation
\begin{align}
G_{\omega \bm{k}}^{-1}
&= (g_{\omega} + g_{\omega} \tilde{\Sigma}_{\omega\bm{k}} g_{\omega})^{-1} + \Delta_{\omega} - \epsilon_{\bm{k}}.
\end{align}
Here, we introduced the dual self-energy
$\tilde{\Sigma}_{\omega\bm{k}}=(\tilde{G}^0_{\omega\bm{k}})^{-1} - \tilde{G}_{\omega\bm{k}}^{-1}$.
It is clear from this expression that $\tilde{\Sigma}_{\omega\bm{k}}=0$ leads to the DMFT formula for the lattice Green's function.
The formalism presented above is still exact.
In the following, two approximations will be made.
Firstly, we retain only two-body interactions in $\tilde{V}$.
Secondly, we perform a perturbation expansion with respect to $\tilde{V}$ to sum up a certain set of diagrams for $\tilde{\Sigma}_{\omega\bm{k}}$.
These approximations rely on the idea that the DMFT is a good starting point for Mott insulators, and hence the spatial correlations may be dealt with perturbatively.
We can also endorse this treatment by arguments based on the $1/d$ expansion, which will be discussed later.
\subsection{Interaction vertex for dual fermions}
We retain only two-body interactions in $\tilde{V}$ neglecting terms involving more than three particles.
Thus, $\tilde{V}$ reads
\begin{align}
\tilde{V}
=-\frac{1}{4} \sum_{kk'q} \sum_{\sigma_1 \sigma_2 \sigma_3 \sigma_4}
\gamma_{\omega\omega'; \nu}^{\sigma_1 \sigma_2 \sigma_3 \sigma_4}
f_{k\sigma_1}^* f_{k'+q,\sigma_2}^* f_{k'\sigma_3} f_{k+q,\sigma_4},
\label{eq:V_tilde}
\end{align}
where $k=(\omega, \bm{k})$ and $q=(\nu, \bm{q})$ with $\nu$ being the bosonic Matsubara frequency.
The interaction coefficient $\gamma$ corresponds to the vertex evaluated in the effective impurity system. It is defined through
\begin{align}
\langle c_1 c_2 c_3^* c_4^* \rangle_{\rm imp}
&= g_1 g_2 ( \delta_{14} \delta_{23} - \delta_{13} \delta_{24})
+ T g_{1} g_{2} \gamma_{1234} g_{3} g_{4}.
\end{align}
Here, a simplified notation is used such as $1\equiv (\omega_1, \sigma_1)$.
Using energy conservation, we parameterize the frequency dependence of $\gamma$ as
\begin{align}
\gamma_{\omega\omega'; \nu}^{\sigma_1 \sigma_2 \sigma_3 \sigma_4}
&\equiv \gamma_{(\omega, \sigma_1), (\omega'+\nu, \sigma_2), (\omega', \sigma_3), (\omega+\nu, \sigma_4)}.
\end{align}
The antisymmetric nature of $\gamma$ leads to the relation
$\gamma_{\omega\omega'; \nu}^{\sigma_1 \sigma_2 \sigma_3 \sigma_4}
=-\gamma_{\omega,\omega+\nu; \omega'-\omega}^{\sigma_1 \sigma_2 \sigma_4 \sigma_3}$.
The interaction $\tilde{V}$ is represented by the diagram in Fig.~\ref{fig:diagrams}(a).
We consider the spin dependence of $\gamma$. Using the Pauli matrix $\sigma^{\xi}$ ($\xi=0,x,y,z$, including the unit matrix $\sigma^0$), we transform $\gamma$ as
\begin{align}
&\gamma^{\sigma_1 \sigma_2 \sigma_3 \sigma_4}
= \frac{1}{2} \sum_{\xi\xi'} \gamma^{\xi\xi'}
\sigma^{\xi}_{\sigma_1\sigma_4} \sigma^{\xi'}_{\sigma_2\sigma_3}.
\label{eq:gamma-pauli}
\end{align}
Without magnetic field, $\gamma^{\xi\xi'}$ is diagonal and there are only two independent components:
$\gamma=\text{diag}(\gamma^{\rm ch}, \gamma^{\rm sp}, \gamma^{\rm sp}, \gamma^{\rm sp})$.
By inverting Eq.~(\ref{eq:gamma-pauli}), we obtain
\begin{align}
&\gamma^{\rm ch} \equiv \gamma^{00}
=\frac{1}{2} \sum_{\sigma\sigma'} \gamma^{\sigma\sigma'\sigma'\sigma},
\\
&\gamma^{\rm sp} \equiv \gamma^{zz}
=\frac{1}{2} \sum_{\sigma\sigma'} \sigma\sigma' \gamma^{\sigma\sigma'\sigma'\sigma},
\end{align}
which corresponds to interactions in charge and longitudinal-spin channels, respectively.
The transverse-spin channel
$\gamma^{\rm \perp} \equiv
\gamma^{\uparrow\downarrow\uparrow\downarrow}
=(\gamma^{xx}-i\gamma^{xy})$
is equivalent to $\gamma^{\rm sp}$,
since we are considering the paramagnetic state.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.95\linewidth]{diagrams.eps}
\end{center}
\caption{Diagrammatic representations of (a) bare interaction $\tilde{V}$ for dual fermions, (b) the dual self-energy $\tilde{\Sigma}$ in the ladder approximation, and (c) the equation for the renormalized vertex $\Gamma$.}
\label{fig:diagrams}
\end{figure}
\subsection{Self-energy}
We evaluate the dual self-energy $\tilde{\Sigma}_{\omega\bm{k}}$
taking the two-body interaction $\tilde{V}$ in Eq.~(\ref{eq:V_tilde}) into account.
In principle, one can apply any numerical method as well as approximations for this purpose.
Here, we use a perturbation theory and sum up certain diagrams based on physical considerations and a $1/d$ analysis.
We first discuss the choice of diagrams from a physical point of view.
Near the Mott insulator, the important ingredient are spin fluctuations, which can be taken into account by ladder-type diagrams.
Indeed, the ladder approximation gives a magnetic spectrum which exhibits low-energy spin excitations (so-called paramagnons), as a consequence of strong AFM fluctuations~\cite{Hafermann09, Hafermann-book}.
In the following, we present the self-energy formula in the ladder approximation, stressing on the SU(2) symmetry for the spin indices.
We first evaluate the renormalized vertex $\Gamma$ collecting successive particle-hole excitations.
Since the propagator $\tilde{G}$ is independent of the spin component, the vertex $\Gamma^{\alpha}$ for each channel $\alpha=\text{ch}, \text{sp}$ independently obeys the Bethe-Salpeter equation:
\begin{align}
\Gamma_{\omega\omega';\nu\bm{q}}^{\alpha}
=\gamma_{\omega\omega'; \nu}^{\alpha}
+ T \sum_{\omega''}
\gamma_{\omega\omega''; \nu}^{\alpha}
\tilde{\chi}^0_{\omega''; \nu\bm{q}}
\Gamma_{\omega''\omega'; \nu\bm{q}}^{\alpha},
\label{eq:Gamma}
\end{align}
where $\tilde{\chi}^0$ is defined by
\begin{align}
\tilde{\chi}^0_{\omega; \nu \bm{q}}
= -\frac{1}{N} \sum_{\bm{k}}
\tilde{G}_{\omega \bm{k}} \tilde{G}_{\omega+\nu, \bm{k}+\bm{q}}.
\end{align}
Figure~\ref{fig:diagrams}(c) shows the diagrammatic representation of the above equation.
The self-energy is evaluated with the renormalized vertex $\Gamma$ as follows:
\begin{align}
\tilde{\Sigma}_{\omega\bm{k}}
&= -\frac{T}{N} \sum_{\omega'\bm{k}'}
\gamma^{\rm ch}_{\omega\omega'; 0}
\tilde{G}_{\omega'\bm{k}'}
\nonumber \\
&+ \frac{T}{4N} \sum_{\nu\bm{q}}
\tilde{G}_{\omega+\nu, \bm{k}+\bm{q}}
( V^{\rm ch} + 3V^{\rm sp})_{\omega\omega; \nu\bm{q}},
\label{eq:self-ladder}
\end{align}
where the effective interaction $V^{\alpha}$ is defined by
\begin{align}
V^{\alpha}_{\omega\omega'; \nu\bm{q}}
&=
T\sum_{\omega''}
\gamma^{\alpha}_{\omega\omega''; \nu} \tilde{\chi}^0_{\omega'';\nu\bm{q}}
\left[ 2\Gamma^{\alpha}_{\omega''\omega'; \nu\bm{q}}
-\gamma^{\alpha}_{\omega''\omega'; \nu} \right].
\label{eq:V_phi}
\end{align}
The corresponding diagram for the dual self-energy is shown in Fig.~\ref{fig:diagrams}(b).
\subsection{A perspective from a $1/d$ expansion}
The momentum dependence of the self-energy disappears in the limit of $d=\infty$ dimensions.
This means that DMFT provides the exact solution of fermionic models in $d=\infty$~\cite{Metzner89, Georges96}.
Since the dual-fermion approach offers an expansion around DMFT, it is reasonable to classify diagrams in terms of $1/d$. In this view, we reconsider the self-energy diagram presented above.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.95\linewidth]{diagrams-aux.eps}
\end{center}
\caption{Self-energy diagrams: (a) Example of a leading order diagram in terms of $1/d$ but not included in the present approximation for physical reasons (see text), (b) an example of the next-leading contributions in the $1/d$ expansion, and (c) zero contribution by the self-consistency condition.}
\label{fig:diagrams-aux}
\end{figure}
We consider the large-$d$ limit with the scaling $t \propto 1/\sqrt{d}$~\cite{Metzner89, Georges96}.
The local Green's function $G_{\bm{r}=0}$ of the original electrons is of zeroth order in $1/d$, while its dual counterpart vanishes, $\tilde{G}_{\bm{r}=0}=0$, by the self-consistency condition given later (we omit the $\omega$ index for simplicity).
Hence, the dual Green's function has only intersite components $\tilde{G}_{\bm{r}\neq0}$ which scale as $\tilde{G}_{\bm{r}\neq0} \sim G_{\bm{r}\neq0} \sim {\cal O}(1/\sqrt{d})$.
The second-order diagram for $\tilde{\Sigma}_{\omega\bm{k}}$ (the second term in Fig.~\ref{fig:diagrams}(b) with the renormalized vertex replaced by the bare interaction) has a contribution of order ${\cal O}(1/\sqrt{d})$.
The ladder diagrams summed up in Fig.~\ref{fig:diagrams}(c) are of the same order, because the factor $1/d$ arising from the two propagators is canceled by the lattice summation.
Therefore, all diagrams included in the second-term of Fig.~\ref{fig:diagrams}(b) provide the leading contributions of order $1/\sqrt{d}$.
Actually, ladder diagrams in the particle-particle channel also have contributions of the same order (e.g., the diagram in Fig.~\ref{fig:diagrams-aux}(a)). However, we may neglect them since particle-particle fluctuations only have a minor effect in the doped Mott insulator.
Diagrams containing higher-order vertices only appear at second-to-leading order [e.g., Fig.~\ref{fig:diagrams-aux}(b)], since the local diagram like Fig.~\ref{fig:diagrams-aux}(c) vanishes as explained later.
It means that the three-body and higher-order interactions do not enter to leading order of the $1/d$ expansion.
Indeed, it has been numerically confirmed that the ladder-type diagrams dominate over diagrams built from the three-particle vertex~\cite{Hafermann09}.
In conclusion, the dual-ladder self-energy in Eq.~(\ref{eq:self-ladder}) constitutes the leading correction to the DMFT around the $d=\infty$ limit.
\subsection{Self-consistency condition}
So far, the hybridization function $\Delta_{\omega}$ is arbitrary.
We discuss here how to determine $\Delta_{\omega}$.
The condition is that the scheme should reduce to DMFT if no self-energy corrections are taken into account: $\tilde{\Sigma}_{\omega\bm{k}}=0$.
The following self-consistency condition fulfills this requirement~\cite{Rubtsov08, Rubtsov09, Hafermann-book}:
\begin{align}
\sum_{\bm{k}} \tilde{G}_{\omega\bm{k}} = 0.
\end{align}
It is clear from Eq.~(\ref{eq:G0_dual}) that when $\tilde{\Sigma}_{\omega\bm{k}}=0$, this condition leads to the DMFT self-consistency condition.
Furthermore, it eliminates the contribution from the Hartree diagram (the first term in Fig.~\ref{fig:diagrams}(b)). Similarly, all diagrams which have a propagator connecting the same site (local loops) give no contribution [e.g., Fig.~\ref{fig:diagrams-aux}(c)].
\subsection{Technical details}
We solve the effective impurity problem using the hybridization-expansion solver (CT-HYB)~\cite{Werner06} of the continuous-time quantum Monte Carlo method~\cite{Rubtsov05, Gull-RMP}.
The vertex $\gamma_{\omega\omega'; \nu}$ as well as $g_{\omega}$ are computed. We applied an efficient implementation for the vertex calculation~\cite{Hafermann12}.
The vertex $\gamma_{\omega \omega'; \nu}$ is computed in a small energy window, while the energy cutoff for $g_{\omega}$ can be taken sufficiently large ($10^3$--$10^4$ Matsubara frequency points) in the CT-HYB algorithm. To be concrete, we restrict the frequencies of $\gamma_{\omega \omega'; \nu}$ to $|\omega|, |\omega'| \leq (2n_{\rm c}+1) \pi T$ and $|\nu| \leq 2m_{\rm c} \pi T$.
Typically, we take $n_{\rm c}=m_{\rm c}=20$ for $T \gtrsim 0.1$, and up to $n_{\rm c}=m_{\rm c}=60$ for lower temperatures.
Such a small cutoff compared to the one for $g_{\omega}$ is possible because the frequency summation of $\gamma_{\omega \omega'; \nu}$ is always taken with $\tilde{G}_{\omega\bm{k}}$, which decays faster than the ordinary Green's function, $\tilde{G}_{\omega\bm{k}} \sim -\epsilon_{\bm{k}}/\omega^2$.
We note that the negative bosonic frequencies, $\nu<0$, need not be computed since we have the relation
$\gamma_{\omega \omega'; \nu}^{\alpha}=(\gamma_{-\omega -\omega'; -\nu}^{\alpha})^*$.
The quantities $g_{\omega}$ and $\gamma_{\omega\omega'; \nu}$ are plugged into the dual-lattice calculations.
The momentum summation (convolution) in Eq.~(\ref{eq:self-ladder}) is evaluated in the real space. Here we can use FFT to reduce ${\cal O}(N^2)$ calculation into ${\cal O}(N\log N)$.
On the other hand, we simply add the frequency $\nu$ in Eq.~(\ref{eq:self-ladder}), since the frequency summation does not simplify considerably in the imaginary-time domain due to the full frequency dependence of $\gamma_{\omega\omega'; \nu}$.
The lattice size $N$ is fixed at $N=32\times32$ (excepting Fig.~\ref{fig:T-afm-scaling}). This size is sufficiently large for our purpose of revealing possible phase transitions.
A larger system size is necessary to observe the critical behavior, which will be discussed in the next section.
We compute $\tilde{\Sigma}_{\omega\bm{k}}$ and $\tilde{G}_{\omega\bm{k}}$ iteratively until they are converged. To get convergence, we mix new and old data of $\tilde{\Sigma}_{\omega\bm{k}}$.
The weight of the new data ranges from 0.5 down to 0.02 by checking the tendency toward convergence.
After $\tilde{\Sigma}_{\omega\bm{k}}$ is obtained, we update the bath $\Delta_{\omega}$ and go back to the impurity problem.
We use the formula~\cite{Rubtsov09}
$\Delta_{\omega}^{\rm new}
=\Delta_{\omega}^{\rm old}
+\xi \tilde{G}_{\omega, \bm{r}=0} / [g_{\omega} ( g_{\omega} + \tilde{G}_{\omega, \bm{r}=0})]$.
Here, $\xi$ is the mixing parameter and typically we take $\xi=0.5$.
When there exist strong AFM fluctuations, i.e., near the half-filling, the iteration for $\Delta_{\omega}$ is unstable.
In this case, we need an elaborate treatment of $\tilde{\Sigma}_{\omega\bm{k}}$ to avoid the numerical instability (see Appendix~\ref{app:stabilize} for details).
\section{Antiferromagnetic Susceptibility}
\label{sec:AFM}
We first present numerical results for the AFM susceptibilities at half filling, $n=1$.
We shall show data for lower temperatures than in the previous calculation of the ladder approximation~\cite{Hafermann09}, and discuss the point of our calculations to achieve convergence in the critical regime.
The results may also be regarded as a benchmark of our calculations.
The spin and charge susceptibilities of the original electrons, $\chi_{\nu\bm{q}}^{\alpha}$, are connected to the reducible vertex part of the dual fermions by an exact relation~\cite{Li08, Brener08, Hafermann-book}.
In the ladder approximation, we use $\Gamma^{\alpha}$ in Eq.~(\ref{eq:Gamma}) for the reducible vertex to obtain explicit expression for $\chi_{\nu\bm{q}}^{\alpha}$ as
\begin{align}
\chi_{\nu\bm{q}}^{\alpha} = \chi_{\nu\bm{q}}^0
+ T^2 \sum_{\omega\omega'} X_{\omega; \nu\bm{q}} \Gamma_{\omega\omega'; \nu\bm{q}}^{\alpha} X_{\omega'; \nu\bm{q}},
\label{eq:chi_lat}
\end{align}
where
\begin{align}
\chi_{\nu\bm{q}}^0 &= -\frac{T}{N} \sum_{\omega \bm{k}}
G_{\omega\bm{k}} G_{\omega+\nu, \bm{k}+\bm{q}},
\\
X_{\omega; \nu\bm{q}} &= -\frac{1}{N} \sum_{\bm{k}}
G_{\omega\bm{k}} G_{\omega+\nu, \bm{k}+\bm{q}}
R_{\omega\bm{k}} R_{\omega+\nu, \bm{k}+\bm{q}},
\end{align}
with
$R_{\omega\bm{k}} = g_{\omega}^{-1} (\Delta_{\omega} - \epsilon_{\bm{k}})^{-1}$.
Equation~(\ref{eq:chi_lat}) reduces to the DMFT formula if the DMFT limit, $\tilde{\Sigma}_{\omega\bm{k}}=0$, is taken.
Figure~\ref{fig:T-afm}(a) shows the temperature dependence of the inverse of the static AFM susceptibility $\chi^{\rm sp}_{\nu=0, \bm{Q}}$, where $\bm{Q}=(\pi,\pi)$ is the nesting vector.
The DMFT result is also plotted for comparison.
From these data, we find that the susceptibility does not diverge in the ladder approximation, while the DMFT susceptibility diverges and obeys the Curie-Weiss law.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\linewidth]{T-afm.eps}
\end{center}
\caption{(Color online) Temperature dependence of (a) the static AFM susceptibility $\chi^{\rm sp}_{\nu=0, \bm{Q}}$ and (b) the largest eigenvalue $\lambda^{\rm sp}$ of the matrix $\hat{A}$ at half-filling $n=1$. The closed symbols and open symbols show results in the present approximation and within DMFT, respectively.}
\label{fig:T-afm}
\end{figure}
In the regime where fluctuations are strong, or more precisely, at $T \lesssim T_{\rm N}^{\rm DMFT}$ with $T_{\rm N}^{\rm DMFT}$ being the DMFT N\'eel temperature, the method presented in Appendix~\ref{app:stabilize} is essential to achieve convergence. Here we only mention that this method does not change the equations, and is simply a way of obtaining a converged solution.
A spurious divergence of $\chi^{\rm sp}$, which may arise during the iteration, is removed.
In this procedure, the main quantity we need to check is the dimensionless matrix $\hat{A}$ defined by
$(\hat{A})_{\omega \omega'} = \gamma^{\rm sp}_{\omega \omega'; 0} \tilde{\chi}^0_{\omega'; 0, \bm{Q}}$.
The condition for the divergence of $\Gamma^{\rm sp}$ in Eq.~(\ref{eq:Gamma}), and hence of $\chi^{\rm sp}$ is $\lambda^{\rm sp}=1$ where $\lambda^{\rm sp}$ denotes the largest eigenvalue of $\hat{A}$~\cite{footnote-lambda}.
The temperature dependence of $\lambda^{\rm sp}$ is shown in Fig.~\ref{fig:T-afm}(b).
It turns out that $\lambda^{\rm sp}$ approaches 1 with decreasing $T$ in the ladder approximation.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.9\linewidth]{T-afm-scaling.eps}
\end{center}
\caption{(Color online) A scaling plot: $1-\lambda^{\rm sp}$ as a function of $1/T$. Results for different system sizes, $N=32\times32$, $64\times64$ and $128\times128$, are shown for comparison. The solid lines indicate the scaling $1-\lambda^{\rm sp} \propto \exp(-\Delta/T)$.}
\label{fig:T-afm-scaling}
\end{figure}
In the critical regime, the susceptibility diverges exponentially toward $T=0$: $\chi \sim e^{\beta \Delta}$~\cite{Chakravarty88, Hasenfratz91}. It follows that $\lambda^{\rm sp}$ approaches 1 according to $1-\lambda^{\rm sp} \propto e^{-\beta \Delta}$.
In order to check this behavior, we plot $1-\lambda^{\rm sp}$ as a function of $1/T$ in Fig.~\ref{fig:T-afm-scaling}.
Results for larger system sizes, $N=64\times64$ and $128\times128$, are plotted as well.
It turns out that the data for different system sizes deviate from each other at low temperatures such that $1-\lambda^{\rm sp} \lesssim 10^{-2}$. It indicates that the slow decays for $N=32\times32$ and $64\times64$ observed at $1/T \gtrsim 7$ are artifacts due to a finite-size effect.
Apart from the finite-size effect, the results agree with the expected scaling $1-\lambda^{\rm sp} \propto e^{-\beta \Delta}$ indicated by the solid lines.
We thus conclude that our approximation correctly reproduces the N\'eel temperature of $T_{\rm N}=0$ required from the Mermin-Wagner theorem.
\section{Superconductivity}
\label{sec:SC}
\subsection{Formulas for pairing susceptibilities}
In this section, we discuss the superconductivity in the doped regime.
We first derive a formula for the pairing susceptibility of the dual fermions.
The susceptibilities of the dual fermions can be transformed to those of the original electrons~\cite{Li08, Brener08, Hafermann-book}.
Actually, numerical transformations cannot be performed in the case of unconventional (momentum-dependent) order parameters because the susceptibility matrix is too large to store in memory [see Eq.~(\ref{eq:BS-pp})].
However, since the diverging point is common to both susceptibilities, we can determine the transition temperature from the dual-fermion susceptibility without transforming to the electron susceptibility.
We consider Cooper pairs with opposite spin directions of the constituent electrons.
With a form factor $\phi_{k}$ which depends on both $\bm{k}$ and $\omega$,
the order parameter $\Phi$ is expressed as
$\Phi = \sum_k \phi_{k} \langle f_{k\uparrow}f_{-k\downarrow} \rangle_{\tilde{\cal S}}$.
The static susceptibility for this pairing is defined by
$\sum_{kk'} \phi_{k} \tilde{P}_{kk'} \phi_{k'}^*$
where
\begin{align}
\tilde{P}_{kk'}
= \langle f_{k\uparrow} f_{-k \downarrow} f^*_{-k'\downarrow} f^*_{k'\uparrow} \rangle_{\tilde{\cal S}}.
\end{align}
The Bethe-Salpeter equation for this Green's function is written as
\begin{align}
\tilde{P}_{kk'} = \tilde{P}^0_{k} \delta_{kk'}
- \frac{T}{N}\sum_{k''} \tilde{P}^0_{k} \Gamma^{\rm pp}_{kk''} \tilde{P}_{k''k'},
\label{eq:BS-pp}
\end{align}
where
\begin{align}
\tilde{P}^0_{k} = \tilde{G}_k \tilde{G}_{-k}.
\end{align}
For the irreducible vertex part $\Gamma^{\rm pp}$, we take account of effective interactions mediated by the spin and charge fluctuations. Hence, $\Gamma^{\rm pp}$ is given in terms of the renormalized vertex in Eq.~(\ref{eq:Gamma}) as~\cite{Hafermann-book,Hafermann09-2}
\begin{align}
\Gamma^{\rm pp}_{kk'}
= &-\Gamma^{\uparrow\downarrow\downarrow\uparrow}_{\omega, -\omega'; \omega'-\omega, \bm{k}'-\bm{k}}
+\Gamma^{\uparrow\downarrow\uparrow\downarrow}_{\omega, \omega'; -\omega-\omega', -\bm{k}-\bm{k}'}
\nonumber\\
&+ \gamma^{\uparrow\downarrow\downarrow\uparrow}_{\omega, -\omega'; \omega'-\omega}.
\label{eq:Gamma-pp}
\end{align}
The first term in Eq.~(\ref{eq:Gamma-pp}) incorporates the charge and longitudinal spin fluctuations, and the second term the transverse spin fluctuations. The third term subtracts their double counting.
A diagrammatic representation for $\Gamma^{\rm pp}$ is shown in Fig.~\ref{fig:pairing}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\linewidth]{vertex-pairing.eps}
\end{center}
\caption{The pairing interaction (the irreducible vertex for the pairing susceptibility) $\Gamma^{\rm pp}$ in the ladder approximation. The box with stripes stands for the renormalized vertex $\Gamma$ in Fig~\ref{fig:diagrams}(c).}
\label{fig:pairing}
\end{figure}
Without magnetic field,
the pairing susceptibility is classified according to the total spin of the pair.
For this purpose, we replace the pair operator by its symmetrized or anti-symmetrized form:
\begin{align}
f_{k\uparrow} f_{-k \downarrow} \to \frac{1}{\sqrt{2}} (f_{k\uparrow} f_{-k \downarrow} \mp f_{k\downarrow} f_{-k \uparrow}).
\end{align}
Here, $-$ corresponds to the spin singlet and $+$ to the spin triplet.
The corresponding pairing susceptibility is expressed as
\begin{align}
\tilde{P}_{k,k'}^{\pm}
= \tilde{P}_{k,k'} \pm \tilde{P}_{k,-k'}.
\end{align}
Hence, the inversion of the fermionic frequency and momentum, $k=(\omega, \bm{k}) \to -k=(-\omega, -\bm{k})$,
transforms $\tilde{P}_{kk'}^{\pm}$ as
$P^{\pm}_{k,k'} =\pm P^{\pm}_{k,-k'} =\pm P^{\pm}_{-k,k'} = P^{\pm}_{-k,-k'}$.
From Eq.~(\ref{eq:BS-pp}), we obtain the equation for $\tilde{P}_{kk'}^{\pm}$,
\begin{align}
\tilde{P}_{kk'}^{\pm} = \tilde{P}^0_{k} (\delta_{k,k'} \pm \delta_{k,-k'})
- \frac{T}{N}\sum_{k''} \tilde{P}^0_{k} \Gamma^{\rm pp\pm}_{kk''} \tilde{P}_{k''k'}^{\pm},
\label{eq:BS-pp-2}
\end{align}
where the (anti-)symmetrized vertex $\Gamma^{\rm pp\pm}_{kk'}$ is defined by
$\Gamma^{\rm pp\pm}_{kk'} = (\Gamma^{\rm pp}_{k,k'} \pm \Gamma^{\rm pp}_{k,-k'})/2$.
Their explicit expressions read
\begin{align}
\Gamma^{\rm pp+}_{kk'}
&= \frac{1}{4} \left[
(3\Gamma^{\rm sp}-\Gamma^{\rm ch})_{\omega, -\omega'; \omega'-\omega, \bm{k}'-\bm{k}}
- 2\gamma^{\rm sp}_{\omega, -\omega'; \omega'-\omega} \right]
\nonumber \\
&+ ( \omega' \to -\omega' ),
\\
\Gamma^{\rm pp-}_{kk'}
&= \frac{1}{4} \left[
-(\Gamma^{\rm sp}+\Gamma^{\rm ch})_{\omega, -\omega'; \omega'-\omega, \bm{k}'-\bm{k}}
+ 2\gamma^{\rm sp}_{\omega, -\omega'; \omega'-\omega} \right]
\nonumber \\
&- ( \omega' \to -\omega' ),
\end{align}
where $( \omega' \to -\omega' )$ is symbolic for the terms appearing before it with $\omega'$ replaced by $-\omega'$.
The dimension of the matrices is too large to solve Eq.~(\ref{eq:BS-pp-2}) numerically. We instead deal with an eigenvalue problem to determine the transition temperature and to extract the dominant pairing fluctuations.
Near the transition temperature,
we may neglect the first term in Eq.~(\ref{eq:BS-pp-2}) to obtain the linear equation
\begin{align}
\label{eq:eigen-sc}
\hat{K}^{\pm} \phi = \lambda^{\rm SC} \phi,
\quad
(\hat{K}^{\pm})_{kk'}
= -\frac{T}{N} \tilde{P}^{0}_{k} \Gamma^{\rm pp\pm}_{kk'}.
\end{align}
We can demonstrate from the explicit form of $\Gamma^{{\rm pp}\pm}_{kk'}$ that the eigenvalues $\lambda^{\rm SC}$ are purely real. The condition for the divergence of the susceptibility is $\lambda^{\rm SC}_{\rm max}=1$ with $\lambda^{\rm SC}_{\rm max}$ being the largest eigenvalue.
The corresponding eigenfunction $\phi_k$ gives the form factor of the order parameter.
\subsection{Numerical results}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.9\linewidth]{pair-sym.eps}
\end{center}
\caption{(Color online) Momentum dependence of the eigenfunctions $\phi_{\omega_0,\bm{k}}$ of Eq.~(\ref{eq:eigen-sc}) for $U=8$, $\delta=0.14$ and $T=0.1$, where $\omega_0=\pi T$.
Either the even-frequency part $\phi^{\rm even}_{\omega_0, \bm{k}}$ or the odd-frequency part $\phi^{\rm odd}_{\omega_0, \bm{k}}$ is plotted depending on which is allowed by the Pauli principle.}
\label{fig:pair-sym}
\end{figure}
We evaluated the largest eigenvalues $\lambda^{\rm SC}_{\rm max}$ of Eq.~(\ref{eq:eigen-sc}) by a kind of power method.
In this calculation, we enforced a particular spatial symmetry to pick up an eigenfunction belonging to a certain irreducible representation (see Appendix~\ref{app:pair_sym} for details).
In this way, we computed 10 types of pairings (2 spin symmetries $\times$ 5 spatial symmetries), which have the largest eigenvalue in each symmetry class.
The phase of the eigenfunction is arbitrary in the linear equation.
We determined the phase factor so that the component which has the largest absolute value becomes a real number.
Then, all components of $\phi_k$ become real.
Finally, we define even- and odd-frequency parts,
$\phi_{\omega \bm{k}}^{\rm even} = \phi_{\omega \bm{k}} + \phi_{-\omega \bm{k}}$ and
$\phi_{\omega \bm{k}}^{\rm odd} = \phi_{\omega \bm{k}} - \phi_{-\omega \bm{k}}$,
to see the frequency dependence.
We have confirmed that either $\phi^{\rm even}$ or $\phi^{\rm odd}$ vanishes to fulfill the Pauli principle, e.g., $\phi^{\rm odd}=0$ for the spin-singlet with symmetry A$_{\rm 1g}$.
We first show eigenfunctions $\phi_{\omega \bm{k}}$ obtained in the way described above.
Figure~\ref{fig:pair-sym} shows the momentum dependence of $\phi_{\omega \bm{k}}$ with the lowest Matsubara frequency, $\omega_0=\pi T$.
The main feature is that some functions have only minimal nodes required from the symmetry and the rest have additional nodes. In the A$_{\rm 1g}$ symmetry, for example, there is no node for the triplet, while a line node exists on the Fermi level for the singlet (i.e., extended s-wave symmetry, $\cos k_x + \cos k_y$).
Which type of superconductivity actually occurs is examined from the temperature dependence of $\lambda^{\rm SC}$.
It can be seen from Fig.~\ref{fig:lambda_sc} that $\lambda^{\rm SC}$ for the spin-singlet B$_{\rm 1g}$ (d$_{x^2-y^2}$) symmetry crosses 1 as expected.
The transition temperature $T_{\rm c}$ is estimated to be $T_{\rm c} \simeq 0.030$ for these parameters. The doping dependence of $T_{\rm c}$ is plotted in the phase diagram in Fig.~\ref{fig:phase-dope}.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\linewidth]{T-sc.eps}
\end{center}
\caption{(Color online) Temperature dependence of the eigenvalues $\lambda^{\rm SC}$ of Eq.~(\ref{eq:eigen-sc}) for $U=8$ and $\delta=0.14$.}
\label{fig:lambda_sc}
\end{figure}
\section{Phase separation}
\label{sec:PS}
Our next interest lies in the paramagnetic state above $T_{\rm c}$ and near the Mott insulator.
In this regime, we found an instability of the uniform charge fluctuations.
Figure~\ref{fig:mu} shows the temperature dependence of the chemical potential $\mu$ for several values of doping $\delta=1-n$ for $U=8$.
The decrease of $\mu$ below $T\simeq 1$ is due to the development of a Mott gap. At around $T=0.1$, some lines for different doping levels intersect.
It means that $\mu$ is a non-monotonic function of $\delta$ at low temperatures as shown in the inset of Fig.~\ref{fig:mu}. This behavior indicates a phase separation as explained below.
At $T=0.1$ in the inset of Fig.~\ref{fig:mu}, there exists two solutions with different doping, say $\delta_1$ and $\delta_2$. Actually, the Mott insulator with $\delta=0$ is also a solution in this case.
Hence, there are three solutions ($\delta_0=0<\delta_1<\delta_2$), two of which ($\delta_0$ and $\delta_2$) are thermodynamically stable and one ($\delta_1$) is unstable.
In order to make the average doping $\bar{\delta}$ at $0 < \bar{\delta} < \delta_2$, the system becomes spatially inhomogeneous between the Mott insulator with $\delta=0$ and the metallic state with $\delta=\delta_2$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\linewidth]{T-mu-2.eps}
\end{center}
\caption{(Color online) Temperature dependence of the chemical potential $\mu$. The doping $\delta$ is varied from 0 to 0.2 in 0.02 steps. The inset shows $\mu$ as a function of $\delta$ for fixed $T$.}
\label{fig:mu}
\end{figure}
We define the temperature $T_{\rm PS}$ for the phase separation by the point where two lines intersect in Fig.~\ref{fig:mu}.
It corresponds to the so-called spinodal point where the uniform charge susceptibility diverges.
The result for $T_{\rm PS}$ is plotted in the phase diagram of Fig.~\ref{fig:phase-dope}.
Below $T_{\rm PS}$, the homogeneous solution is thermodynamically unstable.
Before concluding the paragraph, we comment on a technical issue related to these observations.
The uniform charge susceptibility can be computed in two different ways: Either from the chemical potential as discussed above or from the correlation function as presented in Section~\ref{sec:AFM} for the spin channel. In the present approximation, the two results are not consistent.
Indeed, we found no divergence of the uniform charge susceptibility computed from the correlation function.
In this case, the one computed from the chemical potential is more reliable in the sense that derivative of the self-energy with respect to $\mu$ is taken strictly, while the correlation function incorporates only a part of the corresponding diagrams.
To improve consistency, i.e., to obtain divergence in the correlation function, we need more elaborated treatment of the irreducible vertex to satisfy the Ward identity~\cite{Hafermann-arXiv}.
\section{Unconventional density waves}
\label{sec:DW}
In the previous section, we discussed the phase separation taking place near the Mott insulator.
In this section, we examine the possibility of another phase, which has been discussed extensively, namely, the staggered flux state or the d-DW state~\cite{Chakravarty01, Kotliar88, Honerkamp02, Macridin04, Lu12}.
To make our formulation general, we consider both spin and charge channels ($\alpha=\text{sp}, \text{ch}$), arbitrary wave vectors $\bm{q}$, and arbitrary spatial symmetry.
The order parameter $\Psi^{\alpha\eta}_{\bm{q}}$ is defined by
\begin{align}
\Psi^{\alpha\eta}_{\bm{q}}
=\sum_{\omega\bm{k}\sigma}
\sigma^{\alpha}_{\sigma\sigma}
\psi_{\bm{k}}^{\eta}
\langle
f_{\omega\bm{k}\sigma}^{*}
f_{\omega\bm{k}+\bm{q}\sigma}
\rangle_{\tilde{\cal S}},
\end{align}
where $\sigma^{\rm ch}=\sigma^0$ and $\sigma^{\rm sp}=\sigma^z$.
The index $\eta$ labels different form factors $\psi_{\bm{k}}^{\eta}$.
The d-DW corresponds to $\Psi_{\bm{Q}}^{\rm ch, d}$ with the form factor
$\psi_{\bm{k}}^{\rm d}=i(\cos k_x - \cos k_y)$, while the ordinary DW is given by $\psi_{\bm{k}}^{\rm s}=1$.
In the real-space representation, the d-DW exhibits a local current
$i\langle f_{\bm{r}}^* f_{\bm{r}+\bm{x}} \rangle - i\langle f_{\bm{r}+\bm{x}}^* f_{\bm{r}} \rangle$
which aligns as in Fig.~\ref{fig:stag_flux}.
Following the same reasoning as for the pairing correlations, we consider susceptibilities of the dual fermions.
The susceptibility corresponding to $\Psi^{\alpha\eta}_{\bm{q}}$ is given by
$\sum_{kk'} \psi_{k}^{\eta} \tilde{\chi}_{kk';q}^{\alpha} (\psi_{k'}^{\eta})^*$ with
\begin{align}
\tilde{\chi}_{kk';q}^{\alpha}
=
\frac{1}{2} \sum_{\sigma\sigma'}
\sigma^{\alpha}_{\sigma\sigma} \sigma^{\alpha}_{\sigma'\sigma'}
\langle f^*_{k\sigma} f_{k+q \sigma} f^*_{k'+q\sigma'} f_{k'\sigma'} \rangle_{\tilde{\cal S}}.
\end{align}
The susceptibility formula in Eq.~(\ref{eq:chi_lat}) does not give rise to the unconventional DW, since the irreducible vertex is local in this formula.
Higher-order processes need to be taken into account.
\begin{figure}[tb]
\begin{center}
\includegraphics[scale=0.7]{stag_flux.eps}
\end{center}
\caption{The local-current configuration in the d-DW state.}
\label{fig:stag_flux}
\end{figure}
In order to see which processes enhance unconventional DWs, it is instructive to analyze influences of intersite interactions $H_{\rm int}$ in the mean-field approximation or the random-phase approximation (RPA)~\cite{Ozaki92, Ikeda98}.
We consider the nearest-neighbor repulsion $V$ and the AFM exchange interaction $J$:
\begin{align}
H_{\rm int}
= \frac{1}{2} \sum_{\langle i,j\rangle} \sum_{\sigma \sigma'}
\left[
V
c_{i\sigma}^{\dag} c_{i\sigma}
c_{j\sigma'}^{\dag} c_{j\sigma'}
+J
c_{i\sigma}^{\dag} c_{i\sigma'}
c_{j\sigma'}^{\dag} c_{j\sigma}
\right].
\end{align}
The staggered susceptibility $\chi^{\alpha\eta}_{\bm{Q}}$ in RPA is given by
\begin{align}
\chi^{\alpha\eta}_{\bm{Q}}=[(\chi^{0\eta}_{\bm{Q}})^{-1} - I_{\bm{Q}}^{\alpha\eta}]^{-1},
\end{align}
where
$\chi^{0\eta}_{\bm{Q}}=-(T/N)\sum_{\bm{k}} |\psi_{\bm{k}}^{\eta}|^2 G_{\omega\bm{k}} G_{\omega\bm{k}+\bm{Q}}$.
The sign of $I_{\bm{Q}}^{\alpha\eta}$ determines whether the fluctuations are enhanced ($I_{\bm{Q}}^{\alpha\eta}>0$) or suppressed ($I_{\bm{Q}}^{\alpha\eta}<0$).
Contributions to $I_{\bm{Q}}^{\alpha\eta}$ from $U$, $V$ and $J$ are summarized in Table~\ref{tab:RPA}~\cite{Ozaki92}.
There are two types of diagrams in the RPA: The bubble-type (upper table row) and the ladder-type (lower).
It turns out that the ladder-type diagram for $V$ and $J$ may cause an unconventional CDW and/or SDW~\cite{footnote-uDW}.
For the Hubbard model without $V$ and $J$, the local repulsion $U$ gives rise to AFM fluctuations, which effectively have an effect similar to that of the $J$-term. Hence, there is a chance that the unconventional CDW is induced by processes beyond RPA.
This spin-fluctuation mediated interaction is taken into account, for example, by the diagram in Fig.~\ref{fig:vertex-dw}(a).
This process is indeed included in the irreducible vertex derived by a functional derivative of the Luttinger-Ward functional in the FLEX~\cite{FLEX1}.
\begin{table}[t]
\begin{center}
\caption{Effect of different diagrams and interaction terms on the staggered susceptibilities in the RPA. The signs $+$ and $-$ indicate enhancement and suppression, respectively.}
\label{tab:RPA}
\begin{tabular}{c|ccc}
\hline
& $U$ & $V$ & $J$ \\
\hline
\parbox[c]{3.5cm}{\includegraphics[width=\linewidth]{rpa1.eps}}
& \parbox[c]{40pt}{$+$SDW \\ $-$CDW} & $+$CDW & \parbox[c]{40pt}{$+$SDW \\ $+$CDW}
\\
\parbox[c]{3.5cm}{\includegraphics[width=\linewidth]{rpa2.eps}}
& $+$SDW & \parbox[c]{40pt}{$+$uSDW \\ $+$uCDW} & $+$uCDW
\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[tb]
\begin{center}
\includegraphics[]{vertex-dw.eps}
\end{center}
\caption{(a) An exemplary susceptibility diagram which contributes to the unconventional CDW. (b) A diagram for the dual-fermion irreducible vertex $\Gamma'$ in Eqs.~(\ref{eq:vertex-dw-ch}) and (\ref{eq:vertex-dw-sp}). The box with stripes stands for the renormalized vertex $\Gamma$ in Fig~\ref{fig:diagrams}(c).}
\label{fig:vertex-dw}
\end{figure}
In the dual-fermion approach, the effective interaction mediated by spin fluctuations can be constructed using the renormalized vertex $\Gamma$ in Eq.~(\ref{eq:Gamma}) and Fig.~\ref{fig:diagrams}(c).
We note that $\Gamma$ contains both bubble-type and ladder-type diagrams and more if it is written with $U$, since the interaction vertex $\gamma$ is fully antisymmetrized.
Linearizing the Bethe-Salpeter equation for the static susceptibility as in the case of superconductivity, we obtain the eigenvalue equation
\begin{align}
\hat{L}^{\alpha}_{\bm{q}} \psi = \lambda^{\rm DW} \psi,
\quad
(\hat{L}^{\alpha}_{\bm{q}})_{kk'} = -\frac{T}{N} G_{\omega\bm{k}} G_{\omega,\bm{k}+\bm{q}}
\Gamma'^{\alpha}_{kk'}.
\label{eq:eigen-dw}
\end{align}
The irreducible vertex part $\Gamma'$ is given in terms of $\Gamma$ as
\begin{align}
\Gamma'^{\rm ch}_{kk'}
&= -\frac{1}{2} ( 3 \Gamma^{\rm sp} + \Gamma^{\rm ch} )_{\omega, \omega; \omega'-\omega, \bm{k}'-\bm{k}},
\label{eq:vertex-dw-ch}
\\
\Gamma'^{\rm sp}_{kk'}
&= \frac{1}{2} ( \Gamma^{\rm sp} - \Gamma^{\rm ch} )_{\omega, \omega; \omega'-\omega, \bm{k}'-\bm{k}}.
\label{eq:vertex-dw-sp}
\end{align}
Figure~\ref{fig:vertex-dw}(b) shows a diagram corresponding to the vertex $\Gamma'$.
The matrix $\hat{L}$ in Eq.~(\ref{eq:eigen-dw}) is non-hermitian so that the eigenvalues $\lambda^{\rm DW}$ are complex numbers in general. By numerical calculations, we found that $\lambda^{\rm DW}$ consists of purely real numbers as well as complex numbers.
We have computed eigenvalues which has the largest real part by means of the Arnoldi method~\cite{Arpack}.
As in the case of superconductivity, we obtained 5 eigenvalues with different spatial symmetries for each channel $\alpha=\text{sp, ch}$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=\linewidth]{T-dw_stag.eps}
\end{center}
\caption{(Color online) Temperature dependence of the eigenvalues $\lambda^{\rm DW}$ of Eq.~(\ref{eq:eigen-dw}) with $\bm{q}=\bm{Q}$. The parameters are $U=8$ and $\delta=0.08$.}
\label{fig:T-dw}
\end{figure}
Figure~\ref{fig:T-dw} shows thus obtained eigenvalues for $\bm{q}=\bm{Q}$ as a function of temperature.
The largest fluctuation is the ordinary antiferromagnetism (sp-A$_{\rm 1g}$) as expected. The cusp around $T=0.1$ is due to a cross of eigenvalues between a purely real number on the high-$T$ side and a complex number on the low-$T$ side. On the other hand, at low temperatures, there is a significant enhancement of the leading eigenvalue corresponding to the eigenfunction with B$_{\rm 1g}$ (d$_{x^{2}-y^{2}}$) symmetry in the charge channel. This fluctuation corresponds to the staggered flux state or the d-DW state. This eigenvalue may exceed the one in the AFM spin channel at lower temperatures. An extrapolation of $\lambda^{\rm DW}$ to lower temperature indicates that the transition to the d-DW state takes place at $T\sim 0.01$.
However, the d-SC has a larger transition temperature as shown in Fig.~\ref{fig:phase-dope}. Therefore, the d-DW state actually does not realize in our approximation. We did not find any parameter regime where the d-DW state is superior to d-SC in the parameters we examined.
\section{Discussions and Summary}
\label{sec:summary}
We have applied the dual-fermion approach to the two-dimensional Hubbard model.
The AFM fluctuations have been taken into account by the ladder diagrams, which constitute the leading correction to the DMFT in terms of a $1/d$ expansion.
Practically, there was a convergence issue which prevents calculations below the mean-field critical temperature.
By solving this technical problem, we are able to extend the applicability of the approach to significantly lower temperatures.
Possible phase transitions have been investigated from susceptibilities in both the particle-hole and particle-particle channels.
For both calculations, we used a physically equivalent irreducible vertex representing spin-fluctuation mediated interactions.
Thus, we compared fluctuations of spin/charge DW and singlet/triplet superconductivity for all spatial symmetries including d-SC and d-DW.
The conclusion obtained is summarized in the phase diagrams in Section~\ref{sec:overview}.
The leading instability under doping is the d-SC as expected.
The d-DW fluctuations also show a clear tendency toward divergence at low temperatures.
However, the estimated transition temperature for the d-DW is below the $T_{\rm c}$ of d-SC, for all parameters considered.
Our result hence supports the absence of a d-DW, which was yielded by several approximations~\cite{Honerkamp02, Macridin04, Lu12, Yokoyama-commun}.
At temperatures above $T_{\rm c}$ and low doping, we observed phase separation between the Mott insulator with $\delta=0$ and metallic region with $\delta\neq0$.
The existence of the phase separation agrees with other numerical calculations~\cite{Capone06,Macridin06,Aichhorn07, Yokoyama13, Misawa-arXiv}, but conflicts with QMC results~\cite{Moreo91,Becca00}.
Provided that the instability is not an artifact, a possible reason for the discrepancy is the system size: We employ $N=32 \times 32 = 1024$ lattice sites, while the QMC calculations were performed for smaller size, $N<200$, because of a sign problem.
The region of the phase separation extends to $\delta \simeq 0.15$ for $U=8$, and therefore a pure d-SC occurs only in the limited region $0.15 \lesssim \delta \lesssim 0.18$.
This estimation is in quantitative agreement with the cluster DMFT~\cite{Capone06}.
The dual-fermion approach is complemental to the cluster DMFT among theories based on DMFT.
The ladder approximation in the dual fermion, in particular, aims at incorporating long-range fluctuations, while the cluster DMFT incorporates only short-range correlations.
Hence, it would be informative to summarize consistency and inconsistency between those results to clarify characteristics of two complemental approaches.
The instabilities reported from the cluster DMFT are consistent with ours: The phase separation as well as d-SC take place under doping~\cite{Capone06,Macridin06,Aichhorn07} and the d-DW is predominated by the d-SC~\cite{Macridin04}.
What can be reproduced by the dual-fermion approach but not by cluster DMFT is the critical behavior of the susceptibilities~\cite{Antipov14}, since a feedback of low-energy two-particle excitations to the self-energy is essential for it.
Our results for the AFM susceptibility exhibit a strong departure from the Curie-Weiss law, and are consistent with a critical temperature of $T_{\rm N}=0$ expected from the Mermin-Wagner theorem.
This aspect will be considered in more detail elsewhere.
The short-range correlations, on the other hand, play an important role near the Mott insulator.
Its influence may arise in the doping dependence of $T_{\rm c}$.
In our dual-fermion calculations, it turned out that $T_{\rm c}$ computed by neglecting the phase separation, namely, computed with the thermodynamically unstable solution, show no downturn as approaching the Mott insulator from finite doping.
In contrast, the cluster DMFT yields the dome shape of the d-SC phase~\cite{Lichtenstein00,Gull13}.
Further development beyond the dual-fermion approach has recently been attempted~\cite{Rubtsov12}.
The so-called dual boson theory introduces a bosonic counterpart of the dual fermion for the purpose of treating intersite interactions beyond mean-field theory~\cite{Loon-arXiv2} and collective excitations~\cite{Loon-arXiv1, Hafermann-arXiv}.
Furthermore, we expect that the dual boson in the spin channel yields formation of a intersite singlet, resulting in a reduction of $T_{\rm c}$ near the Mott insulator.
This effect may be brought about by the coupling between spins and the vector bosonic field, which can be treated exactly by the recently developed algorithm~\cite{Otsuki13} based on the CT-QMC method.
\section*{Acknowledgments}
We acknowledge useful discussions with Y. Kuramoto, H. Yokoyama, H. Tsunetsugu, N. Tsuji, and M. Kitatani.
A part of the computations was performed in the ISSP Supercomputer Center, the University of Tokyo.
Two of us (JO and HH) acknowledge hospitality of the ISSP during the NHSCP2014 workshop.
This work was supported by JSPS KAKENHI Grant Number 26800172.
HH acknowledges support from the FP7/ERC, under Grant Agreement No. 278472-MottMetals.
|
1,108,101,565,674 | arxiv | \section{Introduction}
\label{section:introduction}
Radio relics appear as diffuse synchrotron radio sources in the peripheral regions of some merging or cool-core clusters, typically showing an elongated morphology with a linear extent and strong polarization up to the 30\% level in integrated linear measurement \citep{Feretti, Brunetti2014_CReReview, vanweeren_review}.
It has been found that about 30\% of the observed relics appear in pairs in merging clusters, which are located on the opposite sides of the cluster core along the merger axis, such as the double relics in Abell 3667, Abell 3376, and ZWCl 2341.1+0000 \citep{Hindson2014_A3667,George2015_3367,Weeren2009_zwcl2341}.
The radio spectral index of a typical relic tends to stay constant along the length and steepen gradually along the width \citep{DiGennaro2018_Sausage,Ensslin1998_RR}, which is presumably the result of the decay of relativistic electrons behind the shock front.
Since the first detection (the radio source 1253+275 in the Coma cluster; \citealt{Giovannini_1985_Coma, Giovannini_1991_Coma}), 50 radio relics have been detected in the range of 100 MHz to 1.5 GHz, and 25 of them have been confirmed spatially coincident with shocks caused by either a minor or a major merger \citep{Feretti}.
Different from radio halos, which are exclusively discovered in massive systems \citep{vanweeren_review}, radio relics are hosted by clusters with a wide range of mass.
The largest linear size (LLS) of a relic often exceeds 1 Mpc (e.g., A115, A1240, and CIZAJ2242.8+5301; \citealt{Botteon2016a_A115, Hoang2018,Storm2018}), and the LLS of the newly detected relic in CIG 0217+70 even reaches about 3.5 Mpc \citep{Hoang2021}, making it the most extended one ever found.
Among the models proposed to explain the origin of radio relics, the ones based on diffuse shock acceleration (DSA), which was introduced by \citet{Ensslin1998_RR} who assumed that electrons producing the synchrotron radio emission are accelerated by merger-induced shocks $in\ situ$, seems to be most persuasive because they can naturally explain the elongated morphology of the relics, the relic-merger connection and the co-existence of X-ray shocks and relics in, e.g., Abell 115 \citep{Botteon2016a_A115}, RX J0603.3+4212 \citep{Ogrean_2013_toothbrush} and 1E 0657-56 \citep{Shimwell2014_Bullet}). During a typical DSA process electrons can increase their energy via the first-order Fermi acceleration, i.e., electrons trapped in a shock zone gain significant energy through multiple reflections when they encounter moving magnetic inhomogeneities \citep{Fermi_1949} with a rate proportional to their energy, which yields a power-law and time-independent electron energy spectrum. Detailed descriptions of the DSA theory can be found in \citet{Blandford_1987_DSA_review,Drury_1983} and \citet{Malkov_2001}. Recently the term ``standard DSA theory'' has been introduced by \citet{Botteon2020} to refer to the specific case in which only thermal electrons participate in the DSA process, which has been frequently applied in both observational and numerical studies.
Although the scenario based on the DSA has been supported by increasing observational evidence, some problems remain unsolved. One of the most severe challenges is that within the frame of standard DSA theory, most observed shocks seem to be too weak to generate detectable radio emission. For example, in the study of a sample of 10 radio relics associated with X-ray shocks, \citet{Botteon2020} found that in order to reproduce the measured radio luminosities of the relics over 10\% of the shock energy is needed to be transferred to the electrons, if all the accelerated electrons are extracted from the thermal pool. Apparently this acceleration efficiency is unreasonably high, and can be mitigated only by introducing other mechanisms.
\citet{Vazza2020} also used the standard DSA model to simulate radio relics at 150 MHz/1.4 GHz and found that the radio radiation are powerful enough to reconcile the observed $P_{\mathrm{1400}}-M_{\mathrm{vir}}$ relation, but the acceleration efficiency adopted in calculation ($\eta_{\mathrm{e}}=1\%$) is much higher than estimations from other works (e.g., \citealt{Hoeft2007_RRsimu, Kang2017}).
Another long-standing challenge is how to explain the inconsistency between the X-ray Mach number $\mathcal{M}_{\mathrm{X}}$, which is obtained by measuring the differences of both X-ray surface brightness and X-ray gas temperature across the shock front, and the radio Mach number $\mathcal{M}_{\mathrm{radio}}$, which is estimated based on the relation between the shock strength and the radio spectral index assuming the standard DSA theory \citep{Brunetti2014_CReReview}. For most relics $\mathcal{M}_{\mathrm{X}} < \mathcal{M}_{\mathrm{radio}}$ is found \citep{vanweeren_review}.
Both challenges can be mitigated by adding the fossil relativistic electrons into the shock acceleration model, which is a population of preexisting mildly relativistic electrons that have been previously accelerated by, e.g., merger-induced shocks, active galactic nuclei (AGN), and/or turbulence and then experienced a period time of decay \citep{Kang2012b, Pinzke2013, Bonafede_2014, Shimwell2015, VanWeeren2017_AGN}. When the fossil relativistic electrons encounter a shock, they are advected into the shock from upstream. Thermal electrons, on the other hand, are injected isotropically from the environment \citep{Drury_1983}.
Compared with thermal electrons, fossil relativistic electrons become sufficiently energetic after reacceleration and are capable of producing detectable emissions in a relatively broader band even in a relatively weak shock \citep{Kang_2002}. Furthermore, when fossil electrons are taken into account in the model, the calculated $\mathcal{M}_{\mathrm{radio}}$ tends to reflect more about the characteristics of fossil electrons than those of shocks (i.e., the dynamic properties of the intracluster medium (ICM); \citet{vanweeren_review}; see also Section~\ref{section:case_fossil}), which helps explain its deviation from $\mathcal{M}_{\mathrm{X}}$.
\edited{It is worth noting that besides the potential contribution of fossil electrons, the difference between $\mathcal{M}_{\mathrm{X}}$ and $\mathcal{M}_{\mathrm{radio}}$ might also come from the different parts of the underlying Mach number distribution they follow \citep{Hong2015}, or the fact that $\mathcal{M}_{X}$ is more sensitive to the relic's orientation than $\mathcal{M}_{\mathrm{radio}}$ \citep{Wittor2021_Mach_number}.
}
In this work we attempt to establish a new semi-analytical model to investigate the origin and evolution of radio relics by taking into account the contributions of fossil relativistic electrons in a shock propagation model.
We also investigate the properties of the radio relics simulated in a \edited{$20^{\circ} \times 20^{\circ}$} sky patch and estimate their influence on the low frequency radio observations in the future.
Our model is built upon the following three assumptions (a) only two clusters participate in each merger, (b) during each merger only one pair of merger shocks and up to one pair of radio relics are generated, and (c) fossil relativistic electrons are randomly distributed in the regions where the shocks are observed.
This paper is organized as follows.
In Section~\ref{section:RR_ob_samples} we describe the sample of observed relics, which is built based on the data quoted from literature, used to constrain the semi-analytical model.
In Section~\ref{section:Method} we present the method to establish the semi-analytical model.
In Section~\ref{section:results} we constrain the parameters in the model using the observed relic sample, simulate the radio relics population in the \edited{$20^{\circ} \times 20^{\circ}$} sky field, and study the properties of the simulated radio relics.
In Section~\ref{section:discussion} we discuss the possibility of AGNs acting as the source of seed relativistic electrons,
the influence of radio relics as a contaminating foreground component on the detection of the Epoch of Reionization (EoR) signals in future observation, as well as
the contribution of Coulomb collision
as an energy loss mechanism for the relativistic electrons.
Finally, we summarize our results in Section~\ref{section:conclusion}. Throughout this work we adopt a flat $\Lambda$CDM cosmology with \edited{$H_{0}=100h\,\mathrm{km}\,\mathrm{s}^{-1}\, \mathrm{Mpc}^{-1}=71\,\mathrm{km}\,\mathrm{s}^{-1}\, \mathrm{Mpc}^{-1}$}, $\Omega_{\mathrm{m}} = 0.27$, $\Omega_{\Lambda}=0.73$, $\Omega_{b}=0.046$, $n_{s}=0.96$ and $\sigma_{8}=0.81$.
\section{Observation Sample of Radio Relics}
\label{section:RR_ob_samples}
In order to constrain the properties of fossil electrons in the model to be outlined in Section~\ref{section:Method}, we select a sample of well-studied radio relics, which satisfy the following criteria: (1) a shock is confirmed in the X-ray observations at the position of the relic, (2) the relic and the shock are tightly associated with each other, and (3) there are reliable X-ray and radio observation studies of the shock and the relics.
\edited{
We list the properties of the selected radio relics along with those of their corresponding hosting clusters in Table~\ref{table:relics_samples}, and present the Mach numbers $\mathcal{M}_{\mathrm{X}}$ and $\mathcal{M}_{\mathrm{radio}}$ in Table~\ref{table:shock_samples}.
Within the samples we select, the well-known double radio relics in A3667 have integrated radio spectral indices $\lesssim 1$ \citep{Hindson2014_A3667}, which cannot be explained by DSA theory and makes it impossible to calculate the corresponding $\mathcal{M}_{\mathrm{radio}}$ in our model. This might come from the large observation error or the fact that turbulence in the post-shock region alleviates the radiation cooling \citep{Kang2017}. Therefore, we do not include these two radio relics in our following simulation.}
\begin{table*}
\caption{Sample of the observed radio relics}
\label{table:relics_samples}
\begin{tabular}{cccccccccc}
\hline
Name & Position $^{\rm a}$ & Redshift & $M_{\mathrm{vir}}$ $^{\rm b}$ & $\log(P_{\nu})$ $^{\rm c}$ & $\nu$ $^{\rm d}$ & \edited{$r_{\mathrm{relic}}$} $^{\rm e}$ & LLS$^{\rm f}$ & \edited{$\log(\mathcal{P}_{\mathrm{CRe}}/\mathcal{P}_{\mathrm{total}})$} $^{\rm g}$ & $\log(f_{\text{N,fo}}/f_{\text{N,th}})$ $^{\rm h}$\\
--- & --- & --- & M$_{\odot}$ & W/Hz & MHz & Mpc & Mpc & --- & --- \\
\hline
A115 & N & 0.197 & $1.05\times 10^{15}$ & 25.81 & 1400 & 1.32 & 2.44 &\edited{$-4.02$} & \edited{$+0.50$} \\
A521 & SE & 0.253 & $1.71\times 10^{15}$ & 24.41 & 610 & 0.93 & 1.00 &\edited{$-6.51$} & \edited{$-1.40$} \\
A1240 & N & 0.159 & \edited{${5.39\times 10^{14}}^{\dagger}$} & 23.59 & 1400 & 0.70 & 0.65 & \edited{$-5.58$} & \edited{$-0.57$}\\
A1240 & S & 0.159 & \edited{${5.39\times 10^{14}}^{\dagger}$} & 23.81 & 1400 & 1.10 & 1.25 &\edited{$-4.63$} & \edited{$+0.29$}\\
A2255 & NE & 0.081 & $7.90\times 10^{14}$ & 23.25 & 1400 & 0.90 & 0.70 &\edited{$-5.80$} &\edited{$-0.78$}\\
A2744 & NE & 0.308 & $1.39\times 10^{15}$ & 23.42 & 1400 & 1.56 & 1.62 & \edited{$-5.01$} &\edited{$0.25$} \\
A3376 & E & 0.046 & $4.74\times 10^{14}$ & 23.79 & 1400 & 0.52 & 0.95 & \edited{$-5.88$} & \edited{$-0.84$}\\
A3376 & W & 0.046 & $4.74\times 10^{14}$ & 23.88 & 1400 & 1.43 & 0.80 &\edited{$-4.40$} &\edited{$0.00$}\\
A3667 & N & 0.056 & $8.00\times 10^{14}$ & 25.21 & 1400 & 2.05 & 1.86 &\edited{$--^{\rm i}$} & \edited{$--^{\rm i}$}\\
A3667 & S & 0.056 & $8.00\times 10^{14}$ & 24.12 & 1400 & 1.36 & 1.30 & \edited{$--^{\rm i}$} & \edited{$--^{\rm i}$} \\
1E 0657-5655 & E & 0.296 & $1.81 \times 10^{15}$ & 22.20 & 2100 & 1.00 & 0.93 & \edited{$-7.39$} &\edited{$-2.07$} \\
ACT-CL J0102-4915 & NW & 0.870 & $2.16\times 10^{15}$ & 24.56 & 610 & 1.10 & 0.56 &\edited{$-4.94$} & \edited{$0.00$}\\
CIZA J2242.8+5301 & N & 0.192 & $2.50\times 10^{15}$ & 24.41 & 1400 & 1.57 & 1.70 &\edited{$-6.50$} &\edited{$-1.98$}\\
CIZA J2242.8+5301 & S & 0.192 & $2.50\times 10^{15}$ & 23.06 & 1400 & 1.06 & 1.45 & \edited{$-7.35$} & \edited{$-1.88$} \\
RXCJ1314-2515 & W & 0.247 & $9.70\times 10^{14}$ & 24.57 & 325 & 0.55 & 1.10 &\edited{$-6.89$} &\edited{$-2.03$}\\
RX J0603.3+4212 & N & 0.225 & $1.00\times 10^{15}$ & 25.78 & 1400 & 1.00 & 1.87 &\edited{$-4.28$} &\edited{$+0.41$}\\
\hline
\end{tabular}
\begin{justify}
$^{\rm a}$ The position of the radio relics in the cluster.\\
$^{\rm b}$
\edited{The virial mass $M_{\mathrm{vir}}$ are quoted from \citet{PlanckCollaboration2014}, \citet{Botteon2020}, \citet{Botteon2020_A2255}, \citet{Jee_2015}, and \citet{Jee_2016}, with the exception of Abell 1240 marked with an $\dagger$, which is calculated based on the method presented in \citet{Zhu2016}.}\\
$^{\rm c,d,e,f}$ Radio powers $P_{\nu}$ and the corresponding frequencies $\nu$, distances of the relic to the cluster center $R_{\mathrm{relic}}$, and LLS \citep{Feretti,Botteon2020}.\\
$^{\rm g}$ Ratios of the pressure generated by fossil relativistic electrons $P_{\mathrm{CRe}}$ to the total pressure $P_{\mathrm{total}}$ (see Section \ref{section:case_fossil}).\\
$^{\rm h}$ Density ratios between fossil electrons $f_{\mathrm{N,fo}}$ and thermal electrons $f_{\mathrm{N,th}}$ at $p = p_{\mathrm{inj}}$ (see Section~\ref{section:case_fossil}).\\
\edited{$^{\rm i}$ The pressure and density ratio for the radio relics in Abell 3667 are not provided since their integrated radio spectral indices $\lesssim 1$ \citep{Hindson2014_A3667}, which cannot be explained by DSA theory and makes it impossible to calculate the corresponding $\mathcal{M}_{\mathrm{radio}}$ and do the following simulation based on our model.}
\end{justify}
\end{table*}
\begin{table*}
\caption{Mach numbers of the observed radio relics}
\label{table:shock_samples}
\setlength{\tabcolsep}{13pt}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{cccccc}
\hline
Name & Position & $\mathcal{M}_{\mathrm{X}}$ &
$\mathcal{M}_{\mathrm{radio}}$ & Reference (X-ray) & Reference (radio) \\
\hline
A115 & N & $1.87_{-0.4}^{+0.5}$ & \edited{${\sim 4.58}^{*}$} & \citet{Botteon2016a} & \citet{Govoni2001} \\
A521 & SE & $2.13_{-1.13}^{+1.13}$ & $2.33_{-0.04}^{+0.05}$ & \citet{Botteon2020} & \citet{Macario2013} \\
A1240 & N & \edited{${\sim 2}^{*}$} & $2.3_{-0.1}^{+0.1}$ & \citet{Hoang2018} & \citet{Hoang2018}\\
A1240 & S & \edited{${\sim 2}^{*}$} & $2.4_{-0.1}^{+0.1}$ & \citet{Hoang2018} & \citet{Hoang2018}\\
A2255 & NE & $1.36_{-0.16}^{+0.16}$ & \edited{$2.77_{-0.35}^{+0.35}$} & \citet{Akamatsu2017} & \edited{\citet{Pizzo2009_A2255}} \\
A2744 & NE & $1.7_{-0.3}^{+0.5}$ & $2.05_{-0.19}^{+0.31}$ & \citet{Hattori2017} &\citet{Pearce2017} \\
A3376 & E & $1.5_{-0.1}^{+0.1}$ & $2.53_{-0.23}^{+0.23}$ & \citet{Urdampilleta2018} &\citet{George2015} \\
A3376 & W & $2.94_{-0.6}^{+0.6}$ & $3.57_{-0.58}^{+0.58}$ &\citet{Akamatsu2012} & \citet{George2015} \\
A3667 & N & $1.68_{-0.16}^{+0.16}$ &\edited{$--^{\rm \star}$} &\citet{Finoguenov2010} &\citet{Hindson2014}\\
A3667 & S & $1.75_{-0.13}^{+0.13}$ &\edited{$--^{\rm \star}$} & \citet{Akamatsu_and_Kawahara2013} & \citet{Hindson2014} \\
1E 0657-5655 & E & $1.87_{-0.13}^{+0.16}$ & $2.01_{-0.14}^{+0.19}$ & \citet{Botteon2016a} & \citet{Shimwell2015} \\
ACT-CL J0102-4915 & NW & $2.78_{-0.38}^{+0.63}$ & $2.53_{-0.41}^{+1.04}$ & \citet{Botteon2020} & \citet{Botteon2020} \\
CIZA J2242.8+5301 & N & $2.7_{-0.4}^{+0.7}$ & $4.58_{-0.19}^{+0.19}$ & \citet{Akamatsu2015} & \citet{Storm2018} \\
CIZA J2242.8+5301 & S & $1.7_{-0.3}^{+0.4}$ & $1.9_{-0.08}^{+0.08}$ & \citet{Akamatsu2015} & \citet{Hoang2017} \\
RXCJ1314-2515 & W & $1.7_{-0.28}^{+0.40}$ & $3.18_{-0.45}^{+0.87}$ & \citet{Botteon2020} &\citet{George2017} \\
RX J0603.3+4212 & N & $1.7_{-0.42}^{+0.41}$ & $3.78_{-0.2}^{+0.3}$ & \citet{Ogrean2013} & \citet{Rajpurohit2017} \\
\hline
\end{tabular}%
\edited{
\begin{justify}
$^{\rm \ast}$ The uncertainties of $\mathcal{M}_{\mathrm{radio}}$ for A115 and $\mathcal{M}_{\mathrm{X}}$ for A1240 are not provided in the corresponding reference.\\
$^{\rm \star}$ The $\mathcal{M}_{\mathrm{radio}}$ are not given for the radio relics in Abell 3667 since their integrated radio spectral indices $\lesssim 1$ \citep{Hindson2014_A3667}, which cannot be explained by DSA theory and makes it impossible to calculate the corresponding $\mathcal{M}_{\mathrm{radio}}$.
\end{justify}}
\end{table*}
\section{Semi-Analytic Model}
\label{section:Method}
\subsection{Cluster Model}
\label{section:Cluster_model}
\subsubsection{Mass Function and Merger History}
\label{section:PS}
\edited{Closely following \citet{Li2019} and \citet{Vazza2020}}, we employ a standard method to simulate the mass function of galaxy clusters by adopting Press-Schechter formalism \citep{PStheory_1974} and the cold dark matter (CDM) models, according to which the number of clusters with virial mass between $M$ and $M+dM$ per unit of comoving volume at redshift $z$ is
\begin{equation}
\begin{gathered}
n(M, z) d M=\sqrt{\frac{2}{\pi}} \frac{\langle\rho\rangle}{M} \frac{\delta_{\text{c}}(z)}{\sigma^{2}(M)}\left|\frac{d \sigma(M)}{d M}\right|\\ \times \exp \left[-\frac{\delta_{\text{c}}^{2}(z)}{2 \sigma^{2}(M)}\right] d M,
\end{gathered}
\end{equation}
where $\left<\rho\right>$ is the average density of our universe at present, $\delta_{\mathrm{c}}(z)$ is the critical linear overdensity at redshift $z$, and $\sigma(M)$ is the current root-mean-square (rms) density fluctuation within a sphere enclosing a total mass of $M$. We choose to use the power-law expression of $\sigma(M)$ considering the mass range of typical clusters \citep{Sarazin2002,Randall_2002}
\begin{equation}
\sigma(M)=\sigma_{8}\left(\frac{M}{M_{8}}\right)^{-\alpha},
\end{equation}
where $\sigma_{8}$ is the rms density fluctuation on the scale of $8\,h^{-1}$Mpc, $M_{8} = \left(4\pi/3\right)\left(8\,h^{-1}\mathrm{Mpc}\right)^{3}\left<\rho\right>$ is the total mass enclosed in a sphere with a radius of $8\,h^{-1} \mathrm{Mpc}$, and $\alpha = (n+3)/6$ with $n=-7/5$ is related to the primordial power spectrum.
\edited{We use a maximum redshift cutoff $z_{\mathrm{max}} = 4$ with the interval $\Delta z = 0.01$. We assume this redshift range is large enough considering the furthest radio relic observed by now is at $z=0.870$ (ACT-CL J0102-4915; \citealt{Lindner_2014}) and our simulation results in Section~\ref{section:results} show all the detectable relics are located at $z<2$.}
We use the extended Press-Schechter theory outlined in \citet{ExtendedPS1993} to describe the merger history of clusters. Given its present mass and redshift for each cluster, Monte-Carlo simulation is run to trace the mass growth history by identifying merger events and generating a specific merger tree. We set the minimum mass change in a merger to be $\Delta M_{\mathrm{c}}=2 \times 10^{13}\,\text{M}_{\odot}$, thus a mass growth with $\Delta M \leq \Delta M_{\mathrm{c}}$ is treated as an accretion event rather than a merger.
Now consider a cluster that has a mass of $M_{\mathrm{t2}}$ at time $t_{2}$. The probability that this cluster has a mass $M_{\mathrm{t1}}$ $(M_{\mathrm{t1}}<M_{\mathrm{t2}})$ at an earlier time $t_{1}$ $(t_{1} < t_{2})$ is
\begin{equation}
\begin{gathered}
\operatorname{Pr}\left(M_{\text{t1}}, t_{1} \mid M_{\text{t2}}, t_{2}\right) d M_{\mathrm{t1}}=\frac{1}{\sqrt{2 \pi}} \frac{M_{\mathrm{t2}}}{M_{\mathrm{t1}}} \frac{\delta_{\text{c1}}-\delta_{\text{c2}}}{\left(\sigma_{1}^{2}-\sigma_{2}^{2}\right)^{3 / 2}}\\
\times\left|\frac{d \sigma_{1}^{2}}{d M_{\mathrm{t1}}}\right| \exp \left[-\frac{\left(\delta_{c 1}-\delta_{c 2}\right)^{2}}{2\left(\sigma_{1}^{2}-\sigma_{2}^{2}\right)}\right] d M_{t1},
\end{gathered}
\end{equation}
where $i=1,2$ is used to denote parameters at time $t_{1}$ and $t_{2}$, respectively. By introducing $\Delta S\equiv \sigma^{2}_{2}(M_{\mathrm{t2}}) - \sigma^{2}_{1}(M_{\mathrm{t1}})$ and $\Delta \omega \equiv \delta_{\mathrm{c2}}(t_2) - \delta_{\mathrm{c1}}(t_1)$, the equation can be simplified as
\begin{equation}\label{equ:sim_extend_PS}
\operatorname{Pr}(\Delta S, \Delta \omega) d \Delta S=\frac{1}{\sqrt{2 \pi}} \frac{\Delta \omega}{(\Delta S)^{3 / 2}} \exp \left[-\frac{(\Delta \omega)^{2}}{2 \Delta S}\right] d \Delta S.
\end{equation}
\edited{Note that in equation~\ref{equ:sim_extend_PS}, $\Delta \omega$} should satisfy the condition
\begin{equation}
\Delta \omega \lesssim \Delta \omega_{\max }=\left[S\left|\frac{d \ln \sigma^{2}}{d \ln M}\right|\left(\frac{\Delta M_{c}}{M}\right)\right]^{1 / 2}
\end{equation}
in order to resolve the minimum mass change $\Delta M_{\mathrm{c}}$ \citep{ExtendedPS1993}; in this work we adopt $\Delta \omega=\Delta \omega_{\mathrm{max}}/2$ \citep{Randall_2002}. When $\Delta \omega$ is given, \edited{$\Delta S$} can be drawn randomly from a cumulative probability distribution of
\begin{equation}
\begin{gathered}
\operatorname{Pr}(<\Delta S, \Delta \omega)=\int_{0}^{\Delta S} \operatorname{Pr}\left(\Delta S^{\prime}, \Delta \omega\right) d \Delta S^{\prime}\\
=1-\operatorname{erf}\left(\frac{\Delta \omega}{\sqrt{2 \Delta S}}\right),
\end{gathered}
\end{equation}
where $\operatorname{erf}(x)=(2 / \sqrt{\pi}) \int_{0}^{x} e^{-t^{2}} d t$ is the error function.
\begin{figure}
\includegraphics[width=\columnwidth]{figure/PS_cluster.pdf}
\caption{\edited{Cumulative number counts of clusters simulated in a $20^{\circ} \times 20^{\circ}$ sky field with the cut-off redshifts of $z=4$}. }
\label{fig:massfunction}
\end{figure}
We limit our simulation to a \edited{$20^{\circ} \times 20^{\circ}$} sky patch, which is several times larger than the field of view \edited{(i.e., the amount of sky the telescope can image at once)} of the planned Square Kilometers Array (SKA) \citep{SKA_design}, in order to make sure that we have included sufficient number of clusters to carry out statistical analysis.
We set the lower limit of cluster mass to be $M_{\mathrm{min}} = 1\times 10^{14}\, \mathrm{M}_{\odot}$ since less massive systems are found to be incapable of generating detectable signals (see Section~\ref{section:results}).
The halo mass function is shown in Figure~\ref{fig:massfunction}, \edited{which is validated by the observed results given by \citet{Bohringer2017_massfunction}.}
Because radio relics emit through synchrotron radiation and their lifetimes ($\lesssim$ 0.1 Gyrs; \citealt{Brunetti2014_CReReview}) are short compared with the duration of typical merger events (about a few Gyr; e.g., \citealt{Hu_2021}), we trace the evolution of each cluster back 3 Gyr from the cluster's age at its redshift.
Under these conditions we obtain \edited{8704} clusters in the simulation and \edited{5107} of them have experienced a merger.
\begin{figure}
\includegraphics[width=\columnwidth]{figure/cluster_model.pdf}
\caption{Normalized distribution of gas density $\rho_{th}$, gas temperature $T$ and magnetic field $B$. \label{fig:cluster_profile}}
\end{figure}
\subsubsection{Properties of ICM and Merger Shocks}
\label{section:cluster_para}
In our model the ICM mass density $\rho (r)$ is represented by a $\beta$ model \citep{Beta_model_Cava1976}
\begin{equation}\label{equ:beta_model}
\rho(r)=\rho(0)\left[1+\left(\frac{r}{r_{\mathrm{c}}}\right)^{2}\right]^{-3 \beta / 2},
\end{equation}
where $r_{\mathrm{c}}$ is the core radius and is fixed to $r_{\mathrm{c}} = 0.1R_{\mathrm{vir}}$ ($R_{\mathrm{vir}}$ is the virial radius of the cluster; \citealt{Sanderson2003}), and $\rho(0)$ is the central gas density.
\edited{The widely adopted value for the slope parameter $\beta$ is $2/3$ \citep{Jones1984, Li2019, Vazza2020}, which fits the gas density profile well at the central part of the clusters, while cannot give good description at $r>R_{\mathrm{500}}$ \footnote{$r_{500}$ is the characteristic radius of the cluster that the mean enclosed mass density is 500 times the critical density of the universe at the given redshift, and $M_{500}$ represents the total gravitating mass within $r_{\text{500}}$. Similarly, $M_{200}$ is the total gravitating mass within $r_{200}$.}, where the profile is steeper \citep{Vikhlinin_2006, Ghirardini_2019}. Since the radio relics are usually observed at cluster outskirts, the gas density profile in this region should be treated more carefully.
We set $\beta=5/6$ for $r\geq 0.5\,R_{\mathrm{200}}$, i.e., $\rho(r) \sim r^{-2/5}$ \citep{Zhang2019}, and $\beta=2/3$ for $r< 0.5\,R_{\mathrm{200}}$.
$\rho(0)$ can be constrained by the total ICM mass
$M_{\mathrm{gas}} = f_{\mathrm{gas}}M_{\mathrm{\mathrm{vir}}}$, where $f_{\mathrm{gas}}=\Omega_{b}/\Omega_{m}$.
Although this piecewise $\beta$ model is not differential at $0.5\,R_{\mathrm{200}}$, the shock begins to produce radio relics at $R_{\mathrm{sh,min}} = 0.5\,R_{\mathrm{200}}$ (see Section~\ref{section:shock_model}), i.e., we use $\beta=5/6$ for the whole process of shock evolution. We only require the gas density in the central part ($r < 0.5\,R_{\mathrm{200}}$) to calculate the $\rho(0)$.
}
The number density of the thermal electrons is then given by $n_{\mathrm{th,e}} = \rho \left(r\right)/\mu m_{\text{e}}$, where the mean molecular weight $\mu \sim 1.155$ \citep{Ettori2013_clustermassprofile} and $m_{\text{e}}$ is the electron mass.
We assume that the gas temperature profiles in the clusters follow the form given in \citet{Vikhlinin_2006}, i.e., \begin{equation}\label{equ:T_profile}
\frac{T(r)}{T_{\mathrm{mg}}}=1.35 \frac{(x / 0.045)^{1.9}+0.45}{(x / 0.045)^{1.9}+1} \frac{1}{\left(1+(x / 0.6)^{2}\right)^{0.45}},
\end{equation}
where $x=r/r_{500}$, $T_{\mathrm{mg}}$ is the mass weighted average temperature for the whole cluster (except for the central region $r < 0.05\,R_{\mathrm{vir}}$), which scales with $M_{500}$
using the following relation \citep{Finoguenov2001_massTscale}
\begin{equation}\label{equ:M_T_scale}
M_{500}=\left(3.57_{-0.35}^{+0.41}\right) \times 10^{13} \times k T_{\mathrm{mg}}^{1.58_{-0.07}^{+0.06}}.
\end{equation}
Within the frame of the Navarro-Frenk-White (NFW) model, the total mass within the radius $r=s\,r_{\mathrm{vir}}$ is \citep{Lokas2001}
\begin{equation}
M\left(<s\,r_{\text {vir }}\right)=M_{\text {vir }} \frac{\ln (1+c s)-c s /(1+c s)}{\ln (1+c)-c /(1+c)},
\end{equation}
where $s$ can be seen as the radius in units of $r_{\mathrm{vir}}$.
The concentration parameter $c$ is related to the virial mass $M_{\mathrm{vir}}$, which can be approximated by $M_{200}$ \citep{Ettori2009}
\begin{equation}
c=A\left(\frac{M_{200}}{M_{\text {pivot }}}\right)^{B}(1+z)^{C},
\end{equation}
where $M_{\mathrm{pivot}}=2\times 10^{12}\ h^{-1}\text{M}_{\odot}$, $A=5.71$, $B=-0.084$ and $C=0.47$ \citep{Duffy2008}.
The magnetic field in the ICM is less constrained with present data or models. The magnetic field has been measured only for a few radio relics, and the results show that $B \sim \mu G$. In this work we adopt the approach of \citet{Beck2005_mag_equipartition} by assuming that the energy density of the magnetic field $\epsilon_{B}$ has reached equipartition with that of the cosmic rays (CR)
\begin{equation}\label{equ:B_profile}
\epsilon_{B} = \frac{B^{2}}{8\pi} = \chi_{\mathrm{cr}}\epsilon_{\mathrm{th}},
\end{equation}
where the coefficient $\chi_{\mathrm{cr}} = 0.015$ \citep{Li2019,Vazza2020} and $\epsilon_{\mathrm{th}}=3n_{\mathrm{th}}k_{B}T(r)/2$ is the thermal energy density of ICM. This yields a radial profile $B(r)$ that peaks at the cluster center and decays with radii, which is usually believed to be close to the lower limit of the true values \citep{Vazza2020}.
The normalized profiles of gas density, gas temperature, and magnetic field are shown in Figure~\ref{fig:cluster_profile}.
The jumps of the gas density and the temperature across the shock front are both described by the Rankine-Hugoniot relation
\begin{equation}\label{equ:RH}
\left\{\begin{array}{l}
\rho_{2}/\rho_{1}=4 \mathcal{M}_{\mathrm{X}}^{2}/(\mathcal{M}_{\mathrm{X}}^{2}+3),
\\
\\
T_{2}/T_{1}=(5 \mathcal{M}_{\mathrm{X}}^{4}+14 \mathcal{M}_{\mathrm{X}}^{2}-3)/16 \mathcal{M}_{\mathrm{X}}^{2},
\end{array}\right.
\end{equation}
where $i=1,2$ represents upstream and downstream regions, respectively.
The postshock magnetic field is then boosted as
\begin{equation}
B_{2} = B_{1}\sqrt{\frac{1}{3}+\frac{2\sigma^{2}}{3}},
\label{equ:post_Bfield}
\end{equation}
where $\sigma = \rho_{2}/\rho_{1}$ \citep{Kang2017}.
\subsection{Evolution of Merger Shocks}
\label{section:shock_model}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/mach_dis.pdf}
\caption{Distribution of the Mach numbers of the simulated merger shocks. \label{fig:mach_dis}}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/shock_model.pdf}
\caption{Radial evolutions of the shock speed $u_{sh}$, sound speed $c_{s}$ and Mach number $\mathcal{M}$ for a binary merger with $M_{1} = M_{2} = 5\times 10^{14}\, \text{M}_{\odot}$ at $z=0.2$. \label{fig:shock_evol}}
\end{figure}
The properties of the merger shocks vary remarkably from case to case, depending on both merger parameters (e.g., mass ratio, initial velocity, and impact parameter) and evolution stage of the shock.
Although a few radio relics are detected in cool-core systems, which are possibly related to off-axis minor mergers (i.e., the mergers with non-zero impact parameter and mass ratio $\gtrsim3$; \citealt{Feretti}), observational evidences show that most radio relics are likely to appear in mergers with a small impact parameter \citep{Hoang2018},
so we limit our study to binary head-on mergers, i.e., we always set the impact parameter to be zero. Therefore, two shocks will be produced along the merger axis, for which the initial Mach numbers can be given in terms of the mass ratio between the subcluster and the main cluster $\eta_{\mathrm{m}}=M_{2}/M_{1}$ ($M_1>M_2$) \citep{Takizawa_1999, Gabici2003_RRmodel}
\begin{equation}\label{equ:shock_mach}
\left\{\begin{array}{l}
\mathcal{M}_{\mathrm{main}}^{2}=\frac{4(1+\eta_{\mathrm{m}})}{\gamma(\gamma-1)}\left[\frac{1}{1+\eta_{\mathrm{m}}^{1 / 3}}-\frac{1}{4(1+\eta_{\mathrm{m}})^{1 / 3}}\right],
\\
\\
\mathcal{M}_{\mathrm{sub}}^{2}=\eta_{\mathrm{m}}^{-2 / 3} \mathcal{M}_{\mathrm{main}}^{2},
\end{array}\right.
\end{equation}
where $\mathcal{M}_{\mathrm{main}}$ and $\mathcal{M}_{\mathrm{sub}}$ denote the Mach numbers of the shocks propagating in the main cluster and subcluster, respectively. \edited{The calculated Mach numbers are plotted in Figure~\ref{fig:mach_dis} for our simulated shocks.}
On the other hand, because radio relics are rare in the central region of clusters, possibly due to the fact that
the comoving kinetic power through the shock surface
is generally small in central regions \citep{Vazza2012_rare_centraRR}, we assume that the shocks start to accelerate electrons at $R_{\mathrm{sh,min}} = 0.5\,R_{\mathrm{vir}}$ \citep{Feretti}, \edited{and} the acceleration begins $\Delta t \sim 0.5\,R_{\mathrm{vir}}/u_{\mathrm{sh,0.5}}$ after the merger occurs, where \edited{$u_{\mathrm{sh,0.5}}=\mathcal{M}\cdot c_{\mathrm{s}}$} is the shock velocity at $r=0.5\,R_{\mathrm{vir}}$ \edited{with the mach number $\mathcal{M}$ given in equation~\ref{equ:shock_mach} and the sound speed $c_{\mathrm{s}} = 150\,\text{km}\,\text{s}^{-1}\left(T/10^{6}\text{K}\right)^{1/2}$ \citep{Kang_2010}, which is dependent on the ICM temperature $T$.}
To describe the evolution of the shock's strength, we apply the analytical model provided in \citep{Zhang2019} and assume that shock velocity $u_{\mathrm{sh}}$ measured in the rest frame of the upstream ICM decreases with shock's position $R_{\mathrm{sh}}$ following a power law form \edited{
\begin{equation}
u_{\mathrm{sh}}= u_{\mathrm{sh,0.5}} \left(\frac{R_{\mathrm{sh}}}{0.5R_{\mathrm{vir}}}\right)^{\eta},
\end{equation}
}where $\eta=\omega / 4 -1$ and $\omega = - d\ln{\rho}/d\ln{r}$ ($\simeq 2$ for the $\beta$ model). As an example,
in Figure~\ref{fig:shock_evol} we show the radial evolution of shock speed $u_{\mathrm{sh}}$, sound speed $c_{\mathrm{s}}$, and Mach number $\mathcal{M} = u_{\mathrm{sh}}/c_{\mathrm{s}}$ \edited{at $0.5R_{\mathrm{vir}}$} for a binary merger with $M_{1} = M_{2} = 5\times 10^{14}\,\text{M}_{\odot}$, and $z=0.2$.
\edited{The plots in this section following (Figure \ref{fig:fp_atshock} and Figure \ref{fig:diff_R}) are all based on this example.}
We assume that during the propagation a shock stays as part of a spherical surface and has a constant solid angle $\Omega$ with the cluster center being the apex. Thus the surface area of the shock grows with the radius as $A \propto r^{2}$.
\subsection{Shock Acceleration}
\label{section:DSA_solution}
Within the frame of DSA theory, the behavior of electrons in both real space and momentum space is described by the diffusion-convection equation \citep{Skilling1975}
\begin{equation}\label{equ:diffusion-convection}
\begin{gathered}
\frac{\partial f(r,p)}{\partial t} + u\frac{\partial f(r,p)}{\partial r}=
\frac{1}{3} \frac{\partial}{\partial r}\left[up\frac{\partial f(r,p)}{\partial p}\right]\\
+ \frac{\partial}{\partial r}\left[\kappa(r, p) \frac{\partial f(r,p)}{\partial r}\right]
+\frac{1}{p^{2}} \frac{\partial}{\partial p}\left[p^{2} D_{pp}\frac{\partial f(r,p)}{\partial p}\right],
\end{gathered}
\end{equation}
where $f(r,p)$ is the number of electrons with momentum $p$ per unit volume, $u$ is the background flow velocity, $r$ is the distance to the cluster center, $\kappa(r,p)$ and $D_{pp}$ are the spatial and momentum diffusion coefficients, respectively.
Note that the equation is expressed in the rest frame of the shock and the energy loss of relativistic electrons (e.g., via radio synchrotron) is not included. On the right side of the equation the first term describes the influence of the ICM's bulk flow, which corresponds to the convection process,
the second term the diffusion of electrons in real space, and the third term the diffusion in the momentum space that is caused by the turbulence behind the shock.
By applying the gas dynamic conservation equations and considering the feedback of CR on the shock and ICM, we may numerically solve equation~\ref{equ:diffusion-convection} \citep{Kang2012a, Kang2012b}, which is, however, usually computationally expensive. In our case only weak shocks are considered, therefore the CR feedback is insignificant and can be ignored in the calculation (i.e., the test-particle limit). Since the thermal and fossil relativistic electron populations are accelerated independently at the shock front, we describe how to calculate the number distributions of the electrons for the two populations in Section~\ref{section:DSA_thermal} and~\ref{section:DSA_fossil} separately.
\subsubsection{Contribution of the Thermal Electron Population}
\label{section:DSA_thermal}
We follow the approach of, e.g., \citet{Hoeft2007_RRsimu} and \citet{Vazza2020}, and ignore the contribution of turbulence acceleration, which may spatially widen the relics to some extent but without adjusting the radio properties of the relics significantly \citep{Kang2017}.
For thermal electrons equation~\ref{equ:diffusion-convection} can be analytically solved after the convection and diffusion processes achieve a balance for a one-dimension shock front
\begin{equation}\label{equ:DSA_thermal_acc}
f(p)_{\mathrm{th,acc}} = f_{\text{N,th}}\left(\frac{p}{p_{\mathrm{inj}}}\right)^{-q} \exp \left(-\frac{p^{2}}{p_{\mathrm{eq}}^{2}}\right),
\end{equation}
where $f_{\text{N,th}}$ is the normalization factor, $q = 3\sigma/(\sigma-1)$ \citep{Drury_1983} with the density compression factor $\sigma=\rho_{2}/\rho_{1}$ given in equation~\ref{equ:RH}, and $p_{\mathrm{inj}}$ and $p_{\mathrm{eq}}$ are the injection momentum and cutoff momentum, respectively.
Since the amount of the energy gained by the electrons from a single passage across the shock front is small \citep{Fermi_1949}, the electrons responsible for the radio relics must have recrossed the shock repeatedly, which requires that the gyroradii of the electrons in the magnetic field be at least several times of the shock's thickness.
Since electrons with lower energies have smaller gyroradii, there should exist a threshold momentum $p_{\mathrm{inj}}$ for the electrons to participate in the DSA
\citep{Kang_2002}.
According to hybrid simulation of \citet{Kang2017} and \citet{Caprioli_2014}, $p_{\mathrm{inj}}\sim 150\,p_{\mathrm{th,e}}$, where $p_{\mathrm{th,e}} = \sqrt{2m_{\mathrm{e}}kT_{2}}$. The cutoff momentum $p_{\mathrm{eq}}$, on the other hand, can be determined by letting the acceleration rate equal to the loss rate caused by synchrotron emission and inverse Compton scattering off the CMB photons \citep{Kang2011}
\begin{equation}\label{equ:p_eq}
p_{\mathrm{eq}}=\frac{m_{\mathrm{e}}^{2} c^{2} u_{\mathrm{sh}}}{\sqrt{4 e^{3} q / 27}}\left(\frac{B_{1}}{B_{\mathrm{e}, 1}^{2}+B_{\mathrm{e}, 2}^{2}}\right)^{1 / 2} k_{\mathrm{Bohm}}^{-1},
\end{equation}
where $i=1,2$ represents the upstream and downstream regions, respectively, $B_{\mathrm{e}}^{2} = B^{2} + B_{\mathrm{rad}}^{2}$ is the total effective magnetic field ($B$ is the magnetic field of ICM and $B_{\mathrm{rad}}^{2}/8\pi$ is the energy density of ambient radiation field \cite{longair_1994}), and $k_{\mathrm{Bohm}}\sim 1$ corresponds to Bohm diffusion coefficient \citep{Kang2017}.
We determine $f_{\text{N,th}}$ in equation~\ref{equ:DSA_thermal_acc} by assuming that the energy spectrum of the electrons is continuous at $p_{\mathrm{inj}}$, i.e. $f_{\mathrm{th}}(p_{\mathrm{inj}}) = f_{\mathrm{th,acc}}(p_{\mathrm{inj}})$, where $f_{\mathrm{th}}$ and $f_{\mathrm{th,acc}}$ are the number density distributions of the thermal electrons before and after being accelerated, respectively.
Meanwhile, because additional pre-acceleration processes such as shock drift acceleration (SDA) and electrons firehose instability (EFI) \citep{Kang_2019_preacc, Trotta2019_preacc} that may occur ahead of the shock tend to change the initial energy distribution of the thermal electrons, which can cause a large high energy tail in the energy spectrum, we use a $\kappa$ distribution instead of the Maxwell distribution to present the thermal electron population \citep{Kang2017}
\edited{
\begin{equation}\label{equ:kappa_dist}
f(p)_{\kappa} \propto (1+\frac{p^{2}}{\kappa\,p_{\mathrm{th,e}}^{2}})^{-(\kappa +1)} \exp \left(-\frac{p^{2}}{p_{\mathrm{inj}}^{2}}\right),
\end{equation}}
where $\kappa$ is a constant.
\edited{The cut-off term $ \exp \left(-p^{2}/p_{\mathrm{inj}}^{2}\right)$ is included to make sure the thermal population could hardly has electrons with $p>p_{\mathrm{inj}}$ since when the electrons in front of the shock are accelerated to the $p_{\mathrm{inj}}$, most of them would join the DSA process and then would not reflect back to the preshock region.}
SDA and EFI also happen in the solar winds, the CRe (CR electron) spectrum of which is usually well fitted with a $\kappa$ distribution with $2<\kappa<5$ \citep{Pierrard2010}, and we choose to take $\kappa =2$ because $\kappa >2$ the resulting radio relics are generally too dim.
\begin{figure}
\includegraphics[width=\columnwidth]{figure/ene_spec_kappa.pdf}
\caption{$p^{2}f(p)$ distributions of the thermal electron population (blue) and the fossil relativistic electron population (red) before and immediately after shock acceleration (solid and dash, respectively).
The vertical black dash lines indicate $p_{\mathrm{inj}}$ and $p_{\mathrm{c,fo}}$.
X-axis: electron energy and corresponding characteristic frequency of synchrotron emission.
\edited{
The shaded area marks 50 $\sim$ 350 MHz, which is the target band of the SKA1-Low.}
\label{fig:fp_atshock}}
\end{figure}
\subsubsection{Contribution of Fossil Electrons Population}
\label{section:DSA_fossil}
Following \citet{Kang2017} we assume that the fossil electrons initially present a power-law spectrum with an exponential cutoff at $p_{\mathrm{c,fo}}$
\begin{equation}\label{equ:fossil_init}
f_{\mathrm{fo,init}}(p)=f_{\text{N,fo}} \cdot p^{-s} \exp \left[-\left(\frac{p}{p_{\mathrm{c,fo}}}\right)^{2}\right],
\end{equation}
where $f_{\text{N,fo}}$ is the normalization, $s$ is the initial energy spectral index, and the cutoff momentum $p_{\mathrm{c,fo}} = 5 \times 10^{4}\ m_{\mathrm{e}}c$, \edited{which could reproduce the observed flux density $S_{\nu}$ and spectral index $\alpha$ of CIZA J2242.8+5301 \citep{Kang2016} and RX J0603.3+4212 \citep{Kang2017}.
According to our test, the resulting simulated power of relics is insensitive to the value of $p_{\mathrm{inj}}$.}
Under the same conditions applied to obtain equation~\ref{equ:DSA_thermal_acc}, i.e., the effect of turbulence is neglected and a balance between convention and diffusion processes has been established, a solution can be found in a integral form for the shock reaccelerated fossil electrons \citep{Drury_1983}
\begin{equation}\label{equ:DSA_solution_fossil}
f_{\text {fo,acc}}\left( p\right)=q \cdot p^{-q} \int_{p_{\text {inj }}}^{p} p^{\prime q-1} f_{\text{fo,init}}\left(p^{\prime}\right) d p^{\prime}.
\end{equation}
We show the derived energy spectra \edited{at $0.5R_{\mathrm{vir}}$} of the two electron populations in Figure~\ref{fig:fp_atshock}, \edited{whose magnetic field strength is given by equation \ref{equ:B_profile} and equation \ref{equ:post_Bfield}, and $s=4.6$, $f(p_{\mathrm{inj}})_{\mathrm{fo,init}}/f(p_{\mathrm{inj}})_{\mathrm{th,init}}=0.86447$}.
We also plot the characteristic frequency ($v_{\mathrm{obs}}$) of the synchrotron radiation corresponding to a given electron momentum using the relation \citep{Kang2012b}
\begin{equation}\label{equ:syn_gamma_nu}
\gamma \approx 1.26 \times 10^{4}\left(\frac{v_{\mathrm{obs}}}{1 \mathrm{GHz}}\right)^{\frac{1} {2}}\left(\frac{B}{5\,\mu \mathrm{G}}\right)^{-\frac{1}{2}}(1+z)^{\frac{1}{2}},
\end{equation}
and the band of the SKA1-low array (50 $\sim$ \edited{350 MHz}).
\subsection{Evolution of Electron Energy Spectra}
\label{section:spe_evo}
\edited{Besides computing the spectra of the accelerated electrons injected at the shock front, we also consider the influence of the radiation cooling, i.e., the aged CRe population downstream.}
We have assumed that both thermal and fossil electrons are accelerated instantaneously at shock front since the acceleration timescale is much shorter than the decay timescale ( $\lesssim$ 0.1 Gyr; \citealt{Brunetti2014_CReReview}). Thus we may apply the solutions of equations~\ref{equ:DSA_thermal_acc} and \ref{equ:DSA_solution_fossil} as the initial conditions to investigate the energy loss rate of relativistic electrons $(d\gamma/dt)$ behind the shock, which is attributed to three major processes, i.e., radio synchrotron radiation, inverse Compton scattering (IC) and Coulomb collision. In the first two processes the energy loss rates are \citep{Sarazin1999}
\begin{equation}\label{equ:syn_loss_rate}
\left(\frac{d \gamma}{d t}\right)_{\mathrm{syn}}=-4.10 \times 10^{-5} \gamma^{2}\left(\frac{B_2}{1\,\mu \mathrm{G}}\right)^{2}\quad\left[\mathrm{Gyr}^{-1}\right],
\end{equation}
where $B_{2}$ is the magnetic field in the postshock region, and
\begin{equation}\label{equ:IC_loss_rate}
\left(\frac{d \gamma}{d t}\right)_{\mathrm{IC}}=-4.32 \times 10^{-4} \gamma^{2}(1+z)^{4} \quad\left[\mathrm{Gyr}^{-1}\right],
\end{equation}
respectively, showing that the two processes have comparable contributions and dominate the energy loss when the electrons energy is high ($\gamma \gtrsim 200$).
In the case of Coulomb collision, the energy loss rate is given by \citep{Sarazin1999}
\begin{equation}\label{equ:CC_loss_rate}
\begin{gathered}
\left(\frac{d \gamma}{d t}\right)_{\mathrm{Coul}}=-3.79 \times 10^{4}\left(\frac{n_{\mathrm{th}}}{1 \mathrm{~cm}^{-3}}\right)\\ \times\left[1+\frac{1}{75} \ln \left(\gamma \frac{1 \mathrm{~cm}^{-3}}{n_{\mathrm{th}}}\right)\right] \quad\left[\mathrm{Gyr}^{-1}\right] .
\end{gathered}
\end{equation}
Since the energy loss rates due to synchrotron emission and inverse Compton scattering both have the form of $d\gamma/dt = b\gamma^{2}$ in a steady environment, where $b$ is a constant, the corresponding electron energy spectra will have an analytical solution at any time if the effect of the Coulomb collision is negligible; otherwise the equation needs to be solved numerically. Considering that the impact of Coulomb collision is usually limited to $\gamma \lesssim 10^{3}$ (see Figure~\ref{fig:loss_rate} in Section~\ref{section:CC}) with which the electrons can barely produce detectable radio emission, it is reasonable to ignore its contribution in our model. The energy spectrum at any time $t$ after the shock acceleration is then given by \citep{Wang_2010}
\begin{equation}
f\left(\gamma, t\right) = \frac{f\left(\gamma_{0}, t_{0}\right)}{\left(1-b\gamma t\right)^{2}},
\end{equation}
where $\gamma_{0}=\gamma/(1-b\gamma t)$ is the initial electron energy.
\begin{figure}
\includegraphics[width=\columnwidth]{figure/J_diffR.pdf}
\caption{The radio synchrotron emissivity of different radii immediately after shock fronts
\label{fig:diff_R}}
\end{figure}
\subsection{Radio Synchrotron Emission of Radio Relics}
\label{section:radioemi}
Given the number density distribution $f(\gamma,t)$, the synchrotron emissivity of the shock accelerated electrons in the radio relic is given by
\begin{equation}\label{equ:syn_emit}
J(\nu)=\frac{\sqrt{3} e^{3} B}{m_{\mathrm{e}} c^{2}} \int_{\gamma_{\min}}^{\gamma_{\max}} \int_{0}^{\frac{\pi}{2}} F_{\mathrm{syn}}\left(\frac{\nu}{\nu_{\mathrm{c}}}\right) f(\gamma, t) \sin ^{2} \theta \mathrm{d} \theta \mathrm{d} \gamma,
\end{equation}
where $\theta$ is the pitch angle of electrons with respect to the magnetic field, $\nu_{\text{c}} = 3\,\gamma^{2}\,\nu_{\text{L}}\,\text{sin}\theta/2$ is the critical frequency for a given Larmor frequency $\nu_{\text{L}} = eB/(2\pi m_{\text{e}}c)$, and $F(x)$ is the synchrotron kernel
\begin{equation}\label{equ:syn_kernel}
F_{\text {syn }}(x)=x \int_{x}^{\infty} K_{5 / 3}(y) \mathrm{d} y,
\end{equation}
where $K_{5/3}(y)$ is the modified Bessel function of order 5/3 \citep{Rybicki1979}.
Because we assume that shock acceleration starts to occur at $0.5\,R_{\mathrm{vir}}$ and the solid angle of the shock front always stays the same during the propagation, the volume element swept by the shock during $dt$ is $dV = \Omega\,R_{\mathrm{sh}}^{2}\, dR_{\mathrm{sh}} = \Omega\,R_{\mathrm{sh}}^{2} u_{\mathrm{sh}}dt$. Although the shock may travel a long distance in a cluster's outskirts \citep{Zhang2019}, it is not clear to what extend can effective electron acceleration and radio synchrotron emission persist given the lower gas density and magnetic field. We calculate the spatial evolution of shocks in the range of $0.25\,R_{\mathrm{vir}} \sim 1.7\,R_{\mathrm{vir}}$, and find that at $1.2\,R_{\mathrm{vir}}$ the radio emission is only about $0.1\%$ of that produced at $0.5\,R_{\mathrm{vir}}$, \edited{which is shown in the Figure \ref{fig:diff_R}}. Thus it is reasonable to set a maximum radius $R_{\mathrm{sh,max}}=1.2\,R_{\mathrm{vir}}$ (i.e., $0.5\,R_{\mathrm{vir}}<R_{\mathrm{sh}}<1.2\,R_{\mathrm{vir}}$).
When a shock travels to $R_{\mathrm{sh}}>1.2\,R_{\mathrm{vir}}$, only radiative cooling is taken into account in the calculation since the shock acceleration is quenched.
Also we define the evolution time of radio relics as $t_{\mathrm{evol}} = t_{\mathrm{obs}} - t_{\mathrm{merge}}$, where $t_{\mathrm{merger}}$ is the cosmological age when the merger begins to occur.
We assume that $\Omega$ does not vary with frequency drastically since $\Omega$ is mainly decided by the spatial distribution of the accelerated electrons right behind the shock front.
\section{Calculation and Results}
\label{section:results}
\subsection{Observational constrains}
\label{section:real_case}
In this section we apply the observational constrains obtained from the radio relics sample of \citet{Feretti} and that in Table~\ref{table:relics_samples} on our model to determine the solid angle $\Omega$ of the shock, the normalization $f_{\text{N,fo}}$ and the initial energy spectral index $s$ of the fossil electrons (equation~\ref{equ:fossil_init}).
\subsubsection{Solid Angles of Radio Relics}
\label{section:solid_angle}
We use the data of 50 radio relics analyzed in \citet{Feretti} who listed the LLS and the distance to the cluster center $r_{\mathrm{relic}}$ for each radio relic, both observed at 1.4 GHz, to constrain $\Omega$. \edited{We treat radio relics as spherical crowns since we only study the binary head-on mergers in this work, which are symmetric along the merger axes, and we illustrate the geometry model in Figure \ref{fig:geo_model}.}
We calculate the solid angle as $\Omega = 2\pi h/R_{\mathrm{sh}}$, where $R_{\mathrm{sh}} = \sqrt{r_{\mathrm{relic}}^{2}+(\text{LLS}/2)^{2}}$ and \edited{$h = R_{\mathrm{sh}}-r_{\mathrm{relic}}$} is the height of the spherical crown.
Based on the results (Figure~\ref{fig:omega_dis}) we randomly choose $\Omega$ for each simulated shock.
\begin{figure}
\includegraphics[width=0.75\columnwidth]{figure/geo_model.pdf}
\caption{\edited{The geometry model for a typical relic example.} \label{fig:geo_model}}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/omega_list.pdf}
\caption{Distribution of radio relics' solid angle $\Omega$ obtained from the sample of \citet{Feretti}. The dash line indicates the average value ($\Omega_{\mathrm{mean}}=0.22\pi$). \label{fig:omega_dis}}
\end{figure}
\subsubsection{Fossil Electron Properties}
\label{section:case_fossil}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/mach_list.pdf}
\caption{Distribution of $\mathcal{M}_{\mathrm{X}}$ and $\mathcal{M}_{\mathrm{radio}}$ of our radio relic sample. The red dash line indicates where $\mathcal{M}_{\mathrm{X}}$ = $\mathcal{M}_{\mathrm{radio}}$. \label{fig:machlist}}
\end{figure}
Although in the calculation of $\mathcal{M}_{\mathrm{radio}}$ the fossil electrons population is not included, making $\mathcal{M}_{\mathrm{radio}}$ inappropriate for representing the velocity of the shock, it can be used to calculate the energy spectral index of the relativistic electrons. Considering the synchrotron emissivity $J_{\mathrm{syn}}(\nu) \propto \nu^{-\alpha}$ and the spectral index \edited{at the shock front $\alpha_{\mathrm{inj}}=(\delta_{\mathrm{inj}}-3)/2$, where $\delta_{\mathrm{inj}}$ is the power law index of the electrons' energy distribution in three-dimensional momentum space}, the radio Mach number is given by \edited{$\mathcal{M}_{\mathrm{radio}} = \sqrt{1+4/(2\alpha_{\mathrm{inj}}-1)} = \sqrt{1+4/(\delta_{\mathrm{inj}}-4)}$}.
When only thermal electrons are considered (equation~\ref{equ:DSA_thermal_acc}),
$\delta_{\mathrm{inj}} = q$, in which case $\mathcal{M}_{\mathrm{X}}=\mathcal{M}_{\mathrm{radio}}$.
For fossil electrons,
if we ignore the exponent cutoff in equation~\ref{equ:fossil_init}, which only plays a role at $p\sim p_{\mathrm{inj}}$, equation~\ref{equ:DSA_solution_fossil} can be solved analytically
\edited{
\begin{equation}\label{equ:weak_shock_model}
f_{\mathrm{fo,acc}}(p)=\left\{\begin{array}{l}
\frac{q}{(q-s)} f_{\text{N,fo}}\left[\left(\frac{p}{p_{\text{inj}}}\right)^{-s}-\left(\frac{p}{p_{\text{inj}}}\right)^{-q}\right],
\\
\hfill \text { if } q \neq s ; \\
\\
q f_{\text{N,fo}} \frac{p^{-q}}{p_{\text{inj}}^{s}} \ln \frac{p}{p_\text{inj}},
\hfill \text { if } q=s .\\
\end{array}\right.
\end{equation}
}
Therefore, for the fossil electron population with \edited{a flatter initial energy spectrum than that of the accelerated thermal electrons}, i.e., $s<q$, the dominant spectral index of resulting $f_{\mathrm{fo,acc}}$ is $s$, and $\mathcal{M}_{\mathrm{radio}} = \sqrt{1+4/(s-4)}$, which provides a possible explanation for $\mathcal{M}_{\mathrm{radio}} > \mathcal{M}_{\mathrm{X}}$. Adopting
radio Mach number provided in Table~\ref{table:shock_samples}, we can calculate $s$ ($= 4\mathcal{M}_{\mathrm{radio}}^{2}/(\mathcal{M}_{\mathrm{radio}}^{2}-1)$) \edited{and further determine the shape of energy spectra of the potential fossil electrons in these observed samples}.
We have to point out among our sample, there is one cluster: ACT-CL J0102-4915 with $\mathcal{M}_{\mathrm{X}}>\mathcal{M}_{\mathrm{radio}}$, which cannot be explained by equation~\ref{equ:weak_shock_model}. Considering the large observational error, we still include it in calculation to constrain the parameters.
\edited{
Since the emission generated by fossil electrons ($P_{\mathrm{fossil}}$) is proportional to the particle density: $P_{\mathrm{fossil}}\propto f_{\mathrm{fo,acc}}(p)\propto f_{\mathrm{N,fo}}$ for a given cluster, with the energy spectra of fossil electrons whose shape has been determined by the $\mathcal{M}_{\mathrm{radio}}$, we could calculate $P_{\mathrm{fo, unnorm}}$ based on our model (equation~\ref{equ:DSA_solution_fossil}), which equals to $P_{\mathrm{fossil}}/f_{\mathrm{N,fo}}$.
Then the normalized factor $f_{\mathrm{N,fo}}$ could be determined by
\begin{equation}
f_{\mathrm{N,fo}} = \frac{P_{\text{fossil}}}{P_{\text{fo,unnorm}}} = \frac{P_{\text{observed}} - P_{\text{thermal}}}{P_{\text{fo,unnorm}}},
\end{equation}
where the $P_{\mathrm{observed}}$ is the observed power listed in Table~\ref{table:relics_samples}, and $P_{\mathrm{thermal}}$ is that produced by the thermal electrons, which could be computed with the equation~\ref{equ:DSA_thermal_acc}$\sim$\ref{equ:kappa_dist}.
}
We also calculate \edited{$\mathcal{P}_{\mathrm{CRe}}/\mathcal{P}_{\mathrm{total}}$}, the pressure ratio between the fossil electrons and thermal electrons, and list them in Table~\ref{table:relics_samples} along with $f_{\mathrm{N,fo}}/f_{\mathrm{N,th}}$.
We show the distribution of $-s$ and $f_{\mathrm{N,fo}}/f_{\mathrm{N,th}}$, the two parameters needed in our model, in Figure~\ref{fig:fossil_ratio}. In the simulation of radio relics, we apply \edited{$f_{\text{N,fo}} = 0.86447 f_{\text{N,th}}$}, which is the average value of our observed radio relics sample, and randomly choose $s$ between 4 and 5.
\edited{It is worth noting that the $f_{\mathrm{N,fo}}/f_{\mathrm{N,th}}$ does not present the number density ratio between the two electron population since there are much more thermal electrons at $p< p_{\mathrm{inj}}$.}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/fossil_CRe.pdf}
\caption{Distribution of $-s$ against $f_{\mathrm{N,fo}}/f_{\mathrm{N,th}}$ calculated based on our model for the radio relics sample listed in Table~\ref{table:relics_samples}.}\label{fig:fossil_ratio}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/Relic_frac.pdf}
\caption{\edited{Cumulative fraction of clusters hosting one or two radio relics whose flux larger than 1 $\mu$Jy as a function of the virial mass.}
\label{fig:Relic_frac}}
\end{figure}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/P2M.pdf}
\caption{1.4 GHz radio power versus virial mass for the simulated radio relics (dots; $t_{\mathrm{evol}}$ are marked with different colors). Black stars: observed radio relics listed in Table~\ref{table:relics_samples}. Black line: the $P_{\mathrm{1400}} \propto M_{\mathrm{vir}}^{\mathrm{2.8}}$ relation presented in \citet{DeGasperin2014_PM_scale}.
}\label{fig:P_1400_M}
\end{figure}
\begin{figure*}
\includegraphics[width=\columnwidth]{figure/P_hist.pdf}
\includegraphics[width=\columnwidth]{figure/S_hist.pdf}
\caption{ Distributions of radio powers (a) and fluxes (b) of simulated relics at 50 MHz, 158 MHz, 1.4 GHz. The dash lines ($S = 5\,\mu$Jy and $50\,\mu$Jy) mark the SKA and HERA sensitivities (100-hour observation), respectively.
}\label{fig:Results_P_S_dis}
\end{figure*}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/geo_dist.pdf}
\caption{Distribution of LLS and the $R_{\mathrm{relic}}$ of the simulated radio relics produced by total electrons (thermal electrons and fossil electrons) and solely by thermal electrons, respectively, with $P_{1400} > 10^{20}$ W/Hz.
\label{fig:geo_dis}}
\end{figure}
\subsection{Simulated Radio Relics}
\label{section:simulated_cata}
By assuming that fossil electron population exists in 50\% of the galaxy clusters, which matches the fact that about 30\% of the massive clusters ($M_{\mathrm{vir}} > 5\times 10^{14}\,\text{M}_{\odot}$) host at least one radio relic \citep{Gleser2008_foreground} \edited{at 158 MHz},
which is shown in Figure~\ref{fig:Relic_frac}. \edited{Our model predicts that $9.6\%$ and $7.1\%$ clusters with $M_{\mathrm{vir}} > 1.2\times 10^{14}\,\mathrm{M}_{\odot}$ have one or more relics with $S>1\mu$Jy at 50 MHz and 158 MHz, respectively, which is consistent with the result of $10 \pm 6\%$ given by the Second Data Release of the LOFAR Two-meter Sky Survey (LoTSS DR2), whose target band is 120-168 MHz.}
In Figure~\ref{fig:P_1400_M} we show the $P_{1400}-M_{\mathrm{vir}}$ relation for both the simulated radio relics, \edited{which are colored by the injection radio spectral indices $\alpha_{\mathrm{inj}}$}, and the relics included in the observation sample (Table~\ref{table:relics_samples}). This shows that \edited{the upper limits of the power for a given $M_{\mathrm{vir}}$ in} our results agree very well with the relation $P_{1400} \propto M_{\mathrm{vir}}^{2.8}$ obtained by \citet{DeGasperin2014_PM_scale} based on the observations of 15 clusters. For clusters with \edited{smaller luminosity, there is a considerable scattering}
and a tendency of deviation from $P_{1400} \propto M_{\mathrm{vir}}^{2.8}$,
possibly caused by Malmquist bias \citep{Vazza2020}, i.e., only brightest radio relics are observed and included in the analysis in \citet{DeGasperin2014_PM_scale}.
Along with Figure~\ref{fig:Results_P_S_dis}, this indicates that much more radio relics are to be found once the detection sensitivity is increased.
\edited{The $\alpha_{\mathrm{inj}}$ are calculated by fitting straight power-laws on the simulation results at shock fronts at 50 MHz, 158 MHz and 1400 MHz.}
\edited{We find that out of 5107 merging clusters simulated in the $20^{\circ}\times 20^{\circ}$ sky patch, 1137, 1018 and 907 clusters host one or two radio relics brighter than $10^{15}$ W/Hz} at 50 MHz and 158 MHz, the characteristic frequencies that will be covered by the next-generation radio arrays, as well as 1.4 GHz, respectively (Figure~\ref{fig:Results_P_S_dis}).
\edited{
The drop in the power distribution around $P\sim 10^{17}$ W/Hz comes from the fact that only a few percent of small clusters, which generally correspond to less luminosity radio relics, are able to produce detectable relics, which could be seen in Figure~\ref{fig:Relic_frac}.
}
\edited{In Figure~\ref{fig:geo_dis} we plot the distribution of the size (LLS) and the position ($R_{\mathrm{relic}}$) for the simulated radio relics with the power larger than $10^{20}$ W/Hz at 1400 MHz that are generated by all the electrons (thermal electrons and fossil electrons), and solely by thermal electrons, respectively. We exclude the relics with lower luminosity because many of them are relics stacking at $R_{\mathrm{sh,max}}$, losing most of their power after a long time of radiation cooling.
It is obvious that there are more relics at larger radius when the fossil electrons are included, which is consistent with the fact that many relics are observed at large distance to the cluster center, and this could come from the smaller shock speed at large radius (Figure~\ref{fig:shock_evol}), which allows the shocks stay longer in this region and be more likely to be observed. On the other hand, when only thermal electrons are considered, the lower gas density and weak magnetic field at cluster outskirts make it less likely to host detectable radio relics, and the $R_{\mathrm{relic}}$ is generally evenly distributed.
}
In Figure~\ref{fig:fossil2thermal} (a) we plot the relic distribution as a function of the ratio between $P_{\mathrm{fossil}}$ and $P_{\mathrm{thermal}}$ (the power produced exclusively by thermal or fossil electrons respectively) at 50 MHz, 158 MHz, and 1.4 GHz. \edited{$P_{\mathrm{fossil}}$ is larger than $P_{\mathrm{thermal}}$ by about} one order of magnitude for most relics, while in some cases the ratio can reach \edited{four} orders of magnitude.
In Figure~\ref{fig:fossil2thermal} (b) we show $P_{\mathrm{fossil,158\,MHz}}$ versus $P_{\mathrm{thermal,158\,MHz}}$, which is remarkably dependent on the spectral index $-s$, and this, again indicates that for radio relics reaccelerated fossil relativistic electrons are an important source of the radio radiation.
\begin{figure*}
\includegraphics[width=\columnwidth]{figure/fossil2thermal.pdf}
\includegraphics[width=\columnwidth]{figure/P_fossil_thermal.pdf}
\caption{(a) Distributions of the simulated relics as a function of the ratio between the power generated by the fossil electrons to that by the thermal electrons
(b) Distribution of the simulated relics as measured with $P_{\mathrm{fossil,158}}$ and $P_{\mathrm{thermal,158}}$. The red dash line indicates where $P_{\mathrm{fossil,158}} = P_{\mathrm{thermal,158}}$. The color represents the value of the spectral index of initial fossil electrons, i.e., $s$.
}\label{fig:fossil2thermal}
\end{figure*}
In Figure~\ref{fig:P_massratio} we show the power at 1.4 GHz ($P_{1400}$) versus the mass ratio of each merger, where the color represents the total mass ($M_1+M_2$). Although the scatter is large, it still can be found that
\edited{a larger fraction of relics that hosted by minor merger system ($M_{1}/M_{2} > 3$) are powerful ones ($ P_{\mathrm{1400}} > 10^{20}$ W/Hz).}
which is consistent with the results of \citet{Vazza2020}, who suggested that unlike radio halos, radio relics are not limited to be in the major merger systems \citep{Buote_2001}. This is not unexpected since radio halos are caused by radio synchrotron emission of electrons accelerated by merger-induced turbulence, which must be more powerful in major mergers.
\begin{figure}
\includegraphics[width=\columnwidth]{figure/P2Mratio.pdf}
\caption{1.4 GHz radio power versus the mass ratio $M_{1}/M_{2}$ ($M_1 > M_2$).
The virial masses of the clusters are marked with different colors.
\label{fig:P_massratio}}
\end{figure}
\section{Discussion}
\label{section:discussion}
\subsection{Origin of the Fossil Relativistic Electrons}
\label{section:CRe_sources}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/AGN_prob.pdf}
\caption{Numbers of AGNs with different 1.4 GHz luminosities that a shock may encounter during its propagation from $0.5R_{\mathrm{vir}}$ to $1.2R_{\mathrm{vir}}$, assuming different cluster masses (colors). The case of Abell 115 is also shown (blue).
}\label{fig:AGN_prob}
\end{figure}
The fossil relativistic electrons may have several possible origins, which include previous merger-caused shock acceleration, turbulence acceleration, and AGN injection \citep{Kang2017}, as found in Abell 3411-3412 \citep{VanWeeren2017_AGN}. For example, \citet{zuhone2021b,zuhone2021A} revealed in their simulations that the CRe from AGN jets are likely transported to large radii ($\sim 0.6$ Mpc) and form bubbles with morphologies similar to radio relics.
Here we focus only on the contribution of the AGN by presenting a quantitative estimation although other possibilities are also worthy of carefully investigations.
We calculate the probability of a merger-induced shock encountering active AGNs during their outward propagation by modeling the galaxy-AGN distribution in clusters.
First, using the NFW model with a concentration parameter $c=3.59$ \citep{Hennig2017} to describe the density profile of cluster and assuming only one galaxy is formed in each subhalo, the number distribution of member galaxies as a function of subhalo's mass in a cluster is given by \citet{Jinag2016_halostat}
\begin{equation}
dN/d\ln{\psi}=\gamma\,\psi^{\alpha}\,\text{exp}(-\beta\,\psi^{\omega}
\end{equation}
where $\psi \equiv m/M_{H}$ is the ratio between the mass of subhalos $m$ and the mass of cluster halo $M_{\mathrm{H}}$, and $\gamma = 0.22,\ \alpha=-0.91,\ \beta = 6.00,\ \omega = 3.00$ are the parameters constrained by the observations.
Since dwarf-to-giant ratio (DGR) of galaxies does not change significantly with radius in the cluster \citep{Kopylova2013_DGR}, it is reasonable to assume that the spatial distribution of galaxies are uncoupled with galaxies' mass.
On the other hand, by investigating the properties of 2215 galaxies each hosting a radio-loud AGN, \citet{Best2005} found that the fraction of galaxies showing AGN activity depends on the mass of the galaxy and can be described by a broken power-law
\begin{equation}\label{AGN_fraction}
f_{\text {radio-loud }}=A\left(\frac{m_{\mathrm{BH}}}{10^{8}\, \text{M}_{\odot}}\right)^{\alpha}\left[\left(\frac{L_{\mathrm{1.4\, GHz}}}{L_{*}}\right)^{\beta}+\left(\frac{L_{\mathrm{1.4\, GHz}}}{L_{*}}\right)^{\gamma}\right]^{-1}
\end{equation}
where $L_{\mathrm{1.4\, GHz}}$ is AGN's luminosity observed at 1.4 GHz, and the best fit parameter are $A = 0.0055\pm 0.0004,\ \beta = 0.35 \pm 0.03,\ \gamma = 1.54\pm 0.11,\ L_{*}=(2.5\pm0.4)\times 10^{24}\ \mathrm{W}\mathrm{Hz}^{-1}$. $m_{\mathrm{BH}}$ is the mass of the black hole in the subhalo, which scales with $m$ \citep{Aversa2015}:
\begin{equation}
m_{\mathrm{BH}}=N \times\left[\left(\frac{m}{M_{\mathrm{b}}}\right)^{\alpha}+\left(\frac{m}{M_{\mathrm{b}}}\right)^{\omega}\right]^{-1}
\end{equation}
where $\log{N}=8.0,\ \alpha=-1.10,\ \omega=-0.80$, and $M_{\mathrm{b}} = 10^{11.90}\, \text{M}_{\odot}$. With these relations it will be straightforward to estimate how many AGNs are expected at a specific radius in a cluster with a given mass.
Next, assuming that the shock possesses $\Omega=0.22\pi$, which is the average value of the observation sample mentioned in Section~\ref{section:solid_angle},
for cluster with a mass of
$2\times 10^{14}\,\text{M}_{\odot}$, $5\times 10^{14}\,\text{M}_{\odot}$, and $8\times 10^{14}\,\text{M}_{\odot}$ at $z=0.2$, we calculate the numbers of AGNs with different radio luminosities that a shock may encounter when it travels from $0.5\,R_{\mathrm{vir}}$ to $1.2\, R_{\mathrm{vir}}$, and show the results in Figure~\ref{fig:AGN_prob}, where different colors mark the predictions for different cluster mass.
The results indicate that a typical shock does have the chance to interact with the AGN activity during its propagation, especially for those with a relatively low $L_{\mathrm{1.4GHz}}$, and the more massive the cluster is, the bigger are the chances for the shock to encounter active AGNs. This matches the observation of Abell 115 very well where
a giant radio relic is currently coincident with a radio galaxy
(0053+26B, $L_{\mathrm{1.5GHz}} = 1.0\times 10^{32}\ \mathrm{erg}\ \mathrm{ s}^{-1} \mathrm{ Hz}^{-1}$; \citealt{Gregorini1989}).
Detailed investigation, however, requires magnetohydrodynamics (MHD) simulations of AGN jets and shock acceleration, which is beyond the scope of this paper.
\subsection{Impact on the Detection of EoR signals}
\label{section:radioarray}
\begin{table}
\centering
\caption{SKA1-Low and HERA Arrays}
\label{table:radioarray_para}
\begin{tabular}{ccc}
\hline
& SKA1-Low & HERA \\
\hline
Frequency range/MHz &50$\sim$ 350 & 50$\sim$200 \\
Field of view/deg$^{2}$ & $20.77$ & $9^{\circ}\times 9^{\circ}$ \\
Survey area & - & $8^{\circ}\times 180^{\circ}$ \\
Sensitivity & $5\,\mu$Jy/1000hr & $50\,\mu$Jy/100hr \\
Spatial resolution & $7''$ & $11'$ \\
\hline
\end{tabular}
\begin{justify}
Data sources: \citet{Deboer2017} for HERA and \citet{SKA_design, Zheng2020_SKAdeepfield} for SKA1-Low.
\end{justify}
\end{table}
The spatially extended low frequency radio radiation of radio relics may cause severe foreground contamination in the observations of the cosmic 21 cm signals from the EoR with next generation instruments such as the SKA \citep{SKA_design} and the Hydrogen Epoch of Reionization Array (HERA) \citep{Deboer2017}. Although galaxy clusters are treated as foreground contaminating sources, the studies have been focused on the intergalactic medium located at cluster outskirts \citep{KESHET20041119} and radio halos \citep{Li2019}), while the impact of radio relics are barely constrained \citep{Jelic2008_foreground,Gleser2008_foreground}. In this subsection we incorporate the designs of SKA1-low and HERA arrays with our radio relic model to present a estimate of the low frequency radio contamination caused by radio relics in the 21 cm observation.
\edited{By rescaling our simulation to the required size of sky patch, whose cumulative flux distributions are shown in Figure~\ref{fig:radio_array}},
we find that typically \edited{34.89} (50 MHz) and \edited{22.64} (158 MHz) radio relics with $S > 5\mu $Jy will appear in the $20.77\ \text{deg}^{2}$ field of view of SKA1-low deep field. For HERA, \edited{70.86} (50 MHz) and \edited{37.87} (158 MHz) radio relics with $S > 50\mu $Jy will appear in the $9^{\circ}\times 9^{\circ}$ field of view, which are all undetectable because of its low spatial resolution ($11'$).
Their root-mean-square (rms) brightness temperature at 158 MHz calculated on a $10^{\circ} \times 10^{\circ}$ sky map with a pixel size of $20''$, together with a comparison with other contaminating sources, are summarized in Table~\ref{table:rms_Tb}. Clearly, radio relics do provided non-negligible contamination in the detection of the EoR signals, the level of which is \edited{about $1/6$ of radio halos \citep{Li2019}, and similar to galactic free-free emission, which indicates they need} to be treated very carefully in disentangling the EoR signals from the overwhelming foreground.
\begin{figure}
\includegraphics[width=\columnwidth]{figure/Radio_array.pdf}
\caption{Cumulative flux distributions of the simulated relics at 50 and 158 MHz, which are obtained \edited{by rescaling our simulation to} a $20.77\ \text{deg}^{2}$ sky patch, (i.e., the size of the planned SKA deep field). Dash lines: the SKA1-Low ($5\,\mu$Jy) and HERA ($50\,\mu$Jy) sensitivity for a 100-hour observation.
\label{fig:radio_array}}
\end{figure}
\begin{table}
\centering
\caption{The rms Brightness Temperature at 158 MHz of Several EoR Foreground Components (Unit: mK)}
\label{table:rms_Tb}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{cc}
\hline
Component & $T_{\mathrm{158\,MHz}}$ \\
\hline
Radio Relics & \ \edited{312} \\
Radio Halos & $1.81^{+5.28}_{-1.13} \times 10^{3}$ \\
Galactic synchrotron & $2.52 \times 10^{5}$ \\
Galactic free-free & 200 \\
Point sources & $5.90 \times 10^{7}$\\
EoR signal & 11.3 \\
\hline
\end{tabular}
\begin{justify}
The $T_{\mathrm{158\,MHz}}$ of radio relics is from the simulation in this work, and results for other foreground components come from \citet{Li2019}. All of them are calculated on a $10^{\circ} \times 10^{\circ}$ sky map with a pixel size of $20''$.
\end{justify}
\end{table}
\subsection{Coulomb Collision}
\label{section:CC}
\begin{figure}
\includegraphics[width=\columnwidth]{figure/loss_rate.pdf}
\caption{Energy loss rates of inverse Compton scattering, synchrotron radiation, and Coulomb collision. Data are calculated for $z=0.2$ and $B=1\,\mu G$. \label{fig:loss_rate}}
\end{figure}
For a given critical frequency $\nu_{\mathrm{c}}$, the radio synchrotron emission is mainly concentrated in the frequency band of $\nu<100\nu_{\mathrm{c}}$, because the Bessel function $K_{5/3}(y)$ decreases rapidly as $y$ increases (equations~\ref{equ:syn_emit} and \ref{equ:syn_kernel}). Since $\nu_{\text{c}} = 3\,\gamma^{2}\,\nu_{\text{L}}\,\text{sin}\theta/2$, this gives the lower energy limit of the Lorentz factors $\gamma$
\begin{equation}
\gamma \geq \sqrt{\frac{\nu}{150 \nu_{\text{L}}}} \sim 4.88\times 10^{-5}\sqrt{\frac{\nu}{B}},
\end{equation}
Let $\nu_{\mathrm{min}}=50$ MHz, the typical lower limit frequency for the future radio interferometers such as SKA and HERA, we have $\gamma \gtrsim 100$ considering $B \sim \mu G$ in ICM. Incorporating these conditions with the total energy loss rate
\begin{equation}
\begin{gathered}
\left(\frac{d\gamma}{dt}\right)_{\text{total}} = -4.10 \times 10^{-5}\gamma^{2}\left(\frac{B}{1\,\mu G}\right)\\
-4.32 \times 10^{-4} \gamma^{2}(1+z)^{4}\\
-3.79 \times 10^{4}\left(\frac{n_{\mathrm{th}}}{1 \mathrm{~cm}^{-3}}\right) \times\left[1+\frac{1}{75} \ln \left(\gamma \frac{1 \mathrm{~cm}^{-3}}{n_{\mathrm{th}}}\right)\right],
\end{gathered}
\end{equation}
where the first two terms are same with those in equations~\ref{equ:syn_loss_rate} and \ref{equ:IC_loss_rate}, representing the energy loss caused by synchrotron emission and IC, and the third term corresponds to Coulomb collision (equation~\ref{equ:CC_loss_rate}), we may evaluate the impact of energy loss via Coulomb collision by switching on and off the third term. The results are summarized in Figure~\ref{fig:loss_rate}, where we show the loss rate calculated for the three energy loss mechanisms, and in Figure~\ref{fig:CC_compare}, where we show the evolution of the electrons number densities obtained by taking into account or ignoring the Coulomb collision, assuming $z=0.2$ and $B=1\,\mu G$. As can be seen in both figures, the effect of the Coulomb collision is actually not important.
\begin{figure}
\includegraphics[width=\columnwidth]{figure/CC_compare.pdf}
\caption{For $z=0.2$ and $B=1\,\mu G$, evolution of the electron energy spectra when Coulomb collision is switched on (solid) or off (dash) in the calculation.
Black dash line is the lower limit $\gamma = 100$ for $\nu \geq$ 50 MHz obtained in Section~\ref{section:CC}.
\label{fig:CC_compare}}
\end{figure}
\section{Conclusion}
\label{section:conclusion}
In this work we establish a semi-analytic model for radio relics by including the contribution of fossil relativistic electrons, whose properties are constrained by a sample of well-observed radio relics.
Using Press-Schechter formalism to simulate galaxies clusters and their merger history, we obtain a radio relics catalog based on our model in a \edited{$20^{\circ} \times 20^{\circ}$} sky patch at 50 MHz, 158 MHz, and 1.4 GHz, with which
the observed $P_{1400}-M_{\mathrm{vir}}$ relation can be successfully reconciled.
\edited{We predict that $9.6\%$ and $7.1\%$ clusters with $M_{\mathrm{vir}} > 1.2\times 10^{14}\,\mathrm{M}_{\odot}$ would host one or more relics at 50 MHz and 158 MHz, respectively, which are consistent with the result of $10 \pm 6\%$ given by the LoTSS DR2, whose target band is 120-168 MHz.}
We explore the probability of AGNs providing seed relativistic electrons for shocks to form radio relics by calculating how many radio-loud AGNs are expected to be encountered by the shock during its propagation from $0.5\ R_{\mathrm{vir}}$ and $1.2\ R_{\mathrm{vir}}$.
By comparing the rms brightness temperature of radio relics with those of other EoR foreground components, we find that radio relics are severe contaminating sources to EoR observations which needs serious treatment in future experiments.
We also show the effect of the Coulomb collision is not important in the calculation of the emission of radio relics.
\section*{Acknowledgements}
We would like to thank Congyao Zhang for helpful discussions \edited{and thank an anonymous referee for very constructive suggestions that helped to significantly improve the presentation of the paper}. This work was supported by the Ministry of Science and Technology of China (grant Nos. 2020SKA0110201, 2018YFA0404601, 2020SKA0110102), the National Natural Science Foundation of China (grant Nos. 12233005, 11973033, 11835009, 12073078, 11621303, U1531248, U1831205).
\section*{Data Availability}
The data underlying this article will be shared on reasonable request to the corresponding author.
\bibliographystyle{mnras}
|
1,108,101,565,675 | arxiv | \section{#1}}
\newcommand{\sect}[1]{{\bf{#1.} }}
\begin{document}
\title{Categorization of two-loop Feynman diagrams in the $\mathcal O(\alpha^2)$ correction to $e^+e^- \rightarrow ZH$
\thanks{This work was supported by the National Natural Science Foundation of China under Grant No. 11675185 and 12075251.}
}
\author{
Zhao Li$^{1,2)}$ \email{[email protected]} \quad
Yefan Wang$^{1,2)}$ \email{[email protected]} \quad
Quan-feng Wu$^{1,2)}$ \email{[email protected]}
}
\maketitle
\address {$^1$ Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China \\
$^2$ School of Physics Sciences, University of Chinese Academy of Sciences, Beijing 100039, China}
\begin{abstract}
The $e^+e^- \rightarrow ZH$ process is the dominant process for the Higgs boson production at the future Higgs factory.
In order to match the analysis on the Higgs properties with the highly precise experiment data,
it will be crucial to include the theoretical prediction to the next-to-next-to-leading order electroweak effect in
the production rate $\sigma(e^+e^-\rightarrow ZH)$.
In this inspiring work, we categorize the two-loop Feynman diagrams of the $\mathcal O(\alpha^2)$ correction to $e^+e^- \rightarrow ZH$
into 6 categories according to the relevant topological structures.
Although 25377 diagrams contribute to the $\mathcal O(\alpha^2)$ correction in total,
the number of the most challenging diagrams with seven denominators is 2250,
which contain only 312 non-planar diagrams with 151 independent types.
This categorization could be a valuable reference for the complete calculation in the future.
\end{abstract}
\begin{multicols}{2}
\section{Introduction}
The discovery of the Higgs boson in 2012 at the Large Hadron Collider (LHC) has opened a new era in the particle physics. In particular, the Higgs boson is regarded as the key to solve some challenging problems such as the problem of hierarchy, the origin of the neutrino mass and the dark matter problem. Since then, the precise measurements of the Higgs boson properties have become top priorities of both experimental and theoretical particle physics.
Although the LHC can produce a lot of Higgs bosons, the enormously complicated QCD backgrounds make the sufficiently precise measurements hard to achieve.
To precisely measure the properties of the Higgs boson, the next generation $e^+e^-$ colliders have been proposed as the Higgs factories aiming at much higher accuracy of measurements. Compared with the LHC, the $e^+e^-$ colliders will have cleaner experimental conditions and higher luminosity. The candidates of the next generation $e^+e^-$ colliders include Circular Electron Positron Collider (CEPC) \cite{CEPCStudyGroup:2018ghi,CEPCStudyGroup:2018rmc}, International Linear Collider (ILC) \cite{Baer:2013cma,Behnke:2013xla,Bambade:2019fyw} and Future Circular Collider (FCC-ee) \cite{Gomez-Ceballos:2013zzn,Abada:2019zxq,Abada:2019lih}. All of them are designed to operate at the center-of-mass energy $\sqrt{s} \sim 240 - 250$ GeV to precisely measure the Higgs boson. In this energy range, the processes to produce Higgs bosons are $e^+e^- \rightarrow ZH$ (Higgsstrahlung), $e^+e^- \rightarrow \nu_e \bar \nu_e H$ ($W$ fusion) and $e^+e^- \rightarrow e^+e^- H$ ($Z$ fusion). The dominant Higgs production process is the Higgsstrahlung. And the recoil mass method can be applied to identify the Higgs boson candidates \cite{McCullough:2013rea}. Then the Higgs boson production can be disentangled in a model-independent way.
At the CEPC, over one million Higgs boson events are planned to produce in total with the expected integrated luminosity of $5.6 \text{ ab}^{-1}$ \cite{An:2018dwb}. With these sizable events, many important properties of the Higgs boson can be measured in a high precision. For example, the cross section $\sigma(e^+e^-\rightarrow ZH)$ can be measured to the extremely high precision of $0.51\%$. Since the Higgs boson candidate events can be identified by recoil mass method, the measurements of $HZZ$ coupling mainly depend on the precise measurement of $\sigma(e^+e^-\rightarrow ZH)$. Consequently, the $HZZ$ coupling can be measured with an accuracy of $0.25\%$ at the CEPC, which is much better than at the HL-LHC \cite{ATLAS:2018jlh,CMS:2013xfa}. Such precise measurements give the CEPC unprecedented reach into the new physics scenarios which are difficult to probe at the LHC \cite{Craig:2014una}.
In the natural supersymmetry (SUSY), typically dominant effect on Higgs precision from colored top partners may have blind spots \cite{Ellwanger:2009dp,Fan:2014axa}. The blind spots can be filled in by the measurement of $\sigma(e^+e^-\rightarrow ZH)$, which is sensitive to loop-level corrections to the tree-level $HZZ$ coupling \cite{Craig:2014una}. When the $\delta \sigma_{ZH}$ approaches to $0.2\%$, the more constraints for $m_{\tilde t_1}$ and $m_{\tilde t_2}$ can be observed \cite{Essig:2017zwe}. And the $HZZ$ coupling play a important role in the study of Electroweak Phase Transition (EWPT). In the real scalar singlet model \cite{Choi:1993cv,McDonald:1993ey,Profumo:2007wc}, the first order phase transition tend to predict a large suppression of the $HZZ$ coupling ranging from $1\%$ to as much as $30\%$ \cite{Huang:2016cjm}. With the expected sensitivity of $\delta g_{HZZ}$ at the CEPC, the models with a strongly first order phase transition can be tested.
Besides the improvement of the experimental accuracy, the higher precision theoretical prediction for $\sigma(e^+e^-\rightarrow ZH)$ is also demanded to match the precision of experimental measurements. Since the theoretical uncertainties in the CEPC is not considered now, it is necessary to pay attention to the theoretical uncertainties of $\sigma(e^+e^-\rightarrow ZH)$. The next-to-leading-order (NLO) electroweak (EW) corrections to $\sigma(e^+e^-\rightarrow ZH)$ have been calculated two decades before \cite{Fleischer:1982af,Kniehl:1991hk,Denner:1992bc}. And the next-to-next-to-leading-order (NNLO) EW-QCD corrections have also been calculated in recent years \cite{Gong:2016jys,Sun:2016bel,Chen:2018xau}. The results show that the NNLO EW-QCD corrections increase the cross section more than one percent, which is larger than the expected experimental accuracies of the CEPC. On the other hand, it indicates the NNLO EW corrections can be still significant. And it is necessary to emphasize that the corrections depend crucially on the renormalization schemes. In $\alpha(0)$ scheme, the NNLO EW-QCD corrections are about $1.1\%$ of the Leading-order (LO) cross section. But in $G_\mu$ scheme, the NNLO EW-QCD corrections only amount to $0.3\%$ of LO cross section \cite{Sun:2016bel}. And the sensitivity to the different scheme is reduced at NNLO EW-QCD corrections compared to NLO EW corrections. Consequently the EW-QCD $\sigma_{\text{NNLO}}$ ranges from $231\text{ fb}$ to $233\text{ fb}$.
Hence, the missing two-loop corrections to $\sigma(e^+e^-\rightarrow ZH)$ can still lead to an intrinsic uncertainty of $\mathcal O (1\%)$ \cite{Freitas:2019bre}, which is still larger than the experimental accuracy. Since $\sigma(e^+e^-\rightarrow ZH)$ is proportional to the square of the $HZZ$ coupling, the theoretical uncertainties also have a significant impact on the extracting of $HZZ$ coupling. Then the accuracy of $HZZ$ coupling ($0.25\%$) in the CEPC may not be achieved due to the large theoretical uncertainties. We believe that if the full NNLO EW corrections to $\sigma(e^+e^-\rightarrow ZH)$ can be calculated, the scheme dependence can be further reduced. More importantly, the theoretical uncertainties will be sufficiently small. Consequently, the accuracy of $HZZ$ coupling ($0.25\%$) in the CEPC can be achieved.
Due to the complicacy of EW interaction, there are more than 20 thousand Feynman diagrams contribute to the $\mathcal O(\alpha^2)$ correction of $e^+e^- \rightarrow ZH$. The complete calculations of these Feynman diagrams are huge projects. Therefore, in this paper we focus on the categorization of these Feynman diagrams. This categorization will be helpful for future calculations and analyses. In next section we will categorize the Feynman diagrams into six categories and dozens of subcategories. Then the conclusion is made.
\section{Categorization}
In the Feynman gauge, we obtain 25377 diagrams contributing to the $\mathcal O(\alpha^2)$ corrections of $e^+e^- \rightarrow ZH$ by using QGRAF \cite{Nogueira:1991ex}. And we use \textsc{FeynArts} \cite{Hahn:2000kx} to check the correctness to this procedure.
To categorize these diagrams, firstly we put the diagrams which can be factorized into two one-loop diagrams to the category $\mathcal C_1$. Then according to the number of denominators of each diagram, we categorize the remaining non-factorization two-loop diagrams into five categories $\mathcal C_2,\dots,\mathcal C_6$. Furthermore, according to the topologies of loop structures we categorize $\mathcal C_i$ into several subcategories $\{\mathcal C_{i,j}\}$. Since the quarks are regarded as massless except the top-quark, we use $\mathcal C_{i,j,a}$ to denote the diagrams without the top-quark in $\mathcal C_{i,j}$ and $\mathcal C_{i,j,b}$ with the top-quark. In this paper we use the Nickel index \cite{Batkovich:2014bla,nickel,Nagle:1966} to describe the topologies of loop structures. For reader's convenience here we briefly explain the Nickel notation and Nickel index.
Nickel notation is a labelling algorithm to describe connected undirected graphs with "simple" edges and vertices such as the topological structures of the Feynman diagrams. First, we should consider a connected graph with $n$ vertices and label these $n$ vertices by the integers $0$ through $n-1$ at random. Therefore, we can construct the sequence according to \cite{nickel, Batkovich:2014bla}:
\begin{equation}\begin{aligned}
& \text{vertices connected to vertex 0 } | \\
& \text{vertices connected to 1 excluding 0 } |\ \cdots\ |\\
& \text{vertices connected to vertex } i \text{
excluding 0 through } i-1\ | \\
& \cdots\ |.
\end{aligned}\end{equation}
Such as Fig.\ref{fig:nickel-a} can be represented as $12|223|3|$. Otherwise, we can use the label "$e$"s to describe the external lines in the diagrams. For example, the Nickel notation of Fig.\ref{fig:nickel-b} is $ee11|ee|$. With different labeling strategy, one diagram can be represented as some different Nickel notations, which describe the same diagram up to a topological homeomorphism. For simplicity, the Nickel index algorithm is to find the "minimal" Nickel notation, which is called Nickel index. Consequently, the diagram and its Nickel index are in one-to-one correspondence. For example, the Nickel index of Fig.\ref{fig:nickel-a} is $1123|23|||$. The package \textsc{GraphState} \cite{Batkovich:2014bla} is one useful tool for constructing the Nickel index. The detail of the Nickel index algorithm can be found in Ref. \cite{Batkovich:2014bla}.
\begin{figure}[H]
\begin{center}
\subfigure[]{
\centering
\includegraphics[width=2.6cm]{Nickel_a_.pdf}
\label{fig:nickel-a}
}\hspace{3mm}
\subfigure[]{
\centering
\includegraphics[height=2.6cm]{Nickel_b_.pdf}
\label{fig:nickel-b}
}
\caption{Nickel notation and Nickel index}
\end{center}
\end{figure}
In this paper, the topological structures of the diagrams with one vertex connecting to one or two external legs are regarded as equivalent. For example, the topological structures of two diagrams in Fig.\ref{fig:equiv-topo} can be regarded as equivalent.
\begin{figure}[H]
\begin{center}
\subfigure[]{
\centering
\includegraphics[width=4cm]{new_f1.pdf}
}\hspace{-4mm}
\subfigure[]{
\centering
\includegraphics[width=4cm]{new_f2.pdf}
}
\caption{The example of equivalent topological structures}
\label{fig:equiv-topo}
\end{center}
\end{figure}
\subsection{Category $\mathcal C_1$}
The category $ \mathcal C_1$ includes the Feynman diagrams which can be factorized into two one-loop diagrams. So the calculations of diagrams in $ \mathcal C_1$ can be regarded as the one-loop calculations. There are 7908 diagrams in $\mathcal C_1$.
The subcategory $\mathcal C_{1,1}$ includes diagrams which contain at least one one-loop vacuum bubble diagram. $\mathcal C_{1,1}$ includes 2117 diagrams, some of which contain top-quark. $\mathcal C_{1,1,a}$ includes 2055 diagrams and $\mathcal C_{1,1,b}$ includes 62 diagrams. We put diagram \#47 as the representative of $\mathcal C_{1,1,a}$ and diagram \#4418 as the representative of $\mathcal C_{1,1,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_47.pdf}
\caption{Diagram \#47 (representative of $\mathcal C_{1,1,a}$) }
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.45\textwidth]{new_4418.pdf}
\caption{Diagram \#4418 (representative of $\mathcal C_{1,1,b}$) }
\end{figure}
The subcategory $\mathcal C_{1,2}$ includes diagrams which contain self-energy corrections. And the diagrams $\mathcal C_{1,2}$ do not contain vacuum bubble diagrams. $\mathcal C_{1,2}$ includes 5513 diagrams, some of which contain top-quark. $\mathcal C_{1,2,a}$ includes 4775 diagrams and $\mathcal C_{1,1,b}$ includes 738 diagrams. We put diagram \#36 as the representative of $\mathcal C_{1,1,a}$ and diagram \#1035 as the representative of $\mathcal C_{1,1,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.45\textwidth]{new_36.pdf}
\caption{Diagram \#36 (representative of $\mathcal C_{1,2,a}$) }
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_1035.pdf}
\caption{Diagram \#1035 (representative of $\mathcal C_{1,2,b}$) }
\end{figure}
The subcategory $\mathcal C_{1,3}$ includes diagrams which contain two vertex corrections. $\mathcal C_{1,3}$ includes 278 diagrams, some of which contain top-quark. $\mathcal C_{1,3,a}$ includes 260 diagrams and $\mathcal C_{1,3,b}$ includes 18 diagrams. We put diagram \#6983 as the representative of $\mathcal C_{1,3,a}$ and diagram \#23660 as the representative of $\mathcal C_{1,3,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_6983.pdf}
\caption{Diagram \#6983 (representative of $\mathcal C_{1,3,a}$) }
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_23660.pdf}
\caption{Diagram \#23660 (representative of $\mathcal C_{1,3,b}$) }
\end{figure}
\subsection{Category $\mathcal C_2$}
The Category $ \mathcal C_2$ includes non-factorizable two-loop Feynman diagrams with three denominators. We found that all diagrams in $\mathcal C_{2}$ are two-loop self-energy diagrams. $\mathcal C_2$ includes 18 diagrams, none of which contains top-quark. We put diagram \#519 as the representative of $ \mathcal C_2$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_519.pdf}
\caption{Diagram \#519 (representative of $\mathcal C_{2}$)}
\end{figure}
\subsection{Category $\mathcal C_3$}
The category $ \mathcal C_3$ includes non-factorizable two-loop Feynman diagrams with four denominators. According to the topologies of loop structures in $ \mathcal C_3$, we categorize them into 3 subcategories. There are 593 diagrams in $\mathcal C_3$.
The subcategory $\mathcal C_{3,1}$ includes diagrams which can be separated into tree-level diagrams and two-loop vacuum bubble diagrams. The topology of their loop structures can be represented as $112|2||$ in Nickel index. The calculation of two-loop vacuum bubble diagram has been well studied \cite{Davydychev:1992mt}. $\mathcal C_{3,1}$ includes 142 diagrams. We put diagram \#3961 as the representative of $ \mathcal C_{3,1}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_3961.pdf}
\caption{Diagram \#3961 (representative of $\mathcal C_{3,1}$)}
\end{figure}
The subcategory $\mathcal C_{3,2}$ includes two-loop self-energy diagrams. $\mathcal C_{3,2}$ includes 337 diagrams, none of which contains top-quark. We put diagram \#1 as the representative of $ \mathcal C_{3,2}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_1.pdf}
\caption{Diagram \#1 (representative of $\mathcal C_{3,2}$)}
\end{figure}
The subcategory $\mathcal C_{3,3}$ includes two-loop vertex correction diagrams. The topology of their loop structures can be represented as $e112|e2|e|$ in Nickel index. And the denominators of diagrams in $\mathcal C_{3,3}$ only depend on two external momenta.
$\mathcal C_{3,3}$ includes 114 diagrams, none of which contains top-quark. We put diagram \#191 as the representative of $ \mathcal C_{3,3}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_191.pdf}
\caption{Diagram \#191 (representative of $\mathcal C_{3,3}$)}
\end{figure}
\subsection{Category $\mathcal C_4$}
The category $ \mathcal C_4$ includes non-factorizable two-loop Feynman diagrams with five denominators. According to the topologies of loop structures in $ \mathcal C_4$, we categorize them into 3 subcategories. There are 4773 diagrams in $\mathcal C_4$.
The subcategory $\mathcal C_{4,1}$ includes two-loop self-energy diagrams. $\mathcal C_{4,1}$ includes 3266 diagrams, some of which contain top-quark. $\mathcal C_{4,1,a}$ includes 2565 diagrams and $\mathcal C_{4,1,b}$ includes 701 diagrams. We put diagram \#603 as the representative of $\mathcal C_{4,1,a}$ and diagram \#611 as the representative of $\mathcal C_{4,1,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_603.pdf}
\caption{Diagram \#603 (representative of $\mathcal C_{4,1,a}$)}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_611.pdf}
\caption{Diagram \#611 (representative of $\mathcal C_{4,1,b}$)}
\end{figure}
The subcategory $\mathcal C_{4,2}$ includes two-loop vertex correction diagrams. The topology of their loop structures can be represented as $e12|e23|3|e|$ in Nickel index. And the denominators of diagrams in $\mathcal C_{4,2}$ only depend on two external momenta. $\mathcal C_{4,2}$ include 637 diagrams, none of which contains top-quark. We put diagram \#2676 as the representative of $\mathcal C_{4,2}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_2676.pdf}
\caption{Diagram \#2676 (representative of $\mathcal C_{4,2}$)}
\end{figure}
The subcategory $\mathcal C_{4,3}$ includes two-loop vertex correction diagrams. The topology of their loop structures can be represented as $e112|3|e3|e|$ in Nickel index. And the denominators of diagrams in $\mathcal C_{4,3}$ only depend on two external momenta. $\mathcal C_{4,3}$ includes 870 diagrams, none of which contains top-quark. We put diagram \#3063 as the representative of $\mathcal C_{4,3}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_3063.pdf}
\caption{Diagram \#3063 (representative of $\mathcal C_{4,3}$)}
\end{figure}
\subsection{Category $ \mathcal C_5$}
The category $ \mathcal C_5$ includes non-factorizable two-loop Feynman diagrams with six denominators. According to the topologies of loop structures in $ \mathcal C_5$, we categorize them into six subcategories. There are 9835 diagrams in $\mathcal C_5$.
The subcategory $\mathcal C_{5,1}$ includes two-loop planar triangle diagrams. The topology of their loop structures can be represented as $e12|e3|34|4|e|$ in Nickel index. And the denominators of diagrams in $\mathcal C_{5,1}$ only depend on two external momenta. $\mathcal C_{5,1}$ includes 4897 diagrams, some of which contain top-quark. $\mathcal C_{5,1,a}$ includes 3966 diagrams and $\mathcal C_{5,1,b}$ includes 931 diagrams. We put diagram \#1325 as the representative of $\mathcal C_{5,1,a}$ and diagram \#16206 as the representative of $\mathcal C_{5,1,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_1325.pdf}
\caption{Diagram \#1325 (representative of $\mathcal C_{5,1,a}$)}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_16206.pdf}
\caption{Diagram \#16206 (representative of $\mathcal C_{5,1,b}$)}
\end{figure}
The subcategory $\mathcal C_{5,2}$ includes two-loop planar diagrams. The topology of their loop structures can be represented as $e12|e23|4|e4|e|$ in Nickel index. $\mathcal C_{5,2}$ includes 184 diagrams, none of which contains top-quark.
We put diagram \#3613 as the representative of $ \mathcal C_{5,2}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_3613.pdf}
\caption{Diagram \#3613 (representative of $\mathcal C_{5,2}$)}
\end{figure}
The subcategory $\mathcal C_{5,3}$ includes two-loop planar diagrams. The topology of their loop structures can be represented as $e12|e3|e4|44||$ in Nickel index. And the denominators of diagrams in $\mathcal C_{5,3}$ only depend on two external momenta. $\mathcal C_{5,3}$ includes 4067 diagrams, some of which contains top-quark. $\mathcal C_{5,3,a}$ includes 3260 diagrams and $\mathcal C_{5,3,b}$ includes 807 diagrams. We put diagram \#14794 as the representative of $\mathcal C_{5,3,a}$ and diagram \#14812 as the representative of $\mathcal C_{5,3,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_14794.pdf}
\caption{Diagram \#14794 (representative of $\mathcal C_{5,3,a}$)}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_14812.pdf}
\caption{Diagram \#14812 (representative of $\mathcal C_{5,3,b}$)}
\end{figure}
The subcategory $\mathcal C_{5,4}$ includes two-loop planar diagrams. The topology of their loop structures can be represented as $e112|3|e4|e4|e|$ in Nickel index. $\mathcal C_{5,4}$ includes 116 Feynman diagrams, none of which contains top-quark.
We put diagram \#3845 as a representative of $\mathcal C_{5,4}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_3845.pdf}
\caption{Diagram \#3845 (representative of $\mathcal C_{5,4}$)}
\end{figure}
The subcategory $\mathcal C_{5,5}$ includes two-loop nonplanar triangle diagrams. The topology of their loop structures can be represented as $e12|34|34|e|e|$ in Nickel index. And the denominators of diagrams in $\mathcal C_{5,5}$ only depend on two external momenta. $\mathcal C_{5,5}$ includes 560 diagrams, some of which contain top-quark. $\mathcal C_{5,5,a}$ includes 442 diagrams and $\mathcal C_{5,5,b}$ includes 118 diagrams. We put diagram \#1267 as the representative of $\mathcal C_{5,5,a}$ and diagram \#11100 as the representative of $\mathcal C_{5,5,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_1267.pdf}
\caption{Diagram \#1267 (representative of $\mathcal C_{5,5,a}$)}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.45\textwidth]{new_11100.pdf}
\caption{Diagram \#11100 (representative of $\mathcal C_{5,5,b}$)}
\end{figure}
The subcategory $\mathcal C_{5,6}$ includes two-loop nonplanar diagrams. The topology of their loop structures can be represented as $e12|e34|34|e|e|$ in Nickel index. $\mathcal C_{5,6}$ includes 11 diagrams, none of which contains top-quark.
We put diagram \#3602 as the representative of $ \mathcal C_{5,6}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_3602.pdf}
\caption{Diagram \#3602 (representative of $\mathcal C_{5,6}$)}
\end{figure}
\subsection{Category $ \mathcal C_6$}
The category $ \mathcal C_6$ includes non-factorizable two-loop Feynman diagrams with seven denominators. According to the topologies of loop structures in $ \mathcal C_6$, we categorize them into 4 subcategories. There are 2250 diagrams in $\mathcal C_6$.
The subcategory $\mathcal C_{6,1}$ includes two-loop planar double-box diagrams. The topology of their loop structures can be represented as $e12|e3|34|5|e5|e|$ in Nickel index. $\mathcal C_{6,1}$ includes 446 diagrams, some of which contain top-quark. $\mathcal C_{6,1,a}$ includes 424 diagrams and $\mathcal C_{6,1,b}$ includes 22 diagrams. We put diagram \#23202 as the representative of $\mathcal C_{6,1,a}$ and diagram \#23228 as the representative of $\mathcal C_{6,1,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_23202.pdf}
\caption{Diagram \#23202 (representative of $\mathcal C_{6,1,a}$)}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_23228.pdf}
\caption{Diagram \#23228 (representative of $\mathcal C_{6,1,b}$)}
\end{figure}
The subcategory $\mathcal C_{6,2}$ includes two-loop planar diagrams. The topology of their loop structures can be represented as $e1|22|3|e4|e5|e6||$ in Nickel index. $\mathcal C_{6,2}$ includes 688 diagrams, some of which contain top-quark. $\mathcal C_{6,2,a}$ includes 580 diagrams and $\mathcal C_{6,2,b}$ includes 108 diagrams. We put diagram \#24690 as the representative of $\mathcal C_{6,2,a}$ and diagram \#24708 as the representative of $\mathcal C_{6,2,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_24690.pdf}
\caption{Diagram \#24690 (representative of $\mathcal C_{6,2,a}$)}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_24708.pdf}
\caption{Diagram \#24708 (representative of $\mathcal C_{6,2,b}$)}
\end{figure}
The subcategory $\mathcal C_{6,3}$ includes two-loop planar diagrams. The topology of their loop structures can be represented as $e12|e3|e4|45|5|e|$ in Nickel index. $\mathcal C_{6,3}$ includes 804 diagrams, some of which contain top-quark. $\mathcal C_{6,3,a}$ includes 733 diagrams and $\mathcal C_{6,2,b}$ includes 71 diagrams. We put diagram \#23886 as the representative of $\mathcal C_{6,3,a}$ and diagram \#23907 as the representative of $\mathcal C_{6,3,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_23886.pdf}
\caption{Diagram \#23886 (representative of $\mathcal C_{6,3,a}$)}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_23907.pdf}
\caption{Diagram \#23907 (representative of $\mathcal C_{6,3,b}$)}
\end{figure}
The subcategory $\mathcal C_{6,4}$ is the most challenging subcategory which includes two-loop nonplanar double-box diagrams. The topology of their loop structures can be represented as $e12|e3|45|45|e|e|$ in Nickel index. $\mathcal C_{6,4}$ includes 312 diagrams, some of which contain top-quark. $\mathcal C_{6,4,a}$ includes 301 diagrams and $\mathcal C_{6,4,b}$ includes 11 diagrams.
Since some amplitudes of diagrams can be obtained by replacing coupling factors or masses from other diagrams, $\mathcal C_{6,4}$ can be reduced a subset $\mathcal C_{6,4}^{ind}$ which only includes "independent" diagrams.
By this way we find that in $\mathcal C_{6,4,a}^{ind}$ there are 142 independent diagrams including 48 diagrams of weak correction and 94 diagrams of mixed weak-QED correction.
In $\mathcal C_{6,4,b}^{ind}$ there are 9 independent diagrams including 3 diagrams of weak correction and 6 diagrams of mixed weak-QED correction.
We put diagram \#22890 as the representative of $\mathcal C_{6,4,a}$ and diagram \#22909 as the representative of $\mathcal C_{6,4,b}$.
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_22890.pdf}
\caption{Diagram \#22890 (representative of $\mathcal C_{6,4,a}$)}
\end{figure}
\begin{figure}[H]
\centering
\includegraphics[width=0.4\textwidth]{new_22909.pdf}
\caption{Diagram \#22909 (representative of $\mathcal C_{6,4,b}$)}
\end{figure}
\section{Conclusion}
In this paper, we categorize the two-loop Feynman
diagrams contributing to the $\mathcal O(\alpha^2)$ corrections of $e^+e^- \rightarrow ZH$
into 6 categories and dozens of subcategories. The most challenging subcategory is $\mathcal C_{6,4}$, which includes 312 two-loop non-planner double-box diagrams. And there are only 151 independent diagrams in $\mathcal C_{6,4}$.
We believe that the calculations of these Feynman diagrams can be implemented conveniently because of this categorization.
\vspace{1ex}This work was supported by the National Natural Science Foundation of China under Grant No. 11675185 and 12075251.
The authors want to thank Ayres Freitas, Hao Liang, Tao Liu for helpful discussions.
\end{multicols}
\begin{multicols}{2}
|
1,108,101,565,676 | arxiv | \section{Introduction}
The following theorem is one of the main results of this paper.
Our proof of Theorem \ref{f-thm1.1}
uses the framework of the toric Mori theory developed by
\cite{reid}, \cite{fujino-notes},
\cite{fujino-equiv}, \cite{fujino-osaka}, \cite{fujino-sato}, and so on.
\begin{thm-a}[Theorem \ref{f-thm4.2.3} and
Corollary \ref{f-cor4.2.4}]\label{f-thm1.1}
Let $X$ be a $\mathbb Q$-Gorenstein
projective toric $n$-fold and let $D$ be an ample Cartier divisor on $X$.
Then $K_X+(n-1)D$ is pseudo-effective if and only if $K_X+(n-1)D$ is
nef.
In particular, if $X$ is Gorenstein,
then \[H^0(X, \mathcal O_X(K_X+(n-1)D))\ne 0\]
if and only if the complete linear system $| K_X+(n-1)D|$ is
basepoint-free.
\end{thm-a}
This theorem was inspired by Lin's paper (see \cite{lin}).
Our proof of Theorem \ref{f-thm1.1} depends on the following new
estimates of lengths
of extremal rays of birational type for toric varieties.
\begin{thm-a}[Theorem \ref{f-thm3.2.1}]\label{f-thm1.2}
Let $f:X\to Y$ be a projective toric morphism with $\dim X=n$.
Assume that $K_X$ is $\mathbb Q$-Cartier.
Let $R$ be a $K_X$-negative
extremal ray of $\NE(X/Y)$ and let $\varphi_R:X\to W$
be the contraction morphism associated to $R$. We put
\[l(R)=\min_{[C]\in R} (-K_X\cdot C).\]
and call it the length of $R$.
Assume that $\varphi_R$ is birational.
Then we obtain
\[
l(R)<d+1,
\]
where \[d=\max_{w\in W} \dim \varphi^{-1}_R(w)\leq n-1.\]
When $d=n-1$, we have a sharper inequality
\[
l(R)\leq d=n-1.
\]
In particular, if
$l(R)=n-1$,
then $\varphi_R:X\to W$ can be described as follows.
There exists a torus invariant smooth point $P\in W$ such that
$\varphi_R:X\to W$
is a weighted blow-up at $P$ with the weight $(1, a, \cdots, a)$ for some
positive integer $a$.
In this case, the exceptional locus $E$ of $\varphi_R$ is a torus invariant
prime divisor and is isomorphic to $\mathbb P^{n-1}$.
Moreover, $X$ is $\mathbb Q$-factorial in a neighborhood of
$E$.
\end{thm-a}
Theorem \ref{f-thm1.2} supplements \cite[Theorem 0.1]{fujino-notes}
(see also \cite[Theorem 3.13]{fujino-equiv}).
We will see that the estimates obtained in Theorem \ref{f-thm1.2}
are the best by constructing some examples explicitly
(see Examples \ref{f-ex3.3.1} and \ref{f-ex3.3.2}).
For lengths of extremal rays for non-toric varieties,
see \cite{kawamata}. As an application
of Theorem \ref{f-thm1.2}, we can prove the following theorem
on lengths of extremal rays for $\mathbb Q$-factorial toric varieties.
\begin{thm-a}[Theorem \ref{f-thm3.2.9}]\label{f-thm1.3}
Let $X$ be a $\mathbb Q$-Gorenstein
projective toric $n$-fold with $\rho (X)\geq 2$.
Let $R$ be a $K_X$-negative extremal ray of
$\NE(X)$ such that
\[
l(R)=\min _{[C]\in R}(-K_X\cdot C)>n-1.
\]
Then the extremal contraction $\varphi_R:X\to W$ associated to
$R$ is a $\mathbb P^{n-1}$-bundle over $\mathbb P^1$.
\end{thm-a}
As a direct easy consequence of Theorem \ref{f-thm1.3}, we obtain
the following corollary, which supplements Theorem \ref{f-thm1.1}.
\begin{cor-a}[Corollary \ref{f-cor4.2.5}]\label{f-cor1.4}
Let $X$ be a $\mathbb Q$-Gorenstein projective
toric $n$-fold and let $D$ be an ample Cartier divisor
on $X$.
If $\rho (X)\geq 3$, then $K_X+(n-1)D$ is always nef.
More precisely,
if $\rho (X)\geq 2$ and $X$ is not a $\mathbb P^{n-1}$-bundle
over $\mathbb P^1$, then $K_X+(n-1)D$ is nef.
\end{cor-a}
In this paper, we also give some generalizations of
Fujita's freeness and very ampleness for toric varieties
based on our powerful vanishing
theorem (see \cite{fujino-vanishing} and \cite{fujino-toric}).
As a very special case of our generalization of Fujita's freeness for toric varieties
(see Theorem \ref{f-thm4.1.1}),
we can easily recover some parts of Lin's theorem (see \cite[Main Theorem A]{lin}).
\begin{thm-a}[Corollary \ref{f-cor4.1.2}]\label{f-thm1.5}
Let $X$ be an $n$-dimensional projective toric variety
and let $D$ be an ample Cartier divisor on $X$. Then
the reflexive sheaf $\mathcal O_X(K_X+(n+1)D)$ is generated
by its global sections.
\end{thm-a}
By the same way, we can obtain a generalization of Fujita's very ampleness
for toric varieties (see Theorem \ref{f-thm4.1.8}).
\begin{thm-a}[Theorem \ref{f-thm4.1.6}]\label{f-thm1.6}
Let $f:X\to Y$ be a proper surjective toric
morphism, let $\Delta$ be a reduced torus invariant
divisor on $X$ such that $K_X+\Delta$ is Cartier,
and let $D$ be an $f$-ample Cartier divisor on $X$.
Then $\mathcal O_X(K_X+\Delta+kD)$ is $f$-very ample for
every $k\geq \max_{y\in Y} \dim f^{-1}(y) +2$.
\end{thm-a}
For the precise statements of our generalizations of
Fujita's freeness and very ampleness for toric varieties,
see Theorems \ref{f-thm4.1.1} and \ref{f-thm4.1.8}.
We omit them here since they are technically complicated.
This paper is organized as follows.
In Section \ref{f-sec2}, we collect some basic definitions and
results. In subsection \ref{f-subsec2.1}, we
explain the basic concepts of the toric geometry.
In subsection \ref{f-subsec2.2}, we recall the definitions of
{\em{the Kleiman--Mori cone}}, {\em{the nef cone}}, {\em{the ample cone}},
and {\em{the pseudo-effective cone}} for toric projective morphisms,
and some related results. Section \ref{f-sec3} is the main part of this paper.
After recalling the known estimates of lengths of extremal rays
for projective toric varieties in
subsection \ref{f-subsec3.1}, we give
new estimates of lengths of extremal rays of toric birational contraction
morphisms in subsection \ref{f-subsec3.2}.
In subsection \ref{f-subsec3.3}, we see that the estimates
obtained in subsection \ref{f-subsec3.2} are the best by
constructing some examples explicitly.
Section \ref{f-sec4} treats
Fujita's freeness and very ampleness for toric varieties.
The results in subsection \ref{f-subsec4.1} depend
on our powerful vanishing theorem
for toric varieties and are independent of our estimates of lengths of
extremal rays for toric varieties. Therefore,
subsection \ref{f-subsec4.1} is independent of
the other parts of this paper.
In subsection \ref{f-subsec4.2}, we discuss Lin's problem (see \cite{lin}) related to
Fujita's freeness for toric varieties. We use our new estimates of
lengths of extremal rays in this subsection.
Subsection \ref{f-subsec4.3} is a supplement to Fujita's paper:~\cite{fujita}.
This paper contains various supplementary results for \cite{fujita},
\cite{fulton}, \cite{lin}, and so on.
\begin{ack}\label{f-ack}
The first author was partially supported by JSPS KAKENHI Grant
Numbers JP16H03925, JP16H06337.
If the first author remembers correctly, he prepared
a preliminary version of this paper around 2006 in Nagoya.
Then his interests moved to the minimal model program.
In 2011, he revised it in Kyoto.
The current version was written in Osaka.
He thanks the colleagues in Nagoya, Kyoto, and
Osaka very much.
\end{ack}
We will work over an arbitrary algebraically closed field throughout this
paper. For the standard notations of the minimal model program,
see \cite{fujino-fundamental} and \cite{fujino-foundation}.
For the toric Mori theory, we recommend the reader to see \cite{reid},
\cite[Chapter 14]{matsuki}, \cite{fujino-notes}, and \cite{fujino-sato} (see
also \cite{cls}).
\section{Preliminaries}\label{f-sec2}
This section collects some basic definitions and results.
\subsection{Basics of the toric geometry}\label{f-subsec2.1}
In this subsection, we recall the basic notion of toric varieties
and fix the notation. For the details, see \cite{oda}, \cite{fulton},
\cite{reid}, or \cite[Chapter 14]{matsuki} (see also \cite{cls}).
\begin{say}\label{f-say2.1.1}
Let $N\simeq \mathbb Z^n$ be a lattice of rank $n$.
A toric variety $X(\Sigma)$ is associated to a {\em{fan}} $\Sigma$,
a correction of convex cones $\sigma\subset N_\mathbb R =
N\otimes _{\mathbb Z}\mathbb R$ satisfying:
\begin{enumerate}
\item[(i)]
Each convex cone $\sigma$ is a rational polyhedral cone in the sense that
there are finitely many
$v_1, \cdots, v_s\in N\subset N_{\mathbb R}$ such
that
\[
\sigma=\{r_1v_1+\cdots +r_sv_s; \ r_i\geq 0\}=
:\langle v_1, \cdots, v_s\rangle,
\]
and
it is strongly convex in the sense that
\[
\sigma \cap -\sigma=\{0\}.
\]
\item[(ii)] Each face $\tau$ of a convex cone $\sigma\in \Sigma$
is again an element in $\Sigma$.
\item[(iii)] The intersection of two cones in $\Sigma$ is a face of
each.
\end{enumerate}
\begin{defn}\label{f-def2.1.2}
The {\em{dimension}} $\dim \sigma$ of a cone $\sigma$ is
the dimension of the linear space
$\mathbb R\cdot \sigma=\sigma +(-\sigma)$ spanned
by $\sigma$.
We define the sublattice $N_{\sigma}$ of
$N$ generated (as a subgroup) by $\sigma\cap N$ as
follows:
\[
N_{\sigma}:=\sigma\cap N+(-\sigma\cap N).
\]
If $\sigma$ is a $k$-dimensional simplicial
cone, and $v_1,\cdots, v_k$ are the
first lattice points along the edges of $\sigma$,
the {\em{multiplicity}} of $\sigma$ is defined
to be the {\em{index}} of the lattice
generated by the $\{v_1, \cdots, v_k\}$ in the lattice $N_{\sigma}$;
\[
\mult (\sigma):=[N_{\sigma}:\mathbb Zv_1+\cdots +
\mathbb Zv_k].
\]
We note that the affine toric variety
$X(\sigma)$ associated to the cone $\sigma$ is smooth if and only
if $\mult (\sigma)=1$.
\end{defn}
\end{say}
The following is a well-known fact. See, for example,
\cite[Lemma 14-1-1]{matsuki}.
\begin{lem}\label{f-lem2.1.3}
A toric variety $X(\Sigma)$ is $\mathbb Q$-factorial
if and only if each cone $\sigma\in \Sigma$ is simplicial.
\end{lem}
\begin{say}\label{f-say2.1.4}
The {\em{star}} of a cone $\tau$ can be defined abstractly
as the set of cones $\sigma$ in $\Sigma$ that
contain $\tau$ as a face. Such cones $\sigma$ are
determined by their images in
$N(\tau):=N/{N_{\tau}}$, that is, by
\[
\overline \sigma=\sigma+(N_{\tau})_{\mathbb R}/
(N_{\tau})_{\mathbb R}\subset N(\tau)_{\mathbb R}.
\]
These cones $\{\overline \sigma ; \tau\prec \sigma\}$
form a fan in $N(\tau)$, and we denote this fan by
$\Star(\tau)$.
We set $V(\tau)=X(\Star (\tau))$, that is, the toric variety associated to
the fan $\Star (\tau)$.
It is well known that $V(\tau)$ is an $(n-k)$-dimensional
closed toric subvariety of $X(\Sigma)$, where $\dim \tau=k$.
If $\dim V(\tau)=1$ (resp.~$n-1$), then we call $V(\tau)$
a {\em{torus invariant curve}} (resp.~{\em{torus invariant
divisor}}).
For the details about the correspondence between $\tau$ and
$V(\tau)$, see \cite[3.1 Orbits]{fulton}.
\end{say}
\begin{say}[Intersection theory for $\mathbb Q$-factorial
toric varieties]\label{f-say2.1.5}
Assume that $\Sigma$ is simplicial.
If $\sigma, \tau\in \Sigma$ span $\gamma\in \Sigma$ with
$\dim \gamma=\dim \sigma +\dim \tau$,
then
\[
V(\sigma)\cdot V(\tau)=\frac{\mult (\sigma)\cdot \mult(\tau)}
{\mult (\gamma)} V(\gamma)
\]
in the {\em{Chow group}} $A^{*}(X)_{\mathbb Q}$.
For the details, see \cite[5.1 Chow groups]{fulton}.
If $\sigma$ and $\tau$ are contained in no cone of
$\Sigma$, then $V(\sigma)\cdot V(\tau)=0$.
\end{say}
\subsection{Cones of divisors}\label{f-subsec2.2}
In this subsection, we explain various cones of divisors
and some related topics.
\begin{say}\label{f-say2.2.1}
Let $f:X\to Y$ be a proper toric
morphism;
a $1$-cycle of $X/Y$ is a formal sum $\sum a_iC_i$ with
complete curves $C_i$ in the fibers of $f$, and
$a_i\in \mathbb Z$.
We put
\[
Z_1(X/Y):=\{1\text{-cycles of} \ X/Y\},
\]
and
\[
Z_1(X/Y)_{\mathbb R}:=
Z_1(X/Y)\otimes \mathbb R.
\]
There is a pairing
\[
\Pic (X)\times Z_1(X/Y)_{\mathbb R}
\to \mathbb R
\]
defined by
$(\mathcal L, C)\mapsto \deg _C\mathcal L$, extended
by bilinearity.
We define
\[
N^1(X/Y):=(\Pic (X)\otimes \mathbb R)/\equiv
\]
and
\[
N_1(X/Y):= Z_1(X/Y)_{\mathbb R}/\equiv,
\]
where the {\em numerical equivalence} $\equiv$
is by definition the smallest equivalence relation which
makes $N^1$ and $N_1$ into dual
spaces.
Inside $N_1(X/Y)$ there is a distinguished cone of effective
$1$-cycles of $X/Y$,
\[
{\NE}(X/Y)=\{\, Z\, | \ Z\equiv
\sum a_iC_i \ \text{with}\ a_i\in \mathbb R_{\geq 0}\}
\subset N_1(X/Y),
\]
which is usually called the {\em{Kleiman--Mori cone}} of $f:X\to Y$.
It is known that $\NE(X/Y)$ is a rational polyhedral cone.
A face $F\prec {\NE}(X/Y)$ is called an {\em{extremal face}}
in this case.
A one-dimensional extremal face is called an {\em{extremal
ray}}.
We define the {\em{relative Picard number}} $\rho(X/Y)$ by
\[
\rho (X/Y):=\dim _{\mathbb Q}N^1(X/Y)< \infty.
\]
An element $D\in N^1(X/Y)$ is called {\em{$f$-nef}} if
$D\geq 0$ on ${\NE}(X/Y)$.
If $X$ is complete and $Y$ is a point, then we
write ${\NE}(X)$ and $\rho(X)$
for ${\NE}(X/Y)$ and $\rho(X/Y)$, respectively.
We note that $N_1(X/Y)\subset N_1(X)$, and $N^1(X/Y)$
is the corresponding
quotient of $N^1(X)$.
From now on, we assume that $X$ is complete.
We define the {\em{nef cone}} $\Nef (X)$, the
{\em{ample cone}} $\Amp(X)$, and
the
{\em{pseudo-effective cone}} $\PE(X)$ in $N^1(X)$ as follows.
\[
\Nef(X)=\{D\, |\, D \ \text{is nef}\},
\]
\[
\Amp(X)=\{D\, | \, D\ \text{is ample}\}
\]
and
\[
\PE(X)=\left\{D\, \left| \begin{array}{l}
D\equiv \sum a_i D_i \ \text{such that
$D_i$ is an effective}\\ \text{Cartier divisor
and $a_i\in \mathbb R_{\geq 0}$ for every $i$
}\end{array}\right. \right\}.
\]
It is not difficult to see that $\PE(X)$ is a rational polyhedral cone in
$N^1(X)$ since $X$ is toric.
It is easy to see that \[\Amp(X)\subset \Nef (X)\subset \PE(X).\]
The reader can find various examples of
cones of divisors and curves in \cite{kleiman}, \cite{fujino-payne},
and \cite{fujino-sato2}. Although we do not use it explicitly in this paper,
Lemma \ref{f-lem2.2.2} is a very important property of toric varieties.
\begin{lem}\label{f-lem2.2.2}
Let $X$ be a projective toric variety and let $D$ be a
$\mathbb Q$-Cartier $\mathbb Q$-divisor on $X$.
Then $D$ is pseudo-effective if and only if $\kappa (X, D)\geq 0$,
that is, there exists a positive integer $m$ such that $mD$ is Cartier and that
\[H^0(X, \mathcal O_X(mD))\ne 0. \]
\end{lem}
\begin{proof}
It is sufficient to prove that $\kappa (X, D)\geq 0$ when
$D$ is pseudo-effective.
Let $f:Y\to X$ be a projective toric birational morphism
from a smooth projective toric variety $Y$.
By replacing $X$ and $D$ with $Y$ and $f^*D$, we may assume that
$X$ is a smooth projective toric variety.
In this case, it is easy to see that $\PE(X)$ is spanned
by the numerical equivalence classes
of torus invariant prime divisors.
Therefore, we can write $D\equiv \sum _i a_i D_i$ where
$D_i$ is a torus invariant prime divisor
and $a_i\in \mathbb Q_{>0}$ for every $i$ since $D$ is
a $\mathbb Q$-divisor.
Thus, we obtain $D\sim _{\mathbb Q} \sum _i a_i D_i\geq 0$.
This implies $\kappa (X, D)\geq 0$.
\end{proof}
The following lemma is well known and is very important.
We will use it in the subsequent sections repeatedly.
\begin{lem}\label{f-lem2.2.3}
Let $f:X\to Y$ be a toric proper morphism
and let $D$ be an $f$-nef Cartier divisor on $X$. Then $D$ is $f$-free,
that is,
\[f^*f_*\mathcal O_X(D)\to \mathcal O_X(D)\]
is surjective.
\end{lem}
\begin{proof}
See, for example, \cite[Chapter VI.~1.13.~Lemma]{nakayama}.
\end{proof}
We close this section with an easy example.
It is well known that $\NE(X)$ is spanned by the numerical equivalence classes of
torus invariant irreducible curves. However, the
dual cone $\Nef (X)$ of $\NE(X)$ is not
always spanned by the numerical equivalence classes of torus invariant prime
divisors.
\begin{ex}\label{f-ex2.2.4}
We consider $\mathbb P^1\times \mathbb P^1$.
Let $p_i:\mathbb P^1\times \mathbb P^1\to \mathbb P^1$ be
the $i$-th projection for $i=1, 2$.
Let $D_1, D_2$ (resp.~$D_3, D_4$) be the
torus invariant curves in the
fibers of $p_1$ (resp.~$p_2$).
Let $X\to \mathbb P^1\times \mathbb P^1$ be the
blow-up at the point $P=D_1\cap D_3$ and
let $E$ be the exceptional curve on $X$.
Let $D'_i$ denote the strict transform of $D_i$ on $X$ for
all $i$.
Then $\NE(X)$ is spanned by
the numerical equivalence classes of $E, D'_1$,
and $D'_3$.
On the other hand, $\Nef (X)\subset N^1(X)$ is spanned
by $D'_2, D'_4$, and $D'_1+D'_3+E$.
Therefore, the extremal ray of $\Nef (X)$ is not
necessarily spanned by a torus invariant
prime divisor.
\end{ex}
\end{say}
\section{Lengths of extremal rays}\label{f-sec3}
In this section, we discuss some estimates of lengths of
extremal rays of toric projective morphisms.
\subsection{Quick review of the known estimates}\label{f-subsec3.1}
In this subsection, we recall the known estimates of lengths of
extremal rays for toric varieties.
The first result is \cite[Theorem 0.1]{fujino-notes} (see
also \cite[Theorem 3.13]{fujino-equiv}).
\begin{thm}\label{f-thm3.1.1}
Let $f:X\to Y$ be a projective toric morphism with $\dim X=n$
and let $\Delta=\sum \delta_i\Delta_i$ be an $\mathbb R$-divisor
on $X$ such that $\Delta_i$ is a torus
invariant prime divisor and $0\leq \delta_i\leq 1$ for every $i$.
Assume that $K_X+\Delta$ is $\mathbb R$-Cartier.
Let $R$ be an extremal ray of $\NE(X/Y)$.
Then there exists a curve $C$ on $X$ such
that $[C]\in R$ and \[-(K_X+\Delta)\cdot C\leq n+1.\]
More precisely, we can choose $C$ such that
\[-(K_X+\Delta)\cdot C\leq n\] unless $X\simeq \mathbb P^n$
and $\sum \delta_i<1$.
We note that if $X$ is complete then we can make $C$ a torus invariant
curve on $X$.
\end{thm}
Our proof of Theorems \ref{f-thm3.1.1} and
\ref{f-thm3.2.1} below heavily depends on
Reid's description of toric extremal contraction morphisms
(see \cite{reid} and \cite[Chapter 14]{matsuki}).
\begin{say}[Reid's description of toric extremal
contraction morphisms]\label{f-say3.1.2}
Let $f:X\to Y$ be a projective surjective toric morphism from a complete
$\mathbb Q$-factorial toric $n$-fold and let $R$ be
an extremal ray of $\NE(X/Y)$.
Let $\varphi_R:X\to W$ be the extremal contraction associated to
$R$. We write
\[
\begin{matrix}
& A & \longrightarrow &B \\
& \cap & &\cap\\
\varphi_R: & X& \longrightarrow &W, \\
\end{matrix}
\]
where $A$ is the exceptional locus of $\varphi_R$ and
$B$ is the image of $A$ by $\varphi_R$. Then there
exist torus invariant prime divisors $E_1, \cdots, E_{\alpha}$ on $X$ with
$0\leq \alpha
\leq n-1$ such that $E_i$ is negative on $R$ for $1\leq i \leq \alpha$
and that $A$ is $E_1\cap \cdots\cap E_{\alpha}$. In particular, $A$ is
an irreducible torus invariant subvariety of $X$ with
$\dim A=n-\alpha$.
Note that $\alpha =0$ if and only if $A=X$,
that is, $\varphi_R$ is a Fano contraction.
There are torus invariant prime divisors $E_{\beta+1}, \cdots, E_{n+1}$ on
$X$ with $\alpha \leq \beta \leq n-1$ such
that $E_i$ is positive on $R$ for $\beta+1\leq i\leq n+1$.
Let $F$ be a general fiber of $A\to B$.
Then $F$ is a $\mathbb Q$-factorial toric Fano variety with
$\rho (F)=1$ and $\dim F=n-\beta$.
The divisors $E_{\beta+1}|_F, \cdots,
E_{n+1}|_F$ define all the torus invariant prime divisors on $F$.
In particular, $B$ is an irreducible torus invariant
subvariety of $W$ with $\dim B=\beta-\alpha$.
When $X$ is not complete, we can reduce it to the case
where $X$ is complete by the equivariant completion theorem
in \cite{fujino-equiv}. For the details, see \cite{sato}.
\end{say}
\begin{say}\label{f-say3.1.3}
We quickly review the idea of the
proof of Theorem \ref{f-thm3.1.1} in \cite{fujino-notes}.
We will use the same idea in the proof of Theorem \ref{f-thm3.2.1} below.
By replacing $X$ with its projective $\mathbb Q$-factorialization,
we may assume that $X$ is $\mathbb Q$-factorial.
Let $R$ be an extremal ray of $\NE(X/Y)$.
Then we consider the extremal contraction $\varphi_R:X\to W$ associated
to $R$. If $X$ is not projective, then we can
reduce it to the case where $X$ is projective
by the equivariant completion theorem (see \cite{fujino-equiv}).
By Reid's combinatorial description of $\varphi_R$,
any fiber $F$ of $\varphi_R$ is a $\mathbb Q$-factorial projective
toric variety with $\rho (F)=1$.
By (sub)adjunction, we can compare $-(K_X+\Delta)\cdot C$ with
$-K_F\cdot C$, where
$C$ is a curve on $F$.
So, the key ingredient of the proof of Theorem \ref{f-thm3.1.1} is the following
proposition.
\end{say}
\begin{prop}\label{f-prop3.1.4}
Let $X$ be a $\mathbb Q$-factorial projective
toric $n$-fold with $\rho (X)=1$.
Assume that $-K_X\cdot C>n$ for every
integral curve $C$ on $X$.
Then $X\simeq \mathbb P^n$.
\end{prop}
For the proof, see \cite[Proposition 2.9]{fujino-notes}.
Our proof heavily depends on the
calculation described in \ref{f-say3.1.8} below.
\begin{say}[Supplements to \cite{fujino-osaka}]\label{f-say3.1.5}
By the same arguments as in the proof of Proposition \ref{f-prop3.1.4},
we can obtain the next proposition,
which is nothing but \cite[Proposition 2.1]{fujino-osaka}.
\begin{prop}\label{f-prop3.1.6}
Let $X$ be a $\mathbb Q$-factorial projective
toric $n$-fold with $\rho (X)=1$ such that $X\not\simeq \mathbb P^n$.
Assume that $-K_X\cdot C\geq n$ for every
integral curve $C$ on $X$.
Then $X$ is isomorphic to the weighted projective space
$\mathbb P(1,1,2,\cdots, 2)$.
\end{prop}
The following proposition, which is missing in \cite{fujino-osaka},
may help us understand \cite{fujino-osaka}.
This proposition says that the results in \cite{fujino-osaka}
are compatible with \cite[Theorem 2 (a)]{fujita}.
\begin{prop}\label{f-prop3.1.7}
Let $X$ be a projective toric $n$-fold with $n\geq 2$.
We assume that $-K_X\equiv nD$ for some Cartier divisor $D$ on $X$ and
$\rho(X)=1$.
Then $D$ is very ample and $\Phi_{|D|}:X\hookrightarrow \mathbb P^{n+1}$ embeds
$X$ into $\mathbb P^{n+1}$ as a hyperquadric.
\end{prop}
\begin{proof}
By \cite[Theorem 3.2, Remark 3.3, and Theorem 3.4]{fujino-osaka},
there exists a crepant toric resolution $\varphi:Y\to X$,
where $Y=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}
\oplus \cdots \oplus \mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(2))$ or
$Y=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}
\oplus \cdots\oplus
\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(1)\oplus
\mathcal O_{\mathbb P^1}(1))$.
We note that $X=\mathbb P(1, 1, 2, \cdots, 2)$ when
$Y=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}
\oplus \cdots \oplus \mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(2))$.
We also note that $X$ is not $\mathbb Q$-factorial
if $Y=\mathbb P_{\mathbb P^1}(\mathcal O_{\mathbb P^1}
\oplus \cdots \oplus \mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(1)\oplus
\mathcal O_{\mathbb P^1}(1))$.
Let $\mathcal O_Y(1)$ be the tautological
line bundle of the $\mathbb P^{n-1}$-bundle $Y$ over
$\mathbb P^1$.
Then we have $\mathcal O_Y(-K_Y)\simeq
\mathcal O_Y(n)$.
We can directly check that $\dim H^0(Y, \mathcal O_Y(1))=n+2$.
We consider $\Phi_{|\mathcal O_Y(1)|}: Y\to\mathbb P^{n+1}$.
By construction,
\[
\Phi_{|\mathcal O_Y(1)|}: Y\overset{\varphi}{\longrightarrow}
X\overset{\pi}{\longrightarrow}\mathbb P^{n+1}
\]
is the Stein factorization of $\Phi_{|\mathcal O_Y(1)|}: Y\to \mathbb P^{n+1}$.
By construction again, we have $\mathcal O_Y(1)\simeq
\varphi^*\mathcal O_X(D)$.
Since we can directly check that
\[
\mathrm{Sym}^kH^0(Y, \mathcal O_Y(1))\to H^0(Y, \mathcal O_Y(k))
\]
is surjective for every $k\in \mathbb Z_{>0}$, we see that
\[
\mathrm{Sym}^kH^0(X, \mathcal O_X(D))\to H^0(X, \mathcal O_X(kD))
\] is
also surjective for every $k\in \mathbb Z_{>0}$.
This means that $\mathcal O_X(D)$ is very ample.
In particular, $\pi:X\to \mathbb P^{n+1}$ is nothing but the
embedding $\Phi_{|D|}: X\hookrightarrow \mathbb P^{n+1}$.
Since $D^n=(\mathcal O_Y(1))^n=2$, $X$ is a hyperquadric in $\mathbb P^{n+1}$.
\end{proof}
\end{say}
As was mentioned above, the following calculation plays
an important role in the proof of Proposition \ref{f-prop3.1.4}.
\begin{say}[Fake weighted projective spaces]\label{f-say3.1.8}
Now we fix $N\simeq \mathbb Z ^n$. Let $\{v_1,\cdots,v_{n+1}\}$
be a set of primitive vectors of $N$ such that $N_{\mathbb R}=\sum _i
\mathbb R_{\geq 0}v_i$.
We define $n$-dimensional cones
\[
\sigma_i:=\langle v_1,\cdots,v_{i-1},v_{i+1},\cdots,v_{n+1}\rangle
\]
for $1\leq i\leq n+1$.
Let $\Sigma$ be the complete fan generated by $n$-dimensional
cones $\sigma_i$ and their faces for all $i$. Then
we obtain a complete toric variety $X=X(\Sigma)$ with
Picard number $\rho (X)=1$.
We call it a {\em{$\mathbb Q$-factorial
toric Fano variety with Picard number one}}.
It is sometimes called a {\em{fake weighted projective
space}}.
We define $(n-1)$-dimensional cones $\mu_{i,j}=\sigma _i\cap \sigma _j$
for $i\ne j$.
We can write $\sum _i a_i v_i=0$, where $a_i\in \mathbb Z_{>0}$,
$\gcd(a_1,\cdots,a_{n+1})=1$, and $a_1\leq a_2\leq\cdots\leq a_{n+1}$
by changing the order.
Then we obtain
\[
0< V({v_{n+1}})\cdot V(\mu_{n,n+1})=\frac{\mult {(\mu_{n,n+1})}}
{\mult {(\sigma_{n})}}\leq 1,
\]
\[
V({v_{i}})\cdot V(\mu_{n,n+1})=\frac{a_i}{a_{n+1}}\cdot
\frac{\mult {(\mu_{n,n+1})}}
{\mult {(\sigma_{n})}},
\]
and
\begin{eqnarray*}
-K_{X} \cdot V(\mu_{n,n+1})&=&
\sum _{i=1}^{n+1} V({v_i})\cdot V(\mu_{n,n+1})\\
& =&
\frac {1}{a_{n+1}}
{\left(\sum_{i=1}^{n+1} a_i\right)}
\frac{\mult {(\mu_{n,n+1})}}
{\mult {(\sigma_{n})}}\leq n+1.
\end{eqnarray*}
We note that
\[
\frac{\mult(\sigma_n)}{\mult(\mu_{n, n+1})}\in \mathbb Z_{>0}.
\]
For the procedure to compute intersection numbers,
see \ref{f-say2.1.5} or \cite[p.100]{fulton}.
If $-K_{X} \cdot V(\mu_{n,n+1})=n+1$, then $a_i=1$ for every
$i$ and $\mult (\mu_{n,n+1})=\mult (\sigma_{n})$.
We note that the above calculation plays
crucial roles in \cite{fujino-notes}, \cite{fujino-osaka},
\cite{fujino-ishitsuka}, and this paper (see the
proof of Theorem \ref{f-thm3.2.1}, and so on).
\begin{lem}\label{f-lem3.1.9}
We use the same notation as in {\em{\ref{f-say3.1.8}}}.
We consider the sublattice
$N'$ of $N$ spanned by
$\{v_1, \cdots, v_{n+1}\}$.
Then the natural inclusion
$N'\to N$ induces a finite
toric morphism $f:X'\to X$ from a
weighted projective space $X'$ such
that $f$ is {}\'etale in codimension one.
In particular, $X(\Sigma)$ is a weighted projective
space if and only if $\{v_1, \cdots, v_{n+1}\}$
generates $N$.
\end{lem}
We note the above elementary lemma.
Example \ref{f-ex3.1.10} shows that there are
many fake weighted projective spaces which are not weighted projective
spaces.
\begin{ex}\label{f-ex3.1.10}
We put $N=\mathbb Z^3$. Let $\{e_1, e_2, e_3\}$ be the standard
basis of $N$.
We put $v_1=e_1$, $v_2=e_2$, $v_3=e_3$,
and $v_4=-e_1-e_2-e_3$.
The fan $\Sigma$ is the subdivision of $N_{\mathbb R}$
by $\{v_1, v_2, v_3, v_4\}$.
Then $Y=X(\Sigma)\simeq \mathbb P^3$.
We consider a new lattice \[N^\dag
=N+\left(\frac{1}{2}, \frac{1}{2}, 0\right)\mathbb Z.\]
The natural inclusion $N\to N^\dag$ induces a finite toric morphism $Y\to
X$, which is {}\'etale in codimension one. It is easy to see that
$K_X$ is Cartier and $-K_X\sim 4D_4$, where
$D_4=V(v_4)$ is not Cartier but $2D_4$ is Cartier.
Since $\{v_1, v_2, v_3, v_4\}$ does not span the lattice $N^\dag$, $X$ is not a
weighted projective space. Of course, $X$
is a fake weighted projective space.
\end{ex}
\end{say}
\subsection{New estimate of lengths of extremal rays}\label{f-subsec3.2}
The following theorem is one of the main theorems of this paper, in which
we prove new estimates of $K_X$-negative
extremal rays of birational type.
We will see that they are the best by Examples \ref{f-ex3.3.1}
and \ref{f-ex3.3.2}.
\begin{thm}[Lengths of extremal rays of birational type, Theorem \ref{f-thm1.2}]
\label{f-thm3.2.1}
Let $f:X\to Y$ be a projective toric morphism with $\dim X=n$.
Assume that $K_X$ is $\mathbb Q$-Cartier.
Let $R$ be a $K_X$-negative
extremal ray of $\NE(X/Y)$ and let $\varphi_R:X\to W$
be the contraction morphism associated to $R$. We put
\[l(R)=\min_{[C]\in R} (-K_X\cdot C).\]
and call it the length of $R$.
Assume that $\varphi_R$ is birational.
Then we obtain
\[
l(R)<d+1,
\]
where \[d=\max_{w\in W} \dim \varphi^{-1}_R(w)\leq n-1. \]
When $d=n-1$, we have a sharper inequality
\[
l(R)\leq d=n-1.
\]
In particular, if
$l(R)=n-1$,
then $\varphi_R:X\to W$ can be described as follows.
There exists a torus invariant smooth point $P\in W$ such that
$\varphi_R:X\to W$
is a weighted blow-up at $P$ with the weight $(1, a, \cdots, a)$ for some
positive integer $a$.
In this case, the exceptional locus $E$ of $\varphi_R$ is a torus invariant
prime divisor and is isomorphic to $\mathbb P^{n-1}$.
Moreover, $X$ is $\mathbb Q$-factorial in a neighborhood of
$E$.
\end{thm}
The idea of the proof of Theorem \ref{f-thm3.2.1} is the same as
that of Theorem \ref{f-thm3.1.1}.
\begin{proof}[Proof of Theorem \ref{f-thm3.2.1}]
In Step \ref{f-3.2.1step1}, we will explain how to
reduce problems to the case where $X$ is $\mathbb Q$-factorial.
Then we will prove the inequality $l(R)<d+1$ in Step
\ref{f-3.2.1step2}.
In Step \ref{f-3.2.1step3}, we will treat the case where $X$ is $\mathbb Q$-factorial
and $l(R)\geq n-1$.
Finally, in Step \ref{f-3.2.1step4}, we will treat the case where
$l(R)\geq n-1$ under the assumption that $X$ is not necessarily
$\mathbb Q$-factorial.
\begin{step}\label{f-3.2.1step1}
In this step, we will explain how to reduce problems to the case where $X$ is
$\mathbb Q$-factorial.
Without loss of generality, we may assume that $W=Y$.
Let $\pi:\widetilde X\to X$ be a small projective
$\mathbb Q$-factorialization (see, for example,
\cite[Corollary 5.9]{fujino-notes}).
Then we can take an extremal ray $\widetilde R$ of $\NE(\widetilde X/W)$
and construct the following commutative diagram
\[
\xymatrix{
\widetilde X \ar[r]^{\varphi_{\widetilde R}} \ar[d]_\pi& \widetilde W \ar[d]\\
X \ar[r]_{\varphi_R} & W
}
\]
where $\varphi_{\widetilde R}$ is the contraction morphism
associated to $\widetilde R$.
We note that $\varphi_{\widetilde R}$ must be small
when $\varphi_R$ is small.
We write
\[
\begin{matrix}
& \widetilde A & \longrightarrow &\widetilde B \\
& \cap & &\cap\\
\varphi_{\widetilde R}: & \widetilde X& \longrightarrow &\widetilde W, \\
\end{matrix}
\]
where $\widetilde A$ is the exceptional locus of
$\varphi_{\widetilde R}$ and $\widetilde B$ is the image of $\widetilde A$.
Let $\widetilde F$ be a general fiber of $\widetilde A\to \widetilde B$.
Then $\widetilde F$ is a fake weighted projective
space as in \ref{f-say3.1.2}, that is, $\widetilde F$ is a
$\mathbb Q$-factorial
toric Fano variety with Picard number one.
Since $\rho(\widetilde F)=1$, $\pi:\widetilde F\to F:=\pi(\widetilde F)$ is
finite.
Therefore, by definition, $\dim \widetilde F=\dim F\leq d$ since
$\varphi_R(F)$ is a point.
Let $\widetilde C$ be a curve in $\widetilde F$ and let $C$ be the
image of $\widetilde C$ by $\pi$ with the reduced scheme structure.
Then we obtain
\[
-K_{\widetilde X}\cdot \widetilde C=-\pi^*K_X\cdot \widetilde C=-mK_X\cdot
C,
\]
where $m$ is the mapping degree of $\widetilde C\to C$.
Thus, if $-K_{\widetilde X}\cdot \widetilde C$ satisfies the
desired inequality, then $-K_X
\cdot C$ also satisfies the same inequality.
Therefore, for the proof of $l(R)<d+1$,
we may assume that $X$ is $\mathbb Q$-factorial and $W=Y$
by replacing $X$ and $Y$ with $\widetilde X$ and $\widetilde W$, respectively.
\end{step}
\begin{step}\label{f-3.2.1step2}
In this step, we will prove the desired inequality $l(R)<d+1$
under the assumption that $X$ is $\mathbb Q$-factorial.
We write
\[
\begin{matrix}
& A & \longrightarrow &B \\
& \cap & &\cap\\
\varphi_R: & X& \longrightarrow &W, \\
\end{matrix}
\]
where $A$ is the exceptional locus of $\varphi_R$ and $B$ is the
image of $A$. It is well known that $A$ is irreducible.
We put $\dim A=n-\alpha$ and $\dim B=\beta-\alpha$ as in \ref{f-say3.1.2}.
Let $F$ be a general fiber of $A\to B$.
Then, it is known that $F$ is a $\mathbb Q$-factorial toric Fano variety with
Picard number one. It is also well known that there exist torus invariant prime divisors
$E_1,\cdots, E_{\alpha}$ on $X$ such that $E_i$ is negative
on $R$ for every $i$ and
$A$ is $E_1\cap \cdots \cap E_{\alpha}$.
By (sub)adjunction, we have
\[
(K_X+E_1+\cdots +E_\alpha)|_A=K_A+D
\]
for some effective $\mathbb Q$-divisor $D$ on $A$.
Note that $D$ is usually called a {\em{different}}.
Let $C$ be a curve in $F$.
Then
\begin{equation}\label{f-eq3.1}
\begin{split}
-K_X\cdot C&=-(K_A+D)\cdot C+E_1\cdot C+\cdots
+E_{\alpha}\cdot C\\&<-(K_A+D)\cdot C\leq -K_F\cdot C.
\end{split}
\end{equation}
By \cite[Proposition 2.9]{fujino-notes} (see also \ref{f-say3.1.8}),
there exists a torus invariant curve $C$ on $F$ such that
$-K_F\cdot C\leq \dim F+1=n-\beta+1$.
Therefore, we obtain \[-K_X\cdot C<n-\beta+1
=d+1\leq
n\] since $\beta\geq \alpha \geq 1$.
This means that $l(R)<d+1$.
By combining it with Step \ref{f-3.2.1step1},
we have $l(R)<d+1$ without assuming that $X$ is $\mathbb Q$-factorial.
We close this step with easy useful remarks.
\begin{rem}\label{f-rem3.2.2}
We note that if $F\not\simeq \mathbb P^{n-\beta}$
in the above argument, then we can choose $C$ such that
$
-K_F\cdot C\leq \dim F=n-\beta
$
(see Theorem \ref{f-thm3.1.1}).
\end{rem}
\begin{rem}\label{f-rem3.2.3}
If $X$ is Gorenstein, then $-K_X\cdot C<n$
implies $-K_X\cdot C\leq n-1$.
Therefore, by combining it with
Step \ref{f-3.2.1step1}, we can easily see that
the estimate $l(R)\leq n-1$ always holds for Gorenstein
(not necessarily $\mathbb Q$-factorial) toric varieties.
If $\varphi_R$ is small, then we can find $C$ such that
$-K_X\cdot C<n-1$ and $[C]\in R$ since
we know $\beta\geq \alpha \geq 2$.
Therefore, by combining it with Step \ref{f-3.2.1step1}, the estimate
$l(R)<n-1$ always holds for
(not necessarily $\mathbb Q$-factorial) toric
varieties, when $\varphi_R$ is small.
\end{rem}
\end{step}
\begin{step}\label{f-3.2.1step3}
In this step, we will investigate the case where
$l(R)\geq n-1$ under the assumption that $X$ is $\mathbb Q$-factorial.
We will use the same notation as in Step \ref{f-3.2.1step2}.
In this case, we see that $-K_X\cdot C\geq n-1$ for every curve
$C$ on $F$. Then, we see that
$\dim A=\dim F=n-1$, $F\simeq \mathbb P^{n-1}$ and $\dim B=0$
(see Remark \ref{f-rem3.2.2}).
Equivalently, $\varphi_R$ contracts $F\simeq \mathbb P^{n-1}$ to a torus invariant point
$P\in W$. Let $\langle e_1, \cdots, e_n\rangle$ be the
$n$-dimensional cone corresponding to $P\in W$.
Then $X$ is obtained by the star subdivision of $\langle e_1, \cdots, e_n\rangle$
by $e_{n+1}$, where $be_{n+1}=a_1e_1+\cdots +a_ne_n$, $b\in \mathbb Z_{>0}$ and
$a_i\in \mathbb Z_{>0}$ for all $i$.
We may assume that $\gcd (b, a_1,\cdots, a_n)=1$,
$\gcd (b, a_1, \cdots, a_{i-1}, a_{i+1}, \cdots a_n)=1$ for all $i$,
and $\gcd (a_1, \cdots, a_n)=1$. Without loss of generality, we may assume that
$a_1\leq \cdots \leq a_n$ by changing the order.
We write $\sigma_i=\langle e_1, \cdots, e_{i-1}, e_{i+1}, \cdots,
e_{n+1}\rangle$ for all $i$ and $\mu_{k,l}=\sigma_k\cap \sigma_l$ for
$k\ne l$. Then
\begin{equation}\label{f-eq3.2}
-K_X\cdot V(\mu _{k,n})=\frac{1}{a_n}
\left(\sum _{i=1}^{n}
a_i-b\right)\frac{\mult(\mu_{k,n})}{\mult(\sigma_k)}\geq n-1
\end{equation} for
$1\leq k\leq n-1$. Then $\mult (\mu_{k,n})=\mult (\sigma_k)$ for
$1\leq k\leq n-1$. Thus, $a_k$ divides $a_n$ for $1\leq k\leq n-1$.
\begin{case}\label{f-case1}If $a_1=a_n$, then $a_1=\cdots =a_n=1$.
In this case $-K_X\cdot V(\mu _{k,n})\geq
n-1$ implies $b=1$. And we have $\mult (\mu _{k,l})=\mult
(\sigma _k)$ for $1\leq k\leq n$, $1\leq l\leq n$, and $k\ne l$. In particular,
$\mult (\sigma _1) =\mult (\mu_{1, l})$ for $2\leq l\leq n$.
This implies $\mult (\sigma _1)=1$. Since $e_{n+1}=e_1+\cdots
+e_n$, $\langle e_1, \cdots, e_n\rangle$ is a nonsingular cone.
Therefore, $\varphi_R:X\to W$ is a blow-up at a smooth point $P$.
Of course, $l(R)=n-1$.
\end{case}
\begin{case}\label{f-case2}
Assume that $a_1\ne a_n$. If $a_2\ne a_n$,
then $\frac{a_1}{a_n}\leq \frac{1}{2}$ and
$\frac{a_2}{a_n}\leq \frac{1}{2}$. This contradicts $-K_X\cdot
V(\mu_{k,l})\geq n-1$.
Therefore, $a_1=1$ and $a_2=\cdots =a_n=a$ for some positive integer
$a\geq 2$.
The condition $-K_X\cdot V(\mu _{k,n})\geq n-1$ implies $b=1$.
Thus, $\mult (\mu_{k,l})=\mult (\sigma_k)$ for $1\leq
k\leq n$, $2\leq l\leq n$, and $k\ne l$. In particular,
$\mult (\sigma _1)=\mult (\mu _{1, l})$ for
$2\leq l\leq n$.
Thus, $\mult (\sigma _1)=1$.
Since \[e_{n+1}=e_1+ae_2+\cdots +ae_n, \]
$\langle e_1, \cdots, e_n\rangle$ is a nonsingular cone.
Therefore, $\varphi_R:X\to W$ is a weighted blow-up at a smooth point $P\in W$ with
the weight $(1, a, \cdots, a)$.
In this case, $K_X=\varphi^*_RK_W+(n-1)aE$, where
$E\simeq \mathbb P^{n-1}$ is the exceptional divisor
and $l(R)=n-1$ (see Proposition \ref{f-prop3.2.6} below).
\end{case}
Anyway, when $X$ is $\mathbb Q$-factorial,
we obtain that $l(R)\geq n-1$ implies $l(R)=n-1$.
Therefore, the estimate $l(R)\leq n-1$ always holds when $X$ is $\mathbb Q$-factorial
and $\varphi_R$ is birational.
\end{step}
\begin{step}\label{f-3.2.1step4}
In this final step, we will treat the case where $l(R)\geq n-1$ under the
assumption that $X$ is not necessarily $\mathbb Q$-factorial.
Let $\pi:\widetilde X\to X$ be a small projective
$\mathbb Q$-factorialization as in Step \ref{f-3.2.1step1}.
By the argument in Step \ref{f-3.2.1step1},
we can find a $K_{\widetilde X}$-negative extremal ray $\widetilde R$
of $\NE (\widetilde X/W)$ such that
$l(\widetilde R)\geq n-1$. Therefore, by Step \ref{f-3.2.1step3},
the associated contraction
morphism $\varphi_{\widetilde R}:\widetilde X\to \widetilde W$ is a weighted
blow-up at a smooth point $\widetilde P\in \widetilde W$ with
the weight $(1, a, \cdots, a)$ for some positive integer $a$.
Let $\widetilde E$ $(\simeq \mathbb P^{n-1})$ be the
$\varphi_{\widetilde R}$-exceptional
divisor on $\widetilde X$. We put $E=\pi(\widetilde E)$.
Then it is easy to see that $E\simeq \mathbb P^{n-1}$ and that
$\pi:\widetilde E\to E$ is an isomorphism.
\begin{lem}\label{f-lem3.2.4}
$\pi: \widetilde X\to X$ is an isomorphism over some open neighborhood of $E$.
\end{lem}
\begin{proof}[Proof of Lemma \ref{f-lem3.2.4}]
We will get a contradiction by assuming that $\pi:\widetilde X\to X$ is not
an isomorphism over any open neighborhood of $E$.
Since $\varphi_{\widetilde R}$ is a weighted blow-up as
described in the case where $X$ is $\mathbb Q$-factorial
(see Step \ref{f-3.2.1step3}) and
$\pi$ is a crepant small toric morphism by construction,
the fan of $\widetilde{X}$ contains $n$-dimensional cones
\[
\sigma_i:=\langle\{e_1,\ldots,e_{n+1}\}\setminus\{e_i\}\rangle,
\]
for $1\le i\le n$,
where $\{e_1,\ldots,e_n\}$ is the standard basis of $N=\mathbb Z^n$
and
$e_{n+1}:=e_1+ae_2+\cdots+ae_n$ with $a\in\mathbb{Z}_{>0}$.
Since we assume that $\pi:\widetilde
X\to X$ is not an isomorphism over any open
neighborhood of $E$, there exists at least one non-simplicial
$n$-dimensional cone
$\sigma$ in the fan of $X$
such that $\sigma$ contains one of the above $n$-dimensional cones.
By symmetry, it is sufficient to consider the two cases where
$\sigma$ contains $\sigma_n$ or $\sigma_1$.
First, we suppose $\sigma_n\subset\sigma$. Let $x=x_1e_1+\cdots +x_ne_n
\in N$ be the primitive generator
for some one-dimensional
face of $\sigma$ which is not contained in $\sigma_n$.
Then, by considering the facets of $\sigma_n$, we have the inequalities
$ax_1-x_n\ge 0$, $x_i-x_n\ge 0$ for $2\le i\le n-1$, and $x_n<0$.
If $x_1-x_n<0$, then $x_1<x_n<0$.
This means that $ax_1-x_n\leq x_1-x_n<0$.
This is a contradiction. Therefore, the inequality $x_1-x_n\geq 0$ also holds.
\begin{claim}\label{f-claim}
$x_i\leq 0$ for every $i\ne n$.
\end{claim}
\begin{proof}[Proof of Claim]
Suppose $x_i>0$ for some $i\ne n$.
We note that $x$ must be contained
in the hyperplane passing through
the points $e_1,\ldots,e_{n-1},e_{n+1}$ since
$\pi$ is crepant, that is, $K_{\widetilde X}=
\pi^*K_X$. So the equality
\begin{equation*}
\begin{split}
1&=x_1+\cdots+x_{n-1}-(n-2)x_n
\\&=(x_1-x_n)+\cdots+(x_{i-1}-x_n)+x_i+(x_{i+1}-x_n)+\cdots+(x_{n-1}-x_n)
\end{split}
\end{equation*}
holds. Therefore, $x_j-x_n=0$ must hold for every $j\neq i$, and
$x_i=1$. If $i\ne 1$, then we have $a=1$ since $ax_1-x_n=(a-1)x_n\ge 0$ and
$x_n<0$.
However, the linear relation
\[x+(-x_n)e_{n+1}=(1-x_n)e_i\] means that $\pi$
contracts a divisor $V(e_i)$.
This is a contradiction because $\pi$ is small by construction.
If $i=1$, then we have $ax_1-x_n=a-x_n>0$ since $a>0$ and $-x_n>0$.
However, the linear relation
\[
ax+(-x_n)e_{n+1}=(a-x_n)e_1
\]
means that $\pi$ contracts a divisor $V(e_1)$. This is a
contradiction because $\pi$ is small by construction.
In any case, we obtain that $x_i\leq 0$ holds for $1\le i \le n-1$.
\end{proof}
Therefore, the linear relation
\[
(-x_1)e_1+\cdots+(-x_{n})e_{n}+x=0
\]
says that the cone $\langle e_1,\ldots,e_n,x\rangle$
contains a positive dimensional linear subspace of
$N_{\mathbb R}$ because
$-x_i\ge 0$ for $1\le i\le n-1$ and $-x_n>0$.
Since $\langle e_1,\ldots,e_n,x\rangle$ must be contained in a strongly
convex cone in the fan of $W$, this is a contradiction.
Next, we suppose $\sigma_1\subset\sigma$.
We can apply the same argument as above.
Let $x=x_1e_1+\cdots +x_ne_n
\in N$ be the primitive generator
for some one-dimensional
face of $\sigma$ which is not contained in $\sigma_1$.
In this case, we can obtain the
inequalities $x_i-ax_1\ge 0$ for $2\le i\le n$, and $x_1<0$ by considering
the facets of $\sigma_1$, and
the equality $(1-(n-1)a)x_1+x_2+\cdots+x_n=1$ by the fact that
$\pi$ is crepant.
If $x_i>0$ for some $2\le i\le n$, then the equality
\begin{equation*}
\begin{split}
1&=(1-(n-1)a)x_1+x_2+\cdots+x_n
\\&=(1-a)x_1+(x_2-ax_1)+\cdots+(x_{i-1}-ax_1)+x_i+(x_{i+1}-ax_1)
+\cdots+(x_{n}-ax_1)
\end{split}
\end{equation*}
tells us that $a=1$ because $x_1<0$, and that
$x_j-x_1=0$ for every $j\neq i$ and $x_i=1$.
Therefore, as in the proof of Claim,
we get a contradiction by the linear relation
\[
x+(-x_1)e_{n+1}=(1-x_1)e_i.
\]
So we obtain that $x_i\leq 0$ holds for
$2\leq i\leq n$.
Thus we get a linear relation
\[
(-x_1)e_1+\cdots+(-x_{n})e_{n}+x=0
\]
as above, where $-x_i\geq 0$ for $2\leq i\leq n$ and $-x_1>0$.
This means that the cone $\langle e_1, \cdots, e_n, x\rangle$ contains
a positive dimensional linear subspace of $N_{\mathbb R}$.
This is a contradiction as explained above.
In any case, we get a contradiction. Therefore,
$\pi:\widetilde X\to X$ is an isomorphism
over some open neighborhood of $E$.
\end{proof}
Since $\pi:\widetilde X\to X$ is an isomorphism over some
open neighborhood of $E$ by Lemma \ref{f-lem3.2.4},
we see that $E$ is $\mathbb Q$-Cartier.
Therefore, the exceptional locus of $\varphi_R$ coincides with
$E\simeq \mathbb P^{n-1}$. Thus $\varphi_R:X\to W$ is
a weighted blow-up
at a torus invariant smooth
point $P\in W$ with the weight $(1, a, \cdots, a)$ for some positive integer $a$.
\end{step}
So, we complete the proof of Theorem \ref{f-thm3.2.1}.
\end{proof}
\begin{rem}\label{f-rem3.2.5}
If $B$ is complete, then we can make $C$ a torus invariant curve on $X$ in
Theorem \ref{f-thm3.2.1}.
For the details, see the proof of \cite[Theorem 0.1]{fujino-notes}.
\end{rem}
We explicitly state the basic properties
of the weighted blow-up in Theorem
\ref{f-thm3.2.1} for the reader's convenience.
\begin{prop}\label{f-prop3.2.6}
Let $\varphi:X\to \mathbb A^n$ be the weighted blow-up at $0\in
\mathbb A^n$ with the weight $(1, a, \cdots, a)$ for some
positive integer $a$.
If $a=1$, then $\varphi$ is the standard blow-up at $0$.
In particular, $X$ is smooth.
If $a\geq 2$, then $X$ has only canonical Gorenstein singularities which are not
terminal singularities. Furthermore, the exceptional locus $E$ of $\varphi$
is isomorphic to
$\mathbb P(1, a, \cdots, a)\simeq \mathbb P^{n-1}$ and
\[K_X=\varphi ^*K_{\mathbb A^n}+(n-1)aE. \]
We note that $E$ is not Cartier on $X$ if $a\ne 1$.
However, $aE$ is a Cartier divisor on $X$.
\end{prop}
\begin{proof}
We can check the statements by direct calculation.
\end{proof}
Let us see an important related example.
\begin{ex}\label{f-ex3.2.7}
We fix $N=\mathbb Z^n$ and let $\{e_1, \cdots, e_n\}$ be the
standard basis of $N$. We consider the cone $\sigma =\langle e_1,
\cdots, e_n\rangle$ in $N'=N+\mathbb Z e_{n+1}$, where $e_{n+1}=
\frac{1}{b}(1, a, \cdots, a)$.
Here, $a$ and $b$ are positive integers such that $\gcd(a,b)=1$.
We put $Y=X(\sigma)$ is the associated affine toric variety which has only one
singular point $P$.
We take a weighted blow-up of $Y$ at $P$ with
the weight $\frac{1}{b}(1, a, \cdots, a)$.
This means that we divide $\sigma$ by $e_{n+1}$
and obtain a fan $\Sigma$ of $N'_{\mathbb R}$.
We define $X=X(\Sigma)$. Then the induced toric projective birational
morphism $f:X\to Y$ is the desired weighted blow-up. It is obvious that $X$
is $\mathbb Q$-factorial and $\rho (X/Y)=1$. We can easily
obtain \[K_X=f^*K_Y+\left(\frac{1+(n-1)a}{b}-1\right)E, \] where $E=V(e_{n+1})\simeq
\mathbb P^{n-1}$ is the exceptional divisor of $f$,
and \[-K_X\cdot C=(n-1)-\frac{b-1}{a},\]
where $C=V(\langle e_2, \cdots, e_{n-1}, e_{n+1}\rangle)$ is
a torus invariant irreducible curve on $E$.
We note that \[-(K_X+\delta E)\cdot C>n-1\]
if and only if
\[
\delta>\frac{b-1}{b}
\]
since $E\cdot C=-\frac{b}{a}$.
\end{ex}
In subsection \ref{f-subsec3.3}, we will see more sophisticated examples
(see Examples \ref{f-ex3.3.1} and \ref{f-ex3.3.2}), which show the
estimates obtained in Theorem \ref{f-thm3.2.1} are the best.
By the proof of Theorem \ref{f-thm3.2.1}, we can prove the following
theorem.
\begin{thm}\label{f-thm3.2.8}
Let $f:X\to Y$ be a projective toric morphism with $\dim X=n$ and
let $\Delta=\sum \delta_i \Delta_i$ be an effective $\mathbb R$-divisor on $X$,
where $\Delta_i$ is a torus invariant prime divisor and $0\leq \delta_i\leq 1$ for
every $i$.
Let $R$ be an extremal ray of $\NE(X/Y)$ and let $\varphi _R: X\to W$
be the extremal contraction morphism associated to $R$.
Assume that $X$ is $\mathbb Q$-factorial and $\varphi_R$ is
birational.
If \[\min _{[C]\in R}(-(K_X+\Delta)\cdot C)>n-1, \]
then
$\varphi_R:X\to W$ is the weighted blow-up described in Example \ref{f-ex3.2.7}
and $\xSupp \Delta\supset E$, where $E\simeq \mathbb P^{n-1}$ is the
exceptional divisor of $\varphi_R$.
\end{thm}
\begin{proof}
We use the same notation as in Step \ref{f-3.2.1step2} in the proof of
Theorem \ref{f-thm3.2.1}.
Since \[(E_1+\cdots +E_\alpha-\Delta)\cdot C\leq 0,\]
we obtain
\begin{equation*}
\begin{split}
-(K_X+\Delta)\cdot C &= -(K_A+D)\cdot C +
(E_1+\cdots +E_\alpha-\Delta)\cdot C\\
&\leq -(K_A+D)\cdot C
\\&\leq -K_F\cdot C
\end{split}
\end{equation*}
(see \eqref{f-eq3.1}).
By assumption,
$-(K_X+\Delta)\cdot C>n-1$.
This implies that
$n-1<-K_F\cdot C$.
Therefore, we obtain $\dim A=\dim F=n-1$, $F\simeq \mathbb P^{n-1}$ and
$\dim B=0$.
In this situation,
\[
-(K_X+\Delta)\cdot C\leq -(K_X+A)\cdot C
\]
always holds. Thus we have
\begin{equation*}
-(K_X+A)\cdot V(\mu _{k,n})=\frac{1}{a_n}
\left(\sum _{i=1}^{n}
a_i\right)\frac{\mult(\mu_{k,n})}{\mult(\sigma_k)}> n-1
\end{equation*} for
$1\leq k\leq n-1$ (see \eqref{f-eq3.2}).
We note that $A=V(e_{n+1})$.
Thus, by the same arguments as in the proof of
Theorem \ref{f-thm3.2.1},
we see that $\varphi_R$ is the weighted blow-up
described in Example \ref{f-ex3.2.7}.
More precisely, we obtain that
$(a_1, \cdots, a_n)=(1, \cdots, 1)$ or $(1, a, \cdots, a)$ and
that $\sigma_1$ is a nonsingular cone.
However, $b$ is not necessarily $1$ in the proof of Theorem \ref{f-thm3.2.1}.
By direct calculation,
we have
$\xSupp \Delta\supset E$, where
$E(=A=F)$ is the exceptional divisor of $\varphi_R$.
\end{proof}
Finally, we prove
the following theorem.
\begin{thm}[Theorem \ref{f-thm1.3}]\label{f-thm3.2.9}
Let $X$ be a $\mathbb Q$-Gorenstein
projective toric $n$-fold with $\rho (X)\geq 2$.
Let $R$ be a $K_X$-negative extremal ray of
$\NE(X)$ such that
\[
l(R)=\min _{[C]\in R}(-K_X\cdot C)>n-1.
\]
Then the extremal contraction $\varphi_R:X\to W$ associated to
$R$ is a $\mathbb P^{n-1}$-bundle over $\mathbb P^1$.
\end{thm}
\begin{proof}
We divide the proof into several steps.
From Step \ref{f-3.2.9step1} to Step \ref{f-3.2.9step4},
we will prove this theorem under the extra assumption that $X$ is
$\mathbb Q$-factorial. In Step \ref{f-3.2.9step5}, we will
prove that $X$ is always $\mathbb Q$-factorial if there exists an
extremal ray $R$ with $l(R)>n-1$.
\setcounter{step}{0}
\begin{step}\label{f-3.2.9step1}
We consider the contraction morphism
$\varphi_R:X\to W$ associated to $R$.
By Theorem \ref{f-thm3.2.1},
$\varphi_R$ is a Fano contraction, that is,
$\dim W<\dim X$.
Let $F$ be a general fiber of $\varphi_R$ and let $C$ be
a curve on $F$.
Then, by adjunction, we have
\[
-K_X\cdot C=-K_F\cdot C.
\]
We note that $F$ is a fake weighted projective
space. By Theorem \ref{f-thm3.1.1},
$F\simeq \mathbb P^{n-1}$, $W=\mathbb P^1$, and
$\rho(X)=2$.
\end{step}
\begin{step}\label{f-3.2.9step2}
Without loss of generality, $\varphi_R:X\to W$ is induced by
the projection $\pi:N=\mathbb Z^n\to \mathbb Z,
(x_1, \cdots, x_n)\mapsto x_n$.
We put
\begin{align*}
v_1 &= (1,0,\cdots, 0), & v_2&=(0,1,0, \cdots, 0), & &\quad \quad \cdots , \\
v_{n-1}& =(0,\cdots, 0, 1, 0), & v_n &= (-1,\cdots, -1, 0),
&v_+&=(b_1,\cdots, b_{n-1}, a_+), \\ v_-&=(c_1, \cdots, c_{n-1}, -a_-), &
\end{align*}
where $a_+$ and $a_-$ are positive integers.
More precisely, $v_i$ denotes the vector
with a $1$ in the $i$th coordinate and $0$'s elsewhere for
$1\leq i\leq n-1$.
We may assume that the fan $\Sigma$ corresponding to the toric
variety $X$ is the subdivision of $N_{\mathbb R}$ by $v_1, \cdots, v_n,
v_+$, and $v_-$. We note that the following equalities
\begin{equation}\label{f-ex3.3}
\begin{cases}
D_1-D_n+b_1D_++c_1D_-=0\\
D_2-D_n+b_2D_++c_2D_-=0\\
\quad \quad \quad \quad \vdots\\
D_{n-1}-D_n+b_{n-1}D_++c_{n-1}D_-=0\\
a_+D_+-a_-D_-=0
\end{cases}
\end{equation}
hold, where $D_i=V(v_i)$ for
every $i$ and $D_{\pm}=V(v_{\pm})$.
We note that it is sufficient to prove that $a_+=a_-=1$.
\end{step}
\begin{step}\label{f-3.2.9step3}
In this step, we will prove that $a_+=1$ holds.
By taking a suitable coordinate change, we may assume that
\[
0\leq b_1, \cdots, b_{n-1}<a_+
\] holds.
If $b_i=0$ for every $i$, then $a_+=1$ since
$v_+$ is a primitive vector of $N$.
From now on, we assume that $b_{i_0}\ne 0$ for some $i_0$.
Without loss of generality, we may assume that
$b_1\ne 0$.
We put
\[
C=V(\langle v_2, \cdots, v_{n-1}, v_+\rangle).
\]
Then $C$ is a torus invariant curve contained in a fiber of $\varphi_R:X\to W$.
We have
\[
D_1\cdot C=\frac {\mult (\langle v_2, \cdots, v_{n-1}, v_+\rangle)}
{\mult (\langle v_1, \cdots, v_{n-1}, v_+\rangle)}=\frac
{\gcd(a_+, b_1)}{a_+}
\] (see \ref{f-say2.1.5})
and $
D_+\cdot C=D_-\cdot C=0
$.
We note that
$D_i\cdot C=D_1\cdot C$ for every $i$
by \eqref{f-ex3.3}.
Therefore,
we obtain
\[
-K_X\cdot C=\frac{n\gcd (a_+, b_1)}{a_+}.
\]
Since $0<b_1<a_+$, we see $\gcd(a_+, b_1)<a_+$.
Thus, the following inequality
\[
\frac{\gcd(a_+, b_1)}{a_+}\leq \frac{1}{2}
\]
holds.
This means that
\[
-K_X\cdot C\leq \frac{n}{2}\leq n-1.
\]
This is a contradiction.
Therefore, $b_i=0$ for every $i$ and $a_+=1$.
\end{step}
\begin{step}\label{f-3.2.9step4}
By the same argument, we get $a_-=1$.
Thus, we see that $\varphi_R:X\to W$ is a
$\mathbb P^{n-1}$-bundle over $\mathbb P^1$.
\end{step}
\begin{step}\label{f-3.2.9step5}
In this step, we will prove that $X$ is $\mathbb Q$-factorial.
We assume that $X$ is not $\mathbb Q$-factorial. Let $\pi:\widetilde
X\to X$ be a small projective $\mathbb Q$-factorialization.
We note that $\rho(\widetilde X)>\rho (X)\geq 2$ since
$X$ is not $\mathbb Q$-factorial.
By the argument in Step \ref{f-3.2.1step1} in the proof of Theorem \ref{f-thm3.2.1},
there exists an extremal ray $\widetilde R$ of $\NE(\widetilde X/W)$
with $l(\widetilde R)>n-1$.
Let $\varphi_{\widetilde R}: \widetilde X\to \widetilde W$ be the contraction morphism
associated to $\widetilde R$. Then, by the argument in Step \ref{f-3.2.9step1},
we see that $\rho(\widetilde X)=2$ and that $\varphi_{\widetilde R}$ is
nothing but $\varphi_R\circ \pi$.
This is a contradiction because $\rho(\widetilde X)>2$.
This means that $X$ is always $\mathbb Q$-factorial.
\end{step}
Therefore, we get the desired statement.
\end{proof}
We close this subsection with an easy
example, which shows that Theorem \ref{f-thm3.2.9} is sharp.
\begin{ex}\label{f-ex3.2.10}
We consider $N=\mathbb Z^2$,
$
v_1=(0, 1), v_2=(0, -1), v_+=(2, 1), v_-=(-1, 0)
$
and the projection $\pi:N=\mathbb Z^2\to \mathbb Z, (x_1, x_2)\mapsto x_1$.
Let $\Sigma$ be the fan obtained by subdividing $N_\mathbb R$ by
$\{v_1, v_2, v_+, v_-\}$.
Then $X=X(\Sigma)$ is a projective
toric surface with $\rho (X)=2$.
The map $\pi:N\to \mathbb Z$ induces a Fano contraction morphism
$\varphi:X\to \mathbb P^1$.
Let $R$ be the corresponding extremal ray of $\NE(X)$.
Then $l(R)=1=2-1$.
Note that $X$ is not a $\mathbb P^1$-bundle over $\mathbb P^1$.
\end{ex}
\subsection{Examples}\label{f-subsec3.3}
In this subsection, we will see that the
estimates in Theorem \ref{f-thm3.2.1} are the best by the following
examples.
\begin{ex}\label{f-ex3.3.1}
We use the same notation as in Example \ref{f-ex3.2.7}.
In Example \ref{f-ex3.2.7}, we put $a=k^2$ and $b=mk+1$ for any positive integers
$k$ and $m$.
Then it is obvious that $\gcd(a,b)=1$.
So, we can apply the construction in Example \ref{f-ex3.2.7}.
Then we obtain a toric projective birational morphism
$f:X\to Y$ such that
$X$ is
$\mathbb Q$-factorial and $\rho (X/Y)=1$.
We can easily check that
\[K_X=f^*K_Y+\left(\frac{1+k^2(n-1)}{mk+1}-1\right)E\] and
\[-K_X\cdot C=n-1-\frac{m}{k}. \]
Therefore, we see that the minimal lengths of extremal rays do not
satisfy the ascending chain condition in this birational setting.
More precisely,
the minimal lengths of extremal rays can take any values in
$\mathbb Q\cap (0,n-1)$. For a related topic, see \cite{fujino-ishitsuka}.
\end{ex}
Let us construct small contraction morphisms with a
long extremal ray.
\begin{ex}\label{f-ex3.3.2}
We fix $N=\mathbb Z^n$ with $n\geq 3$.
Let $\{v_1, \cdots, v_n\}$ be the standard basis of $N$.
We put
\[
v_{n+1}=(\underbrace{a, \cdots, a}_{n-k+1},
\underbrace{-1, \cdots, -1}_{k-1})
\]
with $2\leq k\leq n-1$, where $a$ is any positive integer.
Let
$\Sigma^+$ be the fan in $\mathbb R^n$ such that the set of maximal
cones of $\Sigma^+$ is
\[
\left\{\left\langle \{v_1, \cdots, v_{n+1}\}\setminus\{v_i\}\right\rangle
\,\left|\,n-k+2\leq i\leq n+1\right\}\right..
\]
Let us consider the smooth toric variety $X^+=X(\Sigma^+)$ associated
to the fan $\Sigma^+$.
We note that the equality
\[
v_{n-k+2}+\cdots + v_n+v_{n+1}=av_1+\cdots +av_{n-k+1}
\]
holds.
We can get an antiflipping contraction
$\varphi^+: X^+\to W$,
that is, a $K_{X^+}$-positive small contraction morphism,
when
\[
a>\frac{k}{n-k+1}.
\]
In this case, we have the following flipping diagram
\[
\xymatrix{
X\ar[dr]_{\varphi}\ar@{-->}[rr]^\phi&& X^+\ar[dl]^{\varphi^+}\\
&W&
}
\]
By construction, $\varphi:X\to W$ is a flipping contraction whose
exceptional locus is isomorphic to $\mathbb P^{n-k}$.
The exceptional locus of $\varphi^+$ is isomorphic
to $\mathbb P^{k-1}$. Of course, $\varphi$ (resp.~$\varphi^+$)
contracts $\mathbb P^{n-k}$ (resp.~$\mathbb P^{k-1}$)
to a point in $W$. We can directly check that
\[
-K_X\cdot C=n-k+1-\frac{k}{a}
\]
for every torus invariant curve $C$ in the $\varphi$-exceptional locus
$\mathbb P^{n-k}$.
\end{ex}
Example \ref{f-ex3.3.2} shows that the estimate for small contractions in
Theorem \ref{f-thm3.2.1} is sharp.
\begin{rem}\label{f-rem3.3.3}
If $(n,k)=(3,2)$ and $a\geq 2$ in Example \ref{f-ex3.3.2},
then $\varphi:X\to W$ is a threefold toric flipping contraction
whose length of the extremal ray is $\geq 3-2=1$.
We note that the lengths of extremal rays of three-dimensional
terminal (not necessarily toric) flipping contractions are less than one.
\end{rem}
\section{Basepoint-free theorems}\label{f-sec4}
This section is a supplement to Fujita's freeness conjecture for toric varieties (see
\cite{fujino-notes}, \cite{fujita}, \cite{laterveer}, \cite{lin}, \cite{mustata}, and
\cite{payne}) and Fulton's book:~\cite{fulton}.
\subsection{Variants of Fujita's conjectures for toric varieties}\label{f-subsec4.1}
One of the most general formulations of
Fujita's freeness conjecture for
toric varieties is \cite[Corollary 0.2]{fujino-notes}.
However, it does not cover the first part of
\cite[Main theorem A]{lin}.
So, we give a generalization here with a
very simple proof.
It is an easy application of the vanishing theorem (see \cite{fujino-vanishing} and
\cite{fujino-toric}).
\begin{thm}[Basepoint-freeness]\label{f-thm4.1.1}
Let $g:Z\to X$ be a proper toric morphism
and let $A$ and $B$ be reduced torus invariant Weil divisors
on $Z$ without common irreducible components.
Let $f:X\to Y$ be a proper surjective toric morphism
and let $D$ be an $f$-ample Cartier divisor on $X$.
Then
\[
R^qg_*(\widetilde
{\Omega}^a_Z(\log (A+B))(-A))\otimes \mathcal O_X(kD)\]
is $f$-free, that is,
\begin{equation*}
f^*f_*(R^qg_*(\widetilde
{\Omega}^a_Z(\log (A+B))(-A))\otimes \mathcal O_X(kD))
\to
R^qg_*(\widetilde
{\Omega}^a_Z(\log (A+B))(-A))\otimes \mathcal O_X(kD)
\end{equation*}
is surjective
for every $a\geq 0$, $q\geq 0$, and $k\geq \max_{y\in Y}\dim f^{-1}(y) +1$.
\end{thm}
As a very special case, we can recover the following result.
\begin{cor}[{cf.~\cite[Main theorem A]{lin}}]\label{f-cor4.1.2}
Let $X$ be an $n$-dimensional
projective toric variety and let $D$ be an ample Cartier divisor on $X$. Then
the reflexive sheaf $\mathcal O_X(K_X+(n+1)D)$ is generated
by its global sections.
\end{cor}
\begin{proof}
In Theorem \ref{f-thm4.1.1},
we assume that
$g:Z\to X$ is the identity, $A=B=0$,
$a=\dim X$, $q=0$, and $Y$ is a point.
Then we obtain the desired statement.
\end{proof}
\begin{ex}\label{f-ex4.1.3}
Let us consider $X=\mathbb P(1, 1, 1, 2)$.
Let $P$ be the unique $\frac{1}{2}(1, 1, 1)$-singular point of $X$
and let $D$ be an ample Cartier divisor on $X$.
We can find a torus invariant curve $C$ on $X$ such
that \[K_X\cdot C\in \frac{1}{2}\mathbb Z\setminus \mathbb Z. \]
Therefore, for every effective Weil divisor $E$ on $X$ such that
$E\sim K_X+4D$,
we have $P\in \xSupp E$.
On the other hand, by Corollary \ref{f-cor4.1.2},
the reflexive sheaf $\mathcal O_X(K_X+4D)$ is generated by its
global sections.
\end{ex}
Before proving Theorem \ref{f-thm4.1.1}, let us recall the definition of
the reflexive sheaf $\widetilde {\Omega}^a_{X}(\log (A+B))(-A)$ and
the vanishing theorem
in \cite{fujino-toric}.
\begin{defn}\label{f-def4.1.4}
Let $W$ be any Zariski open set of $Z$ such that $W$ is smooth and
$\codim _Z(Z\setminus W)\geq 2$.
In this case, $A+B$ is a simple normal crossing divisor on
$W$. On this assumption,
$\Omega^a_{W}(\log (A+B))$
is a well-defined locally free sheaf on $W$.
Let $\iota:W\hookrightarrow Z$ be the natural
open immersion.
Then we put \[\widetilde \Omega^a_Z(\log(A+ B))(-A)=
\iota_*(\Omega^a_W(\log (A+B))\otimes \mathcal O_W(-A))\] for every
$a\geq 0$.
It is easy to see that the reflexive sheaf
\[\widetilde \Omega^a_{Z}(\log (A+B))(-A)\] on $Z$ does not
depend on the choice of $W$.
\end{defn}
The next theorem is one of the vanishing
theorems obtained in \cite{fujino-toric}.
For the proof and other vanishing theorems,
see \cite{fujino-vanishing} and \cite{fujino-toric}.
\begin{thm}[{\cite[Theorem 4.3]{fujino-toric}}]\label{f-thm4.1.5}
Let $g:Z\to X$ be a proper toric morphism and let $A$ and $B$ be reduced
torus invariant Weil divisors on $Z$ without
common irreducible components.
Let $f:X\to Y$ be a proper surjective toric morphism
and let $L$ be an $f$-ample line bundle on $X$.
Then
\[
R^pf_*(
R^qg_*(\widetilde
{\Omega}^a_Z(\log (A+B))(-A))\otimes L)=0\]
for every
$p>0$, $q\geq 0$, and $a\geq 0$.
\end{thm}
Let us prove Theorem \ref{f-thm4.1.1}.
\begin{proof}[Proof of Theorem \ref{f-thm4.1.1}]
By the vanishing theorem:~Theorem \ref{f-thm4.1.5},
we have
\[R^pf_*(
R^qg_*(\widetilde
{\Omega}^a_Z(\log (A+B))(-A))\otimes \mathcal O_X((k-p)D))=0
\]
for every $p>0$, $q\geq 0$, $a\geq 0$, and
$k\geq \max_{y\in Y}\dim f^{-1}(y)+1$.
Since $\mathcal O_X(D)$ is $f$-free, we obtain that
\[
R^qg_*(\widetilde
{\Omega}^a_Z(\log (A+B))(-A))\otimes \mathcal O_X(kD)\]
is $f$-free
by the Castelnuovo--Mumford
regularity (see, for example, \cite[Example 1.8.24]{positive}).
\end{proof}
Here, we treat some generalizations of
Fujita's very ampleness for toric
varieties as applications of Theorem \ref{f-thm4.1.5}.
For the details of Fujita's very ampleness for toric
varieties, see \cite{payne}.
\begin{thm}\label{f-thm4.1.6}
Let $f:X\to Y$ be a proper surjective toric
morphism, let $\Delta$ be a reduced torus invariant
divisor on $X$ such that $K_X+\Delta$ is Cartier,
and let $D$ be an $f$-ample Cartier divisor on $X$.
Then $\mathcal O_X(K_X+\Delta+kD)$ is $f$-very ample for
every $k\geq \max_{y\in Y} \dim f^{-1}(y) +2$.
\end{thm}
\begin{proof}
It follows from the Castelnuovo--Mumford
regularity by the vanishing theorem:~Theorem
\ref{f-thm4.1.5}. For the details,
see \cite[Example 1.8.22]{positive}.
\end{proof}
The following corollary is a special case of the above theorem.
\begin{cor}[{cf.~\cite[Main Theorem B]{lin}}]\label{f-cor4.1.7}
Let $X$ be an $n$-dimensional projective Gorenstein toric variety
and let $D$ be an ample Cartier
divisor on $X$.
Then $\mathcal O_X(K_X+(n+2)D)$ is very ample.
\end{cor}
We think that the following theorem has not been
stated explicitly
in the literature.
\begin{thm}[Very ampleness]\label{f-thm4.1.8}
Let $g:Z\to X$ be a proper toric morphism
and let $A$ and $B$ be reduced torus invariant Weil divisors
on $Z$ without common irreducible components.
Let $f:X\to Y$ be a proper surjective
toric morphism and
let $D$ be an $f$-ample
Cartier divisor on $X$.
Assume that $
R^qg_*(\widetilde
{\Omega}^a_Z(\log (A+B))(-A))
$ is locally
free.
Then \[R^qg_*(\widetilde
{\Omega}^a_Z(\log (A+B))(-A))\otimes \mathcal O_X(kD)\] is $f$-very ample
for every $k\geq \max _{y\in Y} \dim
f^{-1}(y) +2$.
\end{thm}
\begin{proof}
The proof of Theorem \ref{f-thm4.1.6} works
for this theorem since \[
R^qg_*(\widetilde
{\Omega}^a_Z(\log (A+B))(-A))
\]
is locally free by assumption (see \cite[Example 1.8.22]{positive}).
\end{proof}
\subsection{Lin's problem}\label{f-subsec4.2}
In this subsection, we treat Lin's problem raised in \cite{lin}.
In \cite[Lemma 4.3]{lin}, she claimed the following lemma,
which is an exercise in \cite[p.90]{fulton}, without
proof.
\begin{lem}\label{f-lem4.2.1}
Let $X$ be a complete Gorenstein toric variety and let $D$ be an ample
$($Cartier$)$ divisor. If $\Gamma (X, K+D)\ne 0$ then
$K+D$ is generated by its global sections.
In fact, $P_{K+D}$ is the convex hull of $\Int P_D\cap M$.
\end{lem}
Sam Payne pointed out that Lemma \ref{f-lem4.2.1} does not seem
to have a known valid proof (see \cite[p.500 Added in proof]{lin}).
Unfortunately, the following elementary
example is a counterexample to
Lemma \ref{f-lem4.2.1}.
So, Lemma \ref{f-lem4.2.1} is NOT true.
Therefore, the alternative proof of Theorem A in \cite{lin}
does not work.
\begin{ex}\label{f-ex4.2.2}
Let $Y=\mathbb P^n$ and let $P\in Y$ be a torus invariant
closed point. Let $f:X\to Y$ be the blow-up
at $P$.
We put $B=\sum _{i=1} ^{n+1}B_i$, where $B_i$ is a torus
invariant prime divisor on $Y$ for every $i$.
Then it is well known that
$\mathcal O_Y(K_Y)\simeq \mathcal O_Y(-B)$.
We define $D=f^*B-E$, where $E$ is
the exceptional divisor of $f$.
In this case, we have \[K_X=f^*K_Y+(n-1)E\] and
it is not difficult to see that $D$ is ample.
Therefore, \[K_X+(n-1)D=f^*(K_Y+(n-1)B)\] is nef,
that is, $\mathcal O_X(K_X+(n-1)D)$ is generated
by its global sections.
We note that $H^0(X, \mathcal O_X(K_X+aD))\ne 0$ for
every positive integer $a$. However, $K_X+aD$ is not nef for
any real number $a<n-1$. In particular,
$H^0(X, \mathcal O_X(K_X+D))\ne 0$ but
$\mathcal O_X(K_X+D)$ is not generated by its global
sections.
\end{ex}
The following theorem is the main theorem of this section.
It follows from Theorem \ref{f-thm3.2.1}.
\begin{thm}[see Theorem \ref{f-thm1.1}]\label{f-thm4.2.3}
Let $X$ be a $\mathbb Q$-Gorenstein
projective toric $n$-fold and let $D$ be an ample Cartier divisor on $X$.
Then $K_X+(n-1)D$ is pseudo-effective if and only if $K_X+(n-1)D$ is
nef.
\end{thm}
\begin{proof}
If $K_X+(n-1)D$ is nef, then $K_X+(n-1)D$ is obviously
pseudo-effective.
So, all we have to do is to see that $K_X+(n-1)D$ is
nef when it is pseudo-effective.
From now on, we assume that $K_X+(n-1)D$ is pseudo-effective.
We take a positive rational number $\tau$ such that
$K_X+\tau D$ is nef but not ample.
In some
literature, $1/\tau$
is called the {\em{nef threshold of $D$ with respect to $X$}}. It is not difficult to see that
$\tau$ is rational since the Kleiman--Mori cone is
a rational polyhedral cone in our case.
If $\tau \leq n-1$, then the theorem is obvious since
\[K_X+(n-1)D=K_X+
\tau D+(n-1-\tau )D\] and $D$ is ample. Therefore,
we assume that $\tau >n-1$.
We take a sufficiently large positive integer $m$ such
that $m(K_X+\tau D)$ is Cartier.
We consider the toric morphism $f:=\Phi _{|m(K_X+\tau D)|}: X\to
Y$.
By the definition of $\tau$, $f$ is not an isomorphism.
Let $R$ be an extremal ray of $\NE(X/Y)$.
Let $C$ be any integral
curve on $X$ such that
$[C]\in R$. Since $(K_X+\tau D)\cdot C=0$, we
obtain
$-K_X\cdot C=\tau D\cdot C>n-1$.
Therefore, $f$ is not birational by Theorem \ref{f-thm3.2.1}.
Equivalently, $K_X+\tau D$ is not big.
Thus, the numerical equivalence class of $K_X+\tau D$ is on the boundary
of the pseudo-effective cone $\PE(X)$
of $X$.
So, \[K_X+(n-1)D=K_X+\tau D-(\tau -(n-1))D\] is outside
$\PE(X)$. This is a contradiction.
Therefore, $K_X+(n-1)D$ is nef when
$K_X+(n-1)D$ is pseudo-effective.
\end{proof}
As a corollary, we obtain the following result,
which is a correction of Lemma \ref{f-lem4.2.1}.
It is a variant of Fujita's freeness conjecture for toric
varieties. Example \ref{f-ex4.2.2} shows that
the constant $n-1$ in Corollary \ref{f-cor4.2.4} is the
best.
\begin{cor}[see Theorem \ref{f-thm1.1}]\label{f-cor4.2.4}
Let $X$ be a
Gorenstein projective toric $n$-fold and let $D$ be an
ample Cartier divisor on $X$.
If $H^0(X, \mathcal O_X(K_X+(n-1)D))\ne 0$,
then $\mathcal O_X(K_X+(n-1)D)$ is generated
by its global sections.
\end{cor}
\begin{proof}
If $H^0(X, \mathcal O_X(K_X+(n-1)D))\ne 0$, then
$K_X+(n-1)D$ is obviously pseudo-effective.
Then, by Theorem \ref{f-thm4.2.3}, $K_X+(n-1)D$ is
nef.
If $K_X+(n-1)D$ is a nef Cartier divisor, then
the complete linear system $|K_X+(n-1)D|$ is basepoint-free by
Lemma \ref{f-lem2.2.3}.
\end{proof}
By Theorem \ref{f-thm3.2.9}, we can check the
following result.
\begin{cor}[Corollary \ref{f-cor1.4}]\label{f-cor4.2.5}
Let $X$ be a $\mathbb Q$-Gorenstein projective
toric $n$-fold and let $D$ be an ample Cartier divisor
on $X$.
If $\rho (X)\geq 3$, then $K_X+(n-1)D$ is always nef.
More precisely,
if $\rho (X)\geq 2$ and $X$ is not a $\mathbb P^{n-1}$-bundle
over $\mathbb P^1$, then $K_X+(n-1)D$ is nef.
\end{cor}
\begin{proof}
By Theorem \ref{f-thm3.2.9}, $K_X+(n-1)D$ is nef
since $\rho (X)\geq 2$ and $X$ is not a
$\mathbb P^{n-1}$-bundle over $\mathbb P^1$.
\end{proof}
\subsection{Supplements to Fujita's paper}\label{f-subsec4.3}
This subsection supplements Fujita's paper:~\cite{fujita}.
We have never seen Corollary \ref{f-cor4.2.4} in the literature.
However, we believe that Fujita could prove Corollary \ref{f-cor4.2.4} without
any difficulties (see Theorem \ref{f-thm4.3.2} below).
We think that he was not interested in the toric geometry when
he wrote \cite{fujita}.
If he was familiar with the toric geometry, then
he would have adopted Example \ref{f-ex4.3.1} in
\cite[(3.5) Remark]{fujita}.
This example supplements Fujita's remark:~\cite[(3.5) Remark]{fujita}.
We think that our example is much simpler.
\begin{ex}\label{f-ex4.3.1}
We fix $N=\mathbb Z^2$.
We put $e_1=(1,0)$, $e_2=(0,1)$,
$e_3=(-1,-1)$, and $e_4=(1,2)$.
We consider the fan $\Sigma$ obtained by subdividing
$N_{\mathbb R}$ with $e_1$, $e_2$, $e_3$,
and $e_4$.
We write $X=X(\Sigma)$, the associated toric variety.
Then $X$ is Gorenstein and $-K_X$ is ample.
We put $D=-K_X$. It is obvious that $K_X+D\sim 0$.
It is easy to see that the Kleiman--Mori cone $\NE(X)$
is spanned by the two torus invariant curves
$E=V(e_4)$ and $E'=V(e_2)$.
So, we have two extremal contractions.
By removing $e_4$ from $\Sigma$, we obtain a
contraction morphism $f:X\to \mathbb P^2$.
In this case, $E$ is not Cartier although $2E$ is Cartier.
We note that $-K_X\cdot E=1$.
The morphism $f$ is the weighted blow-up with the
weight $(1,2)$ described in Proposition \ref{f-prop3.2.6}.
Another contraction is obtained by
removing $e_2$.
It is a contraction morphism from $X$ to $\mathbb P(1,1,2)$.
Note that $E'$ is a Cartier divisor on $X$.
\end{ex}
We close this subsection with the following theorem.
In Theorem \ref{f-thm4.3.2}, we treat normal Gorenstein projective
varieties defined over
$\mathbb C$ with
only rational singularities, which are not necessarily
toric.
So, the readers who are interested only in the toric
geometry can skip this final theorem.
\begin{thm}[see \cite{fujita}]\label{f-thm4.3.2}
Let $X$ be a normal projective variety defined over $\mathbb C$
with only rational Gorenstein singularities.
Let $D$ be an ample Cartier divisor on $X$.
If $K_X+(n-1)D$ is pseudo-effective with
$n=\dim X$,
then $K_X+(n-1)D$ is nef.
\end{thm}
\begin{proof}
We take a positive rational number $\tau$ such that
$K_X+\tau D$ is nef but not ample.
It is well known that $\tau\leq n+1$ (see \cite[Theorem 1]{fujita}).
If $\tau\leq n-1$, then the theorem is obvious.
Therefore, we assume that $n-1<\tau \leq n+1$.
If $\tau =n+1$, then $X\simeq \mathbb P^n$ and $\mathcal O_X
(D)\simeq \mathcal O_{\mathbb P^n}(1)$.
In this case, $K_X+(n-1)D$ is not pseudo-effective.
Thus, we have $n-1<\tau \leq n$ by
\cite[Theorem 1]{fujita}.
By \cite[Theorem 2]{fujita} and its proof, it can be checked easily that $\tau=n$
and $K_X+\tau D=K_X+nD$ is nef but is not big.
Therefore, $K_X+nD$ is on the boundary of the pseudo-effective
cone of $X$.
So, $K_X+(n-1)D=K_X+nD-D$ is not pseudo-effective.
This is a contradiction.
Anyway, we obtain that
$K_X+(n-1)D$ is nef if $K_X+(n-1)D$ is pseudo-effective.
\end{proof}
|
1,108,101,565,677 | arxiv | \section{#1}}
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\title{Opposite product system for the multiparameter CAR flows}
\author{Anbu Arjunan}
\newcommand{\insfig}[2]{
\begin{figure}[hbpt]
\centerline{\input{#1}}
\caption{#2\label{f-#1}}
\end{figure}
}
\begin{document}
\maketitle
\begin{abstract}
We consider the multiparameter CAR flows and describe its opposite. We also characterize the symmeticity of CAR flows in terms of associated isometric representations.
\end{abstract}
\noindent {\bf AMS Classification No. :} {Primary 46L55; Secondary 46L99.} \\
{\textbf{Keywords :}}$E_0$-semigroups, CCR flow, CAR flow, opposite product system.
\section{Introduction}
Let $P$ be a closed convex cone in $\mathbb{R}^d$. We assume that $P-P=\mathbb{R}^d$ and $P\cap -P=\{0\}$. Let $V$ be a pure isometric representation of $P$ and let $\alpha$ be the CCR flow associated to the isometric representation $V$.
The author in \cite{R19} have shown that the CCR flow is not cocycle conjugate to the CAR flow when the isometric representation $V$ is proper.
The product system associated with the CAR flow is not decomposable in general; see \cite{arjunan2020decomposability}.
It was shown in \cite{sundar2020asymmetric} that $\alpha$ is cocycle conjugate to $ \alpha^{\text{op}}$ if and only if $V$ is unitary equivalent to its opposite $V^{\text{op}}$.
This result uses the characterization of decomposable product system which admits a unit; see \cite{sundar2019arvesons}.
It is natural to ask whether the analogous result holds true for the multiparameter CAR flows. In this article we answer this question affirmatively; see Theorem \ref{main}.
We will achieve this by identifying the opposite of the product system for a CAR flow with the product system for an appropriate CAR flow. Also we will also use this to study the symmetricity of the CAR flows.
\section{Preliminaries}
Let $H$ be a Hilbert space and let $\Gamma_a(H)$ be the antisymmetric Fock space over $H$.
For $f\in H$, define a bounded operator $a(f)^*$ on $\Gamma_a(H)$ as
\begin{align*}
a(f)^*(\Omega)&=f\text{ and }\\
a(f)^*(h_1\wedge h_2\wedge...\wedge h_n)&=f\wedge h_1\wedge h_2\wedge...\wedge h_n
\end{align*}
where $\Omega$ is the vacuum vector of $\Gamma_a(H)$ and $h_1\wedge h_2\wedge...\wedge h_n$ is an arbitrary antisymmetric elementary tensor element with $h_1, h_2,..., h_n\in H$ and $n\geq 1$. Let $a(f)$ be the adjoint of $a(f)^*$. The operators $a(f)^*$ and $a(f)$ are called the creation and the annihilation operator associated to a vector $f$.
By an isometric representation of $P$ on a Hilbert space $H$, we mean a strongly continuous map $V:P\to B(H)$ such that each $V_x$ is an isometry and $V_xV_y=V_{x+y}$ for each $x,y\in P$.
For a given isometric representation $V:P\to B(H)$, there exists a unique $E_0$-semigroup, denoted by $\beta^V$, on $\Gamma_a(H)$ satisfying
\[\beta^V_x(a(f))=a(V_xf)\text{ for each }f\in H.\]
This $E_0$-semigroup $\beta^V$ is called the CAR flow associated to the isometric representation $V$; see \cite{R19}.
Let $H$ and $K$ be Hilbert spaces. For an isometry $W:H\to K$, there exists a unique bounded operator $\Gamma_a(W)$, called the second quantization of $W$, from $\Gamma_a(H)$ to $\Gamma_a(K)$, satisfying
\begin{align*}
\Gamma_a(W)(\Omega)&=\Omega, \text{ and }\\
\Gamma_a(W)(f_1\wedge f_2\wedge...\wedge f_n)&=Wf_1\wedge Wf_2\wedge...\wedge Wf_n ,
\end{align*}
where $\Omega$ is the vacuum vector in the appropriate antisymmetric Fock space and $f_1\wedge f_2\wedge...\wedge f_n$ is any antisymmetric elementary tensor element with $f_1, f_2,..., f_n\in H$ and $n\geq 1$.
\section{Opposite product sysem for a CAR flow}
Let $V$ be a pure isometric representation of $P$ on a Hilbert space $H$. Let $\beta^V$ be the CAR flow associated to the isometric representation $V$ and denote its concrete product system by $\mathcal{E}_{\beta^V}$.
Set $E^V(x)=\Gamma_a(\text{Ker}(V_x^*))$.
Consider the set $E^V$ as
\[E^V=\{(x,f):x\in \Omega\text{ and }f\in E^V(x)\}.\]
Since $E^V$ is a Borel subset of $\Omega\times \Gamma_a(H)$, $E^V$ is a standard Borel space. Define a multiplication $.$ on $E^V$ as
\[(x,f).(y,f):=(x+y, f\otimes \Gamma_a(V_x)g)\]
for every $(x,f),(y,f)\in E^V$. $E_V$ equipped with the above multiplication defines a product system structure over $\Omega$. We define another multiplication $\circ$ on $E^V$ as
\[(x,f)\circ(y,f):=(x+y, g\otimes \Gamma_a(V_y)f).\]
Then the pair $(E^V,\circ)$ also has a structure of product system over $\Omega$, called the opposite product system for $(E^V,.)$, denoted by $(E^V)^{\text{op}}$.
Let $x\in \Omega$ and let $f\in E^V(x)$ be given. Define a bounded operator $T_f$ on $\Gamma_a(H)$ as
\[T_f\eta=f\otimes \Gamma_a(V_x)\eta,\text{ for every }\eta\in \Gamma_a(H).\]
Then we have the following lemma.
\begin{lmma}\label{keylemma} The map $\theta:E^V\ni (x,f)\mapsto (x,T_{f})\in \mathcal{E}_{\beta^V}$ is an isomorphism as product systems.
\end{lmma}
\begin{prf}
Let $(x,f),(y,g)\in E^V$ be given. Since $T_fT_g=T_{f\otimes \Gamma_a(V_x)g}$, it follows that $\theta(x,f)\theta(y,g)=\theta((x,f)(y,g))$. For each $x\in \Omega$, the restriction of $\theta$ to $E^V(x)$, $\theta|_{E^V(x)}:E^V(x)\to \mathcal{E}_{\beta^V}(x)$ is a unitary. For let $f,g\in E^V(x)$ be given. Note that $T_g^*T_f=\langle f,g\rangle 1_{E^V(x)}$ and $T_f\in \mathcal{E}_{\beta^V}(x)$.
This implies that the map $E^V(x)\ni f\mapsto T_f\in \mathcal{E}_{\beta^V}(x)$ is an isometry. To prove that the map is a unitary it suffices to show that whenever $T\in \mathcal{E}_{\beta^V}(x)$ such that $\langle T_f,T\rangle=0$ for all $f$, then $T=0$.
Since the linear span of the set $\{f\otimes \Gamma_a(V_x)\eta: f\in E^V(x)\text{ and }\eta\in \Gamma_a(H)\}$ is dense in $\Gamma_a(H)$, we see that $T=0$.
Since $E^V$ and $\mathcal{E}_{\beta^V}$ are standard Borel spaces and the restriction of $\theta$ to each fibre is a unitary, it follows that the map $\theta$ is a Borel isomorphism and hence it is a isomorphism as product systems by \cite{ArvCA}.
\hfill $\Box$
\end{prf}
Let us recall the opposite isometric representation $V^{\text{op}}$ for the given isometric representation $V$ considered in \cite{sundar2019arvesons}.
Let $U$ be a minimal unitary dilation of $V$. More precisely, there exists a Hilbert space $\widetilde{H}$ containing $H$ as a subspace and a unitary representation $U$ of $\mathbb{R}^d$ on a Hilbert $\widetilde{H}$ such that the following conditions hold.
\begin{enumerate}
\item For $x\in P$, $U_x\xi=V_x\xi$.
\item The set $\cup_{x\in P}U_x^*H$ is dense in $\widetilde{H}$.
\end{enumerate}
Note that for $x\in P$, $K=H^{\perp}$ is invariant under $U_x$. For $x\in P$, define $V^{\text{op}}_x$ on $K$ to be the restriction of $U_{-x}$ to $K$ i.e. $V^{\text{op}}_x:=U_{-x}|_{K}$. Then $V^{\text{op}}:=\{V^{\text{op}}_x\}_{x\in P}$ is an isometric representation of $P$, called the opposite isometric representation for $V$. This isometric representation $V^{\text{op}}$ is pure \cite[Proposition 3.2]{sundar2020asymmetric}.
\begin{ppsn}\label{keyprop}
The map $\phi:(E^V)^{\text{op}}\ni (x,f)\mapsto (x,\Gamma_a(U_{-x})f)\in E^{V^{\text{op}}}$ is an isomorphism as product systems.
\end{ppsn}
\begin{prf}
For each $x\in \Omega$, the map $\text{Ker}(V_x^*)h\mapsto U_{-x}h\in \text{Ker}((V_x^{\text{op}})^*)$ is a unitary; see the proof of \cite[Proposition 3.2]{sundar2020asymmetric}. Then it follows that the map $\phi:(E^V)^{\text{op}}\ni (x,f)\mapsto (x,\Gamma_a(U_{-x})f)\in E^{V^{\text{op}}}$ is a continuous bijection and its inverse is given by $E^{V^{\text{op}}}\ni (x,\xi)\mapsto (x,\Gamma_a(U_{x})\xi)\in(E^V)^{\text{op}}$. Hence it is a Borel isomorphism by \cite{ArvCA}. Now it remains to show that $\phi$ follows product system structure.
Let $(x,f),(y,g)\in E^V$ be given. Then we have
\begin{align*}
\phi((x,f)(y,g))&=\phi(x+y,f\otimes \Gamma_a(V_x)g)\\
&=(x+y,\Gamma_a(U_{-(x+y)})(f\otimes \Gamma_a(V_x)g))\\
&=(x+y,\Gamma_a(U_{-(x+y)}) \Gamma_a(V_x)g\otimes \Gamma_a(U_{-(x+y)})f)\\
&=(x+y,\Gamma_a(U_{-y})g\otimes \Gamma_a(U_{-y})\Gamma_a(U_{-x})f)\\
&=(x+y,\Gamma_a(U_{-y})g\otimes \Gamma_a(V_y^{\text{op}})\Gamma_a(U_{-x})f)\\
&=(y,\Gamma_a(U_{-y})g)(x,\Gamma_a(U_{-x})f)\\
&=\phi(y,g)\phi(x,f).
\end{align*}
Hence the map $\phi$ is an isomorphism as product systems.
\hfill $\Box$
\end{prf}
Let $\mathcal{E}_{\beta^V}$ be the concrete product system for $\beta^V$ and let $\mathcal{E}_{\beta^V}^{\text{op}}$ be its opposite product system. By \cite[Theorem 3.14]{MS}, there exists an $E_0$-semigroup denoted by $(\beta^V)^{\text{op}}$ such that $\mathcal{E}_{\beta^V}^{\text{op}}$ is isomorphic to $\mathcal{E}_{(\beta^V)^{\text{op}}}$.
\begin{crlre}\label{oppositecar}
An $E_0$-semigroup $(\beta^V)^{\text{op}}$ is cocycle conjugate to $\beta^{V^{\text{op}}}$.
\end{crlre}
\begin{prf}
By Proposition \ref{keyprop} and Lemma \ref{keylemma}, we conclude that $(E^V)^{\text{op}}$ is isomorphic to $\mathcal{E}_{\beta^{V^{\text{op}}}}$. This implies that the product system $\mathcal{E}_{(\beta^V)^{\text{op}}}$ is isomorphic to $\mathcal{E}_{\beta^{V^{\text{op}}}}$ by Lemma \ref{keylemma}.
Then by \cite[Theorem 2.9]{MS}, we have $(\beta^V)^{\text{op}}$ is cocycle conjugate to $\beta^{V^{\text{op}}}$.
\hfill $\Box$
\end{prf}
\begin{rmrk}
The above corollary implies that the opposite of a CAR flow over $P$ is again a CAR flow over $P$.
\end{rmrk}
\begin{thm} \label{main} Let $\beta^V$ be the CAR flow associated to an isometric representation $V$. Then the following are equivalent.
\begin{enumerate}
\item The CAR flow $\beta^V$ is cocycle conjugate to its opposite $(\beta^V)^{\text{op}}$
\item The isometric representation $V$ is unitary equivalent to its opposite $V^{\text{op}}$.
\end{enumerate}
\end{thm}
\begin{prf}
Proof follows from \cite[Proposition 4.7]{R19} and Corollary \ref{oppositecar}.
\hfill $\Box$
\end{prf}
\section{Examples for symmetric and asymmetric CAR flows}
By a $P$-module we mean a non-empty closed subset $A$ of $\mathbb{R}^d$ such that $A+P\subseteq A$.
Let $A$ be a $P$-module. For $x\in P$, define an operator $V^A_x$ on $L^2(A)$ as
\[(V_x^{A}f)(y)=\begin{cases}
f(y-x) & \mbox{ if } y-x\in A,\cr
0 & \mbox{ if } y-x\notin A.
\end{cases}\]
for each $f\in L^2(A)$. Then the family $\{V_x^A\}_{x\in P}$ is an isometric representation of $P$.
\begin{ppsn}\label{sundarresult} (See \cite[Proposition 3.4]{sundar2020asymmetric})
We have the following.
\begin{enumerate}
\item The isometric representation $(V^A)^{\text{op}}$ is unitary equivalent to $V^A$.
\item There exists an element $z\in \mathbb{R}^d$ such that $A=-(\text{int}(A)^c)+z$.
\end{enumerate}
Here $\text{int}(A)$ is the interior of $A$ and $\text{int}(A)^c$ is the complement of $\text{int}(A)$ in $\mathbb{R}^d$.
\end{ppsn}
Let $\beta^A$ be the CAR flow associated to the isometric representation $V^A$. It follows from Theorem \ref{main} and Proposition \ref{sundarresult} that the CAR flow $\beta^A$ is cocycle conjugate to its opposite $(\beta^A)^{\text{op}}$ if and only if $A=-(\text{int}(A)^c)+z$ for some $z\in\mathbb{R}^d$.
\begin{rmrk}
By considering the existence of such $P$-modules, we can see that there are uncountably many symmetric CAR flows as well as asymmetric CAR flows over $P$.
\end{rmrk}
\section*{Acknowledgment}
The author would like to thank The Institute of Mathematical Sciences for the Institute Postdoctoral fellowship.
|
1,108,101,565,678 | arxiv | \section{Introduction}
We introduce a random graph $\ensuremath{\mathcal G}$, called \emph{Poisson $h$--generalized Boolean model}, with vertexes in ${\ensuremath{\mathbb R}} ^d$ where
$d\geq 2$,
whose construction depends on a structural symmetric function $h(\cdot, \cdot)$ with real entries, a parameter $\l>0$ and a probability measure $\nu$ on ${\ensuremath{\mathbb R}} $.
Given a homogeneous Poisson point process (PPP) $\xi$ on ${\ensuremath{\mathbb R}} ^d$ with \textcolor{black}{intensity} $\l$, we mark points $x$ of $\xi$ by i.i.d. random variables $E_x$ with common distribution $\nu$. Then the vertexes of $\ensuremath{\mathcal G}$ are the points in $\xi$, while edges of $\ensuremath{\mathcal G}$ are given by unordered pairs of vertexes $\{x,y\}$ with $x\not =y$ and $|x-y|\leq h(E_x,E_y)$. \textcolor{black}{When $E_x\in [1,+\infty)$ and $h(\cdot, \cdot)$ is non--decreasing in each entry, then $\ensuremath{\mathcal G}$ belongs to the class of \emph{weight-dependent random connection models} introduced in \cite{GHMM}}.
\smallskip
When $\nu $ has support inside $[0,+\infty)$ and $h(a,b)=a+b$, one recovers the so called Poisson Boolean model \cite{MR}. Another relevant example is related to the Miller-Abrahams (MA) random resistor network. This resistor network has been introduced in \cite{MA} to study
the anomalous conductivity at low temperature in amorphous materials as doped semiconductors, in the regime of Anderson localization and at low density of impurities. It has been further investigated in the physical literature (cf. \cite{AHL},
\cite{POF} and references therein), where percolation properties have been heuristically analyzed.
A fundamental target has been
to get a more robust derivation of the so called Mott's law, which is a physical law predicting the anomalous decay of electron conductivity at low temperature (cf. \cite{Fhom2,F_final,FM,FSS,POF} and references therein).
When built on a marked PPP $\{(x,E_x)\,:x\in \xi\}$ as above, the MA random resistor network has nodes given by points in $\xi$ and electrical filaments connecting each pair of nodes. The electrical conductivity of the filament between $x$ and $y$ is given by (cf. \cite[Eq. (3.7)]{AHL})
\begin{equation}\label{condu}
c(x,y):=\exp\Big\{ - \frac{2}{\gamma} |x-y| -\frac{\b}{2} ( |E_x|+ |E_y|+ |E_x-E_y|) \Big\}\,,
\end{equation}
where $\b$ is the inverse temperature and $\g$ is the so--called localization length. The physically relevant distributions $\nu$ (for inorganic materials) are of the form $\nu_{\rm phys}(dE)\propto
\mathds{1} ( |E| \leq a_0) |E|^\a dE $ with $\a\geq 0 $ and $a_0>0$.
Note that the conductivity of the filaments is smaller than $1$. Fixed $c_0\in (0,1)$, the resistor subnetwork $\ensuremath{\mathcal G}$ given by the filaments with conductivity lower bounded by $c_0$ is a Poisson $h$--generalized Boolean model
with $h(a,b):=-(\g/2) \ln c_0 -(\g \b/4) (|a|+|b|+|a-b|)$. We point out that, in the low temperature regime (i.e. when
$\b \uparrow \infty$), the conductivity of the MA random resistor network is mainly supported by the subnetwork $\ensuremath{\mathcal G}$ associated to a suitable constant $c_0=c_0(\b)$ (cf. \cite{AHL,F_final}).
Hence, to shorten the terminology and with a slight abuse, in the rest we call $\ensuremath{\mathcal G}$ itself ``MA random resistor network". Moreover, without loss of generality, we take $\b=\g=2$.
\smallskip
For the supercritical Bernoulli site and bond percolations on ${\ensuremath{\mathbb Z}} ^d$ with $d\geq 2$, it is known that the maximal number of
vertex-disjoint left-right crossings of a box $[-L,L]^d$
is lower bounded by $C L^{d-1}$ \textcolor{black}{apart from} an event with exponentially small probability (cf. \cite[Thm.~(7.68), Lemma~(11.22)]{G} and \cite[Remark~(d)]{GM}). A similar result is proved for the supercritical Poisson Boolean model with deterministic radius in \cite{T}. Our main result is that, under general suitable conditions,
the same behavior holds for the $h$--generalized Boolean model (cf. Theorem \ref{teo1}). This result for the MA random resistor network is relevant when studying the low--temperature conductivity in amorphous solids (cf. \cite{AHL,F_final}).
\smallskip
We comment now some technical aspects in the derivation of our contribution.
To prove Theorem \ref{teo1} we first show that it is enough to derive a similar result (given by Theorem
\ref{teo2} in Section \ref{sec_discreto})
for a suitable random graph ${\ensuremath{\mathbb G}} _+$ with vertexes in $\e \bbZ^d$, defined in terms of i.i.d. random variables parametrized by points in $\e \bbZ^d$.
The
proof of Theorem \ref{teo2} is then inspired by the renormalization procedure developed by Grimmett and Marstrand in \cite{GM} for site percolation on ${\ensuremath{\mathbb Z}} ^d$ and by a construction
presented by Tanemura in \cite[Section 4]{T}. We recall that in \cite{GM} it is proved
that the critical probability of a slab in ${\ensuremath{\mathbb Z}} ^d$ converges to the critical probability of ${\ensuremath{\mathbb Z}} ^d$ when the thickness of the slab goes to $+\infty$.
We point out that the renormalization method developed in \cite{GM}
does not apply verbatim to our case. In particular the adaptation of Lemma 6 in \cite{GM} to our setting presents several obstacles due to spatial correlations in our model.
A main novelty here is to build, by a Grimmett-Marstrand-like renormalization procedure, an increasing family of quasi-clusters in our graph ${\ensuremath{\mathbb G}} _+$. We use here the term ``quasi-cluster'' since usually these sets are not connected in ${\ensuremath{\mathbb G}} _+$ and can present some cuts at suitable localized regions. By
expressing the PPP of \textcolor{black}{intensity} $\l$ as superposition of two independent PPP's with \textcolor{black}{intensity} $\l -\d$ and $\d\ll 1 $, respectively, a quasi-cluster is built only by means of points in the PPP with \textcolor{black}{intensity} $\l-\d$. On the other hand, we will show that, with high probability, when superposing the PPP with \textcolor{black}{intensity} $\d$ we will insert a family of points linked with the quasi-cluster, making the resulting set connected in
${\ensuremath{\mathbb G}} _+$. \textcolor{black}{This construction relies on the idea of ``sprinkling" going back to \cite{ACCFR} (see also \cite{G} and references therein)}.
We remark that in \cite{MT} Martineau and Tassion have extended, with some modifications and simplifications, the Grimmett-Marstrand renormalization scheme. Nevertheless, we have followed here the construction in \cite{GM} since it is more suited to be combined with Tanemura's algorithm, which prescribes step by step to build clusters in ${\ensuremath{\mathbb G}} _+$ along the axes of ${\ensuremath{\mathbb R}} ^d$ (while in \cite{MT} there is a good building direction, which is not explicit). On the other hand, the fundamental steps in our Grimmett-Marstrand-like renormalization scheme are slightly simplified w.r.t. the original ones in \cite{GM}.
As special \textcolor{black}{applications} of Theorem \ref{teo1} we consider, \textcolor{black}{for non-negative marks, the
cases $h(a,b):=(a+b)^\g$ with $\g>0$, $h(a,b):=\max(a,b)$, $h(a,b):=\min(a,b)$ and $h(a,b):= \z - (|a|+|b|+|a-b|)$ with $\z>0$ (see Corollaries \ref{cor1} and \ref{cor2})}.
The first case, with $\g=1$, generalizes Tanemura's result to the Poisson Boolean model with random radius. \textcolor{black}{The second and third cases correspond respectively to the min-kernel and the max-kernel random connection model (see \cite{GHMM} and references therein)}.
The \textcolor{black}{fourth} case corresponds to the MA random resistor network with non-negative energy marks. Although this result does not cover the mark distributions $\nu_{\rm phys}$ mentioned above, it is suited to distributions $\nu(dE)\propto
\mathds{1} ( 0\leq E \leq a_0) |E|^\a dE $ with $\a\geq 0 $ and $a_0>0$, which have similar scaling properties to the physical ones $\nu_{\rm phys}$. We stress that these scaling properties are relevant in the heuristic derivation of Mott's law as well as in its rigorous analysis \cite{F_phys,F_final}. The restriction to non-negative marks
in the MA random resistor network comes from the fact that the Grimmett-Marstrand method (as well as its extension in \cite{MT})
relies on the FKG inequality. As \textcolor{black}{one can check}, when considering the MA random resistor network with marks having different signs, the FKG inequality can fail.
For the reader's convenience, an outline of the paper is provided in Section \ref{pigolano} after the presentation of models and main results.
Finally, we point out that percolation results for the subcritical MA random resistor network have been obtained in \cite{FMim1}.
\section{Model and main results}\label{moda}
We introduce a class of random graphs built by means of a symmetric \emph{structural function}
\begin{equation}
h:\D\times \D \to {\ensuremath{\mathbb R}} \,,\qquad \D\subset {\ensuremath{\mathbb R}} \,.
\end{equation}
To this aim we call $\O$ the space of locally finite sets of marked points in ${\ensuremath{\mathbb R}} ^d$, $d\geq 2$, with marks in $\D$. More precisely, a generic element $\o\in \O$ has the form $\o= \{ (x, E_x)\,:\, x\in \xi\}$, where $\xi$ is a locally finite subset of ${\ensuremath{\mathbb R}} ^d$ and $E_x \in \D $ for any $x\in \xi$ ($E_x$ is thought of as the mark of point $x$). It is standard (cf. \cite{DV}) to define on $\O$ a distance such that the $\s$--algebra of Borel sets $\ensuremath{\mathcal B}$ of $\O$
is generated by the sets $\{ |\o \cap A|=n\}$, $A$ varying among the Borel subsets of ${\ensuremath{\mathbb R}} ^d$ and $n $ varying in ${\ensuremath{\mathbb N}} $. We assume that $(\O,\ensuremath{\mathcal B})$ is endowed with a probability measure $P$, thus defining a marked simple point process.
\begin{Definition}[\textcolor{black}{Graph $\ensuremath{\mathcal G}[\o]$}]
To each $\o= \{ (x, E_x)\,:\, x\in \xi\}$ in $ \O$ we associate the unoriented graph $\ensuremath{\mathcal G}[\o]=\bigl( \ensuremath{\mathcal V}[\o], \ensuremath{\mathcal E}[\o]\bigr)$ with vertex set $\ensuremath{\mathcal V}[\o]:=\xi$ and edge set $\ensuremath{\mathcal E}[\o]$ given by the unordered pairs $\{x,y\}$ with $x\not =y$ in $\xi$ and
\begin{equation}\label{pico}
|x-y| \leq h(E_x,E_y) \,.
\end{equation}
We call $h$--generalized Boolean model the resulting random graph $\ensuremath{\mathcal G}=\ensuremath{\mathcal G}[\o]$ defined on $(\O,\ensuremath{\mathcal B},P)$.
\end{Definition}
When $\D:={\ensuremath{\mathbb R}} _+$ and $h(a,b):=a+b$, one has indeed the so--called Boolean model \cite{MR}. As discussed in the Introduction, another relevant example is given by the MA random resistor network with lower bounded conductances: in this case $\D={\ensuremath{\mathbb R}} $ and, fixed the parameter $\z>0$, the structural function $h: {\ensuremath{\mathbb R}} \times {\ensuremath{\mathbb R}} \to {\ensuremath{\mathbb R}} $ is given by
\begin{equation} \label{hMA}
h(a,b):=\z-\bigl( |a|+|b|+|a-b|\bigr)\,.
\end{equation}
We focus here on the left-right crossings of the graph $\ensuremath{\mathcal G}=\ensuremath{\mathcal G}[\o]$:
\begin{Definition}[\textcolor{black}{LR crossing and $\ensuremath{\mathcal R}_L(\ensuremath{\mathcal G})$}] \label{def_LR}
Given \textcolor{black}{$L\in (0,+\infty)$}, a left-right (LR) crossing of the box $[-L,L]^d$ in the graph $\ensuremath{\mathcal G}$ is any sequence of distinct points $x_1,x_2 , \dots, x_n \in \xi$ such that
\begin{itemize}
\item $\{x_i ,x_{i+1}\} \in \ensuremath{\mathcal E}$ for all $i=1,2,\dots, n-1$;
\item $x_1 \in (-\infty,-L)\times [-L,L]^{d-1} $;
\item $x_2, x_3,\dots, x_{n-1}\in [-L,L]^d$;
\item $x_n \in (L, +\infty )\times [-L,L]^{d-1} $.
\end{itemize}
We also define \textcolor{black}{$\ensuremath{\mathcal R}_L(\ensuremath{\mathcal G})$} as the maximal number of vertex-disjoint LR crossings of $[-L,L]^d$ in $\ensuremath{\mathcal G}$.
\end{Definition}
In what follows, given $\l>0$ and a probability measure $\nu $ with support contained in $\D$, we consider the marked Poisson point process (PPP) obtained by sampling $\xi$ according to a homogeneous PPP with \textcolor{black}{intensity} $\l$ on ${\ensuremath{\mathbb R}} ^d$ ($d\geq 2$) and marking each point $x\in \xi $ independently with a random variable $E_x$ having distribution $\nu$ (conditioning to $\xi$,
the marks $(E_x)_{x\in\xi}$ are i.i.d. random variables with distribution $\nu$). The above marked point process is called \emph{$\nu$-randomization} of the PPP with \textcolor{black}{intensity} $\l$ \textcolor{black}{(cf.~\cite[Chp.~12]{Kal})}. The resulting random graph $\ensuremath{\mathcal G}$, whose construction depends also on the structural function $h$, will be denoted by $\ensuremath{\mathcal G}(h,\l)=\ensuremath{\mathcal G}(h,\l)[\o]$ when necessary (note that $\nu$ is understood).
\smallskip
To state our main assumptions, we recall that, given a generic graph with vertexes in ${\ensuremath{\mathbb R}} ^d$, one says that
it percolates if it has an unbounded connected component. We also fix, once and for all, a constant \textcolor{black}{$\l>0$} and a probability measure $\nu $ on $\D$. We write ${\rm supp}(\nu)$ for the support of $\nu$.
\medskip
{\bf Assumptions}:
\emph{
\begin{itemize}
\item[(A1)] $P$ is the $\nu$--randomization of a PPP with \textcolor{black}{intensity} $\l$ on ${\ensuremath{\mathbb R}} ^d$, $d\geq 2$.
\item[(A2)] There exist $\l_*<\l$ and $\ell _* >0$ such that $\ensuremath{\mathcal G}(h-\ell_*, \l_*)$ percolates a.s..
\item[(A3)] $\sup \left\{ h(a,b)\,:\, a,b \in \text{supp}(\nu) \right\} \in (0,+\infty)$.
\item[(A4)] For any $\d>0$ there exists
a Borel subset $U_*(\d)\subset \D$ with $\nu(U_*(\d))>0$ such that
\textcolor{black}{ \begin{equation}\label{indigestione}
\inf_{ a\in U_*(\d) } h(a,b)
\geq \sup_{a'\in {\rm supp}(\nu)} h (a',b) -\d \,, \qquad \forall b\in {\rm supp}(\nu)\,.
\end{equation}}
\item[(A5)] As $a,b $ vary in ${\rm supp}(\nu)$, $h(a,b)$ is weakly decreasing both in $a$ and in $b$ (shortly, $h \searrow$), or $h(a,b)$ is weakly increasing both in $a$ and in $b$ (shortly, $h \nearrow$).
\end{itemize}
}
Let us comment our assumptions.
The definition of $h$ is relevant only for entries in ${\rm supp(\nu)}$, hence one could as well restrict to the case $\D={\rm supp}(\nu)$.
Since the $h$--generalized Boolean model presents spatial correlations by its own definition, Assumptions (A1) avoids further spatial correlations inherited from the marked point process.
By a simple coupling argument, (A2) implies that $\ensuremath{\mathcal G}(h-\ell', \l')$ percolates a.s. for any $\ell'\in [0,\ell_*]$ and $\l'\in [\l_*, \l]$. Hence, (A2) assures some form of ``stable supercriticality'' of the graph $\ensuremath{\mathcal G}(h, \l)$.
Due to \eqref{pico}, (A3) both excludes the trivial case $h\leq 0$ on ${\rm supp}(\nu)$ (which would imply that
$\ensuremath{\mathcal G}(h,\l)$ has no edges) and guarantees that the length of the edges of $\ensuremath{\mathcal G}(h,\l)$ is a.s. bounded by some
deterministic constant.
We move to (A4). By definition of supremum and due to (A3), for any $\d>0$ and for any $b\in {\rm supp}(\nu) $ there exists $a\in {\rm supp}(\nu)$ such that $h(a,b)\geq \sup_{a' \in {\rm supp}(\nu) }h(a,b)-\d$. Assumption (A4) enforces this free inequality requiring that it is satisfies uniformly in $a$ varying in some subset $U_*(\d)$ with positive $\nu$--measure. For example, if $h$ is continuous, ${\rm supp}(\nu)$ is bounded and (A5) is satisfied, then (A4) is automatically satisfied. Indeed, if e.g. $h \nearrow$, then $
\sup_{a' \in {\rm supp}(\nu) }h(a',b)=h(A,b)$ with $A:=\max \bigl( \text{supp}(\nu)\bigr)$ and the claim follows by uniform continuity.
We move to (A5). This assumption implies that we enlarge the graph when reducing the marks if $h \searrow $ or increasing the marks if $h \nearrow$. Moreover, (A5) guarantees the validity of the FKG inequality (cf. Section \ref{scremato}), which in general can fail
\medskip
Our main result is the following one:
\begin{TheoremA} \label{teo1}
Suppose that the \textcolor{black}{intensity} \textcolor{black}{$\l>0$} and the mark probability distribution $\nu$ satisfy the above Assumptions (A1),...,(A5). Then
there exist
positive constants $c, c'$ such that
\begin{equation}\label{stimetta}
P \left( \textcolor{black}{\ensuremath{\mathcal R}_L(\ensuremath{\mathcal G})} \geq c L^{d-1} \right) \geq 1- e^{- c' L^{d-1}} \,,
\end{equation}
for $L$ large enough, where $\ensuremath{\mathcal G}= \ensuremath{\mathcal G}(h,\l)$.
\end{TheoremA}
The proof of Theorem \ref{teo1} is localized as follows: by Proposition \ref{fragolino} to get Theorem \ref{teo1} it is enough to prove Theorem \ref{teo2}. By Proposition \ref{carletto} to get Theorem \ref{teo2} it is enough to prove \eqref{problema2d} \textcolor{black}{in Proposition \ref{carletto}}. The proof of \eqref{problema2d} \textcolor{black}{in Proposition \ref{carletto}} is given in Section \ref{sec_ginepro}.
\begin{Remark} We point out that \eqref{stimetta} cannot hold for all $L>0$, but fixed $L_0>0$ one can play with the constants $c,c'>0$ to extend \eqref{stimetta} to all $L\geq L_0$. Let us explain this issue. Calling $\ell_0$ the supremum in (A3), we get that all edges of $\ensuremath{\mathcal G}$ have length at most $\ell_0$. The event in the l.h.s. of \eqref{stimetta} implies that $\ensuremath{\mathcal R}_L(\ensuremath{\mathcal G})\geq 1$ and therefore the PPP must contain some point both in $[-L-\ell_0,-L) \times [-L,L]^{d-1}$ and in $(L,L+\ell_0] \times [-L,L]^{d-1}$. Hence the probability in \eqref{stimetta} is upper bounded by $( 1- e^{-\l \ell_0 (2L)^{d-1}})^2$, which is of order $L^{2d-2}$ as $L\downarrow 0$. As $d\geq 2$ we conclude that \eqref{stimetta} cannot hold for $L$ too small.
On the other hand, fix $L_0>0$ and suppose that \eqref{stimetta} holds for all $L\geq L_1$ for some $L_1>L_0$. Take now $L\in [L_0,L_1]$.
Our assumptions imply that there exists a set $W\subset {\ensuremath{\mathbb R}} $ such that $\nu(W)>0$ and $r:=\inf _{a,b\in W} h(a,b)>0$ (cf. \eqref{mango}). Set $e_1:=(1,0,\dots, 0)$, $\ell:=\min(r,L_0)$ and let $N$ be the minimal positive integer such that $ N \ell /2 > L_1$. Then $\ensuremath{\mathcal R}_L(\ensuremath{\mathcal G})\geq 1$ whenever each ball of radius $\ell/10$ and centered at $k(\ell/2) e_1$ contains some point $x $ of the Poisson process with mark $E_x\in W$, where $k$ varies from $-N-1$ to $N+1$. Note that this last event does not depend on $L$, and therefore the same holds for its positive probability. At this point, by playing with $c,c'$ in \eqref{stimetta}, one can easily extend \eqref{stimetta} to all $L\geq L_0$.
\end{Remark}
We now discuss some applications.
\smallskip
Let $d\geq 2$, $\l >0$ and let $h:{\ensuremath{\mathbb R}} _+\times{\ensuremath{\mathbb R}} _+\rightarrow{\ensuremath{\mathbb R}} $ be \textcolor{black}{one of the following functions (for the first one $\g$ is a fixed positive constant):
\begin{equation} \label{hgamma}
h(a,b):=(a+b)^\g\,,\qquad h(a,b):= \min (a,b)\, \qquad
h(a,b):=\max(a,b)\,.\end{equation}}
Consider the graph $\ensuremath{\mathcal G}=\ensuremath{\mathcal G}(h,\l)$ built on the $\nu$--randomization of a PPP on ${\ensuremath{\mathbb R}} ^d$ with \textcolor{black}{intensity} $\l$.
As proved in Section \ref{bin_MA} (cf. Lemma \ref{silvestro}),
if $\nu $ has bounded support and $\nu(\{0\})\not =1$, then there exists a critical \textcolor{black}{intensity} \textcolor{black}{$\l_c$} \textcolor{black}{in $(0,+\infty)$} such that
\begin{equation}\label{selva0}
P \bigl( \ensuremath{\mathcal G} \text{ percolates})=
\begin{cases}
1 & \text{ if } \l>\l_c \,,\\
0 &\text{ if } \l < \l_c\,.
\end{cases}
\end{equation}
\begin{Corollary}\label{cor1} Let $d\geq 2$ \textcolor{black}{and let $h$ be given by one of the functions in \eqref{hgamma}}.
Consider the graph $\ensuremath{\mathcal G}=\ensuremath{\mathcal G}(h,\l)$ built on the $\nu$--randomization of a PPP on ${\ensuremath{\mathbb R}} ^d$ with \textcolor{black}{intensity} $\l>\l_c$, where the law $\nu$ has bounded support and $\nu(\{0\})\not =1$. Then there exist $c,c'>0$ such that \eqref{stimetta} is fulfilled for $L$ large enough.
\end{Corollary}
We postpone the proof of the above corollary to Section \ref{bin_MA}.
\textcolor{black}{Note that we recover the Poisson Boolean model \cite{MR} when $h(a,b):=(a+b)^\g$ and $\g=1$. We point out that, according to the notation in \cite{GHMM}, the above graph $\ensuremath{\mathcal G}(h,\l)$ corresponds to the min-kernel (or max-kernel) weight--dependent random connection model by taking $h(a,b)=\max(a,b)$ (respectively $h(a,b)=\min(a,b)$) and by defining the weight of the point $x$ as a suitable rescaling of our $E_x$.}
\smallskip
Let now $d\geq 2$ and $\z,\l>0$.
Consider the MA random resistor network $\ensuremath{\mathcal G}=\ensuremath{\mathcal G}(h,\l)$ with parameter $\z$ (i.e. with structural function $h$ given by \eqref{hMA}), built on the $\nu$--randomization of a PPP on ${\ensuremath{\mathbb R}} ^d$ with \textcolor{black}{intensity} $\l $. \textcolor{black}{When $\nu$ has bounded support it is trivial to check} that there exists a critical length $\z_c$ \textcolor{black}{in $(0,+\infty)$} such that
\begin{equation}\label{selva}
P \bigl( \ensuremath{\mathcal G} \text{ percolates})=
\begin{cases}
1 & \text{ if } \z>\z_c \,,\\
0 &\text{ if } \z < \z_c\,.
\end{cases}
\end{equation}
\textcolor{black}{Indeed, if $\nu$ has support in $[-A,A]$, then $\ensuremath{\mathcal G}$ contains (is contained in) a random graph distributed as a Poisson Boolean model with intensity $\l$ and deterministic radius $\z/2-2A$ ($\z/2$ respectively)}.
Equivalently, one could keep $\z$ fixed and play with the \textcolor{black}{intensity} $\l$ getting a phase transition as in \eqref{selva0} \textcolor{black}{under suitable assumptions (see \cite[Prop.~2.2]{FMim1})}. For physical reasons it is more natural to vary $\z$ while keeping $\l$ fixed.
Then we have:
\begin{Corollary} \label{cor2} Let $d\geq 2$ and $\l>0$. Consider the MA random resistor network with parameter $\z>\z_c$ built on the $\nu$--randomization of a PPP on ${\ensuremath{\mathbb R}} ^d$ with \textcolor{black}{intensity} $\l$.
Suppose that $\nu$ has bounded support contained in $[0,+\infty)$ or in $(-\infty,0]$. Then there exist
positive constants $c, c'$ such that \eqref{stimetta} is satisfied for $L$ large enough.
\end{Corollary}
We postpone the proof of the above corollary to Section \ref{bin_MA}. We point out that
given $a,b \in {\ensuremath{\mathbb R}} $, it holds
\begin{equation}\label{semplice}
|a|+|b|+|a-b| =
\begin{cases}
2 \max \bigl( |a|, |b|\bigr) & \text{ if } a b \geq 0 \,,\\
2|a-b| & \text{ if } a b < 0 \,.
\end{cases}
\end{equation}
Hence, the structural function $h$ defined by \eqref{hMA} reads $h(a,b)= \z-2 \max(|a|,|b|)$ if $a b\geq 0$, and $h(a,b)=\z- 2|a-b|$ if $ab <0$. As a consequence, if the support of $\nu$ intersects both $(-\infty,0)$ and $(0,\infty)$, then Assumption (A5) fails.
\subsection{Outline of the paper}\label{pigolano}
The proof of Theorem \ref{teo1} consists of two main steps: 1) reduction to the analysis of the LR crossings inside a 2d slice of a suitable graph approximating $\ensuremath{\mathcal G}$ and having vertexes in a lattice, 2) combination of Tanemura's algorithm in \cite[Section 4.1]{T} and Grimmett-Marstrand renormalization scheme in \cite{GM} to perform the above reduced analysis.
For what concerns the first part,
in Section \ref{sec_discreto} we show that it is enough to prove the analogous of Theorem \ref{teo1} for a suitable graph ${\ensuremath{\mathbb G}} _+$ whose vertexes lie inside $\e {\ensuremath{\mathbb Z}} ^d$, with $\e>0$ small enough. By a standard argument (cf. \cite[Remark (d)]{GM}), we then show that it is indeed enough
to have a good control from below of the number of vertex--disjoint LR crossings of ${\ensuremath{\mathbb G}} _+$ contained in a 2d slice (this control is analogous to \eqref{stimetta} for $d=2$, cf. Eq. \eqref{problema2d}).
We then move to the second part.
In Section \ref{sec_japan} we recall (with some extension) Tanemura's algorithm in \cite{T} to exhibit a maximal set of vertex-disjoint LR crossings for a generic subgraph of ${\ensuremath{\mathbb Z}} ^2$, where edges are given by pairs of nearest--neighbor points. Under suitable conditions on the random subgraph of ${\ensuremath{\mathbb Z}} ^2$, this algorithm allows to stochastically dominate the maximal number of the vertex-disjoint LR crossings by the analogous quantity for a site percolation. If the latter is supercritical, then one gets an estimate for the random subgraph of ${\ensuremath{\mathbb Z}} ^2$ as \eqref{problema2d}.
We then apply Tanemura's results by taking as random subgraph of ${\ensuremath{\mathbb Z}} ^2$ a suitable graph built from ${\ensuremath{\mathbb G}} _+$ by a renormalization procedure similar to the one developed by Grimmett \& Mastrand in \cite{GM}. The combination of the two methods to conclude the proof of the bound \eqref{problema2d}, and hence of Theorem \ref{teo1}, is provided in Section \ref{sec_ginepro}, where the renormalization scheme and its main properties are only roughly described.
A detailed treatment of the renormalization scheme is given in Sections \ref{sec_RN_tools} and \ref{moto_GP}.
For the reader's convenience, in order to allow a better comprehension of the main structure of the proof of Theorem \ref{teo1}, we have postponed several technical proofs in the last sections \textcolor{black}{(see Sections \ref{sio5}, \ref{trieste65}, \ref{patroclo}, \ref{puffo1}). Corollaries \ref{cor1} and \ref{cor2} are proven in Section \ref{bin_MA}.}
Finally, in Appendix \ref{app_tanemura} we illustrate Tanemura's algorithm in a specific example and in Appendix \ref{app_locus} we collect some straightforward but cumbersome geometric bounds used in the renormalization scheme. In Appendix \ref{app_ultimatum} we gather some technical arguments for the proof of \eqref{problema2d}.
\section{Discretization and reduction to 2d slices}\label{sec_discreto}
\begin{Warning}\label{aaah} Without loss of generality we take $\D={\rm supp}(\nu)$ and assume that $ \sup _{a,b\in \D} h(a,b)= 1$ (cf. (A3)). In particular, a.s. the length of the edges of $\ensuremath{\mathcal G}$ will be bounded by $1$.
\end{Warning}
In this section we show how to reduce the problem of estimating the probability $P \left( \textcolor{black}{\ensuremath{\mathcal R}_L(\ensuremath{\mathcal G})} \geq c L^{d-1} \right) $ in \eqref{stimetta} to a similar problem for a graph ${\ensuremath{\mathbb G}} _+$
with vertexes contained in the rescaled lattice $\e {\ensuremath{\mathbb Z}} ^d$. Afterwards, we show that, for the second problem, it is enough to have
a good control on the LR crossings of ${\ensuremath{\mathbb G}} _+$ contained in 2d slices.
\smallskip
We need to introduce some notation since we will deal with several couplings:
\begin{itemize}
\item We write PPP($\rho$) for the Poisson point process with \textcolor{black}{intensity} $\rho$.
\item We write PPP($\rho,\nu$) for the marked PPP obtained as $\nu$--randomization of a PPP($\rho$).
\item Given a sequence of i.i.d. random variables $(X_n)_{n\geq 1} $ with law $\nu$ and, independently, a Poisson random variable $N$ with parameter $\rho$, we
write $\ensuremath{\mathcal L}(\rho,\nu)$ for the law of $\inf \{X_1,X_2, \dots, X_N\}$
if $h\searrow $ and the law of $\sup \{X_1,X_2, \dots, X_N\}$ if $h\nearrow $.
When $N=0$, the set $ \{X_1,X_2, \dots, X_N\}$ is given by $\emptyset$ and we use the convention that $\inf\emptyset:=+\infty$ and $\sup \emptyset := -\infty$.
\end{itemize}
Recall the constants $\l_*, \ell_*$ appearing in Assumption (A2) and the set $U_*(\d)$ appearing in Assumption (A4).
\begin{Definition}[Parameters $\a,\e, K $, set $U_*$ and boxes $R_z$'s] \label{vinello}
We fix a constant $\a>0$ small enough such that $10 \a\leq \ell_* $, $10 \a\leq 1$ and $\sqrt{d}/\a\in {\ensuremath{\mathbb N}} _+$. We define $\e$ by
$\e \sqrt{d}:= \a/100$ (note that $1/\e\in {\ensuremath{\mathbb N}} _+$). For each $z\in \e \bbZ^d$ we set $R_z:= z+[0,\e)^d $. We fix a positive integer $K$, very large. In Section \ref{francia} we will explain how to choose $K$. Finally, we set $U_*:= U_*(\a/2)$.
\end{Definition}
\begin{Definition}[\textcolor{black}{Fields $(A_z)$, $(T^{(j)}_z)$ and \textcolor{black}{$(A_z^{\rm au})$}}] \label{cavallo}
We introduce the following $K+1$ independent random fields defined on a common probability space $(\Theta, {\ensuremath{\mathbb P}} )$: \begin{itemize}
\item Let $(A_z)_{z\in \e \bbZ^d}$ be i.i.d. random variables with law $\ensuremath{\mathcal L}\bigl( \l_* \e^d , \nu\bigr)$.
\item Given $j\in \{1,2,\dots ,K\}$,
let $(T^{(j)}_z)_{z\in \e \bbZ^d}$ be i.i.d. random variables with law $\ensuremath{\mathcal L}\bigl( (\l-\l_*)\e^d/K , \nu\bigr)$.
\end{itemize}
By means of the above fields we build an \emph{augmented random field} given by the i.i.d. random variables $(A_z^{\rm au})_{z\in \e \bbZ^d}$, where
\begin{equation}\label{augmentin}
A_z^{\rm au}:=
\begin{cases}
A_z\wedge \min _{1\leq j \leq K} T_z^{(j)} & \text{ if } h \searrow\\
A_z\lor \max _{1\leq j \leq K} T_z^{(j)} & \text{ if } h \nearrow
\end{cases}
\quad \,.
\end{equation}
\end{Definition}
Note that $A_z$ and $A_z^{\rm au}$ have value in $\D\cup \{+\infty\}$ if $h \searrow$, and in $\D\cup \{-\infty\}$ if $h \nearrow$.
Let us clarify the relation of the random fields introduced in Definition \ref{cavallo} with the PPP$(\l,\nu)$. We observe that
a PPP$(\l,\nu)$ can be obtained as follows.
Let
\begin{align}
& \{(x,E_x)\,:\, x\in \s\} \,, \label{mela71}\\
& \{ (x,E_x)\,:\, x\in \xi^{(j)} \} \qquad j=1,2,\dots, K\,, \label{mela72}
\end{align} be independent marked PPP's, respectively with law PPP$(\l_*,\nu)$ and PPP$( (\l-\l_*)/K,\nu)$. \textcolor{black}{In particular,
the random sets $\s$ and $\xi^{(j)}$, with $1\leq j \leq K$, are a.s. disjoint and correspond to PPP's with intensity $\l_*$ and $ (\l-\l_*)/K$ respectively.}
The number of points in $\s\cap R_z$ (respectively $\xi^{(j)}\cap R_z $) is a Poisson random variable with parameter $\l_* \e^d$ (respectively $(\l-\l_*)\e^d /K$).
Then, setting $\xi := \s \cup \bigl( \cup _{j=1}^K \xi^{(j)} \bigr)$, the marked point process $\{(x,E_x)\,:\, x\in \xi\} $ is a PPP$(\l, \nu)$.
Let us first suppose that $h\searrow$. We define
\begin{align}
B_z &:= \inf\{ E_x\,:\, x \in \s \cap R_z\}\,, \qquad \;\;\;z \in \e{\ensuremath{\mathbb Z}} ^d\,,\label{bibo}\\
B_z^{(j)}&:= \inf \{ E_x\,:\, x \in \xi^{(j)} \cap R_z\}\,, \qquad z \in \e{\ensuremath{\mathbb Z}} ^d\,,\;j=1,2,\dots,K\,.\label{biboj}
\end{align}
Trivially, we have
\begin{equation}\label{lince}
B_z^{\rm au}:=B_z\wedge \min _{1\leq j \leq K} B_z^{(j)} = \inf\{ E_x\,:\, x \in \xi \cap R_z\}\,, \qquad z \in \e{\ensuremath{\mathbb Z}} ^d\,.
\end{equation}
The above fields in \eqref{bibo} and \eqref{biboj} are independent (also varying $j$) and moreover we have the following identities between laws:
\begin{align}
& \left(B_z\right)_{z\in \e {\ensuremath{\mathbb Z}} ^d} \stackrel{\ensuremath{\mathcal L}}{=} \left(A_z\right)_{z\in \e {\ensuremath{\mathbb Z}} ^d}\,, \\
& \left(B^{(j)} _z\right)_{z\in \e {\ensuremath{\mathbb Z}} ^d} \stackrel{\ensuremath{\mathcal L}}{=} \left (T^{(j)}_z\right)_{z\in \e {\ensuremath{\mathbb Z}} ^d} \text{ for } j=1,2,\dots, K\,,\\
& \left(B_z^{\rm au}\right)_{z\in \e {\ensuremath{\mathbb Z}} ^d} \stackrel{\ensuremath{\mathcal L}}{=} \left(A_z^{\rm au}\right)_{z\in \e {\ensuremath{\mathbb Z}} ^d}\,.
\end{align}
When $h \nearrow $ the above observations remain valid by replacing $\inf, \min$ with $\sup, \max$, respectively.
\begin{Definition}[\textcolor{black}{Graph ${\ensuremath{\mathbb G}} _+=({\ensuremath{\mathbb V}} _+,{\ensuremath{\mathbb E}} _+) $}] \label{vichinghi_+}
On the probability space $(\Theta,{\ensuremath{\mathbb P}} )$ we define the graph
${\ensuremath{\mathbb G}} _+=({\ensuremath{\mathbb V}} _+,{\ensuremath{\mathbb E}} _+) $ as
\begin{align}
{\ensuremath{\mathbb V}} _+&:=\{z\in \e \bbZ^d\,:\, A^{\rm au}_z \in {\ensuremath{\mathbb R}} \} \,,\\
{\ensuremath{\mathbb E}} _+& :=\left \{ \{z,z'\}: z\not = z' \text{ in } {\ensuremath{\mathbb V}} _+\,, \;\;\; |z-z'| \leq h(A^{\rm au}_z,A^{\rm au}_{z'}) -\a \right\}\,.
\end{align}
\end{Definition}
The plus suffix comes from the fact that in Sections \ref{scremato} and \ref{sec_RN_tools} we will introduce two other graphs, ${\ensuremath{\mathbb G}} _-,$ and ${\ensuremath{\mathbb G}} $ respectively, such that
${\ensuremath{\mathbb G}} _-\subset {\ensuremath{\mathbb G}} \subset {\ensuremath{\mathbb G}} _+$.
\begin{Definition}[\textcolor{black}{LR crossing in ${\ensuremath{\mathbb G}} _+$ and ${\ensuremath{\mathbb R}} _L({\ensuremath{\mathbb G}} _+)$}] \label{def_LR_bis}
Given $L>0$, a left-right (LR) crossing of the box $\D_L:=[-L-2,L+2] \times [-L,L]^{d-1}$ in the graph ${\ensuremath{\mathbb G}} _+$ is any sequence of distinct vertexes $x_1,x_2 , \dots, x_n $ of ${\ensuremath{\mathbb G}} _+$
such that
\begin{itemize}
\item $\{x_i ,x_{i+1}\} \in {\ensuremath{\mathbb E}} _+$ for all $i=1,2,\dots, n-1$;
\item $x_1 \in (-\infty,-L-2)\times [-L,L]^{d-1} $;
\item $x_2, x_3,\dots, x_{n-1}\in \D_L$;
\item $x_n \in (L+2, +\infty )\times [-L,L]^{d-1} $.
\end{itemize}
We also define ${\ensuremath{\mathbb R}} _L({\ensuremath{\mathbb G}} _+)$ as the maximal number of vertex-disjoint LR crossings of $\D_L$ in ${\ensuremath{\mathbb G}} _+$.
\end{Definition}
\begin{TheoremA}\label{teo2} Let ${\ensuremath{\mathbb G}} _+$ be the random graph given in Definition \ref{vichinghi_+}. Then
there exist
positive constants $c, c'$ such that
\begin{equation}\label{fiorfiore}
{\ensuremath{\mathbb P}} \left( {\ensuremath{\mathbb R}} _L({\ensuremath{\mathbb G}} _+)\geq c L^{d-1} \right) \geq 1- e^{- c' L^{d-1}}
\end{equation}
for $L$ large enough.
\end{TheoremA}
To get Theorem \ref{teo1} it is enough to prove Theorem \ref{teo2}:
\begin{Proposition}\label{fragolino} Theorem \ref{teo2} implies Theorem \ref{teo1}.
\end{Proposition}
\begin{proof} We restrict to the case $h\searrow $ as the case $h\nearrow $ can be treated by similar arguments.
By the above discussion concerning \eqref{bibo}, \eqref{biboj} and \eqref{lince},
${\ensuremath{\mathbb G}} _+$ has the same law of the following graph $\bar {\ensuremath{\mathbb G}} _+$ built in terms of the
random field \eqref{lince}. The vertex set of $\bar {\ensuremath{\mathbb G}} _+$ is given by $\{z\in \e \bbZ^d\,:\, B^{\rm au}_z <+\infty\}$. The edges of $\bar {\ensuremath{\mathbb G}} _+$ are given by the unordered pairs $\{z,z'\}$ with $z\not =z'$ in the vertex set and
\begin{equation} \label{castello}
|z-z'| \leq h(B_z^{\rm au}, B_{z'}^{\rm au}) -\a\,.
\end{equation}
Due to \eqref{lince} for each
vertex $z$ of $\bar {\ensuremath{\mathbb G}} _+$ we can fix a point $x(z)\in \xi\cap R_z $ such that $E_{x(z)}= B^{\rm au}_z$. Hence, if $\{z,z'\}$ is an edge of $\bar {\ensuremath{\mathbb G}} _+$, then $x(z)$ and $x(z')$ are defined
and it holds
$|z-z'| \leq h( E_{x(z)}, E_{x(z')})-\a$. As $x(z) \in R_z$ it must be $|x(z) -z| \leq \sqrt{d}\e =\a/100$ and, similarly, $|x(z')-z'| \leq \a/100$. It then follows that $|x-y| \leq h (E_x, E_y)$ where $x=x(z)$ and $y=x(z')$. As $x\in R_z$, $y\in R_{z'}$ and $R_z\cap R_{z'}=\emptyset$, it must be $x\not =y$. Due to the above observations $\{x,y\}$ is an edge of $\ensuremath{\mathcal G}(h,\l)$.
We extend Definition \ref{def_LR_bis} to $\bar {\ensuremath{\mathbb G}} _+$ (it is enough to replace ${\ensuremath{\mathbb G}} _+$ by $\bar {\ensuremath{\mathbb G}} _+$ there). Due to the above discussion,
if $z_1, z_2, \dots, z_n$ is a LR crossing of the box $\D_L$ for $\bar {\ensuremath{\mathbb G}} _+$, then we can extract from $x(z_1), x(z_2), \dots, x(z_n)$ a LR crossing of the box $[-L-1,L+1]^d$ for $\ensuremath{\mathcal G}(h,\l)$ (we use that $x(z_1), x(z_2), \dots, x(z_n)$ itself is a path for $\ensuremath{\mathcal G}(h,\l)$, $|x(z_i)-z_i| \leq \a/100$, and
edges of $\bar {\ensuremath{\mathbb G}} _+$ have length at most $1-\a$). Since disjointness is preserved, we deduce that
$R_{L+1}\bigl( \ensuremath{\mathcal G}(h,\l)\bigr) \geq {\ensuremath{\mathbb R}} _L(\bar {\ensuremath{\mathbb G}} _+)$. Due to this inequality Theorem \ref{teo2} implies Theorem \ref{teo1} (by changing the constants $c,c'$ when moving from Theorem \ref{teo2} to Theorem \ref{teo1}).
\end{proof}
Finally, we show that, to prove Theorem \ref{teo2}, it is enough to have
a good control on the LR crossings contained in 2d slices:
\begin{Proposition}\label{carletto} Fixed a positive integer $k$, we call $\textcolor{black}{{\ensuremath{\mathbb R}} ^*_L({\ensuremath{\mathbb G}} _+)}$
the maximal number of vertex-disjoint LR crossings of the box $\D_L=[-L-2,L+2]\times [-L,L]^{d-1}$ for the graph ${\ensuremath{\mathbb G}} _+$ whose vertexes, \textcolor{black}{apart from} the first and last one, are included in the slice
\begin{equation}\label{rondine}
[-L-2,L+2] \times [-L,L] \times [-k,k)^{d-2} \,,
\end{equation}
while the first and last one are included, respectively, in
$ (-\infty, -L-2) \times [-L,L] \times [-k,k)^{d-2}$ and $(L+2,+\infty) \times [-L,L] \times [-k,k)^{d-2} $.
If there exist positive constants $c_1,c_2$ such that
\begin{equation}\label{problema2d}
{\ensuremath{\mathbb P}} ( \textcolor{black}{{\ensuremath{\mathbb R}} ^*_L({\ensuremath{\mathbb G}} _+)} \geq c_1 L) \geq 1- e^{-c_2 L}
\end{equation}
for $L$ large enough,
then the claim of Theorem \ref{teo2} is fulfilled (i.e. $\exists c,c'>0$ such that \eqref{fiorfiore} is true for $L$ large enough).
\end{Proposition}
\begin{proof}
For each $z\in 2k {\ensuremath{\mathbb Z}} ^{d-2}$ we consider the slice
\[ S(z):=[-L-2,L+2] \times [-L,L] \times \left ( z+ [-k,k)^{d-2} \right) \,.
\] Note that, when varying $z$ in $2k {\ensuremath{\mathbb Z}} ^{d-2}$, the above slices are disjoint and that
$\D_L$ contains at least $\lfloor 2 L/ 2k
\rfloor^{d-2}\asymp c_0 L^{d-2}$ slices of the above form.
Let us assume \eqref{problema2d}.
By translation invariance and independence of the random variables \textcolor{black}{$A^{\rm au}_z$ with $z\in \e \bbZ^d$ appearing in \eqref{augmentin}}, the number \textcolor{black}{$X$} of disjoint slices $S(z) \subset \D_L$ \textcolor{black}{with $z\in 2k {\ensuremath{\mathbb Z}} ^{d-2}$} including at least $c_1 L$ vertex-disjoint LR crossings of $\D_L$ for ${\ensuremath{\mathbb G}} _+$ stochastically dominates a binomial random variable $Y$ with parameters $ n\asymp c_0 L^{d-2}$ and $p:=1- e^{-c_2 L}$ (at cost to enlarge the probability space $(\Theta,{\ensuremath{\mathbb P}} )$ we can think $Y$ as defined on $\Theta$). Setting $\d := e^{-c_2 L}$ we get
\begin{multline*}
{\ensuremath{\mathbb P}} ( \textcolor{black}{X}< n/2) \leq
{\ensuremath{\mathbb P}} ( Y < n/2)= {\ensuremath{\mathbb P}} ( \d ^{Y} > \d^{\frac{n}{2}})\leq \d^{-\frac{n}{2}} {\ensuremath{\mathbb E}} \bigl[ \d^{Y}\bigr] =
\d^{-\frac{n}{2}} [ \d p + 1-p] ^n \\= \d^{-\frac{n}{2}} [ \d- \d^2 +\d] ^n\leq \d^{-\frac{n}{2}} [ 2\d ] ^n= 2^{c_0(1+o(1)) L^{d-2}} e^{- c_0 c_2 (1+o(1))L^{d-1}/2} \,.
\end{multline*}
\textcolor{black}{As the event $X\geq n/2$ implies that ${\ensuremath{\mathbb R}} ({\ensuremath{\mathbb G}} _+)\geq c_1 L n/2 \asymp (c_0 c_1 /2) L^{d-1}$, we get} \eqref{fiorfiore} in Theorem \ref{teo2}.
\end{proof}
\subsection{Properties of ${\ensuremath{\mathbb G}} _+$} \label{scremato}
In this subsection we want to isolate the properties of ${\ensuremath{\mathbb G}} _+$ that follow from the main assumptions and that will be crucial to prove Theorem \ref{teo2}.
\begin{Definition}[\textcolor{black}{Graph ${\ensuremath{\mathbb G}} _-= ({\ensuremath{\mathbb V}} _-, {\ensuremath{\mathbb E}} _-)$}] \label{vichinghi_-}
On the probability space $(\Theta,{\ensuremath{\mathbb P}} )$ we define the graph ${\ensuremath{\mathbb G}} _-= ({\ensuremath{\mathbb V}} _-, {\ensuremath{\mathbb E}} _-)$
as
\begin{align}
& {\ensuremath{\mathbb V}} _-:=\{z\in \e \bbZ^d\,:\, A_z \in {\ensuremath{\mathbb R}} \} \,,\\
& {\ensuremath{\mathbb E}} _- :=\left \{ \{z,z'\}: z\not= z' \text{ in } {\ensuremath{\mathbb V}} _-\,, \; |z-z'| \leq h(A_z,A_{z'}) -3\a \right\}\,\,. \end{align}
\end{Definition}
\begin{Lemma}\label{john}
The graph ${\ensuremath{\mathbb G}} _-$ percolates ${\ensuremath{\mathbb P}} $--a.s.
\end{Lemma}
\begin{proof} We restrict to the case $h\searrow $ as the case $h\nearrow $ can be treated similarly.
By the discussion following Definition \ref{cavallo} (recall the notation there)
it is enough to prove that the graph $\bar{{\ensuremath{\mathbb G}} }_-$ percolates a.s., where
$\bar{{\ensuremath{\mathbb G}} }_-$ is defined as ${\ensuremath{\mathbb G}} _-$ with $A_z$ replaced by $B_z$.
Let $x\not =y$ be points of $ \s$ such that
\begin{equation}\label{risorgive} |x-y| \leq h(E_x, E_y) -\ell_* \,.\end{equation}
Equivalently, $\{x,y\}$ is an edge of the graph $\ensuremath{\mathcal G}(h-\ell_* , \l_*)$ built by means of the marked PPP $\{ (x,E_x)\,:\, x\in \s\}$.
Let $z(x)$ and $z(y)$ be the points in $\e {\ensuremath{\mathbb Z}} ^d$ such that $x\in R_{z(x)} $ and $y\in R_{z(y)}$. Trivially, $|z(x)-x|\leq \e\sqrt{d}$, $|z(y) -y|\leq \e\sqrt{d}$, $B_{z(x)} \leq E_x$ and $B_{z(y)}\leq E_y$. Then, from Assumption (A5), \eqref{risorgive} and since $10 \a \leq \ell_*$ in Definition \ref{vinello},
we get
\[ | z(x)-z(y)| \leq |x-y|+ 2 \e \sqrt{d}\leq h(E_x, E_y) -\ell_*+\a/50\leq h(B_{z(x)} , B_{z(y)}) -3 \a \,.
\]
As a consequence, for each edge $\{x,y\}$ in $\ensuremath{\mathcal G}(h-\ell_*, \l_*)$, either we have $z(x)=z(y)$ or we have that $\{z(x), z(y)\}$ is an edge of $\bar{{\ensuremath{\mathbb G}} }_-$.
Since $\ensuremath{\mathcal G}( h-\ell_*, \l_*) $ a.s. percolates by (A2), due to the above observation we conclude that $\bar{{\ensuremath{\mathbb G}} }_-$ a.s. percolates.
\end{proof}
We conclude this section by treating the FKG inequality. On the probability space $(\Theta, {\ensuremath{\mathbb P}} )$ we introduce the partial ordering $\preceq$ as follows: given $\theta_1, \theta_2\in \Theta$ we say that $\theta_1 \preceq \theta_2 $ if, for all $z \in \e \bbZ^d$ and $j\in \{1,2,\dots, K\}$, it holds
\begin{equation}
\begin{cases}
A_z (\theta_1 ) \geq A_z (\theta _2) \; \text{ and } T^{(j)}_z (\theta_1) \geq T^{(j)}_z(\theta_2) & \text{ if } h \searrow\,,\\
A_z (\theta_1 ) \leq A_z (\theta _2) \;\text{ and } T^{(j)}_z (\theta_1) \leq T^{(j)}_z(\theta_2) & \text{ if } h \nearrow\,.
\end{cases}
\end{equation}
We point out that, due to Assumption (A5), if $\theta_1 \preceq \theta_2$ then ${\ensuremath{\mathbb G}} _-(\theta_1) \subset {\ensuremath{\mathbb G}} _-(\theta_2)$ and ${\ensuremath{\mathbb G}} _+(\theta_1) \subset {\ensuremath{\mathbb G}} _+(\theta_2)$.
Since dealing with i.i.d. random variables, we also have that the partial ordering $\preceq$ satisfies the FKG inequality: if $F,G$ are increasing events for $\preceq$, then ${\ensuremath{\mathbb P}} (F\cap G)\geq {\ensuremath{\mathbb P}} (F){\ensuremath{\mathbb P}} (G)$.
\emph{At this point, we can disregard the original problem and the original random objects. One could start afresh keeping in mind only Assumptions (A2), (A3), (A4), (A5),
Definitions \ref{vinello}, \ref{cavallo}, \ref{vichinghi_+}, \ref{def_LR_bis}, \ref{vichinghi_-}, Lemma \ref{john} and the FKG inequality for the partial order $\preceq$ on $(\Theta, {\ensuremath{\mathbb P}} )$. It then remains to prove \eqref{problema2d} in Proposition \ref{carletto} for some fixed positive integer $k$ and $L$ large enough.}
\section{Tanemura's algorithm for LR crossings in ${\ensuremath{\mathbb Z}} ^2$}\label{sec_japan}
In this section we recall a construction introduced by Tanemura in \cite[Section 4]{T} to control the number of LR crossings in a 2d box by stochastic domination with a 2d Bernoulli site percolation. We point out that we had to extend one definition from \cite{T} to treat more general cases (see below for details).
In order to have a notation close to the one in \cite[Section 4]{T}, \textcolor{black}{given a positive integer $M$} we consider the box
\begin{equation}\label{scooby}
\L :=( [0,M+1] \times [0, M-1] )\cap {\ensuremath{\mathbb Z}} ^2\,.
\end{equation} $\L$ has a graph structure, with unoriented edges between points at distance one.
Let $(x_1, x_2, \dots, x_n)$ be a string of points in $\L$, such that $\{x_1, x_2, \dots,x_n\}$ is a connected subset of $\L$.
We \textcolor{black}{now}
introduce a total order \textcolor{black}{$\prec$} on $\D \{x_1, \dots, x_n\}$
(in general, given $A\subset {\ensuremath{\mathbb Z}} ^2$, $\D A:= \{ y\in {\ensuremath{\mathbb Z}} ^2 \setminus A\,:\, |x-y|=1\text{ for some } x\in A\}$). \textcolor{black}{Note that we} have to modify the definition in \cite[Section 4]{T} which is restricted there to the case that $(x_1, x_2, \dots, x_n)$ is a path in ${\ensuremath{\mathbb Z}} ^2$.
For later use, it is more convenient to describe the ordering \textcolor{black}{$\prec$} from the largest to the smallest element. We denote by $\Psi $ the anticlockwise rotation of $\pi/2$ around the origin in ${\ensuremath{\mathbb R}} ^2$ (in particular,
$\Psi (e_1)= e_2$ and $\Psi (e_2)=-e_1$). We first introduce an order $\prec_k$ on the sites in ${\ensuremath{\mathbb Z}} ^2$ neighboring $x_k$ as follows. Putting $x_0:= x_1-e_1$, for $k=1,2,\dots, n$ we set
\[ x_k + \Psi (v) \succ_k x_k + \Psi ^2 (v) \succ_k x_k + \Psi ^3 (v) \succ_k x_k + \Psi^4 (v)=x_{a(k)}\,,\]
where $v:=x_{a(k)}-x_k $ and $a(k):= \max \{j: 0\leq j\leq n \text{ and } |x_k-x_j|=1\} $.
The order \textcolor{black}{$\prec$} on $\D \{x_1, \dots, x_n\}$ is obtained as follows. The largest elements are the sites of $\D \{x_1, \dots, x_n\}$ neighboring $x_n$ (if any), ordered according to $\succ_n$.
The next elements, in decreasing order, are the sites $\D \{x_1, \dots, x_n\}$ neighboring $x_{n-1}$ but not $x_n$ (if any), ordered according to $\succ_{n-1}$. As so on, in the sense that in the generic step one has to consider the elements of $\D \{x_1, \dots, x_n\}$ neighboring $x_k$ but not $x_{k+1}, \dots, x_n$ (if any), ordered according to $\succ_k$ (see Figure \ref{tanemura1}).
\begin{figure}
\includegraphics[scale=0.4]{Tanemura1.pdf}
\captionsetup{width=0.9\linewidth}
\caption{Left: ordering $\prec_1$ on $\D\{x_1\}=\{y_1,y_2,y_3, y_4\}$. We have $a(1):=0$, $v=-e_1$ and $y_1\prec y_2\prec y_3\prec y_4$.
Right: ordering on the boundary of $\D\{x_1,x_2,x_3,x_4\}=\{y_1,\ldots,y_8\}$. We have $y_1 \prec y_2\prec \cdots \prec y_8$. A dotted segment is drawn between two points $y_i$ and $x_j$ if the point $y_i$ is a neighbor of $x_j$ but not of $x_{j+1}, \dots, x_4$.
}
\label{tanemura1}
\end{figure}
Suppose that we have a procedure to decide, given $\theta \in \Theta$ (cf.~Definition \ref{cavallo}), $x_1,x_2, \dots, x_n, x_{n+1}\in \L$ and $x\in \{x_1, \dots, x_n\}$, if the point $x_{n+1}\in \L$ is occupied (for $n=0$) and if it is occupied and linked to $x$ (for $n\geq 1$),
knowing the occupation state of the points $x_1,x_2, \dots, x_n\in \L$ and the presence or absence of links between them.
The precise definition of occupation and link is not relevant now, here we assume that these properties have been previously defined and that can be checked knowing $\theta$.
We now define a random field $\z=\left(\z(x)\,:\, x \in \L\right)$ with $\z(x)\in \{0,1\}$ on the probability space $\bigl( \Theta, {\ensuremath{\mathbb Q}} \bigr)$. ${\ensuremath{\mathbb Q}} $ is a probability measure on $\Theta$, which can differ from the probability ${\ensuremath{\mathbb P}} $ appearing in Definition \ref{cavallo}
(in the application ${\ensuremath{\mathbb Q}} $ will be a suitable conditioning of ${\ensuremath{\mathbb P}} $).
To define the field $\z$, we \textcolor{black}{build below} the sets $C^s_j= ( E^s_j, F^s_j)$, with $s \in \{0,1,\dots, M-1\}$ and $j=1,2,\dots, M^2 $.
The construction will fulfill the following properties:
\begin{itemize}
\item$E^s_j$ will be a connected subset of $\L$;
\item there exists $J^s \in \{1,2\dots,M^2\}$ such that, for $j< J^s$,
$C^s_{j+1}=( E^s_{j+1}, F^s_{j+1})$ will be obtained from $C^s_j=( E^s_j, F^s_j)$ by adding exactly a point (called $x^s_{j+1}$) either to $E^s_j$ or to $F^s_j$; while, for $j \geq J^s$, $C^s_{j+1}=( E^s_{j+1}, F^s_{j+1})$ will equal $C_j^s=( E^s_j, F^s_j)$;
\item $\z \equiv 1$ on $E^s_j$ and $\z\equiv 0$ on $F^s_j$.
\end{itemize}
Note that, because of the above rules, $E^s_j \cup F^s_j =\{x^s_1, x^s_2, \dots ,x^s_j\}$ for $1\leq j \leq J^s$.
In order to make the construction clearer, in Appendix \ref{app_tanemura} we illustrate the construction in a particular example.
We now start with the definitions involved in the construction. In what follows, the index $s$ will vary in $\{0,1,\dots, M-1\}$. We also set $x^s_1:=(0,s)$. We build the sets $C^0_1$, $C^1_1$,...,$C^{M-1}_1$ as follows.
If the point $x^s_1$ is occupied, then we set
\begin{equation}
\z(x^s_1):=1 \text{ and } C^s_1 :=\left( E^s_1, F^s_1 \right):= (\{x^s_1\},\emptyset) \,,\end{equation}
otherwise we set
\begin{equation}
\z(x^s_1):=0 \text{ and } C^s_1 :=\left( E^s_1, F^s_1 \right):= (\emptyset, \{x^s_1\}) \,.
\end{equation}
We then define iteratively
\begin{equation}\label{cannolo42}
C^0_2, \;C^0_3, \dots,\; C^0_{M^2},\; C^1_2,\; C^1_3,\dots,\; C^1_{M^2},\dots,
\; C^{M-1}_2,\; C^{M -1}_3, \dots,\; C^{M-1}_{M^2}
\end{equation}
as follows. If $E^s_1=\emptyset$, then we declare $J^s:=1$, thus implying that $C_1^s=C_{2}^s=\cdots= C_{M^2}^s$.
We restrict now to the case $E^s_1\not =\emptyset$.
Suppose that we have defined all the sets preceding $C^s_{j+1}$ in the above string \eqref{cannolo42} {(i.e. up to $C^s_j$), that we have not declared that $J^s$ equals some value in $\{1,2,\dots, j\}$ and that we want to define $C^s_{j+1}$.
We call $W^s_j$ the points of $\L$ involved in the construction up to this moment, i.e. \[
W^s_j = \{x_1^k:0\leq k \leq M-1\} \cup \{ x^{s'}_r : 0 \leq s'< s, \, 1< r \leq M^2\}
\cup \{ x^{s}_r : 1< r \leq j\}
\,.
\]
As already mentioned, it must be $E^s_0 \subset E^s_1\subset \cdots \subset E^s_j$ and at each inclusion either the two sets are equal or the second one is obtained from the first one by adding exactly a point. We then write $\bar{E}^s_j$ for the non--empty string obtained as follows: the entries of $\bar{E}^s_j$ are the elements of $E^s_j$, moreover if $x^s_a, x^s_b \in E^s_j$ and $a<b$, then $x^s_a$ appears in $\bar{E}^s_j$ before $x^s_b$. Note that the above property ``$x^s_a, x^s_b \in E^s_j$ and $a<b$'' simply means that the point $x^s_a$ has been added before $x^s_b$ to one of the sets $E^s_0 \subset E^s_1\subset \cdots \subset E^s_j$. Then, on $\D E^s_j$ we have the ordering $\prec$ (initially defined) associated to the string $\bar{E}^s_j$.
We call $\ensuremath{\mathcal P}^s_j$ the following property: $E^s_j$ is disjoint from the right vertical face of $\L$, i.e. $ E^s_j \cap \left(\{M+1\} \times \{0,1,\dots, M-1\}\right)=\emptyset$, and $(\L\cap \textcolor{black}{\D{E}^s_j}) \setminus W^s_j \not = \emptyset$. If property $\ensuremath{\mathcal P}^s_j$ is satisfied, then we denote by $x^s_{j+1} $ the last element of $(\L \cap \textcolor{black}{\D{E}^s_j}) \setminus W^s_j $ \textcolor{black}{w.r.t. $\prec$}. We define $k$ as the largest integer $k$ such that $x_k^s\in E^s_j$ and $|x^s_{j+1} -x^s_k |=1$. If $x^s_{j+1}$ is occupied and linked to $x^s_k$,
then we set
\begin{equation}
\z(x^s_{j+1}):=1 \text{ and } C^s_{j+1} :=\left( E^s_j \cup \{ x^s_{j+1}\}, F^s_j \right) \,,\end{equation}
otherwise we set
\begin{equation}
\z(x^s_{j+1}):=0 \text{ and } C^s_{j+1} :=\left( E^s_j, F^s_j \cup \{ x^s_{j+1}\} \right)\,.
\end{equation}
On the other hand, if property $\ensuremath{\mathcal P}^s_j$ is not verified, then we declare $J^s:=j$, thus implying that $C_j^s=C_{j+1}^s=\cdots= C_{M^2}^s$.
It is possible that the set $\cup _{s=0}^{M-1} \cup _{j=1}^{M^2} \left( E^s_j\cup F^s_j\right)$ does not fill all $\L$. In this case we set $\z\equiv 0$ on the remaining points.
This completes the definition of the random field $\z$.
Above we have constructed the sets $C^s_j$ in the following order:
$C^0_1$, $C^1_1$, $\dots$, $C^{M-1}_1$,
$C^0_2$, $C^0_3$, $\dots,$ $ C^0_{M^2},$ $ C^1_2,$ $ C^1_3,$ $\dots,$ $ C^1_{M^2},$ $\dots,$ $C^{M-1}_2,$ $ C^{M -1}_3,$ $ \dots,$ $ C^{M-1}_{M^2}$.
We make the following assumption:
\medskip
{\bf Assumption (A)}: \emph{For some $p\in [0,1]$ at every step in the above construction the probability to add
a point to a set of the form $E^s_j$, conditioned to the construction performed before such a step, is lower bounded by $p$.
}
\medskip
Call $N_M$ the maximal number of vertex-disjoint LR crossings of the box $\L$ for $\z$, where $\{x,y\}$ is an edge if $x$ and $y$ are distinct, linked, occupied sites. Here crossings are the standard ones for percolation on \textcolor{black}{${\ensuremath{\mathbb Z}} ^2$} \cite{G}. Note that $N_M$ also equals the number of indexes $s\in \{0,1,\dots,M-1\}$ such that $E^s_{M^2} $ intersects the right vertical face of $\L$.
By establishing a stochastic domination on a 2--dimensional site percolation in the same spirit of \cite[Lemma 1]{GM} (cf. \cite[Lemma 4.1]{T}), \textcolor{black}{due to Assumption (A) the following holds:}
\begin{Lemma}
Under Assumption (A) $N_M$ stochastically dominates the
maximal number of vertex-disjoint LR crossings in $\L$ for a site percolation on ${\ensuremath{\mathbb Z}} ^2$ of parameter $p$.
\end{Lemma}
Due to the above lemma and the results on LR crossings for the Bernoulli site percolation (cf. \cite[Remark~(d)]{GM}), we get:
\begin{Corollary}\label{cor_pistacchio}
If Assumption (A) is fulfilled with $p>p_c(2)$, where $p_c(2)$ is the critical probability for the site percolation on ${\ensuremath{\mathbb Z}} ^2$, then
there exist constants $c,c'>0$ such that
${\ensuremath{\mathbb Q}} ( N_M\geq c M) \geq 1- e^{-c' M}$ for \textcolor{black}{any positive integer $ M$}.
\end{Corollary}
\section{Proof of Eq. \eqref{problema2d} as a byproduct of Tanemura's algorithm and renormalization}\label{sec_ginepro}
In this section we prove Eq. \eqref{problema2d} by combining Tanemura's algorithm and a renormalization scheme inspired by the one in \cite{GM}. For the latter we will not discuss here all technical aspects, and postpone a detailed treatment to the next sections.
Given $m\in {\ensuremath{\mathbb N}} _+$ and $z\in \e \bbZ^d$ we set
\begin{equation}\label{mentolino} B(m):=[-m,m]^d \cap \e \bbZ^d \text{ and } B(z,m):= z+ B(m)\,.
\end{equation}
Recall that $U_*:= U_*(\a/2)$ (cf. Assumption (A4) and Definition \ref{vinello}) and that $\D={\rm supp}(\nu)$ and $\sup h=1$ (cf. Warning \ref{aaah}).
Note also that, since the structural function $h$ is symmetric, by applying twice \eqref{indigestione} we get
$h(a, \tilde{a}) \geq h(b, \tilde{a}) -\a/2\geq h(b, \tilde{b}) -\a$ for any $b, \tilde{b}\in \D$ and $a,\tilde a \in U_*$.
Hence, we have
\begin{equation}\label{mango}
h(a, \tilde{a}) \geq \sup_{ b, \tilde{b}\in \D} h(b, \tilde b) - \a =1-\a \qquad \forall a, \tilde a \in U_*\,.
\end{equation}
\begin{Definition}[\textcolor{black}{Seed}] \label{def_seed}
Given $z\in \e \bbZ^d$ and $m\in {\ensuremath{\mathbb N}} _+$,
we say that
$B(z,m)$ is a \emph{seed} if $A_x\in U_* $ for all $x\in B(z,m)$.
\end{Definition}
Recall the graph ${\ensuremath{\mathbb G}} _-=({\ensuremath{\mathbb V}} _-,{\ensuremath{\mathbb E}} _-)$ introduced in Definition \ref{vichinghi_-}.
Note that any seed is a subset of ${\ensuremath{\mathbb V}} _-$. Moreover,
a seed is a region of ``high connectivity'' in the minimal graph ${\ensuremath{\mathbb G}} _-$:
\begin{Lemma}\label{ironman}
If $B(z,m)$ is a seed, then $B(z,m)$ is a connected subset in the graph ${\ensuremath{\mathbb G}} _-$. \end{Lemma}
\begin{proof} It is enough to show that $|x-y|\leq h(A_x,A_y) -3 \a$ for any $x,y \in B(z,m)$ with $|x-y|=\e$.
Since $\e = \a/100 \sqrt{d}$, we get $|x-y| \leq \a/100$.
On the other hand, by \eqref{mango} and since $A_x,A_y \in U_*$, we have
$h(A_x,A_y)\geq 1- \a $. To conclude it is enough to recall that $ 10 \a\leq 1$ (cf. Definition \ref{vinello}).
\end{proof}
Recall that $p_c(2)$ is the critical probability for site percolation on ${\ensuremath{\mathbb Z}} ^2$.
We now fix some relevant constants and recall the definitions of others:
\begin{itemize}
\item We fix $\e'\in (0,1)$ such that $ 1-6\e'\geq 3/4>p_c(2)$.
\item We fix positive integers $m\leq n$ satisfying the properties stated in Lemma \ref{pierpilori} in Section \ref{sec_RN_tools} (their precise value is not relevant to follow the arguments below).
\item Recall Definition \ref{vinello} for $\e$.
\item We let $N:= n+m+\e$.
\item $L$ is a positive number as in \eqref{problema2d}.
\item Given $L$, the constant $M$ (introduced in \eqref{scooby}) is defined as the smallest positive integer such that $ 4N (M+1) > 2L +5 +m+N$ \textcolor{black}{(in particular, $M$ and $L$ are of the same order).}
\end{itemize}
In the rest of this section we explain how to get \eqref{problema2d} with $k=4N$.
\textcolor{black}{As we will see in a while, such a choice of $k$ guarantees independence properties in the construction of left-right crossings.}
\smallskip
As in Tanemura's algorithm we take $ \L :=( [0,M+1] \times [0, M-1] )\cap {\ensuremath{\mathbb Z}} ^2$ (in general we will use the notation introduced in Section \ref{sec_japan}). We recall that $x_1^s:= (s,0)$ for $s=0,1,\dots, M-1$.
We set
\begin{equation}\label{deep}
\bar x:= (x,0,0, \dots,0)\in {\ensuremath{\mathbb Z}} ^d \text{ for } x \in {\ensuremath{\mathbb Z}} ^2\,.
\end{equation} Below, we will naturally associate to each point $x\in \L$ the point $ 4N \bar x $ in the renormalized lattice $4N {\ensuremath{\mathbb Z}} ^d\subset \e \bbZ^d$.
\begin{Definition}[\textcolor{black}{Conditional probability ${\ensuremath{\mathbb Q}} $}] We set ${\ensuremath{\mathbb Q}} (\cdot) :={\ensuremath{\mathbb P}} (\cdot \,|\, D)$ where
\begin{equation}\label{dedalo} D:=\{ B(4N \bar{x}_1^s ,m) \text{ is a seed } \forall x_1^s \in \L\}
\end{equation}
\end{Definition}
To run Tanemura's algorithm we first need to define when
the point $x_1^s$ is occupied. Our definition will imply that the graph \textcolor{black}{${\ensuremath{\mathbb G}} \subset {\ensuremath{\mathbb G}} _+$} contains a cluster centered at $4N \bar{x}_1^s$ as in \textcolor{black}{Fig.~\ref{hobbit1} (left)}, when $d=2$. In particular, the seed $ B(4N \bar{x}_1^s ,m) $ is connected in ${\ensuremath{\mathbb G}} _+$ by a cluster of points lying inside the box $ B(4N \bar{x}_1^s ,n) $ to four seeds adjacent to the faces of such a box in the directions $\pm e_1$, $\pm e_2$ ($e_1,e_2,\dots, e_d$ being the canonical basis of ${\ensuremath{\mathbb R}} ^d$).
The precise definition (for all $d\geq 2$) of the occupation of $x_1^s$ is rather technical and explained in Section \ref{moto_GP} (it corresponds to Definition \ref{0occ}, when $\bar{x}_1^s$ is thought of as the new origin of $\e \bbZ^d$).
We point out that the cluster appearing in \textcolor{black}{Fig.~\ref{hobbit1} (left)} is contained in a box of radius $N+m<2N$ centered at \textcolor{black}{at $4N \bar{x}^s_1$.
So, to verify that
$\bar x_1^s$ is occupied,
it is enough to know the random variables $A_z$ with $z$ in the interior
of the box $B( 4N \bar{x}^s_1, 2N)$. Since the above interior parts are disjoint when varying $s$, we conclude that} the events $\{ x_1^s \text{ is occupied}\}$ are ${\ensuremath{\mathbb Q}} $--independent when varying $s$ among $\{0,1,\dots,M-1\}$. Moreover, by Proposition \ref{prop_occ_origin}, ${\ensuremath{\mathbb Q}} ( x_1^s \text{ is occupied})\geq 1-4\e'$.
\begin{figure}
\includegraphics[scale=0.07]{Fig1.pdf}
\captionsetup{width=.9\linewidth}
\caption{Left: cluster contained in ${\ensuremath{\mathbb G}} _+$ when $x_1^s$ is occupied for $d=2$, the centered grey circle corresponds to $4N\bar x_1^s$, the large box has radius $n$, the smaller boxes have radius $m$ and are seeds. Right: cluster of ${\ensuremath{\mathbb G}} _+$ related to the definition of $(0,0)\to (1,0)$ knowing that $x^0_1=(0,0)$ is occupied, the two grey circles are given by $0$ and $4N e_1$.}
\label{hobbit1}
\end{figure}
\medskip
Knowing the occupation state of $x_1^s$, we can
define the sets $C_1^0, C_1^1,\dots, C_1^{M-1}$ in Tanemura's algorithm. Let us move to $C^0_2$. Let us assume for example that $x_1^0=(0,0)$ is occupied, hence $C^1_0=(\{x^0_1\},\emptyset)$. Then, by Tanemura's algorithm, one should check if $(1,0)$ is occupied and linked to the origin $x_1^0$ or not. We have first to define this concept. To this aim
we need to explore the graph ${\ensuremath{\mathbb G}} _+$ in the direction $e_1$ from the origin. Roughly, when $d=2$, we say that $ (1,0)$ is linked to $(0,0)$ and occupied (shortly, $(0,0) \to (0,1) $) if the graph ${\ensuremath{\mathbb G}} _+$ contains a cluster similar to the one in \textcolor{black}{Fig.~\ref{hobbit1} (right)} and Fig.~\ref{hobbit3}, extending the cluster appearing in \textcolor{black}{Fig.~\ref{hobbit1} (left)}.
Note that there is a seed (called $s_5$ in Fig. \ref{hobbit3}) in the proximity of $ 4N e_1 $ (the grey circle on the right in \textcolor{black}{Fig.~\ref{hobbit1} (right)} and Fig.~\ref{hobbit3}) connected to four seeds neighboring the box of radius $n$ concentric to $s_5$, one for each face in the directions $\pm e_1, \pm e_2$. Hence, we have a local geometry similar to the one of \textcolor{black}{Fig.~\ref{hobbit1} (left)}. Moreover, see Fig.~\ref{hobbit3}, the cluster turns in direction $e_1$ and connects inside ${\ensuremath{\mathbb G}} _+$ the seed $s_1$ at $4N\bar x^1_0=0$ to the seed $s_5$ around $4N e_1$ by passing through the intermediate seeds $s_2,s_3,s_4$. Note that, in order to assure that $s_5$ lays around $4Ne_1$ the intermediate seeds have to be located alternatively up and down.
The precise definition (for all $d\geq 2$) of the event $\{(0,0) \to (1,0) \}$, knowing that $x_1^0=(0,0)$ is occupied, is given by Definition \ref{1occ} in Section \ref{moto_GP}. Having defined this concept, we can build the set $C^0_2$
in Tanemura's algorithm.
We move to $C^0_3$.
Suppose for example that $(0,0) \to (1,0) $ occurs, hence $C^0_2=\bigl( \{(0,0), (1,0) \}, \emptyset\bigr)$. Then,
according to Tanemura's algorithm, we need to check if $(2,0)$ is linked to $(1,0)$
and occupied (shortly, $(1,0) \to (2,0) $). The definition of the last event is similar to the one of ``$(0,0) \to (1,0) $''. Roughly, by means of three intermediate seeds (the first one given by $s_6$ in Fig.~\ref{hobbit3}) there is a cluster in ${\ensuremath{\mathbb G}} _+$ similar to the one in \textcolor{black}{Fig.~\ref{hobbit1} (right)} connecting the seed in the proximity of $4N e_1$ to a seed in the proximity $8Ne_1$ and this last seed is connected to four seeds adjacent to the faces in the directions $\pm e_1, \pm e_2$ of a concentric box of radius $n$.
Let us suppose for example that $ (1,0) \not \to (2,0) $. Then $C^0_3=\bigl( \{(0,0), (1,0)\}, \{(2,0)\}\bigr)$.
\begin{figure}
\includegraphics[scale=0.32]{Fig3.pdf}
\captionsetup{width=.9\linewidth}
\caption{\textcolor{black}{Same cluster as in Figure \ref{hobbit1} (right). $s_1,s_2, \dots, s_7$ are seeds used to progressively extend the cluster along the construction.}}\label{hobbit3}
\end{figure}
We move to $C^0_4$. According to Tanemura's algorithm, we need to check if $(1,0)$ is linked to $(1,1)$ and occupied, shortly $(1,0)\to (1,1)$. This last concept is roughly defined as follows:
by means of three intermediate seeds (the first one given by $s_7$ in Fig.~\ref{hobbit3}) there is a cluster in ${\ensuremath{\mathbb G}} _+$ connecting the seed $s_5$ in the proximity of $4N e_1$ to a seed in the proximity $4Ne_1+4Ne_2$ and this last seed is connected to four seeds adjacent to the faces in the directions $\pm e_1, \pm e_2$ of a concentric box of radius $n$ (see Fig.~\ref{hobbit4}). We can then build the set $C^0_4$.
In general,
for any $d\geq 2$ and $v\in \{\pm (1,0)\,,\, \pm (0,1) \}$, the precise definition of ``$(a,b)\to (a,b)+v$ knowing that $(a,b)$ is occupied" is given by Definition \ref{1occ} \textcolor{black}{apart from} changing origin and direction. We point out that in Definition \ref{1occ}, and in general in Section \ref{moto_GP}, we work with ${\ensuremath{\mathbb Z}} ^2 \times\{0\}\subset {\ensuremath{\mathbb Z}} ^d$ instead of ${\ensuremath{\mathbb Z}} ^2$ (${\ensuremath{\mathbb Z}} ^2 \times\{0\}$ and ${\ensuremath{\mathbb Z}} ^2$ are naturally identified).
Proceeding in this way one defines the whole sequence $C^0_2, \;C^0_3, \dots,\; C^0_{M^2}$. Then one has to build the sequence $C^1_2,\; C^1_3,\dots,\; C^1_{M^2}$. If $x_1^1=(0,1)$ is not occupied, i.e. $C^1_1=(\emptyset, \{(0,1)\})$, then one sets $C^1_1=C^1_2=C^1_3\dots=C^1_{M^2}$. Otherwise one starts to build a cluster in ${\ensuremath{\mathbb G}} _+$ similarly to what done above, with the only difference that $x_1^1$ replaces $x_1^0$. As the reader can check, after reading the detailed definitions in Section \ref{moto_GP}, the region of $\e \bbZ^d$ explored when
checking linkages and occupations for the cluster blooming from $x_1^1$ is far enough from the region explored for the cluster blooming from $x_1^0$, and no spatial correlation emerges.
One proceeds in this way to complete Tanemura's algorithm.
In Section \ref{moto_GP} we will analyze in detail the basic steps of the above construction and show (see the discussion in Section \ref{francia}) the validity of Assumption (A) of Section \ref{sec_japan}
with $p=1-6 \e'> p_c(2)$. As a consequence we can apply Corollary \ref{cor_pistacchio} and get that
there exist constants $c_1,c_2>0$ such that
${\ensuremath{\mathbb Q}} ( N_M\geq c _1 M) \geq 1- e^{-c_2 M}$\textcolor{black}{, where} $N_M$ is the maximal number of vertex-disjoint LR crossings of $\L$ for the graph with vertexes given by occupied sites and having edges between nearest--neighbor linked occupied sites.
As rather intuitive and detailed
in Appendix \ref{app_ultimatum}, there is a constant $ C>0$ (independent from $M$) such that that event $\{N_M\geq c _1 M\}$ implies that the graph ${\ensuremath{\mathbb G}} _+$ has at least $C M$ vertex-disjoint LR crossings of a 2d slice of size $O(L)\times O(L) \times O(N)^{d-2}$ (recall that $L \asymp M$). In particular, as discussed in Appendix \ref{app_ultimatum}, the bound ${\ensuremath{\mathbb Q}} (N_M\geq c _1 M) \geq 1- e^{-c_2 M}$ implies the estimate \eqref{problema2d} with $k=4N$ and new positive constants $c_1,c_2$ there.
\begin{figure}
\includegraphics[scale=0.35]{Fig4.pdf}
\captionsetup{width=.9\linewidth}
\caption{Further extension of the cluster in ${\ensuremath{\mathbb G}} _+$ when $(1,0)\to (1,1)$.}
\label{hobbit4}
\end{figure}
\section{Renormalization: preliminary tools}\label{sec_RN_tools}
In the rest we will often write ${\ensuremath{\mathbb P}} (E_1,E_2, \dots, E_n)$ instead of ${\ensuremath{\mathbb P}} (E_1\cap E_2 \cap \cdots \cap E_n)$, also for other probability measures.
For the readers convenience we recall Definitions \ref{vichinghi_+} and \ref{vichinghi_-} of the graphs ${\ensuremath{\mathbb G}} _+= ({\ensuremath{\mathbb V}} _+, {\ensuremath{\mathbb E}} _+)$ and ${\ensuremath{\mathbb G}} _-=({\ensuremath{\mathbb V}} _-,{\ensuremath{\mathbb E}} _-)$:
\begin{align*}
{\ensuremath{\mathbb V}} _-&:=\{z\in \e \bbZ^d\,:\, A_z \in {\ensuremath{\mathbb R}} \} \,,\\
{\ensuremath{\mathbb E}} _- &:=\left \{ \{z,z'\}: z\not= z' \text{ in } {\ensuremath{\mathbb V}} _-\,, \; |z-z'| \leq h(A_z,A_{z'}) -3\a \right\}\,,\\
{\ensuremath{\mathbb V}} _+&:=\{z\in \e \bbZ^d\,:\, A^{\rm au}_z \in {\ensuremath{\mathbb R}} \} \,,\\
{\ensuremath{\mathbb E}} _+& :=\left \{ \{z,z'\}: z\not = z' \text{ in } {\ensuremath{\mathbb V}} \,, \;\;\; |z-z'| \leq h(A^{\rm au}_z,A^{\rm au}_{z'}) -\a \right\}\,.
\end{align*}
We also introduce the intermediate graph ${\ensuremath{\mathbb G}} $ (trivially, we have ${\ensuremath{\mathbb G}} _-\subset {\ensuremath{\mathbb G}} \subset {\ensuremath{\mathbb G}} _+$):
\begin{Definition}[\textcolor{black}{Graph ${\ensuremath{\mathbb G}} = ({\ensuremath{\mathbb V}} , {\ensuremath{\mathbb E}} )$}] \label{vichinghi}
On the probability space $(\Theta,{\ensuremath{\mathbb P}} )$ we define the graph ${\ensuremath{\mathbb G}} = ({\ensuremath{\mathbb V}} , {\ensuremath{\mathbb E}} )$
as
\begin{align}
& {\ensuremath{\mathbb V}} :=\{z\in \e \bbZ^d\,:\, A_z \in {\ensuremath{\mathbb R}} \} \,,\\
& {\ensuremath{\mathbb E}} :=\left \{ \{z,z'\}: z\not= z' \text{ in } {\ensuremath{\mathbb V}} _-\,, \; |z-z'| \leq h(A_z,A_{z'}) -2\a \right\}\,\,. \end{align}
\end{Definition}
We introduce the following conventions:
\begin{itemize}
\item Given $x\in {\ensuremath{\mathbb V}} $ and $C\subset {\ensuremath{\mathbb V}} $ with $x\not \in C$, we say that $x$ is \textcolor{black}{adjacent} to $C$ inside ${\ensuremath{\mathbb G}} $ if there exists $y\in C$ such that $\{x,y\}\in {\ensuremath{\mathbb E}} $.
\item Given $A,B ,C \subset \e \bbZ^d$, we say that ``$A \leftrightarrow B$ in $C$ for ${\ensuremath{\mathbb G}} $'' if there exist $x_1,x_2, \dots, x_k \in C \cap {\ensuremath{\mathbb V}} $ such that $x_1 \in A$, $x_k \in B$ and $\{ x_i , x_{i+1}\} \in {\ensuremath{\mathbb E}} $ for all $i: 1\leq i <k$.
\item Given a bounded set $A\subset {\ensuremath{\mathbb R}} ^d$ we say that ``$A\leftrightarrow \infty $ for ${\ensuremath{\mathbb G}} $'' if there exists an unbounded path in ${\ensuremath{\mathbb G}} $ starting at some point in $A$.
\end{itemize}
Similar definitions hold for the graphs ${\ensuremath{\mathbb G}} _-= ({\ensuremath{\mathbb V}} _-, {\ensuremath{\mathbb E}} _-)$ and ${\ensuremath{\mathbb G}} _+=({\ensuremath{\mathbb V}} _+,{\ensuremath{\mathbb E}} _+)$.
\begin{Definition}[\textcolor{black}{Sets $B(m)$, $T(n)$, $T(m,n)$, $K(m,n)$}] \label{sambinaA}
For $m \leq n\in{\ensuremath{\mathbb N}} _+$ we define the following sets:
\begin{align*}
& B(m):=[-m,m]^d \cap \e \bbZ^d \text{ and } B(z,m):= z+ B(m)\,,\\
& T(n):=\{ x\in \e \bbZ^d : n-1 < \|x\|_\infty \leq n , \,0 \leq x_i \leq x_1 \; \forall i=1,2,\dots, d\}\,, \\
& T(m,n):= \bigl( [n+\e ,n+\e + 2m] \times [0,n]^{d-1}\bigr)\cap \e \bbZ^d
\,,\\
& \textcolor{black}{K(m,n):=\{
x \in {\ensuremath{\mathbb V}} \cap T(n) : \text{$x$ is adjacent in ${\ensuremath{\mathbb G}} $ to a seed included in $T(m,n)$}\}\,.}
\end{align*}
\end{Definition}
\textcolor{black}{For the reader's convenience, in the above definition we have also recalled \eqref{mentolino}.
We refer to Figure \ref{messicano1}-(left) for some illustration. We recall that seeds have been introduced in Definition \ref{def_seed}. The definition of $K(m,n)$ can be restated as follows:
$K(m,n)$ is given by the points $x\in {\ensuremath{\mathbb V}} \cap T(n)$ such that, for some
$z \in \e \bbZ^d$, the box $B(z,m) \subset T(m,n)$ is a seed and
$\exists y \in B(z,m)$ with $ \{ x, y\}\in {\ensuremath{\mathbb E}} $.}
\begin{figure}
\includegraphics[scale=0.3]{santa_pazienza.pdf}
\captionsetup{width=.9\linewidth}
\caption{Left: sets $T(n)$ and $T(m,n)$. Right: sets $T^*(n)$ and $T^*(m,n)$. }\label{messicano1}
\end{figure}
The following proposition is the analogous of \cite[Lemma 5]{GM} in our \textcolor{black}{context}. It will enter in the proof of Lemma \ref{pierpilori} and in the proof of Proposition \ref{prop_occ_origin} (which lower bounds from below the probability that in the first step of the renormalization scheme we enlarge the cluster of occupied sites).
\begin{Proposition}\label{cinquina}
Given $\eta \in (0,1)$, there exist positive integers $m=m(\eta)$ and $n=n(\eta)$ such that $m>2$, $2m < n$, $2m|n$ and
\begin{equation}\label{maggiolino}
{\ensuremath{\mathbb P}} \bigl( B(m) \leftrightarrow K (m,n) \text{ in $B(n)$ for } {\ensuremath{\mathbb G}} \bigr) > 1-\eta\,.
\end{equation}
\end{Proposition}
This proposition is a consequence of Lemma \ref{john} concerning the percolation of graph ${\ensuremath{\mathbb G}} _-$. Even if
having a seed in a specific place is an event of small probability, the big number
of possible configurations for the seed entering in the definition of $K(m,n)$ makes the event in \eqref{maggiolino} of high probability.
We postpone the proof of Proposition \ref{cinquina} to Section \ref{sio5}.
\medskip
It is convenient to introduce the function $h_*: \D \to {\ensuremath{\mathbb R}} $ defined as
\begin{equation}\label{quiquoqua}
h_*(a): =\sup_{b \in \D} h(a,b)\,.
\end{equation}
Moreover, given a finite set $R\subset \e \bbZ^d$, we define the non--random boundary set
\begin{equation}
\partial R\,:=\{ y \in \e \bbZ^d \setminus R \,:\, d(y,R ) \leq 1-2\a\}\,,
\end{equation}
where $d(\cdot, \cdot)$ denotes the Euclidean distance. Note that the edges of ${\ensuremath{\mathbb G}} $ have length bounded by $1-2\a$.
To avoid ambiguity, we point out that in what follows the set $\partial R\cap B(n)$ has to be thought of as $(\partial R)\cap B(n)$ and not as $\partial (R\cap B(n))$.
The following lemma (which is the analogous of \cite[Lemma 6]{GM} in our \textcolor{black}{context})
will be crucial in estimating from below the probability to enlarge the occupied cluster at a generic step of the renormalization scheme (cf. Section \ref{sec_ginepro}). More specifically, it will allow to prove Proposition \ref{prop_occ_e1} and to control the further steps in the renomalization scheme as explained in Section \ref{francia}. Recall Definition \ref{vinello} of $U_*$.
\begin{Lemma}\label{pierpilori}
Fix $\e'\in (0,1)$. Then there exist positive integers
$m$
and $n$, with $m>2$, $2m < n$ and $2m|n$,
satisfying the following property.
Consider the following sets (see Figure \ref{emma}):
\begin{itemize}
\item Let $R $ be a finite subset of $ \e \bbZ^d$ satisfying
\begin{equation}\label{mare100}
B(m) \subset R\,,\qquad
\left( R\cup \partial R\right)\cap \left( T(n) \cup T(m,n)\right) =\emptyset\,.
\end{equation}
\item For any $x\in R\cup \partial R$, let $\L(x)$ be a subset of $\{1,2,\dots, K\} $. We suppose that there exists
$k_*\in \{1,2,\dots, K\}$ such that
\begin{equation}\label{monti100}
k_*\not \in \cup_{x\in D} \L(x) \,,
\end{equation}
where
$D\subset \e \bbZ^d$ is defined as
\begin{equation}\label{campana}
\begin{split} D:= & \left(
\partial R \cap B(n)
\right)\\& \cup \left\{ x\in R\,:\, \exists y\in \partial R \cap B(n)
\text{ with } |x-y| \leq 1-2\a\right\}\,.
\end{split}
\end{equation}
\end{itemize}
Consider the following events:
\begin{itemize}
\item
Let $H$ be any event in the $\s$--algebra $\ensuremath{\mathcal F}$ generated by the random variables
$(A_x)_{x\in R \cup \partial R}$ and $ (T_x^{(j)})_{x \in R \cup \partial R\,, \, j\in \L(x) }$.
\item
Let $G$ be the event that there exists a string $(z_0,z_1, z_2, \dots, z_\ell)$ in ${\ensuremath{\mathbb V}} $ such that
\begin{itemize}
\item[(P1)] $z_0\in R$;
\item[(P2)] $z_1 \in \partial R \cap B(n) $;
\item[(P3)] $z_2, \dots, z_\ell \in B(n) \setminus \bigl( R \cup \partial R\bigr)$;
\item[(P4)] $ z_2,\dots, z_\ell$ is a path in ${\ensuremath{\mathbb G}} $;
\item[(P5)] $z_\ell \in K (m,n)$;
\item[(P6)] \textcolor{black}{$T^{(k_*)} _{z_0}\in U_*$} and \textcolor{black}{$T^{(k_*)} _{z_1}\in U_*$};
\item[(P7)] $|z_0-z_1|\leq 1-2\a$;
\item[(P8)] $|z_1-z_2|\leq \textcolor{black}{h_*( A_{z_2})}-2\a$.
\end{itemize}
\end{itemize}
Then ${\ensuremath{\mathbb P}} ( G\,|\, H) \geq 1-\e'$.
\end{Lemma}
\begin{figure}
\includegraphics[scale=0.2]{fig_barrio2}
\captionsetup{width=.9\linewidth}
\caption{$\partial R$ is the very dark grey contour. $R$ is given by the light/dark grey region around the origin. $D\setminus \bigl( \partial R \cap B(n)\bigr)$ is the dark grey subset of $R$. }\label{emma}
\end{figure}
We postpone the proof of Lemma \ref{pierpilori} to Section \ref{trieste65}.
We point out that the above properties (P6), (P7), (P8) (which can appear a little exotic now) will be crucial to derive the $\textcolor{black}{{\ensuremath{\mathbb G}} _+}$--connectivity issue stated in Lemma \ref{piccolino} \textcolor{black}{below}. Indeed, although $(z_0,z_1, z_2, \dots, z_\ell)$ could be not a path in ${\ensuremath{\mathbb G}} $, one can prove that it is a path in $\textcolor{black}{{\ensuremath{\mathbb G}} _+}$.
\subsection{The sets \textcolor{black}{$E\bigl[C,B,i\bigr] $} and $F\bigl[C,B,B',i\bigr]$}
\label{lenticchie}
In the next section we will iteratively construct random subsets of $\e \bbZ^d$ sharing the property to be connected in ${\ensuremath{\mathbb G}} _+$. We introduce here the fundamental building blocks, which are given by the sets $E\bigl[C,B,i\bigr] $ and $F\bigl[C,B,B',i\bigr]$ (they will appear again in Definition \ref{legoland}):
\begin{Definition}[\textcolor{black}{Sets $E\bigl[C,B,i\bigr] $ and $F\bigl[C,B,B',i\bigr]$}] \label{def_triade}
Given three sets $C, B, B'\subset \e \bbZ^d $ and given $i\in \{1,2,\dots, K\} $, we define the random subsets
$E,F\subset \e \bbZ^d$ as follows:
\begin{itemize}
\item $E $ is given by the points $z_1$ in $ B \cap \partial C$ such that $T_{z_1}^{(i)}\textcolor{black}{\in U_*}$ and there exists $z_0\in C $ with $|z_0-z_1|\leq 1-2\a$ and $T_{z_0}^{(i)}\textcolor{black}{\in U_*}$;
\item $F$ is given by the points $z\in B'$ such that there exists a path $(z_2,\dots,z_k)$ inside ${\ensuremath{\mathbb G}} $ where $z_k=z$, all points $z_2,\cdots,z_k$ are in $B'\setminus (C\cup\partial C)$ and \textcolor{black}{$|z_1-z_2|\leq h_*(A_{z_2})-2\a$} for some $z_1\in E$.
\end{itemize}
To stress the dependence from $C,B, B', i$, we will also write $E\bigl[C,B,i\bigr] $ and $F\bigl[C,B,B',i\bigr]$.
\end{Definition}
The proof of the next two lemmas is given in Section \ref{patroclo}.
\begin{Lemma}\label{ciak2011} Let
\textcolor{black}{$E=E\bigl[C,B,i\bigr] $} and $F=F\bigl[C,B,B',i\bigr]$ be as in Definition \ref{def_triade}. Let $\hat E,\hat F \subset \e \bbZ^d$.
\begin{itemize}
\item[(i)] \textcolor{black}{If the event $\{E=\hat E\}\cap \{F=\hat F\}$ \textcolor{black}{occurs}, then
$\hat E,\hat F$ satisfy} \begin{equation}\label{chip}
\hat E \cap \hat F=\emptyset \,, \qquad
\hat E\subset \bigl( B \cap \partial C\bigr)\,, \qquad \hat F\subset B'\setminus (C\cup\partial C)\,.
\end{equation}
\item[(ii)]
If $\hat E,\hat F$ satisfy \eqref{chip}, then the event $\{E=\hat E\}\cap \{F=\hat F\}$
belongs to the $\s$--algebra generated by
\begin{itemize}
\item[$\bullet$] $T^{(i)}_z$ with $z\in \bigl( B \cap \partial C \bigr) \cup D$, where \[D:=\left\{ x\in C\,:\, \exists y\in B \cap \partial C
\text{ with } |x-y| \leq 1-2\a\right\}\,;\]
\item[$\bullet$] $A_z$ with $z$ belonging to some of the following sets:
\[ \hat F\,, \quad \bigl( B'\setminus (C\cup\partial C) \bigr) \cap \partial \hat F\,, \quad \bigl( B'\setminus (C\cup\partial C) \bigr) \cap \partial \hat E\,.
\] \end{itemize}
\item[(iii)]
As a consequence, given $R\subset \e \bbZ^d$, the event $\{E\cup F= R\}$ belongs to the $\s$--algebra generated by $\{ T^{(i)}_z\,:\, z\in \bigl( B \cap \partial C \bigr) \cup D\} \cup \{A_z\,:\, z \in R\cup \partial R\}$.
\end{itemize}
\end{Lemma}
\begin{Lemma}\label{piccolino}
Given sets $C, B, B'\subset \e \bbZ^d $ and an index $i\in \{1,2,\dots, K\}$, we define
$E:= \textcolor{black}{E\bigl[C,B,i\bigr]}$ and $F:=F\bigl[C,B,B',i\bigr]$.
If $C\subset \textcolor{black}{{\ensuremath{\mathbb V}} _+}$ is connected in the graph $\textcolor{black}{{\ensuremath{\mathbb G}} _+=({\ensuremath{\mathbb V}} _+, {\ensuremath{\mathbb E}} _+)}$, then the set $C':=C\cup E\cup F$
is contained in $\textcolor{black}{{\ensuremath{\mathbb V}} _+}$ and is connected in the graph $\textcolor{black}{{\ensuremath{\mathbb G}} _+}$.
\end{Lemma}
\section{Renormalization scheme}\label{moto_GP}
\textcolor{black}{As in Section \ref{sec_ginepro}, we set $N:=n+m+\e$.
From now \textcolor{black}{on} $\e'$ is a fixed constant in $(0,1)$ such that $1-6\e'\geq 3/4>p_c(2)$, $p_c(2)$ being the critical probability for Bernoulli site percolation on ${\ensuremath{\mathbb Z}} ^2$. Moreover, we choose $m,n$ as in Lemma \ref{pierpilori}. }
\textcolor{black}{Recall Definition \ref{sambinaA} of $T(n)$, $T(m,n)$. We define
\begin{equation} \label{tittina}
T^*(m,n):= f\left( T(m,n)\right)\text{ and } T^*(n):= f\left( T(n) \right)\,,
\end{equation} where $f:{\ensuremath{\mathbb R}} ^d\to {\ensuremath{\mathbb R}} ^d$ is the isometry $f(x_1,x_2, \dots, x_d) := (x_1,- x_2, \dots, -x_d)$ (see Figure \ref{messicano1}).
Given $a \in {\ensuremath{\mathbb R}} ^d$, we define $g(\cdot|a) : {\ensuremath{\mathbb R}} ^d \to {\ensuremath{\mathbb R}} ^d$ as the isometry
\begin{equation} \label{gigi} g(x|a):= ( x_1, -\text{sgn}(a_2) x_2, \dots, - \text{sgn}(a_d) x_d )\,,
\end{equation} where $\text{sgn}(\cdot)$ is the sign function, with the convention that $\text{sgn}(0)=+1$. Note that $f(x)=g(x|0)$.
}
Let $e_1, e_2, \dots, e_d$ be the canonical basis of ${\ensuremath{\mathbb R}} ^d$.
We denote by $L_1, L_2,L_3,L_4$ the isometries of ${\ensuremath{\mathbb R}} ^d$ given respectively by $\mathds{1}, \theta, \theta^2, \theta ^3$, where $\mathds{1}$ is the identity and $\theta$ is the unique rotation such that
$\theta(e_1)= e_2$, $\theta (e_2)=-e_1$, $\theta (e_i)=e_i$ for all $i=3,\dots,d$.
We define $B_0'\subset \e \bbZ^d$ as
\begin{equation}\label{gommina}
B_0':= B(n) \cup \left( \cup_{j=1}^{4} L_j \bigl( T(m,n) \bigr) \right)\,.
\end{equation}
\textcolor{black}{Hence, for $d=2$, $B_0' $ is the region of $\e \bbZ^d$ given by the largest square and the four peripheral rectangles in Figure \ref{fig_C1_2}--(left).}
For $j=1,2,3,4$ we call $K^{(j)} (m,n)$ the random set of points defined similarly to $K (m,n)$
(cf. \textcolor{black}{Definition \ref{sambinaA}})
but with $T(m,n)$ and $T(n)$ replaced by $ L_j \bigl( T(m,n) \bigr) $ and $L_j\bigl( T(n)\bigr)$, respectively.
\begin{Definition}[\textcolor{black}{Set $C_1$ and success-events $S_0$, $S_1$}]\label{gandalf}
We define $C_1$ as the set of points $x \in B_0'$ such that \[
\{x\} \leftrightarrow B(m) \text{ in $B_0'$ for } {\ensuremath{\mathbb G}} \,.\]
Furthermore, we define the success-events $S_0$ and $S_1$ as
\begin{align*}
& S_0:=\{\,\text{$B(m)$ is a seed}\, \}\,,\\
& S_1:=\{\,\text{$C_1$ contains a point of $ K^{(j)}(m,n)$
for each $j=1,2,3,4$}\,\}\,.
\end{align*}
\end{Definition}
\begin{Definition}[\textcolor{black}{Occupation of the origin}] \label{0occ}
We say that the origin $0\in \textcolor{black}{\e \bbZ^d}$ is occupied if the event \textcolor{black}{$S_0\cap S_1 $} takes place.
\end{Definition}
We refer to Figure \ref{fig_C1_2}--(left) for an example of the set $C_1$ when $\textcolor{black}{S_0\cap S_1}$ occurs. We note that the event $S_0$ implies that $B(m) \subset {\ensuremath{\mathbb V}} $, hence $B(m) \subset C_1 $.
\begin{figure}
\includegraphics[scale=0.50]{C23fin.pdf}
\captionsetup{width=.9\linewidth}
\caption{Left: the set $C_1$ when \textcolor{black}{$S_0\cap S_1$} occurs. Right: The set $C_2$ when \textcolor{black}{$S_0\cap S_1\cap S_2$} occurs. Points in $C_1$ correspond to circles, while points in $C_2\setminus C_1$ correspond to \textcolor{black}{red rings if in $E_1$ and blue triangles if in $F_1$.}}
\label{fig_C1_2}
\end{figure}
\begin{Remark}\label{gomitolo1} If the event $S_0$ occurs \textcolor{black}{(e.g. if the origin is occupied)}, then $C_1$ is a connected subset of ${\ensuremath{\mathbb G}} $ (and therefore of $\textcolor{black}{{\ensuremath{\mathbb G}} _+}$) by Lemma \ref{ironman}.
\end{Remark}
\textcolor{black}{When the origin is occupied}, for $i=1,2,3,4$ we define \textcolor{black}{$c^{(i)}$} as the minimal (w.r.t. the lexicographic order) point $z$ in $\e \bbZ^d$ such that $B(z,m)$ is a seed contained in $C_1\cap L_i\bigl( T(m,n)\bigr)$. We point out that such a seed exists by Lemma \ref{ironman} and the definition of $S_1$.
It is simple to check that, when $S_1$ takes place,
\begin{equation}\label{povo}
\textcolor{black}{ c^{(1)} _1}=N \text{ and }\textcolor{black}{c^{(1)} _j}\in [m,n-m] \text{ for }2\leq j \leq d\,,
\end{equation}
where \textcolor{black}{$c^{(1)} _j$} denotes the $j$--th coordinate of \textcolor{black}{$c^{(1)}$}. Similar formulas hold for \textcolor{black}{$c^{(i)}$}, $i=2,3,4$.
\textcolor{black}{For later use, we set}
\begin{equation} \textcolor{black}{b^{(1)}:= c^{(1)}\,.} \end{equation}
\begin{Proposition}\label{prop_occ_origin}
\textcolor{black}{It holds ${\ensuremath{\mathbb P}} ( 0\text{ is occupied}\, |\,S_0) \geq 1-4\e'$.}
\end{Proposition}
We postpone the proof of the above proposition to Section \ref{puffo1}. \textcolor{black}{If the origin is not occupied, then we stop our construction. Hence, from now on we assume that $0$ is occupied without further mention. We fix a unitary vector, that we take equal to $e_1$ without loss of generality, and we explain how we attempt to extend $C_1$ in the direction $e_1$. In order to shorten the presentation, we will define geometric objects only in the successful cases relevant to continue the construction (in the other cases, the definition can be chosen arbitrarily). Figure \ref{aquile} will be useful to locate objects. }
Below, for \textcolor{black}{$i=2,\dots, 7 $}, we will iteratively define points $b^{(i)}$. Moreover,
for \textcolor{black}{$i=1,2,3$},
we will iteratively define sets $T_i(n)$ and $T_i(m,n)$ obtained from $T(n)$ and $T(m,n)$ by an $i$--parametrized orthogonal map. \textcolor{black}{Apart from} the case \textcolor{black}{$i=4$}, many objects will be defined similarly. Hence, we isolate some special definitions to which we will refer in what follows. We stress that we collect these generic definitions below, but we will apply them only when describing the construction step by step in the next subsections. \textcolor{black}{Recall Definition \ref{def_triade}}.
\begin{iDefinition}[\textcolor{black}{Sets $K_i(m,n)$, $B_i' $}] \label{coca?} Given $b^{(i)}$, $T_i(n)$ and $T_i(m,n)$,
we define $K _i (m,n)$ as the set of points $x \in b^{(i)}+T_i(n) $ which are \textcolor{black}{adjacent} inside ${\ensuremath{\mathbb G}} $ to a seed contained in $b^{(i)}+ T_i (m,n)$. Moreover, we define $B_i' := b^{(i)}+ \bigl( B(n)\cup T_i(m,n)\bigr)$.
\end{iDefinition}
\begin{iDefinition}[\textcolor{black}{Sets $E_i,F_i,C_{i+1}$}] \label{legoland}
We set
\begin{align*}
& E_i:=\textcolor{black}{E\bigl[ C_i,\, B\bigl( b^{(i)}, n\bigr), i]}\,, \\
& F_i:=F\bigl[ C_i,\, B\bigl( b^{(i)}, n\bigr),B_i' ,i]\,,\\
& C_{i+1}:=C_i\cup E_i \cup F_i\,.
\end{align*}
\end{iDefinition}
\begin{iDefinition}[\textcolor{black}{Success-event $S_{i+1}$}] \label{ara}
We call $S_{i+1}$ the success-event that $C_{i+1}$ contains at least one
vertex in $K_i(m,n)$.
\end{iDefinition}
\begin{iDefinition}[\textcolor{black}{Property $\mathfrak{p}_i$}] \label{lattino}
We say that property $\mathfrak{p}_i$ is satisfied if the sets
$C_i \cup \partial C_i$ and $b^{(i) }+ \bigl (T_i(n) \cup T_i(m,n) \bigr)$ are disjoint.
\end{iDefinition}
In several steps below we will claim without further comments that property $\mathfrak{p}_i$ is satisfied. This property will correspond to the second property in \eqref{mare100} in the applications of Lemma \ref{pierpilori} in Section \ref{puffo1}, which (when not immediate) will be checked in Section \ref{puffo1} and Appendixes \ref{natale45}, \ref{natale46}. \begin{Remark}\label{aiko}
If $S_{i+1}$ occurs, then $C_{i+1}$ contains a point $x \in b^{(i)}+T_i(n) $ which is \textcolor{black}{adjacent} inside ${\ensuremath{\mathbb G}} $ to a seed $B(z,m)$ contained in $b^{(i)}+ T_i (m,n) \subset B_i' $.
Let us suppose that also property $\mathfrak{p}_i$ in Definition \ref{lattino} is satisfied. Then $(C_i \cup E_i )\subset (C_i \cup \partial C_i)$ does not intersect $b^{(i)}+T_i(n) $, thus implying that $x\in F_i$ \textcolor{black}{($E_i$ and $F_i$ were defined in Definition \ref{legoland})}. Since the above seed $B(z,m)$ is contained in $b^{(i)} + T_i(m,n)$, which is contained in $B_i' \setminus (C_i\cup \partial C_i)$ due to property $\mathfrak{p}_i$, by Lemma \ref{ironman} and Definition \ref{def_triade} we conclude that $F_i \subset C_{i+1}$ contains the above seed $B(z,m)$.
\end{Remark}
We now continue with the construction of increasing clusters and success-events.
\begin{figure}
\begin{center}
\includegraphics[scale=0.42]{mappamondo.pdf}
\end{center}
\captionsetup{width=.9\linewidth}
\caption{Colored small boxes are the seeds $B(m)$ and $B(b^{(i)}, m)$, while bigger boxes are given by $B(n)$ and $B(b^{(i)}, n)$. }
\label{aquile}
\end{figure}
\subsection{\textcolor{black}{Case $i=1$}}
\textcolor{black}{
We define $T_1(n) := T^*(n) = g\bigl( T(n)| b^{(1)}\bigr) $ and $T_1 (m,n)= T^*(m,n)= g\bigl( T(m,n)|b^{(1)}\bigr) $ (cf. \eqref{tittina} and \eqref{gigi}).
We apply (i)--Definition \ref{coca?}, (i)--Definition \ref{legoland}, (i)--Definition \ref{ara} and (i)--Definition \ref{lattino} for $i=1$. In particular, this defines the sets $K_1(m,n), B_1', E_1$, $F_1$, $C_2$ and the success-event $S_2$. See Figure \ref{fig_C1_2}--(right)}.
When \textcolor{black}{$S_0\cap S_1\cap S_2$} occurs, the set \textcolor{black}{$C_{2}$ intersects the box $B(2Ne_1, N)$ as we now show. Indeed, one can prove property $\mathfrak{p}_1$ using \eqref{povo}. Hence, by Remark \ref{aiko}, the event $S_0\cap S_1\cap S_2 $ implies that $F_1 \subset C_{2}$ contains a seed inside $ b^{(1)}+ T^*(m,n)$ (see Figure \ref{fig_C1_2}--(right)). One can easily check that
$ b^{(1)}+ T^*(m,n) \subset B(2N e_1,N)$. To this aim we observe that, by \eqref{povo}, if $x\in b^{(1)}+ T^*(m,n) $,} then
$x_1\in[2N-m,2N+m]\subset2N+[-N,N]$ and
$x_j\in [b_j^{(1)}-n, b_j^{(1)}]\subset[m-n,n-m]\subset[-N,N]$ for $2\leq j \leq d$.
We define \textcolor{black}{$b^{(2)}$} as the minimal point $z\in\e{\ensuremath{\mathbb Z}} ^d$ such that $B(z,m)$ is a seed contained in $\textcolor{black}{C_{2}}\cap \bigl( b^{(1)}+T^*(m,n)\bigr)$. By the above discussion, $b^{(1)}+T^*(m,n) \subset B(2N e_1, N)$.
By \eqref{povo} and since $\textcolor{black}{b^{(2)}_j}\in
[b^{(1)}_j-n,b_j^{(1)}]$ for $j\not=1$, we get for \textcolor{black}{$i=2$}
\begin{equation}\label{povoino}
b^{(i)} _1=\textcolor{black}{i N}, \qquad b^{(i)}_j\in
[-n+m, n-m]
\text{ for }j\not =1\,.
\end{equation}
\subsection{Case \textcolor{black}{$i=2$}} We assume that $S_0\cap S_1 \cap S_2$ occurs.
We set
$ \textcolor{black}{T_2}(m,n):=g\left(T(m,n)| \textcolor{black}{b^{(2)}} \right) $ and $\textcolor{black}{T_2} (n):=g\left(T(n)| \textcolor{black}{b^{(2)}}\right)$.
We apply (i)--Definition
\ref{coca?}, (i)--Definition \ref{legoland}, (i)--Definition \ref{ara} and (i)--Definition \ref{lattino} for \textcolor{black}{$i=2$.}
In particular this defines \textcolor{black}{$K_2(m,n), B_2',E_2,F_2, C_3, S_3$}. It is simple to check that property \textcolor{black}{$\mathfrak{p}_2$} is satisfied. If also \textcolor{black}{$S_3$} occurs, by Remark \ref{aiko} we can define
\textcolor{black}{$b^{(3)}$} as the minimal point in $\e{\ensuremath{\mathbb Z}} ^d$ such that $B(z,m)$ is a seed contained in \textcolor{black}{$C_3\cap\left( b^{(2)}+T_2 (m,n)\right)$}.
Let us localize some objects. Due to Claim \ref{locus2-3} in Appendix \ref{app_locus},
$b^{(\textcolor{black}{2})}+T_{\textcolor{black}{2}}(m,n)\subset B(3Ne_1,N)$ (see Figure~\ref{aquile}). In particular, if $S_{\textcolor{black}{3}}$ occurs, then $C_{\textcolor{black}{3}} \cap B(3Ne_1,N)$ contains the above seed $B(z,m)$. If $S_{\textcolor{black}{3}}$ occurs, due to \eqref{povoino} with \textcolor{black}{$i=2$,} for $j\not =1$ we have
\begin{equation}\label{matiz}
\begin{cases}
b_j^{(\textcolor{black}{3})}\in [b^{(\textcolor{black}{2})}_j-n+m,b_j^{(\textcolor{black}{2})}-m]\subset [-n+m,n-2m] & \text{ if $b^{(\textcolor{black}{2})}_j\geq 0$}\,,\\
b_j^{(\textcolor{black}{3})}\in [b^{(\textcolor{black}{2})}_j+m,b_j^{(\textcolor{black}{2})}+n-m]\subset [-n+2m,n-m], & \text{ if $b^{(\textcolor{black}{2})}_j< 0$}\,.
\end{cases}
\end{equation}
Due to the above bounds and since $b^{(\textcolor{black}{3})}_1=3 N$, we get \textcolor{black}{that \eqref{povoino} holds also for $i=\textcolor{black}{3}$}.
\subsection{Case $\textcolor{black}{i=3}$} \textcolor{black}{We assume that the event $\textcolor{black}{S_0\cap S_1\cap S_2 \cap S_3}$ occurs.}
We define $ T_{\textcolor{black}{3}}(m,n):=g\bigl(T(m,n)\,|\, b^{(\textcolor{black}{3})}\bigr) $ and $T_{\textcolor{black}{3}}(n):=g\bigl(T(n)\,|\, b^{(\textcolor{black}{3})})$.
We apply (i)--Definition
\ref{coca?}, (i)--Definition \ref{legoland}, (i)--Definition \ref{ara} and (i)--Definition \ref{lattino} for $i=\textcolor{black}{3}$. In particular, this defines \textcolor{black}{$K_3(m,n), B_3', E_3,F_3, C_4, S_4$}. Property \textcolor{black}{$\mathfrak{p}_3$} is satisfied.
If also \textcolor{black}{$S_4$} occurs, by Remark \ref{aiko} we can define $b^{(\textcolor{black}{4})}$ as minimal point in $\e{\ensuremath{\mathbb Z}} ^d$ such that $B(z,m)$ is a seed in $b^{(\textcolor{black}{3})}+ T_{\textcolor{black}{3}}(m,n)$.
Let us localize some objects. Due to Claim \ref{locus2-3} in Appendix \ref{app_locus}, $b^{(\textcolor{black}{3})}+T_{\textcolor{black}{3}}(m,n)\subset B(4Ne_1,N)$ (see Figure \ref{aquile}). In particular, if \textcolor{black}{$S_4$} occurs, then $\textcolor{black}{C_4} \cap B(4Ne_1,N)$ contains the above seed $B(z,m)$. When \textcolor{black}{$S_4$} occurs, due to \eqref{povoino} for \textcolor{black}{$i=3$} and
reasoning as in \eqref{matiz}, we get that \eqref{povoino} holds also for \textcolor{black}{$i=4$}.
\subsection{Case \textcolor{black}{$i=4$}}
We assume that the event \textcolor{black}{$S_0\cap S_1\cap \cdots \cap S_4$} occurs.
The idea now is to connect the cluster $C_{\textcolor{black}{4}}$ to seeds adjacent to
the remaining three faces of the cube $b^{(\textcolor{black}{4})}+B(n)$ in directions $e_1$ and $\pm e_2$ \textcolor{black}{(note that we have already the seed $B\bigl(b^{(3)},m\bigr)$ in the direction $-e_1$)}.
To this aim we set
\begin{align*}
& \hat T_1(n):= g\bigl(T(n)\,|\,b^{(\textcolor{black}{4} )} \bigr) \qquad \qquad \hat T_1(m,n):= g\bigl(T(m,n)\,|\,b^{(\textcolor{black}{4} )} \bigr)
\\
& \hat T_2(n):=( h\circ \theta) \bigl( \hat T_1(n) \bigr)
\qquad \;\;\;\;\;\hat T_2(m,n):=( h\circ \theta) \bigl( \hat T_1(m,n) \bigr)
\\
& \hat T_3(n):= ( h\circ \theta^3) \bigl( \hat T_1(n) \bigr) \qquad \;\;\;\; \hat T_3(m,n):=( h\circ \theta^3) \bigl( \hat T_1(m,n) \bigr)
\end{align*}
where $\theta$ is the rotation introduced before \eqref{gommina} and the map $h$ is defined as $h(x_1,x_2, \dots,x_n):= (|x_1|,x_2,\dots,x_n)$.
\textcolor{black}{The use of the above sets $\hat T_j(n), \hat T_j(m,n)$ is due to the fact that we want to avoid to construct seeds in regions that have
already been explored during the constructions of the previous sets of type $C_r$.
Indeed points in the sets $\hat T_j(n), \hat T_j(m,n)$ have first coordinate not smaller than $4N=b^{(4)}_1$ and this assures the desired property.}
We also set
\begin{equation}\label{thorin} B_{\textcolor{black}{4}}':= b^{(\textcolor{black}{4})}+\bigl(B(n)\cup \bigl[\cup_{j=1,2,3} \hat T_j(m,n)\bigr]\bigr)\,.\end{equation}
For $j=1,2,3$ we
call $K^{(\textcolor{black}{4}+j)}(m,n)$ the set of points $x \in b^{(\textcolor{black}{4})}+ \hat T_j(n) $ which are \textcolor{black}{adjacent} inside ${\ensuremath{\mathbb G}} $ to a seed contained in $b^{(\textcolor{black}{4})}+ \hat T_j(m,n)$ \textcolor{black}{(the sets $K^{(1)}(m,n)$,..., $K^{(4)}(m,n)$ had already been introduced to define the success-event $S_1$ in Definition \ref{gandalf})}. \textcolor{black}{Note that we do not apply (i)--Definition \ref{coca?} for $i=4$, since the set $B_{\textcolor{black}{4}}'$ has already been introduced in \eqref{thorin} in a different form}.
We apply \textcolor{black}{here} only (i)--Definition \ref{legoland} with $\textcolor{black}{i=4}$ to define \textcolor{black}{the sets $E_4,F_4,C_5$}.
\begin{Definition}[\textcolor{black}{Success-event $S_{5}$ and points $b^{(5)}$, $b^{(6)}$, $b^{(7)}$}]
We call
$S_{5}$ the success-event that $C_{5}$ contains at least one vertex inside $K^{(4+j)}$ for all $j=1,2,3$. When $S_{5}$ occurs, for $j=1,2,3$ we define $b^{(4+j)}$ as the minimal point in $\e \bbZ^d$ such that $b^{(4+j)}+ B(m)$ is a seed in $C_{5} \cap \bigl( b^{(4)}+\hat T_j(m,n)\bigr)$. The existence of such a seed can be derived by the same arguments of Remark \ref{aiko} since $C_{4}\cup \partial C_{4}$ and $b^{(4) }+\bigl (\hat T_j(n) \cup \hat T_j(m,n) \bigr)$ are disjoint for $j=1,2,3$ (as \textcolor{black}{discussed} in Section \ref{sole88}).
\end{Definition}
Let us localize the above objects \textcolor{black}{when also $S_5$ occurs}. Due to \eqref{povoino} for \textcolor{black}{$i=4$} and
reasoning as in \eqref{matiz}, we get that \eqref{povoino} holds also for $i=\textcolor{black}{5}$.
For $i=\textcolor{black}{6,7}$ we have
\begin{equation}\label{suono}
b_1^{(i)}\in 4N+[m,n-m]\,,\quad
\begin{cases} b_2^{(\textcolor{black}{6})} =b_2^{(\textcolor{black}{4})}+ N \,,\\
b_2^{(\textcolor{black}{7})} =b_2^{(\textcolor{black}{4})}- N \,,
\end{cases}
b^{(i)}_j \in [-n+m,n-m] \text{ for }j\geq 3\,
\end{equation}
(for $j\geq 3$ one has to argue as in \eqref{matiz}).
Moreover, due to Claim \ref{locus4} in Appendix \ref{app_locus}, if
\textcolor{black}{$S_0\cap S_1\cap \cdots \cap S_5$}
occurs, then $b^{(\textcolor{black}{4})}+\hat T_1(m,n) \subset B( 5N e_1, N)$,
$b^{(\textcolor{black}{4})}+\hat T_2(m,n) \subset B(4N e_1+ N e_2, N)$ and $b^{(\textcolor{black}{4})}+\hat T_3(m,n) \subset B(4N e_1- Ne_2, N)$. In particular, the same inclusions hold for the seeds $b^{(\textcolor{black}{5})}+ B(m)$,
$b^{(\textcolor{black}{6})}+ B(m)$ and $b^{(\textcolor{black}{7})}+ B(m)$, respectively.
\begin{Definition}[\textcolor{black}{Occupation and \textcolor{black}{linkage} of $e_1$}] \label{1occ}
Knowing that the origin $0\in \textcolor{black}{\e \bbZ^d} $ is occupied, we say that the site $e_1$ is linked to $0$ and occupied \textcolor{black}{(shortly, $0\to e_1$)}
if \textcolor{black}{also the event $ \cap _{i=2}^{ 5} S_i $} takes place.
\end{Definition}
\begin{Proposition}\label{prop_occ_e1} If $0$ is occupied and $e_1$ is linked to $0$ and occupied, then the sets \textcolor{black}{$C_2,C_3, \dots, C_5$} are connected in $\textcolor{black}{{\ensuremath{\mathbb G}} _+}$. Moreover,
\begin{equation}\label{soglia2} {\ensuremath{\mathbb P}} ( e_1\text{ is linked to $0$ and occupied}\, |\, \text{$0$ is occupied} ) \geq \textcolor{black}{1-6 \e'}\,.
\end{equation}
\end{Proposition}
We postpone the proof of the above proposition to Section \ref{puffo1}.
\subsection{Further comments on the construction of the occupied clusters}\label{francia}
\textcolor{black}{We start by explaining what to do in the case of a non-success by treating an example. Suppose that we have a success until the definition of $C_5$: $0$ is occupied and $e_1$ is linked to $0$ and occupied. Suppose that, according to Tanemura's algorithm, we want to extend $C_5$ along the first direction in order to get a cluster with a seed in the proximity of $8N e_1$. To this aim we set
$T_5(n):= g ( T(n)| b^{(5)})$ and apply (i)-Definitions \ref{coca?}, \ref{legoland} and \ref{ara} with $i=5$ (see Fig. \ref{aquile}). This defines the sets $E_5,F_5, C_6$ and the success-event $S_6$. If $S_6$ does not occur, then we extend $C_6$ trying to develop the cluster along the route from $4N e_1$ to $4Ne_1 +4N e_2$ (if $e_2$ is the direction prescribed by Tanemura's algorithm). To do this we use the seed centered at $b^{(6)}$. We define
\[
g_2 (x|a):= \left(-\text{sgn}(a_1) x_1, x_2, -\text{sgn}(a_3) x_3,\dots, -\text{sgn}(a_d) x_d\right)
\]
and set
$T_6(n):=g_2\bigl(T(n) | b^{(6)}\bigr)$, $T_6(m,n):=g_2\bigl(T(m,n) | b^{(6)}\bigr)$. We then apply (i)-Definitions \ref{coca?}, \ref{legoland} and \ref{ara} with $i=6$. This defines the sets $E_6,F_6, C_7$ and the success-event $S_7$, and we proceed in this way.}
\textcolor{black}{In order to check the validity of Assumption (A) (cf. Section \ref{sec_japan}) for the construction outlined in Section \ref{sec_ginepro}, one applies iteratively Lemma \ref{pierpilori} as done in the proof of Proposition \ref{prop_occ_e1}.
Since we explore uniformly bounded regions, by taking $K$ large enough in Definition \ref{cavallo}, we can apply iteratively Lemma \ref{pierpilori} assuring condition \eqref{monti100} to be fulfilled simply by using some index $k_*\in \{1,2,\dots, K\}$ not already used in the region under exploration.}
\section{Proof of \eqref{selva0}, Corollary \ref{cor1} and Corollary \ref{cor2}}\label{bin_MA}
In this section we prove Corollaries \ref{cor1} and \ref{cor2}.
Before, we state and prove the phase transition mentioned in \eqref{selva0}:
\begin{Lemma}\label{silvestro} Let $d\geq 2$, \textcolor{black}{$\l>0$ and let $h(a,b)$ be given by $h(a,b)=(a+b)^\g$ with $\g>0$, or $h(a,b)=\min(a,b)$ or $h(a,b)=\max(a,b)$.} Consider the graph $\ensuremath{\mathcal G}=\ensuremath{\mathcal G}( h,\l)$ built on the $\nu$--randomization of a PPP on ${\ensuremath{\mathbb R}} ^d$ with \textcolor{black}{intensity} $\l$, where $\nu$ has bounded support and $\nu(\{0\})\not =1$. Then there exists \textcolor{black}{ $\l_c\in (0,+\infty)$} such that \eqref{selva0} holds.
\end{Lemma}
\begin{proof} Since the superposition of independent PPP's is a new PPP with \textcolor{black}{intensity} given by the sum of the original \textcolor{black}{intensities},
by a standard coupling argument the map $\l \mapsto P\left( \ensuremath{\mathcal G}( h ,\l) \text{ percolates} \right)\in [0,1] $ is non-decreasing. By the 0-1 law, the above map takes value $0,1$. Hence, to prove \eqref{selva0}, it is enough to prove that the above map equals $0$ for $\l$ small and equals $1$ for $\l$ large.
We first prove that $P\left( \ensuremath{\mathcal G}( h ,\l) \text{ percolates} \right)=1$ for $\l$ large. To this aim,
we fix a positive constant $c$ such that $ c< \sup\bigl ({\rm supp}(\nu)\bigr) $ (this is possible since $\nu(\{0\})\not =1$).
Note that $\ensuremath{\mathcal G}( h,\l)$ contains the subgraph $\tilde \ensuremath{\mathcal G}$ with vertex set $\tilde \xi :=\{ x\in \xi\,:\, E_x \geq c\}$ and edges $\{x,y\}$ with $x\not = y$ in $\tilde \xi$ such that $|x-y| \leq h(E_x,E_y)$. Given $x\not =y$ in $ \tilde \xi$ with $|x-y| \leq \textcolor{black}{h(c,c)}$, $\{x,y\}$ is an edge of $\tilde \ensuremath{\mathcal G}$ as $h(E_x,E_y) \geq \textcolor{black}{h(c,c)}$. Hence, $\ensuremath{\mathcal G}( h,\l)$ contains the Boolean graph model built on $\tilde \xi$ with deterministic radius
$\left( \textcolor{black}{h(c,c)}\right)/2>0$. As $\tilde \xi$ is a PPP with \textcolor{black}{intensity} $\l p$ where $p:=\nu ( [c,+\infty) )>0$, we get that (see \cite{MR}) there exists \textcolor{black}{$\tilde\l_c\in(0,+\infty)$} such that $\tilde\ensuremath{\mathcal G}$ percolates a.s. if $\l p>\tilde\l_c$. Hence, if \textcolor{black}{$\l>\tilde\l_c/p$}, the graph $\ensuremath{\mathcal G}(h,\l)$ percolates a.s..
We now prove that $P\left( \ensuremath{\mathcal G}( h,\l) \text{ percolates} \right)=0$ for $\l$ small.
Since $\nu$ has bounded support, we can now fix a finite constant
$ c'> \sup\bigl ({\rm supp}(\nu)\bigr) $. Then
$\ensuremath{\mathcal G}( h,\l)$ is contained in the Boolean graph model built on \textcolor{black}{$ \xi$} with deterministic radius
$\left( \textcolor{black}{h(c',c')}\right)/2$.
Since, for $\l$ small, the latter a.s. does not percolate, we get the thesis.\end{proof}
\subsection{Proof of Corollary \ref{cor1}} Due to Theorem \ref{teo1} we only need to check Assumptions (A1),...,(A5).
Note that Assumptions (A1), (A3) and (A5) follow immediately from the hypotheses of Corollary \ref{cor1} and the definition of $h$. As
pointed out in Section \ref{moda}, if $h$ is continuous, ${\rm supp}(\nu)$ is bounded and (A5) is satisfied (as in the present setting), then (A4) is automatically satisfied by compactness.
It remains to prove Assumption (A2). To this aim,
we fix $\l_1\in (\l_c, \l)$. Given $c>0$ we define
$p_0:=\nu(\{0\})$, $p_1=p_1(c):=\nu\left( (0,c)\right)$ and
$p_2=p_2(c):= \nu( [c,+\infty) )$, $\nu_1:= \nu\left( \cdot | \, (0,c) \right)$ and
$\nu_2:= \nu\left( \cdot | \, [c,+\infty)\right)$. Then $\nu=p_0 \d_0 +p_1 \nu_1 + p_2 \nu_2$ ($\d_0$ is the standard Dirac measure at $0$). Trivially, $\lim _{c\downarrow 0} p_1=0$ and $\lim _{c\downarrow 0} p_2=1-p_0$.
We choose $c>0$ small enough to have $\frac{1-p_0}{p_2} \l_1<\l$.
Call $\hat \ensuremath{\mathcal G}= \ensuremath{\mathcal G}( \l_1,h;\mu)$ the graph with structural function
\textcolor{black}{$h$}
built on the $\mu$--randomization of a PPP $\xi$ with \textcolor{black}{intensity} $\l_1$,
where
\[ \mu:=p_0\d_0+ (1-p_0) \nu_2=p_0\d_0+ (p_1+p_2) \nu_2\,. \]
We have that $\mu $ stochastically dominates $\nu$ since, for all $a\geq 0$, it holds
\[
\mu\left( [a, +\infty)\right) =
\begin{cases}
1= \nu \left( [a, +\infty)\right)
& \text{ if } a=0\,,\\
1-p_0\geq \nu \left( [a, +\infty)\right) & \text{ if } 0<a<c \,, \\
(1- p_0) \frac{ \nu \left( [a, +\infty)\right)}{ 1-p_0-p_1}\geq \nu \left( [a, +\infty)\right) & \text{ if } a\geq c\,.
\end{cases}
\]
Due to the above stochastic domination there exists a coupling between the $\mu$-randomization of the PPP $\xi$ with \textcolor{black}{intensity} $\l_1$ and the $\nu$-randomization of the PPP $\xi$ with \textcolor{black}{intensity} $\l_1$ such that the \textcolor{black}{marks} in the former are larger than or equal to the \textcolor{black}{marks} in the latter.
As $\ensuremath{\mathcal G}(h, \l_1)$ percolates a.s. since $\l_1>\l_c$, by the above coupling and since $h(\cdot, \cdot)$ is jointly increasing, $\hat \ensuremath{\mathcal G}$ percolates a.s..
Trivially, given $\rho\in(0,1) $, $\hat \ensuremath{\mathcal G}$ can be described also as the graph with vertex set $\xi$ as above and edge set given by the unordered pairs $\{x,y\}\subset \xi $ with $x\not=y$ and
\begin{equation} \label{bagnoli1}
\bigl | \rho x -\rho y \bigr |
\leq
\textcolor{black}{\rho h(E_x,E_y) = h(E_x,E_y)- (1-\rho) h(E_x,E_y) }\,,
\end{equation}
where the marks $E_x$'s have law $\mu$.
Note that, as $x\not =y$, at least one between the marks $E_x,E_y$ is nonzero \textcolor{black}{(and both of them are non zero if $h(a,b)=\min(a,b)$)} and therefore lower bounded by $c$ \textcolor{black}{a.s. (by definition of $\mu$)}.
This implies that
\textcolor{black}{$(1-\rho) h(E_x,E_y) \geq (1-\rho) c^\g=: \ell_*$ if $h(a,b)=(a+b)^\g$ and $(1-\rho) h(E_x,E_y) \geq (1-\rho) c=: \ell_*$ if $h(a,b)= \min(a,b)$ or $h(a,b)=\max(a,b)$. At this point,}
by applying the map $x\mapsto \rho x$, we get that the image of $\hat \ensuremath{\mathcal G} $ is contained in the graph $\bar \ensuremath{\mathcal G}:=\ensuremath{\mathcal G}(h-\ell_*, \l_2; \mu ) $ with structural function $h-\ell_*$ built on the $\mu$--randomization of a PPP with \textcolor{black}{intensity} $ \l_2:=\l_1 \rho^{-d}$. As $\hat \ensuremath{\mathcal G}$ percolates a.s., the same holds for its $\rho$-rescaling contained in $\bar \ensuremath{\mathcal G}$. This proves that $\bar \ensuremath{\mathcal G} $ percolates a.s..
Recall that $\frac{1-p_0}{p_2} \l_1<\l$. We now choose $\rho $ very near to $1$ to have
$\l_*:=\frac{1-p_0}{p_2}\l_2=\frac{1-p_0}{p_2} \l_1 \rho^{-d}$ smaller than $\l$.
To conclude with (A2) we only need to show that $\ensuremath{\mathcal G}(h-\ell_*, \l_*)$ percolates a.s.. To this aim we observe that
\[
\nu=\frac{p_2}{p_1+p_2}\mu + \frac{1-p_2}{p_1+p_2}\bar \mu\,, \qquad \bar \mu:=\frac{p_1}{1-p_2} p_0 \d_0 + \frac{1-p_0}{1-p_2}p_1 \nu_1\,.
\]
Note that $\nu$ is a convex combination of the probability measures $\mu$ and $\bar \mu$.
Hence,
the marked vertex set of $\ensuremath{\mathcal G}(h-\ell_*, \l_*)$, which is the $\nu$--randomization of a PPP with \textcolor{black}{intensity} $\l_*$, can be obtained as superposition of two independent marked point processes given by
the $\mu$--randomization of a PPP with \textcolor{black}{intensity} $\frac{p_2}{p_1+p_2} \l_*=\l_2$ and the $\bar \mu $-randomization of a PPP with \textcolor{black}{intensity} $\frac{1-p_2}{p_1+p_2} \l_*$.
The subgraph of $\ensuremath{\mathcal G}(h-\ell_*, \l_*)$ given by the points in the first marked PPP and their edges has the same law of $\bar \ensuremath{\mathcal G}$. As $\bar \ensuremath{\mathcal G}$ percolates a.s. we get that $\ensuremath{\mathcal G}(h-\ell_*, \l_*)$ percolates a.s., i.e. (A2) is satisfied.
\subsection{Proof of Corollary \ref{cor2}} Due to Theorem \ref{teo1} we only need to check that Assumptions (A1),...,(A5) are satisfied.
Assumptions (A1) and (A3) follow immediately from the hypothesis of the corollary. Also Assumption (A5) is trivial as $h(a,b)=\z- 2\max\{ |a|,|b|\}$ for $ab\geq 0$. As
pointed out in Section \ref{moda}, since $h$ is continuous, ${\rm supp}(\nu)$ is bounded and (A5) is satisfied, then (A4) is automatically satisfied by compactness.
Let us prove Assumption (A2).
To this this aim we fix $\ell>0$ such that $\z-\ell>\z_c$ (this is possible as $\z>\z_c$). Then, by \eqref{selva}, $\ensuremath{\mathcal G}(h-\ell ,\l )$ percolates a.s..
Moreover,
given $\g \in (0,1)$, $\ensuremath{\mathcal G}(h-\ell ,\l )$ can be described also as the graph with vertex set $\xi$ given by a PPP with \textcolor{black}{intensity} $\l$ and edge set given by the pairs $\{x,y\}\subset \xi $ with $x\not=y$ and
\begin{equation} \label{bagnoli}
\Big | \frac{x}{\g} -\frac{ y}{\g} \Big |
\leq
\frac{\z-\ell}{\g}-\frac{1}{\g} ( |E_x|+ |E_y| +|E_x-E_y|)
\,,
\end{equation}
where the marks come from the $\nu$-randomization of the PPP $\xi$.
Note that the r.h.s. is upper bounded by $(\z-\ell)/\g- ( |E_x|+ |E_y| +|E_x-E_y|)$.
We now fix $\g$ very near to $1$ (from the left) to have $(\z-\ell) /\g<\z$. Hence, we can write $(\z-\ell) /\g=\z-\ell_*$ for some $\ell_*>0$.
Due to the above observations, if $\{x,y\}$ is an edge of $\ensuremath{\mathcal G}(h-\ell ,\l )$, then $|x/\g-y/\g|\leq \z-\ell_*- ( |E_x|+ |E_y| +|E_x-E_y|)$. In other words, the graph $\ensuremath{\mathcal G}(h-\ell ,\l )$ is included in the new graph $\hat \ensuremath{\mathcal G}= (\xi , \hat \ensuremath{\mathcal E})$, with edge set $\hat \ensuremath{\mathcal E}$ given by the pairs $\{x,y\}\subset \xi $ with $x\not=y$ and
\begin{equation} \label{pompei}
\bigl| x/\g - y/\g \bigr | \leq \z-\ell_*- ( |E_x|+ |E_y| +|E_x-E_y|) \,.
\end{equation}
As already observed $\ensuremath{\mathcal G}(h-\ell ,\l )$ percolates a.s., thus implying that $\hat \ensuremath{\mathcal G}$ percolates a.s..
On the other hand, due to \eqref{pompei}, the graph obtained by spatially rescaling $\hat \ensuremath{\mathcal G}$ according to the map $x\mapsto x/\g$
has the same law of the graph $\ensuremath{\mathcal G}( h-\ell_*, \l \g^d )$. Hence, the latter percolates a.s.. To conclude the derivation of (A2) it is enough to take $\l_*:= \l \g^d$.
\section{Proof of Proposition \ref{cinquina}}\label{sio5}
Recall Definition \ref{sambinaA}.
\begin{Definition}[\textcolor{black}{Sets $A(n)$ and $T_{\s,J}(n)$}] \label{sambina}
For $m \leq n\in{\ensuremath{\mathbb N}} _+$, $z \in \e \bbZ^d$, $\s \in \{-1,1\}^d$, $J\in \{1,2,\dots, d\}$ we define the following sets \textcolor{black}{(see Figure
\ref{protex})
\begin{align*}
& A(n):=\{ x\in \e \bbZ^d\,:\, n-1 < \|x\|_\infty \leq n\}\,,\\
& T_{\s,J}(n):=\{ x\in \e \bbZ^d : n-1 < \|x\|_\infty \leq n, \,0 \leq \s_i x_i \leq \s_J x_J \; \forall i=1,2,\dots, d\}
\,.
\end{align*}
\end{Definition}
\begin{figure}
\includegraphics[scale=0.35]{new-fig2.pdf}
\captionsetup{width=.9\linewidth}
\caption{The sets $T_{\s,J}$ for $d=2$. The largest square has radius $n$ and the smallest one has radius $n-1$. The annulus $A(n)$ is the union of the $T_{\s,J}$'s, hence it corresponds to the (dark or light) grey region. }
\label{protex}
\end{figure}
Note that $T_{\underline{1},1}(n)= T(n)$, where
$\underline{1}:=(1,1,\dots, 1)$. The following fact can be easily checked (hence we omit its proof):
\begin{Lemma}\label{euclide}
We have the following properties:
\begin{itemize}
\item[(i)]
$
A(n)= \cup _{\s\in \{-1,1\}^d}\cup_{J=1}^d
T_{\s,J}(n)$;
\item[(ii)] given $(\s,J)$ the map
$ \psi_{\s,J}( x_1,x_2, \dots, x_d):=(y_1,y_2, \dots, y_d)$,
where
\[
y_k:=
\begin{cases}
x_J \s_1 \text{ if } k=1 \,,\\
x_1 \s_J \text{ if } k=J\,, \\
x_k \s_k \text{ otherwise}\,, \end{cases}
\]
is an isometry from $T(n)$ to $T_{\s,J}(n)$ and it is the identity when $\s = \underline{1}$ and $J=1$.
\end{itemize}
\end{Lemma}
The following lemma and its proof are inspired by \cite[Lemma 3]{GM} and its proof. \textcolor{black}{To state it properly we introduce the set $U_n$ (recall the constant $\l_*$ introduced in (A2), the definition of $h_*$ in \eqref{quiquoqua} and that $d(\cdot,\cdot)$ denotes the Euclidean distance):
\begin{Definition}[Set $U_n$]
Given \textcolor{black}{positive integers} $n>m$, we denote by
$U_n$ the set of points $x\in A(n)$ such that $B(m) \leftrightarrow x$ in $ B(n)$ for $ {\ensuremath{\mathbb G}} _-$ and
\begin{equation}\label{rotta}
d\bigl( x, B(n)^c \bigr) \leq h_*(A_x)-3\a\,.
\end{equation}
\end{Definition}}
\begin{Lemma}\label{analogo3} Let $m $ and $ n$ be positive integers such that $n>m$. Then, for each integer $k$, it holds
\begin{equation}\label{LP}
\sum_{n=m+1}^\infty {\ensuremath{\mathbb P}} (|U_n|<k,\,B(m) \leftrightarrow \infty\text{ for } \textcolor{black}{{\ensuremath{\mathbb G}} _-})<e^{ c(d) \l_* k}
\end{equation}
for a positive constant $c(d)$ depending only on the dimension.
\end{Lemma}
\begin{proof} We claim that the event $\{B(m) \leftrightarrow \infty\text{ for } \textcolor{black}{{\ensuremath{\mathbb G}} _-}\}$ implies that $|U_n|\geq 1$.
To prove our claim we observe that,
since the edges in $\textcolor{black}{{\ensuremath{\mathbb G}} _-}$ have length at most $1-3\a $ \textcolor{black}{(see Warning \ref{aaah})},
the event $\{B(m) \leftrightarrow \infty\text{ for } \textcolor{black}{{\ensuremath{\mathbb G}} _-} \}$ implies that there exists $x \in A(n) $ such that $B(m) \leftrightarrow x \text{ in } B(n) $ for $ \textcolor{black}{{\ensuremath{\mathbb G}} _-}$ and $\{x , y\}\in \textcolor{black}{{\ensuremath{\mathbb E}} _-}$ for some $y\in B(n)^c\cap \textcolor{black}{{\ensuremath{\mathbb V}} _-}$. Indeed, it is enough to take
any path from $B(m)$ to $\infty$ for $ \textcolor{black}{{\ensuremath{\mathbb G}} _-} $ and define $y$ as the first visited point in $B(n)^c$ and $x$ as the
point visited before $y$. Note that the property $\{x,y\}\in \textcolor{black}{{\ensuremath{\mathbb E}} _-} $ implies \eqref{rotta}\textcolor{black}{.} Hence $x\in U_n$. This concludes the proof of our claim.
Due to the above claim we have
\begin{equation}\label{torino1}
{\ensuremath{\mathbb P}} (|U_n|<k,\,B(m) \leftrightarrow \infty\text{ for } \textcolor{black}{{\ensuremath{\mathbb G}} _-}) \leq {\ensuremath{\mathbb P}} ( 1\leq |U_n | < k) \,.
\end{equation}
\smallskip
We now want to estimate ${\ensuremath{\mathbb P}} ( U_{n+1}=\emptyset\,|\, 1\leq |U_n | <k)$ from below (the result will be given in \eqref{torino2} below).
For each $x \in U_n$ we denote by
$I_{n+1}(x)$ the set of points $y$ in $A(n+1)$ such that $|x-y|\leq 1-3\a$.
We call $G_n$ the event that $\textcolor{black}{{\ensuremath{\mathbb V}} _-}$ has no points in $\cup _{x \in U_n} I_{n+1}(x)$.
We now claim that $G_{n} \subset \{U_{n+1}=\emptyset\}$. To prove our claim let $z$ be in
$ U_{n+1}$. Then there is a path in $\textcolor{black}{{\ensuremath{\mathbb G}} _-}$ from $z$ to some point in $B(m)$ visiting only points in $B(n+1)$. We call $v$ the last point in the path inside $A(n+1)$ and $x$ the next point in the path. Then $x\in A(n)$ and all the points visited by the path after $x$ are in $B(n)$. Hence, $B(m) \leftrightarrow x$ in $ B(n)$ for $ \textcolor{black}{{\ensuremath{\mathbb G}} _-}$. Moreover, since $\{ x, v\}\in \textcolor{black}{{\ensuremath{\mathbb E}} _-}$, property \eqref{rotta} is verified. Then $x\in U_n$ and $\textcolor{black}{{\ensuremath{\mathbb V}} _-} $ has some point (indeed $v$) in $I_{n+1}(x)$.
In particular, we have shown that, if $U_{n+1}\not=\emptyset$, then $G_n$ does not occur, thus proving our claim.
Recall that
the graph $\textcolor{black}{{\ensuremath{\mathbb G}} _-}$ depends only on the random field $(A_z)_{z\in \e \bbZ^d}$ and that ${\ensuremath{\mathbb P}} (\textcolor{black}{A_z\not \in {\ensuremath{\mathbb R}} })= e^{-\l_* \e^d}$ for any $z\in \e \bbZ^d$.
We call $\ensuremath{\mathcal F}_n$
the $\s$--algebra
generated by the random variables $A_z$ with $z\in B(n)$.
Note that the set $\cup_{x\in U_n} I_{n+1}(x)$
and the event $\{1\leq |U_n |< k\}$ are $\ensuremath{\mathcal F}_n$--measurable. Moreover,
on the event $\{1\leq |U_n |< k\}$, the set $\cup_{x\in U_n} I_{n+1}(x)$ has cardinality bounded by $c(d) k \e^{-d} $, where $c(d)$ is a positive constant depending only on $d$. By the independence of the $A_z$'s we conclude that that ${\ensuremath{\mathbb P}} $--a.s. on the event $\{1\leq |U_n | < k\}$ it holds
\begin{equation}\label{girino}
\begin{split}
{\ensuremath{\mathbb P}} ( G_n\,| \, \ensuremath{\mathcal F}_n)& ={\ensuremath{\mathbb P}} ( \textcolor{black}{A_z\not\in {\ensuremath{\mathbb R}} } \; \forall z\in \cup_{x\in U_n} I_{n+1}(x) \,|\, \ensuremath{\mathcal F}_n)
\\
& \geq {\ensuremath{\mathbb P}} (\textcolor{black}{A_0\not\in {\ensuremath{\mathbb R}} })^{c(d) k \e^{-d}}= e^{- c(d) \l_* k } \,.
\end{split}\end{equation}
Hence, since $G_n \subset \{ U_{n+1}=\emptyset\}$, by \eqref{girino} we conclude that
\begin{equation}\label{torino2}
{\ensuremath{\mathbb P}} ( U_{n+1}=\emptyset\,|\, 1\leq |U_n | < k) \geq {\ensuremath{\mathbb P}} ( G_n \,|\, 1\leq |U_n |< k) \geq \exp \{ - c(d) \l_* k\}\,.
\end{equation}
As a byproduct of \eqref{torino1} and \eqref{torino2} we get
\begin{multline}
e^{ - c(d) \l_* k} {\ensuremath{\mathbb P}} (|U_n|<k,\,B(m) \leftrightarrow \infty\text{ for }\textcolor{black}{{\ensuremath{\mathbb G}} _-})
\leq e^{ - c(d) \l_* k} {\ensuremath{\mathbb P}} ( 1\leq |U_n | < k)\\
\leq
{\ensuremath{\mathbb P}} ( U_{n+1}=\emptyset\,|\, 1\leq |U_n | < k) {\ensuremath{\mathbb P}} ( 1\leq |U_n | < k)\\
={\ensuremath{\mathbb P}} ( U_{n+1}=\emptyset\,,\, 1\leq |U_n | <k) \,.
\end{multline}
Since the events $\{ U_{n+1}=\emptyset\,,\, 1\leq |U_n |<k\}$ are disjoint, we get \eqref{LP}.
\end{proof}
We now present the analogous of \cite[Lemma 4]{GM}. \textcolor{black}{To this aim we introduce the set $V_n$:
\begin{Definition}[Set $V_n$] Given \textcolor{black}{positive integers} $n>m$, we call $V_n$
the set of points $x\in T(n)$ satisfying \eqref{rotta} and such that $B(m) \leftrightarrow x$ in $ B(n)$ for $ {\ensuremath{\mathbb G}} _-$.
\end{Definition}}
\begin{Lemma}\label{analogo4} Let $w:= 2^d d$. Then, for any $\ell \in {\ensuremath{\mathbb N}} $, it holds
\begin{equation}\label{minerva}
\liminf _{n \to \infty}{\ensuremath{\mathbb P}} ( |V_n | \geq \ell ) \geq 1- {\ensuremath{\mathbb P}} (B(m)\not \leftrightarrow \infty \text{ for } \textcolor{black}{{\ensuremath{\mathbb G}} _-} )^{1/w } \,.
\end{equation}
\end{Lemma}
\begin{proof}
Let $\s,J$ be as in Definition \ref{sambina}.
If in the definition of $V_n$ we take $T_{\s, J}(n) $ instead of $T(n) $, then we call $V_{\s,J,n}$ the resulting set. Note that $V_{\underline{1}, 1,n}=V_n$.
By Lemma \ref{euclide}--(i) we get that $|U_n| \leq \sum _{(\s,J) } |V_{\s,J,n}|$, hence
\begin{equation}
\{ |U_n| < w \ell\} \supset \cap _{(\s,J)} \{ |V_{\s, J,n} | < \ell \} \,.
\end{equation}
By the FKG inequality \textcolor{black}{(cf. Section \ref{scremato})} and since each event $\{ |V_{\s, J,n} | < \ell \} $ is decreasing, and by the isometries given in Lemma \ref{euclide}--(ii), we have \[ {\ensuremath{\mathbb P}} ( |U_n| < w \ell ) \geq \prod _{(\s,J)} {\ensuremath{\mathbb P}} ( |V_{\s, J,n} | < \ell )={\ensuremath{\mathbb P}} ( |V _n | < \ell )^w\,. \]
The above bound implies that $ {\ensuremath{\mathbb P}} ( |V_n | \geq \ell )\geq 1- {\ensuremath{\mathbb P}} ( |U_n|< w \ell ) ^{1/w}$.
On the other hand we have
\begin{equation}
\begin{split} {\ensuremath{\mathbb P}} ( |U_n| < w\ell) & \leq {\ensuremath{\mathbb P}} ( |U_n| < w\ell\,,\, B(m) \leftrightarrow \infty \text{ for } \textcolor{black}{{\ensuremath{\mathbb G}} _-} )\\
&+ {\ensuremath{\mathbb P}} (B(m)\not \leftrightarrow \infty \text{ for } \textcolor{black}{{\ensuremath{\mathbb G}} _-})
\end{split}
\end{equation}
and by Lemma \ref{analogo3} the first term in the r.h.s. goes to zero as $n \to \infty$, thus implying the thesis.
\end{proof}
We can finally give the proof of Proposition \ref{cinquina}.
\begin{proof}[Proof of Proposition \ref{cinquina}]
By Lemma \ref{john} $\textcolor{black}{{\ensuremath{\mathbb G}} _-}$ percolates ${\ensuremath{\mathbb P}} $--a.s., hence we can fix an integer $m>2$ such that
\begin{equation} \label{bracciano}
{\ensuremath{\mathbb P}} ( B(m) \not \leftrightarrow \infty \text{ for } \textcolor{black}{{\ensuremath{\mathbb G}} _-} )< (\eta/2)^w\,, \qquad w:=d2^d\,.
\end{equation}
Then, by Lemma \ref{analogo4},
for any $\ell \in {\ensuremath{\mathbb N}} $ we have
\begin{equation}\label{crimine} \liminf _{n \to \infty}{\ensuremath{\mathbb P}} ( |V_n | \geq \ell ) \geq 1- {\ensuremath{\mathbb P}} (B(m)\not \leftrightarrow \infty \text{ for } \textcolor{black}{{\ensuremath{\mathbb G}} _-} )^{1/w} > 1-\eta/2\,.
\end{equation}
We set $\rho := {\ensuremath{\mathbb P}} ( B(m) \text{ is a seed}) \in (0,1)$ and fix an integer $M$ large enough that $(1-\rho)^M<\eta/2$.
We set $\ell:= (2m)^{d-1} 3^{d-1}M\e^{-d}$ and, by \eqref{crimine}, we can fix $n$
large enough that ${\ensuremath{\mathbb P}} ( |V_n | \geq \ell ) >1-\eta/2$, $2m<n$ and $2m|n$.
\textcolor{black}{The main idea behind the proof is the following: by the above choice of constants and since points in $\e \bbZ^d$ have distance at least $\e$, points in $V_n\subset T(n)$ must be enough spread that with high probability some point $x\in V_n$ is in the proximity of a seed $S$ contained in the slice $\{ z\in \e \bbZ^d\,:\, n+\e\leq z_1 \leq n+\e+2m\}$. Then, since $x $ satisfies \eqref{rotta}, we will show that $x$ must be adjacent inside ${\ensuremath{\mathbb G}} $ to the seed $S$ and hence $x\in K(m,n)$. Using that
$B(m) \leftrightarrow x$ in $ B(n)$ for $ {\ensuremath{\mathbb G}} _-\subset {\ensuremath{\mathbb G}} $, we will then conclude that $B(m) \leftrightarrow K (m,n)$ in $B(n)$ for ${\ensuremath{\mathbb G}} $. }
\textcolor{black}{Let us implement the above scheme}.
Since $2m|n$ we can partition $[0,n]^{d-1}$ in non--overlapping $(d-1)$--dimensional closed boxes $D_i^*$, $i \in \ensuremath{\mathcal I}$, of side length $2m$ (by ``non--overlapping'' we mean that the interior parts are disjoint). We set $D_i:= D_i^*\cap \e \bbZ^d $. Note that
$T(n) \subset \cup _{i \in \ensuremath{\mathcal I}} (n-1,n]\times D_i$ and $T(m,n) = \cup _{i \in \ensuremath{\mathcal I}} ([n+\e, n+\e+2m]\cap \e {\ensuremath{\mathbb Z}} ) \times D_i$.
By construction, any set $(n-1,n]\times D_i$ contains at most $(2m)^{d-1} \e^{-d}$ points $x\in T(n)$. Since $\ell= (2m)^{d-1} 3^{d-1}M\e^{-d}$, the event $\{|V_n | \geq \ell\}$ implies that there exists $\ensuremath{\mathcal I}_*\subset \ensuremath{\mathcal I}$ with $|\ensuremath{\mathcal I}_*| =3^{d-1}M$ fulfilling the following property:
for any $k\in \ensuremath{\mathcal I}_*$ there exists $x\in V_n$ with $x\in(n-1,n]\times D_k$. We can choose univocally $\ensuremath{\mathcal I}_*$ by defining it as the set of the first (w.r.t. the lexicographic order) $M$ indexes $k\in \ensuremath{\mathcal I} $ satisfying the above property. We now thin $\ensuremath{\mathcal I}_*$ since we want to deal with disjoint sets $D_k$'s. To this aim we observe that each $D_k$ can intersect at most $3^{d-1}-1$ other sets of the form $D_{k'}$. Hence, there must exists $\ensuremath{\mathcal I}_\natural \subset \ensuremath{\mathcal I}_*$ such that $D_k\cap D_{k'}=\emptyset $ for any $k\not = k'$ in $\ensuremath{\mathcal I}_\natural$ and such that $|\ensuremath{\mathcal I}_\natural| = M$ (again $\ensuremath{\mathcal I}_\natural$ can be fixed deterministically by using the lexicographic order). We introduce the events
\begin{equation}\label{schiavo} G_k:=\{ ([n+\e,n+\e+ 2m] \cap \e {\ensuremath{\mathbb Z}} ) \times D_k \text{ is a seed} \}\,.\end{equation}
We claim that
\begin{equation}\label{cedric}
{\ensuremath{\mathbb P}} \left(\, \{ |V_n| \geq \ell \} \cap (\cup_{ k \in \ensuremath{\mathcal I}_\natural} G_k) \,\right) \geq 1-\eta\,.
\end{equation}
\textcolor{black}{Before proving our claim we show that the event in \eqref{cedric} implies the event in \eqref{maggiolino}, thus allowing to conclude. Hence,
let} us now suppose that $ |V_n | \geq \ell $ and that the event $ G_k $ takes place for some $k \in \ensuremath{\mathcal I}_\natural$. We claim that necessarily
$B(m) \leftrightarrow K (m,n)$ in $B(n)$ for ${\ensuremath{\mathbb G}} $. Note that the above claim and \eqref{cedric} lead to \eqref{maggiolino}.
We prove our claim. As discussed before \eqref{schiavo}, since $k\in \ensuremath{\mathcal I}_\natural$ there exists $x\in V_n\cap( (n-1,n]\times D_k)$.
Let $S $ be the seed $([n+\e,n+\e+ 2m] \cap \e {\ensuremath{\mathbb Z}} ) \times D_k $.
By definition of $V_n$, \textcolor{black}{$ d\bigl( x, B(n)^c \bigr) \leq h_*(A_x) -3\a$} and $B(m) \leftrightarrow x$ in $ B(n)$ for $ \textcolor{black}{{\ensuremath{\mathbb G}} _-}$.
Call $x'$ the point in $\partial B(n)$
such that $|x-x'| = d\bigl(x, B(n)^c\bigr)$. Note that $x'\in \{n\}\times D_k $ as $V_n \subset T(n)$. Let $y:=x'+\e e_1$. Then $y\in S$ and therefore \textcolor{black}{$A_y \in U_*=U_*(\a/2)$} (as $S$ is a seed) and $|x'-y|=\e \leq \a/100$ \textcolor{black}{(cf. Definition \ref{vinello})}.
Then, using \eqref{indigestione} with $\d=\a/2$, the symmetry of $h$ and that $A_y \in U_*$, we get $h_*(A_x)=\sup _{a \in \D} h(A_x,a ) \leq h (A_x,A_y)+\a/2$. Hence, we obtain
\begin{multline}
|x-y| \leq |x-x'| +|x'-y|
\leq d\bigl(x, B(n)^c\bigr)+ \a/100 \leq h_*(A_x)-3\a+ \a/100 \\
\leq h(A_x,A_y)+ \a/2 -3\a+\a/100 \leq h(A_x,A_y)-2\a\,.
\end{multline}
We have therefore shown that $B(m) \leftrightarrow x$ in $ B(n)$ for $ \textcolor{black}{{\ensuremath{\mathbb G}} _-}$ for some $x\in T(n)$ with $\{x,y\}\in {\ensuremath{\mathbb E}} $ for some $y \in S$ \textcolor{black}{(cf. Definition \ref{vichinghi})}. As a consequence, $x\in K(m,n)$. Since $ \textcolor{black}{{\ensuremath{\mathbb G}} _-} \subset {\ensuremath{\mathbb G}} $, we get that $B(m) \leftrightarrow K (m,n)$ in $B(n)$ for $ {\ensuremath{\mathbb G}} $.
\textcolor{black}{It remains now to prove \eqref{cedric}}.
To this aim we call $\ensuremath{\mathcal F}_n$ the $\s$--algebra
generated by the r.v.'s $A_z$ with $z\in B(n)$. We observe that the event $\{ |V_n | \geq \ell \}$ belongs to $\ensuremath{\mathcal F}_n$, the set $\ensuremath{\mathcal I}_\natural $ is $\ensuremath{\mathcal F}_n$--measurable and w.r.t. ${\ensuremath{\mathbb P}} (\cdot| \ensuremath{\mathcal F}_n)$ the events $\{ G_k:k \in \ensuremath{\mathcal I}_\natural \}$ are independent (recall that $D_k\cap D_{k'}=\emptyset $ for any $k\not = k'$ in $\ensuremath{\mathcal I}_\natural$)
and each $G_k$ has probability $\rho:= {\ensuremath{\mathbb P}} ( B(m) \text{ is a seed})$. Hence, ${\ensuremath{\mathbb P}} $--a.s. on the event $\{ |V_n | \geq \ell\}$ we can bound
\begin{equation}\label{curdo}
{\ensuremath{\mathbb P}} ( \cup_{ k \in \ensuremath{\mathcal I}_\natural} G_k \,|\, \ensuremath{\mathcal F}_n) \geq 1-(1-\rho)^M> 1-\eta/2\,. \end{equation} Note that the last bound follows from our choice of $M$.
Since, by our choice of $n$, ${\ensuremath{\mathbb P}} ( |V_n | \geq \ell ) >1-\eta/2$, we conclude that
the l.h.s. of \eqref{cedric} is lower bounded by $(1-\eta/2)^2>1-\eta$. This concludes the proof of \eqref{cedric}.
\end{proof}
\section{Proof of Lemma \ref{pierpilori}}\label{trieste65}
\textcolor{black}{Note that $ \g:= {\ensuremath{\mathbb P}} (T^{(j)}_0\in U_*)>0$ due to Assumption (A4) and Definition \ref{cavallo}}.
We can fix a positive constant $c(d)$ such that the ball $\{y \in {\ensuremath{\mathbb R}} ^d\,:\, |y| \leq 2\}$ contains at most $c(d) \e^{-d}$ points of $\e \bbZ^d$.
We then choose $t$ large enough that $
(1- \g^2) ^{(t \e^d /c(d)) -1 }
\leq \e'/2$. Afterwards
we choose $\eta>0$
small enough so that $ (1-p) ^{-t} \eta \leq \e'/2 $, where
\begin{equation} \label{pippi}
p:={\ensuremath{\mathbb P}} (\textcolor{black}{A_x\in {\ensuremath{\mathbb R}} })=1-\exp\{- \l_* \e^d\}<1\,.
\end{equation}
Then we take $m=m( \eta)$ and $n=n( \eta)$ as in Proposition \ref{cinquina}. In particular, \eqref{maggiolino} holds and moreover
\begin{equation}\label{danza}
[1- (1-p) ^{-t} \eta ]\, [ 1- (1- \g^2 ) ^{(t \e^d /c(d)) -1} ]\geq (1-\e'/2)^2 >1 -\e'\,.
\end{equation}
\begin{Remark}\label{caffeina}
As $\eta \leq \e'/2$, from \eqref{maggiolino} we get that
\begin{equation} \label{fenec}
{\ensuremath{\mathbb P}} \bigl( B(m) \leftrightarrow K (m,n) \text{ in $B(n)$ for } {\ensuremath{\mathbb G}} \bigr) > 1-\e'\,.
\end{equation}
This will be used in other sections.
\end{Remark}
\begin{Lemma}\label{falchetto}
In the same context of Lemma \ref{pierpilori}
let \begin{equation*}
\begin{split}
V_{R}:= \{x \in \partial R \cap B(n) \,:\, & \exists y\in B(n) \setminus ( R\cup \partial R )
\text{ such that } \\ &
|x-y| \leq \textcolor{black}{h_*(A_y)}-2\a \text{ and } \\
&\{y\} \leftrightarrow K(m,n) \text{ in $B(n) \setminus ( R\cup \partial R )$ for ${\ensuremath{\mathbb G}} $}\}\,.
\end{split}
\end{equation*}
Then we have (recall \eqref{pippi})
\begin{equation}\label{navetta}
{\ensuremath{\mathbb P}} \bigl( |V_{R}| > t\bigr) \geq 1- (1-p) ^{-t} \eta\,.
\end{equation}
\end{Lemma}
We postpone the proof of Lemma \ref{falchetto} to Subsection \ref{sec_moka18}. \textcolor{black}{We point out that our set $V_R$ plays the same role of the set $U$ in \cite[Lemma 2]{GM}. As detailed in the proof of Lemma \ref{falchetto}, given a path $(x_0,x_1, \dots, x_k)$ from $B(m)$ to $K(m,n)$ inside ${\ensuremath{\mathbb G}} $ with all vertexes in $B(n)$ and called $x_\ell$ the last vertex of the path inside $\partial R$, it must be $x_\ell\in V_R$, while $x_{\ell+1}$ plays the role of $y$ in the definition of $V_R$ for $x:=x_\ell$.}
\begin{Remark}\label{iacopo}
The random set $V_R$ depends only on $ A_x$ with $x\in B(n) \setminus ( R\cup \partial R )$ and $A_x$ with $x\in T(m,n)$. Indeed, to determine $K(m,n)$, one needs to know the seeds inside $T(m,n)$.
\end{Remark}
Given $x\in \partial R $ we define $x_*$ as the minimal (w.r.t. lexicographic order) point $y\in R$ such that $|x-y|\leq 1-2\a$. Note that $x_*$ exists for any $x\in V_R$ since $V_R \subset \partial R$.
Let us show that $F\subset G$, where
\[ F:=\{
\exists x \in V_R \text{ with } \textcolor{black}{T^{(k_*)} _x \in U_*\,,\; T_{x_*}^{(k_*)} \in U_*}
\}
\,.
\]
To this aim, suppose the event $F$ to be fulfilled and take $ x \in V_R$ with \textcolor{black}{$ T^{(k_*)} _x \in U_*$ and $ T_{x_*}^{(k_*)} \in U_*
$}. Since $x\in V_R$, by definition of $V_R$ there exists $ y\in B(n) \setminus ( R\cup \partial R )$
such that
$|x-y|\leq\textcolor{black}{ h_*( A_y)} -2\a$ and there exists a path $(y,z_3, z_4, \dots, z_\ell)$ inside ${\ensuremath{\mathbb G}} $ connecting $y$ to
$ K(m,n)$ with vertexes in
$B(n) \setminus ( R\cup \partial R )$. We set
$z_0:=x_*$, $z_1=x$, $z_2:=y$. Then the event $G$ is satisfied by the string $(z_0, z_1, \dots, z_\ell)$. This proves that $F\subset G$.
Since $F\subset G$ we can estimate
\begin{equation}\label{alba}
{\ensuremath{\mathbb P}} ( G\,|\ H) \geq {\ensuremath{\mathbb P}} \bigl( |V_R| > t \,,\, F \,|\, H \bigr) = \sum_{\substack{B \subset \partial R \cap B(n) :\\ |B|>t }}
{\ensuremath{\mathbb P}} \bigl( V_R =B\,,\, F _B\,|\, H \bigr) \,,
\end{equation}
where
\[ F_B:= \{
\exists x \in B \text{ with } \textcolor{black}{T_x ^{(k_*)}\in U_* \,,\; T_{x_*}^{(k_*)} \in U_*}
\}\,.
\]
The event $F_B$ is determined by the random variables $\{ T^{(k_*)}_x \}_{ x\in D}$.
In particular (cf. Remark
\ref{iacopo}) the event $\{ V_R =B\}\cap F_B$ is determined by \[
\begin{cases}
T^{(k_*)}_x & \text{ with } x\in D\,,\\
A_x & \text{ with } x\in B(n) \setminus ( R\cup \partial R ) \text{ and with } x \in T(m,n)\,.
\end{cases}
\]
Since by assumption $H$ is $\ensuremath{\mathcal F}$--measurable, and due to conditions \eqref{mare100} and \eqref{monti100}, we conclude that the event
$\{ V_R =B\}\cap F_B$ and $H$ are independent. Hence $ {\ensuremath{\mathbb P}} \bigl( V_R =B, F _B\,|\, H \bigr)= {\ensuremath{\mathbb P}} \bigl( V_R =B,F _B\bigr) $. In particular, coming back to \eqref{alba}, we have
\begin{equation}\label{olleboiccic}
{\ensuremath{\mathbb P}} ( G\,|\ H ) \geq \sum_{\substack{B \subset \partial R \cap B(n) :\\ |B|>t }}
{\ensuremath{\mathbb P}} ( V_R=B\,,\, F_B)
\,.
\end{equation}
To deal with $ {\ensuremath{\mathbb P}} ( V_R=B, F_B)$ we observe that the events $\{ V_R=B\}$ and $ F_B$ are independent
(see Remark \ref{iacopo}), hence we get
\begin{equation}\label{theverde}
{\ensuremath{\mathbb P}} ( V_R=B\,,\, F_B )=
{\ensuremath{\mathbb P}} ( V_R=B){\ensuremath{\mathbb P}} ( F_B )\,.
\end{equation}
It remains to lower bound ${\ensuremath{\mathbb P}} (F_B)$. We first show that there exists a subset $\tilde B\subset B$ such that
\begin{equation} \label{stima}
|\tilde B | \geq |B| \e^d /c(d) -1
\end{equation}
and such that all points of the form $x$ or $x_*$, with $x\in \tilde B$, are distinct. We recall that the positive constant $c(d)$ has been introduced at the beginning of \textcolor{black}{Section} \ref{trieste65}.
To build the above set $\tilde B$ we recall that $B \subset \partial R $ and that, for any $x \in B$, it holds $|x-x_*|\leq 1- 2\a$ and $x_* \in R$. As a consequence, given $x,x'\in B$, $x_* $ and $x'_*$ are distinct if $|x-x'| \geq 2 $ and moreover any point of the form $x_*$ with $x\in B$ cannot coincide with a point in $B$. Hence it is enough to exhibit a subset $\tilde B \subset B$ satisfying \eqref{stima} and such that all points in $\tilde B$ have reciprocal distance at least $2$. We know that the ball ${\ensuremath{\mathbb B}} $ of radius $2$ contains at most $c(d) \e^{-d}$ points of $\e \bbZ^d$. The set $\tilde B$ is then built as follows: choose a point $a_1 $ in $B_1:=B$ and define $B_2:= B_1\setminus (a_1+{\ensuremath{\mathbb B}} )$, then choose a point $a_2 \in B_2$ and define $B_3:= B_2\setminus (a_2+{\ensuremath{\mathbb B}} )$ and so on until possible (each $a_k$ can be chosen as the minimal point w.r.t. the lexicographic order). We call $\tilde B:=\{a_1, a_2, \dots, a_s\}$ the set of chosen points. Since $|B_k| \geq |B| - (k-1) c(d)\e^{-d}$, we get that $s=|\tilde B|$ is bounded from below by the maximal integer $k$ such that $|B| >(k-1) c(d)\e^{-d}$, i.e. $\lfloor |B| \e^d/c(d)\rfloor >k-1$.
Hence, $s=|\tilde B|\geq \lfloor |B| \e^d/c(d)\rfloor $. By the above observations, $\tilde B$ fulfills the desired properties.
Using $\tilde B$ and independence, we have
\begin{equation}\label{agg}
\begin{split}
{\ensuremath{\mathbb P}} (F_B)&=1-{\ensuremath{\mathbb P}} (\cap_{x\in B}\{ \textcolor{black}{T_x^{(k_*)}\in U_*,\,T_{x_*}^{(k_*)}\in U_*}\}^c) \\
& \geq 1-{\ensuremath{\mathbb P}} (\cap_{x\in \tilde B}\{\textcolor{black}{
T_x^{(k_*)}\in U_* ,\,T_{x_*}^{(k_*)}\in U_*}\}^c)
\\ &=1-\prod_{x\in \tilde B}(1-{\ensuremath{\mathbb P}} (\textcolor{black}{T_x^{(k_*)}\in U_*}){\ensuremath{\mathbb P}} (\textcolor{black}{T_{x_*}^{(k_*)}\in U_*}))
\\&=1-(1-\g^2)^{|\tilde B|}\,.
\end{split}
\end{equation}
As a byproduct of \eqref{olleboiccic},
\eqref{theverde}, \eqref{stima} and \eqref{agg} and finally using \eqref{navetta} in Lemma \ref{falchetto} we get
\begin{equation}
\begin{split}
{\ensuremath{\mathbb P}} ( G\,|\ H ) & \geq \sum_{\substack{B \subset \partial R \cap B(n) :\\ |B|>t }} {\ensuremath{\mathbb P}} ( V_R=B) \left(1-(1- \g^2) ^{|\tilde B|}\right)
\\
& \geq \left(1-(1- \g^2) ^{(t \e^d /c(d) ) -1 }\right) \sum_{\substack{B \subset \partial R \cap B(n) :\\ |B|>t }} {\ensuremath{\mathbb P}} ( V_R=B)
\\
&= \left(1-(1- \g^2) ^{(t \e^d /c(d)) -1 }\right) {\ensuremath{\mathbb P}} ( |V_R|>t) \\
&\geq \left[ 1- (1- \g^2 ) ^{(t \e^d /c(d) )-1} \right]\, \left [ 1- (1-p) ^{-t} \eta
\right] \,.
\end{split}
\end{equation}
Finally, using \eqref{danza} we conclude the proof of Lemma \ref{pierpilori}.
\subsection{Proof of Lemma \ref{falchetto}}\label{sec_moka18}
Suppose that $B(m) \leftrightarrow K (m,n)$ in $B(n)$ for $ {\ensuremath{\mathbb G}} $. Take a path $(x_0,x_1, \dots, x_k)$ from $B(m)$ to $ K (m,n)$ inside $ {\ensuremath{\mathbb G}} $ with all vertexes $x_i$ in $B(n)$.
Recall that $K (m,n) \subset T(n)$ and $R\cup \partial R $ is disjoint from $T(n)$ by \eqref{mare100}. In particular, $R\cup \partial R $ is disjoint from $K (m,n)$.
Since $B(m)\subset R$, the path starts at $R$. Let $x_r$ be the last point of the path contained in $R$. Since $R$ is disjoint from $K (m,n)$ and $x_k\in K (m,n)$, it must be $r<k$. Necessarily, $x_{r+1}\in \partial R $. Call $x_\ell$ the last point of the path contained in $\partial R$. It must be $\ell< k$ since
$\partial R$ is disjoint from $K (m,n)\ni x_k$.
We claim that $x_\ell \in V_R$ and \textcolor{black}{$A_{x_\ell} \in {\ensuremath{\mathbb R}} $}. To prove our claim we observe that the last property follows from the fact that
all points $x_0,x_1, \dots, x_k$ are in $ {\ensuremath{\mathbb V}} $. Recall that these points are also in $B(n)$. Hence $x_\ell \in \partial R \cap B(n)$. Since $\{x_\ell, x_{\ell+1}\}\in {\ensuremath{\mathbb E}} $, we have \textcolor{black}{$|x-y|\leq h_*( A_y )-2\a$} with $x:= x_{\ell}$ and $y:= x_{\ell+1}$. Finally, it remains to observe that
$(x_{\ell+1}, \dots, x_k)$ is a path connecting $x_{\ell+1}$ to $x_k \in K(m,n)$ in
$B(n) \setminus ( R\cup \partial R )$ for ${\ensuremath{\mathbb G}} $. Hence, $x_\ell \in V_R$.
We have proved that if $B(m) \leftrightarrow K (m,n)$ in $B(n)$ for $ {\ensuremath{\mathbb G}} $, then $V_R$ contains at least a vertex of ${\ensuremath{\mathbb V}} $. As a byproduct with
\eqref{maggiolino} \textcolor{black}{we} therefore have
\begin{equation}\label{lalla1}
\eta> {\ensuremath{\mathbb P}} \bigl( B(m) \not \leftrightarrow K (m,n) \text{ in $B(n)$ for } {\ensuremath{\mathbb G}} \bigr) \geq {\ensuremath{\mathbb P}} ( V_R\cap {\ensuremath{\mathbb V}} =\emptyset)
\,\textcolor{black}{.}
\end{equation}
On the other hand, we can bound
\begin{equation} \label{lalla2}
{\ensuremath{\mathbb P}} ( V_R\cap {\ensuremath{\mathbb V}} =\emptyset ) \geq {\ensuremath{\mathbb P}} ( V_R\cap {\ensuremath{\mathbb V}} =\emptyset\,,\, |V_R| \leq t)\,.
\end{equation}
Note that $V_R$ and $(A_x)_{x\in \partial R}$ are independent (see Remark \ref{iacopo}).
Hence
\begin{equation}\label{lalla3}
\begin{split}
& {\ensuremath{\mathbb P}} ( V_R\cap {\ensuremath{\mathbb V}} =\emptyset \,,\, |V_R| \leq t)= \sum _{
\substack{
B\subset \partial R\,:\,\\
|B|\leq t}} {\ensuremath{\mathbb P}} ( V_R=B\,,\, \textcolor{black}{A_x\not\in {\ensuremath{\mathbb R}} } \; \forall x\in B)\\
& =\sum _{
\substack{
B\subset \partial R\,:\,\\
|B|\leq t}} {\ensuremath{\mathbb P}} ( V_R=B) {\ensuremath{\mathbb P}} ( \textcolor{black}{A_x \not\in {\ensuremath{\mathbb R}} } \; \forall x\in B)\\
& =\sum _{
\substack{
B\subset \partial R\,:\,\\
|B|\leq t}} {\ensuremath{\mathbb P}} ( V_R=B) (1-p)^{|B|}
\\
&\geq \sum _{
\substack{
B\subset \partial R\,:\,\\
|B|\leq t}} {\ensuremath{\mathbb P}} ( V_R=B) (1-p)^{t}={\ensuremath{\mathbb P}} ( |V_R|\leq t ) (1-p)^{t}\,.
\end{split}
\end{equation}
By combining \eqref{lalla1}, \eqref{lalla2} and \eqref{lalla3} we get that $\eta \geq {\ensuremath{\mathbb P}} ( |V_R|\leq t ) (1-p)^{t}$, which is equivalent to \eqref{navetta}.
\section{Proof of Lemma \ref{ciak2011} and Lemma \ref{piccolino}}\label{patroclo}
\subsection{Proof of Lemma \ref{ciak2011}}
Item (i) is trivial and Item (iii) follows from Items (i) and (ii). Let us assume \eqref{chip} and prove Item (ii). We claim that $\{E=\hat E\}\cap \{F=\hat F\}$ equals $\{E=\hat E\}\cap W$, where $W$ is the event that
\begin{itemize}
\item[(a)] for any $z\in \hat F$ there are points $z_2,\cdots,z_{k-1},z_k=z$ in $ \hat F$
such that \textcolor{black}{$A_{z_i}\in {\ensuremath{\mathbb R}} $ for $i=2,\dots,k$}, \textcolor{black}{$|z_i-z_{i+1}| \leq h (A_{z_i},A_{z_{i+1}}) -2\a$} for $i=2,\dots, k-1$ and such that
\textcolor{black}{$|z_1-z_2|\leq h_*(A_{z_2})-2\a$} for some $z_1\in \hat E $;
\item[(b)] if $z \in B'\setminus (C\cup\partial C) $ is such that
$\exists z'\in \hat E$ with \textcolor{black}{$|z-z'|\leq h_*(A_{z})-2\a$}, then $z\in \hat F$;
\item[(c)] for any $z\in\bigl( B'\setminus (C\cup\partial C)\bigr) \cap \partial \hat F$
there is no $z' \in \hat F$ such that \textcolor{black}{$A_{z'}\in {\ensuremath{\mathbb R}} $ and $|z-z'| \leq h(A_z,A_{z'})-2\a$}.
\end{itemize} Before proving our claim, we observe that it allows to conclude the proof of the lemma.
\textcolor{black}{Indeed, due to the definition of $E$,
the event $\{E=\hat E\}$ belongs to the $\s$--algebra in Item (ii) of the lemma. Moreover, }
as the point $z$ appearing in Item (b) must be in $\partial \hat E$, the event $\{E=\hat E\}\cap W$ belongs to the \textcolor{black}{same $\s$--algebra} due to the explicit description \textcolor{black}{of $W$} given above.
It remains to derive our claim. Due to Item (a), the event $ \{E=\hat E\}\cap W $ implies that $\{E=\hat E\}\cap \{F\supset \hat F\}$. On the other hand, suppose that the event $ \{E=\hat E\}\cap W$ takes place and let $z\in F$. By Definition \ref{def_triade} there exists a path $(z_2,\dots,z_k)$ inside ${\ensuremath{\mathbb G}} $ where $z_k=z$, all points $z_2,\cdots,z_k$ are in $B'\setminus (C\cup\partial C)$ and \textcolor{black}{$|z_1-z_2|\leq h_*(A_{z_2})-2\a$} for some $z_1\in \hat E$.
By Item (b), $z_2 \in \hat F$. Let $j$ be the maximal index in $\{2,3,\dots, k\}$ such that $z_2,z_3, \dots, z_j \in \hat F$. Suppose that $j<k$. As $\{z_j,z_{j+1}\}\in {\ensuremath{\mathbb E}} $, we get that $z_{j+1}\in \partial \hat F$. Since $z_j \in \hat F$, $z_{j+1}\in \bigl( B'\setminus (C\cup\partial C)\bigr) \cap \partial \hat F$ and
$\{z_j,z_{j+1}\}\in {\ensuremath{\mathbb E}} $, we get a contradiction with Item
(c). Then, it must be $j=k$, thus implying that $z=z_j$ and therefore $z\in \hat F$.
Up to now, we have proved that $ \{E=\hat E\}\cap W \subset \{E=\hat E\}\cap \{F= \hat F\}$.
We observe that, given $z, z_1,z_2,\dots, z_k$ as in Definition \ref{def_triade}, it must be $z_2,\dots, z_k\in F$. This observation and the above Items (a), (b), (c) imply the opposite inclusion
$\{E=\hat E\}\cap \{F= \hat F\} \subset \{E=\hat E\}\cap W $.
\subsection{Proof of Lemma \ref{piccolino}}
Recall Definitions \textcolor{black}{\ref{vichinghi_+} and} \ref{vichinghi}.
If $z\in E$, then \textcolor{black}{$T_{z}^{(i)}\in{\ensuremath{\mathbb R}} $} and therefore $z\in \textcolor{black}{{\ensuremath{\mathbb V}} _+}$. If $z\in F$, then $z\in {\ensuremath{\mathbb V}} $ (by definition of $F$) and therefore $z\in \textcolor{black}{{\ensuremath{\mathbb V}} _+}$. This implies that $E, F\subset \textcolor{black}{{\ensuremath{\mathbb V}} _+}$, hence $C'\subset \textcolor{black}{{\ensuremath{\mathbb V}} _+}$.
Since $C$ is connected in $ \textcolor{black}{{\ensuremath{\mathbb G}} _+}$ and since ${\ensuremath{\mathbb G}} \subset \textcolor{black}{{\ensuremath{\mathbb G}} _+}$ (in particular the string $(z_2, \dots, z_k)$ appearing in the definition of $F$ is a path in $ \textcolor{black}{{\ensuremath{\mathbb G}} _+}$), to prove the connectivity of $C'$ in $ \textcolor{black}{{\ensuremath{\mathbb G}} _+}$ it is enough to show the following:
\begin{itemize}
\item[(i)]
if $z_0,z_1 \in \textcolor{black}{ {\ensuremath{\mathbb V}} _+}$ satisfy $ \textcolor{black}{T^{(i)} _{z_0}\in U_*}$, $ \textcolor{black}{ T^{(i)} _{z_1}\in U_*}$ and $|z_0-z_1|\leq 1-2\a$, then $\{z_0,z_1\}\in \textcolor{black}{{\ensuremath{\mathbb E}} _+}$;
\item[(ii)]
if $z_1,z_2 \in \textcolor{black}{{\ensuremath{\mathbb V}} _+}$ satisfy $ \textcolor{black}{T^{(i)} _{z_1}\in U_*}$, \textcolor{black}{$A_{z_2}\in{\ensuremath{\mathbb R}} $} and $\textcolor{black}{|z_1-z_2|\leq h_*( A_{z_2}) -2\a}$, then $\{z_1,z_2\}\in \textcolor{black}{{\ensuremath{\mathbb E}} _+}$.
\end{itemize}
Using the assumptions of Item (i) we get \textcolor{black}{(recall \eqref{mango} and Assumption (A5))}
\begin{equation}\label{lorenzo18}
\textcolor{black}{ |z_1-z_0| \leq 1-2\a \leq h( T_{z_1}^{ (i)} ,T_{z_0}^{ (i)})-\a\leq h( A^{\rm au}_{z_1} ,A ^{\rm au}_{z_0} )-\a\,.
}
\end{equation}
\textcolor{black}{The above estimate implies that $\{z_0,z_1\}\in {\ensuremath{\mathbb E}} _+$}.
Using the assumptions of Item (ii) we get
\begin{equation}\label{pierpaolo18}
|z_1-z_2| \leq \textcolor{black}{h_*( A_{z_2}) -2\a \leq h( T^{(i)}_{z_1},A_{z_2}) +\a/2-2\a \leq h( A^{\rm au}_{z_1} ,A ^{\rm au}_{z_2} )-\a }\,.
\end{equation}
\textcolor{black}{Note that in the second bound we have used that $T^{(i)}_{z_1}\in U_*=U_*(\a/2)$ (see Assumption (A4)), while in the third bound we have used Assumption (A5). Eq.~\eqref{pierpaolo18} implies that $\{z_1,z_2\}\in {\ensuremath{\mathbb E}} _+$.}
\section{Proof of Propositions \ref{prop_occ_origin} and \ref{prop_occ_e1}}\label{puffo1}
By iteratively applying Lemma \ref{piccolino} and using Remark \ref{gomitolo1} as starting point, we get that \textcolor{black}{$C_2, \dots, C_{5}$} are connected subsets in $\textcolor{black}{{\ensuremath{\mathbb G}} _+}$, if the associated success-events are satisfied.
The lower bounds ${\ensuremath{\mathbb P}} ( 0\text{ is occupied}\, |\,S_0) \geq 1-4 \e'$ and \eqref{soglia2} follow from the inequalities
\begin{equation}\label{catenina}
{\ensuremath{\mathbb P}} ( S_{i+1}| S_0, S_1, \cdots, S_i)
\geq
\begin{cases}
1- 4 \e' & \text{ for }i=0\,,\\
1-\e' & \text{ for } i\in
\textcolor{black}{\{1,2,3\}} \,,\\
1- 3 \e '& \text{ for }\textcolor{black}{i=4}\,,
\end{cases}
\end{equation}
by applying the chain rule and the Bernoulli's inequality (i.e. $(1-\d)^k \geq 1- \d k$ for all $k\in {\ensuremath{\mathbb N}} $ and $\d\in [0,1]$).
\textcolor{black}{Apart from} the case $i=0$, the proof of \eqref{catenina} can be obtained by applying Lemma \ref{pierpilori}. Below we will treat in detail the cases $\textcolor{black}{i=0,1}$. For the other cases we will
give some comments, and show the validity of conditions \eqref{mare100} and \eqref{monti100} in Lemma \ref{pierpilori}.
In what follows we will introduce points $\tilde b_1, \tilde b_2,\dots$. We stress that the subindex $k$ in $\tilde b_k$ does not refer to the $k$-th coordinate. We write $(\tilde b_k)_a$
for the $a$--th coordinate of $\tilde b_k$.
\subsection{Proof of \eqref{catenina} with $i=0$} \label{svezia
We want to show that
${\ensuremath{\mathbb P}} (S_1|S_0) \geq 1-4 \e'$. Since $S_0$ and $S_1$ are increasing events w.r.t. $\preceq$, by the FKG inequality (see Section \ref{scremato}) we have ${\ensuremath{\mathbb P}} (S_1|S_0) \geq {\ensuremath{\mathbb P}} (S_1)$. To show that ${\ensuremath{\mathbb P}} (S_1) \geq 1-4 \e'$, we note that the event $W_j:=\{B(m) \leftrightarrow K ^{(j)} (m,n) \text{ in $B(n)$ for } {\ensuremath{\mathbb G}} \}$ implies that $C_1$ contains a point of $ K^{(j)} (m,n) $. Hence,
$\cap_{j=1}^{4} W_j\subset S_1$ and therefore (see Remark \ref{caffeina} \textcolor{black}{which is based on Proposition \ref{cinquina}})
$ {\ensuremath{\mathbb P}} (S_1^c) \leq {\ensuremath{\mathbb P}} \left( \cup_{j=1}^{ 4 } W_j^c\right) \leq 4 \e'$.
\subsection{Proof of \eqref{catenina} with $i=1$}
\label{venezia} We want to show that
${\ensuremath{\mathbb P}} (S_2|S_0, S_1) \geq 1- \e'$.
\begin{Lemma}\label{barrio1}
Given $B(m)\subset R_1\subset B_0'$, the event $S_0\cap \{C_1=R_1\}$ is determined by the random variables $\{A_x\}_{x\in R_1\cup \partial R_1}$.
\end{Lemma}
\begin{proof}
The claim is trivially true for the event $S_0$. It is therefore enough to show that, if $S_0$ takes place, then the event $\{C_1=R_1\}$ is equivalent to the following: (i)
for any $x\in R_1$ there is a path from $x$ to $B(m)$ inside $R_1$ for ${\ensuremath{\mathbb G}} $ and (ii) any $x\in \partial R_1\cap B_0'$ is not \textcolor{black}{adjacent} to $R_1$ in ${\ensuremath{\mathbb G}} $, i.e. there is no $y\in R_1$ such that $\{x,y\}\in {\ensuremath{\mathbb E}} $. Trivially the event $\{C_1=R_1\} $ implies (i) and (ii). On the other hand, let us suppose that (i) and (ii) are satisfied, in addition to $S_0$. Then (i) implies that $R_1\subset C_1$. Take, by contradiction, $x\in C_1\setminus R_1$. By definition of $C_1$ there exists a path from $x$ to $B(m) $ in $B_0'$ for ${\ensuremath{\mathbb G}} $. Since $x\not \in R_1$ and $B(m) \subset R_1$, there exists a last point $x'$ in $R_1^c$ visited by the path. Since the path ends in $B(m) \subset R_1$, after $x'$ the path visits another point $y$ which must belong to $R_1$. Hence we have $\{x',y\}\in {\ensuremath{\mathbb E}} $ (and therefore $x'\in \partial R_1 \cap B_0'$) and $y\in R_1$, thus contradicting (ii).
\end{proof}
We proceed with the proof that ${\ensuremath{\mathbb P}} (S_2|S_0, S_1)\geq 1-\e'$ by applying Lemma \ref{pierpilori}. Recall that $T_1(n)=T^*(n)$, $T_1(m,n)= T^*(m,n)$ and recall Definition \ref{coca?} of $K_1(m,n)$.
We can write
\begin{multline} \label{sam0}
{\ensuremath{\mathbb P}} (S_2|S_0,
S_1)\\
=\sum _{R_1, \tilde b_1} {\ensuremath{\mathbb P}} (S_2| S_0, S_1, C_1=R_1, b^{(1)}=\tilde b _1 ) {\ensuremath{\mathbb P}} ( C_1=R_1, b^{(1)}=\tilde b_1 |S_0, S_1)\,,
\end{multline}
where in the above sum $R_1 \subset \e \bbZ^d$ and $\tilde b _1 \in \e \bbZ^d$ are taken such that
${\ensuremath{\mathbb P}} ( S_0, S_1, C_1=R_1, b^{(1)}=\tilde b _1)>0$.
We now apply Lemma \ref{pierpilori} (with the origin there replaced by $\tilde b_1$) to lower bound ${\ensuremath{\mathbb P}} (S_2|H_1)$ by $1- \e'$, where \textcolor{black}{the event $H_1$ is given by $S_0\cap S_1\cap\{ C_1=R_1\}\cap\{ b^{(1)}=\tilde b _1\}$}.
\smallskip
We first check condition \eqref{mare100}. Note that
$B(m)\subset R_1\subset B_0'$ and $B(\tilde{b}_1,m)\subset R_1\cap T(m,n)$ as ${\ensuremath{\mathbb P}} (H_1)>0$. Hence we have $(R_1 \cup \partial R_1) \subset
(B_0'\cup \partial B_0')$.
We point out that, given $x\in B_0'\cup \partial B_0'$, it must be $x_1 \leq n+ \e + 2m + 1-2\a $. On the other hand, given
$x\in \tilde b_1+ \bigl( T^*(n) \cup T^*(m,n)\bigr)$, it must be $x_1\geq 2n+m+\e-1$.
As $2m < n$ and $2m|n$, we have $n\geq 4m$ and therefore $n>m+2$. Hence $x$ cannot belong to both sets. In particular,
we have the analogous of \eqref{mare100}, i.e.
$ B(\tilde b_1 , m) \subset R_1$
and $ \left( R_1\cup \partial R_1 \right)\cap \bigl(
\tilde{b}_1+\bigl( T^*(n) \cup T^*(m,n)\bigr)
\bigr)=\emptyset$.
Condition \eqref{monti100} is trivially satisfied by taking $\L_1(x) := \emptyset $ for all $x\in R_1 \cup \partial R_1$ and $k_*=1$.
\smallskip
We now prove that $H_1$ belongs to the $\s$--algebra $\ensuremath{\mathcal F}_1$ generated by $(A_x)_{x\in R_1\cup \partial R_1}$.
Due to Lemma \ref{barrio1}, the event $S_0\cap \{C_1=R_1\}$ is determined by $\{ A_x\}_{x\in R _1\cup \partial R_1}$. If the event $S_0\cap \{C_1=R_1\}$ takes place, then the event $S_1\cap \{ b^{(1)}=\tilde b_1\}$ becomes equivalent to the following:
(1)
$B(\tilde b_1, m)$ is a seed and, for any other seed $B(z, m)\subset R_1\cap T(m,n)$, $\tilde b_1$ is lexicographically smaller than $z$;
(2) for any $j=2,3,4$ the set $R_1$ contains a point $x\in L_j(T(n))$ \textcolor{black}{adjacent} for ${\ensuremath{\mathbb G}} $ to a seed contained in $R_1\cap L_j\bigl( T(m,n)\bigr)$.
Note that in Item (2) we have used Lemma \ref{ironman}, thus implying that if a seed $B(z,m)$ is \textcolor{black}{adjacent} for ${\ensuremath{\mathbb G}} $ to a point $x\in R_1$, then any point of $B(z,m)$ is connected for ${\ensuremath{\mathbb G}} $ to $x$, and therefore $B(z,m)\subset R_1$ as $C_1=R_1$.
The above properties $(1),(2)$ can be checked when knowing $\{A_x\}_{x\in R_1\cup \partial R_1}$. Hence, $H_1$ belongs to the $\s$--algebra $\ensuremath{\mathcal F}_1$.
\smallskip
Due to the above observations,
we can apply Lemma \ref{pierpilori} with conditional event $H_1$, $\tilde b_1 $ as new origin, $R_1$ as new set $R$, $\L_1(x):=\emptyset $ for any $x\in R_1 \cup \partial R_1$ as new function $\L(x)$
and $k_*:=1$. We get that $ {\ensuremath{\mathbb P}} ( G_1|H_1) \geq 1-\e'$, where
$G_1$ is the event corresponding to the event $G$ appearing in Lemma \ref{pierpilori} (replacing also $K(m,n)$ by $K_1(m,n)$).
To show that $ {\ensuremath{\mathbb P}} ( S_2|H_1) \geq 1-\e'$, and therefore that ${\ensuremath{\mathbb P}} (S_2|S_0, S_1)\geq 1-\e'$ by \eqref{sam0}, it is enough to show that $ G_1 \cap H _1\subset S_2\cap H_1$.
To this aim
let us suppose that $G_1 \cap H_1 $ takes place. Let (P1),...,(P8) be the properties entering in the definition of $G$ in Lemma \ref{pierpilori}, when replacing $R$, $B(n)$ and $K(m,n)$ by $R_1$, $B(\tilde b_1,n)$ and $K_1(m,n)$, respectively.
To \textcolor{black}{conclude} it is enough to show that $z_\ell\in C_2$ since $z_\ell \in K_1 (m,n) $ by (P5). Note that by $H_1$, (P1), (P2), (P6) and (P7) we have that $z_0\in C_1$ and $z_1\in E_1$, while by $H_1$, (P3), (P4) and (P8) we get that $z_2, \dots, z_\ell \in F_1$. Since $C_2:=C_1\cup E_1\cup F_1$, we have that $ z_\ell\in C_2$.
\subsection{Generalized notation}
In order to define objects once and for all, given $\textcolor{black}{1}\leq i \leq \textcolor{black}{4}$ and given sets $R_1, R_2, \dots, R_i$ and points $\tilde b_1, \tilde b_2, \dots, \tilde b_{i}$ we set
\begin{align}
& H_{i}:=
\left(\cap _{k=0} ^{ i} S_k\right)
\cap \left( \cap _{k=1} ^{i} \bigl\{ C_k = R_k \bigr\} \right)
\cap
\left( \cap_{k=1} ^{i} \bigl\{ b ^{(k)}=\tilde{b}_{k} \bigr\}\right)\,,
\label{acca}\\
& \L_i(x) :=\bigl\{ k\,:\, 1\leq k \leq i-1 \,, \; x\in B( \tilde b_k ,n+1) \bigr\}\,, \quad \forall x\in R_i \cup \partial R_i \,, \label{lambada}\\
& \ensuremath{\mathcal F}_{i}:= \s\left( \{A_x\}_{x\in R_{i}\cup\partial R_{i}} , \{T_x^{(k)}\}_{ x\in R_{i}\cup \partial R_{i},\, k\in \L_{i}(x)}\right)\label{sigmund}\,.
\end{align}
Note that the above definitions cover also the objects $H_1, \L_1, \ensuremath{\mathcal F}_1$ introduced in Section \ref{venezia}.
For later use, it will be convenient to write also $H_i[R_1, \dots, R_i;\tilde b_1, \dots, \tilde b_{i}]$ (instead of $H_i$) to stress the dependence on $R_1, \dots, R_i,\tilde b_1, \dots, \tilde b_{i}$.
\subsection{Proof of \eqref{catenina} with \textcolor{black}{$i= 2,3$}}\label{sole87}
The proof follows the main arguments presented for the case \textcolor{black}{$i=1$}. One has to condition similarly to \textcolor{black}{\eqref{sam0}} and afterwards apply Lemma \ref{pierpilori}
with $k_*:=i$ and $B(m)$, $B(n)$, $T(n)$, $T(n,m)$, $K (m,n)$, $R$, $H$ and $\L(x)$ replaced by
$B(\tilde b_i,m)$, $B(\tilde b_i,n)$, $ \tilde b_i +T_i(n)$, $\tilde b_i+T_i(m,n)$, $K_i (m,n)$, $R_i$, $H_i$ and $\L_i(x)$, respectively. The fact that $H_i\in \ensuremath{\mathcal F}_i$ can be obtained as for the case \textcolor{black}{$i=1$} by writing $H_i= H_{i-1}\cap S_i \cap \{C_i=R_i\}
\cap
\{ b^{(i)}= \tilde b _i \}$ and using the iterative result that $H_{i-1}\in \ensuremath{\mathcal F}_{i-1}$ together with Lemma \ref{ciak2011}.
To check \eqref{mare100} is straightforward but cumbersome and we refer to Appendix \ref{natale45} for the details.
\subsection{Proof of \eqref{catenina} with \textcolor{black}{$i=4$}}\label{sole88}
For $j=1,2,3$, we define the event $W_j:=\{\text{\textcolor{black}{$C_5$} contains at least one vertex inside $K^{(\textcolor{black}{4}+j)}$}\}$. Then $\textcolor{black}{S_5}= \cap_{j=1}^3 W_j$ and
\begin{equation}
\label{eq1}
{\ensuremath{\mathbb P}} (\textcolor{black}{S_5}\,|\, \textcolor{black}{S_0,S_1,\cdots,S_{4}}) \geq 1-\sum _{j=1}^3 {\ensuremath{\mathbb P}} (W_j^c\,|\, \textcolor{black}{S_0,S_1,\cdots,S_{4}})\,.
\end{equation}
Hence, we only need to show that ${\ensuremath{\mathbb P}} (W_j\,|\, \textcolor{black}{S_0,S_1,\cdots,S_{4}})\geq 1- \e'$. We use Lemma \ref{pierpilori} and the same strategy used in the previous steps.
Recall \eqref{acca}, \eqref{lambada}, \eqref{sigmund}.
We have to lower bound by $1-\e'$ the conditional probability ${\ensuremath{\mathbb P}} (W_j\,|\,H_{\textcolor{black}{4}})$ when ${\ensuremath{\mathbb P}} (H_{\textcolor{black}{4}})>0$.
To this aim we apply Lemma \ref{pierpilori} with $R:=R_{\textcolor{black}{4}}$, $\L(x):=\L_{\textcolor{black}{4}}(x)$ and with $B(n)$, $T(n)$, $T(m,n)$, $k_*$ replaced by $B(\tilde b_{\textcolor{black}{4}},n)$, $\tilde b_{\textcolor{black}{4}}+\hat T_j(n)$, $\tilde b_{\textcolor{black}{4}}+ \hat T_j(m,n)$ and $\textcolor{black}{4}$, respectively. The validity of \eqref{monti100} is trivial. To check \eqref{mare100} is straightforward but cumbersome and we refer to Appendix \ref{natale46} for the details.
|
1,108,101,565,679 | arxiv | \section{Introduction}
Point processes serve as canonical models for dealing with support estimation. Poisson point processes (PPP) appear in the continuous limit of nonparametric regression models with one-sided or irregular error variables, cf. Meister and Rei\ss\ \cite{Meister13}, and thus form counterparts of the Gaussian white noise (GWN) model. In this paper we consider the observation of a PPP on $[0,1]\times\R$ with intensity function
\begin{equation}\label{EqIntensity}
\lambda_g(x,y)=n\mathbbm{1}(y\ge g(x)),\qquad x\in[0,1],y\in\R,
\end{equation}
where $g$ is an unknown support boundary curve and $n\in\N$. A prototypical regression model corresponding to this PPP model with $n$ replaced by $\alpha n$ is given by
\begin{equation}\label{EqRegression} Y_i=g(i/n)+\eps_i,\quad i=1,\ldots,n,
\end{equation}
with one-sided i.i.d. error variables $\eps_i\ge 0$, satisfying $\PP(\eps_i\le x)= \alpha x+O(x^2)$ as $x\downarrow 0$, cf. the discussion in Rei\ss\ and Selk \cite{ReissSelk14}. The important point to keep in mind is that in these support boundary models, the standard parametric rate is $n^{-1}$ due to the behaviour of extreme value statistics.
In Korostelev and Tsybakov \cite[Chapter 8]{KorTsy93} the problem of estimating functionals of a binary image boundary from noisy observations has been studied. Although the noise is regular, the Hellinger metric is an $L^1$-distance exactly as in our PPP model. In both models, the minimax rate of convergence for estimating linear functionals of the form $\langle g,\psi\rangle=\int_0^1 g(x)\psi(x)\,dx$, $\psi\in L^2([0,1])$, is $n^{-(\beta+1/2)/(\beta+1)}$ over the H\"older ball
\[
\mathcal{C}^\beta(R)=\{g:[0,1]\rightarrow\R: |g(x)-g(y)|\le R|x-y|^\beta\,\forall
x,y\in[0,1]\}
\]
with $\beta\in(0,1]$ and radius $R>0$. For the PPP model Rei\ss{} and Selk \cite{ReissSelk14}
build up a nonparametric maximum-likelihood approach and
construct an unbiased estimator achieving this rate. Besides minimax
optimality, their estimator has the striking property of being UMVU (uniformly of
minimum variance among all unbiased estimators) over $\mathcal{C}^\beta(R)$.
\begin{table}
\vspace{2mm}
\begin{center}
\begin{tabular}{|l|l|l|} \hline
Rate & PPP& GWN\\\hline
estimate $g(x)$ & $n^{-\beta/(\beta+1)}$ & $n^{-\beta/(2\beta+1)}$\\\hline
estimate $\langle g,\psi\rangle$ & $n^{-(\beta+1/2)/(\beta+1)}$ & $n^{-1/2}$\\\hline
estimate $\norm{g}_p^p$ & $n^{-(\beta+1/2)/(\beta+1)}$ & $p=2$: $n^{-4\beta/(4\beta+1)}\vee n^{-1/2}$\\\hline
estimate $\norm{g}_p$ & $n^{-(\beta+1/(2p))/(\beta+1)}$ & $p$ even: $n^{-\beta/(2\beta+1-1/p)}$\\\hline
testing & $n^{-(\beta+1/(2p))/(\beta+1)}$ & $n^{-\beta/(2\beta+1/2+(1/2-1/p)_+)}$\\
\hline
\end{tabular}
\end{center}
\vspace{2mm}
\caption{Minimax rates for regularity $\beta$ in the Poisson point process
(PPP) and Gaussian white noise (GWN) model.}\label{tab1}
\vspace{-5mm}
\end{table}
Here, we consider the problem of estimating and testing non-linear functionals of the form
\begin{equation}\label{EqFunctional}
F_{\Phi}(g)=\int_0^1\Phi(g(x))\, dx,
\end{equation}
where $\Phi:\R\to\R$ is a known weakly differentiable function with derivative $\Phi'\in L_{loc}^1(\R)$ (i.e. $\Phi(u)=\Phi(0)+\int_0^u\Phi'(v)\,dv$, $u\in\R$, holds). An important class of functionals of the form \eqref{EqFunctional} is given by $p$-th powers $\norm{g}_p^p$ of $L^p$-norms using $\Phi(u)=|u|^p$, $p\ge 1$.
We show that it is still possible to construct an unbiased estimator of $F_{\Phi}(g)$ which is UMVU over $\mathcal{C}^\beta(R)$. Moreover, under weak assumptions on $\Phi'$, we compute the minimax risk of estimation over small neighbourhoods of $g$ and show that the estimator achieves the local minimax rate of convergence $\|\Phi'\circ g\|_2n^{-(\beta+1/2)/(\beta+1)}$. For the special case of estimating $\|g\|_p^p$ and the $L^p$-norm $\|g\|_p$, we prove that the minimax rates of convergence over $\mathcal{C}^\beta(R)$ are $n^{-(\beta+1/2)/(\beta+1)}$ and $n^{-(\beta+1/(2p))/(\beta+1)}$, respectively.
Based on these results we consider the testing problem $H_0:g=g_0$ versus $H_1:g\in \{g_0+h\in \mathcal{C}^\beta(R):\|h\|_p\ge r_n\}$, where the nonparametric alternative is separated by a ball of radius $r_n>0$ in $L^p$-norm. We show that the minimax separation rate is $n^{-(\beta+1/(2p))/(\beta+1)}$ and that this rate can be achieved by a plug-in test, using a minimax optimal estimator of the $L^p$-norm of $g$.
In particular, the minimax rates of testing and estimation coincide, and they are located strictly between the parametric rate $n^{-1}$ and the rate $n^{-\beta/(\beta+1)}$, corresponding to the problem of estimating the function $g$ itself (see e.g. Jirak, Meister and Rei\ss{} \cite{Moritz14} and the references therein).
These fundamental questions have been studied extensively in the mean regression and Gaussian white noise (GWN) model. In the latter, we observe a realisation of
\[
dX(t)=g(t)\,dt+ n^{-1/2}\,dW(t),\quad t\in[0,1],
\]
where $g$ is the unknown regression function and $(W(t):t\in[0,1])$ is a standard Brownian motion. Significant differences appear. Consider, for instance, the case $\Phi(u)=|u|^p$ with $p\in\N$. For $p$ even and $\beta$ large enough, the smooth functional \eqref{EqFunctional} can be estimated with the parametric rate of convergence $n^{-1/2}$, using the method from Ibragimov, Nemirovski and Khasminski \cite{INK86} (see Table \ref{tab1} for the case $p=2$ and the monograph by Nemirovski \cite{Nem00} for more general functionals). Estimation of the $L^p$-norm has been considered by Lepski, Nemirovski and Spokoiny \cite{LepNemSpo99}. For $p$ even, the optimal rate of convergence is $n^{-\beta/(2\beta+1-1/p)}$, while for $p$ odd, the standard nonparametric rate $n^{-\beta/(2\beta+1)}$ can only be improved by $\log n$ factors. In Table~\ref{tab1} we compare these GWN estimation rates with the PPP rates. A structural difference is that for vanishing regularity $\beta\downarrow 0$ the GWN convergence rates become arbitrarily slow, while in the PPP case the rates always remain faster than $n^{-1/2}$ and $n^{-1/(2p)}$, respectively. This phenomenon will be further discussed at the beginning of Section \ref{SecEst}. More generally, the PPP rates hold universally for all $1\le p<\infty$, while the GWN rates depend on $p$ in a very delicate way, showing that $L^p$-norm estimation is to some extent a regular estimation problem in the otherwise rather irregular PPP statistical model.
\begin{figure}[t]
\includegraphics[width=0.9\textwidth,height=0.25\textheight]{PPPGWNTestRates.png}
\caption{Testing rate exponents for the Poisson point process
(PPP) and Gaussian white noise (GWN) model as a function of the regularity
$\beta$.}
\label{fig1}
\end{figure}
Further differences arise in the testing problem, which for the GWN model is the topic of the monograph by Ingster and Suslina \cite{IngsterSuslina03}. The testing problem $H_0:g=0$ versus $H_1:g\in\{h\in L^2([0,1]):\|h\|_p\ge r_n \text{ and } \|h\|_{\beta,q}\le R\}$ is considered, where $\|\cdot\|_{\beta,q}$ is a Sobolev or Besov norm with smoothness measured in $L^q$-norm. For instance, in the case $1\le p\le 2$ and $q=\infty$, the minimax separation rate is $n^{-2\beta/(4\beta+1)}$ which coincides with the minimax rate for estimating the $L^p$-norm if $p=2$ but not if $p=1$. The general minimax GWN separation rates for the case $q\ge p$ are given in the last row of Table \ref{tab1} (for the cases $1\le p \le 2$, $p\le q\le \infty$ and $2<p=q<\infty$), results for the case $q<p$ can be found in Lepski and Spokoiny \cite{LepskiSpokoiny99}. Figure~\ref{fig1} visualises the differences between the GWN and the PPP case by plotting the separation rate exponents for the range of $p\in[1,\infty)$ as a function of the regularity $\beta$. In the GWN model the rates become arbitrarily slow when $\beta$ approaches zero and they do not change for $p\in[1,2]$ ({\it elbow effect}), which is not the case in the PPP case. The absence of an elbow effect in the PPP model may be understood by a different Hellinger geometry: the Hellinger distance is given by an $L^1$-distance between the curves, while it is based on the $L^2$-distance in the GWN model.
In the next Section \ref{SecEst} we construct the
estimator, compute its mean and variance using the underlying point process
geometry and martingale arguments, and we derive the (local) minimax rates of convergence. In Sections
\ref{SecTesting} and \ref{SecEstLp}, we focus on the special case where $\Phi(u)=|u|^p$ and apply our
results to the problem of estimating the $L^p$-norm and to the corresponding hypothesis testing problem.
\section{Estimation of non-linear functionals}\label{SecEst}
\subsection{The estimator}
Let $(X_j,Y_j)_{j\ge 1}$ be the observed support points of a Poisson point process on $[0,1]\times\R$ with intensity function given by \eqref{EqIntensity}. The support boundary curve $g$ is supposed to lie in the H\"older ball $\mathcal{C}^\beta(R)$ with $\beta\in(0,1]$. The aim is to estimate the functional in \eqref{EqFunctional}. Similarly to \cite{ReissSelk14}, our estimation method can be motivated as follows. Suppose that we know a deterministic function $\bar g\in \mathcal{C}^\beta(R)$ with $\bar
g(x)\ge g(x)$ for all $x\in[0,1]$. Then the sum
\begin{equation}\label{EqSumBar}
\frac1n\sum_{j\ge 1}\Phi'(Y_j)\mathbbm{1}\big(\bar g(X_j)\ge Y_j\big)
\end{equation}
is a.s. finite, has expectation equal to
\begin{align*}
&\frac1n\int_0^1\int_{\R}\Phi'(y)\mathbbm{1}\big(\bar g(x)\ge y\big)\lambda_g(x,y)\,dydx
=\int_0^1\big(\Phi(\bar g(x))-\Phi(g(x))\big)\,dx
\end{align*}
and variance equal to
\begin{align}
&\frac1{n^2}\int_0^1\int_{\R}\Phi'(y)^2\mathbbm{1}\big(\bar g(x)\ge y \big)\lambda_g(x,y)\,dydx\nonumber\\
&=\frac1n\int_0^1\int_{\R}\Phi'(y)^2\mathbbm{1}\big(\bar g(x)\ge y\ge g(x)\big)\,dydx\label{EqVarianceDet},
\end{align}
provided the last integral is finite (see e.g. \cite[Lemma 1.1]{Kut98} or \cite[Theorem 4.4]{LastPenrose2016}). Thus,
\[ \hat F^{pseudo}_\Phi=\int_0^1\Phi(\bar g(x))\,dx-\frac1n\sum_{j\ge 1}\Phi'(Y_j)\mathbbm{1}\big(\bar g(X_j)\ge Y_j\big)
\]
forms an unbiased pseudo-estimator (relying on the knowledge of $\bar g$) of $F_{\Phi}(g)$ whose variance is given by \eqref{EqVarianceDet}. The closer $\bar g$ is to $g$, the smaller the variance. Concerning the rate results for $p$-th powers of $L^p$-norms in Table \ref{tab1} note that already the very minor knowledge of some upper bound of $g$ suffices to construct an estimator with convergence rate $n^{-1/2}$, which explains why in the PPP case even for $\beta\downarrow 0$ estimation and testing rates remain consistent.
The main idea is now to find a data-driven upper bound of $g$ which is as small as possible. A solution to this problem is given by
\begin{equation}\label{EqMLE}
\hat g^{MLE}(x)=\min_{k\ge 1}\big(Y_k+R|x-X_k|^{\beta}\big),\qquad x\in[0,1],
\end{equation}
which is the maximum-likelihood estimator over $\mathcal{C}^\beta(R)$ \cite[Section 3]{ReissSelk14}. Indeed, $\hat g^{MLE}$ is an upper bound for $g$ noting that $Y_k\ge g(X_k)$ and $g(X_k)+R\abs{x-X_k}^\beta\ge g(x)$ for all $k\ge 1$, where the latter follows from $g\in\mathcal{C}^\beta(R)$.
The idea is now that the sum
\[
\frac1n\sum_{j\ge 1}\Phi'(Y_j)\mathbbm{1}\big(\hat{g}^{MLE}(X_j)\ge Y_j\big)
\]
is a.s. finite and satisfies
\begin{align*}
&\E \Big[\frac1n\sum_{j\ge 1}\Phi'(Y_j)\mathbbm{1}\big(\hat{g}^{MLE}(X_j)\ge Y_j\big)\Big]\\
&=\frac1n\int_0^1\int_{\R}\Phi'(y)\E\big[\mathbbm{1}\big(\hat{g}^{MLE}(x)\ge y\big)\big]\lambda_g(x,y)\,dydx\\
&=\int_0^1\E\big[\Phi(\hat{g}^{MLE}(x))\big]\,dx-\int_0^1\Phi(g(x))\,dx,
\end{align*}
provided that the integral in the second line is well-defined. For the first equality observe that
\[
\mathbbm{1}\big(\hat{g}^{MLE}(X_j)\ge Y_j\big)=\mathbbm{1}\Big(\min_{k\ge 1:k\neq j}\big(Y_k+R|X_j-X_k|^{\beta}\big)\ge Y_j\Big)
\]
where the term $j=k$ can be dropped. This implies that the observation $(X_j,Y_j)$ can be integrated out, by following the usual arguments for computing sums with respect to a Poisson process (see e.g. \cite[Theorem 4.4]{LastPenrose2016}). To summarise, we propose the following estimator
\begin{equation}\label{EqEstimator}
\hat F_\Phi=\int_0^1\Phi(\hat{g}^{MLE}(x))\,dx-\frac{1}{n}\sum_{j\ge 1}\Phi'(Y_j)\mathbbm{1}\big(\hat{g}^{MLE}(X_j)\ge Y_j\big),
\end{equation}
which is indeed an unbiased estimator of $F_{\Phi}(g)$ under the appropriate integrability condition.
\begin{proposition}\label{PropUnbiased} Suppose that
\begin{equation}\label{EqCondUnbiased}
\int_0^1\int_{0}^\infty|\Phi'(g(x)+u)|\PP\big(\hat{g}^{MLE}(x)-g(x)\ge u\big)\,dudx<\infty.
\end{equation}
Then $\hat F_\Phi$ from \eqref{EqEstimator} is an unbiased estimator of $F_{\Phi}(g)$.
\end{proposition}
\begin{remark}
The above argument can be worked out for more general functionals of the form $\int_0^1\dots\int_0^1\Phi(x_1,\dots,x_m,g(x_1),\dots,g(x_m))\,dx_1\dots dx_m$, but then involves complex expressions in mixed partial derivatives of $\Phi$. We therefore focus on estimation of the basic functional $F_{\Phi}$.
\end{remark}
\begin{remark}
For the one-sided regression model \eqref{EqRegression} the discrete functional $\hat F_\Phi^{(n)}=(1/n)\sum_{i=1}^n\Phi(g(i/n))$ can be estimated analogously by
\[ \frac1n\sum_{i=1}^n\Phi(\hat g^{rMLE}(i/n))-\frac{1}{\alpha n}\sum_{i=1}^n\Phi'(Y_i)\mathbbm{1}(\hat g^{rMLE}(i/n)\ge Y_i)\]
with the regression analogue $\hat g^{rMLE}(x)=\min_i(Y_i+R\abs{x-i/n}^\beta)$ of $\hat g^{MLE}$. This estimator can be analysed with the martingale arguments of the next section, compare the results in \cite{ReissSelk14} for the linear case.
\end{remark}
\subsection{The martingale approach}
We pursue a martingale-based analysis of the estimator $\hat F_\Phi$ in \eqref{EqEstimator}. The following result extends \cite[Theorem 3.2]{ReissSelk14} to non-linear functionals.
\begin{thm}\label{ThmGenRes} Suppose that the right-hand side in \eqref{EqPhiVar} below is finite. Then the estimator $\hat F_\Phi$ is UMVU over $g\in\mathcal{C}^\beta(R)$ with variance
\begin{equation}\label{EqPhiVar}
\operatorname{Var}(\hat F_\Phi)=\frac{1}{n}\int_0^1\int_{0}^\infty(\Phi'(g(x)+u))^2\PP\big(\hat{g}^{MLE}(x)-g(x)\ge u\big)\,dudx.
\end{equation}
\end{thm}
\begin{remark}
If the right-hand side in \eqref{EqPhiVar} is finite, then Condition \eqref{EqCondUnbiased} holds since $\PP(\hat g^{MLE}(x)-g(x)\ge u)$ is integrable in $u$, see also \eqref{EqTails} below.
\end{remark}
\begin{proof}
We first show the formula for the variance.
Let $\lambda=(\lambda_t)$ be the process defined by $\lambda_t=n\int_0^1 \mathbbm{1}\big(g(x)\le t\le \hat{g}^{MLE}(x)\big)\,dx$, $t\in\R$.
Making a linear change of variables, the right-hand side in \eqref{EqPhiVar} can be written as
\begin{equation*}
n^{-2}\E\Big[\int_{t_0}^\infty\Phi'(s)^2\lambda_s\,ds\Big],
\end{equation*}
where $t_0$ is a lower bound for $g$.
In the proof of Theorem 3.2 in \cite{ReissSelk14}, it is shown that the pure counting process $N=(N_t)$ defined by
\[
N_t=\sum_{j\ge 1}\mathbbm{1}\big(Y_j\le t\wedge \hat{g}^{MLE}(X_j)\big),\quad t\ge t_0,
\]
has compensator $A=(A_t)$ given by $A_t=\int_{t_0}^t\lambda_s\,ds$
and that $M=N-A$ is a square-integrable martingale with respect to the filtration ${\mathcal F}_t=\sigma((X_j,Y_j)\mathbbm{1}(Y_j\le t), j\ge 1)$. Its predictable quadratic variation is
\[
\langle M \rangle_t=\int_{t_0}^t\lambda_s\,ds
\]
(see also \cite[Proposition 2.32]{Karr91}). We conclude (e.g. via \cite[Theorem 26.2]{Kallenberg02}) that
\[
(\Phi'\cdot M)_t=\int_{t_0}^t\Phi'(s)\,dM_s=\sum_{j\ge 1}\Phi'(Y_j)\mathbbm{1}\big(Y_j\le t\wedge \hat{g}^{MLE}(X_j)\big)-\int_{t_0}^t\Phi'(s)\lambda_s\,ds
\]
is an $L^2$-bounded martingale with
\[
\langle \Phi'\cdot M \rangle_t=\int_{t_0}^t\Phi'(s)^2\lambda_s\,ds,
\]
noting that $\E[\langle \Phi'\cdot M \rangle_t]$ is bounded by the right-hand side in \eqref{EqPhiVar}, which is finite by assumption.
For $t\to\infty$ the process $((\Phi'\cdot M)_t)$ converges almost surely to
\begin{align*}
&(\Phi'\cdot M)_\infty\\
&=\sum_{j\ge 1}\Phi'(Y_j)\mathbbm{1}\big(Y_j\le \hat{g}^{MLE}(X_j)\big)-\int_{t_0}^\infty\Phi'(s)\lambda_s\,ds\\
&=\sum_{j\ge 1}\Phi'(Y_j)\mathbbm{1}\big(Y_j\le \hat{g}^{MLE}(X_j)\big)-n\int_0^1\Phi(\hat{g}^{MLE}(x))\,dx+n\int_0^1\Phi(g(x))\,dx.
\end{align*}
Moreover, the process $ (\langle \Phi'\cdot M \rangle_t)$ converges almost surely and in $L^1$ to
\[
\langle \Phi'\cdot M \rangle_\infty=\int_{t_0}^\infty\Phi'(s)^2\lambda_s\,ds.
\]
Hence, unbiasedness and \eqref{EqPhiVar} follow from
\begin{equation}\label{EqMartingaleForm}
\E[(\Phi'\cdot M)_\infty]=0\ \ \text{ and }\ \ \E[(\Phi'\cdot M)_\infty^2-\langle \Phi'\cdot M \rangle_\infty]=0,
\end{equation}
which holds due to the $L^2$-convergence of $\Phi'\cdot M$ \cite[Corollary 6.22]{Kallenberg02}.
Finally, the fact that $\hat F_\Phi$ is UMVU follows from the Lehmann-Scheffé theorem and \cite[Proposition 3.1]{ReissSelk14} which says that $(\hat{g}^{MLE}(x):x\in[0,1])$ is a sufficient and complete statistic for $\mathcal{C}^\beta(R)$.
\end{proof}
\subsection{Rates of convergence}
In this section, we derive convergence rates for the estimator $\hat F_\Phi$. Using the argument leading to \cite[Equation (3.3)]{ReissSelk14}, we have the following deviation inequality for $x\in[0,1]$:
\begin{equation}\label{EqTails}
\mathbb{P}\big(\hat{g}^{MLE}(x)-g(x)\ge u\big)\le
\begin{cases}
\exp\big(-\frac{n\beta(2R)^{-\frac{1}{\beta}}u^{\frac{\beta+1}{\beta}}}{\beta+1}\big),& \text{if }u\in[0,2R],\\
\exp\big(-n\big(u-\frac{2R}{\beta+1}\big)\big),&\text{if }u>2R.
\end{cases}
\end{equation}
Thus, the right-hand side in \eqref{EqPhiVar} is finite if $(\Phi')^2$ has at most exponential growth with parameter strictly smaller than $n$. In particular, this holds for $\Phi(u)=|u|^p$, $p\ge 1$, in which case we have $\Phi'(u)=p|u|^{p-1}\operatorname{sgn}(u)$. A more detailed analysis gives:
\begin{corollary}\label{CorPowInt} Let $p\ge 1$ be a real number and consider $\Phi(u)=|u|^p$, $g\in\mathcal{C}^\beta(R)$. Then
\begin{equation}\label{EqFhatLp} \hat F_p=\int_0^1|\hat{g}^{MLE}(x)|^p\,dx-\frac{1}{n}\sum_{j\ge 1}p|Y_j|^{p-1}\sgn(Y_j)\mathbbm{1}\big(\hat{g}^{MLE}(X_j)\ge Y_j\big)
\end{equation}
is an unbiased estimator of $\| g\|_p^p$ with
\begin{equation}\label{EqPowInt}
\mathbb{E}\big[(\hat F_p-\| g\|_p^p)^2\big]\le C\max\Big(\|g\|_{2p-2}^{2p-2}n^{-\frac{2\beta+1}{\beta+1}},n^{-\frac{2\beta p+1}{\beta+1}}\Big),
\end{equation}
where $C$ is a constant depending only on $R$, $\beta$ and $p$. Here, we use the notation $\|\cdot\|_q$ also for $q<1$ with $\|g\|_{0}^{0}:=1$.
\end{corollary}
\begin{remark}
In the proof, a more precise upper bound is derived in which the dependence on the constant $R$ is explicit, see \eqref{EqBinomFormula}. For an asymptotically more precise result see Corollary \ref{CorFrechet} below.
\end{remark}
\begin{remark}\label{RemPosPart}
Since $\Phi(u)=|u|^p$ is non-negative, the positive part $(\hat F_p)_+$ of $\hat F_p$ always improves the estimator. This means that $\hat F_p$ is not an admissible estimator in the decision-theoretic sense, while $(\hat F_p)_+$ on the other hand is no longer unbiased.
\end{remark}
\begin{proof}
Throughout the proof $C>0$ denotes a constant depending only on $\beta$ and $p$ that may change from line to line.
By Theorem \ref{ThmGenRes} and the discussion above, we have
\[
\mathbb{E}\big[(\hat F_p-\| g\|_p^p)^2\big]=\frac{1}{n}\int_0^1\int_0^\infty p^2|u+g(x)|^{2p-2}\mathbb{P}\big(\hat{g}^{MLE}(x)-g(x)\ge u\big)\,dudx.
\]
Applying \eqref{EqTails} and the inequality $|u+g(x)|^{2p-2}\le 2^{2p-2}(u^{2p-2}+|g(x)|^{2p-2})$, the last term is bounded by
\begin{align*}
&\frac{p^22^{2p-2}}{n}\int_0^{2R}(\|g\|_{2p-2}^{2p-2}+u^{2p-2})\exp\Big(-\frac{n\beta(2R)^{-\frac{1}{\beta}}u^{\frac{\beta+1}{\beta}}}{\beta+1}\Big)\,du\\
&+\frac{p^22^{2p-2}}{n}\int_{2R}^\infty (\|g\|_{2p-2}^{2p-2}+u^{2p-2})\exp\Big(-n\Big(u-\frac{2R}{\beta+1}\Big)\Big)\,du=:(I)+(II).
\end{align*}
By a linear substitution, we have for $q\ge 0$
\begin{align}
&\int_0^{2R}u^q\exp\Big(-\frac{n\beta(2R)^{-\frac{1}{\beta}}u^{\frac{\beta+1}{\beta}}}{\beta+1}\Big)\,du\nonumber\\
&\le\Big(\frac{\beta+1}{\beta}\Big)^{\frac{\beta(q+1)}{\beta+1}}(2R)^{\frac{q+1}{\beta+1}}n^{-\frac{\beta(q+1)}{\beta+1}}\int_0^{\infty}v^{q}\exp\big(-v^{\frac{\beta+1}{\beta}}\big)\,dv\nonumber\\
&= \Big(\frac{\beta+1}{\beta}\Big)^{\frac{\beta q-1}{\beta+1}}(2R)^{\frac{q+1}{\beta+1}}\Gamma\Big(\frac{\beta(q+1)}{\beta+1}\Big)n^{-\frac{\beta(q+1)}{\beta+1}}\label{EqPowIntAsympLimit}
\end{align}
with the Gamma function $\Gamma$. Consequently,
\[
(I)\le CR^{\frac{1}{\beta+1}}\|g\|_{2p-2}^{2p-2}n^{-\frac{2\beta+1}{\beta+1}}+CR^{\frac{2p-1}{\beta+1}}n^{-\frac{2\beta p+1}{\beta+1}}.
\]
Next, consider the remainder term $(II)$. We have
\[
\int_{2R}^\infty \exp\Big(-n\Big(u-\frac{2R}{\beta+1}\Big)\Big)\,du= n^{-1}e^{-\frac{2\beta Rn}{\beta+1}}
\]
and
\begin{align*}
&\int_{2R}^\infty u^{2p-2}\exp\Big(-n\Big(u-\frac{2R}{\beta+1}\Big)\Big)\,du\le \int_{2R}^\infty u^{2p-2}\exp\Big(-\frac{n\beta u}{\beta+1}\Big)\,du\\
&\le \Big(\frac{\beta+1}{n\beta}\Big)^{2p-1}\int_{2\beta Rn/(\beta+1)}^\infty v^{2p-2}\exp(-v)\,dv\le C n^{-2p+1}e^{-\frac{\beta Rn}{\beta+1}}.
\end{align*}
Note that the last integral can be computed using partial integration. Thus
\[
(II)\le C\|g\|_{2p-2}^{2p-2}n^{-2}e^{-\frac{2\beta Rn}{\beta+1}}+Cn^{-2p}e^{-\frac{\beta Rn}{\beta+1}}.
\]
Summarising, we have
\begin{align}
\mathbb{E}\big[(\hat F_p-\| g\|_p^p)^2\big]&\le CR^{\frac{1}{\beta+1}}\|g\|_{2p-2}^{2p-2}n^{-\frac{2\beta+1}{\beta+1}}+CR^{\frac{2p-1}{\beta+1}}n^{-\frac{2\beta p+1}{\beta+1}}\nonumber\\
&+C\|g\|_{2p-2}^{2p-2}n^{-2}e^{-\frac{2\beta Rn}{\beta+1}}+Cn^{-2p}e^{-\frac{\beta Rn}{\beta+1}},\label{EqBinomFormula}
\end{align}
and the claim follows.
\end{proof}
One might wonder whether $\hat F_p$ achieves the rate $n^{-(\beta+1/2)/(\beta+1)}$ uniformly over $g\in{\mathcal C}^\beta(R)\cap B_p(R)$ with the $L^p$-ball $B_p(R)=\{g\in L^p([0,1]):\norm{g}_p\le R\}$. For $1\le p\le 2$ this follows from the inclusion $B_p(R)\subseteq B_{2p-2}(R)$. For $p>2$ this holds as well and is a consequence of
the following useful Lemma (with $q=2p-2$) providing a simple interpolation result. Results of this type are well known (cf. Bergh and L\"ofstr\"om \cite{BL76}), but since only H\"older semi-norms appear, we provide a self-contained proof in the appendix.
\begin{lemma}\label{LemNormComp} Let $1\le p\le q\le \infty$ and $f\in \mathcal{C}^\beta(R)$. Then we have
\begin{equation*}
\|f\|_q\le C\|f\|_p\max(1,R/\|f\|_{p})^{\frac{1/p-1/q}{\beta+1/p}},
\end{equation*}
where $C>0$ is a constant depending only on $\beta$, $p$ and $q$ and the right-hand side is understood to be zero for $f=0$.
\end{lemma}
Let us come to another corollary of Theorem \ref{ThmGenRes} which provides a local asymptotic upper bound for the minimax risk under weak assumptions on the functional:
\begin{corollary}\label{CorFrechet} Suppose that there is a constant $C>0$ such that $|\Phi'(u)|\le C\exp(C|u|)$ for all $u\in\R$. Let $f\in \mathcal{C}^\beta(R)$. Suppose that $\norm{\Phi'\circ f}_2\neq 0$ and that the map $F':\mathcal{C}^\beta(R)\subset L^2([0,1])\rightarrow L^2([0,1])$, $F'(g)= \Phi'\circ g$ is continuous at $g=f$ with respect to the $L^2$-norms. Then the estimator $ \hat{F}_{\Phi,n}=\hat F_\Phi$ satisfies the local asymptotic upper bound
\begin{equation*}
\lim_{\delta\rightarrow 0}\limsup_{n\rightarrow\infty}\sup_{\substack{g\in\mathcal{C}^\beta(R):\\ \|f-g\|_2\le \delta}}n^{\frac{2\beta+1}{\beta+1}}\mathbb{E}_g\big[(\hat{F}_{\Phi,n}-F_{\Phi}(g))^2\big]\le \Gamma\big(\tfrac{\beta}{\beta+1}\big)\big(\tfrac{2R\beta}{\beta+1}\big)^{\frac{1}{\beta+1}}\|\Phi'\circ f\|_2^2
\end{equation*}
with the Gamma function $\Gamma$.
\end{corollary}
\begin{proof}
By Theorem \ref{ThmGenRes} and Equation \eqref{EqTails}, we have
\begin{align*}
\mathbb{E}_g\big[(\hat F_\Phi-F_{\Phi}(g))^2\big]
&\le \frac{1}{n}\int_0^{2R}\|\Phi'\circ(u+g)\|_2^2\exp\Big(-\frac{n\beta(2R)^{-\frac{1}{\beta}}u^{\frac{\beta+1}{\beta}}}{\beta+1}\Big)\,du\\
&+\frac{1}{n}\int_{2R}^\infty\|\Phi'\circ(u+g)\|_2^2\exp\Big(-\frac{n\beta u}{\beta+1}\Big)\,du.
\end{align*}
By Lemma \ref{LemNormComp}, applied to $f-g$ and with $p=2$, $q=\infty$, we infer from $g\in\mathcal{C}^\beta(R)$ with $\|f-g\|_2\le \delta$ that
\begin{equation}\label{EqSupBound}
\|f-g\|_\infty\le C'R^{1/(2\beta+1)}\delta^{2\beta/(2\beta+1)}
\end{equation}
holds with some constant $C'$, provided that $\delta\le R$.
Using that $\Phi'$ has at most exponential growth, we get that $\|\Phi'\circ(u+g)\|_2^2\le C\exp(C|u|)$ uniformly over all $g\in\mathcal{C}^\beta(R)$ with $\|f-g\|_2\le \delta$ (adjusting $C$ appropriately). This shows that the second term is of smaller order than $n^{-2}$ and thus asymptotically negligible for our result. Similarly, for every fixed $\delta'>0$ the first integral from $\delta'$ to $2R$ becomes exponentially small in $n$.
Thus, for any $\delta'>0$ the left-hand side in Corollary \ref{CorFrechet} is bounded by
\begin{equation}\label{EqCorFrechetPr}
\lim_{\delta\to 0}\limsup_{n\rightarrow\infty}\sup_{\substack{g\in\mathcal{C}^\beta(R):\\ \|f-g\|_2\le \delta}}n^{\frac{\beta}{\beta+1}} \int_0^{\delta'}\|\Phi'\circ(u+g)\|_2^2\exp\Big(-\frac{n\beta(2R)^{-\frac{1}{\beta}}
u^{\frac{\beta+1}{\beta}}}{\beta+1}\Big)\,du.
\end{equation}
By the continuity of $F_{\Phi}'$ at $f$ and the fact that $\norm{\Phi'\circ f}_2\neq 0$, for every $\eps>0$ there exist $\delta,\delta'>0$ such that $\|\Phi'\circ(u+g)\|_2\le (1+\eps)\|\Phi'\circ f\|_2$ for all $|u|\le \delta'$ and $g\in\mathcal{C}^\beta(R)$ with $\|f-g\|_2\le \delta$. We conclude that \eqref{EqCorFrechetPr} is bounded by (using the computation in \eqref{EqPowIntAsympLimit} for $q=0$)
\[
\Gamma\Big(\frac{\beta}{\beta+1}\Big)\Big(\frac{2R\beta}{\beta+1}\Big)^{\frac{1}{\beta+1}}\|\Phi'\circ f\|_2^2,
\]
and the claim follows.
\end{proof}
\begin{remark}
By Lemma \ref{LemNormComp} continuity of $F_{\Phi}'$ on $\mathcal{C}^\beta(R)$ with respect to $L^2$-norm implies continuity with respect to supremum norm. Under the assumptions of Corollary \ref{CorFrechet}, one can indeed show that the functional $F_{\Phi}$ is Fr\'{e}chet-differentiable in $f$ along $\mathcal{C}^\beta(R)$ with derivative $F_{\Phi}'(f)=\Phi'\circ f$.
\end{remark}
\begin{remark}
Local asymptotic minimax results for estimating smooth functionals in the GWN model can be found in Nemirovski \cite[Chapter 7]{Nem00}. The rate is different (see the discussion in the introduction), but the term $\|F_{\Phi}'(g)\|_2^2$ appears as well. The latter fact can be explained by linearising $F_{\Phi}$ at~$g$.
\end{remark}
\begin{remark}
The estimators are non-adaptive in the sense that they rely on the knowledge of the regularity parameters $\beta$ and $R$. In \cite{ReissSelk14} the Lepski method has been employed to construct adaptive estimators in the linear case, based on a blockwise estimator. We conjecture that this approach would also give an adaptive rate-optimal estimator here. Note also the restriction $\beta\le 1$ on the regularity parameter. The reason is that the MLE for $g\in C^\beta(R)$ with $\beta>1$ does not necessarily provide a pointwise upper bound for $g$ such that the present approach may fail.
\end{remark}
\subsection{Lower bounds}
In this section we establish lower bounds corresponding to Corollaries \ref{CorPowInt} and \ref{CorFrechet}. We will apply the method of two fuzzy hypotheses (see \cite[Chapter 2.7.4]{Tsyb09}) with a prior corresponding to independent non-identical Bernoulli random variables. Our main result states a local asymptotic lower bound in the case that $\Phi$ is continuously differentiable. Possible extensions are discussed afterwards.
\begin{thm}\label{PropLowBoundDiff}
Let $\Phi$ be continuously differentiable and $f\in \mathcal{C}^\beta(R)$ with $\norm{\Phi'\circ f}_2\neq 0$. Then there is a constant $c_1>0$, depending only on $\beta$, such that
\begin{equation*}
\lim_{\delta\rightarrow 0}\liminf_{n\rightarrow\infty}\inf_{\tilde{F}_n}\sup_{\substack{g\in\mathcal{C}^\beta(R):\\ \|f-g\|_2\le \delta}}n^{\frac{2\beta+1}{\beta+1}}\mathbb{E}_g\big[(\tilde{F}_n -F_{\Phi}(g))^2\big]> c_1R^{\frac{1}{\beta+1}}\|\Phi'\circ f\|_2^2.
\end{equation*}
The infimum is taken over all estimators in the PPP model with intensity \eqref{EqIntensity}.
\end{thm}
\begin{proof}
We want to apply the $\chi^2$-version of the method of two fuzzy hypotheses as described in \cite[Theorem 2.15]{Tsyb09}. Consider the functions
\[
g_\theta=\sum_{k=1}^{m}\theta_kg_k\ \ \text{ with } \ \ \theta_k\in\{0,1\}
\]
and \[
g_k(x)=cRh^{\beta}K\left(\frac{x-(k-1)h}{h}\right)=cRh^{\beta+1}K_h(x-(k-1)h)
\]
with $h=1/m$, triangular kernel $K(u)=4(u\wedge (1-u))\mathbbm{1}_{[0,1]}(u)$, $K_h(\cdot)=K(\cdot/h)/h$ and $c>0$ sufficiently small such that $g_\theta\in\mathcal{C}^\beta(R)$ for all $m$ and $\theta$. Let $\pi_n$ be the probability measure on $\{0,1\}^m$ obtained when $\theta_1,\dots,\theta_m$ are independent (non-identical) Bernoulli random variables with success probabilities $p_1,\dots,p_m$. Let $P_g$ denote the law of the observations in the PPP model with intensity function \eqref{EqIntensity}. We set $\mathbf{P}_{0,n}=P_f$ and
\[
\mathbf{P}_{1,n}(\cdot)=\int P_{f+g_\theta}(\cdot)\pi_n(d\theta).
\]
In order to obtain the result, it suffices to find $m\ge 1$ and probabilities $p_1,\dots,p_m$ (both depending on $n$) as well as a constant $c_1>0$, only depending on $\beta$, and an absolute constant $c_2<\infty$, such that
\begin{enumerate}
\item[(i)] For each fixed $\delta>0$ the inequality $\|g_\theta\|_2\le \delta$ holds for all $n$ sufficiently large and for $n\to\infty$ the prior satisfies
\begin{equation*}
\pi_n\Big( F_{\Phi}(f+g_\theta)\ge F_{\Phi}(f)+2c_1\|\Phi'\circ f\|_2R^{\frac{1/2}{\beta+1}}n^{-\frac{\beta+1/2}{\beta+1}}\Big)\rightarrow 1;
\end{equation*}
\item[(ii)] $\limsup_{n\to\infty} \chi^2(\mathbf{P}_{1,n},\mathbf{P}_{0,n})\le c_2$.
\end{enumerate}
We start with the following lemma on the $\chi^2$-distance.
\begin{lemma}\label{LemChiSquare} Suppose that the success probabilities satisfy $\sum_{k=1}^m p_k^2=1$. Then
\begin{equation*}
\chi^2(\mathbf{P}_{1,n},\mathbf{P}_{0,n})= \int\left(\frac{d\mathbf{P}_{1,n}}{d\mathbf{P}_{0,n}}\right)^2\,d\mathbf{P}_{0,n}-1\le
\exp\bigg(\exp\Big( n\int_{I_1}g_1(x)\,dx\Big)-1\bigg)-1
\end{equation*}
holds, where $I_1=[0,h)$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{LemChiSquare}]
We abbreviate $\int g_k=\int_{I_k}g_k(x)\,dx$, where $I_k=[(k-1)h,kh)$ for $k<m$ and $I_m=[1-h,1]$. Let us first see that
\begin{equation}\label{EqLikelihoodFuzzyH}
\frac{d\mathbf{P}_{1,n}}{d\mathbf{P}_{0,n}}=\prod_{k=1}^m\Big(1-p_k+p_ke^{n\int
g_k} \mathbbm{1}\big(\forall X_j\in I_k:Y_j\ge f(X_j)+g_k(X_j)\big)\Big).
\end{equation}
Indeed, by definition the left hand side is equal to
\begin{align*}
&\sum_{\theta\in\{0,1\}^m}\bigg(\prod_{k:\theta_k=0}(1-p_k)\prod_{k:\theta_k=1}p_k\bigg)\frac{dP_{f+g_\theta}}{dP_{f}}\\
&=\sum_{\theta\in\{0,1\}^m}\bigg(\prod_{k:\theta_k=0}(1-p_k)\prod_{k:\theta_k=1}p_ke^{n\int
g_k} \mathbbm{1}\big(\forall X_j\in I_k:Y_j\ge f(X_j)+g_k(X_j)\big)\bigg)\\
&=\prod_{k=1}^m\Big(1-p_k+p_ke^{n\int
g_k} \mathbbm{1}\big(\forall X_j\in I_k:Y_j\ge f(X_j)+g_k(X_j)\big)\Big),
\end{align*}
where we used the formula (see \cite[Theorem 1.3]{Kut98} or \cite[Section 3]{ReissSelk14})
\begin{align*}
\frac{dP_{f+g_\theta}}{dP_f}&=e^{n\int g_\theta}\mathbbm{1}\big(\forall
j:Y_j\ge f(X_j)+g_\theta(X_j)\big)\\
&=\prod_{k:\theta_k=1}e^{n\int g_k}\mathbbm{1}\big(\forall X_j\in I_k:Y_j\ge f(X_j)+g_k(X_j)\big)
\end{align*}
in the first equality. By the defining properties of the PPP, under $\mathbf{P}_{0,n}$, the right-hand side in \eqref{EqLikelihoodFuzzyH} is a product of independent random variables and the corresponding indicators have success probabilities $e^{-n\int g_k}$. Thus we obtain
\begin{align*}
\int\left(\frac{d\mathbf{P}_{1,n}}{d\mathbf{P}_{0,n}}\right)^2\,d\mathbf{P}_{0,n}&=\prod_{k=1}^m((1-p_k)^2+2p_k(1-p_k)+p_k^2e^{n\int
g_k})\\
&=\prod_{k=1}^m(1+p_k^2(e^{n\int g_k}-1))\\
&\le \prod_{k=1}^me^{p_k^2(e^{n\int g_1}-1)} = e^{e^{n\int g_1}-1},
\end{align*}
where we used the bound $1+x\le e^x$ and the assumption $\sum_{k=1}^m p_k^2=1$.
\end{proof}
Using Lemma \ref{LemChiSquare} and the identity
\[
n\int_{I_1}g_1(x)\,dx=cRnh^{\beta+1},
\]
we get (ii) provided that we choose $m=1/h$ of size $(Rn)^{1/(\beta+1)}$ and $p_1,\dots,p_m$ such that $\sum_{k=1}^mp_k^2=1$. Thus it remains to choose the $p_k$ such that the second convergence in (i) is satisfied.
We first consider the case that $\Phi'\circ f\ge 0$. Let $\eps>0$ be a small constant to be chosen later. Since $\Phi'$ is uniformly continuous on compact intervals, there is a $\delta'>0$ such that
\[
\int_0^1\Phi(f(x)+g(x))\,dx- \int_0^1\Phi(f(x))\,dx\ge \int_0^1\Phi'(f(x))g(x)\,dx-\eps\int_0^1|g(x)|\,dx
\]
for all $g\in\mathcal{C}^\beta(R)$ with $\|f-g\|_2\le \delta'$ (using \eqref{EqSupBound} above). Thus, for $n$ sufficiently large, we get
\begin{align*}
F_{\Phi}(f+g_\theta)-F_{\Phi}(f)&\ge \langle \Phi'\circ f,g_\theta\rangle-\eps\langle 1,g_\theta\rangle\\
& =\sum_{k=1}^m \theta_k\langle \Phi'\circ f,g_k\rangle-\eps \sum_{k=1}^m \theta_k\langle 1,g_k\rangle\\
&=cRh^{\beta+1}\bigg(\sum_{k=1}^m \theta_k\langle \Phi'\circ f,K_h(\cdot-(k-1)h)\rangle-\eps \sum_{k=1}^m \theta_k\bigg).
\end{align*}
Setting $a_k=\langle \Phi'\circ f,K_h(\cdot-(k-1)h)\rangle$, this can be written as
\begin{equation}\label{EqSepDiff}
F_{\Phi}(f+g_\theta)-F_{\Phi}(f)\ge cRh^{\beta+1}\bigg(\sum_{k=1}^m a_k\theta_k-\eps \sum_{k=1}^m \theta_k\bigg),
\end{equation}
The first sum is a weighted sum of independent non-identical Bernoulli random variables and the maximising choice for the success probabilities is
\begin{equation}\label{EqChoicePrior}
p_k=\frac{a_k}{\|a\|_2}
\end{equation}
(the $a_k$ satisfy $a_k\ge 0$ since we assumed $\Phi'\circ f\ge 0$).
By the mean value theorem and the fact that $\Phi'\circ f$ is continuous, we get $a_k=\Phi'(f(x_k))$ with $x_k\in [(k-1)h,kh]$ and also
\begin{equation}\label{EqIntConv}
\frac{1}{m}\|a\|_q^q=\frac{1}{m}\sum_{k=1}^m a_k^q\rightarrow \int_0^1 (\Phi'(f(x))^q\,dx=\|\Phi'\circ f\|_q^q\qquad\text{as } n\rightarrow \infty
\end{equation}
for each $q\ge 1$. Using the Chebyshev inequality we get
\begin{align*}
\pi_n\left(\sum_{k=1}^m a_k\theta_k< \|a\|_2/2\right)&= \pi_n\left(\sum_{k=1}^m a_k(\theta_k-p_k)< -\|a\|_2/2\right)\\
&\le \frac{4\sum_{k=1}^ma_k^2p_k(1-p_k)}{\|a\|_2^2}\le 4\left(\frac{\|a\|_3}{\|a\|_2}\right)^3
\end{align*}
and the latter converges to $0$ as $n\rightarrow\infty$ by \eqref{EqIntConv}. Similarly,
\begin{align*}
\pi_n\left(\sum_{k=1}^m\theta_k> 2\|a\|_1/\|a\|_2\right)
&=\pi_n\left(\sum_{k=1}^m(\theta_k-p_k)> \|a\|_1/\|a\|_2\right)\\
&\le \frac{\|a\|_2^2\sum_{k=1}^mp_k(1-p_k)}{\|a\|_1^2}\le \frac{\|a\|_2}{\|a\|_1}
\end{align*}
and the latter converges to $0$ as $n\rightarrow\infty$ by \eqref{EqIntConv}. Combining these two bounds with \eqref{EqSepDiff} we get
\begin{equation}\label{EqAsympSep}
\pi_n\left( F_{\Phi}(f+g_\theta)-F_{\Phi}(f)\ge cRh^{\beta+1/2}\left(\frac{1}{2\sqrt{m}}\|a\|_2-\eps\frac{2}{\sqrt{m}}\frac{\|a\|_1}{\|a\|_2}\right)\right)\rightarrow 1
\end{equation}
as $n\rightarrow\infty$. This implies (i) if $\eps$ is chosen small enough since $\|a\|_2/\sqrt{m}$ and $\|a\|_1/(\sqrt{m}\|a\|_2)$ have non-zero limits by \eqref{EqIntConv} and the assumption $\|\Phi'\circ f\|_2\neq 0$. This completes the proof in the case $\Phi'\circ f\ge 0$.
If $\Phi'\circ f\le 0$, then we may follow the same line of arguments where (ii) is replaced with a left-deviation inequality (which corresponds to apply the above arguments to the functional $F_{-\Phi}$). Next, if $\Phi'\circ f$ takes both, positive and negative values, then we may choose $p_k=a_{k+}/\|a_+\|_2$ (resp. $p_k=a_{k-}/\|a_-\|_2$) leading to a lower bound with $\|\Phi'\circ f\|_2^2$ replaced by $\|(\Phi'\circ f)_+\|_2^2$ (resp. $\|(\Phi'\circ f)_-\|_2^2$). Summing up both lower bounds gives the claim in the general case.
\end{proof}
\begin{remark}\label{RemConvexity} If $\Phi$ is convex, then we can replace \eqref{EqSepDiff} by
\[
F_{\Phi}(f+g_\theta)-F_{\Phi}(f)\ge \langle \Phi'\circ f,g_\theta\rangle=cRh^{\beta+1}\sum_{k=1}^m a_k\theta_k,
\]
leading to a shortening of the above proof.
In this case the lower bound also holds without continuity of $\Phi'$. The arguments, however, must be adapted slightly since the convergence in \eqref{EqIntConv} may not hold in this case.
\end{remark}
\begin{remark}\label{RemLowBoundNonasymp}
By making the constants in the proof of Theorem \ref{PropLowBoundDiff} explicit, one can also establish non-asymptotic lower bounds which include lower-order terms. Consider for instance $\Phi(u)=|u|^p$, $p\in\N$ and $f\equiv a>0$. Then we have
\begin{align}
F_{\Phi}(a+g_\theta)-a^p&
=\bigg(\sum_{k=1}^{m}\theta_k\bigg)\sum_{j=1}^p\binom{p}{j}a^{p-j}c^jR^jh^{\beta j +1}\|K\|_j^j\nonumber\\
&\ge\bigg(\sum_{k=1}^{m}\theta_k\bigg) \max(pa^{p-1}cR h^{\beta+1},c^pR^p\|K\|_p^ph^{\beta p+1})\label{EqSepPol}.
\end{align}
We choose
\[
p_1=\dots=p_m=1/\sqrt{m}\ \ \text{ and } \ \ m=\lfloor 2(cRn)^{1/(\beta+1)}\rfloor.
\]
In order to ensure $\|g_\theta\|_2 \le \delta$, it suffices that $m\ge 1$ and $2cRh^{\beta}\le\delta$ hold, which is satisfied if $n\ge c_1$ with $c_1$ depending only on $c$, $R$ and $\delta$. Now, by Lemma \ref{LemChiSquare} and the choice of $m$ we have $\chi^2(\mathbf{P}_{0,n},\mathbf{P}_{1,n})\le e^{e-1}-1$. Moreover, using the simplification of Remark \ref{RemConvexity}, \eqref{EqAsympSep} becomes
\begin{equation*}
\pi_n\Big( F_{\Phi}(a+g_\theta)\ge a^p+\frac{1}{2}\max(pa^{p-1}cR h^{\beta+1/2},c^pR^p\|K\|_p^ph^{\beta p+1/2})\Big)\ge 1-4/\sqrt{m}.
\end{equation*}
Inserting the value of $h$ and applying \cite[Theorem 2.15 (iii)]{Tsyb09}, we get
\begin{align*}
&\inf_{\tilde{F}_n}\sup_{\substack{g\in\mathcal{C}^\beta(R):\\ \|a-g\|_2\le \delta}}\mathbb{P}_g\Big(|\tilde{F}_n-F_{\Phi}(g)|\ge \max(c_2pa^{p-1}R^{\frac{1/2}{\beta+1}}n^{-\frac{\beta+1/2}{\beta+1}},c_3R^{\frac{p-1/2}{\beta+1}}n^{-\frac{\beta p+1/2}{\beta+1}})\Big)\\
& \ge \frac{1}{4}\exp(-(e^{e-1}-1))-2/\sqrt{m},
\end{align*}
provided that $n\ge c_1$, where $c_2$ is a constant depending only on $\beta$ and $c_3$ is a constant depending only on $\beta$ and $p$. Thus we obtain a lower bound which has the form of the upper bound in Corollary \ref{CorPowInt} (resp. \eqref{EqBinomFormula}).
\end{remark}
\begin{remark}
In the case of linear functionals the above proof can be used to obtain the lower bound in \cite[Theorem 2.6]{ReissSelk14}. Instead of using the method of fuzzy hypothesis, one can also try to apply the method used in Rei\ss{} and Selk \cite{ReissSelk14} and Korostelev and Tsybakov \cite{KorTsy93} which is based on a comparison of the minimax risk with a Bayesian risk. This works for instance for the special case $\Phi(u)=|u|^p$, $p\in\N$, and $f\equiv a>0$, but it is not clear whether this structurally different prior can produce the correct lower bounds more generally.
\end{remark}
\section{Hypothesis testing}
\label{SecTesting}
\subsection{Main result}
In this section we use the previous results to address the hypothesis testing problem
\begin{equation*}
H_0:g=g_0\qquad\text{vs.}\qquad H_1:g\in g_0+\mathcal{G}_n,
\end{equation*}
where $g_0$ is a known function and
\[
\mathcal{G}_n=\mathcal{G}_n(\beta,R,p,r_n)=\{g\in \mathcal{C}^\beta(R):\|g\|_p\ge r_n\}.
\]
In the sequel, we restrict to the case $g_0=0$, since the general case can be reduced to this one by a simple shift of the observations.
We propose the following plug-in test
\begin{equation}\label{EqPlugInTest}
\psi_{n,p} = \mathbbm{1}\big( \hat F_p \ge r^p_n/2\big),
\end{equation}
with the estimator $\hat F_p$ from \eqref{EqFhatLp}.
We follow a minimax approach to hypothesis testing, see e.g. \cite[Chapter 2.4]{IngsterSuslina03}.
Our main result of this section states that $\psi_{n,p}$ achieves the minimax separation rates:
\begin{thm}\label{TheoremTesting}
Let $p\ge 1$ be a real number and
\[
r_n^*=n^{-\frac{\beta+1/(2p)}{\beta+1}}.
\]
Then, the following holds as $n\rightarrow\infty$:
\begin{itemize}
\item[(a)] If $r_n/r^*_n\rightarrow \infty$, then the tests
$\psi_{n,p}$ from \eqref{EqPlugInTest} satisfy
\[
\mathbb{E}_0[\psi_{n,p}] + \sup_{g \in\mathcal{G}_n}
\mathbb{E}_g[1-\psi_{n,p}]\rightarrow 0.
\]
\item[(b)] If $r_n/r^*_n\rightarrow 0$, then we have
\[
\inf_{\psi_n}\big(\mathbb{E}_0[\psi_n] + \sup_{g \in\mathcal{G}_n}
\mathbb{E}_g[1-\psi_n ]\big)\rightarrow 1,
\]
where the infimum is taken over all tests in the PPP model with intensity \eqref{EqIntensity}.
\end{itemize}
\end{thm}
\subsection{Proof of the upper bound}
Throughout the proof $C>0$ denotes a constant depending only on $R$, $\beta$ and $p$ that may change from line to line. Under the null hypothesis we have, using the Chebyshev inequality and Corollary \ref{CorPowInt},
\begin{align}\label{EqTypeIError}
\mathbb{E}_0[\psi_{n,p}]
= \mathbb{P}_0(\hat F_p \ge r^p_n/2)
&\le \frac{4\mathbb{E}_0[\hat F_p^2]}{r^{2p}_n}\le C\dfrac{n^{-\frac{2\beta p + 1}{\beta +1} }}{r^{2p}_n}=C\left(\dfrac{r^*_n}{r_n}\right)^{2p}
\end{align}
and by assumption the right-hand side tends to zero as $n\rightarrow\infty$. Next, consider the type-two error $\mathbb{E}_g[1-\psi_{n,p} ]$ with $g\in\mathcal{G}_n$. Let $k\in\N$ be such that $2^{k-1}r_n^p\le \|g\|_p^p< 2^kr_n^p$ and set $r_{n,k}=2^{k/p}r_n$. By the Chebyshev inequality, we have
\begin{align}
\mathbb{E}_g[1-\psi_{n,p}]= \mathbb{P}_g(\hat F_p < r^p_n/2)
&= \mathbb{P}_g(\|g\|_p^p -\hat F_p > \|g\|_p^p - r^p_n/2) \nonumber\\
&\le \mathbb{P}_g(\|g\|_p^p -\hat F_p > r_{n,k}^p/4) \nonumber\\
&\le \frac{16\mathbb{E}_g[(\hat F_p-\|g\|_p^p)^2]}{r^{2p}_{n,k}}.\label{EqChevyshev}
\end{align}
Now, we may restrict ourselves to the case that
\begin{equation}\label{EqNonzeroRate}
\|g\|_{2p-2}^{2p-2}n^{-\frac{2\beta+1}{\beta+1}}\ge n^{-\frac{2\beta p+1}{\beta+1}}.
\end{equation}
Indeed, if \eqref{EqNonzeroRate} does not hold, then the maximal type-two error is also bounded by $C(r^*_n/r_n)^{2p}$, as can be seen by the same argument as in \eqref{EqTypeIError}. By \eqref{EqChevyshev}, \eqref{EqNonzeroRate} and Corollary \ref{CorPowInt}, we obtain
\begin{equation}\label{EqTypeII}
\mathbb{E}_g[1-\psi_{n,p}]\le C\Vert g \Vert^{2p-2}_{2p-2}\dfrac{n^{-\frac{2\beta+1}{\beta+1}}}{r^{2p}_{n,k}}.
\end{equation}
Let us consider the cases $1\le p \le 2$ and $p>2$ separately.
If $1< p \le 2$, then we have $\|g\|_{2p-2}\le\|g\|_p\le r_{n,k}$ by the H\"older inequality and the definition of $k$. Thus, for $1\le p \le 2$, we get
\begin{align*}
\mathbb{E}_g[1-\psi_{n,p}]
\le C\frac{n^{-\frac{2\beta+1}{\beta+1}}}{r_{n,k}^{2} }\le C\left(\dfrac{r^*_n }{r_{n,k}}\right)^2 n^{- \frac{2\beta+1}{\beta+1}+
\frac{2\beta+1/p}{\beta+1}}\le C\left(\dfrac{r^*_n }{r_n}\right)^2.
\end{align*}
Taking the supremum over all $g\in\mathcal{G}_n$, the right-hand side tends to zero as $n\rightarrow \infty$. Next, consider the case $p> 2$.
Applied with $q=2p-2>p$, Lemma \ref{LemNormComp} gives
\begin{equation}\label{EqLemNormComp}
\Vert g \Vert^{2p-2}_{2p-2}\le C\|g\|_p^{2p-2}\max(1, \|g\|_p^{-1})^{ \frac{1-2/p}{\beta +1/p}}.
\end{equation}
If $\|g\|_p>1$, then the claim follows as in the case $1\le p \le 2$. If $\|g\|_p\le 1$, then by \eqref{EqTypeII} and \eqref{EqLemNormComp}, we have
\begin{align*}
\mathbb{E}_g[1-\psi_{n,p}]
&\le C {r_{n,k}^{ -2-\frac{1-2/p}{\beta +1/p}}} n^{- \frac{2\beta+1}{\beta+1}}\\
&= C\left(\frac{r^*_n}{r_{n,k}}\right)^{ \frac{ 2\beta +1}{\beta +1/p}}
n^{\frac{2\beta+1}{\beta+1/p}\frac{\beta+1/2p}{\beta+1} - \frac{2\beta+1}{\beta+1}
}\le C\left(\dfrac{r^*_n}{r_n}\right)^{ \frac{ 2\beta +1}{\beta +1/p}}.
\end{align*}
Again, taking the supremum over all $g\in\mathcal{G}_n$, the right-hand side tends to zero as $n\rightarrow \infty$. This completes the proof of (i). \qed
\subsection{Proof of the lower bound}\label{SecLowbound}
We set $\mathbf{P}_{1,n}(\cdot)=\int P_{g_\theta}(\cdot)\pi_n(d\theta)$ and $\mathbf{P}_{0,n}=P_0$ with $g_\theta$ and $\pi_n$ as in the proof of Theorem \ref{PropLowBoundDiff} with the choice
\begin{equation}\label{EqIdPrior}
p_1=\dots=p_m=1/\sqrt{m}.
\end{equation}
By \cite[Proposition 2.9 and Proposition 2.12]{IngsterSuslina03}, in order that Theorem \ref{TheoremTesting} (ii) holds, we have to show that as $n\rightarrow \infty$,
\begin{enumerate}
\item[(i)] $\pi_n(g_\theta\in\mathcal{G}_n)\rightarrow 1$;
\item[(ii)] $\chi^2(\mathbf{P}_{1,n},\mathbf{P}_{0,n})\rightarrow 0$.
\end{enumerate}
For (i), note that
\[
\|g_\theta\|_p=\Big(\sum_{k=1}^m\theta_k\Big)^{1/p}cRh^{\beta+1/p}\|K\|_p.
\]
By the Chebyshev inequality, we have
\begin{align*}
\pi_n\bigg(\Big(\sum_{k=1}^m\theta_k\Big)^{1/p}\le 2^{-1/p} m^{1/(2p)}\bigg)&=\pi_n\bigg(\sum_{k=1}^m(\theta_k-1/\sqrt{m})\le -\sqrt{m}/2\bigg)\\
&\le \frac{4m(1/\sqrt{m})(1-1/\sqrt{m})}{m},
\end{align*}
where the right-hand side tends to zero as $m\rightarrow\infty$. Thus (i) holds provided that we choose $m^{-1}=h$ of size
\begin{equation*}
c_1r_n^{\frac{1}{\beta+1/(2p)}}
\end{equation*}
with $c_1>0$ depending only on $R$ and $p$. Moreover, by Lemma \ref{LemChiSquare} and \eqref{EqIdPrior}, we have
\begin{equation*}
\chi^2(\mathbf{P}_{1,n},\mathbf{P}_{0,n})\le \exp\big(\exp(cRnh^{\beta+1})-1\big)-1.
\end{equation*}
Inserting the above choice of $h$, the last expression goes to zero as $n\rightarrow\infty$, since
\[
nr_n^{\frac{\beta+1}{\beta+1/(2p)}}=(r_n/r_n^*)^{\frac{\beta+1}{\beta+1/(2p)}}\rightarrow 0.
\]
This completes the proof.\qed
\section{Estimating the $L^p$-norm}\label{SecEstLp}
Finally let us consider the problem of estimating the $L^p$-norm of $g$. We define the estimator $\hat{T}$ of $\|g\|_p$ by
\[
\hat{T}=\big(\max(\hat F_p, 0)\big)^{1/p}=(\hat F_p)_+^{1/p}.
\]
Our main result of this section is as follows:
\begin{thm}\label{TheoremEstNorm}
Let $p\ge 1$ be a real number. Then we have
\[
\sup_{g\in \mathcal{C}^\beta(R)}\mathbb{E}_g[|\hat{T}-\|g\|_p|]\le Cn^{-\frac{\beta+1/(2p)}{\beta+1}}
\]
with a constant $C>0$ depending only on $R$, $\beta$ and $p$.
On the other hand, we have
\[
\liminf_{n\rightarrow\infty}n^{\frac{\beta+1/(2p)}{\beta+1}}\inf_{\tilde T_n}\sup_{g\in \mathcal{C}^\beta(R)}\mathbb{E}_g[|\tilde{T}_n-\|g\|_p|]>0,
\]
where the infimum is taken over all estimators in the PPP Model with intensity \eqref{EqIntensity}.
In particular, the minimax rate of estimation over $\mathcal{C}^\beta(R)$ is $n^{-(\beta+1/(2p))/(\beta+1)}$.
\end{thm}
\begin{proof}
The lower bound follows from the lower bound in Theorem \ref{TheoremTesting}. To see this, let $r^{est}_n=\inf_{\tilde T_n}\sup_{g\in \mathcal{C}^\beta(R)}\mathbb{E}_g[|\tilde{T}_n-\|g\|_p|]$ be the minimax risk. If the lower bound in Theorem \ref{TheoremEstNorm} was false, then $r^{est}_{n_k}/r^*_{n_k}\rightarrow 0$ along a subsequence $(n_k)$. Now construct $(r_{n_k})$ such that $r_{n_k}/r_{n_k}^*\rightarrow 0$ and $r_{n_k}/r_{n_k}^{est}\rightarrow \infty$. Using \cite[Proposition 2.17]{IngsterSuslina03} and the fact that $r_{n_k}/r_{n_k}^{est}\rightarrow \infty$, we would get $\mathbb{E}_0[\psi_{n_k}] + \sup_{g \in\mathcal{G}_{n_k}}
\mathbb{E}_g[1-\psi_{n_k}]\rightarrow 0$ for suitable plug-in tests $\psi_{n_k}$ based on minimax optimal estimators, contradicting the lower bound in Theorem \ref{TheoremTesting} and the fact that $r_{n_k}/r_{n_k}^*\rightarrow 0$.
It remains to prove the upper bound.
Throughout the proof $C>0$ denotes a constant depending only on $R$, $\beta$ and $p$ that may change from line to line. Since the case $p=1$ is covered in Corollary \ref{CorPowInt}, we restrict to the case $p>1$. By the convexity of $y\mapsto y^p$, we have (for non-negative real numbers $a\neq b$ the inequality $(b^p-a^p)/(b-a)\ge \max(a,b)^{p-1}$ holds)
\[
|\hat{T}-\|g\|_p|\le \frac{|\hat{T}^p-\|g\|_p^p|}{\|g\|_p^{p-1}}.
\]
Hence,
\begin{equation}\label{EqSmoothNorm}
\mathbb{E}_g[|\hat{T}-\|g\|_p|]\le \frac{\mathbb{E}_g[(\hat{T}^p-\|g\|_p^p)^2]^{1/2}}{\|g\|_p^{p-1}}\le \frac{\mathbb{E}_g[(\hat F_p-\|g\|_p^p)^2]^{1/2}}{\|g\|_p^{p-1}},
\end{equation}
where we also used the fact that $\hat T^p=(\hat F_p)_+$ improves $\hat F_p$ (see also Remark \ref{RemPosPart}).
On the other hand, we also have $|\hat{T}-\|g\|_p|\le |\hat{T}|+\|g\|_p$, which leads to
\begin{align}
\mathbb{E}_g[|\hat{T}-\|g\|_p|]&\le \mathbb{E}_g[\hat{T}^p]^{1/p}+\|g\|_p\nonumber\\&\le \mathbb{E}_g[|\hat{T}^p-\|g\|_p^p|]^{1/p}+2\|g\|_p \nonumber\\
& \le \mathbb{E}_g[(\hat F_p-\|g\|_p^p)^2]^{1/(2p)}+2\|g\|_p,\label{EqSingNorm}
\end{align}
where we applied the Hölder inequality and the concavity of the function $y\mapsto y^{1/p}$ (for non-negative real numbers $a\neq b$ the inequality $(a+b)^{1/p}\le a^{1/p}+b^{1/p}$ holds).
If $\norm{g}_p\le n^{-(\beta+1/(2p))/(\beta+1)}$, then by \eqref{EqSingNorm} and Corollary \ref{CorPowInt} it suffices to show
\[ \max\Big(\|g\|_{2p-2}^{2p-2}n^{-\frac{2\beta+1}{\beta+1}},n^{-\frac{2\beta p+1}{\beta+1}}\Big)^{1/(2p)}\le Cn^{-\frac{\beta+1/(2p)}{\beta+1}},
\]
which itself follows from $\|g\|_{2p-2}\le Cn^{-\beta/(\beta+1)}$. For $p\le 2$ the latter holds because of $\|g\|_{2p-2}\le\|g\|_p\le n^{-(\beta+1/(2p))/(\beta+1)}$. For $p>2$ this is implied by Lemma \ref{LemNormComp}:
\[ \|g\|_{2p-2}\le C\max(\|g\|_p,\,\|g\|_p^{(\beta+1/(2p-2))/(\beta+1/p)})\le C\|g\|_p^{\beta/(\beta+1/(2p))}\le Cn^{-\beta/(\beta+1)},
\]
using first $\|g\|_p\le 1$ and then $1/(2p-2)\ge 1/(2p)$.
In the opposite case $\norm{g}_p>n^{-(\beta+1/(2p))/(\beta+1)}$ we apply \eqref{EqSmoothNorm}, Corollary \ref{CorPowInt} and obtain the result if
\[ \max\Big(\|g\|_{2p-2}^{p-1}n^{-\frac{\beta+1/2}{\beta+1}},n^{-\frac{\beta p+1/2}{\beta+1}}\Big)\le C \|g\|_p^{p-1} n^{-(\beta+1/(2p))/(\beta+1)}.
\]
For $p\le 2$ this follows again by $\|g\|_{2p-2}\le\|g\|_p$. For $p>2$ Lemma \ref{LemNormComp} yields the bound
\[ \|g\|_{2p-2}^{p-1}\le C\|g\|_{p}^{p-1} \max(1,\|g\|_p^{(1/2-(p-1)/p)/(\beta+1/p)}\,) \le C\|g\|_p^{p-1}n^{(1/2-1/(2p))/(\beta+1)},
\]
using $((p-1)/p-1/2)(\beta+1/(2p))=(1/2-1/p)(\beta+1/(2p))<(1/2-1/(2p))(\beta+1/p)$. Inserting the bound thus gives the result also for $p>2$.
\end{proof}
\begin{remark}
For the problem of estimating $g$ in $L^\infty$-norm, Drees, Neumeyer and Selk \cite{DNS14} established the rate $(n^{-1}\log n)^{\beta/(\beta+1)}$ (in a boundary regression model). This result is then used to analyse goodness-of-fit tests for parametric classes of error distributions.
\end{remark}
\begin{remark}
Note that we can consider the minimax risk over the whole Hölder class $\mathcal{C}^\beta(R)$ in the case of estimating the norm $\|g\|_p$. In distinction to Corollary \ref{CorPowInt}, the upper bound does not depend on any $L^q$-norm of $g$.
Inspecting the proof, we see more precisely that the minimax rate is driven by functions whose $L^p$-norm is smaller than $n^{-(\beta+1/(2p))/(\beta+1)}$. For functions which have a substantially larger norm we get the rate of convergence $n^{-(\beta+1/2)/(\beta+1)}$ corresponding to a smooth functional. This is explained by the fact that the $L^p$-norm is a non-smooth functional at $g=0$.
\end{remark}
\begin{remark} There is a close connection between Theorem \ref{TheoremEstNorm} and Theorem \ref{TheoremTesting}. First, the upper bound in Theorem \ref{TheoremTesting} follows from Theorem \ref{TheoremEstNorm} by using e.g. \cite[Proposition 2.17]{IngsterSuslina03}. Second, the lower bound in Theorem \ref{TheoremEstNorm} is a consequence of the lower bound in Theorem \ref{TheoremTesting}.
\end{remark}
|
1,108,101,565,680 | arxiv | \section{Introduction}
In the NLP literature, neural networks generally conduct a fixed number of computations over all words in a sentence, regardless of whether they are easy or difficult.
In terms of both efficiency and ease of learning, it is preferable to dynamically vary the numbers of computations according to the hardness of input words \cite{UT_2019}.
\citeauthor{ACT_2016} (\citeyear{ACT_2016}) firstly proposes adaptive computation time (ACT) to improve efficiency of neural networks.
Specifically, ACT employs a halting unit upon each word when processing a sentence, then this halting unit determines a probability that computation should continue or stop layer-by-layer.
Its application to sequence processing is attractive and promising.
For instance, ACT has been extended to reduce computations either by exiting early or by skipping layers for the ResNet \cite{ACT_resnet_2017}, the vanilla Transformer \cite{depth_ada_transformer_2020}, and the Universal Transformer \cite{UT_2019}.
However, there is no explicit supervision to directly train the halting unit of ACT, and thus how to measure the hardness of input words and decide required depths is the key point. Given a task, previous works generally treat the loss from different layers as a measure to implicitly estimate the required depths, {\em e.g.,} gradient estimation in ACT, or reinforcement rewards in SkipNet \cite{skipnet_2018}. Unfortunately, these approaches may lead to inaccurate depth selections with high variances, and thus unstable performance.
More recently, the depth-adaptive Transformer \cite{depth_ada_transformer_2020} directly trains the halting unit with the supervision of `pseudo-labels', which is generated by comparing task-specific losses over all layers. Despite its success, the
depth-adaptive Transformer still relays on a halting unit, which brings additional computing costs for depth predictions, hindering its potential performance.
In this paper, we get rid of a halting unit when building our model, and thus no additional computing costs need to estimate depth. Instead, we propose two approaches to explicitly estimate the required depths in advance, which yield a faster depth-adaptive Transformer.
Specifically, the MI-based approach calculates the mutual dependence between a word and all categorical labels. The larger the MI value of the word is, the more information of labels is obtained through observing this word, thus fewer depths are needed to learn an adequate representation for this word, and vice versa. Due to the MI-based approach is purely conducted in the data preprocessing stage, the computing cost is ignorable when compared with training a neural model in the depth-adaptive Transformer.
The reconstruction loss based approach measures the hardness of learning a word by reconstructing it with its contexts in a sentence. The less reconstruction loss of the word is, the more easily its representation is learned. Therefore the index of the layer with minimum reconstruction loss can be regarded as an approximation for required depths.
As a by-product, the reconstruction loss based approach is easy to apply to unsupervised scenarios, as it needs no task-specific labels.
Both of the above approaches aim to find a suitable depth estimation. Afterward, the estimated depths are directly used to guide our model to conduct corresponding depth for both training and testing.
Without loss of generality, we base our model on the Transformer encoder.
Extensive experiments are conducted on the text classification task (24 datasets in various sizes and domains). Results show that our proposed approaches can accelerate the vanilla Transformer up to 7x, while preserving high accuracy. Furthermore, we improve the efficiency and robustness of previous depth-adaptive models.
Our main contributions are as follows\footnote{Codes will appear at https://github.com/Adaxry/Adaptive-Transformer}:
\begin{itemize}
\item We are the first to estimate the adaptive depths in advance and do not rely on a halting unit to predict depths.
\item We propose two effective approaches to explicitly estimate the required computational depths for input words. Specifically, the MI-based approach is computing efficient and the reconstruction loss based one is also applicable in unsupervised scenarios.
\item Both of our approaches can accelerate the vanilla Transformer up to 7x, while preserving high accuracy. Furthermore, we improve previous depth-adaptive models in terms of accuracy, efficiency, and robustness.
\item We provide thorough analyses to offer more insights and elucidate properties of our approaches.
\end{itemize}
\section{Model}
\subsection{Depth Estimation}
\label{how_to_estimate_sec}
In this section, we introduce how to quantify the hardness of learning representations for input words and obtain corresponding estimated depths.
\paragraph{Mutual Information Based Estimation.}
Mutual Information (MI) is a general concept in information theory. It measures the mutual dependence between two random variables $X$ and $Y$. Formally, the MI value is calculated as:
\begin{equation}
\mathrm{MI}(X ; Y)=
\sum_{y \in Y} \sum_{x \in X} p_{(X, Y)}
\cdot \log (\frac{p_{(X, Y)}(x, y)}{p_{X}(x) \cdot
p_{Y}(y)} )
\end{equation}
where $p_{(X,Y)}$ is the joint probability of $X$ and $Y$, and $p_{X}$ and $p_{Y}$ are the probability functions of $X$ and $Y$ respectively.
MI has been widely used for feature selection in the statistic machine learning literature \cite{mi_feature_selection_2005}.
In our case of text classification, $X$ is the set of all words, and $Y$ is the set of predefined labels. Given a word $x \in X$ and a label $y \in Y$, the value of $\mathrm{MI}(x,y)$ measures the degree of dependence between them. The larger $\mathrm{MI}(x,y)$ is, the greater certainty between this word $x$ and label $y$ is, and thus fewer computations are needed to learn an adequate representation for $x$ to predict $y$. For example, the word `terrible' can decide a `negative' label with high confidence in sentiment analysis tasks, and thus it is unnecessary to conduct a very deep transformation when processing words with high MI values, and vice versa.
Namely, we force our models not to merely focus on a few `important' words and pay more attention to overview contexts when learning the representation of a sentence.
In this way,
our models avoid overfilling limited `important' words, which also takes an effect of regularization, and thus improve generalization and robustness.
Based on the above assumptions, it is intuitive and suitable to choose MI to quantify the difficulty of learning a word.
Formally, given a dataset with vocab $X$ and label set $Y$, the MI value $\mathrm{MI}(x)$ for word $x$ is calculated as:
\begin{equation}
\begin{split}
\mathrm{MI}(x) = &
\sum_{y \in\{Y \}}
\sum_{i_{x} \in\{0,1 \}}
\sum_{i_{y} \in\{0,1 \}}
P(i_{x}, i_{y}) \\ & \cdot \log \left(\frac{
P\left(i_{x}, i_{y}\right)} {P\left(i_{x}\right) \cdot P\left(i_{y}\right)}
\right)
\end{split}
\label{eqn_mi_cls}
\end{equation}
where $i_{x}$ is a boolean indicator whether word $x$ exists in a sentence. Similarly, $i_{y}$ indicates the existence of label $y$.
In practice, the probability formulas $P(\cdot)$ in Equation (\ref{eqn_mi_cls}) are calculated by frequencies of words, labels, or their combinations. A smooth factor (0.1 in our experiments) is introduced to avoid zero division. To avoid injecting information of golden labels when testing, we only use the training set to calculate MI values,
Once the MI value $\mathrm{MI}(x)$ for each word is obtained, we proceed to generate the pesudo-label of depth distribution $d(x)$ accordingly. As the histogram of MI values shown in Figure \ref{long_tail} (the upper part), there is an obvious long tail phenomenon, which manifests that the distribution is extremely imbalanced. To alleviate this issue, we perform a logarithmic scaling for the original $\mathrm{MI}(x)$ as:
\begin{equation}
\begin{split}
\operatorname{MI}_{log}(x) = -\log \left(
\operatorname{MI}(x)
\right)
\end{split}
\end{equation}
Next, according to the scaled $\mathrm{MI}_{log}(x)$, we uniformly divide all words into $N$ bins \footnote{We set $N$ to 12 for the compatibility of BERT.} with fixed-width margin, where $N$ denotes a predefined number of bins ({\em i.e.,} maximum depth). Consequently, the estimated depth value $d(x)$ for word $x$ is the index of corresponding bins.
The MI-based approach is purely calculated at the data preprocessing stage, thus it is highly efficient in computation and does not rely on additional trainable parameters.
\paragraph{Reconstruction Loss Based Estimation.}
Generally in a sentence, several words may bring redundant information that has been included by their contexts. Thus if we mask out these trivial words, it would be easier to reconstruct them than others. Namely, The less reconstruction loss of a word is, the more easily its representation is learned.
Based on the above principle, we utilize this property of reconstruction loss to quantify the hardness of learning the representation for input words and then estimate their required depths.
Firstly, we finetune BERT \cite{bert_2019} with a masked language model task (MLM) on datasets of downstream tasks. Note that we modify BERT to make predictions at any layers with a shared classifier, which is also known as {\em anytime prediction} \cite{huang2017multi,depth_ada_transformer_2020}.
The losses from all layers are summed up \footnote{We experimented with different weights ({\em e.g.,} random sample, or linearly decaying with the number of layers) for different layers, and finally choose the simple equal weights.} to the final loss.
After finetuning the MLM, given an input sentence $\boldsymbol{x}$ with $|\boldsymbol{x}|$ words, we sequentially replace each word $\boldsymbol{x}_t$ ($t \in [1,|\boldsymbol{x}|]$) with a special symbol {\tt <MASK>}, and then feed the sentence with a {\tt <MASK>} into the MLM.
Finally, the index of a layer with the minimum loss is selected as the estimated depth value $d(\boldsymbol{x}_t)$:
\begin{equation}
\label{log_scale_mi}
\begin{split}
d(\boldsymbol{x}_t) = \mathop{\arg\min}_{n}(loss_{n} - \lambda n)
\end{split}
\end{equation}
where $n \in N$ is the index of layer, $loss_{n}$ is the loss of {\tt <MASK>} in the $n$-th layer, and $\lambda n$ is the penalty factor to encourage a lower selection
\footnote{We elaborate the effect and choice of $\lambda$ in the following analytical Section.}.
Specifically, we train MLMs following the experimental setup of BERT \cite{bert_2019} with two major differences: 1) We make predictions at every layer with a shared classifier instead of only at the final layer in BERT; 2) We remove the next sentence prediction task following RoBERTa \cite{liu2019roberta}.
\begin{figure}[t!]
\begin{center}
\scalebox{0.46}{
\includegraphics[width=1\textwidth]{long_tail_clipped_aaai21.pdf}
}
\caption{
The histogram of MI values of partial words from IMDB (the upper part), and corresponding depths of these words by using the reconstruction loss based estimation (the bottom part).
} \label{long_tail}
\end{center}
\end{figure}
\paragraph{Comparisons Between the Two Approaches.}
Although the above approaches perform differently, they both serve as a measure to estimate required depths for input words from the perspective of learning their representations. We proceed to make a detailed comparison between the two approaches.
In the term of computational cost, the MI-based approach calculates MI values, and then stores the word-depth pairs that resemble word embeddings. The above procedures merely happen at the stage of data preprocessing, which requires trivial computational cost and does not rely on additional trainable parameters, and thus the MI-based approach is highly efficient in computation. In contrast, the reconstruction loss based approach needs to train several MLMs with \textit{anytime prediction}, which yields extra computational costs. Considering the MLMs are dependent on the main model, the calculation of depths can be conducted in advance in a piplined manner.
As the histogram shown in Figure \ref{long_tail}, we observe different preferences between the two estimations. Firstly, the MI-based approach (upper part) tends to assign higher MI values to label-relevant words ({\em e.g.,} opinion words `perfect' and `horrible' in IMDB). After the scaling function described by Equation (3), these opinion words are assigned a lower number of depths, namely fewer computational steps.
Such operations make our models not only focus on a few `important' words, but also pay more attention to the overview contexts, which takes an effect of regularization, and thus improve generalization and robustness.
Unlike the bias for label-related words in the MI-based approach, the reconstruction based approach (bottom part in Figure \ref{long_tail}) purely relies on the unsupervised context to measures the hardness of learning, which is good at recognizing common words ({\em e.g.,} `today', `one' and `me'), and assigns a smaller number of computations, and vice versa.
As a by-product, the reconstruction loss based approach is applicable to unsupervised scenarios, as it needs no task-specific labels.
\begin{figure}[t!]
\begin{center}
\scalebox{0.4}{
\includegraphics[width=1\textwidth]{explicit_AAAI2021.pdf}
}
\caption{The overview of our depth-adaptive Transformer.
Once a word $\boldsymbol{x}_t$ achieve its own depth $d(\boldsymbol{x}_t)$, it will simply copy states to upper layers. }
\label{overview}
\end{center}
\end{figure}
\begin{table*}[t!]
\begin{center}
\scalebox{1}{
\begin{tabular}{l|c c c c c c c}
\hline
\textbf{Dataset} & Classes & Type & \makecell{Average \ \ \\ \ \ Lenghts \ \ \ } & \makecell{\ \ Max \ \ \\ \ \ Lengths} &
\makecell{Train \ \ \\ \ \ Sample \ \ } & \makecell{Test \ \ \\ \ \ Sample \ \ } \\
\hline
TREC \cite{trec_2002} & 6 & Question & 12 & 39 & 5,952 & 500 \\
AG’s News \cite{char_cnn_2015} & 4 & Topic & 44 & 221 & 120,000 & 7,600 \\
DBPedia \cite{char_cnn_2015} & 14 & Topic & 67 & 3,841 & 560,000 & 70,000 \\
Subj \cite{subj_2004} & 2 & Sentiment & 26 & 122 & 10,000 & CV \\
MR \cite{MR_2005} & 2 & Sentiment & 23 & 61 & 10,622 & CV \\
Amazon-16 \cite{16_cls_data_17} & 2 & Sentiment & 133 & 5,942 & 31,880 & 6,400 \\
IMDB \cite{IMDB_2011} & 2 & Sentiment & 230 & 2,472 & 25,000 & 25,000 \\
Yelp Polarity \cite{char_cnn_2015} & 2 & Sentiment & 177 & 2,066 & 560,000 & 38,000 \\
Yelp Full \cite{char_cnn_2015} & 5 & Sentiment & 179 & 2,342 & 650,000 & 50,000 \\
\hline
\end{tabular}}
\end{center}
\caption{Dataset statistics. `CV' refers to 5-fold cross-validation. There are 16 subsets in Amazon-16.
}
\label{data_statistics}
\end{table*}
\subsection{Depth-Adaptive Mechanism}
As the overview shown in Figure \ref{overview}, we stack $N$ layers of the Transformer encoder to model a sentence. The Transformer encoder consists of two sub-layers in each layer. The first sub-layer is a multi-head dot-product self-attention and the second one is a position-wise fully connected feed-forward network. We refer readers to the original paper \cite{Transformer_2017} for more details.
To make sure all hidden states of the same layer are available to compute self-attention, once a word $\boldsymbol{x}_t$ reaches its own maximal layer $d(\boldsymbol{x}_t)$, it will stop computation, and simply copy its states to the next layer until all words stop or the
maximal layer $N$ is reached. Formally, at the $n$-th layer, for the word $x_t$, its hidden state $\boldsymbol{h}_i^{n}$ are updated as follows:
\begin{equation}
\boldsymbol{h}_t^{n} =
\begin{cases}
\boldsymbol{h}_{t}^{n-1} \text{ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if $ n > d(x_t) $} \\
\operatorname{Transformer}( \boldsymbol{h}_{t}^{n-1}) \text{\ \ \ \ \ \ \ \ \ \ else}
\end{cases}
\end{equation}
where $n \in [1, N]$ refers to the index of the layer.
Especially, $\boldsymbol{h}_{t}^{0}$ is initialized by the BERT embedding.
\subsection{Task-specific Settings}
After dynamic steps of computation for each word position, we make task-specific predictions upon the maximal stop layer $n_{max} \in [1, N]$ among all word positions. The feature vector $\boldsymbol{v}$ consists of mean and max pooling of output hidden states $\boldsymbol{h}^{n_{max}}$, and is activated by ReLU. Finally, a softmax classifier are built on $\boldsymbol{v}$. Formally, the above-mentioned procedures are computed as follows:
\begin{equation}
\begin{split}
& \boldsymbol{v} = \operatorname{ReLU}([\mathop{\max}(\boldsymbol{h}^{n_{max}}); \operatorname{mean} ({\boldsymbol{h}}^{n_{max}})]) \\
& P(\widetilde{y} | \boldsymbol{v}) = \operatorname{softmax}(\boldsymbol{W} \boldsymbol{v} + \boldsymbol{b})
\label{equ_cls_pred}
\end{split}
\end{equation}
where $\boldsymbol{W} \in \mathbb{R}^{d_{model} \times |S|}$ and $\boldsymbol{b} \in \mathbb{R}^{|S|}$ are parameters of the classifier, $|S|$ is the size of the label set, and $P(\widetilde{y} | \boldsymbol{v})$ is the probability distribution.
At the training stage, we use the cross-entropy loss computed as:
\begin{equation}
\begin{split}
& Loss = - \sum\limits_{i=1}^{|S|}{y_i} log(P_i(\widetilde{y} | \boldsymbol{v}))
\end{split}
\label{equ_task_loss}
\end{equation}
where $y_i$ is the golden label.
For testing, the most probable label $\hat{y}$ is chosen from above probability distribution described by Equation (\ref{equ_cls_pred}):
\begin{equation}
\begin{split}
& \hat{y} = \mathop{\arg\max} P (\widetilde{y} | \boldsymbol{v} ) \\
\end{split}
\label{background_intent_pred}
\end{equation}
\section{Experiments}
\subsection{Task and Datasets}
Text classification aims to assign a predefined label to text \cite{char_cnn_2015}, which is a classic task for natural language processing and is generally evaluated by accuracy score. Generally, The number of labels may range from two to more, which corresponds to binary and fine-grained classification. We conduct extensive experiments on the 24 popular benchmarks collected from diverse domains ({\em e.g.,} \textit{topic}, \textit{sentiment}) ranging from modestly sized to large-scaled. The statistics of these datasets are listed in Table \ref{data_statistics}.
\begin{table*}[t!]
\begin{center}
\scalebox{0.85}{
\begin{tabular}{l|c c c c c c c}
\hline
Data / Model & \makecell{Multi-Scale \ \ \\ \ \ Transformer \ \ \ } & \makecell{Star- \ \ \\ \ \ Transformer \ \ \ } & Transformer & \makecell{Transformer \ \ \\ \ \ w/ halting unit \ \ \ } & \makecell{Transformer \ \ \\ \ \ w/ MI estimation \ \ \ } & \makecell{Transformer \ \ \\ \ \ w/ reconstruction \ \ \ } \\
\hline
Apparel & 86.5 & 88.7 & \textbf{91.9} & 91.6 & 91.4 & 91.8 \\
Baby & 86.3 & 88.0 & 88.8 & 88.1 & \textbf{90.6} & 88.4 \\
Books & 87.8 & 86.9 & 89.5 & 88.3 & \textbf{89.6} & 89.5 \\
Camera & 89.5 & 91.8 & 91.8 & 92.7 & \textbf{93.8} & 92.9 \\
Dvd & 86.5 & 87.4 & 88.3 & \textbf{91.7} & 91.4 & 91.5 \\
Electronics & 84.3 & 87.2 & \textbf{90.8} & 89.3 & 90.6 & 90.2 \\
Health & 86.8 & 89.1 & \textbf{91.7} & 91.3 & 91.6 & 88.4 \\
Imdb & 85.0 & 85.0 & 88.3 & 89.8 & 89.5 & \textbf{90.6} \\
Kitchen & 85.8 & 86.0 & 87.6 & 88.1 & 89.2 & 86.8 \\
Magazines & 91.8 & 91.8 & 94.2 & \textbf{94.8} & 94.6 & 94.7 \\
Mr & 78.3 & 79.0 & \textbf{83.7} & 81.6 & 82.3 & 82.5 \\
Music & 81.5 & 84.7 & \textbf{89.9} & 89.3 & 89.5 & 87.2 \\
Software & 87.3 & 90.9 & 91.2 & 92.9 & 92.3 & \textbf{93.8} \\
Sports & 85.5 & 86.8 & 87.1 & 89.2 & 88.4 & \textbf{89.8} \\
Toys & 87.8 & 85.5 & 89.7 & 90.3 & \textbf{90.9} & 89.7 \\
Video & 88.4 & 89.3 & 93.4 & \textbf{94.3} & 93.1 & 93.5 \\
\hline
Avg & 86.2 & 87.4 & 89.9 & 90.2 & \textbf{90.5} & 90.1 \\
\hline
\end{tabular}}
\end{center}
\caption{Accuracy scores (\%) on the Amazon-16 datasets. Best results on each dataset are bold. The results of Multi-Scale Transformer \cite{ms_transformer} is cited from the original paper, and other results are our implementations with several recent advanced techniques ({\em e.g.,} BERT initialization) under the unified setting.
}
\label{amazon16_result}
\end{table*}
\subsection{Implementation Details}
For the MI-based estimation approach, we calculate word-depth pairs on the training set in advance and then calculate depths for words in the test set. For the reconstruction based approach, we calculate word-depth pairs for both train and test set without using label information.
The penalty factor $\lambda $ in the reconstruction loss based approach is set to 0.1.
Dropout \cite{dropout_2014} is applied to word embeddings, residual connection , and attention scores with a rate of 0.1. Models are optimized by the Adam optimizer \cite{Adam_2014} with gradient clipping of 5 \cite{gradient_clip_2013}.
$BERT_{base}$ is used to initialize the Transformer encoder. Long sentences exceed 512 words are clipped.
\subsection{Main Results}
\begin{table*}[t!]
\begin{center}
\scalebox{0.9}{
\begin{tabular}{l|c c c c c c c c |c}
\hline
{Models / Dataset} & TREC & MR & Subj & IMDB & AG. & DBP. & Yelp P. & Yelp F. & Avg.\\
\hline
RCRN \cite{RCRN_2018} & 96.20 & -- & -- & 92.80 & -- & -- & -- & -- & -- \\
Cove \cite{Cove_2017} & 95.80 & -- & -- & 91.80 & -- & -- & -- & -- & -- \\
Text-CNN \cite{textcnn_2014} & 93.60 & 81.50 & 93.40 & -- & -- & -- & -- & -- & -- \\
Multi-QT \cite{MR_efficient_2018} & 92.80 & 82.40 & 94.80 & -- & -- & -- & -- & -- & -- \\
AdaSent \cite{subj_self_2015} & 92.40 & 83.10 & 95.50 & -- & -- & -- & -- & -- & -- \\
CNN-MCFA \cite{subj_trans_2018} & 94.20 & 81.80 & 94.40 & -- & -- & -- & -- & -- & -- \\
Capsule-B \cite{capsule_2018} & 92.80 & 82.30 & 93.80 & -- & 92.60 & -- & -- & -- & -- \\
DNC+CUW \cite{less_memory_2019} & -- & -- & -- & -- & 93.90 & -- & 96.40 & 65.60 & -- \\
Region-Emb \cite{region_emb_2018} & -- & -- & -- & -- & 92.80 & 98.90 & 96.40 & 64.90 & -- \\
Char-CNN \cite{char_cnn_2015} & -- & -- & -- & -- & 90.49 & 98.45 & 95.12 & 62.05 & -- \\
DPCNN \cite{DPCNN_2017} & -- & -- & -- & -- & 93.13 & 99.12 & 97.36 & 69.42 & -- \\
DRNN \cite{DRNN_2018} & -- & -- & -- & -- & 94.47 & 99.19 & 97.27 & 69.15 & -- \\
SWEM-concat \cite{swem_2018} & 92.20 & 78.20 & 93.00 & -- & 92.66 & 98.57 & 95.81 & 63.79 & -- \\
Star-Transformer \cite{star_transformer_2019} $\dagger$ & 93.00 & 79.76 & 93.40 & 94.52 & 92.50 & 98.62 & 94.20 & 63.21 & 88.65 \\
BERT \cite{bert_2019} & -- & -- & -- & 95.49 & -- & 99.36 & 98.11 & 70.68 & -- \\
XLNet \cite{XLNet_2019} & -- & -- & -- & \textbf{96.80} & 95.55 & \textbf{99.40} & \textbf{98.63} & 72.95 & -- \\
\hline
Transformer \cite{Transformer_2017} $\dagger$ & 96.00 & 83.75 & 96.00 & 95.58 & 95.13 & 99.22 & 98.09 & 69.80 & 91.69 \\
\ \ \ \ w/ Halting unit \cite{depth_ada_transformer_2020} $\dagger$ & 95.80 & 83.23 & 96.00 & 95.80 & 95.50 & 99.30 & 98.25 & 69.75 & 91.70 \\
\ \ \ \ w/ MI estimation (ours) $\dagger$ & \textbf{96.50} & \textbf{84.20} & 96.00 & 96.72 & \textbf{95.90} & 99.32 & 98.10 & \textbf{72.98} & \textbf{92.46} \\
\ \ \ \ w/ Reconstruction estimation (ours) $\dagger$ & 96.20 & 83.90 & \textbf{96.30} & 96.60 & 95.65 & 99.25 & 98.00 & 69.58 & 91.93 \\
\hline
\end{tabular}}
\end{center}
\caption{Accuracy scores (\%) on modestly sized and large-scaled datasets. `AG.', `DBP.', `Yelp P.' and `Yelp F.' are the abbreviations of `AG's News`, `DBPedia', `Yelp Polarity' and `Yelp Full', respectively. $\dagger$ is our implementations with several recent advanced techniques and \textit{analogous} parameter sizes. `Transformer' is the Transformer encoder initialized by $BERT_{base}$ with 12 fixed layers.
}
\label{cls_result}
\end{table*}
\begin{figure*}[t!]
\begin{center}
\scalebox{1}{
\includegraphics[width=1\textwidth]{acc_speed.png}
}
\caption{Accuracy scores (a) and speed (b) for each model on IMDB when $N \in [1,12]$. The solid line indicates the mean performance, and the size of the colored area indicates variance (used to measure robustness). `speed' is the number of samples calculated in ten-second on one Tesla P40 GPU with the batch size of 1.
} \label{fig_acc_speed}
\end{center}
\end{figure*}
\paragraph{Results on Amazon-16.}
Amazon-16 consists of consumer comments from 16 different domains ({\em e.g.,} Apparel).
We compare our approaches with different baseline models in Table \ref{amazon16_result}. The Multi-Scale Transformer \cite{ms_transformer} is designed to capture features from different scales, and the Star-Transformer \cite{star_transformer_2019} is a lightweight Transformer with a star-shaped topology. Due to the absence of a powerful contextual model ({\em e.g.,} BERT), their results underperform others by a margin. The Transformer model is finetuned on BERT and conducts fixed 12 layers for every instance, which yields a strong baseline model. Following the setup of depth-adaptive Transformer, we add a halting unit on the bottom layer of the Transformer encoder, and generate a `pesudo-label' for the halting unit by classification accuracy. Our approaches (the last two columns in Table \ref{amazon16_result}) achieve better or comparable performance over these strong baseline models. The MI-based approach also takes a regularization effect, and thus it achieves better performance than the reconstruction counterpart.
\paragraph{Results on Larger Benchmarks.}
Although the Amazon-16 benchmark is challenging, its small data size makes the results prone to be unstable, therefore we conduct experiments on larger benchmarks for a more convincing conclusion. In this paragraph, we only focus on the classification accuracy listed in Table \ref{cls_result}, and more detailed results about computing speed and model robustness will be discussed in the next section.
The upper part of Table \ref{cls_result} lists several high-performance baseline models. Their detailed descriptions are omitted here. In terms of accuracy, our approaches achieve comparable performance with these state-of-the-art models.
At the bottom part of Table \ref{cls_result}, we finetune BERT as our strong baseline model. Results show that this baseline model performs on par with the state-of-the-art XLNet \cite{XLNet_2019}. Then we build a halting unit at the bottom of the baseline model under the same setup with the depth-adaptive Transformer. Results show that applying a halting unit has no obvious impact on accuracy. The last two rows list results of our estimation approaches, where the MI-based approach brings in consistent improvements over the baseline and the Transformer w/ a halting unit, by +0.77\% and +0.76\% on average, respectively. We speculate the improvements mainly come from the additional deep supervision and regularization effect of the MI-based approach. In contrast, the reconstruction based approach only show improvements over the baseline model (+0.24\%) and the Transformer w/ a halting unit (+0.23\%) by a small margin.
\section{Analysis}
We conduct analytical experiments on the modestly sized IMDB to offer more insights and elucidate the properties of our approaches.
\subsection{Effect of the maximum number of layer}
\label{speed_acc_section}
Firstly, we train several fixed-layer Transformers with $N$ ranging from one to twelve, and then build a halting unit on the above Transformers to dynamically adjust the actual number of layers to conduct. Meanwhile, we respectively utilize our two approaches on the fixed-layer Transformer to activate dynamic layers. Note that each model is trained with different random initialization three times and we report the mean and variance. Here, we take the variance value to measure the robustness against the random initialization and different depth selections. As drawn in Figure \ref{fig_acc_speed}, solid lines are the mean performance, and the size of the colored areas indicate variances.
\begin{figure}[t!]
\begin{center}
\scalebox{0.45}{
\includegraphics[width=1\textwidth]{speed_batch.png}
}
\caption{
Speed for each model on IMDB when $batch size \in [1,15]$. The solid line indicates the mean performance, and the size of the colored area indicates variance (used to measure robustness). `speed' is the number of samples calculated in ten seconds on one Tesla P40 GPU.
} \label{speed_batch}
\end{center}
\end{figure}
\paragraph{Accuracy and Robustness.}
Results of accuracy and robustness are drawn in Figure \ref{fig_acc_speed} (a). In the lower layers ($N \in [1,6]$), as the searching space for depth selection is small, the depth-adaptive models perform worse than the Transformer baseline. In contrast, when $N \in [6,12]$, the depth-adaptive models come up with the baseline. Due to the additional depth supervision and the regularization effect, the application of our approaches can further significantly improve accuracy and robustness over both the Transformer and w/ a halting unit. (green and blue lines vs. purple line in Figure \ref{fig_acc_speed} (a))
\paragraph{Speed and Robustness.}
Figure \ref{fig_acc_speed} (b) shows the speed and robustness of each model. The speed of vanilla Transformer almost linearly decays with the growth of the number of layers. As $N \in [1,3]$, due to the additional prediction for depths, models w/ halting unit runs a bit slower than the baseline. However, the superiority of adaptive depths becomes apparent with the growth of the number of layers. In particular, as $N = 12$, the model w/ halting run 3.4x faster than the fixed-layer baseline (pure lines vs. red line in Figure \ref{fig_acc_speed} (b)). As our approaches free the dependency for depth prediction and can further speed up the model, both of our approaches run about 7x faster than the fixed-layer baseline (green and blue lines vs. red line in Figure \ref{fig_acc_speed} (b)). In addition, our approaches perform more robust in the term of speed gains than the Transformer w/ a halting unit.
\begin{figure}[t!]
\begin{center}
\scalebox{0.45}{
\includegraphics[width=1\textwidth]{case_study_aaai21.pdf}
}
\caption{
The histogram of the depth distribution of a case from IMDB, which is estimated by our approaches.
} \label{case}
\end{center}
\end{figure}
\subsection{Speed on different batch size}
The depth-adaptive models conduct dynamic computations at each word position, and thus the actually activated depths are decided by the maximal depth value. As a result, when the batch size gets larger, the final activated depth may potentially become larger as well, which may hurt the effectiveness of depth-adaptive models. In this section, we fix the maximal number of layer $N$ to 12, and then compare the speed of each model. As shown in Figure \ref{speed_batch}, the speed gain of the depth-adaptive models (green, blue and purple lines in Figure \ref{speed_batch}) grows slower than the vanilla Transformer (red line in Figure \ref{speed_batch}). However, the absolute speed of depth-adaptive models is still much faster than the vanilla Transformer. We leave the further improvement of depth-adaptive models on larger batch sizes to future works.
\subsection{Effect of penalty factor $\lambda$}
\label{sec_effect_of_lambda}
If no constraints are applied on the depth selection, the reconstruction loss based approach tends to choose a layer as deep as possible, and thus an extra penalty factor $\lambda$ is necessary to encourage a lower choice. We simply search $\lambda \in [0,0.2]$, and finally set it to 0.1 for a good accuracy-speed trade-off.
The detailed results are list in Table \ref{effect_of_lambda}.
\subsection{Case Study}
We choose a random sentence from the IMDB dataset, and show the estimated depths outputted by both approaches in Figure \ref{case} (upper part). We observe that the MI-based estimation tends to assign a smaller number of depths for opinion words, {\em e.g.,} `anticipated' and `thriller'. While the reconstruction loss based estimation is prone to omit common words, e.g., `and'.
\begin{table}[t!]
\begin{center}
\scalebox{0.95}{
\begin{tabular}{l|c c c c c c c}
\hline
$\lambda$ & 0 & 0.05 & 0.10 & 0.15 & 0.20 \\
\hline
accuracy & 96.54 & 96.31 & 96.55 & 96.27 & 96.29 \\
speed & 23 & 33 & 48 & 54 & 58 \\
average depth & 9.5 & 6.3 & 4.5 & 3.9 & 3.6 \\
\hline
\end{tabular}}
\end{center}
\caption{Effect of penalty factor $\lambda$. The definition of `speed' is same as that in Figure \ref{fig_acc_speed}. `average depth' is the average predicted depth of words in test set.}
\label{effect_of_lambda}
\end{table}
\section{Related Work}
Our work is mainly inspired by ACT \cite{ACT_2016}, and we further explicitly train the halting union with the supervision of estimated depths. Unlike Universal Transformer \cite{UT_2019} iteratively applies ACT on the same layer, we dynamically adjust the amount of both computation and model capacity.
A closely related work named `Depth-Adaptive Transformer' \cite{depth_ada_transformer_2020} uses task-specific loss as an estimation of depth selection.
Our approaches are different from it in three major aspects: 1) We get rid of the halting unit and remove the additional computing cost for depths, thus yield a faster depth-adaptive Transformer; 2) our MI-based estimation does not need to train an extra module, and is highly efficient in computation; 3) our reconstruction loss based estimation is unsupervised, and can be easily applied on general unlabeled texts. Another group of works also aims to improve efficiency of neural network through reducing the entire layers, {\em e.g.,} DynaBERT \cite{dynabert_2020}, LayerDrop \cite{layerdrop_2019} and MobileBERT \cite{mobilebert_2020}. In contrast, our approaches perform adaptive depths in the fine-grained word level.
\section{Conclusion}
We get rid of the halting unit and remove the additional computing cost for depths, thus yield a faster depth-adaptive Transformer. Specifically, we propose two effective approaches 1) mutual information based estimation and 2) reconstruction loss based estimation.
Experimental results confirm that our approaches can speed up the vanilla Transformer (up to 7x) while preserving high accuracy.
Moreover, we significantly improve previous depth-adaptive models in terms of accuracy, efficiency, and robustness.
We will further explore the potential improvement of the depth-adaptive Transformer when facing larger batch size in future work.
\section{Acknowledgments.}
The research work described in this paper has been supported by the National Key R\&D Program of China (2020AAA0108001) and the National Nature Science Foundation of China (No. 61976015, 61976016, 61876198 and 61370130). The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve this paper.
|
1,108,101,565,681 | arxiv | \section{Introduction}
Methods for prediction of instability and transition have evolved considerably during the past several decades. Advances, driven by increases in computer speed and memory, include the availability of high-fidelity DNS and LES solutions for canonical wall-bounded flows \citep{sayadi2013direct}, the recognition of transient growth (non-modal instability) as a key mechanism and mathematical formulations for {\it optimal} disturbances in linear and nonlinear frameworks \citep{SchmidHenningson2001,schmid2007nonmodal,kerswell2018nonlinear}, and generalization of parallel-flow analysis to global approaches to flows that are inhomogeneous in two or more directions \citep{theofilis}.
Most of the stability studies concern linearised evolution of perturbations. For stable base flows, the physical mechanisms associated to linear growth mechanisms (modal and non-modal) and receptivity can be clarified by finding initial conditions in the time domain, or volumetric forcings, in the frequency domain, that maximize, for example, the kinetic energy of perturbations \citep{SchmidHenningson2001}. The frequency-space problem is also called linear resolvent analysis or input/output analysis in the literature. In these analyses, adjoint methods are used to maximize a specific cost function. \citet{trefethen1993hydrodynamic,jovanovic2005componentwise} showed that the computation of the optimal forcings and responses of the resolvent operator extracts the pseudo-resonances of a flowfield, that is the frequencies and spatial distributions of forcings that optimally trigger linear responses in a system. In a setup where the streamwise direction is also discretized (in addition to the cross-stream direction), accurate methods to extract the optimal features from the global resolvent have first been carried out with time-stepper approaches by \cite{blackburn2008convective,aakervik2008global,monokrousos2010global} and more recently with sparse direct LU methods by \cite{sipp2010dynamics,brandt2011effect,rigas2017one,SchmidtJFM2017}, among others.
Determining the growth of finite-amplitude perturbations is, of course, more challenging. In practice, the direct solution of the 3D Navier-Stokes equations in the time domain is most commonly employed. For example, \cite{rist1995direct} and \cite{bake2002turbulence} reproduced experimental results evidencing different forms of transition in the flat-plate boundary layer. More recently nonlinear transitional mechanisms have been studied by employing gradient-based techniques to find the smallest amplitude optimal initial conditions that trigger transition to turbulence \citep{cherubini2010rapid,cherubini2011minimal,pringle2012minimal,monokrousos2011nonequilibrium,kerswell2018nonlinear}. The optimal perturbation is calculated over a finite time interval and the one with the lowest energy is known as the minimal seed in the time domain. The results still depend on the specific metric (cost function) used to measure the growth; common choices include perturbation kinetic energy \citep{pringle2012minimal,cherubini2011minimal}, integral skin friction coefficient \citep{jahanbakhshi2018nonlinearly}, dissipation \citep{monokrousos2011nonequilibrium}, and mean shear \citep{karp2017secondary}.
The search for the minimal seed, while theoretically interesting, has no direct experimental counterpart. By analog with the linear approaches, it is experimentally more natural to model transition from laminar to turbulent flow as a stationary process where disturbances are continually supplied to the system from the environment, i.e. to consider the receptivity problem. For linear growth, this results in the aforementioned resolvent (or input/output) analysis that provides, in the frequency domain, a transfer function between inputs, for example environmental noise characterized by spatially localized spectral co-variance tensors, and outputs, for example the structure of the resulting amplified flow structures, and the net gain between them.
In order to deal with finite-amplitude perturbations in the frequency domain, the stability and numerical tools have to be extended to account for nonlinearity. Previous attempts in this regard have been limited to the nonlinear parabolized stability equations \citep[NPSE,][]{bertolotti1992linear,chang1994oblique}. While such calculations showed good agreement with DNS for the very early stages of transition, they require specific inlet conditions to be specified and these are typically based on modal solutions to the local (parallel) spatial stability problem. Furthermore, numerical instabilities and robustness issues, associated with the minimum step restriction, have limited the applicability of both PSE and NPSE \citep{towne2019critical}, and cast doubt on whether PSE can be used to identify optimal inlet conditions or volumetric forcing. The aforementioned work on non-modal mechanisms relies on cooperative amplification of modes with disparate wavelengths, which raises further questions about the appropriateness of PSE ansatz.
A natural generalization in order to calculate finite-amplitude perturbations in the frequency domain is to seek solutions to the full Navier-Stokes equations under the form of an expansion consisting of a mean-flow solution, a fundamental mode and $p$ harmonics of the fundamental, but without the parabolizing approximations inherent to PSE. Such an approach, known in literature as the harmonic balance method \citep[HBM,][]{khalil2002nonlinear,fabrepractical} is a general method to find periodic or quasi-periodic solutions. HBM has been used previously in fluid mechanics primarily in the context of turbomachinery \citep{hall2002computation,gopinath2007three, sicot2012time}, where one seeks a mean flow and harmonics associated with the externally imposed blade passing frequency. When used with $p=0$, HBM also recovers the \emph{self-consistent model} introduced by \citet{MLugo2014} and \citet{mantivc2016self} for the cylinder wake and backstep flow, respectively.
In this paper, using HBM we explore the optimal nonlinear amplification problem in the frequency domain, and we use the method to identify and analyze transition scenarios for the flat plate boundary layer. We begin in \S\ref{sec:bl} by briefly reviewing the literature on boundary layer transition. In \S\ref{sec:theory}, we propose a solution strategy for the following optimization problem. Given an amplitude $ A $, a time-period and spanwise-wavelength associated respectively to the fundamental frequency $ \omega $ and fundamental wavenumber $ \beta $, we look for a spatial distribution of a time-periodic (of period $ 2\pi/\omega $) and spanwise-periodic (of period $2\pi/\beta$) volumetric forcing of amplitude $ A $ that triggers a solution maximising the mean skin friction coefficient (integrated over the wall). In \S\ref{sec:Rist} we validate the HBM solver by reproducing a K-type transition scenario previously studied using DNS \citep{rist1995direct}, while in \S\ref{sec:linear_resolvent}, we validate the optimization procedure by reproducing previously reported linear optimal solutions. Finally, in section~\ref{sec:nonlinear_resolvent} we calculate nonlinear optimal reponses and forcings that maximize the skin friction coefficient. By varying $A$, $\omega$, $\beta$ and the forcing component combinations, we identify a range of optimal transition scenarios. We summarize our results in \S\ref{sec:conclusions}, and discuss prospects for transition prediction using HBM.
\section{Boundary layer transition: a brief review}
\label{sec:bl}
Early studies on zero-pressure gradient boundary layer transition have been mainly focused on the modal amplification of Tollmien-Schlichting (TS) waves. The primary TS-waves develop three-dimensional secondary instabilities, and subsequently break down to turbulence. The analysis of transition mechanisms resulting from the secondary instability of TS-waves has identified two main routes:
\begin{enumerate}
\item The fundamental K-type transition, which involves a 2D TS-wave $(\omega,0)$ and two oblique waves of the same frequency $ (\omega,\pm \beta)$. Such a resonance has first been evidenced by \cite{klebanoff1962three}.
\item The subharmonic H-type transition, triggered by a 2D TS-wave $(\omega,0)$ and two subharmonic oblique waves $ (\omega/2,\pm \beta)$. It has been experimentally observed by \citet{kachanov1977nonlinear,kachanov1984resonant}.
\end{enumerate}
In both cases, the oblique waves are strongly amplified, leading to $\Lambda$-shaped patterns composed of strong longitudinal vortices \citep{rist1995direct,berlin1999numerical,sayadi2013direct}. In the case of $H-$type transition, the $\Lambda-$patterns are staggered while they are aligned in the case of $K-$type transition \cite{herbert1988secondary, kachanov1994physical}. Many of the above transition characteristics can be explained by linear modal stability analysis. \cite{herbert1988secondary} examined the secondary stability characteristics of the periodic flow (Blasius flow with superimposed TS waves) using linear Floquet analysis in a local framework. The analysis showed that the growth of three-dimensional subharmonic frequency waves (seen for H-type) is favoured over fundamental waves (K-type).
More recent work shows that disturbances can undergo significant transient growth that leads to faster transition to turbulence, even at subcritical Reynolds numbers, and potentially bypassing transition through TS waves. A linear resolvent analysis for the Blasius boundary layer has been performed by \cite{monokrousos2010global} to identify optimal forcing in the frequency domain. Peaks of the optimal gain in the frequency/spanwise wavenumber space were linked to modal and non-modal instabilities. The analysis showed that maximum energy amplification is due to steady three-dimensional disturbances. The optimal forcing consists of streamwise vortices (rolls) and the response of streamwise elongated vortices, known as streaks. The amplification is a purely non-modal mechanism through the linear lift-up mechanism \citep{landahl1980note,butler1992three}. The non-modal analysis also shows that oblique TS waves are more amplified than the 2D ones, though these are linearly suboptimal to the aforementioned lift up mechanism.
Due to early observations that streaks can be significantly amplified and provide an alternative bypass route to turbulence, various studies have focused on the secondary instability of boundary layers distorted by streaks. \cite{andersson2001breakdown} performed an inviscid, secondary instability analysis of the optimally amplified boundary-layer streaks in a linear framework. Depending on the symmetries of the perturbed flow, varicose or sinuous oscillations of the low-speed streaks are possible, with the latter being the most unstable one. Once the streaks reach certain amplitude and become unstable, breakdown to a turbulent flow is observed \citep{brandt2002transition}. The sinuous mode has been linked to the spanwise shear which leads to the formation of streamwise vortices around the low-speed streaks. On the other hand, the varicose mode has been associated with wall-normal shear and the formation of symmetric hairpin vortices \citep{asai2002instability}.
An alternative bypass scenario for transition relies on oblique waves \citep{schmid1992new}. In this scenario, streamwise-aligned vortices are generated by non-linear interaction between a pair of oblique waves with equal angle but opposite sign in the flow direction \citep[]{schmid1992new,reddy1998stability,berlin1999numerical}. These vortices, in turn, induce streamwise streaks through the lift-up mechanism. The subsequent stages of transition to turbulence are similar to the ones described above for the streak breakdown. The initial stages of the nonlinear interaction of the oblique waves have been described also using NLPSE. \cite{chang1994oblique} showed that the oblique waves are a dominant mechanism at low supersonic speeds. Similarly to the incompressible regime, the nonlinear interaction of a pair of oblique waves results in the evolution of a streamwise vortex. This stage was described by a wave–vortex triad consisting of the oblique waves and a streamwise vortex whereby the oblique waves grow linearly while nonlinear effects result in the rapid growth of the vortex mode.
\section{Nonlinear input/output analysis: theory and algorithms} \label{sec:theory}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{BLsetup2.pdf}
\caption{Schematic of the zero-pressure gradient flat-plate set-up. Transition of the laminar boundary layer is triggered here either by boundary forcing (here wall blowing and suction) or volumetric momentum forcing. }
\label{fig:cartoon}
\end{figure}
In order to extend the linear input/output (resolvent) analysis to finite-amplitude perturbations, we need to proceed in two steps:
\begin{enumerate}
\item Devise a method to find, for a given time- and spanwise-periodic finite amplitude forcing, a time- and spanwise-periodic solution with the same periods that is solution to the forced nonlinear Navier-Stokes equations. For this, we will follow the HBM framework. The theory and numerical algorithms are presented in \S \ref{sec:HBM}.
\item Devise a method to search, over a fixed set of forcing and response frequencies, for an optimal forcing with a finite overall amplitude, $ A $, that maximizes a given cost-functional. Similarly to the optimization strategies followed in the time-domain \citep{kerswell2018nonlinear}, we use gradient-based strategies to find local maxima and optimal solutions in a few iterations (\S \ref{sec:optim}).
\end{enumerate}
\subsection{Nonlinear input/output relation in frequency space with Harmonic Balance Method} \label{sec:HBM}
The flow under consideration is the zero-pressure gradient boundary layer flow, shown schematically in figure~\ref{fig:cartoon}. The spanwise direction $z$ is treated as homogeneous and, without loss of generality, we will assume that the forcing and response are $z$-periodic, in addition to being $t$-periodic.
We consider the forced three-dimensional incompressible Navier-Stokes equations:
\begin{subeqnarray} \label{eq:goveq}
\partial_t \mathbf{u}+\mathbf{u}\cdot\nabla\mathbf{u}&=&-\nabla p+\nu\Delta\mathbf{u}+\mathbf{f}(\mathbf{x},t) \\
\nabla \cdot \mathbf{u}&=& 0 \\
\mathbf{u}&=&\mathbf{g}(\mathbf{x},t) \mbox{ on } \partial \Omega_f,
\end{subeqnarray}
where $ \mathbf{f} $ is a volumetric time-dependent momentum forcing and $ \mathbf{g} $ a time-dependent forcing on some boundary $ \partial \Omega_f $.
We apply no-slip boundary conditions along the plate and zero stress conditions at the outlet. At the inlet and at the upper boundary, we impose the Blasius profile.
The governing equations are discretised in the $x$ and $y$ spatial directions, using the finite-element method, while $z$ and $t$ are treated as continuous homogeneous directions. The discretization is carried out with the FreeFem++ software \citep{freefem}, with first-order $[P_{1b},P_{1b},P_{1b},P_{1}] $ (Mini) elements \citep{arnold1984stable} for a $\mathbf{w}=[u,v,w,p]$ element. In the discrete state space, the forcing and state variables are then vectors depending only on $z$ and $t$, while the explicit dependence on $x$ and $y$ defines the degrees of freedom of the vectors. If we consider the compound state vector $ \mathbf{w}=[\mathbf{u},p] $, where $ \mathbf{u}=[u, v, w]$ refers to the $x$, $y$ and $z$ velocity components,
the semi-discretized governing equations \eqref{eq:goveq} may be recast in the following form:
\begin{subeqnarray} \label{eq:gov1}
\mathbf{M} \partial_t \mathbf{w} + \mathbf{L}\mathbf{w} + \frac{1}{2} \mathbf{N}(\mathbf{w},\mathbf{w})&=& \mathbf{M}\mathbf{P} \mathbf{f}(z,t) \\
\mathbf{w} &=& \mathbf{P} \mathbf{{g}}(z,t) \mbox{ on } \partial \Omega_f,
\end{subeqnarray}
where $ \mathbf{P}$ is the prolongation matrix mapping a $ [u,v,w] $ velocity vector into a $ [u,v,w,0] $ velocity-pressure vector. The matrices $ \mathbf{M}$, $ \mathbf{L} $ and the bilinear operator $ \mathbf{N}$ are defined as:
\begin{eqnarray*}
\mathbf{M}=\left( \begin{array}{cc} \mathbf{M}' & \mathbf{0} \\ \mathbf{0} & 0 \end{array} \right),
\quad
\mathbf{L}=\left( \begin{array}{cc} -\nu \Delta () & \nabla() \\ \nabla \cdot () & 0 \end{array} \right),
\quad
\mathbf{N}(\mathbf{w}_1,\mathbf{w}_2)= \left( \begin{array}{c} \mathbf{u}_1 \cdot \nabla \mathbf{u}_2+\mathbf{u}_2 \cdot \nabla \mathbf{u}_1 \\ 0 \end{array}\right),
\end{eqnarray*}
where $ \mathbf{M} $ and $ \mathbf{M}' $ are the mass matrices associated to the spatial discretization, $ \mathbf{L}$ the Stokes operator and $ \mathbf{N}$ the symmetrized nonlinear convection operator.
The volume $ \mathbf{f}(z,t) $ and boundary $ \mathbf{g}(z,t) $ forcings are assumed to be $z$-periodic of wavelength
$ \lambda = 2\pi/ \beta $ and $t$-periodic of period $T=2\pi/\omega$. We assume that the state vector $\mathbf{w}(z,t)$ behaves the same way.
When considering boundary layers in early-stage transition, \emph{i.e} with weak external forcing amplitude, it is reasonable to assume that the response of the system follows the time-periodicity and spatial symmetries of the external forcing. For high forcing amplitude, quasi-periodic limit-cycles may appear, an investigation which is beyond the scope of the present paper.
A Fourier expansion is introduced for the periodic forcing and state variables, which is truncated at $M+1$ harmonics in $ z $ and $ N+1 $ harmonics in $ t $.
Hence
\begin{equation} \label{eq:expansion}
\mathbf{w}(z,t) =\sum_{\substack{-M \leq m \leq M\\ -N \leq n \leq N}} e^{i(m\beta z+n\omega t)} \mathbf{\hat{w}}_{mn},
\end{equation}
with similar expansions (not shown here) for $ \mathbf{f}(z,t) $ and $\mathbf{g}(z,t) $.
Term $ \mathbf{\hat{w}}_{mn}$ (resp. $ \mathbf{\hat{f}}_{mn}$, $ \mathbf{\hat{g}}_{mn}$
) represents the harmonic associated to $ e^{im\beta z+in\omega t} $ for $\mathbf{\hat{w}}$ (resp. $\mathbf{\hat{f}}$ and $\mathbf{\hat{g}}$).
For these variables to be real, the following symmetry holds:
$$\mathbf{\hat{w}}_{-m,-n}=\overline{\mathbf{\hat{w}}_{mn}}$$ for all $(m,n)$, which
induces that $\mathbf{\hat{w}}_{00}$ is real. The overbar $\overline{(\cdot)}$ denotes the complex conjugate.
The discrete Fourier-transform of \eqref{eq:gov1} yields the Harmonic-Balanced Navier-Stokes (HBNS), described by the following system of coupled equations
\begin{subequations} \label{eq:nleqs}
\begin{align}
\left[ in\omega \mathbf{M} + \mathbf{L}_m + \gamma_{00}^{mn} \mathbf{N}_0^m (\mathbf{\hat{w}}_{00}, \cdot) \right]\mathbf{\hat{w}}_{mn}
& +\sum_{{\cal S}(m,n)} \gamma_{m_1n_1}^{m_2n_2}\mathbf{N}_{m_1}^{m_2} (\mathbf{\hat{w}}_{m_1n_1},\mathbf{\hat{w}}_{m_2n_2}) = \mathbf{M}\mathbf{P}\mathbf{\hat{f}}_{mn}, \label{eq:nleqs1} \\
\mathbf{\hat{w}}_{mn} & =\mathbf{P}\mathbf{\hat{g}}_{mn}, \mbox{ on }\partial \Omega_f \label{eq:nleqs2},
\end{align}
\end{subequations}
for all $(m,n)$ such that $-M\leq m \leq M $ and $ -N \leq n \leq N$, and the sum is over the set of indices
\begin{align}
{\cal S}(m,n) = \left. \begin{cases}
m= m_1 + m_2 & -M \le m_1 \le m_2 \le M \\
n = n_1+n_2 & -N \le n_1 \le n_2 \le N \end{cases} \ \right| \ (m_1,n_1) \ne (0,0), (m_2,n_2) \ne (0,0)
\end{align}
The coefficients $ \gamma_{m_1n_1}^{m_2n_2}=0.5 $ if $(m_1=m_2,n_1=n_2)$ and $1$ in the other cases. The linear matrix $\mathbf{L}_{m}$ and bilinear operator $\mathbf{N}_{m_1}^{m_2}$ are deduced from $\mathbf{L}$ and $\mathbf{N} $ by replacing $\partial_z$ derivatives by $im\beta z$. We define the solution and forcing vectors, $\mathbf{\hat{w}}$, $ \mathbf{\hat{f}} $
and $ \mathbf{\hat{g}}$ whose elements correspond to the $(2M+1)\times(2N+1)$ complex unknowns.
Then, \eqref{eq:nleqs} may be rewritten in compact form:
\begin{subeqnarray} \label{eq:nlfinal}
\mathbf{R}(\mathbf{\hat{w}})&=&\mathbf{M}\mathbf{P}\mathbf{\hat{f}} \\
\mathbf{\hat{w}}&=&\mathbf{P}\mathbf{\hat{g}}, \quad \mbox{ on }\partial \Omega_f,
\end{subeqnarray}
where we reuse the symbols $ \mathbf{M} $ and $ \mathbf{P} $ to now refer to block matrices composed from the individual equations.
For given forcing terms $\mathbf{\hat{f}} $ and $\mathbf{\hat{g}} $, equations~(\ref{eq:nlfinal}) are $ (2M+1)\times(2N+1)$ complex nonlinear equations for the unknowns $\mathbf{\hat{w}}$. Due to the fact that the equation governing the $ (m,n) $ harmonic of $ \mathbf{\hat{w}} $ corresponds to the complex conjugate of the equation governing the $(-m,-n)$ harmonic, the solution will be symmetric, $\mathbf{\hat{w}}_{-m,-n}=\overline{\mathbf{\hat{w}}_{mn}}$, whenever the forcing is.
\subsubsection{Special cases} \label{sec:sc}
In order to get some insight into the structure of the governing equations, we consider two particular cases where the boundary forcing term, $\mathbf{\hat{g}}$, is set to zero for simplicity.
In the case where $M=N=1$, equations \eqref{eq:nleqs} reduce to:
\begin{subequations} \label{eq:nleqssc2}
\begin{align}
\left[ \mathbf{L}_0 + \frac{1}{2}\mathbf{N}_0^0 (\mathbf{\hat{w}}_{00}, \cdot) \right]\mathbf{\hat{w}}_{00}
&+ \mathbf{N}_{-1}^{1} (\overline{\mathbf{\hat{w}}_{10}},\mathbf{\hat{w}}_{10})+\mathbf{N}_{0}^{0} (\overline{\mathbf{\hat{w}}_{01}},\mathbf{\hat{w}}_{01})+ \mathbf{N}_{-1}^{1} (\overline{\mathbf{\hat{w}}_{11}},\mathbf{\hat{w}}_{11}) \nonumber \\
& =\mathbf{M}\mathbf{P}\mathbf{\hat{f}}_{00}, \label{eq:nleqssc2_a}\\
\left[ \mathbf{L}_1 + \mathbf{N}_0^1 (\mathbf{\hat{w}}_{00}, \cdot) \right]\mathbf{\hat{w}}_{10}
&
=\mathbf{M}\mathbf{P}\mathbf{\hat{f}}_{10}, \\
\left[ i\omega \mathbf{M} + \mathbf{L}_0 + \mathbf{N}_0^0 (\mathbf{\hat{w}}_{00}, \cdot) \right]\mathbf{\hat{w}}_{01}
&=\mathbf{M}\mathbf{P}\mathbf{\hat{f}}_{01}, \\
\left[ i\omega \mathbf{M} + \mathbf{L}_1 + \mathbf{N}_0^1 (\mathbf{\hat{w}}_{00}, \cdot) \right]\mathbf{\hat{w}}_{11}
& =\mathbf{M}\mathbf{P}\mathbf{\hat{f}}_{11}.
\end{align}
\end{subequations}
For a boundary layer, the terms $\mathbf{\hat{w}}_{10}e^{i\beta z}$, $\mathbf{\hat{w}}_{01}e^{i\omega t}$ and $\mathbf{\hat{w}}_{11}e^{i\beta z+i\omega t}$ may represent, respectively, a streak, a 2D Tollmien-Schlichting wave and an oblique wave. In this case, these components are {\it linearly} triggered by the forcing terms $\mathbf{\hat{f}}_{10}$, $\mathbf{\hat{f}}_{01}$ and $\mathbf{\hat{f}}_{11}$, whereupon they deform the mean flow through the nonlinear interactions in \eqref{eq:nleqssc2_a} (in addition to any mean flow forcing, $\mathbf{\hat{f}}_{00}$).
The linear operators $in\omega \mathbf{M} + \mathbf{L}_m + \mathbf{N}_0^m (\mathbf{\hat{w}}_{00}, \cdot)$ are strictly damped and thus invertible. Connections with the Restricted Nonlinear Model (RNL) introduced for the study of transition in streamwise invariant configurations \citep{waleffe1997RNL,biau2008optimalRNL,farrell2012dynamicsRNL} become apparent. The equations governing the steady harmonics $\mathbf{\hat{w}}_{00}$ and $\mathbf{\hat{w}}_{10}$ (which comprise the streaks and the rolls), are related to the equation governing the streamwise averaged component of the flow in the RNL equation, while those
governing $\mathbf{\hat{w}}_{01}$ and $\mathbf{\hat{w}}_{11}$ are related to the streamwise fluctuating part (one harmonic in $\omega$ being equivalent to one streamwise wavenumber). In the present approach the spanwise direction is treated as homogeneous while the streamwise direction is solved for, while for RNL model, the opposite is true. But for both models, nonlinear interactions only appear in the mean flow equation due to the low-order truncation.
In the case $M=0,N=2$, nonlinear interactions also appear at the fluctuation level:
\begin{subequations} \label{eq:nleqssc}
\begin{align}
\left[ \mathbf{L}_0 + \frac{1}{2}\mathbf{N}_0^0 (\mathbf{\hat{w}}_{00}, \cdot) \right]\mathbf{\hat{w}}_{00}
&+ \mathbf{N}_{0}^{0} (\overline{\mathbf{\hat{w}}_{01}},\mathbf{\hat{w}}_{01})+ \mathbf{N}_{0}^{0} (\overline{\mathbf{\hat{w}}_{02}},\mathbf{\hat{w}}_{02})&=&\mathbf{M}\mathbf{P}\mathbf{\hat{f}}_{00},\\
\left[ i\omega \mathbf{M} + \mathbf{L}_0 + \mathbf{N}_0^0 (\mathbf{\hat{w}}_{00}, \cdot) \right]\mathbf{\hat{w}}_{01}
&+ \mathbf{N}_{0}^{0} (\overline{\mathbf{\hat{w}}_{01}},\mathbf{\hat{w}}_{02})&=&\mathbf{M}\mathbf{P}\mathbf{\hat{f}}_{01}, \label{eq:nlinterterm}\\
\left[ 2i\omega \mathbf{M} + \mathbf{L}_0 + \mathbf{N}_0^0 (\mathbf{\hat{w}}_{00}, \cdot) \right]\mathbf{\hat{w}}_{02}
&+ \frac{1}{2}\mathbf{N}_{0}^{0} ({\mathbf{\hat{w}}_{01}},\mathbf{\hat{w}}_{01})
&=&\mathbf{M}\mathbf{P}\mathbf{\hat{f}}_{02}.
\end{align}
\end{subequations}
They correspond to the extension at second order of the self-consistent model \citep{mantivc2016self} for backward-facing step flow. We recognize the dynamics of the three harmonics $\mathbf{\hat{w}}_{00}$, $e^{i\omega t}\mathbf{\hat{w}}_{01}$ and $e^{2i\omega t}\mathbf{\hat{w}}_{02}$, the nonlinear interactions ($\mathbf{N}_{0}^{0} (\overline{\mathbf{\hat{w}}_{01}},\mathbf{\hat{w}}_{01})+ \mathbf{N}_{0}^{0} (\overline{\mathbf{\hat{w}}_{02}},\mathbf{\hat{w}}_{02})$) and forcing term ($\mathbf{\hat{f}}_{00}$) generating the mean-flow deformation, the nonlinear interactions ($\mathbf{N}_{0}^{0} (\overline{\mathbf{\hat{w}}_{01}},\mathbf{\hat{w}}_{02})$ and $1/2\mathbf{N}_{0}^{0} ({\mathbf{\hat{w}}_{01}},\mathbf{\hat{w}}_{01})$) and forcing terms ($\mathbf{\hat{f}}_{01}$ and $\mathbf{\hat{f}}_{02}$) affecting the first and second harmonics ($e^{i\omega t}\mathbf{\hat{w}}_{01}$ and $e^{2i\omega t}\mathbf{\hat{w}}_{02}$).
If higher order truncations are considered, the complexity is increased by additional nonlinear interaction terms that affect both the mean-flow and the fluctuating harmonics (see for example the term $\mathbf{N}_{0}^{0} (\overline{\mathbf{\hat{w}}_{01}},\mathbf{\hat{w}}_{02})$ in eq. \ref{eq:nlinterterm}).
\subsubsection{Algorithms and numerical methods} \label{sec:HBMalg}
In order to solve the coupled nonlinear equations \eqref{eq:nlfinal} and calculate the response $\mathbf{\hat{w}}$, we use an iterative Newton algorithm. An initial guess $\mathbf{\hat{w}}_i$ may be improved according to $\mathbf{\hat{w}}_{i+1}=\mathbf{\hat{w}}_i-\mathbf{\delta\hat{w}}_i$ with:
\begin{subeqnarray}\label{eq:newton}
\mathbf{A}\mathbf{\delta\hat{w}}_i&=&\mathbf{R}(\mathbf{\hat{w}}_i)-\mathbf{M}\mathbf{P}\mathbf{\hat{f}} \\
\delta\mathbf{\hat{w}}_i&=& \mathbf{\hat{w}}_i-\mathbf{P}\mathbf{\hat{g}} \mbox{ on }\partial \Omega_f,
\end{subeqnarray}
where $ \mathbf{A}=\partial \mathbf{R}/\partial \mathbf{\hat{w}} $
is the Jacobian of operator $ \mathbf{R}$, given by
\begin{equation}
\left(
\begin{array}{cccc}
\mathbf{L}_0 + \mathbf{N}_0^0 (\mathbf{\hat{w}}_{00},\cdot) & \mathbf{N}_0^0 (\mathbf{\hat{w}}_{0,-1},\cdot) & \mathbf{N}_0^0 (\mathbf{\hat{w}}_{01},\cdot) & \cdots \\
\mathbf{N}_0^0 (\mathbf{\hat{w}}_{01},\cdot) & i \omega \mathbf{M} + \mathbf{L}_0 + \mathbf{N}_0^0 (\mathbf{\hat{w}}_{00},\cdot) & \mathbf{N}_0^0 (\mathbf{\hat{w}}_{02},\cdot) & \cdots \\
\mathbf{N}_0^0 (\mathbf{\hat{w}}_{0,-1},\cdot) &\mathbf{N}_0^0 (\mathbf{\hat{w}}_{0,-2},\cdot) &
-i \omega \mathbf{M} + \mathbf{L}_{0} + \mathbf{N}_0^0 (\mathbf{\hat{w}}_{00},\cdot) & \cdots \\ \vdots & \vdots & \vdots &\ddots
\end{array}\right),
\end{equation}
where the off-diagonal blocks stem from non-linear interactions between harmonics, while the diagonal blocks correspond to Navier-Stokes equations linearized around the current mean-flow $ \hat{\mathbf{w}}_{00} $. This matrix is also known in the literature as the finite-dimensional block Hill-matrix \citep{lazarus2010harmonic}.
The linear problem \eqref{eq:newton} involves a large number of unknowns, equal to the number of harmonics $ (2N+1)(2M+1)$ times the number of degrees of freedom in a velocity-pressure vector on a two-dimensional computational mesh. If the number of retained harmonics is large, solution of the linear system becomes the pacing item, primarily due to associated computer memory limitations rather operation counts, when a direct LU method is used. Iterative solvers for HBM problems partially bypass these limitations \citep{hall2002computation,gopinath2007three, sicot2012time}. In order to decrease the computational cost, we follow \cite{moulin2019augmented} and use a preconditioned
Generalized Minimal Residual (GMRES) algorithm that only requires matrix-vector products. We use a block-Jacobi preconditioner, where the blocks correspond to the harmonics: $ \mathbf{\hat{w}}_{00}$, $ (\mathbf{\hat{w}}_{01},\mathbf{\hat{w}}_{0,-1})$, etc. The block-Jacobi preconditioner is very efficient when the diagonal blocks of matrix $\mathbf{A}$ are dominant, that is when the nonlinear interactions between harmonics remain reasonably weak. This occurs when the amplitude $ A $ of the forcing remains small.
The code is parallel with each processor handling a block. In the block-Jacobi preconditioner, the linear system associated to the diagonal block of a given harmonic, for example
\begin{equation} \label{eq:blockdiag}
\left(
\begin{array}{cc}
i n \omega \mathbf{M} + \mathbf{L}_m + \mathbf{N}_0^m (\mathbf{\hat{w}}_{00},\cdot) & {\mathbf{A}'} \\
\overline{\mathbf{A}'} &
-i n \omega \mathbf{M} + \mathbf{L}_{-m} + \mathbf{N}_0^{-m} (\mathbf{\hat{w}}_{00},\cdot)
\end{array}
\right),
\end{equation}
is solved by the processor handling the harmonic $(\hat{\mathbf{w}}_{mn},\hat{\mathbf{w}}_{-m,-n}) $ with a sparse direct LU method \citep{MUMPS:1}. For an efficient distributed implementation, we use the PETSc software \citep{petsc-web-page} with the scalable linear equation solver component (KSP). Since a single processor solves for a system involving matrix \eqref{eq:blockdiag}, the size of the mesh needs to remain reasonable. Should larger meshes be required, domain decomposition could be used to distribute each harmonic over several processors.
To obtain a good initial guess, we solve the linear problem, which uncouples the equations and may be solved with a direct LU. For larger $A$, we continue in steps from smaller $A$. Likewise, we may increment $M$ and $N$ as the iteration proceeds.
\newcommand{\elim}[1]{
Note that the implementation can also be carried out with just real variables.
In such a case, instead of the structure \eqref{eq:structcmplx}, vector $ \mathbf{\hat{w}} $ contains
the $ \mathbf{\hat{w}}_{00} $
harmonic and the real and imaginary parts of the other harmonics:
\begin{equation}
\mathbf{\hat{w}}=\left( \begin{array}{c} \mathbf{\hat{w}}_{00} \\
\mathbf{\hat{w}}_{01r} \\
\mathbf{\hat{w}}_{01i} \\
\mathbf{\hat{w}}_{02r} \\
\mathbf{\hat{w}}_{02i} \\
\vdots \\
\mathbf{\hat{w}}_{MNr} \\
\mathbf{\hat{w}}_{MNi}
\end{array} \right).
\end{equation}
The size of the new real unknown $ \mathbf{\hat{w}} $
is therefore the same as in the complex case.
The nonlinear $ \mathbf{R} $ operator and the $ \mathbf{A} $ matrix may be rewritten in real formats.
Doing so, the size of the $ \mathbf{A} $ matrix is left unchanged and it therefore seems advantageous to switch to a real formalism (a complex scalar multiplication is equivalent to four real scalar multiplications).
This is only partly true since the diagonal block of the real $ \mathbf{A} $ matrix is now:
\begin{equation} \label{eq:blockdiagreal}
\left(
\begin{array}{cc}
\mathbf{L}_m^r + \mathbf{N}_0^{m,r} (\mathbf{\hat{w}}_{00},\cdot) + {\mathbf{A}'^{r}} & - n \omega \mathbf{M}-\mathbf{L}_m^i-\mathbf{N}_0^{m,i} (\mathbf{\hat{w}}_{00},\cdot) + {\mathbf{A}'^{i}} \\
n \omega \mathbf{M}+\mathbf{L}_m^i+\mathbf{N}_0^{m,i} + {\mathbf{A}'^{i}} &
\mathbf{L}_m^r + \mathbf{N}_0^{m,r} (\mathbf{\hat{w}}_{00},\cdot) -{\mathbf{A}'^{r}}
\end{array}
\right).
\end{equation}
As can be seen, the number of non-zero elements of the matrix has roughly doubled, so that a real implementation may only reduce the cost by a small amount.
}
\subsubsection{Reflectional symmetry in $z$}\label{sec:HBMsym}
For a reflectionally symmetric solution with respect to $z=0$, we restrict the forcing so that
\begin{subequations}\label{eq:symmetryz}
\begin{align}
f_x(-z,t) = f_x(z,t) \ &\implies \ \hat{f}_{x}(-m,n) = {\hat{f}_{x}(m,n)}, \\
f_y(-z,t) = f_y(z,t) \ &\implies \ \hat{f}_{y}(-m,n) = {\hat{f}_{y}(m,n)}, \\
f_z(-z,t) = -f_z(z,t) \ &\implies \ \hat{f}_{z}(-m,n) = -{\hat{f}_{z}(m,n)}.
\end{align}
\end{subequations}
Imposing symmetry on $ \mathbf{f} $ and $\mathbf{g}$ requires that the spanwise velocity component must be set to zero at the inlet boundary. Imposing the same symmetries on the solution reduces the number of unknowns by about a factor of 2. These symmetric solutions, it must be stressed, may be unstable to asymmetrical disturbances.
\subsection{Optimal forcings (nonlinear resolvent)} \label{sec:optim}
For brevity, we only consider optimal volumetric forcings $ \mathbf{\hat{f}} $. Inlet- and wall-forcings $ \mathbf{\hat{g}} $ can be handled in an analogous manner. We pose a procedure to find the forcing $ \mathbf{\hat{f}} $ that maximizes a positive, real-valued cost-functional $ J(\mathbf{\hat{w}}) $, under the constraint that $\mathbf{\hat{w}}$ is a solution to the HBNS nonlinear problem forced by $ \mathbf{\hat{f}} $ with finite amplitude $ A $. To solve the constrained optimization, we consider the Lagrangian functional
\begin{equation}
\mathcal{L}(\mathbf{\hat{w}},[\mathbf{\tilde{w}},\lambda],\mathbf{\hat{f}})=J(\mathbf{\hat{w}})-\mathbf{\tilde{w}}^*\left( \mathbf{R}(\mathbf{\hat{w}})-\mathbf{M}\mathbf{P}\hat{\mathbf{f}}\right)-\lambda\left(\hat{\mathbf{f}}^*\mathbf{Q}\hat{\mathbf{f}} -A^2 \right),
\end{equation}
where $ \mathbf{\tilde{w}} $ and $ \lambda $ are Lagrange multipliers enforcing the constraints. The $\lambda$-constraint is that the forcing $\mathbf{\hat{f}}$ must exhibit a prescribed amplitude $ A $:
\begin{equation} \label{eq:surface}
\mathbf{\hat{f}}^*\mathbf{Q}\mathbf{\hat{f}}=A^2,
\end{equation}
where $\mathbf{Q}$ is a positive-definite Hermitian matrix defining a norm on the forcing space $\mathbf{\hat{f}}$. Proceeding in the usual way by zeroing the variations of ${\cal L}$ with respect to $\mathbf{\tilde{w}}$ and $\lambda$ yields the constraints, whereas variations w.r.t. $\mathbf{\tilde{w}}$ gives an equation for the adjoint state,
\begin{equation}
\mathbf{A}^*\mathbf{\tilde{w}}=\frac{dJ}{d\hat{\mathbf{w}}},
\end{equation}
and variations w.r.t. $\mathbf{\hat{f}}$ lead to a relation
\begin{eqnarray}
\underbrace{\mathbf{Q}^{-1}\mathbf{P}^*\mathbf{M}\mathbf{\tilde{w}}}_{\mathbf{\tilde{w}'}} -2\lambda \mathbf{\hat{f}} & = 0,
\end{eqnarray}
that shows that $\mathbf{\hat{f}}$ needs to be parallel to $\mathbf{\tilde{w}'}$. A convergence criteria (to a local maximum) is that the angle $ \theta $ between these two vectors vanishes
\begin{equation}
\cos(\theta) = \frac{\mathbf{\hat{f}}^*\mathbf{Q} \mathbf{\tilde{w}'}}{A \gamma} = 1,
\end{equation}
where $\gamma=\sqrt{\mathbf{\tilde{w}'^*}\mathbf{Q} \mathbf{\tilde{w}'}}$.
Following \citet{kerswell2018nonlinear}, the algorithm for the update of $\mathbf{\hat{f}}$ is based on steepest ascent:
$$\mathbf{\hat{f}}_{\mbox{new}}=\mathbf{\hat{f}}+A \epsilon (\mathbf{\tilde{w}'} - 2\lambda \mathbf{\hat{f}}),$$
where the Lagrange parameter $\lambda$ is chosen such that it constraints the forcing energy $\mathbf{\hat{f}}_{\mbox{new}}^*\mathbf{Q}\mathbf{\hat{f}}_{\mbox{new}}=A^2$, and $ \epsilon $ governs the amplitude change between
$\mathbf{\hat{f}}$ and $\mathbf{\hat{f}}_{\mbox{new}}$.
The parameter $ \epsilon $ may be chosen as $\epsilon=c/\gamma$ where $0 < c \leq 1 $ to allow a solution for $ \lambda$.
The explicit steps of the iterative procedure are detailed in algorithm \ref{nlalgorith}.
The parameter $c$ can be fixed to 1 if the guess $ \hat{\mathbf{f}}$ is close to the optimum.
If not, large derivatives of the cost functional (i.e transition) can lead to large drifts of $ \hat{\mathbf{f}}$, which may destabilize the Newton algorithm. In such a case, lower values of $c$ need to be imposed. In the present study, a good compromise was found with $c = 0.5$, for which
most of the cases converged, without penalizing too much the number of iterations for the Newton method to converge. In a few cases, we had to decrease the value of $c$ down to $ c=0.2$. The stopping criterion was chosen so that the alignment $\theta$ is less than $\theta_c=1^\circ$.
\begin{algorithm}[t!]
\caption{Nonlinear Optimization using HBNS}\label{nlalgorith}
\begin{algorithmic}[1]
\setstretch{1.0}
\State
Initialize. Set stopping criterion $\theta_c$. Let $\mathbf{\hat{f}}_n$ be an approximation of a maximum of $
J(\mathbf{\hat{w}})$ such that
$$
\mathbf{\hat{f}}_n^* \mathbf{Q}\mathbf{\hat{f}}_n=A^2.
$$
\State \label{step:HBM}
Solve the nonlinear HBNS system \eqref{eq:nlfinal} to determine the state $\mathbf{\hat{w}}_n$, using the iterative Newton method and the iterative preconditioned GMRES algorithm (\S \ref{sec:HBMalg})
$$
\mathbf{R}(\mathbf{\hat{w}}_n)=\mathbf{M}\mathbf{P}\hat{\mathbf{f}}_n.
$$
\State
Solve the linear system for the adjoint state $\mathbf{\tilde{w}}_n$, using the same iterative preconditioned GMRES algorithm (\S \ref{sec:HBMalg})
$$
\mathbf{A}^*\mathbf{\tilde{w}}_n=\left.\frac{dJ}{d\mathbf{\hat{w}}}\right|_{\mathbf{\hat{w}}_n}.
$$
\State
Set
$\mathbf{\tilde{w}'}_n=\mathbf{Q}^{-1}\mathbf{P}^*\mathbf{M}\mathbf{\tilde{w}}_n $, compute the norm $ \gamma_n=\sqrt{\mathbf{\tilde{w}}_n^{'*}\mathbf{Q} \mathbf{\tilde{w}}'_n}$ and evaluate alignment angle
$$
\cos(\theta_n)=\mathbf{\hat{f}}_n^*\mathbf{Q} \mathbf{\tilde{w}}'_n/(A \gamma_n).
$$
\IF{$|\cos(\theta_n)|>\cos(\theta_c)$}
\State
Break.
Return $(\mathbf{\hat{f}_n},\mathbf{\hat{w}_n}) $, which is a reasonable approximation of an extremum.
\ELSE
\State
Update $\mathbf{\hat{f}}$:
\begin{eqnarray} \lambda_n=\frac{1+\epsilon_n \gamma_n \cos\theta_n -\sqrt{1-\epsilon_n^2\gamma_n^2\sin^2\theta_n}}{2A\epsilon_n} \label{eq:enforceampl}, \quad
\epsilon_n=\frac{c }{\gamma_n}. \label{eq:cond}
\end{eqnarray}
$$
\mathbf{\hat{f}}_{n+1} = \mathbf{\hat{f}}_{n}+ A\epsilon_n (\mathbf{\tilde{w}}'_n -2\lambda_n \mathbf{\hat{f}}_n),
$$
\State
Go to \ref{step:HBM}.
\ENDIF
\end{algorithmic}
\end{algorithm}
\section{HBNS: validation for controlled transition}\label{sec:Rist}
In this section, we validate the HBNS implementation described above against the DNS of controlled K-type transition by \citet{rist1995direct}.
We consider the free-stream velocity $ U_\infty $ and $ \nu/U_\infty $ as reference velocity and length scales throughout the manuscript. For this specific choice we have $x \mapsto Re_x$.
The computational domain for the zero-pressure flat-plate configuration is rectangular with the plate located at $y=0$, the upper boundary at $y=\num{1.2 e5} $, the inlet at $ x_i=\num{0.30e5}$ and the outlet at $ x_o=\num{2.52e5} $.
The volumetric forcing $\mathbf{\hat f}$ is set to zero and perturbations are triggered through $\mathbf{\hat g}$ which is chosen to represent wall-normal forcing by local time-dependent blowing and suction within a narrow strip at the wall.
Thus, in accordance with \citet{rist1995direct}, we impose $u=w=0$ and
\begin{eqnarray}
v(x,z,t)&=& \num{5e-3} \sin(\omega t) v_a(x) + \num{1.3e-4} \cos(\beta z) v_s(x),
\end{eqnarray}
which represents a superposition of a 2D planar TS wave $(0,\omega)$ of frequency $ \omega=\num{11e-5} $ and a steady oblique wave $(\beta,0)$ of wavenumber $\beta=\num{42.3e-5}$. The specific profiles of the wall-normal velocity of the unsteady and steady waves, which are localized between $x_1$ and $x_2$ on the wall boundary, are given by:
\begin{eqnarray}
v_a(x)&=&\left\{ \begin{array}{ccc} 0 & , & x \leq x_1 \\
15.1875 \xi^5-35.4375\xi^4+20.25\xi^3 & , &
x_1 < x \leq x_m \\
-v_a(2x_m-x) & , &
x_m < x \leq x_2 \\
0 & , & x_2 < x
\end{array}\right. \\
v_s(x)&=&\left\{ \begin{array}{ccc} 0 & , & x \leq x_1 \\
-3 \xi^4+4\xi^3 & , &
x_1 < x \leq x_m \\
v_s(2x_m-x) & , &
x_m < x \leq x_2 \\
0 & , & x_2 < x
\end{array}\right.
\end{eqnarray}
Here: $x_1=\num{1.3438e5}$, $ x_2=\num{1.5532e5}$, $ x_m=(x_1+x_2)/2$ and $\xi=\frac{x-x_1}{x_m-x_1}$.
Due to the symmetry of the wall forcing, spanwise reflectional symmetry was assumed enforcing equations \eqref{eq:symmetryz}. The mean flow harmonic $\hat{\mathbf{w}}_{00}$ was initialized with the base-flow solution and the other harmonics were set to zero except the $(0,\omega)$ and $(\beta,0)$ harmonics, which were initialized with the linearized responses. For $M=N=2$ (9 harmonics in total) the solution of the HBNS system converged after 9 Newton iterations (residuals of the order of $ 10^{-10} $).
The $M=N=3$ (16 harmonics) solution was obtained using as initial guess the $M=N=2$ solution and converged after 4 iterations, whereas the $M=N=4$ (25 harmonics) solution was obtained from the $M=N=3$ one in 4 iterations.
\begin{figure}
\centering
\hspace{0.5cm}
\includegraphics[width=0.7\textwidth]{Harm2.pdf}
\caption{K-type controlled transition. Comparison between DNS~\citep{rist1995direct} and Harmonic-Balanced Navier-Stokes retaining $M=N=2$ (HBNS$_{22}$) and $M=N=4$ (HBNS$_{44}$) harmonics in spanwise/frequency. The grey region denotes the streamwise extent of the wall blowing and suction region that triggers K-type transition. Note that to ease representation, we have plotted one fifth of the amplitude of harmonics $(0,\omega)$ and $(2\beta,\omega)$.}
\label{fig:comp}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth,trim={0cm 28cm 0cm 15cm},clip]{figs/Rist/Qu_double_44.jpg}
\caption{K-type controlled transition with $\mathit{HBNS_{44}}$. Isosurfaces of pertubation velocity $u'=\pm 0.04$ (red: high speed, blue: low speed) and of the second invariant of the velocity gradient tensor, $Q$, coloured by the vertical distance from the wall ($Q = \num{2e-9}$).}
\label{fig:Qu_Rist}
\end{figure}
In figure~\ref{fig:comp}, we compare the amplitude of the first few harmonics from the HBNS against the DNS results obtained by \citet{rist1995direct}. A sensitivity analysis of the domain length and of the finite element discretization is given in appendix~\ref{app:sensitivityKtype}. For plotting the $(0,0)$ harmonic component, we have subtracted the base-flow solution, which leads to the mean flow deformation (MFD). The definition of the amplitudes of the different harmonics are described in appendix \ref{app:HarmonicAmplitudes}. The wall-normal forcing excites initially planar TS waves $(0,\omega)$ and streamwise vortices/streaks $(\beta,0)$ at a given frequency and spanwise wavelength. Oblique waves $(\beta,\pm\omega)$ and higher harmonics are generated through nonlinear interactions. Similarly, the self-interaction of the modes when they reach sufficiently high amplitudes, generates $(0,0)$ components that cause departure of the mean-flow harmonic from the base-flow solution.
Even with $M=N=2$, good agreement is obtained for the fundamental $(0,\omega)$ and $(\beta,0)$ harmonics and for the oblique wave $(\beta,\omega)$. As the perturbations grow in the streamwise direction, the $M=N=4$ results are in slightly better with the DNS for the higher harmonic $(2\beta,\omega)$.
In figure~\ref{fig:Qu_Rist}, isosurfaces of streamwise velocity show low-speed velocity streaks (blue) developing in the streamwise direction. Isosurfaces of the $Q$-criterion, colored based on the normal distance from the wall, show $\Lambda$-vortices sitting on low-speed streaks. They are elongated and move away from the wall as they propagate downstream, in accordance with \citet{rist1995direct}.
\begin{figure}
\vspace{0.5cm}
\centering
\includegraphics[width=0.5\textwidth]{figs/linear/LinearGain_Full.jpg}
\vspace{0.5cm}
\\{\scriptsize $|f'_v|_{max}=\pm0.5$ \hspace{1.5cm} {\bf A: streak} \hspace{1.5cm} $|u'|_{max}=\pm0.8$ }\\
\vspace{0cm}
\includegraphics[width=0.85\textwidth,trim={15cm 11.4cm 15cm 8cm},clip]{figs/linear/Streaks_top.jpg}
\vspace{0.5cm}
\includegraphics[width=0.85\textwidth,trim={15cm 15cm 15cm 10cm},clip]{figs/linear/Streaks_side.jpg}
\vspace{0.5cm}
\\{\scriptsize $|f'_u|_{max}=\pm0.8$ \hspace{1cm} {\bf B: oblique wave} \hspace{1cm} $|u'|_{max}=\pm0.5$ }\\
\includegraphics[width=0.85\textwidth,trim={15cm 29cm 15cm 25cm},clip]{figs/linear/Oblique_top.jpg}
\includegraphics[width=0.85\textwidth,trim={15cm 5cm 15cm 10cm},clip]{figs/linear/Oblique_side.jpg}
\caption{Linear input/output (resolvent) analysis. Optimal gain (top). The two local maxima correspond to the amplification of streaks (A at $(\beta,\omega) =(100,0)\times10^{-5}$) and oblique waves (B at $(\beta,\omega) =(30,10)\times10^{-5}$). Optimal forcing (left; light gray:positive, dark grey: negative) and optimal response (right; red: positive, blue negative) for streaks and obliques waves. Top and side views are shown. The x-axis has been scaled by a factor of 4.}
\label{fig:LinearGain}
\end{figure}
\section{Linear input/output (resolvent) analysis}\label{sec:linear_resolvent}
Before performing nonlinear optimization, we briefly recall here results obtained by \citet{monokrousos2010global,brandt2011effect} concerning linear optimal forcing in the frequency domain that aim at maximizing energetic gains (resolvent analysis). Such results are important to understand and analyse the forthcoming nonlinear optimizations.
For this, we consider a generic volumetric forcing and no-slip boundary conditions on the wall. The cost function for the linear optimization is the input/output kinetic energy gain of the fluctuations over the whole domain:
\begin{equation}
J^{\mbox{lin}} \equiv \lambda=
\frac{\mathbf{\hat{w}}^*\mathbf{Q}'\mathbf{\hat{w}}}{\mathbf{\hat{f}}^*\mathbf{Q}\mathbf{\hat{f}}},
\end{equation}
where
\begin{eqnarray} \label{eq:Q}
{\mathbf{f}}^*\mathbf{Q}{\mathbf{f}}=\iint (|{f}_x|^2+|{f}_y|^2|+|{f}_z|^2)\; d\Omega, \\
\mathbf{Q}'=\left( \begin{array}{cc}
\mathbf{Q} & 0 \\ 0 & 0
\end{array}
\right). \label{eq:Qp}
\end{eqnarray}
Such a linear optimization problem is efficiently solved by iterative methods \citep{sipp2013characterization}.
The mesh extends here from $ x_i=0.30 \times 10^5$ to $ x_o=3.60 \times 10^5 $. It comprises 116806 triangles, yielding 586178 degrees of freedom. The same mesh will be used in the next section dealing with nonlinear optimization (\S \ref{sec:nonlinear_resolvent}).
The linear optimal amplitude gain ($\sigma=\sqrt{\lambda}$) is shown in figure~\ref{fig:LinearGain}, as a function of frequency $\omega$ and spanwise wavenumber $\beta$. Two local maxima are observed, in agreement with \cite{monokrousos2010global}. The forcing and response mode shapes of the two linear optimal mechanisms are shown in the same figure.
The first local maximum at $(\beta,\omega) =(100,0)\times10^{-5}$, point A, is associated with the nonmodal lift-up mechanism. The optimal forcing corresponds to steady streamwise rolls ($v$, $w$ components; for the optimal forcing the $v$ component is shown), and the optimal response to streamwise streaks located further downstream ($u$ component).
The second local maximum at $(\beta,\omega) =(30,10)\times10^{-5}$, point B, corresponds to the amplification of oblique TS waves. The planar TS waves are not the most amplified ones due to the cooperative non-modal amplification through the Orr and lift-up mechanisms. It is clearly noticed that the optimal forcing is tilted upstream, against the mean shear so that the response takes advantage of the algebraic amplification through the Orr mechanism.
\section{Nonlinear input/output analysis}\label{sec:nonlinear_resolvent}
To uncover the optimal nonlinear mechanisms that promote transition, the nonlinear interactions of the modes and their impact on the mean flow is now incorporated in the analysis through the optimization approach developed in \S\ref{sec:optim}.
We choose as cost function the (squared) shear-stress of the mean-flow deviation, integrated over the wall. With the notation introduced above, this is:
$$ J(\hat{\mathbf{w}})=J(\hat{\mathbf{w}}_{00})=(\hat{\mathbf{w}}_{00}-\mathbf{w}_b)^*\mathbf{C}^*\mathbf{C}(\hat{\mathbf{w}}_{00}-\mathbf{w}_b),$$
with $ \mathbf{C}\mathbf{w} = \int_{y=0} \frac{\partial u}{\partial y} dx$ and $ \mathbf{w}_b$ is the base-flow. For this choice of cost function, we have:
$$
\frac{dJ}{d\hat{\mathbf{w}}_{00}}=2\mathbf{C}^*\mathbf{C}(\hat{\mathbf{w}}_{00}-\mathbf{w}_b)
$$
and $0$ for the other harmonics. This cost function can be directly linked to the drag change exerted on the plate,
\begin{equation}
\Delta C_D = \frac{\nu J^{0.5}}{\frac{1}{2} U_\infty^2 L_p}, \label{eq:deltacd}
\end{equation}
where $ L_p=x_o$ is the plate length.
In other words, by maximizing the specific cost function $J$, we maximize the drag on the plate.
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{cases_schematic.pdf}
\caption{Fundamental (left) and superharmonic (right) cases. The nonlinear optimization is restricted to forcing components $1\beta,\pm1\omega$, or $1\beta,2\beta,\pm1\omega,\pm2\omega$, and their oblique combinations, respectively. Other forcing and response harmonics in the $(m,n)$ plane may be deduced from the real-value constraint, e.g. $ \mathbf{\hat{w}}_{-m-n}=\overline{\mathbf{\hat{w}}_{mn}}$. In case of reflectional symmetry in $z$, the $-n \omega$ components are linked to the $+n\omega$ ones.}
\label{fig:cases}
\end{figure}
The entries of matrix $\mathbf{P}$ allow selection of the forced equations and of a subset of forced harmonics.
As in the linear case, we will restrict the forcing to the momentum equations and exclude mass sources.
In order to preserve the mean-flow harmonic $\mathbf{\hat{w}}_{00}$ from direct modifications induced by steady forcing terms,
we set $\mathbf{\hat{f}}_{00}=0$ and exclude this mode from the optimization process.
Two types of forcing are then considered, which we refer to as \emph{fundamental} and \emph{superharmonic} cases, as depicted in figure~\ref{fig:cases}. For the first case, forcing is restricted to components $(m,n)=(\beta,\pm\omega)$, $(\beta,0)$, $(0,\pm \omega)$; we call this \emph{fundamental}, since forcing is allowed only at the primary forcing frequency and spanwise wavenumber. Each of these forcing components, can potentially lead to the amplification of a pair of unsteady oblique waves, steady streamwise streaks or vortices, and planar TS waves, respectively. For the superharmonic forcing case, we allow also the second forcing harmonics to be optimized, $ |m|\leq 2$ and $ |n|\leq 2$, except $m=n=0$. This allows forcings of fundamental harmonic and superharmonic components. For example, forcing $\hat{\mathbf{f}}_{02}$ is at twice the frequency of forcing $\hat{\mathbf{f}}_{11}$. If the perturbation satisfies reflectional symmetry in $z$, all forcing and response harmonics $n<0$ are directly linked to those satisfying $ n > 0 $. We note that in both cases, we solve separate optimization cases over a wide range of the fundamental forcing frequency, $\omega$ and $\beta$.
\subsection{Identification of optimal transition mechanisms: $z$-symmetric case with $ A=\num{7.07e-5}$ and $M=N=2$} \label{sec:toy}
\begin{figure}
\includegraphics[width=0.52\textwidth]{figs/fundamental/J_2_2_5e-5_3f_v2.pdf}
\includegraphics[width=0.52\textwidth]{figs/subharmonic/J_2_2_5e-5_8f_v2.pdf}
\caption{Optimal drag change from nonlinear input/output analysis with fundamental (left) and superharmonic (right) forcing, $M=N=2$ and $A=\num{7.07e-5}$. Fundamental maximum at point C: $\Delta C_{D,max}=\num{2.8e-5}$ at $(\beta,\omega) =(33.4,11.7)\times 10^{-5}$. Superharmonic maximum at point D: $\Delta C_{D,max}=\num{41.6e-5}$ at $(\beta,\omega) =(50,11.7)\times10^{-5}$. Both are close to the optimal linear amplification of the oblique waves (point B).}
\label{fig:J}
\end{figure}
We first consider a case with imposed spanwise symmetry on the forcing and response and $M=N=2$. For small enough amplitude $A$, we expect that the forcing and perturbation should exhibit the $z$-reflectional symmetry of the configuration. The small number of resulting modes allows for more expensive parametric studies over $\omega$ and $ \beta$, but, strictly speaking, the results are converged for sufficiently small $A$ so that higher-order harmonics may be neglected. We select here $A=\num{7.07e-5}$ and in a subsequent sections we examine convergence as $A$ is increased and with retaining more modes, and verify {\it a posteriori} that the present results are reasonably well converged.
The cost function (expressed as mean drag perturbation via \eqref{eq:deltacd}) is shown in figure \ref{fig:J} for both the fundamental- and superharmonic-type forcings. For the fundamental case, maximum drag increase
is observed at $(\beta,\omega) =(33.4,11.7)\times10^{-5}$, whereas for the superharmonic case
the maximum occurs at the same frequency but a slightly higher wavenumber, $(\beta,\omega) =(50,11.7)\times10^{-5}$. For the superharmonic case, the drag increase is approximately 14 times higher compared to the fundamental forcing. In both cases, the overall optimal frequency/wavenumber pairs are close to the point marked $B$ on the linear amplifcation plot (figure~\ref{fig:LinearGain}), which represents the local maximum in linear amplification of oblique waves. While those waves are linearly less amplified than streaks (point A), they are nonlinearly superior. As will be shown in detail below, the nonlinear fundamental mechanism C and superharmonic mechanism D initially harness oblique wave amplification, and eventually lead, through nonlinearity, to redistribution of energy near A and a strong response related to the lift-up mechanisms producing streaks.
\subsubsection{Symmetric fundamental forcing}
\begin{figure}
\hspace{-0.8cm}
\includegraphics[width=1.1\textwidth,trim={0cm 0cm 0cm 0},clip]{fundamental_A.pdf}
\caption{Nonlinear optimization for fundamental symmetric forcing with $M=N=2$. Amplitudes of optimal forcing (left) and response (right) for each individual harmonic component $(m,n)$, as depicted in figure~\ref{fig:cases}a. Values have been normalized with the the total forcing amplitude $A=\num{7.07e-5}$. The circle marks the frequency/wavenumber that maximum drag increase is observed. Also, isolines of the cost function (dashed lines) have been added on the forcing components.}
\label{fig:A_fundamental}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/fundamental/22/Af_22_5e-05_b200_o70.pdf}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/fundamental/22/Au_22_5e-05_b200_o70.pdf}
\includegraphics[width=1\textwidth,trim={30cm 2cm 30cm 0},clip]{figs/fundamental/22/fq22_5e-05_b200_o70.jpg}
\caption{Optimal oblique fundamental case with $M=N=2$ at $(\beta,\omega)$=($33.3,11.7)\times10^{-5}$ for $A=\num{7.07e-5}$. This forcing results to the maximum amplification of shear stress for fundamental forcing over all forcing frequencies and wavenumbers (point C in figure \ref{fig:J}a). Isosurfaces of streamwise perturbations $f'_u= \pm \num{8.3e-9}$ (bottom left) and $u'= \pm 0.07$ (bottom right), blue negative iso-value and red positive one. One fundamental wavelenghth is shown in $z$.}
\label{fig:Oblique}
\end{figure}
Focusing on the fundamental case first (figure~\ref{fig:cases}a) with $z$-reflectional symmetry, we now delve into the optimal forcing and response in greater detail. To simplify the discussion, we define in appendix \ref{app:HarmonicAmplitudes}, a scalar amplitude $A(m,n)$ of each forcing/response mode, which represents an integral over the spatial domain.
These amplitudes are shown in figure~\ref{fig:A_fundamental}. Note that upon summation of the forcing modes, this yield the overall forcing amplitude (here $A=\num{7.07e-5}$), and all amplitudes in the plot are normalized by this value. Based on the dominant regions of the amplitude response on the $\beta-\omega$ planes, three distinct mechanisms can be identified:
\begin{enumerate}
\item
{\bf Oblique waves}.
The maximal drag increase occurs for $(\beta,\omega)$=($33.3,11.7)\times10^{-5}$, and only involves significant forcing of the oblique wave component $(\beta,\pm\omega)$. In the response, there is some amplification to the response component $(\beta,\pm\omega)$, as expected in a linear framework, but the $(2\beta,0)$ component, which arises from nonlinear interactions between $(\beta,\omega)$ and $(\beta,-\omega)$ components, is highly amplified. The mean flow modification is clearly associated with the amplification of $(2\beta,0)$ streaks via oblique forcing.
\item
{\bf Streamwise vortices}. For high spanwise wavenumbers, $\beta>\num{100e-5}$, the optimal forcing is a streamwise vortex $(m,n)=(\beta,0)$. For these frequencies, the linear amplification of obliques waves is weak and thus the generated streaks through nonlinearity would also be weak. Consequently, for high enough frequencies and wavenumbers, i.e. those that are far from the linear optimal of the oblique waves, the optimal forcing mechanism is the direct amplification of streaks through the lift-up mechanism.
\item
{\bf K-type mechanism}.
Finally, at $(\beta,\omega)=(16,15)\times10^{-5}$, the optimal forcing is a combination of all three components. Main forcing component is the TS wave followed by the oblique waves. This mechanism is similar to the Klebanov one, describing fundamental K-type transition.
\end{enumerate}
\begin{figure}
\centering
\includegraphics[width=1\textwidth,trim={30cm 41cm 30cm 20cm},clip]{figs/fundamental/22/11u.jpg}
\includegraphics[width=1\textwidth,trim={30cm 28cm 30cm 20cm},clip]{figs/fundamental/22/20.jpg}
\caption{Harmonic response components for the optimal oblique fundamental case shown in figure \ref{fig:Oblique}. The response is dominated by the growth of $(\beta,\pm \omega)$ oblique waves ($u'$ shown), followed by the nonlinear generation of $(2\beta,0)$ streamwise vortices ($\omega'_x$ shown) and the linear growth of streaks ($u'$ shown).}
\label{fig:vortex}
\end{figure}
Since the oblique waves are the most dangerous mechanism in terms of drag increase, we examine the structure of the forcing and response fields in greater detail in figure~\ref{fig:Oblique}. Initially, $(\beta,\pm\omega)$ oblique waves\footnote{Only oblique waves with positive wavenumber and frequency are shown due to $z$-symmetry.} are amplified due to linear instability. The quadratic nonlinearity redistributes the energy of the oblique waves into the $(2\beta,0)$ mode in the form of streamwise vortices with streamwise vorticity $\omega_x$ ($v,w$ components with $(2\beta,0)$). In turn, the streamwise vortices lead to the growth of the streaks ($u$ component with $(2\beta,0)$) through the linear lift-up mechanism. The spatial shapes of the the harmonic components mentioned above are shown in figure \ref{fig:vortex}, showing the transition sequence from oblique waves to streamwise vortices and streaks.
These observations are in agreement with previous studies on oblique transition \citep{schmid1992new,berlin1994spatial}. The link between the nonlinear gain map and that obtained from linear analysis is evident now: the most dangerous nonlinear forcing exploits both linear amplification mechanisms, specifically 3D unsteady oblique waves and 3D steady rolls-streaks, through the redistribution of energy from the first linear mechanism to the second linear mechanism via nonlinearity. The fundamental frequency and spanwise wavenumber, $(\beta,\omega)$=($33.3,11.7)\times10^{-5}$, is very close to the linearly optimal oblique ones, $(\beta,\omega)$=($30,10)\times10^{-5}$, and then nonlinearly generated steady vortices are formed at twice the spanwise wavenumber $(\beta,\omega)$=($66.6,0)\times10^{-5}$. The latter part does not coincide with the maximum linearly amplified lift-up wavenumber, $(\beta,\omega)$=($100,0)\times10^{-5}$, but it is close enough to take advantage of this mechanism in an optimal synergistic way.
\subsubsection{Symmetric superharmonic forcing}
Here, forcing is expanded to consider both the fundamental its first harmonic in both frequency and spanwise wavenumber, see figure \ref{fig:cases}b. Thus, forcing is allowed in 8 forcing components arising from all the combinations of fundamental and first harmonic components.
We retain for now $M=N=2$ and $A=\num{7.07e-5}$, and recall (figure~\ref{fig:J}b) that maximum amplification of shear stress is observed for forcing at $(\beta,\omega)=(50,11.7)\times10^{-5}$.
\begin{figure}
\hspace{-0.8cm}
\includegraphics[width=1.1\textwidth,trim={0cm 0cm 0cm 0},clip]{subharmonic_A.pdf}
\caption{Nonlinear optimization for superharmonic symmetric forcing $M=N=2$. Amplitudes of optimal forcing (left) and response (right) for each individual harmonic component $(m,n)$, as depicted in figure~\ref{fig:cases}b. Values are normalized with the the total forcing amplitude $A=\num{7.07e-5}$. The square marks the frequency/wavenumber that maximum drag increase is observed.}
\label{fig:A_subharmonic}
\end{figure}
The optimum in the $(\beta,\omega)$ plane for the superharmonic case is close to the one found for the fundamental case where the oblique waves were the optimal forcing mechanism. However, now the TS waves at twice the frequency of the oblique waves share similar amplitude to the oblique waves. As can be observed in figure~\ref{fig:A_subharmonic}, only two major forcing components exist at the optimal $(\beta,\omega)$ pair. The optimal forcing corresponds to a three-dimensional oblique wave $(\beta,\omega)$ and a superharmonic two-dimensional TS wave at twice the frequency, $(2\beta,0)$. The optimal superharmonic forcing is in agreement with the typical scenario for H-type transition. In the literature describing H-type transition, typically the TS is called the fundamental wave and the oblique the subharmonic, but our description is equivalent.
\begin{figure}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/subharmonic/22_8f/Af_22_5e-05_b300_o70.pdf}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/subharmonic/22_8f/Au_22_5e-05_b300_o70.pdf}
\includegraphics[width=1\textwidth,trim={30cm 2cm 30cm 0cm},clip]{figs/subharmonic/22_8f/fq_22_5e-05_b300_o70.jpg}
\caption{Optimal H-type superharmonic case $M=N=2$ at $(\beta,\omega)=(50,11.7)\times 10^{-5}$ for $A=\num{7.07e-5}$. This forcing results to the maximum amplification of shear stress for superharmonic forcing over all forcing frequencies and wavenumbers (point D in figure \ref{fig:J}b). Isosurface of $f'_u= \pm \num{8.3e-9}$ (bottom left) and $u'= \pm 0.07$ (bottom right), blue negative iso-value and red positive one. One fundamental wavelenghth is shown in $z$.}
\label{fig:Htype}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\textwidth,trim={30cm 41cm 30cm 20cm},clip]{figs/subharmonic/22_8f/combined.jpg}
\includegraphics[width=1\textwidth,trim={30cm 41cm 30cm 20cm},clip]{figs/subharmonic/22_8f/vortex.jpg}
\caption{Harmonic response components for the optimal superharmonic case shown in figure \ref{fig:Htype}. The response is dominated by the growth of $(\beta,\pm \omega)_{OW}$ oblique waves and planar TS waves $(0,2\pm \omega)_{TS}$ at twice the fundamental frequency ($u'$ overlaid for OW and TS) characteristic of H-type resonance. The oblique waves generate nonlinearly $(2\beta,0)$ streamwise vortices ($\omega'_x$ shown) which promote the linear growth of streaks ($u'$ shown). Also, initial stages of streak instability are observed near the domain outlet.}
\label{fig:vortex2}
\end{figure}
The streamwise evolution of each forcing and response harmonic for the optimal superharmonic case is shown in figure \ref{fig:Htype}. The forcing is dominated by the superharmonic two-dimensional TS wave at twice the fundamental frequency, $(0,2\omega)$, and the three-dimensional oblique waves, $(\beta,\pm\omega)$. Since spanwise reflectional symmetry has been enforced, the amplitudes of the $(\beta,+\omega)$ and $(\beta,-\omega)$ oblique waves are equal and only one of those is shown. Despite the differences in the forcing components when compared to the fundamental case where only the oblique waves are present, the amplitude response is qualitatively similar and dominated by streaks. However, the nonlinearly generated streaks have almost twice as high amplitude when compared to the fundamental case, due to the efficient amplification of the parent oblique waves through the resonant interaction with the planar TS waves (see figure \ref{fig:vortex2}). The subsequent stages are similar to the ones of the fundamental case where streamwise vortices are generated from the nonlinear interactions between $(\beta,\omega)$ and $(\beta,-\omega)$ components, which in turn produce streaks. Finally, towards the end of the domain, low- and high-speed streaks start to undergo streamwise oscillations. These oscillations are stronger than in the fundamental oblique case (compare with figure \ref{fig:Oblique}), since for a given amplitude, the resonant H-type forcing leads to higher streak amplification through the stages described above.
For the optimal superharmonic forcing (oblique and TS waves), low levels of energy are observed in two other forcing components, the $(2\beta,0)$ and $(2\beta,2\omega)$ components. In figure \ref{fig:analysis_H}, we analyse their importance by plotting the total amplitude of each forcing and response harmonic. Also, to ensure converged results, we have increased the number of response harmonics and lowered the forcing amplitude. The left column shows the superharmonic optimized forcing and response amplitude for each harmonic, when forcing is allowed in all 8 forcing components as above. The $(0,2\omega)$ TS and $(\beta,\omega)$ oblique waves are the dominant forcing components, and the $(2\beta,0)$ and $(4\beta,0)$ streaky structures, the $(\beta,\omega)$ and $(3\beta,\omega)$ oblique waves and the (0,0) MFD in the response. The second column corresponds to an optimization restricted solely to the $(0,2\omega)$ and $(\beta,\omega)$ harmonics: it is seen that it reproduces closely the more complex optimization of the left-column (the reached $\Delta C_D$ is nearly the same). On the contrary (right column), it is seen that if the $(0,2\omega)$ TS wave is replaced by the $(2\beta,0)$ streaky component in the forcing (this also corresponds to a superharmonic forcing, but in spanwise wavenumber), then the optimal response only achieves weak drag increase, in agreement with those shown for the fundamental optimization at similar amplitudes. This validates that it is the interplay between the $(0,2\omega)$ TS and the $(\beta,\omega)$ oblique harmonic that accounts for the strong amplification observed in H-type transition.
The catalytic role of the TS waves in the superharmonic H-type case be also evidenced from a weakly-nonlinear analysis based on scaling arguments. The analysis (see appendix \ref{sec:WNL}) shows that the drag increase, for the two optimal fundamental and superharmonic cases, scales as
\begin{align}
\Delta C_D &=& \Delta C_{D,2}A^2&+&\Delta C_{D,3}A^3&+&\Delta C_{D,4}A^4&+&\cdots && \mbox{ for superharmonic forcing} \\
\Delta C_D &=& \Delta C_{D,2}A^2&& &+&\Delta C_{D,4}A^4&+&\cdots && \mbox{ for fundamental forcing}.
\end{align}
Hence, superharmonic resonant forcing allows the presence of additional odd terms in the expansion.
For example, the $A$-order $(0,2\omega)$ TS and $(\beta,-\omega)$ oblique waves generate the $ A^2$-order $(\beta,\omega)$ oblique wave, which may interact with the $ A$-order $(-\beta,-\omega)$ oblique wave to promote an $ A^3$-order (0,0) MFD. Hence, in the case of superharmonic forcing, it is possible to take advantage of the odd-orders to optimize the drag increase, while for fundamental oblique forcing, only even orders exist in the expansion.
\begin{figure}
\begin{minipage}{0.33\textwidth}
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\caption{Optimal forcing (top) and response (bottom) amplitudes for superharmonic cases $M=4$, $N=2$ at $(\beta,\omega)=(50,11.7)\times 10^{-5}$ for $A=\num{5.13e-5}$. Forcing has been optimised in all fundamental and superharmonic components (left); only at $(\beta,\omega)$ and $(\beta,2\omega)$ (middle); only at $(\beta,\omega)$ and $(\beta,0)$.}
\label{fig:analysis_H}
\end{figure}
\subsection{Fundamental forcing for higher $ A $ and the effects of truncation} \label{sec:fundconv}
The results shown above were obtained with a truncated expansion with $M=N=2$ response modes. The impact of the truncation can be preliminary assessed by examining the amplitude of higher or truncated wavenumber/frequency components. In figure~\ref{fig:A_fundamental}, we observe that the second frequency harmonics, $(m \beta,2\omega)$, have a much smaller amplitude than the fundamental ones, $(m \beta,\omega)$. However, this is not the case for the truncation in $\beta$ harmonics. As we saw above, a strong response was obtained at $(2\beta,0)$ component through nonlinear interactions.
\begin{figure}
\includegraphics[width=1\textwidth]{fundamental_V_Cf.pdf}
\caption{Maximum drag increase as a function of forcing amplitude for different truncations in spanwise (M) and frequency (N) components (left). Results shown for the optimal oblique fundamental case $(\beta,\omega)$=($33.3,11.7)\times10^{-5}$, with reflectional symmetry in the spanwise direction. Skin friction coefficient for $M=N=4$ as a function of streamwise distance for different forcing amplitudes (right). For $A>\num{8.5e-5}$, varicose transition of the low-speed streaks is observed.}
\label{fig:convergence_optim_fundamental}
\end{figure}
To directly assess the truncation error, calculations were performed with larger $M$ and $N$. The resulting maximum in the cost function is shown, as a function of forcing amplitude, in figure~\ref{fig:convergence_optim_fundamental}a for various orders of truncation. Apart from the most highly truncated case, we see a tendency towards convergence for forcing amplitudes $A<\num{7e-05}$. The $M=N=1$ case is clearly too highly truncated--this can be understood physically since the nonlinear amplification mechanisms described above require the generation of streaks at $(2\beta,0)$.
As discussed above, during the initial stages of transition and for a small forcing amplitude, the second and higher $\omega$-harmonics are not as strongly amplified as the $\beta$-harmonics, meaning that the energy spreading occurs faster in $\beta$ than $\omega$. For example, the $M=2,N=1$ case is almost identical to the $M=N=2$. Similarly, $M=4,N=2$ is close to $M=N=4$. The dominance of the $\beta$-cascade has been observed in various DNS and experimental transition studies (\cite{breuer1997late} and \cite{yeo2010dns} triggered transition with an impulse wavepacket, or K-type controlled transition \citep{rist1995direct}) and it is consistent with the results presented here.
\subsubsection{Streak breakdown}
Increasing further the number $M$ of $\beta$-harmonics, a sudden change is observed in the drag values for $A\approx8\times 10^{-5}$ and for $M\ge 4$, see figure~\ref{fig:convergence_optim_fundamental}a. The skin friction coefficient for various amplitudes is shown in figure~\ref{fig:convergence_optim_fundamental}b for $M=N=4$. The spanwise averaged skin-friction coefficient is calculated from the $(0,0)$ streamwise velocity component:
\begin{equation*}
C_f = \frac{\tau_{\textrm{wall}}}{\frac{1}{2} U_\infty^2},
\quad \textrm{with} \quad
\tau_{\textrm{wall}} = \nu \left( \frac{\partial \hat{u}_{00}}{\partial y} \right)_{y=0}.
\end{equation*}
For comparison, the values of the laminar skin friction coefficient ($C_f^{\mathrm{lam}} = 0.664/ \sqrt{Re_x}$) and the empirical one corresponding to fully developed turbulence ($C_f^{\mathrm{turb}} = 0.455 / \ln^2 ( 0.06 Re_x )$) are shown with dashed lines \citep{white2006viscous,yeo2010dns}. The transition is accompanied by an overshoot of the skin friction coefficient up to the empirical turbulent values, for sufficiently high forcing amplitudes. Increasing $M$, the transition threshold moves to lower forcing amplitudes, suggesting that the flow has transitioned to a more complex regime, for which a large number of harmonics would be required to capture quantitatively accurately the solution, as will be discussed in greater detail below.
\begin{figure}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/fundamental/44/Af_44_8e-05_b200_o70.pdf}\vspace{1cm}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/fundamental/44/Au_44_8e-05_b200_o70.pdf}
\includegraphics[width=1\textwidth,trim={35cm 12cm 35cm 25cm},clip]{figs/fundamental/44/fq_44_8e-05_b200_o70.jpg}
\includegraphics[width=1\textwidth,trim={30cm 0cm 30cm 13cm},clip]{figs/fundamental/44/Q_44_8e-05_b200_o70_v2.jpg}
\caption{Laminar-turbulent transition for optimal oblique fundamental case (symmetry in $z$) with $M=N=4$ at $(\beta,\omega)$=($33.3,11.7)\times10^{-5}$ for $A=\num{11.3e-5}$. Amplitude for forcing (top left) and response (top right) for each individual harmonic component $(m,n)$. Isosurfaces of streamwise perturbations for $f'_u= \pm \num{8.3e-9}$ (middle left) and $u'= \pm 0.16$ (middle right). Vortical structures visualized with the $Q$-criterion (iso-$Q=\num{1.4e-9}$; green) and low speed streaks ($u'=-0.16$; blue). Two fundamental wavelenghts are shown in $z$ to facilitate the presence of staggered $\Lambda$-structures and hairpins.}
\label{fig:Oblique62_transition}
\end{figure}
The amplitudes of the forcing and response components are shown in figure~\ref{fig:Oblique62_transition} for $A=\num{11.3e-5}$ and the $M=N=4$ case, again for the optimal fundamental forcing. The forcing reaches maximum amplitude further downstream at $Re_x=200,000$, when compared to the lower amplitude case, and also a second distinct region of forcing appears for $Re_x>250,000$. For all the cases examined, we noticed that the second region of forcing triggers streak oscillations in the streamwise direction and they subsequently break down. Regarding the response, once the $(2\beta,0)$ streaks reach sufficiently high amplitude, the harmonic component $(4\beta,0)$ increases up to $Re_x \approx 320,000$ along with the $(3\beta,\omega)$ harmonic. The latter is responsible for the generation and progressive elevation of hairpins from the wall. The MFD increases monotonically in agreement with the increase in skin friction coefficient. A cascade of nonlinear interactions makes the amplitude of all the harmonic components to increase significantly toward the end of the domain, where the skin friction has exceeded the empirical turbulent value.
For all the cases presented above we have imposed symmetry in $z$. Under this restriction, the low-speed streaks undergo varicose oscillations in $x$ whereas the high-speed streaks undergo sinuous oscillations (subharmonic varicose case in \cite{andersson2001breakdown}) creating a staggered pattern of $\Lambda$-structures and the emergence of hairpin vortices further downstream \citep{asai2002instability}. Similar behavior has been observed in DNS simulations \citep{berlin1999numerical} where a pair of oblique waves was introduced in the domain inlet and reflectional symmetry in spanwise was imposed. Initial stages of this process are visualized using the Q-criterion. The emergence of the hairpin vortices coincides with the final regime during the transition process and the overshoot of the skin friction coefficient to the turbulent values.
\subsubsection{Breaking the $z$-reflectional symmetry}
\begin{figure}
\centering
\includegraphics[width=1\textwidth]{fundamental_S_Cf.pdf}
\caption{Maximum drag increase for optimal oblique fundamental forcing with no imposed symmetry in $z$ (solid lines) and $z$-reflectional symmetry (SYM,
dashed lines) for various orders of truncation $M,N$ (left). Skin friction coefficient as a function of $Re_x$ for various forcing amplitudes for $M=3,N=2$ (right). For the non-symmetric cases, sinuous-like transition of the low-speed streaks is observed.}
\label{fig:Cf_amplitude}
\end{figure}
\begin{figure}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/fundamental/32GEN/Af_32_10e-05_b200_o70.pdf}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/fundamental/32GEN/Au_32_10e-05_b200_o70.pdf}\vspace{1cm}
\includegraphics[width=1\textwidth,trim={36cm 12cm 36cm 25cm},clip]{figs/fundamental/32GEN/fq_32_10e-05_b200_o70.jpg}
\includegraphics[width=1\textwidth,trim={30cm 0cm 30cm 13cm},clip]{figs/fundamental/32GEN/Q_32_10e-05_b200_o70.jpg}
\caption{Laminar-turbulent for optimal oblique fundamental forcing (no symmetry in $z$) with $M=3,N=2$, $(\beta,\omega)$=($33.3,11.7)\times10^{-5}$, $A=\num{14.1e-5}$. Maximum amplitudes of optimal forcing (top left) and response (top right) for each individual harmonic component $(m,n)$. Isosurfaces of streamwise perturbations for $f'_u$ and $u'$ (middle). Vortical structures visualized with the $Q$-criterion along with low-speed-streaks (bottom). Two fundamental wavelenghts are shown in $z$ ($f'_u= \pm \num{8.3e-9}, u'= \pm 0.2, Q=10^{-9}$).}
\label{fig:ObliqueGEN}
\end{figure}
In this section we relax the reflectional symmetry assumption in $z$ that was imposed above. The computational cost increases since we have to account for almost twice the number of harmonics. We focus again here on the optimal fundamental forcing at $(\beta,\omega)$=($33.3,11.7)\times10^{-5}$ that is initiated through a pair of equal amplitude oblique waves.
The dependence of the maximum drag increase on the forcing amplitude with and without $z$-reflectional symmetry is shown in figure~\ref{fig:Cf_amplitude}a for $M=N=2$ and $M=3,N=2$. The dashed lines correspond to the values obtained in the previous section imposing reflectional symmetry (SYM cases). We repeated the optimization for each forcing amplitude and restricted the forcing to act only on the oblique $(\beta,\omega)$ component without imposing symmetry in $z$. The initial guess was the symmetric solution with random noise of 10\% of the maximum value of each forcing component added to break the symmetry. Up to a critical forcing amplitude, $A_{c}=\num{18e-5}$ for $M=N=2$ and $A_{c}=\num{9.2 e-5}$ for $M=3,N=2$, the solution converges to the one satisfying the reflectional symmetry. At the critical amplitude the solution bifurcates to a different equilibrium with approximately two times higher drag increase than the one for the symmetric case.
In figure~\ref{fig:Cf_amplitude}b, the skin friction coefficients of the two cases with and without $z$-reflectional symmetry are shown for $M=3,N=2$ for various forcing amplitudes. For the symmetric cases, the skin friction values saturate to values close and above the laminar curve for low forcing amplitudes. Only the highest amplitude shows a tendency for departuture from the trend of the lower amplitude curves, indicating that the streaks are on the verge of symmetric breakdown. Relaxing the symmetry assumption, and for the same amplitudes as the symmetric case, the skin friction reaches values significantly higher than the turbulent ones. For the two highest amplitudes, after the overshoot to the turbulent values, the skin friction drops. In contrast, for the symmetric case a monotonic increase for similar values beyond the threshold of the turbulent skin friction values was observed (see figure \ref{fig:convergence_optim_fundamental}b).
The amplitude of the forcing and response harmonic components is shown in figure~\ref{fig:ObliqueGEN} for the $M=3,N=2$ case. The oblique forcing components, $(\beta,+\omega)$ and $(\beta,-\omega)$, break their symmetry and are characterized by different amplitudes now. Also, two new local maxima appear for $Re_x>\num{2e5}$ in the amplitude forcing curves. This is similar to the one that appeared in the symmetric case that promoted the varicose streak breakdown, but here it is more pronounced. The amplitude response of the different harmonic components shows that the initial stages are similar to the ones observed in the case with imposed spanwise symmetry. The oblique waves $(\beta,\pm\omega)$ interact nonlinearly to promote the growth of rolls-streaks at twice the spanwise wavenumber, $(2\beta,0)$. The $(3\beta,\pm\omega)$ components are amplified as well, similar to the symmetric case. Immediately after that, all the harmonics appear to attain high energy values, due to the more effective energy spread through the symmetry break of the forcing.
Despite the similaties in the amplitude response, the flow is qualitatively different to the symmetric case. The reflectional symmetry break of the forcing can be observed in the isorfurfaces of the streamwise velocity perturbation. Towards the decaying phase of the forcing, the dominance of the left-travelling $(1,-1)$ oblique wave is evident. This mechanism promotes in an optimal way the sinuous-like breakdown of the low-speed streaks. The sinuous low-speed streak breakdown occurs for lower forcing thresholds compared to the varicose breakdown. This is in accordance with previous results in the literature examining streak breakdown \citep{andersson2001breakdown}. This regime is not associated with hairpin vortices, but with quasi-streamwise vortices of alternate sign, in accordance with the findings of \cite{asai2002instability}. Visualization of the vortices using the $Q$-criterion shows longitudinal vortices staggered on each side of the low speed streaks up to $Re_x=300,000$. At this location the the low-speed streaks come close together in an alternate staggered manner and merge. In the same time they break and then create individual $\Lambda$-like staggered structures. Exactly at this stage, the skin friction coefficient has reached the turbulent value.
\subsection{Superharmonic forcing for high $\textit{MN}$ and high $ A $} \label{sec:superconv}
\begin{figure}
\includegraphics[width=1\textwidth]{subharmonicJ.pdf}
\caption{Maximum drag increase for optimal H-type superharmonic forcing at $(\beta,\omega)=(50,11.7)\times10^{-5}$ with $z$-symmetry as a function of forcing amplitude (left). Various orders of truncation $\mathit{MN}$ are shown. Skin friction coefficient (right) as a function of $Re_x$ for $M=6,N=3$.}
\label{fig:J_sub}
\end{figure}
A convergence study of the truncated HBM expansion was performed for the superharmonic case with imposed $z$-reflectional symmetry. Up to a forcing amplitude $A=\num{3e-05}$, the solution appears converged, for the $M=N=2$ case. Increasing the forcing amplitude, the flow transitions. For $A> \num{5.13e-05}$ and $M=6,N=3$, the skin friction coefficient overshoots towards the turbulent values, see figure~\ref{fig:J_sub}b. Similarly to the symmetric fundamental case, a monotonic increase of the skin friction coefficient is observed by increasing the forcing amplitude.
\begin{figure}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/subharmonic/63_8f/Af_63_4e-05_b300_o70.pdf}
\includegraphics[width=0.5\textwidth,trim={1cm 0 1cm 0},clip]{figs/subharmonic/63_8f/Au_63_4e-05_b300_o70.pdf}\vspace{1cm}
\includegraphics[width=1\textwidth,trim={36cm 12cm 36cm 32cm},clip]{figs/subharmonic/63_8f/fq_63_4e-05_b300_o70.jpg}
\includegraphics[width=1\textwidth,trim={30cm 10cm 30cm 24cm},clip]{figs/subharmonic/63_8f/Q_63_4e-05_b300_o70_v2.jpg}
\caption{Laminar-turbulent transition for optimal H-type superharmonic forcing with $M=6,N=3$, $(\beta,\omega)=(50,11.7)\times10^{-5}$, $A=\num{5.65e-5}$. Total energy for forcing (top left) and response (top right) for each individual harmonic component $(m,n)$. Isosurfaces of streamwise perturbations $f'_u$ (middle left) and $u'$ (middle right). Vortical structures visualized with the $Q$-criterion along with low-speed-streaks (bottom). Two fundamental wavelenghts are shown in $z$ ($f'_u= \pm \num{6.2e-9}, u'= \pm 0.2$, Q=\num{5.5e-9}).}
\label{fig:Htype_highampl63}
\end{figure}
The amplitude of the forcing and response components are shown in figure~\ref{fig:Htype_highampl63}, for the high amplitude forcing case with $M=6,N=3$. The dominant forcing component is the $(0,2\omega)$ mode followed by the $(\beta,\pm\omega)$ components. The nonlinear interaction of $(\beta,\pm\omega)$ response components create a strong response in the $(2\beta,0)$ component. This process continues resulting in the progression of energy along the $\beta$-axis and the emergence of $(4\beta,0)$ and $(6\beta,0)$ components. Although higher harmonics are also created by nonlinear interactions, they are less energetic since they are not amplified by the transient growth to the same degree as the low-wavenumber modes \citep{breuer1997late}.
The low-speed streaks undergo symmetric varicose type of oscillations, whereas the high-speed streaks oscillate in a sinuous mode in the streamwise direction. The response appears similar to the one for the fundamental oblique forcing, where spanwise reflectional symmetry is imposed. The low-speed streaks attain a $\Lambda$-shape, which creates a staggered pattern of $\Lambda$ vortices. These vortices are identified using the $Q$-criterion.
\subsection{Summary and implications for turbulent dynamics}
Three high-amplitude forcing cases have been identified above as the worst case nonlinear disturbances that reach values of the skin friction coefficient that are close to and above the empirical turbulent values. These cases were obtained by restricting the forcing to specific harmonic components, with or without spanwise symmetry. For the three cases considered,
we plot the mean velocity profile at various streamwise locations for the highest forcing amplitude in figure \ref{fig:velocity_profiles}. Distinct regimes can be identified in accordance with the transition sequences observed in the previous sections.
\begin{itemize}
\item
At the very early stages of transition up to $Re_x=200,000$ the velocity profiles obey the linear wall law $u^+=y^+$ for all three cases. This stage is characterised by linear growth of perturbations. Transition has been triggered optimally with a pair of oblique waves (fundamental cases). In the case of subharmonic instability (superharmonic case), the TS waves are also excited.
\item
The second stage of transition is associated with the generation of streaks through nonlinear interactions of the oblique waves. At this regime the skin friction coefficient departs from the laminar Blasius values. This new regime is reflected as well through the modification of the local velocity profile outside of the viscous sublayer for $y^+>5$, in accordance with the increase of the skin friction coefficient (recall that $u_\infty^+=\sqrt{2/C_f}$). Depending on the symmetry of the forcing, varicose $\Lambda$-shaped (symmetry in $z$) or sinuous (no symmetry in $z$) low-speed streaks have been clearly identified for $Re_x>260,000$.
\item
A third regime is observed where a distinct plateau is formed in the buffer region, $15<y^+<30$ for all three cases. In the symmetric cases, hairpin-like vortical structures grow around the $\Lambda$-shaped low-speed streaks at $Re_x\approx 330000$. In the case without symmetry, alternate quasi-streamwise vortices grow around the sinuous low-speed streaks, i.e. $Re_x\approx 300000$. Immediately after the vortical structures are formed, the skin friction coefficient overshoots to the turbulent values.
\item
The final transition regime is associated with the breakdown. At this regime, the skin friction coefficient reaches the empirical turbulent values and energy is transferred to all the higher harmonics.
\end{itemize}
\begin{figure}
\vspace{0.5cm}
{\scriptsize \hspace{0.7cm} (a) Fundamental ($z$-symmetry) \hspace{0.7cm} (b) Fundamental (no symmetry) \hspace{0.7cm} (c) Superharmonic ($z$-symmetry) }\\
\includegraphics[width=0.35\textwidth,trim={0.cm 0cm 1cm 0cm},clip]{figs/fundamental/44/loglayer_44_8e-05_b200_o70_v2.pdf}
\includegraphics[width=0.31\textwidth,trim={1.5cm 0cm 1cm 0cm},clip]{figs/fundamental/32GEN/loglayer_32_10e-05_b200_o70_v2.pdf}
\includegraphics[width=0.31\textwidth,trim={1.5cm 0cm 1cm 0cm},clip]{figs/subharmonic/63_8f/loglayer_63_4e-05_b300_o70_v2.pdf}
\caption{Mean velocity profiles during transition in innner units based on the local friction velocity $u_\tau$. Linear ($u^+=y^+$; dashed) and log laws ($u^+=\frac{1}{0.41}\log y^+ + 5$; dashed-dotted) are also shown. Then insets show the skin friction coefficient as a function of $Re_x$ and the location where the velocity profiles are plot are marked with circles.}
\label{fig:velocity_profiles}
\end{figure}
When $z$-reflectional symmetry is imposed, the velocity profiles show qualitatively similar characteristics as $Re_x$ increases, both for the fundamental and superharmonic cases. A monotonic decrease of the local streamwise velocity is observed in accordance with the monotonic increase of the skin friction coefficient. For the fundamental case with symmetry, a small logarithmic region is observed for $Re_x=360000$; however, the velocities are lower than those associated with the turbulent profile, which is in accordance with the increased skin friction coefficient beyond the turbulent values. For the superharmonic case and the forcing amplitudes examined here, the behavior is similar without the observation of logarithmic region. The symmetric high amplitude solutions show striking similarities with the optimal nonlinear solutions calculated by \cite{cherubini2011minimal,duguet2012self} in the time domain. However, this is not surprising since their calculations were obtained by using a symmetric initial condition as a guess for the optimization \citep{cherubini2011minimal} or spanwise symmetry was imposed \citep{duguet2012self}.
Interestingly, for the fundamental case with no symmetry in $z$ (figure \ref{fig:velocity_profiles}b), the velocity profiles at the final stages of transition show characteristics similar to the ones observed in turbulent boundary layers. Specifically, the velocity profile appears to develops a nascent logarithmic region, $u^+=\frac{1}{0.41}\log y^+ + 5$, that extends in $y^+$ as $Re_x$ increases. The skin friction coefficient, after an initial overshoot above the turbulent empirical values, drops to values close to the turbulent ones. For this specific case, we observed sinuous low-speed streaks and quasi-streamwise staggered vortices, which are fundamental building blocks in the self-sustaining process (SSP) in a variety of streamwise homogeneous flows \citep{waleffe1997self}, in contrast to the varicose streak instability and the hairpins that were observed for the two symmetric cases.
\section{Conclusion}\label{sec:conclusions}
The nonlinear optimal mechanisms for wall-bounded laminar-turbulent transition have been investigated through solution of the frequency-domain Harmonic-Balanced Navier-Stokes equations by projecting the governing Navier-Stokes equations on to a limited number of harmonics whose triadic interactions are considered. The new framework complements previous methods that seek nonlinear optimal initial conditions in the time domain within a finite time horizon. The proposed \emph{nonlinear input/output analysis} identifies the most dangerous nonlinear forcing mechanisms that trigger transition and can be viewed as \emph{the minimal forcing seed in the frequency domain}.
Optimal nonlinear forcing mechanisms that lead to transition and maximize the skin-friction coefficient have been identified based on a variational method using direct-adjoint looping. By increasing the finite forcing amplitude, we identified the key-mechanisms that distort the laminar flow and lead to transition. We showed that for fundamental forcing, the most amplified disturbances correspond to a pair of oblique waves with frequency and spanwise wavenumber close to the linear optimal one. Nonlinearity is responsible for redistributing the energy to the streamwise uniform vortex component which leads to the amplification of streaks through the lift-up mechanism. Depending on the imposed spanwise symmetry, the low-speed streaks break down to turbulence through varicose oscillations (imposing reflectional symmetry in spanwise) or sinuous-like ones (no symmetry in spanwise), with the latter being more efficient in promoting transition. In each case, hairpin vortices and quasi-streamwise vortices are observed prior to breakdown. When multi-harmonic forcing is allowed, the resonant interaction between oblique waves and TS waves at twice the frequency allows for even more rapid transition. At the final stages of transition, the skin-friction coefficient reaches the empirical turbulent values and the velocity profiles depart from the law of the wall, for all cases examined here. However, only for the non-symmetric sinuous-like streak breakdown the velocity profiles develop a clear logarithmic region similar to the one observed for turbulent boundary layers.
\vspace{0.5cm}
We would like to thank U. Rist for providing the details for the boundary conditions used in the DNS \citep{rist1995direct}. This work was initiated while D. Sipp was Visiting Associate at Caltech. G.R. and T.C. also acknowledge the support of the Boeing Company through a Strategic Research and Development Relationship Agreement CT-BA-GTA-1.
|
1,108,101,565,682 | arxiv | \section{Discussion}
\label{sec:conclusion}
In this paper, we propose a novel self-supervised prior with a joint-level spatial-temporal layer for recovering human motions from monocular occluded videos. For better generalization ability, we represent the human motion in 2D maps, and thus we can employ a lot of non-occluded 2D and 3D data for training a model that is robust to different occlusion types and various motions. The proposed method can obtain accurate and coherent motions from monocular occluded videos. To reduce the gap between synthetic and real occlusion data, we further build the first 3D occluded motion dataset~(OcMotion), which can be used for training and evaluating video-based methods for occlusion scenarios. We hope the dataset will promote future research on video-based human mesh recovery.
However, there also exist some limitations. Although the current implementation can obtain satisfactory results from a single-person occluded video, the 2D detection still affects the accuracy of 2D motion capture. When an incorrect detected joint has high confidence, it cannot be removed in the 2D occluded motion map and may cause a jittering 3D motion. The case often appears in multi-person scenarios since the inter-person occlusions are more ambiguous, and the 2D detector may predict erroneous joints with high confidence for closely interacting people. Future works may integrate more low-level vision features in the estimation and filter the noises with motion prior knowledge to prevent the undesirable impact.
\section{OcMotion Dataset}\label{sec:Dataset}
\begin{figure*}
\vspace{-8mm}
\begin{center}
\includegraphics[width=1.0\linewidth]{Figures/dataset.pdf}
\end{center}
\vspace{-8mm}
\caption{Samples from the proposed OcMotion dataset. The dataset contains 300K images captured at 10 FPS~(frame per second) with accurate 3D motion annotations.}\label{fig:dataset}
\vspace{-8mm}
\end{figure*}
Although 3D human datasets are exponentially increasing in recent years, only a few datasets are particularly designed for occlusion problems. AGORA~\cite{patel2021agora} contains frequent occlusions, but it is a synthetic dataset. 3DOH50K~\cite{zhang2020object} is the first 3D human dataset that explicitly considers object occlusion. However, it is an image dataset and cannot be used for evaluating video-based methods. In this work, we extend the 3DOH50K dataset to have complete motion annotations. We obtain human motions with~\cite{huang2021dynamic} and the samples in 3DOH50K are also used for constraining the optimization. For severely occluded cases, we manually adjust the 3D motion. To evaluate the accuracy of the dataset, we randomly select 5K images and manually annotate 2D poses. The re-projection error on these images is 7.3 pixels, which is sufficient for motion capture tasks. Finally, the dataset has 300K images captured at 10 FPS, 43 sequences with 6 viewpoints, 3D motion annotations represented by SMPL, 2D poses, and camera parameters.
\begin{table}
\caption{Comparison with commonly used 3D human datasets. OcMotion is the first motion dataset that contains diverse real object occlusions with complete and accurate annotations.}
\label{tab:dataset}
\vspace{-5mm}
\begin{center}
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{l|c|c|c|c|c|r|r}
\noalign{\hrule height 1.5pt}
Dataset &Occlusion Data &Sequence &Real Data &3D Pose &Mesh &Frames &Views \\
\noalign{\hrule height 1pt}
Human3.6M~\cite{ionescu2013human3} &-- &\checkmark &\checkmark &\checkmark &-- &3.6M &4 \\
AIST++~\cite{li2021ai} &-- &\checkmark &\checkmark &\checkmark &\checkmark &10.1M &9 \\
HUMBI~\cite{yu2020humbi} &-- &\checkmark &\checkmark &\checkmark &\checkmark &17.3M &107 \\
MPI-INF-3DHP~\cite{mehta2017monocular} &+ &\checkmark &\checkmark &\checkmark &-- &1.3M &14\\
3DPW~\cite{von2018recovering} &++ &\checkmark &\checkmark &\checkmark &\checkmark &50K &1 \\
MuPoTs-3D~\cite{mehta2018single} &++ &\checkmark &\checkmark &\checkmark &-- &8K &1\\
Panoptic Studio~\cite{joo2017panoptic} &++ &\checkmark &\checkmark &\checkmark &-- &1.5M &480 \\
GPA~\cite{wang2019geometric} &++ &\checkmark &\checkmark &\checkmark &-- &700K &5 \\
3DOH50K~\cite{zhang2020object} &++++ &-- &\checkmark &\checkmark &\checkmark &50K &6 \\
AGORA~\cite{patel2021agora} &++++ &-- &-- &\checkmark &\checkmark &17K &-- \\
\textbf{OcMotion} &++++ &\checkmark &\checkmark &\checkmark &\checkmark &300K &6\\
\noalign{\hrule height 1.5pt}
\end{tabular}
}
\end{center}
\vspace{-10mm}
\end{table}
We show the comparison with commonly used 3D human datasets in~\tabref{tab:dataset}. Despite a large number of samples or the wide variety of actions, most existing datasets pay little attention to the occlusion problem. MPI-INF-3DHP~\cite{mehta2017monocular}, MuPoTs-3D~\cite{mehta2018single} and Panoptic Studio~\cite{joo2017panoptic} include only a few occluded cases. GPA~\cite{wang2019geometric} has limited types of occluders. In addition, AGORA~\cite{patel2021agora} and 3DOH50K~\cite{zhang2020object} are image-based datasets and cannot be applied to video-based methods. In contrast, our dataset contains complete motion annotations and explicitly considers object-occluded scenarios, which may promote future research on object-occluded human mesh recovery.
\section{Experiments}
\label{sec:experiments}
\subsection{Implementation details}
The spatial-temporal layer consists of 3 dilated convolution layers with a dilation rate of 1, 2, and 5, respectively. We also adopt two separate transformers~\cite{dosovitskiy2020image} to model long-term spatial and temporal information. The decoder of the 2D branch only contains a LayerNorm~\cite{ba2016layer} and a linear layer, thus the features encoded by the prior can represent a full motion. The lifting network is also a transformer. We rely on PyTorch~\cite{paszke2019pytorch} to implement the model and use AdamW~\cite{loshchilov2017decoupled} optimizer with a learning rate of 1e-4 for training. The batchsize of all experiments is 32. The model is trained on a single NVIDIA RTX 3090 GPU with 24GB memory for 45 epochs. We use a joint regressor in LSP format~\cite{johnson2010clustered} to obtain 3D joints and calculate errors for the predicted mesh.
\begin{table}
\caption{Quantitative comparison with state-of-the-art methods. Our method obtains good results and achieves the best performance in some metrics on occluded datasets. $^\ast$ means the image-based method. $^\dagger$ denotes the method that explicitly considers the occlusion problem. Total Params is the total number of model parameters including 2D detection~\cite{cao2017realtime} or feature extraction~\cite{kolotouros2019learning}.}
\label{tab:experiment}
\vspace{-3mm}
\begin{center}
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{l|c|c c c|c c c| c c c}
\noalign{\hrule height 1.5pt}
\begin{tabular}[l]{l}\multirow{2}{*}{Method}\end{tabular} &\begin{tabular}[l]{l}\multirow{2}{*}{\begin{tabular}[1]{c}Total\\Params\end{tabular}}\end{tabular} &\multicolumn{3}{c|}{OcMotion} &\multicolumn{3}{c|}{3DPW} &\multicolumn{3}{c}{3DPW-OC}\\
& &MPJPE &PA-MPJPE &Accel. &MPJPE &PA-MPJPE &PVE &MPJPE &PA-MPJPE &Accel.\\
\noalign{\hrule height 1pt}
HMMR~\cite{kanazawa2019learning} &29.8M &-- &-- &-- &116.5 &72.6 &139.3 &-- &-- &-- \\
$^\ast$SPIN~\cite{kolotouros2019learning} &27.0M &88.2 &56.7 &47.0 &96.9 &59.2 &116.4 &105.0 &71.3 &44.6 \\
$^\ast$$^\dagger$OCHMR~\cite{khirodkar2022occluded} &-- &-- &-- &-- &89.7 &58.3 &107.1 &-- &-- &--\\
$^\ast$$^\dagger$LASOR~\cite{yang2022lasor} &-- &-- &-- &-- &-- &57.9 &-- &-- &-- &-- \\
VIBE~\cite{kocabas2020vibe} &48.3M &89.6 &58.6 &44.5 &93.5 &56.5 &113.4 &98.3 &69.7 &39.0 \\
TCMR~\cite{choi2021beyond} &108.9M &95.8 &62.6 &24.3 &95.0 &55.8 &111.5 &90.3 &63.0 &\textbf{8.0} \\
$^\ast$$^\dagger$OOH~\cite{zhang2020object} &33.0M &83.0 &55.0 &48.6 &86.7 &55.2 &105.2 &90.4 &57.0 &45.3 \\
MEVA~\cite{luo20203d} &92.0M &88.8 &59.9 &29.0 &86.9 &54.7 &-- &91.4 &63.5 &17.8 \\
$^\ast$$^\dagger$ROMP~\cite{sun2021monocular} &29.0M &79.4 &48.1 &57.2 &85.5 &53.3 &103.1 &-- &66.5 &--\\
$^\ast$$^\dagger$PARE~\cite{kocabas2021pare} &32.9M &81.1 &52.0 &43.6 &\textbf{82.9} &52.3 &\textbf{99.7} &90.5 &56.6 &40.9 \\
Wan~\etal~\cite{wan2021encoder} &-- &-- &-- &-- &88.8 &50.7 &104.5 &-- &-- &-- \\
Chen~\etal~\cite{chen2021self} &51.4M &-- &-- &-- &85.8 &\textbf{50.4} &100.6 &-- &-- &-- \\
\textbf{$^\dagger$Ours}~(w/o OcMotion) &56.3M &72.1 &44.9 &24.2 &83.7 &51.8 &110.4 &90.1 &54.5 &16.6 \\
\textbf{$^\dagger$Ours} &56.3M &\textbf{58.3} &\textbf{36.1} &\textbf{23.2} &83.7 &51.7 &110.1 &\textbf{89.4} &\textbf{53.4} &16.6 \\
\noalign{\hrule height 1.5pt}
\end{tabular}
}
\end{center}
\vspace{-10mm}
\end{table}
\noindent\textbf{Metrics}. We adopt the metrics in previous works~\cite{kocabas2020vibe,luo20203d,choi2021beyond} to evaluate our method. The Mean Per Joint Position Error (MPJPE) and the MPJPE after rigid alignment of the prediction with ground truth using Procrustes Analysis (PA-MPJPE) are used for measuring joint positions. The Per Vertex Error (PVE) and acceleration error (Accel.) are applied to evaluate mesh quality and motion smoothness.
\subsection{Dataset}
We follow previous works~\cite{kocabas2020vibe,choi2021beyond} to set the training data for a fair comparison. The spatial-temporal prior is trained on 2D~(PoseTrack~\cite{andriluka2018posetrack}, InstaVariety~\cite{kanazawa2019learning} and PennAction~\cite{zhang2013actemes}) and 3D human motion datasets~(Human3.6M~\cite{ionescu2013human3} and MPI-INF-3DHP~\cite{mehta2017monocular}). We use Human3.6M and MPI-INF-3DHP to train the lifting network. The method is evaluated on 3DPW~\cite{von2018recovering}, Human3.6M, and MPI-INF-3DHP. In addition, we use 3DPW-OC, the occluded sequences from the entire dataset selected by~\cite{zhang2020object}, to demonstrate the effectiveness of our approach in occluded cases. We also report the results with and without OcMotion training. The sequences \textit{0013, 0015, 0017, 0019} in OcMotion with 6 views are used for testing, and the rest are adopted for training.
\begin{figure*}
\begin{center}
\includegraphics[width=1\linewidth]{Figures/qualitative_results.pdf}
\end{center}
\vspace{-6mm}
\caption{Our method achieves good performance in both occluded and non-occluded cases.}
\label{fig:quali_results}
\vspace{-7mm}
\end{figure*}
\subsection{Comparison to state-of-the-art results}
We first compared our method to state-of-the-art approaches on OcMotion dataset. To the best of our knowledge, OcMotion is the first video dataset designed for the object-occluded human mesh recovery task. We conducted experiments on this dataset to demonstrate the superiority of our method in occluded cases. As shown in~\tabref{tab:experiment}, since previous methods do not explicitly consider the occlusion problem, our method significantly outperforms previous video-based methods on all metrics. In addition, PARE~\cite{kocabas2021pare} and OOH~\cite{zhang2020object} are image-based methods and are designed for the occluded human reconstruction task. Benefited from the spatial-temporal information, our method can obtain more robust results. Moreover, OOH~\cite{zhang2020object} uses UV map representation, though it can explicitly describe an occluded human, the resampled meshes show a lot of artifacts~\figref{fig:results}~(e). Furthermore, we found that video-based methods show more motion jitters and have higher acceleration errors in occluded cases. In contrast, the spatial-temporal prior obtains features of a full motion and avoids the ambiguities induced by occlusions. With the prior, our method achieves the best performance in terms of acceleration error and produces more temporally coherent results~\figref{fig:qualitative_video} on the occlusion dataset.
\begin{figure*}
\vspace{-4mm}
\begin{center}
\includegraphics[width=1\linewidth]{Figures/results.pdf}
\end{center}
\vspace{-8mm}
\caption{Qualitative comparison among the methods that utilize temporal information~(b, c, d) and explicitly consider the occlusion problem~(e, f). Our method is more robust to occlusions than other methods.}
\label{fig:results}
\vspace{-8mm}
\end{figure*}
Another strength of our method is the good generalization ability. Since we represent the human motion in a 2D map, the background of the image cannot affect the model performance, thus the trained model is insensitive to environmental changes. Besides, the self-supervised training on a large amount of synthetic data makes the model robust to various occlusions. We conduct some experiments on 3DPW dataset to demonstrate the advantages. Our model is trained on indoor 3D datasets but can also achieve satisfactory performance in outdoor scenarios. Specifically, \cite{wan2021encoder} and \cite{chen2021self} also develop attention modules to exploit temporal cues and produce the best results. The results in~\tabref{tab:experiment} demonstrate that our method can also obtain similar results as state-of-the-arts on the non-occluded dataset. To further show the performance in more challenging occluded environments, we follow~\cite{zhang2020object} to evaluate the method on 3DPW-OC dataset, which is a subset of 3DPW that contains occlusions. We have the same training data as VIBE, MEVA, and TCMR. Although VIBE, MEVA, and TCMR achieve excellent results on 3DPW, they are sensitive to the occlusion~\figref{fig:results}~(b,c,d), while our method is robust. Moreover, TCMR relies on the past and future features to regress SMPL parameters for the current frame, and it may produce over-smoothed motion~(refer to supplemental video). With the spatial-temporal prior, our method can obtain coherent and highly dynamic motions.
We conduct experiments on Human3.6M dataset to further demonstrate the effectiveness of our method. As shown in~\tabref{tab:h36m}, our method achieves state-of-the-art in protocol 1 of Human3.6M in terms of MPJPE, which outperforms VIBE by 4.1mm. In addition, in~\tabref{tab:experiment} and~\tabref{tab:h36m}, LASOR~\cite{yang2022lasor} and Pose2Mesh~\cite{choi2020pose2mesh} also recover human mesh from 2D detections, and our method can obtain more accurate results.
We also report the number of model parameters to show the efficiency of our approach. The results in~\tabref{tab:experiment} show the total parameters including 2D pose detection~\cite{cao2017realtime} and feature extraction~\cite{kolotouros2019learning} of different methods. We have fewer parameters than TCMR and MEVA. Specifically, the lifting network in our method contains 19.2M parameters. The 2D detection consumes most of the computations, which can be replaced with more compact models. To further demonstrate the runtime efficiency, we report the model parameters without the 2D detection and feature extraction among the two-stage methods in~\tabref{tab:h36m}. Our model can achieve the best performance with the fewest parameters. For current implementation with~\cite{cao2017realtime}, the inference FPS is 32 on a single NVIDIA RTX 3090 GPU, which is acceptable for real-time applications. Furthermore, when regressing from the 2D pose inputs, our model runs at 286 FPS, which is significantly faster than common 2D pose detectors. The inference speed will not be the bottleneck for real-world implementation.
\begin{table}
\vspace{-5mm}
\caption{We conduct quantitative comparisons on Human3.6M following the protocols defined in~\cite{kanazawa2018end}. In the column of \#Params, we also compare the number of model parameters without 2D detection and feature extraction among two-stage methods. MACs is the estimated multiply–accumulate operations. Our method achieves competitive performance as state-of-the-art methods with higher runtime efficiency.}
\label{tab:h36m}
\vspace{-2mm}
\begin{center}
\resizebox{0.65\linewidth}{!}{
\begin{tabular}{l|c|c|c c|c c c}
\noalign{\hrule height 1.5pt}
\begin{tabular}[l]{l}\multirow{2}{*}{Method}\end{tabular} &\begin{tabular}[l]{l}\multirow{2}{*}{\#Params}\end{tabular} &\begin{tabular}[l]{l}\multirow{2}{*}{MACs}\end{tabular} &\multicolumn{2}{c|}{Protocol 1} &\multicolumn{3}{c}{Protocol 2}\\
& & &MPJPE &PA-MPJPE &MPJPE &PA-MPJPE &Accel.\\
\noalign{\hrule height 1pt}
HMMR~\cite{kanazawa2019learning} &-- &-- &-- &-- &-- &56.9 &--\\
MEVA~\cite{luo20203d} &65.0M &1.31G &73.4 &51.9 &76.0 &53.2 &15.3\\
$^\ast$$^\dagger$PARE~\cite{kocabas2021pare} &-- &-- &78.5 &55.1 &71.6 &49.9 &32.3 \\
$^\ast$Pose2Mesh~\cite{choi2020pose2mesh} &76.3M &3.81G &-- &-- &64.9 &47.0 &-- \\
DSD-SATN~\cite{sun2019human} &-- &-- &-- &-- &59.1 &42.4 &--\\
VIBE~\cite{kocabas2020vibe} &21.3M &0.45G &68.8 &49.5 &65.9 &41.5 &27.3 \\
$^\ast$$^\dagger$OOH~\cite{zhang2020object} &-- &-- &74.7 &53.3 &61.8 &41.2 &35.3 \\
TCMR~\cite{choi2021beyond} &81.9M &1.29G &79.8 &56.8 &62.3 &41.1 &\textbf{5.3} \\
Chen~\etal~\cite{chen2021self} &-- &-- &-- &-- &58.9 &\textbf{38.7} &--\\
Wan~\etal~\cite{wan2021encoder} &-- &-- &-- &-- &\textbf{56.3} &\textbf{38.7} &--\\
\textbf{$^\dagger$Ours}~(w/o OcMotion) &\textbf{19.2M} &0.49G &\textbf{64.7} &\textbf{46.3} &59.7 &40.1 &13.0 \\
\noalign{\hrule height 1.5pt}
\end{tabular}
}
\end{center}
\vspace{-10mm}
\end{table}
\subsection{Ablation}
\noindent\textbf{Self-supervised prior.}
We ablate the self-supervised spatial-temporal prior to reveal the properties of this module in occluded human motion capture. We found that the occlusion token can promote the prior to learn better motion representation in self-supervised learning. In \tabref{tab:Ablation}, we remove the token and use constant value 0 to represent the occluded parts as~\cite{zhang2020object}. Using the prior learned with the constant value even produces a worse performance. The experimental results demonstrate the importance of the occlusion token. We then compared the model with and without the prior. The two strategies also use the occlusion augmentation technique~\cite{sarandi2018robust}. The results show that the prior significantly improves the joint and vertex accuracy. Directly recovering 3D human motion from occluded 2D poses without the prior is a highly ill-posed problem, and the same occluded 2D pose can map to various 3D poses. However, with the self-supervised training on many 2D motions and synthetic occlusions, the model learns the prior knowledge of converting an occluded motion to a highly dynamic full motion. The results show that with the assistance of the learned prior, the occluded human motion capture can obtain more accurate and coherent motions.
\begin{table}
\vspace{-6mm}
\caption{Ablation studies on different key components on OcMotion dataset. The spatial-temporal layer~(ST layer) and the self-supervised prior improve the motion capture in both joint accuracy and motion smoothness in occluded cases. + denotes adding the corresponding module on the temporal model.}
\label{tab:Ablation}
\vspace{-3mm}
\begin{center}
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{l|c|c|c c c c}
\noalign{\hrule height 1.5pt}
Method &\#Params &MACs &MPJPE&PA-MPJPE &PVE &Accel.\\
\noalign{\hrule height 1pt}
temporal &19.1M &0.46G &70.5 &44.8 &79.6 &40.6 \\
+ smoothness loss &19.1M &0.46G &72.6 &45.4 &80.1 &27.4 \\
+ spatial &19.1M &0.47G &65.4 &41.1 &76.4 &40.4 \\
+ ST layer &19.2M &0.49G &63.6 &39.8 &73.7 &38.6 \\
+ ST layer + self-supervised prior (w/o occlusion token) &19.2M &0.49G &66.7 &43.2 &78.9 &37.1 \\
+ ST layer + self-supervised prior &19.2M &0.49G &58.2 &36.1 &67.5 &36.4 \\
+ ST layer + self-supervised prior + smoothness loss &19.2M &0.49G &58.3 &36.1 &67.1 &23.2 \\
+ ST layer + self-supervised prior + smoothness loss + gt 2D pose &19.2M &0.49G &40.5 &23.8 &45.5 &16.3 \\
\noalign{\hrule height 1.5pt}
\end{tabular}
}
\end{center}
\vspace{-10mm}
\end{table}
\noindent\textbf{Spatial-temporal layer.}
The joint-level spatial-temporal information is essential for the occlusion problem. We first compared the temporal model without the spatial relations in~\tabref{tab:Ablation}. In addition, VIBE, TCMR, and MEVA also rely on temporal information and do not consider the kinematic features. The two comparisons in~\tabref{tab:Ablation} and~\tabref{tab:experiment} turn out the same conclusion that the model which only focuses on the temporal relation cannot achieve good results in the occluded cases. We then added a separate transformer module like~\cite{zheng20213d} in the temporal model to exploit the kinematic relation to assist the motion capture. Since the model cannot consider joint-level spatial-temporal correlation in the same stage, the performance is inferior to the spatial-temporal layer.
\begin{figure*}
\vspace{-6mm}
\begin{center}
\includegraphics[width=1\linewidth]{Figures/qualitative_video.pdf}
\end{center}
\vspace{-8mm}
\caption{Qualitative results on consecutive frames in occlusion scenario. More results can refer to our supplemental video.}
\label{fig:qualitative_video}
\vspace{-8mm}
\end{figure*}
\noindent\textbf{Sensitivity to occlusions.}
We analyzed the sensitivity to the occlusion of our method to further demonstrate the effectiveness of the proposed components by synthesizing additional occlusions on OcMotion dataset. As shown in~\figref{fig:curve}, we first synthesize occluded images with different occlusion ratios to simulate severely occluded cases. Since the OcMotion dataset contains a lot of real occlusions, we synthesize realistic occlusions among the occlusion ratios from 0\% to 50\%. We use OpenPose~\cite{cao2017realtime} to detect visible 2D poses and conduct evaluations of different models on the synthetic data. The results show that the temporal approach without joint-level spatial-temporal correlation is sensitive to the occlusions, while the model with self-supervised prior is robust. With the proposed modules, our method is insensitive to various occlusions.
\begin{figure}
\vspace{-4mm}
\begin{center}
\includegraphics[width=0.5\linewidth]{Figures/curve.pdf}
\end{center}
\vspace{-8mm}
\caption{We synthesize additional realistic occlusions on OcMotion with different occlusion ratios. The curve on different occlusion data shows that our method is more robust to variant occlusion proportions and different occlusion types.}
\label{fig:curve}
\vspace{-6mm}
\end{figure}
\section{Introduction}
\label{sec:introduction}
Recovering 3D human motion from monocular images is a long-standing problem, which has wide applications such as computer animation, human behavior understanding, and human well-being. Recently, researches in this area have gained significant progress~\cite{pavllo20193d,luo20203d,kocabas2020vibe,zheng20213d,wang2017outdoor}, but most of them do not consider the occlusion scenarios that are very common in the real world.
Only a few works explicitly focus on 3D pose estimation from occluded images. Although~\cite{huang2009estimating,rafi2015semantic,sarandi2018robust,zhang2020object,kocabas2021pare,yang2022lasor,sun2021monocular,Pose2UV} can estimate 3D human poses from single occluded images, they do not utilize temporal information, and the results in highly ambiguous occlusion cases are unreliable. Some latest works~\cite{huang2021dynamic,rempe2021humor} incorporate the temporal information in a variational motion prior and can predict the full human motion from partial observations, but the optimization process in their methods is particularly time-consuming.
\begin{figure*}
\begin{center}
\includegraphics[width=1.0\linewidth]{Figures/pip.pdf}
\end{center}
\vspace{-8mm}
\caption{The pipeline of our method. The 2D and 3D motions are represented with three 2D maps~($I_{2d}^o$, $I_{2d}$, and $I_{3d}$). We first train a joint-level spatial-temporal prior for occluded motions based on dilated convolutions and transformer blocks via self-supervised learning~(light blue box). The prior is then adopted to assist the lifting network for occluded human motion capture. The overall training can be conducted with synthetic occlusions on non-occluded data~(dark blue box). When testing on real occlusion data, the occluded map ($I_{2d}^o$) obtained with detectors is fed into the network to regress the 3D motion.}
\label{fig:pipeline}
\vspace{-6mm}
\end{figure*}
Training a neural network to regress 3D human motions from monocular occluded videos with temporal relations can significantly improve the runtime efficiency. However, the intuitive task faces two challenging obstacles. Due to occlusions and the loss of depth information, the problem is highly ambiguous as multiple 3D poses can map to the same 2D observations. On the other side, the limited amount of real occlusion 3D human data is still a bottleneck for training a robust temporal model. Thus, the networks cannot obtain stable and accurate results. Strong prior knowledge with compact motion representation is a feasible solution to address the challenges.
Our key-idea is \textbf{to employ non-occluded human data to learn a joint-level spatial-temporal prior for occluded human motion with a self-supervised strategy.} We then build a 3D occluded motion dataset~(\textbf{OcMotion}) to reduce the gap between synthetic and real occlusion data, which contains 43 motions and 300K frames with accurate 3D annotations. To the best of our knowledge, OcMotion is the first video-based dataset explicitly designed for the occlusion problem. To achieve the occluded motion capture, inspired by~\cite{zhang2020object} which represents a single occluded human in a 2D map, we propose an occluded motion map to simultaneously encode spatial and temporal information. As shown in~\figref{fig:representation}, with the map representation, we can synthesize occlusions on non-occluded data and eliminate the effect of image appearance. The intermediate representation also improves the human motion capture in generalization ability and accuracy~\cite{pavllo20193d,li2021exploiting,zhang2021learning}. Nevertheless, due to the loss of depth information and ambiguities of the occluded parts, it is difficult for the network to directly recover 3D human motion from the occluded 2D motion map. Thus, we learn a joint-level spatial-temporal prior to assist the 3D motion capture. We construct a 2D self-supervised task that regresses the full motion map from the synthetic occluded input. Different from the simple augmentation techniques~\cite{sarandi2018robust,biggs20203d,rockwell2020full,kocabas2021pare}, the self-supervised strategy recovers original 2D inputs from the occluded motion map, which formulates the problem as a masked image modeling~(MIM) task~\cite{bao2021beit,he2021masked} and can learn an expressive motion representation for the occluded problem. With the self-supervised training on a large amount of 2D data, the network also generalizes well on various occlusion types. To lift 3D human motion from the encoded motion features can alleviate the ambiguities induced by occlusions and produce more accurate results. In addition, previous video-based methods~\cite{kocabas2020vibe,choi2021beyond,pavllo20193d,li2021exploiting} all rely on image features~\cite{kolotouros2019learning} or pose features~\cite{li2021exploiting} to model the temporal relations among different frames, which ignore the joint-level kinematic information that is important for the occlusion problem. In contrast, with the map representation, we can further design a spatial-temporal layer based on dilated convolution and vision transformer~\cite{dosovitskiy2020image} to simultaneously consider the local joint-level correlations and global motion dependencies. The learned joint-level correlations can improve the prediction accuracy for occluded parts. In the inference phase, we detect the visible and reliable 2D joint coordinates with state-of-the-art detectors~\cite{cao2017realtime,fang2017rmpe,sun2019deep} to obtain the occluded 2D map, and then the full 3D human motion can be estimated with the trained model.
To sum up, the contributions of this paper are as follows:
\begin{itemize}
\vspace{-1mm}
\item We propose an occluded motion prior learned with self-supervised strategy on non-occluded data, which reduces the ambiguities of occluded observations and improves the motion capture in both generalization ability and accuracy.
\item We propose a spatial-temporal layer with a map representation to simultaneously model local joint-level correlations and global motion dependencies for reliable occluded human motion capture.
\item We build the first 3D occluded motion dataset, OcMotion, which contains 300K images captured in real occlusion scenarios. The dataset can be used for both training and testing. The dataset and code are publicly available.
\end{itemize}
\section{Method}\label{sec:method}
Our goal is to recover human motion from monocular occluded images. We first represent the human motions in 2D maps and synthesize occluded motions using a large amount of non-occluded data~(\secref{sec:representation}). Since directly recovering 3D human motion from occluded 2D poses is a highly ambiguous problem, we then design a prior with a spatial-temporal layer to exploit joint-level correlations for occluded motions via self-supervised learning~(\secref{sec:prior}). The learned prior is then adopted to improve occluded 3D human motion capture~(\secref{sec:lifting}). Finally, we can obtain 3D human motion from an occluded video with 2D pose detectors in real-time~(\secref{sec:mocap}).
\subsection{Motion Representation}~\label{sec:representation}
To model joint-level spatial-temporal correlations for the occluded inputs, we represent the 2D motion and 3D motion in 2D maps. Different from previous map representations~\cite{sun2021monocular,zhang2020object} that encode body cues for single images, the motion map is more flexible in exploiting skeletal and temporal information. The 2D motion map $I_{2d}\in \mathcal{R} ^{F\times K\times 2}$ records the 2D joint coordinates of a motion sequence, which can be obtained by state-of-the-art 2D pose detectors~\cite{cao2017realtime,chen2018cascaded,fang2017rmpe,sun2019deep}. The $F$ and $K$ are frame length and number of joints. We normalize the 2D poses with the bounding-box for better generalization ability. The 3D motion map $I_{3d}\in \mathcal{R} ^{F\times N\times 6}$ is represented with SMPL parameters~\cite{loper2015smpl}. $N$ is the number of joints for SMPL. A 6D representation~\cite{zhou2019continuity} for the 3D rotation is adopted, thus the dimension of a joint rotation is 6.
The motion map can be further used to represent the occluded motion by modifying the values in image pixels. Previous works~\cite{zhang2020object,zhang2021learning} represent the occluded joints with constant values~(\eg, 0) and inpaint all pixels of the target region. Although they can also obtain satisfactory 2D full maps via an inpainting task, we found the strategy cannot learn good representation for downstream tasks in self-supervised learning. Thus, we store a learned occlusion token in the pixel of an occluded joint in the motion map. In the inference phase, the occluded 2D motion map can be generated by replacing the detected low confidence joints with the learned token. With the occluded motion representation, we can train a joint-level spatial-temporal correlation prior for the occlusion problem and improve the generalization ability.
\begin{figure}
\begin{center}
\vspace{-4mm}
\includegraphics[width=1.0\linewidth]{Figures/representation.pdf}
\end{center}
\vspace{-8mm}
\caption{We record 2D and 3D poses from consecutive frames in rows of 2D maps~(c,d). The dark green denotes occluded joints, which are represented with a learnable token. Since the human motion maps are not affected by image appearance, we can synthesize occlusions on a large amount of non-occluded data~(\eg, Human3.6M~\cite{ionescu2013human3}) to train a robust model.}
\label{fig:representation}
\vspace{-10mm}
\end{figure}
\subsection{Joint-level spatial-temporal correlation prior}~\label{sec:prior}
Due to the severe ambiguities and insufficient occlusion training data, it is challenging to learn an accurate and generalized model for the occluded human body capture. To address these obstacles, we use synthetic occlusion data to train a joint-level spatial-temporal correlation prior via self-supervised learning. The trained prior can learn the compact and expressive representation for the occluded 2D motion. The pre-trained network parameters with the learned prior knowledge are then used in the downstream lifting task to improve the occluded pose estimation.
\textbf{Occlusion data synthesis.} Since existing occluded human data is insufficient, we synthesize occlusions on non-occluded data~\cite{ionescu2013human3} and enforce the prior to exploit the essential motion information from the visible parts. To learn good representation from visible observations, recent works in masked image modeling tasks~\cite{bao2021beit,he2021masked} use random masks with a specific masking ratio to construct a self-supervised learning task. However, the strategy can hardly simulate real occlusion cases. Thus, we synthesize occlusions to generate an occluded 2D motion $I_{o2d}\in \mathcal{R} ^{F\times K\times 2}$ for the prior to learn more expressive occluded 2D motion representation. The procedures are shown in~\figref{fig:representation}~(a, b, c). We add a random occluder on the non-occluded image and store the visible 2D joint coordinates in the 2D motion map. Different from previous works~\cite{zhang2020object,zhang2021learning}, we use a learned occlusion token to represent the occluded joints in the map~(dark green region). The token promotes the network to learn better motion representation in self-supervised training. In addition, benefiting from our intermediate representation, the synthetic occlusions on the RGB images do not cause domain gap~\cite{zhang2020object}, thus we can learn a generalized prior with a large amount of data.
\textbf{Self-supervised training.} With the synthetic data, estimating accurate 3D human bodies from the occluded inputs is still a challenging problem. Previous works \cite{zhang2020object,kocabas2021pare,sarandi2018robust} apply a simple occlusion augmentation in 2D images and regress the 3D poses. However, directly recovering 3D motion from 2D poses itself is an ill-posed problem, and the occlusions further increase the uncertainty for the estimation, thus the same 2D observation can be mapped to various different 3D poses. They can still not obtain stable results for occluded cases. Thus, we propose a self-supervised strategy to learn a motion prior to reduce the ambiguities and fully exploit the motion information from visible cues. Different from previous works~\cite{zhang2020object,kocabas2021pare} that directly estimate 3D poses from 2D images, the self-supervised learning enforces the prior to recover the original full 2D map $I_{2d}$ from the occluded input $I_{o2d}$. With the self-supervised learning, the prior learns the expressive representation for the human motion~\cite{he2021masked,bao2021beit} with only the partial observations.
For the occluded pose estimation, the kinematic information and temporal dependencies are bases to infer an occluded body part. The joint-level spatial-temporal correlations are essential cues. Conventional methods~\cite{kocabas2020vibe,choi2021beyond,pavllo20193d,li2021exploiting} model the temporal relations among different frames based on image or pose features, which ignore the local kinematic information. Although a recent work~\cite{zheng20213d} considers both the spatial and temporal aspects with distinct transformer modules~\cite{dosovitskiy2020image}, it explores spatial and temporal relations in different stages. The temporal information cannot be considered in the spatial stage, thus some joint-level temporal features may be lost. Therefore, we design a dilated convolutional layer to extract local joint-level spatial-temporal features from the occluded 2D map. As shown in~\figref{fig:pipeline}, the joint-level spatial-temporal layer consists of 3 convolutional layers with different dilations. The convolutional kernel can simultaneously model joint relations from the kinematic structure and temporal features, which is essential for inferring occluded joints.
Since the dilated convolutional layers are limited in temporal connectivity, we concatenate the 3 output feature maps from the joint-level spatial-temporal layer and use two stacked transformers~\cite{dosovitskiy2020image} to model global information. The concatenated map $f\in \mathcal{R} ^{F\times K\times D}$ has the same resolution as the occluded map, and $D$ is the dimension of the vectors in feature map pixels. We then add a learnable spatial positional embedding $E_{SPos}\in \mathcal{R} ^{K\times D}$ in the skeletal dimension of the feature map. The resulting skeletal features are fed into the first transformer encoder to exploit information across the skeletal structure. We flatten the output features to $f_T\in \mathcal{R} ^{F\times (K \cdot D)}$ and add a learnable temporal positional embedding $E_{TPos}\in \mathcal{R} ^{F\times (K \cdot D)}$. The features are then encoded with the second transformer module. Finally, a regression head with an MLP block and a Layer norm is used to obtain a full 2D motion map $I_{2d}$.
As shown in the light blue box in \figref{fig:pipeline}, the training does not require extra 3D annotations, and we can employ diverse 2D motion data~\cite{andriluka2018posetrack,zhang2013actemes} to learn a robust prior. A large amount of synthetic data also relieves the data-hungry problem for the transformer-based prior in the occluded pose estimation.
\textbf{Loss function.} We use the following loss function for the self-supervised training:
\begin{equation}
\mathcal{L}_{self}=M_{2d}\left\| I_{2d}-I_{2d}^{gt}\right\|_1,
\end{equation}
The L1 loss is applied to the masked region of the predicted motion map. $M_{2d}$ is the synthetic mask, where the occluded part is 1, and the rest are 0. We found that the L1 loss without any other smoothness term can promote the prior to learn better motion representation in the self-supervised task.
When the training is completed, we use the pre-trained prior in the lifting task to assist the 3D motion capture.
\subsection{Occluded motion lifting}~\label{sec:lifting}
The encoder of the trained prior is then used in the lifting network to improve the occluded motion estimation. The lifting network is a transformer with two heads for 3D motion map and shape parameters. The transformer module has the same structure as the first transformer in the prior and receives the feature map $f$ from the convolutional layer. We then add the output features to the features from the prior and feed them to the heads to obtain 3D motion map $I_{3d}$ and shape parameters $\beta$. With the learned motion representation, the lifting network can achieve better accuracy and generalization, and we also optimize the network parameters of the prior in the lifting stage. Since the 2D occluded motion maps are not affected by image appearance, the training of the lifting network can also be conducted on synthetic non-occluded data~\cite{ionescu2013human3,sarandi2018robust}. The loss function is:
\begin{equation}
\mathcal{L}_{motion}= \mathcal{L}_{rec} + \mathcal{L}_{shape} + \mathcal{L}_{smo}.
\end{equation}
The reconstruction term is:
\begin{equation}
\mathcal{L}_{rec}= \left\| I_{3d}-I_{3d}^{gt}\right\|_2 + \left\| V_{3d}-V_{3d}^{gt}\right\|_2 + \left\| J_{3d}-J_{3d}^{gt}\right\|_2.
\end{equation}
The $I_{3d}$, $V_{3d}$ and $J_{3d}$ are predicted 3D motion map, vertex positions and joint positions. $gt$ denotes the ground-truth. To prevent the jitters and obtain coherent motions, we further add a smoothness loss on the predicted 3D motion map.
\begin{equation}
\mathcal{L}_{smo}= \sum_{i=0}^{F-1} \left\| I_{3d}^i - I_{3d}^{i+1} \right\|_2,
\end{equation}
where $i$ denotes the $i$th row of $I_{3d}$. Additional shape and regularization terms are applied to prevent abnormal body shape:
\begin{equation}
\mathcal{L}_{shape}= \left\| \beta-\beta^{gt}\right\|_2 + \left\| \beta\right\|_2.
\end{equation}
\subsection{Occluded motion estimation from monocular videos}~\label{sec:mocap}
With the trained spatial-temporal motion prior and the lifting network, we can recover full 3D motion from occluded images. We first adopt~\cite{cao2017realtime} to detect 2D poses and corresponding confidence maps. The joint with a confidence lower than 0.6 will be regarded as the occluded joint, which will be replaced with the learned token. Thus, the occluded 2D map can be obtained. We then regress the full 3D motion map through the prior and lifting network from the occluded 2D map. Since the 3D motion resampled from the map is in the local coordinate system, we obtain global positions by solving the least square function.
\begin{equation}
\mathop{\arg\min}_{\left(\mathcal{T}\right)_{0: F-1}} \mathcal{L}=\sum^{F-1}_{t=0}{ (w_c \left\| K(J_{3d}^t+\mathcal{T}^t)-P^{t}\right\| + \left\| \mathcal{T}^{t+1} - \mathcal{T}^{t} \right\| )},
\end{equation}
where $K$ is intrinsic camera parameters, $ P$ and $w_c$ are detected 2D pose and corresponding confidence. $\mathcal{T}$ is the translation of the SMPL model. Finally, the 3D motion with absolute positions can be obtained by adding the translations to the estimated SMPL models.
\section{Related Work}\label{sec:relatedwork}
\subsection{Occluded 3D human pose estimation}
Although the 3D human pose estimation has progressively developed in recent years, it still cannot achieve satisfactory performance in occlusion scenarios. Historically, there are few works~\cite{huang2009estimating,rafi2015semantic,sarandi2018robust,zhang2020object,kocabas2021pare,yang2022lasor} that explicitly focus on occluded human pose estimation. \cite{rafi2015semantic} relies on the depth information to infer the occluded body parts. To capture occluded human poses from RGB images, \cite{huang2009estimating} uses a linear combination of training samples, but it has limited expressive capabilities for various occlusion types and poses. Recently, ROMP~\cite{sun2021monocular,sun2021putting} regresses the occluded human from a single image by encoding the SMPL parameters in a 2D map, but it is not flexible enough to exploit temporal information. Other works~\cite{sarandi2018robust,biggs20203d,rockwell2020full,yang2022lasor,Pose2UV,wang2022best} also adopt neural networks for occluded pose estimation. However, no sufficient real occluded human data can be used to train a robust network, which has been the bottleneck of the regression-based methods for a long time. To improve occlusion-robustness, \cite{sarandi2018robust,biggs20203d,rockwell2020full} use synthetic occlusion data during training. To reduce the gap between synthetic and real occlusions, \cite{zhang2020object} represents the occluded human in the UV map and builds the first image-based object-occluded human dataset. Nonetheless, previous works that do not employ temporal information cannot obtain reliable results. Without the motion data for training, existing methods~\cite{rempe2021humor,huang2021dynamic} can only utilize the temporal information based on motion priors via a time-consuming optimization. In this work, we extend the dataset proposed by~\cite{zhang2020object} to have 43 real occluded 3D motions with complete and accurate annotations, thus we can train a temporal model for real-world occlusion problems. We also synthesize occlusions on existing non-occluded data to fully employ diverse motions. Different from the augmentation techniques in the previous methods~\cite{sarandi2018robust,biggs20203d,rockwell2020full,kocabas2021pare}, we propose a self-supervised strategy by recovering original 2D inputs from the occluded motion map, which can learn an expressive motion representation for the occluded problem via the MIM task~\cite{bao2021beit,he2021masked}. With the pre-trained motion prior, the model can regress full 3D motion from monocular occluded images with high accuracy and runtime efficiency.
\subsection{Human mesh recovery from monocular video}
Conventional methods~\cite{gall2009motion,wang2017outdoor,xu2018monoperfcap,arnab2019exploiting} fit predefined models to 2D image features to realize video-based human mesh recovery via solving a time-consuming constrained optimization. With the development of deep learning, temporal neural networks~\cite{kanazawa2019learning,sun2019human,liu2019temporally,zhao2021travelnet} are applied to learn the dependencies among frames. However, due to the limited training data, they use pseudo-ground-truth labels for training, which are unreliable for modeling accurate 3D human motion. \cite{kocabas2020vibe,luo20203d,choi2021beyond,yuan2022glamr} follow~\cite{kanazawa2019learning,sun2019human} to use the static features generated by image-based methods~\cite{kanazawa2018end,kolotouros2019learning} for modeling temporal relations. These methods depend heavily on the static features and ignore the kinematic information, which results in severe temporal inconsistency~\cite{kocabas2020vibe} and motion oversmoothness~\cite{choi2021beyond}. \cite{wan2021encoder} models the kinematic and temporal relations by attention mechanism, which has been proved useful for human mesh recovery from occluded inputs~\cite{wan2021encoder,chen2021self}. Although previous methods achieve competitive results on specific datasets, the generalization ability remains an obstacle due to the effect of image appearance and occlusions. Our method intermediately represents human motions in 2D maps and generalizes well to the change of environments. With the motion map, the local joint-level spatial-temporal correlations can be easily explored, which improves the human mesh recovery in occluded part inference.
\subsection{2D-to-3D lifting}
Due to the significant development of 2D pose detectors~\cite{cao2017realtime,chen2018cascaded,fang2017rmpe,sun2019deep}, 2D-to-3D lifting approaches generally outperform direct estimation methods in generalization ability and accuracy. Although many works~\cite{chen20173d,martinez2017simple,tekin2017learning,zhou2017towards,jiang20103d} achieve comparable results from single 2D poses, they can hardly be leveraged in real-world applications due to the loss of temporal dependencies. In recent years, some works~\cite{lee2018propagating,pavllo20193d,wang2020motion,li2021exploiting} add spatial-temporal information in 3D pose estimation, which greatly improve the accuracy and stability. \cite{lee2018propagating,hossain2018exploiting} rely on recurrent neural networks, but they cannot achieve parallel processing of multiple frames. Temporal convolution networks~\cite{pavllo20193d} with confidence heatmaps~\cite{cheng2019occlusion}, attention mechanism~\cite{liu2020attention} and body prior~\cite{chen2021anatomy} are also applied to explore spatial-temporal relations. Like graph convolutional networks~\cite{wang2020motion,cai2019exploiting}, dilated temporal convolutions are also inherently limited in temporal connectivity. Recently, transformer-based 3D human pose estimation~\cite{li2021exploiting,zheng20213d} show superior performance on long-term motions. However, they can only model information from one dimension. The joint-level spatial and temporal relations cannot be simultaneously considered, which is not suitable for the occlusion problem. Thus, we design a spatial-temporal layer with dilated convolutions to improve the current transformer-based structure in occluded human motion capture. The model can consider joint-level kinematic information and temporal dependencies in the same stage, which reduces ambiguities of occluded motions and consequently improves the accuracy and occlusion-robustness.
|
1,108,101,565,683 | arxiv |
\subsection{Authentication in Disk Drive}
\vspace{-0.15in}
\subsection{Disk-level Access-Control}
\label{sec:auth_disk}
{\textsc{Key-SSD}}
implements a disk-level access-control mechanism where unauthorized requests are denied by the disk drive.
Access authentication is performed using
such computational resources as an ARM-based, multi-core storage controller on SSDs.
SSD communicates with a host through various I/O protocols such as SATA protocol.
In this paper, we implemented {\textsc{Key-SSD}} for SATA-based SSD.
{\bf {\textsc{Key-FTL}}:}
{\textsc{Key-SSD}} can extend the FTL to manage the key per block, for which we call {{\em \textsc{Key-FTL}}}.
{\textsc{Key-FTL}} can be implemented statically or dynamically, depending on how the key per block is managed in the SSD's internal memory.
In a static method, a key field can be added to each mapping table entry to have a unique key for each {disk} block.
This method, called {\textsc{Key-FTL(S)}} is advantageous in that the implementation is simple and the key search time for block access is O(1) because FTL is a linear page-table, as shown in Figure~\ref{fig:key-ftl}(a).
However,
the drawback is that a large memory space is required
because blocks of files that do not need to be protected with a key also require memory space for the key value.
In order to reduce memory space overhead, we propose a dynamic method of managing only LPNs protected by keys in the memory of an SSD ({\textsc{Key-FTL(D)}}).
In our implementation, this method used
a red-black tree for managing LPNs that is protected by the corresponding key because the red-black tree data structure is appropriate for fast searching of LPNs corresponding to keys.
When a block request arrives with LPNs and key,
the key value of the request is hashed to search for the red-black tree of the corresponding LPNs.
The search for LPNs corresponding to the key is performed.
{\bf Access Control Mechanism:}
A disk drive that receives a key along with an I/O request from the host performs access control on the data block by comparing its key value in the {\textsc{Key-FTL}} with the key value in the requested block when performing a read/write operation.
\begin{figure}[!t]
\begin{center}
\begin{tabular}{@{}c@{}c@{}}
\includegraphics[width=0.45\textwidth]{./key-ftl} &
\vspace{-0.1in}
\end{tabular}
\caption{
(a) and (b) depict{\textsc{Key-FTL(S)}} and {\textsc{Key-FTL(D)}}. In (b), K-Node is a pointer to a red-black-tree for LPNs corresponding to a key, and a node in the red-black-tree denotes a LPN.
(c) shows sequence of read/write operations with access-codes.
}
\label{fig:key-ftl}
\vspace{-0.3in}
\end{center}
\end{figure}
When the device receives a write/read command, the authentication is granted as follows.
In case of a write,
when an SSD receives a write request,
the SSD firmware first reads the key from the {\textsc{Key-FTL}} and compares it with the key sent by the host.
If the value of a key is different, it sends an error message back to host without performing the write operation because it is not authenticated.
If the key value is the same as that in the {\textsc{Key-FTL}}, write operation is allowed.
Also, the value of the key field of the {\textsc{Key-FTL}} may be NULL. In this case, it means that write to the LBA has not yet occurred. Therefore, in this case, the key value received from the host is added to the key field and write is performed.
Read also operates the same way as write.
But, in read, if the key field of the LBA referenced in the {\textsc{Key-FTL}} is NULL,
the key is not stored and is granted.
Figure~\ref{fig:key-ftl}(c) shows the read write sequence with keys and how a read request generated by ransomware is denied. Refer to $<$READ, LPN=2, KEY=0xFFFFFF$>$ in the sequence. The key value of LPN=2 is 0x000018 in {\textsc{Key-FTL(S)}} in (a).
In particular, the way to access {\textsc{Key-FTL(D)}} is divided into insert and search steps. Insert step is to add a new (key, LPN) to the {\textsc{Key-FTL(D)}}. When the host attempts to write to the new page on the SSD with the key, the corresponding LPN and key are inserted into the {\textsc{Key-FTL(D)}}. Search step is used for authentication by searching the {\textsc{Key-FTL(D)}} with (key, LPN). In our implementation, in order to minimize the overhead of sequential I/Os for authenticating every access to LPNs, we only allowed the first LPN access during search step.
\begin{comment}
Figure~\ref{fig:key-ftl} describes how the {\textsc{Key-FTL}} is implemented and the read/write sequence with reference to the {\textsc{Key-FTL}}.
Consider examples in Figure~\ref{fig:key-ftl}.
For the host command of the first
request,
$<$WRITE, LPN = 0, KEY = 0x000033$>$, the firmware confirms that the key in the {\textsc{Key-FTL}} is NULL, adds the key value 0x000033 to the {\textsc{Key-FTL}} and performs write.
The same operation is performed on the host command of $<$WRITE, LPN = 2, KEY = 0x000018$>$.
Next, for the command $<$READ, LPN = 0, KEY = 3$>$, the firmware checks the key value of LPN = 0 in {\textsc{Key-FTL}}.
The corresponding key value is also 0x000033, so read is performed.
For {the next} command, $<$WRITE, LPN = 4, KEY = 0x000027$>$, the key value, 0x000027 is added to the {\textsc{Key-FTL}} as its corresponding key value is NULL.
For the last command, $<$READ, LPN = 2, KEY = 0xFFFFFF$>$, which is actually generated by ransomware, the firmware compares to the key value at point LPN = 2 in the {\textsc{Key-FTL}}.
As we will describe in the following section,
in our current implementation,
read/write with no key uses 0xFFFFFF as key value by default.
Since the current key value is 0x000018, an authentication message is sent to the host without performing the read because it is not authenticated.
\end{comment}
{\bf Selective Flush for {\textsc{Key-FTL}}:}
{\textsc{Key-FTL}} is loaded into volatile memory in SSD. Sudden power-failures can cause all {\textsc{Key-FTL}} entries in the memory to be lost, so they need to be synchronized with flash, the permanent storage space. This operation is called {\em flush} operation.
The {\textsc{Key-FTL}} flush function is called from the SSD firmware.
For the SATA protocol, HOST sends the command ATA\_CMD\_FLUSH
and the SSD firmware calls the {\textsc{Key-FTL}} flush
function.
Calling a flush operation can affect SSD performance and the performance overhead is proportional to the size of the mapping table.
The {\textsc{Key-FTL}} additionally manages the key information
and the size of the {\textsc{Key-FTL}} is larger than normal FTL.
In order to minimize the increased flushing overhead due to large FTL, we
propose a method, called {\em Selective Flush} which flushes only the changed entries in {\textsc{Key-FTL}}, rather than updating all the {\textsc{Key-FTL}} entries in batches. This can greatly reduce the overhead of writing FTL to the flash.
\begin{comment}
\sout{
In the selective flush,
the {\textsc{Key-FTL}} is divided into page units and flushes only the corresponding mapping table page when at least one of the mapping entries in the page area is changed.
An additional bitmap-based data structure (flush bitmap) is used to track which mapping entry pages in the {\textsc{Key-FTL}} have been modified.
That is, each page of the mapping table is managed in a single bit unit. If the value of this entry is 1, it means that there is a change in the corresponding page in the mapping table. If it is 0, it means that there is no change in the page in the mapping table.
For example, if the entry in the i-th page of the mapping table changes, the i-th bit of the flush table changes to 1.
Since the entry change in the mapping table only occurs during write or garbage collection, we track the page that contains the entry of the mapping table that changes when write is executed and update the corresponding bit of the flush bitmap to 1.
Then, when flush occurs, it searches the flush bitmap and flushes only pages with bit 1.
When the flush is done, the values of the flush bitmap are changed to 0.
As we will see in the experimental section, we observe that the flush overhead of {\textsc{Key-FTL}} is negligible when using selective flushing.
}
\end{comment}
\begin{comment}
The authentication mechanism is integrated with the process of mapping a logical page to a physical page.
In the proposed technique, an additional field is added to the table.
A new column maintains a key for the corresponding logical page.
When GC is running, it selects a victim block to erase, and moves all valid pages in the block to prevent loss of data.
When a page is moved by GC, the mapping table should be updated accordingly to maintain keys.
\end{comment}
\begin{comment}
A typical scenario of processing write and read requests is illustrated in Figure~\ref{fig:ssd_flow}.
When a write request arrives with a requested logical block address (LBA) and a key, \texttt{KeyIn}, the FTL computes the logical page number (LPN) of the requested LBA, and looks up the logical-to-physical page mapping table to find the pre-assigned key, \texttt{KeyPre}.
If the two keys are identical, the request is granted.
If not, the FTL checks if \texttt{KeyPre} is null or not.
If it is null, it means the incoming data is new, or this file is not under protection of the proposed technique.
In this case, \texttt{KeyIn} is stored to the mapping table.
Otherwise, the request is denied.
In a similar vein, a read request is serviced only if keys are matched.
\end{comment}
\begin{comment}
If data is written to an empty space where a key is not found, it is considered as new data, and the given key is stored.
Thus, if data is added to a file (append operation), the key is not checked, but stored in the table.
It is allowed for an application (potentially ransomware) to append data to a file which belongs to another application.
Since it does not destroy the original data, theoretically, the original data can be restored from ransomware attacks.
However, since one file has more than one key, the benign application cannot read the entire file, which may cause inconvenience, though the original data is still retained.
To address this issue, when the file system needs to allocate a new data block (which means new data is appended to the file), the file system attempts to read any existing block of the file with the key given by the application.
Note that the access code is compared only by the disk drive.
The file system does not know whether the access code given by the application is correct or not.
Only if the attempt succeeds, which means the access code is correct, the file system proceeds to writing the new block to the disk drive.
In this way, we can make sure all blocks belonging to a file are associated with the same access code.
\end{comment}
\section{Acknowledgments }
\label{sec:ack}
\vspace{-0.1in}
We are grateful to Mr. Yung-woo Ko for proof-reading this paper. This work was supported by Samsung Semiconductor research grant.
\section{Background}
\label{sec:back}
\vspace{-0.1in}
\subsection{Threat Model: Ransomware Attacks}
\label{sec:threat}
\vspace{-0.05in}
The primary target of ransomware is data files created by a user through an application (e.g. Word processor).
As will be discussed in more detail in Section~\ref{sec:sec_anal}, typical ransomware goes through infection, persistence, removing backup copies, encryption, and notice.
Specifically, we consider that ransomware may exhibit following behaviors.
\squishlist
\item While encrypting a user file, ransomware may overwrite it or create a new file after deleting the original file.
Though file deletion only removes the metadata that keeps data blocks in the disk, the data blocks may be overwritten by other write operations.
Thus, we consider both cases to ensure no data loss.
\item Ransomware may access files through a regular file system or directly access the raw disk drive without going through the file system.
Traditional access-control mechanisms are implemented in the file system.
If ransomware bypasses the file system, there is no more barrier in the disk drive.
Since the proposed access-control mechanism is implemented in the disk drive, it can also prevent direct access attacks.
\item Ransomware may be a user-level application with a root privilege.
Ransomware can acquire a root privilege by exploiting vulnerabilities in applications or operating system.
Once it has a root privilege, it can access files of other users.
The proposed technique can defend files against this type of ransomware.
\item Ransomware may hijack system calls, but cannot access kernel data structures in file system layers.
System call hijacking is one of the most popular techniques to implement a rootkit (kernel-level malware).
A system call may be replaced by a malicious one.
However, it is assumed that kernel data structures
cannot be tampered because it requires recompilation of the kernel, which is more challenging than hijacking.
\squishend
\vspace{-0.15in}
\subsection{Access Control in Disk Drive}
\label{sec:justification}
\vspace{-0.05in}
Data files created by a user cannot be restored unless they have backup copies.
Loss of data incurs not only financial damages but also interruption of operations.
In 2015, a zero-day ransomware, WannaCry, attacked computers in more than 150 countries, and caused U.K. National Health Service hospitals and Honda Motor Company to shut down~\cite{chen_robert}.
This catastrophic damage implies that existing countermeasures were not effective.
The application may protect data files by a password or encryption, but a password and encryption are intended to protect \textit{contents} of a file from being revealed, not a file itself.
Thus, ransomware is still able to read and encrypt the file, though the ransomware may not interpret the contents of the file.
The proposed approach is to integrate an access-control mechanism with a disk drive.
The access-control mechanism allows access to files under protection only for authorized applications.
This access-control mechanism specifically targets at preventing ransomware.
In fact, this access-control mechanism can be implemented in the file system.
But,
Compared to file system implementation, disk-level implementation has the following advantages.
\textbf{Security:}
Since a disk drive is a separate system from a host, it is not easy to compromise both host and disk drive at the same time.
Especially, if the disk drive does not allow firmware update by the host, it is very difficult to compromise the disk drive unless the disk drive is physically accessible.
\textbf{Compatibility:}
The implementation of the disk-level access-control mechanism is independent of a file system.
To support {\textsc{Key-SSD}}, an additional kernel module needs to be inserted to the file system, but it does not change any structures in the file system.
Therefore, {\textsc{Key-SSD}} can be adopted without changing the file system.
\section{Concluding Remarks}
\label{sec:conc}
\vspace{-0.1in}
This paper proposes a fundamental countermeasure to ransomware attacks.
{\textsc{Key-SSD}}, a disk drive using an access-control mechanism can be
the last barrier of data breach.
Even if ransomware acquires an administrative privilege
and bypasses the access control mechanism in the file system,
it cannot avoid the disk-level access control.
To support disk-level access control,
we slightly modify the Linux kernel to transfer the keys to access-control drives.
Our extensive experimental results demonstrate that the performance overhead for {\textsc{Key-SSD}} with {\textsc{Key-FTL(S)}} is negligible
and file data has been successfully protected from attacks by real ransomware samples.
\subsection{Host to Device Communication}
The key transferred from the kernel to the SATA device driver is transmitted to the disk drive in compliance with the SATA protocol.
SATA protocol consists of 5 layers (Application, Command, Transport, Link, Physical layers).
In the Application layer, the host stores the disk command in the shadow command register. Next, the Command layer uses the Command Sequence State Machine to create a frame information structure (FIS) transfer protocol for transmitting commands of the host.
There are a total of 14 FIS packets in the transport layer and the FIS used for disk read/write includes Register FIS and Data FIS.
In the Transport layer, the FIS is transmitted in the order specified by the Command layer, and the corresponding FIS is encoded in the Link layer and reaches the disk drive through the physical layer. In our implementation,
the key is transferred to the disk using the transport layer of the SATA protocol.
In particular, we use the reserved space of the Register FIS to transmit the key.
Figure~\ref{fig:key_device_transmission}(a) shows the structure of Register FIS.
It consists of five words, but the last word (32 bit) is not used.
Since the Register FIS is transmitted to disk drive firstly including LBA before data transfer,
Register FIS can transmit the key synchronously with LBA and can make the key mapping table on the {\textsc{Key-SSD}} before data transfer.
In addition, since the key is allocated to the reserved space of the Register FIS, there is no need to modify the SATA protocol itself for key transmission.
32-bit access can be vulnerable to brute force attacks (exhaustive search). However, {\textsc{Key-SSD}} can implement a mechanism that counts the number of invalid attempts and rejects these brute force attacks by blocking requests if the request exceeds a predefined threshold.
And if the SATA protocol is extended, a longer key can be used.
As illustrated in Figure~\ref{fig:key_device_transmission}(b), the Register FIS is involved at the beginning and end of the protocol in both write and read operations and a key is piggybacked in the Register FIS from the host to \textsc{Key-SSD}.
\begin{comment}
\textcolor{blue}{
{\em \bf Software Event Queue in Device:}
In the SATA device, the command in the registerFIS sent through the SATA protocol is first assigned to the NCQ. NCQ is a command queue for receiving multiple commands in a single drive at the same time. Currently, the Jasmine OpenSSD platform can accommodate a total of 32 commands and operates on the FIFO (First Input First Output) method. From the viewpoint of HOST, commands that have not yet been transmitted are stored in the NCQ. When the data transmission is completed, the command is released from the NCQ and stored in the event queue. The event queue is responsible for queuing commands that have completed data transmission from the HOST point of view. SATA event queues can accommodate up to 128 instructions simultaneously, and operate in FIFO mode. These commands are not yet I / O to the actual flash, and when FTL takes a command to perform I / O to flash, the command is removed from the event queue. However, the event queue implemented in the current Jasmine OpenSSD platform is implemented as a hardware queue. The problem is the event queue. In the event queue, only the command, lba, and size are stored automatically in hardware, so there is no way to store the key here. Therefore, we created a new key event queue that operates with software and stored command, LBA, size, and key in the corresponding queue. In NCQ, when the HOST command is transferred in the form of registerFIS, the contents of registerFIS are stored in the NCQ as a whole, so the key is also automatically stored. When saving to the event queue in NCQ, we store the command, LBA, and size together with the key in the newly created key event software queue.
Figure~\ref{fig:software_queue} illustrates the implementation of software event queue in {\textsc{Key-SSD}}.
}
\begin{figure}[!t]
\centering
\begin{tabular}{@{}c@{}}
\includegraphics[width=0.46\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./omni/software_queue}
\end{tabular}
\vspace{-0.1in}
\caption{Description of software queue implementation for {\textsc{Key-SSD}}.}
\vspace{-0.2in}
\label{fig:software_queue}
\end{figure}
\end{comment}
\begin{comment}
\clearpage
In order to take full advantage of \textsc{Key-SSD}, a new file system should be designed, which takes significant time and efforts.
Since the focus of this paper is to demonstrate the effectiveness of disk-level authentication against ransomware attacks, the file system is minimally modified to support \textsc{Key-SSD}.
Design of a new file system for \textsc{Key-SSD} will be studied in our future work.
To support \textsc{Key-SSD}, the file system needs to take a key from an application, and transfer it to the disk drive along with a request.
If a key is sent separately from a request, it may introduce a vulnerability allowing an adversary to access data immediately after the key is sent but before the legitimate request is placed.
Otherwise, the atomic operation of sending a key and placing a request should be guaranteed by spin-locks.
However, spin-locks have an adverse impact on performance because multiple accesses to a drive are not allowed, which will drastically limits the throughput of disk access.
In our implementation, four system calls are added to handle a key of an application, and two kernel-level hash tables are added to transfer the key.
The added system calls are \texttt{sys\_open\_key, sys\_write\_key, sys\_read\_key}, and \texttt{sys\_close\_key}.
By calling \texttt{sys\_open\_key}, the application opens a file, and pass the key to the file system.
The file system registers the key to one of the hash table (details of the hash tables are presented below).
The application accesses the opened file by calling \texttt{sys\_write\_key} and \texttt{sys\_read\_key}.
When the file is closed by \texttt{sys\_close\_key}, all corresponding entries are removed from the hash tables.
The two hash tables use the same data structure which is depicted by Figure~\ref{fig:hash_table}.
The purpose of this data structure is to maintain mapping of an input to an output.
When an input is given, its corresponding output is searched for by looking up this table.
The table is implemented with linked list structures (e.g. \texttt{hlist\_head} and \texttt{hlist\_node}) provided by the Linux kernel.
An array of \texttt{hlist\_head} is created, each of which corresponds to a bucket.
A bucket index is calculated through a hash function.
If a collision occurs in a bucket, multiple nodes can be added as a linked list.
For data consistency by concurrent accesses, \texttt{rcu\_lock} is used for each bucket.
Thus, concurrent accesses are supported as long as they are not on the same bucket.
\begin{figure}
\centering
\includegraphics[width=0.49\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./fig/hash_table.pdf}
\vspace{-0.1in}
\caption{Data structure of the hash tables.}
\label{fig:hash_table}
\vspace{-0.1in}
\end{figure}
We use two hash tables to transfer a key with a request: \texttt{KeyInode} and \texttt{KeyLBA} tables.
\texttt{KeyInode} table maintains mapping from an \texttt{inode} to a key, and \texttt{KeyLBA} table does from LBA to a key.
Their workflow is illustrated in Figure~\ref{fig:tables}.
When a file is open by calling \texttt{sys\_open\_key}, a file descriptor and a key are given from an application.
In the virtual file system (VFS) layer, the corresponding \texttt{inode} is retrieved by the file descriptor.
The \texttt{inode} and the input key (\texttt{KeyIn}) pair is inserted to the \texttt{KeyInode} table.
When a read or write request is placed by the application, the file descriptor and offset are provided.
The LBA of the request is calculated by the VFS layer.
In the generic block layer, the \texttt{inode} of the request can be found by the \texttt{bio} data structure.
The pair of LBA and \texttt{inode} is passed to the \texttt{KeyLBA} table.
By looking up the \texttt{KeyInode} table, the key of the \texttt{inode} is identified and inserted to the \texttt{KeyLBA} table with the requested LBA.
In the SATA device driver, when the requested LBA is given, its key can be retrieved by looking up the \texttt{KeyLBA} table.
\begin{figure}[!t]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.22\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./fig/sata_write.pdf} &
\includegraphics[width=0.22\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./fig/sata_read.pdf} \\
\small (a) Write &
\small (b) Read \\
\vspace{-0.2in}
\end{tabular}
\caption{Write and read operations in the SATA protocol.}
\vspace{-0.1in}
\label{fig:sata_protocol}
\end{figure}
\begin{figure}[!t]
\centering
\includegraphics[width=0.45\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./fig/tables.pdf}
\vspace{-0.3in}
\caption{Illustration of block key transfer with {\em KeyInode}
and {\em KeyLBA}
in OS kernel.}
\vspace{-0.15in}
\label{fig:key}
\end{figure}
\begin{figure}[!t]
\centering
\begin{tabular}{@{}c@{}}
\includegraphics[width=0.4\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./fig/fis.pdf}
\end{tabular}
\vspace{-0.15in}
\caption{Register FIS -- host to device. Word 4 is used to deliver an access code.}
\vspace{-0.2in}
\label{fig:fis}
\end{figure}
\end{comment}
\subsection{Security Analysis}
\label{sec:sec_anal}
\vspace{-0.05in}
In this section, we discuss the steps that ransomware typically performs and the effectiveness of Key-SSD, explaining how Key-SSD can defend against ransomware attacks at each step. The steps are as follows: Infection $\rightarrow$ Persistence $\rightarrow$ Removing backup copies $\rightarrow$ Encryption $\rightarrow$ Notice, illustrated in Table~\ref{tab:security_anal}.
{\bf Infection:}
Ransomware is a type of malware that exploits vulnerabilities to damage victim's computers.
For example, CryptoLocker exploits vulnerabilities in Internet Explorer and Adobe Flash to control the computer~\cite{7886569}.
The approach proposed in this study is that the application provides the key.
If the application is compromised, the files of the compromised application can be infected by the ransomware.
However, its impact is confined only
to
the compromised application.
If an application is compromised, its key
is very likely to
be revealed.
In this case, files belonging to the compromised application
can be encrypted by the ransomware.
However, since the revealed key is only for the compromised application, ransomware cannot use the key to access other files that belong to other applications.
For example, even if CryptoLocker compromises Internet Explorer, \texttt{docx} files and \texttt{pptx} files that belong to MS Word and PowerPoint cannot be accessed by CryptoLocker.
{\bf Persistence:}
Ransomware can survive after rebooting.
It may also include a self-propagating mechanism that infects other computers in the network.
For example, WannaCry propagates itself by exploiting vulnerability of Window's Server Message Block (SMB) protocol~\cite{chen_robert}.
This persistence mechanism often
modifies system files.
Therefore, if system files
are protected by {\textsc{Key-SSD}},
it will be much harder for ransomware to modify system files.
For example,
{\textsc{Key-SSD}} can enforce system files to be updated only by the legitimate updater.
{\bf Removing backup copies:}
Ransomware destroys backup copies.
This step is unique to ransomware compared to other types of malware.
If backup copies are available, the victim can recover files without paying for ransom.
Thus, ransomware finds and destroys
all backup copies.
Again, if backup copies are under the protection of
{\textsc{Key-SSD}}, ransomware will not be able to delete them.
\begin{table}[!t]
\scriptsize
\begin{tabular}{|M{28mm}@{}||M{11mm}@{}|M{13mm}@{}|M{4mm}|M{7mm}|} \hline
\textbf{Step} & \textbf{Infection} & \textbf{Persistence} & \textbf{RBC} & \textbf{Encrypt} \\ \hline\hline
{Malware Migration~\cite{7778160,6620049}} & X & & & \\ \hline
{Renaming~\cite{7924925}} & & X & & \\ \hline
{App. Behavior Monitor~\cite{7536529,7600214,7336353}} & & & & X \\ \hline
{Net. Behavior Monitor~\cite{7764294,7387902,7981522}} & & & & X \\ \hline
{Crypto. Library~\cite{Kolodenker:2017:PDA:3052973.3053035,Lee2017b,Maiorca:2017:RAP:3019612.3019793}} & & & & X \\ \hline
{File Backup~\cite{cldsafe_an_efficient_file_backup,Continella:2016:SSR:2991079.2991110}} & & & & X \\ \hline
{ \cellcolor{gray!20}{\textsc{Key-SSD}}} & \cellcolor{gray!20}& \cellcolor{gray!20}X & \cellcolor{gray!20}X & \cellcolor{gray!20}X \\ \hline
\end{tabular}
\vspace{-0.1in}
\caption{Step-by-step analysis from Infection to Encryption of ransomware attacks and comparison with existing defense techniques and {\textsc{Key-SSD}}.
RBC denotes removing back copies.
}
\vspace{-0.2in}
\label{tab:security_anal}
\end{table}
{\bf Encryption:}
Ransomware finds the data file and encrypts it
with an encryption key.
Before encrypting data files, ransomware (e.g. CryptoLock)
can
exchange encryption keys with a remote command and control (C\&C) server.
If data files are under the protection of
{\textsc{Key-SSD}},
ransomware cannot read and write (overwrite)
the file
unless
the application of the file is compromised.
Most (if not all) of the existing ransomware mitigation techniques work
at
this step.
They detect ransomware by monitoring specific behaviors pertaining to key exchanges and encryption.
{\textsc{Key-SSD}} prevents ransomware not only at this step, but also previous steps as explained above.
It should be noted that several prior works for
malware mitigation techniques\cite{7778160,6620049}
investigate the problems of detecting and preventing ransomware
at the first two steps because ransomware is also a kind of malware.
{\bf Notice:}
Eventually, ransomware informs the victim of ransom payment. Once it is received, ransomware will restore the hostage file and erase all forensic evidence.
\begin{comment}
\begin{figure}
\centering
\includegraphics[width=0.48\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./fig/steps.png}
\caption{Primary targets of techniques to prevent common steps of ransomware.}
\label{fig:steps}
\end{figure}
\begin{table*}
\caption{Primary targets of techniques to prevent common steps of ransomware.}
\label{tab:steps}
\centering
\begin{tabular}{|l|p{0.6\textwidth}|}
\hline
Step & Existing mitigation techniques \\
\hline
\hline
(Step 1) Infection & Malware mitigation techniques~\cite{7778160} \\
\hline
\multirow{2}{*}{(Step 2) Persistence} & Malware mitigation techniques~\cite{7778160} \\
& \textbf{\textsc{Key-SSD}} \\
\hline
\multirow{2}{*}{(Step 3) Removing backup copies} & Renaming backup service~\cite{7924925} \\
& \textbf{\textsc{Key-SSD}} \\
\hline
\multirow{5}{*}{(Step 4) Encryption} & Application behavior monitoring~\cite{7536529,7600214,7336353} \\
& Network behavior monitoring~\cite{7764294,7387902,7981522} \\
& Identifying or hooking cryptographic library~\cite{Kolodenker:2017:PDA:3052973.3053035,Lee2017b,Maiorca:2017:RAP:3019612.3019793} \\
& File backup~\cite{cldsafe_an_efficient_file_backup,Continella:2016:SSR:2991079.2991110} \\
& \textbf{\textsc{Key-SSD}} \\
\hline
(Step 5) Notice & N/A \\
\hline
\end{tabular}
\end{table*}
\end{comment}
\section{Evaluation}
\label{sec:eval}
\vspace{-0.1in}
In this section, first we show the overhead analysis of {\textsc{Key-FTL}} and the effectiveness of selective flushing of the {\textsc{Key-FTL}}.
Second, we analyze the OS kernel overhead for key transmission to {\textsc{Key-SSD}} and the end-to-end overhead including the OS kernel and {\textsc{Key-FTL}} implementation overhead.
Third, we perform a step-by-step analysis of the usefulness of the {\textsc{Key-SSD}} for a typical ransomware attack.
\vspace{-0.15in}
\subsection{Experimental Setup}
\label{sec:expr_setup}
\vspace{-0.05in}
{\bf Implementation:}
In order to prototype the {\textsc{Key-SSD}}, we modified 360 lines of C code in the Linux VFS and the generic block I/O layers, 180 lines of C code in the firmware of the Jasmine OpenSSD platform~\cite{jasmine}.
We also modified 70 lines of C code in the SATA device driver.
Specifically, {\textsc{Key-FTL}} has extended a page-based FTL with greedy GC on
the Jasmine OpenSSD platform.
{\bf Software I/O Event Queue:}
In the Jasmine OpenSSD platform,
the command in the Register FIS sent through the SATA protocol is first assigned
to the I/O event queue,
which is responsible for queuing commands.
This event queue can accommodate up to 128
I/O commands simultaneously,
operating in FIFO mode.
Jasmine OpenSSD platform uses a hardware event queue.
In the hardware event queue, only the command, LBA, and size are stored automatically in hardware, so there is no way to store the key here.
Therefore, we created a new key event queue that operates with software and stored command, LBA, size and key in the queue.
Figure~\ref{fig:software_queue} illustrates the implementation of software event queue for {\textsc{Key-SSD}}.
\begin{figure}[!t]
\centering
\begin{tabular}{@{}c@{}}
\includegraphics[width=0.38\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./software_queue}
\end{tabular}
\vspace{-0.05in}
\caption{
Software queue implementation for {\textsc{Key-SSD}}.}
\vspace{-0.2in}
\label{fig:software_queue}
\end{figure}
\begin{figure*}[!t]
\begin{center}
\begin{tabular}{@{}c@{}c@{}c@{}c@{}}
\includegraphics[width=0.25\textwidth]{./expr2-seqwrite} &
\includegraphics[width=0.25\textwidth]{./expr2-randwrite} &
\includegraphics[width=0.25\textwidth]{./expr1-seqread} &
\includegraphics[width=0.25\textwidth]{./expr1-randread} \\
(a) Sequential Write & (b) Random Write &
(c) Sequential Read & (d) Random Read\\
\end{tabular}
\vspace{-0.15in}
\caption{Performance comparisons of {\textsc{Key-FTL(S)}} in Key OpenSSD and normal FTL in OpenSSD for
write and read workloads.
SW, RW, SR, and RR denote Sequential Write, Random Write, Sequential Read and Random Read respectively. Queue depth is denoted Q$i$ where $i$ is the number of outstanding I/O requests.
AF and SF denote All Flush and Selective Flush respectively.}
\label{fig:key-FTL-eval}
\vspace{-0.35in}
\end{center}
\end{figure*}
{\bf Test-bed:}
All experiments were performed on a single server with 16 GB of RAM and an Intel (R) Core (TM) i5-7500 CPU @ 3.40GHz. The operating system is Linux with kernel 4.10.16.
We examined two storage devices that are detailed in Table~\ref{tab:ssd_spec}.
We selected the Micron 250GB MLC SSD as a baseline to measure Linux kernel implementation overhead for {\textsc{Key-SSD}}. In order to include the implementation overhead of internal FTL and the SATA target driver on the SSD,
we modified the firmware of the Jasmine OpenSSD development platform~\cite{jasmine}.
{\bf Workloads:}
We examined the key transmission and processing overhead on the SSD in terms of I/O bandwidth. To measure the overhead of disk-side implementation, we used an I/O Benchmark Suite, fair-lio~\cite{olcf_benchmark_suit} that uses the \texttt{libaio} asynchronous I/O library on Linux~\cite{libaio},
performing reads and writes on raw block devices.
We have also made an in-house multi-process based I/O benchmark program to evaluate the overhead of Linux kernel implementations for key transfer and management
in the kernel.
To evaluate {\textsc{Key-SSD}}, this benchmark program use \texttt{sys\_open\_key()} and \texttt{sys\_close\_key()} to pass keys to the kernel during I/Os.
We used two representative file sizes to have different file groups
because the data center workload consists of many small files and a few large files~\cite{DBLP:conf/fast/KimAVS15}.
We used 4~KB {\em small files}
and
512~MB {\em big files}.
We also used SQlite~\cite{sqlite} and DBbench~\cite{dbbench} for the end-to-end performance analysis of {\textsc{Key-SSD}} for realistic experiments.
An actual ransomware sample is used to show that
{\textsc{Key-SSD}} can prevent from both normal file I/O path attack and Direct I/O attack bypassing file system layers in the OS.
In particular, for every experiment to measure the performance overhead of kernel and FTL implementations,
we did a page cache flush to rule out the OS page cache effect .
\begin{table}[!t]
\centering
\scriptsize
\begin{tabular}{|M{30mm}||M{12mm}|M{20mm}|M{4mm}|M{4mm}|M{6mm}|} \hline
\textbf{SSD Spec} & \textbf{Jasmine} & \textbf{MX200 SSD} \\ \hline\hline
{Company} & Indilinx & Crucial Micron \\ \hline
{Type} & MLC & MLC \\ \hline
{Interface} & SATA & SATA \\ \hline
{Capacity} & 64~GB & 250~GB \\ \hline
{Read (MB/s)} & 270 & 555 \\ \hline
{Write (MB/s)} & 90 & 556 \\ \hline
\end{tabular}
\vspace{-0.1in}
\caption{SSD Specification.
Read and write bandwidth was measured using I/O Benchmark Suite~\cite{olcf_benchmark_suit}.
}
\label{tab:ssd_spec}
\vspace{-0.2in}
\end{table}
{\bf Threat Model:}
According to our threat model in Section~\ref{sec:back}, a system call may be hijacked. If an application sends a plain key to OS, it could be revealed by a hijacked system call. Since the focus of this paper is on demonstrating the effectiveness of integration of an access-control mechanism with a disk drive, we used a plain key in our current implementation.
But, a signature that is encryption of a hash value of the address and data of each request can be used instead of the plain
key.
\vspace{-0.15in}
\subsection{Overhead Analysis of {\textsc{Key-FTL}}}
\vspace{-0.05in}
In this section, we show the overhead analysis of {\textsc{Key-FTL(S)}} in {\textsc{Key-SSD}} versus normal page-based FTL in SSD.
Firstly,
we compared the performance of I/O hardware and software queues using the fair-lio I/O benchmark suite.
We observed that there is negligible performance difference between hardware and our software queue implementations, but owing to space constraints here, we do not show results.
Secondly, we evaluated the disk-level performance overhead of {\textsc{Key-FTL(S)}} for {\textsc{Key-SSD}}. We compared the write and read I/O performance of {\textsc{Key-SSD}} with {\textsc{Key-FTL}}, called {\em Key OpenSSD} and normal SSD with normal FTL (with no key), called {\em OpenSSD} for a variety of I/O workload patterns (sequential and random) by varying I/O queue depth.
Figure~\ref{fig:key-FTL-eval}(a)(b) show the results to compare {\textsc{Key-SSD}} with {\textsc{Key-FTL(S)}} and normal SSD for write only workloads.
We evaluated the performance comparison by changing the FTL flushing methods ({\em all flush (AF)} or {\em selective flush (SF)}).
Note that all flush means all entries of the mapping table are all synchronized no matter what entries are changed while selective flush means only changed entries from the table are synchronized.
In case of all flush, it is observed that the throughput of {Key OpenSSD}
is less than 20-30\% of that of {OpenSSD}.
It is because the size of the {\textsc{Key-FTL(S)}} in Key OpenSSD is
nine times larger than that of the normal FTL. A page size in OS is 4~KB whereas a page size in the Jasmine OpenSSD is 32~KB. Thus, an entry of FTL has eight key values for each 4~KB page.
However, in case of selective flush, we observe that there is little difference between Key OpenSSD and OpenSSD due to small flush overhead.
\begin{comment}
\begin{figure}[!t]
\begin{center}
\begin{tabular}{@{}c@{}c@{}c@{}c@{}}
\includegraphics[width=0.25\textwidth]{./gnuplot/expr3-lppf} &
\includegraphics[width=0.25\textwidth]{./gnuplot/expr3-lppf-rand}\\ (a) Sequential Write & (b) Random Write\\
\vspace{-0.2in}
\end{tabular}
\caption{Results for average page writes (\#) per flush.}
\label{fig:page_write_per_flush}
\vspace{-0.3in}
\end{center}
\end{figure}
To investigate the reason in more detail,
we compared the average number of page write operations at each flush operation.
Figure~\ref{fig:page_write_per_flush} shows the results
with respect to the increased number of queue depth in the I/O benchmark program.
For sequential writes in Figure~\ref{fig:page_write_per_flush}(a), we see the average number of write operations at each flush is slightly less than 1.
Consider there are 4096 key mapping entries in one page (32 KB) for {\textsc{Key-FTL}} where as there are 8192 key mapping entries in one page for OpenSSD. We found that the workload is sequential writes and more than 4096 pages were not modified every time flush occurred.
On the other hand, random write result is shown in Figure~\ref{fig:page_write_per_flush}(b), overall we observe that the number of write operations at every flush increases as the queue depth increases. It is because random writes modifies multiple pages and increased queue depth increases the number of modified pages to be synchronized at flush. Interestingly, we observe there is little difference in the number of write operations till queue depth is 16, however, Key OpenSSD starts to write more after the queue depth is 16. It is because before the queue depth is 16, Key OpenSSD could exploit the locality characteristic of key and physical page number (PPN) adjacent to each other and the number of modified page entries is less than half of total number of page entries in OpenSSD, however, modified pages increased after queue depth is 16.
\end{comment}
Figure~\ref{fig:key-FTL-eval}(c)(d) show the results for read only workloads.
Read workloads rarely flush, so there is little difference in performance between {\textsc{Key-SSD}} and normal SSDs.
\begin{figure}[!t]
\begin{center}
\begin{tabular}{@{}c@{}c@{}c@{}c@{}}
\includegraphics[width=0.25\textwidth]{./flush_read} &
\includegraphics[width=0.25\textwidth]{./flush_write}\\ (a) Read & (b) Write\\
\vspace{-0.3in}
\end{tabular}
\caption{Performance analysis of {\textsc{Key-FTL(D)}} by varying the percentage of blocks locked by keys. }
\label{fig:static_and_dynamic}
\vspace{-0.3in}
\end{center}
\end{figure}
{\bf Performance analysis of {\textsc{Key-FTL(D)}} and {\textsc{Key-FTL(S)}}:}
In {\textsc{Key-FTL(D)}}, a key is dynamically allocated. It is advantageous in terms of space compared to {\textsc{Key-FTL(S)}}, but the performance of {\textsc{Key-FTL(D)}} will depend on the portion of blocks locked by keys.
For performance evaluation, we partitioned 4GB and measured direct I/O performance by varying the percentage of blocks locked among all blocks to
0\% (0GB), 25\% (1GB), 50\% (2GB), 75\% (3GB), and 100\% (4GB).
We ran the fair-lio I/O benchmark suite for a variety of I/O patterns such as sequential read and write and random read and write by increasing queue depth.
0\% means there is no protected blocks by keys, which is a similar case to a baseline FTL without keys. On the other hand, 100\% means all data blocks are protected by keys.
Figure~\ref{fig:static_and_dynamic}(a) shows the results for reads. We observe read bandwidth is around 250 MB/s regardless of the percentage of locked blocks. It means performance is almost equivalent to the baseline without keys and {\textsc{Key-FTL(S)}}.
{\textsc{Key-FTL(D)}} has a search overhead for granting access to the LPNs corresponding to the key, however, with our results, the search overhead can be said almost negligible.
Figure~\ref{fig:static_and_dynamic}(b) shows the results for writes.
The baseline without keys and {\textsc{Key-FTL(S)}} show very similar performance while
{\textsc{Key-FTL(D)}} shows decreased throughputs.
This is because the overhead of inserting key nodes is larger than the search overhead. In our implementation for {\textsc{Key-FTL(D)}},
it is necessary to register the key in {\textsc{Key-FTL}} and lock it for all LPNs of the request sent from the host.
However, these results are for worse case when all LPNs are for first writes.
If they are update-writes on blocks already locked, it only involves the search overhead of LPNs for granting access with the key, so performance will get better.
\begin{figure*}[!t]
\begin{center}
\begin{tabular}{@{}c@{}c@{}c@{}c@{}}
\includegraphics[width=0.25\textwidth]{./expr4-kernel-seqwrite} &
\includegraphics[width=0.25\textwidth]{./expr4-kernel-seqread} &
\includegraphics[width=0.25\textwidth]{./expr4-kernel-seqwrite-big} &
\includegraphics[width=0.25\textwidth]{./expr4-kernel-seqread-Big} \\
(a) Write (Small Files) & (b) Read (Small Files) & (c) Write (Big Files) & (d) Read (Big Files) \\
\vspace{-0.25in}
\end{tabular}
\caption{Analysis of the kernel implementation overhead for {\textsc{Key-SSD}} with Micron MLC SSD.
Small Files and Big Files in the parenthesis denote workload type according to file size.
The error bar depicts min. and max. deviation from the average of 3 iterations.
}
\label{fig:kernel_overhead_ssd}
\vspace{-0.25in}
\end{center}
\end{figure*}
\begin{figure*}[!t]
\begin{center}
\begin{tabular}{@{}c@{}c@{}c@{}c@{}}
\includegraphics[width=0.25\textwidth]{./expr5-fullpath-seqwrite} &
\includegraphics[width=0.25\textwidth]{./expr5-fullpath-seqread} &
\includegraphics[width=0.25\textwidth]{./expr5-fullpath-seqwrite-Big} &
\includegraphics[width=0.25\textwidth]{./expr5-fullpath-seqread-Big} \\
(a) Write (Small Files) & (b) Read (Small Files) &
(c) Write (Big Files) & (d) Read (Big Files)\\
\vspace{-0.25in}
\end{tabular}
\caption{
Analysis of the end-to-end performance experiment including the kernel and firmware implementation overhead using the Jasmine OpenSSD.
The experimental environment is the same as that for Figure~\ref{fig:kernel_overhead_ssd}.
}
\label{fig:kernel_overhead_jasmine}
\vspace{-0.3in}
\end{center}
\end{figure*}
\vspace{-0.15in}
\subsection{Kernel Implementation Overhead}
\vspace{-0.05in}
In this experiment, we analyze the kernel implementation overhead for key transmission and management for {\textsc{Key-SSD}}.
The experiments were performed by increasing the number of processes, each generating the same workload.
Figure~\ref{fig:kernel_overhead_ssd}
compares the performance results of {\textsc{Key-SSD}} (Key MicronSSD) with normal SSD (Micron SSD).
This experiment was performed using a Micron MLC SSD to analyze only the kernel implementation overhead, excluding the {\textsc{Key-FTL}} implementation overhead.
The SATA driver sends the key in the Register FIS, but the SSD ignores the key.
In particular, in order to analyze the performance cost of key search in {\textsc{KeyInode}}, we performed two experiments -- (i) to delete key from {\textsc{KeyInode}} when closing (Close) the file and (ii) that does not delete key when closing the files (NoClose).
From Figure~\ref{fig:kernel_overhead_ssd},
we observe the results for small files show lower throughputs than big files workload.
Small workload performs I/O operations on a large number of small files and involves large delays caused by opening and closing files each time for many files, as opposed to doing continuous I/Os on large files with large workloads.
From Figure~\ref{fig:kernel_overhead_ssd}(a)(b), we observe increased throughput as the number of processes increases in all three cases (MicronSSD, Key MicronSSD (Close), Key MicronSSD (NoClose)).
Comparing the results for MicronSSD and Key MicronSSD (Close),
{\textsc{Key-SSD}} (Key MicronSSD) seems to have slightly lower throughput than MicronSSD when I/O loads are high (referring to 32P or 64P).
Since the OS kernel runs the read-verify method on the {\textsc{Key-SSD}}, it requires additional disk block access for key authentication on the first page when opening the file.
When we compare the results for Key MicronSSD (Close) and Key MicronSSD (NoClose),
we again see the performance difference between the two is almost negligible, unlike the expectation that the key retrieval time will take longer for all key searches because the keys are kept in the {\textsc{KeyInode}} table without being deleted.
For example, in Key MicronSSD (NoClose), I/O is performed for 64,000 files in 64 processes. If the key is not removed from the {\textsc{KeyInode}} at the time of file close, up to 64,000 entries may accumulate in the {\textsc{KeyInode}}, which may degrade table search performance. However, in our experiment, the search time overhead is too small to degrade the overall I/O performance.
Figure~\ref{fig:kernel_overhead_ssd}(c)(d) show the results for big files workloads.
Unlike the results for small workloads, we see there is very little performance difference between three. This is because the additional disk access overhead of the read-verify method is not noticeably large.
\vspace{-0.15in}
\subsection{End-to-End Performance Analysis}
\vspace{-0.05in}
In this experiment, we perform an end-to-end experiment to analyze the overhead including the kernel and
{\textsc{Key-FTL(S)}} implementation overhead.
For the Key OpenSSD, we deleted keys from
{\textsc{KeyInode}} when closing.
Figure~\ref{fig:kernel_overhead_jasmine} shows the performance comparison between OpenSSD and Key OpenSSD.
Overall bandwidths were observed lower than those from Micron experiments because overall read/write bandwidths of OpenSSD are lower than MicronSSD as in Table~\ref{tab:ssd_spec}.
Comparing the performance of OpenSSD and Key OpenSSD, we see little difference in performance between them, which is slightly different from
Figure~\ref{fig:kernel_overhead_ssd}.
In particular, in Figure~\ref{fig:kernel_overhead_ssd},
the kernel overhead caused by the read-verify method is noticeable when I/O loads are very high, but it is not shown here. This is because the performance of the OpenSSD is so low that the overhead of the read-verify method is hidden.
For another realistic end-to-end experiment, we have modified the SQLite~\cite{sqlite} source code and protected DB files with a key.
SQLite could pass a user-defined key when accessing DB files using \texttt{sys\_open\_key()} and \texttt{sys\_close\_key()} system call.
Table~\ref{tab:sqlite_test} presents the results of comparing the performance of OpenSSD and Key OpenSSD.
Performance was measured using the Database Benchmark Tool~\cite{dbbench} in SQLite~\cite{sqlite} with two representative DB workloads.
Insert workload is write heavy and intersection workload is read heavy.
For the insert workload, we observe OpenSSD shows 16.48 TPS (Transactions Per Second) while Key OpenSSD shows 16.42 TPS, which is little difference between them. For the intersection workloads, we have similar observation.
\begin{table}[!t]
\centering
\footnotesize
\begin{tabular}{|M{20mm}||M{14mm}|M{20mm}|M{4mm}|M{4mm}|M{6mm}|} \hline
\textbf{Workload} & {\bf Key OpenSSD} & {\bf OpenSSD}\\ \hline\hline
{Insert} & 16.42 & 16.48 \\ \hline
{Intersection} & 880.80 & 892.23 \\ \hline
\end{tabular}
\vspace{-0.05in}
\caption{Performance comparison of OpenSSD and Key OpenSSD for database application.
}
\label{tab:sqlite_test}
\vspace{-0.2in}
\end{table}
\begin{figure*}[!t]
\begin{center}
\begin{tabular}{@{}c@{}c@{}c@{}c@{}}
\includegraphics[width=0.96\textwidth]{./direct_io} \\
\vspace{-0.35in}
\end{tabular}
\caption{
Verification of protection by {\textsc{Key-SSD}} for direct I/O ransomware attacks.
}
\label{fig:direct_io}
\vspace{-0.35in}
\end{center}
\end{figure*}
\vspace{-0.15in}
\subsection{Protection against Real Ransomware}
\vspace{-0.05in}
In this experiment, we show {\textsc{Key-SSD}} can protect against attacks performed through the normal file I/O path of the OS and by bypassing the OS directly from the user application.
{\bf Normal file I/O path attack:}
To evaluate whether {\textsc{Key-SSD}} can prevent actual ransomware attacks, we run an ransomware sample~\cite{ransomware} on the SQLite DB file (test.db) generated in the previous experiment.
The ransomware sample works as follows.
The ransomware first reads test.db, the target file, and encrypts it with AES 256 encryption. Then, it creates a new file, infected.db and write the encrypted data on it, and then deletes the original db file, test.db.
Ransomware opens test.db without a key.
When attempting to read data (test.db), ransomware does not have a key, so the block request can not find the correct key when referring to the {\textsc{KeyInode}} in the kernel block layer, and eventually the access is blocked at the disk end.
{\bf Direct I/O attack:}
We formatted OpenSSD as EXT4 and created a 1M file with a key. We also used a fair-lio benchmark to directly access the device and perform I/O on the data blocks on the disk. Then, we write to the 1MB file data locked by the key on the disk in 4 KB units without the key, and write 24 KB sequentially.
Figure~\ref{fig:direct_io} shows the host's kernel log and OpenSSD's device log.
Currently a 1M file is written beginning with LBA: 0x43000.
Figure~\ref{fig:direct_io}(a) shows the kernel and OpenSSD device logs on the host for a 24 KB write request without a key.
Figure~\ref{fig:direct_io}(b)(c) show the kernel and device's logs when attempting to write on the locked blocks with the proper key.
Figure~\ref{fig:direct_io}(b) shows kernel logs,
kernel block layer and device driver send the 24 KB write requests without keys.
And Figure~\ref{fig:direct_io}(b), Figure~\ref{fig:direct_io}(c) shows {\textsc{Key-SSD}}
blocks LBA's requests on the blocks from 43000 to 43028.
Therefore, this experiment confirmed that {\textsc{Key-SSD}} protects direct I/O access of unauthorized applications.
{\bf Attacks using page cache:}
There could be a ransomware attack that reads the file data in the page cache and deletes the original file after creating a new file.
As mentioned in Section~\ref{sec:issue}, {\textsc{Key-SSD}} can prevent these attacks through the read-verity method. We have experimentally proved that this attack can be successfully defended. However, owing to space constraints here, we do not show results.
\input{eval_security_anal}
\section{Introduction}
\label{sec:intro}
\vspace{-0.05in}
Ransomware is a type of malware that encrypts data files of a victim computer and
requires the victim to pay a ransom to regain file access.
Very recently, in June 2017, more than 12,000 computers were attacked worldwide by ransomware including those of at least 80 large companies~\cite{a_new_ransomware_outbreak}.
Only a month before, a similar massive ransomware attack happened~\cite{a_new_ransomware_outbreak}.
According to a recent report~\cite{meeting_the_threat_of,roundtable_ransomware}, it is estimated that \$200 million were paid for ransomware only in the first quarter of 2016.
Since ransomware incurs immediate financial damages, it is one of {the} growing concerns in information security.
A typical ransomware attack reads and encrypts files and takes encrypted files as hostages.
These attacks can be performed through the normal file I/O path of the OS or by bypassing the OS directly in the user application.
The techniques to detect and prevent ransomware has been researched and developed, but it can not be perfect as ransomware evolves.
Ransomware can be detected by monitoring the behavior of applications (potential ransomware) in the operating system, network, or file system~\cite{7536529,7600214,7336353,7784627,7736455,7764294,7387902,Continella:2016:SSR:2991079.2991110}.
For example, if an application exhibits frequent renaming, frequent access to cryptographic library, and communicating with known malicious servers, it is considered as ransomware.
However, if adversaries are aware of these techniques, they may manage to develop a new type of ransomware that does not exhibit the behaviors recognized by these techniques.
Another way to mitigate ransomware is to back up data.
If a file is infected by ransomware,
version control systems
can be
used to track the history of a file and recover it.
However, maintaining backup copies of files requires additional storage and may incur the performance overhead on host computers and network traffic.
Furthermore, there is a risk of intelligent ransomware destroying backup files~\cite{7924925,7886569}.
To address the problems of these existing techniques, we present a fundamental solution of ransomware that is not based on signature-based behavioral monitoring (which can be circumvented by intelligent ransomware) and does not cause excessive storage or performance overhead.
In this paper, we present {\textsc{Key-SSD}}, an {\em access-control drive} to protect the files from ransomware attacks.
{\textsc{Key-SSD}} can shield
aforementioned attacks through I/O paths to the disk drive.
Our primary contribution is that {\textsc{Key-SSD}} implements a {\em disk-level access-control mechanism} where unauthorized requests are denied by a \textit{disk drive}.
Even if ransomware bypasses the traditional access-control mechanism in the file system by exploiting the vulnerabilities of applications, it cannot bypass the disk-level access-control mechanism.
Unauthorized ransomware cannot even read
a file data on the disk drive, and consequently, cannot take files hostage.
Traditional SSDs are block-devices that do not have capability of granting or denying read/write requests.
On the other hand, {\textsc{Key-SSD}}
controls block-level access to block-level read/write requests, thereby blocking unauthorized data read/write access.
Object storage~\cite{1612479} includes an \textit{object-level} access-control mechanism.
Since it adopts a higher level of abstraction,
it requires a significant modification to the system software to take advantage of it.
But, {\textsc{Key-SSD}}
requires only minimal modification of existing system software because it maintains the traditional abstraction level.
While traditional access-control mechanisms are implemented in a \textit{file system} and control access to \textit{files}, our focus is on controlling access to \textit{blocks} by a \textit{disk drive}.
Modern SSDs use low-power and
multi-core
controllers to provide significant computing performance~\cite{tiwari_active_2013, cho2013,
Gu:2016:BFN:3007787.3001154, Qin:2006, boboila2012active, sim:2015:sc}.
For {\textsc{Key-SSD}}, we implemented the access control mechanism using
these powerful computation resources
on the disk drive.
Specifically,
the disk-level access control mechanism is implemented in the flash translation layer (FTL) on the SSD.
Blocks in any file that need protection from Ransomware attacks can be assigned an access code (key) from the application. The FTL of an SSD determines whether to grant access to each block while retaining key information per block.
Since the key must be delivered to disk through the OS kernel, it is necessary to modify the OS, but it is crucial to implement it with little performance degradation.
{We conducted a comprehensive evaluation
for {\textsc{Key-SSD}}
by implementing the LBA-key map table in Linux kernel while keeping the existing SATA protocols with real commodity SSD and the Jasmine OpenSSD platform~\cite{jasmine}.
We compared the performance of
{\textsc{Key-SSD}}
with a variety of file I/O patterns using a mix of synthetic and realistic workloads.
Specifically, in our evaluation with both real SSD and Jasmine OpenSSD platform, we observed that {\textsc{Key-SSD}} yields negligible overhead compared to a baseline with normal SSDs without the access-control mechanism.
We have also verified that {\textsc{Key-SSD}} can block unauthorized ransomware I/O access by running actual ransomware code.}
\subsection{Access Code Transfer to Device}
\vspace{-0.15in}
\subsection{Block Key Transfer and OS Support}
To perform key authentication per block request on a disk drive, it is essential to transfer the key to the disk drive.
The key corresponding to the LBA requested by the host must be correctly registered in the {\textsc{Key-FTL}} of the \textsc{Key-SSD}, and the block access authentication control must be properly performed.
Importantly, the key must be sent synchronously with the LBA block request.
After the block request is completed, the key must be deleted from the OS kernel to minimize space overhead.
{\bf Key Management in OS:}
Specifically, the LBA key must pass through OS kernel including the file system, generic block layer, and block device driver.
LBA keys must be managed by the OS kernel.
We create and manage two hash tables -- KeyInode table ({\textsc{KeyInode}}) and KeyLBA table ({\textsc{KeyLBA}}) in the kernel for this purpose.
In the Linux kernel, the information accessed at each layer is limited. That is, the LBA information of the file is not known at the VFS
layer, but the file \texttt{inode} information of the VFS layer can be accessed at the generic block layer.
Therefore, in this study, we implement the two tables in the kernel.
In Linux, each file is represented by an \texttt{inode}.
The {\textsc{KeyInode}} table is implemented in the VFS layer of the kernel and manages the key value assigned to each file.
The {\textsc{KeyLBA}} manages the key value assigned to each LBA and is implemented at the generic block layer in the kernel.
The {\textsc{KeyInode}} is referenced by the LBA's \texttt{inode} at the generic block layer, and it builds the {\textsc{KeyLBA}}.
When making a request to the device from the driver, it consults with the {\textsc{KeyLBA}} to find the appropriate key to the LBA and send the LBA and key together at the same time in a request to the device.
\begin{figure}
\centering
\includegraphics[width=0.4\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./key_transfer_}
\vspace{-0.05in}
\caption{
Illustration of block key transfer with
{\textsc{KeyInode}} and {\textsc{KeyLBA}}
in OS kernel.
\texttt{KeyIn} is an input key of a file specified by users.
}
\vspace{-0.15in}
\label{fig:key_transfer}
\end{figure}
An application has key information for each file.
In order to transfer the key of the file from the application to the disk drive, we define the following system calls:
\texttt{sys\_open\_key()}, and \texttt{sys\_close\_key()}.
Each system call is similar to existing system calls, \texttt{sys\_open()} and \texttt{sys\_close()}.
The \texttt{sys\_open\_key()} system call can insert a new key into the {\textsc{KeyInode}} by sending a key from the application to the kernel when opening the file.
The \texttt{sys\_close\_key()} system call removes dynamically allocated inode-key elements from {\textsc{KeyInode}}
when the file is closed.
This makes the kernel free from the threat of hackers because it does not have the key information of the closed file.
Our specific implementation is as follows:
We extended \texttt{do\_sys\_open} kernel function
by adding a key function parameter and name it \texttt{do\_sys\_open\_key}.
The \texttt{do\_sys\_open\_key} function is the first kernel function to be called when the \texttt{sys\_open\_key} system call is called.
It uses file path to fetch file metadata such as \texttt{inode} from disk, link it to file descriptor, return it, and open file.
In the \texttt{do\_sys\_open\_key} function,
we can create an element with key and \texttt{inode} values for the file and insert it to the
{\textsc{KeyInode}}.
We also implemented \texttt{\_\_close\_fd\_key} to free the element corresponding to the current file from the
{\textsc{KeyInode}}. The \texttt{\_\_close\_fd\_key} function is the first kernel function that is called when the \texttt{sys\_close\_key} system call is called, and performs file close by freeing file metadata mapped to the file descriptor. Note that the file key information is removed from the kernel by deleting the element from the {\textsc{KeyInode}} when the file closes.
The generic block layer creates a
{\textsc{KeyLBA}}
to transfer the keys to the device driver.
Each element in the
{\textsc{KeyLBA}}
is created when reading or writing a block of related files and is deleted after
the key is sent from the device driver (SATA driver) to the disk drive.
Our specific implementation is as follows:
In the generic block layer, the
{\textsc{KeyLBA}}
is constructed using the \texttt{make\_generic\_request} function. The \texttt{make\_generic\_request} function is used to create an I/O request using the \texttt{bio} structure, which is called from the generic block layer after the EXT4 file system.
The \texttt{bio} structure has the inode information of the corresponding block.
Therefore, it is suitable to create
{\textsc{KeyLBA}}
in this layer, and it can refer to the existing
{\textsc{KeyInode}}
in this function, find the LBA key, and insert the element with LBA and key information together into the
{\textsc{KeyLBA}}.
As we will describe in the next section, the elements of the
{\textsc{KeyLBA}}
are deleted from the table after the device driver completes the I/O request to the device.
Figure~\ref{fig:key_transfer} illustrates the implementation of {\textsc{KeyInode}} and {\textsc{KeyLBA}} in the OS kernel stack.
The {\textsc{KeyLBA}} managed by the kernel can protect only the user data of the file. That is,
the key value of {\textsc{KeyLBA}} is to protect only the data block defined by the users.
Therefore, in this study, the block type
(data blocks requested through normal file I/O operations and those requested by directly I/O operations)
are
distinguished in the block layer.
At the block layer in Linux, the block accesses
are managed by \texttt{address\_space} structure of \texttt{inode}.
If the block is mapped to the address space of the \texttt{inode} of its corresponding file, it means block requests from normal file I/O operations.
Keys are assigned accordingly by consulting with {\textsc{KeyLBA}}.
Else if the block is mapped to the address space of the unique \texttt{inode} within the block device structure,
it is block requests by direct I/O operations, thus keys are not assigned to those blocks.
Secure key management is orthogonal to the proposed approach.
In this paper, we assume the key of user data is managed by the application.
To enhance key security, we may consider using a remote authentication server or Trusted Computing Module (TPM) to manage keys for {\textsc{Key-SSD}}.
\section{{{\textsc{Key-SSD}}}: Access Control Drive}
\label{sec:overview}
\vspace{-0.1in}
\begin{comment}
A
typical attack by ransomware is to encrypt the data in the file so that users can not view the data.
{\textsc{Key-SSD}} allows file access only for authorized applications. This access is enforced by the disk drive, not by the application or the OS file system. However, some malicious applications may bypass the OS and attempt to access data blocks directly.
{\textsc{Key-SSD}} can block this attack because it determines the access permissions of the application at the disk-level that can build the last barrier of data access.
\end{comment}
\subsection{Goals}
\vspace{-0.1in}
In this section, we discuss
our key design principles.
{\bf Selective Disk-level Block Access Control:}
Data blocks must be given an access key to grant access to the data block. This access control must be implemented within the disk drive.
Not all data blocks need to be protected with an access key. The user/application should be able to specify data blocks of the files to be protected, and the data blocks must be selectively protected by the access key according to the user's request.
It is also necessary to minimize the performance and space overhead of managing the key per block in the disk drive.
In addition, the cost of keeping this information persistent should be minimized.
{\bf Key Transmission in OS:}
Since the disk drive controls the block-level key access, the OS must be able to transfer the key corresponding to the block to the disk drive.
Also, data blocks of the protected files and their corresponding keys must be transferred to the disk drive at the same time.
There is a possibility for ransomware to access the disk without a key if data block and key transfer occur separately.
{\bf Minimal OS support for {\textsc{Key-SSD}}:}
Ransomware can attack through normal file I/O paths in OS or bypassing the OS file system to access file data directly.
In order to differ normal file I/O requests and direct I/O requests, the OS must be able to assign the keys only to the block requests of normal file I/O operations.
In addition, most OS speed up disk access through the file system page cache. Because the page cache does not require disk access, malicious attacks can not only read data using cached data, but also delete files without accessing disk data.
To protect against attacks using data in the page cache, an OS level implementation is required.
To this end, OS kernel code modifications should be minimized.
\vspace{-0.1in}
\subsection{{{\textsc{Key-SSD}}} Overview}
\vspace{-0.05in}
\begin{figure}[t!]
\centering
\includegraphics[width=0.33\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./overview_.pdf}
\vspace{-0.05in}
\caption{An overview for {\textsc{Key-SSD}} with OS stack.
}
\label{fig:overview}
\vspace{-0.15in}
\end{figure}
We envision {\textsc{Key-SSD}} to be the last firewall to block ransomware attacks.
Figure~\ref{fig:overview} presents a bottom-up description of the system for each component necessarily implemented at every level of OS stack and device drive.
{\bf Access Control Drive:} The lowest level is the access-control drive ({\textsc{Key-SSD}}) that is a solid-state drive, capable of running {disk}-level access-control. Flash translation layer (FTL), which is a firmware in the SSD, manages the key per block.
To authorize a request, an application needs to put a request along with a key.
If a key is sent
separately from the request, the key can be exploited by ransomware by putting a malicious request between the key and the valid request.
Therefore, we guarantee a request arrives at the disk drive at the same time with a key by piggy-backing the key with the request in the SATA protocol.
{\bf OS Support for Access Control by {\textsc{Key-SSD}}:}
OS manages the key and passes the key assigned by the application to the access control drive.
The key stays in the OS temporarily.
In particular, when a block request is transmitted to {\textsc{Key-SSD}} in a block layer, the type of block (normal file I/O operation or direct I/O access)
can be distinguished and a key can be assigned accordingly.
Moreover, it also protects against attacks using the OS's page cache and deleting files without authentication by {\textsc{Key-SSD}}.
{\bf Application Interface Passing Keys to OS:}
The user must be able to assign keys to blocks of files to be protected. Especially, applications need a mechanism to transfer block keys to OS. In other words, it should not go beyond the existing system call interface design principles and it should be able to pass the file block keys to the OS when performing I/Os.
{\em Together, these construct to build a last-level protection against data attacks by malicious applications.}
\begin{comment}
The proposed technique, {{\textsc{Key-SSD}}}, is to grant file access only to an authorized application.
This is enforced by the disk drive.
When a file is protected by the proposed technique, the file can be accessed only through one authorized application.
For example, \texttt{a.docx} file can be accessed only through MS Word.
\textcolor{red}{Kim: is it okay to name a file type of MS software? Because we may need to be careful to mention the specific file type as an example.}
If a user wants to send the file to somewhere via another application (e.g. Explorer),
\textcolor{red}{Kim: again this is another MS example. I am afraid if reviewers are are big fan of Microsoft. Can it be oaky? If we can't make a better example, I don't object to use this example.}
the file should be copied to another file by the application (e.g. using ``save as'') and the new file should be released from the proposed protection.
The proposed approach is to integrate an authentication mechanism in a disk drive as illustrated in Figure~\ref{fig:bypass}.
\textcolor{red}{Kim: can we improve the figure? I think this figure is necessary. But the figure does not look professional...}
In order for an application to access files under the protection of the proposed technique, a unique key should be given.
We assume the key is given by the user like a password, but it can also be automatically given by the operating system, or by the disk drive (KEY-SSD) through calling \texttt{ioctl}.
\textcolor{red}{Kim: Who can give the key? The current writing is that application, OS, and disk can all give the key all. But the way we implement is only to allow users/applications to give the keys. Isn't it? What do you think? We had better to not confuse readers.}
When the application accesses a file, the key is attached to the request and passed to the disk drive.
The key is stored in {{\textsc{Key-SSD}}} when the file is created.
The disk drive grants subsequent accesses only if the key is correct.
Thus, the file can be accessed by its creator.
\textcolor{red}{Kim: If OS or disk drive can give the keys, then, they can be all creators?}
If another application (e.g. Ransomware) tries to access the file without a correct key, it will be denied by the disk drive.
Thus, ransomware cannot even read the file, which eliminates possibility of the file being a hostage of ransomware.
Ransomware may compromise the application.
In this case, ransomware can access the file created by the compromised application.
However, it cannot access files of other applications.
Therefore, even if an application is compromised by ransomware, its impact is confined to within that application.
\textcolor{red}{Kim: Figure~\ref{fig:bypass} describes the followings: how a key is transferred in OS to the disk drive (file system, block layer, and device driver than finally the key SSD), an attack bypassing OS components and directly attempting to access data on the SSD, existing access control happening on the layer of file system, the key-SSD denying data access without a proper key..
I think these are all story components that need to be addressed well for readers to understand well how our proposed ideas are novel, and the reason why we implement this being detailed in the following section. Can you improve this section too? Some contents such as threat model and ransom attack scenarios used in the background can be re-stated, which should not a problem...}
\begin{figure}
\centering
\includegraphics[width=0.40\textwidth, trim={0.5cm, 0.5cm, 0.5cm, 0.5cm}]{./fig/bypass.pdf}
\caption{Integration of an authentication mechanism with a disk drive.}
\label{fig:bypass}
\end{figure}
To realize this technique, the disk drive needs to maintain and compare keys, and the operating system should support key transfer.
{\bf Kim: I moved the following paragraph from next section.}
\textcolor{blue}{
The purpose of existing encryption/decryption schemes is not to prevent ransomware, but to prevent exposure of data.
We propose an authentication mechanism to be implemented by disk drives in a way to
completely block data encryption due to ransomware attacks.\footnote{A drive that implements block-level access control is called an access control drive (ACD) hereafter.}
ACD can perform access control of data by performing authentication before performing I/O to disk drive. Data can be protected by blocking data access from ransomware attacks without block-level data access.
}
\end{comment}
\subsection*{Abstract}
\input{abstract}
\vspace{-0.15in}
\input{intro}
\vspace{-0.15in}
\input{back}
\vspace{-0.3in}
\input{overview}
\vspace{-0.2in}
\input{proposed}
\vspace{-0.2in}
\input{eval}
\vspace{-0.2in}
\input{related}
\vspace{-0.4in}
\input{conc}
\vspace{-0.15in}
\input{ack}
\vspace{-0.15in}
\begin{comment}
\section{Introduction}
A paragraph of text goes here. Lots of text. Plenty of interesting
text. \\
More fascinating text. Features\endnote{Remember to use endnotes, not footnotes!} galore, plethora of promises.\\
\section{This is Another Section}
Some embedded literal typset code might
look like the following :
{\tt \small
\begin{verbatim}
int wrap_fact(ClientData clientData,
Tcl_Interp *interp,
int argc, char *argv[]) {
int result;
int arg0;
if (argc != 2) {
interp->result = "wrong # args";
return TCL_ERROR;
}
arg0 = atoi(argv[1]);
result = fact(arg0);
sprintf(interp->result,
return TCL_OK;
}
\end{verbatim}
}
Now we're going to cite somebody. Watch for the cite tag.
Here it comes~\cite{Chaum1981,Diffie1976}. The tilde character (\~{})
in the source means a non-breaking space. This way, your reference will
always be attached to the word that preceded it, instead of going to the
next line.
\section{This Section has SubSections}
\subsection{First SubSection}
Here's a typical figure reference. The figure is centered at the
top of the column. It's scaled. It's explicitly placed. You'll
have to tweak the numbers to get what you want.\\
\begin{figure}[t]
\begin{center}
\begin{picture}(300,150)(0,200)
\put(-15,-30){\special{psfile = fig1.ps hscale = 50 vscale = 50}}
\end{picture}\\
\end{center}
\caption{Wonderful Flowchart}
\end{figure}
This text came after the figure, so we'll casually refer to Figure 1
as we go on our merry way.
\subsection{New Subsection}
It can get tricky typesetting Tcl and C code in LaTeX because they share
a lot of mystical feelings about certain magic characters. You
will have to do a lot of escaping to typeset curly braces and percent
signs, for example, like this:
``The {\tt \%module} directive
sets the name of the initialization function. This is optional, but is
recommended if building a Tcl 7.5 module.
Everything inside the {\tt \%\{, \%\}}
block is copied directly into the output. allowing the inclusion of
header files and additional C code." \\
Sometimes you want to really call attention to a piece of text. You
can center it in the column like this:
\begin{center}
{\tt \_1008e614\_Vector\_p}
\end{center}
and people will really notice it.\\
\noindent
The noindent at the start of this paragraph makes it clear that it's
a continuation of the preceding text, not a new para in its own right.
Now this is an ingenious way to get a forced space.
{\tt Real~$*$} and {\tt double~$*$} are equivalent.
Now here is another way to call attention to a line of code, but instead
of centering it, we noindent and bold it.\\
\noindent
{\bf \tt size\_t : fread ptr size nobj stream } \\
And here we have made an indented para like a definition tag (dt)
in HTML. You don't need a surrounding list macro pair.
\begin{itemize}
\item[] {\tt fread} reads from {\tt stream} into the array {\tt ptr} at
most {\tt nobj} objects of size {\tt size}. {\tt fread} returns
the number of objects read.
\end{itemize}
This concludes the definitions tag.
\subsection{How to Build Your Paper}
You have to run {\tt latex} once to prepare your references for
munging. Then run {\tt bibtex} to build your bibliography metadata.
Then run {\tt latex} twice to ensure all references have been resolved.
If your source file is called {\tt usenixTemplate.tex} and your {\tt
bibtex} file is called {\tt usenixTemplate.bib}, here's what you do:
{\tt \small
\begin{verbatim}
latex usenixTemplate
bibtex usenixTemplate
latex usenixTemplate
latex usenixTemplate
\end{verbatim}
}
\subsection{Last SubSection}
Well, it's getting boring isn't it. This is the last subsection
before we wrap it up.
\section{Acknowledgments}
A polite author always includes acknowledgments. Thank everyone,
especially those who funded the work.
\section{Availability}
It's great when this section says that MyWonderfulApp is free software,
available via anonymous FTP from
\begin{center}
{\tt ftp.site.dom/pub/myname/Wonderful}\\
\end{center}
Also, it's even greater when you can write that information is also
available on the Wonderful homepage at
\begin{center}
{\tt http://www.site.dom/\~{}myname/SWIG}
\end{center}
Now we get serious and fill in those references. Remember you will
have to run latex twice on the document in order to resolve those
cite tags you met earlier. This is where they get resolved.
We've preserved some real ones in addition to the template-speak.
After the bibliography you are DONE.
\end{comment}
{
\bibliographystyle{acm}
\section{Design and Implementation}
\label{sec:proposed}
\vspace{-0.1in}
In this section, we describe
the implementation of the access control mechanism in the SSD and the implementation in the OS kernel to pass keys from the application to the {\textsc{Key-SSD}}.
Specifically, we have implemented a LBA-key management framework in OS kernel and the FTL extension for {disk}-level access control management on the SSD with the following main design goals:
(i) efficient implementation of FTL performing {disk}-level key management and access control on the SSD,
(ii) lightweight key management in OS kernel, and (iii) no modification of existing SATA protocol to communicate between host OS and the SSD.
\input{acd}
\input{key_transfer}
\input{device_comm}
\vspace{-0.15in}
\subsection{OS Security Issues}
\label{sec:issue}
\textbf{Inode protection:}
In {\textsc{Key-SSD}}, only the user data of the file is protected by the key, and the metadata of the file is not protected. Therefore, the file open which reads only the \texttt{inode} of the file can succeed without the file key.
To solve this problem, we propose a read-verify method. The read-verify method generates a request to read data directly from the disk and confirms whether the data can be actually read. That is, it does not read only the \texttt{inode} of the file when the file is opened, but verifies that the actual data of the file can be read by using the read-verify method.
\textbf{Page cache:}
Most OS implement disk cache, called page cache, to improve disk access time. However, page cache can cause security problems to access files cached in the page cache without a key.
Consider a case where a process with the correct key reads a file, and another process without an valid key or key accesses the file from the page cache.
In particular, a ransomware attack can read a file's original data from the page cache, encrypt it, create a new file, and delete the original file.
To solve this problem, we have implemented the read-verify method described as a solution to the \texttt{inode} protection security problem.
Previously, the read system call first checks whether there is data in the page cache of the file, and then reads the data from the disk only if it does not exist. On the other hand, the read-verify method unconditionally reads the first page data of the file from the disk, and verifies that the data can be actually read from {\textsc{Key-SSD}} when reading the file.
\textbf{File deletion with invalid key:}
When deleting a file, we used the read-verify method by calling a function that checks if the data can be read from {\textsc{Key-SSD}}. It can not delete a file without a valid key.
\begin{comment}
In this subsection, implementation issues are discussed.
Note that these issues are of our current implementation, but not of our proposed approach.
We expect these issues can be resolved by designing a new file system, which is left as our future work.
\textbf{Inode protection:}
The page cache is not fully utilized.
Since requests are authenticated by the disk drive, if a request hits the page cache, the file system cannot determine whether the request is legitimate or not.
Though a file can be accessed by one application at a time, it is enforced only to those files under the protection of the proposed technique.
For other files which are not under the protection, they can be accessed without a key through normal system calls.
There might be a situation where ransomware and a legitimate application are concurrently access a same file.
Since the legitimate application has a correct key, it can access the file.
If another application tries to access the same file through the new system calls (i.e. \texttt{sys\_read\_key} or \texttt{sys\_write\_key}), the request is pending until the current application closes the file.
However, if another application tries to access the file through normal system calls, the request should be denied by the disk drive.
In this situation, if the request hits the page cache, it can be accessed without checking the key.
To address this issue, we do ... \textcolor{blue}{[Could you explain how we address this issue?]}
\begin{comment}
In this subsection, implementation issues are discussed.
Note that these issues are of our current implementation, but not of our proposed approach.
We expect these issues can be resolved by designing a new file system, which is left as our future work.
\textbf{Concurrent file access:}
A file can be read by one application at a time.
While a file is open, another application cannot access the file.
However, it does not limit other applications from accessing their own files.
In our implementation, the device driver figures out the access code given by the application looking up the table.
The requested block address is the only information that the device driver can use to look up the table.
Thus, if a file is open by multiple applications, the device driver cannot figure out which application placed the request.
In a normal situation, however, a file should be accessed only by one application if the file is under the protection of the proposed technique.
Therefore, even though a file cannot be concurrently accessed by multiple applications, it should not happen under a normal situation.
\textbf{Inode protection:}
Since ransomware usually changes the name and location of files before or after encrypting them, updating the metadata should be protected.
In our current implementation, all \texttt{inode} data structures are protected by a single key managed by the file system.
Whenever the \texttt{inode} is updated and written to the disk, the file system itself provides the key along with the request.
Whether the update is for metadata (\texttt{inode}) or for file data is distinguished at the generic block layer by the difference of page mapping.
Since the block addresses of file data are managed by \texttt{address\_space} structure of \texttt{inode}, if the requested address is found in this structure, it means data update.
Otherwise, it is metadata update, and, thus, a key is given by the file system.
\textbf{Page cache:}
The page cache is not fully utilized.
Since requests are authenticated by the disk drive, if a request hits the page cache, the file system cannot determine whether the request is legitimate or not.
Though a file can be accessed by one application at a time, it is enforced only to those files under the protection of the proposed technique.
For other files which are not under the protection, they can be accessed without a key through normal system calls.
There might be a situation where ransomware and a legitimate application are concurrently access a same file.
Since the legitimate application has a correct key, it can access the file.
If another application tries to access the same file through the new system calls (i.e. \texttt{sys\_read\_key} or \texttt{sys\_write\_key}), the request is pending until the current application closes the file.
However, if another application tries to access the file through normal system calls, the request should be denied by the disk drive.
In this situation, if the request hits the page cache, it can be accessed without checking the key.
To address this issue, we do ... \textcolor{blue}{[Could you explain how we address this issue?]}
\textbf{Key length:}
The length of a key is one word (32 bits) because we use the reserved word to deliver a key.
The 32-bit access might be vulnerable under a bruteforce attack (exhaustive search).
Thus, \textsc{Key-SSD} is equipped with an addtional defense mechanism to prevent it.
However, this is only a limitation of our current implementation.
If the SATA protocol is further extended to accommodate our proposed technique, a longer key can be used.
\end{comment}
\section{Related Work}
\label{sec:related}
\vspace{-0.1in}
Since ransomware is a kind of malware, existing malware mitigation techniques can be used to detect and prevent ransomware~\cite{7778160}.
Ransomware exhibits specific behaviors such as
searching target files,
accessing
cryptographic libraries
frequently, and exchanging encryption keys with a remote command-and-control
servers.
There are studies
to prevent ransomware by detecting these known
behaviors~\cite{7536529,7600214,7336353, 7764294,7387902,7981522, Kolodenker:2017:PDA:3052973.3053035,Lee2017b}.
\begin{comment}
Ransomwares are often required to contact a C\&C server to receive an encryption key.
At this step, by monitoring the network traffic, ransomware can be detected and prevented~\cite{7764294,7387902,7981522}.
When an application accesses a cryptographic library,
the function call can be intercepted and the encryption key, which is passed by the function call, can be stored in a safe place.
If the application turns out to be ransomware, the intercepted key can be used to restore the encrypted files~\cite{Kolodenker:2017:PDA:3052973.3053035,Lee2017b}.
Static analysis of the binary to find a specific cryptographic library
can be used to detect ransomware~\cite{Maiorca:2017:RAP:3019612.3019793}.
All of these techniques assume specific behaviours of ransomware.
\end{comment}
While these techniques increase the difficulty of successful infections, evolved ransomware can circumvent these techniques~\cite{7536529}.
Data backup is another category of mitigation techniques for ransomware attacks~\cite{cldsafe_an_efficient_file_backup}.
ShieldFS is a filesystem that automatically recovers files from backup copies, when the files are infected by ransomware~\cite{Continella:2016:SSR:2991079.2991110}.
However, data backup requires extra storage, and
intelligent ransomware can erase backup copies~\cite{7924925,7886569}.
\begin{comment}
Table~\ref{tab:related} summaries existing ransomware mitigation techniques.
\begin{table*}
\caption{classification of existing ransomware mitigation techniques.}
\label{tab:related}
\centering
\begin{tabular}{|l|l|}
\hline
Category & Existing techniques \\
\hline
\hline
Application behavior monitoring & \cite{7536529,7600214,7336353,7784627,7736455,Andronio2015,Kharraz2015,502676,Young2006,197235,Lee2017,the_effective_ransomware_prevention,design_of_quantification_model,Al-rimy2018,Mbol2016,Zheng2017,8004894,7422770} \\
\hline
Network behavior monitoring & \cite{7764294,7387902,7981522} \\
\hline
Identifying or hooking cryptographic library & \cite{Kolodenker:2017:PDA:3052973.3053035,Lee2017b,Maiorca:2017:RAP:3019612.3019793} \\
\hline
Backup & \cite{cldsafe_an_efficient_file_backup,Continella:2016:SSR:2991079.2991110} \\
\hline
\end{tabular}
\end{table*}
\end{comment}
Mutual authentication techniques~\cite{Gotzfried:2014:MAT:2689660.2663348,4249828} are often
used
to protect disk drives.
The disk grants access
only if the host is authenticated by the protocol.
The host also accesses the disk only if the disk is authenticated.
The mutual authentication techniques are not intended to protect individual files
on the disk
from malware.
Authentication entities are disks, not individual files.
{\textsc{Key-SSD}}
Encryption~\cite{6951337,7428036,1598131,6307756,6237012,Young:2015:DWE:2694344.2694387,6881505,7544407,6662530,7428037,7484726} is a popular technique
for protecting data stored in
disk drives.
The entire disk can be encrypted (full-disk encryption), or files are encrypted selectively.
Ransomware can still read files even if they are encrypted.
Ransomware cannot interpret the contents of the file, but can re-encrypt the file
with its own encryption key.
Trusted Computing Module (TPM) is often used
with authentication or encryption for key management~\cite{5689500}.
This is because
keys need to be stored in a safe place.
Since key management is orthogonal to our approach, TPM can also be
used
to manage keys for {\textsc{Key-SSD}}.
|
1,108,101,565,684 | arxiv | \section{Introduction} \label{SEC:intro}
Charles Babbage was an English mathematician, philosopher, inventor, mechanical engineer, and ``irascible genius'' who pioneered computing machines \cite{babbagebio,beyer,grabiner,moseley,mactutor,odonnell}.
Although he held the Lucasian Chair of Mathematics at Cambridge University from $1828$ to $1839$,
during that period he never resided in Cambridge or delivered a lecture \cite{blackwood}, \cite[p.~$7$]{dubbey}.\\
\centerline{\includegraphics[width=.9in]{YoungBabbage}} \vspace{-.3 cm} \begin{center}{Charles Babbage (1791--1871)}\end{center}
In $1819$ he published his only work on number theory, a short paper \cite{babbage} that begins:
\begin{quotation}
The singular theorem of Wilson respecting Prime Numbers, which was first
published by Waring in his {\em Meditationes Analyticae} \cite[p.~218]{waring}, and to which neither himself
nor its author could supply the demonstration, excited the attention of the
most celebrated analysts of the continent, and to the labors of Lagrange \cite{lagrange} and Euler
we are indebted for several modes of proof $\dotso.$
\end{quotation}
Babbage formulated {\bf Wilson's theorem} as a criterion for primality:
{\em an integer $p>1$ is a prime if and only if $(p-1)! \equiv -1 \!\pmod{p}$}. (For a modern proof, see Moll \cite[p.~66]{moll}.)
He then introduced several such criteria, involving congruences for binomial coefficients (see Granville \cite[Sections~1 and 4]{granville}). However, some of his claims were unproven or even wrong
(as Dubbey points out in \cite[pp.~139--141]{dubbey}). One of his valid results is a necessary and sufficient condition for primality, based on a number of simultaneous congruences.
Henceforth let $n$ denote an integer.
\begin{theorem}[Babbage's Primality Test] \label{THM:test}
An integer $p>1$ is a prime if and only if
\begin{equation} \label{EQ:test}
\binom{p+n}{n}\equiv 1 \pmod{p}
\end{equation}
for all $n$ satisfying $0 \le n \le p-1.$
\end{theorem}
This is of only theoretical interest, the test being slower than trial division.
The ``only if'' part is an immediate consequence of the beautiful {\bf theorem of Lucas} \cite{lucas} (see \cite{fine,granville,mestrovic0,mestrovic2} and \cite[p.~70]{moll}), which
asserts that {\em if $p$ is a prime and the non-negative integers
$a = \alpha_0 + \alpha_1 \, p + \dotsb + \alpha_r \, p^r$ and
\mbox{$b = \beta_0 + \beta_1 \, p + \dotsb + \beta_r \, p^r$}
are written in base $p$ $($so that \mbox{$0 \le \alpha_i, \beta_i \le p-1$} for all~$i)$, then}
\begin{equation} \label{EQ:Lucas}
\binom{a}{b} \equiv
\prod_{i=0}^r \binom{\alpha_i}{\beta_i} \pmod{p}.
\end{equation}
(Here the convention is that $\binom{\alpha}{\beta} =0$ if $\alpha < \beta$.) The congruence \eqref{EQ:test} follows if \mbox{$0 \le n \le p-1$}, for then all the binomial coefficients formed on the right-hand side of \eqref{EQ:Lucas} are of the form $\binom{\alpha}{\alpha} =1,$ except the last one, which is $\binom{1}{0} =1.$
However, the theorem was not available to Babbage, because when it was published in $1878$ he had been dead for seven years.
Lucas's theorem implies more generally that {\em for $p$ a prime and $m$ a power of $p,$ the congruences
\begin{equation} \label{EQ:test2}
\binom{m+n}{n}\equiv 1 \pmod{p} \qquad (0 \le n \le m-1)
\end{equation}
hold.} A converse was proven
in $2013$:
{\bf Me\v{s}trovi\'{c}'s theorem} \cite{mestrovic2} states that {\em if $m>1$ and $p>1$ are integers such that \eqref{EQ:test2} holds,
then $p$ is a prime and $m$ is a power of~$p.$}
To begin the proof, Me\v{s}trovi\'{c} noted that for $n = 1$ the hypothesis gives
\begin{equation*} \label{EQ:Mestpf}
\binom{m+1}{1} = m+1 \equiv 1 \pmod{p}\quad \implies\quad p\mid m.
\end{equation*}
The rest of the proof involves combinatorial congruences modulo prime powers.
As Me\v{s}trovi\'{c} pointed out,
\noindent
``the `if' part of Theorem~\ref{THM:test} is an immediate consequence of [his theorem] (supposing a~priori [that $m = p$]). Accordingly, [his theorem] may be considered as a generalization of Babbage's criterion for primality.''
Here we offer another generalization of Babbage's primality test.
\begin{theorem}[Least-Prime-Factor Test] \label{THM:LPF}
The least prime factor of an integer $m>1$ is the smallest natural number $\ell$ satisfying
\begin{align} \label{EQ:LPF2}
\binom{m+\ell}{\ell} \not \equiv 1 \pmod{m}.
\end{align}
For that value of $\ell,$ the least non-negative residue of
$\binom{m+\ell}{\ell}$ modulo $m$ is $\frac{m}{\ell} + 1.$
\end{theorem}
The proof is given in Section~\ref{SEC:proof}.
Babbage's primality test is an easy corollary of the least-prime-factor test. Indeed, Theorem~\ref{THM:LPF} implies a sharp version of Theorem~\ref{THM:test} noticed by Granville \cite{granville} in $1995.$
\begin{corollary}[Sharp Babbage Primality Test] \label{THM:short}
Theorem~\ref{THM:test} remains true if the range for $n$ is shortened to $0 \le n\le \sqrt{p}.$
\end{corollary}
\begin{proof}
An integer $m>1$ is a prime if and only if its least prime factor $\ell$ exceeds
$\sqrt{m}.$ The corollary follows by setting $m=p$ in Theorem~\ref{THM:LPF}. \begin{flushright}\vspace{-1.85em}\Square\end{flushright}
\end{proof}
To see that {\em Corollary \ref{THM:short} is sharp in that the range for $n$ cannot be further shortened to} $0\le n \le \sqrt{p}-1$, let $q$ be any prime and set $p=q^2$. Then $p$ is not a prime, but the least-prime-factor test with $m=p$ and $\ell=q$ implies \eqref{EQ:test} when \mbox{$0\le n \le q-1$}.
\begin{problem}
Since the ``if'' part of Babbage's primality test is a consequence both of Me\v{s}trovi\'{c}'s theorem and of the least-prime-factor test, one may ask, {\em Is there a common generalization of Me\v{s}trovi\'{c}'s theorem and Theorem~\ref{THM:LPF}?}
(Note, though, that the modulus in the former is~$p,$ while that in the latter is~$m.$)
\end{problem}
Actually, the incongruence \eqref{EQ:LPF2} holds more generally
if the {\em least} prime factor $\ell \mid m$ is replaced with {\em any} prime factor $p \mid m$. The following extension of the least-prime-factor test is proven in Section~\ref{SEC:proof}. See also Sondow \cite[Part (a)]{sondow}.
\begin{theorem} \label{THM:pft}
$(i)$ Given a positive integer $m$ and a prime factor $p\mid m$, we have
\begin{equation} \label{EQ:primedivisor}
\binom{m+p}{p} \not\equiv 1 \pmod{m}.
\end{equation}
$(ii)$ If in addition $p^r\mid m$ but $p^{r+1}\nmid m$, where $r\ge1$, then
\begin{equation} \label{EQ:power}
\binom{m+p}{p} \equiv \frac{m}{p} + 1 \not\equiv 1 \pmod{p^r}.
\end{equation}
\end{theorem}
Part ($i$) is clearly equivalent to the statement that {\em if $d>1$ divides $m$ and $\binom{m+d}{d} \equiv 1 \pmod{m}$, then $d$ is composite.}
As an example, for $m=260$ and $d=10$ we have
$$ \binom{m+d}{d} = \binom{270}{10} = 479322759878148681 \equiv 1 \pmod{260}.$$
The sequence of integers $m>1,$ for which some integer $d$ (necessarily composite) satisfies
\begin{equation} \label{EQ:equiv 1}
d>1 , \qquad d \mid m, \qquad \binom{m+d}{d} \equiv 1 \pmod{m},
\end{equation}
begins \cite[Seq. A290040]{oeis}
$$
m=260, 1056, 1060, 3460, 3905, 4428, 5000, 5060, 5512, 5860, 6372, 6596,\dotso
$$
and the sequence of smallest such divisors $d$ is, respectively, \cite[Seq. A290041]{oeis}
\begin{equation} \label{EQ:d}
d=10, 264, 10, 10, 55, 18, 20, 10, 52, 10, 18, 34,\dotso.
\end{equation}
\begin{problem}
Does Theorem~\ref{THM:pft} extend to prime power factors, i.e., does \eqref{EQ:primedivisor} also hold when $p$ is replaced with $p^k$, where $p^k\mid m$ and $k>1$? In particular, in the sequence \eqref{EQ:d}, is any term $d$ a prime power?
\end{problem}
See \cite[Part (c)]{sondow}.
Babbage also claimed a necessary and sufficient condition for primality based on a {\em single} congruence. But he proved only necessity, so we call it a test for non-primality.
\begin{theorem}[{\bf Babbage's Non-Primality Test}] \label{THM:bab}
An integer $m\ge3$ is composite if
\begin{equation} \label{EQ:babcong}
\binom{2m-1}{m-1} \not\equiv 1 \pmod{m^2}.
\end{equation}
\end{theorem}
Our version of his proof is given in Section~\ref{SEC:non-primality}.
Not only did Babbage not prove the claimed converse, but in fact it is false. Indeed,
{\em the numbers \mbox{$m_1=p_1^2=283686649$} and \mbox{$m_2=p_2^2=4514260853041$} are composite but do not satisfy} \eqref{EQ:babcong}, where \mbox{$p_1=16843$} and \mbox{$p_2=2124679$} are primes.
Here $p_1$ (indicated by Selfridge and Pollack in $1964$) and $p_2$ (discovered by Crandall, Ernvall and Mets\"{a}nkyl\"{a} in $1993$) are {\em Wolstenholme primes}, so called by Mcintosh \cite{mcintosh} because, while {\bf Wolstenholme's theorem} \cite{wolstenholme} (see \cite{granville,mestrovic,tw} and \cite[p.~73]{moll}) of $1862$ guarantees that {\em every prime $p\ge5$ satisfies}
\begin{equation} \label{EQ:Wolstenholme}
\binom{2p-1}{p-1} \equiv 1 \pmod{p^3},
\end{equation}
in fact $p_1$ and $p_2$ satisfy the congruence in \eqref{EQ:Wolstenholme} modulo $p^4$, not just $p^3$ (see Guy \cite[p.~131]{guy} and Ribenboim \cite[p.~23]{ribenboim}).
Note that \eqref{EQ:Wolstenholme} strengthens Babbage's non-primality test, as Theorem~\ref{THM:bab} is equivalent to the statement that {\em the congruence in \eqref{EQ:Wolstenholme} holds modulo $p^2$ for any prime} $p\ge3$.
In their solutions to a problem by Segal in the \textit{Monthly}, Brinkmann \cite{sb} and Johnson \cite{sj} made Babbage's and Wolstenholme's theorems more precise by showing that {\em every prime $p\ge5$ satisfies the congruences}
$$\binom{2p-1}{p-1} \equiv 1-\frac23 p^3B_{p-3} \equiv \binom{2p^2-1}{p^2-1} \pmod{p^4},$$
where $B_k$ denotes the $k$th \textit{Bernoulli number}, a rational number. (See also Gardiner \cite{gardiner} and Mcintosh \cite{mcintosh}.) Thus, {\em a prime \mbox{$p\ge5$} is a Wolstenholme prime if and only if} $B_{p-3} \equiv 0 \pmod{p}$. (The congruence means that $p$~divides the numerator of $B_{p-3}$.)
In that case, the square of that prime, say \mbox{$m=p^2$}, is composite but must satisfy
$$\binom{2m-1}{m-1} \equiv 1\pmod{m^2},$$ thereby providing a counterexample to the converse of Babbage's non-primality test.
Johnson \cite{sj} commented that ``interest in [Wolstenholme primes] arises from the fact that in $1857$, Kummer proved that the first case of [Fermat's Last Theorem] is true for all prime exponents $p$ such that $p\nmid B_{p-3}$.''
We have seen that the converse of Babbage's non-primality test is false.
The converse of Wolstenholme's theorem is the statement that {\em if $p\ge5$ is composite, then \eqref{EQ:Wolstenholme} does not hold.}
It is not known whether this is generally true.
A proof that it is true for {\em even} positive integers was outlined by Trevisan and Weber \cite{tw} in $2001$.
In Section \ref{SEC:non-primality}, we fill in some details omitted from their argument and extend it to prove the following stronger result.
\begin{theorem}[Converse of Babbage's Non-Primality Test for Even Numbers] \label{THM:even}
If a positive integer $m$ is even, then
\begin{equation} \label{EQ:even}
\binom{2m-1}{m-1} \not\equiv1\pmod{m^2}.
\end{equation}
\end{theorem}
\section{Proofs of the least-prime-factor test and its extension} \label{SEC:proof}
We prove Theorems~\ref{THM:LPF} and \ref{THM:pft}. The arguments
use only mathematics available in Babbage's time.
\begin{proof}[Theorem~\ref{THM:LPF}]
As $\ell$ is the smallest prime factor of $m,$ if $0<k < \ell$ then $k!$ and $m$ are coprime. In that case, {\bf B\'{e}zout's identity}
(proven in $1624$ by Bachet in a book with the charming title {\em Pleasant and Delectable Problems} \cite[p. 18, Proposition XVIII]{bachet}\textemdash see \cite[Section 4.3]{chabert}) gives integers $a$ and $b$ with
\mbox{$ak!+bm=1$}. Multiplying B\'{e}zout's equation by the number \mbox{$\binom{m}{k} = m(m-1)\dotsb(m-k+1)/k!$} yields
$$ a m(m-1)\dotsb(m-k+1) + bm\binom{m}{k} = \binom{m}{k},$$
so $\binom{m}{k} \equiv0\!\pmod{m}$
if $1 \le k \le \ell-1.$
Now, for \mbox{$n=0,1,\dotsc, \ell-1$}, {\bf Vandermonde's convolution} \cite{vandermonde} (see \cite[p.~164]{moll}) of $1772$ gives
\begin{equation*} \label{EQ:Chu}
\binom{m+n}{n} = \sum_{k=0}^n \binom{m}{k} \binom{n}{n-k}
\equiv \binom{m}{0} \binom{n}{n} \pmod{m}.
\end{equation*}
(To see the equality, equate the coefficients of $x^n$ in the expansions of \mbox{$(1+x)^{m+n}$} and $(1+x)^m(1+x)^n.$)
Thus, we arrive at the congruences
\begin{equation} \label{EQ:step1}
\binom{m+n}{n}\equiv 1 \pmod{m} \qquad (0 \le n \le \ell-1).
\end{equation}
On the other hand, from the identity
\begin{equation} \label{EQ:identity}
\binom{a}{b} =
\frac{a}{b} \binom{a-1}{b-1}
\end{equation}
(to prove it, use factorials),
the congruence \eqref{EQ:step1} for $n=\ell-1$, the integrality of \mbox{$\frac{m+\ell}{\ell}=\frac{m}{\ell}+1$},
and the inequality $\ell>1$ (as $\ell$ is a prime),
we deduce that
\begin{equation*} \label{EQ:lpftest}
\binom{m+\ell}{\ell} = \frac{m+\ell}{\ell}\binom{m+\ell-1}{\ell-1}\equiv \frac{m}{\ell}+1 \not\equiv 1 \pmod{m}.
\end{equation*}
Together with \eqref{EQ:step1}, this implies the least-prime-factor test. \begin{flushright}\vspace{-1.85em}\Square\end{flushright}
\end{proof}
\begin{proof}[Theorem~\ref{THM:pft}]
It suffices to prove (ii).
Set
$$g\DefEqDisp\gcd((p-1)!,m) \qquad \text{and}\qquad m_p\DefEqDisp\frac{m}{g}.$$
Note that
\begin{equation} \label{EQ:implies}
p \ \text{prime} \implies p\nmid g \implies p^r\mid m_p,
\end{equation}
since $p^r\mid m$. B\'{e}zout's identity
gives integers $a$ and $b$ with
\mbox{$a(p-1)!+bm=g$}. When $0<k<p$, multiplying B\'{e}zout's equation by $\binom{m}{k}$
yields
$$ a m(m-1)\dotsb(m-k+1)\frac{(p-1)!}{k!} + bm\binom{m}{k} = g\binom{m}{k}$$
with $(p-1)!/k!$ an integer, so $g\binom{m}{k} \equiv0\!\pmod{m}$. Dividing by $g$ gives
$$\binom{m}{k} \equiv0\!\pmod{m_p}\quad (1\le k \le p-1).$$
Combining this with \eqref{EQ:identity} and Vandermonde's convolution, we get
\begin{align} \label{EQ:Chu2}
\begin{split}
\binom{m+p}{p} = \frac{m+p}{p}\binom{m+p-1}{p-1} &= \frac{m+p}{p}\sum_{k=0}^{p-1} \binom{m}{k} \binom{p-1}{p-1-k}\\
&\equiv \frac{m}{p}+1 \pmod{m_p}.
\end{split}
\end{align}
As $p^{r+1}\nmid m$, we have $p^r\nmid \frac{m}{p}$. Now, \eqref{EQ:implies} and \eqref{EQ:Chu2} imply \eqref{EQ:power}, as required. \begin{flushright}\vspace{-1.85em}\Square\end{flushright}
\end{proof}
\section{Proofs of Babbage's non-primality test and its converse for even numbers} \label{SEC:non-primality}
The following proof is close to the one Babbage gave.
\begin{proof}[Theorem \ref{THM:bab}]
Suppose on the contrary that $m$ is prime. If we have $1 \le n \le m-1$, then $m$ divides the numerator of $\binom{m}{n} = m!/n!(m-n)!$ but not the denominator, so
\mbox{$\binom{m}{n}\equiv 0 \pmod{m}$}.
Thus, by
\eqref{EQ:identity} and a famous case of Vandermonde's convolution,
\begin{equation} \label{EQ:special}
2\binom{2m-1}{m-1} =\binom{2m}{m} = \sum_{n=0}^m\binom{m}{n}^2 \equiv 1^2+1^2 \equiv 2 \pmod{m^2}.
\end{equation}
But as $m\ge3$ is odd, \eqref{EQ:special} contradicts
\eqref{EQ:babcong}. Therefore, $m$ is composite. \begin{flushright}\vspace{-1.85em}\Square\end{flushright}
\end{proof}
Before giving the proof of Theorem~\ref{THM:even}, we establish two lemmas.
For any positive integer $k,$ let $2^{v(k)}$ denote the highest power of $2$ that divides $k.$
\begin{lemma} \label{LEM:power}
If $m\ge n\ge1$ are integers satisfying $n\le2^{v(m)},$ then the formula $v(\binom{m}{n}) = v(m)-v(n)$ holds.
\end{lemma}
\begin{proof}
Let $m=2^rm'$ with $m'$ odd. Note that $v(2^rm'-k) = v(k)$ if \mbox{$0<k < 2^r$}.
({\em Proof.} Write $k =2^tk',$ where $0\le t = v(k) \le r-1$ and $k'$ is odd. Then \mbox{$2^{r-t}m'-k'$} is also odd, so $v(2^rm'-k) =v(2^t(2^{r-t}m'-k'))=t=v(k).$)
The logarithmic formula \mbox{$v(ab)=v(a)+v(b)$} then implies that when $1 \le n\le2^r$ the exponent of the highest power of $2$ that divides the product
$$n!\binom{m}{n}= 2^rm'(2^rm'-1)(2^rm'-2)\dotsb(2^rm'-(n-1))$$
is
$v(n!)+v(\binom{m}{n}) = r +v(1\cdot2\dotsb(n-1))$, so $v(\binom{m}{n}) = r-v(n)$. As \mbox{$r=v(m)$}, this proves the desired formula. \begin{flushright}\vspace{-1.85em}\Square\end{flushright}
\end{proof}
Lemma \ref{LEM:power} is sharp in that the hypothesis $n\le2^{v(m)}$ cannot be replaced with the weaker hypothesis $v(n) \le v(m).$ For example, $v(\binom{10}{6}) = v(210) = 1$, but $v(10)-v(6) =0.$
\begin{lemma} \label{LEM:powerof2}
A binomial coefficient $\binom{2m-1}{m-1}$ is odd if and only if $m=2^r$ for some $r\ge0.$
\end{lemma}
\begin{proof}
{\bf Kummer's theorem} \cite{kummer} (see \cite[p.~78]{moll} or \cite{pomerance}) for the prime~$2$ states that $v(\binom{a+b}{a})$ equals
the number of carries when adding $a$ and $b$ in base~$2$ arithmetic.
Hence $v(\binom{m+m}{m})$ is the number of ones in the binary expansion of $m$, and so \mbox{$v(\binom{2m}{m})=1$} if and only if $m=2^r$ for some $r\ge0$.
As $\binom{2m}{m} = 2 \binom{2m-1}{m-1}$ by \eqref{EQ:identity}, we are done. \begin{flushright}\vspace{-1.85em}\Square\end{flushright}
\end{proof}
We can now prove the converse of Babbage's non-primality test for even numbers.
\begin{proof}[Theorem~\ref{THM:even}]
For $m\ge2$ not a power of $2,$ Lemma~\ref{LEM:powerof2} implies that $\binom{2m-1}{m-1}$ is even, so $\binom{2m-1}{m-1}$ is congruent modulo~$4$ to either $0$ or $2$.
For $m\ge2$ a power of $2$, say $m=2^r$, the equalities in \eqref{EQ:special} and the symmetry $\binom{m}{n} =\binom{m}{m-n}$ yield
$$\binom{2m-1}{m-1} = 1 + \frac12\binom{2^r}{2^{r-1}}^2+ \sum_{k=1}^{2^{r-1}-1} \binom{2^r}{k}^2, $$
and Lemma \ref{LEM:power} implies that $\frac12\binom{2^r}{2^{r-1}}^2 \equiv2\!\!\pmod{4}$
and that $ \binom{2^r}{k}^2\equiv0\!\!\pmod{4}$ when \mbox{$0<k<2^{r-1}$}; thus, by addition $\binom{2m-1}{m-1} \equiv 3\!\! \pmod{4}$.
Hence for all $m\ge2$ we have $\binom{2m-1}{m-1} \not\equiv 1\!\! \pmod{4}$. Now as $4$ divides $m^2$ when $m$ is even, \eqref{EQ:even} holds a fortiori.
This completes the proof. \begin{flushright}\vspace{-1.85em}\Square\end{flushright}
\end{proof}
|
1,108,101,565,685 | arxiv | \section{Security Analysis}
\subsection{Outline}
We begin our security analysis by describing how we model the honest and dishonest scenarios, i.e.~what assumptions are made on the devices used to execute the protocol. In Section~\ref{sec:fault-tolerant-rbc} we describe a generalized protocol, in which the states emitted by Alice in the commit phase do not need to be \emph{exactly} the four BB84 states and Bob's answer in the open phase does not need to match \emph{exactly} the string encoded by Alice. For this fault-tolerant RBC protocol we present a security proof and compute an explicit security bound as a function of the number of qubits exchanged in the commit phase. This allows us to deal with errors on the communication channel in the experiment, but not yet with losses. Note that we do not simply extend the analysis of~\cite{Kaniewski2013a}, but in fact our analysis is based on entirely new techniques that can - as a byproduct - also be used to obtain better parameters for the idealised setting. In Section~\ref{sec:fault-tolerant-rbc-with-backreporting} we finally make our protocol robust against errors \emph{and} losses.
In particular, we show how the fault-tolerant RBC can be extended to deal with the presence of losses. We investigate how good the devices owned by the honest parties need to be to result in a protocol that is both robust and secure in the asymptotic limit. Finally, we show how to compute explicit security bounds for a finite-length protocol.
\subsection{Modelling the experiment}
\label{sec:modelling-the-experiment}
Our model hinges on the following assumptions:
\begin{enumerate}
\item Alice uses a weak-coherent source with phase randomization.
\item Every photon is subject to independent noise and loss processes.
\item The detectors used by Bob for different bases have the same efficiency (independent of the state).
\end{enumerate}
In the following we elaborate on what thes assumptions mean for the honest and dishonest scenarios.
\subsubsection{The honest scenario}
Assuming that Alice uses a weak-coherent source with phase randomisation the ensemble emitted by the source is described by
\begin{gather*}
\rho = \sum_{r = 0}^{\infty} p_{r} \ketbraq{r},\\
\nbox{where} p_{r} = e^{- \mu} \cdot \frac{\mu^{r}}{r!},
\end{gather*}
$\mu$ is the average number of photons per pulse and $\ket{r}$ is the Fock state of $r$ photons. To model noise and losses between Alice and Bob for every photon we introduce an independent, binary random variable. Let $\eta$ be the detection efficiency: every photon emitted by the source gets detected on Bob's side with probability $\eta$, otherwise it is lost. Applying such a process to the original state simply affects the probability distribution: the probability of Bob detecting $r$ photons equals
\begin{equation}
\label{eq:pr-definition}
p_{r}(\mu, \eta) = e^{-\mu \eta} \cdot \frac{(\mu \eta)^{r}}{r!}.
\end{equation}
Now, suppose that $n$ rounds are played (these are independent due to Assumption 2) and let $N_{r}$ be the random variable counting the number of rounds in which exactly $t$ photons were received. Then
\begin{equation}
\label{eq:nt-and-expectation-value}
\Pr[N_{r} = k] = {n \choose k} p_{r}^{k} (1 - p_{r})^{n - k} \nbox{and} \amsbb{E}[N_{r}] = p_{r} n.
\end{equation}
We model bit-flip errors in a similar way. For every photon detected on Bob's side and measured in the correct basis we introduce a bit flip error with probability $Q$.
\subsubsection{Dishonest scenarios}
In the case of one party being dishonest we need to assume that his/her devices are perfect\,---\,we want to achieve security against all-powerful adversaries. On the other hand, our assumptions still apply to the honest party.
If Alice is honest then we model her source as described before but
now no losses or errors are guaranteed. Hence, what Bob sees are light
pulses containing a Poisson-distributed (with parameter $\mu$) number
of photons. Moreover, we assume that Bob can do a perfect
photon-counting experiment, which might help him to cheat.
Bob can locate any number of agents anywhere in space-time, send
classical or quantum information, including the quantum states
sent by Alice, securely between his agents at
light speed, and carry out arbitrary quantum operations on Alice's
states and any ancillae.
If Bob is honest then we make no assumptions on the state emitted by
Alice but we assume that the detectors Bob is using for the two
different bases have the same efficiency (and that it does not depend
on the state prepared by Alice).
\subsection{Notation}
\label{sec:notation}
We also fix some notation.
Let $[n] = \{1, 2, \ldots, n\}$ and $\mathcal{C}_{n} = \{0, 1\}^{n}$. For $\theta \in \mathcal{C}_{n}$ define the following sets
\begin{align*}
S = \{k \in [n] : \theta_{k} = 0\},\\
T = \{k \in [n] : \theta_{k} = 1\}.
\end{align*}
Note that $S \cup T = [n]$ and $S \cap T = \varnothing$. The (normalised) Hamming distance between two strings $x, y \in C_{n}$ is
\begin{equation*}
\dham(x, y) := \frac{1}{n} \abs[\big]{\{k \in [n]: x_{k} \neq y_{k}\}}.
\end{equation*}
The Hamming weight is defined as the distance from the string of all zeroes:
\begin{equation*}
\wham(x) := \dham(x, 00 \ldots 0).
\end{equation*}
\subsection{Fault-tolerant RBC}
\label{sec:fault-tolerant-rbc}
The protocol presented in this section is essentially the original protocol in~\cite{Kent2012a} with the following two extensions:
\begin{itemize}
\item In the original protocol Alice is required to create random
BB84 states. In other words, she is required to generate two
uniformly random bits, $x, \theta \in \{0, 1\}$, and create the
state $H^{\theta} \ket{x}$ of a qubit. We relax this constraint and
simply require the two states for the same value of $\theta$ to be
orthogonal: if $\ket{\psi_{x}^{\theta}}$ is the state corresponding
to the random variables being $x$ and $\theta$ then we require
\begin{equation*}
\braket{\psi_{0}^{0}}{\psi_{1}^{0}} = \braket{\psi_{0}^{1}}{\psi_{1}^{1}} = 0,
\end{equation*}
which for qubit states implies
\begin{equation}
\label{eq:constraint-on-states}
\ketbraq{\psi_{0}^{0}} + \ketbraq{\psi_{1}^{0}} = \ketbraq{\psi_{0}^{1}} + \ketbraq{\psi_{1}^{1}} = \mathbb{1}.
\end{equation}
The assumption that each pair forms a basis means that any prepare-and-measure scheme can be turned into an entanglement-based one. It will become evident from the security proof that the maximally incompatible states (e.g.~the BB84 states) are the optimal choice from the security point of view.
\item The strings that Bob's agents, Bill ($B_{1}$) and Brian ($B_{2}$), unveil in the open phase do not have to match Alice's string exactly. The opening will be accepted as long as Bill ($B_{1}$) and Brian ($B_{2}$) unveil the same string and it is close (in terms of Hamming distance) to the one measured by Alice.
\end{itemize}
\subsubsection{The protocol}
Let $\delta$ be the error threshold and $n$ be the number of rounds that Alice and Bob agree to play.
\printprotocol{0}
\subsubsection{Security for honest Alice}
\label{sec:security-proof}
A dishonest Bob can locate agents anywhere in space-time and transmit
classical and quantum information between them securely at light
speed. However, the relativistic signalling constraints implied
by the timings in our protocol mean that the quantum states
controlled respectively by Bill and Brian at the unveiling stage
belong to disjoint Hilbert spaces that cannot have interacted
since the commitment time $t_c$. We can thus restrict attention
to the Hilbert spaces defining Bill and Brian's possible
states at unveiling and to Bill and Brian's actions on
these spaces.
We use the security model discussed in~\cite{Kaniewski2013a}, in
particular, we employ the global command scenario.
Suppose that at the beginning of the open phase Bill and Brian receive a message asking them to unveil the bit $b$ and let $p_{b}$ be the probability that they succeed. We will show that for any strategy
\begin{equation}
\label{eq:security-definition}
p_{0} + p_{1} \leq 1 + \varepsilon,
\end{equation}
where $\varepsilon$ decays exponentially in $n$, the number of rounds played.
The most general strategy by cheating Bob is to apply a quantum
channel that ``splits up'' the quantum state into two parts: one of
which is sent to Bill at or before unveiling, while the other is sent
to Brian at or before unveiling.
In the open phase, each of the agents will know which bit they are
trying to unveil and will perform an arbitrary measurement on their
respective systems to find out what answers they should send to
Alice's agents.
Hence, the cheating strategy consists of a splitting channel and four
POVMs (each agent has two different measurements depending on what
they are trying to unveil). Let the measurement operators
corresponding to Bill (Brian) trying to unveil $b$ be
$\{P_{y}^{b}\}_{y \in \mathcal{C}_{n}}$ ($\{Q_{y}^{b}\}_{y \in \mathcal{C}_{n}}$).
Note that we can without loss of generality assume that these
measurements are projective (by Neumark's dilation theorem since we
make no assumptions on the dimension of the quantum state).
Moreover, let us purify the protocol. Generating one of the four
states satisfying~\eqref{eq:constraint-on-states} uniformly at random
is equivalent to creating an EPR pair, and measuring one half of it in
one of the two bases with equal probability. It is clear that the two
protocols are equivalent from the security point of view but the
purified version turns out to be easier to analyse.
Let $\theta \in \mathcal{C}_{n}$ be the basis string that Alice picks
uniformly at random. Then her measurement operators take the form
$\ket{x^{\theta}} = \bigotimes_{k = 1}^{n}
\ket{\psi_{x_{k}}^{\theta_{k}}}$ for $x \in \mathcal{C}_{n}$. Now, the checks
that Alice performs can be written as projectors. Let
$\Pi_{b}^{\theta}$ be the projector corresponding to Alice accepting
the unveiling of $b$ given that her basis string is $\theta$. Then
\begin{align*}
\Pi_{0}^{\theta} &= \sum_{x \in \mathcal{C}_{n}} \ketbraq{x^{\theta}} \otimes \sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{S}, y_{S}) \leq \delta}} P_{y}^{0} \otimes Q_{y}^{0},\\
\Pi_{1}^{\theta} &= \sum_{x \in \mathcal{C}_{n}} \ketbraq{x^{\theta}} \otimes \sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{T}, y_{T}) \leq \delta}} P_{y}^{1} \otimes Q_{y}^{1}.
\end{align*}
Let $\rho$ be the tripartite state shared between Alice, Bill and Brian. Then the probability of successfully unveiling $b$ can be written as
\begin{equation*}
p_{b} = 2^{-n} \sum_{\theta \in \mathcal{C}_{n}} \tr(\rho \Pi_{b}^{\theta}).
\end{equation*}
We want to bound $p_{0} + p_{1}$, which corresponds to
\begin{equation}
\label{eq:p0-plus-p1}
p_{0} + p_{1} = 2^{-n} \sum_{\theta \in \mathcal{C}_{n}} \tr \big( \rho [\Pi_{0}^{\theta} + \Pi_{1}^{\theta}] \big).
\end{equation}
Adding the two projectors up yields
\begin{equation*}
\Pi_{0}^{\theta} + \Pi_{1}^{\theta} = \sum_{x \in \mathcal{C}_{n}} \ketbraq{x^{\theta}} \otimes \Bigg[ \sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{S}, y_{S}) \leq \delta}} P_{y}^{0} \otimes Q_{y}^{0} \quad + \sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{T}, y_{T}) \leq \delta}} P_{y}^{1} \otimes Q_{y}^{1} \Bigg].
\end{equation*}
The terms in the square bracket can be upper bounded by replacing one of the measurement operators by the identity matrix. Therefore,
\begin{gather*}
\sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{S}, y_{S}) \leq \delta}} P_{y}^{0} \otimes Q_{y}^{0} \quad + \sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{T}, y_{T}) \leq \delta}} P_{y}^{1} \otimes Q_{y}^{1} \leq \Big( \sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{S}, y_{S}) \leq \delta}} P_{y}^{0} \Big) \otimes \mathbb{1} \quad + \mathbb{1} \otimes \Big( \sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{T}, y_{T}) \leq \delta}} Q_{y}^{1} \Big)\\
\leq \mathbb{1} \otimes \mathbb{1} \quad + \sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{S}, y_{S}) \leq \delta}} P_{y}^{0} \otimes \sum_{\substack{z \in \mathcal{C}_{n}\\ \dham(x_{T}, z_{T}) \leq \delta}} Q_{z}^{1},
\end{gather*}
where we applied the following operator inequality in the last step:
\begin{equation*}
A \otimes \mathbb{1} + \mathbb{1} \otimes B = \mathbb{1} \otimes \mathbb{1} + A \otimes B - (\mathbb{1} - A) \otimes (\mathbb{1} - B)\leq \mathbb{1} \otimes \mathbb{1} + A \otimes B,
\end{equation*}
for any $0 \leq A, B \leq \mathbb{1}$. Therefore,
\begin{equation*}
\Pi_{0}^{\theta} + \Pi_{1}^{\theta} \leq \mathbb{1} + \Pi_{c}^{\theta},
\end{equation*}
where $\Pi_{c}^{\theta}$ is the projector for the ``cross-game'', in which Bill has to unveil $0$ and Brian has to unveil $1$,
\begin{equation*}
\Pi_{c}^{\theta} = \sum_{x \in \mathcal{C}_{n}} \ketbraq{x^{\theta}} \otimes \sum_{\substack{y \in \mathcal{C}_{n}\\ \dham(x_{S}, y_{S}) \leq \delta}} P_{y}^{0} \otimes \sum_{\substack{z \in \mathcal{C}_{n}\\ \dham(x_{T}, z_{T}) \leq \delta}} Q_{z}^{1}.
\end{equation*}
Clearly, $\tr \big( \rho [\Pi_{0}^{\theta} + \Pi_{1}^{\theta}] \big) \leq 1 + \tr \big( \rho \Pi_{c}^{\theta} \big)$, which combined with~\eqref{eq:p0-plus-p1} gives
\begin{equation*}
p_{0} + p_{1} \leq 1 + 2^{-n} \sum_{\theta \in \mathcal{C}_{n}} \tr \big( \rho \Pi_{c}^{\theta} \big) = 1 + \tr \big( \rho \ave{\Pi_{c}^{\theta}} \big) \leq 1 + \norm{\ave{\Pi_{c}^{\theta}}},
\end{equation*}
where $\ave{\cdot}$ denotes averaging over $\theta$, i.e.~$\ave{\Pi_{c}^{\theta}} = 2^{- n} \sum_{\theta \in \mathcal{C}_{n}} \Pi_{c}^{\theta}$, and $\norm{\cdot}$ denotes the Schatten $\infty$-norm. Changing the order of summation in $\Pi_{c}^{\theta}$ gives
\begin{equation*}
\Pi_{c}^{\theta} = \sum_{y, z \in \mathcal{C}_{n}} \sum_{\substack{x \in \mathcal{C}_{n}\\ \dham(x_{S}, y_{S}) \leq \delta\\ \dham(x_{T}, z_{T}) \leq \delta}} \ketbraq{x^{\theta}} \otimes P_{y}^{0} \otimes Q_{z}^{1}.
\end{equation*}
Now, it is clear that only the $x$-dependent part needs to be averaged:
\begin{equation*}
\ave{\Pi_{c}^{\theta}} = 2^{-n} \sum_{\theta \in \mathcal{C}_{n}} \Pi_{c}^{\theta} = \sum_{y, z \in \mathcal{C}_{n}} B_{yz} \otimes P_{y}^{0} \otimes Q_{z}^{1},
\end{equation*}
where
\begin{equation*}
B_{yz} = 2^{-n} \sum_{\theta \in \mathcal{C}_{n}} \sum_{\substack{x \in \mathcal{C}_{n}\\ \dham(x_{S}, y_{S}) \leq \delta\\ \dham(x_{T}, z_{T}) \leq \delta}} \ketbraq{x^{\theta}}.
\end{equation*}
The product $P_{y}^{0} \otimes Q_{z}^{1}$ yields orthogonal projectors so the norm of $\ave{\Pi_{c}^{\theta}}$ can be written as
\begin{equation}
\label{eq:cross-projector-norm}
\norm{\ave{\Pi_{c}^{\theta}}} = \max_{y, z \in \mathcal{C}_{n}} \norm{B_{yz}}.
\end{equation}
To see which values of $y$ and $z$ maximize the norm we need to look more closely at the matrices $B_{yz}$. For every $\theta$ define $u(\theta) \in \mathcal{C}_{n}$ to be the string that satisfies $[u(\theta)]_{S} = y_{S}$ and $[u(\theta)]_{T} = z_{T}$. Relabelling $x \to x \oplus u(\theta)$ yields
\begin{equation*}
B_{yz} = 2^{-n} \sum_{\theta \in \mathcal{C}_{n}} \sum_{\substack{x \in \mathcal{C}_{n}\\ \wham(x_{S}) \leq \delta\\ \wham(x_{T}) \leq \delta}} \ketbraq{(x \oplus u(\theta))^{\theta}}.
\end{equation*}
The constraints on the second sum can be relaxed by noting that $\wham(x_{S}) \leq \delta$ and $\wham(x_{T}) \leq \delta$ implies $\wham(x) \leq \delta$. Hence,
\begin{equation*}
B_{yz} \leq B_{yz}' = 2^{-n} \sum_{\theta \in \mathcal{C}_{n}} \sum_{\substack{x \in \mathcal{C}_{n}\\ \wham(x) \leq \delta}} \ketbraq{(x \oplus u(\theta))^{\theta}}.
\end{equation*}
Now the second sum is independent of $\theta$, hence, the summation over $\theta$ can be performed first. The product structure simplifies the summation
\begin{equation*}
\ketbraq{x^{\theta}} = \bigotimes_{k = 1}^{n} \ketbraq{\psi_{x_{k}}^{\theta_{k}}} \nbox{and} \sum_{\theta \in \mathcal{C}_{n}} \ldots \iff \bigotimes_{k = 1}^{n} \sum_{\theta_{k} \in \{0, 1\}} \ldots,
\end{equation*}
which implies that
\begin{equation*}
2^{-n} \sum_{\theta \in \mathcal{C}_{n}} \ketbraq{(x \oplus u(\theta))^{\theta}} = \bigotimes_{k = 1}^{n} \rho_{x_{k} \oplus y_{k}, x_{k} \oplus z_{k}}.
\end{equation*}
where
\begin{equation*}
\rho_{b, c} = \frac{1}{2} (\ketbraq{\psi_{b}^{0}} + \ketbraq{\psi_{c}^{1}}).
\end{equation*}
for $b, c \in \{0, 1\}$. Note that $\rho_{b, c} + \rho_{1 - b, 1 - c} = \mathbb{1}$ so they are diagonal in the same eigenbasis. Therefore, without loss of generality we can write
\begin{equation*}
\rho_{b, c} = \sum_{t \in \{0, 1\}} \lambda_{t}^{b \oplus c} \ketbraq{e_{t}^{b \oplus c}},
\end{equation*}
for $b, c \in \{0, 1\}$, where $\lambda_{0}^{b \oplus c} + \lambda_{1}^{b \oplus c} = 1$. In particular, we have
\begin{equation*}
\Big( \bigotimes_{k = 1}^{n} \rho_{x_{k} \oplus y_{k}, x_{k} \oplus z_{k}} \Big) \Big( \bigotimes_{k = 1}^{n} \ket{e_{v_{k}}^{y_{k} \oplus z_{k}}} \Big) = \bigotimes_{k = 1}^{n} \lambda_{x_{k} \oplus y_{k} \oplus v_{k}}^{y_{k} \oplus z_{k}} \ket{e_{v_{k}}^{y_{k} \oplus z_{k}}}.
\end{equation*}
Therefore, we also know the eigenbasis of
\begin{equation*}
B_{yz}' = \sum_{\substack{x \in \mathcal{C}_{n}\\ \wham(x) \leq \delta}} \bigotimes_{k = 1}^{n} \rho_{x_{k} \oplus y_{k}, x_{k} \oplus z_{k}},
\end{equation*}
and the highest eigenvalue equals simply
\begin{equation*}
\norm{B_{yz}'} = \max_{v \in \mathcal{C}_{n}} \sum_{\substack{x \in \mathcal{C}_{n}\\ \wham(x) \leq \delta}} \prod_{k = 1}^{n} \lambda_{x_{k} \oplus y_{k} \oplus v_{k}}^{y_{k} \oplus z_{k}}.
\end{equation*}
Recall from~\eqref{eq:cross-projector-norm} that the expression we want to bound is
\begin{equation*}
\max_{y, z \in \mathcal{C}_{n}} \norm{B_{yz}'} = \max_{v, y, z \in \mathcal{C}_{n}} \sum_{\substack{x \in \mathcal{C}_{n}\\ \wham(x) \leq \delta}} \prod_{k = 1}^{n} \lambda_{x_{k} \oplus y_{k} \oplus v_{k}}^{y_{k} \oplus z_{k}} = \max_{a, b \in \mathcal{C}_{n}} \sum_{\substack{x \in \mathcal{C}_{n}\\ \wham(x) \leq \delta}} \prod_{k = 1}^{n} \lambda_{x_{k} \oplus b_{k}}^{a_{k}} .
\end{equation*}
Solving this maximization is easy (see Appendix~\ref{app:simple-maximization-problem} for details) and yields
\begin{equation*}
\max_{y, z \in \mathcal{C}_{n}} \norm{B_{yz}'} = \sum_{\substack{x \in \mathcal{C}_{n}\\ \wham(x) \leq \delta}} \prod_{k = 1}^{n} \lambda_{x_{k}} = \sum_{k = 0}^{ \lfloor \delta n \rfloor } {n \choose k} \lambda_{0}^{n - k} \lambda_{1}^{k},
\end{equation*}
where $\lambda_{0} = \max_{b, c} \lambda_{b}^{c}$ and $\lambda_{1} = 1 - \lambda_{0}$. Finally, as we know that
\begin{equation*}
\norm{\ave{\Pi_{c}^{\theta}}} = \max_{y, z \in \mathcal{C}_{n}} \norm{B_{yz}} \leq \max_{y, z \in \mathcal{C}_{n}} \norm{B_{yz}'}
\end{equation*}
we obtain the security guarantee of the form presented in~\eqref{eq:security-definition} with
\begin{equation*}
\varepsilon = \sum_{k = 0}^{ \lfloor \delta n \rfloor } {n \choose k} \lambda_{0}^{n - k} \lambda_{1}^{k}.
\end{equation*}
In the noiseless case ($\delta = 0$) we obtain $\varepsilon = \lambda_{0}^{n}$, while for $\delta > 0$ we use the Chernoff bound~\eqref{eq:chernoff-bound-below} for the binomial distribution to get
\begin{equation}
\label{eq:theory-final-result}
\varepsilon \leq \exp \bigg(- \frac{1}{2} \Big( \sqrt{\lambda_{1}} - \frac{\delta}{\sqrt{\lambda_{1}}} \Big)^{2} n \bigg),
\end{equation}
for any $\delta < \lambda_{1}$. It is also clear that the cross-game becomes easy for the dishonest parties if $\delta > \lambda_{1}$. Hence, $\lambda_{1}$ is the error threshold below which the cross-game becomes secure (asymptotically).
It is not clear whether the bit commitment game can be secure for any $\delta > \lambda_{1}$ but if that is the case it cannot be proven using the cross-game reduction.
Finally, we want to connect $\lambda_{1}$ to some measure of complementarity of the states emitted by Alice. We show in Appendix~\ref{app:explicit-diagonalisation} that
\begin{equation*}
\lambda_{1} = \frac{1 - t}{2},
\end{equation*}
where $t = \max_{b, c \in \{0, 1\}} \abs{\braket{\psi_{b}^{0}}{\psi_{c}^{1}}}$. For example for the perfect BB84 states we get $t = 1/\sqrt{2}$ and $\lambda = \lambda_{1} = 1/2 - 1/(2 \sqrt{2}) \approx 0.146$.
\subsubsection{Security for honest Bob}
Since Alice does not receive any information from Bob before the open phase, she remains fully ignorant about the value of his commitment.
\subsection{Fault-tolerant RBC with backreporting}
\label{sec:fault-tolerant-rbc-with-backreporting}
If Alice were to implement the protocol presented in Section~\ref{sec:fault-tolerant-rbc} using a Poisson-type source she would need to set the mean number of photons very high so that the honest Bob detects clicks in the majority of rounds. However, then the probability of multi-photon emissions becomes significant which allows dishonest Bob to cheat. Here, we present a modified protocol, in which Bob is required to backreport the rounds he missed, which closes this security loophole.
\subsubsection{The protocol}
The new protocol admits three parameters: the error allowance, $\delta$, the detection threshold, $\gamma$, and the mean number of photons per pulse emitted by Alice, $\mu$. We will see later that these paremeters determine the trade-off between robustness and security.
\printprotocol{1}
The security for honest Bob follows straightforwardly from Assumption 3. If the probability of detecting a photon does not depend on the basis chosen by Bob then there is no correlation between the valid set and Bob's measurement choice. Therefore, despite having received the valid set Alice remains fully ignorant about Bob's measurement basis.
The remainder of the paper investigates the trade-off between robustness and security (for honest Alice). First, we consider the asymptotic limit (the number of rounds goes to infinity) and we establish simple conditions that the hardware and protocol parameters must satisfy for the protocol to be both robust and secure. These conditions define a region in the space of parameters, which we call the \emph{feasible region}, and essentially tell us how good the equipment of the honest parties must be in order to implement the protocol securely. Finally, we show how to compute explicit security bounds for finite $n$.
\subsubsection{Asymptotic analysis}
In the asymptotic limit all the random variables are concentrated around their average values. Therefore, to see for what values of the parameters that the protocol is in principle feasible it is enough to calculate the relevant averages.
\paragraph*{Robustness.}
In the honest scenario if Bob has received at least one photon he will consider the round valid. For the protocol to go through he must have received at least $\gamma n$ non-vacuum rounds. Hence, what we care about in the asymptotic limit is that the expected number of non-vacuum rounds (as observed by Bob) exceeds the detection threshold, $\sum_{t = 1}^{\infty} \amsbb{E}[N_{r}] > \gamma n$. Expressing the expectation values through probabilities~\eqref{eq:nt-and-expectation-value} gives
\begin{equation*}
\sum_{t = 1}^{\infty} p_{r} > \gamma.
\end{equation*}
Replacing the sum by $1 - p_{0}$ and evaluating it using~\eqref{eq:pr-definition} we get
\begin{equation*}
e^{-\mu \eta} + \gamma < 1.
\end{equation*}
If the first condition is satisfied then we only need to ensure that Bob's error rate is below the threshold. As the probability of making an error (obtaining the wrong outcome despite measuring in the right basis) in each round equals $Q$, the expected Hamming distance between what Alice encoded and what Bob measured also equals $Q$. Therefore, the second condition for robustness is $
Q < \delta$.
\paragraph*{Security for honest Alice}
In the case of dishonest Bob we assume that his fibre and detectors are perfect ($\eta = 1$, $Q = 0$) and, also, let us make the following modification that gives (slightly) more power to Bob but also makes the analysis simpler: we will replace all the vacuum rounds by single-photon emissions. Then, $N_{m} := \sum_{k = 2}^{\infty} N_{k}$, the number of multi-photon emissions, becomes the only relevant parameter. If Bob receives two (or more) photons he will measure the first photon in the $Z$ basis and the second one in the $X$ basis and he will obtain valid answers to both questions (namely opening $0$ and opening $1$). This means that whichever bit he decides to unveil his answer will always be accepted by Alice. Therefore, the multi-photon rounds Bob can ``win'' for free and such rounds do not contribute to security. Hence, dishonest Bob's optimal strategy will be to discard as many single-photon rounds as possible. It is clear that if Bob can discard all of them, he is left with multi-photon emissions only and no security is possible. Therefore, the first condition for security is: the number of multi-photon rounds is lower than Bob's detection threshold
\begin{equation*}
N_{m} < \lceil \gamma n \rceil.
\end{equation*}
After using up the entire backreporting allowance the number of valid rounds is $m = \lceil \gamma n \rceil$ but there are only $\lceil \gamma n \rceil - N_{m}$ (which is now guaranteed to be a positive number) single-photon rounds among them. Therefore, honest Alice thinks that they are playing a standard bit commitment game of $\lceil \gamma n \rceil$ rounds but there is a certain number of multi-photon ones, which Bob can win ``for free''. Hence, he can use his ``error allowance'' for the single-photon rounds and the security we achieve is that of playing a game of $\lceil \gamma n \rceil - N_{m}$ rounds with the non-fractional error allowance of $\delta \gamma n$, which gives the effective (fractional) error allowance of
\begin{equation*}
\delta' = \frac{\delta \gamma n}{\lceil \gamma n \rceil - N_{m}}.
\end{equation*}
In the proof (Section~\ref{sec:security-proof}) we find that we can only prove security if the effective (fractional) error allowance, $\delta'$, satisfies
\begin{equation*}
\delta' < \lambda_{1},
\end{equation*}
where $\lambda$ measures how complementary the bases used by Alice are. Hence, in our case we require
\begin{equation}
\label{eq:security-bound}
\delta \gamma n < (\lceil \gamma n \rceil - N_{m}) \lambda_{1}.
\end{equation}
In the asymptotic limit we look at the expectation value:
\begin{equation*}
\amsbb{E}[N_{m}] = [1 - e^{-\mu} (1 + \mu)] n,
\end{equation*}
which applied to~\eqref{eq:security-bound} gives
\begin{equation*}
e^{-\mu} (1 + \mu) + (1 - \frac{\delta}{\lambda_{1}}) \gamma > 1.
\end{equation*}
\paragraph*{Asymptotically feasible region.}
The asymptotically feasible region is defined as the region in the space of parameters that satisfies the robustness and security constraints, namely:
\begin{gather*}
e^{-\mu \eta} + \gamma < 1,\\
Q < \delta,\\
e^{-\mu} (1 + \mu) + (1 - \frac{\delta}{\lambda_{1}}) \gamma > 1.
\end{gather*}
It is clear that the first two inequalities (the robustness conditions) can be assumed to be equalities, which fixes $\delta$ and $\gamma$ and leaves us with only one, but fairly complicated, condition:
\begin{equation*}
e^{-\mu} (1 + \mu) + (1 - \frac{Q}{\lambda_{1}}) (1 - e^{-\mu \eta}) > 1.
\end{equation*}
This expression allows to verify whether for devices of given quality (quanitified by $\lambda_{1}$, $Q$ and $\eta$) there exists $\mu$ that makes the protocol both robust and secure.
\subsubsection{Finite-size effects}
The asymptotic analysis is relevant as $n \to \infty$ but in any practical scenario the number of rounds will be finite. Alice and Bob, having performed the protocol, will want explicit security guarantees (as a function of $n$). Note that robustness is verified experimentally, hence, there is no need to calculate it and security for honest Bob is perfect by assumption. Therefore, we are only left with security for honest Alice.
Let $E$ denote the event that Bob wins the cross-game. As shown in Section~\ref{sec:security-proof}, $\Pr[E]$ is an upper bound on the security parameter of the bit commitment protocol. Conditioning on the number of multi-photon emissions yields
\begin{equation}
\Pr[E] = \sum_{k = 0}^{n} \Pr[E | N_{m} = k] \Pr[N_{m} = k].
\end{equation}
Equation~\eqref{eq:theory-final-result} allows us to bound $\Pr[E | N_{m} = k]$ as long as the number of multi-photon emissions is below the threshold, $k_{t} = \gamma n (1 - \delta / \lambda_{1})$. For $k < k_{t}$ we have
\begin{equation}
\Pr[E | N_{m} = k] \leq \exp \bigg(- \frac{1}{2} \Big( \sqrt{(\gamma n - k) \lambda_{1}} - \frac{\delta \gamma n}{\sqrt{(\gamma n - k) \lambda_{1}}} \Big)^{2} \bigg),
\end{equation}
whereas for $k \geq k_{t}$ there is no security and the trivial bound, $\Pr[E | N_{m} = k] \leq 1$, is the best we can hope for. Noting that the random variable $N_{m}$ is Poisson-distributed with the parameter $p_{m} = 1 - e^{-\mu} (1 + \mu)$ allows us to evaluate the bound numerically.
\section{A simple maximization problem}
\label{app:simple-maximization-problem}
Suppose one is given two coins and is required to make $n$ independent coin flips. The goal is to maximize the probability of guessing correctly at least a certain fraction of the outcome string. Which coin should be used in every round and what is the optimal guess? Intuitively, it is clear that one should pick the more biased coin and always guess its more probable outcome. For completeness we provide a rigorous proof of this statement.
For $b, c \in \{0, 1\}$ let $\lambda_{b}^{c} \in [0, 1]$ be such that $\sum_{b} \lambda_{b}^{c} = 1$. In other words, $b$ corresponds to the outcome, while $c$ tells us which coin has been used. Let $y \in \mathcal{C}_{n}$ be the string denoting which coin was used in each round and let $X$ be the random variable corresponding to the string of outcomes. Then for $x \in \mathcal{C}_{n}$
\begin{equation*}
\Pr[X = x] = \prod_{k = 1}^{n} \lambda_{x_{k}}^{y_{k}}.
\end{equation*}
For an arbitrary string $z \in \mathcal{C}_{n}$ the probability of being correct on at least $n (1 - \delta)$ position equals
\begin{equation*}
\Pr[\dham(X, z) \leq \delta] = \sum_{\substack{x \in \mathcal{C}_{n}\\ \dham(x, z) \leq \delta}} \prod_{k = 1}^{n} \lambda_{x_{k}}^{y_{k}} = \sum_{\substack{x \in \mathcal{C}_{n}\\ \wham(x) \leq \delta}} \prod_{k = 1}^{n} \lambda_{x_{k} \oplus z_{k}}^{y_{k}}.
\end{equation*}
Now, we want to see what values of $y, z \in \mathcal{C}_{n}$ maximize this expression. Let us write down the summation corresponding to the $j$-th bit explicitly
\begin{equation*}
\sum_{\substack{x \in \mathcal{C}_{n}\\ \wham(x) \leq \delta}} \prod_{k = 1}^{n} \lambda_{x_{k} \oplus z_{k}}^{y_{k}} = \lambda_{z_{j}}^{y_{j}} \Big[ \sum_{\substack{x \in \mathcal{C}_{n - 1}\\ \wham(x) \leq n \delta/(n - 1)}} \prod_{k \neq j} \lambda_{x_{k} \oplus z_{k}}^{y_{k}} \Big] + (1 - \lambda_{z_{j}}^{y_{j}}) \Big[ \sum_{\substack{x \in \mathcal{C}_{n - 1}\\ \wham(x) \leq (n \delta - 1)/(n - 1)}} \prod_{k \neq j} \lambda_{x_{k} \oplus z_{k}}^{y_{k}} \Big].
\end{equation*}
As the first square bracket is never smaller than the second one (because all the terms counted in the second bracket are also counted in the first one) we should always maximize $\lambda_{z_{j}}^{y_{j}}$. This observation is always true, regardless of the choices made for other rounds, and it applies to every round in the same way. Therefore, if $\lambda_{0} = \max_{b, c} \lambda_{b}^{c}$ and $\lambda_{1} = 1 - \lambda_{0}$ we find
\begin{equation*}
\max_{z \in \mathcal{C}_{n}} \Pr[\dham(X, z) \leq \delta] = \sum_{\substack{v \in \mathcal{C}_{n}\\ \wham(v) \leq \delta}} \prod_{k = 1}^{n} \lambda_{v_{k}} = \sum_{k = 0}^{ \lfloor \delta n \rfloor } {n \choose k} \lambda_{0}^{n - k} \lambda_{1}^{k},
\end{equation*}
which recovers the intuitively correct solution.
\section{Explicit diagonalisation of a mixture of two pure states}
\label{app:explicit-diagonalisation}
Let $\ket{\psi_{0}}$ and $\ket{\psi_{1}}$ be two pure states such that $\braket{\psi_{0}}{\psi_{1}} = c e^{i \phi}$. Then the eigenvalue decomposition of $\rho = \frac{1}{2} (\ketbraq{\psi_{0}} + \ketbraq{\psi_{1}})$ is
\begin{equation*}
\rho = \lambda_{+} \ketbraq{e_{+}} + \lambda_{-} \ketbraq{e_{-}},
\end{equation*}
where
\begin{gather*}
\lambda_{\pm} = \frac{1 \pm c}{2},\\
\ket{e_{\pm}} = \frac{1}{\sqrt{2 (1 + c)}} \Big[\ket{\psi_{0}} \pm e^{-i \phi} \ket{\psi_{1}} \Big].
\end{gather*}
\section{Chernoff bounds for the binomial distribution \cite{chernoff52}}
Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent random variables taking on values 0 or 1. Then let $X = \sum_{i = 1}^{n} X_{i}$ and $\mu$ be the expectation value of $X$. Then for any $\delta > 0$ the following inequalities hold.
\begin{align*}
\Pr[X < (1 - \delta) \mu] &< \bigg(\frac{e^{- \delta}}{(1 - \delta)^{1 - \delta}} \bigg)^{\mu} \leq \exp \bigg(- \frac{\mu \delta^{2}}{2} \bigg), \nonumber\\
\Pr[X > (1 + \delta) \mu] &< \bigg(\frac{e^{\delta}}{(1 + \delta)^{1 + \delta}} \bigg)^{\mu} \leq \exp \bigg(- \frac{\mu \delta^{2}}{3} \bigg).
\end{align*}
The bounds can be expressed more conveniently as
\begin{align}
\label{eq:chernoff-bound-below}
\Pr[X < s] &< \exp \bigg(- \frac{1}{2} \Big( \sqrt{\mu} - \frac{s}{\sqrt{\mu}} \Big)^{2} \bigg),\\
\label{eq:chernoff-bound-above}
\Pr[X > s] &< \Big( \frac{\mu}{s} \Big)^{s} e^{s - \mu}.
\end{align}
\section{Remarks on Quantum Advantage} \label{quantumadvantage}
\subsection{Remarks 1}
Note that our protocol requires no communication from $A_1$ or $A_2$
at commitment or unveiling. These agents need
trusted spacetime coordinates and reliable classical data storage, but
do not need secure sites to keep data secret before or during the
protocol: Alice may keep the secret data about her quantum
states at some other secure site for later cross-checking.
No unconditionally secure classical bit commitment
protocol (relativistic or otherwise) allows this freedom, which
is potentially valuable in the technologically asymmetric case where Alice is less able
than Bob to maintain a network of secure sites.
Conversely, if Alice shares the quantum state information securely with
$A_1$ and $A_2$ and they keep it securely, Bob's cheating possibilities are
quite constrained.
If Bob cheats at commitment, i.e. $B_1$ and $B_2$ send different bit
values for $b'$, he will be exposed as soon as $A_1$ and $A_2$ crosscheck, whether or not
he unveils. If they send the same $b'$ and Bob unveils at any time
between $t_c$ and $t_c + (d/2c)$, but tries to cheat at this stage,
at least one of the $A_i$ will learn instantly --
without needing to cross-check -- since he must send inconsistent information about
$b$ to at least one of them. Again, no unconditionally secure
classical bit commitment protocol has this last feature,
which allows (in a relativistically appropriate sense) instant action
to be taken in the event of cheating at the unveiling stage.
\subsection{Remarks 2}
In the main part of our manuscript, we already briefly touched upon
different security models \cite{Kent2012b,Kaniewski2013a}.
In particular, one can see \cite{Kaniewski2013a} our two site protocol as
achieving security in a setting where communication is not allowed
during certain phases of the protocol. For a bit commitment protocol,
one can thereby distinguish several natural phases, namely the commit
phase, the wait phase, and finally the open and verification phase.
One can then ask whether security is possible if communication is not
possible in any of these phases.
This model is interesting because, while imposing minimum communication
constraints necessary to evade the impossibility
result~\cite[Sec.~III.A]{Kaniewski2013a}, it is also sufficiently
strong to allow for secure quantum bit commitment. One can show that
no classical protocol implemented in this model can be
secure~\cite[Sec.~IV.B]{Kaniewski2013a}. From the above description
of the setup with multiple agents occupying different locations (see
Fig.~\ref{protocols}-b), we can readily deduce that the communication
restraints are satisfied. However, the setup enforces even stronger
communication constraints.
For an in-depth discussion and
illustrations concerning the different phases of bit commitment and
communication models we refer to~\cite{Kaniewski2013a}.
In~\cite{Kaniewski2013a} it was shown that there is a strict quantum
advantage when we take the minimum number of phases in which
communication is not allowed, namely the wait and open phase.
Communication during the open phase, however, was possible. As
mentioned before, the significant distance between Singapore and
Geneva naturally guarantees such a lack of communication and thus
enables secure bit commitment.
However, since of course the locations of our experimental labs are
fixed to be in Geneva and Singapore, the distance also guarantees a
lack of communication already during the commit phase itself. This
scenario, when communication is never allowed, has long been studied
in computer science~\cite{benor:bc}, where in particular it is known
that there exists a classical protocol for bit commitment that is
secure as long as communication is \emph{never} possible at all (see
e.g.~\cite{Kaniewski2013a} for further background). Similar protocols
were derived using relativistic assumptions~\cite{Kent1999,Kent2005}, whose first
round is closely related to~\cite{benor:bc}.
The classical protocol of~\cite{benor:bc} is secure against all
quantum attacks~(\cite{Kent2005}; see also ~\cite{bcThesis}),
essentially because a successful quantum attack would imply
the possibility of communication between the separate sites.
However, these classical protocols would not be
secure even against classical attacks if communication were possible during the commit phase, and so
the two site quantum protocol implemented here would offer a theoretical advantage in such a
setting.
It is a very interesting open question, whether there exists an
natural experimentally implementable physical scenario in which
communication during the commit phase is allowed, but
communication during the wait and open phase is forbidden.
Looking at the protocol
of~\cite{benor:bc} one could imagine that supplementing special
relativity with the assumption that only a finite amount of
communication is possible within a given time frame, and that an
infinitesimally small point in space can, by the holographic
priniciple, not store an arbitrary amount of randomness\,---\,required
to execute the protocol of~\cite{benor:bc}\,---\,can possibly lead to
such a scenario. However, these are only clues we can derive from a
particular classical protocol~\cite{benor:bc} and they do not by itself answer this interesting open question.
\section{Extension for three agents}
\label{three-agents}
Suppose that Bob sites his third agent, $B_0$, equidistant from $B_1$ and $B_2$ on the
shorter arc of a great circle joining them and preshare
$b$ and $r^{(b)}$ among all three agents, while Alice
locates her third agent, $A_0$, right next to $B_0$. Then, instead of
$B_1$ and $B_2$ committing the bit $a = b \oplus b'$, $B_0$
can do so by sending $b'$ to $A_0$ at $t_c$, while $B_1$ and $B_2$
unveil by sending $b$ and $r^{(b)}$ at $t_c + d/(2c)$.
This allows $B_0$ to commit to a bit as soon as he learns it,
without presharing it with $B_1$ and $B_2$, while Alice is
guaranteed that Bob's agents have collectively committed him
from $t_c$ until $t_c + d/(2c)$.
This freedom to commit bits in real time could be made more practical
if the $B_i$ preshare a large number of quantumly generated random bits $b_j$ and
commitment data $r^{(b_j )}$ and if $B_1$ and $B_2$ release
these in rapid sequence at coordinated times pre-agreed with $B_0$.
$B_0$ can then commit a bit $a= b_j \oplus b'$ at any time
of his choice, by sending $A_0$ both $b'$ and the value $j$
(chosen so that the unveiling times of $r^{(b_j)}$ guarantee
commitment for roughly $t_c + (d/ (2c))$.
Adding a third site adds logistical complexity but no significant
technological challenge beyond those addressed in our implementation, which
thus demonstrates the practicality of secure real
time relativistic quantum commitment of bits acquired at a single location.
\end{document}
|
1,108,101,565,686 | arxiv | \section{Introduction}\label{Intro}
Normal polytopes show up in various contexts, including algebraic geometry (via toric varieties), integer programming (integral Carath\'eodory property), combinatorial commutative
algebra (quadratic Gr\"ob\-ner bases of binomial ideals), and geometric combinatorics (Ehr\-hart theory). Section \ref{ClassesofPolytopes} below introduces all relevant classes of polytopes and provides background material which also serves as motivation.
The weakest of the general properties, distinguishing a polytope from random lattice polytopes, is \emph{very ampleness}. In geometric terms, very ample polytopes
correspond to normal projective but not necessarily projectively normal embeddings of toric varieties. The finest of the algebraic properties a lattice polytope can have is
the \emph{Koszul} property, which means that the polytope in question is homologically wonderful; the name draws its origin from Hilbert's Syzygy Theorem. It says that (the algebra of) a unit lattice simplex, i.e., the polynomial algebra with the vertices of the simplex as variables, is Koszul with the standard Koszul complex on the vertices as a certificate.
Koszul algebras were discovered by topologists in the early 1970s and, with the advent of powerful theoretical, computational, and practical tools, the topic of
Koszul polytopal algebras became a popular topic in algebraic combinatorics starting from the early 1990s.
By contrast, the very ample polytopes, formally treatable
within the framework of elementary additive number theory, came under spotlight more recently. Partly this can be explained by the traditional inclusion of the
normality property in the definition of toric varieties, so a property weaker than normality seemed unnatural. The very question of existence of very ample non-normal polytopes became interesting only in the late 1990s.
There is a whole panorama of interesting classes of lattice polytopes sandwiched between the very ample and Koszul ones, but we are not delving into them. In this
paper, by exploring a simple polytopal construction called \emph{lattice segmental fibrations} (Definition \ref{fibration} below), we detect very ample lattice polytopes, arbitrarily far away from the normality property (Theorem \ref{highgaps}), and construct a new large class of Koszul polytopes (Theorem \ref{koszulclass}), containing many examples of smooth polytopes. A subclass of Koszul polytopes was described in \cite{DHZ} and in Section \ref{Koszul} we explain that the argument there works for our general class as well.
\bigskip\noindent\emph{Acknowledgement.} We thank Winfried Bruns and Serkan Ho\c{s}ten for helpful comments and providing us with an invaluable set
of examples of very ample polytopes. We also thank Milena Hering for pointing out the overlap of our work with \cite{DHZ} and two anonymous referees for helpful
comments. The last author also thanks Mathematisches Forschungsinstitut Oberwolfach for the great working atmosphere and hosting.
\section{Normal, very ample, and Koszul polytopes}\label{ClassesofPolytopes}
In this section we introduce three polytopal classes and give a characterization of the very ampleness condition (Proposition \ref{veryample}), which we were not able to find in the literature in the given generality.
\subsection{Normal polytopes}\label{NormalPolytopes}
A \textbf{(convex) polytope} $\mathcal P$ is the convex hull of a finite subset of ${\Bbb R}^d$. The inclusion-minimal such set
$\operatorname{vert}(\mathcal P)$ consists of the \textbf{vertices} of $\mathcal P$.
If $\operatorname{vert}(\mathcal P)$ is affinely independent, $\mathcal P$ is a \textbf{simplex}.
A polytope $\mathcal P\subset{\Bbb R}^d$ is \textbf{lattice} if $\operatorname{vert}(\mathcal P$)$\subset{\Bbb Z}^d$.
For a lattice polytope $\mathcal P\subset{\Bbb R}^d$ we denote by $\L(\mathcal P)$ the subgroup of ${\Bbb Z}^d$ that is affinely generated by the lattice points in $\mathcal P$, i.e.,
$$
\L(\mathcal P)=\sum_{\mathbf x, \, \mathbf y\in \mathcal P\cap{\Bbb Z}^d}{\Bbb Z}(\mathbf x-\mathbf y)\subset{\Bbb Z}^d.
$$
A lattice polytope $\mathcal P\subset\mathbb{R}^d$ is called
\begin{enumerate}[(a)]
\item \textbf{integrally closed} if for every natural number $c$ and every point $\mathbf z\in c \, \mathcal P\cap{\Bbb Z}^d$ there exist
$\mathbf x_1,\mathbf x_2,\ldots,\mathbf x_c\in \mathcal P\cap{\Bbb Z}^d$ such that $\mathbf x_1+\mathbf x_2+\dots+\mathbf x_c=\mathbf z$;
\item \textbf{normal} if for some (equivalently, every) point $\mathbf t\in \mathcal P\cap \mathbb{Z}^d$ the following condition is satisfied: for every natural number
$c$ and every point $\mathbf z\in c \, \mathcal P\cap(c\mathbf t+\L(\mathcal P))$ there exist $\mathbf x_1,\mathbf x_2,\ldots,\mathbf x_c\in \mathcal P\cap{\Bbb Z}^d$ such that $\mathbf x_1+\mathbf x_2+\dots+\mathbf x_c=\mathbf z$.
\end{enumerate}
\noindent(For $\mathbf t$ in (b), we have $\mathcal P\cap(\mathbf t+\L(\mathcal P))=\mathcal P\cap{\Bbb Z}^d$.)
One easily observes that a lattice polytope $\mathcal P\subset{\Bbb R}^d$ is integrally closed if and only if it
is normal and $\L(\mathcal P)$ is a direct summand of ${\Bbb Z}^d$. In particular, a normal polytope $\mathcal P$ becomes full-dimensional
and integrally closed if one changes the ambient space ${\Bbb R}^d$ to ${\Bbb R}\L(\mathcal P)$ and the lattice of reference to $\L(\mathcal P)$.
This explains why the difference between \emph{normal} and \emph{integrally closed} is often blurred in the literature.
An \textbf{empty} simplex (i.e., a lattice simplex that contains no lattice points besides its vertices) of large volume
provides an example of a normal but not integrally closed polytope. The classification of empty simplices is an
active area of research.
One class of empty simplices is important for this paper: a lattice $n$-simplex $\conv(\mathbf x_0,\mathbf x_1,\ldots,\mathbf x_n)\subset{\Bbb R}^d$ is \textbf{unimodular} if $\{\mathbf x_1-\mathbf x_0,\ldots,\mathbf x_n-\mathbf x_0\}$ is a part of a basis of ${\Bbb Z}^d$. Unimodular
simplices are integrally closed, and if a lattice polytope is a union of unimodular simplices, i.e., admits a \textbf{unimodular cover}, then it is integrally closed. Not
all lattice 4-polytopes with a unimodular cover admit triangulations into unimodular simplices---an example \cite[Proposition 1.2.4(c)]{BrGuTr} has been shown by
effective methods to have a unimodular cover---and not all integrally closed 5-polytopes are covered by unimodular simplices~\cite{uncovered}.
\subsection{Point configurations and Proj}\label{Proj}
Our next two classes of polytopes are best explained by providing an algebraic context.
We denote the \textbf{conical hull} of $X\subset{\Bbb R}^d$ by ${\Bbb R}_{\ge0}X$. An \textbf{affine monoid} is a finitely
generated additive submonoid of ${\Bbb Z}^d$ for some $d\in{\Bbb N}$. For a finite subset $X=\{\mathbf x_1,\ldots,\mathbf x_n\}\subset{\Bbb Z}^d$ the affine monoid generated by $X$ will be denoted by
\[
{\Bbb Z}_{\ge0}X={\Bbb Z}_{\ge0}\, \mathbf x_1+\cdots+{\Bbb Z}_{\ge0}\, \mathbf x_n \, .
\]
The group of differences of an affine monoid $M\subset{\Bbb Z}^d$ (i.e., the subgroup of ${\Bbb Z}^d$ generated by $M$) is denoted by $\operatorname{gp}(M)$, and the \textbf{normalization} of
$M$ is the following affine monoid
$$
\overline M:=\operatorname{gp}(M)\cap{\Bbb R}_{\ge0}M=\{\mathbf x\in\operatorname{gp}(M)\ |\ n \, \mathbf x\in M\ \text{for some}\ n\in{\Bbb N}\} \, .
$$
For a field ${\Bbb K}$ and an affine monoid $M\subset{\Bbb Z}^d$, the normalization of the monoid ring ${\Bbb K}[M]$ (i.e., its integral closure in the field of fractions) equals
${\Bbb K}[\overline M]$ (see, e.g., \cite[Section 4.E]{Kripo}). The monoid algebra ${\Bbb K}[M]$ can be thought of as the monomial subalgebra of ${\Bbb K}[X_1^{\pm1},\ldots,X_d^{\pm1}]$ spanned by the Laurent monomials with exponent vectors in~$M$.
We will refer to a finite subset $\operatorname{\mathcal A}\subset{\Bbb Z}^d$ as a \textbf{point configuration}. For a point configuration $\operatorname{\mathcal A}$ and a field ${\Bbb K}$, we let ${\Bbb K}[\operatorname{\mathcal A}]$ denote the monoid ${\Bbb K}$-algebra of the affine monoid
\[
M_{\operatorname{\mathcal A}}:=\sum_{\mathbf x \in \operatorname{\mathcal A}}{\Bbb Z}_{\ge0}(\mathbf x,1)\subset{\Bbb Z}^{d+1}.
\]
It is natural to call the last coordinate of a point $\mathbf y \in \overline{M_{\operatorname{\mathcal A}}}$ the \textbf{height} of~$\mathbf y$.
A \textbf{grading} on an affine monoid $M$ is a partition $M=\bigcup_{i \in {\Bbb Z}_{\ge0}}M_i$, where $M_0=\{0\}$ and $M_i+M_j\subset
M_{i+j}$. For a field ${\Bbb K}$ and a graded affine monoid $M$, the monoid algebra ${\Bbb K}[M]$ is graded in the natural way, where the $0$th component equals ${\Bbb K}$.
The monoids of type $M_{\operatorname{\mathcal A}}$, where $\operatorname{\mathcal A}$ is a point configuration, are naturally graded with respect to the last
coordinate, and they are generated in degree 1. In particular, for a field ${\Bbb K}$ and a point configuration $\operatorname{\mathcal A}$, the graded algebra ${\Bbb K}[\operatorname{\mathcal A}]$ is \textbf{homogeneous}:
\[
{\Bbb K}[\operatorname{\mathcal A}]={\Bbb K}\oplus A_1\oplus A_2\oplus\cdots,\qquad {\Bbb K}[\operatorname{\mathcal A}]={\Bbb K}[A_1] \, , \qquad A_1={\Bbb K}(\operatorname{\mathcal A},1) \, .
\]
For a lattice polytope $\mathcal P\subset{\Bbb R}^d$, the corresponding \textbf{polytopal monoid} is defined by $M_\mathcal P:=M_{\operatorname{\mathcal A}}$ where $\operatorname{\mathcal A}=\mathcal P\cap{\Bbb Z}^d$.
We denote $\L(\operatorname{\mathcal A}):=\sum_{\mathbf x,\mathbf y\in\operatorname{\mathcal A}}{\Bbb Z}(\mathbf x-\mathbf y)\subset{\Bbb Z}^d$. Thus, using the previous notation, for a lattice polytope $\mathcal P\subset{\Bbb Z}^d$, we have $\L(\mathcal P)=\L(\mathcal P\cap{\Bbb Z}^d)$.
For an algebraically closed field ${\Bbb K}$ and a point configuration $\operatorname{\mathcal A}\subset{\Bbb Z}^d$, we have the projective variety
\[
X_{\operatorname{\mathcal A}}:=\operatorname{Proj}({\Bbb K}[\operatorname{\mathcal A}])\subset\PP^{N-1}_{\Bbb K} \, ,
\]
where $N=\#\operatorname{\mathcal A}$, and the resulting very ample line bundle $\operatorname{\mathcal L}_{\operatorname{\mathcal A}}\in\operatorname{Pic}(X_{\operatorname{\mathcal A}})$ (see, e.g., \cite[Ch.~II,~\S7]{Algeo}).
The following proposition is a semi-folklore result.
\begin{proposition}\label{veryample}
Let ${\Bbb K}$ be an algebraically closed field, $\operatorname{\mathcal A}\subset{\Bbb Z}^d$ a point configuration, and $\mathcal P:=\conv(\operatorname{\mathcal A})$. Then the following are equivalent:
\begin{enumerate}[{\rm (a)}]
\item $X_{\operatorname{\mathcal A}}$ is normal;
\item $\bigoplus_{i\ge0}H^0(X_{\operatorname{\mathcal A}},\operatorname{\mathcal L}_{\operatorname{\mathcal A}}^{\otimes i})={\Bbb K}\big[{\Bbb R}_{\ge0}(\mathcal P,1)\cap\operatorname{gp}(M_{\operatorname{\mathcal A}})\big]$;
\item ${\Bbb R}_{\ge0}(\mathcal P-\mathbf v)\cap\L(\operatorname{\mathcal A})={\Bbb Z}_{\ge0}(\operatorname{\mathcal A}-\mathbf v)$ for every $\mathbf v\in\operatorname{vert}(\mathcal P)$;
\item $\#({\Bbb R}_{\ge0}(\mathcal P,1)\cap\operatorname{gp}(M_{\operatorname{\mathcal A}}))\setminus M_{\operatorname{\mathcal A}}<\infty$;
\item $X_{\operatorname{\mathcal A}}$ is a projective toric variety.
\end{enumerate}
\end{proposition}
\begin{proof} (a)$\Longleftrightarrow$(b) holds because the left-hand side of (b) is the normalization of
${\Bbb K}[\operatorname{\mathcal A}]$---a general fact for a normal projective variety and a very ample line bundle on it \cite[Ch. II, Exercise 5.14]{Algeo}, while the right-hand side is ${\Bbb K}[\overline{M_{\operatorname{\mathcal A}}}]$.
\medskip \noindent (a)$\Longleftrightarrow$(c) follows from the open affine cover
$$
X_{\operatorname{\mathcal A}}=\bigcup_{\mathbf v \in \operatorname{vert}(\mathcal P)} \Spec({\Bbb K}[{\Bbb Z}_{\ge0}(\operatorname{\mathcal A}-\mathbf v)])
$$
\cite[Proposition 2.1.9]{TORICvarieties} and the fact that, for every vertex $\mathbf v\in \mathcal P$, the normalization of
${\Bbb K}[{\Bbb Z}_{\ge0}(\operatorname{\mathcal A}-\mathbf v)]$ equals ${\Bbb K}[{\Bbb R}_{\ge0}(\mathcal P-\mathbf v)\cap\L(A)]$ \cite[Section 4.E]{Kripo}.
The aforementioned affine cover, in turn, follows from the standard
dehomogenization with respect to the degree-1 generating set $(\operatorname{\mathcal A},1)\subset{\Bbb K}[\operatorname{\mathcal A}]$ and the observation that
the affine charts $\Spec({\Bbb K}[{\Bbb Z}_{\ge0}(\operatorname{\mathcal A}-\mathbf x)])$ with $\mathbf x\in\operatorname{\mathcal A}\setminus\operatorname{vert}(\mathcal P)$ are redundant: if $\mathbf x\in\operatorname{int}(F)$ for a positive-dimensional face $F\subset \mathcal P$, then we have
\begin{align*}
{\Bbb K}[{\Bbb Z}_{\ge0}(\operatorname{\mathcal A}-\mathbf x)]={\Bbb K}[{\Bbb Z}_{\ge0}(\operatorname{\mathcal A}-\mathbf z)-{\Bbb Z}_{\ge0}((\operatorname{\mathcal A}\cap F)-\mathbf z)] \, ,
\end{align*}
where $\mathbf z$ is any vertex of $F$ and the right-hand side is the localization with respect to the monomial multiplicative subset
${\Bbb Z}_{\ge0}((\operatorname{\mathcal A}\cap F)-\mathbf z)\subset {\Bbb K}[{\Bbb Z}_{\ge0}(\operatorname{\mathcal A}-\mathbf z)]$.
\medskip \noindent (a)$\Longleftrightarrow(d)$ This is Theorem 13.11 in \cite{STURMpol}.
\medskip \noindent (a)$\Longleftrightarrow$(e) This is, essentially, a matter of convention: some sources (e.g., \cite{Kripo,Toric}) include the normality in the
definition of a toric variety, whereas the recent comprehensive reference in the field \cite{TORICvarieties} relaxes this assumption.
\end{proof}
\subsection{Very ample polytopes}\label{VeryAmplePolytopes}
In view of Proposition \ref{veryample}, it is natural to call a point configuration $\operatorname{\mathcal A}\subset{\Bbb Z}^d$ \textbf{very
ample} if it satisfies the equivalent conditions in the proposition. If $\operatorname{\mathcal A}$ is very ample, the elements of $\overline{M_{\operatorname{\mathcal A}}}\setminus M_{\operatorname{\mathcal A}}$ will be called \textbf{gaps}, and the maximal possible degree of a gap will be denoted by $\gamma(\operatorname{\mathcal A})$, i.e.,
$$
\gamma(\operatorname{\mathcal A})=
\begin{cases}
&0,\ \text{if}\ (M_{\operatorname{\mathcal A}})_i=(\overline{M_{\operatorname{\mathcal A}}})_i\ \text{for all}\ i\in{\Bbb Z}_{\ge0},\\
&\max\left(i\ |\ (M_{\operatorname{\mathcal A}})_i\subsetneq(\overline{M_{\operatorname{\mathcal A}}})_i\right)\in{\Bbb N},\ \text{otherwise}.
\end{cases}
$$
For $\operatorname{\mathcal A}$ very ample, $\gamma(\operatorname{\mathcal A})$ is a higher-dimensional analog of the \emph{Frobenius number of a numerical
monoid} (see, e.g., \cite{DIOPHANTINE}): there are no gaps in the degrees $>\gamma(\operatorname{\mathcal A})$. The analogy is limited though---the monoid $M_{\operatorname{\mathcal A}}$ is generated in degree
1, whereas the classical Frobenius number of a numerical monoid $M\subset{\Bbb Z}_{\ge0}$ is not defined exactly in the situation when $M$ is generated in degree 1, i.e., when $M={\Bbb Z}_{\ge0}$.
Since $\gamma(\operatorname{\mathcal A})$ depends only on the monoid $M_{\operatorname{\mathcal A}}$ and not on how $\operatorname{\mathcal A}$ sits in ${\Bbb Z}^d$, without loss of
generality we can assume $\L(\operatorname{\mathcal A})={\Bbb Z}^d$.
This is achieved by changing the original ambient lattice ${\Bbb Z}^d$ to $\L(\operatorname{\mathcal A})$. The upshot of this assumption is that the Hilbert function of ${\Bbb K}[\overline{M_{\operatorname{\mathcal A}}}]$ is now the \textbf{Ehrhart polynomial} of $\mathcal P=\conv(\operatorname{\mathcal A})$, i.e.,
$$
\dim_{\Bbb K}({\Bbb K}[\overline{M_{\operatorname{\mathcal A}}}])_j)= \# \left( j\mathcal P \cap {\Bbb Z}^d \right) =: \operatorname{ehr}_\mathcal P(j),\qquad j\in{\Bbb N}.
$$
Next we observe that $\gamma(\operatorname{\mathcal A})$ can be made arbitrarily large by varying $\operatorname{\mathcal A}$ without changing $X_{\operatorname{\mathcal A}}$. In
fact, for the `rarified' very ample configurations
$$
\operatorname{\mathcal A}_c=\bigcup_{\mathbf v \in \operatorname{vert}(\mathcal P)}((c-1)\mathbf v+\operatorname{\mathcal A}),\quad c\in{\Bbb N},
$$
we have
\begin{align*}
&\conv(\operatorname{\mathcal A}_c)=c\cdot\conv(\operatorname{\mathcal A})\, , \\
&X_{\operatorname{\mathcal A}_c}\cong X_{\operatorname{\mathcal A}},\quad c\in{\Bbb N} \, , \\
&\gamma(\operatorname{\mathcal A}_c)\to\infty\ \ \text{as}\ \ c\to \infty \, .
\end{align*}
These observations explain why one needs to restrict to very ample polytopes in the quest for upper bounds for $\gamma(\operatorname{\mathcal A})$:
a \textbf{very ample} polytope is a lattice polytope $\mathcal P\subset{\Bbb R}^d$ such that the point configuration
$\mathcal P\cap{\Bbb Z}^d\subset{\Bbb Z}^d$ is very ample and $\L(\mathcal P)$ is a direct summand of ${\Bbb Z}^d$.
For a very ample polytope $\mathcal P\subset{\Bbb R}^d$ we denote $\gamma(\mathcal P)=\gamma(\mathcal P\cap{\Bbb Z}^d)$. The number $\gamma(\mathcal P)$ measures how far the embedding
$X_{\mathcal P\cap{\Bbb Z}^d}\hookrightarrow\PP^{N-1}_{\Bbb K}$, where again $N=\#(\mathcal P\cap{\Bbb Z}^d)$, is from being projectively normal. Alternatively, the
number $\gamma(\mathcal P)$ can be defined as the maximal degree beyond which the Hilbert function of ${\Bbb K}[\mathcal P\cap{\Bbb Z}^d]$ equals
its Hilbert polynomial.
For a lattice polytope $\mathcal P\subset{\Bbb R}^d$ the smallest generating set of the normalization
$\overline{M_{\mathcal P\cap{\Bbb Z}^d}}$ is called the \textbf{Hilbert basis}. It is
concentrated in degrees $<\dim(\mathcal P)$ \cite[Theorem 2.52]{Kripo}. For $\mathcal P$ very ample, the elements of the Hilbert basis of $\overline{M_{\mathcal P\cap{\Bbb Z}^d}}$
represent gaps in $\overline{M_{\mathcal P\cap{\Bbb Z}^d}}$. One might expect that, similarly, there is a dimensionally uniform upper bound for the degrees of \emph{all} gaps.
However, we show in Section \ref{Class} that this is false already in dimension three, even for very ample lattice polytopes with eight lattice points (see
Theorem \ref{highgaps} below).
Our extremal examples suggest that it might be of interest to study the \textbf{gap vector}
$\operatorname{gv}(\mathcal P)$ of a very ample polytope $\mathcal P$, with entries
\[
\operatorname{gv}_k(\mathcal P) := \# \text{ gaps in $M_\mathcal P$ at height $k$,}
\]
stopping at the largest height $\gamma(\mathcal P)$ that contains gaps in $M_\mathcal P$.
As an indication that this might be an interesting concept, we offer a conjecture on unimodality of gap vectors (see
Conjecture \ref{gapvectorconj} below) and verify it for the gap vectors for a
family of polytopes that play a central role in Section~\ref{Class}.
In the proof of Proposition \ref{veryample} we described the monomial affine charts of $\operatorname{Proj}({\Bbb K}[\operatorname{\mathcal A}])$. This description implies
that, for a lattice polytope $\mathcal P\subset{\Bbb R}^d$ with $\L(\mathcal P)$ a direct summand of ${\Bbb Z}^d$, the variety $\operatorname{Proj}({\Bbb K}[M_\mathcal P])$
is smooth if and only if the primitive edge vectors at every vertex of $\mathcal P$ define a part of a basis of ${\Bbb Z}^d$ \cite[Exercise 4.25]{Kripo}.
Correspondingly, one calls such polytopes \textbf{smooth}. Clearly, smooth polytopes are very ample. Much effort went into the study whether $\gamma(\mathcal P)=0$ for a
smooth polytope $\mathcal P$, i.e., whether smooth polytopes are integrally closed. This is the well-known \textbf{Oda question}, still wide open, even in dimension three \cite{mfo}.
\subsection{Koszul polytopes}\label{KoszulPolytopes}
Let ${\Bbb K}$ be a field. A finitely generated graded ${\Bbb K}$-algebra $\Lambda={\Bbb K}\oplus\Lambda_1\oplus\cdots$ is \textbf{Koszul} if the minimal free graded resolution of ${\Bbb K}$ over $\Lambda$ is linear:
\[
\cdots\mathrel{\mathop{\longrightarrow}\Lambda^{\beta_2}}
\mathrel{\mathop{\longrightarrow}^{\partial_2}}\Lambda^{\beta_1}\mathrel{\mathop{\longrightarrow}^{\partial_1}}\Lambda\mathrel{\mathop{\longrightarrow}^{\partial_0}}{\Bbb K}
\mathrel{\mathop{\longrightarrow}}0,\quad\deg(\partial_i)=1,\quad i>0.
\]
The condition $\deg(\partial_1)=1$ is equivalent to $\Lambda$ being homogeneous and the condition $\deg(\partial_1)=\deg(\partial_2)=1$
is equivalent to $\Lambda$ being \textbf{quadratically defined}, i.e.,
\begin{align*}
\Lambda={\Bbb K}[X_1,\ldots,X_N]/(F_1,\ldots,F_n),\quad N=\dim_{\Bbb K}\Lambda_1,
\end{align*}
for some homogeneous quadratic polynomials $F_1,\ldots,F_n$.
When $\Lambda$ is a quadratically defined graded monoid algebra ${\Bbb K}[M]$, then the polynomials
$F_1,\ldots,F_n$ can be chosen to be of the form $m-m'$ for some degree-2 monomials $m,m'\in{\Bbb K}[X_1,\ldots,X_N]$; see \cite[Sections 4.A,B,C]{Kripo} for generalities on monoid algebras.
A well-known sufficient (but in general not necessary) criterion for the Koszul property, already detected in
Priddy's pioneering work on Koszul algebras \cite{Priddy}, is the existence of a quadratic Gr\"obner basis; for a proof using Gr\"obner-bases terminology see, e.g., \cite{Vetter}. Namely, a ${\Bbb K}$-algebra
$\Lambda={\Bbb K}[X_1,\ldots,X_N]/I$, where $I$ is a homogeneous ideal, is Koszul if $I$ admits a
quadratic Gr\"obner basis with respect to some term order on ${\Bbb K}[X_1,\ldots,X_N]$.
Oda's question, mentioned above, corresponds to the degree-$1$ part of \textbf{B{\o}gvad's conjecture}, which claims that for every
smooth lattice polytope $\mathcal P\subset{\Bbb R}^d$, the algebra ${\Bbb K}[{\Bbb R}_{\ge0}(\mathcal P,1)\cap{\Bbb Z}^{d+1}]$ is Koszul.
We call a lattice polytope $\mathcal P$ \textbf{quadratically defined} or \textbf{Koszul} if the graded monoid algebra
${\Bbb K}[\mathcal P\cap{\Bbb Z}^d]$ is quadratically defined or Koszul, respectively, for every field ${\Bbb K}$. The property of being quadratically defined is independent of ${\Bbb K}$,
but whether ${\Bbb K}[\mathcal P\cap{\Bbb Z}^d]$ being Koszul depends on ${\Bbb K}$ is an open question \cite[Question 8.5.6]{Peeva}. In
particular, if Oda's question has a positive answer, then B{\o}gvad's conjecture is equivalent to the claim that smooth polytopes are Koszul.
Examples of Koszul polytopes (or point configurations) include:
\begin{enumerate}[{\rm$\bullet$}]
\item The dilated lattice polytopes $c \, \mathcal P$ for $c\ge\dim \mathcal P$ \cite[Theorem 1.3.3]{BrGuTr}; for
sharper lower bounds for $c$, depending on $\mathcal P$, see \cite[Section 4]{Hering}.
\item Lattice polytopes cut out by root systems of classical type and their Cayley sums \cite{PAYNEroot}; type $A$ was considered before in \cite[Theorem 2.3.10]{BrGuTr}, using different methods; see Example \ref{Example2} below.
\item The non-polytopal point configuration $\operatorname{\mathcal A}=\conv(0,3\mathbf e_1,3\mathbf e_2,3\mathbf e_3)\setminus\{(1,1,1)\}$ \cite{Caviglia}.
\end{enumerate}
Recently, Oda's question and B{\o}gvad's conjecture (and extensions to more general point configurations) have been
attracting considerable interest in the community of algebraic combinatorics \cite{aim,mfo}. For an effective approach to a potential counterexample
to B{\o}gvad's conjecture, see~\cite{BRUNSquest}. The surveys
\cite{CONCA,Froberg,Peeva} include much relevant general background material.
In Section \ref{Koszul}, we derive a new large
class of Koszul polytopes in arbitrary dimensions. In particular, when our examples of very ample 3-polytopes in
Section \ref{Class} below happen to be smooth, then they are normal and Koszul as well.
Note that if $\mathcal P$ is integrally closed (resp.\ normal, very ample, Koszul) then so are the faces of $\mathcal P$; for the Koszul property one uses \cite[Proposition 1.4]{Ohsugi}.
One can also show that if $\mathcal P$ and $\mathcal Q$ are integrally closed (resp.\ normal, very ample, Koszul) then so is $\mathcal P\times\mathcal Q$; the ring ${\Bbb K}[M_{\mathcal P\times\mathcal Q}]$ is the \textbf{Segre product} of ${\Bbb K}[\mathcal P]$ and ${\Bbb K}[Q]$ and the Koszul property transfers from factor algebras to their Segre product \cite[Theorem 2(ii)]{Froberg}. In particular, since there are 3-dimensional non-normal very ample polytopes (see Section \ref{Class}), the direct product with the unit segment $[0,1]$ yields the existence of non-normal very ample polytopes in all dimensions $d\ge3$. Classically, all lattice $d$-polytopes
with $d\le2$ are integrally closed (see, e.g., \cite[Proposition 1.2.4]{BrGuTr}). Moreover, by \cite[Corollary 3.2.5]{BrGuTr} a lattice polygon is Koszul if and only if either it is a unimodular triangle or it has at least $4$ lattice points in the boundary.
\section{Very ample 3-polytopes with gaps of arbitrarily large degrees}\label{Class}
The polytopal construction announced in the introduction and central to this paper is as follows:
\begin{definition}\label{fibration}
An affine map $f:\mathcal P\to \mathcal Q$ between lattice polytopes $\mathcal P\subset{\Bbb Z}^{d_1}$ and $\mathcal Q\subset{\Bbb Z}^{d_2}$ is a \textbf{lattice segmental fibration} if it satisfies the following conditions:
\begin{enumerate}[{\rm (i)}]
\item $f^{-1}(\mathbf x)$ is a lattice segment, i.e., a one-dimensional lattice polytope or a lattice point, for every $\mathbf x\in \mathcal Q\cap{\Bbb Z}^{d_2}$,
\item $\dim(f^{-1}(\mathbf x))=1$ for at least one $\mathbf x\in \mathcal Q\cap{\Bbb Z}^{d_2}$,
\item $\mathcal P\cap{\Bbb Z}^{d_1}\subset\bigcup_{\mathcal Q\cap{\Bbb Z}^{d_2}}f^{-1}(\mathbf x)$.
\end{enumerate}
\end{definition}
\begin{figure}[h!]
\caption{A lattice segmental fibration.}
\vspace{.2in}
\includegraphics[trim = 0mm 1.25in 0mm 1.5in, clip, scale=.2]{Figure1.png}
\end{figure}
It follows from this definition that a lattice segmental fibration $f:\mathcal P\to\mathcal Q$ is a surjective map and we have the isomorphism of groups $\L(\mathcal P)\cong\L(\mathcal Q)\oplus{\Bbb Z}$.
In this section, using certain small lattice segmental fibrations, we show that there is no uniform upper bound for $\gamma(\mathcal P)$ even for 3-dimensional very ample polytopes with a few lattice points.
The following class of 3-polytopes was introduced in \cite[Exercise 2.24]{Kripo}. The first explicit representatives of the class showed up already in Bruns' report
\cite[p.~2290]{mfo}. Let
$I_k=[a_k,b_k]\subset{\Bbb R}$ be lattice segments for $1\le k\le 4$, none of them degenerated to a point. Let
\[
\mathcal P(I_1,I_2,I_3,I_4):=\conv\big((0,0,I_1),(1,0,I_2),(0,1,I_3),(1,1,I_4)\big)\subset{\Bbb R}^3 .
\]
Thus the map
\begin{align*}
\mathcal P(I_1,I_2,I_3,I_4)\to\conv\big((0,0),(1,0),(0,1),(1,1)\big),\quad (x,y,z)\mapsto(x,y),
\end{align*}
is a lattice segmental fibration.
\begin{lemma}\label{Kripoexercise}
\begin{enumerate}[{\rm (a)}]
\item $\mathcal P(I_1,I_2,I_3,I_4)$ is very ample \cite[Exercise 2.24]{Kripo}.
\item $\mathcal P(I_1,I_2,I_3,I_4)$ is smooth if and only if $a_1+a_4=a_2+a_3$ and $b_1+b_4=b_2+b_3$.
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{Kripoexercise}] (a)
Acting by translations and lattice automorphisms we can assume $I_1=[0,b_1]$ and we only need to check that
\begin{equation}\label{veryampleequality}
C\cap{\Bbb Z}^3={\Bbb Z}_{\ge0}(1,0,a_2)+{\Bbb Z}_{\ge0}(0,1,a_3)+{\Bbb Z}_{\ge0}(1,1,a_4)+{\Bbb Z}_{\ge0}\mathbf e_3 \, ,
\end{equation}
where $C={\Bbb R}_{\ge0}\mathcal P(I_1,I_2,I_3,I_4)$ and $\mathbf e_3=(0,0,1)$.
There are two possibilities: either
\begin{align*}
&C={\Bbb R}_{\ge0}\mathbf e_3+{\Bbb R}_{\ge0}(1,0,a_2)+{\Bbb R}_{\ge0}(0,1,a_3),\ \text{or}\\
&C=\big({\Bbb R}_{\ge0}\mathbf e_3+{\Bbb R}_{\ge0}(1,0,a_2)+{\Bbb R}_{\ge0}(1,1,a_4)\big)\ \cup\ \big({\Bbb R}_{\ge0}\mathbf e_3+{\Bbb R}_{\ge0}(0,1,a_3)+{\Bbb R}_{\ge0}(1,1,a_4)\big).
\end{align*}
In the first case, (\ref{veryampleequality}) holds because $\{\mathbf e_3,(1,0,a_2),(0,1,a_3)\}$ is a basis of ${\Bbb Z}^3$:
\[
\det\left[ \begin{array}{ccc}
1&0&a_2\\
0&1&a_3\\
0&0&1
\end{array} \right]=1.
\]
In the second case, (\ref{veryampleequality}) holds because the two cones on the right-hand side are spanned by bases of~${\Bbb Z}^3$:
\[
\det\left[ \begin{array}{ccc}
1&0&a_2\\
1&1&a_4\\
0&0&1
\end{array} \right]=1,
\qquad
\det\left[ \begin{array}{ccc}
0&1&a_3\\
1&1&a_4\\
0&0&1
\end{array} \right]=-1,
\]
and therefore
\[
C\cup{\Bbb Z}^d=\big({\Bbb Z}_{\ge0}(1,0,a_2)+{\Bbb Z}_{\ge0}(1,1,a_4)+{\Bbb Z}_{\ge0}\mathbf e_3\big)\, \cup\, \big({\Bbb Z}_{\ge0}(0,1,a_3)+{\Bbb Z}_{\ge0}(1,1,a_4)+{\Bbb Z}_{\ge0}\mathbf e_3\big).
\]
\medskip \noindent (b) The argument in part (a) shows that if a vertex of $\mathcal P(I_1,I_2,I_3,I_4)$ is simple then the primitive edge vectors at this vertex form a basis of ${\Bbb Z}^3$. So the polytope $\mathcal P(I_1,I_2,I_3,I_4)$ is smooth if and only if it is simple. On the other hand, $\mathcal P(I_1,I_2,I_3,I_4)$ being simple means the `bottom' vertices
$(0,0,a_1)$, $(1,0,a_2)$, $(0,1,a_3)$, $(1,1,a_4)$ align in a plane (i.e., they span a facet) and so do the `top' vertices $(0,0,b_1)$, $(1,0,b_2)$, $(0,1,b_3)$, $(1,1,b_4)$. This is equivalent to the desired equalities.
\end{proof}
An extension of (\ref{veryampleequality}) in the proof of Lemma \ref{Kripoexercise}(a) for $d$-dimensional
rational cones $C\subset{\Bbb R}^d$ with $d+1$ extremal vectors was given in \cite[Proposition 8.1]{BRUNSquest}. The class of smooth polytopes of type $\mathcal P(I_1,I_2,I_3,I_4)$ will be extended in Example \ref{Example} below.
We now specialize to the family of polytopes
\begin{align*}
\mathcal P_m &:= \mathcal P \left( [0,1], \, [0,1], \, [0,1], \, [m,m+1] \right) .
\end{align*}
\def.7{.7}
\begin{figure}[h!]
\caption{The polytope $\mathcal{P}_m$.}
\input{motherex}
\end{figure}
\noindent The underlying point configuration, written as a matrix, is
\[
\mathcal P_m\cap{\Bbb Z}^3 = \left[ \begin{array}{cccccccc}
0 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 \\
0 & 0 & 0 & 1 & 1 & 1 & m & m+1
\end{array} \right] .
\]
The same family is featured in \cite[Example 15]{8authors} in connection with the (lack of the) Gorenstein property. The monomial realizations of the corresponding monoid rings are
$$
{\Bbb K}\left[\mathcal P_m\cap{\Bbb Z}^3\right]\cong{\Bbb K}[Z,XZ,YZ,WZ,XWZ,YWZ,XYW^mZ,XYW^{m+1}Z] \, .
$$
Recall that the gap vector $\operatorname{gv}(\mathcal P)$ of a very ample polytope $\mathcal P$ has entries
\[
\operatorname{gv}_k(\mathcal P) := \# \text{ gaps in $M_\mathcal P$ at height $k$,}
\]
stopping at the largest height $\gamma(\mathcal P)$ that contains gaps in $M_\mathcal P$.
We can give an explicit formula for the gap vectors $\operatorname{gv}(\mathcal P_m)$ for all $m$.
\begin{theorem}\label{gapvectorofPm}
Let $m\geq 3$.
The gap vector of $\mathcal P_m$ has the entries
\[
\operatorname{gv}_k(\mathcal P_m) = {k+1\choose3}(m-k-1).
\]
In particular, $\gamma(\mathcal P_m) = m-2$ and
\[
\operatorname{gv}_1(\mathcal P_m)\le\cdots\le\operatorname{gv}_j(\mathcal P_m)\ge\operatorname{gv}_{j+1}(\mathcal P_m)\ge\cdots\ge\operatorname{gv}_{m-2}(\mathcal P_m)
\]
for $j=\lceil{\frac{3m-5}4}\rceil$.
\end{theorem}
Since for $\mathcal P$ very ample, $\mathcal P\times[0,1]$ is also very ample, the polytopes $\mathcal P_m$ imply the existence of non-normal very ample polytopes in all dimensions $\ge3$ with an arbitrarily large number of degree-2 gaps. Similar topics are discussed in \cite{Akihiro}.
In addition, \cite{katthan} gives examples of very ample 3-polytopes $\mathcal P$ with arbitrarily deep
gaps, measured via lattice distance relative to the facets of the cone ${\Bbb R}_{\ge 0}M_{\mathcal P}$. However, the gaps in all examples constructed in \cite{Akihiro,katthan} are concentrated in degree~2.
For the proof of Theorem \ref{gapvectorofPm} we will need several auxiliary results.
\begin{lemma}\label{highgaps}
The Ehrhart polynomial of the polytope $P_m$ equals
$$\operatorname{ehr}_{ \mathcal P_m } (j) = \left( \tfrac m 6 + 1 \right) j^3 + 3 \, j^2 + \left( 3 - \tfrac m 6 \right) j + 1 \, .$$
\end{lemma}
\begin{proof}
The volume of $\mathcal P_m$ is easily seen to be $\frac m 6 + 1$;
furthermore, $\mathcal P_m$ has four unimodular-triangle facets and four square facets that are unimodularly equivalent to
a unit square, and so
$
\operatorname{ehr}_{ \mathcal P_m } (j) = \left( \frac m 6 + 1 \right) j^3 + 3 \, j^2 + c \, j + 1
$ (see, e.g., \cite[Theorem 5.6]{Discretely}).
Since $\operatorname{ehr}_{ \mathcal P_m } (1) = 8$, we compute $c = 3 - \frac m 6$. The lemma follows.
\end{proof}
\begin{lemma}\label{dimensionstop}
Let $\mathcal P$ be a (not necessarily very ample) lattice polytope of dimension $d$ and $k_0\geq d-1$ be an integer. If $(M_\mathcal P)_{k_0}=(\overline{M_\mathcal P})_{k_0}$, then $(M_\mathcal P)_k=(\overline{M_\mathcal P})_k$ for all $k\geq k_0$.
\end{lemma}
\begin{proof}
The lemma follows from the fact that $\overline{M_\mathcal P}$ is generated as an \textbf{$M_\mathcal P$-module} by elements of degree at most $d-1$, i.e.,
\[
\overline{M_\mathcal P} \ =\bigcup_{
\begin{matrix}
\mathbf m\in\overline{M_\mathcal P}\\
\deg\mathbf m\le d-1
\end{matrix}
}(\mathbf m+M) \, ;
\]
see \cite[Corollary 1.3.4]{BrGuTr} and its proof.
\end{proof}
\begin{proof}[Proof of Theorem \ref{gapvectorofPm}]
First we prove the following formula for the Hilbert function:
\[
\#\left(M_{\mathcal P_m}\right)_j=(j+1){j+3\choose 3},\qquad 1\leq j\leq m-1.
\]
We fix $1\leq j\leq m-1$. Consider a point $S=\sum_{i=1}^j T_i=(e,f,g)\in{\Bbb Z}^3$ for some lattice points $T_i\in P_m$. We use the following notation:
$$A_0=(0,0,0),\quad A_1=(0,0,1),\quad B_0=(1,0,0),\quad B_1=(1,0,1)$$
$$C_0=(0,1,0),\quad C_1=(0,1,1),\quad D_0=(1,1,m),\quad D_1=(1,1,m+1).$$
Let $d$ be the number of times the points $D_i$ occur in some decomposition of $S$. We have $dm\leq g\leq dm+j$, so $d=\lfloor \frac{g}{m}\rfloor$. Thus the number
of times the points $B_i$ and $C_i$ occur in the decomposition equals respectively $b:=e- \lfloor \frac{g}{m}\rfloor$ and $c:=f-\lfloor \frac{g}{m}\rfloor$. Hence,
the points $A_i$ must occur $a:=j-b-c-d$ times. In particular, each decomposition of $S$ has the same number of occurrences of the points in each group $A_i$, $B_i$,
$C_i$ and $D_i$. Moreover, the points $A_1,B_1,C_1,D_1$ must occur $(g\mod m)\leq j$ times. The numbers $a,b,c,d$ and $(g\mod m)$ uniquely determine the point $S$.
We obtain a bijection between the points of $(M_{\mathcal P_m})_j$ and points of the form $(a,b,c,d,h)\in{\Bbb Z}^5$ where $a,b,c,d,h\geq 0, \ a+b+c+d=j, \ h\leq j$.
Thus $\#\left(M_{\mathcal P_m}\right)_j$ equals the number of ordered partitions of $j$ into four parts, for the choice of $a,b,c,d$, times $j+1$ for the choice of $(g\mod m)$. This equals $(j+1){j+3\choose 3}$.
So by Lemma \ref{highgaps}, for $j<m$ we obtain
\begin{align*}
\operatorname{gv}_j(\mathcal P_m)&=\operatorname{ehr}_{P_m}(j)-\#\left(M_{\mathcal P_m}\right)_j\\
&=\left( \tfrac m 6 + 1 \right) j^3 + 3 \, j^2 + \left( 3 - \tfrac m 6 \right) j + 1-(j+1){j+3\choose 3}\\
&={j+1\choose 3}(m-j-1) \, .
\end{align*}
Now the formula for $\operatorname{gv}_k(\mathcal P_m)$ follows by Lemma \ref{dimensionstop} because $\operatorname{ehr}_{P_m}(m-1)-\#\left(M_{\mathcal P_m}\right)_{m-1}=0$ and $m\geq 3=\dim(\mathcal P_m)$. The rest of Theorem \ref{gapvectorofPm} follows easily from this formula.
\end{proof}
Based on Theorem \ref{gapvectorofPm} and several other gap vectors of very ample polytopes we computed by effective methods, we offer the following conjecture.
\begin{conjecture}\label{gapvectorconj}
The gap vector of any very ample polytope $\mathcal P$ that has normal facets is unimodal, i.e., there exists $j$ such that
\[
\operatorname{gv}_1(\mathcal P) \le \operatorname{gv}_2(\mathcal P) \le \dots \le \operatorname{gv}_j(\mathcal P) \ge \operatorname{gv}_{ j+1 }(\mathcal P) \ge \dots \ge \operatorname{gv}_{ \gamma(\mathcal P) }(\mathcal P) \, .
\]
\end{conjecture}
The reason we require that the facets of $\mathcal P$ are normal in Conjecture \ref{gapvectorconj} (which in this case is equivalent to the facets of $\mathcal P$ being
integrally closed and automatically satisfied for very ample $3$-polytopes) is that if the gap vectors of the facets of $\mathcal P$ contribute to $\operatorname{gv}(\mathcal P)$ in a nontrivial
way, then---because the former are relatively independent of each other---the resulting interference
can cause oscillation in $\operatorname{gv}(\mathcal P)$. Very recently explicit examples of this phenomenon appeared in
\cite{JaMichalVeryAmple}: there exist very ample polytopes with gap vectors having only two nonzero entries at two arbitrary indices.
\section{Koszul segmental fibrations}\label{Koszul}
For a field ${\Bbb K}$, a narrower class of Koszul ${\Bbb K}$-algebras than those admitting quadratic Gr\"obner bases is formed by the homogeneous ${\Bbb K}$-algebras for which the defining ideal $I$ admits a square-free quadratic Gr\"obner basis. In the special case of algebras of the form ${\Bbb K}[\operatorname{\mathcal A}]$, where $\operatorname{\mathcal A}\subset{\Bbb Z}^d$ is a point
configuration, the existence of such Gr\"obner bases is a purely combinatorial condition due to Sturmfels (see \cite{STURMpol} or \cite[Sections 7.A,B]{Kripo})---Theorem \ref{Sturmfels} below.
To describe the connection with polytopal combinatorics, we recall the relevant terminology. We refer the reader
to \cite[Sections 1.D,E,F]{Kripo} for background on polytopal complexes, regular subdivisions and triangulations, stars and links in simplicial
complexes, etc.
A polytopal subdivision $\Delta$ of a polytope $\mathcal P\subset{\Bbb R}^d$ is \textbf{regular} if there is a convex function
$h:\mathcal P\to{\Bbb R}$ (i.e., $h(\lambda \mathbf x+\mu \mathbf y)\le \lambda h(\mathbf x)+\mu h(\mathbf y)$ for all $\mathbf x,\mathbf y\in \mathcal P$ and
$\lambda,\mu\in{\Bbb R}_{\ge0}$ with $\lambda+\mu=1$) whose domains of linearity are exactly the facets of $\Delta$,
i.e., the maximal faces of $\Delta$ are the maximal subsets of $\mathcal P$ on which $h$ restricts to an affine map. We
say that $h$ is a \textbf{support function} for~$\Delta$.
A simplicial complex $\Delta$ is \textbf{flag} if the minimal non-faces of $\Delta$ are pairs of vertices of $\Delta$.
(A \textbf{non-face} of the simplicial complex $\Delta$ with vertex set $V$ is a subset $W\subset V$ with $W\notin\Delta$.)
In general, the \textbf{degree} of a simplicial complex $\Delta$ on a vertex set $V$ is
\[
\deg(\Delta):=\max\big(\#W\ |\ W\ \text{a minimal non-face of}\ \Delta\big) \, ;
\]
see \cite{BrGuTr}. Thus, flag simplicial complexes are exactly the simplicial complexes of degree~2.
A triangulation of a polytope is thought of as the corresponding geometric simplicial complex, as opposed to the underlying abstract simplicial complex, i.e., the elements of the triangulation are simplices in the ambient Euclidean space.
A triangulation $\Delta$ of a lattice polytope $\mathcal P\subset{\Bbb R}^d$ is \textbf{unimodular} if the simplices in $\Delta$ are all unimodular.
Observe that, in exploring ring-theoretical properties of ${\Bbb K}[\operatorname{\mathcal A}]$, there is no loss of generality in assuming that
$\L(\operatorname{\mathcal A})={\Bbb Z}^d$: the isomorphism class of ${\Bbb K}[\operatorname{\mathcal A}]$ is independent of the ambient lattice for $\operatorname{\mathcal A}$ and,
therefore, it can be chosen to be~$\L(\operatorname{\mathcal A})$.
\begin{theorem}[Sturmfels \cite{STURMpol}]\label{Sturmfels}
For a point configuration $\operatorname{\mathcal A} = \{\mathbf a_1,\ldots,\mathbf a_N\} \subset{\Bbb Z}^d$ with $\L(\operatorname{\mathcal A})={\Bbb Z}^d$, the binomial ideal
\begin{align*}
I_{\operatorname{\mathcal A}}:=\Ker\left(
\begin{array}{rcl}
{\Bbb K}[X_1,\ldots,X_N] & \to & {\Bbb K}[\operatorname{\mathcal A}] \\
X_i & \mapsto & \mathbf a_i
\end{array}
\right)\subset{\Bbb K}[&X_1,\ldots,X_N]
\end{align*}
admits a square-free quadratic Gr\"obner basis if and only if there is a regular unimodular flag triangulation of
$\conv(\operatorname{\mathcal A})$ with the vertex set $\operatorname{\mathcal A}$.
\end{theorem}
By \cite[Ch. III]{Toroidal}, for every lattice polytope $\mathcal P$, the dilated polytope $c \, \mathcal P$ has a regular unimodular
triangulation for some $c\in{\Bbb N}$. By \cite[Theorem 1.4.1]{BrGuTr}, for a
lattice polytope $\mathcal P\subset{\Bbb R}^d$, the ideal $I_{c \mathcal P\cap{\Bbb Z}^d}$ has a quadratic (but possibly not square-free)
Gr\"obner basis whenever $c\ge\dim \mathcal P$. Thus, informally speaking, there is no algebraic obstruction to the
existence of regular unimodular flag triangulations of the dilated lattice polytopes $c \, \mathcal P$ for $c\ge\dim \mathcal P$. However, currently even the existence of dimensionally uniform lower
bounds for the factors $c$ such that the polytopes $c \, \mathcal P$ have unimodular triangulations is a major open problem \cite{Santos}.
Theorem \ref{koszulclass} below leads to a large class of polytopes admitting triangulations with all the nice properties. As we explain later on, this theorem could have been included in
\cite{DHZ}---the argument in \cite[Section 4.2]{DHZ}, used there for a rather
special case of \emph{Nakajima polytopes}, works also in the general case. A
related discussion can be found in Haase--Paffenholz's report in \cite{mfo}.
\begin{theorem}\label{koszulclass}
Let $f:\mathcal P\to \mathcal Q$ be a lattice segmental fibration of lattice polytopes. Assume $\Delta$ is a regular unimodular flag triangulation of $\mathcal Q$ such that the image $f(F)$ of every face $F\subset \mathcal P$ is a union of faces of $\Delta$. Then $\mathcal P$ has a regular unimodular flag triangulation; in particular, $\mathcal P$ is integrally closed and Koszul.
\end{theorem}
Observe that the polytope $\mathcal P_m$ in Theorem \ref{highgaps} satisfies the additional condition in Theorem
\ref{koszulclass} (with respect to both triangulations of the unit square as the polytope $\mathcal Q$ simultaneously) if and only if $m=0$.
Before outlining the proof of Theorem \ref{koszulclass} we discuss some explicit classes of polytopes this theorem leads to.
\begin{example}[\emph{Nakajima polytopes}]\label{Example}
Assume $\mathcal Q\subset{\Bbb R}^d$ is a lattice polytope and $\alpha,\beta:\mathcal Q\to{\Bbb R}$ are affine maps such that $\alpha(\mathbf x),\beta(\mathbf x)\in{\Bbb Z}$ for all $x\in\mathcal Q\cap{\Bbb Z}^d$ and $\alpha\le\beta$ on $\mathcal Q$. Consider the lattice polytope
$$
\mathcal Q(\alpha,\beta):=\conv\big((\mathbf x,y)\ |\ \mathbf x\in \mathcal Q,\ \alpha(\mathbf x)\le y\le\beta(\mathbf x)\big)\subset{\Bbb R}^{d+1}.
$$
Then the orthogonal projection $f:\mathcal Q(\alpha,\beta)\to \mathcal Q$ and any regular unimodular flag triangulation of $\mathcal Q$ satisfy the conditions in Theorem \ref{koszulclass}. It is easily seen that $\mathcal Q(\alpha,\beta)\cong\mathcal Q(0,\beta-\alpha)$ as lattice polytopes.
Iteratively using the $\mathcal Q(\alpha,\beta)$-construction, starting with a point, we get exactly the class of polytopes characterized in \cite{Nakajima} as the polytopes $\mathcal P$ for which the (complex) \emph{affine} toric variety $\Spec({\Bbb C}[M_\mathcal P])$ is a local compete intersection. In \cite{DHZ} these polytopes are called \emph{Nakajima polytopes}.
It is clear that the polytopes $\mathcal P$ in Theorem \ref{koszulclass} can have arbitrarily more complicated shapes than the ones resulting from the $\mathcal Q(\alpha,\beta)$-construction.
The smooth polytopes of type $\mathcal P(I_1,I_2,I_3,I_4)$ in Section \ref{Class} are of type $\mathcal Q(\alpha,\beta)$ and, therefore, integrally closed and Koszul. More generally, if $Q$ is any smooth polytope and $\alpha,\beta:\mathcal Q\to{\Bbb R}$ are as above, satisfying the stronger condition $\alpha<\beta$ on $\mathcal Q$, then $Q(\alpha,\beta)$ is smooth as well.
In particular, iteratively using the $\mathcal Q(\alpha,\beta)$-construction with $\alpha<\beta$ on $\mathcal Q$, starting with a point, we get smooth Nakajima polytopes in arbitrary dimensions, all combinatorially equivalent to cubes but representing infinitely many affine equivalence classes.
Starting the iteration with other smooth polytopes that admit triangulations with the desired properties, we get richer classes of higher-dimensional smooth Koszul polytopes. For instance, by \cite[Corollary 3.2.5]{BrGuTr}, all lattice smooth polygons can serve as the initial input of this machine.
\end{example}
\begin{example}[\emph{Lattice $A$-fibrations}]\label{Example2} Let $\mathcal P\subset{\Bbb R}^d$ be a lattice polytope cut out by a root system of type $A$. In other words, $\mathcal P$
is bounded by hyperplanes parallel to hyperplanes of the form $X_i=0$ and $X_i=X_j$, $1\le i\not=j\le d$ (in the language of \cite{lampostnikov}, $\mathcal P$ is an
\emph{alcoved polytope}). Then $\mathcal P$ has a \emph{canonical} nice triangulation $\Delta(\mathcal P)$ satisfying the compatibility condition: if $\dim \mathcal P>0$ then its orthogonal projection $\mathcal Q\subset{\Bbb R}^{d-1}$ is also cut out by a root system of type $A$ and the projection $f:\mathcal P\to \mathcal Q$ satisfies the condition in Theorem~\ref{koszulclass} with respect to $\Delta(\mathcal Q)$.
In fact, it was shown in \cite[Section 2]{BrGuTr} that each polytope from the above-mentioned class is nicely triangulated by cutting it along the integer translates
of the coordinate hyperplanes and the hyperplanes of the form $X_i-X_j$. So it is enough to show the following:
\medskip\noindent\emph{Claim.} Let $\mathcal P\subset{\Bbb R}^d$ be a lattice polytope cut out by a root system of type $A$. Then its
orthogonal projection $\mathcal Q$ in ${\Bbb R}^{d-1}$ is also a lattice polytope cut out by a root system of type $A$.
\medskip Any facet of $\mathcal Q$ is the orthogonal projection of either a facet of $\mathcal P$ or a codimension-2 face of
$\mathcal P$. In the first case the corresponding support hyperplane of $\mathcal P$ is of the form $X_i=a$ or $X_i-X_j=b$ for some $a,b\in{\Bbb Z}$ and $1\le i\not=j\le d-1$. In particular,
the same equality defines the image facet of $\mathcal Q$. In the second case the codimension-2 face of $\mathcal P$ in question corresponds to a system of type either $X_d=a$ and
$X_d-X_i=b$ or $X_d-X_i=a$ and $X_d-X_j=b$, where $a,b\in{\Bbb Z}$ and $1\le i\not= j\le d-1$. Correspondingly, the image facet of $\mathcal Q$ is defined by either $X_i=a-b$ or $X_i-X_j=b-a$. \qed
\medskip One can extend the notion of lattice segmental fibrations as follows: assume $\mathcal P\subset{\Bbb R}^{d_1}$ and $\mathcal Q\subset{\Bbb R}^{d_2}$ are lattice polytopes and $f:\mathcal P\to\mathcal Q$ is an affine map; call $f$ a \textbf{lattice $A$-fibration} if it satisfies the conditions
\begin{enumerate}[{\rm (i)}]
\item $f^{-1}(\mathbf x)$ is a lattice polytope for every $\mathbf x\in\mathcal Q\cap{\Bbb Z}^{d_2}$,
\item $\mathcal P\cap{\Bbb Z}^{d_1}\subset\bigcup_{\mathcal Q\cap{\Bbb Z}^{d_2}}f^{-1}(\mathbf x)$,
\item there is a full-rank affine map $\pi:\mathcal P\to{\Bbb R}^{\dim\mathcal P-\dim\mathcal Q}$, injective on $f^{-1}(\mathbf x)$ for every $\mathbf x\in\mathcal Q$ and such that $\pi$ induces a surjective group homomorphism onto
${\Bbb Z}^{\dim\mathcal P-\dim\mathcal Q}$ and $\pi\big(f^{-1}(\mathbf x)\big)$ is a lattice polytope, cut out by a root system of type $A$, for every $\mathbf x\in\mathcal Q\cap{\Bbb Z}^{d_2}$.
\end{enumerate}
The class of $A$-fibrations is considerably larger than that of segmental fibrations and, in general, $\mathcal P$ is very different from a Nakajima polytope even for simple $\mathcal Q$ (e.g., a segment). Let $f:\mathcal P\to\mathcal Q$ be a lattice $A$-fibration, where $\mathcal P\subset{\Bbb R}^{d_1}$ and $\mathcal Q\subset{\Bbb R}^{d_2}$. In view of the claim above, applied iteratively to the fibers over the lattice points of $\mathcal Q$, the map $f$ factors through segmental fibrations
\begin{equation}\label{sequenceoffibrations}
\mathcal P\mathrel{\mathop{\longrightarrow}^{\phi_0}}\mathcal P_1\mathrel{\mathop{\longrightarrow}^{\phi_1}}\cdots\mathrel{\mathop{\longrightarrow}^{\phi_{k-1}}}
\mathcal P_k\mathrel{\mathop{\longrightarrow}^{\phi_k}}\mathcal Q \qquad \text{ where } \qquad k=\max\big(\dim f^{-1}(\mathbf x)\ |\ \mathbf x\in\mathcal Q\cap{\Bbb Z}^{d_2}\big).
\end{equation}
So one can ask whether Theorem \ref{koszulclass} can be extended to lattice $A$-fibrations. The
obstruction to iteration of Theorem \ref{koszulclass} is that it is not clear whether the condition
on faces in that theorem can be kept under control at each step from $\phi_k$ to $\phi_0$. In fact,
the triangulation of $\mathcal P$ resulting from the proof of Theorem \ref{koszulclass}, when both $\mathcal P$ and $\mathcal Q$ are cut out by a root system of type $A$, is \emph{not} the same as $\Delta(\mathcal P)$, not even if $\dim\mathcal P=\dim\mathcal Q+1$.
However, there is a big subclass of lattice $A$-fibrations for which Theorem \ref{koszulclass} can be iterated along the corresponding sequences (\ref{sequenceoffibrations}). Call a lattice $A$-fibration a \emph{lattice cubical fibration} if in the condition (iii) above we require that the polytope $f^{-1}(\mathbf x)$ has the facets parallel to coordinate hyperplanes.
One can easily check that the proof of Theorem \ref{koszulclass} allows one to control the condition on the faces in the theorem at each step from $\phi_k$ to $\phi_0$ when $f:\mathcal P\to\mathcal Q$ is cubical. Therefore, Theorem \ref{koszulclass} extends to lattice cubical fibrations.
\end{example}
\begin{proof}[Sketch of the proof of Theorem \ref{koszulclass}]
Following the approach in \cite[Section 4.2]{DHZ}, we first take the regular polytopal subdivision $R:=(f^{-1}(\delta))_{\delta\in\Delta}$ of $\mathcal P$ and then refine it to a triangulation, using successive stellar subdivisions by the lattice points in $\mathcal P$ in any linear order. The regularity, flag, and unimodularity properties of the final outcome are checked exactly the same way as in \cite{DHZ}.
Our original approach (the one we used before we learned about the overlap with \cite{DHZ}) produces different triangulations of $\mathcal P$, also refining the polytopal subdivision $R$ but without involving stellar subdivisions. Below we describe the construction.
For a closed subset $Y\subset{\Bbb R}^{d+1}$, we put
\begin{align*}
&Y^+:=\{\mathbf y\in Y\ |\ \mathbf y\ \text{has the largest}\ (d+1)\text{st coordinate within}\ f^{-1}(f(\mathbf y))\},\\
&Y^-:=\{\mathbf y\in Y\ |\ \mathbf y\ \text{has the smallest}\ (d+1)\text{st coordinate within}\ f^{-1}(f(\mathbf y))\}.
\end{align*}
There is no loss of generality in assuming that $\mathcal Q\subset{\Bbb R}^d$, $\dim \mathcal Q=d$, $\mathcal P\subset{\Bbb R}^{d+1}$, and
$f$ is the projection onto the first $(d+1)$-coordinates.
We can assume $(\mathcal P\setminus \mathcal P^-)\cap{\Bbb Z}^{d+1}=\{\mathbf y_1,\ldots,\mathbf y_r\}$, where
$$
f(\mathbf y_i)=f(\mathbf y_j)\quad \text{ and } \quad (\mathbf y_i)_{d+1}< (\mathbf y_j)_{d+1}\ \quad \text{ imply } \quad i < j \, .
$$
Define the sequence of polytopal complexes $\Pi_0,\Pi_1,\ldots,\Pi_r$ inductively as follows:
\begin{enumerate}[{\rm$\bullet$}]
\item $\Pi_0=\{f^{-1}(\delta)\cap \mathcal P^-\}_\Delta$
\item $\Pi_k=\big\{\conv(\mathbf y_k,F)\ |\ F\in\operatorname{star}^+_{\Pi_{k-1}}(\mathbf y_k-\mathbf e_{d+1})\big\}\cup\Pi_{k-1}$, where
\begin{align*}
\operatorname{star}^+_{\Pi_{k-1}}(\mathbf y_k-\mathbf e_{d+1})=\{\tau\in\operatorname{star}_{\Pi_{k-1}}(\mathbf y_k-\mathbf e_{d+1})\ |\ \tau&\subset|\Pi_{k-1}|^+\},\\
&k=1,\ldots,r.
\end{align*}
\end{enumerate}
\noindent($|\dots|$ denotes the support of the polytopal complex in question.) That $\Pi_r$ is a triangulation of $\mathcal P$ with the desired properties can be shown along the same lines as for the triangulations in \cite{DHZ} (only the regularity needs a minor change in the argument).
\end{proof}
It is interesting to notice that the triangulations $\Pi_r$ are usually \emph{different} from those in \cite[Section 4.2]{DHZ} when the fibers $f^{-1}(\mathbf x)$ contain at least $4$ lattice points for several $\mathbf x\in\mathcal Q\cap{\Bbb Z}^d$.
\clearpage
\begin{figure}[h!]
\flushleft
\caption{Two triangulations $\Pi_r$ of $\mathcal P$ for two different enumerations of $(\mathcal P\setminus\mathcal P^-)\cap{\Bbb Z}^{d+1}$.}
\includegraphics[trim = 0mm 0in 0mm 1.5in, clip, scale=.4]{Figure3}
\end{figure}
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1,108,101,565,687 | arxiv | \section{Introduction}
Ever since Black, Merton, and Scholes introduced their celebrated option valuation formula~\cite{black:scholes:paper,merton:bspaper} a great deal of effort has been dedicated to reproduce that result using more realistic stochastic models for the underlying asset. In the discrete time modeling setup, the GARCH parametric family~\cite{engle:arch, bollerslev:garch} has been a very popular and successful choice for which the option pricing and hedging problem has been profusely studied; see for example~\cite{duan:GARCH:pricing, heston:nandi, Badescu:option:pricing, chorro:guegan:ielpo, ortega:garch:pricing}, and references therein. Even though the GARCH specification can accommodate most stylized features of financial returns like leptokurticity, volatility clustering, and autocorrelation of squared returns, there are mathematical relations between some of their moments that impose undesirable constraints on some of the parameter values. For example, a well-known phenomenon~\cite{carnero:pena:ruiz} has to do with the relation that links the kurtosis of the process with the parameter that accounts for the persistence of the volatility shocks (the sum of the ARCH and the GARCH parameters, bound to be smaller than one in order to ensure stationarity). This relation implies that for highly volatility persistent time series like the ones observed in practice, the ARCH coefficient is automatically forced to be very small, which is in turn incompatible with having sizeable autocorrelation (ACF) for the squares (the ACF of the squares at lag $1$ is linear in this coefficient). This situation aggravates when innovations with fat tails are used in order to better reproduce leptokurticity.
The structural rigidities associated to the finiteness of the fourth moment are of particular importance when using quadratic hedging methods for there is an imperative need to remain in the category of square summable objects and finite kurtosis is a convenient way to ensure that (see, for example~\cite[Theorem 3.1 (iv)]{ortega:garch:pricing}).
The auto-regressive stochastic volatility (ARSV) models~\cite{Taylor:Book1, Taylor:Book2} that we will carefully describe later on in Section~\ref{Auto-regressive stochastic volatility (ARSV) models} are a parametric family designed in part to overcome the difficulties that we just explained. The defining feature of these models is the fact that the volatility (or a function of it) is determined by an autoregressive process that, unlike the GARCH situation, is exclusively driven by past volatility values and by designated innovations that are in principle independent from those driving the returns. The use of additional innovations introduces in the model a huge structural leeway because, this time around, the same constraint that ensures stationarity guarantees the existence of finite moments of arbitrary order which is of much value in the context of option pricing and hedging.
The modifications in the ARSV prescription that allow an enriching of the dynamics come with a price that has to do with the fact that conditional volatilities are not determined, as it was the case in the GARCH situation, by the price process. This causes added difficulties at the time of volatility and parameter estimation; in particular, a conditional likelihood cannot be written down in this context (see later on in Section~\ref{Volatility and model estimation}). This also has a serious impact when these models are used in derivatives pricing and hedging because the existence of additional noise sources driving the dynamics makes more acute the incomplete character of the corresponding market; this makes more complex the pricing and hedging of contingent products.
\emph{The main goal of this paper is showing that appropriate implementations of the local risk minimization scheme developed by F\"ollmer, Schweizer, and Sondermann~\cite{foellmer:sondermann, foellmer:schweizer, schweizer:2001} tailored to the ARSV setup provide an appropriate tool for pricing and hedging contingent products that use these models to characterize their underlying assets}. The realization of this hedging scheme in the ARSV context requires two main tools:
\begin{itemize}
\item {\bf Use of a volatility estimation technique:} as we already said, volatility for these models is not determined by the price process and hence becomes a genuinely hidden variable that needs to be estimated separately. In our work we will use a {\bf Kalman filtering} approach advocated by~\cite{harvey:ruiz:shephard} and the so called {\bf hierarchical likelihood} ($h$-likelihood) strategy~\cite{lee:nelder:1, lee:nelder:2, delcastillo:lee:1, delcastillo:lee:2} which, roughly speaking, consists of carrying out a likelihood estimation while considering the volatilities as unobserved parameters that are part of the optimization problem. Even though both methods are adequate volatility estimation techniques, the $h$-likelihood technique has a much wider range of applicability for it is not subjected to the rigidity of the state space representation necessary for Kalman and hence can be used for stochastic volatility models with complex link functions or when innovations are non-Gaussian.
\item {\bf Use of a pricing kernel:} this term refers to a probability measure equivalent to the physical one with respect to which the discounted prices are martingales. The expectation of the discounted payoff of a derivative product with respect to any of these equivalent measures yields an arbitrage free price for it. In our work we will also use these martingale measures in order to devise hedging strategies via local risk minimization. This is admittedly just an approximation of the optimal local risk minimization scheme that should be carried out with respect to the physical measure that actually quantifies the local risk. However there are good reasons that advise the use of these equivalent martingale measures mainly having to do with numerical computability and interpretation of the resulting value processes as arbitrage free prices. These arguments are analyzed in Section~\ref{Local risk minimization with respect to a martingale measure}.
\end{itemize}
Regarding the last point, there are two equivalent martingale measures that we will be using. The first one is inspired by the {\it Extended Girsanov Principle} introduced in~\cite{elliott:madan:mcmm}. This principle allows the construction of a martingale measure in a general setup under which the process behaves as its ``martingale component'' used to do under the physical probability; this is the reason why this measure is sometimes referred to as the {\bf mean-correcting martingale measure}, denomination that we will adopt. This construction has been widely used in the GARCH context (see~\cite{Badescu:option:pricing, ortega:garch:pricing} and references therein) where it admits a particularly simple and numerically efficient expression in terms of the probability density function of the model innovations. As we will see, in the ARSV case, this feature is not anymore available and we will hence work with the expression coming from the predictable situation that even though in the ARSV case produces a measure that does not satisfy the Extended Girsanov Principle, it is still a martingale measure.
Secondly, in Theorem~\ref{introduction minimal martingale measure} we construct the so called {\bf minimal martingale measure} $Q_{{\rm min}}$ in the ARSV setup; the importance in our context of this measure is given by the fact that the value process of the local risk-minimizing strategy {\it with respect to the physical measure} for a derivative product coincides with its arbitrage free price when using $Q_{{\rm min}}$ as a pricing kernel. A concern with $Q_{{\rm min}}$ in the ARSV setup is that {\it this measure is in general signed}; fortunately, the occurrence of negative Radon-Nikodym derivatives is extremely rare for the usual parameter values that one encounters in financial time series. Consequently, the bias introduced by censoring paths that yield negative Radon-Nikodym derivatives and using $Q _{{\rm min}} $ as a well-defined positive measure is hardly noticeable. A point that is worth emphasizing is that even though the value processes obtained when carrying out local risk minimization with respect to the physical and the minimal martingale measures are identical, the hedges are in general {\it not} the same and consequently so are the hedging errors; this difference is studied in Proposition~\ref{hedges under change of measure}.
A general remark about local risk minimization that supports its choice in applications is its {\bf adaptability to different hedging frequencies}. Most sensitivity based (delta) hedging methods for time series type underlyings are constructed by discretizing a continuous time hedging argument; consequently, when the hedging frequency diminishes, the use of such techniques loses pertinence. As we explain in Section~\ref{Local risk minimization and changes in the hedging frequency}, the local risk minimization hedging scheme can be adapted to prescribed changes in the hedging frequency, which regularly happens in real life applications.
The paper is organized in four sections. Section~\ref{Auto-regressive stochastic volatility (ARSV) models} contains a brief introduction to ARSV models and their dynamical features of interest to our study; this section contains two subsections that explain the volatility estimation techniques that we will be using and the martingale measures that we mentioned above. The details about the implementation of the local risk minimization strategy are contained in Section~\ref{Local risk minimization for ARSV options}. Section~\ref{Empirical study} contains a numerical study where we carry out a comparison between the
hedging performances obtained by implementing the local risk minimization scheme using the different volatility estimation techniques and the martingale measures introduced in Section~\ref{Auto-regressive stochastic volatility (ARSV) models}. In order to enrich the discussion, we have added to the study two standard sensitivity based hedging methods that provide good results in other contexts, namely Black-Scholes~\cite{black:scholes:paper} and an adaptation of Duan's static hedge~\cite{duan:GARCH:pricing} to the ARSV context. As a summary of the results obtained in that section it can be said that {\it local risk minimization outperforms sensitivity based hedging methods, especially when maturities are long and the hedging frequencies are low}; this could be expected due to the adaptability feature of local risk minimization with respect to modifications in this variable. As to the different measures proposed, {\it it is the minimal martingale measure that yields the best hedging performances when implemented using a Kalman based estimation of volatilities, in the case of short and medium maturities and $h$-likelihood for longer maturities and low hedging frequencies}.
\medskip
\noindent {\bf Conventions and notations:} The proofs of all the results in the paper are contained in the appendix in Section~\ref{Appendix}.
Given a filtered probability space $(\Omega, P, \mathcal{F}, \{ \mathcal{F} _t\}_{t \in \mathbb{N}})$ and $X,Y $ two random variables, we will denote by $E _t[X]:=E[X| \mathcal{F} _t]$ the conditional expectation, ${\rm cov} _t(X,Y):= {\rm cov}(X,Y| \mathcal{F} _t):=E_t[XY]-E_t[X]E _t[Y]$ the conditional covariance, and by ${\rm var} _t(X):=E_t[X ^2]-E _t[X] ^2 $ the conditional variance. A discrete-time stochastic process $\{X _t\}_{t \in \mathbb{N}} $ is predictable when $X _t$ is $\mathcal{F} _{t-1} $-measurable, for any $t \in \mathbb{N} $.
\medskip
\noindent {\bf Acknowledgments:} we thank Alexandru Badescu for insightful comments about our work that have significantly improved its overall quality.
\section{Auto-regressive stochastic volatility (ARSV) models}
\label{Auto-regressive stochastic volatility (ARSV) models}
The auto-regressive stochastic volatility (ARSV) model has been introduced in~\cite{Taylor:Book1} with the objective of capturing some of the most common stylized features observed in the excess returns of financial time series: volatility clustering, excess kurtosis, and autodecorrelation in the presence of dependence; this last feature can be visualized by noticing that financial log-returns time series exhibit autocorrelation at, say lag $1$, close to zero while the autocorrelation of the squared returns is significantly not null. Let $S _t $ be the price at time $t$ of the asset under consideration, $r$ the risk-free interest rate, and $y _t:=\log\left(S _t/S_{t-1} \right) $ the associated log-return. In this paper we will be considering the so-called {\bf standard ARSV} model which is given by the prescription
\begin{equation}
\label{arsv model}
\left\{
\begin{array}{rcl}
y _t &= &r+ \sigma _t \epsilon _t, \qquad \{ \epsilon _t\} \sim {\rm IIDN}(0,1)\\
b _t&= & \gamma+ \phi b_{t-1}+ w _t, \qquad \{ w _t\} \sim {\rm IIDN}(0,\sigma _w^2)
\end{array}
\right.
\end{equation}
where $b _t:= \log (\sigma_t^2)$, $\gamma $ is a real parameter, and $\phi \in (-1,1) $. Notice that in this model, the volatility process $\{ \sigma _t\} $ is a non-traded stochastic latent variable that, unlike the situation in GARCH like models~\cite{engle:arch, bollerslev:garch} is not a predictable process that can be written as a function of previous returns and volatilities.
It is easy to prove that the unique stationary returns process induced by~(\ref{arsv model}) available in the presence of the constraint $\phi \in (-1,1) $ is a white noise (the returns have no autocorrelation) with finite moments of arbitrary order. In particular, the marginal variance and kurtosis are given by
\begin{equation}
\label{variance and kurtosis}
{\rm var}(y _t)= {\rm E} [ \sigma _t^2]= \exp\left[ \frac{\gamma}{1- \phi}+ \frac{1}{2} \sigma _b ^2\right], \qquad {\rm kurtosis}\,(y _t)= 3\exp \left( \sigma _b ^2 \right)
\end{equation}
Moreover, let $\sigma _b ^2 $ be the marginal variance of the stationary process $\{ b _t \} $, that is,
\begin{equation*}
\sigma _b ^2= \frac{\sigma_w ^2}{1- \phi^2}.
\end{equation*}
It can be shown~\cite{Taylor:Book1} that whenever $\sigma _b ^2 $ is small and/or $\phi$ is close to one then the autocorrelation $\gamma (h) $ of the squared returns at lag $h$ can be approximated by
\begin{equation*}
\gamma (h)\simeq \frac{\exp(\sigma_b ^2)-1}{3\exp(\sigma_b ^2)-1}\phi ^h.
\end{equation*}
The existence of finite moments is particularly convenient in the context of quadratic hedging methods. For example, in Theorem 3.1 of~\cite{ortega:garch:pricing} it is shown that the finiteness of kurtosis is a sufficient condition for the availability of adequate integrability conditions necessary to carry out pricing and hedging via local risk minimization.
Let $(\Omega, P) $ be the probability space where the model~(\ref{arsv model}) has been formulated and let $\mathcal{F}_t $ be the information set generated by the observables $\{S _0, S _1, \ldots, S_t \} $. This statement can be mathematically coded by setting $\mathcal{F}_t= \sigma\left(S _0, S _1, \ldots, S_t \right) $, where $\sigma\left(S _0, S _1, \ldots, S_t \right) $ is the sigma algebra generated by the prices $\{S _0, S _1, \ldots, S_t\} $. As several equivalent probability measures will appear in our discussion, we will refer to $P$ as the {\bf physical} or {\bf historical} probability measure.
Unless we indicate otherwise, in the rest of the discussion we will not assume that the innovations $\{ \epsilon _t\} $ are Gaussian.
We will also need the following marginal and {\bf conditional cumulant functions} of $\epsilon _t $ with respect to the filtration $\mathcal{F}=\{ \mathcal{F} _t\}_{t \in \{0, \ldots, T\}} $.
\begin{eqnarray*}
L_{\epsilon _t}^P(z)&=&\log E^P\left[e^{z \epsilon _t}\right], \qquad z \in \mathbb{R},\\
K_{\epsilon _t}^P(u)&=&\log E^P\left[e^{u \epsilon _t}| {\cal F}_{t-1}\right], \qquad \text{with } u \mbox{ a random variable}. \quad
\end{eqnarray*}
If the innovations $\{ \epsilon _t\} $ are Gaussian, we obviously have $L_{\epsilon _t}^P(z)=z ^2/2 $. When the random variables $u$ and $\epsilon_t $ are $P$-independent, the following relation between $L_{\epsilon _t}^P $ and $K_{\epsilon _t}^P $ holds.
\begin{lemma}
\label{relation cumulant functions}
Let $u:( \Omega,P) \rightarrow \mathbb{R}$ be a random variable independent of $\epsilon _t $ and $\mathcal{F}=\{ \mathcal{F} _t\}_{t \in \{0, \ldots, T\}} $ the filtration introduced above. Then
\begin{equation}
\label{equality with Ks}
K_{\epsilon _t}^P(u)=\log E^P\left[e^{L^P_{\epsilon _t}(u)}\mid {\cal F}_{t-1}\right].
\end{equation}
\end{lemma}
The cumulant functions that we just introduced are very useful at the time of writing down the conditional means and variances of price changes with respect to the physical probability; these quantities will show up frequently in our developments later on. Before we provide these expressions we introduce the following notation for discounted prices:
\begin{equation*}
\widetilde{S _t}:=S _te^{-rt},
\end{equation*}
where $r$ denotes the risk-free interest rate. The tilde will be used in general for discounted processes. Using this notation and Lemma~\ref{relation cumulant functions}, a straightforward computation using the independence at any given time step $t$ of the innovations $\{\epsilon_t \}$ and the volatility process $\{ \sigma _t \} $ defined by ~(\ref{arsv model}), shows that
\begin{eqnarray}
E^P\left[\widetilde{S _t}-\widetilde{S} _{t-1}\mid \mathcal{F}_{t-1}\right] &=&\widetilde{S} _{t-1} \left(e^{K_{\epsilon _t}^P (\sigma _t)}-1\right)=\widetilde{S} _{t-1}E^P\left[e^{L^P_{\epsilon _t}(\sigma _t)}\mid {\cal F}_{t-1}\right],\label{expected increase 1}\\
{\rm var}\left[\widetilde{S _t}-\widetilde{S} _{t-1}\mid \mathcal{F}_{t-1}\right] &= & \widetilde{S} _{t-1} ^2 \left[e^{K_{\epsilon _t}^P(2\sigma _t)}-e^{2K_{\epsilon _t}^P(\sigma _t)}\right].\label{expected increase 2}
\end{eqnarray}
\subsection{Volatility and model estimation}
\label{Volatility and model estimation}
The main complication in the estimation of ARSV models is due to the fact that $\mathcal{F}_t $ does not determine the random variable $\sigma _t $ and hence makes impossible the writing of a likelihood function in a traditional sense. Many procedures have been developed over the years to go around this difficulty based on different techniques: moment matching~\cite{Taylor:Book1}, generalized method of moments~\cite{melino:turnbull, jacquier:polson:rossi, andersen:sorensen}, combinations of quasi-maximum likelihood with the Kalman filter~\cite{harvey:ruiz:shephard}, simulated maximum likelihood~\cite{danielsson;simulated:ml}, MCMC~\cite{jacquier:polson:rossi} and, more recently, hierarchical-likelihood~\cite{lee:nelder:1, lee:nelder:2, delcastillo:lee:1, delcastillo:lee:2} (abbreviated in what follows as h-likelihood). An excellent overview of some of these methods is provided in Chapter 11 of the monograph~\cite{Taylor:Book2}.
In this paper we will focus on the Kalman and h-likelihood approaches since both are based on numerically efficient estimations of the volatility, a point of much importance in our developments.
\medskip
\noindent {\bf State space representation and Kalman filtering.} This approach is advocated by~\cite{harvey:ruiz:shephard} and consists of writing down the model~(\ref{arsv model}) using a state space representation by setting
\begin{equation*}
l _t=\log \left(|y _t-r|\right), \qquad L _t=\log \left(\sigma _t\right).
\end{equation*}
We assume in this paragraph that the innovations $\{\epsilon _t \}$ are Gaussian. Using these variables,~(\ref{arsv model}) can be rephrased as the observation and state equations:
\begin{equation}
\label{arsv model kalman form}
\left\{
\begin{array}{rcl}
l _t &= &L _t+ \xi_t, \qquad \xi _t=\log \left(| \epsilon _t|\right),\\
(L _t - \alpha _K)&= &\phi (L _{t-1} - \alpha _K)+ \eta_t, \qquad \eta_t= \frac{1}{2}w _t,
\end{array}
\right.
\end{equation}
where $ \alpha _K= E [L _t]= \frac{1}{2}E[b _t]= \frac{\gamma}{2(1- \phi)} $. Kalman filtering cannot be directly applied to~(\ref{arsv model kalman form}) due to the non-Gaussian character of the innovations $\xi _t=\log \left(| \epsilon _t|\right) $ hence, the approximation in~\cite{harvey:ruiz:shephard} consists of using this procedure by thinking of $\xi _t $ as a Gaussian variable with mean $\mu _{\xi}:=E[\xi _t]=-0.63518 $ and variance $\sigma^2_{\xi }= {\rm Var}[\xi_t] = \pi ^2/8$. The Kalman filter provides an algorithm to iteratively produce one-step ahead linear forecasts $l _{t,1}:=P[l _t| \mathcal{F} _{t-1}]$; given that $l_t= \log (\sigma _t)+\log \left(| \epsilon _t|\right) $, these forecasts can be used to produce predictable estimations $\sigma_t^k $ of the volatilities $\sigma _t $ by setting $\log \left(\sigma_t^k\right):= l_{t-1,1}- \mu_\xi $ and hence
\begin{equation}
\label{kalman vola}
\sigma_t^k=\exp (l_{t-1,1}- \mu_\xi).
\end{equation}
\medskip
\noindent {\bf The $h$-likelihood approach.} This method~\cite{lee:nelder:1, lee:nelder:2, delcastillo:lee:1, delcastillo:lee:2} consists of carrying out a likelihood estimation while considering the volatilities as unobserved parameters that are part of the optimization problem. More specifically, let $\alpha:=(\gamma, \phi, \sigma _w) $ be the parameters vector, $b=(b _1, \ldots, b _T)$ the vector that contains the unobserved $b _t:=\log (\sigma_t^2) $, and let $z _t:= y _t- r $; if $f(z _t|b _t)$ and $f(b| \alpha ) $ denote the corresponding probability density functions, the associated $h$-(log)likelihood is defined as
\begin{eqnarray}
h(z;b, \alpha) &= &\sum_{t=1}^T\log f(z _t|b _t)+\log f(b| \alpha )\notag\\
&= &- \frac{1}{2}\sum_{t=1}^T \left(z _t ^2\exp (-b _t) + b _t+ \frac{1}{\sigma _w ^2}(b _t- \gamma- \phi b _{t-1})^2+\log \sigma _w ^2\right).\label{h-likelihood f}
\end{eqnarray}
The references listed above provide numerically efficient procedures to maximize~(\ref{h-likelihood f}) with respect to the variables $b$ and $\alpha$ once the sample $z$ has been specified. An idea of much importance in our context is that, once the coefficients $\alpha $ have been estimated, this procedure has a natural one-step ahead forecast of the volatility associated, similar to the one obtained using the Kalman filter. The technique operates using the variables $b _t:=\log (\sigma_t^2) $ alternating prediction and updating; more specifically, we initialize the algorithm using the mean of the unique stationary solution of~(\ref{arsv model}), that is, $b _0:= \gamma/(1- \phi) $. We use the second equation in~(\ref{arsv model}) to produce a linear prediction $ b_{1p} $ of $b _1 $ based on the value of $b _0$, namely $b_{1p}= \gamma+ \phi b _0 $. Once the quote $z _1 $ is available, the h-likelihood approach can be used to update this forecast to the value $b_{1u} $ by solving the optimization problem
\begin{equation*}
b_{1u}=\mathop{\rm arg\, min}_{b _1\in \mathbb{R}} \left(z _1 ^2\exp (-b _1) + b _1+ \frac{1}{\sigma _w ^2}(b _1- \gamma- \phi b _{0})^2 \right).
\end{equation*}
This value can be used in its turn to produce a forecast $b_{2p} $ for $b _2 $ by setting $b_{2p}= \gamma+ \phi b _{1u} $. If we continue this pattern we obtain a succession of prediction/updating steps given by the expressions
\begin{eqnarray*}
b_{tp}&=& \gamma+ \phi b _{(t-1)u},\\
b_{tu}&=&\mathop{\rm arg\, min}_{b _t\in \mathbb{R}} \left(z _t ^2\exp (-b _t) + b _t+ \frac{1}{\sigma _w ^2}(b _t- \gamma- \phi b _{(t-1)u})^2 \right).
\end{eqnarray*}
These forecasts can be used to produce predictable estimations $\sigma_t^h $ of the volatilities $\sigma _t $ by setting
\begin{equation*}
\sigma_t^h:=\exp \left(\frac{1}{2} b _{tp}\right).
\end{equation*}
\begin{remark}
\normalfont
Even though both the Kalman and the $h$-likelihood methods constitute adequate estimation and volatility forecasting methods, the $h$-likelihood technique has a much wider range of applicability for it is not subjected to the rigidity of the state space representation and hence can be generalized to stochastic volatility models with more complex link functions than the second expression in~(\ref{arsv model}), to situations where the innovations are non-Gaussian, or there is a dependence between $\{ \epsilon _t\} $ and $\{ w _t\} $.
\end{remark}
\subsection{Equivalent martingale measures for stochastic volatility models}
\label{Equivalent martingale and quasi-martingale measures for stochastic volatility models}
Any technique for the pricing and hedging of options based on no-arbitrage theory requires the use of an equivalent measure for the probability space used for the modeling of the underlying asset under which its discounted prices are martingales with respect to the filtration generated by the observables. Measures with this property are usually referred to as risk-neutral or simply martingale measures and the Radon-Nikodym derivative that links this measure with the historical or physical one is called a pricing kernel.
A number of martingale measures have been formulated in the literature in the context of GARCH-like time discrete processes with predictable conditional volatility; see~\cite{Badescu:option:pricing} for a good comparative account of many of them. Those constructions do not generalize to the SV context mainly due to the fact that the volatility process $\{\sigma _t \} $ is not uniquely determined by the price process $ \{S _t\} $ and hence it is not predictable with respect to the filtration $ \mathcal{F}=\{\mathcal{F}_t \}$ generated by $ \{S _t\} $.
In this piece of work we will explore two solutions to this problem that, when applied to option pricing and hedging, yields a good combination of theoretical computability and numerical efficiency. The first one has to do with the so called minimal martingale measure and the second approach is based on an approximation inspired in the Extended Girsanov Principle~\cite{elliott:madan:mcmm}.
\medskip
\noindent {\bf The minimal martingale measure.} As we will see in the next section, this measure is particularly convenient when using local risk minimization with respect to the physical measure~\cite{foellmer:sondermann, foellmer:schweizer, foellmer:schied:book} as a pricing/hedging technique, for it provides the necessary tools to interpret the associated value process as an arbitrage free price for the contingent product under consideration.
The minimal martingale measure $Q_{{\rm min}}$ is a measure equivalent to $P $ defined by the following property: every $P$-martingale $M \in L ^2(\Omega,P)$ that is strongly orthogonal to the discounted price process $\widetilde{S}$, is also a $Q_{{\rm min}}$-martingale. The following result spells out the specific form that $Q _{{\rm min}} $ takes when it comes to the ARSV models.
\begin{theorem}
\label{introduction minimal martingale measure}
Consider the price process $S=\{S _0, S _1, \ldots, S_T\} $ associated to the ARSV model given by the expression~(\ref{arsv model}). In this setup, the minimal martingale measure is determined by the Radon-Nikodym derivative $dQ _{{\rm min}}/ d P $ that is obtained by evaluating at time $T$ the $P$-martingale $\{Z _t\}_{t \in \{1, \ldots, T\}} $ defined by
\begin{equation}
\label{mmm sv expression}
Z _t:=\prod_{k=1}^t \left(1+ \frac{\left(e^{K_{\epsilon_t}^P(\sigma _k)}-1\right)\left(e^{\sigma _k \epsilon _k}-e^{K_{\epsilon_t}^P(\sigma _k)}\right)}{e^{2K_{\epsilon_t}^P(\sigma _k)}-e^{K_{\epsilon_t}^P(2\sigma _k)}}\right).
\end{equation}
\end{theorem}
\noindent \noindent\textbf{Proof.\ \ } By Corollaries 10.28 and 10.29 and Theorem 10.30 in~\cite{foellmer:schied:book}, the minimal martingale measure, when it exists, is unique and is determined by the Radon-Nikodym derivative obtained by evaluating at $T$ the $P$-martingale
\begin{equation}
\label{mmm general expression}
Z _t:=\prod_{k=1}^t 1+ \lambda _k\left(\widetilde{Y} _k- \widetilde{Y}_{k-1}\right), \quad \mbox{where} \quad \lambda _k:=- \frac{E^P_{k-1}\left[\widetilde{S} _k- \widetilde{S}_{k-1}\right]}{{\rm var}_{k-1} \left(\widetilde{S} _k- \widetilde{S}_{k-1}\right)},
\end{equation}
and $\widetilde{Y} _k $ is the martingale part in the Doob decomposition of $\widetilde{S} _k $ with respect to $P$. Hence we have
\begin{equation}
\label{doob part sv}
\widetilde{Y} _k- \widetilde{Y}_{k-1}=\widetilde{S} _k- E^P_{k-1}\left[\widetilde{S}_{k}\right]=\widetilde{S} _{k-1} \left(e^{\sigma _k \epsilon _k}-e^{K^P_{\epsilon _k}(\sigma _k)}\right).
\end{equation}
The equality~(\ref{mmm sv expression}) follows from~(\ref{mmm general expression}),~(\ref{doob part sv}),~(\ref{expected increase 1}), and~(\ref{expected increase 2}). \quad $\blacksquare$
\begin{remark}
\label{minimal is signed}
\normalfont
The measure $Q _{{\rm min}} $ obtained by using~(\ref{mmm sv expression}) {\it is in general signed}. Indeed, as the random variable $ \sigma_k \epsilon_k $ is $\mathcal{F} _k $-adapted and $K_{\epsilon _k}^P(\sigma _k) $ is $\mathcal{F} _k $-predictable, the term $\left(e^{\sigma _k \epsilon _k}-e^{K^P_{\epsilon _k}(\sigma _k)}\right) $ can take arbitrarily negative values that can force $Z _t $ to become negative. We will see in our numerical experiments that even though this is in general the case, negative occurrences are extremely unlikely for the usual parameter values that one encounters in financial time series. Consequently, the bias introduced by censoring paths that yield negative Radon-Nikodym derivatives and using $Q _{{\rm min}} $ as a well-defined positive measure is not noticeable.
\end{remark}
\begin{remark}
\normalfont
The quantity $K_{\epsilon _t}^P(\sigma _t)=\log E^P\left[e^{\sigma_t \epsilon _t}| {\cal F}_{t-1}\right]$ in~(\ref{mmm sv expression}) is in general extremely difficult to compute. Lemma~\ref{relation cumulant functions} allows us to rewrite it as a conditional expectation of a function that depends only on $\sigma _t $ and hence it is reasonable to use the estimations $\sigma_t^k $ and $\sigma_t^h $ introduced in Section~\ref{Volatility and model estimation} to approximate $K_{\epsilon _t}^P(\sigma _t) $. For example, if the innovations $\{ \epsilon _t\} $ are Gaussian and we hence have $L_{\epsilon _t}^P(z)=z ^2/2 $, we will approximate
\begin{equation*}
K_{\epsilon _t}^P(\sigma _t)=\log E^P_{t-1}\left[e^{\frac{\sigma_t^2}{2}}\right]
\end{equation*}
by $(\sigma_t^k) ^2/2 $ or $(\sigma_t^h) ^2/2 $ depending on the technique (Kalman or $h$-likelihood, respectively) used to estimate the conditional volatility.
\end{remark}
\medskip
\noindent {\bf The Extended Girsanov Principle and the mean correcting martingale measure.} This construction has been introduced in~\cite{elliott:madan:mcmm} as an extension in discrete time and for multivariate processes of the classical Girsanov Theorem. This measure is designed so that when the process is considered with respect to it, its dynamical behavior coincides with that of its martingale component under the original historical probability measure. These martingale measures are widely used in the GARCH context (see for example~\cite{Badescu:option:pricing, ortega:garch:pricing} and references therein).
In this paragraph we will work with models that are slightly more general than~(\ref{arsv model}) in the sense that we will allow for predictable trend terms and generalized innovations, that is, our ARSV model will take the form
\begin{equation}
\label{arsv model general}
\left\{
\begin{array}{rcl}
y _t &= &m _t+ \sigma _t \epsilon _t, \qquad \{ \epsilon _t\} \sim {\rm IID}(0,1)\\
b _t&= & \gamma+ \phi b_{t-1}+ w _t, \qquad \{ w _t\} \sim {\rm IID}(0,\sigma _w^2)
\end{array}
\right.
\end{equation}
with $m _t $ an $\mathcal{F}_{t-1} $-measurable random variable.
\begin{theorem}
\label{mcmm arsv expression}
Given the model specified by~(\ref{arsv model general}), let $f_{\sigma _t \epsilon _t} ^P$ be the conditional probability density function under $P$ of the random variable $\sigma _t \epsilon _t $ given $\mathcal{F} _{t-1}$ and let $\{ N _t \} $ be the stochastic discount factors defined by:
\begin{equation*}
N _t:=\frac{f^P_{\sigma_t \epsilon _t}(\sigma _t\epsilon _t+ m _t -r+\log E^P_{t-1}\left[e^{\sigma _t\epsilon _t}\right])}{f^P_{\sigma_t \epsilon _t}(\sigma _t\epsilon _t)}.
\end{equation*}
The process $Z _t:=\prod_{k=1}^t N _k $, is a $(\mathcal{F}_t,P)$-martingale such that $E ^P[Z _t]=1 $ and $Z _T$ defines in the ARSV setup the martingale measure $Q _{{\rm EGP}}$ associated to Extended Girsanov Principle via the identity $Z _T= d Q_{{\rm EGP}}/d P$.
\end{theorem}
\begin{remark}
\normalfont
As $\sigma_t $ and $\epsilon _t $ are independent processes, Rohatgi's formula yields:
\begin{equation*}
f^P_{\sigma_t \epsilon _t} (z)=\int_{-\infty}^\infty \frac{1}{|x|}f^P_{\sigma_t} (x) f^P_{\epsilon _t} \left(\frac{z}{x}\right) d x.
\end{equation*}
Unfortunately, there is no closed form expression for this integral even in the Gaussian setup. Though {\it ad hoc} numerical have been developed to treat it (see for instance~\cite{glenetal:product:distributions}), the computation of the stochastic discount factors $N _t $ for each time step $t$ may prove to be a computationally heavy task.
\end{remark}
The point that we just raised in the remark leads us to introduce yet another martingale measure that mimics the good analytical properties of the Extended Girsanov Principle in the GARCH case but that, in the ARSV setup, only shares approximately its other defining features. In order to present this construction we define the {\bf market price of risk} process:
\begin{equation}
\label{market prices of risk}
\rho_t= \frac{m _t+K_{\epsilon _t}^P(\sigma _t)-r}{\sigma _t},
\end{equation}
to which we can associate what we will call the {\bf stochastic discount factor}:
\begin{equation*}
N _t(\epsilon _t, \rho _t ) = \frac{f_t ^P(\epsilon_t+ \rho _t)}{f_t ^P(\epsilon _t)},
\end{equation*}
where $f_t ^P $ is the conditional probability density function of the innovations $\{ \epsilon _t\} $ with respect to the measure $P$ given $ \mathcal{F} _{t-1}$.
\begin{theorem}
\label{quasi mean correcting martingale measure}
Using the notation that we just introduced, we have that
\begin{description}
\item [(i)]The process $Z _t:=\prod_{k=1}^t N _k $, is a $(\mathcal{F}_t,P)$-martingale such that $E ^P[Z _t]=1 $.
\item [(ii)] $Z _T$ defines an equivalent measure $Q _{{\rm mc}}$ such that $Z _T= d Q_{{\rm mc}}/d P$ under which the discounted price process $\left\{S _0, S _1, \ldots, S _T\right\}$ determined by~(\ref{arsv model general}) is a martingale. By an abuse of language we will refer to $Q_{{\rm mc}} $ as the ARSV {\bf mean correcting martingale measure}.
\end{description}
\end{theorem}
\section{Local risk minimization for ARSV options}
\label{Local risk minimization for ARSV options}
In this section we use the local risk minimization pricing/hedging technique developed by F\"ollmer, Schweizer, and Sondermann (see~\cite{foellmer:sondermann, foellmer:schweizer, schweizer:2001}, and references therein) as well as the different measures and volatility estimation techniques introduced in the previous section to come up with prices and hedging ratios for European options that have an ARSV process as model for the underlying asset.
As we will see, this technique provides simultaneous expressions for prices and hedging ratios that, even though require Monte Carlo computations in most cases, admit convenient interpretations based on the notion of hedging error minimization that can be adapted to various models for the underlying asset. This hedging approach has been studied in~\cite{ortega:garch:pricing} in the context of GARCH models with Gaussian innovations and in~\cite{Badescu:Ortega} for more general innovations. Additionally, the technique can be tuned in order to accommodate different prescribed hedging frequencies and hence adapts very well to realistic situations encountered by practitioners. In the negative side, like any other quadratic method, local risk minimization penalizes equally shortfall and windfall hedging errors, which may sometimes lead to inadequate hedging decisions.
\subsection{Generalized trading strategies and local risk minimization}
We now briefly review the necessary concepts on pricing by local risk minimization that we will needed in the sequel. The reader is encouraged to check with Chapter 10 of the excellent monograph~\cite{foellmer:schied:book} for a self-contained and comprehensive presentation of the subject.
As we have done so far, we will denote by $S _t $ the price of the underlying asset at time $t$. The symbol $r _t $ denotes the continuously composed risk-free interest rate paid on the currency of the underlying in the period that goes from time $t-1 $ to $t$; we will assume that $\{ r _t\} $ is a predictable process. Denote by
\begin{equation*}
R _t:=\sum_{j=0}^t r _j.
\end{equation*}
The price at time $t$ of the riskless asset $S ^0 $ such that $S _0 ^0=1 $, is given by $S _t ^0= {\rm e}^{R _t}$.
Let now $H(S _T)$ be a European contingent claim that depends on the terminal value of the risky asset $S _t $. In the context of an incomplete market, it will be in general impossible to replicate the payoff $H$ by using a self-financing portfolio. Therefore, we introduce the notion of {\bf generalized trading strategy}, in which the possibility of additional investment in the riskless asset throughout the trading periods up to expiry time $T$ is allowed. All the following statements are made with respect to a fixed filtered probability space $(\Omega, P, \mathcal{F}, \{ \mathcal{F} _t\}_{t \in\{0, \ldots, T\}})$:
\begin{itemize}
\item A {\bfseries\itshape generalized trading strategy} is a pair of stochastic processes $(\xi^0, \xi)$ such that $\{\xi^0_t\}_{t \in \{0, \ldots,T\}}$ is adapted and $\{\xi_t\} _{t \in \{1, \ldots,T\}}$ is predictable. The associated {\bfseries\itshape value process} $V $ of $(\xi^0, \xi) $ is defined as
\begin{equation*}
V _0:= \xi_0^0, \quad \mbox{ and } \quad V _t:= \xi_t^0 \cdot S ^0_t+ \xi_t\cdot S _t, \quad t\geq 1.
\end{equation*}
\item The {\bfseries\itshape gains process} $G$ of the generalized trading strategy $(\xi^0, \xi)$ is given by
\begin{equation*}
G _0:=0 \quad \mbox{and } \quad G _t:=\sum_{k=1}^t \xi_k\cdot (S _k-S_{k-1}), \quad t=0, \ldots ,T.
\end{equation*}
\item The {\bfseries\itshape cost process} $C$ is defined by the difference
\begin{equation*}
C _t:=V _t - G _t, \quad t=0, \ldots, T.
\end{equation*}
\end{itemize}
These processes have discounted versions $\widetilde{V} _t $, $\widetilde{G} _t $, and $\widetilde{C} _t $ defined as:
\begin{equation*}
\widetilde{V }_t:= V _t {\rm e}^{-R _t}, \qquad \widetilde{G} _t:=\sum_{k=1}^t \xi_k\cdot (\widetilde{S} _k-\widetilde{S}_{k-1}), \quad \mbox{and} \quad \widetilde{C} _t:=\widetilde{V} _t - \widetilde{G} _t.
\end{equation*}
Assume now that both $H$ and the $\{S _n\}_{n \in\{0, \ldots, T\}}$ are in $L ^2(\Omega, P)$. A generalized trading strategy is called {\bfseries\itshape admissible} for $H$ whenever it is in $L ^2(\Omega,P) $ and its associated value process is such that
\begin{equation*}
V _T=H, \quad {\rm P}\ {\it a.s.},\quad \quad V _t, G _t \in L ^2(\Omega,P), \quad \mbox{for each} \ t.
\end{equation*}
The hedging technique via local risk minimization consists of finding the strategies $(\widehat{\xi}^0, \widehat{\xi})$ that minimize the {\bfseries\itshape local risk process}
\begin{equation}
\label{local risk process}
R _t(\xi^0, \xi):=E^P \left[(\widetilde{C} _{t+1}-\widetilde{C} _t)^2\mid \mathcal{F}_t\right], \quad t=0, \ldots, T-1,
\end{equation}
within the set of admissible strategies $(\xi^0, \xi)$. More specifically,
the admissible strategy $(\widehat{\xi}^0, \widehat{\xi})$ is called {\bfseries\itshape local risk-minimizing} if
\begin{equation*}
R _t(\widehat{\xi}^0, \widehat{\xi})\leq R _t(\xi^0, \xi), \quad {\rm P}\ {\it a.s.}
\end{equation*}
for all $t$ and each admissible strategy $(\xi^0, \xi)$. It can be shown that~\cite[Theorem 10.9]{foellmer:schied:book} an admissible strategy is local risk-minimizing if and only if the discounted cost process is a $P$-martingale and it is strongly orthogonal to $\widetilde{S} $, in the sense that ${\rm cov} _t(\widetilde{S}_{t+1}-\widetilde{S }_t, \widetilde{C}_{t+1}-\widetilde{C} _t)=0 $, $P$-a.s., for any $t=0, \ldots, T-1 $. More explicitly, whenever a local risk-minimizing technique exists, it is fully determined by the backwards recursions:
\begin{eqnarray}
V _T &= & H,\label{hedge general 1}\\
\xi_{t+1} &= & \frac{{\rm cov}^P(\widetilde{V}_{t+1},\widetilde{S}_{t+1}-\widetilde{S }_t\mid \mathcal{F} _t)}{{\rm var}^P(\widetilde{S}_{t+1}-\widetilde{S }_t\mid \mathcal{F} _t)},\label{hedge general 2}\\
\widetilde{V} _t &= &E^P\left[\widetilde{V}_{t+1}\mid \mathcal{F} _t\right]- \xi_{t+1}E^P\left[\left(\widetilde{S}_{t+1}-\widetilde{S }_t\right)\mid \mathcal{F} _t\right].\label{hedge general 3}
\end{eqnarray}
The initial investment $V _0 $ determined by these relations will be referred to as the {\bf local risk minimization option price} associated to the measure $P$.
\subsection{Local risk minimization with respect to a martingale measure}
\label{Local risk minimization with respect to a martingale measure}
As we have explicitly indicated in expressions~(\ref{hedge general 1})--(\ref{hedge general 3}), the option values and the hedging ratios that they provide depend on a previously chosen probability measure and a filtration. In the absence of external sources of information, the natural filtration $\{\mathcal{F}_t\}$ to be used is the one generated by the price process. Regarding the choice of the probability measure, the first candidate that should be considered is the physical or historical probability associated to the price process due to the fact that from a risk management perspective this is the natural measure that should be used in order to construct the local risk~(\ref{local risk process}).
In practice, the use of the physical probability encounters two major difficulties: on one hand, when~(\ref{hedge general 1})--(\ref{hedge general 3}) are written down with respect to this measure, the resulting expressions are convoluted and numerically difficult to estimate due to the high variance of the associated Monte Carlo estimators; a hint of this difficulty in the GARCH situation can be seen in Proposition 2.5 of~\cite{ortega:garch:pricing}. Secondly, unless the discounted prices are martingales with respect to the physical probability measure or there is a minimal martingale measure available, the option prices that result from this technique cannot be interpreted as arbitrage free prices.
These reasons lead us to explore the local risk minimization strategy for martingale measures and, more specifically for the martingale measures introduced in Section~\ref{Equivalent martingale and quasi-martingale measures for stochastic volatility models}. Given that the trend terms that separate the physical measure from being a martingale measure are usually very small when dealing with daily or weekly financial returns, it is expected that the inaccuracy committed by using a conveniently chosen martingale measure will be smaller than the numerical error that we would face in the Monte Carlo evaluation of the expressions corresponding to the physical measure; see~\cite[Proposition 3.2]{ortega:garch:pricing} for an argument in this direction in the GARCH context. In any case, the next section will be dedicated to carry out an empirical comparison between the hedging performances obtained using the different measures in Section~\ref{Equivalent martingale and quasi-martingale measures for stochastic volatility models}, as well as the various volatility estimation techniques that we described and that are necessary to use them.
The following proposition is proved using a straightforward recursive argument in expressions~(\ref{hedge general 1})--(\ref{hedge general 3}) combined with the martingale hypothesis in its statement.
\begin{proposition}
Let $Q$ be an equivalent martingale measure for the price process $\{ S _t\} $ and $H(S _T)$ be a European contingent claim that depends on the terminal value of the risky asset $S _t $. The local risk minimizing strategy with respect to the measure $Q$ is determined by the recursions:
\begin{eqnarray}
V _T &= & H,\label{hedge general martingale 1}\\
\xi_{t+1} &= & \frac{1}{\Sigma_{t+1}^2}E^Q_{t}\left[e^{-(R _T+ R _t)}H(S _T)\left(S_{t+1}e^{-r_{t+1}}-S _t\right)\right],\label{hedge general martingale 2}\\
V _t &= &E^Q_{t}\left[e^{-(R _T-R _t)}H(S _T)\right],\label{hedge general martingale 3}
\end{eqnarray}
where
\begin{equation}
\label{sigma as variance}
\Sigma_{t+1}^2:= {\rm var} ^Q(\widetilde{S} _{t+1}-\widetilde{S} _t\mid \mathcal{F} _t) = e^{-2R _t}E^Q_{t}\left[S _{t+1}^2e^{-2 r_{t+1}}-S _t^2\right]=e^{-2R _t}{\rm var} ^Q(S _{t+1} {\rm e}^{-r_{t+1}}\mid \mathcal{F} _t).
\end{equation}
\end{proposition}
\begin{remark}
\normalfont
When expression~(\ref{hedge general martingale 3}) is evaluated at $t=0 $ it yields the initial investment $V _0 $ necessary to setup the generalized local risk minimizing trading strategy that fully replicates the derivative $H$ and coincides with the arbitrage free price for $H$ that results from using $Q$ as a pricing measure. Obviously, this connection only holds when local risk minimization is carried out with respect to a martingale measure.
\end{remark}
\begin{remark}
\normalfont
Local risk-minimizing trading strategies computed with respect to a martingale measure $Q$ also minimize~\cite[Proposition 10.34]{foellmer:schied:book} the so called {\bf remaining conditional risk}, defined as the process $R ^Q _t (\xi^0, \xi) :=E_t[(\widetilde{C} _T-\widetilde{C} _t)^2]$, $t=0, \ldots, T $; this is in general not true outside the martingale framework (see~\cite[Proposition 3.1]{schweizer:2001} for a counterexample). Analogously, local risk minimizing strategies are also {\bf variance-optimal}, that is, they minimize $E^Q\left[ \left(\widetilde{H}-V _0-\widetilde{G} _T\right)^2\right]$ (see~\cite[Proposition 10.37]{foellmer:schied:book}). This is particularly relevant in the ARSV context in which a standard sufficient condition (see~\cite[Theorem 10.40]{foellmer:schied:book}) that guarantees that local risk minimization with respect to the physical measure implies variance optimality does not hold; we recall that this condition demands for the deterministic character of the quotient
\begin{equation*}
\beta _t:=\frac{\left(E^P_{t-1}\left[S _t-S_{t-1}\right]\right)^2}{{\rm var}_{t-1}^P\left[ S _t-S_{t-1}\right]}.
\end{equation*}
Indeed, expressions~(\ref{expected increase 1}) and~(\ref{expected increase 2}), together with Lemma~\ref{equality with Ks} imply that in our situation
\begin{equation*}
\beta _t:=\frac{e^{2K_{\epsilon _t}^P(\sigma _t)}}{\left[e^{K_{\epsilon _t}^P(2\sigma _t)}-e^{2K_{\epsilon _t}^P(\sigma _t)}\right]},
\end{equation*}
which is obviously not deterministic.
\end{remark}
\begin{remark}
\label{remark about denominator}
\normalfont
The last equality in~(\ref{sigma as variance}) is of much importance when using Monte Carlo simulations to evaluate~(\ref{hedge general martingale 1})--(\ref{hedge general martingale 3}) and suggests the correct estimator that must be used in practice: if we generate under the measure $Q$ a set of $N$ price paths all of which start at $S _t $ at time $t$ and take values $S_{t+1}^1, \ldots, S_{t+1}^N $ at time $t+1 $, then in view of the last equality in~(\ref{sigma as variance}) we write
\begin{equation*}
\Sigma_{t+1}^2=e^{-2R _t}{\rm var} ^Q(S _{t+1} {\rm e}^{-r_{t+1}}\mid \mathcal{F} _t)\simeq \frac{e^{-2R _t}}{N}\sum_{i=1}^N \left(S _{t+1}^i {\rm e}^{-r_{t+1}}-S _t \right)^2.
\end{equation*}
We warn the reader that an estimator for $\Sigma_{t+1}^2 $ based on the Monte Carlo evaluation of
\[e^{-2R _t}E^Q_{t}\left[S _{t+1}^2e^{-2 r_{t+1}}-S _t^2\right] \] would have much more variance and once inserted in the denominator of~(\ref{hedge general martingale 2}) would produce unacceptable results.
\end{remark}
\begin{remark}
\normalfont
The Monte Carlo estimation of the expressions~(\ref{hedge general martingale 1})--(\ref{hedge general martingale 3}) requires the generation of price paths with respect to the martingale measure $Q$. In the GARCH context this can be easily carried out for a variety of pricing kernels by rewriting the process in connection with the new equivalent measure in terms of new innovations (see for example~\cite{duan:GARCH:pricing, ortega:garch:pricing, chorro:guegan:ielpo, Badescu:option:pricing}). This method can be combined with modified Monte Carlo estimators that enforce the martingale condition and that have a beneficial variance reduction effect like, for example, the Empirical Martingale Simulation technique introduced in~\cite{duan:ems1, duan:ems2}. Unfortunately, this is difficult to carry out in our context and, more importantly, it does not produce good numerical results. Given that all the pricing measures introduced in the previous section are constructed out of the physical one, the best option in our setup consists of carrying out the path simulation with respect to the physical measure and computing the expectations in~(\ref{hedge general martingale 1})--(\ref{hedge general martingale 3}) using the corresponding Radon-Nikodym derivative. More specifically, let $Q$ be an equivalent martingale measure that is obtained out of the physical measure $P$ by constructing a Radon-Nikodym derivative of the form:
\begin{equation*}
\frac{dQ}{dP}=\prod_{k=1}^T N _k,
\end{equation*}
such that the process $\{Z _t\} $ defined by $Z _t:=\prod_{k=1}^t N _k $ is a $P$-martingale that satisfies $ E^P\left[Z _t\right]=1 $; notice that all the measures introduced in the previous section are of that form. In that setup we define the process $\{F _t\} $ by $F _t:=\prod_{i=t}^T N _i$ that allows us to rewrite $Q$-expectations in terms of $P$-expectations, that is, for any $\mathcal{F}_T$ measurable function $f$ and any $ t \in \left\{ 0, \ldots, T\right\}$,
\begin{equation*}
E^Q\left[f\mid \mathcal{F}_t\right]=E^P\left[F_{t+1}f\mid \mathcal{F}_t\right].
\end{equation*}
The proof of this equality is a straightforward consequence of~\cite[Proposition A.11]{foellmer:schied:book} and of the specific way in which the measure $Q$ is constructed.
\end{remark}
\subsection{Local risk minimization and changes in the hedging frequency}
\label{Local risk minimization and changes in the hedging frequency}
As we already pointed out one of the major advantages of the local risk minimization hedging scheme when compared to other sensitivity based methods, is its adaptability to prescribed changes in the hedging frequency. Indeed, suppose that the life of the option $H$ with maturity in $T$ time steps is partitioned into identical time intervals of duration $j$; this assumption implies the existence of an integer $k$ such that $kj=T $.
We now want to set up a local risk minimizing replication strategy for $H$ in which hedging is carried out once every $j$ time steps. We will denote by $\xi_{t+j} $ the hedging ratio at time $t$ that presupposes that the next hedging will take place at time $t+j $. The value of such ratios will be obtained by minimizing the $j$-spaced local risk process:
\begin{equation*}
R _t^j(\xi^0, \xi):=E^P \left[(\widetilde{C} _{t+j}^j-\widetilde{C} _t^j)^2\mid \mathcal{F}_t\right], \quad t=0, j, 2j,\ldots, (k-1)j=T-j,
\end{equation*}
where $\{\widetilde{C} _t^j \}$ is a cost process constructed out of value and gains processes, $\{V _t ^j\} $ and $\left\{ G _t ^j\right\}$ that only take into account the prices of the underlying assets at time steps $t=0, j, 2j,\ldots, kj=T $, in particular, given an integer $l$ such that $t=lj $
\begin{equation*}
\widetilde{G} _t^j:=\sum_{r=1}^l \xi_{rj}\cdot (\widetilde{S} _{rj}-\widetilde{S}_{(r-1)j}).
\end{equation*}
A straightforward modification of the argument in~\cite[Theorem 10.9]{foellmer:schied:book} proves that the solution of this local risk minimization problem with modified hedging frequency with respect to a martingale measure is given by the expressions:
\begin{eqnarray}
V _T^j &= & H,\label{hedge general martingale frequency 1}\\
\xi_{t+j} &= & {\rm e}^{-(R _T-R _t)}\frac{E^Q_{t}\left[H(S _T)\left(S_{t+j}e^{-(R_{t+j}-R _t)}-S _t\right)\right]}{E^Q_{t}\left[S_{t+j}^2{\rm e}^{-2(R_{t+j}-R _t)}-S _t ^2\right]},\label{hedge general martingale frequency 2}\\
V _t ^j&= &E^Q_{t}\left[e^{-(R _T- R _t)}H(S _T)\right],\label{hedge general martingale frequency 3}
\end{eqnarray}
for any $t=0, j, 2j,\ldots, (k-1)j=T-j $. As a follow up to what we pointed out in the remark~\ref{remark about denominator}, the denominator in~(\ref{hedge general martingale frequency 2}) should be computed by first noticing that
\begin{equation*}
E^Q_{t}\left[S_{t+j}^2{\rm e}^{-2(R_{t+j}-R _t)}-S _t ^2\right]= {\rm var} _t^Q \left[
S_{t+j}{\rm e}^{-(R_{t+j}-R _t)}\right],
\end{equation*}
and then using the appropriate Monte Carlo estimator for the variance, that is,
\begin{equation}
\label{variance for denominator general}
{\rm var} ^Q(S _{t+1} {\rm e}^{-r_{t+1}}\mid \mathcal{F} _t)\simeq \frac{1}{N}\sum_{i=1}^N \left(S _{t+j}^i {\rm e}^{-(R_{t+j}^i-R _t)}-S _t \right)^2,
\end{equation}
where $S_{t+j}^1, \ldots, S_{t+j}^N $ are $N$ realizations of the price process under $Q$ at time $t+j $, using paths that have all the same origin $S _t$ at time $t$. If the interest rate process $\left\{ r _t\right\}$ is constant and equal to $r$, then~(\ref{variance for denominator general}) obviously reduces to
\begin{equation*}
{\rm var} ^Q(S _{t+1} {\rm e}^{-r_{t+1}}\mid \mathcal{F} _t)\simeq \frac{1}{N}\sum_{i=1}^N \left(S _{t+j}^i {\rm e}^{-jr}-S _t \right)^2.
\end{equation*}
\subsection{Local risk minimization and the minimal martingale measure}
Local risk-minimization requires picking a particular probability measure in the problem. We also saw that it is only when discounted prices are martingales with respect to that measure that the obtained value process for the option in question coincides with the arbitrage free price that one would obtain by using that martingale measure as a pricing kernel. This observation particularly concerns the physical probability which one would take as the first candidate to use this technique since it is the natural measure to be used when quantifying the local risk.
The solution of this problem is one of the motivations for the introduction of the minimal martingale measure $Q_{{\rm min}}$ that we defined in Section~\ref{Equivalent martingale and quasi-martingale measures for stochastic volatility models}. Indeed, it can be proved that (see Theorem 10.22 in~\cite{foellmer:schied:book}) the value process of the local risk-minimizing strategy {\it with respect to the physical measure} coincides with arbitrage free price for $H$ obtained by using $Q_{{\rm min}}$ as a pricing kernel. More specifically, if $V _t ^P $ is the value process for $H$ obtained out of formulas~(\ref{hedge general 1})--(\ref{hedge general 3}) using the physical measure then
\begin{equation}
\label{values with minimal martingale}
V _t ^P=E^{Q_{{\rm min}}}_{t}\left[e^{-(R _T- R _t)}H(S _T)\right].
\end{equation}
We emphasize that, as we pointed out in Remark~\ref{minimal is signed}, the minimal martingale measure in the ARSV setup is in general signed. Nevertheless, the occurrences of negative Radon-Nikodym derivatives are extremely unlikely, at least when dealing with Gaussian innovations, which justifies its use for local risk minimization.
We also underline that even though the value processes obtained when carrying out local risk minimization with respect to the physical and the minimal martingale measures are identical, the hedges are in general {\it not} the same and consequently so are the hedging errors. In the next proposition we provide a relation that links both hedges and identify situations in which they coincide. Before we proceed we need to introduce the notion of {\bf global (hedging) risk process}: it can be proved that once a probability measure $P$ has been fixed, if there exists a local risk-minimizing strategy $(\xi^0, \xi)$ with respect to it, then it is unique (see~\cite[Proposition 10.9]{foellmer:schied:book}) and the discounted payoff $\widetilde{H}$ can be decomposed as (see~\cite[Corollary 10.14]{foellmer:schied:book})
\begin{equation}
\label{decomposition payoff risk minimizing}
\widetilde{H}=V _0+\widetilde{G} _T+\widetilde{L} _T,
\end{equation}
with $\widetilde{G} _t $ the discounted gains process associated to $(\xi^0, \xi)$ and $\widetilde{L} _t:=\widetilde{C} _t-C _0 $, $t=0, \ldots,T $ a sequence $\{ \widetilde{L} _t\}_{t \in \{0, \ldots, T\}}$ that we will call the (discounted) global (hedging) risk process. $\{\widetilde{L} _t\}_{t \in \{0, \ldots, T\}}$ is a square integrable $P$-martingale that satisfies $L _0 =0 $ and that is {\bf strongly orthogonal} to $\widetilde{S}$ in the sense that
\begin{equation*}
{\rm cov} ^P((\widetilde{L}_{t+1}-\widetilde{L} _t)(\widetilde{S}_{t+1}-\widetilde{S} _t)\mid \mathcal{F} _t)=0 \quad \mbox{for any} \quad t=0, \ldots, T-1.
\end{equation*}
The decomposition~(\ref{decomposition payoff risk minimizing}) shows that $\widetilde{L} _T $ measures how far $\widetilde{H} $ is from the terminal value of the self-financing portfolio uniquely determined by the initial investment $V _0 $ and the trading strategy given by $\{\xi _t\} $ (see~\cite[Proposition 1.1.3]{lamberton:lapeyre}).
\begin{proposition}
\label{hedges under change of measure}
Let $H(S _T)$ be a European contingent claim that depends on the terminal value of the risky asset $S _t $ and let $\{ \xi _t^{Q_{{\rm min}}}\}_{t=1}^T $ and $\{ \xi _t^{P}\}_{t=1}^T $ be the local risk minimizing hedges associated to the minimal martingale measure $Q_{{\rm min}} $ and the physical measure $P$, respectively. Let $\left\{ \widetilde{L }_t ^P\right\}_{t=0}^T $ be the associated $P$-global risk process. Then, for any $t \in \left\{ 1, \ldots, T\right\}$:
\begin{equation}
\label{relation hedges}
\xi_t^{Q_{{\rm min}}}= \xi _t ^P+ \frac{E^{Q_{{\rm min}}}_{t-1}\left[ \widetilde{L }_t ^P (\widetilde{S} _t- \widetilde{S}_{t-1})\right]}{{\rm var}_{t-1}^{Q_{{\rm min}}} \left[\widetilde{S} _t- \widetilde{S}_{t-1}\right]}.
\end{equation}
If the processes $\left\{ \widetilde{L }_t ^P\right\}_{t=0}^T $ and $\left\{ \widetilde{S}_t\right\}_{t=0}^T $ are either $ Q_{{\rm min}} $--strongly orthogonal or $Q_{{\rm min}} $--independent, then $\{ \xi _t^{Q_{{\rm min}}}\}_{t=1}^T =\{ \xi _t^{P}\}_{t=1}^T $.
\end{proposition}
\section{Empirical study}
\label{Empirical study}
The goal of this section is comparing the hedging performances obtained by implementing the local risk minimization scheme using the different volatility estimation techniques and martingale measures introduced in Section~\ref{Auto-regressive stochastic volatility (ARSV) models}. In order to enrich the discussion, we will add to the list two standard sensitivity based hedging methods that provide good results in other contexts, namely standard Black-Scholes~\cite{black:scholes:paper} and an adaptation of Duan's static hedge~\cite{duan:GARCH:pricing} to the ARSV context.
The way in which we proceed with the comparison consists of taking a particular ARSV model as price paths generating process and then carrying out the replication of a European call option according to the prescribed list of hedging methods for various moneyness, strikes and hedging frequencies. At the end of the life of each of this option, the square hedging error is recorded. This task is carried out for a number of independently generated price paths which allows us to estimate a mean square hedging error that we will use to compare the different hedging techniques.
\medskip
\noindent {\bf The price generating ARSV process.} The price process chosen for our simulations is obtained by taking the model that has as associated log-returns one of the ARSV prescriptions studied in~\cite{delcastillo:lee:2}. It consists of taking the following parameters in~(\ref{arsv model}):
\begin{equation*}
r=0.1/252, \qquad \gamma=-0.821, \qquad \phi=0.9, \qquad \sigma _w=0.675.
\end{equation*}
The use of formulas~(\ref{variance and kurtosis}) shows that the resulting log-returns are very leptokurtic (${\rm kurtosis}=33.00 $) and volatile (the stationary variance is $9.01 \cdot 10^{-4} $ which amounts to an annualized volatility of 47\%). The intentionality behind this choice is exacerbating the defects of the different hedging methods in order to better compare them.
\medskip
\noindent {\bf The hedging methods.} Given the price paths generated by the ARSV process detailed in the previous paragraph, we will carry out the replication of European call options that have them as underlying asset using local risk minimization with respect to the two martingale measures introduced in Section~\ref{Equivalent martingale and quasi-martingale measures for stochastic volatility models}, namely:
\begin{itemize}
\item {\bf ARSV mean correcting martingale measure introduced in Theorem~\ref{quasi mean correcting martingale measure}}. The hedging method is abbreviated as LRM-MCMM in the figures.
\item {\bf Minimal martingale measure}. Denoted by $Q_{{\rm min}}$ and introduced in Theorem~\ref{introduction minimal martingale measure}. The hedging method is abbreviated as LRM-MMM in the figures. This measure may be signed in the ARSV context but this happens with extremely low probability (see Remark~\ref{minimal is signed}); in our simulations we did not encounter this situation even a single time. Local risk minimization with respect to this measure yields the same value process as the physical probability but not the same hedging ratios; Proposition~\ref{hedges under change of measure} spells out the difference and provides sufficient conditions for them to coincide.
\end{itemize}
The construction of all these measures requires conditional expectations of functions of the conditional variance of the log-returns with respect to the filtration generated by the price process. As this filtration does not determine the conditional volatilities, those expectations cannot be easily computed; we hence proceed by estimating the conditional volatilities via
Kalman filtering and a prediction-update routine based on the $h$-likelihood approach, both briefly described in Section~\ref{Volatility and model estimation}. Different volatility estimation techniques lead to different pricing/hedging kernels and hence to different hedging performances that we compare in our simulations.
We will complete the comparison exercise with two more methods:
\begin{itemize}
\item {\bf Black-Scholes}: we forget that the price process is generated via an ARSV model and we handle the hedging using the standard Black-Scholes delta~\cite{black:scholes:paper} as if the underlying was a realization of a log-normal process with constant drift and volatility given by the drift and marginal volatility (given by the expression~(\ref{variance and kurtosis})) of the ARSV model specified above.
\item {\bf Duan's static delta hedge}: this is a generalization of the Black-Scholes delta hedge that has been introduced in~\cite[Corollary 2.4]{duan:GARCH:pricing} in the GARCH context. In that result the author computes the derivative of a European call price (the delta put can be readily obtained out of the call-put parity) with respect to the price of the underlying in an arbitrary incomplete situation in which that option price is simply obtained as the expectation of the discounted payoff with respect to a given pricing kernel $Q $, that is:
\begin{equation*}
V _t = E^Q\left[e^{-(R _T-R _t)}H(S _T)\mid \mathcal{F}_t\right].
\end{equation*}
In those circumstances:
\begin{equation*}
\xi^{{\rm Duan}}_{t+1}:=\frac{\partial V _t}{\partial S _t}=e^{-(R _T-R _t)} E^Q\left[\frac{S _T}{S _t}\boldsymbol{1}_{S _T\geq K}\mid \mathcal{F} _t\right].
\end{equation*}
In our empirical study we will use this hedging technique using the various pricing kernels previously introduced. A {\it dynamical} refinement of this hedge can be obtained by considering additional dependences of the value process on the underlying price that may occur (like in our case) through the volatility process~\cite{garcia:renault, Badescu:Ortega}. This will not be considered in our study.
\end{itemize}
\medskip
\noindent {\bf Numerical specifications.} The comparisons in performance will be carried out by hedging batches of European call options with different maturities and three different moneyness $S _0/K $ of 1.11, 1, and 0.90, and for a number of independent price paths. The initial price $S _0$ of the underlying is always equal to $100 $. We will assign to each hedging experiment a prescribed hedging frequency. The mean square hedging error associated to each individual hedging method will be computed by taking the mean of the terminal square hedging errors for each of the price paths. In the particular case of the local risk minimization method, where the hedges are obtained by computing the expressions~(\ref{hedge general martingale frequency 1})--(\ref{hedge general martingale frequency 3}), the conditional expectations that they contain will be estimated via Monte Carlo evaluations based on the use of $2500 $ price paths.
\medskip
\noindent {\bf The hedging exercises.} We now explain in detail the different hedging experiments that we have carried out.
\begin{itemize}
\item {\bf Exercise 1}: all the hedging techniques explained in the paper are used. The four maturities considered are $6$, $8$, $10 $, and $12 $ time steps. Hedging is carried out daily. The computation of the mean square hedging error is based on $1000 $ price paths. The first thing that we notice is that, even for these short maturities, the performance of Duan's static hedge is very poor. This is specific to the ARSV situation for in the GARCH case this technique provides very competitive results~\cite{ortega:garch:pricing} even in the presence of processes excited by leptokurtic innovations~\cite{Badescu:Ortega}. Additionally, the Black-Scholes hedging scheme provides remarkably good results, specially for in and at the money options. Regarding the local risk minimization strategy, the best results are obtained when using a combination of a Kalman based estimation of the conditional volatility and the minimal martingale measure $Q_{{\rm min}}$.
\begin{figure}[!htp]
\includegraphics[scale=.25,angle=-90]{results_duan_complete_final.png}
\caption{Experiment 1. All the hedging techniques explained in the paper are used. The four maturities considered are $6$, $8$, $10 $, and $12 $ time steps. Hedging is carried out daily. The computation of the mean square hedging error is based on $1000 $ price paths.}
\label{fig:prices}
\end{figure}
\item {\bf Exercise 2}: we have retaken the specifications of Exercise $1 $, but this time around we have eliminated from the list of techniques under examination Duan's static hedge due to its poor performance. The four maturities considered are $10$, $20$, $30 $, and $40 $ time steps. Hedging is carried out every $10 $ days. The main difference with the previous exercise is the degradation of the performance of the Black-Scholes scheme in comparison with local risk minimization. This is due to the fact that, unlike Black-Scholes, local risk minimization admits adjustment to the change of hedging frequency by using the formulas~(\ref{hedge general martingale frequency 1})--(\ref{hedge general martingale frequency 3}). Regarding local risk minimization it is still the combination of the minimal martingale measure with Kalman based estimation for the volatilities that performs the best.
\begin{figure}[!htp]
\includegraphics[scale=.25,angle=-90]{results_BS_final.png}
\caption{Experiment 2. The four maturities considered are $10$, $20$, $30 $, and $40 $ time steps. Hedging is carried out every $10 $ days. The computation of the mean square hedging error is based on $1000 $ price paths.}
\label{fig:prices}
\end{figure}
\item {\bf Exercise 3}: we experiment with longer maturities and smaller hedging frequencies. We consider maturities of $20 $, $40 $, $60 $, $80 $, $100 $, and $120 $ time steps with hedging carried out every $20 $ days. The computation of the mean square hedging error is based on $600 $ price paths. The novelty in these experiments with respect to the preceding ones is that in some instances the volatility estimation via $h$--likelihood seems more appropriate than Kalman's method in this setup.
\begin{figure}[!htp]
\includegraphics[scale=.25,angle=-90]{results_long_final.png}
\caption{Experiment 6. The hedging techniques used are spelled out in the legend. Six maturities are considered: $40$, $60$, $80 $, and $100$ time steps. Hedging is carried out every $20 $ days. The computation of the mean square hedging error is based on $600 $ price paths.}
\label{fig:prices}
\end{figure}
\end{itemize}
\section{Conclusions}
In this work we have reported on the applicability of the local risk minimization pricing/hedging technique in the handling of European options that have an auto-regressive stochastic volatility (ARSV) model as their underlying asset. The method has been implemented using the following martingale measures:
\begin{itemize}
\item The minimal martingale measure: even though it is in general signed, the probability of occurrence of negative Radon-Nikodym derivatives is extremely low and hence can be used to serve our purposes.
\item Mean correcting martingale measure: it is a measure that shares the same expression and good analytical and numerical properties as the measure obtained in the GARCH context out of the Extended Girsanov Principle. The reason to use this proxy instead of the genuine Extended Girsanov Principle is that in the ARSV case, this procedure produces a measure that is numerically expensive to evaluate.
\end{itemize}
An added difficulty when putting this scheme into practice and that is specific to ARSV models, has to do with the need of estimating the volatility that, even though in this situation is not determined by the observable price process, it is necessary in order to construct the pricing/hedging kernels. Two techniques are used to achieve that: Kalman filtering and hierarchical likelihood ($h$-likelihood). Both methods are adequate volatility estimation techniques, however the $h$-likelihood technique has a much wider range of applicability for it is not subjected to the rigidity of the state space representation necessary for Kalman and hence can be used for stochastic volatility models with complex link functions or when innovations are non-Gaussian.
We have carried out a numerical study to compare the hedging performance of the methods based on different measures and volatility estimation techniques. Overall, the most appropriate technique seems to be the choice that combines the minimal martingale measure with a Kalman filtering based estimation of the volatility. $h$-likelihood based volatility estimation gains pertinence as maturities become longer and the hedging frequency diminishes.
\section{Appendix}
\label{Appendix}
\subsection{Proof of Lemma~\ref{relation cumulant functions}}
Let $Z$ be an arbitrary $\mathcal{F}_{t-1} $ measurable random variable. As $ \epsilon _t $ is independent of $\mathcal{F}_{t-1} $ and by hypothesis it is also independent of $u$, the joint law $P_{\epsilon _t, u,Z} $ of the random variable $( \epsilon _t, u,Z):(\Omega,P)\rightarrow \mathbb{R} ^3 $ can be written as the product $P_{\epsilon _t, u,Z} (x,y,z)=P_{\epsilon_t} (x)P _{u,Z} (y,z)$. Hence, using Fubini's Theorem we have
\begin{eqnarray}
E^P\left[e^{u \epsilon _t}Z\right]&=&\int\int e^{yx}z dP_{\epsilon_t} (x)dP _{u,Z} (y,z)
=\int\left[\int e^{yx} dP_{\epsilon_t} (x)\right]z dP _{u,Z} (y,z)\notag\\
&= &\int e^{L_{\epsilon _t}(y)}z dP _{u,Z} (y,z)= E^P\left[e^{L_{\epsilon _t}(u)}Z\right].\label{intermediate conditional}
\end{eqnarray}
At the same time, by the definition of $K_{\epsilon _t}^P$, for any $\mathcal{F}_{t-1} $ measurable random variable $Z$, the equality~(\ref{intermediate conditional}) implies that:
\begin{equation*}
E^P\left[e^{K_{\epsilon _t}^P (u)}Z\right]=E^P\left[e^{u\epsilon _t}Z\right]=E^P\left[e^{L_{\epsilon _t}(u)}Z\right].
\end{equation*}
As $e^{K_{\epsilon _t}^P (u)} $ is $\mathcal{F}_{t-1} $ measurable and $Z$ arbitrary, the almost sure uniqueness of the conditional expectation implies that
\begin{equation*}
e^{K_{\epsilon _t}^P (u) }=E^P\left[e^{L_{\epsilon _t}(u)}\mid \mathcal{F}_{t-1}\right],
\end{equation*}
as required. \quad $\blacksquare$
\medskip
The result that we just proved is a particular case of the following Lemma that we state for reference. The proof follows the same pattern as in the previous paragraphs.
\begin{lemma}
\label{generalized cumulant functions}
Let $\left(\Omega, \mathcal{F}, P\right) $ be a probability space and $\mathcal{B} \subset \mathcal{F} $ a sub-$\sigma$-algebra. Let $X$ and $Y$ be two random variables such that $X$ is simultaneously independent of $Y$ and $\mathcal{B} $. For any measurable function $\Phi: \mathbb{R}^2 \rightarrow \mathbb{R} $ define the mapping $F: \mathbb{R} \rightarrow \mathbb{R} $ by $F (y)= E\left[\Phi(X,y)\right]$. Then,
\begin{equation*}
E\left[\Phi(X,Y)\mid \mathcal{B}\right]=E[F (Y)\mid \mathcal{B}].
\end{equation*}
\end{lemma}
\subsection{Proof of Theorem~\ref{mcmm arsv expression}}
The proof is a straightforward consequence of Theorem 3.1 in~\cite{elliott:madan:mcmm}. Indeed, it suffices to identify in the ARSV case the two main ingredients necessary to carry out the Extended Girsanov Principle, namely the one period excess discounted return $\mu_t $ defined by:
\begin{equation*}
e^{\mu_t}:= E^P_{t-1}\left[\frac{\widetilde{S} _t}{\widetilde{S} _{t-1}}\right]=e^{m _t-r}E^P_{t-1}\left[e^{\sigma_t \epsilon _t}\right]= \frac{e^{m _t-r}}{K},
\end{equation*}
with $K:=1/E^P_{t-1}\left[e^{\sigma_t \epsilon _t}\right] $, and the process $\{W _t\}$ uniquely determined by the relation:
\begin{equation*}
W _t:= \frac{\widetilde{S} _t}{\widetilde{S}_{t-1}}e^{-\mu _t}=Ke^{\sigma_t \epsilon _t}.
\end{equation*}
The Theorem follows from expression (3.8) in~\cite{elliott:madan:mcmm}. \quad $\blacksquare$
\subsection{Proof of Theorem~\ref{quasi mean correcting martingale measure}}
\noindent\textbf{(i)} We start by noticing that since $\epsilon_t $ is independent of both $ \rho_t $ and $\mathcal{F}_{t-1} $, by Lemma~\ref{generalized cumulant functions},
\begin{equation}
\label{for later 33}
E^P_{t-1}\left[N _t (\epsilon _t, \rho _t )\right]= E^P_{t-1}\left[F(\rho _t )\right],
\end{equation}
where the real function $F: \mathbb{R} \rightarrow \mathbb{R}$ is defined by
\begin{equation*}
F (y)= E^P\left[N _t (\epsilon _t, y)\right]=\int \frac{f ^P(x+ y)}{f ^P(x)}f ^P(x) dx=\int f ^P(x+ y) dx=\int f ^P(x) dx=1,
\end{equation*}
which substituted in~(\ref{for later 33}) yields
\begin{equation}
\label{integral equal to 1}
E^P_{t-1}\left[N _t (\epsilon _t, \rho _t )\right]=1.
\end{equation}
This equality proves immediately the martingale property for $\{Z _t \} $. Indeed,
\begin{equation*}
E^P_{t-1}\left[Z _t \right]=E^P_{t-1}\left[N_t \right]Z _{t-1} =Z _{t-1}.
\end{equation*}
Finally, as $\{Z _t \} $ is a $P$-martingale
\begin{equation*}
E^P\left[Z _t \right]= E^P\left[Z _1 \right]= E^P\left[N _1 \right]=1.
\end{equation*}
\medskip
\noindent {\bf (ii)} $Z _T $ is by construction non-negative and $E[Z _T ]=\mathbb{P}(Z _T >0)=1 $. This guarantees (see, for example, Remarks after Theorem 4.2.1 in~\cite{lamberton:lapeyre}) that $Q _{{\rm mc}}$ is a probability measure equivalent to $P$.
We now start proving that $Q _{{\rm mc}}$ is a martingale measure by showing that:
\begin{equation}
\label{simple return for martingale 1}
E^{Q^1 _{{\rm mc}}}_{t-1}\left[\frac{S _t}{S _{t-1}}\right]= E^P_{t-1}\left[e^{m _t+ \sigma _t \epsilon _t}N _t\right].
\end{equation}
Indeed,
\begin{eqnarray*}
E^{Q^1 _{{\rm mc}}}_{t-1}\left[\frac{S _t}{S _{t-1}}\right]&= & E^{Q^1 _{{\rm mc}}}_{t-1}\left[e^{m _t+ \sigma _t \epsilon _t}\right]= \frac{1}{E^P_{t-1}\left[Z _T\right]}E^P_{t-1}\left[Z _Te^{m _t+ \sigma _t \epsilon _t}\right]= \frac{1}{Z _{t-1}}E^P_{t-1}\left[Z _Te^{m _t+ \sigma _t \epsilon _t}\right]\\
&=&
\frac{1}{Z _{t-1}}E^P_{t-1}\left[E^P_{T-1}\left[Z _Te^{m _t+ \sigma _t \epsilon _t}\right]\right]= \frac{1}{Z _{t-1}}E^P_{t-1}\left[E^P_{T-1}\left[Z _T\right]e^{m _t+ \sigma _t \epsilon _t}\right]\\
&=&\frac{1}{Z _{t-1}}E^P_{t-1}\left[Z _{T-1}e^{m _t+ \sigma _t \epsilon _t}\right]= \cdots
=\frac{1}{Z _{t-1}}E^P_{t-1}\left[Z _{t}e^{m _t+ \sigma _t \epsilon _t}\right]\\
&= &\frac{1}{Z _{t-1}}E^P_{t-1}\left[N _tZ _{t-1}e^{m _t+ \sigma _t \epsilon _t}\right]=E^P_{t-1}\left[e^{m _t+ \sigma _t \epsilon _t}N _t\right].
\end{eqnarray*}
Now, as $\epsilon _t $ is independent of both $\sigma_t $ and $\mathcal{F} _{t-1} $, we can use Lemma~\ref{generalized cumulant functions} to prove that
\begin{equation}
\label{second intermediate simple 1}
E^P_{t-1}\left[e^{m _t+ \sigma _t \epsilon _t}N _t\right] = E^P_{t-1}\left[e^{m _t- \sigma _t\rho _t+L_{\epsilon _t}(\sigma _t)}\right].
\end{equation}
Indeed, we can write
\begin{eqnarray}
\label{second intermediate simple 2}
E^P_{t-1}\left[e^{m _t+ \sigma _t \epsilon _t}N _t\right] &= &e^{m _t} E^P_{t-1}\left[e^{\sigma _t \epsilon _t}N _t(\epsilon _t, \rho _t)\right]=e^{m _t} E^P_{t-1}\left[F(\sigma _t, \rho _t)\right],
\end{eqnarray}
where the function $F: \mathbb{R} ^2\rightarrow \mathbb{R} $ is defined by
\begin{eqnarray*}
F(y,z)&=& E^P\left[e^{y \epsilon _t}N _t(\epsilon _t, z)\right]=\int e^{yx} f ^P(x) \frac{f ^P(x+z)}{f ^P(x)}d x=\int e^{y(s-z)}f ^P(s) d s\\
&= & E^P\left[e^{y(\epsilon _t -z)}\right]= e^{-yz}e^{L_{\epsilon _t}^P (y)},
\end{eqnarray*}
which substituted in~(\ref{second intermediate simple 2}) yields~(\ref{second intermediate simple 1}). Finally, if we insert in~(\ref{second intermediate simple 1}) the explicit expression~(\ref{market prices of risk}) that defines the market price of risk, we obtain that
\begin{equation*}
E^{Q^1 _{{\rm mc}}}_{t-1}\left[\frac{S _t}{S _{t-1}}\right]=E^P_{t-1}\left[e^{m _t- \sigma _t\rho _t+L_{\epsilon _t}(\sigma _t)}\right]= e ^{r-K_{\epsilon _t}^P(\sigma _t)}E^P_{t-1}\left[ e^{L^P_{\epsilon _t}(\sigma_t)}\right]=e ^r,
\end{equation*}
as required. The last equality in the previous expression follows from Lemma~\ref{relation cumulant functions}. \quad $\blacksquare$
\subsection{Proof of Proposition~\ref{hedges under change of measure}}
Consider the unique decomposition~(\ref{decomposition payoff risk minimizing}) of the discounted payoff $\widetilde{H } $ as a sum of the discounted $P$-gains and $P$-risk processes:
\begin{equation}
\label{decomposition at T}
\widetilde{H}=V _0+\sum_{k=1}^T \xi _k^P \cdot \left(\widetilde{S} _k- \widetilde{S}_{k-1}\right)+\widetilde{L} _T^P.
\end{equation}
We recall that the minimal martingale measure is characterized by the property that every $P$-martingale that is strongly orthogonal to the discounted price process $\left\{ \widetilde{S}_t\right\}_{t=0}^T $, is also a $Q_{{\rm min}}$-martingale. Hence, as the $P$-martingale $\left\{ \widetilde{L }_t ^P\right\}_{t=0}^T $ is $P$-strongly orthogonal to $\left\{ \widetilde{S}_t\right\}_{t=0}^T $, it is therefore a $Q_{{\rm min}}$-martingale. Having this in mind, as well as~(\ref{values with minimal martingale}), we take conditional expectations $E^{ Q_{{\rm min}}}_{t-1}$ on both sides of the decomposition~(\ref{decomposition at T}) and we obtain:
\begin{equation*}
\widetilde{V} _t^P=V _0+\sum_{k=1}^t \xi _k^P \cdot \left(\widetilde{S} _k- \widetilde{S}_{k-1}\right)+\widetilde{L} _t^P.
\end{equation*}
As $\widetilde{V} _t^P $ is the local risk minimizing value process for both the $P$ and the $Q_{{\rm min}} $ measures, that is $\widetilde{V} _t^P= \widetilde{V} _t^{Q_{{\rm min}}}$, if we multiply both sides of by $ \left(\widetilde{S} _t- \widetilde{S}_{t-1}\right)$ and take conditional expectations $E^{ Q_{{\rm min}}}_{t-1}$ we have:
\begin{eqnarray}
E^{ Q_{{\rm min}}}_{t-1}\left[\widetilde{V} _t^{ Q_{{\rm min}}} \left(\widetilde{S} _t- \widetilde{S}_{t-1}\right)\right]&=&E^{ Q_{{\rm min}}}_{t-1}\left[V _0\left(\widetilde{S} _t- \widetilde{S}_{t-1}\right)\right]+\sum_{k=1}^t E^{ Q_{{\rm min}}}_{t-1}\left[\xi _k^P \cdot \left(\widetilde{S} _k- \widetilde{S}_{k-1}\right)\left(\widetilde{S} _t- \widetilde{S}_{t-1}\right)\right]\notag\\
& &+E^{ Q_{{\rm min}}}_{t-1}\left[\widetilde{L} _t^P\left(\widetilde{S} _t- \widetilde{S}_{t-1}\right)\right]\notag\\
&= &\xi _t^P E^{ Q_{{\rm min}}}_{t-1}\left[ \left(\widetilde{S} _t- \widetilde{S}_{t-1}\right)^2\right]+E^{ Q_{{\rm min}}}_{t-1}\left[\widetilde{L} _t^P \left(\widetilde{S} _t- \widetilde{S}_{t-1}\right)\right]\notag.
\end{eqnarray}
If we divide both sides of this equality by $E^{ Q_{{\rm min}}}_{t-1}\left[ \left(\widetilde{S} _t- \widetilde{S}_{t-1}\right)^2\right] $ and we use the relation~(\ref{hedge general 2}) we obtain that
\begin{equation*}
\xi_t^{Q_{{\rm min}}}= \xi _t ^P+ \frac{E^{Q_{{\rm min}}}_{t-1}\left[ \widetilde{L }_t ^P (\widetilde{S} _t- \widetilde{S}_{t-1})\right]}{{\rm var}_{t-1}^{Q_{{\rm min}}} \left[\widetilde{S} _t- \widetilde{S}_{t-1}\right]},
\end{equation*}
as required. Finally, we notice that
\begin{equation*}
E^{Q_{{\rm min}}}_{t-1}\left[ \left(\widetilde{L }_t ^P-\widetilde{L }_{t-1} ^P \right) (\widetilde{S} _t- \widetilde{S}_{t-1})\right]=E^{Q_{{\rm min}}}_{t-1}\left[ \widetilde{L }_t ^P (\widetilde{S} _t- \widetilde{S}_{t-1})\right]-E^{Q_{{\rm min}}}_{t-1}\left[ \widetilde{L }_{t-1} ^P (\widetilde{S} _t- \widetilde{S}_{t-1})\right]=E^{Q_{{\rm min}}}_{t-1}\left[ \widetilde{L }_t ^P (\widetilde{S} _t- \widetilde{S}_{t-1})\right],
\end{equation*}
hence, either the $Q_{{\rm min}} $-independence or the orthogonality of $\left\{ \widetilde{L }_t ^P\right\}_{t=0}^T $ and $\left\{ \widetilde{S}_t\right\}_{t=0}^T $ imply that $E^{Q_{{\rm min}}}_{t-1}\left[ \widetilde{L }_t ^P (\widetilde{S} _t- \widetilde{S}_{t-1})\right]=0 $ and hence $\{ \xi _t^{Q_{{\rm min}}}\}_{t=1}^T =\{ \xi _t^{P}\}_{t=1}^T $. \quad $\blacksquare$
\addcontentsline{toc}{section}{Bibliography}
\bibliographystyle{alpha}
|
1,108,101,565,688 | arxiv | \section{\qquad Introduction}
This work was originally motived by mathematical proofs that a positive
frequency field cannot be confined to a finite region of space. According to
the Hegerfeldt theorem a positive frequency field localized in a finite region
for an instant spreads immediately throughout space \cite{Hegerfeldt}. It has
been shown explicitly in the case of one dimensional square wells
\cite{Karpov} and for three dimensional position eigenvectors
\cite{HawtonDebierre,MaxwellQM} that this instantaneous localization is only
apparent since it is due to destructive interference of intrinsically nonlocal
counterpropagating waves. In algebraic quantum field theory (QFT),\ the
Reeh-Schleider theorm states that there are no local annihilation or creation
operators \cite{ReehSchlieder}. However, confinement of real fields to a
finite region is not a problem in classical electromagnetism (EM). It will be
proved here that this use of real fields can be extended to photon quantum
mechanics (QM).
The QM of electrons and other Fermions is well understood but a consistent
first quantized theory of the photon has been elusive. Photons have two
properties not shared with electrons that have made derivation of photon QM
difficult - they are neural and massless. While fields describing charged
particles are intrinsically complex, neutral particles should be described by
real fields. Reality of the photon wave function ensures that photons and
antiphotons, being indistinguishable, are equally probable and that, after
second quantization, their field operators become Hermitian. This property is
problematic in a first quantized theory since the standard relativistic scalar
product is zero for neutral particles. Also, while the Wigner little group
describing massive particles is the set of spatial rotations, the Wigner
little group for massless particles is cylindrically symmetrical. In their
seminal paper titled "Localized states of elementary systems", Newton and
Wigner assumed invariance under spherically symmetrical rotations and
concluded that "for equations with zero mass .. with spin 1 (i.e. Maxwell's
equations) we found that no localized states in the above sense exist. This is
an unsatisfactory .. feature of our work" \cite{NW}. Both of these
difficulties have been overcome
\cite{HawtonPosOp,HawtonDebierre,MostafazadehZamani,BabaeiMostafazadeh,MaxwellQM,WignerLittleGroup}
but here we will extend this work by formulating photon QM in terms of
physically correct real fields and show that this leads to a significant
simplification of the mathematics.
In field theory, particles that transform into themselves are represented by
real fields and Hermitian field operators \cite{GellMann}. These real fields
can be written as linear combinations of the positive frequency particle
terms, $A^{+}$, and negative frequency antiparticle terms, $A^{-}$. The real
fields $A_{c}=\left( A^{+}+A^{-}\right) /\sqrt{2}$ and $A_{s}=\left(
A^{+}-A^{-}\right) /\sqrt{2}i$ are even and odd respectively under
particle/antiparticle exchange (charge conjugation) where the subscripts $c$
and $s$ denote Fourier cosine and sine series respectively. The complex
positive frequency function $A^{+}=A_{c}+iA_{a}$ describes both of these
potentials, so it will be used here to simplify the mathematics.
In Section II the photon equations of motion will be derived from the standard
Lagrangian. The fields will be assumed to be real but for mathematical
convenience these real fields will be identified with the real and imaginary
parts of a complex positive frequency field. The zeroth component of the
standard relativistic four-current will be interpreted as a positive definite
photon number density and used to define a scalar product. Interaction with
polarized matter will be included so that these equations can be applied to
photon propagation in a nonabsorptive medium and emission and absorption of
photons by localized sources and sinks. In Section III the real Hilbert space
will be defined and operators for all the usual physical observables,
including position, will be reviewed to provide a complete first quantized
description of single-photon states. In Section IV the fields will be second
quantized and in the final Section we will summarize and conclude.
\section{Real and complex fields, Lagrangian and scalar product}
In this Section four-vector notation and complex fields whose real and
imaginary parts are even and odd under QFT charge conjugation will be defined.
The photon equations of motion, four-current and positive definite number
density will be derived from the real standard Lagrangian written in terns of
complex fields. A scalar product will be defined in position space and Fourier
transformed to momentum space to complete the Hilbert space.
Relativistic notation and SI units will be used. The contravariant space-time,
wave vector and momentum four-vectors are $x=x^{\mu}=\left( ct,\mathbf{x
\right) ,$ $k=\left( \omega_{k}/c,\mathbf{k}\right) $ and $p=\hbar k$ where
$kx=\omega_{k}t-\mathbf{k\cdot x}$, the four-gradiant is $\partial=\left(
\partial_{ct},-\mathbf{\nabla}\right) $, the four-potential is $A\left(
t,\mathbf{x}\right) =A^{\mu}=\left( \frac{\phi}{c},\mathbf{A}\right) $ or
$a\left( t,\mathbf{k}\right) =\left( a_{0},\mathbf{a}\right) $ and
$a_{\lambda}\left( \mathbf{k}\right) $ will denote a Lorentz invariant
scalar describing a state with definite helicity, $\lambda$. The covariant
four-vector corresponding to $U^{\mu}=\left( U_{0},\mathbf{U}\right) $ is
$U_{\mu}=g_{\mu\nu}U^{\nu}=\left( U_{0},-\mathbf{U}\right) $ where
$g_{\mu\nu}=g^{\mu\nu}$ is a $4\times4$ matrix with diagonal $\left(
1,-1,-1,-1\right) $.
Positive and negative frequency four-potentials will be used here to describe
photons and antiphotons respectively. In Fourier expanded form
\begin{align}
A^{+}\left( x\right) & =\sqrt{\frac{\hbar}{\epsilon_{0}}}\int_{t
\frac{d\mathbf{k}}{\left( 2\pi\right) ^{3}\omega_{k}}a\left( \mathbf{k
\right) e^{-ikx}\label{A+}\\
A^{-}\left( x\right) & =A^{+\ast}\left( x\right) \label{A-
\end{align}
where $d\mathbf{k}\equiv d^{3}k$ is an infinitesimal volume in $\mathbf{k
$-space. This form of was selected because lim$_{V\rightarrow\infty
\Delta\mathbf{n}/V=d\mathbf{k}/\left( 2\pi\right) ^{3}$ where $\Delta
\mathbf{n}$ is the number of states and $\int d^{4}k\delta\left( \omega
_{k}^{2}/c^{2}-\left\vert \mathbf{k}\right\vert ^{2}\right) =\int_{t
\frac{d\mathbf{k}}{\left( 2\pi\right) ^{3}2\omega_{k}/c}$. Since $kx$ and
$\int_{t}\frac{d\mathbf{k}}{\left( 2\pi\right) ^{3}\omega_{k}}$ are
invariants, if $A\left( x\right) $ is a four-vector then $a\left(
\mathbf{k}\right) $ is a four-vector. For complex Fourier coefficient
\begin{equation}
a\left( \mathbf{k}\right) =\frac{a_{R}\left( \mathbf{k}\right)
+ia_{I}\left( \mathbf{k}\right) }{\sqrt{2}}\label{a
\end{equation}
the positive frequeny four-potential can be written a
\begin{equation}
A^{+}\left( x\right) =A_{c}\left( x\right) +iA_{s}\left( x\right)
\label{Acs
\end{equation}
where
\begin{align}
A_{c}\left( x\right) & =\frac{A^{+}\left( x\right) +A^{-}\left(
x\right) }{\sqrt{2}}\label{Ac}\\
& =\sqrt{\frac{2\hbar}{\epsilon_{0}}}\int_{t}\frac{d\mathbf{k}}{\left(
2\pi\right) ^{3}\omega_{k}}a_{R}\left( \mathbf{k}\right) \cos\left(
kx\right) ,\nonumber\\
A_{s}\left( x\right) & =\frac{A^{+}\left( x\right) -A^{-}\left(
x\right) }{\sqrt{2}i}\label{As}\\
& =\sqrt{\frac{2\hbar}{\epsilon_{0}}}\int_{t}\frac{d\mathbf{k}}{\left(
2\pi\right) ^{3}\omega_{k}}a_{I}\left( \mathbf{k}\right) \sin\left(
kx\right) \nonumber
\end{align}
are even and odd with respect to interchange of photons and antiphotons (QFT
charge conjugation). The real field is
\begin{equation}
2A\left( x\right) =A^{+}\left( x\right) +A^{-}\left( x\right)
=A_{c}\left( x\right) +A_{s}\left( x\right) .\label{Areal
\end{equation}
The factor $2$ is included for consistency with the definitions in QFT where
division is by $2\omega_{k}$. The positive frequency electric and magnetic
fields are $\mathbf{E}^{+}=-\partial_{t}\mathbf{A}^{+}-\mathbf{\nabla}\phi
^{+}$ and $\mathbf{B}^{+}=\mathbf{\nabla}\times\mathbf{A}^{+}$ and the
antisymmetric Faraday tensor is
\begin{equation}
\mathcal{F^{+\mu\nu}=}\partial^{\mu}A^{+\nu}-\partial^{\nu}A^{+\mu
}\label{Ftensor
\end{equation}
where $\mathcal{F}^{+00}=\mathcal{F}^{+ii}=0$, $\mathcal{F}^{+i0
=-\mathcal{F}^{+0i}=E_{i}^{+}/c$, $\mathcal{F}^{+ij}=-\mathcal{F
^{+ji}=\epsilon_{ijk}B_{k}^{+}$ and $\epsilon_{ijk}$ is the Levi-Civita symbol.
The Lagrangian describing two real fields can be written in complex form
provided this field and its complex conjugate are treated as formally
independent \cite{CT}. In the presence of a matter four-current density
$J_{m}^{+\nu}+c.c,$ the real Lagrangian density will be written a
\begin{align}
\mathcal{L} & =\mathcal{L}_{std}+\mathcal{L}_{int}\label{L}\\
\mathcal{L}_{std} & =-\epsilon_{0}\left( \mathbf{E}^{+}\mathbf{\cdot
E}^{+\ast}-c^{2}\mathbf{B}^{+}\mathbf{\cdot B}^{+\ast}\right) \label{Lstd}\\
& =-\frac{1}{2}\epsilon_{0}c^{2}\mathcal{F_{\mu\nu}^{+}F^{+\mu\nu\ast
}\nonumber\\
\mathcal{L}_{int} & =-J_{m}^{+\nu\ast}A_{\nu}^{+}-J_{m}^{+\nu}A_{\nu}^{+\ast
}\label{Lint
\end{align}
where $c$ is the speed of light and $\epsilon_{0}$ is the dielectric
permittivity. The Lagrange equations of motion are then
\begin{equation}
\partial_{\mu}\left[ \frac{\partial\mathcal{L}}{\partial\left( \partial
_{\mu}A_{\nu}^{^{+\ast}}\right) }\right] =\frac{\partial\mathcal{L
}{\partial A_{\nu}^{^{+\ast}}}\label{Lagrange
\end{equation}
where the momentum conjugate to $A_{\nu}^{^{+\ast}}$ i
\begin{equation}
\Pi^{+\mu\nu}\equiv\frac{\partial\mathcal{L}}{\partial\left( \partial_{\mu
}A_{\nu}^{^{+\ast}}\right) }=\epsilon_{0}c^{2}\mathcal{F}^{+\mu\nu
}.\label{Pi
\end{equation}
Eq. (\ref{Lagrange}) then gives
\begin{equation}
\epsilon_{o}c^{2}\partial_{\mu}\mathcal{F^{+\mu\nu}}=\mu_{0}J_{m}^{+\nu
},\label{Leqn
\end{equation}
that can be written as the Maxwell equations (ME
\begin{equation}
\mathbf{\nabla}\cdot\mathbf{D}^{+}=\rho_{m}^{+},\ \ \mathbf{\nabla
\times\mathbf{H}^{+}\mathbf{-}\partial_{t}\mathbf{D}^{+}=\mathbf{J}_{m
^{+}\label{MEs
\end{equation}
where $\mathbf{D}^{+}=\epsilon_{0}\mathbf{E}^{+}$, $\mathbf{B}^{+}=\mu
_{0}\mathbf{H}^{+}$ and $\mu_{0}\epsilon_{0}=1/c^{2}$. Substitution of
(\ref{Ftensor}) in (\ref{Leqn}) gives the wave equation
\begin{equation}
\partial_{\mu}\partial^{\mu}A^{+\nu}-\nabla^{2}A^{+\nu}=\partial^{\nu
\Lambda^{+}+\mu_{0}J_{m}^{+\nu}\label{Awave
\end{equation}
where
\begin{equation}
\partial_{\mu}\partial^{\mu}=\partial_{ct}^{2}-\mathbf{\nabla}^{2
\equiv\square.\label{del2
\end{equation}
The gauge is determined by
\begin{equation}
\Lambda^{+}=c^{-2}\partial_{t}\phi^{+}+\mathbf{\nabla}\cdot\mathbf{A^{+
.}\label{Lambda
\end{equation}
In the Coulomb gauge $\nabla\cdot\mathbf{A}^{+}=0$, so there are no
longitudinal modes and the scalar field satisfying $\nabla^{2}\phi^{+
=-\rho^{+}/\epsilon_{0}$ responds instantaneously to changes in charge
density. Only the transverses modes propagate at the speed of light and are
second quantized in quantum electrodynamics (QED) to allow creation and
annihilation of physical photons. In the Lorenz gauge $\Lambda=0$ inserted
into (\ref{Awave}) gives $\partial_{\mu}\partial^{\mu}A^{+\nu}=\mu_{0
J_{m}^{+\nu}$. In this gauge all four components of $A$ describing the scalar,
longitudinal and transverse photon modes propagate at the speed of light and
are second quantized in QED. Each complex equation in this paragraph is
equivalent to two real equations; one for the even potentials and one for the
odd potentials.
Using these complex fields a positive definite photon number density and a
scalar product can be derived from the global phase change $A^{+
\longrightarrow e^{i\alpha}A^{+}$, $A^{+\ast}\longrightarrow e^{-i\alpha
}A^{+\ast}$ \ that is a symmetry of the free space Lagrangian $\mathcal{L
_{std}$. For an infinitesimal change in $A$, $\delta A^{+}\simeq i\alpha
A^{+}$ and $\delta A^{+\ast}\simeq-i\alpha A^{+\ast}$ so, using $\Pi^{+\mu\nu
}=\epsilon_{0}c^{2}\mathcal{F}^{+\mu\nu}$, the Noether four-current density
become
\begin{equation}
J^{\mu}\left( x\right) =\frac{i\epsilon_{0}c^{2}}{2\hbar}\left[
\mathcal{F}^{+\mu\nu\ast}A_{\nu}^{+}-\mathcal{F}^{+\mu\nu}A_{\nu}^{+\ast
}\right] .\label{J
\end{equation}
Since $\partial^{\mu}\cos\left( kx\right) =-k^{\mu}\sin\left( kx\right) $
and $\partial_{\mu}\sin\left( kx\right) =k^{\mu}\cos\left( kx\right) $,
differentiation of the complex potential $A^{+}$ with respect to any component
of $x^{\mu}$ yields a factor $-i$ and changes $\cos\left( kx\right) $ to
$-\sin\left( kx\right) $ and $\sin\left( kx\right) $ to $\cos\left(
kx\right) $ so that $A_{c}+iA_{s}\rightarrow i\mathcal{F}_{s}+\mathcal{F
_{c}$ gives $\frac{1}{2}\mathcal{F}^{+\ast}A^{+}+c.c.=\mathcal{F}_{c
A_{c}+\mathcal{F}_{s}A_{s}$ an
\begin{equation}
J^{\mu}\left( x\right) =\frac{\epsilon_{0}c^{2}}{\hbar}\left(
\mathcal{F}_{c}^{\mu\nu}A_{c\nu}+\mathcal{F}_{s}^{\mu\nu}A_{s\nu}\right)
.\label{Jcs
\end{equation}
If $J_{m}=0$, $J$ satisfies a continuity equation and the spatial integral of
the number density $J^{0}\left( x\right) $ is conserved. The four-current
density $J^{\mu}\left( x\right) \propto\mathcal{F}^{\mu\nu}\left( x\right)
A_{\nu}\left( x\right) $ was first obtained in \cite{HawtonMelde} where it
was used to derive a Hermitian number density operator.
The four-current (\ref{J}) or (\ref{Jcs}) is not gauge invariant due to its
dependence on $A$. In the Coulomb gauge only transverse waves propagate, while
in the Lorenz gauge longitudinal and transverse photons exist, but their
contributions to the scalar product cancel in free space \cite{MaxwellQM}.
Writing $\mathcal{F}^{+0\nu}$ as an electric field and including only
transverse photons with helicity $\lambda=\pm1$, the scalar product at a fixed
time $t$ will be defined as
\begin{align}
\left( A_{1},A_{2}\right) _{t} & \equiv\frac{i\epsilon_{0}}{2\hbar
\sum_{\lambda=\pm1}\int_{t}d\mathbf{xA}_{1\lambda}^{+\ast}\left( x\right)
\cdot\mathbf{E}_{2\lambda}^{+}\left( x\right) +c.c.\label{EA}\\
& =\frac{\epsilon_{0}}{\hbar}\sum_{\lambda=\pm1}\int_{t}d\mathbf{x}\left[
\mathbf{A}_{1c\lambda}\left( x\right) \cdot\mathbf{E}_{2c\lambda}\left(
x\right) \right. \label{EAcs}\\
& \left. +\mathbf{A}_{1s\lambda}\left( x\right) \cdot\mathbf{E
_{2s\lambda}\left( x\right) \right] .\nonumber
\end{align}
If $A_{2}=A_{1}$ (\ref{EA}) reduces to the spatial integral of number density
$J^{0}\left( t,\mathbf{x}\right) $. Only free transverse photons are
counted. In the presence of sources and sinks this photon number is not conserved.
Fourier transforms and the bra-ket notation will be defined
nonrelativistically as in Schr\"{o}dinger QM. At a fixed tim
\begin{align}
f\left( \mathbf{x}\right) & \equiv\int\frac{d\mathbf{k}}{\left(
2\pi\right) ^{3}}\widetilde{f}\left( \mathbf{x}\right) \exp\left(
i\mathbf{k\cdot x}\right) ,\label{invFT}\\
\widetilde{f}\left( \mathbf{k}\right) & \equiv\int d\mathbf{x}f\left(
\mathbf{x}\right) \exp\left( -i\mathbf{k\cdot x}\right) \text{
and}\label{FT}\\
\left\langle f|g\right\rangle & \equiv\int d\mathbf{x}f^{\ast}\left(
\mathbf{x}\right) g\left( \mathbf{x}\right) =\int\frac{d\mathbf{k}}{\left(
2\pi\right) ^{3}}\widetilde{f}^{\ast}\left( \mathbf{k}\right)
\widetilde{g}\left( \mathbf{k}\right) \label{braket}\\
\left\langle \mathbf{f}\cdot\mathbf{g}\right\rangle & \equiv\sum
_{j=1,3}\left\langle \mathbf{f}_{j}|\mathbf{g}_{j}\right\rangle .\label{dot
\end{align}
Using this notation the scalar product (\ref{EA}) i
\begin{align}
\left( A_{1},A_{2}\right) _{t} & =\frac{i\epsilon_{0}}{2\hbar}\sum
_{\lambda=\pm1}\left\langle \mathbf{A}_{1\lambda}^{+}\cdot\mathbf{E
_{2\lambda}^{+}\right\rangle +c.c.\label{EAbracket}\\
& =\frac{\epsilon_{0}}{\hbar}\sum_{\lambda=\pm1}\left( \left\langle
\mathbf{A}_{1c\lambda}\cdot\mathbf{E}_{2c\lambda}\right\rangle +\left\langle
\mathbf{A}_{1s\lambda}\cdot\mathbf{E}_{2s\lambda}\right\rangle \right)
\label{EAreal
\end{align}
where from (\ref{A+}), using $a=\left( a_{0},\mathbf{a}\right) $ and
$\mathbf{E}_{\lambda}=-\partial_{t}\mathbf{A}_{\lambda}$
\begin{equation}
\mathbf{E}_{\lambda}^{+}\left( x\right) =i\sqrt{\frac{\hbar}{\epsilon_{0}
}\int_{t}\frac{d\mathbf{k}}{\left( 2\pi\right) ^{3}}\mathbf{a}_{\lambda
}\left( \mathbf{k}\right) e^{-ikx}.\label{E+
\end{equation}
By inspection of (\ref{A+}) and (\ref{E+}) the Fourier transforms of
$\mathbf{A}_{\lambda}$ and $\mathbf{E}_{\lambda}$ are $\sqrt{\frac{\hbar
}{\epsilon_{0}}}\mathbf{a}_{\lambda}\left( \mathbf{k}\right) /\omega_{k}$
and $i\sqrt{\frac{\hbar}{\epsilon_{0}}}\mathbf{a}_{\lambda}\left(
\mathbf{k}\right) $ respectively so, using the Perseval-Pancherel identity
(\ref{braket}),
\begin{align}
\left( A_{1},A_{2}\right) _{t} & =\sum_{\lambda=\pm1}\int\frac{d\mathbf{k
}{\left( 2\pi\right) ^{3}}\frac{\mathbf{a}_{1\lambda}\left( \mathbf{k
\right) }{\omega_{k}}\cdot\mathbf{a}_{2\lambda}\left( \mathbf{k}\right)
\label{EAkspace}\\
& =\sum_{\lambda=\pm1}\left\langle \frac{\mathbf{a}_{1\lambda}}{\omega_{k
}\cdot\mathbf{a}_{2\lambda}\right\rangle .\label{EAbraket
\end{align}
Inspection of (\ref{EA}) to (\ref{EAbraket}) shows that these expressions for
the scalar product involve both the potential and the electric field rather
than a single function. In the next Section we will show that $\left\langle
\mathbf{E\cdot A}\right\rangle $ implies a biorthogonal basis so this
formulation is called biorthogonal QM \cite{Brody,HawtonDebierre}. The
relationship between the notation used here and that in \cite{MaxwellQM} is
that here $a\left( \mathbf{k}\right) $ is a Fourier transform while in
\cite{MaxwellQM} $\mathbf{c}\left( \mathbf{k}\right) $ is the probability
amplitude for a covariantly normalized plane wave.
\section{Hilbert space and observables}
In this Section the Hilbert space and the momentum, position and angular
momentum operators and their eigenvectors are defined. Covariant normalization
that leads to expressions of the classical form is used. The probability
amplitude to find a photon at $\mathbf{x}$ on the $t$-hyperplane is calculated
and it is verified that the Born rule is satisfied.
The real Hilbert space is the vector space of all $A_{c}$ and $A_{s}$ and
their derivatives with the scalar product (\ref{EAreal}). For mathematical
convenience the positive frequency potentials $A^{+}$ and scalar product
(\ref{EAbracket}) will be used. Since $A^{+\mu}\left( \mathbf{x},t\right) $
is a four-vector, it follows from (\ref{A+}) that $a^{\mu}\left(
\mathbf{k},t\right) $ must be a four-vector. With the mutually orthogonal
polarization unit vectors $e^{\mu}$ defined such that 0 is time-like, 1 and 2
are transverse and 3 is longitudual, $e_{0}=n^{\mu}=\left( 1,0,0,0\right)
,\ \mathbf{e}_{3}\left( \mathbf{k}\right) =\mathbf{e}_{\mathbf{k
}=\mathbf{k}/\left\vert \mathbf{k}\right\vert $ and the definite helicity
transverse unit vectors ar
\begin{equation}
\mathbf{e}_{\lambda}=\frac{1}{\sqrt{2}}\left( \mathbf{e}_{\theta
+i\lambda\mathbf{e}_{\phi}\right) \label{transverseeigenvectors
\end{equation}
for $\lambda=\pm1\ $where $\mathbf{e}_{\theta},$ $\mathbf{e}_{\phi}$ and
$\mathbf{e}_{\mathbf{k}}$ are $\mathbf{k}$-space spherical polar unit vectors
on the $t$-hyperplane. Since $e^{\mu}$ is a four-vector its coefficient in any
expression for $a^{\mu}$ should be a Lorentz invariant scalar.
Momentum is an observable. It can be verified by substitution in
(\ref{EAkspace}) that the plane waves with definite momentum $\hbar
\mathbf{k}^{\prime}$ defined covariantly a
\begin{equation}
\mathbf{a}_{\mathbf{k}^{\prime}\lambda^{\prime}}\left( \mathbf{k}\right)
=\left( 2\pi\right) ^{3}\omega_{k}\delta\left( \mathbf{k}-\mathbf{k
^{\prime}\right) \mathbf{e}_{\lambda^{\prime}}\left( \mathbf{k}^{\prime
}\right) \label{PlaneWaves
\end{equation}
are biorthogonal in the sense tha
\begin{equation}
\left( A_{\mathbf{k\lambda}},A_{\mathbf{k}^{\prime}\lambda^{\prime}}\right)
=\delta_{\lambda\lambda^{\prime}}\left( 2\pi\right) ^{3}\omega_{k
\delta\left( \mathbf{k}-\mathbf{k}^{\prime}\right) .\label{kBiorthogonal
\end{equation}
This normalization of the plane wave basis is invariant as can be seen from
$\int\frac{d\mathbf{k}}{\omega_{k}}\omega_{k}\delta\left( \mathbf{k-k
^{\prime}\right) =1$. In position space in the Heisenberg picture (HP)
(\ref{A+}) gives
\begin{equation}
\mathbf{A}_{\mathbf{k}\lambda}^{+}\left( \mathbf{x},t\right) =\sqrt
{\frac{\hbar}{\epsilon_{0}}}e^{-ikx}\mathbf{e}_{\lambda}\left( \mathbf{k
\right) .\label{xPlaneWaves
\end{equation}
In a general state $A$ the probability amplitude for wave vector $\mathbf{k}$
i
\begin{equation}
\left( A_{\mathbf{k}\lambda},A\right) =a_{\lambda}\left( \mathbf{k}\right)
\label{a(k)
\end{equation}
If $\omega_{k}$ is replace with $\omega_{k}^{1/2}\ $in (\ref{PlaneWaves}) the
noncovariant Newton-Wigner \cite{NW} normalization, $\left( A_{\mathbf{k
\lambda},A_{\mathbf{k}^{\prime}\lambda^{\prime}}\right) _{NW}=\delta
_{\lambda\lambda^{\prime}}\left( 2\pi\right) ^{3}\delta\left(
\mathbf{k}-\mathbf{k}^{\prime}\right) $, is obtained. The position space
plane waves (\ref{xPlaneWaves}) are eigenvectors of the four-momentum operato
\begin{equation}
\widehat{P}=\left( \widehat{p}_{0},\widehat{\mathbf{p}}\right) =\left(
\frac{1}{c}\widehat{H},-i\hbar\mathbf{\nabla}\right) \label{Pop
\end{equation}
with eigenvalues $\hbar\left( \omega_{k_{j}}/c,\mathbf{k}\right) $. The
Hamiltonian operator
\begin{equation}
\widehat{H}=\hbar c\sqrt{-\nabla^{2}}\label{H
\end{equation}
with eigenvalues $\hbar\omega_{k}$ generates unitary transformations according
to the Schr\"{o}dinger equation $i\hbar\partial_{t}A^{+}\left( t\right)
=\widehat{H}A^{\ast}\left( t\right) $. In $\mathbf{k}$-space the four
momentum operator is $\widehat{P}=\left( \hbar kc,\hbar\mathbf{k}\right) .$
Position is also an observable. According to (\ref{FT}) the Fourier transform
of the localized state $\delta\left( \mathbf{x}-\mathbf{x}^{\prime}\right) $
is the plane wave $\exp\left( -i\mathbf{k\cdot x}^{\prime}\right) $ so
covariant HP photon position eigenvectors should be of the for
\begin{equation}
\mathbf{a}_{x^{\prime}\lambda^{\prime}}\left( \mathbf{k}\right)
=\mathbf{e}_{\lambda^{\prime}}\left( \mathbf{k}\right) e^{ikx^{\prime
}.\label{evecs
\end{equation}
The projection of an arbitrary state $A$ onto the $A_{x\lambda}$ basis
\begin{equation}
\phi_{\lambda}^{+}\left( x\right) =\left( A_{x\lambda},A\right) _{t
=\int_{t}\frac{d\mathbf{k}}{\left( 2\pi\right) ^{3}\omega_{k}}a_{\lambda
}\left( \mathbf{k}\right) e^{-ikx},\label{phi
\end{equation}
has the mathematical form of a Lorentz invariant scalar potential that
satisfies the zero mass Klein-Gordon (KG) equation. The photon position
operator with commuting components can be derived by rotating $\mathbf{e
_{1}+i\lambda\mathbf{e}_{2}$ about $\mathbf{e}_{2}$ by $\theta$, then about
$\mathbf{e}_{3}$ by $\phi$ to give $\mathbf{e}_{\theta}+i\lambda
\mathbf{e}_{\phi}$ using the operator $\widehat{R}$ so that $\mathbf{e
_{\theta}+i\lambda\mathbf{e}_{\phi}=\widehat{R}\left( \mathbf{e}_{1
+i\lambda\mathbf{e}_{2}\right) $ and $i\mathbf{\partial}_{\mathbf{k}}$
transforms to $\widehat{\mathbf{x}}=\widehat{R}i\mathbf{\partial}_{\mathbf{k
}\widehat{R}^{-1}$ \cite{HawtonBaylis}. Alternatively it can be obtained by
covariant differentiation \cite{Covariant}. It was originally obtained by
brute force subtraction of the $\mathbf{k}$-space gradient of $\mathbf{e
_{\lambda_{j}}\left( \mathbf{k}\right) $ to give \cite{HawtonPosOp}
\begin{equation}
\widehat{\mathbf{x}}=i\mathbf{\partial}_{\mathbf{k}}-i\alpha\frac{\mathbf{k
}{\left\vert \mathbf{k}\right\vert ^{2}}+\frac{1}{\left\vert \mathbf{k
\right\vert ^{2}}\mathbf{k\times}\widehat{\mathbf{S}}-\widehat{\lambda
\frac{\cos\theta}{k\sin\theta}\mathbf{e}_{\phi}\label{xop
\end{equation}
for position eigenvectors proportional to $\left( \omega_{k}\right)
^{\alpha}$. The helicity $\lambda$ photon position eigenvectors have a
definite component of total angular momentum in the fixed but arbitrary
direction $\mathbf{e}_{3}$ with indefinite spin and orbital contributions
\cite{HawtonBaylis}. The total angular momentum operator is
\begin{align}
\widehat{\mathbf{J}} & =\widehat{\mathbf{x}}\times\widehat{\mathbf{P
}+\widehat{\mathbf{J}}_{int},\label{Jtotal}\\
\widehat{\mathbf{J}}_{int} & =\hbar\lambda\left( \frac{\cos\theta
{\sin\theta}\mathbf{e}_{\theta}\mathbf{+e}_{\mathbf{k}}\right) .\label{Jint
\end{align}
where $\widehat{\mathbf{J}}_{int}$ is the internal angular momentum operator
and $\hbar\widehat{\mathbf{x}}\times\widehat{\mathbf{k}}$ describes external
angular momentum. The position and angular momentum operators are reviewed in
more detail in \cite{HawtonBaylis,MaxwellQM} where rotation about $\mathbf{k}$
through an Euler angle that gives a more general expression for $\mathbf{J}$
is included.
Setting $\mathbf{a}_{\lambda}\left( \mathbf{k}\right) $ equal to an
$\alpha=0$ position eigenvector $\mathbf{a}_{x^{\prime}\lambda^{\prime
}\left( \mathbf{k}\right) $ in (\ref{phi}) with $\Delta t\equiv t-t^{\prime
}$ and $r\equiv\left\vert \mathbf{x}-\mathbf{x}^{\prime}\right\vert $ an
explicit expression for its time evolution can be obtained by integrating
(\ref{phi}) to giv
\begin{equation}
\phi_{x^{\prime}\lambda^{\prime}}^{+}\left( x\right) =\frac{1}{4\pi^{2
r}\sum_{\gamma=\pm}\left[ i\gamma\pi\delta\left( r-\gamma c\Delta t\right)
+P\left( \frac{1}{r-\gamma c\Delta t}\right) \right] \label{phix
\end{equation}
for $\lambda^{\prime}=\pm1$. Both the real and imaginary parts of (\ref{phix})
satisfy the photon wave equation, but only its imaginary part is odd under QFT
charge conjugation and couples to charged matter. Thus, if the real part of
$\phi_{\lambda}\left( t,\mathbf{x}\right) $ is discarded on physical
grounds, (\ref{phix}) is reduced to its odd sine series imaginary part,
\begin{equation}
\phi_{sx^{\prime}\lambda^{\prime}}\left( x\right) =\frac{1}{4\pi r}\left[
\delta\left( r+c\Delta t\right) -\delta\left( r-c\Delta t\right) \right]
\label{Homo
\end{equation}
that satisfies the homogeneous wave equation $\square\phi_{\lambda}\left(
x\right) =0$. Schweber \cite{Schweber} inverted $\square$ and found that the
unique Green's function solving $\square\phi_{\lambda}\left( x\right)
=\delta\left( \mathbf{x}-\mathbf{x}^{\prime}\right) \delta\left(
t-t^{\prime}\right) $ is $\frac{1}{4\pi r}\left[ \delta\left( r+\Delta
t\right) +\delta\left( r-c\Delta t\right) \right] $ where $t^{\prime
}=t-r<0$ is the retarded time and $t^{\prime}=t+r>0$ is the advanced time. He
concluded that the retarded potential is determined by boundary conditions and
equals the sum of his unique contour integral independent Green's function and
a solution to the homegeneous wave equations of the form (\ref{Homo}). This
retarded potential solution is important in classical EM and, according to
(\ref{Homo}), it can be applied to photon QM with the significant advantage
that $\phi_{\lambda}\left( x\right) $ is a scalar.
In Schr\"{o}dinger quantum mechanics, the non-square integrable $\delta
$-functions satisfy a completeness relation and form a very convenient basis
\cite{CTQM1} so $\delta$-localized photon states will be defined here. The
usual approach in photon QM and QED is to introduce a factor $\omega_{k
^{1/2}$ into (\ref{evecs}) to give $\mathbf{a}_{x^{\prime}\lambda^{\prime
}\left( \mathbf{k}\right) =\omega_{k}^{1/2}\mathbf{e}_{\lambda^{\prime
}\left( \mathbf{k}^{\prime}\right) e^{ikx^{\prime}}$ and the
Newton-Wigner-like noncovariant normalization $\left( A_{\mathbf{x}\lambda
},A_{\mathbf{x}^{\prime}\lambda^{\prime}}\right) _{NW}=\delta_{\lambda
\lambda^{\prime}}\delta\left( \mathbf{x}-\mathbf{x}^{\prime}\right) $. With
this choice $\phi_{\lambda}^{+}\left( x\right) =\int_{t}\frac{d\mathbf{k
}{\left( 2\pi\right) ^{3}}\omega_{k}^{-1/2}a_{\lambda}\left( \mathbf{k
\right) e^{-ikx}$ is not covariant. Here, to retain covariance and avoid the
inconvenient factor $\omega_{k}^{1/2}$, the time derivative of (\ref{phi})
that is the zeroth component of a four-vector will be used to define the
localized basis. Sinc
\begin{equation}
\psi_{\lambda}^{+}\left( x\right) =i\partial_{t}\phi_{\lambda}^{+}\left(
x\right) =\int_{t}\frac{d\mathbf{k}}{\left( 2\pi\right) ^{3}}a_{\lambda
}\left( \mathbf{k}\right) e^{-ikx},\label{psi
\end{equation}
with $kx=\omega_{k}t-\mathbf{k\cdot x}$, for $a_{\lambda}\left(
\mathbf{k}\right) =a_{x^{\prime}\lambda^{\prime}}\left( \mathbf{k}\right)
$, at $t=t^{\prime}$
\begin{equation}
\psi_{x^{\prime}\lambda}^{+}\left( x\right) =\delta\left( \mathbf{x
-\mathbf{x}^{\prime}\right) .\label{xbasis
\end{equation}
Using (\ref{EAkspace}) and expanding $\psi_{\lambda}^{+}$ in the $\delta
$-basis a
\begin{equation}
\psi_{\lambda}^{+}\left( t,\mathbf{x}\right) =\int_{t}d\mathbf{x}^{\prime
}\delta\left( \mathbf{x}-\mathbf{x}^{\prime}\right) \psi_{\lambda
^{+}\left( \mathbf{x}^{\prime}\right) ,\label{psixbasis
\end{equation}
it can be seen that $\psi_{\lambda}^{+}\left( x\right) $ is the probability
amplitude for a photon to be in the state $\delta\left( \mathbf{x
-\mathbf{x}^{\prime}\right) $ on the $t$-hyperplane. (Only its odd
sine-series part is truely localized.) Its integrated squared norm,
\begin{align}
\left( A,A\right) & =\sum_{\lambda=\pm1}\int d\mathbf{x}\left\vert
\psi_{\lambda}^{+}\left( x\right) \right\vert ^{2}\label{PsiNorm2}\\
& =\sum_{\lambda=\pm1}\int d\mathbf{x}\left[ \left\vert \psi_{c\lambda
}\left( x\right) \right\vert ^{2}+\left\vert \psi_{s\lambda}\left(
x\right) \right\vert ^{2}\right] ,\nonumber
\end{align}
satisfies the Born rule. $\psi_{c\lambda}\left( x\right) $ and
$\psi_{s\lambda}\left( x\right) $ are the real probability amplitudes for a
$\lambda$-helicity photon to be at $\mathbf{x}$ on the $t$-hyperplane. The
$\lambda$-helicity $\mathbf{x}$-space probability density is
\begin{equation}
\rho_{\lambda}^{\epsilon}\left( x\right) =\left\vert \psi_{c\lambda}\left(
x\right) \right\vert ^{2}+\left\vert \psi_{s\lambda}\left( x\right)
\right\vert ^{2}.\label{photonxdensity
\end{equation}
If $\left( A,A\right) $ is finite, $A$ is normalizable as $\left(
A,A\right) =1$. Since $a_{\lambda}\left( \mathbf{k}\right) $ is the Fourier
transform of $\psi_{\lambda}\left( x\right) $, using (\ref{braket}) the
$\mathbf{k}$-space Born rule i
\begin{equation}
\left( A,A\right) =\sum_{\lambda}\int\frac{d\mathbf{k}}{\left( 2\pi\right)
^{3}}\left[ \left\vert a_{c\lambda}\left( \mathbf{k}\right) \right\vert
^{2}+\left\vert a_{s\lambda}\left( \mathbf{k}\right) \right\vert
^{2}\right] .\label{kNorm2
\end{equation}
Covariance was maintained here by defining $\psi_{\lambda}^{+}\left(
t,\mathbf{x}\right) $ as the time-like component of a four-vector.
Only the odd $\psi_{s\lambda}$ density is localizable in a finite region and
has the correct transformation properties under QFT charge conjugation
\cite{GellMann}. The cosine solutions to the wave equation have been retained
in (\ref{PsiNorm2}) to (\ref{kNorm2}) for generality because they satisfy ME
and the requirement that $A$ should be odd is not usually imposed.
Quantum mechanics requires state vectors to describe physical systems and
operators representing observables such that the only possible result of a
measurement is one of their eigenvalues \cite{CTQM1}. Eqs. (\ref{psi}) to
(\ref{kNorm2}) provide a scalar Schr\"{o}dinger-like description of a photon
with helicity $\lambda$ in which $\psi_{\lambda}^{+}\left( t,\mathbf{x
\right) $ is the probability amplitude for a photon to be at $\mathbf{x}$ on
the $t$-hyperplane and its FT $a_{\lambda}\left( \mathbf{k}\right) $ is the
probability amplitude for it to have momentum $\hbar\mathbf{k}$.
\section{Second quantization}
A first quantized photon cannot be created or destroyed - creation and
annihilation of photons and the description of $n$-photon states requires
second quantization. In QFT, QED and QO fields are second quantized by raising
them to the status of operators. For an arbitrary first quantized state in
which (\ref{A+}) and (\ref{Areal}) are generalized to include a factor
$\omega_{k}^{\alpha}$ to accomodate the factor $\omega_{k}^{1/2}$ commonly
used,
\begin{equation}
\widehat{\mathbf{A}}\left( x\right) =\sqrt{\frac{\hbar}{\epsilon_{0}}
\sum_{\lambda=\pm1}\int_{t}\frac{d\mathbf{k}}{\left( 2\pi\right) ^{3
}\left( 2\omega_{k}\right) ^{\alpha-1}\left( \widehat{a}_{\mathbf{k
\lambda}e^{-ikx}+\widehat{a}_{\mathbf{k}\lambda}^{\dagger}e^{ikx}\right)
,\label{Aop
\end{equation}
where the operator $\widehat{a}_{\mathbf{k}\lambda}$ annihilates a photon with
wave vector $\mathbf{k}$ and helicity $\lambda$ and $\widehat{a
_{\mathbf{k}\lambda}^{\dagger}$ creates one. For plane wave states satisfying
the commutation relations
\begin{align}
\left[ \widehat{a}_{\lambda}\left( \mathbf{k}\right) ,\widehat{a
_{\lambda^{\prime}}\left( \mathbf{q}\right) \right] & =0,\ \left[
\widehat{a}_{\lambda}^{\dagger}\left( \mathbf{k}\right) ,\widehat{a
_{\lambda^{\prime}}^{\dagger}\left( \mathbf{q}\right) \right]
=0,\nonumber\\
\left[ \widehat{a}_{\lambda}\left( \mathbf{k}\right) ,\widehat{a
_{\lambda^{\prime}}^{\dagger}\left( \mathbf{q}\right) \right] &
=\delta_{\lambda,\lambda^{\prime}}\left( 2\pi\right) ^{3}\left( 2\omega
_{k}\right) ^{1-2\alpha}\delta\left( \mathbf{k}-\mathbf{q}\right)
\label{kcommutation
\end{align}
it can be verified by substitution that the operators $\widehat{\mathbf{A
}_{\lambda}\left( x\right) $ and $\widehat{\mathbf{E}}_{\lambda}\left(
x\right) =-\partial\widehat{\mathbf{A}}_{\lambda}\left( x\right) $ satisfy
the commutation relations
\begin{equation}
\sum_{j=1,3}\left[ \widehat{A}_{\lambda j}\left( t,\mathbf{x}\right)
,\widehat{E}_{\lambda^{\prime}j}\left( t,\mathbf{x}^{\prime}\right) \right]
=-\frac{i\hbar}{\epsilon_{0}}\delta_{\lambda\lambda^{\prime}}\delta\left(
\mathbf{x}-\mathbf{x}^{\prime}\right) .\label{Commutation
\end{equation}
on the $t$-hyperplane. The usual text book choice is $\alpha=\frac{1}{2}$ but
$\alpha=0$ for which (\ref{Aop}) and (\ref{kcommutation}) are covariant is
used here and in \cite{ItzyksonZuber}.
In QED microcausality is a consequence of the commutation relations
(\ref{Commutation}). (On a general spacelike hyperplane $t$ is not necessarily
equal to $t^{\prime}$ but time ordering is observer dependent.) Defining the
one photon state
\begin{equation}
\left\vert \mathbf{A}_{x\lambda}\right\rangle =\widehat{\mathbf{A}}_{x\lambda
}\left\vert 0\right\rangle ,\ \left\vert \mathbf{E}_{x\lambda}\right\rangle
=\widehat{\mathbf{E}}_{x\lambda}\left\vert 0\right\rangle ,\label{OnePhoton
\end{equation}
the vacuum expectation value of (\ref{Commutation}) multiplied by
$i\epsilon_{0}/2\hbar$ is
\begin{equation}
\psi_{y\lambda}\left( x\right) =\frac{i\epsilon_{0}}{2\hbar}\left[
\left\langle \mathbf{A}_{x\lambda}\cdot\mathbf{E}_{x^{\prime}\lambda
}\right\rangle +\left\langle \mathbf{A}_{x^{\prime}\lambda}\cdot
\mathbf{E}_{x\lambda}\right\rangle \right] .\label{VacuumExpectation
\end{equation}
It implies that creation at $\left( t^{\prime},\mathbf{x}^{\prime}\right) $
followed by annihilation at $\left( t,\mathbf{x}\right) $ and creation at
$\left( t,\mathbf{x}\right) $ followed by annihilation at $\left(
t^{\prime},\mathbf{x}^{\prime}\right) $ are equally likely. This provides a
physical interpretation of the real probability amplitude and implies that we
don't know whether the photon was created or destroyed at $\left(
t,\mathbf{x}\right) $ as discussed in \cite{HawtonDebierre,MaxwellQM}.
\section{Summary and Conclusion{}}
In its covariant $\alpha=0$ version, photon quantum mechanics as described
here preserves the classical form of the EM potential and fields when first
quantized and second quantized. Only the interpretation need be changed - from
real observable classical fields, to probability amplitudes, and then to
operators that create and annihilate photons. The real potentials are even and
odd under QFT charge conjugation, but only those that are odd can be localized
in a finite region and coupled to charged matter. These even and odd fields
are real and imaginary parts of a positive frequency field whose use simplifes
the mathematics and facilites use of the standard Lagrangian and relativistic
scalar product. Propagation of finite pulses is as in classical EM and
mathematical techniques such as finite difference time domain (FDTD)
\cite{FDTD} developed to handle problems in classical EM theory can be applied
directly to single photons. Projection onto momentum and position bases gives
a covariant Schr\"{o}dinger-like description of photon QM. Eqs. (\ref{phi})
and (\ref{psi}) provide a new scalar description of single photon states with
a well defined physical interpretation that may prove to be useful in applications.
|
1,108,101,565,689 | arxiv | \section{ALGORITHM}\label{sec:algo}
The key assumption that motivates RNM is a smoothness condition
that goes beyond the standard assumptions in the optimization literature, where smoothness would be characterized by a symmetric quadratic with the radius $L$. Instead, Assumption \ref{ass:smooth} below is tighter, allowing for more refined analysis, and can be related to the standard assumption by $L = \lambda_{\max}(\bM)$.
\begin{assumption}[Smoothness]\label{ass:smooth}
There exists a symmetric p.d.~matrix $\mathbf{M} \in \mathbb{R}^{d \times d}$ such that $\forall x,h \in \mathbb{R}^d$,
\begin{equation}\label{eq:smooth}
f(x+h)\leq f(x)+\braket{\nabla f(x),h}+\frac{1}{2}\braket{h,\mathbf{M}h}.
\end{equation}
\end{assumption}
This assumption is satisfied for quadratic problems such as ridge regression with squared loss,
\(
y = \bA^\top w + \epsilon,
\)
where $\bA \in \mR^{n\times d}$ is the data matrix, and $y$ is the vector of responses, which is corrupted via the noise $\epsilon \in \mR^n$.
In this case, Assumption \ref{ass:smooth} holds with $\bM$ being the offset covariance matrix $\bA^\top \bA + \lambda\bI $, where $\lambda$ is the regularization parameter. Beyond quadratic problems, it holds for many common problems such as logistic regression, where $\bM = \frac{1}{4} \bA^\top \bA$. Section~\ref{sec:ridge} provides examples in the dual formulation.
\subsection{Randomized Newton Method}
Let $k$ be the iteration count and $x_0$ be the initial point. The Randomized Newton Method algorithm is defined via the following update rule:
\begin{equation}\label{eq:update}
x_{k+1} = x_k - \left(\bM_{S_k}\right)^+\nabla f(x_k),
\end{equation}
where $S_k \subseteq [d]$ is a subset of coordinates chosen at iteration $k$ from random sampling $\hat{S}$ to be defined. Notice that since $\bM_{S_k}$ is a sparse $d \times d$ matrix with only a $|S_k| \times |S_k|$ principal submatrix that is non-zero, its inversion costs $\mO(|S_k|^3)$ arithmetic operations. Moreover, only $|S_k|$ elements of $\nabla f(x_k)$ are needed for the update.
Note that if $|S_k|=1$ then we are in the classical case of coordinate descent, while if $S=[d]$, then we are performing a Newton step (with $\bM$ in place of the true Hessian).
\subsection{Sampling}
The strategy with which one chooses blocks $S_k \subseteq [d]$ in \eqref{eq:update} is of great importance and it influences the algorithm significantly.
This strategy, called a \emph{sampling} and denoted $\hat{S}$, is a random set-valued mapping with values being subsets of $[d]$. A \emph{proper sampling} is such that $p_i \eqdef \pP(i \in \hat{S}) > 0$ for all $i$.
The most popular are \emph{uniform} samplings, i.e., those for which the marginal probabilities are equal:
\[ \pP(i \in \hat{S}) = \pP(j \in \hat{S}) \quad \forall i,j \in [d]. \]
This class includes $\tau$-nice and $\tau$-list samplings \citep{Qu2016}. The $\tau$-nice sampling considers all elements of a power set of $[d]$ with a fixed cardinality s.t.~$|\hat{S}| = \tau$. There are $\binom{d}{\tau}$ of such subsets and each of them is equally probable. Consequently, the probability $\pP(i \in \hat{S}) = \frac{\tau}{n}$. On the other hand, the $\tau$-list sampling is restricted to ordered and consecutive subsets of the power set, with cardinality fixed to $\tau$.
Data dependent (and potentially non-uniform) samplings, which sample according to the diagonal elements of $\bM$, have been analyzed in the context of coordinate descent \citep{Qu2016,Allen-Zhu2016,Hanzely2018,Richtarik2015a}.
\section{CONVERGENCE ANALYSIS}\label{sec:analysis}
In this section, we analyze the convergence properties of the
update scheme \eqref{eq:update} with determinantal sampling
defined by \eqref{eq:sample_def}. In order to establish linear
rate of convergence, we need to assume strong convexity.
\begin{assumption}[Strong Convexity]\label{ass:strongconvex}
Under Assumption \ref{ass:smooth}, there exists a $\kappa > 0$ such that $\forall x,h \in \mathbb{R}^d$,
\begin{equation*}\label{eq:strgcnvx}
f(x)+\braket{\nabla f(x),h}+\frac{\kappa}{2}\braket{h,\mathbf{M}h}\leq f(x+h)
\end{equation*}
\end{assumption}
Intuitively, the parameter $\kappa \in (0,1]$ measures the
degree of accuracy of our quadratic approximation. For a
quadratic function $\kappa = 1$.
\begin{lemma}[\cite{Qu2015Feb}]\label{theorem:1}
Under Assumptions \ref{ass:smooth} and \ref{ass:strongconvex},
let $\{x^k\}_{k\geq0}$ be a sequence of random vectors produced by the
Algorithm with a proper sampling $\hat{S}$, and let $x^*$ be the optimum
of $f$. Then,
\begin{equation*
\mE\big[f(x^{k+1})-f(x^*)\big]\leq
\big(1-\sigma(\hat{S})\big)\mE\big[f(x^k)-f(x^*)\big] ,
\end{equation*}
where
\begin{equation}\label{eq:sigma_1}
\sigma (\hat{S}) \eqdef \kappa\cdot
\lambda_{\min}\big(\mathbf{M}^{1/2}
\mathbb{E}[(\mathbf{M}_{\hat{S}})^{+}]\mathbf{M}^{1/2}\big).
\end{equation}
\end{lemma}
Strong convexity is not necessary to run RNM (\ref{eq:update}). In the
cases where the function is only convex, we recover the standard sublinear
rate depending on $\sigma$.
\begin{lemma}[\cite{Karimireddy2018a}]\label{thm:convex}
Let $f$ be convex and satisfy Assumption \ref{ass:smooth}. Then using
the update scheme in \eqref{eq:update} with any proper
sampling,
\[\mE[f(x^k) - f(x^*)] \leq \frac{2D}{\sigma(\hat{S}) k} \]
where $\sigma(\hat{S})$ is as in \eqref{eq:sigma_1}, and $D = \max_{x}
\{ (x^* -x)^\top \bM (x^* -x) | f(x) \leq f(x^0) \}$ is the set
diameter in $\bM$ geometry at the initial level sets.
\end{lemma}
The preceding two lemmas introduced the quantity $\sigma(\hat{S})$ characterizing
the theoretical convergence rate of the method. By
applying our new expectation formula (Theorem \ref{t:main}) we obtain
a simple form for this
quantity under DPP sampling.
\begin{theorem}\label{corr:recurence}
Under Assumption \ref{ass:smooth}, given $\alpha>0$:
\begin{equation}\label{eq:corr}
\sigma(\hat{S}) = \kappa \frac{ \lambda_d }{ \lambda_d +
\alpha}\quad\text{for}\quad\hat{S} \sim \DPP\big(\tfrac{1}{\alpha} \bM \big),
\end{equation}
where $\lambda_d = \lambda_{\min}(\bM)$.
\end{theorem}
Note that $\sigma(\hat{S})$ depends solely on the
smallest eigenvalue and the parameter $\alpha$ controlling the
expected size. This is not the case for other samplings, and other
closed forms are not known in general \citep{Qu2015Feb}.
Recall that the smaller the $\alpha$ the bigger the subsets. The
closed form expression from Theorem \ref{corr:recurence} combined with
Lemma~\ref{lemma:lambdas_exp} allows us to formulate a
recurrence relation between the convergence rates with different
expected set sizes.
\begin{proposition}[Recurrence relation]\label{prop:recurence}
Let $\{\lambda_i\}_{i=1}^d$ be the eigenvalues of $\bM$ in a
decreasing order. Let $k<d$ be a positive integer, $\alpha(k) = \sum_{i > k-1}\lambda_i$, and $\sigma(k) = \frac{\lambda_d}{\lambda_d + \alpha(k)}$. Then,
\[ \sigma(k) = \frac{\sigma(k+1)}{1 + \frac{\lambda_{k}}{\lambda_d}\sigma(k+1)} \]
and $\sigma(d) = 1$ while $\mE[|S|] < k$.
\end{proposition}
This result allows us to further improve the theoretical
bounds from \cite{Qu2015Feb} on the parameter
$\sigma$. Namely, it has been previously established that
$\sigma$ grows at least linearly with the increasing subset
size of $\tau$-uniform sampling, i.e., $\tau\sigma(1) \leq
\sigma(\tau)$. We can establish more informative bounds
depending on the eigenvalue decay. Specifically, for a
decreasing sequence of eigenvalues $\{\lambda_i
\}_{i=1}^d$,
\begin{equation}\label{eq:tighter}
\left(1 + \sum_{j=1}^{\tau-1}\frac{\lambda_j}{\lambda_d}\right)\sigma(1) \leq \sigma(\tau).
\end{equation}
For example, given exponentially decaying eigenvalues
$\lambda_i = \gamma^i$ where $\gamma < 1$, the increase is at
least exponential, and the convergence rate is at least $(1+(\tau-1)\gamma^{\tau-d})$ bigger. The case with linear speed-up is recovered when all eigenvalues are equal.
\section{OPTIMAL BLOCK SIZE}\label{sec:decay}
Our results such as Proposition \ref{prop:recurence} and inequality
\eqref{eq:tighter} describe the convergence speedup of using larger
coordinate blocks for RNM with determinantal sampling as a function of
the eigenvalues of $\bM$. In this
section, we demonstrate that covariance matrices arising in kernel
ridge regression have known asymptotic eigenvalue decays, which allows
for a precise characterization of RNM performance.
\subsection{Kernel Ridge Regression} \label{sec:ridge}
The motivating example for our analysis is the dual formulation of
kernel ridge regression which is a natural application for block
coordinate descent because of its high dimensionality.
Suppose our (primal) regression problem is defined by the following
objective:
\[
\min_{x \in \mR^d} \frac{1}{n} \sum_{i=1}^{n} \frac{1}{2}(\Phi(a_i)^\top x - y_i)^2 + \frac{\lambda}{2} \norm{x}^2_2 ,
\]
where $\Phi(\cdot)$ represents the kernel feature mapping and $\lambda$ is the
regularization parameter. Due to the Fenchel duality theorem \citep{Borwein2005},
the dual formulation of this problem is:
\begin{equation}\label{eq:dual}
\min_{\alpha \in \mR^n} \frac{1}{2n} \alpha^\top \bK \alpha + \frac{\lambda}{2}\sum_{i=1}^{n}\left(\alpha_i^2 + 2\alpha_i y_i\right),
\end{equation}
where $\bK_{ij} = \Phi(a_i)^\top\Phi(a_j)$. It is easy to see that the
minimization problem \eqref{eq:dual} is exactly in the right form
for RNM to be applied with the matrix $\bM = \frac{1}{n}\bK + \lambda
\bI$. Notice that $\bM$ is an $n\times n$ matrix and sampling
sub-matrices of $\bM$ has the interpretation of subsampling the
dataset. However, to keep the notation consistent with earlier
discussion, w.l.o.g.~we will let $d=n$ for the remainder of this
section so that $\bM$ is $d\times d$. We will also assume that the
minimization problem is solved with the RNM update where each
coordinate block is sampled as $\hat{S}\sim\DPP(\frac1\alpha\bM)$.
\begin{figure*}
\centering
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width =
1\textwidth]{pics/analysis/squared_exp_time_2.png}
\vspace{2mm}
\caption{The left column varies
$\lambda$ and the right one varies
$\gamma$. The vertical line
corresponds to the value of $q =
\frac{\log(\lambda)}{\log(\gamma)}$.}
\label{fig:gamma}
\end{subfigure}
\hspace{2mm}
\begin{subfigure}[b]{0.48\textwidth}
\includegraphics[width = 1\textwidth]{pics/analysis/expected_size_2.png}
\caption{(left) $\mE[|S|]$ vs
parameter p for exponential kernel.\\
~\quad(right) Numerical effort for polynomial decay.}
\label{fig:esize}
\end{subfigure}
\vspace{1mm}
\caption{In (a) we consider exponentially decreasing eigenvalues.
In (b) (left) we plot the relationship
between $p$ and $\mE[\hat{S}]$ for exponential decay. In (b) (right)
we show the numerical effort for polynomial decay.}
\label{fig:exponential}
\end{figure*}
\subsection{Exponentially decreasing spectrum}\label{sec:exponential_decay}
Let $\{\lambda_i\}_{i=1}^d$ be the eigenvalues of $\bM$ in decreasing order. Suppose that the eigenvalue decay is exponentially decreasing:
\[ \lambda_i = C \gamma^i + C\lambda \text{ for } \gamma<1. \]
\paragraph{Motivation }
A classical motivating example for the exponential eigenvalue decay
is the \emph{squared exponential kernel}, where an
analytical form of the decay can be derived for normally
distributed data \citep{Rasmussen2006a}. In particular,
assuming $x \sim \mN(0,\eta^2)$, and using the kernel
function $k(x,y) =
\exp\big(-\frac{(x-y)^2}{2l^2}\big)$ in one dimension, the
eigenvalues satisfy $\lambda_k \leq
C\gamma^k$ for a general constant $C$ independent of $k$,
where
\begin{equation}\label{eq:se_decay}
\gamma = \frac{2\eta^2}{l^2 + 2\eta^2 + \sqrt{l^2 + 2\eta^2}}.
\end{equation}
\paragraph{Complexity} For the ease of exposition, suppose that
$\gamma^{q+1} \leq \lambda \leq \gamma^q$, where $\lambda$ is
the regularization constant and $q \in [d]$. Intuitively, this means that the
regularization parameter flattens the decay at $\gamma^{q+1}$, which will play a role in the analysis.
To control the expected size $\mE[|\hat{S}|]$ of determinantal
sampling, let $\alpha(p) = C\gamma^p$, where $p \in [1,d]$.
We get:
\[ \mE[|\hat S|] \stackrel{\eqref{eq:expected_size}} \leq
p + R_d(p,q)\quad\text{for}\quad\hat S\sim\DPP\big(\tfrac1{\alpha(p)}\bM\big), \] where $R_d(p,q) = \sum_{i=1}^{d-p}
\frac{\gamma^i + \gamma^{q-p}}{\gamma^i +
\gamma^{q-p}+1}$. Asymptotically, if
$p\ll q$, i.e., the parameter $\alpha(p)$ dominates the
regularization $C\lambda$, then the expected
subset is $\mE[|\hat S|] \approx p$. However, in the regime
where $p=\Omega(q)$, the expected
subset size rapidly goes up to $d$ (see Figure \ref{fig:esize}
(left)).
We now derive the convergence rate of RNM under determinantal
sampling:
\[1 - \sigma(\hat{S}) \stackrel{\eqref{eq:corr}} =
\frac{C\gamma^p}{C\gamma^d + C\lambda + C\gamma^p} \geq
\frac{1}{1 + \gamma^{-p}(\gamma^{d} + \gamma^{q}) }. \]
Likewise one can see that the convergence rate improves exponentially
with $p$.
\paragraph{Numerical effort} From Theorem \ref{theorem:1}, we know that in order to reach an $\epsilon$ accurate solution from the initial
accuracy $\epsilon_0 = f(x^0)-f(x^*)$ under the convergence rate $ \epsilon \leq
(1-\sigma)^T \epsilon_0$, the number of needed steps can be bounded by
\begin{equation}\label{eq:12}
T \leq \frac{\log\left({\epsilon_t}\right) -\log\left({\epsilon_0}\right) }{\log(1-\sigma)}.
\end{equation}
Using the bound derived for $1-\sigma(\hat S)$, we obtain \(T \leq
\big(\log(\frac{1}{\epsilon_t})-\log(\frac{1}{\epsilon_0})\big)
\big(\frac{\gamma^{p}}{\gamma^{d} + \gamma^{q}} + 1 \big)\). Since,
the computation step is dominated by the inversion operation
$\mO(\mE[|\hat S|]^3)$, the number of arithmetic
operations is
\[ \mO\left(\mE[|S|]^3\cdot T\right)\leq \mO\left( \big(p + R_d(p,q)\big)^3 \frac{\gamma^{p}}{\gamma^{d} + \gamma^{q}} \right). \]
The upper bound on the numerical effort in the previous
equation has two regimes. At first, for small subset sizes it is
increasing, but then exponential decay starts to dominate and
using larger blocks significantly improves the convergence
rate. Finally it flattens around $\mE[|\hat S|] = q$. Note that when $\lambda
\approx \gamma^q$, i.e., for $q=\frac{\log(\lambda)}{\log(\gamma)}$, this
phenomenon is visualized in Figure
\ref{fig:gamma} where the vertical bars correspond to $q$. In
the regime where $d \approx q$, inverting the whole matrix
seems to be the best option. When $q<d$, the term $\gamma^q$
dominates the term $\gamma^d$, and the best subset size is
either 1 or on the order of $q$, depending on the value of $\lambda$.
These observations are contrary to the intuition from the
previous works. We suspect that, due to fixed memory fetching
costs, for small sizes the initial phase is unobserved but the
second phase should be observed. Figure \ref{fig:gamma}
suggests that for sufficiently small values of $\lambda$ the
numerical performance is maximized at the attenuation point
$q$ and the predicted optimal block size is
$\frac{\log(\lambda)}{\log(\gamma)}$.
\subsection{Polynomially decaying spectrum}
Suppose that the eigenvalues $\{\lambda_i\}_{i=1}^d$
are decreasing polynomially, i.e., so that $\lambda_i
= Ci^{-s} + C\lambda$ for $s > 1$.
\paragraph{Motivation} For example, consider a
Mat\'ern kernel of order $s$, which has the form
$k(x,y)=C_2B_s(\norm{x-y})\exp(-C_1 \norm{x-y})$, where
$C_1$, $C_2$ are constants, and $B_s(d)$ is a modified
Bessel function of order $s$
\citep[see][]{Rasmussen2006a}. This class of kernels
exhibits asymptotically polynomial decay of eigenvalues
\citep[see][]{Seeger2008}.
\paragraph{Complexity} Suppose, $\lambda = q^{-s}$ for
$q \in [1,\infty)$. To control the expected size let
us parameterize the tuning parameters as $\alpha
=Cp^{-s}$, where $C$ is a suitable general
constant. Then the convergence rate becomes:
\[1-\sigma(\hat{S}) = \frac{p^{-s}}{d^{-s}+p^{-s} + q^{-s} }= \frac{1}{(p/d)^{s} + (p/q)^{s}+ 1 } \]
and $ \mE[|\hat S|] = \sum_{i=1}^{d} \frac{ i^{-s} + q^{-s}}{i^{-s} + p^{-s}+q^{-s}}$. If $p \ll q$, we can establish by integral approximation that $\mE[|S|] \approx \mO(p)$, otherwise the expected size grows faster. Additionally, with increasing $p$ the convergence rate always improves.
\paragraph{Numerical Effort} When $p \ll q$, similarly as in the preceding subsection,
the numerical cost becomes \( \mO\big( p^3
(\frac{d^sq^s}{p^s(q^s+d^s)} ) \big) \). This suggests that for $s \geq 3$ the
total numerical cost decreases for larger subsets, while for the problems with smaller $s$,
the cost increases. In general, it is difficult to obtain general
insights from the formulas, but the visalization in Figure
\ref{fig:exponential}b (right) suggests that if the regularization constant is
large (small $q$), even problems with large $s$ might incur more cost
as the subset size increases.
This suggests that \emph{small block sizes matching the memory
fetching costs} should be optimal if either the
regularization is large or if $s$ is small. With the same assumption,
if the desired accuracy is very high, performing \emph{full matrix
inversion} can be more efficient, corresponding to $\mE[|\hat S|]
\rightarrow d$ in Figure~\ref{fig:exponential}b (right). Note that
increasing the accuracy to which we optimize the problem shifts the
curves up in the logarithmic plot, while keeping the end point fixed.
\begin{figure*}
\centering
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width = 1\textwidth]{pics/comp/sparse.png}
\vspace{-3mm}
\caption{Sparse spectrum, rank of $\bK$ shown.}
\label{fig:sparse}
\end{subfigure}
\begin{subfigure}[t]{0.49\textwidth}
\includegraphics[width = 1\textwidth]{pics/comp/time_ridge_gamma_sq_exp_both.png}
\vspace{-3.5mm}
\caption{Exponential decay, varying lengthscale $\gamma$, for $\lambda = 10^{-7}$}
\label{fig:gamma_exp}
\end{subfigure}
\caption{For Gaussian data, RNM exhibits
similar behavior under DPP and uniform ($\tau$-nice) samplings.}
\end{figure*}
\subsection{Sparse spectrum}
Suppose that only $s$ out of the $d$ eigenvalues are
relatively large, while the remaining ones are very
small. This scenario occurs with a linear kernel where
the number of large eigenvalues corresponds to the number of features,
and the remaining ones are proportional to the
regularization parameter $\lambda$.
\paragraph{Complexity} For the ease of exposition, let the
large eigenvalues all be equal to $\mu \gg \lambda$.
Lemma~\ref{eq:expected_size} implies that if $\alpha = \sum_{i> k-1}^{d}
\lambda_i$ then $\mE[|\hat S|] \leq k$.
The convergence rate can be split to two cases:
\begin{equation*}
1- \sigma(\hat{S}) = \begin{cases}
\frac{d-k}{d-k+1} & \text{when } k \in [s,d-1],\\
1 - \frac{\lambda}{\mu}\frac{1}{d-k} + \mO\left(\frac{\lambda^2}{\mu^2}\right)& \text{when } k \in [0,s).
\end{cases}
\end{equation*}
We see that once $k\geq s$ a discontinuity in the spectrum
implies a discontinuity in the convergence rate. Consequently,
the \emph{optimal subset size is of the order of $s$} as long as
$\frac{\lambda}{\mu}$ is sufficiently small.
\section{DETERMINANTAL SAMPLING}\label{sec:dpp}
Our proposed sampling for the Randomized Newton Method is based
on a class of distributions called {\em Determinantal Point Processes
(DPPs)}. Originally proposed by \cite{dpp-physics}, DPPs have found numerous
applications in machine learning \citep{dpp-ml} as well as optimization
\citep{dpp-minibatch,Borsos2019}, for their variance reduction properties and the ability to produce diverse samples.
\begin{definition}
For a $d\times d$ p.s.d.~matrix $\bM$, we define $\DPP(\bM)$ as a distribution over all subsets $S\subseteq [d]$, so
that
\begin{equation}\label{eq:sample_def}
\pP(S)\propto \det(\bM_{SS}).
\end{equation}
\end{definition}
Even though this is a combinatorial distribution, the
normalization constant can be computed exactly. We state this well
known fact (e.g., see \cite{dpp-ml}) separately because it is crucial
for proving our main result.
\begin{lemma}[Normalization]\label{l:normalization}
For a $d\times d$ matrix $\bM$,
\begin{align*}
\sum_{S\subseteq [d]}\det(\bM_{SS}) = \det(\bI+\bM).
\end{align*}
\end{lemma}
Note that the distribution samples out of a power set of $[d]$.
While cardinality constrained versions have also been used,
they lack certain properties such as a simple normalization
constant. Even though the subset size of $\DPP(\bM)$ is a random
variable, it is highly concentrated around its mean, and it can also
be easily adjusted by replacing the matrix with a rescaled version
$\frac1\alpha\bM$, where $\alpha>0$. This only affects the
distribution of the subset sizes, with the expected size given by the following
lemma \citep[see][]{dpp-ml}.
\begin{lemma}[Subset Size]\label{prop:size}
If $\hat S\sim \DPP\left(\frac{1}{\alpha}\bM\right)$, then
\begin{equation} \label{eq:expected_size}
\mE[|\hat{S}|] = \Tr(\bM(\alpha \bI + \bM)^{-1}).
\end{equation}
\end{lemma}
By varying the value of $\alpha$, we can obtain any desired expected
subset size between $0$ and $d$. As we increase $\alpha$, the subset
size decreases, whereas if we take $\alpha\rightarrow 0$, then in the
limit the subset size becomes $d$, i.e., always selecting the $[d]$.
While the relationship between $\alpha$ and $\mE[|\hat{S}|]$ cannot
be easily inverted analytically, it still provides a convenient way of
smoothly interpolating between the full Newton and coordinate
descent. To give a sense of what $\alpha$ can be used to ensure
subset size bounded by some $k$, we give the following lemma.
\begin{lemma}\label{lemma:lambdas_exp}
Let $\{\lambda_i\}_{i=1}^d$ be the eigenvalues of $\bM$ in a decreasing order.
If $\alpha = \sum_{j\geq k} \lambda_j$, then $\mE[|\hat{S}|] < k$.
\end{lemma}
\subsection{New expectation formula}
We are now ready to state our main result regarding DPPs, which is a new
expectation formula that can be viewed as a matrix
counterpart of the determinantal identity from Lemma
\ref{l:normalization}.
\begin{theorem}\label{t:main}
If $\bM\succ \mathbf{0}$ and $\hat S\sim \DPP\left(\frac{1}{\alpha}\bM\right)$, then
\begin{equation}
\mE\big[ (\bM_{\hat S})^+\big] = (\alpha\bI+\bM)^{-1}.\label{eq:expectation}
\end{equation}
\end{theorem}
\begin{remark}
If we let $\bM\succeq \mathbf{0}$, then the equality
in \eqref{eq:expectation} must be replaced by a p.s.d.~inequality $\preceq$.
\end{remark}
We postpone the proof to the appendix. The remarkable simplicity of our result
leads us to believe that it is of interest not only in the context of the Randomized Newton Method, but also to the broader DPP community. While some
matrix expectation formulas involving the pseudoinverse have been
recently shown for some special DPPs
\citep[e.g.,][]{unbiased-estimates-journal},
this result for the first time relates an \emph{unregularized} subsampled
pseudoinverse with a $\alpha\bI$-\emph{regularized} inverse of the full matrix
$\bM$. Moreover, the amount of regularization that appears in the
formula is directly related to the expected sample size.
\subsection{Efficient sampling}
Efficient DPP sampling has been an active area of research over the
past decade. Several different approaches have been developed,
such as an algorithm based on the eigendecomposition of $\bM$
\citep{dpp-independence,dpp-ml} as well as an approximate MCMC sampler
\citep{rayleigh-mcmc} among others. For our problem, it is important
to be able to sample
from $\DPP(\bM)$ without having to actually construct the entire
matrix $\bM$, and much faster than it takes to compute the full
inverse $\bM^{-1}$. Moreover, being able to rapidly generate multiple independent
samples is crucial because of the iterative nature of the Randomized
Newton Method. A recently proposed DPP sampler satisfies all of these
conditions. We quote the time complexity of this method (the bounds
hold with high probability relative to the randomness of the algorithm).
\begin{lemma}[\cite{dpp-sublinear}]\label{t:dpp-algo}
For a $d\times d$ p.s.d.~matrix $\bM$ let $k=\mE[|\hat{S}|]$ where
$\hat{S} \sim\DPP(\bM)$. Given $\bM$, we can sample
\begin{enumerate}
\item the first $\hat S$ in:\quad
$d\cdot\mathrm{poly}(k)\,\mathrm{polylog}(d)$ time,
\item each next sample of $\hat S$ in:\hspace{6mm} $\mathrm{poly}(k)$ time.
\end{enumerate}
\end{lemma}
Note that the time it takes to obtain the first sample (i.e., the
preprocessing cost) is $o(d^2)$,
meaning that we do not actually have to read the entire matrix
$\bM$. Moreover, the cost of producing repeated samples only depends
on the sample size $k$, which is typically small. The key idea behind
the algorithm of \cite{dpp-sublinear} is to produce a
larger sample of indices drawn i.i.d. proportionally to the marginal probabilities
of $\DPP(\bM)$. For any $i\in [d]$, the marginal probability of $i$ in
$\hat{S} \sim \DPP(\frac1\alpha\bM)$ is:
\begin{align*}
\pP(i\in \hat{S}) = \big[\bM(\alpha\bI+\bM)^{-1}\big]_{ii}.
\end{align*}
In the randomized linear algebra literature, this quantity is often
called the $i$th $\alpha$-ridge leverage score
\citep{ridge-leverage-scores}, and sampling i.i.d.~according to ridge
leverage scores is known to have strong
guarantees in approximating p.s.d.~matrices.
Approximate ridge leverage score sampling incurs a
smaller preprocessing cost compared to a DPP \citep{Calandriello2017},
and basically no resampling cost. Motivated
by this, we propose to use this sampling as
a fast approximation to $\DPP(\frac1\alpha\bM)$ and our experiments
demonstrate that it exhibits similar convergence properties for
Randomized Newton.
We numerically compare the sampler from Lemma~\ref{t:dpp-algo}
against leverage score sampling in Appendix \ref{a:leverage}.
\section{INTRODUCTION}
We study unconstrained optimization of the form:
\[ \min_{x \in \mR^d} f(x), \]
where we assume that the function $f:\mR^d \rightarrow \mR$ is smooth, convex, and potentially high dimensional. This problem commonly arises in empirical risk minimization \citep[ERM, see][]{Shalev-Shwartz2014}.
State-of-the-art approaches for minimization of convex ERM objectives with large numbers of data points include variants of stochastic gradient descent (SGD) such as SVRG \citep{Johnson2013}, SARAH \citep{Nguyen2017} and a plethora of others. Alternatively, one can approach the ERM problem via a dual formulation, where fast coordinate minimization techniques such as SDCA \citep{Shalev-Shwartz2013}, or parallel coordinate descent \citep{PCDM,Richtarik2015a} can be applied. This is especially desirable in distributed and parallel environments \citep{HYDRA,Ma2015,Duenner2016}. These approaches are closely related to methods that subsample the Hessian \citep{Pilanci2015, subsampled-newton, Roosta-Khorasani2016}.
We study a block coordinate descent algorithm
first introduced by \cite{Qu2015Feb}. In each iteration of this algorithm, we sample a block of coordinates and then solve a Newton step on the chosen coordinate subspace.
However, in place of the true Hessian, a fixed over-approximation matrix $\bM$ is used for the sake of efficiency. The Newton step is computed on a sparsified version of this matrix with all but the selected coordinates set to zero, denoted $\bM_{\hat{S}}$ (see Section \ref{s:notation} for the complete notation).
Originally, \cite{Qu2015Feb} called this method Stochastic Dual Newton Ascent (SDNA), appealing to the fact that it operates in a dual ERM formulation. Later, it was also called a Stochastic Newton method \citep{Mutny2018a}, while we use the name \emph{Randomized Newton Method} (RNM) following \cite{Gower2019}\footnote{\citet{Gower2019} consider a more general algorithm, relying on the novel assumptions of \emph{relative} smoothness and convexity. We discuss this setting in Appendix \ref{appendix:relative}.}.
The sampling strategy for the coordinate blocks has a dramatic impact on the convergence rate \citep{Qu2016}. \cite{Gower2015} demonstrate that by optimizing the sampling probabilities one can obtain very significant speedups, however this optimization is a semidefinite program which may be even more challenging than the original optimization problem itself. Even when using a basic sampling strategy (such as uniform), the convergence analysis of RNM is challenging because it hinges on deriving the \emph{expected pseudoinverse} of $\bM_{\hat{S}}$, henceforth denoted $\mE[(\bM_{\hat{S}})^+]$. Prior to this work, no simple closed form expression was known for this quantity.
To overcome this challenge, we focus on a strategy of randomly sampling blocks of coordinates proportionally to the determinant of the corresponding submatrix of $\bM$, which we call \emph{determinantal sampling}. Similar sampling schemes have been analyzed in the context of stochastic optimization before \citep{dpp-minibatch, Borsos2019}. Recently, \citet{Rodomanov2019} analyzed determinantal sampling for randomized block coordinate descent, however they imposed cardinality constraints on the block size, and as a result were unable to obtain a simple expression for $\mE[(\bM_{\hat{S}})^+]$.
We use determinantal sampling with randomized block size, which allows us to obtain a simple closed form expression for the expected pseudoinverse:
\[ \mE[(\bM_{\hat{S}})^+] = (\alpha\bI + \bM)^{-1}, \]
where $\alpha$ is a tunable parameter that is used to control the expected block size.
With the use of this new expectation formula, we establish novel bounds on the convergence rate of RNM depending on the spectral properties of the over-approximating matrix $\bM$. For many instances of the problem, the matrix coincides with the data covariance, and spectral decays of such covariances are well understood \citep{Blanchard2007}. This allows us to predict the decay-specific behavior of RNM with determinantal sampling and recommend the optimal block size.
The cost of each iteration of RNM scales cubically with the size of the block due to matrix inversion. \citet{Qu2015Feb} demonstrate numerically that for small blocks the optimization time decreases but at some point it starts to increase again. They surmise that the improvement is obtained only as long as the inversion cost is dominated by
the other fixed per-iteration costs such as fetching from memory. However, whether the only possible speedup stems from this has remained unclear. We answer this question for determinantal sampling by deriving the optimal subset size in the case of kernel ridge regression. We show that when the eigenvalue decay is sufficiently rapid, then the gain in convergence rate can dominate the cost of inversion even for larger block sizes.
\subsection{Contributions}
The main contributions of this paper can be summarized as follows:
\begin{itemize}
\item We obtain a novel and remarkably simple expectation formula for determinantal sampling that allows us to derive a simple and closed form expression for the convergence rate of the Randomized Newton Method.
\item This allows us to improve the previous bounds on the theoretical speedup of using coordinate blocks of larger sizes. For example, we show that in the case of kernel regression with a covariance operator that has exponentially decreasing spectrum, the theoretical speedup is \emph{exponential}.
\item We take into account the actual per iteration cost, and analyze not only the convergence rate of the algorithm, but also its numerical effort to solve a problem up to some relative precision. This allows us to classify the problems into categories where the optimal block size is one, the full matrix, or somewhere in between.
\item We numerically validate the discovered theoretical properties of \emph{determinantal sampling}, and demonstrate cases when it improves over uniform sampling, and when it performs similarly. If the two perform similarity, our analysis serves as a more interpretable proxy for the convergence analysis of uniform sampling.
\end{itemize}
\subsection{Notation}
\label{s:notation}
Let $S$ be a non-empty subset of $[d]:=\{1,2, \dots, d\}$. We let $\bI_{:S}$ be the $d \times |S|$ matrix composed of columns $i \in S$ of the $d\times d$ identity matrix $\bI$. Note that $\bI_{:S}^\top \bI_{:S}$ is the $|S|\times |S|$ identity matrix.
Given an invertible matrix $\bM\in \mR^{d\times d}$, we can extract its principal $|S| \times |S|$ sub-matrix with the corresponding rows and columns indexed by $S$ via \( \mathbf{M}_{SS} \eqdef \bI_{:S}^\top \mathbf{M} \bI_{:S}\),
and additionally keeping the sub-matrix in its canonical place we can define the following operation,
\begin{equation}\label{eq:slice}
\bM_S \eqdef \bI_{:S}\bM_{SS} \bI_{:S}^\top.
\end{equation}
Note that $\mathbf{M}_{S}$ is the $n\times n$ matrix obtained from $\bM$ by retaining elements $\bM_{ij}$ for $i \in S$ and $j \in S$; and all the other elements set to zero. By $(\cdot)^+$ we denote the Moore-Penrose pseudoinverse. The matrix $(\bM_{S})^{+}$ can be calculated by inverting $\bM_{SS} \in \mR^{|S|\times |S|}$, and then placing it back into the $d \times d$ matrix.
\section{EXPERIMENTS}
We numerically validate the theoretical findings from the previous sections.
Our main objective is to demonstrate that the convergence behavior of
RNM under DPP sampling aligns well with the behavior of RNM under
uniform sampling (called $\tau$-nice), which is more commonly used. This
would suggest that our convergence analysis under DPP sampling is also
predictive for other standard samplings.
In addition to providing evidence for this claim, we also show that
there are cases where DPP sampling leads to superior performance of RNM.
Even though there exist efficient algorithms for DPP sampling, we chose
to use approximate ridge leverage score sampling as a cheaper proxy for DPP
sampling, as suggested in a recent line of work
\citep{dpp-intermediate,dpp-sublinear}. The real data experiments were
performed with sampling according to the $\frac{1}{2}$-approximate
ridge leverage scores \citep{Calandriello2017}. We always report the mean value of $10$ reruns of the experiment with the given parameters.
\paragraph{Gaussian Data}
The first experiment deals with data sampled from a Gaussian
distribution. The optimization using a kernel $\bK$ with sparse
spectrum (Figure \ref{fig:sparse}) verifies the theoretical findings
that the optimal block size should be of the same order as
$\operatorname{rank}(\bK)$.
Using similarly generated data, and the relation in
\eqref{eq:se_decay} to relate lengthscale $l$ and $\gamma$ of squared
exponential kernel, we reproduce the prediction of the theory that for
sharper decays the optimal expected size should be larger (see Figure
\ref{fig:gamma_exp}, compared with theory, Figure \ref{fig:gamma}). The
performance of DPP and uniform sampling
is on par as the intuition suggests, since for normally distributed
data even a uniform subsample provides good summary statistics.
\paragraph{Gaussian Mixture Data}
\begin{figure}
\centering
\includegraphics[width = 0.49\textwidth]{pics/comp/cluster.png}
\caption{Gaussian mixture data - sparse spectrum, rank of kernel $\bK$ shown.}
\label{fig:cluster}
\end{figure}
Akin to results from the sketching literature
\citep[e.g., see][]{unbiased-estimates-journal}, we suspect that the
superior convergence of DPP sampling over uniform presents itself primarily
if the dataset is heterogeneous. By heterogeneity we mean that a
uniform subsampling of the points is likely not a
good summary of the dataset. Consider a dataset where the points
are sampled from a Gaussian Mixture Model with $8$ clusters that are
equally likely. In order to have a good summary, a point from each
cluster should be present in the sample. DPP samples are generally
more diverse than uniform samples which makes it more likely that they
will cover all the clusters. In Figure \ref{fig:cluster}, we see that
DPP significantly outperforms uniform sampling for this dataset because it allows RNM
to solve more representative subproblems.
\paragraph{Real Data Experiments}
We perform two real data experiments on standard UCI datasets where we
optimize until statistical precision. In Figure~\ref{fig:real}a, we
optimize linear ridge regression on the \emph{superconductivity}
dataset. Next, in Figure \ref{fig:real}b we fit
kernel ridge regression with squared exponential kernel on the
\emph{cpusmall} dataset. For both datasets, the optimal subset size
under DPP sampling roughly matches the optimal size under uniform
sampling. Moreover, in the case of the superconductivity dataset, as
suggested by the theory for linear kernels, the optimal size is of
the same order as the feature dimensionality.
\begin{figure}
\centering
\begin{subfigure}[t]{0.23\textwidth}
\includegraphics[width = 1\textwidth]{pics/comp/superconductivity_time.png}
\caption{$n=1460$ and $d = 81$.}
\label{fig:house}
\end{subfigure}
\begin{subfigure}[t]{0.23\textwidth}
\includegraphics[width = 1\textwidth]{pics/comp/cpusmall_time.png}
\caption{$n=2191$ and $d = 12$.}
\label{fig:cpusmall}
\end{subfigure}
\vspace{2mm}
\caption{Experiments on real data.}
\vspace{1mm}
\label{fig:real}
\end{figure}
\subsubsection*{\bibname}}
\hypersetup{
colorlinks,
linkcolor={red!40!gray},
citecolor={blue!40!gray},
urlcolor={blue!70!gray}
}
\bibliographystyle{apalike}
\begin{document}
\renewcommand*{\thefootnote}{\fnsymbol{footnote}}
\twocolumn[
\aistatstitle{Convergence Analysis of Block Coordinate Algorithms with Determinantal Sampling}
\aistatsauthor{ Mojm\'ir Mutn\'y\footnote[2]{Equal contribution.} \And Micha\l{} Derezi\'nski\footnotemark[2] \And Andreas Krause }
\aistatsaddress{ Department of Computer Science\\
ETH Zurich, Switzerland \\ \texttt{[email protected]} \And Department of Statistics \\ University of California, Berkeley \\ \texttt{[email protected]} \And Department of Computer Science\\
ETH Zurich, Switzerland \\ \texttt{[email protected]} }
]
\begin{abstract}
We analyze the convergence rate of the randomized Newton-like method
introduced by \cite{Qu2015Feb} for smooth and convex
objectives, which uses random coordinate
blocks of a Hessian-over-approximation matrix
$\bM$ instead of the true Hessian. The convergence analysis of the algorithm is
challenging because of its complex dependence on the structure
of $\bM$. However, we show that when the coordinate blocks are
sampled with probability
proportional to their determinant, the convergence rate depends solely on the eigenvalue distribution of matrix $\bM$, and has an analytically tractable form. To
do so, we derive a fundamental new expectation formula for
determinantal point processes. We show that determinantal
sampling allows us to reason about the optimal subset size of
blocks in terms of the spectrum of $\bM$. Additionally, we
provide a numerical evaluation of our analysis,
demonstrating cases where determinantal sampling is superior
or on par with uniform sampling.
\end{abstract}
\input{intro}
\input{algo}
\input{dpp}
\input{analysis}
\input{decay}
\input{numerics}
\section{CONCLUSION}
We analyzed a sampling strategy for the Randomized Newton Method, where
coordinate blocks of the Hessian over-approximation are sampled
according to their determinant. This sampling allows for a simple
interpretation of the convergence rate of the algorithm, which was
previously not well understood. We demonstrated that for empirical
risk minimization this convergence analysis allows us to predict the
optimal size for the sampled coordinate blocks in order to minimize
the total computational cost of the optimization.
\paragraph{Acknowledgments} This work was supported by SNSF grant
407540\_167212 through the NRP 75 Big Data program. Also, MD thanks
the NSF for funding via the NSF TRIPODS program.
\section{PROOFS}
\subsection{DPPs}
\begin{proof}[Proof of Theorem \ref{t:main}]
First, assume that $\alpha=1$. Since $\bM\succ \mathbf{0}$, we have $\det(\bM_{SS})>0$ for
all $S\subseteq[d]$. We will next use the following
standard determinantal formula which holds for any $v\in\mR^d$ and any
invertible matrix $\bM$:
\begin{align}
\det(\bM)v^\top\bM^{-1}v = \det(\bM+vv^\top)
- \det(\bM).\label{eq:pr0}
\end{align}
Applying this formula to the submatrices of $\bM$ and denoting by $v_S$
the sub-vector of $v$ indexed by $S$, we show that for any $v\in\mR^d$:
\begin{align*}
v^\top
&\mE\big[(\bM_S)^+
\big]v =
\sum_{S\subseteq[d]}\frac{\det(\bM_{SS})}{\det(\bI+\bM)}v_S^\top\bM_{SS}^{-1}v_S\\
{\scriptsize\eqref{eq:pr0}}\
&=\sum_{S\subseteq[d]}\frac{\det(\bM_{SS}+v_Sv_S^\top)-\det(\bM_{SS})}{\det(\bI+\bM)}\\
&=\frac{\sum_S \det([\bM+vv^\top]_{SS}) - \sum_S\det(\bM_{SS})}{\det(\bI+\bM)}\\
\text{\scriptsize(Lemma~\ref{l:normalization})}\
&=\frac{\det(\bI+\bM+vv^\top) - \det(\bI+\bM)}{\det(\bI+\bM)}\\
{\scriptsize\eqref{eq:pr0}}\
&=\frac{\det(\bI+\bM)\,v^\top(\bI+\bM)^{-1}v}{\det(\bI+\bM)} \\
&=
v^\top(\bI+\bM)^{-1}v.
\end{align*}
Since the above holds for all $v$, the equality also holds for the pd.
matrices. To obtain the result with $\alpha\neq 1$, it suffices to
replace $\bM$ with $\frac1\alpha\bM$.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lemma:lambdas_exp}]
The eigenvalues of $\bM(\alpha\bI+\bM)^{-1}$ are
$\frac{\lambda_i}{\lambda_i+\alpha}$ so
\begin{align*}
\mE[|S|] & = \sum_{i=1}^{d} \frac{\lambda_i}{\lambda_i + \alpha } =
\sum_{i=1}^{d} \frac{\lambda_i}{\lambda_i + \sum_{j\geq k} \lambda_j } \\
& = \sum_{i < k}^{d} \frac{\lambda_i}{\lambda_i +
\sum_{j\geq k} \lambda_j } + \sum_{i \geq k}^{d}
\frac{\lambda_i}{\lambda_i + \sum_{j\geq k} \lambda_j }\\
&< (k-1)+ 1 = k,
\end{align*}
which concludes the proof.
\end{proof}
\subsection{Convergence Analysis}
\begin{proof}[Proof of Theorem \ref{corr:recurence}]
\begin{eqnarray}
\sigma_1 & \stackrel{\eqref{eq:sigma_1}}= & \lambda_{\min}\left( \bM^{1/2} \left({\alpha}\bI + \bM\right)^{-1} \bM^{1/2} \right) \\
& = & \lambda_{min} \left( \left( {\alpha}\bM^{-1} + \bI \right)^{-1} \right)\\
& = & \frac{1}{\lambda_{\max}\left(
{\alpha} \bM^{-1} + \bI \right)} = \frac{1}{1 + {\alpha}\lambda_{\max}(\bM^{-1})} \\
& = & \frac{ \mu }{ \mu + \alpha} \\
\end{eqnarray}
where $\mu = \lambda_{\min}(\bM)$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:recurence}]
By definition,
\[ \frac{1}{\sigma(k+1)} = 1 + \frac{\sum_{i>k}^{d}\lambda_i}{\lambda_d} = 1 + \frac{\sum_{i>{k-1}}^{d}\lambda_i - \lambda_{k}}{\lambda_d} = \frac{1}{\sigma(k)} - \frac{\lambda_{k}}{\lambda_d} \]
Rearranging,
\[ \frac{1}{\sigma(k)} = \frac{1}{\sigma(k+1)} + \frac{\lambda_{k}}{\lambda_d} \implies \sigma(k) = \frac{\sigma(k+1)\lambda_d}{\lambda_d + \lambda_k \sigma(k+1)} \]
Dividing the denominator and the numerator by $\lambda_d$ finishes the proof.
\end{proof}
\subsection{Dual convergence rate}
The dual convergence rate established in \cite{Qu2015Feb} relies on the
notion of expected separable over-approximation. Namely, the existence of
$v \in \mR^d$ s.t. $\mE[\bM_S] \preceq \bD(p \circ v)$, where $p$ is
the vector of marginal probabilities. In case of DPP sampling, one can
choose $v = \diag(\bM)\circ \diag(\bM(\bM + \alpha \bI)^{-1})^{-1}$,
and apply dual convergence results established in this literature. By
$\circ$ we denote element-wise product.
\section{LEVERAGE SCORE SAMPLING VS DPP SAMPLING}
\label{a:leverage}
We perform a simple experiment on the Gaussian Mixtures dataset where the matrix has a sparse spectrum. In Figure \ref{fig:comparison} we see that the optimization process is influenced minimally.
\begin{figure}[H]
\centering
\includegraphics[width = 0.5\textwidth]{pics/comp/comparison.png}
\caption{Comparison of leverage score sampling and DPP}
\label{fig:comparison}
\end{figure}
\section{RELATIVE SMOOTHNESS AND RELATIVE STRONG CONVEXITY}\label{appendix:relative}
Recent works such as \citep{Gower2019} and \citep{Karimireddy2018a} introduce the concepts of relative-smoothness, relative strong convexity and $c$-stability. These are weaker conditions than assumed in this paper. With these conditions, the proof techniques used to analyze coordinate descent algorithms are applicable to Newton-like algorithms, where instead of a fixed matrix $\bM$, the actual Hessian $\bH(x)$ can be used. The extension to $c-$stability is done trivially in Theorem 2 of \cite{Karimireddy2018a}, here we focus on a slightly more elaborate connection with relative smoothness and relative strong-convexity.
\begin{assumption}[\cite{Gower2019}]\label{def:assumption}
There exists a constant $\tilde{L}\geq\tilde{\mu}$ such that for all $x,y \in \mQ \subseteq \mR^d$, where $\mQ:=\{x \in \mR^d : f(x)\leq f(x_0)\}$:
\begin{equation}\label{eq:rsmooth}
f(x) \leq f(y) + \braket{\nabla f(y),x} + \frac{\tilde{L}}{2}\norm{x-y}_{\bH(y)}
\end{equation}
and
\begin{equation}\label{eq:rconvex}
f(x) \geq f(y) + \braket{\nabla f(y),x} + \frac{\tilde{\mu}}{2}\norm{x-y}_{\bH(y)}.
\end{equation}
\end{assumption}
Now the task is to analyze the algorithm with the following update rule, which is identical to general Newton rule when $S = [d]$,
\begin{equation}\label{eq:update2}
x_{k+1} = x_k - \gamma (\bH(x_k)_{S_k})^+ \nabla f(x_k).
\end{equation}
We fix a particular choice of $\gamma = 1/\tilde{L}$. This should be contrasted with the update rule $\eqref{eq:update}$.
Now given these assumption, we are able to show that the constant akin to $\sigma(\hat{S})$ appears in the analysis of this algorithm by utilizing the notions from \citep{Gower2019}. We sacrifice generality for the sake of brevity, and assume that range of $\bH(x)$ spans whole $\mR^d$ for each $x \in \mQ$. Then, the following quantities resembling $\sigma(\hat{S})$ appear in the convergence analysis of the update rule \eqref{eq:update2}
\begin{equation}\label{eq:sigmahat}
\hat{\sigma}(\hat{S},x) = \lambda_{\min} \left( \mE_{\hat{S}} \left[ \bH^{1/2}(x)(\bH(x)_{\hat{S}})^+ \bH(x)^{1/2} \right] \right)
\end{equation}
and \[\hat{\sigma}(\hat{S}) = \min_{x\in \mQ } \hat{\sigma}(\hat{S},x) \]
\begin{theorem}[Theorem 3.1 of \cite{Gower2019}, modified]\label{thm:rel}
Let $f$ satisfy Assumption \ref{def:assumption}, and let $\bH(x)$ be the Hessian at $x$ having range that spans whole $\mR^d$ for all $x$. Then
\[ \mE_{\hat{S}}[ f(x_{k+1}) - f(x^*) ] \leq \left( 1 - \frac{\hat{\sigma}(\hat{S},x_k)\mu}{L} \right) (f(x_k) - f(x^*), \]
and
\[ \mE_{\hat{S}}[ f(x_{k}) - f(x^*) ] \leq \left( 1 - \frac{\hat{\sigma}(\hat{S})\mu}{L} \right)^k (f(x_0) - f(x^*), \]
where $ \hat{\sigma}(\hat{S}) = \min_{x\in \mQ } \hat{\sigma}(\hat{S},x)$ as in Equation \eqref{eq:sigmahat}.
\end{theorem}
\begin{proof}
Minimizing the upper bound in \eqref{eq:rsmooth} restricted to coordinates in $S_k$, we arrive at,
\begin{eqnarray*}
f(x_{k+1}) - f(x_k) & \stackrel{\eqref{eq:smooth}} \leq & - \frac{1}{2\tilde{L}} \braket{\nabla f(x_k), (\bH(x_k)_{S_k})^+ \nabla f(x_k)} \\
\mE[f(x_{k+1}) - f(x_k)] & \leq & - \frac{1}{2\tilde{L}} \braket{\nabla f(x_k),\mE_{\hat{S}}[(\bH(x_k)_{S_k})^+] \nabla f(x_k)}\\
& \stackrel{\eqref{eq:rconvex}, \eqref{eq:sigmahat}} \leq & -\frac{\mu}{L}\hat{\sigma}(\hat{S},x) (f(x_k) - f(x^*))\\
& \leq & -\frac{\mu}{L}\hat{\sigma}(\hat{S}) (f(x_k) - f(x^*))\\
\end{eqnarray*}
rearranging finishes the proof.
\end{proof}
The following corollary states that with DPP sampling, the update rule in \eqref{eq:update2} can have a more interpretable convergence rate than stated in the Theorem \ref{thm:rel}.
\begin{corollary}[of Theorem \ref{thm:rel}]
Under the assumption of Theorem \ref{thm:rel}, let additionally $S_k$ be a sample from sampling $\hat{S}_k \sim \DPP(\frac{1}{\alpha} \bH(x_k))$, then
\[ \mE_{\hat{S}_k}[ f(x_{k+1}) - f(x^*) ] \leq \left( 1 - \left(\frac{\lambda(x_k) }{\lambda(x_k) + \alpha}\right)\frac{\mu}{L} \right) (f(x_k) - f(x^*), \]
where $\lambda(x_k) = \lambda_{\min}(\bH(x_k))$.
\end{corollary}
The following lemma relates the complexity quantity defined above to the definition of $\sigma(\hat{S})$ used in the main body of this paper. Note that $\hat{\sigma}$ is larger than $\sigma$, even if the fixed over-approximation exists, as previously we assumed the over-approximation to be valid globally not just in $\mQ$.
\begin{lemma} If for all $x \in \mQ$, $\bM \succeq \bH(x) \succeq \kappa \bM \succ 0$, then
\[ \hat{\sigma}(\hat{S}) \geq \kappa \sigma(\hat{S}).\]
The relative smoothness, and strong-convexity can be chosen to be $\tilde{L} = 1$, and $\tilde{\mu} = 1$, respectively.
\end{lemma}
\begin{proof}
\begin{eqnarray*}
\hat{\sigma}(\hat{S}) & = & \min_{x\in \mQ } \min_{v \in \mR^d} \frac{ \braket{v,\mE_{\hat{S}} \left[ \bH^{1/2}(x)(\bH(x)_{\hat{S}})^+ \bH(x)^{1/2} \right]v }}{{\norm{v}_2^2}} = \min_{v \in \mR^d} \min_{x\in \mQ} \frac{ \braket{v,\mE_{\hat{S}} \left[ \bH^{1/2}(x)(\bH(x)_{\hat{S}})^+ \bH(x)^{1/2} \right]v }}{\norm{v}_2^2} \\
& \geq & \min_{v \in \mR^d} \frac{ \braket{v,\mE_{\hat{S}} \kappa \left[ \bM^{1/2}(\bM_{\hat{S}})^+ \bM^{1/2} \right]v }}{\norm{v}_2^2} = \kappa \sigma(\hat{S})
\end{eqnarray*}
\end{proof}
\section{OTHER SAMPLINGS}
The convergence properties of RNM with determinantal sampling depend solely on the spectral properties of $\bM$. This is not true of other common samplings such as $\tau$-nice. Indeed we can improve or worsen the performance of $\tau$-nice sampling when $\bM$ is transformed via spectrum preserving operation such as unitary transformation \[\bM \gets \bR^\top \bM \bR, \text{ where } \bR^\top \bR = \bI.\]
Suppose that we are given an eigenvalues of the matrix $\bM$, for any sampling $\hat{S}$ is it possible to find a spectrum preserving rotation such that $\sigma(\hat{S})$ is at least as small as $\sigma(\hat{S}_{\DPP})$ which corresponds to DPP sampling with the same expected cardinality? The answer turns out to be negative, and we show counter-example.
\begin{remark}[Counter-example]
Let $\hat{S}_1$ be a sampling such that $[n]$ is sampled with $1/2$ probability and $\emptyset$ and $1/2$ probability. The expected size of the subset $\mE[|\hat{S}_1|] = d/2$ and $\sigma(\hat{S}_1) = \frac{1}{2}$ irrespective of the matrix $\bM$.
Suppose matrix $\bM$ has degenerate spectrum such that $\lambda$ is eigenvalue with multiplicity $d/2$ and $\mu$ is eigenvalue with $d/2$ multiplicity where $\lambda < \mu$. In order s.t. $\mE[|S_{\DPP}|] = \frac{d}{2}$, $\alpha = \sqrt{\lambda \mu}$, then $\sigma(\hat{S}_{\DPP}) < \frac{1}{2}$.
\end{remark}
In what circumstances does DPP sampling perform better than a uniform sampling? First, we consider circumstances where uniform sampling is optimal.
\subsection{Uniform sampling}
It is important to allow for variation in the off-diagonal of $\bM$. If we consider only diagonal $\bM$, the optimal sampling is uniform sampling.
\begin{lemma} \label{lemma:uniform_optimal} Let $\bM$ be diagonal. The quantity $\sigma(\hat{S})$ of a sampling over a power set $P([d])$ constrained by $\mE[|\hat{S}|] = k$ is maximized for uniform samplings.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:uniform_optimal} ]
We want to maximize the minimum eigenvalue of a matrix $\bM^{1/2} \mE[(\bM_S)^{-1} ] \bM^{1/2}$. For a diagonal $\bM$ we know that $(\bM_S)^{-1} = (\bM^{-1})_S$. Hence, \(\bM^{1/2} \mE[(\bM_S)^{-1} ] \bM^{1/2} \bD(p)\), where $p$ is a vector of marginals $p_i = P(i \in \hat{S})$. Hence, the minimum eigenvalue is the minimum marginal probability subject to a constraint that $\mE[|S|] = \sum_{j=1}^{d}P(j \in \hat{S}) \leq k$. This leads to an optimum where $P(i \in \hat{S}) = P(j \in \hat{S})$ for all $i,j \in [d]$. Hence the optimal sampling distribution is uniform.
\end{proof}
\subsection{Parallel Sampling}
The parallel extension of the update method \ref{eq:update} has been considered in \cite{Mutny2018a} and \cite{Karimireddy2018a}. Namely, the authors consider a case, when the updates with $c$ machines are aggregated together to form a single update in the form $\approx \frac{1}{b}\sum_{j=1}^{c} (\bM_{S_j})^+$, where $b$ is the aggregating parameter. It is known that for parallel disjoint samplings the convergence rate increases linearly with the number of processors. For independent samplings the aggregating parameter $b$ depends on the quantity,
\[ \theta(\hat{S}) = \lambda_{\max} (\bM^{1/2}\mE[(\bM_{\hat{S}})^+] \bM^{1/2} ) \]
which in the case of DPP sampling is equal to $\theta = \frac{\lambda_1}{\lambda_1 + \alpha}$. The quantity $\theta (\hat{S}) \in [\sigma(\hat{S}),1]$, and as $\theta \rightarrow 1$, the aggregation operation becomes averaging $b \rightarrow c$. For DPP sampling, we can see an inverse relationship between increasing $\sigma(\hat{S})$ by increasing block size, which inherently makes the parallelization problem more difficult by increasing $\theta(\hat{S})$.
|
1,108,101,565,690 | arxiv | \section{Introduction}
Two-player games on finite graphs which admit infinite plays are of fundamental importance in many areas of logic and computer science, especially in the
formal analysis of reactive systems, where they model the non-terminating interaction between a system and its environment.
In such a game, the \emph{objective} or \emph{winning condition} of the player who represents the system specifies
the desired set of behaviours of the system. The most basic classes of such
objectives are \emph{reachability} and \emph{safety} objectives defined by a set of states (positions) that the player should reach, or avoid.
We can assume, without loss of generality, that even though infinite plays are possible in a game with reachability or safety objectives,
they are all won by the same player.
Games with genuine and non-trivial winning conditions for infinite plays are harder to analyse; they include games with arbitrary
$\omega$-regular objectives, such as liveness, Muller, Streett-Rabin, or parity objectives, and many others.
The goal of this paper is to provide a case study of a recent method for strategy analysis, based on semiring semantics,
and we would like to explore its potential for providing detailed information about strategies in genuinely infinite games.
One of the simplest class of games with a non-trivial winning condition for the infinite plays are games with
Büchi objectives, which require that a specific target set $F$ of states is reached infinitely often during the play (see e.g. \cite{GraedelThoWil02} for background).
Büchi games, as well as some of their straightforward generalisations, have many applications in formal methods,
and efficient algorithms for solving them have been studied thoroughly (see e.g. \cite{ChatterjeeDvoHenLoi16,ChatterjeeHen14,ChatterjeeHenPit08}).
They are also of interest from the points of view of topology and logic, because they are among the simplest games where the set of winning plays is neither open nor closed, and where logical definition of the winning region requires a genuine alternation of a greatest and a least fixed point (see Sect.~\ref{sec:TheFormula}).
Strategies in infinite games can be very complicated because, in principle, they may depend on the entire history of a play.
Thus, there exist uncountably many different strategies, even on a finite game graph.
Fortunately, in many cases and in particular for Büchi games, simple strategies are sufficient to win.
A fundamental result in this context is the positional determinacy of parity games (of which Büchi games are a special case),
saying that from each position, one of the two players has a \emph{positional winning strategy}, i.e.
a winning strategy that only depends on the current position and not on the history of the play.
A positional strategy can be viewed as a subgraph of the game graph, and can therefore be represented in
a compact way. As a consequence, the algorithmic analysis of Büchi games has concentrated almost exclusively on the positional strategies. Here we extend this point of view somewhat and take also other kinds of simple strategies into account.
Specifically, we are interested in \emph{absorption-dominant} winning strategies \cite{GraedelTan20} which are strategies without redundant
moves; this means that taking away anything, in the sense of demanding that some specific move is played less often,
makes the strategy non-winning.
Another way to distinguish positional strategies from absorption-dominant ones concerns their minimisation properties:
while positional strategies minimize the \emph{set} of moves that they use, absorption-dominant strategies
take multiplicities into account and minimize the \emph{multiset} of moves.
A further interesting class are the \emph{persistent} strategies \cite{MarcinkowskiTruderung02}, which are positional in each individual play
but not necessarily across distinct plays.
We shall study the relationship between these different classes of simple strategies, and prove that
every positional strategy is absorption-dominant and every absorption-dominant strategy is persistent,
and that these inclusions are strict.
The specific method for strategy analysis that we want to apply to Büchi games in this paper is
based on the logical definability of the winning positions by
a formula in the fixed-point logic LFP, and on the semiring semantics for LFP
developed in \cite{DannertGraNaaTan21}.
In the classical Boolean semantics, a model $\AA$ of a formula $\phi$
assigns to each (instantiated) literal a Boolean value.
$\K$-interpretations $\pi$, for a suitable semiring $\K$, generalize this
by assigning to each such literal a semiring value from $\K$.
We then interpret $0$ as \emph{false} and all other semiring values as \emph{nuances of true}
that provide additional information, depending on the semiring:
For example, the Boolean semiring $\mathbb{B} = (\{0,1\}, \lor, \land, 0, 1)$ corresponds to Boolean semantics, the Viterbi-semiring $\mathbb{V} = ([0,1], \max, \cdot, 0, 1)$ can model \emph{confidence} scores,
the tropical semiring $\mathbb{T}= (\mathbb{R}_{+}^{\infty},\min,+,\infty,0)$
is used for cost analysis, and min-max-semirings $(A, \max, \min, a, b)$ for a totally ordered set $(A,<)$ can model different access levels.
Most importantly, semirings of polynomials, such as $\N[X]$, allow us to \emph{track} certain literals by mapping them to different indeterminates. The overall value of the formula is then a polynomial that describes precisely what combinations of literals prove the truth of the formula.
Semiring semantics has been studied for various logics \cite{BourgauxOzaPenPre20, DannertGra19, DannertGra20,DannertGraNaaTan21,GraedelTan17},
following the successful development of semiring provenance in database theory and related fields (see e.g.\cite{DeutchMilRoyTan14,GeertsPog10,GreenKarTan07,GreenTan17,OzakiPen18,RaghothamanMenZhaNaiSch20,Senellart17}).
While semiring provenance analysis for database queries had largely been confined
to positive query languages such as conjunctive queries, positive relational algebra, and Datalog,
the generalisation to logics such as first-order logic FO and least fixed-point logic LFP -- featuring full negation and unrestricted interaction between least and greatest fixed points -- poses non-trivial mathematical challenges and requires
new algebraic constructions. Specifically, it has turned out that appropriate semirings for LFP should be
absorptive and fully continuous. Fortunately, this is the case for most of the important application semirings such
as $\mathbb{V}, \mathbb{T}$ or min-max-semirings, but not for the natural semiring $\N$, or the general provenance semirings
of polynomials or formal power series, $\N[X]$ and $\N^\infty \ps X$.
Instead, we rely on semirings ${\mathbb S}^{\infty}[X]$ of \emph{generalized absorptive polynomials},
which we explain in Sect.~\ref{sect:semirings}, and which are
the \emph{universal} absorptive, fully-continuous semirings, in the sense that
every mapping $h \colon X \to \K$ into an absorptive, fully-continuous semiring $\K$
uniquely extends to a fully-continuous semiring homomorphism $h \colon {\mathbb S}^{\infty}[X] \to \K$,
see Theorem~\ref{universality}.
From valuations of fixed-point formulae in such semirings we thus can
derive detailed insights into why the formula holds -- and by applying this to
the fixed-point definition of winning positions in Büchi games we obtain compact descriptions of
winning strategies, in particular of all positional strategies and all absorption-dominant ones.
After an analysis of simple winning strategies in Büchi games, and a short introduction to semiring semantics for fixed-point logic, we shall study the semiring valuations of the particular LFP-formula $\mathsf{win}_0(x)$
that defines the winning region for Player~0 in Büchi games. Given that the objective of Player~0 is to ensure
that the play hits the target set $F$
infinitely often, we may informally describe their winning region as the \emph{largest} set
$Y$ of positions from which they can enforce a (further) visit to $Y\cap F$
after $k\geq 1$ moves. On the other side the set of positions from which
Player~0 can enforce a visit to a target set is the \emph{smallest} set
of positions that either are already in the target set, or from which Player~0 can
enforce the play to come closer to it. Thus, the winning region of Player~0
can be described as a greatest fixed point inside of which there is a least fixed point, and it is well-known that
this fixed-point alternation in the treatment of Büchi objectives cannot be avoided, see e.g. \cite{BradfieldWal18}.
We shall prove a Sum-of-Strategies Theorem, saying that for any position $v$ in a Büchi game, the
valuation of the LFP-formula $\mathsf{win}_0(v)$ in an absorptive, fully-continuous semiring
coincides with the sum of the valuations of all absorption-dominant winning strategies from $v$.
Besides being of theoretical interest, this result allows to study a number of interesting questions
concerning the available winning strategies in a given Büchi game:
\smallskip\noindent\textbf{Strategy tracking.}
Introducing indeterminates for all edges in a fixed Büchi game $\Gg$,
the semiring value $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)}$ for a position $v$
is a polynomial whose monomials are concise descriptions of all absorption-dominant strategies.
From these monomials we can derive whether Player 0 wins from $v$ (if there are any monomials)
and which edges are used by each absorption-dominant strategy, and how often they appear in the strategy tree.
In particular, we can immediately identify and count positional strategies from the polynomial.
Going further, we can answer questions such as: can Player~0 still win if we remove edge $e$, or several edges at once?
Can they still win if edge $e$ may only be used finitely often in each play?
\smallskip\noindent\textbf{Repairing a game.} Instead of analysing strategies in a fixed game, we may also reason about modifications or synthesis of (parts of) the game.
For example, assuming Player~0 loses from $v$, what are minimal modifications to the game that would let Player~0 win from $v$?
To answer such questions we have to take into account also negative information (i.e., absent edges in the graph),
so as to find a minimal repair consisting of both moves to delete and moves to add.
Algebraically, this requires to extend our semirings by dual-indeterminates, which leads to quotient semirings
${\mathbb S}^{\infty}[X,\bar X]$ by a construction that has been used before in \cite{GraedelTan17,XuZhaAlaTan18,DannertGraNaaTan21} to deal with
semiring semantics for negation.
We illustrate with the example of minimal repairs that we can indeed derive the desired information from valuations in such semirings.
\smallskip\noindent\textbf{Cost computation.} A typical application of semiring provenance in databases is cost analysis:
assuming that atomic facts are not for free but come with a cost (a non-negative real number),
then the minimal cost of evaluating a query is described by a provenance valuation in
the tropical semiring $\mathbb{T}= (\mathbb{R}_{+}^{\infty},\min,+,\infty,0)$. In a game, we may ask
the analogous question of what is the minimal cost of a winning strategy assuming that
moves come with a cost. For reachability and safety games that admit only finite plays, such an analysis
works in a reasonably straightforward way by means of an appropriate
sum-of-strategies theorem \cite{GraedelTan20} (which is much simpler than the one for Büchi games).
However, as we shall show, this does not generalize in a nice way
to Büchi games, and this seems to be a general limitation of the method of semiring valuations.
\section{Büchi Games and Strategies}
A Büchi game is given by a tuple $\Gg = (V, V_0, V_1, E, F)$ where
$V$ is a set of positions (here assumed to be finite), with a disjoint decomposition
$V=V_0 \dot\cup V_1$ into positions of Player~0 and positions of Player~1.
The relation $E\subseteq V\times V$ specifies the possible moves, and the target set
$F\subseteq V$ describes the winning condition.
We denote the set of immediate successors of a position $v$ by
$vE:=\{w \mid vw\in E\}$ and require that $vE\neq\emptyset$ for all $v$.
A play from an initial position $v_0$ is an infinite path $v_0v_1v_2\dots$
through $\Gg$ where the successor $v_{i+1}\in v_iE$ is chosen by Player~0
if $v_i\in V_0$ and by Player~1 if $v_1\in V_1$.
A play $v_0v_1v_2\dots$ is won by Player~0 if $v_i\in F$ for infinitely many $i<\omega$,
otherwise it is won by Player~1.
The winning region of Player~$\sigma$ is the set
of those positions $v\in V$ such that Player~$\sigma$ has a winning strategy
from $v$, i.e. a strategy that guarantees them a win, no matter what the opponent does.
A strategy for Player~$\sigma$ in $\Gg = (V, V_0, V_1, E, F)$ can be represented in different ways,
for instance as a function $f \colon V^* V_\sigma \to V$ that assigns a next position to each partial play ending in a position of Player $\sigma$,
or simply $f \colon V_\sigma \to V$ if the strategy is positional.
Here we follow an alternative approach and represent strategies as trees,
comprised of all plays that are consistent with the strategy (see, e.g., \cite{GraedelTan20}).
For simplicity, we only consider strategies of Player~0, so unless mentioned otherwise, \emph{strategy} always refers to a strategy for Player~0.
\begin{definition}
Given a Büchi game $\Gg = (V, V_0, V_1, E, F)$, the \emph{tree unraveling} from $v_0$ is the tree $\Tt(\Gg, v_0)$ whose nodes are all finite paths $\rho$ from $v_0$ in $\Gg$ and whose edges are $\rho \to \rho v$ for $v \in V$.
We often denote a node of $\Tt(\Gg, v_0)$ as $\rho v$ to indicate a finite path ending in $v \in V$.
We write $|\rho|$ for the length of $\rho$ and $\rho \sqsubseteq \rho'$ if $\rho$ is a (not necessarily strict) prefix of $\rho'$.
\end{definition}
Strategies can then be defined as subtrees of the tree unraveling, which allows for a more visual way to reason about strategies.
An important detail is that the strategy tree only contains positions (and thus choices for these positions) that
are reachable when following the strategy.
Moreover, we only consider finite Büchi games and hence the tree unraveling and all strategies are finitely branching.
\begin{definition}
\label{def:strategyAsTree}
A \emph{strategy} $\Ss$ (of Player~0) from $v_0$ in $\Gg$ is a subtree of $\Tt(\Gg, v_0)$ induced by a node set $W$ satisfying the following conditions:
\begin{itemize}
\item if $\rho v \in W$, then also $\rho \in W$ (prefix closure);
\item if $\rho v \in W$ and $v \in V_0$, then there is a unique $v' \in vE$ with $\rho v v' \in W$ (unique choice);
\item if $\rho v \in W$ and $v \in V_1$, then $\rho v v' \in W$ for all $v' \in vE$ (all moves of the opponent).
\end{itemize}
The strategy is winning if all plays contained in $\Ss$ are winning.
We commonly write $\rho \in \Ss$ instead of $\rho \in W$,
and we often refer to paths of the form $\rho v \in \Ss$ as \emph{occurrences of $v$} in $\Ss$.
When we depict strategies graphically, we represent finite paths $\rho v$ just by their last position $v$ to ease readability (notice that in the tree unravelling
$\rho$ can be reconstructed from $v$ by following the path to the root).
See \cref{fig:RunningStrategy} for an example.
For $v \in V_0$, we further write $\Ss(\rho v) = w$ if $\rho v w$ is the (unique) successor of $\rho v$ in $\Ss$.
If $\Ss$ is positional, we may also write $\Ss(v)$ to denote the unique successor of $v$ chosen by $\Ss$.
We write $\Strat_\Gg(v)$ and $\WinStrat_\Gg(v)$ to denote the set of all (winning) strategies of Player~0 from position $v \in \Gg$, and we drop $\Gg$ if the game is clear from the context.
\end{definition}
\begin{figure}
\newcommand{\enode}[4][]{\draw [arr] (#2) edge node [#1] {$#4$} (#3);}
\begin{subfigure}[t]{.46\linewidth}
\centering
\begin{tikzpicture}[node distance=1.5cm,framed,baseline]
\node [p1,label={left:$v$}] (0) {};
\node [p1, right of=0, yshift=.7cm] (1) {};
\node [p0, F, right of=1, yshift=-.7cm, label={below:$v'$}] (2) {};
\node [p0, right of=2, yshift=.7cm] (3) {};
\node [p1, F, right of=2, yshift=-.7cm, label={below:$u$}] (4) {};
\node [p0, F, below of=2,label={above:$w$}] (5) {};
\node [p1, left of=5] (6) {};
\draw [arr]
(0) edge node {$a$} (1) (0) edge node {$c$} (2) (1) edge node {$b$} (2)
(2) edge [bend left=20pt] node {$d$} (0)
(2) edge node {$e$} (3)
(3) edge [loop right] node {$i$} (3)
(2) edge node [below left] {$f$} (4)
(3) edge node {$h$} (4)
(4) edge [loop right] node {$g$} (4)
(4) edge node {$k$} (5)
(5) edge [loop below] node {\strut$m$} (5)
(5) edge node {$n$} (6)
(6) edge [loop below] node {\strut$p$} (6)
(6) edge [bend left] node {$q$} (0)
;
\end{tikzpicture}
\caption{Rectangular nodes belong to Pl.~1, round nodes to Pl.~0, dashed nodes are in $F$.}
\label{fig:RunningGame}
\end{subfigure}
\hfill
\begin{subfigure}[t]{.45\linewidth}
\centering
\begin{tikzpicture}[node distance=1.2cm,every node/.style={scale=0.7},framed,baseline]
\node [p1,label={left:$v$}] (0) {};
\node [p1,right of=0,yshift=.8cm] (1) {};
\node [p0,F,right of=1,label={below:$v'$}] (2) {};
\node [p0,right of=2] (3) {};
\node [p1,F,right of=3] (4) {};
\node [marker,right of=4] (4a) {};
\node [marker,below of=4,xshift=.5cm] (4b) {};
\node [p0,F,right of=0,yshift=-.8cm,label={below:$v'$}] (a) {};
\node [p1,F,right of=a] (b) {};
\node [p1,F,below of=b,xshift=.5cm] (c) {};
\node [p1,F,below of=c,xshift=.5cm] (d) {};
\node [marker,right of=d] (d1) {};
\node [marker,below of=d,xshift=.5cm] (d2) {};
\node [p0,F,right of=b] (b1) {};
\node [p0,F,right of=b1] (b2) {};
\node [p0,F,right of=b2] (b3) {};
\node [marker,right of=b3] (b4) {};
\node [p0,F,right of=c] (c1) {};
\node [p0,F,right of=c1] (c2) {};
\node [p0,F,right of=c2] (c3) {};
\node [marker,right of=c3] (c4) {};
\enode 0 1 a
\enode 1 2 b
\enode 2 3 e
\enode 3 4 h
\enode[swap] 0 a c
\enode a b f
\enode[left] b c g
\enode[left] c d g
\enode b {b1} k
\enode {b1} {b2} m
\enode {b2} {b3} m
\enode c {c1} k
\enode {c1} {c2} m
\enode {c2} {c3} m
\draw [densely dotted,shorten <=3pt,shorten >=10pt] (4) edge (4a) (4) edge (4b);
\begin{scope}
\draw [densely dotted,shorten <=3pt,shorten >=10pt] (b3) edge (b4);
\clip (c3.north west) rectangle ($(c3.south east)+(.4cm,-.4cm)$);
\draw [densely dotted,shorten <=3pt,shorten >=10pt] (c3) edge (c4);
\end{scope}
\begin{scope}
\clip (d.north west) rectangle ($(d.south east)+(.4cm,-.4cm)$);
\draw [densely dotted,shorten <=3pt,shorten >=10pt] (d) edge (d1) (d) edge (d2);
\end{scope}
\end{tikzpicture}
\caption{Depiction of an infinite strategy tree of a winning strategy for Pl.~0 from position $v$.}
\label{fig:RunningStrategy}
\end{subfigure}
\caption{Running example of a Büchi game and a winning strategy.}
\label{fig:Running}
\end{figure}
\begin{example}
An example of a Büchi game is depicted in \cref{fig:Running}.
Player~0 has essentially three different positional winning strategies from $v$,
by either choosing edge $d$, or edges $e,h,m$ or $f,m$.
Notice that for the first strategy, we did not specify moves for all positions in $V_0$ as these positions cannot be reached when edge $d$ is played; this is the main reason why we represent strategies as trees.
\Cref{fig:RunningStrategy} depicts such a tree representation of a strategy.
This strategy is a typical example of a winning strategy that is not positional, but still minimal if we take edge multiplicities into account.
\end{example}
\section{Strategies with Minimal Effort}
\begin{aquote}{Antoine de Saint-Exupéry}
La perfection est atteinte, non pas lorsqu’il n’y a plus rien à ajouter, mais lorsqu’il n’y a plus rien
à retirer.\footnote{Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take away.}
\end{aquote}
\noindent
As a measure for the complexity or effort of a strategy,
we consider the set of edges a strategy $\Ss$ uses and how often each of these edges appears in the strategy tree.
Under this measure, the simplest strategies are the ones that do not play redundant edges
-- hence no moves are left to take away.
\begin{definition}
Given an edge $e = vw \in E$ in a Büchi game $\Gg$ and a strategy $\Ss$ in $\Gg$,
we denote by $\ecount \Ss e = | \{ \rho v \in \Ss \mid \rho v \to \rho v w \text{ is an edge in } \Ss \}| \in \N \cup \{ \infty \}$ the number of times (possibly infinite) the edge $e$ occurs in $\Ss$.
With each strategy $\Ss$ we associate its \emph{edge profile}, the vector $\ep \Ss = (\ecount \Ss e)_{e \in E}$.
\end{definition}
\begin{example}\label{ex:redundantMove}
Consider the following Büchi game:
\begin{center}
\begin{tikzpicture}[baseline]
\node [p0,label={below:$v$},anchor=base,yshift=.1cm] (0) {};
\node [p0,F,label={below:$w$},right of=0] (1) {};
\draw [arr]
(0) edge [loop left] node {$a$} (0)
(1) edge [loop right] node {$c$} (1)
(0) edge node {$b$} (1);
\end{tikzpicture}
\end{center}
Player~0 wins by first looping $n$ times at position $v$ (for any fixed $n \in \N$) and then moving to $w$, corresponding to the edge profile $(n,1,\infty)$.
Clearly, looping at $v$ is a redundant move, so we regard the strategy with $n=0$ as the simplest one (that wins with the least effort).
\end{example}
To formalize the intuition of redundant moves, we define an order $\succeq$ on strategies called \textit{absorption}.
This is defined in such a way that the $\succeq$-maximal strategies are the simplest ones
that avoid redundant moves whenever possible.
\begin{definition}
Let $\Ss_1,\Ss_2$ be two strategies in a Büchi game $\Gg = (V,V_0,V_1,E,F)$.
We say that $\Ss_1$ \emph{absorbs} $\Ss_2$, denoted $\Ss_1 \succeq \Ss_2$,
if $\ecount {\Ss_1} e \le \ecount {\Ss_2} e$ for all edges $e \in E$.
If additionally $\ecount {\Ss_1} e < \ecount {\Ss_2} e$ for some $e \in E$,
we say that $\Ss_1$ \emph{strictly absorbs} $\Ss_2$, denoted $\Ss_1 \succ \Ss_2$.
They are \emph{absorption-equivalent}, denoted $\Ss_1 \equiv \Ss_2$, if both $\Ss_1 \succeq \Ss_2$ and $\Ss_2 \succeq \Ss_1$.
A strategy $\Ss \in \Strat(v)$ is \emph{absorption-dominant from position $v$}, if there is no strategy $\Ss' \in \Strat(v)$ with $\Ss' \succ \Ss$.
It is further \emph{strictly} absorption-dominant, if there is no other strategy $\Ss' \in \Strat(v)$ with $\Ss' \succeq \Ss$, so no other strategy is absorption-equivalent to $\Ss$.
\end{definition}
Notice that absorption is simply the inverse pointwise order on the edge profiles.
In particular, $\Ss_1 \equiv \Ss_2$ if, and only if, $\ep {\Ss_1} = \ep {\Ss_2}$.
We next aim at understanding the relation between
(strictly) absorption-dominant strategies and the standard notion of positional strategies.
As a starter, we show that absorption-dominant strategies are not necessarily positional
(cf.\ \cite{GraedelTan20} for a similar example).
\begin{example}\label{ex:Weakpos}
Consider the strategy $\Ss$ as depicted in \cref{fig:RunningStrategy}.
It is not positional, as the choice for position $v'$ is not unique (both $e$ and $f$ occur in $\Ss$).
It is, however, absorption-dominant.
As there are two paths to $v'$, every strategy must either use $e$ or $f$ twice, or use both edges.
If $e$ (or $f$) is used twice, then the strategy cannot absorb $\Ss$, and one can verify that $\Ss$ absorbs all strategies using both $e$ and $f$.
It is not strictly absorption-dominant, as we obtain an absorption-equivalent strategy by switching the two branches in the depiction of $\Ss$, so that $e$ is used after $c$, and $f$ after $b$.
\end{example}
Strategies such as the one in \cref{fig:RunningStrategy} are not positional, but satisfy the weaker property that within each \emph{play}, the strategy makes a unique decision for each position $v \in V_0$.
This notion of strategies has been introduced as \emph{persistent} strategies in \cite{MarcinkowskiTruderung02} in the context of LTL on game graphs and has been further studied in \cite{Duparc03}.
Persistent strategies have also been called \emph{weakly positional} in \cite{GraedelTan20}.
We say that a strategy \emph{plays positionally} from a position $v \in V_0$ if the strategy makes a unique choice at position $v$ (not depending on the history of the play).
A strategy that plays positionally from all positions in $V_0$ is positional.
With this notation, we now clarify the relation between the different notions of strategies; a summary is shown in \cref{figStrategyClasses}.
We first observe that if a strategy $\Ss$ does not play positionally from $v$, we can always obtain a strategy $\Ss'$ with $\Ss' \succeq \Ss$ by swapping the choices at $v$, which leads to:
\begin{figure}[t]
\centering
\begin{tikzpicture}[font=\small, every label/.style={font=\scriptsize}, xscale=0.8,yscale=0.9]
\draw [pattern=crosshatch dots, pattern color=lightgray!30] (0,.4) ellipse (4cm and 1.15cm);
\node[anchor=north] at (0,1.4) {absorption-dominant};
\draw [pattern=north east lines, pattern color=lightgray!60] (0,0) ellipse (2.6cm and .7cm) node [align=center,yshift=0pt] {positional \\[-.3em] = \\[-.2em] strictly abs.-dom.} ;
\draw (0,.8) ellipse (6cm and 1.6cm);
\node[anchor=north] at (0,2.3) {persistent};
\draw (-8,-.85) rectangle (8,2.7);
\node [anchor=north west] at (-8,2.7) {Winning strategies};
\node [dot,label={right:Ex.~\ref{ex:Weakpos}}] at (2.2,0.7) {};
\node [dot,label={right:Ex.~\ref{ex:WeakposNotDominant}}] at (4.2,1.0) {};
\end{tikzpicture}
\vspace{-.3em}
\caption{Venn diagram depicting various classes of winning strategies.}
\label{figStrategyClasses}
\end{figure}
\begin{proposition}
Strictly absorption-dominant strategies coincide with positional strategies.
\end{proposition}
\begin{proof}
Let $\Ss \in \Strat_\Gg(v)$ be a strategy from $v$.
First assume towards a contradiction that $\Ss$ is positional but not strictly absorption-dominant.
That is, there is a different strategy $\Ss' \in \Strat(v)$ with $\Ss' \succeq \Ss$.
Since $\Ss'$ is different from $\Ss$, there is a position $w$ and a path $\rho w$ occurring in both strategies
for which the strategies differ, i.e., we have $w_1 = \Ss(\rho w)$ and $w_2 = \Ss'(\rho w)$ with $w_1 \neq w_2$.
Since $\Ss$ is positional, the edge $ww_2$ does not occur in $\Ss$.
Hence it occurs strictly more often in $\Ss'$, contradicting the assumption $\Ss' \succeq \Ss$.
We prove the other direction by contraposition.
Let $\Ss$ be non-positional, so there is a position $w$ and two paths $\rho w$ and $\rho' w$ such that $\Ss(\rho w) \neq \Ss(\rho' w)$.
Let $\Ss_{\rho w}$ and $\Ss_{\rho' w}$ be the substrategies of $\Ss$ from $\rho w$ and $\rho' w$, respectively.
First assume that $\rho \sqsubseteq \rho'$.
We then consider the strategy $\Ss'$ that behaves like $\Ss$, but switches to $\Ss_{\rho' w}$ at $\rho w$.
As every edge occurring in $\Ss'$ also occurs in $\Ss$, we have $\Ss' \succeq \Ss$ and $\Ss$ is not strictly absorption-dominant.
The case $\rho' \sqsubseteq \rho$ is symmetric.
If, on the other hand, $\rho$ and $\rho'$ are incomparable nodes in $\Ss$,
we consider the strategy $\Ss'$ that behaves like $\Ss$, but plays $\Ss_{\rho' w}$ from $\rho w$ and $\Ss_{\rho w}$ from $\rho' w$, swapping the two substrategies.
Then $\Ss' \equiv \Ss$, so $\Ss$ is not strictly absorption-dominant.
\end{proof}
We next establish the relation to persistent strategies.
To this end, we first show under which circumstances absorption-dominant strategies must make unique choices.
Our proof needs the following combinatorial observation.
\begin{lemma}\label{stratDominantFiniteEquiv}
Let $v \in \Gg$ be a position.
There are only finitely many absorption-dominant strategies from $v$ up to absorption-equivalence.
\end{lemma}
\begin{proof}
Consider the pointwise order on edge profiles induced by the standard order on $\N \cup \{\infty\}$.
By definition, a strategy $\Ss$ is absorption-dominant from $v$ if, and only if, its edge profile $\ep \Ss$ is minimal among all strategies from $v$ (and absorption-equivalent strategies have the same edge profile).
By a simple combinatorial fact known as Dickson's lemma, every set of edge profiles contains only finitely many minimal elements.
\end{proof}
\begin{proposition}\label{stratInfinitePositional}
Let $\Ss \in \WinStrat_\Gg(v)$ be absorption-dominant from $v$, and let $w \in V_0$ be a position.
If $w$ occurs infinitely often in $\Ss$, then $\Ss$ plays positionally from $w$.
\end{proposition}
\begin{proof}
Consider the infinitely many substrategies at occurrences of $w$ in $\Ss$.
By \cref{stratDominantFiniteEquiv}, there is one such substrategy $\Ss_w$ such that infinitely many of the substrategies are absorption-equivalent to $\Ss_w$.
This means that every edge occurring in $\Ss_w$ also occurs in infinitely many substrategies and hence infinitely often in the full strategy $\Ss$.
Notice that $\Ss_w$ is winning from $w$, as it is a substrategy of the winning strategy $\Ss$.
Consider the subgame of $\Gg$ containing only edges occurring in $\Ss_w$.
Clearly, Player~0 wins from $w$ (using $\Ss_w$) and by positional determinacy, there is thus a positional winning strategy $\Ss_{\text{pos}}$ from $w$ using only edges that occur in $\Ss_w$ and hence infinitely often in $\Ss$.
Now consider the strategy $\Ss' \in \WinStrat_\Gg(v)$ that behaves like $\Ss$, but always uses $\Ss_{\text{pos}}$ from $w$. Then $\Ss' \succeq \Ss$ by construction of $\Ss_{\text{pos}}$.
Further, $\Ss_{\text{pos}}$ is positional and makes a unique choice $\Ss_{\text{pos}}(w)$.
If $\Ss$ would not play positionally from $w$, then there would be some path $\rho w$ such that $\Ss(\rho w) = w' \neq \Ss_{\text{pos}}(w)$.
But then the edge $w w'$ never occurs in $\Ss'$, so $\Ss' \succ \Ss$ and $\Ss$ would not be absorption-dominant.
\end{proof}
With this important insight, we can deduce that the absorption-dominant winning strategies (from some position $v$) are a (strict) subset of the persistent strategies:
An absorption-dominant strategy must play positionally from positions that occur infinitely often; repetitions of positions that occur finitely often are always redundant.
\begin{corollary}\label{stratDominantWeakpos}
Every absorption-dominant winning strategy in $\Gg$ is persistent.
\end{corollary}
\begin{proof}
Let $\Ss \in \WinStrat_\Gg(v)$ be absorption-dominant from $v$.
Assume towards a contradiction that $\Ss$ is not persistent, so there is a position $w$ and a play of the form $\rho_1 w \rho_2 w \rho_3$ such that $\Ss$ makes different decisions at $w$, say $\Ss(\rho_1 w) = w_1$ and $\Ss(\rho_1 w \rho_2 w) = w_2$ with $w_1 \neq w_2$.
By \cref{stratInfinitePositional}, $w$ can only occur finitely often in $\Ss$.
Hence the edge $w w_1$ also occurs finitely often, say $n$ times.
Let $\Ss'_w$ be the substrategy of $\Ss$ from $\rho_1 w \rho_2 w$.
Now consider the strategy $\Ss' \in \WinStrat_\Gg(v)$ that behaves like $\Ss$, but switches to $\Ss'_w$ at $\rho_1 w$.
By construction, $\Ss'$ uses each edge at most as often as $\Ss$.
Moreover, one occurrence of the edge $w w_1$ is removed, so this edge occurs at most $n-1$ times in $\Ss'$.
Hence $\Ss' \succ \Ss$, contradicting the absorption-dominance of $\Ss$.
\end{proof}
\begin{example}\label{ex:WeakposNotDominant}
For strictness, consider the following game (a modified part of \cref{fig:RunningGame}):
\begin{center}
\begin{tikzpicture}[scale=0.9,yscale=0.7,xscale=0.6,baseline={(0,0)}]
\node [p1] at (0,0) (1) {$v$};
\node [p1] at (1.5,1) (2a) {};
\node [p0,F] at (3,0) (3) {};
\node [p0,F] at (4.5,1) (4a) {};
\path [arr] (1) edge node {} (2a) (1) edge node [below left] {} (3)
(2a) edge node {} (3)
(3) edge node {$a$} (4a)
(3) edge [in=-60,out=-20,min distance=1cm] node {$b$} (3)
(4a) edge [loop right,min distance=.9cm] node {} (4a)
;
\end{tikzpicture}
\hspace{.5cm}
\parbox[c]{3.5cm}{
$\ecount {\Ss_1} {a,b} = (2,0)$, \\
$\ecount {\Ss_2} {a,b} = (0,\infty)$, \\
$\ecount {\Ss_3} {a,b} = (1,\infty)$.
}
\end{center}
\noindent
Due to the self-loop $b$, only the positional strategies $\Ss_1$ (always take $a$) and $\Ss_2$ (always take $b$) are absorption-dominant from $v$.
The strategy $\Ss_3$ that, depending on Player~1's choice, either takes edge $a$ or loops indefinitely using edge $b$ is persistent, but not absorption-dominant:
it is strictly absorbed by $\Ss_2$.
\end{example}
As a consequence of \cref{stratDominantWeakpos}, all moves after the first repeated position are determined by persistence.
We can thus represent absorption-dominant strategies in a compact way and strengthen \cref{stratDominantFiniteEquiv} as follows.
\begin{corollary}
Let $\Gg$ be a game with $n = |V|$ positions.
Every winning strategy $\Ss \in \WinStrat_\Gg(v)$ that is absorption-dominant from $v$
can be uniquely represented by a subtree of the tree unraveling of height at most $n$.
In particular, the number of absorption-dominant winning strategies is finite.
\end{corollary}
\section{A Whirlwind Tour of Semiring Semantics}
This section gives an overview on semiring semantics for fixed-point logics,
with a focus on the semirings relevant for the case study.
For a complete account, we refer to \cite{DannertGraNaaTan21}.
\subsection{Semirings}\label{sect:semirings}
Semirings are algebraic structures with two binary operations, usually denoted $+$ and $\bcdot$,
which we use to interpret the logical connectives $\lor$ and $\land$.
While semirings are very general structures, we make additional assumptions to ensure well-defined and meaningful semiring semantics for logics with fixed-point operators.
\begin{definition}
A \emph{commutative semiring} is an algebraic structure
$(\K,+,\bcdot,0,1)$, with $0\neq1$, such that $(\K,+,0)$
and $(\K,\bcdot,1)$ are commutative monoids, $\bcdot$
distributes over $+$, and $0\bcdot a=a\bcdot 0=0$.
It is \emph{idempotent} if $a+a = a$ for all $a \in \K$.
\end{definition}
All semirings we consider are commutative, so we omit \emph{commutative} in the following.
Towards fixed-point logic, we compute least and greatest fixed points with respect to the natural order $\le_\K$ (see below) and to ensure that they exist, we require $\le_\K$ to be a complete lattice (in fact, suprema and infima of chains would suffice, but in idempotent semirings this is equivalent).
We additionally impose a natural \emph{continuity} requirement which is crucial to our proofs, but does not seem to be a strong restriction in practice (we are not aware of any natural complete-lattice semirings that are not continuous).
Regarding notation, a \emph{chain} is a totally ordered set $C \subseteq \K$ and we write
$a \circ C = \{ a \circ c \mid c \in C\}$ for $a \in \K$.
\begin{definition}
In an idempotent semiring $(\K,+,\bcdot,0,1)$, the \emph{natural order} $\le_\K$ is the partial order defined by $a \le_\K b \Leftrightarrow a+b=b$.
We say that $\K$ is \emph{fully continuous} if $\le_\K$ is a complete lattice (with supremum $\bigsqcup$ and infimum $\bigsqcap$) and for all non-empty chains $C \subseteq \K$, elements $a \in \K$ and $\circ \in \{+,\bcdot\}$,
\[
\bigsqcup (a \circ C) = a \circ \bigsqcup C,
\quad \text{and} \quad
\bigsqcap (a \circ C) = a \circ \bigsqcap C.
\]
A semiring homomorphism $h \colon \K_1 \to \K_2$ on fully-continuous semirings is \emph{fully continuous} if $h(\bigsqcup C) = \bigsqcup h(C)$ and $h(\bigsqcap C) = \bigsqcap h(C)$ for all non-empty chains $C \subseteq \K_1$.
\end{definition}
By the Knaster-Tarski theorem, every $\le_\K$-monotone function $f \colon \K \to \K$ on a fully-continuous semiring has a least fixed point $\lfp(f)$ and a greatest fixed point $\gfp(f)$ in $\K$,
and this suffices to guarantee well-defined semiring semantics of fixed-point logics.
However, from a provenance perspective we further want this semantics to be meaningful in the sense that the value of a formula provides insights into why the formula holds.
It turns out that this is the case if we additionally require the semiring to be absorptive (see \cite{DannertGraNaaTan21}).
\begin{definition}
A semiring $\K$ is \emph{absorptive} if $a+ab = a$ for all $a,b \in \K$.
\end{definition}
We remark that absorption is equivalent to $\K$ being \emph{0-closed} or \emph{bounded} \cite{Mohri02},
that is, $1+a = 1$.
If $\K$ is idempotent, then absorption is further equivalent to multiplication being decreasing,
that is, $a \bcdot b \le_\K a,b$.
Clearly, every absorptive semiring is idempotent and thus partially ordered by $\le_\K$, with $1$ as top element.
If we additionally assume full continuity, we can extend any absorptive semiring by an infinitary power operation
$a^\infty = \bigsqcap_{n \in \N} a^n$ with natural properties such as $a \bcdot a^\infty = a^\infty$, $(ab)^\infty = a^\infty b^\infty$ and $(a+b)^\infty = a^\infty + b^\infty$.
\begin{example}
Here is a short, non-exhaustive list of semirings used in provenance analysis of databases and logics \cite{GreenKarTan07,GreenTan17,GraedelTan20}.
\begin{itemize}
\item The \emph{Boolean semiring} $\mathbb{B}=(\{\mathbf{0},\mathbf{1}\},\vee,\wedge,\mathbf{0},\mathbf{1})$ is the standard habitat of
logical truth. It is absorptive and (trivially) fully continuous.
\item $\mathbb{N}=(\mathbb{N},+,\cdot,0,1)$ is used for counting evaluation strategies for a logical statement.
It is not absorptive and hence not well suited for fixed-point logics.
\item The \emph{Viterbi} semiring $\mathbb{V}=([0,1],\max,\cdot,0,1)$
is used to compute \emph{confidence scores}
for logical statements. It is isomorphic to the \emph{tropical} semiring $\mathbb{T}=(\mathbb{R}_{+}^{\infty},\min,+,\infty,0)$
which is used for measuring the cost of
evaluation strategies.
Both are absorptive and fully continuous.
\item The \emph{min-max} semiring $(A, \max, \min, a, b)$ on a totally ordered set $(A,\leq)$
with least and greatest elements $a$ and $b$ can be used to model access privileges. It is absorptive and fully continuous. \hfill$\lrcorner$\gdef\ExampleEndMarker{}
\end{itemize}
\end{example}
From now on, all semirings we consider are commutative, absorptive and fully continuous.
Besides the application semirings listed above, we are particularly interested in universal semirings of polynomials to represent abstract information.
We can then use fully-continuous homomorphisms to specialize the computed information to application semirings as needed, as these homomorphisms preserve fixed points.
The common examples of semirings of polynomials $\N[X]$ and formal power series $\N^\infty \ps X$,
as used for provenance analysis of FO and Datalog in \cite{GreenKarTan07, GraedelTan17}, are not absorptive and hence not well-suited for fixed-point logic.
Instead, we rely on semirings of (generalized\footnote{The definition we use here generalizes the one in \cite{DeutchMilRoyTan14} by allowing $\infty$ as exponent.}) \emph{absorptive polynomials} (cf.\ \cite{DannertGraNaaTan21}).
Essentially, an absorptive polynomial such as $ab^3 + c^\infty$ is a sum of monomials over a finite set of variables $X$,
but without coefficients and with exponents from $\N \cup \{\infty\}$.
Monomial multiplication is defined as usual by adding exponents (with $n+\infty = \infty$).
The key ingredient is absorption among monomials:
we say that a monomial $m_1$ \emph{absorbs} $m_2$, if all its exponents are smaller (or equal).
Formally, $m_1 \succeq m_2$ if $m_1(x) \le m_2(x)$ for all $x \in X$, where $m_1(x)$ denotes the exponent of $x$ in $m_1$.
For example, $ab^2 \succeq a^\infty b^2$ and $a \succeq ab$, but $a^2b$ and $ab^2$ are incomparable.
In an absorptive polynomial, we omit all monomials that would be absorbed, so absorptive polynomials
are $\succeq$-\emph{antichains} of monomials. Consequently, addition and multiplication
are defined as usual, but afterwards we only keep the $\succeq$-maximal monomials.
For example, $(ab^2 + a^2b) \bcdot a^\infty = a^\infty b^2 + a^\infty b = a^\infty b$.
We write ${\mathbb S}^{\infty}[X]$ for the semiring of absorptive polynomials over the finite variable set $X$.
The $0$ and $1$-elements are the empty polynomial and the single monomial $1$ (with all zero exponents).
This defines an absorptive, fully-continuous semiring \cite{DannertGraNaaTan21}.
In fact, ${\mathbb S}^{\infty}[X]$ is the most general such semiring:
\begin{theorem}[Universal property, \cite{DannertGraNaaTan21}]
\label{universality}
Every mapping $h \colon X \to \K$ into an absorptive, fully-continuous semiring $\K$
uniquely extends to a fully-continuous semiring homomorphism $h \colon {\mathbb S}^{\infty}[X] \to \K$
(by means of polynomial evaluation).
\end{theorem}
\subsection{Logic}
We consider here the fixed-point logic LFP that extends first-order logic FO by least and greatest fixed-point formulae of the form $\psi(\tup y) = [\lfp R \tup x.\ \phi(R,\tup x)](\tup y)$ and $\psi(\tup y) = [\gfp R \tup x.\ \phi(R,\tup x)](\tup y)$. Here, $R$ is a relation symbol occurring only positively in $\phi$ and $\tup x,\tup y$ are variable tuples of matching arity.
Given a (Boolean) model $\AA$ and a tuple $\tup a$ of elements of $\AA$,
the formula $\psi(\tup a)$ holds in $\AA$, denoted $\AA \models \psi(\tup a)$,
if $\tup a$ is contained in the least (or greatest) fixed point of the operator
$F_\phi \colon R \mapsto \{ \tup a \mid \AA \models \phi(R,\tup a) \}$
that maps a relation $R$ to the relation consisting of those tuples for which $\phi$ holds.
For more background and a precise definition, we refer to \cite{Graedel+07}.
In order to generalize Boolean semantics to semiring semantics,
we first adapt the notion of a model $\AA$.
Instead of determining for each literal whether it is true or false in $\AA$,
we assign to each literal a semiring value, interpreting $0$ as \emph{false} and all other values as \emph{nuances of true}.
Special care is required to ensure that the assignment is consistent with respect to opposing literals (this is not always necessary, but often desirable).
In the following, let $\K$ be a semiring, $A$ a finite universe and $\tau$ a relational signature
(we drop $A$ and $\tau$ if clear from the context).
We denote the set of (instantiated) literals as
\begin{align*}
\Lit_{A,\tau} ={} &\{ R \tup a, \neg R \tup a \mid R \in \tau \text{ of arity $k$}, \tup a \in A^k \}
\,\cup\, \{\tup a = \tup b, \tup a \neq \tup b \mid \tup a,\tup b \in A^k\}.
\end{align*}
Given a literal $\lit$, we write $\neg \lit$ for the opposing literal (identifying $\neg \neg \lit$ and $\lit$).
The role of the Boolean model $\AA$ is then replaced by a semiring interpretation $\pi$ that assigns semiring values to all literals.
\begin{definition}\label{defKInterpretation}
Let $\K$ be a semiring. A \emph{$\K$-interpretation} (over finite $A$ and $\tau$) is an assignment $\pi\colon \Lit_{A,\tau} \to \K$ that maps true (in)equalities to $1$ and false (in)equalities to $0$.
We say that $\K$ is \emph{model-defining}, if for each literal $\lit$, exactly one of $\pi(\lit)$ and $\pi(\neg \lit)$ is $0$.
\end{definition}
For Büchi games, we will always use the signature $\tau = \{ E, F, V_0, V_1 \}$, where $E$ is a binary and $F,V_0,V_1$ are unary relation symbols.
We can then view a game $\Gg = (V, V_0, V_1, E, F)$ as a $\tau$-structure.
Notice that we do not distinguish between the edge relation $E$ of $\Gg$ and the relation symbol $E$; it will always be clear from the context what we refer to.
The set of instantiated literals $\Lit_{V,\tau}$ then contains, e.g., $Ev_1 v_2$ and $\neg F v_1$, where $v_1,v_2 \in \Gg$.
We lift $K$-interpretations $\pi$ from literals to LFP-formulae in negation normal form ($\nnf$), resulting in a semiring value $\pi \ext \psi$, by interpreting the logical connectives as semiring operations.
For fixed-point formulae, we consider the induced operator $F_\phi$ analogous to the Boolean case, but acting on functions $g \colon A^k \to \K$ instead of relations $R \subseteq A^k$ (which can be seen as functions $R \colon A^k \to \mathbb{B}$, justifying our generalisation).
We extend the natural order to such functions by pointwise comparison.
More formally, given a $\K$-interpretation $\pi$ over signature $\tau$, we denote by $\pi[R\mapsto g]$ the $\K$-interpretation over $\tau \cup \{R\}$ obtained from $\pi$ by adding values $g(\tup a)$ for the instantiated atoms $R\tup a$.
The analogue of the Boolean operator $F_\phi$ is then the operator $F_\pi^\phi$ that maps a function $g \colon A^k \to \K$ to the function
\[ F_\pi^\phi(g) \colon \; \tup a\mapsto \pi[R\mapsto g]\ext{\phi(R,\tup a)}. \]
With this in mind, we define the following natural generalization of Boolean semantics.
\begin{definition}
A $\K$-interpretation $\pi\colon \Lit_A(\tau)\rightarrow \K$
(for finite $A$ and $\tau$)
in a fully-continuous semiring $\K$
extends to a $\K$-\emph{valuation}
$\pi\colon {\rm LFP}(\tau)\rightarrow \K$ by mapping an ${\rm LFP}$-sentence $\psi(\tup{a})$ in negation normal form to a value $\pi\ext\psi$ using the rules
\begin{align*}
\pi\ext{\psi\vee\phi}&\coloneqq\pi\ext\psi + \pi\ext\phi, &
\pi\ext{\exists x\psi(x)}&\coloneqq\textstyle\sum_{a\in A}\pi\ext{\phi(a)}, &
\pi\ext{\neg\psi}&\coloneqq\pi\ext{\nnf(\neg\psi)}, \\
%
\pi\ext{\psi\wedge\phi}&\coloneqq\pi\ext\psi \bcdot \pi\ext\phi, &
\pi\ext{\forall x\psi(x)}&\coloneqq\textstyle\prod_{a\in A}\pi\ext{\phi(a)},
\end{align*}
and, for fixed-point formulae,
\begin{align*}
\pi \ext{[\lfp R\tup x. \phi(R,\tup x)](\tup a)} &\coloneqq \lfp(F_\pi^\phi)(\tup a), &
\pi \ext{[\gfp R\tup x. \phi(R,\tup x)](\tup a)} &\coloneqq \gfp(F_\pi^\phi)(\tup a).
\end{align*}
\end{definition}
An important property of the resulting semantics is that it is preserved by fully-continuous semiring homomorphisms,
in particular by polynomial evaluation in ${\mathbb S}^{\infty}[X]$ due to \cref{universality} (but not by polynomial evaluation of $\mathbb{N}[X]$ or formal power series!).
\begin{theorem}[Fundamental property, \cite{DannertGraNaaTan21}]
\label{fundamental}
Let $h \colon \K_1 \to \K_2$ be a fully-continuous semiring homomorphism.
Then for every $\K_1$-interpretation $\pi$, the mapping $h \circ \pi$ is a $\K_2$-interpretation and $h(\pi \ext \phi) = (h \circ \pi) \ext \phi$, for every $\phi\in{\rm LFP}$.
\end{theorem}
\begin{center}
\begin{tikzpicture}[baseline,node distance=1.7cm,font=\small]
\node [baseline, anchor=base] (lit) {$\Lit_A(\tau)$};
\node [below left of=lit] (litS) {$K_1$};
\node [below right of=lit] (litT) {$K_2$};
\node [right=5cm of lit.base, anchor=base] (fol) {\upshape LFP};
\node [below left of=fol] (folS) {$K_1$};
\node [below right of=fol] (folT) {$K_2$};
\node [align=center] at ({$(lit.center)!0.5!(fol.center)$} |- {$(lit.center)!0.5!(litS.center)$}) {$\implies$};
\path[draw,->,shorten <=1pt, shorten >=1pt, font=\scriptsize]
(lit) edge node [above left] {$\pi$} (litS)
(lit) edge node [above right] {$h \circ \pi$} (litT)
(litS) edge node [above] {$h$} (litT)
(fol) edge node [above left] {$\pi$} (folS)
(fol) edge node [above right] {$h \circ \pi$} (folT)
(folS) edge node [above] {$h$} (folT)
;
\end{tikzpicture}
\end{center}
\section{Computing Strategies with Semiring Semantics}
\label{sec:TheFormula}
This section connects the previous sections on semiring semantics and absorption-dominant strategies.
We focus on the formula for the winning region in a Büchi game and show that its value under semiring semantics can be understood in terms of (absorption-dominant) winning strategies.
\subsection{The Semiring Interpretation}
We want to use semiring semantics to analyze moves in winning strategies.
For this reason, we label edges with indeterminates $X$ (cf.\ \cref{fig:RunningGame}) and use an ${\mathbb S}^{\infty}[X]$-interpretation $\pi_{\text{\sffamily\upshape strat}}$ to track moves (i.e., edge literals $Evw$ for positions $v,w$) via their indeterminates.
We assume the game graph to be fixed and do not wish to track information about the target set $F$ or the active player at a certain node, hence we simply map all other literals over $\tau = \{E,F,V_0,V_1\}$, such as $F v$, $V_0 v$ and $\neg Evw$, to $0$ or $1$, depending on whether they are true or false in the fixed game.
The resulting interpretation is almost Boolean and hence behaves very similar to the original game, except that we remember which edges are used in the evaluation of a formula.
\begin{definition}
Let $\Gg = (V, V_0, V_1, E, F)$ be a Büchi game and let $X = \{ X_{vw} \mid vw \in E \}$ be a set of indeterminates for all edges.
We define the ${\mathbb S}^{\infty}[X]$-interpretation $\pi_{\text{\sffamily\upshape strat}} \colon \Lit_{V,\tau} \to {\mathbb S}^{\infty}[X]$ as follows (depending on $\Gg$):
\begin{align*}
\pi_{\text{\sffamily\upshape strat}}(Evw) &= X_{vw} \text{ for all edges } vw \in E, \quad \\
\pi_{\text{\sffamily\upshape strat}}(\lit) &= \begin{cases}
1, &\text{ if } \Gg \models \lit, \\
0, &\text{ if } \Gg \not\models \lit,
\end{cases} \; \text{ for all other literals $\lit \in \Lit_{V,\tau}$.}
\end{align*}
\end{definition}
For the applications in \cref{sec:Applications}, we may consider other interpretations which are defined in a similar way, but also track negative edge literals or the target set $F$.
An overview is given in \cref{fig:Interpretations}.
In this section, we always work with $\pi_{\text{\sffamily\upshape strat}}$.
\begin{figure*}
\centering
\begin{tabular}{r||c|c|c}
Interpretation & $\pi_{\text{\sffamily\upshape strat}}$ & $\pi_{\text{\sffamily\upshape repair}}$ & $\pi_{\text{\sffamily\upshape target}}$ \\
\& application & \textit{strategy tracking} & \textit{reverse analysis of moves} & \textit{target synthesis} \\ \hline
Semiring & ${\mathbb S}^{\infty}[X]$ & ${\mathbb S}^{\infty}[X,\nn{X}]$ or $\mathsf{PosBool}[X,\nn{X}]$ & $\mathsf{PosBool}[X,\nn{X}]$ \\
$\pi(Evw)$ & $X_{vw}/0$ & $X_{vw}$ (or $1/0)$ & $1/0$ \\
$\pi(\neg Evw)$ & $1/0$ & $\nn{X_{vw}}$ (or $1/0$) & $1/0$ \\
$\pi(F v)$ & $1/0$ & $1/0$ & $X_v$ \\
$\pi(\neg F v)$ & $1/0$ & $1/0$ & $\nn{X_v}$ \\
other literals & $1/0$ & $1/0$ & $1/0$ \\
\end{tabular}
\caption{Semiring interpretations used in this paper (the notation $a/0$ indicates the value $a$ if the literal is true,
and $0$ if it is false in $\Gg$).}
\label{fig:Interpretations}
\end{figure*}
\subsection{The Formula}
It is well known that the winning region (of Player~0) in a Büchi game is definable in fixed-point logic.
Intuitively, the winning region is the largest set $Y$ such that from each position in $Y$,
Player~0 can enforce a visit to $Y \cap F$ (after at least one move).
In LFP, we can express the winning region as follows (see, e.g., \cite{CanavoiGraLesPak15,Walukiewicz02}):
\[
\mathsf{win}_0(x) \coloneqq{} \big[\gfp Y y.\ [\lfp Z z.\ \phi(Y,Z,z)](y) \big](x)
\]
where
\begin{align*}
\phi(Y,Z,z) \coloneqq{} &\Big(Fz \;\land\; ((V_0 z \land \exists u (Ezu \land Yu))
\lor (V_1 z \land \forall u (Ezu \to Yu)))\Big) \\
{}\lor{} &\Big(\neg Fz \;\land\; ((V_0 z \land \exists u (Ezu \land Zu))
\lor (V_1 z \land \forall u (Ezu \to Zu)))\Big).
\end{align*}
Given a $\K$-interpretation $\pi$ for a Büchi game $\Gg = (V,V_0,V_1,E,F)$, semiring semantics of the above formula induces%
\footnote{Here we first translate $Ezu \to Yu$ to the formula $\neg Ezu \lor (Ezu \land Yu)$ in negation normal form.}
the following fixed-point computation.
To simplify the presentation, we introduce two families of variables,
$\mathbf Y = (Y_v)_{v \in V}$ and $\mathbf Z = (Z_v)_{v \in V}$ that take values in $\K$.
We can then express the resulting semiring valuation as $\pi \ext {\mathsf{win}_0(v)} = Y^*_v$ where
$\mathbf Y^* = (Y_v^*)_{v \in V}$ is the \emph{greatest} solution to the equation system
\[
\mathbf Y = \mathbf Z^*(\mathbf Y)
\]
where, in turn, $\mathbf Z^*(\mathbf Y)$ is the \emph{least} solution, given values $\mathbf Y = (Y_v)_{v \in V}$, to the equation system consisting of the following equation for all $v \in V$:
\begin{align*}
Z_v ={} &\pi(Fv) \bcdot \Big((\pi(V_0 v) \bcdot \sum_{w \in V} (\pi(Evw) \bcdot Y_w)) + (\pi(V_1 v) \bcdot \prod_{w \in V} (\pi(\neg Evw) + \pi(Evw) \bcdot Y_w))\Big) \\
{}+{} &\pi(\neg Fv) \bcdot \Big((\pi(V_0 v) \bcdot \sum_{w \in V} (\pi(Evw) \bcdot Z_w)) + (\pi(V_1 v) \bcdot \prod_{w \in V} (\pi(\neg Evw) + \pi(Evw) \bcdot Z_w))\Big).
\end{align*}
For most of this paper, we use $\pi_{\text{\sffamily\upshape strat}}$ to track only \emph{moves} of winning strategies.
As $\pi_{\text{\sffamily\upshape strat}}$ maps most of the literals to $0$ or $1$, we can simplify the equations depending on $v$:
\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{c|c|c}
& $v \in F$ & $v \notin F$ \\ \hline
$v \in V_0$ &
$\displaystyle Z_v = \sum_{w \in vE} \pi(Evw) \bcdot Y_w$ &
$\displaystyle Z_v = \sum_{w \in vE} \pi(Evw) \bcdot Z_w$ \\
$v \in V_1$ &
$\displaystyle Z_v = \prod_{w \in vE} \pi(Evw) \bcdot Y_w$ &
$\displaystyle Z_v = \prod_{w \in vE} \pi(Evw) \bcdot Z_w$
\end{tabular}
\end{center}
A good way to think about (and compute) the least and greatest solutions is the fixed-point iteration.
The idea is to start with each $Z_v$ set to the least element $0$ of the semiring, then apply the above equations (i.e., the induced operator $F^\phi_{\pi_{\text{\sffamily\upshape strat}}}$) to compute a next, larger semiring value and repeat this process until a fixed-point is reached (notice that the iteration can also be infinite, the fixed-point is then the supremum or infimum).
\begin{example}\label{exComputation}
Recall the simple game from \cref{ex:redundantMove}
$\big($\!\!
\begin{tikzpicture}[baseline,font=\scriptsize]
\node [p0,label={below:$v$},anchor=base,yshift=.1cm] (0) {};
\node [p0,F,label={below:$w$},right of=0] (1) {};
\draw [arr]
(0) edge [loop left] node {$a$} (0)
(1) edge [loop right] node {$c$} (1)
(0) edge node {$b$} (1);
\end{tikzpicture}
$\!\!\big)$.
\newcommand{\vv}[2]{\begin{pmatrix}#1 \\ #2\end{pmatrix}}
\newcommand{\mathbf Y}{\mathbf Y}
\newcommand{\mathbf Z}{\mathbf Z}
\newcommand{\xmapsto{\!\!\!F_{\pitrack}^\phi\!}}{\xmapsto{\!\!\!F_{\pi_{\text{\sffamily\upshape strat}}}^\phi\!}}
\noindent
Using the interpretation $\pi_{\text{\sffamily\upshape strat}}$ corresponding to the edge labels,
we obtain the following fixed-point iteration.
We write the tuples $\mathbf Y$ and $\mathbf Z$ as vectors $({Y_v} \; {Y_w})^T$ and $({Z_v} \; {Z_w})^T$.
\begin{center}
\begin{tikzpicture}[font=\small,node distance=1.5cm]
\node (Y) {$\mathbf Y:$};
\node (Z) [below=.7cm of Y] {$\mathbf Z:$};
\node [right=0cm of Y] (Y1) {$\vv 1 1$};
\node [xshift=.7cm] at (Z -| Y1) (Z11) {$\vv 0 0$};
\node [right of=Z11] (Z12) {$\vv 0 c$};
\node [right of=Z12] (Z13) {$\vv {bc} c$};
\node [xshift=.7cm] (Y2) at (Y1 -| Z13) {$\vv {bc} c$};
\node [xshift=.7cm] at (Z -| Y2) (Z21) {$\vv 0 0$};
\node [right of=Z21] (Z22) {$\vv 0 {c^2}$};
\node [right of=Z22] (Z23) {$\vv {bc^2} {c^2}$};
\node [xshift=.7cm] (Y3) at (Y1 -| Z23) {$\vv {bc^2} {c^2}$};
\node [xshift=.8cm] at (Z -| Y3) (Z31) {$\;\dots\vphantom{\vv x x}$};
\node [xshift=.9cm] (Y4) at (Y1 -| Z31) {$\vv {bc^n} {c^n}$};
\node [xshift=.8cm] at (Z -| Y4) (Z41) {$\;\dots\vphantom{\vv x x}$};
\path [draw,|->,every node/.style={anchor=base,yshift=4pt}]
(Z11) edge node {$F^\phi$} (Z12)
(Z12) edge node {$F^\phi$} (Z13)
(Z21) edge node {$F^\phi$} (Z22)
(Z22) edge node {$F^\phi$} (Z23)
;
\path [draw,->,short=-3pt]
(Y1) edge (Z11)
(Z13) edge (Y2)
(Y2) edge (Z21)
(Z23) edge (Y3)
(Y3) edge (Z31)
(Z31) edge (Y4)
(Y4) edge (Z41)
;
\end{tikzpicture}
\end{center}
\noindent
We obtain the overall result $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)} = Y_v^* = \bigsqcap_n bc^n = bc^\infty$ corresponding to the unique absorption-dominant strategy using edge $b$ once and $c$ infinitely often (cf.\ \cref{ex:redundantMove}).
\end{example}
\subsection{Connection to Strategies}
By mapping edges to semiring values, we can track edges through the fixed-point computation.
In \cref{exComputation}, the resulting semiring value revealed how often each edge is used in the unique absorption-dominant winning strategy.
We now generalize this observation.
For simplicity, we only consider $\K$-interpretations $\pi$ that are \emph{edge tracking} for a given game $\Gg$.
That is, they may assign arbitrary values to positive edge literals $Evw$, but all other literals are mapped to $0$ or $1$ in accordance with $\Gg$.
To make the connection to strategies explicit, we first define semiring values for strategies based on the appearance of edges.
\begin{definition}
Let $\Ss$ be a strategy in a Büchi game $\Gg = (V, V_0, V_1, E, F)$.
Let $\K$ be an absorptive, fully-continuous semiring and $\pi$ an edge-tracking $\K$-interpretation on $\Gg$.
The \emph{$\K$-value} of $\Ss$ is the product of the values for all edges appearing in $\Ss$.
Formally,
\[
\pi \ext \Ss \coloneqq \prod_{vw \in E} \pi(Evw)^{\ecount \Ss {vw}},
\]
where infinite exponents are interpreted by the infinitary power operation of the semiring.
\end{definition}
The semiring value of $\mathsf{win}_0$ can then be expressed as the sum over the values of all winning strategies.
A direct proof is not completely straightforward, as fixed-point iterations and strategy trees can both be infinite (even if $\Gg$ is finite).
Instead, we make use of a similar sum-of-strategies result for model-checking games for LFP (see \cite{DannertGraNaaTan21}).
\begin{theorem}[Sum of Strategies]
\label{thmSumOfStrategiesTracking}
Let $\Gg$ be a Büchi game and $v$ a position in $\Gg$.
Let $\K$ be an absorptive, fully-continuous semiring and $\pi$ an edge-tracking $\K$-interpretation.
Then,
\[
\pi \ext {\mathsf{win}_0(v)} = \sum \big\{ \pi \ext \Ss \;\big|\;
\text{$\Ss \in \WinStrat_\Gg(v)$ is absorption-dominant from $v$} \big\}.
\]
\end{theorem}
It is in fact this central result that motivated the notion of \emph{absorption-dominant} strategies.
However, as we have already discussed, these may also be interesting in their own right if one is interested in minimal winning strategies.
\begin{example}
For the edge-tracking interpretation $\pi_{\text{\sffamily\upshape strat}}$ induced by the edge labels in \cref{fig:RunningGame}, we obtain
\begin{align*}
\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)} ={}
&(abcd)^\infty +
abc \, e^2 h^2 (gkm)^\infty +
abc \, f^2 (gkm)^\infty +
abc \, ef h (gkm)^\infty.
\end{align*}
There are four monomials, corresponding to four equivalence classes of absorption-dominant strategies.
Each monomial reveals the edges that appear in the corresponding strategies, so we see that the first three monomials belong to positional (and hence uniquely defined) strategies.
The last monomial belongs to the non-positional strategy shown in \cref{fig:RunningStrategy} (and its switched version, see \cref{ex:Weakpos}).
The values of all other strategies are strictly absorbed by one of these monomials.
\end{example}
\subsection{Proof of the Sum-of-Strategies Theorem}
\begingroup
\newcommand{\MC}[2]{\mathsf{MC}(#1,#2)}
\newcommand{\MCP}[2]{\widetilde{\mathsf{MC}}(#1,#2)}
\tikzset{
arr/.style={draw,->,>=stealth',shorten <=2pt,shorten >=2pt,every node/.style={auto,inner sep=2pt,font=\scriptsize}},
gamenode/.style={draw,inner sep=2pt,minimum size=.5cm,font=\small},
lit/.style={gamenode,draw=none},
p0/.style={gamenode,rectangle,rounded corners=.25cm},
p1/.style={gamenode,rectangle},
F/.style={thick, preaction={fill,pattern=north east lines,opacity=0.4},font=\boldmath\small},
dot/.style={circle,draw,fill,black,minimum size=3pt,inner sep=0pt},
marker/.style={draw=none,inner sep=0pt,overlay},
short/.style={ shorten >=#1, shorten <=#1 }
}
\begin{figure*}
\centering
\begin{tikzpicture}[font=\small,node distance=.6cm]
\node [p0,F] (Y1) {$Y v_1$};
\node [p0,right=of Y1] (Z1) {$Zv_1$};
\node [draw,cloud,minimum width=11cm,minimum height=3cm,cloud puffs=30,densely dotted,aspect=1,cloud puff arc=100,right=of Z1,text=gray] (C1) {};
\node [p0,F,below=14cm of Y1] (Yn) {$Y v_n$};
\node [p0,right=of Yn] (Zn) {$Zv_n$};
\node [draw,cloud,minimum width=11cm,minimum height=3cm,cloud puffs=30,densely dotted,aspect=1,cloud puff arc=100,right=of Zn,text=gray] (Cn) {};
\node [p0,F,below=7cm of Y1] (Y) {$Yv_i$};
\node [p0,right=.4cm of Y] (Z) {$Zv_i$};
\node [p0,right=.4cm of Z] (phi) {$\phi(v_i)$};
\node [p1,right=.4cm of phi,yshift=1.8cm] (F) {$Fv_i \land \theta_1(v_i)$};
\node [p0,right=of F] (t1) {$\theta_1(v_i)$};
\node [p1,right=of t1,yshift=.7cm] (V0) {$V_0 v_i \land \dots$};
\node [p1,right=of t1,yshift=-.7cm] (V1) {$V_1 v_i \land \dots$};
\node [p0,right=of V0] (Ex) {$\exists u \dots$};
\node [p1,right=of Ex,yshift=1cm,minimum height=.2cm,align=center] (E1) {\dots};
\node [p1,right=of Ex,yshift=.4cm,minimum height=.2cm,align=center] (E2) {\dots};
\node [p1,right=of Ex,yshift=-.2cm,minimum height=.2cm,align=center] (E3) {\dots};
\node [lit,right=of E1] (Ev1) {$E v_i v_1$};
\node [lit,right=of E2] (Ev2) {$E v_i v_i$};
\node [lit,right=of E3] (Ev3) {$E v_i v_n$};
\node [p1,right=of V1] (All) {$\forall u \dots$};
\node [p1,right=of All,yshift=.2cm,minimum height=.2cm,align=center] (A1) {\dots};
\node [p1,right=of All,yshift=-.4cm,minimum height=.2cm,align=center] (A2) {\dots};
\node [p1,right=of All,yshift=-1cm,minimum height=.2cm,align=center] (A3) {\dots};
\node [lit,right=of A1] (Av1) {$E v_i v_1$};
\node [lit,right=of A2] (Av2) {$E v_i v_i$};
\node [lit,right=of A3] (Av3) {$E v_i v_n$};
\node [p1,right=.4cm of phi,yshift=-1.8cm] (nF) {$\neg Fv_i \land \theta_1(v_i)$};
\node [p0] at (t1 |- nF) (t2) {$\theta_2(v_i)$};
\node [draw,cloud,minimum width=4cm,cloud puffs=30,densely dotted,aspect=3.3,cloud puff arc=100,right=of t2,text=gray,align=center] (analog) {analogous to $\theta_1(v_i)$,\\but move to $Z v_j$ afterwards};
\node [lit,below=.5cm of F] (Fv) {$Fv_i$};
\node [lit,above=.5cm of nF] (nFv) {$\neg Fv_i$};
\node [lit,above=.5cm of V0] (V0v) {$V_0 v_i$};
\node [lit,below=.5cm of V1] (V1v) {$V_1 v_i$};
\draw [arr]
(Y) edge (Z) (Z) edge (phi)
(phi) edge (F.west) (phi) edge (nF.west)
(F) edge (t1)
(t1) edge (V0) (t1) edge (V1)
(V0) edge (Ex)
(V1) edge (All)
(Ex) edge (E1.west) (Ex) edge (E2.west) (Ex) edge (E3.west)
(E1) edge (Ev1) (E2) edge (Ev2) (E3) edge (Ev3)
(All) edge (A1.west) (All) edge (A2.west) (All) edge (A3.west)
(A1) edge (Av1) (A2) edge (Av2) (A3) edge (Av3)
(nF) edge (t2)
(t2) edge (analog)
(Y1) edge (Z1) (Z1) edge (C1)
(Yn) edge (Zn) (Zn) edge (Cn)
(F) edge (Fv) (nF) edge (nFv)
(V0) edge (V0v) (V1) edge (V1v)
;
\draw [densely dotted, thick, short=2pt]
(E1) edge (E2) (E2) edge (E3)
(A1) edge (A2) (A2) edge (A3)
;
\draw [arr,gray,overlay]
(analog.east) .. controls ($(Ev1)+(8,4)$) and ($(Z1)-(0,3)$) .. (Z1);
\draw [arr,gray,overlay]
(analog.east) to [out=-40,in=-90,looseness=.7] (Z);
\draw [arr,gray,overlay]
(analog.east) .. controls ($(analog.east)+(5,-4.5)$) and ($(Zn)+(0,4)$) .. (Zn);
\draw [arr,overlay]
(E1) edge [out=50,in=-90,looseness=0.7] (Y1)
(E2) edge [out=40,in=70,looseness=0.8] (Y)
(A3) .. controls ($(analog.east)+(9,-5)$) and ($(Yn)+(0,5)$) .. (Yn)
;
\draw [densely dotted, short=2pt]
(E3) edge ++(-20:1)
(A1) edge ++(20:1)
(A2) edge ++(15:1)
;
\draw [densely dotted, short=2pt, overlay]
(C1.east) to [out=0,in=90,looseness=.5] ($(C1.east)+(1,-1)$)
(C1.east) to [out=0,in=90,looseness=.5] ($(C1.east)+(.8,-1.1)$)
(C1.east) to [out=0,in=90,looseness=.5] ($(C1.east)+(.6,-1.2)$)
(Cn.east) to [out=0,in=-90,looseness=.5] ($(Cn.east)+(1,1)$)
(Cn.east) to [out=0,in=-90,looseness=.5] ($(Cn.east)+(.8,1.1)$)
(Cn.east) to [out=0,in=-90,looseness=.5] ($(Cn.east)+(.6,1.2)$)
;
\draw [arr,overlay] ($(Y)-(1cm,0)$) to (Y);
%
\begin{scope}[on background layer]
\node [draw=none,fill=lightgray!50,rectangle,inner sep=5pt,rounded corners,fit=(Y)(E1)(Av3)(analog)] (rect) {};
\coordinate (R1) at (Av3.east |- C1);
\coordinate (Rn) at (Av3.east |- Cn);
\node [draw=none,fill=lightgray!50,rectangle,inner sep=5pt,rounded corners,fit=(Y1)(C1.north)(C1.south)(R1)] (rect1) {};
\node [draw=none,fill=lightgray!50,rectangle,inner sep=5pt,rounded corners,fit=(Yn)(Cn.north)(Cn.south)(Rn)] (rectn) {};
\end{scope}
\draw [loosely dotted, very thick, short=.5cm]
(Y.east |- rect1.south) edge (Y.east |- rect.north)
(Y.east |- rectn.north) edge (Y.east |- rect.south)
;
\end{tikzpicture}
\caption{Illustration of the model-checking game $\MC \Gg {v_i}$ for a Büchi game with positions $V = \{v_1,\dots,v_n\}$.
Rounded nodes belong to Verifier, rectangular nodes to Falsifier. Nodes without border are terminal positions representing literals,
dashed nodes are target positions.
}
\label{figModelCheckingFull}
\end{figure*}
To prove \cref{thmSumOfStrategiesTracking}, we show that it follows from a more general sum-of-strategies theorem in \cite{DannertGraNaaTan21}\footnote{A complete proof can be found in the full version \cite{DannertGraNaaTan19}.}.
The general theorem expresses the semiring semantics of arbitrary LFP-formula by means of winning strategies in the corresponding model-checking game;
this does not immediately apply to Büchi games, but it is not difficult to see (and we explain below) that the model-checking game for $\mathsf{win}_0(v)$ on a Büchi game $\Gg$ has the same structure as the original game graph $\Gg$.
Towards the proof, we first sketch the model-checking game $\MC \Gg v$ of $\mathsf{win}_0(v)$ on a Büchi game $\Gg$.
We then briefly recapitulate how semiring values of winning strategies are defined in \cite{DannertGraNaaTan21},
and we show that we can simplify $\MC \Gg v$ without changing these values so that it becomes almost identical to $\Gg$.
This allows us to derive \cref{thmSumOfStrategiesTracking} from the general sum-of-strategies result on $\MC \Gg v$.
\paragraph*{Model-Checking Game}
In general, the model-checking game of an LFP-formula $\psi$ in a structure $\AA$ is a parity game whose positions are pairs of a subformula $\phi(\tup x)$ of $\psi$ and an instantiation of the free variables, conveniently written as $\phi(\tup a)$ for some tuple $\tup a \subseteq \AA$.
Edges allow to move from a position $\phi(\tup a)$ to a direct subformula of $\phi(\tup a)$, or from fixed-point literals (such as $Yu$ in our case) back to the entire fixed-point formula.
The model-checking game we are interested in is shown in \cref{figModelCheckingFull}, so we refrain from a complete definition, see e.g.\ \cite[Chap.~4]{AptGraedel11} for more background.
Here we are only concerned with the model-checking game of $\mathsf{win}_0(v)$ in a Büchi game $\Gg$, and we denote this game as $\MC \Gg v$.
Since $\mathsf{win}_0(v)$ only has a single alternation of fixed-point operators, $\MC \Gg v$ has only two priorities and can thus be equivalently represented as a Büchi game.
Recall the formula from Sect.~\ref{sec:TheFormula} (split into subformulae to ease referencing and with implication already rewritten):
\[
\mathsf{win}_0(x) \coloneqq{} \big[\gfp Y y.\ [\lfp Z z.\ \phi(Y,Z,z)](y) \big](x)
\]
where
\begin{align*}
\phi(z) \coloneqq{} &(Fz \land \theta_1(z)) \lor (\neg Fz \land \theta_2(z)) \\[.5\normalbaselineskip]
\theta_1(z) \coloneqq{} &\big((V_0 z \land \exists u (Ezu \land Yu))
\lor (V_1 z \land \forall u (\neg Ezu \lor (Ezu \land Yu)))\big) \\[.5\normalbaselineskip]
\theta_2(z) \coloneqq{} &\big((V_0 z \land \exists u (Ezu \land Zu))
\lor (V_1 z \land \forall u (\neg Ezu \lor (Ezu \land Zu)))\big).
\end{align*}
A depiction of the complete model-checking game $\MC \Gg v$ is shown in \cref{figModelCheckingFull} (with some unavoidable omissions due to space reasons).
\paragraph*{Semiring Values for the Model-Checking Game}
We define strategies for $\MC \Gg v$ as we did for Büchi games, always taking the perspective of Verifier.
To avoid confusion, we use the letter $\Mm$ to denote strategies in the model-checking game, and $\Ss$ for strategies in $\Gg$.
Leaving literals aside for a moment, we say that a strategy $\Mm$ is \emph{winning} if on each infinite play, target nodes (i.e., nodes of the form $Y v$) occur infinitely often.
We only speak about \emph{winning} strategies for $\MC \Gg v$ in the following.
Now consider literals, the terminal positions of the model-checking game.
As in \cref{defKInterpretation}, we write $\Lit$ (omitting $V$ and $\tau$) for the set of instantiated literals over the signature of $\mathsf{win}_0$ and the set of positions of $\Gg$.
Given a $\K$-interpretation $\pi$, it assigns to each literal $\lit \in \Lit$ the value $\pi(\lit)$.
Following \cite{DannertGraNaaTan21}, we define the value of a winning strategy $\Mm$ by counting literals.
\begin{definition}
Let $\pi$ be a $\K$-interpretation in an absorptive, fully-continuous semiring.
Let $\Gg$ be a Büchi game and $\Mm$ a winning strategy in $\MC \Gg v$.
For a literal $\lit \in \Lit$, we write $\ecount \Mm \lit \in \N \cup \{\infty\}$ for the number of occurrences
of $\lit$ in $\Mm$ (represented as a strategy tree).
The $\K$-value of $\Mm$ is
\[
\pi \ext \Mm \coloneqq \prod_{\lit \in \Lit} \pi(\lit)^{\ecount \Mm \lit}.
\]\end{definition}
Notice that for the Boolean interpretation $\pi \colon \Lit \to \mathbb{B}$ that maps all literals to truth values according to $\Gg$,
we have $\pi \ext \Mm \neq 0$ if, and only if, $\Mm$ is a winning strategy in the classical Boolean model-checking game for $\mathsf{win}_0(v)$ and $\Gg$.
By assigning non-Boolean values for the literals, in particular indeterminates in ${\mathbb S}^{\infty}[X]$, we can track how the literals affect the truth of $\mathsf{win}_0(v)$.
With this terminology, applying the sum-of-strategies theorem in \cite{DannertGraNaaTan21} to $\mathsf{win}_0(v)$ gives:
\begin{theorem}
\label{generalSumOfStrategies}
Let $\pi$ be a $\K$-interpretation into an absorptive, fully-continuous semiring.
Then $\pi \ext {\mathsf{win}_0(v)} = \sum \{ \pi \ext \Mm \mid \Mm \text{ is a winning strategy in } \MC \Gg v \}$.
\end{theorem}
In order to use this result in our context, we will show that the winning strategies $\Mm$ in $\MC \Gg v$ correspond to winning strategies $\Ss$ in $\Gg$.
For edge-tracking interpretations $\pi$, only the edge literals $Evw$ are relevant and thus the corresponding strategies $\Mm$ and $\Ss$ have the same $\K$-value,
so \cref{generalSumOfStrategies} will then imply \cref{thmSumOfStrategiesTracking}.
\paragraph*{From the Model-Checking Game to the Büchi Game}
Let $\pi$ be an edge-tracking $\K$-interpretation for $\Gg$, so most of the literals are mapped to $0$ or $1$.
We can then remove the corresponding terminal positions in $\MC \Gg v$ and in some cases also their predecessors.
For instance, consider a position $\phi(\tup a)$ in $\MC \Gg v$ from which Falsifier can move to a literal $\lit$ with $\pi(\lit) = 0$.
Then every strategy $\Mm$ that visits $\phi(\tup a)$ must also visit $\lit$, thus having value $\pi \ext \Mm = 0$,
so we can ignore this strategy for the sum in \cref{generalSumOfStrategies}.
Hence, replacing the position $\phi(\tup a)$ by its successor $\lit$ does not change the sum.
On the other hand, if $\pi(\lit) = 1$, then visiting $\lit$ does not affect the $\K$-value of $\Mm$ and hence we can remove $\lit$.
Similar reasoning applies to positions of Verifier.
Moreover, we can always skip over non-target positions with a unique successor, as they neither affect gameplay nor the $\K$-values of strategies.
With these insights, we can simplify the model-checking game in \cref{figModelCheckingFull} quite a bit.
If, say, $v_i \in F$ and $v_i \in V_0$, then the central part of the picture simplifies to:
\begin{center}
\begin{tikzpicture}[node distance=1.5cm]
\node [p0,F] (Y) {$Y v_i$};
\node [p0,right=0.7cm of Y] (Z) {$Z v_i$};
\node [p1,minimum height=.2cm,right=of Z,yshift=.7cm] (E1) {\dots};
\node [p1,minimum height=.2cm,right=of Z,yshift=-.7cm] (E2) {\dots};
\node [lit,right=0.7cm of E1] (Ew1) {$Ev_i w_1$};
\node [lit,right=0.7cm of E2] (Ew2) {$Ev_i w_k$};
\draw [dotted, thick, short=5pt] (E1) edge (E2);
\draw [arr] (Y) edge (Z) (Z) edge (E1) (Z) edge (E2);
\draw [arr] (E1) edge (Ew1) (E2) edge (Ew2);
\draw [arr,dotted] (Z) edge ($(E1.west)!0.35!(E2.west)$);
\draw [arr,dotted] (Z) edge ($(E1.west)!0.65!(E2.west)$);
\node [p0,F,right=1.2cm of Ew1] (Y1) {$Y w_1$};
\node [p0,F,right=1.2cm of Ew2] (Y2) {$Y w_k$};
\coordinate [right=.6cm of Y1] (R1);
\coordinate [right=.6cm of Y2] (R2);
\draw [thick,loosely dotted,shorten <=5pt,shorten >=0pt] (Y1) edge (R1) (Y2) edge (R2);
\draw [arr] (E1) edge [out=-50,in=190,looseness=0.7] (Y1);
\draw [arr] (E2) edge [out=50,in=-190,looseness=0.7] (Y2);
\begin{scope}[on background layer]
\node [draw=none,fill=lightgray!50,rectangle,rounded corners,inner sep=5pt,fit=(Y)(Ew1)(Ew2)] {};
\node [draw=none,fill=lightgray!50,rectangle,rounded corners,inner sep=5pt,fit=(Y1)(R1)] {};
\node [draw=none,fill=lightgray!50,rectangle,rounded corners,inner sep=5pt,fit=(Y2)(R2)] {};
\end{scope}
\end{tikzpicture}
\end{center}
Here, $v_i E = \{w_1,\dots,w_k\}$ are the successors of $v_i$ in $\Gg$, so Verifier's moves from $Zv_i$ are only those corresponding to actual edges of $\Gg$.
The other situations are similar, here is the case $v_i \notin F$, $v_i \in V_1$:
\begin{center}
\begin{tikzpicture}[node distance=1.5cm]
\node [p0,F] (Y) {$Y v_i$};
\node [p1,right=0.7cm of Y] (Z) {$Z v_i$};
\node [p1,minimum height=.2cm,right=of Z,yshift=.7cm] (E1) {\dots};
\node [p1,minimum height=.2cm,right=of Z,yshift=-.7cm] (E2) {\dots};
\node [lit,right=0.7cm of E1] (Ew1) {$Ev_i w_1$};
\node [lit,right=0.7cm of E2] (Ew2) {$Ev_i w_k$};
\draw [dotted, thick, short=5pt] (E1) edge (E2);
\draw [arr] (Y) edge (Z) (Z) edge (E1) (Z) edge (E2);
\draw [arr] (E1) edge (Ew1) (E2) edge (Ew2);
\draw [arr,dotted] (Z) edge ($(E1.west)!0.35!(E2.west)$);
\draw [arr,dotted] (Z) edge ($(E1.west)!0.65!(E2.west)$);
\node [p0,F,right=.8cm of Ew1] (Y1) {$Y w_1$};
\node [p0,F,right=.8cm of Ew2] (Y2) {$Y w_k$};
\node [p0,right=.7cm of Y1] (Z1) {$Z w_1$};
\node [p1,right=.7cm of Y2] (Z2) {$Z w_k$};
\draw [arr] (Y1) edge (Z1) (Y2) edge (Z2);
\coordinate [right=.8cm of Z1] (R1);
\coordinate [right=.8cm of Z2] (R2);
\draw [thick,loosely dotted,shorten <=5pt,shorten >=0pt] (Z1) edge (R1) (Z2) edge (R2);
\draw [arr] (E1) edge [out=-50,in=200,looseness=0.6] (Z1);
\draw [arr] (E2) edge [out=50,in=-200,looseness=0.6] (Z2);
\begin{scope}[on background layer]
\node [draw=none,fill=lightgray!50,rectangle,rounded corners,inner sep=5pt,fit=(Y)(Ew1)(Ew2)] {};
\node [draw=none,fill=lightgray!50,rectangle,rounded corners,inner sep=5pt,fit=(Y1)(R1)] {};
\node [draw=none,fill=lightgray!50,rectangle,rounded corners,inner sep=5pt,fit=(Y2)(R2)] {};
\end{scope}
\end{tikzpicture}
\end{center}
where again $\{w_1,\dots,w_k\}$ are the successors of $v_i$ in $\Gg$.
Notice that $Z v_i$ belongs to Falsifier (as a result of skipping several positions).
In general, $Z v_i$ belongs to Verifier precisely if $v_i$ belongs to Player~0 in $\Gg$.
We can now identify the entire subgraph from $Y v_i$ up to the edge literals $E v_i w_j$ (that is, the gray rectangle) with the position $v_i$ in $\Gg$,
and we call this the \emph{gadget for $v_i$}.
Indeed, if $v_i \in V_0$ then Verifier chooses a successor $w \in v_iE$ and moves to the corresponding gadget,
in analogy to Player~0 choosing a successor in $\Gg$.
Similarly, Falsifier chooses a successor if it is Player~1's turn in $\Gg$.
What remains to discuss are the target positions.
Each gadget has two entry points $Y v$ and $Z v$, and only $Y v$ is a target position.
Notice that when we move from a gadget for $v$ to a gadget for $w$, we use the entry point $Y w$ if, and only if, $v \in F$ (where $F$ is the target set of the original game $\Gg$).
Hence any play that visits infinitely many target positions $Y w$ also visits infinitely many gadgets for positions $v \in F$ (the predecessors of $w$).
We can thus change the target set without affecting the winning strategies:
Instead of the positions $Y w$, we set the target set to $\{ Z v \mid v \in F \}$.
The positions $Y w$ are then regular positions with unique successors and can thus be removed.
As an example, say we have $v_i \in F$, $w_1 \in F$ and $w_k \notin F$. The previous picture then becomes:
\begin{center}
\begin{tikzpicture}[node distance=1.5cm]
\node [p1,F] (Z) {$Z v_i$};
\node [p1,minimum height=.2cm,right=of Z,yshift=.7cm] (E1) {\dots};
\node [p1,minimum height=.2cm,right=of Z,yshift=-.7cm] (E2) {\dots};
\node [lit,right=0.7cm of E1] (Ew1) {$Ev_i w_1$};
\node [lit,right=0.7cm of E2] (Ew2) {$Ev_i w_k$};
\draw [dotted, thick, short=5pt] (E1) edge (E2);
\draw [arr] (Z) edge (E1) (Z) edge (E2);
\draw [arr] (E1) edge (Ew1) (E2) edge (Ew2);
\draw [arr,dotted] (Z) edge ($(E1.west)!0.35!(E2.west)$);
\draw [arr,dotted] (Z) edge ($(E1.west)!0.65!(E2.west)$);
\node [p0,F,right=1.5cm of Ew1] (Z1) {$Z w_1$};
\node [p1,right=1.5cm of Ew2] (Z2) {$Z w_k$};
\coordinate [right=.6cm of Z1] (R1);
\coordinate [right=.6cm of Z2] (R2);
\draw [thick,loosely dotted,shorten <=5pt,shorten >=0pt] (Z1) edge (R1) (Z2) edge (R2);
\draw [arr] (E1) edge [out=-50,in=200,looseness=0.6] (Z1);
\draw [arr] (E2) edge [out=50,in=-200,looseness=0.6] (Z2);
\begin{scope}[on background layer]
\node [draw=none,fill=lightgray!50,rectangle,rounded corners,inner sep=5pt,fit=(Z)(Ew1)(Ew2)] {};
\node [draw=none,fill=lightgray!50,rectangle,rounded corners,inner sep=5pt,fit=(Z1)(R1)] {};
\node [draw=none,fill=lightgray!50,rectangle,rounded corners,inner sep=5pt,fit=(Z2)(R2)] {};
\end{scope}
\end{tikzpicture}
\end{center}
\paragraph*{Proof of \protect{\Cref{thmSumOfStrategiesTracking}}}
Let $\MCP \Gg v$ be the game that results from $\MC \Gg v$ by applying all of the above-mentioned simplifications.
Hence $\MCP \Gg v$ contains for each position $v$ a gadget with unique entry point $Z v$
that belongs to Verifier exactly if $v \in V_0$, and $Z v$ is a target position exactly if $v \in F$. Moreover, the gadget for $v$ is directly
connected to the gadget for $w$ if, and only if, the edge $vw$ exists in $\Gg$.
It is now easy to see that every winning strategy $\Ss$ in the original Büchi game $\Gg$
induces a unique winning strategy $\Mm$ in $\MCP \Gg v$: Whenever $\Ss$ visits a position $v$,
$\Mm$ visits the gadget for $v$ (via the unique entry point $Z v$).
Conversely, every winning strategy $\Mm$ uniquely induces a winning strategy $\Ss$,
and we thus say that $\Mm$ and $\Ss$ are \emph{corresponding} winning strategies.
\begin{lemma}
Let $\Mm$ be a winning strategy in $\MCP \Gg v$.
Let $\Ss$ be a winning strategy in $\Gg$ so that $\Mm$ and $\Ss$ are corresponding strategies.
Then, $\pi \ext \Mm = \pi \ext \Ss$.
\end{lemma}
\begin{proof}
As we removed all other literals from $\MC \Gg v$, the only literals occurring in $\Mm$ are edge literals of the form $Evw$,
so $\pi \ext \Mm = \prod_{vw \in E} \pi(Evw)^{\ecount \Mm {Evw}}$.
An edge literal $Evw$ occurs in $\Mm$ whenever $\Mm$ transitions from the gadget for $v$ to the gadget for $w$,
and this happens whenever the edge $vw$ occurs in $\Ss$.
So $\pi \ext \Mm = \prod_{vw \in E} \pi(Evw)^{\ecount \Ss {vw}} = \pi \ext \Ss$.
\end{proof}
This closes the proof of the Sum-of-Strategies Theorem:
since our modifications did not affect the sum over all winning strategies, \cref{generalSumOfStrategies} implies
\begin{align*}
&\pi \ext {\mathsf{win}_0(v)} \\
{}={} &\sum \{ \pi \ext \Mm \mid \Mm \text{ is a winning strategy in } \MC \Gg v \} \\
{}={} &\sum \{ \pi \ext \Mm \mid \Mm \text{ is a winning strategy in } \MCP \Gg v \} \\
{}={} &\sum \{ \pi \ext \Ss \mid \Ss \text{ corresponds to a winning strategy in } \MCP \Gg v \} \\
{}={} &\sum \{ \pi \ext \Ss \mid \Ss \in \WinStrat_\Gg(v) \},
\end{align*}
and restricting the sum to winning strategies $\Ss$ that are absorption-dominant from $v$ does not change the overall value. \hfill $\square$
\endgroup
\section{Case Study: Applications of Semiring Semantics}
\label{sec:Applications}
We now have all of the necessary groundwork to consider applications of semiring semantics for Büchi games.
This section discusses what information the Sum-of-Strategies Theorem provides about winning strategies, how semiring semantics helps to find minimal repairs and why it is not well suited for cost computations.
\subsection{Strategy Analysis}
\label{sec:StrategyTracking}
We begin with the question what information we can derive from the Sum-of-Strategies Theorem.
To this end, we fix a Büchi game $\Gg$ and focus on the ${\mathbb S}^{\infty}[X]$-interpretation $\pi_{\text{\sffamily\upshape strat}}$ with $X = \{ X_{uv} \mid u,v \in \Gg \}$.
The values $\pi_{\text{\sffamily\upshape strat}} \ext \Ss$ are monomials and we can read off the number of occurrences of each edge in $\Ss$ from the exponents, i.e., the monomial is a representation of the edge profile $\ep \Ss$.
In particular, $\pi_{\text{\sffamily\upshape strat}} \ext {\Ss_1} \succeq \pi_{\text{\sffamily\upshape strat}} \ext {\Ss_2}$ if,
and only if, $\Ss_1 \succeq \Ss_2$.
The fact that absorptive polynomials are always finite \cite{DannertGraNaaTan21} is thus another way to see that the number of absorption-dominant strategies is finite.
What can we learn from the polynomial $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)}$?
First, $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)} \neq 0$ holds if, and only if, Player~0 has a winning strategy from $v$.
By \cref{thmSumOfStrategiesTracking}, we can further derive information about all absorption-dominant strategies.
More precisely, we learn which edges each absorption-dominant strategy uses and how often they appear in the strategy tree.
Knowing the edge profile immediately reveals whether the strategy is positional and what the positional choices are.
By counting monomials, we can thus count the positional strategies, as well as the absorption-dominant strategies up to absorption-equivalence.
We can further answer questions such as: can Player~0 still win if we remove edge $e$?
This is the case if, and only if, the polynomial $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)}$ contains a monomial without the variable $X_e$ (if there is a winning strategy without $e$, then there is also an absorption-dominant strategy and hence a monomial without $X_e$).
Going further, a more interesting question is: can Player~0 still win if edge $e$ may only be used finitely often in each play?
The answer is not immediately obvious.
Consider for example the strategy $\Ss$ in \cref{fig:RunningStrategy}.
The edge $k$ occurs infinitely often in the strategy tree and we get $\pi_{\text{\sffamily\upshape strat}} \ext \Ss = abcefh g^\infty k^\infty m^\infty$.
However, $k$ is clearly played only once in each play consistent with $\Ss$, whereas edge $m$ is played infinitely often.
We cannot distinguish edges $k$ and $m$ just from $\pi_{\text{\sffamily\upshape strat}} \ext \Ss$, but we can do so if we compute $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(w)}$ for all positions $w \in V$, by the following criterion (notice that all of these values are computed anyway for the fixed-point iteration).
\begin{proposition}
Let $\Ss \in \WinStrat_\Gg(v)$ be absorption-domi\-nant from $v$, and let $e = uw \in E$ be an edge with $\ecount \Ss e = \infty$.
Then there is a unique (positional) strategy $\Ss_w \in \WinStrat_\Gg(w)$ such that $\pi_{\text{\sffamily\upshape strat}} \ext {\Ss_w} \succeq \pi_{\text{\sffamily\upshape strat}} \ext \Ss$.
Moreover, $\Ss$ admits a play in which $e$ occurs infinitely often if, and only if, $e$ occurs
in $\Ss_w$.
\end{proposition}
\begin{proof}
Consider the strategy tree $\Ss$ and let $\rho w$ be an occurrence of $w$ in $\Ss$.
By assumption, $w$ occurs infinitely often in $\Ss$.
But then, for all successors $\rho w w'$ of $\rho w$, also $w'$ occurs infinitely often in $\Ss$.
Indeed, either $w \in V_1$ and every occurrence of $w$ must be followed by an occurrence of $w'$;
or $w \in V_0$ and $\Ss$ plays positionally from $w$ by \cref{stratInfinitePositional}, so again every occurrence of $w$ is followed by an occurrence of $w'$ in $\Ss$.
By induction, it follows that $\Ss$ plays positionally from $w$ and from all positions occurring below $\rho w$.
In particular, the substrategy $\Ss_w$ that $\Ss$ plays from $\rho w$ (and any other occurrence of $w$) is positional.
As a substrategy, we trivially have $\pi_{\text{\sffamily\upshape strat}} \ext {\Ss_w} \succeq \pi_{\text{\sffamily\upshape strat}} \ext \Ss$.
To see that $\Ss_w$ is unique with this property, notice that a strategy $\Ss' \in \Strat_\Gg(w)$ deviating from $\Ss_w$ must play an edge which does not occur in $\Ss_w$ and hence also not in $\Ss$, so $\pi_{\text{\sffamily\upshape strat}} \ext {\Ss'} \not\succeq \pi_{\text{\sffamily\upshape strat}} \ext \Ss$.
For the second statement, first assume that $\Ss$ admits a play $v v_1 v_2 v_3 \dots$ in which the edge $e=uw$ occurs infinitely often.
Let $v_i = w$ be the first occurrence of $w$.
Then the remaining play $v_i v_{i+1} v_{i+2} \dots$ is consistent with $\Ss_w$ and hence $e$ occurs (infinitely often) in $\Ss_w$.
Conversely, assume that $e$ occurs in $\Ss_w$, so there is some $\rho \in V^*$ such that $w \rho u w \in \Ss_w$.
As $\Ss_w$ is positional, we can repeat $\rho u w$ to obtain the infinite play $w (\rho u w)^\omega$ consistent with $\Ss_w$.
And since $\Ss_w$ is a substrategy of $\Ss$, this induces an infinite play of the form $\rho' w (\rho u w)^\omega$ in $\Ss$ which indeed uses the edge $uw$ infinitely often.
\end{proof}
\begin{example}
Consider the strategy $\Ss$ in \cref{fig:RunningStrategy} with $\pi_{\text{\sffamily\upshape strat}} \ext \Ss = abcefh g^\infty k^\infty m^\infty$ and the edge $k$ from $u$ to $w$.
Since edge $n$ does not occur in $\pi_{\text{\sffamily\upshape strat}} \ext \Ss$, the only winning strategy $\Ss_w$ from $w$ we need to consider is the strategy that always stays at $w$, with $\pi_{\text{\sffamily\upshape strat}} \ext {\Ss_w} = m^\infty \succeq \pi_{\text{\sffamily\upshape strat}} \ext \Ss$.
As $k$ does not occur in $\Ss_w$, we conclude that it occurs only finitely often (and hence at most once) in each play consistent with $\Ss$.
If, on the other hand, we consider edge $m$ (which also leads to position $w$), we see that $m$ occurs in $S_w$ and we can thus infer that $\Ss$ contains a play visiting $m$ infinitely often.
\end{example}
Summarizing the results of this section, we see that semiring semantics in ${\mathbb S}^{\infty}[X]$ is very informative and allows us to derive important information about the winning strategies.
\begin{corollary}\label{corPolynomialInformation}
From the polynomial $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)}$, we can efficiently (in the size of the polynomial) derive the following information:
\begin{itemize}
\item whether Player $0$ wins from $v$,
\item the edge profiles of all absorption-dominant winning strategies from $v$,
\item the number and precise shape of all positional winning strategies from $v$,
\item whether Player $0$ can still win from $v$ if only a subset of the edges is allowed.
\end{itemize}
Given the polynomials $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)}$, for \emph{all} positions $v$,
we can further derive for each (equivalence class of an) absorption-dominant strategy and each edge, how often the edge can occur in a play consistent with the strategy.
\end{corollary}
\subsection{Reverse Analysis}
\label{sec:ReverseAnalysis}
Instead of tracking strategies in a fixed game, we may also ask questions such as:
assuming Player~1 wins from $v$, what are minimal modification to $\Gg$ such that instead Player~0 wins?
The generality of semiring semantics enables us to answer such questions by choosing appropriate semirings and interpretations.
More precisely, let $\Gg = (V,V_0,V_1,E,F)$ be a Büchi game and $v \in \Gg$ a position from which Player~1 wins.
Let $E^- \subseteq E$ and $E^+ \subseteq V^2 \setminus E$ be sets of edges we are allowed to delete or add, respectively.
We call a set of edges in $E^\pm \coloneqq E^- \cup E^+$ a \emph{repair} if Player~0 wins when these edges are deleted or added.
Our goal is to determine all (preferably minimal) repairs.
We achieve this by evaluating $\mathsf{win}_0(v)$ in a modified polynomial semiring, similar to the computation of repairs for database queries in \cite{XuZhaAlaTan18},
except that here we need absorptive polynomials to deal with fixed points.
\bigskip\noindent\textbf{Dual-Indeterminates.}
To track negative information, such as the absence of an edge, we follow the approach in \cite{GraedelTan17,XuZhaAlaTan18,DannertGraNaaTan21} and extend our semiring by dual-indeterminates $\nn{X} = \{ \nn{x} \mid x \in X \}$.
The idea is to label a literal and its negation by corresponding indeterminates $x$ and $\nn{x}$.
We must then avoid monomials such as $x\nn{x}$, as they represent contradictory information.
To this end, we consider the quotient of ${\mathbb S}^{\infty}[X \cup \nn{X}]$ with respect to the congruence generated by $x \cdot \nn{x} = 0$ for $x \in X$
and refer to the resulting quotient semiring as \emph{dual-indeterminate absorptive polynomials} ${\mathbb S}^{\infty}[X,\nn{X}]$.
This semiring inherits most of the properties of ${\mathbb S}^{\infty}[X]$.
Most importantly, any assignment $h \colon X \cup \nn{X}$ that respects dual-indeterminates, i.e., $h(x) \cdot h(\nn{x}) = 0$, lifts to a fully-continuous homomorphism analogous to \cref{universality}.
We then replace $\pi_{\text{\sffamily\upshape strat}}$ by an ${\mathbb S}^{\infty}[X,\nn{X}]$-interpretation $\pi_{\text{\sffamily\upshape repair}}^\pm$ with $X = \{ X_e \mid e \in E^\pm\}$:
if $vw \in E^\pm$, we set $\pi_{\text{\sffamily\upshape repair}}^\pm(Evw) = X_{vw}$ and $\pi_{\text{\sffamily\upshape repair}}^\pm(\neg Evw) = \nn{X_{vw}}$,
all other literals are mapped to $0$ or $1$ according to $\Gg$ (cf.\ \cref{fig:Interpretations}).
Notice that $\pi_{\text{\sffamily\upshape repair}}^\pm$ is neither model-defining nor edge-tracking, but still satisfies $\pi_{\text{\sffamily\upshape repair}}^\pm(\lit) \bcdot \pi_{\text{\sffamily\upshape repair}}^\pm(\neg \lit) = 0$ for all literals $\lit$.
\bigskip\noindent\textbf{Back and Forth between Monomials and Models.}
Let $X^\pm = \{ X_e \mid e \in E^+ \} \cup \{ \nn{X_e} \mid e \in E^- \}$.
Given $Y \subseteq X^\pm$, we further write $E(Y) = \{ e \mid X_e \in Y \text{ or } \nn{X_e} \in Y \}$ for the set of edges mentioned in $Y$.
We denote the set of all (dual-)indeterminates occurring in a monomial $m$ by $\mathsf{var}(m) = \{ x \in X \cup \nn{X} \mid m(x) > 0 \}$.
By examining what combinations of indeterminates from $X^\pm$ occur in the monomials of $\pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)}$, we can read off all minimal repairs as follows.
\begin{proposition}\label{reverseBothDirections}
In the above setting, the following holds:
\begin{enumerate}
\item
Let $m \in \pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)}$ be a monomial.
Then the set $E(\mathsf{var}(m) \cap X^\pm)$ is a repair.
\item
Let $R \subseteq E^\pm$ be a repair.
Then there is a monomial $m \in \pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)}$ such that
$E(\mathsf{var}(m) \cap X^\pm) \subseteq R$.
If $R$ is minimal, then $E(\mathsf{var}(m) \cap X^\pm) = R$.
\end{enumerate}
\end{proposition}
Before proving \cref{reverseBothDirections}, we illustrate the computation of minimal repairs in a small example.
\begin{example}
In the following game, Player~1 wins from $v$.
We are interested in the minimal repairs with $E^+ = \{c\}$ and $E^- = \{a,b\}$.
\begin{center}
\begin{tikzpicture}[node distance=1.5cm, baseline]
\node [p1,label={left:$v$}] (0) {};
\node [p0,F,right of=0] (1) {};
\draw [arr]
(0) edge (1)
(1) edge [bend left] node {$b$} (0)
(1) edge [loop above,densely dotted,gray!80!black] node {$c$} (1)
(0) edge [loop above] node {$a$} (0)
;
\end{tikzpicture}
\hspace{1cm}
$\displaystyle \pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)} = \nn{X_a} X_c^\infty + \nn{X_a}^\infty X_b^\infty$\\[.3cm]
\end{center}
Evaluating $\mathsf{win}_0(v)$ in the ${\mathbb S}^{\infty}[X,\nn{X}]$-interpretation $\pi_{\text{\sffamily\upshape repair}}^\pm$ described above results in two monomials.
The first yields the repair $\{a,c\}$, the second yields the minimal repair $\{a\}$ (notice that $X_b \notin X^\pm$, as edge $b$ is already present).
The reason why we get two monomials is that we track also positive usage of edge $b$ by $X_b$, but are only interested in the negative indeterminate $\nn{X_b}$ for the repairs.
\end{example}
\begin{proof}[Proof of \cref{reverseBothDirections}]
We prove both statements by considering homomorphisms into the Boolean semiring $\mathbb{B}$.
For the first statement, let $m \in \pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)}$ be a monomial and
let $h \colon X \cup \nn{X} \to \mathbb{B}$ be the unique function that respects dual-indeterminates and satisfies
\begin{itemize}
\item $h(x) = \mathbf{1}$, for all $x \in \mathsf{var}(m)$
\item $h(X_e) = \mathbf{0}$, if $X_e, \nn{X_e} \notin \mathsf{var}(m)$ and $e \in E^+$ (do not add $e$ without reason),
\item $h(X_e) = \mathbf{1}$, if $X_e, \nn{X_e} \notin \mathsf{var}(m)$ and $e \in E^-$ (do not remove $e$ without reason).
\end{itemize}
Then, $h$ lifts to a fully-continuous semiring homomorphism $h \colon {\mathbb S}^{\infty}[X,\nn{X}] \to \mathbb{B}$ with $h(m) = \mathbf{1}$.
Moreover, $h \circ \pi_{\text{\sffamily\upshape repair}}^\pm$ is a Boolean interpretation which corresponds to a Boolean model $\Gg'$.
Since semiring semantics are preserved by fully-continuous homomorphisms, we have $h \circ \pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)} = h(\pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)}) \ge h(m) = \mathbf{1}$ and hence $\Gg' \models \mathsf{win}_0(v)$.
By the choice of $h$, the model $\Gg'$ is equal to $\Gg$ except that we add all edges $e \in X^+$ with $X_e \in \mathsf{var}(m)$, and remove all $e \in X^-$ with $\nn{X_e} \in \mathsf{var}(m)$.
Hence $\Gg'$ results from $\Gg$ by adding or deleting the edges $E(\mathsf{var}(m) \cap X^\pm)$, and since $\Gg' \models \mathsf{win}_0(v)$, this set is a repair as claimed.
For the second statement, let $R \subseteq E^\pm$ be a repair and consider the repaired game $\Gg' \models \mathsf{win}_0(v)$.
As $\Gg'$ differs from $\Gg$ only by edges in $E^\pm$, there is a unique assignment $h \colon X \cup \nn{X} \to \mathbb{B}$ such that $h \circ \pi_{\text{\sffamily\upshape repair}}^\pm$ corresponds to $\Gg'$.
Again, $h$ lifts to a fully-continuous homomorphisms and we thus get $\mathbf{1} = h \circ \pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)} = h( \pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)})$.
So there must be a monomial $m \in \pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)}$ with $h(m) = \mathbf{1}$.
Consider the set $\mathsf{var}(m) \cap X^\pm$.
If $X_e \in \mathsf{var}(m) \cap X^\pm$, then $h(X_e) = \mathbf{1}$ and hence $e \in R$ by construction of $h$.
Further, $\nn{X_e} \in \mathsf{var}(m) \cap X^\pm$ implies $h(\nn{X_e}) = \mathbf{1}$ and thus again $e \in R$ by construction of $h$.
This proves $E(\mathsf{var}(m) \cap X^\pm) \subseteq R$.
If $R$ is minimal, we have equality:
otherwise $E(\mathsf{var}(m) \cap X^\pm)$ would be a smaller repair by the first statement, contradicting minimality.
\end{proof}
We remark that these results ignore the exponents of the monomials, so we can drop exponents from ${\mathbb S}^{\infty}[X,\nn{X}]$ (in other words, we use exponents from $\mathbb{B}$ instead of $\N \cup \{\infty\}$) and work in the resulting, simpler semiring $\mathsf{PosBool}[X,\nn{X}]$ (the dual-indeterminate quotient of the semiring $\mathsf{PosBool}[X]$, see e.g.\ \cite{GreenTan17}).
\subsection{Target synthesis}
The reverse analysis approach and the proof technique based on homomorphisms are not limited to questions about edges, but are general concepts of semiring provenance analysis.
As an example of a different application, we consider the synthesis of the target set $F$.
More precisely, we consider a game arena with positions $V = V_0 \dot\cup V_1$ and edges $E$ and want to compute all minimal choices for the set $F$ so that Player~0 wins the resulting Büchi game from some fixed starting position $u \in V$.
Similar to the computation minimal repairs, we can solve this task with an interpretation $\pi_{\text{\sffamily\upshape target}}$ over the dual-indeterminate semiring $\mathsf{PosBool}[X,\nn{X}]$ which interprets most literals, including edge literals, by Boolean values, but tracks the target set $F$ using corresponding pairs of dual-indeterminates $\pi_{\text{\sffamily\upshape target}}(Fv) = X_v$ and $\pi_{\text{\sffamily\upshape target}}(\neg Fv) = \nn{X_v}$ for each position $v \in V$ (cf. \cref{fig:Interpretations}).
We can then derive all possible minimal choices for $F$ from the polynomial $\pi_{\text{\sffamily\upshape target}} \ext {\mathsf{win}_0(u)}$, as illustrated in the following example.
\begin{example}
Consider the following game arena.
What are the minimal choices of the target set $F$ so that Player~0 wins from position $a$, or $b$?\\
\begin{center}
\begin{tikzpicture}[node distance=1.5cm, baseline]
\node [p0,label={below:$\vphantom{b}a$}] (0) {};
\node [p1,right of=0,label={below:$b$}] (1) {};
\node [p1,right of=1,label={below:$\vphantom{b}c$}] (2) {};
\draw [arr]
(0) edge [loop above] (0)
(2) edge [loop above] (2)
(0) edge [bend left=10pt] (1)
(1) edge [bend left=10pt] (0)
(1) edge (2)
(2) edge [bend right] (0);
\end{tikzpicture}
\hspace{1cm}
$\displaystyle \begin{aligned}
\pi_{\text{\sffamily\upshape target}} \ext {\mathsf{win}_0(a)} &= X_a + \nn{X_a} X_b X_c \\
\pi_{\text{\sffamily\upshape target}} \ext {\mathsf{win}_0(b)} &= X_a X_b X_c + X_a \nn{X_b} X_c + \nn{X_a} X_b X_c
\end{aligned}$\\[.3cm]
\end{center}
Using the $\mathsf{PosBool}[X,\nn{X}]$-interpretation $\pi_{\text{\sffamily\upshape target}}$, we can derive that to win from $a$, the target set must contain at least $\{a\}$ or $\{b,c\}$.
To win from $b$, it must contain at least $\{a,b,c\}$, $\{a,c\}$ or $\{b,c\}$ (notice that $\{a,b,c\}$ is a valid choice for $F$, but not minimal due to the presence of negative indeterminates).
This covers all minimal possible choices of the target set.
\end{example}
In general, the positive indeterminates $X_v$ in each monomial of $\pi_{\text{\sffamily\upshape target}} \ext {\mathsf{win}_0(u)}$ induce one possible choice of the target set $F$, and conversely every minimal choice of $F$ occurs as a monomial; this follows by the same arguments as in \cref{reverseBothDirections}.
We remark that we can do slightly better, as we do not need to track negative information for the synthesis problem and hence do not need the negative indeterminates $\nn{X_v}$.
In the above example, we would then obtain the polynomials $X_a + X_b X_c$ and $X_a X_c + X_b X_c$ for positions $a$ and $b$, respectively, which correspond exactly to the minimal choices for $F$.
This can be achieved by setting $\pi_{\text{\sffamily\upshape target}}(\neg Fv) = 1$ for all $v \in V$ and observing that this has the same effect as omitting the subformula $\neg F v$ from $\mathsf{win}_0$.
Since the resulting formula is equivalent in Boolean semantics, the reasoning in the proof of \cref{reverseBothDirections} can be adapted to this setting.
However, here we presented the general dual-indeterminate approach which does not depend on the actual formula we consider and hence also works in many other scenarios of provenance analysis, beyond the analysis of Büchi games.
\subsection{Complexity}
The previous applications show that once we have computed the polynomial $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)}$ or $\pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)}$,
it is easy to derive information about strategies or minimal repairs -- but how efficient is the computation of the polynomial in the first place?
Recall that for a fixed formula, the LFP model-checking problem for a classical structure can be solved in polynomial time in the size of the structure (here the number of positions).
The analogous problem in semiring semantics, to compute the value $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)}$, is more involved since depending on the semiring, fixed-point iterations can be infinite, and contrary to the Boolean case, it is not
trivial that fixed points can be computed efficiently.
However, it has been shown in \cite{Naaf21} that in absorptive, fully-continuous semirings, a least or greatest fixed point of a polynomial system, such as the ones induced by LFP-formulae, can be computed using only a polynomial number of semiring operations (including the infinitary power operation).
By evaluating nested fixed points recursively, we can thus compute the semiring value of a fixed\footnote{In general, this applies to all LFP formulae in which the alternation depth, the arity of fixed-point relations and the number of free variables in any subformula is bounded by a constant.} formula in a number of semiring operations that is polynomial in the size of the structure (here in the number of positions).
Whether this computation is efficient depends on the complexity of the semiring operations and hence on the semiring.
Indeed, although the \emph{number} of semiring operations is polynomial, the resulting polynomial $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)}$ can nevertheless have an exponential number of monomials.
This is, in general, unavoidable as both the number of (positional) winning strategies
as well as the number of minimal repairs can be exponential in the size of the game.
For instance, games of the following form have an exponential number of positional (and hence absorption-dominant) winning strategies:
\begin{center}
\begin{tikzpicture}[node distance=1cm]
\node [p0] (0) {};
\node [p1,right of=0,yshift=7mm] (1a) {};
\node [p1,right of=0,yshift=-7mm] (1b) {};
\node [p0,right of=1a,yshift=-7mm] (2) {};
\node [p1,right of=2,yshift=7mm] (3a) {};
\node [p1,right of=2,yshift=-7mm] (3b) {};
\node [p0,right of=3a,yshift=-7mm] (4) {};
\node [p0,right of=4] (4') {};
\node at ($(4)!0.5!(4')$) {$\,\cdots$};
\node [p1,right of=4',yshift=7mm] (5a) {};
\node [p1,right of=4',yshift=-7mm] (5b) {};
\node [p0,right of=5a,yshift=-7mm] (6) {};
\node [p1,right of=6,yshift=7mm] (7a) {};
\node [p1,right of=6,yshift=-7mm] (7b) {};
\node [p0,F,right of=7a,yshift=-7mm] (8) {};
\draw [arr]
(0) edge node {$a_1$} (1a) (0) edge node [swap] {$b_1$} (1b)
(1a) edge (2) (1b) edge (2)
(2) edge node {$a_2$} (3a) (2) edge node [swap] {$b_2$} (3b)
(3a) edge (4) (3b) edge (4)
(4') edge node {$a_{n-1}$} (5a) (4') edge node [swap] {$b_{n-1}$} (5b)
(5a) edge (6) (5b) edge (6)
(6) edge node {$a_n$} (7a) (6) edge node [swap] {$b_n$} (7b)
(7a) edge (8) (7b) edge (8)
(8) edge [loop right] node {$k$} (8)
;
\end{tikzpicture}
\end{center}
More efficient algorithms can be obtained by tracking only some of the edges:
instead of the interpretation $\pi_{\text{\sffamily\upshape strat}}$ that assigns indeterminates to all edges, we
map edges we are not interested in to $1$.
The resulting ${\mathbb S}^{\infty}[X]$-interpretation remains edge-tracking, so the Sum-of-Strategies Theorem still applies
and we can see which sets of the tracked edges are at least required for a winning strategy.
In particular, $\mathsf{win}_0(v)$ evaluates to $1$ if there is a winning strategy that avoids all tracked edges.
If this information is sufficient, for instance if we know that a certain part of the game must be visited and only care about edges within this part, we can make provenance analysis more efficient by only tracking small parts of a potentially large game.
Notice that the questions adressed in \cref{corPolynomialInformation} can also be solved by direct methods, and
just for finding \emph{some} winning strategy or repair, these will in many cases be more efficient
than computing the polynomial $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0(v)}$ or $\pi_{\text{\sffamily\upshape repair}}^\pm \ext {\mathsf{win}_0(v)}$.
Thus, the main benefit of semiring semantics in ${\mathbb S}^{\infty}[X]$ does not lie in a more efficient method to compute
some specific winning strategy or repair, but rather in providing a general and compact description of all important strategies at once, from which we can directly derive the answers to many different questions
concerning the strategy analysis of a Büchi game.
\subsection{Limitations}
We have seen that semiring semantics in polynomial semirings such as ${\mathbb S}^{\infty}[X]$ reveals useful information about strategies.
However, there are also limitations to this framework when it comes to certain typical applications of provenance analysis
such as cost computation. Given cost annotations of the edges, say in the tropical semiring $\mathbb{T}=(\mathbb{R}_{+}^{\infty},\min,+,\infty,0)$ (which is absorptive and fully continuous), do semiring semantics provide a sensible cost measure for the evaluation of $\mathsf{win}_0(v)$? Or, more intuitively, for the cost that Player~0 has to pay to win?
For first-order logic and acyclic games, this is indeed the case \cite{GraedelTan17,GraedelTan20}, but here fixed-point computations and infinite plays complicate the situation.
To see this, let $\pi$ be an edge-tracking $\mathbb{T}$-interpretation that assigns to each edge $vw$ in $\Gg$ a cost $\pi(Evw) \neq \infty$ (cost $0$ is allowed).
By the Sum-of-Strategies Theorem, we can view $\pi \ext {\mathsf{win}_0(v)}$ as the minimum over the cost of each strategy, and in the tropical semiring, this cost can be expressed as follows (computed over real numbers):
\[
\pi \ext \Ss = \sum_{vw \in E} {\ecount \Ss {vw}} \cdot \pi(Evw).
\]
That is, Player~0 has to pay for each \emph{occurrence} of an edge in the strategy tree.
While this is certainly a possible cost measure for a strategy, it is debatable whether it is an intuitive one.
Many edges will occur infinitely often in a strategy, and then $\infty \cdot \pi(Evw)$ is either $0$ or $\infty$, and the latter leads to the overall value $\pi \ext \Ss = \infty$.
Even if an edge occurs only once per play, but infinitely often in $\Ss$, Player~0 has to pay the cost $\infty \cdot \pi(Evw)$.
Instead, it might be more intuitive to define the cost of a strategy so that
\begin{enumerate}
\item Player~0 only has to pay once for an edge, no matter how often it occurs (think of an ``unlocking fee''),
\item Player~0 only has to pay for the maximal cost of any play consistent with $\Ss$, but not for all plays simultaneously.
\end{enumerate}
We claim that neither of these options is possible without adapting our notion of semiring semantics.
Notice that we can actually solve (1) by first computing the polynomial $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0}$ in ${\mathbb S}^{\infty}[X]$, then dropping exponents (or directly working in $\mathsf{PosBool}[X]$), and then instantiating each variable $X_{vw}$ by its cost to obtain a value in $\mathbb{T}$.
But this is only an indirect solution and computing $\pi_{\text{\sffamily\upshape strat}} \ext {\mathsf{win}_0}$ may incur an exponential blowup even though we only compute a single cost value.
A direct computation, in the sense that $\pi \ext \Ss$ yields the desired cost value for some suitable $\pi$, is not possible.
To see this for (1), note that we multiply the cost for each occurrence of an edge (in $\mathbb{T}$, semiring multiplication is defined as summation on real numbers, but we stick to the general vocabulary).
Say we have two edges $vw$ and $v'w'$ with the same cost $\pi(Evw) = \pi(E v'w') = c$.
To pay only once for $vw$, we would need $\pi(Evw) \bcdot \pi(Evw) = c$, but at the same time $\pi(Evw) \bcdot \pi(Ev'w') = c^2$ for different edges, a contradiction.
The issue is that the information on whether two edges are equal is abstracted away by $\pi$.
A different argument explains why (2) is not possible:
At a position in $V_0$, we want to \emph{minimize} the cost over all possible choices (corresponding to the existential quantification in $\mathsf{win}_0$).
For consecutive edges, we have to \emph{add} costs up, requiring a second operation.
But for positions in $V_1$, to fulfil (2) we must \emph{maximize} the cost over all possible choices, thus requiring a third operation.
Hence this cost measure is not expressible in semirings with only two operations and we would need different algebraic structures.
\section{Conclusion}
Based on a recent line of research on semiring provenance analysis that lead from database theory to semiring semantics for LFP,
we reported here on a case study that puts semiring semantics to use for a strategy analysis in Büchi games.
The choice of Büchi games has been motivated on one side by their relevance for applications in
the synthesis and verification of reactive systems, on the other side because they provide one of the simplest non-trivial
cases of infinite games for which the definability of winning positions requires an alternation between least and greatest fixed points --
and can thus not be treated by simpler classes of semirings such as the $\omega$-continuous ones
used for Datalog and reachability games.
The aim of the case study was to illustrate how semiring semantics can be applied to more complex games,
featuring infinite plays and complicated winning conditions, and what kind of insights it provides (or fails to provide)
about the winning strategies in the game.
This is captured in the central Sum-of-Strategies Theorem and its applications.
This non-trivial result can be seen as a simpler version of the general sum-of-strategies characterization in terms of model-checking games in \cite{DannertGraNaaTan21}
and it essentially identifies the value of the statement that Player~0 wins with the sum of the
valuations of all (absorption-dominant) winning strategies.
While this applies to the class of all absorptive, fully-continuous semirings,
the most important semirings for our analysis are generalized absorptive polynomials ${\mathbb S}^{\infty}[X]$.
Due to their universal property, these provide the most general information,
allowing us to read off the edge profiles of all absorption-dominant strategies.
With this information, we can count positional strategies,
we can determine whether a particular move is needed (once or even infinitely often) for winning the game,
and we can compute minimal ``repairs'' for a game.
The method of semiring valuations is rather flexible; we can use different semirings than ${\mathbb S}^{\infty}[X]$, and we can tailor the set of moves that
we track to make the resulting polynomial smaller and its computation more efficient.
Of course, not all relevant questions about strategies can be answered directly by semiring valuations,
and as an example for a limitation of this method, we have shown that minimal cost computations
provide serious obstacles to a semiring treatment.
The Sum-of-Strategies Theorem motivates the notion of absorption-dominant strategies which is of interest in its own right as it
captures strategies that are minimal with respect to the multiplicities of the edges they use.
To understand these strategies, we discussed how they relate to other classes of simple strategies, namely to positional and persistent strategies,
and we have shown that these form a strict hierarchy.
Finally, we remark that although Büchi games have been chosen as the topic of our case study,
the method of semiring valuations in absorptive semirings is not confined to this case. In principle,
it can be applied to different formulae and generalizes in particular to other games such as parity games,
as long as the winning positions are definable in fixed-point logic.
The win-formula for parity games is more complicated, and is parametrised by the number of priorities,
and so the Sum-of-Strategies Theorem requires different technical
details, but can be established along the same lines.
\bibliographystyle{alphaurl}
|
1,108,101,565,691 | arxiv | \section*{Introduction: a new tool for nanotechnology}
Although based on physical principles it should be possible to assemble matter atom-by-atom, only few attempts at such manipulation have been successful. The first and so far only methods to truly achieve this goal are scanning probe microscopies (SPM), scanning tunneling microscopy (STM) in particular. STM is able to resolve the atomic structure at conducting surfaces by measuring the tunneling current to an atomically sharp tip. Voltage pulses from the tip also allow atoms to be picked up from one potential energy minimum and placed at the location of another one on a flat surface~\cite{Eigler90N}, or moved to a vacancy in a surface layer~\cite{Ebert93PRL}. Similarly, the interaction between the tip of an atomic force microscope (AFM) and surface atoms can lower migration barriers, allowing manipulation even at insulating surfaces~\cite{Custance09NN}. However, despite many impressive demonstrations such as quantum corrals~\cite{Crommie93S}, atomic-scale memories~\cite{Kalff16NN}, and designer atomic lattices~\cite{Drost17NP}, these techniques are by nature limited to relatively weakly bound surface adatoms or vacancies.
In transmission electron microscopy (TEM), on the other hand, the scattering of highly energetic electrons passing through the sample is used for imaging. The mass of electrons makes TEM fundamentally different from optical microscopies: since electrons carry significant momentum, they can eject atoms from the structure being observed. Although the quantum mechanical scattering of relativistic electrons from the Coulomb potential of a nucleus is a complicated process in its fine details, the ejection event can be adequately described as an elastic collision between one imaging electron and one nucleus of the target material. This electron beam damage is typically a detrimental side-effect, prompting a push towards lower acceleration voltages to avoid degrading or destroying conducting samples for which this is the dominant damage mechanism. However, atom-number-conserving changes can be stochastically induced when the transferable kinetic energies are comparable to bond strengths in the material. A crucial advantage compared to SPM is that these energies are on the order of ten electronvolts in light-atom materials even for modest electron acceleration voltages. This potentially allows atoms with strong covalent bonds to be manipulated.
Since the entire field of view is uniformly irradiated in TEM, it is difficult to control the dose at the atomic level. However, in scanning transmission electron microscopy (STEM), pre-specimen condenser lenses are used to focus the electrons into an atomically small beam, which is rastered over the sample while the scattered intensity is recorded to form an image. In modern aberration-corrected instruments, the size of the beam spot (full-width at half-maximum) is only on the order of 1 \AA~\cite{Krivanek99U,Krivanek08UM}, making it possible to essentially direct the dose at individual atomic columns. These developments have given STEM the technological maturity to become a promising candidate for a fundamentally different tool for atomically precise manipulation. Indeed, some of us recently published the very first experimental demonstration of this process~\cite{Susi17UM}.
\section*{Scanning transmission electron microscopy: enabling developments}
While STM has allowed routine atomic resolution imaging since the 1980s, the non-destructive imaging of individual atoms by STEM only became widespread recently. The challenge was a pernicious property of electron optical systems, whereby no rotationally-symmetric electron lens of the type commonly used to form atomic-sized probes can eliminate spherical and chromatic aberrations~\cite{Hawkes09PTRSL}. The development of aberration correctors in the 1990s finally overcame this technical barrier, and electron microscopy instrumentation has seen enormous advances over the last two decades~\cite{Krivanek97Cs,Haider98N,Krivanek99U,Batson2002,Sawada2005,Krivanek08UM,Dahmen2009,Kaiser11U,Hosokawa2013}, making atomic resolution possible in most materials~\cite{Tanaka2008,Pennycook2011a}. Further, the identification of the exact chemical structure is also possible, especially in low-dimensional materials. The elemental composition can be revealed by direct Z-contrast imaging~\cite{Krivanek10N} (since the scattered intensity is proportional to the atomic number) or through energy dispersive X-ray spectroscopy~\cite{lovejoy_single_2012}, whereas electron energy loss spectroscopy (EELS)~\cite{Varela2004, Muller2008, Suenaga09NC, Colliex2012a, Bangert13NL} can additionally reveal the exact bonding configuration. Non-destructive imaging of a range of materials has meanwhile been enabled by lower electron acceleration voltages~\cite{Suenaga09NC,Krivanek10N,Kaiser11U}, yet still retaining atomic resolution.
Early aberration-corrected instruments were, however, too sensitive to focus the electron beam on a single atom for extended periods of time, and typical residual vacuum pressures in the instrument columns led to chemical etching~\cite{Meyer12PRL}. Sample stability is very important for EELS mapping, but particularly crucial for single-atom manipulation where the electron dose needs to be accurately directed. These challenges have been largely addressed in the latest generation of instruments, leading to remarkable sample stability. Stage drift can be as low as 0.1~{\AA} per minute, and, thanks to improved vacuum, beam damage in pristine graphene is completely suppressed at electron acceleration voltages below 80 kV~\cite{Meyer12PRL,Susi16NC}.
\section*{Graphene: advantage of two dimensions}
Concurrently with improvements in STEM, the discovery of graphene~\cite{Novoselov05N} has provided ideal samples for atomic resolution studies~\cite{Meyer08NL}. Not only is monolayer graphene highly conductive, which effectively mitigates other damage mechanisms such as radiolysis and ionization, it also has remarkable electronic and mechanical properties~\cite{Geim07NM}. These are theoretically well understood, and possible to accurately model from first principles using modern computational techniques.
The greatest advantage of graphene for manipulation experiments is its reduced dimensionality. The two-dimensional atomic structure can be directly imaged and atomic species easily identified, both of the material itself as well as any present impurities~\cite{Krivanek10N}. Even in cases where a projected image may be ambiguous, as when a heavier impurity atom such as Si~\cite{Ramasse13NL,Zhou12PRL,Susi14PRL} or P~\cite{Susi172DM} buckles out of the plane, EELS can be used to unambiguously identify the precise nature of the bonding. Further, the electron dose can be predominantly directed at single atoms, apart from probe tails, \textit{i.e.} the spreading of the intensity envelope of the electron beam beyond its sharp maximum~\cite{Kotakoski14NC}. Aberration-corrected beams are so finely focused that if the dose is directed at a single carbon atom in graphene, we estimate that only 0.3\% impinges upon the other two neighbors of an impurity (integrated beam intensity based on the electron probe shape measured in~\cite{Kotakoski14NC}). Since the relevant atom-scale dynamics are caused by electron impacts on a specific carbon atom~\cite{Susi14PRL}, this precision is crucial for control.
The reduced dimensionality also considerably simplifies both the physical interaction between the beam and the sample, and the analysis of the recorded image data. Each electron interacts only with a quasi-2D electron density, and predominantly scatters from the electrostatic potential of a single atomic nucleus. Besides simplifying theoretical descriptions of the interaction, the atomic structure can thus be directly and unambiguously identified from the scattered signal. This stands in stark contrast to three-dimensional crystals, where the complex interactions between the probe electrons and the sample, and the precise real-time reconstruction of its three-dimensional lattice from projected images present daunting challenges for atomically precise control~\cite{Kalinin16N}.
\section*{Atom-scale dynamics: state-of-the-art}
In addition to spatial resolution, understanding dynamics requires collecting information in the time domain. Atomic motion that happens on femto-to-picosecond timescales is several orders of magnitude too fast to capture using conventional instruments, and even dedicated ultrafast techniques are limited to nanosecond timescales for real-space imaging far from atomic resolution~\cite{Plemmons15CM, Vogelsang15NL}. However, if the probability (\textit{i.e.} cross section) of an electron-beam-driven process is low enough compared to the irradiation dose rate, each event can be individually distinguished. This enables their statistical treatment as a Poisson process~\cite{Susi14PRL}, allowing experimental interaction cross sections to be extracted and compared to simulations.
In the case of graphene, there is a growing body of work on the controlled creation~\cite{Robertson12NC} and annihilation~\cite{Kurasch12NL} of defects, and etching and edge shaping directed by the electron beam~\cite{Girit09S,Lin14NN}. However, our focus here is on non-destructive (atom-number-conserving) manipulation. Fundamentally, such reversible beam-driven processes in graphene can be divided into two categories: bond rotation events and impurity dynamics.
Most of the former observations have been of all-carbon structures, ranging from the creation and annihilation of the double pentagon-heptagon defect (controversially~\cite{Monthioux14Carbon} called "Stone-Wales defect" in the current literature) in pristine graphene~\cite{Wakabayashi07NN,Meyer08NL} and bond rotations at a graphene edge~\cite{Gong14ACSNano} to the migration of dislocation cores~\cite{Warner12Science,Lehtinen13NC} and divacancies~\cite{Kotakoski11PRL,Kotakoski14NC,Borner16PRB}, and the transformation of grain boundaries~\cite{Kurasch12NL}. The bond rotation process is also responsible for the healing of the so-called flower defect under electron irradiation~\cite{Kurasch12NL}. Crucially, in all of these cases, the structural changes in the material can be explained through rotated carbon-carbon bonds, understood to occur due to a single impact from an electron to one of the involved carbon atoms~\cite{Kotakoski11PRB}. In Fig.~\ref{fig:rotation}, we present examples for some of these cases, pointing out the individual bonds that have rotated between the frames when possible.
In contrast to observations of bond rotations, only a handful of examples of beam-driven and non-destructive impurity motion in graphene are known. These are the movement of the trivalent Si substitution via an out-of-plane bond inversion process~\cite{Susi14PRL}, the rotation of a Si trimer in a divacancy~\cite{Yang14AC}, the atomic motions in a Si$_6$ cluster in a pore~\cite{Lee13NC}, and the jumping of a bivalent N between two equivalent bonding sites across a single vacancy~\cite{Lin15NL}. Although not previously discussed, we have observed both B and N substitutions to also undergo lattice jumps similar to Si (data from published experiments~\cite{Ramasse13NL,Kepaptsoglou15AN,Lin15NL}). Fig.~\ref{fig:impurity} illustrates each of these processes, with the impurity atoms appearing brighter than the carbon atoms of the lattice due to their higher atomic number (apart from B, which appears dimmer). In the case of the trivalent Si impurity, a physical mechanism to direct the atom-scale dynamics is known~\cite{Susi14PRL}, pointing a way for atom manipulation experiments.
\section*{Electron irradiation: physics of manipulation}
The highly energetic electron can transfer a maximal amount of kinetic energy when it backscatters from the electrostatic potential of an atomic nucleus. For typical relativistic velocities this process occurs on the 10$^{-21}$ s time scale, allowing it to be treated as an instantaneous event even compared to electron dynamics. When the transferred energy is larger than what is required to remove an atom from the immediate vicinity of its lattice site (called the displacement threshold), knock-on damage occurs.
At room temperature, pristine graphene is damaged~\cite{Meyer12PRL,Susi16NC} at electron acceleration voltages above 80 kV (corresponding to an experimentally estimated~\cite{Susi16NC} displacement threshold of 21.14 eV), even leading to the amorphization of the material~\cite{Kotakoski11PRL} during extended experiments at 100 kV. Excitations in highly conductive graphene are damped extremely fast, and each impact can thus be accurately described as a perturbation of the equilibrium state~\cite{Susi16NC}. How well this assumption holds for imperfect structures needs to be confirmed, but early indications show a puzzling discrepancy with theoretical expectations for the rate of bond rotations under 60 kV TEM observation~\cite{Skowron16C}, as well as at impurity sites as we discuss below.
For atomic-scale manipulation of structures with impurities, mass is important since heavier heteroatoms receive less energy in a momentum-conserving interaction with an electron. However, carbon atoms next to an impurity tend to be more weakly bound than atoms of the bulk, leading to their preferential ejection~\cite{Susi12AN}. This effect can impose a direction for atomic motion caused by sub-displacement electron impacts. For three-coordinated Si impurities in graphene irradiated by 60-keV electrons, these considerations result in a striking effect~\cite{Susi14PRL}. Although electrons of such energy are unlikely to cause outright damage, they can momentarily displace the C neighbor of Si. In the resulting dynamical out-of-plane process revealed by modeling~\cite{Susi14PRL} (Fig.~\ref{fig:mechanism}a), the Si relaxes into the vacancy created by the impact on C, which is pulled back into the lattice on the opposite side. This makes the switching of places of the Si and C---reminiscent of flipping a digital bit from one state to the other one---highly directional and enables control over it~\cite{Susi14PRL,Susi17UM}. In the case of bond rotations, the process is very similar~\cite{Kotakoski11PRB} (Fig.~\ref{fig:mechanism}b). For bonds in defective graphene structures~\cite{Kotakoski14NC}, rotations that lead to rare atomic configurations (such as carbon rings with four or fewer atoms) are less likely to occur than those involving only pentagons, hexagons and heptagons. Hence, it should be possible to control the atomic structure by placing the electron beam on top of the bond one wishes to rotate to maximize the electron dose on the two involved atoms. To our knowledge, control of bond rotations has not been attempted. However, we recently showed that by iteratively placing the electron probe on top of a selected carbon neighbor, it is indeed possible to "pull" a silicon impurity through the graphene lattice~\cite{Susi17UM}.
\section*{First principles modeling: pushing the limits}
Modeling atomic dynamics accurately over the picosecond timescales required to extract displacement thresholds is demanding. Most computationally affordable tight-binding (TB) models have failed to yield accurate thresholds for 2D materials with non-carbon atoms, necessitating the use of more expensive density functional theory (DFT) for dynamical simulations (e.g. N-doped graphene~\cite{Loponen06PRB,Susi12AN}, hexagonal boron nitride (hBN)~\cite{Zobelli07PRB,Kotakoski10PRB}). These studies have established DFT-based molecular dynamics (DFT/MD) as the most reliable way to theoretically estimate the displacement threshold energies.
In addition, at electron energies near the threshold, the vibration of nuclei in the direction of the beam become important in activating otherwise energetically prohibited processes~\cite{Meyer12PRL,Susi16NC}. By modeling the atomic motion via a quantum description of lattice vibrations, it is possible to estimate the cross section of the knock-on process. When the velocity distribution in the perpendicular direction was included in the model, the probability of displacements from $^{12}$C and $^{13}$C graphene could be directly predicted from theory~\cite{Susi16NC}. However, all of the tested exchange correlation functionals (including some with a description of the van der Waals interaction) seem to overestimate the displacement threshold, with the closest match within 0.3 eV (1.4\%) of the best-fit experimental threshold. Although this agreement is good enough that the DFT-derived displacement cross section values are within experimental uncertainties, this is achieved by selecting the functional providing the closest match rather than the physically best motivated theoretical model. The explicit phonon calculations required are also only feasible for small numbers of atoms, making it difficult to extend precise analyses to systems with impurities. For their displacement, discussed below, we thus merely correct the mass term in the mean square velocity with the impurity mass.
With these caveats in mind, in Table~\ref{tab:thresholds} we show our reanalysis of published data using the latest out-of-plane vibration model~\cite{Susi16NC}. For the case of N~\cite{Susi12AN} and Si~\cite{Susi14PRL}, we find discrepancies of 1.7 and 4.1 eV in the simulated thresholds for the ejection of C atoms neighboring the impurity. Using data collected on B during the experiments described in~\cite{Kepaptsoglou15AN}, we find discrepancies of 2.1 and 4.5 eV for B and C ejection, with the theoretical thresholds leading to negligible displacement rates. Calculating cross sections with a 50\% increased mean square velocity of vibrations would reduce the discrepancies, but even such a high correction does not bring the consistently underestimated theoretical cross section values into agreement with experiment.
Finally, we have occasionally observed the ejection of the N impurity itself upon intense spot irradiation at 60 keV, hinting at the possibility of dose-rate dependence at impurity sites that is not included in the modeling, or to the occurrence of rare chemical etching events. The most serious discrepancy with current theory, however, results from the observed rate of reversible jumps of a pyridinic N dopant across a vacancy~\cite{Lin15NL}. The thermal barrier for such a transformation is at least 4 eV~\cite{Arenal14NL}, leaving beam activation as the only plausible mechanism even in high temperature experiments (we have once also observed the same event at room temperature). The experimental rate calculated from seven consecutive jumps corresponds to a cross section of over 30 barn, whereas DFT/MD modeling fails to reproduce the event (the jump is only initiated for in-plane momentum transfers that are unphysically large for the experimental geometry), let alone quantify its probability.
The physical reasons behind such differences need to be understood and the theoretical treatments correspondingly improved to provide accurate predictions for manipulation experiments. Apart from neglecting vibrational impurity modes, the remaining inaccuracy of DFT/MD could be due to one or more of the presently necessary approximations: 1) the system size is limited due to the computational cost; 2) the simulation timestep is likewise limited, possibly leading to cumulative errors in the integrated equations of motion~\cite{Lippert07JCP}; and 3) the description of the dynamics may be inadequate either due to the approximation of exchange and correlation, or the lack of time-dependence of the electronic degrees of freedom.
Although the out-of-plane buckling of silicon~\cite{Ramasse13NL} (or phosphorus~\cite{Susi172DM}) provides a strong asymmetry for the atomic motion even for momentum transfers completely perpendicular to the plane, for impurities with planar bonding geometries such as N and B, in-plane components of the momentum either from the electron impact or from phonons (or contraction of bonds at the moment of the impact due to the vibrations) may be required to explain the observed dynamics. A coordinated experimental and theoretical effort should be able to bring light on these issues.
A related but different topic where first-principles or other type of quantitative modeling has not even been attempted is radiation damage by ionization or electronic excitations (also known as radiolysis)~\cite{Egerton10UM}. Although irradiation effects in graphene seem to be adequately described solely by the knock-on effect~\cite{Susi16NC}, as the materials zoo is extended---especially if soft matter and molecular structures are included---radiolysis becomes important. Ionization damage is already the most obvious hindrance for manipulation of non-graphene materials, and our understanding of radiation damage is inevitably incomplete without its quantitative understanding. Isotope-labeled hBN or transition metal dichalcogenides would make for excellent systems to isolate radiolysis from knock-on damage, especially when supplemented with experiments combining these materials with graphene to mitigate ionization~\cite{Algara-Siller13APL,lehnert_electron_2017}.
\section*{Outlook: paths forward}
To develop STEM into a practical manipulation technique, in addition to better modeling of beam-induced dynamics, improvements in sample preparation and the automation of manipulations are required. In our view, Si impurities in graphene provide the most promising system for initial experiments, as shown in Fig.~\ref{fig:automove}. In this case, the process of moving the Si atom in graphene was carried out by manually directing the beam. While this is how also the first successes of STM were achieved, it is impractical to fabricate complex nanostructures in such a manner. Fortunately, the modern computerized STEM is well-suited for automation, and the required software tools can be swiftly developed, along the lines discussed in~\cite{Kalinin16N,Susi17UM}.
Sample quality presents another obstacle for large-scale manipulation. Even a single atomic layer of contamination prevents such experiments, and finding clean areas, especially those that contain heteroatoms~\cite{Bangert13NL,Susi172DM}, is difficult. Even when such areas are found, contamination often gathers under the beam and obscures the surface~\cite{Meyer08APL}. Finally, a sufficient quantity of heteroatoms need to be introduced into the lattice in the first place, but without causing significant structural damage or contamination. Ion implantation is a particularly promising technique, but contamination remains a serious issue~\cite{Bangert13NL,Susi172DM}. Cleaning the samples outside the microscope vacuum~\cite{Lin12NL,Algara-Siller14APL} seems insufficient for ideal samples, since atmospheric exposure will be particularly detrimental to the chemically more reactive impurity sites. \textit{In situ} heating of samples appears more promising, either in a dedicated sample holder with a resistive heating element~\cite{Lin15NL}, via Joule heating~\cite{Lu11NL} or using a laser~\cite{Tripathi17PSSRRL}.
Current literature almost exclusively provides examples of reversible electron-beam driven dynamics in graphene, largely because of its availability and robustness as an electron microscopy specimen. However, there is no fundamental reason why other 2D materials~\cite{Hong17AM} could not host similar processes. If ionization damage can be mitigated, possibilities include 2D SiO$_2$~\cite{Bjorkman13SciRep}, silicene~\cite{Gao13Nanoscale} and phosphorene~\cite{Hu15JPCC}, which all share bond-rotation-type defects with graphene. Other candidates could be directed vacancy or impurity atom migration in transition metal dichalcogenides~\cite{Komsa12PRL}.
Finally, although two-dimensional crystals are ideal for manipulation due to their relative simplicity both experimentally and theoretically, the penetration of an electron beam through a sample offers the possibility of bulk manipulation. The methods and tools developed in 2D would largely translate to materials more generally, allowing us to eventually tackle the full complexities of three-dimensional crystals~\cite{Kalinin16N}.
\section*{Conclusion}
The challenges that we have discussed above are significant, but so is the current rate of progress. The rewards, however, are even greater: all properties of a material are determined by its chemical structure, whose precise control would allow these to be designed at will within the bounds given only by the laws of physics and material stability. Electron beam manipulation will in the near future allow the creation and tailoring of covalently bonded 2D nanostructures, with a further possibility of extending the technique to 3D. Initial experiments could for example target designed molecules embedded within graphene~\cite{Guo14NC}, plasmonic nanoantennas~\cite{Zhou12NN}, and novel quantum corrals~\cite{Susi15FWF}. Once established, atomically precise manipulation in the electron microscope will open a new playground for materials science and engineering at the ultimate limit of control.
\clearpage
|
1,108,101,565,692 | arxiv | \section{Introduction}
Active galactic nuclei (AGNs) are classified as type-1 or type-2, depending on the presence or absence of broad optical emission lines. The existence of different classes of AGNs can be explained by the unified model \citep{Antonucci1993}, which is based on the orientation of the optically thick torus with respect to our line-of-sight. Recently, a new sub-class of AGNs, known as changing look AGNs (CLAGNs), has been identified by optical observations. These objects display the appearence or disappearance of the broad optical emission lines, transitioning from type-1 (or type~1.2/1.5) to type-2 (or type~1.8/1.9) and vice versa. Several nearby galaxies, such as Mrk~590 \citep{Denney2014}, NGC~2617 \citep{Shappee2014}, Mrk~1018 \citep{Cohen1986}, NGC~7582 \citep{Aretxaga1999}, NGC~3065 \citep{Eracleous2001}, have been found to show such a peculiar behaviour. In the X-rays, a different type of changing-look events have been observed, with AGN switching between Compton-thin (line-of-sight column density, $N_{\rm H} < 1.5 \times 10^{24}$ cm$^{-2}$) and Compton-thick (CT; $N_{\rm H} > 1.5 \times 10^{24}$ cm$^{-2}$) states \citep{Risaliti2002,Matt2003}. These X-ray changing-look events have been observed in many AGNs, namely, NGC~1365 \citep{Risaliti2007}, NGC~4388 \citep{Elvis2004}, NGC~7582 \citep{Piconcelli2007,Bianchi2009}, NGC~4395 \citep{Nardini2011}, IC~751 \citep{Ricci2016}, NGC~4507 \citep{Braito2013}, NGC~6300 \citep{Guainazzi2002,AJ2020}.
The origin of the CL events is still unclear. The X-ray changing-look events could be explained by variability of the line-of-sight column density ($N_{\rm H}$) associated with the clumpiness of the BLR or of the circumnuclear molecular torus \citep{Nenkova2008a,Nenkova2008b,Elitzur2012,Yaqoob2015,Guainazzi2016,AJ2020}. On the other hand, optical CL events could be related to changes in the accretion rate \citep{Elitzur2014,MacLeod2016,Sheng2017}, which could be linked with the appearance and disappearance of the broad line regions (BLRs) \citep{Nicastro2000,Korista2004,Runnoe2016}. Some of these optical CL events could also be associated with the tidal disruption of a star by the supermassive black hole (SMBH) at the centre of the galaxy \citep{Eracleous1995,Merloni2015,Ricci2020,Ricci2021}.
NGC~1566 is a nearby (z=0.005), face-on spiral galaxy, classified as a type~SAB(s)bc \citep{devaucouleurs1973,Shobbrook1966}. The AGN was intensively studied over the last 70 years and is one of the first galaxies where variability was detected \citep{Quintana1975,Alloin1985,Alloin1986,Winkler1992}. In the 1960s, NGC~1566~ was classified as a Seyfert~1, with broad H$\alpha$ and H$\beta$ lines \citep{devaucouleurs1961}. Later, the H$\beta$ line was found to be weak, leading to the source being classified as a Seyfert~2 \citep{Pastoriza1970}. In the 1970s and 1980s, NGC~1566~ was observed to be in the low state with weak H$\beta$ emission \citep{Alloin1986}. Over the years, it was observed to change its type again from Seyfert~1.9-1.8 to Seyfert~1.5-1.2 \citep{dasilva2017}, with two optical outbursts in 1962 and 1992 \citep{Shobbrook1966,Pastoriza1970,dasilva2017,Oknyansky2019}.
{\it INTEGRAL} caught NGC~1566~ in the outburst state in hard X-ray band in June 2018 \citep{Ducci18}. Follow-up observations were carried out in the X-ray, optical, ultraviolet (UV), and infrared (IR) bands \citep{Grupe18a,Ferrigno18,Kuin18,Dai18,Cutri18}. The flux of the AGN was found to increase in all wavebands and reached its peak in July 2018 \citep{Oknyansky2019,Oknyansky2020,Parker2019}. Long-term ASAS-SN and NEOWISE light curves showed that the optical and IR flux started to increase from September 2017 \citep{Dai18,Cutri18}. The {\it Swift}/XRT flux increased by about $\sim 25-30$ times \citep{Oknyansky2019} as the source changed to Seyfert~1.2 from Seyfert~1.8-1.9 type \citep{Oknyansky2019,Oknyansky2020}. The source became a type-1, with the appearance of strong, broad emission lines \citep{Oknyansky2019,Ochmann2020}. After reaching their peak, the fluxes declined in all wavebands. After the main outburst, several small flares were observed \citep{Grupe18b,Grupe19}.
In this paper, we explore the timing and spectral properties of NGC~1566 during the 2018 outburst using data from the {\it XMM-Newton}~ and {\it NuSTAR}~ observatories, covering a broad energy range ($0.5-70$~keV). In Section~2, we present the observations and describe the procedure adopted for the data extraction. In Section~3 \& 4, we present results obtained from our timing and spectral analysis, respectively. In Section~5, we discuss our findings. We summarise our results in Section~6.
\begin{table*}
\caption{Log of observations of NGC~1566}
\label{tab:log}
\begin{tabular}{lcccccccccc}
\hline
ID & UT Date & {\it NuSTAR}~ & Exp (ks) & Count s$^{-1}$ & {\it XMM-Newton}~ & Exp (ks) & Count s$^{-1}$ & {\it Swift}/XRT & Exp (ks) & Count s$^{-1}$ \\
\hline
X1 & 2015-11-05 & -- & -- & -- & 0763500201 & 91.0 & $1.21\pm0.01$ & -- & -- & -- \\
O1 & 2018-06-26 & 80301601002 & 56.8 & $1.99\pm0.01$ & 0800840201 & 94.2 & $26.40\pm0.02$ & -- & -- & -- \\
O2 & 2018-10-04 & 80401601002 & 75.4 & $0.55\pm0.003$ & 0820530401 & 108.0& $4.92\pm0.01$ & -- & -- & -- \\
O3 & 2019-06-05 & 80502606002 & 57.2 & $0.19\pm0.002$ & 0840800401 & 94.0 & $3.78\pm0.01$ & -- & -- & -- \\
O4 & 2019-08-08 & 60501031002 & 58.9 & $0.60\pm0.003$ & -- & -- & -- & 00088910001 & 1.9 & $0.55\pm0.02$ \\
O5 & 2019-08-11 & -- & -- & -- & 0851980101 & 18.0 & $4.83\pm0.02$ & -- & -- & -- \\
O6 & 2019-08-18 & 60501031004 & 77.2 &$0.33\pm0.002$ & -- & -- & -- & 00088910002 & 1.7 & $0.27\pm0.01$ \\
O7 & 2019-08-21 & 60501031006 & 86.0 & $0.36\pm0.002$ & -- & -- & -- & 00088910003 & 2.0 & $0.21\pm0.01$ \\
\hline
\end{tabular}
\end{table*}
\label{sec:obs}
\section{Observation and Data Reduction}
In the present work, we used data from {\it NuSTAR}, {\it XMM-Newton}~ and {\it Swift}~ observations of NGC~1566, carried out at different epochs, as reported in Table~\ref{tab:log}. Out of the eight available epochs, X1 and O1 were studied by \citet{Parker2019}. We also included those observations for a complete study of the source at different phases (pre-outbursts, outburst and post-outbursts period).
\label{sec:nustar}
\subsection{{\it NuSTAR}}
{\it NuSTAR}~ is a hard X-ray focusing telescopes, consisting of two identical modules: FPMA and FPMB \citep{Harrison2013}. NGC~1566~ was observed by {\it NuSTAR}~ six times between 2018 June 26 and 2019 August 21, simultaneously with either {\it XMM-Newton}~ or {\it Swift}~ (see Table~\ref{tab:log}). Reprocessing of the raw data was performed with the {\it NuSTAR}~ Data Analysis Software ({\tt NuSTARDAS}, version 1.4.1). Cleaned event files were generated and calibrated by using the standard filtering criteria in the {\tt nupipeline} task and the latest calibration data files available in the NuSTAR calibration database (CALDB) \footnote{\url{http://heasarc.gsfc.nasa.gov/FTP/caldb/data/nustar/fpm/}}. The source and background products were extracted by considering circular regions with radius 60 arcsec and 90 arcsec, respectively. While the source region was centred at the coordinates of the optical counterparts, the background spectrum was extracted from a region devoid of other sources. The spectra and light curves were extracted using the {\tt nuproduct} task. The light curves were binned at 500~s. We re-binned the spectra with 20 counts per bin by using the {\tt grppha} task. No background flare was detected in the {\it NuSTAR}~ observations.
\label{sec:xmm}
\subsection{{\it XMM-Newton}}
NGC~1566 was observed by {\it XMM-Newton}~ \citep{Jansen2001} at five epochs between November 2015 and August 2019. Out of these five observations, the source was observed simultaneously with {\it NuSTAR}~ in three epochs (see Table~\ref{tab:lag}). We used the Science Analysis System ({\tt SAS v16.1.0}\footnote{\url{https://www.cosmos.esa.int/web/xmm-newton/sas-threads}}) to reprocess the raw data from EPIC-pn \citep{Struder2001}. We considered only unflagged events with {\tt PATTERN} $\leq 4$. Particle background flares were observed above 10 keV in X1 and O5. The Good Time Interval (GTI) file was generated considering only intervals with $<0.65$ counts$^{-1}$, using the SAS task {\tt tabgtigen}. No flares were observed in O1, O2 and O3. The source and background spectra were initially extracted from a circular region of 30" centered at the position of the optical counterpart, and from a circular region of 60" radius away from the source, respectively. Then we extracted the spectrum using the SAS task `{\tt especget}'. The observed and expected pattern distributions were generated by applying {\tt epaplot} task on the filtered events. We found that the source spectrum would be affected by photon pile-up and as a result, the expected distributions were significantly discrepant from the observed ones. We therefore considered an annular region of 30" outer radius and different values of inner radii for the source and checked for the presence of pile-up. We found that, with an inner radius of 10", the source would be pile-up free\footnote{\url{https://www.cosmos.esa.int/web/xmm-newton/sas-thread-epatplot}}. We therefore used this inner radius for the spectral extraction. The response files (\textit{arf} and \textit{rmf} files) were generated by using {\tt SAS} tasks {\tt ARFGEN} and {\tt RMFGEN}, respectively. Background-subtracted light curves were produced using the {\tt FTOOLS} task {\tt LCMATH}. It is also to be noted that we ran SAS task `{\tt correctforpileup}' when we generated the pile-up corrected rmf file by using `{\tt rmfgen}'.
\label{sec:swift}
\subsection{{\it Swift}}
{\it Swift}~ monitored NGC~1566~ over a decade in both window-timing (WT) and photon counting (PC) modes. The source was observed simultaneously with {\it Swift}/XRT and {\it NuSTAR}~ three times (see Table~\ref{tab:log} for details). The $0.3-10$~keV spectra were generated using the standard online tools provided by the UK {\it Swift} Science Data Centre \citep{Evans2009} \footnote{\url{http://www.swift.ac.uk/user_objects/}}. For the present study, we used both WT and PC mode spectra in the $0.3-10$~keV range. We also generated long-term light curves in the $0.3-10$~keV band using the online-tools\footnote{\url{http://www.swift.ac.uk/user_objects/}}.
\begin{figure*}
\includegraphics[width=18cm]{xrt-lc.eps}
\caption{The long-term, $0.3-10$~keV {\it Swift}/XRT light curve of NGC~1566~ is shown in the top panel. The shaded region shows the light curve between 2018 and 2020. The inset figure in the top panel shows the expanded light curve of the shaded region from 2018 to 2020 for clarity. The red arrows represent the {\it NuSTAR}, {\it XMM-Newton}~ and {\it Swift}/XRT observations of the source (see Table~\ref{tab:log} for details). In the bottom panel, we illustrate the variation of the hardness ratio (HR). The HR is defined as the ratio between count rates in the $1.5-10$~keV and the $0.3-1.5$~keV band.}
\label{fig:xrt-lc}
\end{figure*}
\label{sec:res}
\section{Results}
\begin{table*}
\caption{Variability in different energy bands are shown here. In some cases, the average error of observational data exceeds the 1$\sigma$ limit, resulting negative excess variance. In such cases, we obtained imaginary $F_{\rm var}$, which are not included in this table.}
\label{tab:var}
\begin{tabular}{lccccccccccccc}
\hline
ID & Energy & N & $F_{\rm max}$ & $F_{\rm min}$& R & Mean & $\sigma$ &$\sigma^2_{\rm XS}$&$\sigma^2_{\rm NXS}$ & $F_{\rm var}$ \\
& (keV) & &(Count $s^{-1}$)&(Count $s^{-1}$) & & &($10^{-3}$) & ($10^{-3}$) & ($10^{-3}$) & (\%) \\
(1) & (2) & (3) & (4) & (5) & (6) & (7) & (8) & (9) & (10) & (11)\\
\hline
X1& 0.5-3& 178& 1.04& 0.63 & 1.66& 0.82 &$ 71\pm2$& 2.74&$ 3.3\pm0.7 $&$ 5.8\pm0.6$\\
X1& 3-10 & 178& 0.19& 0.07 & 2.63& 0.12 &$ 19\pm1$& -- &$ - $&$ - $\\
\hline
O1& 0.5-3& 185&27.84& 16.69& 1.67& 21.89&$ 2573\pm78 $& 6.55&$ 13.7\pm0.2 $&$ 11.7\pm0.6$\\
O1$^*$& 3-10 & 145& 2.35& 1.31 & 1.80& 1.81 &$ 210 \pm11 $& 2.2 &$ 9.7\pm1.0 $&$ 9.8\pm0.8$\\
O1& 10-70& 145& 1.21& 0.54 & 2.23& 0.80 &$ 98 \pm 5 $& 4.94&$ 7.8\pm1.5 $&$ 8.8\pm1.0$\\
\hline
O2& 0.5-3& 213& 5.69& 2.98 & 1.91& 4.05 &$ 628 \pm15 $& 380.2&$ 93.9\pm0.6 $&$ 30.6\pm0.8$\\
O2$^*$& 3-10 & 197& 0.86& 0.29 & 2.99& 0.52 &$ 98\pm3 $& 6.50&$ 24.4\pm2.6 $&$ 15.6\pm1.2$\\
O2& 10-70& 197& 0.47& 0.12 & 3.78& 0.25 &$ 51\pm2$& 1.11&$ 17.2\pm0.4 $&$ 13.1\pm1.6$\\
\hline
O3& 0.5-3& 185& 3.74& 2.52 & 1.48& 3.13 &$ 289 \pm11 $& 72.2&$ 23.1\pm0.5 $&$ 15.2\pm0.4$\\
O3$^*$& 3-10 & 161& 0.48& 0.08 & 6.46& 0.22 &$ 48\pm2$& 0.16&$ 3.2\pm5.4 $&$ 5.7\pm4.8$\\
O3& 10-70& 161& 0.47& 0.04 & 12.5& 0.16 &$ 45\pm2$& 0.28&$ 11.2\pm0.9 $&$ 10.6\pm4.3$\\
\hline
O4& 3-10 & 145& 0.99& 0.36 & 2.78& 0.55 &$ 86\pm3$& 4.76&$ 15.4\pm2.2$&$ 12.4\pm1.1$\\
O4& 10-70& 145& 0.37& 0.15 & 2.42& 0.26 &$ 40\pm1$& 0.39&$ 5.9\pm2.8$&$ 7.7\pm1.8$\\
\hline
O5& 0.5-3& 27 & 3.71& 3.33 & 1.12& 3.52 &$ 109 \pm12 $& 0.46&$ 0.1\pm0.3$&$ 0.6\pm2.2$\\
O5& 3-10 & 27 & 0.55& 0.15 & 3.68& 0.39 &$ 116 \pm3 $& 10.78&$ 71.5\pm14.7$&$ 26.7\pm4.6$\\
\hline
O6& 3-10 & 191& 0.45& 0.16 & 2.77& 0.31 &$ 52\pm2$& 1.27&$ 13.1\pm2.6$&$ 11.4\pm1.3$\\
O6& 10-70& 191& 0.23& 0.08 & 2.83& 0.16 &$ 30 \pm1 $& 0.12&$ 4.9\pm3.8 $&$ 7.0\pm02.7$\\
\hline
O7& 3-10 & 208& 1.46& 0.21 & 6.85& 0.35 &$ 98\pm7$& 2.50&$ 20.3\pm7.4$&$ 14.3\pm2.7$\\
O7& 10-70& 208& 0.73& 0.09 & 8.22& 0.17 &$ 53\pm4$& --&$ - $&$ - $\\
\hline
\end{tabular}
\leftline{Columns in the table represent -- (1) ID of observation, (2) energy range, (3) number of data points or length of the light curve, (4) maximum count of }
\leftline{the light curve, (5) minimum count of the light curve, (6) ratio of maximum count to minimum count, $R=F_{\rm max}/F_{\rm min}$, (7) mean count of the light curve,}
\leftline{(8) standard deviation of the light curve, (9) excess variance, (10) normalized excess variance, (11) fractional rms amplitude of the light curve.}
\leftline{$^{*}$During O1, O2 and O3, we reported only {\it NuSTAR}~ observation in $3-10$~keV energy range, although both {\it XMM-Newton}~ and {\it NuSTAR}~ data}
\leftline{were available in $3-10$~keV energy band.}
\end{table*}
\begin{figure}
\includegraphics[width=8.5cm]{hid1.eps}
\caption{Hardness-intensity diagram (HID) for the first outburst F1 (from MJD 58308 to MJD 58380). The $0.3-10$~keV {\it Swift}/XRT count rate is plotted as a function of hardness ratio (HR). The arrow marks in the figure represent the evolution of the outburst.}
\label{fig:hid}
\end{figure}
\label{sec:timing}
\subsection{Timing Analysis}
We studied the long term {\it Swift}/XRT light curves of NGC~1566~ in the $0.3-10$~keV energy range for the timing analysis. Along with the {\it Swift}/XRT light curves, the $0.5-10$~keV and $3-70$ keV light curves (500~s time-bin) from {\it XMM-Newton}~ and {\it NuSTAR}~ observations were also analysed.
\label{sec:prof}
\subsubsection{Outburst Profile}
NGC~1566~was observed intensively with many observatories for about 70 years, starting from the 1950s. In this time period, NGC~1566~ showed two major outbursts in optical wavebands, in 1962 and 1992, along with several flaring episodes \citep{Alloin1986,dasilva2017}. Before the major X-ray outburst in 2018, a flaring event was observed in 2010 by the {\it Swift}/BAT survey\footnote{\url{https://swift.gsfc.nasa.gov/results/bs105mon/216}} \citep{Oknyansky2018,Oknyansky2019}. Since then, NGC~1566~ remained in the low state with a luminosity of $L \sim 10^{41}$ erg s$^{-1}$~ in 2--10 keV energy band.
NGC~1566~ was monitored by {\it Swift}/XRT over a decade, and it was caught in an outburst in June 2018, when the X-ray intensity increased by $\sim25-30$ times compared to the quiescent state \citep{Oknyansky2019}. During the 2018 X-ray outburst, the source also brightened in the optical, ultraviolet (UV) and infrared (IR) wavebands \citep{Dai18,Cutri18}. The optical and near-infrared (NIR) observations showed that the AGN started to brighten since 2017 September \citep{Dai18}. In the upper panel of Figure~\ref{fig:xrt-lc}, we show the long term $0.3-10$~keV {\it Swift}/XRT light curve. From this figure, it can be seen that the source experienced a major outburst in June 2018 (F1), which was followed by three smaller flaring events in December 2018 \citep[F2;][]{Grupe18b}, May 2019 \citep[F3;][]{Grupe19}, and May 2020 (F4). The smaller outbursts (F2, F3 and F4) were not as bright as the main outburst (F1). In the bottom panel of Figure~\ref{fig:xrt-lc}, we show the evolution of the hardness ratio (HR; i.e. the ratio between the $1.5-10$~keV and $0.3-1.5$~keV count rate) with time. Unlike the {\it Swift}/XRT light curve (top panel), the HR plot did not show any significant long-term variability. In Figure~\ref{fig:hid}, we show hardness-intensity diagram \citep{Homan2001,RM06} for the main outburst (F1), where the {\it Swift}/XRT count rate is plotted as a function of HR. The HID or `q'-diagram appeared to show a `q'-like shape, which is ubiquitous for outbursting Galactic black hole X-ray binaries. Interestingly, we did not observe any clear sign of `q'-shape HID for the next three recurrent outbursts.
\label{sec:variability}
\subsubsection{Variability}
We studied the source variability in different energy bands. As the soft excess in AGNs is generally observed below 3 keV, whereas the primary emission is observed above 3~keV, we analysed the {\it XMM-Newton}~ light curves in $0.5-3$~keV and $3-10$~keV energy ranges. We also studied the {\it NuSTAR}~ light curves in two separate bands ($3-10$~keV and $10-70$~keV).
We calculated the peak-to-peak ratio of the light curves, which is defined as $R=F_{\rm max}/F_{\rm min}$, where $F_{\rm max}$ and $F_{\rm min}$ are the maximum and minimum count rates, respectively. The light curves in different energy bands ($0.5-3$~keV, $3-10$~keV, and $10-70$~keV) showed different magnitude of variability. In all observations, $R$ was higher in the $3-10$~keV energy band than in the $0.5-3$~keV range. In the $10-70$~keV energy band, $R$ was higher than in the lower energy band, except for O4. The mean value of $R$ in $0.5-3$~keV, $3-10$~keV, and $10-70$~keV energy bands are $<R> = 1.34, 3.43$, and $5.33$, respectively. Thus, it is clear that the light curves showed higher variability in the higher energy bands in terms of R. However, this is too simplistic as very high or low count could be generated due to systematic/instrumental error. Hence, we calculated the normalized variance ($\sigma_{\rm NXS}^2$) and fractional variability ($F_{\rm var}$ to study the variability.
We calculated the fractional variability ($F_{\rm var}$) \citep{Edelson1996,Edelson2001,Edelson2012,Nandra1997,Vaughan2003} in the 0.5--3, 3--10 and 10--70 keV bands for a light curve of $x_i$ counts $s^{-1}$ with uncertainties $\sigma_i$ of length $N$, with a mean $\mu$ and standard deviation $\sigma$ is given by,
\begin{equation}
F_{\rm var} = \sqrt{\frac{\sigma^2_{\rm XS}}{\mu^2}},
\end{equation}
where, $\sigma^2_{\rm XS}$ is the excess variance \citep{Nandra1997,Edelson2002} which is given by,
\begin{equation}
\sigma^2_{\rm XS}=\sigma^2 - \sigma^2_{\rm err},
\end{equation}
where, $\sigma^{2}_{\rm err}$ is the mean squared error. The $\sigma^{2}_{\rm err}$ is given by,
\begin{equation}
\sigma^2_{\rm err} = \frac{1}{N}\sum_{i=1}^{N} \sigma^2_{\rm i}.
\end{equation}
The normalized excess variance is given by,
\begin{equation}
\sigma^2_{\rm NXS}=\frac{\sigma^2_{\rm XS}}{\mu^2}.
\end{equation}
The uncertainties in $F_{\rm var}$ and $\sigma_{\rm NXS}$ \citep{Vaughan2003} are given by,
\begin{equation}
{\rm err}(F_{\rm var})= \sqrt{(\sqrt{\frac{1}{2N}}\frac{\sigma_{\rm err}^2}{\mu^2 F_{\rm var}})^2+(\frac{1}{\mu}\sqrt{\frac{\sigma^2_{\rm err}}{N}})^2},
\end{equation}
and
\begin{equation}
{\rm err}(\sigma^2_{\rm NXS}) = \sqrt{(\sqrt{\frac{2}{N}}\frac{\sigma_{\rm err}^2}{\mu^2})^2+(\sqrt{\frac{\sigma_{\rm err}^2}{N}}\frac{2F_{\rm var}}{\mu})^2}.
\end{equation}
For a few observations, we could not estimate excess variance due to large errors in the data. Thus, from the normalized variance ($\sigma_{\rm NXS}^2$), the trend of variability was not clear. We calculated the fractional rms variability amplitude ($F_{\rm var}$) to study the variability. We obtained the highest fractional variability ($F_{\rm var}$) in the $0.5-3$~keV energy range for four observations (X1, O1, O2 \& O3), while the highest variability is observed in $3-10$~keV range for O5. The mean value of the fractional variability in $0.5-3$~keV, $3-10$~keV, and $10-70$~keV energy ranges were $<F_{\rm var}> = 12.8\pm 0.5\%, 12.0\pm 2.8\%$, and $7.9\pm 2.4 \%$, respectively. This indicates that the strongest variability is observed in the $0.5-3$~keV energy range. The variability parameters discussed here are reported in Table~\ref{tab:var}.
\label{sec:correlation}
\subsubsection{Correlation}
The soft excess is generally observed below 3~keV. To investigate the origin of the soft excess, we calculated the time delay between the soft X-ray photons ($0.5-3$~keV range) and the continuum photons ($3-10$~keV) using cross-correlation method from the {\it XMM-Newton}~observations.
We used the $\xi$-transformed discrete correlation function (ZDCF) method \citep{Alexander1997}\footnote{\url{http://www.weizmann.ac.il/particle/tal/research-activities/software}} to investigate the time-delay between the soft-excess and the X-ray continuum. The ZDCF co-efficient was calculated for two cases: omitting the zero lag points and including the zero lag points. In both cases, similar results were obtained. A strong correlation between the $0.5-3$ keV and $3-10$ keV energy bands was observed (amplitude $>0.7$) during observations O1, O2 and O3. However, no significant delay was observed during those observations. The values of ZDCF coefficient with time delay are presented in Table~\ref{tab:lag}.
\begin{table}
\caption{ZDCF results.}
\label{tab:lag}
\begin{tabular}{lcccc}
\hline
& soft excess& & & \\
ID & $\Delta t^{\dagger}$ & $\sigma^{\dagger}$ & amp$^{\dagger}$ \\
& (min) & (min) & \\
\hline
X1& $-10.5\pm13.0$ & $145.2 \pm 8.3$& $0.24\pm0.07$ \\
O1& $13.8 \pm 3.1$ & $103.3\pm 8.3$& $0.84 \pm 0.08$ \\
O2& $11.7 \pm 5.9$ & $222.9 \pm 8.3$& $0.90 \pm 0.07$ \\
O3& $13.9 \pm 7.8$ & $263.4\pm 8.3$& $0.69 \pm 0.07$ \\
O5& -- & -- & -- \\
\hline
\end{tabular}
\leftline{$^{\dagger}$ ZDCF correlation between primary X-ray continuum ($3-10$ energy band) }
\leftline{and soft excess ($0.5-3$~keV energy band). $\sigma$'s and amp's are FWHM and }
\leftline{amplitude of the ZDCF function. Note that the maximum amplitude can be 1.}
\end{table}
\begin{figure*}
\includegraphics[width=8.5cm]{spec-1.eps}
\includegraphics[width=8.5cm]{del-1.eps}
\caption{The left panel shows the unfolded spectra obtained from each observations. Triangles represent the {\it XMM-Newton}~ or {\it Swift}/XRT data, while circles represent the {\it NuSTAR}~ data. The light-green, brown, red, magenta, blue, orange, dark green and black points represent the observation X1, O1, O2, O3, O4, O5, O6 and O7, respectively. The residuals obtained after fitting the source spectra with Model--1 and Model--3 are shown in the right panels. The spectra are re-binned for visual purpose.}
\label{fig:spec1}
\end{figure*}
\label{sec:spec}
\subsection{Spectral Analysis}
We carry out the X-ray spectral analysis using data obtained from the {\it Swift}/XRT, {\it XMM-Newton}~ and {\it NuSTAR}~ observations of NGC~1566~ using {\tt XSPEC} v12.10 package \citep{Arnaud1996}\footnote{\url{https://heasarc.gsfc.nasa.gov/xanadu/xspec/}}. The spectral analysis was performed using simultaneous {\it XMM-Newton}~ and {\it NuSTAR}~ observations in the $0.5-70$~keV energy band for three epochs, simultaneous {\it Swift}/XRT and {\it NuSTAR}~ observations ($0.5-70$~keV) for three epochs, and {\it XMM-Newton}~ observations for two epochs ($0.5-10$~keV), between 2015 November 5 and 2019 August 21 (see Table~\ref{tab:log}).
For the spectral analysis, we used various phenomenological and physical models, namely, {\tt powerlaw} (PL)\footnote{\url{https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node213.html}}, {\tt NTHCOMP}\footnote{\url{https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node205.html}} \citep{Z96,Zycki1999}, {\tt OPTXAGNF}\footnote{\url{https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node206.html}} \citep{Done2012}, and {\tt RELXILL}\footnote{\url{www.sternwarte.uni-erlangen.de/~dauser/research/relxill/}} \citep{Garcia2014,Dauser2014} to understand the X-ray properties of NGC~1566. In general, an X-ray spectrum of an AGN consists of a power-law continuum, a reflection hump at around $15-40$~keV, a Fe K$\alpha$ fluorescent line, and a soft X-ray component below 2 keV \citep{Netzer2015,Padovani2017,Ricci2017}. The observed X-ray emission also suffers from absorption caused by the interstellar medium and the torus. In our analysis, we used two absorption components. For the Galactic interstellar absorption, we used {\tt TBabs}\footnote{\url{https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node265.html}}\citep{Wilms2000} with fixed hydrogen column density of $N_{\rm H} = 7.15 \times 10^{19}$ cm$^{-2}$ \citep{HI4PI2016}. In addition, we also used a ionized absorption model {\tt zxipcf}. For both the absorption components, we used the {\it wilms} abundances \citep{Wilms2000} and the Verner cross-section \citep{Verner1996}. In this work, we used the following cosmological parameters : $H_0= 70$ km s$^{-1}$ Mpc $^{-1}$, $\Lambda_0 = 70$, and $\sigma_M= 0.27$ \citep{Bennett2003}. The uncertainties in each spectral parameters are calculated using the `{\tt error}' command in {\tt XSPEC}, and reported at the 90\% confidence (1.6 $\sigma$).
\begin{figure*}
\includegraphics[angle=270,width=6cm]{nh-nh.eps} \hskip-0.3cm
\includegraphics[angle=270,width=6cm]{xi-cf-1.eps} \hskip-0.3cm
\includegraphics[angle=270,width=6cm]{xi-cf-2.eps}
\caption{Left panel shows the 2D contour between the column density of the low-ionizing absorber ($N_{\rm H,1}$) and high-ionizing absorber ($N_{\rm H,2}$) for O1. The middle and right panel show 2D contour between log $\xi$ and covering fraction for low and high-ionizing absorber, respectively.}
\label{fig:cntr}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=8.5cm]{fe.eps}
\caption{Evolution of Fe complex. The ratio of {\it XMM-Newton}~ and {\it NuSTAR}~ data to the power-law continuum model for every observations are shown. The red triangles and blue circles represent the ratio obtained from {\it XMM-Newton}~ and {\it NuSTAR}~ data, respectively. The vertical dashed line represent E=6.4~keV. The horizontal dashed lines represent the data/model=1 for each observation. The ratios are re-scaled by adding 0.5 in the y-axis separating the observations.}
\label{fig:fe}
\end{figure}
\subsubsection{Model 1: POWERLAW}
We built our baseline spectral model with {\tt power-law} continuum, along with the soft-excess, and Fe K-line emission. We used a blackbody component ({\tt bbody} in {\tt XSPEC}) for the soft-excess, and one or more Gaussian functions to incorporate the Fe~K complex. Out of the eight epochs, two Gaussian lines were required for six observations. Two ionized absorbers were also needed while fitting the data from three observations. The final model (hereafter Model--1) reads in {\tt XSPEC} as,
{\tt TB $\times$ zxipcf1 $\times$ zxipcf2 $\times$ ( zPL1+ zGa + zGa + bbody)}.
We started our analysis with the pre-outburst {\it XMM-Newton}~ observation X1 (in the $0.5-10$~keV energy range), $\sim$2.5 years prior to the 2018 outburst. Model--1 provided a good fit to the {\it XMM-Newton}~ data, with $N_{\rm H} = (3.53 \pm 0.06) \times 10^{21}$ cm$^{-2}$~ and power-law photon index of $\Gamma = 1.72\pm 0.05$, with $\chi^2 = 1073$ for 998 degrees of freedom (dof). Along with this, an iron K$\alpha$ emission line at $\sim$6.4~keV with an equivalent width (EW) of $\sim 206^{+4}_{-18}$ eV was detected.
Next, we analyzed simultaneous observations of NGC~1566~ with {\it XMM-Newton}~ and {\it NuSTAR}~ in the rising phase of the 2018 outburst (O1). First, we included one {\tt zxipcf} component in our spectral model. The fit returned with $\chi^2=2850$ for 2562 degrees of freedom (dof). However, negative residuals were clearly observed in soft X-rays (<1 keV). Thus, we included another {\tt zxipcf} component, and our fit improved significantly with $\Delta \chi^2=106$ for 3 dof. The spectral fitting in $0.5-70$~keV range returned $\Gamma = 1.85\pm0.04$, and $\chi^2 = 2744$ for 2559 dof. The Fe~K$\alpha$ line was detected at 6.38 keV with EW of $114\pm15$~eV, as well as another emission feature at 6.87 keV, with EW $<37$ eV. This line could be associated with Fe~XXVI line. We required two ionized absorber to fit the spectra, one low-ionization absorber ($\xi \sim 10^{1.7 \pm 0.1}$) with $N_{\rm H,1} = (8.1 \pm 2.2) \times 10^{20}$ cm$^{-2}$, and one high-ionization absorber ($\xi \sim 10^{4.7 \pm 0.4}$) with $N_{\rm H,2} = (4.3 \pm 0.4) \times 10^{21}$ cm$^{-2}$. The high-ionizing absorber required a high covering fraction ($CF>0.73$), while the low-ionizing absorber a moderate covering fraction with $CF \sim 0.46 \pm 0.04$.
The next observation (O2) was carried out on 2018 October 10, simultaneously with {\it XMM-Newton}~ and {\it NuSTAR}, covering the $0.5-70$~keV energy range. The source was in the decay phase of the outburst at the time of the observation. We started our fitting with one {\tt zxipcf} component. Although, the model provided a good fit to the data with $\chi^2=2298$ for 2130 dof, an absorption feature was seen in the residuals. Thus, we added a second {\tt zxipcf} component, and our fit returned with $\chi^2$ = 2224 for 2127 dof. The photon index decreased marginally compared to O1 ($\Gamma = 1.78\pm0.02$), while the column density increased slightly for both low and high-ionizing absorbers. The Fe~K$\alpha$ and Fe~XXVI lines were detected at 6.41 keV and 6.89 keV, with EWs of $126^{+3}_{-21}$~eV and $<49$~eV, respectively. The next simultaneous observations of NGC~1566~ with {\it XMM-Newton}~ and {\it NuSTAR}~ (O3) were carried out $\sim$8 months after the end of the outburst. The source was in a low state during the observation. Similar to O1 and O2, adding second absorption component improved the fit $\Delta \chi^2=67$ for 3 dof. The column density increased to $N_{\rm H,1} = (1.24 \pm 0.14) \times 10^{21}$ cm$^{-2}$~ for the low-ionization absorber, while the column density decreased to $N_{\rm H,2} = (8.9\pm0.2) \times 10^{20}$ cm$^{-2}$~ for the high-ionization absorber. The photon index was found to be $\Gamma = 1.68\pm 0.02$ in this observation. We detected both Fe~K$\alpha$ and Fe~k$\beta$ lines at 6.39~keV and 7.04 keV, with EWs of $117\pm 14$~eV and $<92$~eV, respectively.
The last four observations (O4, O5, O6 \& O7) were carried out in the span of 13 days. Observation O4 was made during the second small flare (F2), after the 2018 main outburst. During these four observations, the photon index was nearly constant ($\Gamma \sim 1.67\pm0.07-1.69\pm0.06$), while the column density of the low-ionizing absorber was found to vary in the range of $N_{\rm H,1} \sim 1.2-1.3 \times 10^{21}$ cm$^{-2}$. A high-ionization absorber was not required to fit the spectra of these four observations. We also observed a low covering fraction in these four observations (see Table~\ref{tab:mod1}). The Fe K$\alpha$ line was detected in all four observations, with EW $> 100$ eV (except O5). During our observations, the blackbody temperature ($T_{\rm bb}$) was observed to be remarkably constant with $T_{\rm bb} \sim 110$~eV. The parameters obtained by our spectral fitting results are listed in Table~\ref{tab:mod1}. Model--1 fitted spectra of NGC~1566~ are shown in the left panel of Fig.~\ref{fig:spec1}, whereas the corresponding residuals are shown in the right panels. To test for the presence of degeneracies between the column densities of two ionizing absorbers, we plotted the 2D contour in Fig.~\ref{fig:cntr}a for the observation O1. In Fig.~\ref{fig:cntr}b and Fig.~\ref{fig:cntr}c, we show 2D contours between log($\xi$) and covering fraction for the low and high-ionizing absorbers, respectively, for observation O1. In Fig~\ref{fig:fe}, we show the residuals above the continuum in $4-8$~keV energy range for Fe line emission. In Fig.~\ref{fig:uf-spec}a, we also show the unfolded spectrum fitted with Model--1 for observation O2.
\begin{figure*}
\centering
\includegraphics[width=18cm]{uf-spec.eps}
\caption{Best-fit unfolded spectra using Model--1, Model--3 and Model--4 are shown in the left, middle, and right panel, respectively, for O2. The corresponding residuals are shown in the bottom of each panel. Left panel: the black, yellow, magenta and red lines represent the total, primary emission, soft excess and Fe line emission, respectively. Middle panel: the black, yellow, and red lines represent the total, the AGN emission and Fe line emission, respectively. Right panel: the black, yellow, magenta and red lines represent the total, primary emission, soft excess and reflection component, respectively.}
\label{fig:uf-spec}
\end{figure*}
\begin{table*}
\caption{Best-fit parameters obtained from the spectral fitting of the source spectra with Model--1 ({\tt POWER-LAW}) \& Model--2 ({\tt NTHCOMP}).}
\label{tab:mod1}
\hspace*{-0.5in}
\begin{tabular}{lccccccccccccccc}
\hline
& X1 & O1 & O2 & O3 & O4 & O5 & O6 & O7\\
\hline
$N_{\rm H,1}$ ($10^{21}$ cm$^{-2}$) & $3.53^{+0.05}_{-0.06}$ & $0.81^{+0.13}_{-0.22}$ & $0.96^{+0.15}_{-0.18}$ & $1.24^{+0.14}_{-0.11}$& $1.33^{+0.14}_{-0.10}$ & $1.18^{+0.08}_{-0.09}$& $1.25^{+0.12}_{-0.16}$ & $1.30^{+0.17}_{-0.22}$ \\
\\
log${\rm \xi_{\rm 1}}$ & $-3^{\dagger}$ & $1.71^{+0.12}_{-0.11}$ & $1.81^{+0.10}_{-0.08}$ & $1.37^{+0.08}_{-0.09}$ & $1.10^{+0.04}_{-0.08}$& $0.26^{+0.09}_{-0.05}$& $0.17^{+0.14}_{-0.05}$& $0.21^{+0.03}_{-0.08}$ \\
\\
Cov Frac1 & $0.20^{+0.13}_{-0.05}$ & $0.46^{+0.04}_{-0.03}$ & $0.31^{+0.09}_{-0.04}$ & $0.33^{+0.07}_{-0.12}$ & $<0.12$ & $0.24^{+0.12}_{-0.15}$ & $0.17^{+0.04}_{-0.08}$& $<0.1$ \\
\\
$N_{\rm H,2}$ ($10^{21}$ cm$^{-2}$) & -- & $4.31^{+0.41}_{-0.26}$ & $4.56^{+0.58}_{-0.47}$ &$0.89^{+0.22}_{-0.17}$ & -- & -- & -- & --\\
\\
log${\rm \xi_{\rm 2}}$ & -- & $4.73^{+0.40}_{-0.02}$ & $3.56^{+0.26}_{-0.08}$ & $3.07^{+0.19}_{-0.07}$& -- & -- & --& -- \\
\\
Cov Frac2 & -- &$>0.73$ & $0.61^{+0.26}_{-0.21}$ & $0.54^{+0.23}_{-0.39}$ & -- & --&-- & -- \\
\\
$\Gamma$&$ 1.72^{+0.05}_{-0.05}$ &$ 1.85^{+0.04}_{-0.04}$ &$ 1.78^{+0.02}_{-0.02}$ &$ 1.68^{+0.02}_{-0.02}$ &$ 1.67^{+0.05}_{-0.07}$ &$ 1.68^{+0.03}_{-0.04}$ &$ 1.69^{+0.04}_{-0.06}$ &$ 1.67^{+0.05}_{-0.07}$ \\
\\
PL Norm ($10^{-3}$)&$ 1.59^{+0.16}_{-0.18}$ &$14.6^{+1.43}_{-2.05}$ &$ 5.97^{+0.06}_{-0.10}$ &$ 2.45^{+0.05}_{-0.10}$ &$ 4.71^{+0.07}_{-0.08}$ &$ 2.84^{+0.13}_{-0.09}$ &$ 2.78^{+0.11}_{-0.15}$ &$ 2.58^{+0.10}_{-0.17}$ \\
\\
Fe K$\alpha$ ~~~~~~LE (keV)&$ 6.44^{+0.03}_{-0.04}$ &$ 6.38^{+0.04}_{-0.04}$ &$ 6.41^{+0.03}_{-0.06}$ &$ 6.39^{+0.07}_{-0.10}$ &$ 6.29^{+0.08}_{-0.14}$ &$ 6.28^{+0.07}_{-0.11}$ &$ 6.29^{+0.05}_{-0.09}$ &$ 6.19^{+0.05}_{-0.10}$ \\
\\
~~~~~~~~~~~~~~~~~~EW (eV)&$ 206^{+4}_{-18}$ &$ 114^{+11}_{-15}$ &$ 126^{+3}_{-21}$ &$117^{+14}_{-10 }$ &$ 106^{+1}_{-6}$ &$ <95$ &$ 155^{+12 }_{-16 }$ &$ 108^{+15 }_{-22}$ \\
\\
~~~~~~~~~~~~~~~~~~FWHM (km s$^{-1}$)&$ 2329^*$ &$ 8695^{+924 }_{-1243 }$ &$ 2108^*$ &$ 4796^{+863 }_{-982 }$ &$ 2337^*$ &$ 4461^{+1045}_{-845 }$ &$ 6025^{+946 }_{-1194 }$ &$ 6613^{+1275 }_{-1223 }$ \\
\\
~~~~~~~~~~~~~~~~~~Norm ($10^{-5}$)&$ 6.83^{+0.15}_{-0.22}$ &$11.74^{+1.10}_{-1.22}$ &$ 3.47^{+0.14}_{-0.10}$ &$ 2.56^{+0.18}_{-0.13}$ &$ 1.91^{+0.07}_{-0.09}$ &$ 0.93^{+0.04}_{-0.10}$ &$ 2.14^{+0.10}_{-0.13}$ &$ 2.43^{+0.10}_{-0.08}$ \\
\\
Fe XXVI ~~LE (keV)&$ -$ &$ 6.87^{+0.04}_{-0.04}$ &$ 6.89^{+0.04}_{-0.03}$ &$ 7.04^{+0.05}_{-0.04}$ &$ 6.93^{+0.04}_{-0.03}$ &$ - $ &-- &-- \\
\\
~~~~~~~~~~~~~~~~~~EW (eV)&$ - $ &$ <37$ &$ <49$ &$ <92$ &$ <42$ &$- $ & -- &$ -$ \\
\\
~~~~~~~~~~~~~~~~~~FWHM (km s$^{-1}$) &$ - $ &$ <4452$ &$ <4049$ &$ <4943$ &$ <3913$ &$ - $ & -- & -- \\
\\
~~~~~~~~~~~~~~~~~~Norm ($10^{-6}$)&$- $ &$21.09^{+1.02}_{-0.94}$ &$ 7.88^{+0.13}_{-0.22}$ &$ 8.95^{+1.06}_{-0.94}$ &$ 7.30^{+0.31}_{-0.73}$ &$ - $ & -- & -- \\
\\
$kT_{\rm bb1}$ (eV) & $116^{+7}_{-8}$ & $112^{+6}_{-8}$ &$117^{+11}_{-7}$ & $115^{+5}_{-9}$& $108^{+8}_{-12}$& $114^{+9}_{-6}$ & $117^{+12}_{-14}$& $122^{+17}_{-10}$ \\
\\
$N_{\rm bb1}$ ($10^{-5}$) &$0.49^{+0.03}_{-0.06}$ & $28.6^{+2.29}_{-3.42}$&$8.30^{+1.09}_{-0.65}$ & $4.35^{+0.76}_{-0.96}$& $5.33^{+0.51}_{-1.04}$& $0.98^{+0.24}_{-0.19}$& $4.71^{+0.72}_{-1.13}$& $2.65^{+0.80}_{-0.56}$\\
\\
$\chi^2$/dof&1073/998 &2744/2559 &2224/2127 &1608/1646 & 693/654 & 835/834 & 535/542 & 576/567 \\
\hline
$N_{\rm H,1}$ ($10^{21}$ cm$^{-2}$) & $3.46^{+0.06}_{-0.05}$ & $0.78^{+0.10}_{-0.21}$ & $0.96^{+0.17}_{-0.16}$ & $1.22^{+0.13}_{-0.15}$& $1.32^{+0.12}_{-0.16}$ & $1.19^{+0.10}_{-0.09}$& $1.26^{+0.12}_{-0.18}$ & $1.27^{+0.15}_{-0.19}$ \\
\\
log${\rm \xi_{\rm 1}}$ & $-3^{\dagger}$ & $1.66^{+0.10}_{-0.08}$ & $1.85^{+0.06}_{-0.11}$ & $1.32^{+0.07}_{-0.10}$ & $1.05^{+0.04}_{-0.06}$& $0.26^{+0.05}_{-0.07}$& $0.15^{+0.12}_{-0.07}$& $0.21^{+0.04}_{-0.05}$ \\
\\
Cov Frac1 & $0.20^{+0.07}_{-0.05}$ & $0.46^{+0.03}_{-0.03}$ & $0.32^{+0.07}_{-0.04}$ & $0.34^{+0.05}_{-0.14}$ & $<0.12$ & $0.22^{+0.11}_{-0.16}$ & $<0.15$& $<0.1$ \\
\\
$N_{\rm H,2}$ ($10^{21}$ cm$^{-2}$) & -- & $4.25^{+0.32}_{-0.23}$ & $4.50^{+0.64}_{-0.43}$ &$0.82^{+0.27}_{-0.19}$ & -- & -- & -- & --\\
\\
log${\rm \xi_{\rm 2}}$ & -- & $4.63^{+0.45}_{-0.09}$ & $3.32^{+0.33}_{-0.10}$ & $3.11^{+0.23}_{-0.16}$& -- & -- & --& -- \\
\\
Cov Frac2 & -- &$>0.75$ & $0.59^{+0.22}_{-0.25}$ & $0.56^{+0.23}_{-0.45}$ & -- & --&-- & -- \\
\\
$\Gamma$&$1.74 ^{+0.05}_{-0.07}$ &$ 1.88^{+0.04}_{-0.06}$ &$ 1.75^{+0.03}_{-0.03}$ &$ 1.72^{+0.02}_{-0.03}$ &$ 1.71^{+0.03}_{-0.05}$ &$ 1.69^{+0.04}_{-0.04}$ &$ 1.68^{+0.05}_{-0.05}$ &$ 1.69^{+0.04}_{-0.05}$ \\
\\
$kT_{\rm e}$ (keV) &$101.8^{+4.9}_{-4.3}$ &$60.8^{+5.5}_{-6.7}$ &$85.7^{+7.4}_{-5.9}$ &$92.2^{+8.4}_{-9.4}$ &$75.2^{+8.3}_{-7.6}$ &$104.2^{+5.6}_{-6.9}$ &$105.8^{+6.9}_{-5.5}$ &$96.7^{+7.5}_{-8.2}$ \\
\\
$\tau$ & $1.27^{+0.17}_{-0.11}$ &$1.60^{+0.33}_{-0.18}$ &$1.43^{+0.16}_{-0.15}$ &$1.41^{+0.17}_{-0.12}$ &$1.66^{+0.24}_{-0.17}$ &$1.30^{+0.14}_{-0.12}$ &$1.34^{+0.15}_{-0.13}$ &$1.41^{+0.19}_{-0.13}$ \\
\hline
\end{tabular}
\leftline{$\xi$'s are in the unit ergs cm s$^{-1}$. PL norm is in unit of ph cm$^{-2}$ s$^{-1}$. Fe line norm are in the unit of ph cm$^{-2}$ s$^{-1}$.}
\leftline{$\tau$'s are calculated using Eqn. 1.}
\leftline{$\dagger$ pegged at the lowest value.}
\leftline{$^*$ Gaussian fitted with fixed line width, $\sigma=0.05$~keV.}
\end{table*}
\subsubsection{Model 2: NTHCOMP}
While fitting the source spectra with Model--1 provided us with information on the spectral shape and hydrogen column density, it provided limited information on the physical properties of the Comptonizing plasma. The Comptonizing plasma can be characterized by the electron temperature ($kT_{\rm e}$) and optical depth ($\tau$). In order to understand the properties of the Compton cloud, we replaced the {\tt powerlaw} continuum model with {\tt NTHCOMP} \citep{Z96,Zycki1999} in Model--1. The {\tt NTHCOMP} model provided us the photon index ($\Gamma$) and the hot electron temperature of the Compton cloud ($kT_{\rm e}$). The optical depth can be easily calculated from the information on $\Gamma$ and $kT_{\rm e}$ using the following equation \citep{ST80,Z96},
\begin{equation}
\tau \sim \sqrt{\frac{9}{4}+\frac{m_e c^2}{kT_{\rm e}}\frac{3}{(\Gamma-1)(\Gamma+2)}} -\frac{3}{2}.
\label{eqn:tau}
\end{equation}
This model (hereafter Model--2) reads in {\tt XSPEC} as,
{\tt TB $\times$ zxipcf1 $\times$ zxipcf2 $\times$ (NTHCOMP + zGa + zGa + bbody)}.
We fixed the seed photon temperature at $kT_{\rm s} = 30$~eV, which is reasonable for a BH of mass $\sim 8.3 \times 10^6$ $M_{\odot}$ \citep{SS1973,Makishima2000}. We required two absorption component during O1, O2 and O3. Inclusion of second absorption improved the fit significantly with $\Delta \chi^2=$~108, 84, 78 for 3 dof, during O1, O2 and O3, respectively. The photon indices and column densities obtained are similar to those obtained using Model--1. We found that $kT_{\rm e} = 102 \pm 5$~keV for observation X1. During the rising phase of the outburst (observation~O1), the Compton cloud was found to be relatively cooler, with $kT_{\rm e} = 61\pm 7$~keV. The Compton cloud was hot during the later observations, with the electron temperature increasing up to $106\pm 7$~keV within the eight observations analyzed here. A nearly constant photon index and a variable temperature would imply a variation in the density of the Comptonizing cloud. This would suggest that the optical depth was varying during the observations in the range of $\sim 1.27-1.66$. The results obtained from our spectral fitting with this model are listed in Table~\ref{tab:mod1}.
\begin{figure}
\centering
\includegraphics[angle=270,width=8.5cm]{rcor-astar.eps}
\caption{2D contour plot between $R_{\rm cor}$ and $a^*$ for Model--3 during the observation O2.}
\label{fig:rcor-a}
\end{figure}
\subsubsection{Model 3: OPTXAGNF}
The X-ray spectra of AGN typically show a soft excess below 2 keV \citep{Arnaud1985,Singh1985}. Although the soft excess in AGNs was first detected in the 1980s, its origin is still very debated. In models 1 \& 2, we fitted the soft excess component with a phenomenological {\tt blackbody} model. To shed light on the origin of the soft excess in this source, we used a more physical model, {\tt OPTXAGNF} \citep{Done2012}.
The {\tt OPTXAGNF} model (hereafter Model--3) computes the soft excess and the primary emission self-consistently. In this model, total emission is determined by the mass accretion rate and by the BH mass. The disc emission emerges as a colour temperature corrected blackbody emission at radii $R_{\rm out} > r > R_{\rm cor}$, where $R_{\rm out}$ and $R_{\rm cor}$ are the outer edge of the disc and the corona, respectively. At $r<R_{\rm cor}$, the disc emission emerges as the Comptonized emission from a warm and optically-thick medium, rather than thermal emission. The hot and optically-thin corona is located around the disc and produces the high energy power-law continuum. The total Comptonized emission is divided between the cold and hot corona, and the fraction of the hot-Comptonized emission ($f_{\rm PL}$) can be found from the model fitting. The temperature of the cold corona ($kT_{\rm S}$), temperature of the seed photon, and the optical depth of the cold corona ($\tau$) at $r=R_{\rm cor}$ determine the energy of the up-scattered soft excess emission. The power-law continuum is approximated as the {\tt NTHCOMP} model, with the seed photon temperature fixed at the disc temperature at $r=R_{\rm cor}$, and the electron temperature fixed at 100 keV.
When using the {\tt OPTXAGNF} model, we included two Gaussian components to incorporate the Fe emission lines. The model reads in {\tt XSPEC} as,
{\tt TB $\times$ zxipcf1 $\times$ zxipcf2 $\times$ ( OPTXAGNF + zGa + zGa)}.
While fitting the data with this model, we kept the BH mass frozen at $M_{\rm BH} = 8.3 \times 10^6$ $M_{\odot}$~ \citep{Woo2002}. As recommended, we fixed normalization to unity during our analysis. Initially, we started our analysis with one absorption component. However, an absorption feature was seen during O1, O2 and O3. Thus, we added second absorption component in the spectra of O1, O2 and O3. Adding second {\tt zxipcf} component significantly improved the fit with $\Delta \chi^2 =$~88, 76, 65 for 3 dof, during the observation O1, O2 and O3, respectively. Overall, this model provided a good fit for all the observations. A clear variation in the Eddington ratio and size of the corona were observed in the different observations. In the rising phase of the 2018 outburst (observation O1), a high Eddington ratio ($L/L_{\rm Edd} \sim 0.23$) and a large corona ($R_{\rm cor} = 43\pm 3~R_{\rm g}$) were observed. These values were found to be higher than the pre-outburst observation (X1; $L/L_{\rm Edd} \sim 0.04$; $R_{\rm cor} = 12^{+2}_{-1}$ $R_{\rm g}$). In later observations, both Eddington ratio and size of the X-ray corona decreased to $L/L_{\rm Edd} \sim 0.06-0.07$ and $R_{\rm cor} \sim 15\pm 2-26\pm 2$ $R_g$, respectively. In the observation~O1, the electron temperature of the optically thick Comptonizing region was observed to be $kT_{\rm S} \sim 1.4 \pm 0.1$~keV, along with optical depth $\tau \sim 4.6 \pm 1$. The temperature of the optically-thick medium decreased to $kT_{\rm S} \sim 0.5-0.6$~keV in the later observations (O3 -- O7). However, the optical depth did not change much and varied in the range of $\tau \sim 4-5$. A significant fraction of the Comptonized emission was emitted from the optically thin corona with $f_{\rm PL} > 0.79$ during all the observations. We allowed the spin of the BH to vary. The best-fitted spin parameter fluctuated in the range of $a^* \sim 0.18^{+0.01}_{-0.02} - 0.21^{+0.03}_{-0.02}$, suggesting that the SMBH is spinning slowly. The variation of the column density ($N_{\rm H}$) and of the photon indices ($\Gamma$) was similar to that observed using Model--1. All the parameters obtained from our spectral analysis using Model--3 are presented in Table~\ref{tab:opt}. In Fig.~\ref{fig:uf-spec}b, we show the unfolded spectrum fitted with Model--3 for observation O2. In Fig.~\ref{fig:rcor-a}, we display the contour plot of $R_{\rm cor}$ and $a^*$, which shows that there is no strong degeneracy between those two parameters.
\begin{table*}
\caption{Best-fit parameters obtained from the spectral fitting of the source spectra with Model--3 ({\tt OPTXAGNF}).}
\label{tab:opt}
\begin{tabular}{lccccccccccccc}
\hline
&X1 &O1 &O2 & O3 & O4 &O5 & O6 &O7 \\
\hline
$N_{\rm H,1}$ ($10^{21}$ cm$^{-2}$) & $3.49^{+0.05}_{-0.05}$ & $0.80^{+0.08}_{-0.16}$ &$0.94^{+0.15}_{-0.15}$ &$1.20^{+0.15}_{-0.11}$ &$1.34^{+0.10}_{-0.12}$ &$1.17^{+0.08}_{-0.11}$ &$1.25^{+0.08}_{-0.14}$ &$1.29^{+0.15}_{-0.22}$\\
\\
log${\rm \xi_{\rm 1}}$ &$-3^{\dagger}$ & $1.67^{+0.07}_{-0.06}$ &$1.82^{+0.03}_{-0.06}$ &$1.30^{+0.07}_{-0.09}$ &$1.08^{+0.07}_{-0.10}$ &$0.23^{+0.07}_{-0.10}$ &$0.18^{+0.06}_{-0.04}$ &$0.20^{+0.03}_{-0.04}$\\
\\
Cov Frac2 &$0.18^{+0.08}_{-0.06}$ & $0.45^{+0.03}_{-0.04}$ &$0.34^{+0.09}_{-0.06}$ &$0.35^{+0.05}_{-0.11}$ &$<0.12$ &$0.21^{+0.12}_{-0.17}$ &$<0.14$& $<0.15$ \\
\\
$N_{\rm H,2}$ ($10^{21}$ cm$^{-2}$) & -- & $4.28^{+0.45}_{-0.38}$ &$4.56^{+0.51}_{-0.46}$ &$0.89^{+0.21}_{-0.16}$ & -- & -- & -- & --\\
\\
log${\rm \xi_{\rm 2}}$ &-- & $4.52^{+0.39}_{-0.22}$ &$3.15^{+0.21}_{-0.18}$ &$2.93^{+0.32}_{-0.19}$ & -- & -- & --& -- \\
\\
Cov Frac2 &-- &$>0.79$ &$0.54^{+0.22}_{-0.17}$ &$0.51^{+0.28}_{-0.21}$ & -- & --&-- & -- \\
\\
$L/L_{\rm Edd}$ &$-1.98^{+0.03}_{-0.04}$ &$-0.64^{+0.02}_{-0.02}$ &$-0.78^{+0.03}_{-0.03}$&$-1.16^{+0.03}_{-0.04}$ &$-0.81^{+0.02}_{-0.03}$ &$-0.97^{+0.04}_{-0.05}$ & $-1.20^{+0.03}_{-0.03}$ &$-1.17^{+0.02}_{-0.01}$\\
\\
$a^*$& $0.19^{+0.01}_{-0.01}$ &$0.18^{+0.01}_{-0.02}$ &$0.18^{+0.01}_{-0.01}$ &$0.21^{+0.01}_{-0.02}$ &$0.20^{+0.02}_{-0.03}$ &$0.19^{+0.03}_{-0.02}$ & $0.21^{+0.03}_{-0.02}$ &$0.19^{+0.01}_{-0.01}$\\
\\
$R_{\rm cor}$ ($R_{\rm g}$) &$12^{+2}_{-1}$ &$43^{+3}_{-2}$ &$26^{+2}_{-2}$ &$15^{+1}_{-2}$ &$22^{+2}_{-2}$ &$20^{+2}_{-3}$ & $17^{+2}_{-3}$ &$18^{+1}_{-1}$\\
\\
$kT_{\rm S}$ (keV) &$0.32^{+0.04}_{-0.05}$ &$1.39^{+0.10}_{-0.15}$ &$0.90^{+0.07}_{-0.11}$ &$0.65^{+0.05}_{-0.10}$ &$0.61^{+0.08}_{-0.06}$ &$0.56^{+0.07}_{-0.08}$ & $0.59^{+0.07}_{-0.03}$ &$0.54^{+0.04}_{-0.03}$\\
\\
$\tau$ & $4.16^{+0.28}_{-0.21}$ &$4.56^{+0.12}_{-0.14}$ &$4.30^{+0.27}_{-0.20}$ &$4.33^{+0.18}_{-0.21}$ &$5.23^{+0.30}_{-0.19}$ &$4.92^{+0.34}_{-0.39}$ & $4.29^{+0.39}_{-0.45}$ &$3.54^{+0.34}_{-0.46}$\\
\\
$\Gamma$ &$1.77^{+0.03}_{-0.03}$ &$1.74^{+0.02}_{-0.02}$ &$1.66^{+0.04}_{-0.05}$ &$1.73^{+0.02}_{-0.04}$ &$1.72^{+0.02}_{-0.04}$ &$1.70^{+0.03}_{-0.02}$ & $1.72^{+0.02}_{-0.02}$ &$1.66^{+0.03}_{-0.03}$\\
\\
$f_{\rm PL}$ &$0.84^{+0.01}_{-0.02}$ &$0.79^{+0.01}_{-0.02}$ &$0.81^{+0.03}_{-0.02}$ &$0.84^{+0.03}_{-0.03}$ &$ 0.88^{+0.03}_{-0.03}$ &$0.83^{+0.03}_{-0.04}$ & $0.88^{+0.03}_{-0.04}$ &$0.94^{+0.03}_{-0.03}$\\
\\
$\chi^2$/dof &1070/995 &2784/2551 & 2311/2120 & 1677/1643& 706/649 & 827/831 & 526/535 & 571/560 \\
\hline
\end{tabular}
\leftline{$\xi$'s are in the unit ergs cm s$^{-1}$. $\dagger$ pegged at the lowest value.}
\end{table*}
\subsubsection{Model--4 : RELXILL}
Reprocessed X-ray radiation is a feature often observed in AGN spectra. This reflection component typically consists of a reflection hump at $\sim 15-40$~keV and fluorescent iron lines. In Model 1 -- 3 we did not include a reprocessed X-ray radiation, hence, we add a relativistic reflection component {\tt RELXILL} \citep{Garcia2014,Dauser2014,Dauser2016} to our baseline model.
In this model, the strength of reflection is measured from the relative reflection fraction ($R_{\rm refl}$), which is defined as the ratio between the observed Comptonized emission and the radiation reprocessed by the disc. {\tt RELXILL} assumes a broken power-law emission profile ($E(r) \approx r^{-q}$), where $r$ is the distance from the SMBH, $E(r)$ is the emissivity, and $q$ is the emissivity index. At a larger disc radii, in a non-relativistic domain, the emissivity profile has a form of $E(r) \sim r^{-3}$. However, in the relativistic domain, the emissivity profile is steeper. The break radius ($R_{\rm br}$) separates the relativistic and non-relativistic domains. In our analysis, we fixed $q_2=3$ for emission at $r>R_{\rm br}$.
In our spectral analysis, we used {\tt RELXILL} along with the absorbed {\tt power-law} continuum. We only considered the reflection component from {\tt RELXILL} model by setting $R_{\rm refl}$ to a negative value. The model (hereafter Model--4) read in {\tt XSPEC} as,
{\tt TB $\times$ zxipcf1 $\times$ zxipcf2 $\times$ (zPL1 + RELXILL + bbody)}.
While fitting the data with this model, we tied the photon indices of the {\tt RELXILL} component to that of the {\tt power-law} model. Although, we started our analysis with one absorption component, we required two absorption component during O1, O2 and O3. The second absorption component improved the fit with $\Delta \chi^2 =$~82, 68 and 59 for 3 dof, during O1, O2 and O3 respectively. Throughout the observations, we obtained a fairly unchanged value of the iron abundances with $A_{\rm Fe} \sim 3.7-4.2$ $A_{\sun}$. Disc ionization was also constant during our observation period with $\xi \sim 10^{1.9-2.2}$ erg~cm~s$^{-1}$. The emissivity profile was quite stable with $q_2 \sim 4-5$, although $R_{\rm br}$ was found to change. This parameter reached its maximum during the observation~O1 ($R_{\rm br} = 42^{+3}_{-2}$ $R_g$). In the later observations, it decreased and varied in the range of $R_{\rm br} \sim 16-26$ $R_g$. The inner edge of the disc varied in the range of $R_{\rm in} \sim 4-7$ $R_g$. The best-fit inclination angle of the AGN was obtained in the range of $10.3^{+4.9}_{-6.1}$\textdegree -- $17.7^{+2.9}_{-5.2}$\textdegree. The spin of the BH in NGC~1566~ was observed to be low, with the best-fitted spin parameters found to be in the range of $0.15^{+0.03}_{-0.03} - 0.21^{+0.04}_{-0.02}$, which is consistent with what we found from the {\tt OPTXAGNF} model. In all the observations, the reflection component was found to be relatively weak with reflection fraction varied in the range of, $R_{\rm refl} \sim 0.10-0.18$. The {\tt RELXILL} model fitted spectral analysis results are given in Table~\ref{tab:rel}. We show the unfolded spectrum fitted with Model--4 for observation O2 in Figure~\ref{fig:uf-spec}. In Fig.~\ref{fig:rin-a}, we show the contour plot of $R_{\rm in}$ and $a^*$ for the observation O2.
\begin{figure}
\centering
\includegraphics[angle=270,width=8.5cm]{rel-a-rin.eps}
\caption{2D contour plot between $R_{\rm in}$ and $a^*$ for Model--4 during the observation O2.}
\label{fig:rin-a}
\end{figure}
\begin{table*}
\caption{Best-fit parameters obtained from the spectral fitting of the source spectra with Model--4 ({\tt RELXILL}).}
\label{tab:rel}
\hspace*{-0.5in}
\begin{tabular}{lccccccccc}
\hline
&X1 &O1 &O2 & O3 &O4 &O5 &O6 &O7 \\
\hline
$N_{\rm H,1}$ ($10^{21}$ cm$^{-2}$) & $3.51^{+0.03}_{-0.04}$ & $0.77^{+0.06}_{-0.11}$ &$0.95^{+0.11}_{-0.14}$ &$1.18^{+0.12}_{-0.10}$ &$1.28^{+0.08}_{-0.10}$ &$1.16^{+0.07}_{-0.06}$ &$1.28^{+0.10}_{-0.08}$ &$1.31^{+0.12}_{-0.18}$\\
\\
log${\rm \xi_{\rm 1}}$ &$-3^{\dagger}$ & $1.63^{+0.05}_{-0.04}$ &$1.85^{+0.05}_{-0.04}$ &$1.31^{+0.07}_{-0.08}$ &$1.05^{+0.05}_{-0.08}$ &$0.21^{+0.05}_{-0.06}$ &$0.19^{+0.06}_{-0.03}$ &$0.24^{+0.04}_{-0.03}$\\
\\
Cov Frac2 &$0.17^{+0.05}_{-0.06}$ & $0.44^{+0.04}_{-0.06}$ &$0.36^{+0.05}_{-0.08}$ &$0.33^{+0.06}_{-0.12}$ &$<0.14$ &$<0.18$ &$<0.16$& $<0.13$ \\
\\
$N_{\rm H,2}$ ($10^{21}$ cm$^{-2}$) & -- & $4.36^{+0.34}_{-0.47}$ &$4.58^{+0.37}_{-0.42}$ &$0.95^{+0.19}_{-0.26}$ & -- & -- & -- & --\\
\\
log${\rm \xi_{\rm 2}}$ &-- & $4.58^{+0.41}_{-0.26}$ &$3.10^{+0.19}_{-0.24}$ &$2.96^{+0.26}_{-0.29}$ & -- & -- & --& -- \\
\\
Cov Frac2 &-- &$>0.74$ &$0.52^{+0.18}_{-0.23}$ &$0.53^{+0.31}_{-0.25}$ & -- & --&-- & -- \\
\\
$\Gamma$ &$ 1.77^{+0.03}_{-0.04}$ &$ 1.76^{+0.05}_{-0.03}$ &$ 1.68^{+0.04}_{-0.02}$ &$ 1.73^{+0.03}_{-0.03}$ &$ 1.65^{+0.04}_{-0.03}$ &$ 1.67^{+0.05}_{-0.03}$ &$ 1.71^{+0.02}_{-0.03}$ &$ 1.68^{+0.03}_{-0.04}$ \\
\\
$N_{\rm PL}$ (10$^{-3}$ ph cm$^{-2}$ s$^{-1}$)&$ 1.48^{+0.10}_{-0.14}$ &$ 22.35^{+1.02}_{-0.93}$ &$ 6.15^{+0.45}_{-0.68}$ &$ 2.54^{+0.32}_{-0.47}$ &$ 4.41^{+0.41}_{-0.57}$ &$ 2.82^{+0.26}_{-0.32}$ &$ 2.72^{+0.15}_{-0.22}$ &$ 2.55^{+0.21}_{-0.28}$ \\
\\
$A_{\rm Fe}$ ($A_{\odot}$)&$ 3.68^{+0.22}_{-0.28}$ &$ 3.89^{+0.26}_{-0.35}$ &$ 4.06^{+0.32}_{-0.22}$ &$ 4.15^{+0.34}_{-0.38}$ &$ 3.77^{+0.27}_{-0.38}$ &$ 4.12^{+0.27}_{-0.37}$ &$ 3.95^{+0.38}_{-0.42}$ &$ 4.18^{+0.27}_{-0.39}$ \\
\\
log($\xi$)&$ 1.96^{+0.03}_{-0.02}$ &$2.09^{+0.05}_{-0.04}$ &$ 2.18^{+0.03}_{-0.02}$ &$ 2.11^{+0.02}_{-0.02}$ &$ 2.07^{+0.02}_{-0.03}$ &$2.18^{+0.04}_{-0.03}$ &$ 2.13^{+0.02}_{-0.03}$ &$ 1.98^{+0.02}_{-0.02}$ \\
\\
$\theta_{\rm incl}$ (degree)&$ 13.8^{+2.7}_{-4.1}$ &$11.8^{+4.1}_{-5.5}$ &$ 16.2^{+2.8}_{-4.7}$ &$ 14.3^{+4.4 }_{-3.9 }$ &$ 10.3^{+4.9 }_{-6.1}$ &$ 13.2^{+2.5}_{4.7}$ &$ 17.7^{+2.9}_{-5.2}$ &$ 12.1^{+4.5}_{-5.8}$ \\
\\
$R_{\rm refl}$ &$ 0.11^{+0.02}_{-0.02}$ &$ 0.16^{+0.03}_{-0.04}$ &$ 0.15^{+0.03}_{-0.02}$ &$ 0.13^{+0.02}_{-0.02}$ &$ 0.18^{+0.02}_{-0.03}$ &$0.16^{+0.03}_{-0.02}$ &$ 0.10^{+0.03}_{-0.02}$ &$ 0.12^{+0.03}_{-0.02}$ \\
\\
$q_2$&$ 3.66^{+0.17}_{-0.41}$ &$ 4.69^{+0.68}_{-0.35}$&$ 4.41^{+0.25}_{-0.37}$ &$ 5.35^{+0.45}_{-0.29}$ &$ 5.31^{+0.29}_{-0.41}$ &$4.81^{+0.42}_{-0.63}$ &$ 4.56^{+0.72}_{-0.89}$ &$ 4.94^{+0.60}_{-0.76}$ \\
\\
$R_{\rm br}$ ($R_g$)&$ 12^{+3 }_{-2 }$ &$ 42^{+3}_{-2}$ &$ 26^{+2}_{-3}$ &$ 17^{+3}_{-1}$ &$ 18^{+2}_{-3}$ &$ 21^{+2}_{-3}$ &$ 21^{+2}_{-1}$ &$ 16^{+2}_{-3}$ \\
\\
$a^*$&$0.15^{+0.02}_{-0.03}$ &$0.16^{+0.03}_{-0.03}$ &$0.15^{+0.03}_{-0.03}$ &$0.17^{+0.02}_{-0.03}$ &$0.21^{+0.04}_{-0.02}$ &$0.15^{+0.03}_{-0.03}$ &$0.16^{+0.02}_{-0.03}$ &$0.20^{+0.02}_{-0.04}$ \\
\\
$R_{\rm in}$ ($R_g$)&$ 4.71^{+0.65}_{-0.88}$ &$ 5.78^{+0.43}_{-0.77}$ &$ 6.61^{+0.33}_{-0.37}$ &$ 4.76^{+0.28}_{-0.48}$ &$ 6.73^{+0.59}_{-0.73}$ &$ 4.06^{+0.32}_{-0.62}$ &$ 4.88^{+0.74}_{-0.91}$ &$ 6.92^{+0.65}_{-0.85}$ \\
\\
$N_{\rm rel}$ (10$^{-5}$ ph cm$^{-2}$ s$^{-1}$)&$ 0.62^{+0.08}_{-0.10}$ &$ 7.11^{+0.42}_{-0.62}$ &$ 0.49^{+0.06}_{-0.07}$ &$ 1.18^{+0.03}_{-0.04}$ &$ 1.87^{+0.14}_{-0.23}$ &$ 0.67^{+0.07}_{-0.10}$ &$ 1.10^{+0.12}_{-0.06}$ &$ 0.56^{+0.04}_{-0.07}$ \\
\\
$\chi^2$/dof& 983/992 & 2845/2550 & 2275/2119 &1648/1640 & 685/646 & 829/828 & 541/532 & 533/557 \\
\hline
\end{tabular}
\leftline{$\xi$'s are in the unit ergs cm s$^{-1}$. $\dagger$ pegged at the lowest value.}
\end{table*}
\section{Discussion}
We studied NGC~1566 during and after the 2018 outburst event using data from {\it XMM-Newton}, {\it Swift}~ and {\it NuSTAR}~ in the $0.5-70$~keV energy band. From a detailed spectral and timing analysis, we explored the nuclear properties of the AGN.
\subsection{Black Hole Properties}
NGC~1566~ hosts a supermassive black hole of mass $M_{\rm BH} \approx 8.3 \times 10^6$ $M_{\odot}$ \citep{Woo2002}. We kept the mass of the BH frozen during our spectral analysis with Model--3. Fitting the spectra with Model--3 and Model--4, we estimated the spin parameter ($a^*$) to be in the range $0.18^{+0.01}_{-0.02}-0.21^{+0.03}_{-0.02}$ and $0.15^{+0.2}_{-0.3}-0.21^{+0.04}_{-0.02}$, respectively. Both models favour a low spinning BH with the spin parameter $a^* \sim 0.2$, which is consistent with the findings of \citet{Parker2019}.
The inclination angle was a free parameter in Model--4, and it was estimated to be in the range of $10.3^{+4.9}_{-6.1}$\textdegree -- $17.7^{+2.9}_{-5.2}$\textdegree. \citet{Parker2019} also found a consistent inclination angle ($\theta_{\rm incl} <$ 11\textdegree).
\begin{table*}
\caption{Luminosities of NGC~1566\ in the observations analyzed here.}
\label{tab:lum}
\begin{tabular}{lccccccccc}
\hline
ID & Day & $L_{\rm nuc}$ & $L_{\rm soft}$ & $L_{\rm 0.1-100}$ & $L_{\rm bol}$ & $\lambda_{\rm Edd}$ \\
& & ($10^{42}$ erg s$^{-1}$) & ($10^{41}$ erg s$^{-1}$) & ($10^{42}$ erg s$^{-1}$) & ($10^{43}$ erg s$^{-1}$) & \\
\hline
X1& 57331 &$ 0.34 \pm0.01 $&$ 0.89 \pm0.06 $&$ 0.43 \pm0.01 $&$ 0.09 \pm0.01 $&$ 0.003\pm0.001 $\\
O1& 58295 &$ 13.42 \pm0.08 $&$ 4.89 \pm0.33 $&$ 13.91 \pm0.08 $&$ 7.11 \pm0.04 $&$ 0.066\pm0.001 $\\
O2& 58395 &$ 4.02 \pm0.05 $&$ 2.45 \pm0.14 $&$ 4.26 \pm0.05 $&$ 1.84 \pm0.03 $&$ 0.017\pm0.001 $\\
O3& 58639 &$ 2.36 \pm0.01 $&$ 1.16 \pm0.25 $&$ 2.48 \pm0.03 $&$ 1.19 \pm0.02 $&$ 0.011\pm0.001 $\\
O4& 58703 &$ 4.01 \pm0.05 $&$ 1.56 \pm0.22 $&$ 4.17 \pm0.05 $&$ 2.04 \pm0.03 $&$ 0.019\pm0.001 $\\
O5& 58706 &$ 2.16 \pm0.06 $&$ 1.64 \pm0.21 $&$ 2.32 \pm0.06 $&$ 1.33 \pm0.05 $&$ 0.012\pm0.001 $\\
O6& 58713 &$ 1.96 \pm0.12 $&$ 1.54 \pm0.29 $&$ 2.11 \pm0.12 $&$ 1.29 \pm0.07 $&$ 0.012\pm0.001 $\\
O7& 58716 &$ 2.40 \pm0.03 $&$ 1.59 \pm0.31 $&$ 2.58 \pm0.04 $&$ 1.31 \pm0.03 $&$ 0.012\pm0.001 $\\
\hline
\end{tabular}
\leftline{$L_{\rm nuc}$ and $L_{\rm soft}$ are calculated for the primary power-law and soft excess components, respectively.}
\leftline{Eddington ratio, $\lambda_{\rm Edd}$ is calculated using $L_{\rm bol}/L_{\rm Edd}$ for a BH of mass $8.3 \times 10^6$ $M_{\odot}$.}
\end{table*}
\subsection{Corona Properties}
The X-ray emitting corona is generally located very close to the central BH \citep{Fabian2015}. This corona is characterized by the photon index ($\Gamma$), temperature ($kT_{\rm e}$) and optical depth ($\tau$) of the Comptonizing plasma. While using a simple power-law model gives us information only about the photon index, the {\tt NTHCOMP} model can provide us with information on the electron temperature of the Compton cloud ($kT_{\rm e}$), while the optical depth ($\tau$) is calculated from Equation~\ref{eqn:tau}. We found that the photon index varied within a narrow range of $\Gamma \sim 1.7-1.8$, and can be considered as constant within the uncertainties. To constrain the photon index with more accuracy, we fitted the source spectra from {\it NuSTAR}~ observations in $3-70$~keV and $10-70$~keV energy ranges to approximate only the power-law part. We found similar results as from the simultaneous fitting of the {\it XMM-Newton}~ and {\it NuSTAR}~ data in $0.5-70$~keV range. From the spectral analysis with Model--3, we estimated the Compton cloud radius to be $12-43~R_{\rm g}$.
We calculated the intrinsic luminosity ($L_{\rm 0.1-100}$) of the AGN from Model--1. The intrinsic luminosity was observed to be low during the pre-outburst observation, X1, with $L_{\rm 0.1-100} \sim (4.3 \pm 0.1) \times 10^{41}$ erg s$^{-1}$. During O1, was observed to be maximum with $L_{\rm 0.1-100} \sim (1.39 \pm 0.01) \times 10^{44}$ erg s$^{-1}$. Later, the intrinsic luminosity decreased and varied in the range of $2.1-4.3 \times 10^{43}$ erg s$^{-1}$. We also computed the luminosity for the primary emission ($L_{\rm nuc}$) and soft excess ($L_{\rm soft}$) from the individual components while analyzing the spectra with Model--1. We calculated the bolometric luminosity ($L_{\rm bol}$) using the 2--10\,keV bolometric correction, $\kappa_{\rm bol, 2-10~keV} = 20$ \citep{Vasedevan2009}. The Eddington ratio ($\lambda_{\rm Edd}=L_{\rm bol}/L_{\rm Edd}$), assuming a BH of mass of $8.3 \times 10^6$ $M_{\odot}$ \citep{Woo2002}, was estimated to be $\lambda_{\rm Edd} \sim 0.003-0.066$ in different epoch which is consistent with other nearby Seyfert-1 galaxies \citep{Wu2004,Sikora2007,Koss2017}.
In the pre-outburst observation in November 2015 (X1 : see Table~\ref{tab:lum}), we obtained a bolometric luminosity of $L_{\rm bol}=(0.9\pm 0.1) \times 10^{42}$ erg s$^{-1}$, with corona size $R_{\rm cor} = 12\pm 3~R_{\rm g}$ and hot electron plasma temperature $kT_{\rm e}=102\pm 5$~keV. In the observation during the outburst in June 2018 (O1), the luminosity of the AGN increased by a factor of about $\sim$25, compared to the November 2015 observation (X1). During this observation, the corona was large ($R_{\rm cor} = 43\pm 3~R_{\rm g}$) with hot electron plasma temperature $kT_{\rm e} = 61\pm7$~keV and the observed spectrum was harder. As the outburst progressed, the bolometric luminosity and the corona size decreased. As $R_{\rm cor}$ decreased, the electron plasma temperature increased. Overall, $kT_{\rm e}$ varied in a range of $\sim 61-106$~keV during the observations. In general, the plasma temperature is observed in a wide range, with a median at $kT_{\rm e} \sim 105 \pm 18$~keV \citep{Ricci2018}. Thus, the plasma temperature is consistent with other AGNs. During these observations, the optical depth of the Compton cloud varied within $\sim 1.2-1.7$. Interestingly, the photon index ($\Gamma$) was almost constant, although some of the properties of the corona evolved with time. This appears to imply that both the optical depth and the hot electron temperature changed in such a way that the spectral shape remained the same.
We found several correlations and anti-correlations between the spectral parameters and show them in Fig.~\ref{fig:cor}. We fitted the data points with linear regression method using y = mx + c. The fitted value of the slope (m) and intercept (c) are mentioned in each panel of Fig.~\ref{fig:cor}. We found that the nuclear luminosity ($L_{\rm nuc}$) and the soft excess luminosity ($L_{\rm soft}$) are strongly correlated with the Eddington ratio with the Pearson correlation indices of 0.84 and 0.85, respectively. We also found that the bolometric luminosity ($L_{\rm bol}$) and the Compton cloud temperature ($kT_{\rm e}$) are anti-correlated ($\rho_{\rm s}=-0.85$). The electron temperature is found to be anti-correlated with the Eddington ratio ($\rho_{\rm s}=-0.85$), while the size of the Compton corona and the luminosity are positively correlated ($\rho_{\rm s}=0.93$). We also observed that the electron temperature is anti-correlated with the size of the Compton cloud ($\rho_{\rm s}=-0.84$). The above correlations can be explained thinking that, as the accretion rate increased, the energy radiation increased, thereby, increasing the luminosity. An increase in the mass accretion rate makes the cooling more efficient, leading to a decrease in the electron temperature of the Comptonizing region \citep{HM1991,Done2007}.
\begin{figure*}
\centering
\includegraphics[width=16cm, angle=0]{cor.eps}
\caption{Correlation between different spectral parameters. In each panel, the corresponding Pearson correlation co-efficient ($\rho_{s}$) is quoted. The solid green line in each panel represent the linear fit, y=mx+c. Corresponding fitted values of the slope (m) and intercept (c) are also mentioned in each panels.}
\label{fig:cor}
\end{figure*}
\subsection{Reflection}
The hard X-ray photons from the corona are reflected from cold material in the accretion disc, BLR and torus, producing a reflection hump and a Fe-K emission line \citep{George1991,Matt1991}. When fitted with a simple power-law model, NGC~1566~ showed the presence of Fe K$\alpha$ emission line along with a weak reflection hump at $\sim 15-40$~keV energy range (see Fig~\ref{fig:uf-spec}). Thus, we fitted the spectra with the relativistic reflection model {\tt RELXILL} to probe the reflection component.
\citet{Parker2019} analyzed O1 observation with {\tt RELXILL} and {\tt XILLVER} models in their spectral analysis. They found $\xi = 10^{2.4\pm0.1}$ erg cm$^{-2}$ s$^{-1}$, $A_{\rm Fe} = 3\pm0.2 A_{\odot}$ and $R_{\rm refl} = 0.091\pm0.005$. In the present work, we obtained $\xi= 10^{2.09\pm0.05}$ erg cm$^{-2}$ s$^{-1}$, $A_{\rm Fe} = 3.89\pm0.35$, and $R_{\rm refl} = 0.16\pm0.04$ for O1. The marginal difference between our results and those of \citet{Parker2019} could be ascribed to the different spectral models used. We observed fairly constant ionization ($\xi \sim 10^{1.9-2.2}$ erg cm s$^{-1}$) and iron abundances ($A_{\rm Fe} \sim 4-5$ $A_{\sun}$) across the observations. This is expected within our short period of observation. In all the spectra, we found a very weak reflection with reflection fraction, $R_{\rm refl} < 0.2$. We found a weak correlation between the reflection fraction ($R_{\rm refl}$) and the luminosity with the Pearson correlation coefficient of 0.47. In general, reflection becomes strong with increase in the luminosity \citep{Z99}. The low inclination angle of the source also results in a weak reflection \citep{Ricci2011,Chatterjee2018}. Therefore, the observed weak correlation between the reflection fraction and luminosity in NGC~1566\ could be due to the low inclination angle of the source.
During the observations, a strong Fe K$\alpha$ emission line with equivalent width $EW > 100$~eV was detected, despite a weak reflection component, except for O5. This could be explained by high iron abundances in the reflector. From the spectral analysis with model--4, we found the inner edge of the disc extends up to $\sim 5~R_{\rm g}$. If iron originates from the inner disc, a broad iron line is expected. However, we did not observe a broad iron line. Either the broad line was absent or it was blurred beyond detection. However, during our observation, a narrow iron line was detected, which given its width, originates in the material further out than the accretion disc. Hence, from the full-width at half maximum (FWHM) of the line, we tried to constraint the Fe K$\alpha$ line emitting region. During our observation period, the FWHM of Fe K$\alpha$ line emission was found to be $<8700$ km s$^{-1}$, which corresponds to the region $>1200$~$R_{\rm g}$ from the BH. This corresponds to the distance at which we expect to find the BLR \citep{Kaspi2000}. Thus, the BLR is the most probable Fe K$\alpha$ line emitting region in NGC~1566.
\subsection{Soft Excess}
The origin of the soft excess in AGNs is still very debated, and several models have been proposed to explain it. Relativistic blurred ionized reflection from the accretion disc has been put forward as a likely explanation for the soft excess in many sources \citep{Fabian2002,Ross2005,Walton2013,Garcia2019,Ghosh2020}. An alternative scenario considers Comptonization by a optically thick, cold corona \citep{Done2012}. In this model, the comptonizing region is located above the inner accretion disc as a thin layer. Heating of circumnuclear region by bulk motion Comptonization in AGN with high Eddington ratio is also considered to be the reason for the soft excess \citep{Kaufman2017}. Recently, \cite{Nandi2021} argued from long-term observations and Monte-Carlo simulations (see \cite{Chatterjee2018} and references therein) that the thermal Comptonization of photons which have suffered fewer scatterings could explain the origin of soft excess in Ark 120.
In our analysis, we used a {\tt bbody} component to take into account the soft excess. The blackbody temperature ($kT_{\rm bb}$) was roughly constant with $kT_{\rm bb} \sim 110$~eV during our observations. This is consistent with the observation of other nearby AGNs \citep{Gierlinski2004,Winter2009,Ricci2017,Garcia2019}. A good-fit with the {\tt OPTXAGNF} model favoured soft Comptonization by an optically thick corona as the origin of the soft excess. During observation~O1, the temperature of the optically-thick corona was observed to be $\sim 1.4$~keV. Later, the optically thick corona cooled with decreasing luminosity.
We tried to estimate delay and correlation between the soft excess and X-ray continuum light curves (see Section~\ref{sec:correlation}). In the {\tt OPTXAGNF} scenario, the total energetic depends on the mass accretion rate and is divided between the soft-Comptonization (soft-excess) and hard-Comptonization (nuclear or primary emission). We found a strong correlation between the soft excess luminosity ($L_{\rm soft}$) and nuclear luminosity ($L_{\rm nuc}$). The soft-excess luminosity and the Eddington ratio ($L/L_{\rm Edd}$) were found to be correlated ($\rho_{\rm s}=0.78$). This indicated that the soft excess strongly depended on the accretion rate, supporting the soft-Comptonization as the origin of the soft excess. We also found a delay of $\sim 10$ minutes between the soft excess and primary emission, indicating the origin of the soft excess was beyond the corona, possibly the accretion disc. Higher variability was also observed in the soft excess (see Section~\ref{sec:correlation}) during observations X1, O1, and O3. This could indicate a higher stochasticity in the origin of the soft-excess. It should be noted that, theoretically, infinite scattering produces blackbody which has the lowest variability. As \citet{Nandi2021} suggested, fewer scattering which could generate higher variability than the large number of scatterings, possibly lead to the origin of soft-excess. Overall, the soft excess in NGC~1566~ has a complex origin, including reflection from the accretion disc and soft-Comptonization.
\subsection{Changing-Look Event and its Evolution}
The physical drivers of CL events are still highly debated, and could change from source to source. Tidal disruption events (TDEs), changes in obscuration, and variations in the mass accretion rate could all be possible explanations for CL events.
In the pre-outburst observation, X1, the absorber was not strongly ionized ($\xi_1 <0.001$), and had a column density of $N_{\rm H,1} = (3.53 \pm 0.06) \times 10^{21}$ cm$^{-2}$. Two ionizing absorbers were detected during the observation O1, O2, and O3 which could be associated with an outflow. \citet{Parker2019} also observed two ionized absorbers during O1, and suggested that they could be associated with an outflow. The highest ionization (for both absorbers) was observed in O1, which corresponded to the epoch in which the observed luminosity was the highest. During the 2018 outburst, when the X-ray intensity increased, the emitted radiation may have caused the sublimation of the dust along the line of sight, thereby decreasing the hydrogen column density \citep{Parker2019}. As the X-ray intensity decreased after the main outburst, the dust could have condensed, leading to an increase in the column density. Eventually, during the August 2019 observations, the dust formation was stable as the hydrogen column density was approximately constant ($N_{\rm H} \sim 1.3 \times 10^{21}$ cm$^{-2}$). Generally, the dust clouds can recover in the timescale of several years \citep{Kishimoto2013,Oknyansky2017}. In this case, we observed that the dust clouds already recovered with an increase in $N_{\rm H}$ from $\sim 6 \times 10^{20}$ cm$^{-2}$~ to $1.3\times 10^{21}$ cm$^{-2}$~ in $\sim$14 months time. If this is correct, we would expect the $N_{\rm H}$ to reach at its pre-outburst value of $N_{\rm H} \sim 3.5 \times 10^{21}$ cm$^{-2}$~ in next few months.
The strong correlation between the accretion rate and the bolometric luminosity suggests that the accretion rate is responsible for the CL events in NGC~1566~ during the 2018 outburst. \citet{Parker2019} also discussed several possibilities for the CL event in NGC~1566~ and concluded that the disc-instability is the most likely reason for the outburst. The instability at the outer disc could propagate through the disc and cause the outburst. \citet{Noda2018} explained the flux drop and changing-look phenomena in Mrk~1018 with the disc-instability model where the time-scale for the changing-look event was $\sim 8$~years. The time-scale for changing-look event for NGC~1566 was $\sim 10$~months as the flux started to increase from September 2017 \citep{Dai18,Cutri18}. The time-scales are similar if we consider the mass of the BH. As the mass of Mrk~1018 is $M_{\rm 1018} \sim 10^{7.84}~M_{\odot}$ \citep{Ezhikode2017}, the expected time-scale for NGC~1566 is $\sim 8~{\rm years}/10 \sim 10$~months. The observed `q'-shaped in the HID during the main outburst (F1), also suggest of disc instability as seen in the case of the outbursting black holes. The `q'-shaped HID is very common for the outbursting black holes \citep{RM06} where disc instability is believed to lead the outburst. \citet{Noda2018} also suggested that the soft-excess would decrease much more compared to the hard X-ray with decreasing $L/L_{\rm Edd}$. NGC~1566 also showed the strongest soft-excess emission during the highest $L/L_{\rm Edd}$ (O1), while the soft-excess emission dropped as $L/L_{\rm Edd}$ decreased. In the pre-outburst quiescent state, no soft-excess was observed in NGC~1566. As the soft-excess can produce most of the ionizing photons necessary to create broad optical lines \citep{Noda2018}, the broad line appeared during O1 (when the soft-excess was strong), leading to the changing-look event.
A TDE might be another possible explanation for the 2018 outburst of NGC\,1566. A TDE could supply the accreting matter to the central SMBH, which would lead to an increase in luminosity. Several recurrent outbursts (F2, F3 \& F4) and nearly periodic X-ray variations were observed after the main outburst (F1) as seen in the upper panel of Fig.~\ref{fig:xrt-lc} and Fig~\ref{fig:lc-smooth}. In TDEs, some amount of matter could be left out, and cause recurrent outbursts \citep{Komossa2017}. In the case of a classic TDE, after a star is tidally disrupted by the SMBH, a decay profile of the luminosity with $t^{-5/3}$ is expected \citep{Rees1988,komossa2015,Komossa2017}, which is not observed in this case. During all observations, the source showed a relatively hard spectrum, with $\Gamma \sim 1.6-1.7$. This is clear contrast with classic TDEs, which typically show much softer spectra ($\Gamma \geq 3$) \citep{Komossa2017}. During the June 2018 outburst (F1), the X-ray luminosity changed by about $\sim 25$~times compared to the low state, which is low in comparison to other candidate TDEs. For example, 1ES~1927+654 and RX~J1242-1119 showed a change in luminosity by over 4 orders of magnitude \citep{Ricci2020} and 1500 times \citep{Komossa2004}, respectively. In general, no iron emission line is observed in the X-ray band during the TDE \citep{Saxton2020}, whereas the case of NGC\,1566 a strong Fe K$\alpha$ line was observed. Considering all this, we deem unlikely that the June 2018 outburst of NGC~1566~ was triggered by a TDE.
\begin{figure}
\centering
\includegraphics[angle=0,width=8.5cm]{light-curve_new.eps}
\caption{Lightcurve of NGC 1566 between June 2018 and August 2020 with smooth lines showing near-periodic variation of count rate.}
\label{fig:lc-smooth}
\end{figure}
An alternative explanation could be that a star is tidally disrupted by a merging SMBH binary at the center of NGC~1566. According to \cite{Hayasaki2016}, after the stellar tidal disruption, the debris chaotically moves in the binary potential and is well-stretched. The debris orbital energy is then dissipated by the shock due to the self-crossing, leading to the formation of an accretion disc around each black hole after several mass exchanges through the Lagrange (L1) point of the binary system. If the orbital period of an unequal-mass binary is short enough ($<1000$~days) to lose the orbital energy by gravitational wave (GW) emission, the secondary, less massive SMBH orbits around the center of mass at highly relativistic speed, while the primary SMBH hardly moves. Therefore, the electromagnetic emission from the accretion disc around the secondary black hole would be enhanced periodically by relativistic Doppler boosting. This scenario could explain the recurrent outburst observed in case of NGC~1566 (see Fig~\ref{fig:lc-smooth}). From this figure, the orbital period is estimated to be $P_{\rm orb}\sim 160$ days (between F1 \& F2). Taking into account the SMBH mass ($M_{\rm BH}=8.3\times10^6 ~{\rm M_{\odot}}$), we obtain $a\sim 710~r_{\rm s} \approx 5.6 \times 10^{-4}$~pc as a binary semi-major axis, where $r_{\rm s}=2GM_{\rm BH}/c^2$ is the Schwarzschild radius. The merging timescale of two SMBH with mass ratio $q$ due to GW emission is given by \citep{Peters1964},
\begin{equation}
t_{\rm gw} = \frac{5}{8} \frac{(1-q)^2}{q} \frac{r_s}{c} \Big( \frac{a}{r_s} \Big)\sim 5.0\times10^6 yr.
\end{equation}
This suggests that more than 100 TDEs could occur before the SMBH merger, considering the TDE rate for a single SMBH ($10^{-4}$ to $10^{-5}$ $yr^{-1}$ per galaxy, \citealp{Stone2020}). However, the event rate can be enhanced up to 0.1 $yr^{-1}$ per galaxy due to chaotic orbital evolution and Kozai-Lidov effect in the case of SMBH binaries \citep{Chen2009,Li2015}. Moreover, if stars are supplied by accretion from a circumbinary disc \citep[eg,][]{Hayasaki2007,Amaro2013}, then the TDE rate could be higher up to $\sim0.2~ yr^{-1}$ if the mass supply rate is at the Eddington limit \citep{Wolf2021}. Therefore, the detection of similar, periodic burst events in the next few years to tens of years would support this interpretation.
\section{Summary}
We analyzed the X-ray emission of the changing-look AGN NGC~1566\ between 2015 and 2019. Our key findings are the following.
\begin{enumerate}
\item NGC~1566\ showed a giant outburst in June 2018 when the X-ray luminosity increased by $\sim25-30$ times compared to that during the low state. After the main outburst, several recurrent outbursts were also observed.
\item NGC~1566~ hosts a low-spinning BH with the spin parameter, $a^* \sim 0.2$.
\item The inclination angle is estimated to be in the range of $i \sim 10$\textdegree--$21$\textdegree.
\item The variation of the accretion rate is responsible for the evolution of the Compton corona and X-ray luminosity.
\item A rise in the accretion rate is responsible for the change of luminosity. The HID or `q'-diagram links the CL event of NGC~1566~ with the outbursting black holes.
\item A strong soft excess was observed when the luminosity of NGC~1566~ was high. The origin of the soft excess is observed to be complex.
\item We rule out the possibility that the event was triggered by a classical TDE, where a star is tidally disrupted by the SMBH.
\item We propose a possible scenario where the central core is a merging binary SMBH. This scenario could explain the recurrent outburst.
\end{enumerate}
\section*{Data Availability}
We have used archival data for our analysis in this manuscript. All the models and software used in this manuscript are publicly available. Appropriate links are given in the text.
\section*{Acknowledgements}
We acknowledge the anonymous reviewer for the helpful comments and suggestions which improved the paper. Work at Physical Research Laboratory, Ahmedabad, is funded by the Department of Space, Government of India. PN acknowledges CSIR fellowship for this work. A. C. and K.H. has been supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1A5A1013277 (A.C. and K.H) and 2020R1A2C1007219 (K.H.)). K.H. acknowledges the Yukawa Institute for Theoretical Physics (YITP) at Kyoto University. Discussions during the YITP workshop YITP-T-19-07 on International Molecule-type Workshop ``Tidal Disruption Events: General Relativistic Transients" were useful for this work. This research has made use of data and/or software provided by the High Energy Astrophysics Science Archive Research Center (HEASARC), which is a service of the Astrophysics Science Division at NASA/GSFC and the High Energy Astrophysics Division of the Smithsonian Astrophysical Observatory. This research has made use of the {\it NuSTAR} Data Analysis Software ({\tt NuSTARDAS}) jointly developed by the ASI Space Science Data Center (SSDC, Italy) and the California Institute of Technology (Caltech, USA). This research has made use of the SIMBAD database, operated at CDS, Strasbourg, France.
\bibliographystyle{mnras}
|
1,108,101,565,693 | arxiv | \section{Introduction}
\label{sec:intro}
More and more research institutes and scientists consider attaching camera to Autonomous Underwater Vehicles (AUVs) and Remotely Operated Vehicles (ROVs) in order to perform different underwater tasks, such as marine organism capturing, ecological surveillance and biodiversity monitoring. Underwater object detection is an indispensable technology for AUVs to fulfill these tasks.
In application, once a underwater object detector aiming at certain categories have been trained, we hope this detector can be applied in any underwater circumstances. As a result, it is necessary to build a General Underwater Object Detector (GUOD). A GUOD faces three kinds of challenges:
(1) It is much harder to obtain underwater images, and the annotation task usually need experts to accomplish, which is costly. Therefore, labeled dataset of underwater object detection is extremely limited, inevitably leading to overfitting of deep model. Data augmentation aims at solving the problem of lack of data. There are three types of augmentation. First, geometrical transformations (e.g., horizontal flipping, rotation, patch crop \cite{liu2016ssd}, perspective simulation \cite{huang2019faster}) have been proved effective in various fields. Second, cut-Paste-based methods (e.g., randomly cut and paste \cite{dwibedi2017cut}, Mixup \cite{zhang2017mixup}, CutMix \cite{yun2019cutmix}, PSIS \cite{wang2019psis}) help model learn contextual invariance. Third, domain-transfer based methods (e.g., SIN \cite{geirhos2018imagenet}) force model to focus more semantic information.
(2) The contradiction between speed and performance becomes even more critical. A GUOD should be able to work in real time, which is a common requirement in robotics field. However, it is impractical to equip small AUVs with high performance hardware. Some works focus on the speed of deep learning model but keep good control of performance decrease, such as MobileNet \cite{howard2017mobilenets}, SSD \cite{liu2016ssd}, YOLOv3 \cite{redmon2018yolov3}.
(3) Deep model severely suffers from domain shift, but a GUOD should be invariant of water quality, which can not only work well in oceans, but also in lakes and rivers. This can be seen as a kind of domain generalization problem that a model trains on source domains but evaluates on an unseen domain. Some domain adaptation (DA) (e.g., style consistency \cite{rodriguez2019domain}, DA-Faster RCNN \cite{chen2018domain}) and domain generalization (DG) (e.g., JiGEN \cite{carlucci2019domain}, MMD-AAE \cite{li2018domain}, EPi-FCR \cite{li2019episodic}) technologies are proposed before. Nevertheless, most of DG works focus on object recognition and DA works can not directly transplant to DG task, so their effectiveness are not proved in DG object detection task.
This work aims to use small dataset with limited domains to train a GUOD. To handle challenge (1), a new augmentation method Water Quality Transfer (WQT) is proposed to enlarge the dataset and increase domain diversity. To handle challenge (2) and (3), DG-YOLO is proposed to further boost domain invariance of object detection based on a real-time detector YOLOv3. Our method is implemented on Underwater Robot Picking Contest 2019 (URPC2019) dataset, and achieve performance improvement.
\begin{figure}[!htp]
\centering
\includegraphics[width=8.7cm,height=3.5cm]{wqt}
\caption{Water Quality Transfer.}
\label{fig:wqt}
\end{figure}
In summary, our contributions are listed as follows: (1) We propose a new augmentation WQT specially for underwater condition and analyze its effectiveness and reveal its limitations; (2) Based on WQT, DG-YOLO is proposed to further mine the domain-invariant (semantic) information of underwater image, which realizes domain generalization; (3) A lot of experiments and visualization are conducted to prove the effectiveness of our method.
\section{Method}
\subsection{Water Quality Transfer (WQT)}
As Figure \ref{fig:wqt} shows, we select 8 images with different types of water quality, and use $WCT^2$ \cite{yoo2019photorealistic} to transfer URPC dataset to different types of water quality. The content image is from URPC's training set and validation set. In the following section, this seven types of training set are denoted as type1 to type7 and the corresponding validation set are denoted as \emph{val\_type1} to \emph{val\_type7}. As for type8, only the validation set is transferred to obtain \emph{val\_type8} without corresponding training set.
Since model will never train on type8 domain, \emph{val\_type8} is to test the domain generalization capacity of model.
\subsection{Domain Generalization YOLO (DG-YOLO)}
\label{sec:DG-YOLO}
\textbf{A review of YOLOv3}. Because AUVs with a small processing unit have limited calculation capacity, the real-time detector YOLOv3 \cite{redmon2018yolov3} is a promising choice. YOLOv3 is a one-stage object detector, using Darknet-53 as backbone. Compared with Faster R-CNN \cite{ren2015faster}, YOLOv3 does not use region proposal network. It directly regresses the coordinates of bounding box and class information with a fully convolutional network. YOLOv3 divides an image into $S \times S$ cells, and each cell is responsible for the objects lie in the cell.
The training losses of YOLOv3 consists of the loss of classification $L_{cls}$, the loss of coordination $L_{coord}$, loss of object $L_{obj}$ and loss of no-object $L_{noobj}$:
\begin{equation}\label{eq:yolo-loss}
L_{yolo} = L_{cls}+\lambda_{coord} \cdot L_{coord}+L_{obj}+\lambda_{noobj} \cdot L_{noobj},
\end{equation}
where $\lambda_{coord}$ and $\lambda_{noobj}$ are trade-off parameters.
\begin{figure}[!htp]
\centering
\includegraphics[width=8.7cm,height=3.5cm]{DG-YOLO+wqt}
\caption{The pipeline of WQT and DG-YOLO. \emph{YOLO BackBone} + \emph{YOLO Detector} is original YOLOv3. Domain label comes from WQT.}
\label{fig:dg-yolo}
\end{figure}
\textbf{Domain Invariant Module (DIM)}. Since DA and DG have some similarities, we modify the domain classifier proposed by \cite{ganin2015unsupervised} to apply in our DG task. Given a batch of input images as $\{X_1,X_2,...,X_N\}$ from $K$ different source domains, its corresponding domain labels are $\{d_1,d_2,...,d_N\}$, in which $N$ is the number of batch, $d_i\in \mathbb{R}^{K\times 1}$. Denoting that $G$ is a feature extractor and $D$ is a domain classifier, the domain loss $L_{d}$ is defined as follows:
\begin{equation}\label{eq:domain-loss}
L_d = \sum_{i}^{N} l_{CE}(D(G(X_i)),d_i),
\end{equation}
where $l_{CE}$ means categorical cross entropy. In application, domain label comes from WQT, and $K$ is 7 corresponding to 7 types of water quality that WQT synthesizes. Domain loss for data from original dataset is not calculated.
\textbf{IRM Penalty}.
Inspired by recent study \cite{arjovsky2019invariant}, Invariant Risk Minimization (IRM) help learn an invariant predictor across multiple domain. Given a set of training environments (same meaning as domains) $e \in \mathcal{E}_{tr} $, our final goal is to achieve good performance across a large set of unseen but related environments $\mathcal{E}_{all}$ ($\mathcal{E}_{tr} \in \mathcal{E}_{all}$). However, directly using Empirical Risk Minimization (ERM) \cite{vapnik1992principles} will lead to overfitting on training environment and learn spurious correlation. In order to generalize well on unseen environments, IRM is a better choice to obtain invariance:
\begin{equation}
{min}_{\Phi:\mathcal{X}\sim \mathcal{Y}}\ \sum_{e \in \mathcal{E}_{tr}}R^e(\Phi)+\lambda \cdot ||\nabla_{r|r=1.0}R^e(r\ \cdot \Phi)||^2,
\end{equation}
where $\Phi$ is the entire invariant predictors, $R^e(\Phi)$ is ERM term on environment $e$, $r=1.0$ is a fixed scalar, $||\nabla_{r|r=1.0}R^e(r\ \cdot \Phi)||^2$ is invariance penalty, and $\lambda \in [0,\infty)$ is a trade-off parameter balancing the ERM term and the invariance penalty.
To apply IRM in YOLOv3, the IRM penalty specially for YOLOv3 is designed as follows:
\begin{footnotesize}
\begin{equation}
\begin{aligned}
& Pen_{coord} = \nabla_{r|r=1.0}||\sum_{i=0}^{S^2}\sum_{j=0}^{B}\mathbb{I}^{obj}_{i,j}[(x_i-\hat{x}_i \cdot r)^2+(y_i-\hat{y}_i\cdot r)^2, \\
& \ \ \ \ \ \ \ \ \ \ +(w_i-\hat{w}_i\cdot r)^2+(h_i-\hat{h}_i\cdot r)^2]||^2, \nonumber
\end{aligned}
\end{equation}
\begin{equation}
Pen_{cls}= \nabla_{r|r=1.0}||\sum_{i=0}^{S^2}\mathbb{I}^{obj}_{i}\sum_{c \in classes} l_{CE}(p_i(c),\sigma(a_i(c) \cdot r))||^2, \nonumber
\end{equation}
\begin{equation}
Pen_{obj}=\nabla_{r|r=1.0}||\sum_{i=0}^{S^2}\sum_{j=0}^{B}\mathbb{I}^{obj}_{i,j}l_{CE}(C_i,\sigma(S_i \cdot r))||^2, \nonumber
\end{equation}
\begin{equation}
Pen_{noobj}=\nabla_{r|r=1.0}||\sum_{i=0}^{S^2}\sum_{j=0}^{B}\mathbb{I}^{noobj}_{i,j}l_{CE}(C_i,\sigma(S_i \cdot r))||^2, \nonumber
\end{equation}
\end{footnotesize}
\begin{equation}\label{eq:full-penalty}
P_{IRM} = pen_{coord}+pen_{cls}+pen_{obj}+pen_{noobj},
\end{equation}
where $\mathbb{I}_{i}^{obj}$ denotes if object appears in cell $i$, $\mathbb{I}_{i,j}^{obj}$ denotes that the $j$th bounding box predictor in cell $i$ is responsible for that prediction, $\sigma$ is sigmoid operation, $\hat{a_i}(c)$ is the score of class $c$ before sigmoid operation, $p_i \in \mathbb{R}^{K \times 1}$ is class label, $\hat{S_i}$ is the score of objects before sigmoid operation, $C_i \in \{0,1\}$ is object label. $\{\hat{x_i},\hat{y_i},\hat{w_i},\hat{h_i}\}$ is the bounding box outputted by YOLOv3, whose corresponding ground truth is $\{x_i,y_i,w_i,h_i\}$.
Penalty term is designed based on corresponding losses of YOLOv3. To be specific, $r$ is added to different places of losses. Square gradient of losses to $r$ is the corresponding penalty term.
\textbf{Network overview}.
An overview of our network is shown in Figure \ref{fig:dg-yolo}, we denote it DG-YOLO. Compared to YOLOv3, DIM and IRM penalty are added. In details, the backbone of YOLO darknet-53 can be regarded as a feature extractor. The feature maps extracted from darknet will be fed into Gradient Reversal Layer (GRL) \cite{ganin2015unsupervised} first, which reverses the gradient when backpropagating for adversarial learning. After that, domain classifier distinguish feature maps between domains. With the help of GRL and domain classifier, the backbone will be forced to abandon information of water quality to fool domain classifier. As a result, DG-YOLO can make a prediction depending more on semantic information. Moreover, IRM penalty is calculated simultaneously with YOLO loss. Combining (\ref{eq:yolo-loss}), (\ref{eq:domain-loss}) and (\ref{eq:full-penalty}) ,the total loss of DG-YOLO is:
\begin{equation}\label{eq:total-loss}
L_{total}=L_{yolo}+\lambda_{d} \cdot L_d+\lambda_{p} \cdot P_{IRM},
\end{equation}
$\lambda_{p}$ and $\lambda_{d}$ set to 1 in experiment. In inference stage, because DIM and IRM penalty can be abandoned, DG-YOLO doesn't affect the speed of YOLOv3. It should be emphasized that because domain label comes from WQT, DG-YOLO can not be used alone without WQT.
\section{Experiments and Discussions}
\label{sec:Experiment and discussion}
\subsection{Dataset}
\label{ssec:dataset}
We evaluate WQT and DG-YOLO on a publicly available datasets: URPC2019$\footnote{www.cnurpc.org}$, which consists of 3765 training samples and 942 validation samples over five categories: echinus, starfish, holothurian, scallop and waterweeds. Applying WQT on training set and validation set of URPC2019, we can synthesize \emph{type1-7} for training and \emph{val\_type1-8} for validation. The performance on \emph{val\_type8} represents domain generalization capacity of model.
\subsection{Training details}
\label{ssec:training details}
YOLOv3 and DG-YOLO is trained for 300 epochs and evaluated on original and all synthetic validation sets, with image resizing to 416 $\times$ 416. Models are trained on a Nvidia GTX 1080Ti GPU with PyTorch implementation, setting batch size to 8. Adam algorithm is adopted for optimization and learning rate sets to 0.001, with $\beta_1=0.9$ and $\beta_2=0.999$. IoU, confidence and non-max suppression threshold all set to 0.5. Accumulating gradient is leveraged, which is summing up the gradient and make one step of gradient descent in each two iterations. We do not use any other data augmentation on YOLOv3 and DG-YOLO unless we mention it.
\begin{table*}[!ht]
\scriptsize
\caption{The performance of model augmented by different types of water quality. Evaluation is performed on URPC2019 validation set of different types of water quality. All methods are implemented on YOLOv3.}
\vspace{7pt}
\centering
\setlength{\tabcolsep}{0.8mm}
\begin{tabular}{ccccccccc}
\hline
& & & & Evaluation (mAP) & & & & \\
\cline{2-9}
Method & val\_ori & val\_type1 & val\_type2 & val\_type3 & val\_type4 & val\_type5 & val\_type6 & val\_type7 \\
\hline
baseline & 56.45 & 18.72 & 16.83 & 26.57 &10.71 &23.66 &9.38 &29.04 \\
ori+type1 & 56.66 & 52.39 & 27.66 & 42.07 & 14.25 & 42.79 &20.96 &41.07 \\
ori+type2 & 56.71 & 18.90 & 51.86 & 39.44 & 24.89 & 34.51 & 6.21 & 45.85 \\
ori+type3 & 57.78 & 18.01 & 29.96 & 53.10 & 15.63 & 35.07 & 5.68 & 41.20 \\
ori+type4 & 58.33 & 16.80 & 33.50 & 41.57 & 53.85 & 35.52 & 3.72 & 42.30 \\
ori+type5 & 57.63 & 35.35 & 30.73 & 42.04 & 20.04 & 53.12 & 19.41 & 42.28 \\
ori+type6 & 57.19 & 21.64 & 35.63 & 42.19 & 24.37 & 36.04 & 51.22 & 46.15 \\
ori+type7 & 58.43 & 7.57 & 34.81 & 39.23 & 15.52 & 32.77 & 3.88 & 52.36 \\
\textbf{Full\_WQT} & \textbf{58.56} & \textbf{55.93} & \textbf{53.60} & \textbf{57.48} & \textbf{54.95} & \textbf{56.08} & \textbf{53.51} & \textbf{54.29} \\
\hline
ori+rot+flip & 62.53 & 14.81 & 18.29 & 31.36 & 8.89 & 24.95 & 5.34 & 33.18 \\
\textbf{Full\_WQT+rot+flip} & \textbf{63.83} & \textbf{60.57} & \textbf{57.71} & \textbf{60.38} & \textbf{58.96} & \textbf{59.84} & \textbf{58.43} & \textbf{60.53} \\
\hline
\end{tabular}
\label{tab:wqt performance}
\end{table*}
\subsection{Experiments of WQT}
\label{sec:WQT}
In this subsection, we analyze why WQT works. In Table \ref{tab:wqt performance}, \emph{Ori} means original URPC dataset, \emph{baseline} means YOLOv3 is trained only on original dataset, and \emph{ori+type1} means YOLOv3 is trained with original dataset and type1 dataset. \emph{Full\_WQT} means YOLOv3 is trained across type1 to type7. From Table \ref{tab:wqt performance}, we can find three interesting points:
(1) Compared to \emph{baseline}, it can be concluded that every group of augmentation improves the performance in original validation dataset. WQT can be used together with other data augmentation methods to obtain higher performance (last two rows of Table \ref{tab:wqt performance}), which further proves its effectiveness. Besides, there is a phenomenon that WQT also helps model generalize better on other type of water quality in most of the cases. For example, \emph{ori+type7} evaluates on type3 get mAP 39.23\%, 12.66\% higher than \emph{baseline}.
(2) We believe that there is a correlation between performance and similarity between water qualities. First, we use style loss proposed by \cite{gatys2016image} to represent style distance, and calculate the style distance between different types water quality. We feed style image type1 to type7 into $WCT^2$, and extract the feature maps at certain layers from both encoder and decoder, calculating style loss between any two types and obtaining $H_{dist}$. The result is shown in Table \ref{tab:styleloss}. Second, we take the data from column 3 to 9 (\emph{val\_+type1} to \emph{val\_type7}) and row 2 to 8 (model \emph{ori+type1} to \emph{ori+type7}) in Table \ref{tab:wqt performance}, subtracting each row of this 7 $\times$ 7 matrix to the performance of corresponding type of \emph{baseline}, getting $H_{perf}$. Using Pearson Correlation Coefficient and taking negative, it can be found that the correlation coefficient between style and performance is \emph{0.4634}. From this analysis, it can be inferred that the increase of generalization capacity gaining from WQT is from the similarity between different types of water quality.
(3) To further analyze the finding of (2), model is evaluated on \emph{val\_type8} which is a very different style from type1 to type7. There is no doubt that the WQT-trained model will perform not only better on original dataset, but also across type1 to type7 dataset. However, the model still fails on \emph{val\_type8} (see Table \ref{tab:ablation}), which is far from the requirement of a GUOD. WQT is not enough for domain generalization.
\renewcommand\arraystretch{1.2}
\begin{table}[h]
\scriptsize
\caption{Style distance between types of water quality.}
\vspace{7pt}
\centering
\setlength{\tabcolsep}{0.8mm}
\begin{tabular}{cccccccc}
\hline
& type1 & type2 & type3 & type4 & type5 & type6 & type7 \\
\hline
type1 & 0 & 0.6281 & 0.1105 & 0.6893 & 0.0495 & 0.7239 & 0.6286 \\
type2 & 0.6281 & 0 & 0.2860 & 0.0052 & 0.3311 & 0.0077 & 0.0033 \\
type3 & 0.1105 & 0.2860 & 0 & 0.3435 & 0.0411 & 0.3575 & 0.2977 \\
type4 & 0.6893 & 0.0052 & 0.3435 & 0 & 0.3747 & 0.0074 & 0.0037 \\
type5 & 0.0495 & 0.3311 & 0.0411 & 0.3747 & 0 & 0.4024 & 0.3308 \\
type6 & 0.7239 & 0.0077 & 0.3575 & 0.0074 & 0.4024 & 0 & 0.0094 \\
type7 & 0.6286 & 0.0033 & 0.2977 & 0.0037 & 0.3308 & 0.0094 & 0 \\
\hline
\end{tabular}
\label{tab:styleloss}
\end{table}
\renewcommand\arraystretch{1.2}
\begin{table}[!ht]
\scriptsize
\caption{Comparisons with other object detector and ablation study of DG-YOLO. IoU, NMS and Conf threshold all set to 0.5. The best checkpoints on \emph{ori} and \emph{val\_type8} are selected respectively for different methods.}
\vspace{7pt}
\centering
\setlength{\tabcolsep}{0.5mm}
\begin{tabular}{cccccccc}
\hline
& & & & val\_type8 (mAP) & & & \\
\cline{3-8}
Method & ori & echinus & starfish & holothurian & scallop & waterweeds & ave. \\
\hline
baseline (YOLOv3) & 56.45 & 53.51 & 7.32 & 11.15 & 9.89 & 0 & 16.37 \\
WQT-only & \textbf{58.56} & 60.98 & 17.08 & 33.29 & \textbf{39.02} & 2.38 & 30.55 \\
\hline
Faster-RCNN+FPN & 58.20 & 29.49 & 5.91 & 9.13 & 1.07 & 10.40 & 11.23 \\
SSD512 & 56.51 & 26.62 & 14.44 & 18.07 & 1.41 & 14.5 & 15.22 \\
SSD300 & 50.66 & 27.31 & 14.57 & 13.62 & 3.01 & 2.98 & 12.31 \\
\textbf{WQT+DG-YOLO} & 54.81 & \textbf{63.84} & \textbf{27.37} & \textbf{35.64} & 36.88 & \textbf{5.11} & \textbf{33.77} \\
\hline
WQT+DIM & 58.06 & 58.78 & 18.55 & 26.64 & 21.82 & 4.39 & 26.03 \\
WQT+$P_{IRM}$ & 57.01 & 54.99 & 25.98 & 32.90 & 29.25 & 0 & 30.63 \\
\hline
\end{tabular}
\label{tab:ablation}
\end{table}
\subsection{Experiments of DG-YOLO}
\label{sec:DG-YOLO experiment}
\textbf{The effectiveness of DG-YOLO}. WQT helps YOLOv3 to learn domain-invariant information, but the model still suffers from domain shift severely. In Table \ref{tab:ablation}, it is shown that DG-YOLO further digs domain-invariant information from data, obtaining 3.21\% mAP improvement on \emph{val\_type8} compared to \emph{WQT-only}. Besides, compared with other object detectors on \emph{val\_type8} performance, DG-YOLO shows its much better domain generalization capacity.
\textbf{Ablation study}.
The result of ablation study is shown in Table \ref{tab:ablation}. Compared to \emph{WQT-only} on \emph{val\_type8}, \emph{WQT+DIM} has 4.52\% performance decrease and \emph{WQT+$P_{IRM}$} has little improvement. However, \emph{WQT+DG-YOLO} achieves 3.21\% improvement, which suggests only by combining DIM and $P_{IRM}$ can lead to better performance.
\begin{figure}[h]
\centering
\includegraphics[width=7cm,height=2cm]{smooth_grad}
\caption{SmoothGrad Visualization on "echinus" samples. Images are from original UPRC2019 dataset.}
\label{fig:smoothgrad}
\end{figure}
\textbf{Visualization of DG-YOLO}.
One thing that can not be ignored is that there is performance decrease in original validation dataset of \emph{WQT+DG-YOLO}. It is because \emph{WQT-only} is "cheating", learning spurious correlation to make predictions. For example, waterweeds are green in greenish water, but they may turn black in another type of water. Therefore, color of the object is not a domain-invariant information, although it is convenient to use this spurious correlation to achieve good result in just one domain. The performance decrease of DG-YOLO can be interpreted that the model abandons the domain-related information and tries to learn domain invariant information from dataset. We use SmoothGrad \cite{smilkov2017smoothgrad} visualization technique to prove our hypothesis, finding the area that make model to believe there is echinus with probability higher than 95\%. As is shown in Figure \ref{fig:smoothgrad}, \emph{baseline} focuses on the shadow on the top left of image where there is no echinus. The pixel that \emph{WQT-only} focuses is too dispersed, which means \emph{WQT-only} learns spurious correlation. And the pixel DG-YOLO focuses is concentrated and exactly lie on the place where there is echinus. The visualization shows that DG-YOLO learn more semantic information than \emph{baseline} and \emph{WQT-only}.
\section{Conclusion}
This paper propose a data augmentation method WQT and a novel model DG-YOLO to overcome three challenges a GUOD faces: limited data, real-time processing and domain shift. Leveraging $WCT^2$, WQT is intended to increase domain diversity of original dataset. With DIM and IRM penalty, DG-YOLO can further mine semantic information from dataset. Experiments on original and synthetic URPC2019 dataset prove remarkable domain generalization capacity of our method. However, since the performance of DG-YOLO in an unseen domain can still not reach similar level as that in the seen domains, there is still a lot to explore in this field.
\bibliographystyle{IEEEbib}
|
1,108,101,565,694 | arxiv | \section{Introduction}
We describe FAR (Forward Abstracted Reachability), an algorithm for
fully automatic verification of parameterized software systems. A
parameterized system describes a family of programs such as cache
coherence protocols where the number of processes involved can change
but the algorithm handling their behaviour is the same for all of
them. Thus, the parameter allows to talk about these algorithms
without knowing the actual number of processes that will be involved
and then to prove its safety regardless of this number.
Safety properties state that ``nothing bad happens'' in our
parameterized system. Verifying them
can be reduced to finding an invariant of it. Finding an invariant can
be hard (and even undecidable \cite{Abdulla96}). The standard approach
to find one is to find a formula $\Phi$ such that $\Phi$ is an
inductive invariant of the system (\textit{i.e.} the initial state of
the system satisfies $\Phi$ and taking a transition from a state
satisfying $\Phi$ leads to another state satisfying $\Phi$).
In this paper we describe an algorithm for the automatic construction
of inductive invariants for array-based systems
(Section~\ref{sec:abs}). This algorithm, based on both IC3
\cite{bradley} and Lazy Abstraction \cite{mcmillan}, builds an
inductive invariant by unwinding a graph (Section~\ref{sec:unwinding})
building a forward abstract reachability of our system. This unwinding
is described as a set of non deterministic rules. We then provide an
implementation in Cubicle \cite{thesemebsout},
\cite{cav2012}, \cite{fmcad2013} (Section~\ref{sec:implementation}) of these
rules and test its effectiveness on several cache coherence protocols
(Section~\ref{sec:benchmarks}).
\section{Array-based Systems}
\label{sec:abs}
An array-based system is described in \cite{mcmt-foundations} as
first-order logic formulas on arrays. Such a system can be described
as a set of basic types, a set $X$ of \textit{system variables}
associated to type (built as usual with basic types and standard
constructions), a formula \textit{init} representing the initial
states and a set $\Delta{}$ of transition rules $\tau{}_i(X, X')$
($X'$ is the set $X$ where all the variables are primed which
represents the \emph{next state} reached after the application of a
transition). Since we work on \emph{parameterized} programs, our
arrays are indexed by an infinite type \textit{proc}.
We describe the Dekker mutual exclusion algorithm
as an array-based system. Each process has two boolean variables,
\textit{want} (stating that the process wants to enter in critical
section or not) and \textit{crit} (stating that the process is in
critical section or not). There is a global variable \textit{turn} of
type \textit{proc} that tracks which process can go into the critical
section. Since we work in the array-based systems fragment, we
represent the local variables as arrays indexed by processes and
containing booleans. The set $X$ contains two arrays,
\textit{want[proc] : bool} and \textit{crit[proc] : bool} and the
global variable \textit{turn : proc}. Initially, no process is or
wants to be in critical section. Three transitions can be triggered,
one to require an access to the critical section, one to enter in it
and one to exit it. According to the previous description, we
write this algorithm as in Figure~\ref{fig:dekker_abs}.
\begin{figure}[h]
\scalebox{.8}{
\vbox{%
\centering
\[
\begin{array}{ll}
\mathtt{turn} : proc
& \\
\mathtt{crit}[proc] : bool
& \\
\mathtt{want}[proc] : bool
& \\
& \\
init: \forall p. \hspace*{-0.5em}
&\neg \mathtt{want}[p] ~\land~ \neg \mathtt{crit}[p]\\[1em]
\multirow{2}{*}{$req: \exists p.$} \hspace*{-0.5em}
& \neg \mathtt{want}[p] \\
& \mathtt{want}'[p] \\[1em]
\multirow{2}{*}{$enter: \exists p.$} \hspace*{-0.5em}
& \mathtt{want}[p] ~\land~ \mathtt{turn} = p \\
&\mathtt{crit}'[p] \\[1em]
\multirow{2}{*}{$exit:\exists p_1,~ p_2.$} \hspace*{-0.5em}
& \mathtt{crit}[p_1] \\
& \neg\mathtt{want}'[p_1] \land \neg\mathtt{crit}'[p_1]\\
& \land \mathtt{turn}' = p_2 \\[1em]
\end{array}
\]
}
}
\caption{Dekker algorithm as array-based system}
\label{fig:dekker_abs}
\end{figure}
Since we focus on safety problems (\textit{nothing bad happens}),
we need to define what is considered as \emph{bad states}. In that
case these states would be defined with the following formula :\\[-2.1em]
\[unsafe \equiv \exists p_1,~ p_2.~ p_1 \neq p_2 ~\land~
\mathtt{crit}[p_1] ~\land~ \mathtt{crit}[p_2]\]
Our goal is then to prove that no state represented by $unsafe$ is
reachable from $init$ (which can be seen as : there exists no path
$init = X_0 \xrightarrow{p_1} X_1 \xrightarrow{\dots} \dots
\xrightarrow{p_n} X_n = unsafe$
with $p_i \in \{req, enter, exit\}$). To do so on parameterized
system, one of the main algorithm came from Ghilardi et
al. with MCMT \cite{ghilardiMCMT} and builds the set of all
reachable states by \emph{backward reachability} (\textit{starting,
then, from the unsafe state}) and checks if this set contains an
initial state. In this paper, we implement a different algorithm
which offers a wider range of possibilities in terms of reachabilty
construction.
\begin{figure*}[!t] \scriptsize
\centering
\scalebox{0.8}{
\vbox{
\begin{subfloat}[Extend]{
\begin{tikzpicture}[]
\node (BadN) at (0,0) [noeudex]
{
$\top$
\nodepart{second}$\mathcal{U}$
};
\node (root) at (0,1.8) [noeudex]
{
$\mathcal{I}$
\nodepart{second}$\bot$
};
\path[myarrow] (root) edge node[right]{\te{req}} (BadN);
\end{tikzpicture}
}
\end{subfloat}
\hfill
\begin{subfloat}[Refine]{
\begin{tikzpicture}[]
\node (BadN) at (-1,0) [noeudex]
{
$\top$
\nodepart{second}$\mathcal{U}$
};
\node (root) at (0,1.8) [noeudex]
{
$\mathcal{I}$
\nodepart{second}$\bot$
};
\node (node1) at (1,0) [noeudex]
{
$\neg \mathcal{U}$
\nodepart{second}$\bot$
};
\path[myarrow] (root) edge node[right]{\te{req}} (node1);
\end{tikzpicture}
}
\end{subfloat}
\hfill
\begin{subfloat}[Extend]{
\begin{tikzpicture}[]
\node (BadN) at (-1,0) [noeudex]
{
$\top$
\nodepart{second}$\mathcal{U}$
};
\node (root) at (0,1.8) [noeudex]
{
$\mathcal{I}$
\nodepart{second}$\bot$
};
\node (node1) at (1,0) [noeudex]
{
$\neg \mathcal{U}$
\nodepart{second}$\bot$
};
\path[myarrow] (root) edge node[right]{\te{req}} (node1);
\path[myarrow] (node1) edge[bend right] node[above]{\te{req}} (BadN);
\end{tikzpicture}
}
\end{subfloat}
\hfill
\begin{subfloat}[Cover]{
\begin{tikzpicture}[]
\node (BadN) at (-1,0) [noeudex]
{
$\top$
\nodepart{second}$\mathcal{U}$
};
\node (root) at (0,1.8) [noeudex]
{
$\mathcal{I}$
\nodepart{second}$\bot$
};
\node (node1) at (1,0) [noeudex]
{
$\neg \mathcal{U}$
\nodepart{second}$\bot$
};
\path[myarrow] (root) edge node[right]{\te{req}} (node1);
\path[myarrow] (node1) edge[out=5, in=-5, loop] node[above]{\te{req}} (node1);
\end{tikzpicture}
}
\end{subfloat}
\begin{subfloat}[Final state]{
\begin{tikzpicture}[]
\node (BadN) at (-0.8,0) [noeudex]
{
$\top$
\nodepart{second}$\mathcal{U}$
};
\node (root) at (-0.8,-1) [noeudex]
{
$\mathcal{I}$
\nodepart{second}$\bot$
};
\node (node1) at (0.8,0) [noeudex]
{
$\neg \mathcal{U}$
\nodepart{second}$\gamma_1$
};
\node (node2) at (2.6,0) [noeudex]
{
\nodepart[text width=0.7cm]{one}$\neg \mathcal{U} \land \neg \gamma_1$
\nodepart{second}$\gamma_2$
};
\node (node3) at (1.9,-1.5) [noeudex]
{
\nodepart[]{one}$\neg \mathcal{U} \land \neg \gamma_1 \neg \gamma_2$
\nodepart{second}$\bot$
};
\path[myarrow] (root) edge[bend left] node[pos=0.4,below]{\te{req}} (node3);
\path[myarrow] (node1) edge[loop above] node[above]{\te{req}} (node1);
\path[myarrow] (node1) edge[bend right] node[above]{\te{enter}} (BadN);
\path[myarrow] (node2) edge[bend left] node[below]{\te{req}} (node1);
\path[myarrow] (node3) edge[loop below] node[below]{\te{req}} (node3);
\path[myarrow] (node3) edge[loop left] node[left]{\te{enter}} (node3);
\path[myarrow] (node3) edge[loop right] node[right]{\te{exit}} (node3);
\end{tikzpicture}
}
\end{subfloat}
}
}
\caption{First four steps of the unwinding and final state of the
graph (\textit{sink rules are not shown)}}
\label{fig:dekker_run}
\end{figure*}
\section{Program unwinding}
\label{sec:unwinding}
This algorithm starts also from the $unsafe$ formula but tries to
build an invariant of the system that does not contain it. Before
going into details, we give a brief explanation. This invariant is
iteratively built as an inductive invariant $\Theta$ that does not
contain $unsafe$ :
\begin{itemize}
\item if $\Theta ~\land~ \Delta ~\land~ \neg\Theta'$ is
unsatisfiable then we found an inductive invariant
\item if $\Theta ~\land~ \Delta ~\land~ \neg\Theta'$ is
satisfiable, our candidate invariant is not inductive and we try to
refine it until we either discover that there is no such refinement
or we find some.
\end{itemize}
\noindent
For Dekker's algorithm, for example, let's take $\Theta = \neg unsafe
= \forall p_1\neq p_2. \neg crit[p_1] \lor \neg crit[p_2]$
:
\begin{itemize}
\item $\Theta ~\land~ \Delta ~\land~ \neg\Theta'$ is satisfiable (if
we take, for example, the following state : $\varphi_1 = crit[p_1] ~\land~
want[p_2] ~\land~ turn = p_2 ~\land~ \neg crit[p_2]$, $\varphi_1
\models \Theta$ but if we apply $enter$ to it we obtain the state
$\varphi_2 = crit[p_1] ~\land~ crit[p_2] ~\land~ \dots$ and $\varphi_2
\not\models \Theta$.)
\item We need to create $\Theta' = \Theta ~\land~ \rho$ which is a
refinement of $\Theta$ that does not contain $\varphi_1$.
\end{itemize}
\noindent
To do so, we build an \emph{unwinding} of the algorithm as a quadruple
$\langle V, E, \mathcal{W}, \mathcal{B} \rangle$, where:
\begin{itemize}
\item $\langle V, E\rangle$ is a rooted graph with edges
labeled by transitions from $\Delta$;
\item $\mathcal{W}$ associates a formula (called \emph{world
of the vertex}) to each vertex;
\item $\mathcal{B}$ associates a formula (called \emph{bad
part of the vertex}) to each vertex.
\end{itemize}
This graph contains three initial vertices :
\begin{enumerate}
\item[$\epsilon$] : the root vertex, $\mathcal{W}(\epsilon) = init$ and
$\mathcal{B}(\epsilon) = \bot$;
\item[$\beta$] : the unsafe vertex, $\mathcal{W}(\beta) = \top$ and
$\mathcal{B}(\beta) = unsafe$;
\item[$\omega$] : the sink vertex, $\mathcal{W}(\omega) = \bot$ and
$\mathcal{B}(\omega) = \bot$.
\end{enumerate}
We call $V^\epsilon = \{v \in V,~ \epsilon \xrightarrow{*}v \in E\}$
(\textit{i.e.} the set of vertices that are linked to the
root). $\mathcal{W}(v) \models_\tau \mathcal{W}(v') \equiv \mathcal{W}(v)
~\land~ \tau \models \mathcal{W}(v')$
\\[1em]
\indent The idea behind this unwinding it that if we manage to create a graph
$\mathcal{G}$ of a system $\mathcal{S} =\langle init, \Delta \rangle$
where every vertex in $V^\epsilon$ does not contain a bad part and
from which no more transition can be taken, then the disjunction of
their worlds ($\Theta = \bigvee_{v \in V^\epsilon}\mathcal{W}(v)$) is an
invariant of the system ($init \models \Theta$ and $\Theta
\models_\Delta \Theta$).
We now propose a set of non-deterministic rules for building this
unwinding. Let $\langle X, init, \Delta, unsafe\rangle$ be an
array-based system. Initially, $\mathcal{G}$ is defined as follow :
\begin{enumerate}
\item[-] $V = \{\epsilon, \omega, \beta\}$
\item[-] $E = \emptyset$
\end{enumerate}
The unwinding works by the non-deterministic application of the
following rules :
\begin{myrule}[\textbf{Extend}]
If $\exists v \in V, \tau \in \Delta.~\mathcal{W}(v) \models_\tau \top$
and $\nexists v'. v \xrightarrow{\tau} v' \in E$ then $E =
E \cup \{v \xrightarrow{\tau} \beta\}$
\end{myrule}
\begin{myrule}[\textbf{Refine}]
If $\exists v, v' \in V, \tau \in \Delta.~ v \xrightarrow{\tau} v'
\in E$, $\mathcal{B}(v') \neq \bot$, $\exists \varphi. \mathcal{W}(v)
\models_\tau \varphi$ and $\varphi \models \neg \mathcal{B}(v')$ then we
create a new vertex $v''$ such that $\mathcal{W}(v'') = \mathcal{W}(v')
~\land~ \varphi$ and $E = E \cup \{v \xrightarrow{\tau} v''\}
\setminus \{v \xrightarrow{\tau} v'\}$
\end{myrule}
\begin{myrule}[\textbf{Propagate}]
If $\exists v, v' \in V, \tau \in \Delta.~ v \xrightarrow{\tau} v'
\in E$, $\mathcal{B}(v') \neq \bot$, $\exists \gamma.~ \gamma \models
\mathcal{W}(v)$, and $\gamma \models_\tau \mathcal{B}(v')$ then $B(v)
\leftarrow \gamma$
\end{myrule}
\begin{myrule}[\textbf{Cover}]
If $\exists v, v' \in V, \tau \in \Delta.~ v \xrightarrow{\tau} v'
\in E$, $v'' \in V$ such that $\mathcal{W}(v'') \models \mathcal{W}(v')$ and
$\mathcal{W}(v) \models_\tau \mathcal{W}(v'')$ then $E = E \cup \{v
\xrightarrow{\tau} v''\} \setminus \{v \xrightarrow{\tau} v'\}$
\end{myrule}
\begin{myrule}[\textbf{Sink}]
If $\exists v \in V, \tau \in \Delta.~\mathcal{W}(v) \models_\tau \bot$
and $\nexists v'. v \xrightarrow{\tau} v' \in E$ then $E =
E \cup \{v \xrightarrow{\tau} \omega \}$
\end{myrule}
$S$ is safe if and only if no rule can be applied to $\mathcal{G}$, an
unwinding $\mathcal{S}$ and $\mathcal{B}(\epsilon) = \bot$. Intuitively,
since no more transitions can be taken and all the vertices connected
to the root are not bad, root will never be able to lead to unsafe.
\section{Example}
\label{sec:example}
The example shown in Figure~\ref{fig:dekker_run} describes the first
four runs of the unwinding on the Dekker's algorithm (we decided not
to show $\omega$ since it just serves as a sink for the transitions
that can not be taken from a vertex) :
\begin{itemize}
\item[(a)] initially, the only rule that can be applied is the
\textbf{Extend} rule from $\epsilon$ (with $\mathcal{W}(\epsilon) \equiv
init \equiv \forall p.~ \neg \mathtt{want}[p] ~\land~ \neg
\mathtt{crit}[p]$) with $req$;
\item[(b)] we can only apply, then, the \textbf{Refine} rule because
$init \nvDash_{req} unsafe \equiv \mathcal{B}(\beta)$. We create a new vertex
called $v_1$;
\item[(c)] we can chose, here, to apply the \textbf{Extend} rule from
the new vertex with any transition. We chose to take the transition $req$;
\item[(d)] $\mathcal{W}(v_1) \nvDash_{req} \mathcal{B}(\beta)$ and $\mathcal{W}(v_1)
\models_{req} \mathcal{W}(v_1)$ so we can apply the cover rule;
\item[(e)] if we keep applying these rules, we reach a fixpoint at
the third created vertex.
\end{itemize}
\section{Implementation}
\label{sec:implementation}
We implemented this unwinding in Cubicle
\footnote{\url{cubicle.lri.fr/far}}. To do so we had to chose a
deterministic strategy depending on multiple parameters :
\begin{itemize}
\item the order in which the rules are applied;
\item which vertex and transition should be taken for the
\textbf{Extend} rule;
\item which formula $\varphi$ should we take for the \textbf{Refine}
rule;
\item which formula $\gamma$ should we take for the \textbf{Propagate}
rule;
\item which vertex $v''$ should we take for the \textbf{Cover} rule.
\end{itemize}
Based on this problems, we came out with the following algorithm (we
write $v = (W, B)$ to denote the fact that $\mathcal{W}(v) = W$ and
$\mathcal{B}(v) = B$) :
\begin{algorithm}
\caption{Graph unwinding - main loop}\label{alg:main_loop}
\begin{algorithmic}[]
\Procedure{far-cubicle}{$\mathcal{S} = \langle init, \Delta, unsafe\rangle$}
\State $\epsilon \gets (init, \bot)$
\State $\beta \gets (\top, unsafe)$
\State $\omega \gets (\bot, \bot)$
\State $V\gets \{\epsilon, \beta, \omega\}$
\State $E = \emptyset$
\State \Call{push}{$\mathcal{Q}$, $\epsilon$}
\Comment{$\mathcal{Q}$ is a priority queue}
\While{\Call{not\_empty}{$\mathcal{Q}$}}
\State $v\gets$\Call{pop}{$\mathcal{Q}$}
\ForAll{$\tau \in \Delta$}
\If{$\mathcal{W}(v) \models_\tau \top$}
\State $E = E \cup \{v \xrightarrow{\tau} \beta\}$
\State \Call{unwind}{$v \xrightarrow{\tau} \beta$}
\Else $~E = E \cup \{v \xrightarrow{\tau} \omega\}$
\EndIf
\EndFor
\EndWhile
\State \Return \textcolor{green}{\textbf{safe}}
\EndProcedure
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{Graph unwinding - unwinding procedure}\label{alg:main_loop}
\begin{algorithmic}[]
\Procedure{unwind}{$v \xrightarrow{\tau} v'$}
\If{$\mathcal{B}(v) = \bot ~\land~ \mathcal{B}(v') \neq \bot$}
\Switch{\Call{close}{$v \xrightarrow{\tau} v'$}}
\Case{Covered v''}
\State $E = E \cup \{v \xrightarrow{\tau} v''\} \setminus \{v
\xrightarrow{\tau} v'\}$
\State \Call{unwind}{$v \xrightarrow{\tau} v''$}
\EndCase
\Case{Bad $\varphi$}
\If{$v = \epsilon$} \Return \textcolor{red}{\textbf{unsafe}}
\Else
\State $\mathcal{B}(v) \gets \varphi$
\ForAll{$u \xrightarrow{\tau'} v$}
\State \Call{unwind}{$u \xrightarrow{\tau} v$}
\EndFor
\EndIf
\EndCase
\Case{Refined v''}
\State $E = E \cup \{v \xrightarrow{\tau} v''\} \setminus \{v
\xrightarrow{\tau} v'\}$
\State \Call{push}{$\mathcal{Q}, v''$}
\EndCase
\EndSwitch
\EndIf
\EndProcedure
\end{algorithmic}
\end{algorithm}
This algorithm picks a vertex $v$ from a priority queue (which initially
contains only the root vertex) and for all the transitions, adds an edge
to the graph from this transition to the sink vertex if the formula
represented by $v$ is inconsistent with the transition or to
the unsafe vertex if the transition can be taken. If the edge goes to
a vertex $v'$ that is not the sink, the procedure \textsc{unwind} is called
on it. This procedure checks if $\mathcal{B}(v) \neq \bot$ or if
the $\mathcal{B}(v') = \bot$ and if both these conditions are false it
tries to \emph{close} the edge. An edge is \emph{closed} if :
\begin{itemize}
\item $\mathcal{W}(v) \models_\tau \mathcal{B}(v')$. In this case, all
the edges coming to it must be unwinded again;
\item there exists another vertex $v''$ such that $v \models_\tau v''$
and $\mathcal{v''} \models \mathcal{W}(v')$. In this case, the edge
from $v$ to $v'$ is deleted and a new one from $v$ to $v''$ is
created and unwinded;
\item $\mathcal{W}(v) \not\models_\tau \mathcal{B}(v')$. A counter example $\varphi$ is
found a new node $v''$ is created with $\mathcal{W}(v'') \equiv
\mathcal{W}(v') \land \varphi$ and pushed in the queue.
\end{itemize}
If all the edges are closed and the queue is empty, the system is
safe. If the propagation of bad parts reaches the root vertex, the
system is unsafe.
\begin{algorithm}
\caption{Graph unwinding - closing edge procedure}\label{alg:main_loop}
\begin{algorithmic}[1]
\Procedure{close}{$v \xrightarrow{\tau} v'$}
\If{$\exists v''.~ \mathcal{W}(v'') \models \mathcal{W}(v') ~\land~
\mathcal{W}(v) \models_\tau \mathcal{W}(v'')$}
\State\Return Covered $v''$
\ElsIf{$\mathcal{W}(v) \models_\tau \mathcal{B}(v')$}
\State\Return Bad \Call{pre}{$\mathcal{B}(v'), \tau$}\label{lbl:pre_bad}
\Else
\State $v'' \gets (\mathcal{W}(v') ~\land~$ \Call{generalize}{$\neg \mathcal{B}(v')}, \bot)$\label{lbl:neg_refine}
\State\Return Refined v''
\EndIf
\EndProcedure
\end{algorithmic}
\end{algorithm}
As we can see on line~\ref{lbl:pre_bad} of the \textsc{close}
procedure, the formula $\gamma$ chosen for the \textbf{Propagate} rule
is the pre image of the bad formula of the vertex $v'$.
Also, on line~\ref{lbl:neg_refine} of the \textsc{close}
procedure, the formula $\varphi$ chosen for the \textbf{Refine} rule
is a generalization of the negation of the bad part of the vertex
$v'$. These are, of course,
implementation choices. Other implementation could involve \emph{model
finding}, \emph{interpolants} ... In our case, the generalization is
a naive one consisting in taking the smallest part of the
resulting formula that was not already taken and that still satifies
the conditions of the \textbf{Refine} rule.
\section{Benchmarks}
\label{sec:benchmarks}
We compared our implementation to the backward reachability algorithm
already implemented in Cubicle (without the invariants inference
implemented with BRAB \cite{thesemebsout}, \cite{fmcad2013}) and
obtained the following results (the
timeout was set to 5 minutes and the $\alpha$ version uses an
abstraction engine related to the approximation implemented in BRAB to
get better refinement):\\[1em]
\begin{tabular}[h]{|c|c|c|c|}
Protocol & Cubicle & FAR & FAR-$\alpha$ \\
\hline
\texttt{dekker}& 0.04s & 0.04s & 0.03s\\
\texttt{mux\_sem} & 0.04s & 0.05s & 0.03s \\
\texttt{german}-ish & 0.06s & 0.1s & 0.55s\\
\texttt{german}-ish2 & 0.13s & 0.11s & 0.65s\\
\texttt{german}-ish3 & 1.2s & 8.3s & 0.65s \\
\texttt{german}-ish4 & 3.5s & 2.5s & 0.75s\\
\texttt{german}-ish5 & 1.9s & 8.2s & 0.60s\\
\texttt{german} & 18s & 5.8s & 4.25s \\
\texttt{szymanski\_at} & TO & 13s & 2.60s \\
\texttt{szymanski\_na} & TO & TO & 16s
\end{tabular}
\vspace*{1em}As we can see in this table, this algorithm is
competitive and even better when good refinements can be found.
\section{Related Works}
There has been a lot of research in software model checking and
Property-Driven Reachability. This type of algorithm was first
introduced by Bradley in \cite{bradley} and McMillan revisited his
Lazy Annotation (which shares similarities with PDR algorithms) in
\cite{mcmillan_revisited} or the recent approach from Cimatti et
al. \cite{cimattiIC3} and Z3 with a PDR approach in
\cite{DBLP:conf/sat/HoderB12} and \cite{DBLP:conf/cav/HoderBM11}.
Even though some of these tools are supposed to work on parameterized
systems, we were either not able to find them or they were not able to
prove our examples.
\section{Conclusion}
We presented the problem of parameterized protocol verification and
gave an algorithm to automatically do it. This new algorithm was
implemented in Cubicle and successfully applied to many
cache coherence protocols.
This algorithm could be improved with a better generalisation engine
(allowing to explore less vertices), an incremental approach (the
parameterized aspect of our language makes it hard to
\textit{remember} the state of our SMT solver). Other optimizations
could involve a novel way of refining our formulas (it is clear that
the best refinements are inductive invariants but it is still an open
problem as how to find these).
Some optimizations were not documented in this article such as
\begin{itemize}
\item \emph{Set-theoretic test} : some formulas are trivially
unsatisfiable and don't require call to the SMT solver;
\item \emph{relevant instantiations} : handling universally quantified
formulas can lead to multiple useless instantiations that are
trivially unsatisfiable or valid and do not help the SMT solver to
solve the whole formula. This optimization allows to gain a
significant time in the SMT solver.
\item \emph{selecting good bads} : handling bad parts from the ones
with less processes involved allows to control the number of
processes that have to be instantiated when checking the
satisfiability of formulas. It is mandatory, if we want to have a
competitive algorithm, that we handle the bad parts cleverly (this
can be done in the priority queue).
\end{itemize}
\bibliographystyle{IEEEtran}
|
1,108,101,565,695 | arxiv |
\section{Introduction}
Parity games are games played on a directed graph without leaves by
two players, Even (0) and Odd (1). A node has an owner (a player) and an integer
priority. A play is an infinite path in the graph where the owner of a node
chooses which outgoing edge to follow. A play and its nodes is
won by Even if the highest priority that occurs infinitely often is
even and by Odd otherwise. A parity game is solved when the winner of
every node is determined and proven.
Parity games are relevant for boolean equation
systems~\cite{mCRL2_10.1007/978-3-642-36742-7_15,DBLP:journals/corr/abs-1210-6414},
temporal logics such as LTL, CTL and
CTL*~\cite{DBLP:conf/dagstuhl/2001automata} and µ-calculus~\cite{DBLP:conf/stacs/Walukiewicz96,DBLP:conf/dagstuhl/2001automata}.
Many problems in these domains can be reduced to solving a parity game.
Quasi-polynomial time algorithm for solving them exist
\cite{DBLP:conf/stoc/CaludeJKL017,DBLP:conf/spin/FearnleyJS0W17,DBLP:conf/mfcs/Parys19}.
However, all current state-of-the-art algorithms (Zielonka's
algorithm~\cite{DBLP:journals/tcs/Zielonka98},
strategy-improvement~\cite{DBLP:conf/csl/Schewe08},
priority
promotion~\cite{DBLP:journals/iandc/BenerecettiDM18,DBLP:conf/cav/BenerecettiDM16,DBLP:conf/hvc/BenerecettiDM16}
and tangle learning~\cite{DBLP:conf/cav/Dijk18}) are exponential.
We start the paper with a short description of the role of parity game
solvers in the domain of formal verification
(Section~\ref{sec:verif}).
In Section~\ref{sec:ppg}, we recall the essentials of parity games
and introduce parametrized parity games as a generalization of parity
games.
In Section~\ref{sec:just} we recall justifications, which we
introduced in \cite{DBLP:conf/vmcai/LapauwBD20} to store winning
strategies and to speed up algorithms.
Here we introduce safe justifications and define a {\operation{Justify}}{} operation
and proof its properties.
Next, in Section~\ref{sec:recursive}, we reconstruct three algorithms
for solving parity games by defining different orderings over {\operation{Justify}}{}
operations. We conclude in Section~\ref{sec:conclusion}.
\section{Verification and parity game solving}\label{sec:verif}
Time logics such as LTL are used to express properties of interacting
systems. Synthesis consists of extracting an implementation with the
desired properties. Typically, formulas in such logics are handled by reduction
to other formalisms. LTL can be reduced to
Büchi-automata~\cite{DBLP:conf/lics/VardiW86,DBLP:conf/cav/KestenMMP93}, determinized with
Safra's construction~\cite{DBLP:conf/focs/Safra88}, and transformed to
parity games~\cite{DBLP:conf/lics/Piterman06}. Other modal logics
have similar reductions, CTL* can be reduced to
automata~\cite{DBLP:conf/cav/BernholtzVW94}, to
µ-calculus~\cite{DBLP:journals/tcs/CranenGR11}, and recently to
LTL-formulae~\cite{DBLP:journals/corr/abs-1711-10636}. All are
reducible to parity games.
One of the tools that support the synthesis of implementations for
such formulas is
Strix~\cite{DBLP:journals/acta/LuttenbergerMS20,DBLP:conf/cav/MeyerSL18},
one of the winners of the
SyntComp~2018~\cite{DBLP:journals/corr/abs-1904-07736} and
SyntComp~2019 competition. It reduces LTL formulas on the fly to
parity games. A game has three possible outcomes: (i) the parity game
needs further expansion,
(ii) the machine wins the game, i.e., an implementation is feasible,
(iii) the environment wins, i.e., no implementation exists.
Strix also extracts an implementation with the specified behaviour, e.g., as a Mealy machine.
Consider a formula based on the well-known dining philosophers problem:
\begin{equation}
\label{eq:philo}
\begin{array}{lll}
G (\mathit{hungry}_A \Rightarrow F \mathit{eat}_A) \land && \text{If A is hungry, he will eventually eat}\\
G (\mathit{hungry}_B \Rightarrow F \mathit{eat}_B) \land && \text{If B is hungry, he will eventually eat}\\
G (\lnot eat_A \lor \lnot eat_B) && \text{A and B cannot eat at the same time.} \\
\end{array}
\end{equation}
Here $(G\phi)$ means $\phi$ holds in every future trace and $(F \phi)$
means $\phi$ holds in some future trace where a trace is a succession
of states.
Strix transforms the LTL-formula~\ref{eq:philo} to the parity game of
Figure~\ref{fig:philo}. The
machine (Even) plays in the square nodes and the environment (Odd) in
the diamond nodes.
By playing in state $b$ to $d$, and in state $f$ to $h$, Even wins every node as 2 is then the highest priority that occurs infinitely often in
every play.
From the solution, Strix extracts a 2-state
Mealy machine (Figure~\ref{fig:mealy}). Its behaviour satisfies Formula~\ref{eq:philo}: both philosophers alternate eating regardless of their hunger.
\begin{figure}[tb]
\begin{minipage}{0.55\textwidth}
\centering{
\includegraphics[width=0.9\textwidth]{images/philo.pdf}
\caption{A reduced parity game.\label{fig:philo}}
}
\end{minipage}\begin{minipage}{0.4\textwidth}
\centering{
\includegraphics[width=\textwidth]{images/mealy.pdf}
\caption{The resulting Mealy machine with two
states, alternating $\neg eat_A, eat_B$ and $eat_A, \neg eat_B$ regardless of the input of $hungry_A$ and $hungry_B$.\label{fig:mealy}}
}
\end{minipage}
\end{figure}
\section{Parametrized parity games\label{sec:ppg}}
A parity game~\cite{mostowski1991games,DBLP:conf/focs/EmersonJ91,DBLP:conf/stacs/Walukiewicz96} is a two-player
game of player $0$ (Even) against $1$ (Odd). We use $\alpha \in
\set{0,1} $ to denote a player and $\oppon$ to denote its
opponent. Formally, we define a {\em parity game} as a tuple $\mathcal{PG} =
(V, E, O, Pr)$ with $V$ the set of nodes, $E$ the set of possible
moves represented as pairs $(v,w)$ of nodes, $O:V \to \{0,1\}$
the owner function, and $Pr$ the priority function $V \to \mathbb{N}$
mapping nodes to their priority; $(V,E)$ is also called the game
graph. Each $v\in V$ has at least one possible move.
We use $O_\alpha$ to denote nodes owned by $\alpha$.
A {\em play} (in node $v_1$) of the parity game is an infinite
sequence of nodes $\play{v_1, v_2,\dots, v_n\dots}$ where $\forall i:
v_i \in V \land (v_i, v_{i+1}) \in E$.
We use $\pi$ as a mathematical variable to denote a play. $\pi(i)$ is the $i$-th node $v_i$ of $\pi$.
In a play $\pi$, it is the owner of the node $v_i$ that decides the move $(v_i,v_{i+1})$.
There exists plays in every node.
We call the player $\alpha=(n\mod 2)$ the {\em winner of priority $n$}.
The winner of a play is
the winner of the highest priority $n$ through which the play passes infinitely often. Formally:
$\mathit{Winner}(\pi) = \allied{\lim_{i \to {+\infty}} max\set{Pr(\pi(j)) \middle| j \geq i}}.$
The key questions for a parity game $\mathcal{PG}$ are, for each node $v$: Who is the winner? And how?
As proven by \cite{DBLP:conf/focs/EmersonJ91}, parity games are memoryless determined: every node has a unique winner and a corresponding memoryless winning strategy.
A (memoryless) strategy for player $\alpha$ is a partial function $\sigma_\alpha$ from a subset of $O_\alpha$ to $V$. A play $\pi$ is consistent with $\sigma_\alpha$ if for every $v_i$ in $\pi$ belonging to the domain of $\sigma_\alpha$, $v_{i+1}$ is $\sigma_\alpha(v_i)$.
A strategy $\sigma_\alpha$ for player $\alpha$ is a {\em winning strategy} for a node $v$ if every play in $v$ consistent with this strategy is won by $\alpha$, i.e. regardless of the moves selected by $\oppon$.
As such, a game $\mathcal{PG}$ defines a winning function $W_\mathcal{PG}:V\mapsto
\{0,1\}$.
The set $W_{\mathcal{PG},\alpha}$ or, when $\mathcal{PG}$ is clear from the context,
$W_\alpha$ denotes the set of nodes won by $\alpha$.
Moreover, for both players $\alpha\in \{0,1\}$, there exists a memoryless winning strategy $\sigma_\alpha$ with domain $W_\alpha \cap O_\alpha$ that wins in all nodes won by $\alpha$. A {\em solution} of $\mathcal{PG}$ consists of a function $W':V \to \{0,1\}$ and two winning strategies $\sigma_0$ and $\sigma_1$, with $dom(\sigma_\alpha)=W'_\alpha\cap O_\alpha$, such that every play in $v\in W'_\alpha$ consistent with $\sigma_\alpha$ is won by $\alpha$. Solutions always exist; they may differ in strategy but all have $W'=W_{\mathcal{PG}}$, the winning function of the game. We can say that the pair $(\sigma_0,\sigma_1)$ proves that $W'=W_{\mathcal{PG}}$.
In order to have a framework in which we can discuss different
algorithms from the literature, we define a parametrized parity
game. It consists of a parity game $\mathcal{PG}$ and a parameter function $P$,
a partial function $P:V\rightharpoonup \{0,1\}$ with domain $dom(P) \subseteq
V$. Elements of $dom(P)$ are called parameters, and $P$ assigns a
winner to each parameter. Plays are the same as in a $\mathcal{PG}$ except that
every play that reaches a parameter $v$ ends and is won by $P(v)$.
\begin{definition}[Parametrized parity game\label{def:ppg}]
Let $\mathcal{PG} = (V, E, O, Pr)$ be a parity game and $P:V\rightharpoonup \{0,1\}$ a partial function with domain $dom(P) \subseteq V$. Then $(\mathcal{PG}, P)$ is a {\em parametrized parity game} denoted ${\mathcal{PG}_P}$, with parameter set $dom(P)$. If $P(v)=\alpha$, we call $\alpha$ the assigned winner of parameter $v$. The sets $P_0$ and $P_1$ denote parameter nodes with assigned winner 0 respectively 1.
A play of $(\mathcal{PG}, P)$ is a
sequence of nodes $\play{v_0, v_1, \dots}$ such that for all $i$: if
$v_i \in P_\alpha$ then the play halts and is won by $\alpha$,
otherwise $v_{i+1}$ exists and $(v, v_{i+1}) \in E$. For infinite
plays, the winner is as in the original parity game $\mathcal{PG}$.
\end{definition}
Every parity game $\mathcal{PG}$ defines a class of parametrized parity games
(PPG's), one for each partial function $P$. The original $\mathcal{PG}$
corresponds to one of these games, namely the one without parameters
($dom(P)=\emptyset$); every total function $P:V\to \{0,1\}$
defines a trivial PPG, with plays of length 0 and $P=W_{\mathcal{PG}_P}$.
A PPG ${\mathcal{PG}_P}$ can be reduced to an equivalent PG $G$: in each parameter $v \in dom(P)$ replace the outgoing edges with a self-loop and the priority of $v$ with $P(v)$. We now
have a standard parity game $G$. Every infinite play $\play{v_0, v_1,
\dots}$ in ${\mathcal{PG}_P}$ is also an infinite play in $G$ with the same winner.
Every finite play $\play{v_0, v_1, \dots, v_n}$ with winner $P(v_n)$ in ${\mathcal{PG}_P}$
corresponds to an infinite play $\play{v_0, v_1, \dots, v_n,v_n,\ldots
}$ with winner $P(v_n)$ in $G$. Thus, the two games are equivalent. It follows that any PPG ${\mathcal{PG}_P}$ is a zero-sum game defining a winning function $W$ and having memory-less winning strategies $\sigma_\alpha$ with domain $(W_\alpha \setminus P_\alpha) \cap O_\alpha$ (for $\alpha=0,1$).
PPG's allow us to capture the behaviour of several state of the
art algorithms as a sequence of solved PPG's. In each step,
strategies and parameters are modified and a solution for one PPG is
transformed into a solution for a next PPG and this until a solution for the input
PG is reached.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.6\textwidth]{images/game_para_e_e.pdf}
\end{center}
\caption{A parametrized parity game with nodes $a,\dots,f$, $P_0= \{d\}$ and $P_1 = \{a\}$, and winning strategies for $0$ and $1$. The two parameter nodes are in bold. Square nodes are owned by Even, diamonds by
Odd. The labels inside a node are the name and priority; the label on top of a node is the winner. A bold edge belongs to a winning strategy (of the owner of its start node). A slim edge is one starting in a node that is lost by its owner. All remaining edges are dotted.\label{ex:game_para}
}
\end{figure}
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.6\textwidth]{images/game_para_2_e_e.pdf}
\end{center}
\caption{{A parametrized parity game and strategy, after withdrawing $d$ from the parameter list.}
\label{ex:game_para_2}}
\end{figure}
\begin{example}\label{ex:first}
Figure~\ref{ex:game_para} shows a parametrized parity game and its
winning strategies. The parameter nodes $a$ and $d$ are won by the
assigned winners, respectively 1 and 0. Player 1 owns
node $c$ and wins its priority. Hence, by playing
from $c$ to $c$, 1 wins in this node. Node $b$ is owned by 0 but
has only moves to nodes won by 1, hence it is also won by 1. Player 0
wins node $e$ by playing to node $d$; 1 plays in node $f$ but
playing to $f$ results in an infinite path won by 0, while playing
to node $e$ runs into a path won by 0, so $f$ is won by 0.
Based on this PPG, we can construct a solved PPG where node $d$ is removed from the parameters. The strategy is adjusted accordingly: Odd wins in $d$ by playing to $c$ . However, changing the winner of $d$ breaks the strategies and winners of the nodes $e$ and $f$.
Figure~\ref{ex:game_para_2} shows one way to obtain a solved PPG with further adjustments: nodes $e$ and $f$ are turned into parameters won by $1$. Many other solutions exist, e.g., by turning $e$ into a parameter won by $0$.
\end{example}
\section{Justifications\label{sec:just}}
In Figure~\ref{ex:game_para} and Figure~\ref{ex:game_para_2}, the
solid edges form the subgraph of the game graph that was analysed to
confirm the winners of all nodes. We formalize this subgraph as a
{\em justification}, a concept introduced in
\cite{DBLP:journals/tplp/HouCD10} and described below. In the rest of
the paper, we assume the existence of a parity game
$\mathcal{PG}=(V,E,O,Pr)$ and a parametrized parity game $\mathcal{PG}_P=(\mathcal{PG},P)$
with $P$ a parameter function with set of parameters $dom(P)$. Also,
we use $H: V \to \{0,1\}$ as a function describing a
``hypothesis'' of who is winning in the nodes.
\begin{definition}[Direct justification]\label{def:just_win}
A {\em direct justification} $dj$ for player $\alpha$ to win node $v$ is a
set containing one outgoing edge of $v$ if $O(v)=\alpha$ and all
outgoing edges of $v$ if $O(v)= \oppon$.
A direct justification $dj$ {\em wins $v$ for $\alpha$ under
hypothesis $H$} if for all $(v,w)\in dj$, $H(w)=\alpha$. We also
say: {\em $\alpha$ wins $v$ by $dj$ under $H$}.
\end{definition}
\begin{definition}[Justification]
A {\em justification} $J$ for $\mathcal{PG}$ is a tuple $(V,D,H)$ such that $(V,D)$ is a subgraph of $(V,E)$.
If a node has outgoing edges in $D$, it is {\em justified} in $J$, otherwise it is {\em unjustified}.
\end{definition}
\begin{definition}[Weakly winning]
A justification $(V,D,H)$ is {\em weakly winning} if for all justified nodes $v\in V$ the set of outgoing edges $Out_v$ is a direct justification that wins $v$ for $H(v)$ under $H$.
\end{definition}
We observe that any justification $J = (V,D,H)$ determines a PPG $\mathcal{PG}_{P_J}$ where the parameter function $P_J$ is the restriction of $H$ to unjustified nodes.
If $J$ is weakly winning, the set of edges $\{(v,w)\in D\mid
O(v)=H(v)=\alpha\}$ is a partial function on $O_\alpha$, i.e.,
a strategy for $\alpha$. We denote it as $\sigma_{J,\alpha}$.
\begin{proposition}\label{prop:weaklywinning}
Assume a weakly winning justification $J = (V,D,H)$. Then, (i) For every path $\pi$ in $D$, all nodes $v$ on $\pi$ have the same hypothetical winner $H(v)$. (ii) All finite paths $\pi$ starting in node $v$ in $D$ are won in $\mathcal{PG}_{P_J}$ by $H(v)$. (iii) Every path in $D$ with nodes hypothetically won by $\alpha$ is consistent with $\sigma_{J,\alpha}$. (iv) Every play starting in $v$ of $\mathcal{PG}_{P_J}$ consistent with $\sigma_{J,H(v)}$ is a path in $D$.
\end{proposition}
\begin{proof}
(i) Since any edge $(v,w)\in D$ belongs to a direct justification that wins $v$ for $H(v)$, it holds that $H(v)=H(w)$. It follows that every path $\pi$ in $D$ consists of nodes with the same hypothetical winner. (ii) If path $\pi$ in $v$ is finite and ends in parameter $w$, then $H(v)=H(w)$. The winner of $\pi$ in $\mathcal{PG}_{P_J}$ is $P_J(w)$ which is equal to $H(v)$ as $H$ expands $P_J$. (iii) Every path in $D$ with hypothetical winner $\alpha$, follows $\sigma_{J,\alpha}$ when it is in a node $v$ with owner $\alpha$. (iv) Let $H(v)=\alpha$ and $\pi$ be a play in $v$ of ${\mathcal{PG}_P}$ consistent with $\sigma_{J,\alpha}$. We can inductively construct a path from $v=v_1$ in $D$. It follows from (i) that the $n$'th node $v_n$ has $H(v_n)=H(v_1)=\alpha$. For each non-parameter node $v_n$, if $O(v_n)=\alpha$, then $v_{i+1}=\sigma_{J,\alpha}(v_i)$ which is in $D$. If $O(v_n)=\oppon$ then $D$ contains all outgoing edges from $v_n$ including the one to $v_{n+1}$. \qed
\end{proof}
\begin{definition}[Winning]
A justification $J = (V,D,H)$ is {\em winning} if (i) $J$ is weakly winning and (ii) all
infinite paths $\langle v_1,v_2,\dots\rangle$ in $D$ are plays of
$\mathcal{PG}$ won by $H(v_1)$.
\end{definition}
Observe that, if $J$ is winning and $H(v)=\alpha$, all plays in $\mathcal{PG}_{P_J}$ starting in $v$ and consistent with $\sigma_{(V,D,H),\alpha}$ are paths in
$(V,D)$ won by $\alpha$. Hence:
\begin{theorem}
If $J=(V,D,H)$ is a winning justification for $\mathcal{PG}_{P_J}$ then $H$ is $W_{\mathcal{PG}_{P_J}}$, the
winning function of $\mathcal{PG}_{P_J}$, with corresponding winning strategies $\sigma_{J, 0}$ and $\sigma_{J, 1}$.
\end{theorem}
The central invariant of the algorithm presented below is that its data structure $J=(V,D,H)$ is a winning justification. Thus, in every stage, $H$ is the winning function of $\mathcal{PG}_{P_J}$ and the graph $(V,D)$ comprises winning strategies $\sigma_{J,\alpha}$ for both players. In a sense, $(V,D)$ provides a proof that $H$ is $W_{\mathcal{PG}_{P_J}}$.
\subsection{Operations on weakly winning justifications}
We introduce an operation that modifies a justification $J=(V,D,H)$
and hence also the underlying game $\mathcal{PG}_{P_J}$. Let $v$ be a node in
$V$, $\alpha$ a player and $dj$ either the empty set or a direct
justification. We define $J[v:dj,\alpha]$ as the justification
$J'=(V,D',H')$ where $D'$ is obtained from $D$ by replacing the
outgoing edges of $v$ by the edges in $dj$, and $H'$ is the function
obtained from $H$ by setting $H'(v):=\alpha$.
Modifications for a set of nodes are independent of application
order. E.g., $J[v:\emptyset,H'(v)\mid v\in S]$ removes all
out-going edges of $v$ and sets $H'(v)$ for all $v\in S$.
Multiple operations, like $J[v:dj,\alpha][v':dj',\alpha']$, are applied left to right.
Some useful instances, with their properties, are below.
In the proposition, a cycle in $J$ is a finite sequence of nodes following edges in $J$ that ends in its starting node.
\begin{proposition} \label{prop:operations}
For a weakly winning justification $J$ and a node $v$ with direct justification $dj$ the following holds:
(i) If $H(v)=\oppon$, $v$ has no incoming edges and $dj$ wins $v$ for $\alpha$ under $H$, then $J[v:dj,\alpha]$ is weakly winning and there are no cycles in $J'$ with edges of $dj$.
(ii) Let $S$ be a set of nodes closed under incoming edges (if $v\in S$ and $(w,v)\in D$, then $w\in S$), let $H_f$ be an arbitrary function mapping nodes of $S$ to players. It holds that $J[v:\emptyset,H_f(v) \mid v\in S]$ is weakly winning. There are no cycles in $J'$ with edges of $dj$.
(iii) If $H(v)=\alpha$ and $dj$ wins $v$ for $\alpha$ under
$H$, then $J[v:dj,\alpha]$ is weakly winning. There are no new
cycles when $(v,v)\not\in dj$ and no $w\in range(dj)$ can reach $v$ in
$J$. Otherwise new cycles pass through $v$ and have at least one
edge in $dj$.
\end{proposition}
\begin{proof}
We exploit the fact that $J$ and $J'$ are very similar.
(i) The direct justification $dj$ cannot have an edge ending in $v$ since $H(v)\neq H(w)$ for $(v,w)\in dj$ and no $w\in dj$ can reach $v$ in $J$ since $v$ has no incoming edges, hence $J'$ has no cycles through $dj$. As $J$ is weakly winning and $H$ is updated only in $v$, the direct justification of a justified node $w\neq v$ in $J$ is still winning in $J'$. Since also $dj$ wins $v$ for $\alpha$, $J'$ is weakly winning.
(ii) Setting $H(v)$ arbitrary cannot endanger the weak
support of $J'$ as $v$ has no direct justification and no incoming
edges in $J'$. Hence $J'$ is weakly winning. Also, removing direct
justifications cannot introduce new cycles.
(iii) Let $H(v)=\alpha$ and $dj$ wins $v$ for $\alpha$ under $H$.
Let $J'=J[v:dj,\alpha]$. We have $H'=H$ so the direct justifications of all nodes
$w \neq v$ in $J'$ win $w$ for $H'(w)$. Since $dj$ wins $v$ for $H'(v)$,
$J'$ is weakly winning. Also, new cycles if any, pass through
$dj$ and $v$.
\end{proof}
\subsection{Constructing winning justifications}
The eventual goal of a justification is to create a winning justification without unjustified nodes.
Such a justification contains a solution for the parity game without parameters.
To reach this goal we start with an empty winning justification and iteratively assign a direct justification to one of the nodes.
However, haphazardly (re)assigning direct justifications will violate the intended winning justification invariant.
Three problems appear:
First, changing the hypothesis of a node may violate weakly winning for incoming edges.
The easiest fix is to remove the direct justification of nodes with edges to this node.
Yet removing direct justifications decreases the justification progress.
Thus a second problem is ensuring progress and termination despite these removals.
Third, newly created cycles must be winning for the hypothesis.
To solve these problems, we introduce safe justifications; we start with
some auxiliary concepts.
Let $J$ be a justification.
The set of nodes {\em reaching} $v$ in J, including $v$, is closed
under incoming edges and is denoted with $\reaching{J}{v}$.
The set of nodes {\em reachable} from $v$ in $J$, including $v$, is denoted with
$\reachable{J}{v}$.
We define $\mathit{Par}_J(v)$ as the parameters reachable from the node $v$,
formally $\mathit{Par}_J(v) = \reachable{J}{v} \cap dom(P)$.
The {\em justification level} $\ensuremath{jl}_J(v)$ of a node $v$ is the lowest
priority of all its parameters and $+\infty$ if $v$ has none.
The {\em justification leve}l $jl_J(dj)$ of a direct justification
$dj=\{(v,w_1),\ldots,(v,w_n)\}$ is $min \{\ensuremath{jl}_J(w_1), \ldots, \ensuremath{jl}_J(w_n)\}$,
the minimum of the justification levels of the $w_i$.
We drop the subscript $J$ when it is clear from the context and write
$\mathit{Par}(v)$, $\ensuremath{jl}(v)$ and $\ensuremath{jl}(dj)$ for the above concepts. The {\em
default winner} of a node $v$ is the winner of its priority, i.e.,
$\allied {Pr(v)} $; the {\em default hypothesis} $H_d$ assigns default
winners to all nodes, i.e., $H_d(v)=\allied {Pr(v)}$.
\begin{definition}[Safe justification\label{def:extdom}]
A justification is safe iff (i) it is a winning justification, (ii)
all unjustified nodes $v$ have $H(v)={H_d}(v)$, that is, the winners of the current parameters of the PPG are their default winners, and (iii)
$\forall v \in V : \ensuremath{jl}(v) \geq Pr(v)$, i.e., the justification level of
a node is at least its priority.
\end{definition}
Fixing the invariants is easier for safe justifications. Indeed, for
nodes $w$ on a path to a parameter $v$, $Pr(v) \geq jl(w) \geq Pr(w)$,
so when $v$ is given a direct justification to $w$ then $Pr(v)$ is the
highest priority in the created cycle and $H(v)$ correctly denotes its
winner.
Furthermore, the empty safe justification $(V,\emptyset,{H_d})$ will serve as initialisation of the solving process.
\subsection{The operation Justify}
To progress towards a solution, we introduce a single operation, namely
${\operation{Justify}}$. Given appropriate inputs, it can assign a direct justification
to an unjustified node or replace the direct justification of a
justified node. Furthermore, if needed, it manipulates the justification in order
to restore its safety.
\begin{definition}[{\operation{Justify}}] \label{def:justify}
The operation \ensuremath{\attr(J,v,dj)}{} is {\em executable} if
\begin{itemize}
\item Precondition 1: $J = (V,D,H)$ is a safe justification, $v$ is
a node in $V$, there exists a player $\alpha$ who wins $v$ by $dj$
under $H$.
\item Precondition 2: if $v$ is unjustified in $J$ then $\ensuremath{jl}(dj)
\geq \ensuremath{jl}(v)$
else
$\ensuremath{jl}(dj) > \ensuremath{jl}(v)$. \\
\end{itemize}
Let \ensuremath{\attr(J,v,dj)}{} be executable. If $H(v) = \alpha$ then $\ensuremath{\attr(J,v,dj)}{} = J[v: dj,H(v)]$, i.e., $dj$ becomes the direct justification of $v$.
If $H(v) = \oppon$, then $\ensuremath{\attr(J,v,dj)}{} = {J[w:\emptyset,{H_d}(w)\mid
w\in \reaching{J}{v}][v:dj,\alpha]}$, i.e., $\alpha$ wins $v$ by
$dj$, while all other nodes $w$ that can reach $v$ become unjustified, and
their hypothetical winner $H(w)$ is reset to their default winner.
\end{definition}
If \ensuremath{\attr(J,v,dj)}{} is executable, we say that $v$ is {\em justifiable with $dj$} or {\em justifiable} for short;
when performing the operation, we {\em justify} $v$.
Observe, when {\operation{Justify}}\ modifies the hypothetical winner $H(v)$, then, to preserve weak winning, edges $(w,v)$ need to be removed, which is achieved by removing the direct justification of $w$. Moreover, to preserve (iii) of safety, this process must be iterated until fixpoint and to preserve (ii) of safety, the hypothetical winner $H(w)$ of $w$ needs to be reset to its default winner. This produces a situation satisfying all invariants.
Furthermore, when {\operation{Justify}}\ is applied on a justified $v$, it preserves $H(v)$ but it replaces $v$'s direct justification by one with a strictly higher justification level. As the proof below shows, this ensures that no new cycles are created through $v$ so we can guarantee that all remaining cycles still have the correct winner. So, cycles can only be created by justifying an unjustified node.
\begin{lemma}\label{theo:safe} An executable operation \ensuremath{\attr(J,v,dj)}{} returns a safe justification.
\end{lemma}
\begin{proof}
Assume \ensuremath{\attr(J,v,dj)}{} is executable, $J'=\ensuremath{\attr(J,v,dj)}$ and let $\alpha$ be the player that wins $v$ by $dj$. First, we prove that $J'$ is also a winning justification, i.e., that $J'$ is weakly winning and that the winner of every infinite path in $J'$ is the hypothetical winner $H(w)$ of the nodes $w$ on the path.
The operations applied to obtain $J'$ are the ones that have been analysed in Proposition \ref{prop:operations} and for which it was proven that they preserve weakly winning.
Note that, in case $H(v) = \oppon$, the intermediate justification $J[v: \emptyset, {H_d}(v) \mid v \in \reaching{J}{v}]$ removes all incoming edges of $v$. Hence, $J'$ is weakly winning and all nodes $v, w$ connected in $J$ have $H'(v) = H'(w)$ (*).
If no edge in $dj$ belongs to a cycle, then every infinite path $\play{v_1, v_2, \dots}$ in $J'$ has an infinite tail in $J$ starting in $w \neq v$ which is, since $J$ is winning, won by $H(w)$. By (*), this path is won by $H(v_1) = H(w)$ and $J'$ is winning.
If $J'$ has cycles through edges in $dj$, then, by (i) of Proposition~\ref{prop:operations}, $H(v)$ must be $\alpha$ and we are in case (iii)
of Proposition \ref{prop:operations}.
We analyse the nodes
$n$ on such a cycle. By safety of $J$, $Pr(n) \leq jl_J(n)$; as $n$
reaches $v$ in $J$, $jl_J(n) \leq jl_J(v)$. If $v$ is
unjustified in $J$ then $jl_J(v) = Pr(v) \geq Pr(n)$, hence $Pr(v)$
is the highest priority on the cycle and $H(v)$ wins the cycle. If
$v$ is justified in $J$ and $(v,w) \in dj$ is on the new cycle, then
$jl_J(w) \geq jl_J(dj) > jl_J(v)$ (Precondition 2 of {\operation{Justify}}). But $w$
reaches $v$ so $jl_J(w) \leq jl_J(v)$ , which is a contradiction.
Next, we prove that J' is a safe justification
(Definition~\ref{def:extdom}). (i) We just proved that $J'$ is a winning justification.
(ii) For all unjustified nodes $v$ of $J'$, it holds that $H(v)={H_d}(v)$, its default winner. Indeed, $J$ has this property and whenever the direct justification of a node $w$ is removed, $H'(w)$ is set to ${H_d}(w)$.
(iii) We need to prove that for all nodes $w$, it holds that $\ensuremath{jl}_{J'}(w) \geq Pr(w)$. We distinguish between the two cases of \ensuremath{\attr(J,v,dj)}.
(a) Assume $H(v) = \alpha = H'(v)$ and $J'=J[v:dj,H(v)]$ and let $w$ be an arbitrary node of $V$. If $w$ cannot reach $v$ in $J'$, the parameters that $w$ reaches in $J$ and $J'$ are the same and it follows that $\ensuremath{jl}_{J'}(w)=\ensuremath{jl}_J(w)\geq Pr(w)$. So, (iii) holds for $w$. Otherwise, if $w$ reaches $v$ in $J'$, then $w$ reaches $v$ in $J$ and any parameter $x$ that $w$ reaches in $J'$ is a parameter that $w$ reaches in $J$ or one that an element of $dj$ reaches in $J$. It follows that $\ensuremath{jl}_{J'}(w)$ is
at least
the minimum of $\ensuremath{jl}_J(w)$ and $\ensuremath{jl}_J(dj)$. As $w$ reaches $v$ in $J$, $\ensuremath{jl}_J(w)\leq\ensuremath{jl}_J(v)$. Also, by Precondition 2 of {\operation{Justify}}, $\ensuremath{jl}_J(v)\leq\ensuremath{jl}_J(dj)$. It follows that $\ensuremath{jl}_{J'}(w)\geq\ensuremath{jl}_J(w)\geq Pr(w)$. Thus, (iii) holds for $w$.
(b) Assume $H'(v) \neq H(v) = \oppon$ and $ J'=J[w:\emptyset,{H_d}(w)\mid w\in \reaching{J}{v}][v:dj,\alpha]$ then for nodes $w$ that cannot reach $v$ in $J$, $\mathit{Par}_{J'}(w) = \mathit{Par}_J(w)$ hence $\ensuremath{jl}_{J'}(w) = \ensuremath{jl}_J(w) \geq Pr(w)$ and (iii) holds for $w$. All nodes $w\neq v$ that can reach $v$ in $J$ are reset, hence $\ensuremath{jl}_{J'}(w) = Pr(w)$ and (iii) holds. As for $v$, by construction $\ensuremath{jl}_{J'}(v) = \ensuremath{jl}_J(dj) \geq \ensuremath{jl}_J(v)$; also $\ensuremath{jl}_J(v) \geq Pr(v)$ hence (iii) also
holds. \qed
\end{proof}
\begin{lemma}\label{theo:monotonic} Let $J$ be a safe justification for a
parametrized parity game. Unless $J$ defines the parametrized parity
game $PG_\emptyset=\mathcal{PG}$, there exists a node $v$ justifiable
with a direct justification $dj$, i.e., such that \ensuremath{\attr(J,v,dj)}{} is executable.
\end{lemma}
\begin{proof}
If $J$ defines the parametrized parity game
$PG_\emptyset$ then all nodes are justified and $J$ is a solution for the original $\mathcal{PG}$. Otherwise let $p$ be the minimal priority of all
unjustified nodes, and $v$ an arbitrary unjustified node of priority $p$ and let its owner be
$\alpha$. Then either $v$ has an outgoing edge
$(v,w)$ to a node $w$ with $H(w) = \alpha$, thus a winning direct
justification for $\alpha$, or all outgoing edges are to nodes
$w$ for which $H(w) = \oppon$, thus $v$ has a winning direct justification for
$\oppon$. In both cases, this direct justification $dj$ has a
justification level larger or equal to $p$ since no parameter
with a smaller priority exist, so \ensuremath{\attr(J,v,dj)}{} is executable.
\qed
\end{proof}
To show progress and termination, we need an order over
justifications.
\begin{definition}[Justification size and order over justifications]
Let $1,\ldots,n$ be the range of the priority function of a parity
game $PG$ ( ${+\infty} > n$) and $J$ a winning justification for a parametrized parity
game extending $PG$.
The size of $J$, $s(J)$ is the tuple $(s_{{+\infty}}(J),s_n(J), \ldots s_1(J))$ where for $i \in \{1,\dots,n,+\infty\}$, $s_i(J)$ is the number of
justified nodes with justification level $i$.
The order over justifications is the lexicographic order over their
size: with $i$ the highest index such that $s_i(J) \neq s_i(J')$,
we have $J >_s J'$ iff $s_i(J) > s_i(J')$.
\end{definition}
The order over justifications is a total order which is bounded as $\Sigma_i s_i(J) \leq |V|$.
\begin{figure} [tb]
\begin{center}
\includegraphics[width=0.6\textwidth]{images/game_para_e_e.pdf}
\end{center}
\begin{center}
\includegraphics[width=0.6\textwidth]{images/game_para_3_e_e.pdf}
\end{center}
\caption{Above,
in solid line the edges of the justification graph of
the winning but unsafe justification of Figure~\ref{ex:game_para}
and below the result of justifying node $a$, a non-winning justification.\label{fig:just}}
\end{figure}
\begin{example}\label{ex:second} Let us revisit Example~\ref{ex:first}. The winning justification $J$
of Figure~\ref{ex:game_para} is shown at the top of
Figure~\ref{fig:just}. For the justified nodes of $J$, we have
$\ensuremath{jl}(b)=3$, $\ensuremath{jl}(c)=+\infty$, $\ensuremath{jl}(e)=2$ and $\ensuremath{jl}(f)=2$. The justification is not safe as, e.g., $\ensuremath{jl}(b)=3 <
Pr(b)=4$. Both unjustified nodes $a$ and $d$ have a winning direct justification, the direct justification $\{(a,b)\}$ wins $a$ for player 1 and the direct justification
$\{(d,c)\}$ wins $d$ for 1. The figure at the bottom shows the justification resulting from inserting the direct justification winning $a$. There is
now an infinite path $\play{a,b,a,b,\ldots}$ won by Even but with nodes with hypothetical winner Odd. The justification ${\operation{Justify}}(J,a,\set{(a,b)})$ is not winning. This shows that condition (iii) of safety of $J$ is a necessary
precondition for maintaining the desired invariants.
\end{example}
\begin{lemma}\label{theo:size}
Let $J$ be a safe justification with size $s_J$, $v$ a node
justifiable with $dj$ and $J'=\ensuremath{\attr(J,v,dj)}{}$ a justification with size
$s_{J'}$. Then $s_{J'} > s_J$.
\end{lemma}
\begin{proof}
In case $v$ is unjustified in $J$ and is assigned a $dj$ that wins
$v$ for $H(v)$, $v$ is not counted for the size of $J$ but is
counted for the size of $J'$. Moreover, other nodes keep their
justification level (if they cannot reach $v$ in $J$) or may
increase their justification level (if they can reach $v$ in
$J$). In any case, $s_{J'} > s_J$.
In case $v$ is justified in $J$ and is assigned a $dj$ that wins $v$
for $H(v)$, then $\ensuremath{jl}_J(dj) > \ensuremath{jl}_J(v)$, so
$\ensuremath{jl}_J'(v) > \ensuremath{jl}_J(v)$. Other nodes keep their justification level
or, if they reach $v$, may increase their justification
level. Again, $s_{J'} > s_J$.
Finally, the case where $dj$ wins $v$ for the opponent of
$H(v)$. Nodes can be reset; these nodes $w$ have $\ensuremath{jl}_J(w)\leq
Pr(v)$. As a node cannot have a winning direct justification for
both players, $v$ is unjustified in $J$. Hence, by precondition (2)
of {\operation{Justify}}, $\ensuremath{jl}_J(dj)\geq Pr(v)$. In fact, it holds that $\ensuremath{jl}_J(dj)>
Pr(v)$. Indeed, if some $w\in dj$ would have a path to a parameter
of $v$'s priority, that path would be won by ${H_d}(v)=H(v)$ while
$H(w)$ is its opponent. Thus, the highest index $i$ where $s_i$
changes is $\ensuremath{jl}_J(dj)$, and $s_i$ increases. Hence, $s_{J'}>s_J$.
\qed
\end{proof}
\begin{theorem}\label{theo:basic}
Any iteration of {\operation{Justify}}{} steps from a safe justification, in
particular from $(V, \emptyset,{H_d})$, with ${H_d}$ the default hypothesis, eventually solves $\mathcal{PG}$.
\end{theorem}
\begin{proof}
By induction: Let $\mathcal{PG}=(V,E,O,Pr)$ be a parity game. Clearly, the empty justification $J^0=(V,\emptyset,{H_d})$ is a safe justification. This is the base case.
Induction step: Let $J^i$ be the safe justification after $i$ successful {\operation{Justify}}\ steps and assume that $J^i=(V,D^i,H^i)$
contains an unjustified node.
By Lemma~\ref{theo:monotonic}, there exists a pair $v$ and $dj$ such that $v$ is justifiable with $dj$. For {\em any} pair $v$ and $dj$ such that ${\operation{Justify}}(J^i,v,dj)$ is executable, let $J^{i+1}={\operation{Justify}}(J^i,v,dj)$. By
Lemma~\ref{theo:safe}, $J^{i+1}$ is a safe justification. By
Lemma~\ref{theo:size}, there is a strict increase in size, i.e.,
$s(J^{i+1}) > s(J^i)$.
Since the number of different sizes is bounded, this eventually produces a safe $J^k=(V,D^k,H^k)$ without unjustified nodes.
The parametrized parity game $\mathcal{PG}_{P_{J^k}}$ determined by $J^k$ is $\mathcal{PG}$. Hence, $H^k$ is the winning function of $\mathcal{PG}$, and $J^k$ comprises winning strategies for both players. \qed
\end{proof}
Theorem~\ref{theo:basic} gives a basic algorithm to solve parity games. The algorithm has three features: it is (1) simple, (2) {\em nondeterministic}, and (3) in successive steps it may arbitrarily switch between different priority levels.
Hence, by imposing different strategies, different instantiations of the algorithm are obtained.
Existing algorithms differ in the order in which they (implicitly) justify
nodes. In the next section we simulate such algorithms by
different strategies for selecting nodes to be justified. Another
difference between algorithms is in computing the set $R$ of nodes that is
reset when $dj$ wins $v$ for the opponent of $H(v)$. Some algorithms
reset more nodes; the largest reset set for which the proofs in this
paper remain valid is $\{w \in V \mid \ensuremath{jl}(w) < \ensuremath{jl}(dj)\}$. To the best
of our knowledge, the only algorithms that reset as few nodes as
\ensuremath{\attr(J,v,dj)}{} are the ones we presented in
\cite{DBLP:conf/vmcai/LapauwBD20}. As the experiments presented there
show, the work saved across iterations by using justifications
results in better performance.
\section{A reformulation of three existing algorithms\label{sec:recursive}}
In this section, by ordering justification steps, we obtain basic
versions of different algorithms known from the literature.
In our versions, we represent the parity game $G$ as $(V,E,O,Pr)$ and the justification J as $(V,D,H)$.
All algorithms start with the safe empty
justification $(V, \emptyset, {H_d})$. The recursive algorithms
operate on a subgame $SG$ determined by a set of nodes $V_{SG}$. This
subgame determines the selection of \ensuremath{\attr(J,v,dj)}{} steps that are
performed on $G$. For convenience of presentation, $G$ is considered
as a global constant.
\subsubsection{Nested fixpoint iteration~\cite{DBLP:journals/corr/BruseFL14,DBLP:journals/corr/abs-1909-07659,DBLP:conf/vmcai/LapauwBD20}} is one of the earliest algorithms
able to solve parity games. In Algorithm~\ref{alg:fpi}, we show a
basic form that makes use of our \ensuremath{\attr(J,v,dj)}{} action. It starts from the
initial justification $(V,\emptyset,H_d)$. Iteratively, it determines
the lowest priority $p$ over all unjustified nodes, it selects a node
$v$ of this priority and justifies it. Recall from the proof of
Lemma~\ref{theo:monotonic}, that all unjustified nodes of this priority are
justifiable. Eventually, all nodes are justified and a solution is
obtained. For more background on nested fixpoint algorithms and the
effect of justifications on the performance, we refer to our work in
\cite{DBLP:conf/vmcai/LapauwBD20}.
A feature of nested fixpoint iteration is that it solves a parity game
{\em bottom up}. It may take many iterations before it uncovers that the current hypothesis of some high priority unjustified node $v$ is, in fact,
wrong and so that playing to $v$ is a bad strategy for
$\alpha$. The next algorithms are {\em top down}, they start out from
nodes with the highest priority.
\begin{figure}[t]
{\begin{minipage}{0.46\textwidth}
\begin{algorithm}[H]
\algsetup
\Fn{${\operation{Fixpoint}}(G)$}{
$J \is (V,\emptyset,{H_d})$ the initial safe justification\\
\While{$J$ has unjustified nodes} {
$p \is min\set{Pr(v) \ \middle|\ v \text{ is unjustified} }$ \\
$v \is $ an unjustified node with $Pr(v) = p$ \\
${dj} \is $ a winning direct justification for $v$ under $H$\\
$J \is {\operation{Justify}}(J, v, {dj})$\\
}
\Return{$J$}
}
\caption{A fixpoint algorithm for justifying nodes\label{alg:fpi}}
\end{algorithm}
\end{minipage}~
\begin{minipage}{0.54\textwidth}
\begin{algorithm}[H]
\algsetup
\Input{A parity game $G$}
$J \is {\operation{Zielonka}}((V,\emptyset,{H_d}), V)$ \label{line:zlk_init}\\
\Fn{${\operation{Zielonka}}(J, V_{SG})$}{
$p \is max\set{Pr(v) \ \middle|\ v \in V_{SG}}$ \\
$\alpha \is \allied p$\\
\While{$true$}{
\While {$\exists v \in V_{SG}, dj: v$ is unjustified, $v$ is justifiable with ${dj}$ for $\alpha$ with $jl({dj})\geq p$ \label{line:zlk:greedy}}{
$J \is {\operation{Justify}}(J, v, {dj})$ \label{line:zlk_just}\\
}
$V_{SSG} \is \{ v \in V_{SG}| Pr(v)<p,$\\
~~~~~~~~~~~~ $v$ is unjustified\}\\
\lIf{$V_{SSG} = \emptyset$} {\Return{$J$}}
$J \is {\operation{Zielonka}}(J, V_{SSG})$\\
\While {$\exists v \in V_{SG}, dj: v$ is unjustified, $v$ is justifiable with ${dj}$ for $\oppon$ with $jl({dj})\geq p+1$ \label{line:zlk:oppon:attr}}{
$J \is {\operation{Justify}}(J, v, {dj})$\\
}
}
}
\caption{A {\operation{Justify}}{} variant of Zielonka's algorithm.}\label{alg:ZielJust}
\end{algorithm}
\end{minipage}}
\end{figure}
\subsubsection{Zielonka's algorithm~\cite{DBLP:journals/tcs/Zielonka98},}
one of the oldest algorithms, is recursive and starts with a greedy
computation of a set of nodes, called {\em attracted} nodes, in which the winner $\alpha$ of the top priority $p$ has a strategy to force playing to nodes of top priority $p$.
In our reconstruction, Algorithm~\ref{alg:ZielJust}, attracting nodes is simulated at Line~\ref{line:zlk:greedy} by repeatedly justifying nodes $v$ with a direct justification that wins $v$ for $\alpha$ and has a justification level $\geq p$. Observe that the while test ensures that the preconditions of \ensuremath{\attr(J,v,dj)}{} on the justification level of $v$ are satisfied. Also, every node \ignore{in $V'$}can be justified at most once.
The procedure is called with a set $V_{SG}$ of nodes of maximal level
$p$ that cannot be attracted by levels $>p$. It follows that the
subgraph determined by
$V_{SG}$ contains for each of its nodes an outgoing edge (otherwise the opponent of
the owner of the node would have attracted the node at a level $>p$) , hence
this subgraph determines a parity game.
The main loop invariants are that (1) the justification $J$ is safe; (2) the justification level of all justified nodes is $\geq p$ and (3) $\oppon$ has no direct justifications of justification level $> p$ to win an unjustified node in $V_{SG}$.
The initial justification is safe and it remains so as every ${\operation{Justify}}$
call satisfies the preconditions.
After the attraction loop at Line~\ref{line:zlk:greedy}, no more unjustified nodes of $V_{SG}$ can be attracted to level $p$ for player $\alpha$. Then, the set of $V_{SSG}$ of unjustified nodes of priority $<p$ is determined. If this set is empty, then by Lemma~\ref{theo:monotonic} all unjustified nodes of priority $p$ are justifiable with a direct justification $dj$ with $jl(dj)\geq p$, hence they would be attracted to some level $\geq p$ which is impossible. Thus, there are no unjustified nodes of priority $p$. In this case, the returned justification $J$ justifies all elements of $V_{SG}$. Else, $V_{SSG}$ is passed in a recursive call to justify all its nodes. Upon return, if $\oppon$ was winning some nodes in $V_{SSG}$, their justification level will be $\geq p+1$. Now it is possible that some unjustified nodes of priority $p$ can be won by $\oppon$ and this may be the start of a cascade of resets and attractions for $\oppon$. The purpose of Line~\ref{line:zlk:oppon:attr} is to attract nodes of $V_{SG}$ for $\oppon$.
Note that \ensuremath{\attr(J,v,dj)}{} resets all nodes that depend on nodes that switch to $\oppon$.
When the justification returned by the recursive call shows that $\alpha$ wins
all nodes of $V_{SSG}$, the yet unjustified nodes of $V_{SG}$ are of priority $p$, are justifiable by Lemma~\ref{theo:monotonic} and can be won only by $\alpha$. So, at the next iteration, the call to $Attr_\alpha$ will justify all of them for $\alpha$ and $V_{SSG}$ will be empty. Eventually the initial call of Line~\ref{line:zlk_init} finishes with a safe justification in which all nodes are justified thus solving the game $G$.
Whereas fixpoint
iteration first justifies low priority nodes resulting in low justification levels, Zielonka's algorithm
first justifies nodes attracted to the highest priority. Compared to
fixpoint iteration, this results in large improvements in
justification size which might explain its better performance. However, Zielonka's
algorithm still disregards certain opportunities for increasing
justification size as it proceeds by priority level, only returning to
level $p$ when all sub-problems at level $<p$ are completely
solved. Indeed, some nodes computed at a low level $i<\!\!<p$ may have
a very high justification level, even $+\infty$ and might be useful
to revise false hypotheses at high levels, saving much work, but this
is not exploited. The next algorithm, priority promotion, overcomes
this limitation.
\subsubsection{Priority promotion~\cite{DBLP:conf/cav/BenerecettiDM16,DBLP:conf/hvc/BenerecettiDM16,DBLP:journals/iandc/BenerecettiDM18}}
follows the strategy of Zielonka's algorithm except that, {when it
detects that all nodes for priority $p$ are justified, it does not
make a recursive call but returns the set of nodes attracted to
priority $p$ nodes as a set $R_p$ to a previous level $q$.}
There $R_p$ is added to the attraction set at level $q$ and the attraction process is restarted. In the terminology of \cite{DBLP:conf/cav/BenerecettiDM16}, the set $R_p$ is a {\em closed $p$-region} that is {\em promoted} to level $q$. A {\em
closed $p$-region} of $V_{SG}$, with maximal priority $p$, is a subset $R_p \subseteq V_{SG}$ that includes all
nodes of $V_{SG}$ with priority $p$ and for which $\alpha = \allied{p}$ has
a strategy winning all infinite plays in $R_p$ and for which $\oppon$ cannot escape from $R_p$ unless to nodes of higher $q$-regions won by $\alpha$.
We call the latter nodes the {\em escape nodes} from $R_p$ denote the set of them as $Escape(R_p)$.
The level to which $R_p$ is promoted is the lowest $q$-region that contains an escape node from $R_p$.
It is easy to show that $q$ is a lower bound of the justification level of $R_p$. In absence of escape nodes, $R_p$ is promoted to $+\infty$.
\begin{figure}[tb]
\begin{minipage}{0.45\textwidth}
\begin{algorithm}[H]
\algsetup
\Input{A parity game $G$ }
$J \is (V,\emptyset,{H_d})$\\
\While{$\exists v \in V_G: v$ is unjustified}{
$R_{+\infty} \is \set{ v \ \middle|\ jl(v) = +\infty}$ \\
$V_{SG} \is V \setminus R_{+\infty}$ \\
$(J, \_,\_) \is {\operation{Promote}}(V_{SG},J)$\\
\While {$\exists v \in V_{SG}, dj: v$ is justifiable with ${dj}$ and $jl({dj}) = +\infty$}{
$J \is {\operation{Justify}}(J, v, {dj})$ \label{line:attr_solve2}\\
}
}
\caption{A variant of priority promotion using {\operation{Justify}}. \label{alg:Mmprom}}
\end{algorithm}
\end{minipage}~
\begin{minipage}{0.55\textwidth}
\begin{algorithm}[H]
\algsetup
\Fn{${\operation{Promote}}(V_{SG},J)$}{
$p \is max\set{Pr(v) \ \middle|\ v \in V_{SG}}$ \\
$\alpha \is \allied{p}$ \\
\While{$true$}{
\While {$\exists v \in V_{SG}, dj: v$ is unjustified or $jl(v)<p$, $v$ is justifiable with ${dj}$ for $\alpha$ with $jl({dj})\geq p$ \label{line:greedy}}{
$J \is {\operation{Justify}}(J, v, {dj})$ \\}
$R_p \is \set{v \in V_{SG} \ \middle|\ jl(v) \geq p}$ \\
\If{${\operation{Closed}}(R_p, V_{SG})$}{
$l \is min\{q| R_q$ contains an escape node of $R_p\}$ \label{line:escapelevel}\\
\Return{$(J,R_p,l)$}
}
$V_{SSG} \is V_{SG} \setminus R_p$ \\
$(J,R_{p'},l) \is {\operation{Promote}}(V_{SSG}, J)$\\
\If{$l > p$\label{line:backjump}}{\Return{$(J,R_{p'},l)$}}\label{line:skip}
}
}
\end{algorithm}
\end{minipage}
\end{figure}
Our variant of priority promotion (PPJ) is in Algorithm~\ref{alg:Mmprom}.
Whereas {\operation{Zielonka}}{} returned a complete solution $J$ on $V_{SG}$, {\operation{Promote}}{} returns only a partial $J$ on $V_{SG}$; some nodes of $V_{SG}$ may have an unfinished justification ($jl(v)<+\infty$).
To deal with this, {\operation{Promote}}{} is iterated in a while loop that continues as long as there are
unjustified nodes.
Upon return of {\operation{Promote}}{}, all nodes attracted to the returned $+\infty$-region are justified.
In the next iteration, all nodes with justification level $+\infty$ are removed from the game, permanently.
Note that when promoting to some $q$-region, justified nodes of justification level $<q$ can remain.
A substantial gain can be obtained compared to the original priority promotion algorithm which does not maintain justifications and loses all work stored in $J$.
By invariant, the function {\operation{Promote}}{} is called with a set of nodes $V_{SG}$ that cannot be justified with a direct justification of level larger than the maximal priority $p$.
The function starts its main loop by attracting nodes for level $p$.
The attraction process is identical to Zielonka's algorithm except that leftover justified nodes $v$ with $jl(v)<p$ may be rejustified.
As before, the safety of $J$ is preserved.
Then $R_p$ consists of elements of $V_{SG}$ with justification level $\geq p$.
It is tested ({\operation{Closed}}{}) whether $R_p$ is a closed $p$-region.
This is provably the case if all nodes of priority $p$ are justified.
If so, $J$, $R_p$ and its minimal escape level are returned.
If not, the game proceeds as in Zielonka's algorithm and the
game is solved for the nodes not in $R_p$ which have strictly lower justification level.
Sooner or later, a closed region will be obtained.
{Indeed, at some point, a subgame is entered in which all nodes have
the same priority $p$. All nodes are justifiable
(Lemma~\ref{theo:monotonic}) and the resulting region is closed.}
Upon return from the recursive call, it is
checked whether the returned region ($R_{p'}$) promotes to the current
level $p$.
If not, the function exits as well (Line~\ref{line:skip}).
Otherwise a new iteration starts with attracting nodes of justification level $p$ for $\alpha$. Note that contrary to Zielonka's algorithm, there is no attraction step for $\oppon$:
attracting for $\oppon$ at $p$ is the same as attracting for $\alpha' = \oppon$ at $p' = p+1$.
\subsubsection{Discussion}
Our versions of Zielonka's algorithm and priority promotion use the
justification level to decide which nodes to attract. While
maintaining justification levels can be costly, in these algorithms,
it can be replaced by selecting nodes that are ``forced to play'' to
a particular set of nodes (or to an already attracted node). In the
first attraction loop of {\operation{Zielonka}}, the set is initialised with
all nodes of priority $p$, in the second attraction loop, with the
nodes won by $\oppon$; In {\operation{Promote}}, the initial set consists also of
the nodes of priority $p$.
Observe that the recursive algorithms implement a strategy to reach as soon
as possible the justification level $+\infty$ for a group of nodes
(the nodes won by the opponent in the outer call of {\operation{Zielonka}}, the
return of a closed region ---for any of the players--- to the outer
level in {\operation{Promote}}). When achieved, a large jump in justification size
follows. This may explain why these algorithms outperform fixpoint
iteration.
Comparing our priority promotion algorithm (PPJ) to other variants, we
see a large overlap with region recovery (RR)~\cite{DBLP:conf/hvc/BenerecettiDM16} both algorithms avoid
resetting nodes of lower regions. However, RR always resets the full
region, while PPJ can reset only a part of a region, hence can save more
previous work.
Conversely, PPJ eagerly resets nodes while RR only validates the
regions before use, so it can
recover a region when the reset escape node is easily re-attracted.
The equivalent justification of such a state is winning but unsafe, thus unreachable by applying $\ensuremath{\attr(J,v,dj)}{}$.
However, one likely can define a variant of $\ensuremath{\attr(J,v,dj)}{}$ that can reconstruct RR.
Delayed priority promotion~\cite{DBLP:journals/iandc/BenerecettiDM18} is another variant which prioritises the best
promotion over the first promotion and, likely, can be directly reconstructed.
Tangle learning~\cite{DBLP:conf/cav/Dijk18} is another state of
the art algorithm that we have studied. Space restrictions disallow us
to go in details. We refer to \cite{DBLP:conf/vmcai/LapauwBD20} for a
version of tangle learning with justifications. For a more formal
analysis, we refer to~\cite{LapauwPhD}). Interestingly, the updates of the
justification in the nodes of a tangle cannot be modelled with a
sequence of safe \ensuremath{\attr(J,v,dj)}{} steps. One needs an alternative with a
precondition on the set of nodes in a tangle. Similarly as for \ensuremath{\attr(J,v,dj)}{},
it is proven in~\cite{LapauwPhD} that the resulting justification is
safe and larger than the initial one.
Justification are not only a way to explicitly model (evolving)
winning strategies, they can also speed up algorithms. We have
implemented justification variants of the nested fixpoint algorithm,
Zielonka's algorithm, priority promotion, and tangle learning. For the
experimental results we refer to \cite{DBLP:conf/vmcai/LapauwBD20,LapauwPhD}.
Note that the data structure used to implement the justification graph
matters. Following an idea of Benerecetti et al.\cite{DBLP:conf/cav/BenerecettiDM16}, our
implementations use a single field to represent the direct
justification of a node; it holds either a single node, or
$\mathit{null}$ to represent the set of all outgoing nodes.
To compute the reset set $R$ of a node, we found two efficient methods to encode the graph $J$: (i) iterate over all incoming nodes in $E$ and test if their justification contains $v$, (ii) store for every node a hash set of every dependent node.
On average, the first approach is better, while the second is more efficient for sparse graphs but worse for dense graphs.
\section{Conclusion}\label{sec:conclusion}
This paper explored the use of justifications in parity game
solving.
First, we generalized parity games by adding parameter
nodes. When a play reaches a parameter it stops in favour of one
player. Next, we introduced justifications and proved that a winning
justification contains the solution of the parametrized parity game.
Then, we introduced safe justifications and a {\operation{Justify}}\ operation and
proved that a parity game can be solved by a sequence of {\operation{Justify}}\ steps.
A {\operation{Justify}}\ operation can be applied on a node satisfying its
preconditions, it assigns a winning direct justification to the node,
resets ---if needed--- other nodes as parameters, preserves safety
of the justification, and ensures the progress of the solving process.
To illustrate the power of {\operation{Justify}}, we reconstructed three algorithms:
nested fixpoint iteration, Zielonka's algorithm and priority promotion
by ordering applicable ${\operation{Justify}}{}$ operations differently. Nested
fixpoint induction prefers operations on nodes with the lowest
priorities; Zielonka's algorithm starts on nodes with the maximal
priority and recursively descends; priority promotion improves upon
Zielonka with an early exit on detection of a closed region (a solved
subgame).
A distinguishing feature of a justification based algorithm is that it
makes active use of the partial strategies of both players.
While other algorithms, such as region recovery and tangle learning, use the constructed partial strategies while solving the parity game, we do not consider them justification based algorithms.
For region recovery, the generated states are not always weakly winning, while tangle learning applies the partial strategies for different purposes.
As shown in \cite{DBLP:conf/vmcai/LapauwBD20} where justifications improve tangle learning, combining different techniques can further improve parity game algorithms.
Interesting future research includes: (i) exploring the
possible role of justifications in the quasi-polynomial algorithm of
Parys~\cite{DBLP:conf/mfcs/Parys19}, (ii) analysing the similarity
between small progress measures
algorithms~\cite{DBLP:conf/spin/FearnleyJS0W17,DBLP:conf/stacs/Jurdzinski00}
and justification level, (iii) analysing whether the increase in
justification size is a useful guide for selecting the most
promising justifiable nodes, (iv) proving the worst-case time
complexity by analysing the length of the longest path in the lattice
of justification states where states are connected by \ensuremath{\attr(J,v,dj)}{} steps. |
1,108,101,565,696 | arxiv |
\section{Introduction}
\label{sec:introduction}
Trajectory prediction is a fundamental and challenging task, which needs to forecast the future path of the agents in autonomous applications, such as autonomous vehicles, socially compliant robots, agents in simulators, to navigate in a shared environment.
With multi-agent interaction in these applications, the agents are required to respond timely and precisely to the environment for collision avoidance.
Therefore, the ability of the agents to predict the future paths of their neighbors in an efficient and accurate manner is thus much needed.
Although recent works~\cite{liang2019peeking,sadeghian2019sophie,huang2019stgat,mohamed2020social} have achieved great improvement in modeling complex social interactions among agents to generate accurate future paths, trajectory prediction is still a challenging task, where the deployment of the prediction models in real-world applications is mostly restricted by its high computational cost and long inference time.
For example, some small robots are only equipped with limited computing devices that can not afford the high inference cost with existing solutions.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.65\linewidth]{method_imgs/human-human.pdf}
\end{center}
\caption{
\revise{The illustration of trajectory prediction in a crowd.}
The solid blue lines are the observed trajectories, and the dash blue lines indicate the plausible future path.
The influence levels $\alpha$ between two agents are different based on their relative movement trends, e.g., $\alpha_{ij}$ and $\alpha_{ji}$ between agent $i$, and agent $j$ are different.
}
\label{fig:crowd_illustration}
\end{figure}
In particular, the trajectory prediction is typically modeled in two dimensions, i.e., the temporal dimension and the spatial dimension, which is illustrated in Fig.~\ref{fig:crowd_illustration}.
The temporal dimension models the historical movement dynamics for each agent.
Most of the state-of-the-art approaches~\cite{alahi2016social,gupta2018social,liang2019peeking,huang2019stgat,ivanovic2019trajectron,salzmann2020trajectron++} have focused on Recurrent Neural Networks (RNNs),
to capture such sequence dynamics since RNNs are designed for sequence modeling.
However, besides the training difficulties of gradient vanishing and exploding~\cite{razvan13rnn} in modeling long sequential data, both training and inference of RNN models are notoriously slow compared with their feed-forward counterparts, e.g., Convolutional Neural Networks (CNNs).
This is largely due to the fact that each hidden state of RNNs is dependent on the previous inputs and hidden states.
As a consequence, the prediction of RNNs is produced sequentially, and thus not parallelizable.
The spatial dimension models the human-human interaction, i.e., interactions between the agent and its neighbors.
There are mainly three categories of methods proposed to capture the spatial interaction, including the distance-based~\cite{alahi2016social,gupta2018social,liang2019peeking},
attention-based~\cite{sadeghian2019sophie,fernando2018soft,vemula2018social,ivanovic2019trajectron} and graph-based~\cite{huang2019stgat,kosaraju2019social,zhang2019sr,mohamed2020social} approaches.
Distance-based approaches introduce a social pooling layer to summarize the crowd interactions, while the attention-based approaches instead dynamically generate the importance of neighbors using soft attention.
The graph-based approaches model the agents' representation with a graph and utilize graph neural network, e.g., GCN~\cite{kipf2017semi,zhang2019sr,mohamed2020social} or GAT~\cite{velivckovic2017graph}, to capture the spatial interaction features of agents, which is empirically more intuitive and effective in modeling complex social interactions.
However, existing graph-based approaches are mostly based on the simple aggregation of neighbor features or their absolute geometric distance, which neglects the relative relation between agents in the spatial modeling.
To improve effectiveness and efficiency, we propose a novel \textbf{\underline{Graph}}-based \textbf{\underline{T}}emporal \textbf{\underline{C}}onvolutional \textbf{\underline{N}}etwork
(GraphTCN), to capture the spatial and temporal interaction for trajectory prediction.
In the temporal dimension, different from RNN-based methods, we adopt a modified gated convolutional network (TCN) to capture the temporal dynamics for each agent.
The gated highway mechanism introduced to CNNs dynamically regulate the information flow by focusing on more salient features, and the feed-forward nature of CNN makes it more tractable in training and parallelizable for much higher efficiency both in training and inference.
In the spatial dimension, we propose an edge feature based graph attention network (EFGAT) with skip connections and the gate mechanism for each time step to model the spatial interaction between the agents.
Specifically, nodes in the graph represent agents, and edges between agents denote their relative spatial relation.
EFGAT learns the adjacency matrix, i.e., the spatial interaction, of the graph adaptively.
Together, the spatial and temporal modules of GraphTCN support more effective and efficient modeling of the interactions within each time step between agents and across the time steps for each and every agent.
We summarize our main contributions as follows:
\begin{itemize}
\item We propose an edge feature based graph attention network (EFGAT), which introduces the relative spatial relation as prior knowledge, to capture the spatial interaction adaptively with attention.
\item We propose to model the temporal interactions with a gated convolutional network (TCN), which empirically proves to be more efficient and effective.
\item Our spatial-temporal framework achieves better performance compared with best-performing approaches. Specifically, we reduce the average displacement error by 19.4\% and final displacement error by 13.6\% with 5 times less predicted paths, and achieves up to 5.22x wall-clock time speedup over existing solutions.
\end{itemize}
We organize this paper as follows: in Section~\ref{sec:relaed_work}, we introduce the background and discuss related works in detail. Our GraphTCN framework is introduced in Section~\ref{sec:methodology}. Then in Section~\ref{sec:experiment}, results of GraphTCN measured in both accuracy and efficiency are compared with state-of-the-art approaches. Finally, Section~\ref{sec:conclusion} concludes the paper.
\section{Related Work}
\label{sec:relaed_work}
\noindent
\textbf{Human-Human Interactions.}
Research in the crowd interaction model can be traced back to the Social Force model~\cite{helbing1995social}, which adopts the nonlinearly coupled Langevin equations to represent the attractive and repulsive forces for human movement in the crowed scenarios.
Similar hand-crafted approaches\cite{treuille2006continuum,antonini2006discrete,wang2007gaussian} have proved successful in crowd simulation~\cite{hou2014social,saboia2012crowd}, crowd behavior detection~\cite{mehran2009abnormal}, and trajectory prediction~\cite{yamaguchi2011you}.
Recent works instead investigate deep learning techniques to capture the interaction between agents and neighbors.
The distance-based approaches~\cite{alahi2016social,gupta2018social,mangalam2020not} either adopt the grid-based pooling or symmetric function to aggregate the hidden states from neighbors or encode the geometric relation between the agents.
Different from the distance-based methods, attention-based approaches~\cite{sadeghian2019sophie,vemula2018social,fernando2018soft,zhang2019sr} provide better crowd modeling since they differentiate the importance of neighbors by soft attention or gating mechanisms.
More recent works~\cite{huang2019stgat,kosaraju2019social,mohamed2020social} adopts graph-based networks to learn the social interaction by aggregating neighborhood features adaptively with the adjacency matrix, which provides an effective way to represent the pedestrian's topology in a shared space.
Social-STGCNN~\cite{mohamed2020social} captures the spatial relation by introducing a kernel function on the weighted adjacency matrix;
STGAT~\cite{huang2019stgat,kosaraju2019social} adopts GAT directly on the LSTM hidden states to capture the spatial interaction between pedestrians.
However, Social-STGCNN only focuses on distance features between agents, and STGAT simple aggregates neighbor features.
EGNN~\cite{gong2019exploiting} incorporates the edge feature into the graph attention mechanism to exploit richer graph information.
However, EGNN neglects the relative relation between pedestrians.
We propose to model the pedestrian interactions with a novel edge feature based graph network, which integrates the relative distance feature into graph attention to learn an adaptive adjacency matrix for the most salient interaction information.
\noindent
\textbf{Pattern-based Sequence Prediction.}
Sequence prediction refers to the problem of predicting the future sequence using historical information.
Recently, pattern-based methods prevails for many sequence prediction tasks, e.g., speed recognition~\cite{oord2016wavenet,chorowski2014end,graves2014towards}, activity recognition~\cite{donahue2015long,ibrahim2016hierarchical}, and natural language processing~\cite{cho2014learning,sutskever2014sequence,gehring2017convolutional}.
In particular, trajectory prediction can be formulated as a sequence prediction task, which uses historical movement patterns of the agent to predict the future path.
Most trajectory prediction methods adopt recurrent neural networks (RNNs), e.g., Long Short-Term Memory (LSTM) networks~\cite{hochreiter1997long}, to capture the temporal movement.
However, RNN-based models suffer from gradient vanishing and exploding in training and focus more on recent inputs during prediction, especially for long input sequences.
Many sequence prediction works~\cite{oord2016wavenet,wu2019graph} instead adopt convolutional neural networks (CNNs).
The convolutional networks can effectively capture long-term dependency and greatly improve prediction efficiency.
The superiority of CNN-based methods can be largely attributed to the convolutional operation, which is independent of preceding time-steps and thus can process in parallel.
The recent work~\cite{nikhil2018convolutional} proposes a compact CNN model to capture the temporal information, and the results confirm that the CNN-based model can yield competitive performance in trajectory prediction.
However, it fails to model the spatial interaction between pedestrians.
In this work, we propose to capture the spatial interaction with EFGAT and introduce gated convolutional networks to better capture the temporal dynamics.
\noindent
\textbf{Graph Networks for Trajectory Prediction.}
Many studies adopt spatial-temporal graph neural networks (STGNNs) for the sequence prediction task, such as action recognition~\cite{yan2018spatial,si2019attention},
taxi demand prediction~\cite{yao2018deep}, and traffic prediction~\cite{yao2018modeling}.
Specifically, the sequence can be formulated as a sequence of graphs of nodes and edges, where nodes correspond to agents and edges denote their interactions.
The sequence can then be effectively modeled with the spatial-temporal graph network.
Likewise, the trajectory prediction task can be modeled with the \revise{spatial-temporal graph network~\cite{vemula2018social,wang2019pedestrian,huang2019stgat,liang2020garden,mohamed2020social}}.
In particular, the prediction task needs to be modeled in two dimensions, i.e., the spatial dimension and the temporal dimension.
The spatial dimension models the interaction between the agent and its neighbors, and the temporal dimension models the historical trajectory for each agent.
Specifically, each node in the graph represents one pedestrian of a scene, and each edge between two nodes captures the interaction between the two corresponding pedestrians.
For example, social attention~\cite{vemula2018social} models each node with the location of the agent, and edge with the distance between pedestrians, where the spatial relation is modeled with an attention module and then the temporal with RNNs.
Similarly, ~\cite{wang2019pedestrian} constructs the STGNN with Edge RNN and Node RNN based on the location.
STGAT~\cite{huang2019stgat} adopts GAT to capture the spatial interaction by assigning different importance to neighbors and adopts extra LSTMs to capture the temporal information.
The major limitation of these methods lies in capturing the spatial interaction along the temporal dimension.
Notably, the future path of an agent is not only dependent on the current position but meanwhile the neighbors'.
However, the information of such spatial interaction may be lost during the aggregation of the node features along the temporal dimension using RNN-based models.
Different from RNN-based methods, Social-STGCNN~\cite{mohamed2020social} and
Graph WaveNet~\cite{wu2019graph} adopts CNNs to alleviate parameter inefficiency and demonstrate the capability of CNNs in the temporal modeling of long sequences.
In this paper, we propose an enhanced temporal convolutional network to integrate both the temporal dynamics of the agent and the spatial interactions.
\section{GraphTCN}
\label{sec:methodology}
The goal of trajectory prediction is to predict the future paths of all agents that are present in a scene.
Naturally, the future path of an agent depends on its historical trajectory, i.e., the temporal interaction, and is influenced by the trajectories of neighboring agents, i.e., the spatial interaction.
Therefore, the trajectory prediction model needs to take into account both features when modeling the spatial and temporal interactions for the prediction.
\noindent
\textbf{Problem Formulation}
Formally, the trajectory prediction can be defined as follows:
given N pedestrians observed in a scene with $T_{obs}$ steps,
the position of a single pedestrian $i \in \{1, \dots, N\}$ at the time step $t \in \{1, \dots, T_{obs}\}$ is denoted as $X_{i}^{t} =
\left(x_{i}^{t}, y_{i}^{t}\right)$.
Therefore, the observation positions of the pedestrian $X_i$ can be represented as $X_{i}^{1:T_{obs}}$ = $X_{i}^{1}$, $X_{i}^{2}$,..., $X_{i}^{T_{obs}}$.
The goal of trajectory prediction is then to predict all the future positions $\hat{Y}_{i}^{t}$ ($t \in \{T_{obs+1}, \dots, T_{pred}$\}) concurrently.
\subsection{Overall Framework}
\begin{figure*}
\begin{center}
\begin{subfigure}{0.76\textwidth}
\centering
\includegraphics[height=5.6cm]{method_imgs/graphtcn_overview.pdf}
\caption{GraphTCN Overview}
\label{fig:overview}
\end{subfigure}
\begin{subfigure}{.23\textwidth}
\centering
\includegraphics[height=5cm]{method_imgs/efgat_framework.pdf}
\caption{\revise{EFGAT}}
\label{fig:efgat}
\end{subfigure}
\end{center}
\caption{(a) The overview of GraphTCN, where EFGAT captures the spatial interaction between agents for each time step and is based on the historical trajectory embedding. TCN further captures the temporal interaction across time steps.
The decoder module then produces multiple socially acceptable trajectories for all the agents simultaneously.
(b) EFGAT captures the spatial salient information with graph attentional layers (GAL) and skip connections.
}
\label{fig:overall_framework}
\end{figure*}
As illustrated in Fig.~\ref{fig:overall_framework}(a), GraphTCN comprises three key modules, including the edge feature graph attention (EFGAT) module, temporal convolutional (TCN) module, and a decoder.
First, we embed the absolute positions and relative positions of each pedestrian into a fixed-length hidden space and feed these trajectories features into the EFGAT module.
The residual learning mechanism and skip connections~\cite{he2016deep} are incorporated into the network to facilitate the gradient backpropagation and encourage the intermediate feature reusage.
The TCN module is a feed-forward one-dimensional convolutional network with a gating activation unit~\cite{oord2016wavenet} for capturing the most salient features.
Finally, the decoder module produces future trajectories of all pedestrians.
We elaborate on the details of each module of GraphTCN in the following sections.
\subsection{EFGAT Module for Spatial Interaction}
The EFGAT module shown in Fig.~\ref{fig:overall_framework}(b) is designed to encode the spatial interaction between pedestrians with graph attentional layers and graph residual connections.
Formally, pedestrians within the same time step can be formulated as a directed graph $\mathcal{G}=(\mathcal{V}^t, \mathcal{E}^t)$, where each node $v_{i}^t \in \mathcal{V}^t, i \in \{1, \dots, N\}$ corresponds to the $i$-th pedestrian, and the weighted edge $\left(v_{i}^t, v_{j}^t\right) \in \mathcal{E}^t$ represents the human-human interaction from pedestrian $i$ to $j$.
The adjacency matrix $A^t \in \mathbb{R}^{N \times N}$ of $\mathcal{G}$ thus represents the spatial relationships between pedestrians.
\begin{figure}[!htb]
\begin{center}
\includegraphics[width=0.55\linewidth]{method_imgs/single_layer_egat.pdf}
\end{center}
\caption{An illustration of the graph attentional layer with 5 nodes employed by our EFGAT module. The attention $\alpha_{i j}^{t}$ between node $i$ and its neighbors is learned from their embedding features and relative spatial relation $\hat{A}_{ij}^{t}$.
}
\label{fig:single_layer_egat}
\end{figure}
We represent the spatial relation of nodes as an asymmetric, non-negative matrix in this task since the influence between the agents should be different based on their relative movement behavior.
Therefore, instead of constructing graphs with undirected spatial distance, we introduce relative spatial location as prior edge feature knowledge of the adjacency matrix:
\begin{equation}
\hat{A}_{ij}^{t}=\phi_{s}\left(x_{i}^{t}-x_{j}^{t}, y_{i}^{t}-y_{j}^{t} ; W_{s}\right)
\end{equation}
\noindent
\revise{
where $\phi_s(\cdot)$ embeds the relative distance features to a higher dimension $F_1$ with a linear transformation,} and $\mathbf{W}_{s}$ is the embedding weight.
We feed the learnable edge weights and node features into graph attentional layers illustrated in Fig.~\ref{fig:single_layer_egat} to capture the spatial interaction:
\begin{equation}
\alpha_{i j}^{t}=\frac{\exp \left(\sigma\left(\mathbf{W}_{1} \hat{h}_{i}^{t} + \mathbf{W}_{2} \hat{h}_{j}^{t} + \hat{A}_{i j}^{t}\right)\right)}{\sum_{k \in \mathcal{N}_{i}} \exp \left(\sigma\left(\mathbf{W}_{1} \hat{h}_{i}^{t} + \mathbf{W}_{2} \hat{h}_{k}^{t} + \hat{A}_{i j}^{t}\right)\right)}
\end{equation}
\noindent
where $\hat{h}_i^t \in \mathbb{R}^{F_1}$ is the node input feature for pedestrian $i$ at time step $t$, ${F_1}$ is the dimension of node features, \revise{${\cal N}_i$ is the set of neighbors for node $i$ in the graph}, $\sigma(\cdot)$ is the LeakyReLU activation, and $\mathbf{W}_{1}$, $\mathbf{W}_{2}$ are learnable weights.
In this way, $\alpha_{i j}^{t}$ gives the importance weight of neighbor $j$ to pedestrian $i$ dynamically via the self-attention mechanism.
The gating function empirically proves to be effective in controlling the signal bypassing~\revise{~\cite{oord2016wavenet,xie2018rethinking,dauphin2017language}}.
We therefore adopt the gated activation unit to dynamically regulate the information flow and select salient features:
\begin{equation}
g_{i}^{t}= g\left(\mathbf{W}_{h} \hat{h}_{i}^{t}+b_{h}\right) \odot \left(\mathbf{W}_{h} \hat{h}_{i}^{t}+b_{h}\right)
\end{equation}
\noindent
where $g(\cdot)$ is the \textit{tanh} activation function, $\mathbf{W}_{h}$ is the affine transformation parameters, $b_{h}$ is the bias, and $\odot$ denotes the element-wise multiplication.
This can be understood as a multiplicative skip connection which facilitates gradients
flow through layers~\cite{dauphin2017language}.
To stabilize the self-attention process~\cite{velivckovic2017graph,wu2019graph}, we adopt the multi-head attention mechanism:
\begin{equation}
\revise{
{h}_{i}^{t}= \bigparallel_{k=1}^{K} \sigma\left(\sum_{j \in \mathcal{N}_{i}} \alpha_{k i j}^{t} g_{j}^{t}\right) + \mathbf{R}({\hat{h}}_{i}^{t}; \mathbf{W_{r}})
}
\end{equation}
\noindent
where
$\mathbf{W}_{r}$ is the learnable parameters, $\bigparallel$ denotes concatenation, and $K$ is the number of attention heads.
$\mathbf{R}(\cdot)$ denotes the graph residual term~\cite{oord2016wavenet,velivckovic2017graph,wu2019graph}.
\revise{We name the proposed multi-head graph attention layers as GAL, which
can be stacked multiple times for better modeling spatial relation (e.g., twice as in Fig.~\ref{fig:overall_framework}(b)).
Subsequently, we can obtain final node representations of ${\mathbf{h}}=\left\{{h}_{1}, {h}_{2}, \ldots, {h}_{N} \right\}$, where ${h}_{i} \in \mathbb{R}^{T_{obs} \times (K \cdot F_1)}$ captures the aggregated spatial interaction between pedestrian $i$ and all the neighbors at each time step.}
Therefore, the EFGAT module can learn a self-adaptive adjacency matrix that captures the relative importance for different pedestrians.
\subsection{TCN for Spatial and Temporal Interaction Modeling}
The movement pattern of a pedestrian is greatly influenced by the historical trajectory and the moving patterns of neighboring pedestrians.
We therefore propose to capture the spatial and temporal interaction between pedestrians using a modified temporal convolution network (TCN), which is illustrated in Fig.~\ref{fig:convolutional_layers}.
\begin{figure}[!htb]
\begin{subfigure}{.65\textwidth}
\centering
\includegraphics[height=3.6cm]{method_imgs/tcn_layers_overview.pdf}
\caption{}
\label{fig:tcn_layers}
\end{subfigure}
\begin{subfigure}{.34\textwidth}
\centering
\includegraphics[height=3.6cm]{method_imgs/single_tcn_layer.pdf}
\caption{}
\label{fig:tcn_layer}
\end{subfigure}
\caption{
(a) An illustration of TCN with a stack of 3 convolution layers of kernel size 3.
The input $\mathbf{h}_{in}$ (i.e., $\mathbf{h}^{(0)}$) contains the spatial information captured by the preceding EFGAT modules.
The output of TCN $\vec{\mathbf{h}}$ is collected by concatenating $\mathbf{h}_{out}$ (i.e., $\mathbf{h}^{(L)}$) across time.
(b) The gating function in each of the TCN layers to control the bypass signals.
}
\label{fig:convolutional_layers}
\end{figure}
The network shown in Fig.~\ref{fig:convolutional_layers}(a) can be regarded as a short-term and long-term encoder, where lower convolution layers focus on local short-term interactions, while in higher layers, long-term interactions are captured with a larger receptive field.
For example, if the kernel size of the TCN is $k$, the receptive field size in the $l$-th layer is $(k-1) \cdot l+1$, which increases linearly ascending layers.
Therefore, the top layer of TCN captures interactions within a longer time span.
Since the order of the input is important in the sequence prediction task, we therefore adopt the left padding of size $k-1$ instead of symmetric padding for the convolution, where each convolution output convolves over the input of the corresponding time step and the preceding $k-1$ time steps.
The output size of each convolution then remains the same as the input.
In each layer of TCN~\ref{fig:convolutional_layers} in Fig.(b), the gated activation unit utilizes two non-linear functions to dynamically regulate the information flow formed as:
\begin{equation}
\mathbf{h}^{(l+1)}= g\left(\mathbf{W}_{g}^{(l)} * \mathbf{h}^{(l)} \right) \odot \sigma\left(\mathbf{W}_{f}^{(l)} * \mathbf{{h}}^{(l)}\right)
\end{equation}
\noindent
where $\mathbf{h}^{(0)}$ is the output $\mathbf{h}$ from the EFGAT modules, $\mathbf{h}^{(l)} \in \mathbb{R}^{N \times T_{obs} \times F_2}$, $\mathbf{W}_{g}$ and $\mathbf{W}_{f}$ are the learnable 1D-convolution parameters, and $\sigma(\cdot)$ denotes the \textit{sigmoid} function.
\revise{The final output of the TCN module can be denoted as $\vec{\mathbf{h}} \in \mathbb{R}^{N \times T_{obs} \times F_2}$.}
In this way, the embedding vector $\vec{h}_i$ captures all the spatial-temporal interaction between the $i$-th pedestrian and its neighbors.
We note that TCN can handle much longer input sequences with the dilated convolution~\cite{oord2016wavenet}, which is more efficient than RNN-based methods.
\subsection{Future Trajectory Prediction}
In real-world applications, given the historical trajectory, there are multiple plausible paths of future movements.
We also model such uncertainty of the final movement in our decoder module for the trajectory prediction.
Following STGAT~\cite{huang2019stgat}, the decoder module produces multiple socially acceptable trajectories by introducing \revise{a shared random noise $\boldsymbol{z} \in \mathbb{R}^{T_{obs} \times F_3}$, which is concatenated with the spatial-temporal embedding $\vec{\mathbf{h}}$, as part of the decoder input.}
\revise{Specifically, the input of the decoder can be denoted as $\tilde{\mathbf{h}} \in \mathbb{R}^{N \times T_{obs} \times (F_2+F_3)}$.
We adopt a canonical MLP layer to generate the relative future locations $\Delta \hat{Y} \in \mathbb{R}^{N \times T_{pred} \times 2}$ and denote the architecture with such an MLP decoder as GraphTCN.}
\revise{The predicted relative location $\Delta \hat{Y}$ is the relative position to the origin for all the pedestrians.}
We then convert relative positions to absolute positions $\hat{Y}$ and adopt the variety loss as the loss function for training, which computes the minimum ADE loss among the $M$ plausible trajectories:
\begin{equation}
\mathcal{L}_{\text {ADE}} (\hat{Y}) =\frac{\sum_{i=1}^{N} \sum_{t=1}^{T_{pred}}\left\|\hat{Y}_{i}^{t}-Y_{i}^{t}\right\|_{2}}{N T_{pred}}
\label{eq:ade}
\end{equation}
\begin{equation}
\mathcal{L}_{\text {variety}} = \min _{m}\left(\mathcal{L}_{A D E}\left(\hat{Y}^{(m)}\right)\right)
\end{equation}
\noindent
where ${Y}$ is the ground truth, $\hat{Y}^{(1)}, \dots, \hat{Y}^{(M)}$ are the $M$ plausible trajectories predicted.
Although this loss function may lead to a diluted probability density function~\cite{thiede2019analyzing}, we empirically find that it facilitates better predictions of multiple future trajectories.
We further integrate the deep generative strategy\revise{~\cite{sohn2015learning,ivanovic2019trajectron,mangalam2020not}} adopted widely in multimodal prediction to enhance the decoder of our GraphTCN.
\revise{
Specifically, during training, we concatenate $\overrightarrow{\mathbf{h}}$ with the ground-truth future trajectories encoded by an MLP layer, and then further encode the two features with an MLP to produce $\boldsymbol{\mu}$ and $\boldsymbol{\sigma}$ for the noise distribution $\boldsymbol{\hat{z}} = \mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\sigma}), \boldsymbol{\hat{z}} \in \mathbb{R}^{N \times F_4}$ following CVAE~\cite{sohn2015learning,mangalam2020not}.
Note that $\boldsymbol{\hat{z}}$ is randomly sampled from distribution $\mathcal{N}(\boldsymbol{0}, \mathbf{I})$ during inference.
For the final relative location prediction, we again concatenate $\boldsymbol{\hat{z}}$ with $\vec{\mathbf{h}}$ and feed them into an MLP layer to produce $\Delta \hat{Y}$.
We further introduce the KL divergence regularization term~\cite{Goodfellow-et-al-2016,lee2017desire} to stabilize the training process:}
\begin{equation}
\mathcal{L} = \lambda_{1} \mathcal{L}_{\text {variety}} + \lambda_{2} {D_{K L}(\mathcal{N}(\boldsymbol{\mu}, \boldsymbol{\sigma}) \| \mathcal{N}(\boldsymbol{0}, \mathbf{I}))}
\end{equation}
\noindent
GraphTCN with such a decoder is denoted as GraphTCN-G in the following experiments.
\section{Experiments}
\label{sec:experiment}
In this section, we evaluate our GraphTCN on two world coordinates trajectory prediction datasets, i.e., ETH~\cite{pellegrini2010improving} and UCY~\cite{lerner2007crowds}, and compare the performance of GraphTCN with state-of-the-art approaches.
\subsection{Datasets and Evaluation Metrics}
ETH and UCY datasets comprise five outdoor environments that are recorded from a fixed top-view.
The ETH dataset includes ETH and Hotel, and the UCY dataset consists of UNIV, ZARA1, and ZARA2.
In these datasets, pedestrians exhibit complex behaviors, including nonlinear trajectories, moving from different directions, walking together, walking unpredictably, avoiding collisions, standing, etc.
The crowd density of a single scene in each environment varies from 0 to 51 pedestrians per frame.
All datasets are records at 25 frames per second (FPS), and the pedestrian trajectory is extracted at every 2.5 FPS.
We use two metrics to evaluate model performance:
\textit{Average Displacement Error (ADE)} defined in Equation~\ref{eq:ade}, which is the average Euclidean distance between the predicted trajectory and the ground truth over all prediction time steps, and \textit{Final Displacement Error (FDE)}, which is the Euclidean distance between the predict position and the ground truth position at the final time step $T_{pred}$.
The model is trained with the leave-one-out policy~\cite{alahi2016social,gupta2018social,vemula2018social,huang2019stgat}.
We produced 4 samples for the next 4.8 seconds (12 timesteps) based on 3.2 seconds (8 timesteps) observations.
\subsection{Implementation Details}
We train with Adam optimizer in 50 epochs with a learning rate of 0.0001.
The node feature embedding size is set to 64.
The EFGAT module comprises two graph attention layers with \revise{attention heads K = 2, 1, and a output dimension F1 = 16, 32 for the first and second GAL respectively.}
\revise{$F_2$ is also set to 16, and the noise $\boldsymbol{z}$ has a dimension of $F_3$ = 4.}
For the GraphTCN-G decoder, the ground truth trajectory during training is encoded
into a dimension of 64 for $F_4$.
$M$ is set to 4 and 20 for predicting 4 and 20 sample paths.
All the LeakyReLU in our model has a negative slope of 0.2.
$\lambda_{1}$ is set to 1, and $\lambda_{2}$ is set to 0.5 for the first 15 epochs and 0.2 for the rest epochs for GraphTCN-G.
\subsection{Baselines}
We compare our framework with the baselines and four state-of-the-art approaches:
\textit{LSTM} adopts the vanilla LSTM encoder-decoder model to predict the sequence of every single pedestrian.
\textit{Social LSTM}~\cite{alahi2016social} builds on top of LSTM and introduces a social pooling layer to capture the spatial interaction between pedestrians.
\textit{CNN}~\cite{nikhil2018convolutional} adopts the CNNs to predict the sequence.
\textit{SR-LSTM}~\cite{zhang2019sr} obtains the spatial influence by iteratively refining the LSTM hidden states through the gate and attention mechanism.
\textit{Social GAN}~\cite{gupta2018social} improves over Social LSTM with socially generative GAN to generate multiple plausible trajectories.
\textit{Trajectron}~\cite{ivanovic2019trajectron} utilizes LSTM to capture the spatial and temporal relations and incorporates CVAE \revise{~\cite{sohn2015learning}} to generate the distributions of future paths.
\revise{\textit{SoPhie}~\cite{sadeghian2019sophie} introduces the social and physical attention mechanisms to an LSTM based GAN model.}
\textit{Social-STGCNN}~\cite{mohamed2020social} is one of the SOTA approaches that utilizes CNNs to extract spatio-temporal features.
\textit{STGAT}~\cite{huang2019stgat} is one of the SOTA approaches which adopts vanilla GAT to model the spatial interactions and LSTMs to capture temporal interaction.
\subsection{Quantitative Results}
The results in Table \ref{quantitative_result} show that GraphTCN achieves consistently better performance compared with existing models on these benchmark datasets.
Our model generates multiple trajectories at once for future trajectories.
\revise{We can notice that GraphTCN achieves better prediction performance than other baselines with only four predictions instead of 20 as in most baselines, e.g., STGAT~\cite{huang2019stgat}, with an ADE of 0.36 and an FDE of 0.72 on average.}
These results confirm that our GraphTCN yields competitive results even with less generated paths compared with previous approaches in terms of prediction accuracy, especially on the more complex dataset UNIV, ZARA1, and ZARA2.
\begin{table*}
\begin{center}
\caption{Quantitative results of GraphTCN compared with baseline approaches.
Evaluation metrics are reported in ADE / FDE in meters (the lower numerical result is better).
The $\ast$ mark denotes the deterministic model, and the rest of the baseline approaches are stochastic models with M = 20 prediction samples.
}
\label{quantitative_result}
\begin{tabular}{c||c|c|c|c|c||c}
\hline
Method & ETH & HOTEL & UNIV & ZARA1 & ZARA2 & AVG\\
\hline
LSTM$\ast$~\cite{alahi2016social}& 1.09 / 2.41 & 0.86 / 1.91 & 0.61 / 1.31 & 0.41 / 0.88 & 0.52 / 1.11 &
0.70 / 1.52\\
Social-LSTM$\ast$~\cite{alahi2016social} & 1.09 / 2.35 & 0.79 / 1.76 & 0.67 / 1.40 & 0.47 / 1.00 & 0.56 / 1.17 & 0.72 / 1.54\\
CNN$\ast$~\cite{nikhil2018convolutional} & 1.04 / 2.07 & 0.59 / 1.27 & 0.57 / 1.21 & 0.43 / 0.90 & 0.34 / 0.75 & 0.59 / 1.22\\
SR-LSTM$\ast$~\cite{zhang2019sr} & 0.63 / 1.25 & 0.37 / 0.74 & 0.51 / 1.10 & 0.41 / 0.90 & 0.32 / 0.70 & 0.45 / 0.94\\
\hline
Social-GAN~\cite{gupta2018social} & 0.81 / 1.52 & 0.72 / 1.61 & 0.60 / 1.26 & 0.34 / 0.69 & 0.42 / 0.84 & 0.58 / 1.18\\
Trajectron~\cite{ivanovic2019trajectron} & 0.59 / 1.14 & 0.35 / 0.66 & 0.54 / 1.13 & 0.43 / 0.83 & 0.43 / 0.85 & 0.56 / 1.14 \\
\revise{
SoPhie~\cite{sadeghian2019sophie}} & 0.70 / 1.43 & 0.76 / 1.67 & 0.54 / 1.24 & 0.30 / 0.63 & 0.38 / 0.78&0.54 / 1.15\\
Social-STGCNN~\cite{mohamed2020social} & 0.64 / 1.11 & 0.49 / 0.85 & 0.44 / 0.79 & 0.34 / 0.53 & 0.30 / 0.48 & 0.44 / 0.75\\
STGAT~\cite{huang2019stgat} & 0.65 / 1.12 & 0.35 / 0.66 & 0.52 / 1.10 & 0.34 / 0.69 & 0.29 / 0.60 & 0.43 / 0.83\\
\hline
GraphTCN (M = 4) & {\bf 0.59 / 1.12} & {\bf 0.27 / 0.52 } & { 0.42 / 0.87 } & { 0.30 / 0.62 } & { 0.23 / 0.48 } & {\bf 0.36 / 0.72 }\\
\revise{GraphTCN (M = 20)} & {\bf{0.39 / 0.71}} & {0.21 / 0.44 } & {0.33 / 0.66 } & {0.21 / 0.42 } & { 0.17 / 0.43 } & { 0.26 / 0.51 }\\
GraphTCN-G (M = 4) & { 0.60 / 1.21} & {\bf 0.27 / 0.52 } & {\bf 0.41 / 0.84 } & {\bf 0.28 / 0.58 } & {\bf 0.22 / 0.47 } & {\bf 0.36 / 0.72 }\\
\revise{GraphTCN-G (M = 20)} & {\bf0.39} / 0.75 & {\bf 0.18 / 0.33 } & { \bf 0.30 / 0.60 } & { \bf 0.20 / 0.39 } & { \bf 0.16 / 0.32 } & { \bf 0.25 / 0.48 }\\
\hline
\end{tabular}
\end{center}
\end{table*}
\noindent
\textbf{Ablation Study.}
We evaluate each module of GraphTCN through ablation studies in Table~\ref{ablation_study}.
\textit{w/o EGNN} refers to the model without the spatial module.
\textit{vanilla GAT} refers to the model with GAT as the spatial module while ignoring the relative relation between pedestrians in spatial modeling.
\textit{GraphTCN-G} refers to the model which integrates VAE for multi-modal future path prediction.
The result demonstrates that introducing graph neural networks (GNN) into the framework can reduce ADE and FDE, and adding edge relative relation to GNN leads to further improvement.
However, these spatial interactions can only improve performance mildly.
We further scrutinize the dataset and attribute these findings to the fact that pedestrians seldom change their path suddenly to avoid their neighbors.
As a consequence, the temporal features already contain part of the spatial interactions for the prediction.
Therefore, spatial information is less critical in the prediction.
Meanwhile, compared with RNN-based approaches, GraphTCN can model the whole observed sequence better without losing important temporal information.
\begin{table}
\begin{center}
\caption{Ablation studies of GraphTCN.
}
\label{ablation_study}
\begin{tabular}{c||c|c}
\hline
Method & M = 4 & M = 20 \\
\hline
w/o EGNN & 0.38 / 0.78 & 0.28 / 0.54 \\
vanilla GAT & 0.37 / 0.74 & 0.27 / 0.54 \\
GraphTCN & 0.36 / 0.72 & 0.26 / 0.51 \\
GraphTCN-G & 0.36 / 0.72 & 0.25 / 0.48 \\
\hline
\end{tabular}
\end{center}\vspace{-6mm}
\end{table}
\noindent
\textbf{Inference Speed.}
We compare the inference speed of GraphTCN with state-of-the-art methods, including Social GAN~\cite{gupta2018social}, SR-LSTM~\cite{zhang2019sr}, Social-STGCNN~\cite{mohamed2020social} and STGAT~\cite{huang2019stgat}.
Table \ref{speed_comparison}\footnote{
For a fair comparison, the reported time includes the data processing time since some approaches require extra time to construct the graph during inference.
Note that we use the corresponding official implementations and settings for each model, and the batch size is one in the evaluation.
} reports the model inference time and the speedup factor compared with the Social GAN in wall-clock second.
As can be observed from the results, GraphTCN achieves much faster inference compared with these baseline approaches.
In particular, GraphTCN takes 0.00067 second inference time to generate 4 samples, which is 42.82 times and 5.22 times faster than Social-GAN and the most similar prior approach STGAT respectively.
\begin{table}
\begin{center}
\caption{The inference time and speedup of GraphTCN compared with baseline methods.
The inference time is the average of the total inference steps per pedestrians.
The results are reported on an Intel Core i9-9880H Processor.}
\label{speed_comparison}
\begin{tabular}{c||c|c}
\hline
& Inference Time & Speed-up \\
\hline
Social-GAN~\cite{gupta2018social} & 0.02869 & 1$\times$\\
Social-STGCNN~\cite{mohamed2020social} & 0.00861 & 3.33 $\times$\\
STGAT~\cite{huang2019stgat} & 0.00350 & 8.20 $\times$\\
Trajectron~\cite{ivanovic2019trajectron} & 0.00081 & 35.42 $\times$\\
\hline
GraphTCN (M=4) & 0.00066 & 43.47 $\times$\\
GraphTCN-G (M=4) & 0.00067 & 42.82 $\times$\\
GraphTCN-G (M=20) & 0.00075 & 38.25 $\times$\\
\hline
\end{tabular}
\end{center}
\vspace{-6mm}
\end{table}
\begin{figure}[t]
\begin{center}
\setlength{\tabcolsep}{2pt}
\begin{tabular}{p{0.2cm}ccc}
\rotatebox[origin=lb]{90}{\hspace{1.em} STGAT} &
\frame{\includegraphics[scale=0.1]{exp_imgs/stgat-single-769.pdf}} &
\frame{\includegraphics[scale=0.1]{exp_imgs/stgat_8_5360.pdf}} &
\frame{\includegraphics[scale=0.1]{exp_imgs/stgat_13_5430.pdf}} \\
\rotatebox[origin=lb]{90}{\hspace{1.em} OURS} &
\frame{\includegraphics[scale=0.1]{exp_imgs/graphtcn-g-single-769.pdf}} &
\frame{\includegraphics[scale=0.1]{exp_imgs/graphtcn_8_5360.pdf}} &
\frame{\includegraphics[scale=0.1]{exp_imgs/graphtcn_13_5430.pdf}} \\
&(a)&(b)&(c)\\
\end{tabular}
\caption{Comparison of our GraphTCN (M=4) and STGAT predictions with ground truth trajectories.
To better illustrate the results, only part of the pedestrian trajectories are presented.
The solid red line, solid blue line, and dashed yellow line denote the observed trajectory, ground truth future trajectory, and predicted trajectory, respectively.}
\label{fig_quantitative_result}
\end{center}
\vspace{-3mm}
\end{figure}
\subsection{Qualitative Analysis}
We investigate the prediction results of our GraphTCN by visualizing and comparing the predicted trajectories with the best-performing approach STGAT in Fig.~\ref{fig_quantitative_result}.
We choose three different scenarios in which the complex interactions take place.
The complex interactions include pedestrian standing, pedestrian merging, pedestrian following and pedestrian avoidance.
In Fig.~\ref{fig_quantitative_result}, we can observe that GraphTCN achieves better performance on:
firstly, \textit{direction and speed},
from Fig.~\ref{fig_quantitative_result}(a)(b), we find that trajectories generated by GraphTCN follow the same direction as the ground truth, while predictions of STGAT deviate from the path noticeably.
In Fig.~\ref{fig_quantitative_result}(a), one pedestrian moves in an unexpected direction, and GraphTCN generates an acceptable prediction accordingly.
Besides, GraphTCN generates plausible short trajectories to the stationary pedestrian and the pedestrian who moves slowly.
Secondly, \textit{collision-free future paths},
Fig.~\ref{fig_quantitative_result}(b)(c) show that STGAT may fail to make satisfactory predictions when pedestrians come from different groups, while GraphTCN generates better prediction in scenarios where one pedestrian meets another group.
In Fig.~\ref{fig_quantitative_result}(b), GraphTCN can successfully produce predictions avoiding future collisions when the pedestrian moves in the same direction from an angle.
Further, GraphTCN produces socially acceptable predictions even in the more complex scenario in Fig.~\ref{fig_quantitative_result}(c) when the pedestrian departs for the opposite directions or walks towards the same direction.
\begin{figure}[!htb]
\begin{subfigure}{.32\textwidth}
\centering
\frame{\includegraphics[width=1.\linewidth]{exp_imgs/pic_4_3090_3120_2_new.pdf}}
\frame{\includegraphics[width=1.\linewidth]{exp_imgs/pic_4_3090_3120_0_new.pdf}}
\caption{}
\label{fig:stationary}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\centering
\frame{\includegraphics[width=1.\linewidth]{exp_imgs/pic_4_10260_10290_2_new.pdf}}
\frame{\includegraphics[width=1.\linewidth]{exp_imgs/pic_4_10260_10290_3_new.pdf}}
\caption{}
\label{fig:front_behind}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\centering
\frame{\includegraphics[width=1.\linewidth]{exp_imgs/pic_4_700_730_1_new.pdf}}
\frame{\includegraphics[width=1.\linewidth]{exp_imgs/pic_4_700_730_2_new.pdf}}
\caption{}
\label{fig:opposite_direction}
\end{subfigure}
\caption{\revise{Illustration of EFGAT attention weights.} The solid green/red line is the trajectory, and the arrow indicates the trajectory direction.
The circle color shows the attention at each time step, and the circle size corresponds to the attention weight.
\revise{The green trajectory without circles denotes the target pedestrian.}}
\label{fig:attn}
\end{figure}
\noindent
\textbf{Social attention.}
In Fig.~\ref{fig:attn}, we illustrate the learned attention weights by the EGAT module.
The results show that our model can capture the relative importance of the target's neighbors, and the attention weights between two pedestrians vary along the path.
Further, the attention weight from pedestrian $i$ to pedestrian $j$ and pedestrian $j$ to pedestrian $i$ is different, which is not considered in existing approaches~\cite{huang2019stgat,kosaraju2019social,zhang2019sr,mohamed2020social}.
\textit{Less attention weight}:
in Fig.~\ref{fig:attn}(a), the stationary pedestrian has less impact on its moving neighbors, and the model assigns small importance to the pedestrians far away from the target.
\textit{More significant influence}: our model assigns a higher attention weight to the pedestrian moving toward the target in Fig.~\ref{fig:attn}(a), moving ahead or moving in the rear while having a higher velocity in Fig.~\ref{fig:attn}(b), and moving from the opposite directions before meeting the target Fig.~\ref{fig:attn}(c).
These cases demonstrate that reasonable attention weights are successfully assigned to the target pedestrian's neighbors according to all pedestrian movement patterns in the scene.
\begin{figure}[!htb]
\begin{subfigure}{.32\textwidth}
\centering
\frame{\includegraphics[width=1.\linewidth]{exp_imgs/graphtcn-multi-769.pdf}}
\caption{GraphTCN}
\label{fig:sample_4}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\centering
\frame{\includegraphics[width=1.\linewidth]{exp_imgs/graphtcn-g_multi-769.pdf}}
\caption{GraphTCN-G}
\label{fig:sample_20_graphtcn}
\end{subfigure}
\begin{subfigure}{.32\textwidth}
\centering
\frame{\includegraphics[width=1.\linewidth]{exp_imgs/stgat-multi-60.pdf}}
\caption{STGAT}
\label{fig:sample_20_stgat}
\end{subfigure}
\caption{Visualizations of diverse predicted trajectories.
(a) and (b) show four trajectories produced by GraphTCN and GraphTCN-G, and (c) shows the 20 trajectories generated by STGAT.
}
\label{fig:diverse}
\end{figure}
\noindent
\textbf{Diverse trajectory predictions.}
Fig.~\ref{fig:diverse} is the visualization of diverse predictions.
The result shows that GraphTCN can generate the prediction closer to the ground truth even with a smaller number of samples and can make good predictions for the pedestrian, who have relatively unexpected behaviors.
In this scenario, one pedestrian has the intention to change its direction from the observation, and GraphTCN can generate both the normal and unexpected predictions for it.
And for other pedestrians who have a more consistent observation, the model produces future paths with normal behaviors.
Further, from Fig.~\ref{fig:diverse}(b) and (c), the prediction area of GraphTCN is much smaller and precise than STGAT with 20 predicted trajectories.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we proposed GraphTCN for trajectory prediction, which captures the spatial and temporal interaction between pedestrians effectively by integrating EFGAT to model their spatial interactions, and TCN to model both the spatial and temporal interactions.
The proposed GraphTCN is completely based on feed-forward networks, which is more tractable during training, and achieves better prediction accuracy and higher inference speed compared with existing RNN-based solutions.
Experimental results confirm that our GraphTCN outperforms state-of-the-art approaches on various benchmark datasets.
\noindent
\textbf{Acknowledgement} This research is supported by National Research Foundation (NRF) Singapore (Award NRF2018AU-SG01).
|
1,108,101,565,697 | arxiv | \section{Introduction}
One of the possible ways for building completely integrable quantum systems
is the {\it inverse method of separation of variables} based on
the use of the
so-called {\it multi-parameter spectral equations}. The idea is to
interpret these equations as the result of separating variables in a
certain multi-dimensional completely integrable quantum system and
to reconstruct the form of the latter by eliminating some
of the spectral
parameters considered as separtion constants (for details
see e.g. refs. \cite{Us88,Us89,Us94, MaUsWa97}).
If one wants to obtain this way exactly
solvable completely integrable systems,
one should start with {\it exactly solvable}
multi-parameter spectral equations. In this paper we present one possible
way of building such equations by considering as an example
one-dimensional second-order differential equations.
We show that in this case both the multi-parameter spectral equations
and their solutions can be constructed from one and the same elementary
building blocks called in the paper the {\it $\rho$-functions} which satisfy
special functional relations which we called {\it scalar triangle equations}.
The choice of this terminology is not accidental: The reader having some
elementary acquaintance with the celebrated $r$-matrix approach
(see e.g. \cite{Ji90}) can easily
be convinced that there are many common features between the scalar triangle
equations and ordinary matrix triangle equations\footnote{We mean here
the so-called classical Yang -- Baxter equations.} There is also a deep
relation between the solutions of these equations: the $\rho$-functions
are natural analogues of the classical $r$-matrices. The general form of the
substitution for
solving the multi-parameter spectral equations is very similar
to that of the famous Bethe Ansatz used for solving completely integrable
quantum models. The numerical equations determining the solvability
conditions have the same meaning as the corresponding Bethe Ansatz equations.
In this paper we show that there are two essentially different classes of
exactly solvable multi-parameter spectral equations which can be constructed
from $\rho$-functions. We call them {\it rational} and {\it irrational}
equations stressing the fact that in a certain "canonical" coordinate system
their "potentials" can be expressed respectively via rational and irrational
functions. The rational multi-parameter spectral equations are very well
known in the literature. In 1987 Sklyanin obtained such equations
\cite{Sk87} as a result
of a separation of variables in the so-called completely integrable Gaudin
models \cite{Ga83}
(which are easily constructable in the framework of the $r$-matrix
approach). Applying to the rational multi-parameter spectral equations the
inverse method of separation of variables one recovers the Gaudin models
as shown in refs. \cite{Us88,Us89,Us94}. As to the irrational multi-parameter
spectral equations, these were never discussed in the literature. The reason
for this is that the form of the Bethe Ansatz equations
determining their spectra drastically differs from the standard one usually
obtained in the framework of $r$-matrix method. Moreover, there is no
change of variables which could reduce these equations into standard form.
This leads us to the claim that the class of completely integrable quantum
systems associated with irrational multi-parameter spectral equations is
different from the known
classes of models and therefore its investigation is an
interesting mathematical problem.
\section{The problem}
Second-order linear differential equations play an
important role in many branches of mathematical physics.
By an appropriate homogeneous transformation and a change
of variable any such equation can be reduced to the
following canonical form
\begin{eqnarray}
\left(-\frac{\partial^2}{\partial x^2}+W(x)\right)\psi(x)=0,
\label{1.1}
\end{eqnarray}
in which $W(x)$ and $\psi(x)$ are assumed to be analytic functions of
the complex variable $x$.
There are many different mathematical problems which are
connected with an equation of the form (\ref{1.1}).
The simplest one can be
formulated as follows: for a given function $W(x)$ find
the function $\psi(x)$. The general solution of this problem
obviously is
\begin{eqnarray}
\psi(x)\sim\sin\varphi \cdot\psi_1(x)+ \cos\varphi \cdot\psi_2(x)
\label{1.2}
\end{eqnarray}
where $\psi_1(x)$ and $\psi_2(x)$ are two linearly
independent solutions and $\varphi$ is an arbitrary mixing angle.
The usual way to fix this angle is to impose one additional
constraint on the general solution (\ref{1.2}). This
constraint has to be compatible with the linearity of
equation (\ref{1.1}) and thus should have the form
\begin{eqnarray}
{\cal L}_0[\psi(x)]=0,
\label{1.3}
\end{eqnarray}
where ${\cal L}_0[\psi(x)]$ is some appropriately chosen
linear functional\footnote{In practice, one usually uses
the simplest functionals: ${\cal
L}_0[\psi(x)]\equiv \psi(0)$ and ${\cal L}_0[\psi(x)]\equiv \psi'(0)$.}.
It is well known that, along with this trivial (and
mathematically not very interesting) interpretation of equation (\ref{1.1}),
there are many others which lead to richer sets of solutions and
are of greater theoretical importance. The essence of most of these
interpretations is to allow some freedom in choosing
the function $W(x)$ restricting simultaneously the class of
allowed functions $\psi(x)$. This leads to the so-called
spectral versions of equation (\ref{1.1}).
Consider an example which will play a central role in our
further discussion. Thereby the form of the function $W(x)$ is
restricted to
\begin{eqnarray}
W(x)=W_0(x)+\sum_{n=1}^N e_n W_n(x),
\label{1.4}
\end{eqnarray}
where $W_0(x)$ and $W_1(x),\ldots,W_N(x)$ are some fixed
functions and $e_1,\ldots,e_N$ are arbitrary numerical parameters.
Restrict the class of admissible functions $\psi(x)$ by the following
$N+1$ constraints
\begin{eqnarray}
{\cal L}_0[\psi(x)]={\cal L}_1[\psi(x)]= \ldots
={\cal L}_N[\psi(x)]=0,
\label{1.5}
\end{eqnarray}
where the ${\cal L}_n[\psi(x)],\ n=0,\ldots, N$ are some arbitrarily chosen
linear but linearly independent functionals. Then one can state the problem
of finding those values of the parameters $e_1,\ldots, e_N$ for
which equation (\ref{1.1}) (with $W(x)$ given by formula
(\ref{1.4})) has solutions fulfilling equations (\ref{1.5}).
It is natural to call $e_1,\ldots,e_N$ {\it spectral
parameters} and equation (\ref{1.1}) supplemented by conditions
(\ref{1.4}) and (\ref{1.5}) a {\it multi-parameter spectral equation}.
The set of admissible values for the parameters $e_1,\ldots,e_N$
we shall call the {\it spectrum}. It is easily seen that
the problem (\ref{1.1}), (\ref{1.4}), (\ref{1.5}) is a
natural generalization of an ordinary one-parameter spectral equation
which corresponds to the case $N=1$.
It is not difficult to show that in general the spectrum of equations
(\ref{1.1}), (\ref{1.4}), (\ref{1.5}) is infinite and discrete.
Indeed, let $\psi_1(x,e_1,\ldots,e_N)$ and $\psi_2(x,e_1,\ldots,e_N)$
denote two linearly independent solutions of equation (\ref{1.1})
considered as functions of the spectral parameters $e_1,\ldots,e_N$.
Then the general solution can be written in the form
\begin{eqnarray}
\psi(x)\sim \sin\varphi\cdot \psi_1(x,e_1,\ldots,e_N)+ \cos\varphi
\cdot\psi_2(x,e_1,\ldots,e_N)
\label{1.6}
\end{eqnarray}
where $\varphi$ is an arbitrary parameter (mixing angle).
Substituting (\ref{1.6}) into (\ref{1.5})
leads to a system of $N+1$ numerical equations
\begin{eqnarray}
l_0(\varphi,e_1,\ldots,e_N)=l_1(\varphi,e_1,\ldots,e_N)=
\ldots=l_N(\varphi,e_1,\ldots,e_N)=0,
\label{1.7}
\end{eqnarray}
for $N+1$ quantities $\varphi$ and $e_1,\ldots,e_N$. Here
$l_n(c,e_1,\ldots,e_N)$ denotes the value of the
linear functional ${\cal L}_n[\psi(x)]$ applied to the solution
(\ref{1.6}). Since the number of equations coincides with the
number of unknowns, the spectrum of equation
(\ref{1.1}), (\ref{1.4}), (\ref{1.5})
will be discrete in general. Also generally, the
function (\ref{1.6}) is transcendental
which suggests that the spectrum should be infinite.
It is however quite obvious that the scheme given cannot be considered
a practical way for solving the multi-parameter spectral equations.
This is so because for most functions $W_0(x)$ and $W_1(x),\ldots,W_N(x)$
the explicit form of the general solution (\ref{1.6}) is unknown.
Thus, one can expect that most of the multi-parameter spectral
equations of the form (\ref{1.1}), (\ref{1.4}), (\ref{1.5})
should not be exactly solvable.
Fortunately, there are exceptional cases, and these
we intend to discuss in this paper. We think of
so-called {\it exactly solvable} multi-parameter spectral equations
which can be solved by means of purely algebraic methods
and have many important applications in the theory of
completely integrable quantum systems. One of the standard
and most effective methods for constructing such equations is
the so-called {\it inverse method}. It rests on a very
simple idea: instead of
looking for solutions of equation (\ref{1.1}) for
given functions $W_0(x)$ and $W_1(x),\ldots,W_N(x)$, one should
try to reconstruct the form of these functions starting
with appropriately chosen function $\psi(x)$. An advantage
of the inverse problem in comparison with the direct one is
obvious: the problem of solving differential equations is
replaced by the problem of taking derivatives of known functions.
However, this is only a relative simplicity: the
algebro-analytic part of the inverse problem remains rather
non-trivial which is clearly seen from the following discussion:
Let us fix some $K+1$ functions $\psi_k(x),\ k=0,\ldots,K$
satisfying constraints (\ref{1.5}) and $K+1$ sets of numbers
$\{e_{k1},\ldots,e_{kN}\},\ k=0,\ldots,K$.
Substituting these into (\ref{1.1}), using relation (\ref{1.4}),
taking $e_{k0}=1$ and dividing finally the
$k$th equation by $\psi_k(x)$, we obtain
\begin{eqnarray}
\sum_{n=0}^N e_{kn} W_n(x)=\psi^{-1}_k(x)
\frac{\partial^2\psi_k(x)}{\partial x^2},\quad k=0,1,\ldots,K.
\label{1.8}
\end{eqnarray}
Formula (\ref{1.8}) can be considered a system of $K+1$ linear
inhomogeneous equations for $N+1$ functions $W_0(x)$ and
$W_1(x),\ldots,W_N(x)$. If $K\le N$, then system
(\ref{1.8}) is obviously solvable. In this case one easily finds
the explicit form
of the functions $W_0(x)$ and $W_1(x),\ldots,W_N(x)$ for which
the equations (\ref{1.1}), (\ref{1.4}), (\ref{1.5}) have $K+1$
{\it a priori} known (and, hence, explicit) solutions.
The situation changes, however, if $K>N$ (this is just our case,
because we are looking for multi-parameter spectral equations having
an infinite number of explicit solutions). In this
case, the number of equations (\ref{1.8}) exceeds the
number of unknowns which makes the system (\ref{1.8}) overdetermined
for almost all sets of functions $\psi_k(x)$. The only
way to get rid of this problem is to start with functions
$\psi_k(x)$ for which the compatibility
of the equations forming system (\ref{1.8}) would be guaranteed
from the very beginning. But for this to work
we need some reasonable {\it ansatz}
for the functions $\psi(x)$. It is hardly neccessary to emphasize
that the problem of finding such an ansatz is far from
being trivial.
From a purely practical point of view, it is much more
convenient to deal not with the functions $\psi(x)$ but with their
logarithmic derivatives $P(x)$. In terms of the functions $P(x)$,
the ansatz which we intend to use takes an especially simple form.
The substitution
\begin{eqnarray}
\psi(x)=\exp\left\{\int^x P(x')dx'\right\}
\label{1.9}
\end{eqnarray}
simplifies also the form of equation (\ref{1.1}). The
new equation
\begin{eqnarray}
W(x)=P^2(x)+P'(x),
\label{ste4.3}
\end{eqnarray}
will be considered as starting point for our further considerations.
\section{Separable functions of several variables}
We shall call a function $f(x_1,\ldots,x_k, y_1,\ldots,y_l)$ of $k+l$
variables {\it separable} with respect to the variables
$x_1,\ldots,x_k$ and $y_1,\ldots,y_l$ if
it can be represented in the form of a {\it finite} sum
\begin{eqnarray}
f(x_1,\ldots,x_k,y_1,\ldots,y_l)=\sum_{n=1}^Ng_n(x_1,\ldots,x_k)
h_n(y_1,\ldots,y_l)
\label{ste1.1}
\end{eqnarray}
where $h_n(x_1,\ldots,x_k),\ n=1,\ldots,N$
and $g_n(y_1,\ldots,y_l),\ n=1,\ldots,N$
are some functions of $k$ and $l$ variables, respectively.
For stressing this property we shall denote such a function
by
\begin{eqnarray}
f(x_1,\ldots,x_k,y_1,\ldots,y_l)=f(x_1,\ldots,x_k|y_1,\ldots,y_l)
\label{ste1.2}
\end{eqnarray}
Functions of several variables separable with
respect to all arguments we shall call {\it
completely separable}. Any such function can be represented
in the form
\begin{eqnarray}
f(x_1,x_2,\ldots,x_m)=\sum_{n=1}^Nf_{1,n}(x_1)f_{2,n}(x_2)\ldots
f_{m,n}(x_m)
\label{ste1.3}
\end{eqnarray}
where the $f_{i,n}(x_i),\ i=1,\ldots,m$ are certain functions of one
variable. For such functions we shall use the notation
\begin{eqnarray}
f(x_1,x_2,\ldots,x_m)=f(x_1|x_2|\ldots|x_m)
\label{ste1.4}
\end{eqnarray}
Let us now formulate a simple lemma about separable
functions.
\medskip
{\bf Lemma 1.} Let $r(\xi)$ be a rational function of $\xi$.
Then
\begin{eqnarray}
\frac{r(\xi_1)}{\xi_1-\xi_2}+
\frac{r(\xi_2)}{\xi_2-\xi_1}=f(\xi_1|\xi_2)
\label{ste1.5}
\end{eqnarray}
and
\begin{eqnarray}
\frac{r(\xi_1)}{(\xi_1-\xi_2)(\xi_1-\xi_3)}+
\frac{r(\xi_2)}{(\xi_2-\xi_3)(\xi_2-\xi_1)}+
\frac{r(\xi_3)}{(\xi_3-\xi_1)(\xi_3-\xi_2)}
=f(\xi_1|\xi_2|\xi_3)
\label{ste1.6}
\end{eqnarray}
are symmetric and completely separable functions.
\medskip
{\bf Proof.} Any rational function can be represented as a
linear combination of the so-called {\it elementary
rational functions} having only one singularity in the complex plane.
Thus, in order to prove separability of functions (\ref{ste1.5})
and (\ref{ste1.6}),
it is sufficient to make sure that the positions of
the singularities of the functions $f(\xi_1|\xi_2)$ and
$f(\xi_1|\xi_2|\xi_3)$ with respect to each
argument do not depend on the values of other
arguments. It is clear that the "dangerous" singularities of both
functions (\ref{ste1.5}) and (\ref{ste1.6})
can only arise from their denominators. However, a
simple analysis shows that these singularities (which,
obviously, are present in each of the separate terms)
cancel each other. The absence of these
``dangerous'' singularities completes the proof of the lemma.
\section{Scalar triangle equation}
Let $r(\xi)$ be an arbitrary rational function of a
complex variable $\xi$. Define a complex valued function
$\xi(x)$ by the equation
\begin{eqnarray}
(\xi'(x))^2=r(\xi(x))
\label{ste2.1}
\end{eqnarray}
and construct from it a new function of two complex variables
\begin{eqnarray}
\rho(x,y)=\frac{1}{2}\cdot\frac{\xi'(x)+\xi'(y)}{\xi(x)-\xi(y)}.
\label{ste2.2}
\end{eqnarray}
\medskip
{\bf Lemma 2.} The function $\rho(x,y)$ obeys the
following functional relations:
\begin{eqnarray}
\rho(x,y)+\rho(y,x)=0,
\label{ste2.3}
\end{eqnarray}
\begin{eqnarray}
\rho(x,y)\rho(x,z)+\rho(y,z)\rho(y,x)+\rho(z,x)\rho(z,y)=\omega(x|y|z),
\label{ste2.4}
\end{eqnarray}
\begin{eqnarray}
\frac{\partial}{\partial x}\rho(x,y)+\rho^2(x,y)=\omega(x|x|y),
\label{ste2.5}
\end{eqnarray}
with $\omega(x|y|z)$ a certain separable function.
\medskip
{\bf Proof.} The proof of the anti-symmetry of the function $\rho(x,y)$
immediately follows from definition (\ref{ste2.2}). In order to prove
the separability of the function $\omega(x|y|z)$ we consider the following
chain of equalities
\begin{eqnarray}
4\omega(x|y|z)=4[\rho(x,y)\rho(x,z)+\rho(y,z)\rho(y,x)+\rho(z,x)\rho(z,y)]=
\nonumber\\[0.6cm]
\frac{\xi'(x)+\xi'(y)}{\xi(x)-\xi(y)}\cdot
\frac{\xi'(x)+\xi'(z)}{\xi(x)-\xi(z)}+
\frac{\xi'(y)+\xi'(z)}{\xi(y)-\xi(z)}\cdot
\frac{\xi'(y)+\xi'(x)}{\xi(y)-\xi(x)}+
\frac{\xi'(z)+\xi'(x)}{\xi(z)-\xi(x)}\cdot
\frac{\xi'(z)+\xi'(y)}{\xi(z)-\xi(y)}=\nonumber\\[0.3cm]
\frac{[\xi'(x)]^2}{(\xi(x)-\xi(y))
(\xi(x)-\xi(z))}
+\frac{[\xi'(y)]^2}{(\xi(y)-\xi(z))
(\xi(x)-\xi(x))}
+\frac{[\xi'(z)]^2}{(\xi(z)-\xi(x))
(\xi(x)-\xi(y))}=\nonumber\\[0.3cm]
\frac{r[\xi(x)]}{(\xi(x)-\xi(y))(\xi(x)-\xi(z))}
+\frac{r[\xi(y)]}{(\xi(y)-\xi(z))(\xi(y)-\xi(x))}
+\frac{r[\xi(z)]}{(\xi(z)-\xi(x))(\xi(z)-\xi(y))}.\quad
\label{ste2.6}
\end{eqnarray}
Lemma 1 then shows that the function $\omega(x|y|z)$
is separable.
The last equation (\ref{ste2.5})
can be easily proved if we take $z=x+\epsilon$
in (\ref{ste2.4}), and take the limit
$\epsilon\rightarrow0$ noting that
$\lim_{\epsilon\rightarrow0}\epsilon\rho(x+\epsilon,x)=1$. This
completes the proof.
\medskip
Hereafter we shall call equation (\ref{ste2.4}) the
scalar triangle equation.
\section{$\xi$-functions}
Any rational function of $\xi(x)$ and $\xi'(x)$ we
shall call a $\xi$-function. If $F(x)$ is a $\xi$-function,
it can be represented in the form
\begin{eqnarray}
F(x)=R(\xi(x))+\xi'(x)G(\xi(x))
\label{ste3.1}
\end{eqnarray}
where $R(\xi)$ and $G(\xi)$ are some rational functions.
The sum, difference, product and quotient of two
$\xi$-functions is again a $\xi$-function. The
derivative of a $\xi$-function is again a $\xi$-function. We call
a $\xi$-function $F(x)$ even if $G(\xi)\equiv 0$ in (\ref{ste3.1})
and odd if $R(\xi)\equiv 0$ in (\ref{ste3.1}). The product of two
odd or two even $\xi$-functions is an even $\xi$-function, and
the product of an even and an odd $\xi$-function is an odd $\xi$-function.
This means that the algebra of odd and even $\xi$-functions is
a $z_2$-graded algebra. The differential operator
$\partial/\partial x$ becomes in this case an odd object.
\medskip
{\bf Lemma 3.} If $F(x)$ is a $\xi$-function, then
\begin{eqnarray}
\sigma(x|y)=[F(x)-F(y)]\rho(x,y)
\label{ste3.2}
\end{eqnarray}
is a separable function.
\medskip
{\bf Proof.} Substituting (\ref{ste3.1}) into
(\ref{ste3.2}), we can write
\begin{eqnarray}
\sigma(x|y)=\sigma_R(x|y)+\sigma_G(x|y)
\label{ste3.3}
\end{eqnarray}
where
\begin{eqnarray}
\sigma_R(x|y)=\frac{R(\xi(x))-R(\xi(y))}{\xi(x)-\xi(y)}\cdot
(\xi'(x)-\xi'(y))
\label{ste3.4}
\end{eqnarray}
and
\begin{eqnarray}
\sigma_G(x|y)=\frac{\xi'(x)G(\xi(x))-\xi'(y)G(\xi(y))}
{\xi(x)-\xi(y)}\cdot (\xi'(x)-\xi'(y)).
\label{ste3.5}
\end{eqnarray}
From Lemma 1 it immediately follows that $\sigma_R(x|y)$ is
a separable function. In order to prove separability of
$\sigma_G(x|y)$, let us rewrite (\ref{ste3.5}) in the form
\begin{eqnarray}
\sigma_G(x|y)=\frac{[\xi'(x)]^2-[\xi'(y)]^2}
{\xi(x)-\xi(y)}\cdot (G(\xi(x))+G(\xi(y)))+
\frac{G(\xi(x))-G(\xi(y))}
{\xi(x)-\xi(y)}\cdot (\xi'(x)+\xi'(y))^2.
\label{ste3.6}
\end{eqnarray}
Using now (\ref{ste2.1}) in the first term and applying then
Lemma 1 we find that also $\sigma_G(x|y)$ is
separable. This proves the lemma.
\section{Bethe ansatz}
Let us look for solutions of equation (\ref{ste4.3}) in the
form
\begin{eqnarray}
P(x)=F(x)+\sum_{i=1}^M\rho(x,x_i)
\label{ste4.4}
\end{eqnarray}
where $M$ is an arbitrary non-negative integer,
$x_1,\ldots,x_M$ are still unknown numbers and $F(x)$
is a $\xi$-function. We call
this form the Bethe Ansatz.
Substituting (\ref{ste4.4}) into (\ref{ste4.3}) gives
\begin{eqnarray}
W(x)=F^2(x)+F'(x)+2\sum_{i=1}^M[F(x)-F(x_i)]
\rho(x,x_i)+\nonumber\\
+2\sum_{i=1}^MF(x_i)\rho(x,x_i)
+\sum_{i=1}^M\rho^2(x,x_i)+\sum_{i=1}^M\rho'(x,y)+
\sum_{i\neq k}^M \rho(x,x_i)\rho(x,x_k).
\label{ste4.5}
\end{eqnarray}
Using formulas (\ref{ste2.4}), (\ref{ste2.5}) and (\ref{ste3.2}),
we obtain
\begin{eqnarray}
W(x)=F^2(x)+F'(x)+2\sum_{i=1}^M\sigma(x|x_i)
+\sum_{i=1}^M\omega(x|x|x_i)+\sum_{i\neq k}^M \omega(x|x_i|x_k)+
\nonumber\\
+2\sum_{i=1}^M\rho(x,x_i)\left\{\sum_{k=1,k\neq i}^M \rho(x_i,x_k)
+F(x_i)\right\}.
\label{ste4.6}
\end{eqnarray}
We see that the first three terms in the right hand side of
(\ref{ste4.6}) represent some separable function of $x$ and
$x_i$, while the last sum of the so-called {\it unwanted terms}
is, obviously, non-separable. In order to make
the function $W(x)$ separable, one should require
all the unwanted terms to vanish. This is equivalent to the system
of $M$ equations
\begin{eqnarray}
\sum_{k=1,k\neq i}^M \rho(x_i,x_k)
+F(x_i)=0,\quad i=1,\ldots,M
\label{ste4.7}
\end{eqnarray}
which we shall call the Bethe Ansatz equations. If these
equations are satisfied then
\begin{eqnarray}
W(x)=F^2(x)+F'(x)+2\sum_{i=1}^M\sigma(x|x_i)
+\sum_{i=1}^M\omega(x|x|x_i)+\sum_{i\neq k}^M \omega(x|x_i|x_k)
\label{ste4.8}
\end{eqnarray}
and hence can be written in the form
\begin{eqnarray}
W(x)=W_0(x)+\sum_{n=1}^N e_nW_n(x)
\label{ste4.9}
\end{eqnarray}
where $W_0(x)$ and $W_n(x),\ n=1,\ldots,N$
are certain $\xi$-functions
and $e_n, \ n=1,\ldots, N$ are some numerical coefficients
in which all the dependence on numbers $x_1,\ldots,x_N$ is contained.
We see that equation (\ref{1.1}) takes in this case
the form of a multi-parameter spectral equation. The role
of the spectral parameters is played by the numbers $e_1,\ldots,e_N$.
The admissible values for these parameters are determined
by the solutions
of the Bethe Ansatz equations (\ref{ste4.7}). It is easily seen
that, for any finite $M$, the number of Bethe Ansatz
equations coincides with the number of unknowns and
therefore this system has a discrete set of solutions.
Since this is an algebraic system, it has a finite number of solutions
for any $M$, but $M$ was an arbitrary non-negative integer.
This means that the total number of solutions of system (\ref{ste4.7})
is infinite and thus the corresponding multi-parameter
spectral equation has infinite discrete and algebraically
calculable spectrum.
\section{Some simple examples}
In this section we consider three simple examples of
solutions of the scalar triangle equations and construct the
corresponding classes of $\xi$-functions.
\medskip
{\bf Example 1.} Assume that $r(\xi)$ is a first-order polynomial
\begin{eqnarray}
r(\xi)=a+b\xi.
\label{ste.1}
\end{eqnarray}
Then, from (\ref{ste2.1}) it follows that
\begin{eqnarray}
\xi'(x)=\sqrt{a+b\xi(x)}.
\label{ste.2}
\end{eqnarray}
Solving this differential equation we obtain
\begin{eqnarray}
\xi(x)=\frac{b(x-t)^2}{4}-\frac{a}{b}, \quad \xi'(x)=\frac{b(x-t)}{2}.
\label{ste.3}
\end{eqnarray}
Construction of the function $\rho(x,y)$ by formula (\ref{ste2.2})
gives
\begin{eqnarray}
\rho(x,y)=\frac{1}{x-y}.
\label{ste.4}
\end{eqnarray}
This function obeys all three relations (\ref{ste2.3}) -- (\ref{ste2.5})
with a trivial function $\omega(x|y|z)$:
\begin{eqnarray}
\omega(x|y|z)=0
\label{ste.5}
\end{eqnarray}
In this simple case, the set of $\xi$-functions coincides with
the set of all rational functions of $x$.
\medskip
{\bf Example 2.} Let us now assume that $r(\xi)$ is a second-order
polynomial:
\begin{eqnarray}
r(\xi)=a+b\xi+c\xi^2.
\label{ste.6}
\end{eqnarray}
Then, from (\ref{ste2.1}) it follows that
\begin{eqnarray}
\xi'(x)=\sqrt{a+b\xi(x)+c\xi^2(x)}.
\label{ste.7}
\end{eqnarray}
Solving this differential equation we obtain
\begin{eqnarray}
\xi(x)=\frac{\sqrt{4ac-b^2}}{2c}\mbox{sinh}\
\sqrt{c}(x-t) -\frac{b}{2c},\quad
\xi'(x)=\frac{\sqrt{4ac-b^2}}{2\sqrt{c}}\mbox{cosh}\ \sqrt{c}(x-t).
\label{ste.8}
\end{eqnarray}
Construction of function $\rho(x,y)$ by formula (\ref{ste2.2})
gives in this case
\begin{eqnarray}
\rho(x,y)=\frac{\sqrt{c}}{2}\cdot\mbox{cth} \ \frac{\sqrt{c}}{2}(x-y).
\label{ste.9}
\end{eqnarray}
This function obeys all three relations (\ref{ste2.3}) -- (\ref{ste2.5})
with a constant function $\omega(x|y|z)$:
\begin{eqnarray}
\omega(x|y|z)=\frac{c}{4}.
\label{ste.10}
\end{eqnarray}
In this case, the set of $\xi$-functions coincides with
the set of all hyperbolic functions of $x$ with period
$2\pi i/\sqrt{c}$. For negative $c$ these functions become
trigonometric.
\medskip
{\bf Example 3.} Let now $r(x)$ be a third-order polynomial of the form
\begin{eqnarray}
r(\xi)=a+b\xi+c\xi^2+d\xi^3
\label{ste.11}
\end{eqnarray}
Then, from (\ref{ste2.1}) it follows that
\begin{eqnarray}
\xi'(x)=\sqrt{a+b\xi(x)+c\xi^2(x)+d\xi^3(x)}.
\label{ste.12}
\end{eqnarray}
Solving this differential equation we obtain
\begin{eqnarray}
\xi(x)=\frac{4}{d}{\cal P}(x-t,g_2,g_3)-\frac{c}{3d}, \quad
\xi'(x)=\frac{4}{d}{\cal P}(x-t,g_2,g_3).
\label{ste.13}
\end{eqnarray}
where ${\cal P}(x,g_2,g_3)$ denotes the Weierstrass ${\cal P}$-function
with
\begin{eqnarray}
g_2=\frac{c^2-3bd}{12},\quad
g_3=\frac{2c^3+9bcd-27ad^2}{16\cdot 27}
\label{ste.14}
\end{eqnarray}
Construction of the function $\rho(x,y)$ by formula (\ref{ste2.2})
gives
\begin{eqnarray}
\rho(x,y)=\zeta(x-y,g_2,g_3)-\zeta(x-t,g_2,g_3)+\zeta(y-t,g_2,g_3),
\label{ste.15}
\end{eqnarray}
where $\zeta(x,g_2,g_3)$ denotes the Weierstrass $\zeta$-function.
This function obeys all three relations (\ref{ste2.3}) -- (\ref{ste2.5})
with a non-trivial function $\omega(x|y|z)$:
\begin{eqnarray}
\omega(x|y|z)={\cal P}(x,g_2,g_3)+{\cal P}(y,g_2,g_3)+{\cal P}(z,g_2,g_3)
\label{ste.16}
\end{eqnarray}
In this case, the set of $\xi$-functions coincides with
the set of all elliptic functions of $x$.
\section{An equivalent description}
Note that equation (\ref{ste4.3}) admits an
important equivalence transformation which preserves its
form and its spectrum. This transformation includes
the change of the initial variable $x$
\begin{eqnarray}
\xi=\xi(x)
\label{ste5.1}
\end{eqnarray}
and a linear inhomogeneous tranformation of the functions $P(x)$ and $W(x)$:
\begin{eqnarray}
\bar P(\xi)=\xi'(x)\left[P(x)+\frac{1}{2}\frac{\partial}{\partial
\xi(x)}\ln \xi'(x)\right],
\label{ste5.2}
\end{eqnarray}
\begin{eqnarray}
\bar W(\xi)=
\frac{W(x)}{[\xi'(x)]^2}-\frac{1}{2}\left(\frac{\partial}
{\partial \xi(x)}\right)^2 \ln \xi'(x) -
\frac{1}{4}\left(\frac{\partial}{\partial \xi(x)}\ln \xi'(x)\right)^2.
\label{ste5.3}
\end{eqnarray}
In terms of the new variable $\xi$ and new functions
$\bar W$ and $\bar F$ the equation (\ref{ste4.3})
becomes indeed
\begin{eqnarray}
\bar W(\xi)={\bar P}^2(\xi)+{\bar P}'(\xi).
\label{ste5.4}
\end{eqnarray}
If we make in (\ref{ste5.4}) the substitution
\begin{eqnarray}
{\bar P}(\xi)=\frac{\bar\psi'(\xi)}{\bar\psi(\xi)}
\label{ste5.5}
\end{eqnarray}
then we obtain the transformed version of the initial linear
equation (\ref{1.1})
\begin{eqnarray}
\left(-\frac{\partial^2}{\partial \xi^2}+\bar W(\xi)\right)
\bar\psi(x)=0.
\label{ste5.6}
\end{eqnarray}
We see that the form of equation (\ref{ste5.6}) exactly
coincides with that of the initial equation (\ref{1.1}).
Linearity of the transformation (\ref{ste5.3}) means
that equation (\ref{ste5.6}) is again a multi-parameter spectral
equation having the same spectrum as (\ref{1.1}).
The equations connected by the transformations (\ref{ste5.2}) and
(\ref{ste5.3}) we shall call equivalent.
From the examples discussed in the previous section we know that
the scalar triangle equation has many different solutions which lead
to exactly solvable multi-parameter spectral equations expressable
in terms of rational, trigonometric, elliptic and also more
complicated functions. It is naturally to ask, which of these equations
are equivalent in the sense of the transformations (\ref{ste5.2}) --
(\ref{ste5.3}) and which are not. For this one should solve
the classification problem. The simplest way to do this is to find
some distinguished variable $\xi=\xi(x)$ in terms of which the
solutions of the triangle equation have a more or less unified form. The best
candidate for a variable is obviously the function
$\xi(x)$ obeying equation (\ref{ste2.1}).
Indeed, in this case all functions $\rho(x,x_i)$ used in
the Bethe Ansatz (\ref{ste4.4}) transform
into functions with the same denominator
$\xi-\xi_i$, where $\xi=\xi(x)$ and $\xi_i=\xi(x_i)$. Furthermore,
the condition for vanishing of all unwanted (non-separable) terms in
expression (\ref{ste4.6}) transforms into the condition of
regularity of this expression at the points $\xi=\xi_i$.
The most natural way to perform the neccessary calculations in this
case is to start out immediately from the transformed version (\ref{ste5.4})
of our equation and use only very general informations
on the form of the functions $\bar W(\xi)$ and $\bar P(\xi)$. This
form follows from the results of section 5 and is given as
\begin{eqnarray}
\bar W(\xi)=A(\xi)+\sqrt{r(\xi)}B(\xi)
\label{ste5.7}
\end{eqnarray}
and
\begin{eqnarray}
\bar P(\xi)=a(\xi)+\sqrt{r(\xi)}b(\xi)
\label{ste5.8}
\end{eqnarray}
where $A(\xi)$, $B(\xi)$ and $a(\xi)$, $b(\xi)$ are some
rational functions and $r(\xi)$ as in (\ref{ste2.1}).
Substituting (\ref{ste5.7}) and (\ref{ste5.8}) into
(\ref{ste5.4}) we obtain two independent equations for $a(\xi)$
and $b(\xi)$
\begin{eqnarray}
A(\xi)=a^2(\xi)+a'(\xi)+r(\xi)b^2(\xi)
\label{ste5.9}
\end{eqnarray}
and
\begin{eqnarray}
B(\xi)=b'(\xi)+
\left(2a(\xi)+\frac{r'(\xi)}{2r(\xi)}\right)b(\xi)
\label{ste5.10}
\end{eqnarray}
In the following two sections we demonstrate that these
equations admit two principally different Bethe Ansatz
solutions which we call the {\it rational} and {\it
irrational} ones.
\section{The rational Bethe Ansatz solution}
Before discussing the general case, consider an important
special one. Obviously, the choice
\begin{eqnarray}
b(\xi)=0
\label{ste5.11}
\end{eqnarray}
leads to a simpler form for system (\ref{ste5.9}), (\ref{ste5.10}):
\begin{eqnarray}
A(\xi)=a^2(\xi)+a'(\xi),
\label{ste5.12}
\end{eqnarray}
and
\begin{eqnarray}
B(\xi)=0.
\label{ste5.13}
\end{eqnarray}
which is hence reduced to only one equation (\ref{ste5.12}). The Bethe
Ansatz for this equation reads
\begin{eqnarray}
a(\xi)=\alpha(\xi)+\sum_{i=1}^M \frac{1}{\xi-\xi_i}
\label{ste5.14}
\end{eqnarray}
Substituting this ansatz into (\ref{ste5.12}) and requiring
regularity of the function $A(\xi)$ at the points
$\xi_i,\ i=1,\ldots,M$ we obtain
\begin{eqnarray}
A(\xi)=\alpha^2(\xi)+\alpha'(\xi)+2\sum_{i=1}^M
\frac{\alpha(\xi)-\alpha(\xi_i)}{\xi-\xi_i}
\label{ste5.15}
\end{eqnarray}
where the numbers $\xi_i,\ i=1,\ldots,M$ have to
be solutions of the system of Bethe Ansatz equations
\begin{eqnarray}
\sum_{k=1,k\neq i}^M \frac{1}{\xi_i-\xi_k}+\alpha(\xi_i)=0,
\quad i=1,\ldots,M
\label{ste5.16}
\end{eqnarray}
Substituting (\ref{ste5.15}) and (\ref{ste5.13}) into (\ref{ste5.7})
and using Lemma 1 we can reduce the function $\bar W(\xi)$ to
the form
\begin{eqnarray}
\bar W(\xi)=\bar W_0(\xi)+\sum_{n=1}^N e_n \bar W_n(\xi)
\label{ste5.17}
\end{eqnarray}
with $\bar W_0(\xi)$ and $\bar W_n(\xi),\
n=1,\ldots,N$ certain rational functions and the numbers
$e_n,\ n=1,\ldots,N$ depending on the values of
parameters $\xi_i,\ i=1,\ldots,M$ which fulfill the Bethe
Ansatz equations (\ref{ste5.16}).
\section{The irrational Bethe Ansatz solution}
The most general form for the functions
$a(x)$ and $b(x)$ for the Bethe Ansatz for $P(x)$ reads
\begin{eqnarray}
a(\xi)=\alpha(\xi)+\sum_{i=1}^M \frac{\alpha_i}{\xi-\xi_i}
\label{ste6.1}
\end{eqnarray}
and
\begin{eqnarray}
b(\xi)=\beta(\xi)+\sum_{i=1}^M \frac{\beta_i}{\xi-\xi_i}
\label{ste6.2}
\end{eqnarray}
where $\alpha_i,\beta_i,\xi_i,\ i=1,\ldots,M$ are some
unknown numerical parameters and $\alpha(\xi)$ and $\beta(\xi)$
are fixed rational functions.
Substituting formulas (\ref{ste6.1}) and (\ref{ste6.2})
into equations (\ref{ste5.9}) and (\ref{ste5.10}) and
requiring regularity of the functions $A(\xi)$ and
$B(\xi)$ at the points $\xi_i,\ i=1,\ldots,M$ we obtain
\begin{eqnarray}
A(\xi)=\alpha^2(\xi)+\alpha'(\xi)+r(\xi)\beta^2(\xi)+
\sum_{i=1}^M
\frac{1}{(\xi-\xi_i)^2}\left(
\frac{r(\xi)-r(\xi_i)-(\xi-\xi_i)r'(\xi_i)}
{4r(\xi_i)}\right)+\nonumber\\
+\sum_{i=1}^M \frac{1}
{(\xi-\xi_i)}\left(
\alpha(\xi)-\alpha(\xi_i)+
\frac{r(\xi)\beta(\xi)-r(\xi_i)\beta(\xi_i)}{\sqrt{r(\xi_i)}}+
\frac{r(\xi)-r(\xi_i)}{4}\sum_{i=1}^M
\frac{[r(\xi_i)r(\xi_k)]^{-\frac{1}{2}}}{\xi_i-\xi_k}
\right)
\label{ste6.3}
\end{eqnarray}
and
\begin{eqnarray}
B(\xi)=\beta'(\xi)+\left(2\alpha(\xi)+
\frac{r'(\xi)}{4r(\xi)}\right)\beta(\xi)+\nonumber\\
+\sum_{i=1}^M \frac{1}{\xi-\xi_i}\left(
\beta(\xi)-\beta(\xi_i)+
\frac{\alpha(\xi)-\alpha(\xi_i)}{\sqrt{r(\xi_i)}}+
\frac{r'(\xi)/r(\xi)-r'(\xi_i)/r(\xi)}{4\sqrt{r(\xi_i)}}
\right)
\label{ste6.3'}
\end{eqnarray}
where the numbers $\xi_i,\ i=1,\ldots,M$ are again assumed to
be solutions of the system of Bethe Ansatz equations
\begin{eqnarray}
\sum_{k=1,k\neq i}^M \frac{1}{\xi_i-\xi_k}\left(
1+\sqrt{\frac{r(\xi_i)}{r(\xi_k)}}\right)+
\sqrt{r(\xi_i)}\beta(\xi_i)+\frac{r'(\xi_i)}{2r(\xi_i)}=0,
\quad i=1,\ldots,M
\label{ste6.4}
\end{eqnarray}
For the numbers $\alpha_i,\beta_i,\ i=1,\ldots,M$ we
obtain
\begin{eqnarray}
\alpha_i=\frac{1}{2},\quad
\beta_i=\frac{1}{2\sqrt{r(\xi)}}, \quad i=1,\ldots,M
\label{ste6.5}
\end{eqnarray}
Substituting (\ref{ste6.2}), (\ref{ste6.3}) and (\ref{ste6.3'})
into (\ref{ste5.7})
and using Lemma 1 we can reduce the function $\bar W(\xi)$ to
the form
\begin{eqnarray}
\bar W(\xi)=\bar W_0(\xi)+\sum_{n=1}^N e_n \bar W_n(\xi)
\label{ste6.6}
\end{eqnarray}
where $\bar W_0(\xi)$ and $\bar W_n(\xi),\
n=1,\ldots,N$ are certain irrational functions and the numbers
$e_n,\ n=1,\ldots,N$ depend on the
parameters $\xi_i,\ i=1,\ldots,M$ which satisfy the Bethe
Ansatz equations (\ref{ste6.4}).
|
1,108,101,565,698 | arxiv | \section*{List of Accepted Publications}
\begin{itemize}
\item {\em Shai Bagon}, {\bf Boundary Driven Interactive Segmentation}.
Published in the proceedings of the 3$^{rd}$ International Conference on Information Science and Applications (ICISA), 2012.
\item {\em Sebastian Nowozin, Carsten Rother, Shai Bagon, Toby Sharp, Bangpeng Yao and Pushmeet Kohli},
{\bf Decision Tree Fields}.
Published in the proceedings of the 13$^{th}$ International Conference on Computer Vision (ICCV), 2011.
\item {\em Shai Bagon, Or Brostovsky, Meirav Galun and Michal Irani},
{\bf Detecting and Sketching the Common}.
Published in the proceedings of the 23$^{rd}$ IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2010.
\end{itemize}
}
\end{document}
\chapter*{Acknowledgments}
To Daffy
To Alma, Mica and Ruth
To my Parents
To Michal and Meirav
Eli Schechtmann and Aseem Agrawala - Adobe
Carsten Rother, Sebastian Nowozin, Toby Sharp and Pushmeet Kohli - MSRC
\clearpage
\part{Applications}
\label{part:app}
This part concentrate on the first axis of this thesis.
This direction explores new applications which require arbitrary energies.
We start with an unsupervised clustering objective function, Correlation Clustering (CC).
Chapters~\ref{cp:negaff-sketch} and~\ref{cp:negaff-cc} show several applications all revolving around the correlation clustering energy.
This energy is hard to optimize:
It has both smoothness-encouraging as well as contrastive pair-wise terms;
it has no data term to guide the optimization process,
and the number of discrete labels is not known a-priori.
We analyze an interesting property of the correlation clustering functional: its ability to recover the underlying number of clusters.
This interesting property is due mainly to the usage of terms that enhance contrast, rather than smoothness, in the solution $\mathbf{x}$.
Another application that requires arbitrary energy is 3D reconstruction of surfaces.
We show, in chapter~\ref{cp:lighting}, how under certain conditions 3D reconstruction may be posed as a solution to a partial differential equation (PDE).
Solving this PDE to recover the 3D surface can be done through discretization of the solution space.
The discrete version of the PDE yields pair-wise terms that are beyond semi-metric.
Finally, the parameters of the energy function defining the terms $\varphi_i$ and $\varphi_{ij}$ may not be fixed a-priori.
It may happen that one would like to learn the energy function from training data for various applications.
Chapter~\ref{cp:dtf} shows an example of such an energy learning framework.
The resulting learned energy is no longer guaranteed to be ``well-behaved".
In fact, experiments show that when the learning procedure is not constrained it is often the case that the resulting energy is arbitrary and does not yield any known structure.
\chapter[Sketching the Common]{Sketching the Common\protect\footnotemark{}}\protect\footnotetext{This is joint work with Or Brostovsky, Meirav Galun and Michal Irani. It was published in the 23$^{rd}$ International Conference on Computer Vision and Pattern Recognition (CVPR), \protect\citeyear{Bagon2010}.}
\label{cp:negaff-sketch}
Given very few images containing a common object of interest under
severe variations in appearance, we detect the common object and
provide a compact visual representation of that object, depicted
by a binary sketch. Our algorithm is composed of two stages:
(i)~Detect a mutually common (yet non-trivial) ensemble of
`self-similarity descriptors' shared by all the input images. \
(ii)~Having found such a mutually common ensemble, `invert' it to
generate a compact sketch which best represents this ensemble.
This provides a simple and compact visual representation of the
common object, while eliminating the background clutter of the
query images. It can be obtained from {\em very few} query images.
Such clean sketches may be useful for detection, retrieval,
recognition, co-segmentation, and for artistic graphical purposes.
The `inversion' process that generates the sketch is formulated as
a discrete optimization problem of a binary, non-submodular energy function.
\input{sketching}
\chapter[Negative Affinities]{Negative Affinities\protect\footnotemark{}}\protect\footnotetext{This is joint work with Meirav Galun}
\label{cp:negaff-cc}
Clustering is a fundamental task in unsupervised learning.
The focus of this chapter is the Correlation Clustering (CC) functional which combines positive and negative affinities between pairs of data points.
In this chapter we provide a theoretical analysis of the CC functional.
Our analysis suggests a probabilistic generative interpretation for the functional, and
justifies its intrinsic ``model-selection" capability.
In addition we suggest two new applications that utilize the ``model-selection" capability of CC:
unsupervised face identification and interactive multi-object segmentation by rough boundary delineation.
The resulting CC energy is arbitrary and is very difficult to approximate.
We defer the discussion on our approximate minimization algorithms for the CC energy to chapter~\ref{cp:CC} in part~\ref{part:approx}, which deals with approximation schemes for arbitrary energies.
\input{cc_app}
\chapter[3D Shape Reconstruction by Combining Motion and Lighting Cues]{3D Shape Reconstruction by Combining Motion and Lighting Cues\protect\footnotemark{}}\protect\footnotetext{This is joint work with Meirav Galun and Ronen Basri}
\label{cp:lighting}
In this chapter we consider the problem of reconstructing the 3D shape of a moving object while accounting for the change of intensities due to a change in orientation with respect to the light sources. We assume that both the lighting and motion parameters are given. Two methods are presented. First, for lambertian objects illuminated by a point source we derive a PDE that is quasilinear and implicit in the surface shape. We propose to solve this PDE by continuation (characteristic curves), extending an existing method to allow for large motion. Secondly, we formulate the reconstruction problem as an MRF and solve it using discrete optimization techniques. The latter method works with fairly general reflectance functions and can be applied to sequences of two or more images. It can also incorporate prior information and boundary conditions. We further discuss a method for extracting boundary conditions. We demonstrate the performance of our algorithms by showing reconstructions of smooth shaped objects and comparing these reconstructions to reconstructions with laser scans.
\input{lighting}
\chapter[Learning Discrete Energies]{Learning Discrete Energies\protect\footnotemark{}}\protect\footnotetext{This is joint work with Sebastian Nowozin, Carsten Rother, Toby Sharp, Bangpeng Yao and Pushmeet Kohli. It was published in the 13$^{th}$ International Conference on Computer Vision (ICCV), \citeyear{Nowozin2011}.}
\label{cp:dtf}
This chapter introduces a new formulation for discrete image labeling tasks,
the Decision Tree Field (DTF), that combines and generalizes random forests
and conditional random fields (CRF) which have been widely used in computer
vision.
In a typical CRF model the unary potentials are derived from
sophisticated random forest or boosting based classifiers, however, the
pairwise potentials are assumed to (1) have a simple parametric form
with a pre-specified and fixed dependence on the image data, and (2)
to be defined on the basis of a small and fixed neighborhood. In contrast, in
DTF, local interactions between multiple variables are determined by
means of decision trees evaluated on the image data, allowing the
interactions to be adapted to the image content.
This results in powerful graphical models which are able to represent
complex label structure.
Our key technical contribution is to show that the DTF model
can be trained efficiently and jointly using a convex approximate
likelihood function, enabling us to learn over a million free model
parameters.
We show experimentally that for applications which have a rich and complex
label structure, our model outperforms state-of-the-art approaches.
\input{dtf}
\part{Approximations}
\part{Approximate Optimization}
\label{part:approx}
In the previous part several different applications in the domain of computer vision were presented to demonstrate the
enhanced descriptive power gained by considering arbitrary energies.
However, this gain comes with a price tag: existing optimization algorithms no longer provide good approximations in practice.
This part of my work addresses this issue.
It concentrates around the second axis of this thesis, which focuses on practical methods and approaches for approximating the minimization of challenging arbitrary discrete energies.
In particular, Chapter~\ref{cp:CC} proposes a discrete optimization approach to the correlation clustering functional presented in Chapters~\ref{cp:negaff-sketch} and~\ref{cp:negaff-cc}.
This approach scales gracefully with the number of variables, better than existing approaches (\cite{Vitaladevuni2010}).
In fact, we show that our discrete approach to the CC optimization can handle energies defined over hundreds of thousands of variables, as arise in e.g., image segmentation (Sec.~\ref{sec:cc-app-uimos}).
This is by far more variables than any other existing method can handle.
Chapter~\ref{cp:multiscale} concludes this part with a more general perspective on discrete optimization.
This new perspective is inspired by multiscale approaches and suggests to cope with the NP-hardness of discrete optimization using the {\em multiscale landscape} of the energy function.
Defining and observing this multiscale landscape of the energy, I propose methods to explore and exploit it to derive coarse-to-fine optimization framework.
This new perspective gives rise to a unified multiscale framework for discrete optimization.
The proposed multiscale approach is applicable to a diversity of discrete energies, both smoothness-encouraging as well as arbitrary, contrast-enhancing functions.
\chapter[Correlation Clustering Optimization]{Correlation Clustering Optimization\protect\footnotemark{}}\protect\footnotetext{This is joint work with Meirav Galun}
\label{cp:CC}
The focus of this chapter is the optimization of the Correlation Clustering functional which combines positive and negative affinities between the data points.
The main contribution of this chapter are new optimization algorithms which can cope with large scale problems
($>100K$ variables) that are infeasible using existing methods.
We draw an analogy between optimizing this functional and the well known Potts energy minimization.
This analogy allows us to suggest several new optimization algorithms,
which exploit the intrinsic ``model-selection" capability of the functional
to automatically recover the underlying number of clusters.
We compare our algorithms to existing methods on both synthetic and real data.
\input{cc}
\chapter[Discrete Multiscale Optimization]{Discrete Multiscale Optimization\protect\footnotemark{}}\protect\footnotetext{This is joint work with Meirav Galun. It was published in the $5^{th}$ NIPS workshop on optimization for machine learning, \citeyear{Bagon2012opt}.}
\label{cp:multiscale}
This chapter presents a unified multiscale framework for discrete energy minimization
that directly acts on the energy.
Our approach utilizes algebraic multiscale principles to efficiently explore the discrete solution space.
The main goal of our framework is to improve optimization performance for challenging, non-submodular energies for which current methods provide unsatisfactory approximations.
Furthermore, the ability to derive a multiscale pyramid directly from an energy makes our framework application independent.
Two important implications rise from this independence:
(i)~One no longer needs to tailor a multiscale scheme to suit each different application.
(ii)~Our framework makes it trivial to turn existing single scale optimization algorithms into powerful multiscale methods.
Our framework gives rise to two complementary energy coarsening strategies:
one in which coarser scales involve fewer variables,
and a more revolutionary one in which the coarser scales involve fewer discrete labels.
We empirically evaluated our unified framework on a variety of both non-submodular and submodular energies, including energies from the Middlebury benchmark.
\input{ms}
\section{Introduction}
\label{sec:cc-opt-intro}
Optimizing CC is tightly related to many graph partitioning formulations (\cite{Nowozin2009}), however it is
known to be NP-hard (\cite{Bansal2004}).
Existing methods derive convex continuous relaxations to approximately optimize the CC functional.
However, these algorithms do not scale beyond a few thousands of variables.
See for example, the works of \cite{Nowozin2009,Bagon2010,Vitaladevuni2010,Glasner2011}.
This work suggests a new perspective on the CC functional, showing its analogy to the known {\em Potts model}.
This new perspective allows us to leverage on recent advances in discrete optimization to propose new CC optimization algorithms.
We show that our algorithms scale to large number of variables ($>100K$), and in fact can tackle tasks that were {\bf infeasible in the past}, e.g., applying CC to pixel-level image segmentation.
In addition, we provide a {\em rigorous statistical interpretation} for the CC functional and justify its intrinsic model selection capability.
Our algorithms exploit this ``model selection" property to automatically recover the underlying number of clusters $k$.
\section{CC Optimization: Continuous Perspective}\label{sec:existing-cc-optimization}
Optimizing the correlation clustering functional (Eq.~(\ref{eq:CorrClust})) is NP-hard (\cite{Bansal2004}).
Instead of solving {\bf directly} for a partition $U$, existing methods optimize {\bf indirectly} for the binary adjacency matrix $X=UU^T$, i.e., $X_{ij}=1$ iff $i$ and $j$ belong to the same cluster.
By introducing the binary adjacency matrix the quadratic objective (w.r.t. $U$): $-\sum_{ij} W_{ij}\left[UU^T\right]_{ij}$ becomes linear (w.r.t. $X$): $-\sum_{ij} W_{ij}X_{ij}$.
The connected components of $X$, after proper rounding, are the resulting clusters, and the number of clusters $k$ naturally emerges.
Indirect optimization methods must ascertain that the feasible set consists only of ``decomposable" $X$: $X=UU^T$.
This may be achieved either by posing semi-definite constraints on $X$ (\cite{Vitaladevuni2010}), or by introducing large number of linear inequalities (\cite{Demaine2003,Vitaladevuni2010}).
These methods take a continuous and convex relaxation approach to approximate the resulting functional.
This approach allows for nice theoretical properties due to the convex optimization at the cost of a very restricted scalability.
Solving for $X$ requires $\sim n^2$ variables instead of only \mbox{$\sim n$} when solving directly for $U$.
Therefore, these methods scale poorly with the number of variables $n$, and in fact, they cannot handle more than a few hundreds of variables.
In summary, these methods suffer from two drawbacks: (i)~recovering $U$ from $X$ is highly susceptible to noise and more importantly (ii)~it is {\em infeasible} to solve large scale problems by these methods.
\section{Our New Perspective on CC}
\label{sec:perspective}
Existing methods view the CC optimization in the context of convex relaxation and build upon methods and approaches that are common practice in this field of continuous optimization.
We propose an alternative perspective to the CC optimization:
{\em viewing it as a discrete energy minimization}.
This new perspective allows us to build upon recent advances in discrete optimization and propose efficient and direct CC optimization algorithms.
More importantly, the resulting algorithms solve {\em directly} for $U$, and thus scales significantly better with the number of variables.
We now show how to cast the CC functional of Eq.(\ref{eq:CorrClust}) as a discrete pair-wise conditional random field (CRF) energy.
For ease of notation, we describe a partition $U$ using a labeling vector $L\in\left\{1,2\ldots\right\}^n$: $l_i = c$ iff $U_{ic}=1$.
A general form of pair-wise CRF energy is $E\left(L\right)=\sum_i E_i\left(l_i\right) + \sum_{ij} E_{ij}\left(l_i, l_j\right)$ (\cite{Boykov2001}). Discarding the unary term ($\sum_i E_i\left(l_i\right)$), and taking the pair-wise term to be $W_{ij}$ if $l_i \ne l_j$ we can re-write the CC functional as a CRF energy:
\begin{eqnarray}
\mathcal{E}_{CC}\left(L\right) = \sum_{ij} W_{ij} \cdot \mathbbm{1}_{\left[l_i\ne l_j\right]} \label{eq:CorrClustCRF}
\end{eqnarray}
This is a Potts model.
Optimizing the CC functional can now be interpreted as searching for a MAP assignment for the energy (\ref{eq:CorrClustCRF}).
The resulting Potts energy has three unique characteristics, each posing a challenge to the optimization process:\\*
\noindent(i)~{\bf Non-submodular:} The energy is non-submodular.
The notion of submodularity is the discrete analogue of convexity from continuous optimization (\cite{Lovasz1983}).
Optimizing a non-submodular energy is NP-hard, even for the binary case (\cite{Rother2007}).\\*
\noindent(ii)~{\bf Unknown number of labels:}
Most CRF energies are defined for a fixed and known number of labels.
Thus, the search space is restricted to $L\in\left\{1,\ldots,k\right\}^n$.
When the number of labels $k$ is unknown the search space is by far larger and more complicated.\\*
\noindent(iii)~{\bf No unary term:} There is no unary term in the energy.
The unary term plays an important role in guiding the optimization process (\cite{Szeliski2008}).
Moreover, a strong unary term is crucial when the energy in non-submodular (\cite{Rother2007}).
There exist examples of CRFs in the literature that share some of these characteristics (e.g., non-submodular \cite{Rother2007,kolmogorov2005}, unknown number of labels \cite{Isack2011,Bleyer2010}).
Yet, to the best of our knowledge, no existing CRF exhibits all these three challenges at once.
More specifically, we are the first to handle non-submodular energy that has no unary term.
Therefore, we cannot just use ``off-the-shelf" Potts optimization algorithms,
but rather modify and improve them to cope with the three challenges posed by the CC energy.
\begin{algorithm}[t!]
\caption{Expand-and-Explore}\label{alg:a-expand}
\DontPrintSemicolon
\SetKw{KwInit}{Init}
\SetKwFunction{KwExpand}{Expand}
\SetKwFunction{Weinberg}{$\mathcal{E}_{CC}$}
\KwIn{Affinity matrix $W\in\mathbb{R}^{n\times n}$}
\KwOut{Labeling vector $L\in\left\{1,2,\ldots\right\}^n$}
\BlankLine
\KwInit{$L_i\leftarrow 1$, $i=1,\ldots,n$}\tcp*[f]{initial labeling}\;
\Repeat{$L$ is unchanged}{
\For{$\alpha\leftarrow1$ ; $\alpha\le\#L+1$ ; $\alpha++$}{
$L \leftarrow $ \KwExpand{$\alpha$}\;
}
}
\BlankLine
$\#L$ denotes the number of different labels in $L$.\;
\KwExpand{$\alpha$}: expanding $\alpha$ using QPBOI.\;
By letting $\alpha = \#L+1$ the algorithm ``expand" and explore an empty label. This may affect the number of labels $\#L$.\;
\end{algorithm}
\section{Our Large Scale CC Optimization}
\label{sec:alg}
In this section we adapt known discrete energy minimization algorithms to cope with the three challenges posed by the CC energy.
We derive three CC optimization algorithms that stem from either large move making algorithms ($\alpha$-expand and $\alpha\beta$-swap of \cite{Boykov2001}), or Iterated Conditional Modes (ICM) of \cite{Besag1986}.
Our resulting algorithms scale gracefully with the number of variables $n$, and solve CC optimization problems that were {\em infeasible} in the past.
Furthermore, our algorithms take advantage of the intrinsic model selection capability of the CC functional (Sec.~\ref{sec:cc-theory}) to robustly recover the underlying number of clusters.
\subsection{Improved large move making algorithms}
\cite{Boykov2001} introduced a very effective method for multi-label energy minimization that makes large search steps by iteratively solving binary sub-problems.
There are two large move making algorithms: $\alpha$-expand and $\alpha\beta$-swap that differ by the binary sub-problem they solve.
$\alpha$-expand consider for each variable whether it is better to retain its current label or flip it to label $\alpha$.
The binary step of $\alpha\beta$-swap involves only variables that are currently assigned to labels $\alpha$ or $\beta$, and consider whether it is better to retain their current label or switch to either $\alpha$ or $\beta$.
Defined for submodular energies, the binary step in these algorithms is solved using graph-cut.
We propose new optimization algorithms: {\em Expand-and-Explore} and {\em Swap-and-Explore},
that can cope with the challenges of the CC energy.
(i)~For the binary step we use a solver that handles non-submodular energies.
(ii)~We incorporate ``model selection" into the iterative search to recover the underlying number of clusters $k$.
(iii)~In the absence of unary term, a good initial labeling is provided to the non-submodular binary solver.
Binary non-submodular energies can be approximated by an extension of graph-cuts: QPBO (\cite{Rother2007}).
When the binary energy is non-submodular QPBO is not guaranteed to provide a labeling for all variables.
Instead, it outputs only a partial labeling. How many variables are labeled depends on the amount of non-submodular pairs and the relative strength of the unary term for the specific energy.
When no unary term exists in the energy QPBO leaves most of the variables unlabeled. To circumvent this behavior we use the ``improve" extension of QPBO (denoted by QPBOI): This extension is capable of improving an initial labeling to find a labeling with lower energy (\cite{Rother2007}).
In the context of expand and swap algorithms a natural initial labeling for the binary steps is to use the current labels of the variables and use QPBOI to improve on it, ensuring the energy does not increase during iterations.
To overcome the problem of finding the number of clusters $k$ our algorithms do not iterate over a fixed number of labels, but explore an ``empty" cluster in addition to the existing clusters in the current solution. Exploring an extra empty cluster allows the algorithms to optimize over all solutions with any number of clusters $k$. The fact that there is no unary term in the energy makes it straight forward to perform. Alg.~\ref{alg:a-expand} and Alg.~\ref{alg:ab-swap} presents our {\em Expand-and-Explore} and {\em Swap-and-Explore} algorithms in more detail.
\begin{algorithm}[t!]
\caption{Swap-and-Explore}\label{alg:ab-swap}
\DontPrintSemicolon
\SetKw{KwInit}{Init}
\SetKwFunction{KwSwap}{Swap}
\SetKwFunction{KwEnergy}{$\mathcal{E}_{CC}$}
\KwIn{Affinity matrix $W\in\mathbb{R}^{n\times n}$}
\KwOut{Labeling vector $L\in\left\{1,2,\ldots\right\}^n$}
\BlankLine
\KwInit{$L_i\leftarrow 1$, $i=1,\ldots,n$}\tcp*[f]{initial labeling}\;
\Repeat{$L$ is unchanged}{
\For{$\alpha\leftarrow1$ ; $\alpha\le\#L$ ; $\alpha++$}{
\For{$\beta\leftarrow\alpha$ ; $\beta\le\#L+1$ ; $\beta++$}{
$L \leftarrow $ \KwSwap{$\alpha$, $\beta$}\;
}
}
}
\BlankLine
$\#L$ denotes the number of different labels in $L$.\;
\KwSwap{$\alpha$, $\beta$}: swapping labels $\alpha$ and $\beta$ using QPBOI.\;
By letting $\beta = \#L+1$ the algorithm explore new number of clusters, this may affect the number of labels $\#L$.\;
\end{algorithm}
\subsection{Adaptive-label ICM}
Another discrete energy minimization method that we modified to cope with the three challenges of the CC optimization is ICM (\cite{Besag1986}).
It is a point-wise greedy search algorithm.
Iteratively, each variable is assigned the label that minimizes the energy, conditioned on the current labels of all the other variables.
ICM is commonly used for MAP estimation of energies with a {\em fixed} number of labels.
Here we present an {\em adaptive-label ICM}: using the ICM conditional iterations
we adaptively determine the number of labels $k$.
Conditioned on the current labeling, we assign each point to the cluster it is most attracted to, or to a singleton cluster if it is repelled by all.
\
In this section we proposed a new perspective on CC optimization.
Interpreting it as MAP estimation of Potts energy allows us to propose a variety of efficient optimization methods\footnote{Matlab implementation available at: \protect\url{http://www.wisdom.weizmann.ac.il/~bagon/matlab.html}.}:
\begin{itemize}
\item Swap-and-Explore (with binary step using QPBOI)
\item Expand-and-Explore (with binary step using QPBOI)
\item Adaptive-label ICM
\end{itemize}
Our proposed approach has the following advantages:\\*
\noindent(i)~It solves only for $n$ integer variables.
This is by far less than the number of variables required by existing methods described in Sec.~\ref{sec:existing-cc-optimization}, which require $\sim n^2$ variables of the adjacency matrix $X=UU^T$. It makes our approach capable of dealing with large number of variables ($>100K$) and suitable for pixel-level image segmentation.\\*
\noindent(ii)~The algorithms solve directly for the cluster membership of each point, thus there is no need for rounding scheme to extract $U$ from the adjacency matrix $X$.\\*
\noindent(iii)~The number of clusters $k$ is optimally determined by the algorithm and it does not have to be externally supplied like in many other clustering/segmentation methods.
In their work \cite{elsner2009} proposed a greedy algorithm to optimize the CC functional over complete graphs. Their algorithm is in fact an ICM method presented outside the proper context of CRF energy minimization, and thus does not allow to generalize the concept of discrete optimization to more recent optimization methods.
\section{Experimental Results} \label{sec:synthetic}
This section evaluates the performance of our proposed optimization algorithms using both synthetic and real data.
We compare to both existing discrete optimization algorithms that can handle multi-label non-submodular energies (TRW-S of \cite{Kolmogorov2006} and BP of \cite{Pearl1988}\footnote{Since these two algorithms work only with pre-defined number of clusters $k$, we over-estimate $k$ and report only the number of {\em non empty} clusters in the solution.}), and to existing state-of-the-art CC optimization method of \cite{Vitaladevuni2010}.
Since existing CC optimization methods do not scale beyond several hundreds of variables, extremely small matrices are used in the following experiments.
Interactive segmentation results (Sec.~\ref{sec:cc-app-uimos}) evaluated our method on large scale problems.
\subsection{Synthetic data}
\begin{figure}
\centering
\begin{tabular}{cc}
(a) Energy (lower=better) & (b) Recovered $k$ (GT in dashed) \\
\includegraphics[width=.25\linewidth]{cc/synth750_energy_plot.pdf}&
\includegraphics[width=.25\linewidth]{cc/synth750_nl_plot.pdf}\\
&\\
(c) Purity & (d) Run time \\
\includegraphics[width=.25\linewidth]{cc/synth750_purity_plot.pdf}&
\includegraphics[width=.25\linewidth]{cc/synth750_time_plot.pdf}\\
\multicolumn{2}{c}{Legend:{\color{swap} Swap-and-Explore}, {\color{expand} Expand-and-Explore}, {\color{icm} ICM}, {\color{trws} TRW-S}, {\color{bp} BP}
}
\end{tabular}
\caption{{\bf Synthetic results: }{\em Graphs comparing (a)~Energy at convergence. (b)~Recovered number of clusters. (c)~Purity of resulting clusters. (d)~Run time of algorithms (log scale).
{\color{trws}TRW-S} and {\color{bp}BP} are almost indistinguishable, as are {\color{swap}Swap} and {\color{expand}Expand} in most of the plots.
}}
\label{fig:synth-res}
\end{figure}
This experiment uses synthetic affinity matrices $W$ to compare our algorithms to existing Potts optimization algorithms.
The synthetic data have 750 variables randomly assigned to 15 clusters with different sizes
(ratio between larger to smaller cluster: $\sim\times5$). For each variable we sampled roughly the same number of neighbors: of which $\sim25\%$ are from within the cluster and the rest from the other clusters.
We corrupted the clean ground-truth adjacency matrix with $20\%$ noise affecting both the sign of $W_{ij}$ and the certainty (i.e., $\left|W_{ij}\right|$). Overall the resulting percent of positive (submodular) connections is $\sim30\%$.
We report several measurements for these experiments: run-time, energy ($\mathcal{E}_{CC}$), purity of the resulting clusters and the recovered number of clusters $k$ for each of the algorithms as a function of the sparsity of the matrix $W$, i.e., percent of non-zero entries. Each experiment was repeated $10$ times with different randomly generated matrices.
Fig.~\ref{fig:synth-res} shows results of the synthetic experiments.
Existing multi-label approaches ({\color{trws}TRW-S} and {\color{bp}BP}) do not perform too well: higher $\mathcal{E}_{CC}$, lower purity and incorrect recovery of $k$.
This demonstrates the difficulty of the energy minimization problem that has no unary term and many non-submodular pair-wise terms. These results are in accordance with the observations of \cite{kolmogorov2005} when the energy is hard to optimize.
For our large move making algorithms, {\color{expand}Expand-and-Explore} provides marginally better clustering results than the {\color{swap}Swap-and-Explore}. However, its relatively slow running time makes it infeasible for large CC problems\footnote{This difference in run time between Expand and Swap can be explained by looking at the number of variables involved in each of the binary steps carried out: For the expand algorithm, each binary step involves almost all the variables, while the binary swap move considers only a small subset of the variables.}.
A somewhat surprising result of these experiments shows that for matrices not too sparse (above $10\%$), {\color{icm} adaptive-label ICM} performs surprisingly well. In fact, it is significantly faster than all the other methods and manages to converge to the correct number of clusters with high purity and low energy.
From these experiments we conclude that {\color{swap}Swap-and-Explore} (Alg.~\ref{alg:ab-swap}) is a very good choice of optimization algorithm for the CC functional. However, when the affinity matrix $W$ is not too sparse, it is worth while giving our {\color{icm}adaptive-label ICM} a shot.
\subsection{Co-clustering data}
The following experiment compares our algorithms with a state-of-the-art CC optimization method, R-LP, of \cite{Vitaladevuni2010}.
For this comparison we use affinity matrices computed for co-segmentation.
The co-segmentation problem can be formulated as a correlation clustering problem with super pixels as the variables (\cite{Glasner2011}).
We obtained 77 affinity matrices, courtesy of \cite{Glasner2011}, used in their experiments.
The number of variables in each matrix ranges from 87 to 788,
Their sparsity (percent of non-zero entries) ranges from $6\%$ to $50\%$,
and there are roughly the same number of positive (submodular) and negative (non-submodular) entries.
Table~\ref{tab:comp-daniel} shows the ratio between our energy and the energy of R-LP method. The table also shows the percent of matrices for which our algorithms found a solution with lower energy than R-LP.
The results show the superiority of our algorithms to existing multi-label energy minimization approaches (TRW-S and BP). Furthermore, it is shown that our methods are in par with existing state-of-the-art CC optimization method on small problems.
However, unlike existing methods, our algorithms can be applied to problems {\em two orders of magnitude larger}.
Optimizing directly for $U$ not only did not compromise the performance of our method, but also allows us to handle large scale CC optimization, as demonstrated in the next section.
\begin{table}
\centering
\begin{tabular}{c||c|c|c||c|c}
&\multicolumn{3}{c||}{Ours}& & \\
& Swap & Expand & ICM & TRWS & BP \\
\hline \hline
Energy ratio & $98.6$ & $98.4$ & $77.4$ & $83.8$ & $83.6$ \\
(\%) & $\pm1.4$ & $\pm1.9$ & $\pm23.9$ & $\pm5.4$ & $\pm6.3$ \\
\hline
Strictly lower& \multirow{2}{*}{15\%} & \multirow{2}{*}{11.7\%} & \multirow{2}{*}{0} & \multirow{2}{*}{0} & \multirow{2}{*}{0} \\
$\left(>100\%\right)$ & & & & &\\
\end{tabular}
\caption{{\bf Comparison to \protect\cite{Glasner2011}:}
{\em Ratio between our energy and of Glasner~\mbox{{\em et~al.}}:
Since all energies are negative, higher ratio means lower energy.
Ratio higher than $100\%$ means our energy is better than Glasner~\mbox{{\em et~al.}}.
Bottom row shows the percentage of cases where each method got strictly lower energy than Glasner~\mbox{{\em et~al.}}.}}
\label{tab:comp-daniel}
\end{table}
\section{Conclusion}\label{sec:concl}
This work suggests a new perspective on the functional, viewing it as a discrete Potts energy.
The resulting energy minimization presents three challenges: (i)~the energy is non submodular, (ii)~the number of clusters is not known in advance, and (iii)~there is no unary term.
We proposed new energy minimization algorithms that can successfully cope with these challenges.
\section{Introduction}
One of the fundamental tasks in unsupervised learning is clustering: grouping data points into coherent clusters.
In clustering of data points, two aspects of pair-wise affinities can be measured: (i)~{\em Attraction} (positive affinities), i.e., how likely are points $i$ and $j$ to be in the same cluster, and
(ii)~{\em Repulsion} (negative affinities), i.e., how likely are points $i$ and $j$ to be in different clusters.
Indeed, new approaches for clustering, recently presented by \cite{Yu2001} and \cite{Bansal2004}, suggest to combine attraction and repulsion information.
Normalized cuts was extended by \cite{Yu2001} to allow for negative affinities.
However, the resulting functional provides sub-optimal clustering results in the sense that it may lead to fragmentation of large homogeneous clusters.
The Correlation Clustering functional ({\bf CC}), proposed by \cite{Bansal2004}, tries to maximize the intra-cluster agreement (attraction)
and the inter-cluster disagreement (repulsion).
Contrary to many clustering objectives, the CC functional has an inherent
``model-selection" property allowing to {\em automatically} recover the underlying number of clusters (\cite{Demaine2003}).
Sec.~\ref{sec:cc-theory} focuses on a theoretical probabilistic interpretation of the CC functional.
The subsequent sections (Sec.~\ref{sec:cc-app-uimos} and~\ref{sec:cc-app-faces}) present two new applications.
Both these applications build upon integrating attraction and repulsion information
between large number of points, and require the robust recovery of the underlying number of clusters $k$.
This chapter focuses on the CC functional, its properties and derived applications.
We defer to chapter~\ref{cp:CC} our novel approach to CC optimization.
Experimental results presented in this chapter were produced using our algorithms, which are described in more detail in chapter~\ref{cp:CC}.
\section*{Correlation Clustering (CC) Functional}
Let $W\in\mathbb{R}^{n\times n}$ be an affinity matrix
combining attraction and repulsion: for $W_{ij}>0$ we say that $i$ and $j$ attract each other with certainty $\abs{W_{ij}}$, and for $W_{ij}<0$ we say that $i$ and $j$ repel each other with certainty $\abs{W_{ij}}$. Thus the sign of $W_{ij}$ tells us if the points attract or repel each other and the magnitude of $W_{ij}$ indicates our certainty.
Any $k$-way partition of $n$ points can be written as $U\in\left\{0,1\right\}^{n\times k}$ s.t. $U_{ic}=1$ iff point $i$ belongs to cluster $c$. $\sum_c U_{ic}=1\;\forall i$ ensure that every $i$ belongs to {\em exactly} one cluster.
The CC functional
maximizes the intra-cluster agreement (\cite{Bansal2004}).
Given a matrix $W$\footnote{Note that $W$ may be sparse.
The ``missing" entries are simply assigned ``zero certainty" and therefore they do not affect the optimization.},
an optimal partition $U$ minimizes:
\begin{eqnarray}
\mathcal{E}_{CC}\left(U\right) &=& - \sum_{ij} W_{ij}\sum_c U_{ic}U_{jc} \label{eq:CorrClust}\\
& s.t. & U_{ic}\in \left\{0,1\right\} ,\; \sum_c U_{ic}=1 \nonumber
\end{eqnarray}
Note that $\sum_c U_{ic}U_{jc}$ equals 1 iff $i$ and $j$ belong to the same cluster.
For brevity, we will denote $\sum_c U_{ic}U_{jc}$ by $\left[UU^T\right]_{ij}$ from here on.
\section{Probabilistic Interpretation}
\label{sec:cc-theory}
This section provides a probabilistic interpretation for the CC functional.
This interpretation allows us to provide a theoretic justification for the ``model selection" property of the CC functional.
Moreover, our analysis exposes the underlying implicit prior that this functional assumes.
We consider the following probabilistic generative model for matrix $W$.
Let $U$ be the true unobserved partition of $n$ points into clusters.
Assume that for some pairs of points $i,j$ we observe their pairwise similarity values $s_{ij}$.
These values are random realizations from either a distribution $f^+$ or $f^-$,
depending on whether points $i,j$ are in the same cluster or not. Namely,
\begin{eqnarray*}
p\left(s_{ij}=s\left|\left[UU^T\right]_{ij}=1\right.\right) &=& f^+\left(s\right) \\
p\left(s_{ij}=s\left|\left[UU^T\right]_{ij}=0\right.\right) &=& f^-\left(s\right)
\end{eqnarray*}
Assuming independency of the pairs, the likelihood of observing similarities $\left\{s_{ij}\right\}$ given a partition $U$ is then
\[
\mathcal{L}\left(\left\{s_{ij}\right\}\left|U\right.\right) = \prod_{ij} f^+\left(s_{ij}\right)^{\left[UU^T\right]_{ij}}\cdot f^-\left(s_{ij}\right)^{\left(1-\left[UU^T\right]_{ij}\right)}
\]
To infer a partition $U$ using this generative model we look at the posterior distribution:
\[
Pr\left(U\left|\left\{s_{ij}\right\}\right.\right) \propto \mathcal{L}\left(\left\{s_{ij}\right\}\left|U\right.\right) \cdot Pr\left(U\right)
\]
where $Pr\left(U\right)$ is a prior. Assuming {\em a uniform prior} over all partitions, i.e., $Pr\left(U\right)=const$, yields:
\[
Pr\left(U\left|\left\{s_{ij}\right\}\right.\right) \propto \prod_{ij} f^+\left(s_{ij}\right)^{\left[UU^T\right]_{ij}}\cdot f^-\left(s_{ij}\right)^{\left(1-\left[UU^T\right]_{ij}\right)}
\]
Then, the negative logarithm of the posterior is given by
\begin{eqnarray*}
-\log Pr\left(U\left|\left\{s_{ij}\right\}\right.\right) &=& \hat{C} + \sum_{ij}\log f^+\left(s_{ij}\right)\left[UU^T\right]_{ij} \\
& & + \sum_{ij}\log f^-\left(s_{ij}\right)\left(1-\left[UU^T\right]_{ij}\right)
\end{eqnarray*}
where $\hat{C}$ is a constant not depending on $U$.
Interpreting the affinities as log odds ratios $W_{ij} = \log\left(\frac{f^+\left(s_{ij}\right)}{f^-\left(s_{ij}\right)}\right)$,
the posterior becomes
\begin{eqnarray}
-\log Pr\left(U\left|\left\{s_{ij}\right\}\right.\right) & = &
C - \sum_{ij} W_{ij}\left[UU^T\right]_{ij}
\label{eq:CorrClustPij}
\end{eqnarray}
That is, Eq.~(\ref{eq:CorrClustPij}) estimates the log-posterior of a partition $U$.
Therefore, a partition $U$ that minimizes Eq.~(\ref{eq:CorrClustPij}) is the {\bf MAP} (maximum a-posteriori) partition.
Since Eq.~(\ref{eq:CorrClust}) and Eq.~(\ref{eq:CorrClustPij}) differ only by a constant they share the same minimizer: the MAP partition.
\subsection{Recovering $k$ (a.k.a. ``model selection")}
We showed that the generative model underlying the CC functional has a {\em single} model for all partitions, regardless of $k$.
Therefore, optimizing the CC functional one need not select between different generative models to decide on the optimal $k$.
Comparing partitions with different $k$ is therefore straight forward and does not require an additional ``model complexity" term (such as BIC, MDL, etc.)
As described in the previous section the CC functional assumes a uniform prior over all partitions.
This uniform prior on $U$ induces a prior on the number of clusters $k$,
i.e., what is the a-priori probability of $U$ having $k$ clusters: $Pr\left(k\right)=Pr\left(U\mbox{ has $k$ clusters}\right)$.
We use Stirling numbers of the second kind (\cite{Rennie1969}) to compute this induced prior on $k$.
Fig~\ref{fig:stirling} shows the non-trivial shape of this induced prior on the number of clusters $k$.
\begin{figure}
\centering
\includegraphics[width=.42\linewidth]{cc/Stirling150_plot.pdf}
\caption{ {\bf Prior on the number of clusters $k$:}
{\em Graph shows $-\log Pr\left(k\right)$, for uniformly distributed $U$.
The induced prior on $k$ takes a non-trivial shape:
it assigns very low probability to the trivial solutions of $k=1$ and $k=n$,
while at the same time gives preference to partitions with non-trivial $k$.
The mode of this prior is when $U$ has roughly $\frac{n}{\log n}$ clusters.
}}
\label{fig:stirling}
\end{figure}
\section[Interactive multi-object segmentation]{Interactive multi-object segmentation {\color{red}(Patent Pending)\protect\footnotemark{}}}\footnotetext{This work was published in the 3$^{rd}$ International Conference on Information Science and Applications (ICISA), \citeyear{Bagon2012icisa}.}
\label{sec:cc-app-uimos}
\begin{figure}
\centering
\begin{tabular}{p{.45\linewidth}p{.45\linewidth}}
\includegraphics[width=\linewidth]{cc/img_006_bstrokes.png}&
\includegraphics[width=\linewidth]{cc/img_006_04_swap-qpbo.png}\\
\TabCenter{\linewidth}{(a) Input image and boundary scribbles (red)}&\TabCenter{\linewidth}{(b) Resulting segmentation}
\end{tabular}
\caption{\label{fig:interactive_result006}
{\bf Interactive multi-object segmentation:}
{\em (a)~The user provides only crude and partial indications to the locations of boundaries between the relevant objects in an image (red). (b)~The output of our algorithm correctly segments the image into multiple segments. Image was taken from \protect\cite{alpert2007}.}
}\vspace*{-5mm}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{cc/fig_neg_aff_ala_stein.pdf}
\caption{\label{fig:neg_aff_ala_stein}
{\bf From boundary scribbles to affinity matrix:}
{\em (a)~A boundary scribble is drawn by the user (red), inducing ``figure/ground" regions on its opposite sides (black and white regions). (b)~For each scribble we use the method of \protect\cite{Levin2008} to generate a soft segmentation of the image into two segments: pixel values in the soft segmentation are in the range $\left[-1,1\right]$. Pixels far away from the scribble are assigned 0 as it is uncertain to what segment they should belong to. Each pixel $i$ is described using a segmentation membership vector $v_i$ with an entry corresponding to its assignment at each soft segmentation (red columns). (c)~A non-zero entry $w_{ij}$ in the sparse affinity matrix is the {\em correlation} between normalized vectors $v_i$ and $v_j$: $w_{ij}=v_i^Tv_j/\norm{v_i}\cdot\norm{v_j}$. We also add strong repulsion across each scribble.}
}
\vspace*{-3mm}
\end{figure}
Negative affinities in image segmentation may come very naturally from boundary information: pixels on the same side of a boundary are likely to be in the same segment (attraction), while pixels on opposite sides of a boundary are likely to be in different segments (repulsion). We use this observation to design a novel approach to interactive multi-object image segmentation. Instead of using $k$ different ``strokes" for the different objects (e.g., \cite{Santner2011}), the user applies a {\em single} ``brush" to indicate parts of the boundaries between the different objects. Using these {\em sparse and incomplete} boundary hints we can correctly complete the boundaries and extract the desired number of segments. Although the user does not provide at any stage the number of objects $k$, our method is able to automatically detect the number of segments using only the {\em incomplete} boundary cues.
Fig.~\ref{fig:interactive_result006} provides an example of our novel interactive multi-object segmentation approach.
\noindent{\bf Computing affinities:} Fig.~\ref{fig:neg_aff_ala_stein} illustrates how we use sporadic user-provided boundary cues to compute a {\em sparse} affinity matrix with both positive and negative entries.
Note that this is a modification of the affinity computation presented by \cite{Stein2008}: (i)~We use the interactive boundary cues to drive the computation, rather than some boundaries computed by unsupervised technique. (ii)~We only compute a small fraction of all entries of the matrix, as opposed to the full matrix of Stein~\mbox{{\em et~al.}} (iii)~Most importantly, we end up with both positive and negative affinities in contrast to Stein~\mbox{{\em et~al.}}\ who use only positive affinities.
The sparse affinity matrix $W$ is very large ($\sim100k\times100k$). Existing methods for optimizing the correlation clustering functional are unable to handle this size of a matrix.
Chapter~\ref{cp:CC} describes in detail our novel approach to CC optimization that enables us to optimize such large scale problems.
We applied our Swap-and-Explore algorithm (Alg.~\ref{alg:ab-swap}, described in~\ref{sec:alg}) to this problem and it provides good looking results with only several minutes of processing per image.
Fig.~\ref{fig:uimos} shows input images and user marked boundary cues used for computing the affinity matrix. Our results are shown at the bottom row.
The new interface allows the user to segment the image into several coherent segments without changing brushes and without explicitly enumerate the number of desired segments to the algorithm.
\begin{figure}
\newlength{\imh}
\setlength{\imh}{2cm}
\centering
\begin{tabular}{c|c|c|c|c}
\includegraphics[height=\imh]{cc/img_007_bstrokes.png}&
\includegraphics[height=\imh]{cc/img_008_bstrokes.png}&
\includegraphics[height=\imh]{cc/img_004_bstrokes.png}&
\includegraphics[height=\imh]{cc/img_005_bstrokes.png}&
\includegraphics[height=\imh]{cc/img_001_bstrokes.png}\\
\includegraphics[height=\imh]{cc/img_007_04_swap-qpbo.png}&
\includegraphics[height=\imh]{cc/img_008_04_swap-qpbo.png}&
\includegraphics[height=\imh]{cc/img_004_04_swap-qpbo.png}&
\includegraphics[height=\imh]{cc/img_005_04_swap-qpbo.png}&
\includegraphics[height=\imh]{cc/img_001_04_swap-qpbo.png}\\\hline
\end{tabular}
\begin{tabular}{c|c|c|c}
\hline
\includegraphics[height=\imh]{cc/img_003_bstrokes.png}&
\includegraphics[height=\imh]{cc/img_009_bstrokes.png}&
\includegraphics[height=\imh]{cc/img_010_bstrokes.png}&
\includegraphics[height=\imh]{cc/img_002_bstrokes.png}\\
\includegraphics[height=\imh]{cc/img_003_04_swap-qpbo.png}&
\includegraphics[height=\imh]{cc/img_009_04_swap-qpbo.png}&
\includegraphics[height=\imh]{cc/img_010_04_swap-qpbo.png}&
\includegraphics[height=\imh]{cc/img_002_04_swap-qpbo.png}
\end{tabular}
\caption{ {\bf Interactive segmentation results. }{\em Input image and user boundary cues (top), our result (bottom). Images were taken from \protect\cite{alpert2007}.} }
\label{fig:uimos}
\end{figure}
\section{Clustering and face identification}
\label{sec:cc-app-faces}
This application shows that detecting the underlying number of clusters $k$ can be an important task on its own.
Given a collection of face images we expect the different clusters to correspond to different persons. Identifying the different people requires not only high purity of the resulting clusters but more importantly to {\em correctly discover the appropriate number of clusters}.
This experiment is an extension of existing work on the problem of ``same/not-same" learning. Following recent metric learning approach (e.g., \cite{Guillaumin2009,Guillaumin2010}) we learn a {\em single}
classifier that assigns a probability to each pair of faces: ``how likely is this pair to be of the same person".
Then, using this classifier, we are able to determine {\em the number of persons} and cluster the faces of {\em unseen people}. That is, given a new set of face images of several {\em unseen} people, our clustering approach is able to automatically cluster and identify how many different people are in the new set of face images of {\em never seen before} people.
For this experiment we use PUT face dataset \cite{Kasinskiput2008}. The dataset consists of 9971 images of 100 people (roughly 100 images per person). Images were taken in partially controlled illumination conditions over a uniform background. The main sources of face appearance variations are changes in head pose, and facial expression.
We use the same method as \cite{Guillaumin2009} to describe each face. SIFT descriptors are computed at fixed points on the face at multiple scales. We use the annotations provided in the dataset to generate these keypoints.
Given a training set of labeled faces $\left\{x_i,y_i\right\}_{i=1}^N$ we use a state-of-the-art method by \cite{Guillaumin2010} to learn a Mahalanobis distance $L$ and threshold $b$ such that:
\[
\hspace*{-3mm}
Pr\left(y_i=y_j\vert x_i,x_j;L,b\right)=\sigma\left(b-\left(x_i-x_j\right)^TL^TL\left(x_i-x_j\right)\right)
\]
where $\sigma(z)=(1-e^{-z})^{-1}$ is the sigmoid function.
For each experiment we chose $k$ people for test (roughly $100\cdot k$ images), and used the images of the other $100-k$ people for training.
The learned distance is then used to compute $p_{ij}$, the probability that faces $i$ and $j$ belong to the same person, for all pairs of face images of the $k$ people in the test set.
The affinities are set to $W_{ij}=\log\frac{p_{ij}}{1-p_{ij}}$.
We apply our clustering algorithm to search for an optimal partition, and report the identified number of people $k^\prime$ and the purity of the resulting clusters.
We experimented with $k=15, 20, \ldots, 35$. For each $k$ we repeated the experiments for several different choices of $k$ different persons.
In these settings all our algorithms, described in Sec.~\ref{sec:alg}, performed roughly the same in terms of recovering $k$ and the purity of the resulting clustering. However, in terms of running time adaptive-label ICM completed the task significantly faster than other methods.
We compare Swap-and-Explore (Alg.~\ref{alg:ab-swap} of Sec.~\ref{sec:alg}) to two different approaches: (i)~{\em Connected components:} Looking at the matrix of probabilities $p_{ij}$, thresholding it induces $k^\prime$ connected components. Each such component should correspond to a different person. At each experiment we tried 10 threshold values and reported the best result. (ii)~{\em Spectral gap:} Treating the probabilities matrix as a {\em positive} affinity matrix we use NCuts \cite{shi2000} to cluster the faces. For this method the number of clusters $k^\prime$ is determined according to the spectral gap: Let $\lambda_i$ be the $i^{th}$ largest eigenvalue of the normalized Laplacian matrix, the number of clusters is then $k^\prime=\arg\max_i\frac{\lambda_i}{\lambda_{i+1}}$.
Fig.~\ref{fig:PUT_faces} shows cluster purity and the number of different persons $k^\prime$ identified as a function of the actual number of people $k$ for the different methods. Our method succeeds to identify roughly the correct number of people (dashed black line) for all sizes of test sets, and maintain relatively high purity values.
\begin{figure}
\centering
\includegraphics[width=.48\linewidth]{cc/PUT_nc_plot.pdf}
\includegraphics[width=.48\linewidth]{cc/PUT_pr_plot.pdf}
\caption{
{\bf Face identification:} {\em Graphs showing {\color{Blue} our result (Swap)}, {\color{Green} spectral} and {\color{Red} connected components}. Left: recovered number of people ($k^\prime$) vs. number of people in the test set. Dashed line shows the true number of people. Right: purity of resulting clusters.}}\protect\vspace*{-4mm}
\label{fig:PUT_faces}
\end{figure}
\section{Conclusion}
\label{sec:cc-concl}
This chapter provides a generative probabilistic interpretation for the Correlation Clustering functional,
justifying its intrinsic ``model selection" capability.
Using a generative probabilistic formulation allows
for a better understanding of the functional,
underlying assumptions it makes, including the prior it imposes on the solution.
Optimizing large scale CC and robustly recovering the underlying number of clusters allows us to propose new applications: interactive multi-label image segmentation and unsupervised face identification.
\part{Discussion}
In this work I motivated the use of challenging discrete energies, beyond semi-metric, for different tasks in computer vision.
I conclude that despite the computational challenges posed by stepping away form the well studied smoothness-encouraging energies, the gain in terms of expressiveness and quality of solution is worth while.
For some applications these energies describe better the underlying process and therefore better suited for modeling.
Moreover, practical algorithms and approaches were proposed in this work to cope with the resulting challenging optimization tasks.
Considering the task of unsupervised clustering as an example, using the arbitrary Potts energy of the Correlation Clustering functional allows not only to model the partition of the data into different clusters, but also to recover the underlying number of clusters.
This recovery of the number of cluster is an issue usually evaded by many clustering algorithms that assume this information is somehow given to them as an input.
This thesis showed how the challenging optimization of the CC functional lies at the core of several diverse applications, such as: sketching the common, unsupervised face identification and interactive segmentation.
Focusing on the CC optimization example, this work also proposed several practical approximation algorithms.
The CC functional was casted as a special case of an arbitrary Potts energy minimization.
This discrete formulation of the functional allows to adapt and utilize known methods for discrete optimization resulting with CC optimization algorithms capable of handling large number of variables (two orders of magnitude more than existing methods).
Another example of challenging arbitrary energy is the task of 3D shape reconstruction.
The problem of 3D reconstruction from two images with common lighting conditions and known object motion with respect to static camera is discretized to form a challenging energy.
This energy emerges from the induced integrality constraints on the recovered surface.
It is defined over a very large label space (a few thousand of labels).
Despite the challenging pair-wise terms and the large label space successful approximation scheme was proposed and adapted for this task.
When it comes to learning the parameters of discrete energies for various tasks in computer vision,
it so happens that the underlying data may be better explained by contrast enhancing terms, as in the Chinese characters and the body parts.
In fact when the learning procedure is not restricted it may prefer contrast-enhancing functions as better depictions for the underlying statistical behavior of the training set.
Therefore, restricting the learned model to only submodular or semi-metric energies might severely compromise its ability to generalize to novel exemplars.
Coping with the optimization of the resulting challenging energies is a difficult task.
Yet, this work proposes several methods that provide good empirical approximations for a variety of practical energies.
Our unified discrete multiscale framework succeeds in providing good approximations, in practice, for a variety of challenging energies.
Our multiscale framework exploits the underlying {\em multiscale landscape} of the given energy and exposes it to allow for an efficient and effective coarse-to-fine optimization process.
Formulating the coarsening procedure in a clear algebraic formulation allows us to propose two paths to expose the multiscale landscape of the energy:
one in which coarser scales involve fewer and coarser {\em variables},
and another in which the coarser levels involve fewer {\em labels}.
We also propose adaptive methods for energy-aware interpolation between the scales.
This approach exploits the underlying {\em multiscale landscape} of the given energy and exposes it to allow for an efficient and effective coarse-to-fine optimization process.
Our framework provides the mathematical formulation that ``bridges the gap" and relates multiscale discrete optimization and algebraic multiscale methods used in PDE solvers (e.g., \cite{Brandt1986}).
This connection allows for methods and practices developed for numerical solvers to be applied in multiscale discrete optimization as well.
\section{Introduction}
The last decade has seen the meteoric rise in the use of random
field models in computer vision~\cite{Szeliski2008}. Random
fields have been used to model many problems including
foreground-background (fg-bg)
segmentation~\cite{Blake04,Boykov2001}, semantic
segmentation~\cite{he2004multiscalecrf,shotton2007textonboost,winn2006layoutcrf},
stereo~\cite{Boykov2001}, optical
flow~\cite{Baker:ICCV2007,RothBIJCV09,Horn:optical-flow}, and 3D
reconstruction~\cite{SVZ:CVPR00,Vogiatzis:CVPR05}.
Many of these problems can be cast as an image labeling problem, where we are
given an image $\boldsymbol{x}$ and need to predict labels $\boldsymbol{y}$.
Random fields provide a way of factorizing the joint distribution
$p(\boldsymbol{x},\boldsymbol{y})$ or the posterior distribution $p(\boldsymbol{y}|\boldsymbol{x})$ into a product of local
interactions.
In the classic Markov random field (MRF) we obtain the posterior distribution
$p(\boldsymbol{y}|\boldsymbol{x})$ by integrating a per-pixel likelihood functions with pairwise
consistency potentials ensuring a smooth solution~\cite{Geman:84,li1995mrf}.
One major advance in the field was to make these smoothness cost dependent on
the local image structure~\cite{Boykov2001},
conditioning parts of the model on the input data. In the last decade, these {\em conditional}
random field (CRF) models~\cite{lafferty2001crf,sutton2007crf,he2004multiscalecrf}
have become popular for their improved ability to capture the relationship between
labels and the image.
A lot of research effort has been devoted at the development of efficient
algorithms for estimating the Maximum a Posteriori (MAP) solution of such
models~\cite{Felzenszwalb2006,Szeliski2008,Kolmogorov2006,komodakis2007fastpd},
and the same is true for algorithms for probabilistic
inference~\cite{wainwright2008graphicalmodels,Koller2009}.
Further, a large number of methods have been proposed to learn the parameters
of random field
models~\cite{AnguelovTCKGHNCVPR2005,SKumarAHEMMCVPR05,sutton2007crf,szummer2008learningcrf,ZhangS05}.
A number of higher order random field formulations have also been proposed
that are able to encode interactions between groups of pixels, and have been
shown to produce much better results~\cite{KohliLTCVPR08,RothBIJCV09}.
However, despite these rapid developments, (most) state-of-the-art random
field CRF models continue to suffer from the following limitations: (1) they
are defined on the basis of a fixed neighborhood structure (except the work of
\cite{KolmogorovB05,RothB07}), and (2) the potentials are assumed to have a
simple parametric form with a pre-specified and fixed dependence on the image
data. While it is relatively easy to think of various ways to overcome these
limitations, the key research challenge is to suggest a model for which
efficient and high-quality training is still tractable.
This paper introduces a new graphical model, the Decision Tree
Field (DTF), which overcomes the above-mentioned limitations of
existing models. We take a simple yet radical view: every interaction
in our model depends on the image, and further, the
dependence is non-parametric. It is easy to see that even
representing such a model is extremely challenging, since there are
numerous ways of defining a mapping between the image
and the parameters of a unary or pairwise interaction in the graphical
model.
Our model uses \emph{decision trees} to map the image content to
interaction values. Every node of every decision tree in our model
is associated with a set of parameters, which are used to define the
potential functions in the graphical model.
When making predictions on a novel test instance, the leaf node of the
decision tree determines the effective weights.
There are a number of important reasons for the choice of decision trees
to specify the dependence between potentials and image content.
Firstly, decision trees are non-parametric and can represent rich functional
relationships if sufficient training data is available.
Secondly, the training of decision trees is scalable, both
in the training set size and in that the approach can be parallelized;
recent advances even allows training on graphics processors~\cite{Sharp08}.
Since for most computer vision applications it is well known that the key to
obtaining high predictive performance is the amount of training data, many
recent works use decision trees, or a related variant of it (random
Forests~\cite{breiman2001randomforests}, extremely randomized
trees~\cite{geurts2006extremelyrandomizedtrees}, semantic texton
forest~\cite{shotton2008stf}).
In our context, decision trees give another big advantage: they allow us to efficiently
and jointly learn all parameters in the model. We achieve this
by using a log-concave pseudo-likelihood objective function, which is known to work well
given enough training data because it is a consistent
estimator~\cite{Koller2009}.
\paragraph{Our Contributions}\mbox{ }\\%
(1) To the best of our knowledge, we propose the first graphical model for
image labeling problems which allows all potential functions to have an
arbitrary dependence on the image data.\\%
(2) We show how the dependence between potential functions and image data can
be expressed via decision trees.\\%
(3) We show how the training of the DTF model, which involves learning of a
large number of parameters, can be performed efficiently.\\%
(4) We empirically demonstrate that DTFs are superior to existing models such
as random forest and common CRFs for applications with complex label
interactions and large neighborhood structures.
\section{Related Work}
\label{sec:related}
There has been relatively little work on learning image-dependent
potential functions, i.e. the ``conditional part'' of a random
field.
Most algorithms for learning the parameters of a random field try to learn
a class-to-class energy table that does not depend on the image
content~\cite{AnguelovTCKGHNCVPR2005,batra2008classspecific,nowozin2009connectivity,szummer2008learningcrf,TaskarCKG05}.
However, there have been few attempts at learning the parameters
of conditional
potentials~\cite{ChoJZKSF10,PrasadZFKT06,gould2009semanticregions}.
Recently, Gould {\em et al.}~\cite{gould2009semanticregions} used a
multiclass logistic regression classifier on a set of manually
selected features, such as the length and orientation of region
boundaries to obtain an image-dependent learned model for pairwise
interactions. Even more recently, Cho {\em et al.}~\cite{ChoJZKSF10}
proposed a model for image restoration whose interactions were dependent
on the semantic meaning of the local image content as predicted by
a classifier. Unlike our work, all the above-mentioned models either target
specific tasks, or assume a particular form for the dependence of the
potentials on the image content. Neither of the above-mentioned approaches is able
to learn a dependency model with thousands or even millions of parameters
which our model can achieve.
Decision trees are popularly used to model unary interactions,
e.g.,~\cite{shotton2011real}; but with two exceptions they have not been
used for pairwise or higher-order interactions. The first exception
is the paper of Glesner and Koller~\cite{glesner1995beliefnetworks},
where decision trees are used to model conditional probability
tables over many discrete variables in a Bayesian network. The
difference to our work is that our decision trees evaluate the given
image content, not the state of a random variable, thus requiring no
change to the inference procedure used.
The second exception is the ``random forest random
field''~\cite{payet2010rfrf}. Despite the similarity in name, the approach is
fundamentally different from ours. Instead of defining an explicit model as
we do in~(\ref{eqn:p}), Payet and Todorovic~\cite{payet2010rfrf} define the
model distribution implicitly as the equilibrium distribution of a learned
Metropolis-Hastings Markov chain. The Metropolis-Hastings ratio is estimated
by classification trees. This is a \emph{clever idea} but comes with a number
of limitations,
i) at test-time there is no choice between different inference methods but one
is bound to using inefficient Markov Chain Monte Carlo (MCMC);
in~\cite{payet2010rfrf} superpixel graphs of few hundred regions are used and
inference takes 30 seconds despite using advanced Swendsen-Wang cuts,
and ii) the model remains \emph{implicit}, such that inspecting the learned
interactions as we will do in Section~\ref{sec:bodyparts}
is not possible.
In a broader view, our model has a richer representation of complex label
structure.
Deep architectures, such as~\cite{lee2009convolutionaldbn} and latent variable
CRFs, as in~\cite{schnitzspan2010latentcrf}, have the same goal, but use
hidden variables representing the presence of larger entities such as
object parts. While these models are successful at representing structure,
they are generally difficult to train because their negative log-likelihood
function is no longer convex. In contrast, by learning powerful
non-parametric conditional interactions we achieve a similar expressive power
but retain convexity of the training problem.
\section{Model}
\label{sec:model} We now describe the details of our model.
Throughout we will refer to $\boldsymbol{x} \in \mathcal{X}$ as a given observed image
from the set of all possible images $\mathcal{X}$. Our goal is to infer a
discrete labeling $\boldsymbol{y} \in \mathcal{Y}$, where the labeling is per-pixel,
i.e., we have $\boldsymbol{y}=(y_i)_{i \in \mathcal{V}}$, $y_i \in \mathcal{L}$, where all
variables have the same label set $\mathcal{L}$.
We describe the relationship between $\boldsymbol{x}$ and $\boldsymbol{y}$ by means of an
\emph{energy function $E$} that decomposes into a sum of
energy functions $E_{t_F}$ over \emph{factors} $F$, where $F$ defines a
subsets of variables. For example, for a pairwise factor it is $|F|=2$.
We have
\vspace{-0.15cm}%
\begin{equation}
E(\boldsymbol{y},\boldsymbol{x},\boldsymbol{w}) = \sum_{F \in \mathcal{F}} E_{t_F}(y_F,x_F,w_{t_F}).\label{eqn:E}
\vspace{-0.1cm}%
\end{equation}
By $y_F$ we denote the collection $(y_i)_{i \in F}$, and likewise we
write $x_F$ to denote the parts of $\boldsymbol{x}$ contained in $F$. While
there may be many different subsets in $\mathcal{F}$, we assume they are
of few distinct \emph{types} and denote the type of the factor $F$
by $t_F$. The function $E_{t_F}$ is the same for all factors of
that type, but the variables and image content it acts upon differs.
Furthermore, the function is \emph{parameterized} by means of a
weight vector $w_{t_F}$ to be discussed below.
A visualization of a small factor graph model is shown in
Figure~\ref{fig:nbhd}. It has three pairwise factor types (red,
blue, and green) and two unary factor types (black and turquoise).
All factors depend on the image data $\boldsymbol{x}$.
Figure~\ref{fig:nbhd-unroll} shows the ``unrolled'' factor graph for
an image of size 4-by-3 pixels, where the basic model structure is repeated
around each pixel $i \in \mathcal{V}$, and pairwise factors which reach outside the
image range are omitted. In total we have $|\mathcal{F}|=31$ factors.
The energy function~(\ref{eqn:E}) defines a conditional probability
distribution $p(\boldsymbol{y}|\boldsymbol{x},\boldsymbol{w})$ as
\vspace{-0.15cm}%
\begin{equation}
p(\boldsymbol{y}|\boldsymbol{x},\boldsymbol{w}) = \frac{1}{Z(\boldsymbol{x},\boldsymbol{w})} \exp(-E(\boldsymbol{y},\boldsymbol{x},\boldsymbol{w})),
\label{eqn:p}
\vspace{-0.1cm}%
\end{equation}
where $Z(\boldsymbol{x},\boldsymbol{w})=\sum_{\boldsymbol{y} \in \mathcal{Y}} \exp(-E(\boldsymbol{y},\boldsymbol{x},\boldsymbol{w}))$ is the normalizing
constant.
So far, our model is in the general form of a \emph{conditional random
field}~\cite{lafferty2001crf}.
We now show how to use decision trees for representing $E_{t_F}$
in~(\ref{eqn:E}).
\begin{floatingfigure}[right]{0.5\linewidth}%
\hspace{-0.2cm}\includegraphics[width=0.5\linewidth]{dtf/pathsum.pdf}%
\caption{Summation of all energy tables along the path of visited decision
nodes (shaded blue).}
\label{fig:pathsum}%
\end{floatingfigure}%
With each function $E_{t}$ we associate one decision tree. To evaluate
$E_{t_F}(y_F,x_F,w_{t_F})$, we start at the root of the tree, and perform a
sequence of tests $s$ on the image content $x_F$, traversing the tree to the
left or right. This process is illustrated in Figure~\ref{fig:pathsum}.
When a leaf node has been reached, we collect the \emph{path} of traversed
nodes from the root node to the leaf node. With each node $q$ of the tree we
associate a table of energy values $w_{t_F}(q,y_F)$. Depending on the number
of variables $y_F$ this energy function acts on, the table can be a vector
(unary), a matrix (pairwise), or general $k$-dimensional array (higher order).
We \emph{sum} all the tables along the path taken and compute the energy as
\[E_{t_F}(y_F,x_F,w_{t_F})=\sum_{q \in \textrm{Path}(x_F)} w_{t_F}(q,y_F),\]
where $\textrm{Path}(x_F)$ denotes the set of nodes taken during evaluating
the tree.
By using one set of weights at each node we can regularize the nodes at the
root of the tree to exert a stronger influence, affecting a large number of
leaves; at test-time we can precompute the summation along each root-to-leaf
path and store the result at each leaf.
To compute the overall energy~(\ref{eqn:E}) we evaluate $E_{t_F}$
for all factors $F \in \mathcal{F}$. Although the type $t_F$ might be the
same, the function $E_{t_F}$ depends on $x_F$ through the evaluation
of the decision tree. This allows image-dependent unary, pairwise,
and higher-order interactions. The set $\mathcal{F}$ is determined by
repeating the same local neighborhood structure for each pixel, as shown in
Figures~\ref{fig:nbhd} and~\ref{fig:nbhd-unroll}.
In summary, our model consists of a set of factor types. Each factor type
contains, (i) the number $k$ of variables it acts on and their relative offsets,
(ii) a single decision tree, and (iii) for each node in the decision tree, a
table of energies of size $\mathcal{L}^k$.
Given a new image $\boldsymbol{x}$, for each possible labeling $\boldsymbol{y}$ we can evaluate
$E(\boldsymbol{y},\boldsymbol{x},\boldsymbol{w})$ by the above procedure.
\begin{figure}[t!]
\centering
\begin{minipage}[b]{5cm}
\begin{center}
\includegraphics[width=0.99\linewidth]{dtf/nbhd-struct-wx.pdf}
\end{center}%
\vspace{-0.2cm}%
\caption{Neighborhood structure around each pixel with five different
factor types.}
\label{fig:nbhd}
\end{minipage}%
\hspace{1cm}%
\begin{minipage}[b]{6cm}
\begin{center}
\includegraphics[width=0.99\linewidth]{dtf/nbhd-unroll.pdf}%
\end{center}%
\vspace{-0.2cm}%
\caption{Unrolled factor graph (image size 4-by-3 pixels), dependencies on
$\boldsymbol{x}$ and $\boldsymbol{w}$ are not shown.}
\label{fig:nbhd-unroll}
\end{minipage}%
\vspace{-0.5cm}%
\end{figure}
\subsection{Relation to other Models}
The proposed DTF generalizes a number of popular existing image labeling
methods. If we ignore pairwise and higher-order interactions
in~(\ref{eqn:E}), then the variables are independent and making predictions
for each pixel is the same as evaluating a random forests, as used in
e.g.~\cite{shotton2008stf,TuB10}.
Interestingly, as we will show in the experiments, even in this setting we
still slightly outperform standard random forests since we learn the weights
in each decision node instead of using empirical histograms; this novel
modification improves predictive performance without any test-time overhead
compared to random forests.
For pairwise interactions we generalize simple CRFs with contrast-sensitive
pairwise potentials such as the popular \emph{GrabCut
system}~\cite{rother2004GC} and
TextonBoost~\cite{shotton2007textonboost}.
Finally, if for the pairwise interactions we use decision trees of depth one,
such that these interactions do not depend on the image content, then our
model becomes a classic Markov random field prior~\cite{li1995mrf}.
\section{Learning Decision Tree Fields}
\label{sec:learning_inference}
Learning the model involves selecting the neighborhood structure, the decision
trees, and the weights stored in the decision nodes.
During learning we are given an iid set $\{(\boldsymbol{x}_m,\boldsymbol{y}^*_m)\}_{m=1,\dots,M}$ of
images $\boldsymbol{x}_m$ and ground truth labelings $\boldsymbol{y}^*_m$. Our goal is to estimate
the parameters $\boldsymbol{w}$ of our model such as to predict $\boldsymbol{y}^*_m$ for a given
$\boldsymbol{x}_m$. For simplifying the derivation of the learning method, we can treat
the given set of images as if it would be one large collection of pixels as is
done in~\cite{sutton2007crf}.
\subsection{Maximum Likelihood Learning}
For learning the parameters of our model, we need to elaborate on how the
parameters $\boldsymbol{w}$ define the energy.
One important observation is that for a fixed set of decision trees the energy
function~(\ref{eqn:E}) can be represented such that it is linear in the
parameters $\boldsymbol{w}$. To see this, consider a single $E_{t_F}(y_F,x_F,w_{t_F})$
function and define a binary indicator function
\vspace{-0.2cm}%
\[B_{t_F}(q,z;y_F,x_F)=\left\{\begin{array}{cl}
1 & \textrm{if $n \in \textrm{Path}(x_F)$ and $z=y_F$},\\
0 & \textrm{otherwise.}\end{array}\right.\]
Then, we can write the energy $E_{t_F}(y_F,x_F,w_{t_F})$ equivalently as a
function linear in $w_{t_F}$,
\vspace{-0.15cm}%
\begin{equation}
\sum_{n \in \textrm{Tree}(t_F)} \sum_{z \in \mathcal{Y}_F}
w_{t_F}(q,z) B_{t_F}(q,z;y_F,x_F).
\label{eqn:linearform}
\vspace{-0.1cm}%
\end{equation}
The use of decision trees allows us to represent non-linear functions on $\boldsymbol{x}$.
Although non-linear in $\boldsymbol{x}$, by the representation~(\ref{eqn:linearform}) we
can parameterize this function linearly in $w_{t_F}$.
Then, from~(\ref{eqn:linearform}) we see that the gradient has a simple form,
$\nabla_{w_{t_F}(q,z)} E_{t_F}(y_F,x_F,w_{t_F}) = B_{t_F}(q,z;y_F,x_F)$.
Because~(\ref{eqn:E}) is linear in $\boldsymbol{w}$, the log-likelihood of~(\ref{eqn:p})
is a concave and differentiable function in $\boldsymbol{w}$~\cite[Corollary
20.2]{Koller2009}.
This means that if computing $Z(\boldsymbol{x},\boldsymbol{w})$ and the marginal distributions
$p(y_F|\boldsymbol{x},\boldsymbol{w})$ for all $F \in \mathcal{F}$ would be tractable, then learning the
parameters by maximum likelihood becomes a convex optimization problem.
We now show how to use efficient approximate likelihood methods to learn all
parameters associated to the decision trees from training data.
For now we assume we are given a fixed set of factor types, including decision
trees, but have to learn the weights/energies associated to the nodes of the
trees. We will discuss how to learn trees later.
\subsection{Pseudolikelihood}
The pseudolikelihood~\cite{besag1977pseudolikelihood} defines a surrogate
likelihood function that is maximized. In contrast to the true likelihood
function computing the pseudolikelihood is tractable and very efficient.
The pseudolikelihood is derived from the per-variable conditional
distributions $p(y_i|y^*_{\mathcal{V} \setminus \{i\}},\boldsymbol{x},\boldsymbol{w})$.
By defining $\ell_i(\boldsymbol{w})=-\log p(y_i|y^*_{\mathcal{V} \setminus \{i\}},\boldsymbol{x},\boldsymbol{w})$ we can
write the regularized negative log-pseudolikelihood $\ell_{npl}(\boldsymbol{w})$ as the
average $\ell_i$ over all pixels,
\vspace{-0.15cm}%
\begin{equation}
\ell_{npl}(\boldsymbol{w}) = \frac{1}{|\mathcal{V}|} \sum_{i \in \mathcal{V}} \ell_i(\boldsymbol{w})
- \sum_{t} \log p_t(w_t)\label{eqn:npl},
\vspace{-0.1cm}%
\end{equation}
where $p_t(w_t)$ is a \emph{prior distribution} over $w_t$ used to regularize
the weights. We will use multivariate Normal distributions
$\mathcal{N}(0,\sigma_t I)$, so that $-\log p_t(w_t)$ is of the form
$\frac{1}{2\sigma_t^2} \|w_t\|^2 + C_t(\sigma_t)$ and the constant
$C_t(\sigma_t)$ can be omitted during optimization because it does not depend
on $\boldsymbol{w}$.
For each factor type $t$ the prior hyperparameter $\sigma_t > 0$ controls the
overall influence of the factor and we need to select a suitable value by
means of a model selection procedure such as cross validation.
Function~(\ref{eqn:npl}) is convex, differentiable, and tractably computable.
For optimizing~(\ref{eqn:npl}) we use the L-BFGS numerical optimization
method~\cite{zhu1997lbfgs}. To use L-BFGS we need to iteratively compute
$\ell_i(\boldsymbol{w})$ and the gradient $\nabla_{w_t} \ell_i(\boldsymbol{w})$.
The computation of $\ell_i(\boldsymbol{w})$ and $\nabla_{w_t} \ell_i(\boldsymbol{w})$ is
straightforward
\begin{eqnarray}
\ell_i\left(\boldsymbol{w}\right) & = & \sum_{F \in M(i)} E_F\left(y^*_F,\boldsymbol{x},w_{t_F}\right) \nonumber\\
& & + \log \sum_{y_i \in \mathcal{Y}_i} \exp \left(
- \!\! \sum_{F \in M(i)} \!\! E_F\left(y_i,y^*_{\mathcal{V} \setminus
\{i\}},\boldsymbol{x},w_{t_F}\right)\right)
\label{eqn:npl-i-obj}\\
\nabla_{w_t} \ell_i(\boldsymbol{w}) & = &
\sum_{F \in M_t(i)} \!\!\! \nabla_{w_t} E_F\left(\boldsymbol{y}^*,\boldsymbol{x},w_t\right) \nonumber\\
& & - \mathbb{E}_{y_i \sim p\left(y_i\left|y^*_{\mathcal{V} \setminus \{i\}},\boldsymbol{x},\boldsymbol{w}\right.\right)}\left[
\sum_{F \in M_t(i)} \!\! \nabla_{w_t}
E_F\left(y_i,y^*_{\mathcal{V} \setminus \{i\}},\boldsymbol{x},w_t\right)
\right]
\label{eqn:npl-i-grad}
\end{eqnarray}
where we use $M(i)$ to denote the subset of $\mathcal{F}$ that involves variable
$y_i$, and $M_t(i)$ likewise but restricted to factors of matching type, i.e.,
$M_t(i)=\{F \in M(i): t_F=t\}$.
By summing~(\ref{eqn:npl-i-obj}) and~(\ref{eqn:npl-i-grad}) over all pixels in
all images, we obtain the objective and its gradient, respectively.
When initializing the weights to zero we have approximately $\|\nabla_{\boldsymbol{w}}
\ell_{npl}(\boldsymbol{w})\| \approx 1$. During optimization we stop when
$\|\nabla_{\boldsymbol{w}} \ell_{npl}(\boldsymbol{w})\| \leq 10^{-4}$, which is the case after around
100-250 L-BFGS iterations, even for models with over a million parameters.
\subsection{Learning the Tree Structure}
Ideally, we would like to learn the neighborhood structure and decision trees
jointly with their weights using a single objective function.
However, whereas the weights are continuous, the set of decision trees is a
large combinatorial set. We therefore propose to use a simple two-step
heuristic to determine the decision tree structure: we
learn the classification tree using the training samples and the
\emph{information gain} splitting criterion. This greedy tree construction is
popular and known to work well on image labeling
problems~\cite{shotton2008stf}.
The key parameters are the maximum depth of the tree, the minimum number of
samples required to keep growing the tree, and the type and number of split
features used. As these settings differ from application to application, we
describe them in the experimental section.
Unlike in a normal classification tree, we store weights at every decision
node and initialize them to zero, instead of storing histograms over classes
at the leaf nodes only.
The above procedure is easily understood for unary interactions, but now show
that it can be extended in a straightforward manner to learn decision trees
for pairwise factors as well.
To this end, if we have a pairwise factor we consider the product set
$\mathcal{L} \times \mathcal{L}$ of labels and treat each label pair
$(l_1,l_2)\in\mathcal{L}\times\mathcal{L}$ as a single class. Each training pair of
labels becomes a single class in the product set. Given a set of such
training instances we learn a classification tree over $|\mathcal{L}|^2$ classes
using the information gain criterion. Instead of storing class histograms we
now store weight tables with one entry per element in $\mathcal{L}\times\mathcal{L}$.
The procedure extends to higher-order factors in a straightforward way.
Once the trees are obtained, we set all their weights to zero and
optimize~(\ref{eqn:npl}). During optimization the interaction between
different decision trees is taken into account.
This is important because the tree structures are determined
independently and if we were to optimize their weight independently as well,
then we would suffer from overcounting labels during training.
The same overcounting problem would occur if we would want to use the class
histograms at the leaf nodes directly, for example by taking the negative
log-probability as an energy.
\subsection{Complexity of Training}
The complexity to compute the overall objective~(\ref{eqn:npl}) and its
gradient is $O(|\mathcal{V}|\cdot|\mathcal{L}|\cdot N)$, where $\mathcal{V}$ is the set of
pixels in the training set, $\mathcal{L}$ is the label set, and $N$ is the number of
factors in the neighborhood structure.
Note that this is linear in all quantities, and independent of the order of
the factors. This is possible only because of the pseudolikelihood
approximation. Moreover, it is even more efficient than performing a single
sweep of message passing in loopy belief propagation, which has complexity
$O(|\mathcal{V}|\cdot|\mathcal{L}|^k\cdot N)$ for factors of order $k \geq 2$.
\subsection{Making Training Efficient}
Training a graphical model on millions of pixels is computationally
challenging. We have two principled methods to make training efficient.
First, observe that our training procedure parallelizes in every step: we
train the classification trees in parallel~\cite{Sharp08}.
Likewise, evaluating~(\ref{eqn:npl}) and its gradient is a large summation of
independent terms, which we again compute in parallel with no communication
overhead.
The second observation is that every step in our training procedure can be
carried out on a subsampled training set. For classification trees we can
process a subset of pixels, as in~\cite{shotton2008stf}.
Less obvious, we can do the same thing when optimizing our
objective~(\ref{eqn:npl}). The first term in equation~(\ref{eqn:npl}) takes
the form of an empirical expectation
$\mathbb{E}_{i\sim \mathcal{U}(\mathcal{V})}[\ell_i(\boldsymbol{w})]$ that can be
approximated both deterministically or by means of stochastic approximation.
We use a deterministic approximation by selecting a fixed subset $\mathcal{V}'
\subset \mathcal{V}$ and evaluating
$\ell'_{npl}(\boldsymbol{w}) = \frac{1}{|\mathcal{V}'|} \sum_{i\in\mathcal{V}'} \ell_i(\boldsymbol{w}) - \sum_t \log p_t(w_t)$.
We select $\mathcal{V}'$ to be large enough so this computation remains efficient;
typically $|\mathcal{V}'|$ has a few million elements.\footnote{When sampling
$\mathcal{V}'$ uniformly at random with replacement from $\mathcal{V}$, the \emph{law of
large numbers} guarantees the asymptotic correctness of this approximation.}
\subsection{Inference}
We use different inference methods during test-time. For making maximum
posterior marginal predictions (MPM) we use an efficient Gibbs sampler.
Because the Gibbs sampling updates use the same quantities as used for
computing~(\ref{eqn:npl-i-obj}) we do not have to unroll the graph.
For obtaining approximate MAP predictions, we use the Gibbs sampler with
simulated annealing (SA), again exploiting the model structure.
Both the Gibbs sampler and the SA minimization is explained in the
supplementary materials.
To have a baseline comparison, we also minimize~(\ref{eqn:E}) using
tree-reweighted message passing (TRW) by unrolling the factor graph and using
the implementation of~\cite{Kolmogorov2006}.
\section{Experiments}
\label{sec:exp}
We considered a broad range of applications and report experiments for three
data sets. One more experiment is reported in the supplementary materials.
The aim is to show that the DTF enables improved performance in challenging
tasks, where a large number of interactions and parameters need to be
considered and these cannot be manually tuned. Moreover, we show that
conditional pairwise interactions better represent the data and lead to
improved performance.
As the three datasets are quite diverse, they also show the broad
applicability of our system.
\subsection{Conditional Interactions: Snake Dataset}
\begin{floatingfigure}[right]{0.3\linewidth}%
\hspace{-0.2cm}%
\centering
\fbox{
\includegraphics[width=1.25cm]{dtf/snakes_train.png}%
}%
\fbox{
\includegraphics[width=1.25cm]{dtf/snakes_train_GT.png}%
}%
\vspace{0.1cm}%
\caption{Input (left), labeling (right).}%
\label{fig:snakes-task}%
\end{floatingfigure}%
In this experiment we construct a task that has only very weak local evidence
for any particular label and structural information needs to be propagated at
test-time in order to make correct predictions. Moreover, this structure
is not given but needs to be learned from training data.
Consider Figure~\ref{fig:snakes-task} to the right, illustrating the task.
A ``snake'' shown on the input image is a sequence of adjacent pixels, and the
color in the input image encodes the direction of the next pixel: red means
``go north'', yellow means ``go east'', blue means ``go west'', and green
means ``go south''. Once a background pixel is reached, the snake ends. Each
snake is ten pixels long, and each pixel is assigned its own label, starting
from the head (black) to the tail (white). Knowing about these rules, the
labeling (Figure~\ref{fig:snakes-task}, right) can be perfectly
reconstructed. Here, however, these rules need to be learned from training
instances.
Of course, in a real system the unary interactions typically provide strong
cues~\cite{shotton2007textonboost,batra2008classspecific}, but we believe that the
task distills the limitations of noisy unary interactions: in this task, for
making perfect predictions, the unary would need to learn about all possible
snakes of length ten, of which there are clearly too many.\footnote{In
general, the number of snakes is equal to the number of fixed-length
self-avoiding walks on a lattice, a number which is conjectured to be
exponential in the length.}
We use a standard 4-neighborhood for both the MRF and the DTF models.
The unary decision trees are allowed to look at every pixel in the input
image, and therefore could remember the entire training image. For
experimental details, please see the supplementary materials.
We use a training set of 200 images, and a test set of 100 images.
\begin{figure
\vspace{-0.1cm}%
\begin{center}
\hfill
\begin{minipage}[t]{1.0\linewidth}
\includegraphics[width=0.995\linewidth]{dtf/snakes_all.png}%
\caption{Predictions on a novel test instance.}%
\label{fig:snakes-test}%
\end{minipage}%
\hfill%
\end{center}%
\end{figure}
\begin{table
\vspace{-0.2cm}%
\begin{center}
\begin{small}
\begin{tabular}{lcccc}
& RF & Unary & MRF & DTF\\
\hline
\hline
Accuracy & 90.3 & 90.9 & 91.9 & \textbf{99.4}\\
Accuracy (tail) & 100 & 100 & 100 & 100\\
Accuracy (mid) & 28 & 28 & 38 & 95\\
\hline
\end{tabular}%
\end{small}%
\end{center}%
\vspace{-0.15cm}%
\caption{Test set accuracies for the snake data set.}
\label{tab:snake-results}
\end{table}
The results obtained are shown in Table~\ref{tab:snake-results} and
Figure~\ref{fig:snakes-test}. Here random forests (RF), trained unary
potentials (Unary), and the learned Markov random field (MRF) perform
equally well, at around 91\%. Upon examining this performance
further, we discovered that while the head and tail labels are labeled with
perfect accuracy, towards the middle segments of the snakes the labeling
error is highest, see Table~\ref{tab:snake-results}. This is plausible, as
for these labels the local evidence is weakest.
When using conditional pairwise interactions the performance improves to an
almost perfect 99.4\%. This again makes sense because the pairwise
conditional interactions are allowed to peek at the color-codes at their
neighbors for determining the directionality of the snake.
The predictions are illustrated for a single test instance in
Figure~\ref{fig:snakes-test}. We see that only the DTF makes a perfect
prediction. To show the uncertainty of the unary model, we visualize two
samples from the model.
\subsection{Learning Calligraphy: Chinese Characters}
\begin{figure
\vspace{-0.3cm}%
\begin{center}
\hfill
\begin{minipage}[t]{1.0\linewidth}
\includegraphics[width=0.995\linewidth]{dtf/chinese_all.png}%
\caption{Test set predictions for the large occlusion case.}
\label{fig:chinese-inpaint-test}%
\end{minipage}%
\hfill%
\end{center}%
\end{figure}
\begin{floatingfigure}[right]{0.5\linewidth}%
\hspace{-0.2cm}\includegraphics[width=0.5\linewidth]{dtf/chinese-0227smallA_td1_1e-2.pdf}%
\caption{Pairwise associativity strength. Please see text.}
\label{fig:chinese-weights}%
\end{floatingfigure}%
In the previous experiment we have used a standard 4-connected neighborhood
structure.
In this experiment we show that by using larger conditional neighborhoods we
are able to represent shape.
We use the KAIST Hanja2 database of handwritten Chinese characters.
We occlude each character by grey box centered on the image, but with random
width and height. For more details, please see the supplementary materials.
This is shown in the leftmost
column of Figure~\ref{fig:chinese-inpaint-test}. We consider two datasets,
one where we have a ``small occlusion'' and one with a ``large occlusion''
box.
Note that most characters in the test set have never been observed in the
training set, but a model that has learned about shape structure of Chinese
characters can still find plausible completions of the input image.
To this end we use one unary factor with a decision tree of depth 15.
Additionally, we use a dense pairwise neighborhood structure of 8-connected
neighbors at one and two pixels distance, plus a sparse set of 27 neighbors
at $\{(-9,0),(-9,3),(-9,6),(-9,9),(-6,0),\dots,(9,9)\}$. Therefore, each
variable has $2\cdot(24+4+4)=64$ neighboring variables in the model. For the
pairwise decision trees we use trees of depth one (MRF), or six (DTF).
The results for the large occlusion task are shown in
Figure~\ref{fig:chinese-inpaint-test}.
Qualitatively, they show the difference between a rich connectivity structure
and conditional interactions. Observe, for example, that the MRF essentially
performs only a smoothing of the results while respecting local stroke-width
constraints, as apparent from the MRF MAP prediction in the first row of
Figure~\ref{fig:chinese-inpaint-test}.
In contrast, the DTF predictions hallucinate meaningful structure that may be
quite different from the ground truth but bears similarity to Chinese
characters. Note that we achieve this rich structure without the use of any
latent variables.
Because this task is an inpainting task, the quantitative assessment is more
difficult since the task is truly ambiguous. We therefore report accuracies
only for the small-occlusion case, where a reasonable reconstruction of the
ground truth seems more feasible.
We measure the per-pixel accuracy in the occluded area on the test set. For
the random forest baseline we obtain $67.74\%$. The MRF with dense
neighborhood improves this to $75.18\%$ and the DTF obtains $76.01\%$.
As an example of what structure the model is able to learn, consider the
visualization of the MRF pairwise interactions shown in
Figure~\ref{fig:chinese-weights}.
The figure shows for each pairwise interaction the sum of learned diagonal
energies minus the sum of cross-diagonal entries. If this value is negative
(shown in blue) the interaction is encouraging the pixels to take the same
value. Red marks interactions that encourage pixels to take different values.
The plot shows that there is a strong local smoothing term, but interestingly
it is not symmetric. This can be explained by the fact that horizontal
strokes in Chinese characters are typically slanted slightly
upwards~\cite{chen2011chinesecalligraphy}.
Note that we discovered these regularities automatically from the training
data.
\subsection{Accurate body-part detection}\label{sec:bodyparts}
We consider the task of body part classification from depth images, as
recently proposed in~\cite{shotton2011real}. Given a 2D depth image, and a
foreground mask, the task is to label each pixel as belonging to one of 31
different body parts, as shown in Figure~\ref{fig:KinectEx}.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.995\linewidth]{dtf/KinectEx/figure_all.png}%
\end{center}
\caption{Test recognition results. MRF (top) and DTF (bottom).}
\label{fig:KinectEx}
\end{figure}
Despite the variations in pose and body sizes~\cite{shotton2011real} obtains
high-quality recognition results by evaluating a random forest for each pixel,
testing local and global depth disparities.
In this task, the label set has a large amount of structure, but it is not
clear that a sufficiently complex unary classifier, when given the image,
cannot implicitly represent this structure reasonably well.
Here we show that by adding pairwise interactions we in fact improve the
recognition accuracy. Moreover, once we make the interactions conditional,
accuracy improves even further.
The experimental setup is as follows.
We use a subset of the data used in~\cite{shotton2011real}: 30 depth images
for training, and 150 images for testing.
We train 4 unary decision trees for all models. For the pairwise models, we
use the following neighborhood sizes, (i) ``+1'' for adding a 4-neighborhood
one pixel away, (ii) ``+5'' for an 8-neighborhood five pixels away, and (iii)
``+20'' when adding an 8-neighborhood twenty pixels away. In the ``+1,5,20''
configuration, each variable has 4+8+8=20 neighbors.
For each of the pairwise interactions we train two trees of depth six. A
more detailed description of the exact experimental setup can be found in the
supplementary materials.
We measure the results using the same mean per-class accuracy score as used
in~\cite{shotton2011real}.
The results for 30 training images are shown in
Table~\ref{tab:kinect-results} and one instance is shown in
Figure~\ref{fig:KinectEx}.
Even without adding pairwise interactions, our learned unary weights already
outperform the random forest classifier~\cite{shotton2011real}.
When adding more interactions (+1, +1,20, +1,5,20), the performance increases
because dense pairwise interactions can represent implicit size preferences
for the body parts.
Likewise, when adding conditionality (MRF to DTF), the performance improves.
The best performing model is our DTF with large structure (+1,5,20) and almost
1.5 million free parameters. It is trained in only 22 minutes and achieves
27.35\% mean per-class accuracy.
For the same setup of 30 images, the original work~\cite{shotton2011real}
reports a mean per class accuracy of 14.8\%, while achieving an impressive
56.6\% when scaling to 900k training images, trained for one day on a 1000
core cluster.
An example of a learned pairwise interaction is shown in
Figure~\ref{fig:kinect-horz}, demonstrating that the improved performance of
the DTF can be directly attributed to the more powerful interactions that are
allowed to take the image into account. We report more results in the
supplementary materials.
\begin{table}[!hbt]
\begin{center}
\begin{small}
\begin{tabular}{llrrrrr}
Model & Measure & \cite{shotton2011real} & unary & +1 & +1,20 & +1,5,20\\
\hline
\hline
MRF & avg-acc & 14.8 & 21.36 & 21.96 & 23.64 & 24.05\\
& runtime & 1m & 3m18 & 3m38 & 10m & 10m\\
& weights & - & 176k & 178k & 183k & 187k\\
\hline
DTF & avg-acc & - & - & 23.71 & 25.72 & \textbf{27.35}\\
& runtime & - & - & 5m16 & 17m & 22m\\
& weights & - & - & 438k & 951k & 1.47M\\
\hline
\end{tabular}%
\end{small}%
\end{center}%
\vspace{-0.15cm}%
\caption{Body-part recognition results: mean per-class accuracy, training
time on a single 8-core machine, and number of model parameters.}
\label{tab:kinect-results}
\end{table}
\begin{figure}
\begin{center}
\hfill
\begin{minipage}[b]{1.9cm}%
\includegraphics[height=3.4cm]{dtf/Kinect/horizontal_all_patches.png}%
\end{minipage}%
\begin{minipage}[b]{1.9cm}%
\includegraphics[height=1.7cm]{dtf/Kinect/horizontal_patch.png}%
\vspace{0.05cm}\\
\includegraphics[height=1.7cm]{dtf/Kinect/horizontal_weights.png}
\end{minipage}%
\begin{minipage}[b]{12.5cm}%
\includegraphics[height=3.4cm]{dtf/Kinect/horizontal_pose1.png}%
\hspace{0.15cm}%
\includegraphics[height=3.4cm]{dtf/Kinect/horizontal_pose3.png}%
\hspace{0.15cm}%
\includegraphics[height=3.4cm]{dtf/Kinect/horizontal_pose2.png}%
\hspace{0.15cm}%
\includegraphics[height=3.4cm]{dtf/Kinect/horizontal_pose4.png}%
\end{minipage}
\hfill%
\end{center
\caption{
{\bf Learned horizontal interactions:}
The left figure shows the average depth-normalized silhouette reaching one of
the 32 leaf nodes in the learned decision tree.
We select one leaf (marked red, enlarged) and show the corresponding effective
$32\times32$ weight matrix obtained by summing the learned weights along the
path from the root to leaf node.
The conditional interaction can be understood by visualizing the most
attractive ({\color{Blue}blue}) and most repulsive ({\color{Red}red}) elements
in the matrix. We superimpose arrows for the two most attractive and
repulsive interactions on test images (right).
The first and second pose exemplify how left and right upper parts of the legs
can appear 20 pix to the right of each other in a way that matches the pattern
of the leaf.
While a configuration like shown in the third and fourth pose is plausible, it
does not fit the leaf pattern and thus the interaction is not active.
}%
\label{fig:kinect-horz}%
\end{figure}
\section{Conclusion}
\label{sec:con}
We have introduced Decision Tree Fields as flexible and accurate models for
image labeling tasks. This accuracy is achieved by being able to represent
complex image-dependent structure between labels. Most importantly, this
expressiveness is achieved without the use of latent variables and therefore
we can learn the parameters of our model efficiently by minimizing a convex
function.
\part{Introduction}
In this thesis I explore challenging discrete energy minimization problems that arise mainly in the context of computer vision.
From binary energies for figure-ground segmentations through multi-label semantic segmentation, stereo, denoising, to inpainting and image editing (e.g., \cite{Szeliski2008,Pritch2009,Bagon2012icisa}).
These energies usually involve thousands of variables and dozens of discrete states.
Moreover, most of the energies in this domain are pair-wise energies, that is, they only involve interactions between pairs of neighboring variables.
The optimization of these discrete energies is known to be NP-hard in most cases (\cite{Boykov2001}).
Still, despite this theoretical hardness, instances of these energies that have special properties may give rise to polynomial time global optimal algorithms.
Other instances with slightly different properties allow, in practice, for good and efficient approximation schemes.
The next chapter~(Chap.~\ref{cp:related-work}) reviews previous work related to discrete pair-wise energy functions and their optimization.
It outlines the properties and conditions under which global optimization is feasible, and the conditions required for successful practical approximations.
Chapter~\ref{cp:related-work} also surveys several key approximation algorithms.
It provides a brief outline of the properties of the energy that must be met in order for each algorithm to succeed.
The conclusion of this survey is that discrete pair-wise energies may be broadly classify into two categories: \*
\noindent{\bf smoothness-encouraging} energies: energies that favor configurations with neighboring variables taking the same discrete state.\*
\noindent{\bf contrast-enhancing} energies: energies that encourage solutions where neighboring variables take different states.\*
So far the energies mainly used in computer vision tasks are of the first category: smoothness-encouraging (see e.g., \cite{Szeliski2008}).
These smoothness-encouraging energies allow for efficient approximation schemes.
On the other hand, contrast-enhancing energies are far more challenging when it comes to optimization, and are indeed less popular in practice.
\pagebreak
In this thesis I would like to step outside of this ``comfort-zone" of the smoothness-encouraging energies and explore more challenging discrete energies.
This work revolves around two axes:
\begin{enumerate}
\item {\bf Applications:} The first motivates the use of such ``hard-to-optimize" functionals by introducing new applications.
As the energies become less constrained and structured one gains more expressive power for the objective function achieving more accurate models.
Results show how contrast-enhancing, hard-to-optimize, functionals are more adequate for certain computer vision tasks.
\item {\bf Approximations:} To overcome the resulting challenging optimization tasks the second axis of this thesis proposes methods and algorithms to cope with the NP-hardness of this optimization.
Experiments show that these new methods yield good results for representative challenging problems.
\end{enumerate}
\chapter{Discrete Pair-wise Energies -- a Review}
\label{cp:related-work}
Discrete energy minimization is a ubiquitous task in computer vision.
From binary energies for figure-ground segmentations through multi-label semantic segmentation, stereo, denoising, to inpainting and image editing (e.g., \cite{Szeliski2008,Pritch2009,Bagon2012icisa}).
In my thesis I focus on various types of minimization problems of pair-wise energies as they arise in different computer vision applications.
These discrete optimization problems are, in general, NP-hard.
Yet, there are cases in which the minimization of a pair-wise energy can be solved {\em exactly} in polynomial time.
In this introductory chapter I survey different properties of discrete pair-wise energies.
I show how these properties of the energies relate to the inherent difficulty of the optimization task.
Some properties entail exact optimization algorithms, while other properties admit efficient approximations.
The most important property is ``smoothness-encouraging": an energy that prefers the labels of neighboring variables to be the same.
For these ``smoothness-encouraging" energies there exist efficient approximate minimization algorithms.
In contrast, energies that encourage neighboring variables to have {\em different} labels
are much more challenging to minimize.
For these ``contrast-enhancing" energies existing algorithms provide poor approximations.
This thesis focuses on the optimization of these challenging ``contrast-enhancing" energies.
\section{Pair-wise Energy Function}
Before diving into the minimization task, this section presents the discrete pair-wise energy function and the notations that are used in this thesis.
It also provides some insights and motivation for the use of such energies.
A discrete pair-wise energy is a functional of the form
\begin{equation}
F\left(\bx\right) = \sum_{ij\in\mbox{$\mathcal{E}$}} \varphi_{ij}\left(x_i, x_j\right) + \sum_i \varphi_i\left(x_i\right)
\label{eq:pair-wise-gen}
\end{equation}
It is defined over $n$ variables ($x_i$, $i=1,\ldots,n$), each taking one of $l$ discrete labels ($x_i\in\left\{1,\ldots,l\right\}$), where $\mbox{$\mathcal{E}$}$ represents a set of neighboring variables.
The term $\varphi_i\left(x_i\right)$ is a unary term reflecting the compatibility of label $x_i$ to variable $i$ (also known as a ``data term").
The pair-wise term, $\varphi_{ij}\left(x_i,x_j\right)$ reflects the interaction between labels $x_i$ and $x_j$ assigned to variables $i$ and $j$ respectively.
If we were to discard the pair-wise term, minimizing energy~(\ref{eq:pair-wise-gen}) is simply choosing the label that best fit each variable separately.
However, the presence of the pair-wise term introduces dependencies between the different variables and turns the optimization into a much more complicated process.
Despite the local nature of the pair-wise term -- binding the values of only neighboring variables introduces {\em global} effects on the overall optimization.
Propagating the information from the local pair-wise terms to form a global solution is a major challenge for the optimization of Eq.~(\ref{eq:pair-wise-gen}).
The energy function of Eq.~(\ref{eq:pair-wise-gen}) has an underlying structure defined by the choice of interacting neighbors.
It is common to associate with $F\left(\bx\right)$ a graph $\mbox{$\mathcal{G}$}=\left(\mbox{$\mathcal{V}$},\mbox{$\mathcal{E}$}\right)$, where the set of nodes $\mbox{$\mathcal{V}$}=\left\{1,\ldots,n\right\}$ represents the variables, and the edges $\mbox{$\mathcal{E}$}$ represents the interacting neighboring pairs.
The following example shows how such discrete functional may arise in a well studied computer vision application.
This example also illustrates the relation between discrete optimization and inference in graphical models.
\begin{example}
\hrulefill
\noindent{\bf Example: Stereo reconstruction via MRF representation}
Given a rectified stereo pair of images, the goal is to find the disparity of each pixel in the reference image.
The true disparity of each pixel is a random variable denoted by $x_i$ for the pixel at location $i$.
Each variable can take one of $L$ discrete states, which represent the possible disparities at that point.
For each possible disparity value, there is a cost associated with matching the pixel in the reference image to the corresponding pixel in the other image at that disparity value.
Typically, this cost is based on the intensity differences between the two pixels, $y_i$, which is an observed quantity.
We denote this cost by $\Phi(x_i,y_i)$.
It relates how compatible a disparity value $x_i$ is with the observed intensity difference $y_i$.
A second function $\Psi(x_i,x_j)$ expresses the disparity compatibility between neighboring pixels.
This function usually expresses the {\em prior} assumption that the disparity field should be smooth.
Examples of such prior that are commonly used are the Potts model:
\begin{equation}
\Psi(x_i,x_j) = \left\{
\begin{array}{cl}
1 & \mbox{if $x_i = x_j$} \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{equation}
the $\ell_1$ similarity:
\begin{equation}
\Psi(x_i,x_j) \propto e^{-\left|x_i-x_j\right|}
\end{equation}
or its robust (truncated) version:
\begin{equation}
\Psi(x_i,x_j) \propto e^{-\min\left\{\left|x_i-x_j\right|,\tau\right\}}
\end{equation}
With the two functions $\Phi$ and $\Psi$ the joint probability for an assignment of disparities to pixels can be written as:
\begin{equation}
P(\mathbf{x},\mathbf{y}) \propto \prod_{ij\in\mbox{$\mathcal{E}$}} \Psi(x_i,x_j) \prod_{i} \Phi(x_i,y_i)
\label{eq:intro-exmp-mrf-prob}
\end{equation}
where $\mathbf{x}$ is an assignment of a disparity value for each pixel $i$.
$x_i$ is the hidden variable (disparity) at location $i$ and $y_i$ is the observed variable (intensity difference) at location $i$.
${ij\in\mbox{$\mathcal{E}$}}$ represent a pair of neighboring nodes and is usually taken as a regular 4-connected grid over the image domain.
The resulting graphical model is known as a pairwise Markov Random Field.
Although the compatibility functions only consider adjacent variables, each variable is still able to influence every other variable in the field via these pairwise connections.
We look for a disparity assignment such that the labeling $\mathbf{x}^\star$ maximizes the joint probability, i.e.,
\begin{equation}
\mathbf{x}^\star = \underset{\mathbf{x}}{\operatorname{argmax}} ~~~ P(\mathbf{x},\mathbf{y})
\end{equation}
Assuming uniform prior over all configurations $\mathbf{x}$, the Maximum A Posteriori (MAP) estimator is equivalent to
\begin{equation}
\mathbf{x}^\star = \underset{\mathbf{x}}{\operatorname{argmax}} ~~~ P(\mathbf{x}\vert\mathbf{y})
\end{equation}
Maximizing the posterior probability is equivalent to minimizing an energy functional of the form (\ref{eq:pair-wise-gen}).
Taking $-\log$ of (\ref{eq:intro-exmp-mrf-prob}) yields the following function
\begin{equation}
F\left(\bx\right) = \sum_{ij\in\mbox{$\mathcal{E}$}} -\log\Psi(x_i,x_j) + \sum_{i} -\log\Phi(x_i,y_i)
\end{equation}
This equation can be expressed as
\begin{equation}
\sum_{ij\in\mbox{$\mathcal{E}$}} \varphi_{ij}\left(x_i, x_j\right) + \sum_i \varphi_i\left(x_i\right)
\end{equation}
where $\varphi_{ij}\left(x_i, x_j\right) \triangleq -\log\Psi(x_i,x_j)$ and $\varphi_i\left(x_i\right) \triangleq -\log\Phi(x_i,y_i)$.
Thus MAP (maximum a-posteriori) inference in graphical models such as MRF and conditional random fields (CRF) boils down to the optimization of a discrete pair-wise energy functional of the form Eq.~(\ref{eq:pair-wise-gen}) which is the focus of this thesis.
\hrulefill
\end{example}
Energy functions of the form Eq.~(\ref{eq:pair-wise-gen}) arise in many graphical models (MRFs and CRFs) (see e.g., \cite{Blake2011} and references therein).
However, it is not restricted to that domain and are also encountered in a variety of other domains such as structural learning (e.g., \cite{Nowozin2011structured}), and as the inference part of discriminative models (e.g., structural SVM \cite{Taskar2003,Tsochantaridis2006}).
This thesis explores some challenging instances of these energies and explores new methods for improved minimization approaches for these hard-to-optimize energies.
\section{Minimization of Discrete Pair-wise Energies}
The energy function of Eq.~(\ref{eq:pair-wise-gen}) presented in the previous section is used in many applications to evaluate how compatible is a certain discrete solution $\mathbf{x}$ to a given problem.
It is now desired to find the {\em best} solution for the given problem by finding a solution $\mathbf{x}^\star$ with the lowest energy.
That is, solving the optimization problem
\begin{equation}
\mathbf{x}^\star = \arg\min_{\mathbf{x}} F\left(\bx\right)
\label{eq:intro-optimization-prob}
\end{equation}
over all {\em discrete} solutions $\mathbf{x}$.
In general, the optimization problem~(\ref{eq:intro-optimization-prob}) is NP-hard.
Known hard combinatorial problems such as max-cut, multi-way cut and many others may be formulated in the form of~(\ref{eq:intro-optimization-prob}) (see e.g., \cite{Boykov2001}).
Yet, there are special instances of problem~(\ref{eq:intro-optimization-prob}) which can be optimized exactly in polynomial time.
The main two factors that affect the difficulty of the optimization problem~(\ref{eq:intro-optimization-prob}) are:
\begin{enumerate}
\item The underlying graph structure, $\mbox{$\mathcal{E}$}$.
\item Mathematical properties of the pair-wise interactions, $\varphi_{ij}$.
\end{enumerate}
When the underlying graph $\mbox{$\mathcal{E}$}$ has no cycles (that is, $\mbox{$\mathcal{E}$}$ has a tree structure) optimization of~(\ref{eq:intro-optimization-prob}) is fairly straight-forward:
By propagating information back and forth from the leafs to the root convergence to {\em global} minimum is attained.
This information propagation is often referred to as belief-propagation (BP) (\cite{Pearl1988,Koller2009}).
A cycle-free graph is crucial for this rapid polynomial time convergence of this message-passing scheme:
Every path from root to leaf is unique and therefore consensus along path is attained in a single forward backward pass.
However, when $\mbox{$\mathcal{E}$}$ has cycles, paths between the different variables are no longer unique and may give rise to contradicting messages.
These contradictions introduce an inherent difficulty to the optimization process making it NP-hard in general.
Despite the inherent difficulty of optimizing~(\ref{eq:intro-optimization-prob}) over a cyclic graph $\mbox{$\mathcal{E}$}$,
there are instances of problem~(\ref{eq:intro-optimization-prob}) that can still be efficiently minimized when the pair-wise terms, $\varphi_{ij}$, meet certain conditions.
The next few sections explore these conditions in more detail, providing pointers to existing optimization methods that succeed in exploiting special structures of $\varphi_{ij}$ to suggest methods and guarantees on the optimization process.
For simplicity, we start in Sec.~\ref{sec:intro-binary} with the special case of discrete binary variables, that is $\mathbf{x}\in\left\{0,1\right\}^n$.
Then we move on, in Sec.~\ref{sec:intro-multilabel}, to the multilabel case of discrete variables taking one of multiple possible states, $\mathbf{x}\in\left\{1,\ldots,l\right\}^n$.
\section{Binary problems}
\label{sec:intro-binary}
Binary optimization problems are of the form (\ref{eq:intro-optimization-prob}) where the solution space is restricted to binary vectors only, i.e., $\mathbf{x}\in\left\{0,1\right\}^n$.
The most basic property of binary functionals is {\bf submodularity}.
This property is defined as follows:
\begin{definition}[Binary submodular]
A pair-wise function $F\left(\bx\right)$ defined for binary vectors $\mathbf{x}$ is {\bf submodular} iff $\forall i,j\in\mbox{$\mathcal{E}$}$:
$\varphi_{ij}\left(0,0\right)+\varphi_{ij}\left(1,1\right)\le\varphi_{ij}\left(1,0\right)+\varphi_{ij}\left(0,1\right)$
\label{def:binary-submodular}
\end{definition}
Close inspection of Def.~\ref{def:binary-submodular} reveals that a submodular function assigns lower energy to smooth configurations (i.e. to $x_i=x_j$) than to ``contrastive" configurations (i.e., to $x_i\ne x_j$).
That is the ``smooth" state $\varphi_{ij}\left(0,0\right)+\varphi_{ij}\left(1,1\right)$ has lower energy than the ``contrastive" state $\left(1,0\right)+\varphi_{ij}\left(0,1\right)$.
Fig.~\ref{fig:types-binary} provides an illustration of the space of all pair-wise energies.
\begin{figure}
\centering
\includegraphics[width=.65\linewidth]{intro/binary-types}
\caption{
{\bf types of binary $F\left(\bx\right)$:}
{\em submodular vs. non-submodular. Green color indicates energies for which exact minimization can be done in polynomial time.}
}
\label{fig:types-binary}
\end{figure}
Submodularity is an important property of $\varphi_{ij}$ since exact optimization of binary submodular functions can be done in polynomial time regardless of the graph structure $\mbox{$\mathcal{E}$}$ (\cite{Greig1989}).
One such optimization algorithm identifies a $1:1$ mapping between binary assignments $\mathbf{x}$ and cuts on a specially constructed graph.
Careful choice of weights for the edges on this special graph gives rise to a $1:1$ mapping between the {\em weight} of a cut and $F\left(\bx\right)$ of the appropriate binary assignment $\mathbf{x}$.
This construction and choice of weights is illustrated in Fig.~\ref{fig:intro-binary-graph}.
The appropriate correspondences between assignments and graph-cuts, and between cut weight and energy are illustrated in Fig.~\ref{fig:intro-binary-graph-cuts}.
Once this $1:1$ mapping is established, optimizing $F\left(\bx\right)$ is simply finding a minimum cut of the constructed graph.
This can be done in polynomial time {\em provided that all the weights of the edges are non-negative} (\cite{Cormen2001}).
Examining the details of this construction reveals that edge weights are non-negative iff the function $F\left(\bx\right)$ is submodular.
Details of this construction can be found in e.g., \cite{Greig1989,Boykov2001} and in more detail in \cite{KolmogorovZabih}.
\begin{figure}[t!]
\hspace*{-1cm}
\centering
\begin{tabular}{cccc}
\multirow{2}{*}{\includegraphics[width=.25\linewidth]{intro/binary_graph_mapping}} &
\multicolumn{3}{c}{$F\left(\bx\right)=\varphi_i\left(x_i\right)+\varphi_j\left(x_j\right)+\varphi_{ij}\left(x_i,x_j\right)$} \\
& & & \\
&
\begin{tabular}{c|c|}
& $\varphi_i$ \\\hline
$x_i=0$ & $a$ \\\hline
$x_1=1$ & $b$
\end{tabular} &
\begin{tabular}{c|c|}
& $\varphi_j$ \\\hline
$x_j=0$ & $c$ \\\hline
$x_j=1$ & $d$
\end{tabular} &
\begin{tabular}{c|c|c}
$\varphi_{ij}$ & $x_i=0$ & $x_i=1$\\\hline
$x_j=0$ & $e$ & $f$ \\\hline
$x_j=1$ & $g$ & $h$
\end{tabular} \\
& & & \\
& \multicolumn{3}{c}{\parbox{.73\linewidth}{
Binary energy over two variables $i$ and $j$ parameterized by 8 parameters $a-h$.
}} \\
\end{tabular}
\caption{
{\bf Graph construction:}
{\em
For a binary energy $F\left(\bx\right)$ a weighted graph is constructed in the following way:
Each variable $x_i$ has a corresponding node $i$. In addition, two special nodes (denoted in this figure as \fbox{$0$} and \fbox{$1$}) are added.
Two edges connect each $i^{th}$ node to the special nodes \fbox{$0$} and \fbox{$1$}.
In addition, an edge $(i,j)$ is added for each interacting pair of variables (for which $\varphi_{ij}$ exists).
The weights of the edges are defined according to the parameters of the energy as illustrated in the figure.
Note that the weight of the $(i,j)$ edge, $g+f-e-h$, is exactly $\varphi_{ij}(0,1)+\varphi_{ij}(1,0)-\varphi_{ij}(0,0)-\varphi_{ij}(1,1)$, this weight is non-negative iff $\varphi_{ij}$ is {\em submodular}.
}}
\label{fig:intro-binary-graph}
\
\
\hspace*{-1.5cm}
\centering
\begin{tabular}{c|cccc}
\includegraphics[height=3cm]{intro/binary_graph_mapping} & \multicolumn{4}{c}{\includegraphics[height=3cm]{intro/binary_graph_mapping_4cuts}} \\ \hline \hline
cut & $\left\{0\right\}$, $\left\{1, i, j\right\}$ & $\left\{0, j\right\}$, $\left\{1, i\right\}$ & $\left\{0, i\right\}$, $\left\{1, j\right\}$ & $\left\{0, i, j\right\}$, $\left\{1\right\}$ \\ \hline
assignment $\mathbf{x}$ & $x_i=1$, $x_j=1$ & $x_i=1$, $x_j=0$ & $x_i=0$, $x_j=1$ & $x_i=0$, $x_j=0$ \\ \hline \hline
cut & $(b+f)$ & $(b+f)$ & $(a+e)+(d-f+h)$ & $(a+e)$ \\
weight & $+(d-f+h)$ & $+c$ & $+(g+f-e-h)$ & $+c$ \\ \hline
energy $F\left(\bx\right)$ & $b+d+h$ & $b+f+c$ & $a+g+d$ & $a+e+c$ \\
\end{tabular}
\caption{
{\bf Binary energies and graph-cuts:}
{\em
All four possible cuts of the graph constructed in Fig.~\ref{fig:intro-binary-graph}.
A cut (dashed red) separates the two special nodes, \fbox{$0$} from \fbox{$1$}.
Cut edges (edges pointing from the \fbox{$0$} side to the \fbox{$1$} side) are marked with dash lines.
First two rows show the $1:1$ mapping between a cut and an assignment to $\mathbf{x}$.
Last two rows show the $1:1$ mapping between the weight of the cut (using the weights defined in Fig.~\ref{fig:intro-binary-graph}) and the energy $F\left(\bx\right)$ of the appropriate assignment.}
}
\label{fig:intro-binary-graph-cuts}
\end{figure}
However, when $F\left(\bx\right)$ is not submodular
(i.e., there exists at least one pair-wise term $\varphi_{ij}$ for which the submodularity inequality~(\ref{def:binary-submodular}) does not hold)
the optimization of $F\left(\bx\right)$ becomes NP-hard (\cite{Rother2007}).
In that case exact minimum cannot be guaranteed in polynomial time and an approximation is sought.
One notable approach for approximating non-submodular binary optimization problems is by an extension of the min-cut approach.
This method, called QPBO (Quadratic Pseudo-Boolean Optimization) was proposed by \cite{Hammer1984} and later extended by \cite{Rother2007}.
The main idea behind this approximation scheme is that when the energy function $F\left(\bx\right)$ is non-submodular, the derived graph has edges with negative weights.
Therefore, they propose to construct a redundant graph in which each variable is represented by two nodes (rather than only one as in the original construction).
One node represents the case where the variable is assigned $0$ and the other node represents the case of assigning $1$ to the variable.
This redundant representation eliminates the need for negative edge weights and thus a min-cut of the new graph can be computed in polynomial time.
Looking at the resulting min-cut we can discern two cases for each variable:
The first, in which the cut separates the two complementary nodes representing this variable.
In this case, the cut clearly defines an optimal state for the variable (either $0$ or $1$).
However, there is a second case in which both complementary nodes fall in the same side of the cut.
In this case, we are unable to determine what is the proper assignment for this variable and the variable remains unlabeled.
Therefore, we can conclude that QPBO extends the min-cut approach to handle non-submodular binary energies.
Recovery of the global minimum is no longer guaranteed, but the algorithm may recover a {\em partial} labeling that is guaranteed to be part of some globally optimal solution.
However, in the extreme case it may happen that QPBO is unable to label any variable.
Table~\ref{tab:binary-opt-classification} summarizes the different types of pair-wise binary energy functions and the difficulty they entail on their optimization.
\begin{table}
\centering
\begin{tabular}{r|c|c}
\backslashbox{pair-wise}{structure} & Tree & Cyclic \\\hline
submodular & {\bf Easy:} mincut, BP & {\bf Easy:} mincut \\\hline
non-submodular & {\bf Easy:} QPBO, BP & {\bf Hard} \\
\end{tabular}
\caption{{\bf Hardness of binary optimization:}
{\em the computational ``hardness" of a discrete minimization as a function of the underlying graph structure ($\mbox{$\mathcal{E}$}$), and the class of pair-wise interactions ($\varphi_{ij}$).}}
\label{tab:binary-opt-classification}
\end{table}
\section{Multilabel problems}
\label{sec:intro-multilabel}
A multilabel discrete problem is the optimization of a discrete function $F\left(\bx\right)$ of the form (\ref{eq:pair-wise-gen}) defined over a finite discrete vector $\mathbf{x}\in\left\{1,\ldots,L\right\}^n$.
As was shown in the previous section, properties of the pair-wise terms $\varphi_{ij}$ of $F\left(\bx\right)$ have a crucial effect on the computational complexity of the optimization problem.
For the binary case the only distinctive property was the {\em submodularity} of $F\left(\bx\right)$.
However, when discrete variables over more than two states are considered, there are more subtle types of pair-wise interactions that affect the ability to optimize, or at least efficiently approximate it.
This section describes these various types of $\varphi_{ij}$ and their effect on the discrete optimization task.
The following definition of multilabel submodularity is given in \cite{Schlesinger2006}.
\begin{definition}[Multilabel submodular]
Assume the labels ($\alpha,\beta,\ldots$) are fully ordered.
Then $F\left(\bx\right)$ is {\bf multilabel submodular} iff $\forall i,j\in\mbox{$\mathcal{E}$}$ and $\forall \alpha\le\gamma,\beta\le\delta$
the following inequality holds
\begin{equation}
\varphi_{ij}\left(\alpha,\beta\right) + \varphi_{ij}\left(\gamma,\delta\right) \le
\varphi_{ij}\left(\alpha,\delta\right) + \varphi_{ij}\left(\gamma,\beta\right)
\label{eq:multilabel-submodular}
\end{equation}
\label{def:multilabel-submodular}
\end{definition}
A slightly simpler and equivalent condition for submodularity uses the following condition:
\begin{equation}
\varphi_{ij}\left(\alpha,\beta\right) + \varphi_{ij}\left(\alpha+1,\beta+1\right) \le
\varphi_{ij}\left(\alpha,\beta+1\right) + \varphi_{ij}\left(\alpha+1,\beta\right)
\label{eq:multilabel-submodular-simple}
\end{equation}
$F\left(\bx\right)$ is multilabel submodular iff~(\ref{eq:multilabel-submodular-simple}) holds for all labels $\alpha,\beta$ and for all $\varphi_{ij}$.
The notion of submodularity ~is ~strongly ~related to Monge matrices (\cite{Cechlarova1990}):
A matrix $V$ is a {\em Monge matrix} iff $V_{\alpha,\beta} + V_{\alpha+1,\beta+1} \le V_{\alpha+1,\beta} + V_{\alpha,\beta+1}$, $\forall\alpha,\beta$.
Monge matrices were defined by the French mathematician G. Monge (\citeyear{Monge1781}), and they play a major role in optimal transportation problems and other discrete optimization tasks (see, e.g., \cite{Burkard2007}).
Consider a matrix $V\in\mathbb{R}^{l\times l}$ whose entries are $V_{\alpha\beta} \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} \varphi_{ij}\left(\alpha,\beta\right)$,
then $\varphi_{ij}$ is multilabel submodular iff $V$ is a Monge matrix.
A sub class of multilabel submodular functions are functions where $\varphi_{ij}$ are convex on the set of labels.
Convexity on a discrete set is defined in \cite{Ishikawa2003}, as follows:
\begin{definition}[Convexity on a discrete set]
A real valued function $g\left(x\right)$ is {\bf convex} on a set $\mathcal{A}$ iff
\begin{equation}
g\left(t x + (1-t) y \right) \le t g\left(x\right) + (1-t) g\left(y\right)
\end{equation}
for all $x,y\in\mathcal{A}$ and $t\in\left[0,1\right]$ s.t. $t x + (1-t) y \in \mathcal{A}$
\label{def:multilabel-convex-ishikawa}
\end{definition}
When the set of labels is fully ordered and if $\forall i,j$ $\varphi_{ij}\left(x_i,x_j\right) = g_{ij}\left(x_i - x_j\right)$ and all $g_{ij}$ are convex, then $F\left(\bx\right)$ is convex.
For example $\varphi_{ij}\left(x_i,x_j\right) = \left|x_i - x_j\right|^p$.
Note that a truncated $\ell_p$ norm is no longer convex.
\begin{claim}
Convex is a special case of submodular \cite{Schlesinger2006}.
\end{claim}
\begin{proof}
Let $\varphi\left(\alpha,\beta\right)=g\left(\alpha-\beta\right)$ for some convex $g\left(x\right)$.
Then for all $\alpha,\beta$:
\begin{eqnarray*}
g\left(\frac{1}{2}\left(\alpha-\beta-1\right) + \frac{1}{2}\left(\alpha-\beta+1\right)\right) & \le & \frac{1}{2} g\left(\alpha-\beta-1\right) + \frac{1}{2} g\left(\alpha-\beta+1\right) \\
g\left(\alpha - \beta\right) & \le & \frac{1}{2} g\left(\alpha-\beta-1\right) + \frac{1}{2} g\left(\alpha-\beta+1\right) \\
g\left(\alpha - \beta\right) + g\left(\alpha - \beta\right) & \le & g\left(\alpha-\beta-1\right) + g\left(\alpha-\beta+1\right) \\
g\left(\alpha - \beta\right) + g\left(\alpha + 1 - \left(\beta + 1\right) \right) & \le & g\left(\alpha-\left(\beta+1\right)\right) + g\left(\alpha+1 -\beta\right) \\
\varphi\left(\alpha,\beta\right) + \varphi\left(\alpha+1,\beta+1\right) & \le & \varphi\left(\alpha+1,\beta\right) + \varphi\left(\alpha,\beta+1\right) \\
\end{eqnarray*}
From property (\ref{eq:multilabel-submodular-simple}) it follows that a convex $\varphi$ is also submodular.
\end{proof}
To make these definitions more concrete, we can consider a few examples.
Popular pair-wise terms of the form $\varphi_{ij}\left(x_i,x_j\right)=\left|x_i-x_j\right|$ (also known as $\ell_1$), and the $\ell_2$: $\varphi_{ij}\left(x_i,x_j\right)=\left(x_i-x_j\right)^2$ are both {\em convex} and therefore {\em multilabel submodular}.
However, the robust (or truncated) version of these $\ell_p$ terms: $\varphi_{ij}\left(x_i,x_j\right)=\min\left\{\left|x_i-x_j\right|^p,\tau\right\}$, is no longer multilabel submodular.
An important result regarding the minimization of multilabel submodular functions is presented in \cite{Schlesinger2006}. A reduction is made from submodular minimization to st-mincut on a specially constructed graph.
It is shown that when the original energy is multilabel submodular all weights in the resulting graph are non-negative and hence a {\em global} minimum can be found in polynomial time.
This construction generalizes the construction of \cite{Ishikawa2003} that is specific to convex pair-wise functions.
However, submodularity of $F\left(\bx\right)$ is a very restrictive property.
The well known Potts term, and many other pair-wise interactions are not submodular.
Still, there are other important properties for non-submodular functions $F\left(\bx\right)$.
\cite{Boykov2001} derived important approximations that rely on other properties of $F\left(\bx\right)$.
These properties were further relaxed by \cite{KolmogorovZabih}:
\begin{definition}[Relaxed metric]
A function $F\left(\bx\right)$ is a {\bf relaxed metric} iff $\forall i,j\in\mbox{$\mathcal{E}$}$ and $\forall \alpha,\beta,\gamma$
\begin{equation}
\varphi_{ij}\left(\alpha,\alpha\right) + \varphi_{ij}\left(\beta,\gamma\right) \le \varphi_{ij}\left(\beta,\alpha\right) + \varphi_{ij}\left(\alpha,\gamma\right)
\end{equation}
\label{def:large-move-metric}
\end{definition}
The condition of Def.~\ref{def:large-move-metric} resembles the triangle inequality of metric functions in the case $\varphi_{ij}(\alpha,\alpha) = 0$.
The Potts model and the robust (truncated) $\ell_1$ interaction, mentioned earlier in this section are examples of relaxed-metric pair-wise interaction.
Note that this property of relaxed metric is different than the convexity of Def.~\ref{def:multilabel-convex-ishikawa}.
Another property, less restrictive than relaxed metric is:
\begin{definition}[Relaxed semi-metric]
A function $F\left(\bx\right)$ is a {\bf relaxed semi-metric} iff $\forall i,j\in\mbox{$\mathcal{E}$}$ and $\forall \alpha,\beta$
\begin{equation}
\varphi_{ij}\left(\alpha,\alpha\right) + \varphi_{ij}\left(\beta,\beta\right) \le \varphi_{ij}\left(\beta,\alpha\right) + \varphi_{ij}\left(\alpha,\beta\right)
\end{equation}
\label{def:large-move-semi-metric}
\end{definition}
Examples of relaxed semi-metric functions: $\ell_2$, truncated $\ell_2$.
Clearly, any relaxed metric function is also a relaxed semi-metric.
At this point it may be useful to get some intuition about the meaning of the semi-metric property:
According to Def.~\ref{def:large-move-semi-metric} a function $F\left(\bx\right)$ is semi-metric if the cost of assigning neighboring variables $i$ and $j$ to the {\em same} label (either $\alpha$ or $\beta$) is never greater than the cost of assigning them to different labels.
This property implies that $F\left(\bx\right)$ encourages smoothness of the solution $\mathbf{x}$.
Figure~\ref{fig:large-move-multilabel-types} shows the relation between the different types of functions $\varphi_{ij}$.
The most restrictive type is the convex (Def.~\ref{def:multilabel-convex-ishikawa}) which is a subset of submodular (Def.~\ref{def:multilabel-submodular}).
Regarding relaxed metric and submodular: there are submodular functions that are not relaxed metric (e.g., $\ell_2$), and there are relaxed metric that are not submodular (e.g., Potts).
Table~\ref{tab:multilabel-examples-phi} shows examples of popular pair-wise functions and their properties.
\begin{claim}
In general, multilabel submodular (Def.~\ref{def:multilabel-submodular}) is not metric (Def.~\ref{def:large-move-metric}).
\end{claim}
\begin{proof}
Let $\varphi_{ij}$ be multilabel submodular. Consider three labels $\alpha\le\gamma\le\delta$. Choose $\beta=\gamma$.
We now have $\alpha\le\gamma$ and $\beta\le\delta$.
Submodularity of $\varphi_{ij}$ (Def.~\ref{def:multilabel-submodular}) yields:
\begin{eqnarray*}
\varphi_{ij}\left(\alpha,\beta\right)+\varphi_{ij}\left(\gamma,\delta\right)& \le & \varphi_{ij}\left(\alpha,\delta\right)+\varphi_{ij}\left(\gamma,\beta\right) \\
\varphi_{ij}\left(\alpha,\gamma\right)+\varphi_{ij}\left(\gamma,\delta\right)& \le & \varphi_{ij}\left(\alpha,\delta\right)+\varphi_{ij}\left(\gamma,\gamma\right) \\
\end{eqnarray*}
This inequality is the opposite of the inequality of Def.~(\ref{def:large-move-metric}) (semi-metric).
In general, Equality does not hold and therefore most submodular $\varphi_{ij}$ are not metric.
However, for $\ell_1$, i.e., $\varphi_{ij}\left(x_i,x_j\right)=\left|x_i-x_j\right|$ equality holds and thus $\ell_1$ is a special case of $\varphi_{ij}$ that is both submodular and metric.
\end{proof}
\begin{figure}
\centering
\includegraphics[width=.9\textwidth]{intro/multilabel-types.pdf}
\caption{
{\bf Different types of multilabel $F\left(\bx\right)$:}
{\em The hierarchy and relations between different types of multilabel energies.
Green indicates the existence of global minimization algorithms.
For energies in red there are good approximation algorithms.}
}
\label{fig:large-move-multilabel-types}
\end{figure}
\begin{table}
\centering
\begin{tabular}{cccc}
{\bf convex} & {\bf submodular} & {\bf relaxed metric} & {\bf relaxed semi-metric} \\\hline
$\ell_1$ & $\ell_1$ & truncated $\ell_1$ & $\ell_2$ \\
$\ell_2$ & $\ell_2$ & $\ell_1$ & truncated $\ell_2$ \\
& & Potts & \\
\end{tabular}
\caption{
{\bf examples of different types of pair-wise functions $\varphi_{ij}$}}
\label{tab:multilabel-examples-phi}
\end{table}
Unfortunately, the promising results of \cite{Schlesinger2006} regarding the {\em global} minimization of submodular functions does not hold for more general functions.
When $F\left(\bx\right)$ is no longer submodular one can no longer hope to achieve global optimality in polynomial time.
However, for relaxed metric and relaxed semi-metric functions \cite{Boykov2001} showed large move making approximate algorithms that performs quite well in practice (see e.g., \cite{Szeliski2008}).
Large move making algorithms iteratively seek to improve the energy of a current solution by updating large number of variables at once.
Each such large step is carried out by solving a simple binary submodular minimization via st-mincut.
For relaxed metric functions the large move is $\alpha$-expansion.
At each iteration a binary problem is solved: for each variable it can either retain its current label (0) or switch label to $\alpha$ (1).
The relaxed metric property of $F\left(\bx\right)$ ensures that the resulting binary problem is submodular and thus can be solved globally in polynomial time.
The $\alpha$-expansion algorithm iterates over all labels until it converges.
Convergence after finite number of iterations is guaranteed, and in certain cases some theoretical bounds can be proven on the quality of the approximation (see \cite{Boykov2001} for more details).
For relaxed semi-metric functions a slightly different large move is devised.
For each pair of labels $\alpha$ and $\beta$ the large move is called $\alpha\beta$-swap.
At each iteration a binary problem is solved for all variables currently labeled either $\alpha$ or $\beta$:
for each variable it can pick either $\alpha$ (0) or $\beta$ (1).
The relaxed semi-metric property of $F\left(\bx\right)$ ensures that the resulting binary problem is submodular and thus can be solved globally in polynomial time.
The $\alpha\beta$-swap algorithm iterates over all {\em pairs} of labels until it converges.
Convergence after finite number of iterations is guaranteed, however, theoretical bounds on the approximation no longer exists (see \cite{Boykov2001} for more details).
Table~\ref{tab:ml-opt-classification} shows the resulting types of pair-wise energies and the current results on their minimization.
This thesis focuses on the hard optimization of the contrast-enhancing functionals defined on cyclic graphs.
Part~\ref{part:app} shows how introducing energies that contain contrast-enhancing terms gives rise to new applications.
While Part~\ref{part:approx} deals with the methods of approximating these challenging contrast-enhancing energies.
\begin{table}
\centering
\begin{tabular}{r|c|c}
\backslashbox{pair-wise}{structure} & Tree & Cyclic \\\hline
submodular & {\bf Easy:} mincut, BP & {\bf Easy:} mincut \\\hline
semi-metric & {\bf Easy:} BP & Good Approximations \\\hline
contrast-enhancing & {\bf Easy:} BP & {\bf Hard} \\
\end{tabular}
\caption{{\bf Hardness of optimization (multilabel):}
{\em the computational ``hardness" of discrete optimization as a function of the underlying graph structure and the class of pair-wise interactions.}}
\label{tab:ml-opt-classification}
\end{table}
\section{Relation to Linear Programming (LP)}
\label{sec:intro-lp}
This section establishes a connection between the pair-wise energy minimization problem~(\ref{eq:intro-optimization-prob}) and the field of convex optimization, in particular to Linear Programming (LP).
For the following discussion it is useful to introduce some new notations and definitions.
The first useful representation is the {\em overcomplete representation} of the solution vector $\mathbf{x}$.
This representation is defined as follows \cite{Wainwright2005,wainwright2008graphicalmodels}:
\begin{definition}[Overcomplete representation]
A discrete solution $\mathbf{x}$ can be represented by an extended binary vector $\phi\left(\bx\right)$, s.t.
\begin{eqnarray}
\phi\left(\bx\right)_{i,\alpha} & = & \delta\left(x_i==\alpha\right) \\
\phi\left(\bx\right)_{ij,\alpha\beta} & = & \delta\left(x_i==\alpha\right) \cdot \delta\left(x_j==\beta\right)
\end{eqnarray}
where $\delta(\cdot)$ is the Kronecker delta function.
\label{def:local-update-overcomplete-representation}
\end{definition}
The overcomplete representation projects a discrete vector $\mathbf{x}$ of dimension $n$ into a $d$-dimensional binary vector $\phi\left(\bx\right)$.
The index set of vector $\phi\left(\bx\right)$ is defined as $\mathcal{I}=\left\{i,\alpha\right\}\cup\left\{ij,\alpha\beta\right\}$, with $d=\left|\mathcal{I}\right|$.
With the overcomplete representation in mind, it is useful to parameterize the discrete function $F\left(\bx\right)$ of Eq.~(\ref{eq:pair-wise-gen}) using a parameter vector $\theta$:
\begin{subequations}
\begin{equation}
\theta_{i,\alpha} = \varphi_i\left(\alpha\right)
\end{equation}
\begin{equation}
\theta_{ij,\alpha\beta} = \varphi_{ij}\left(\alpha,\beta\right)
\end{equation}
\label{eq:local-overcomplete-theta}
\end{subequations}
Combining Def.~\ref{def:local-update-overcomplete-representation} with the parametrization of~(\ref{eq:local-overcomplete-theta}), the functional of Eq.~(\ref{eq:pair-wise-gen}) becomes
\begin{eqnarray}
F\left(\bx\right) & = & \sum_{i,\alpha} \theta_{i,\alpha} \phi\left(\bx\right)_{i,\alpha} + \sum_{ij,\alpha\beta} \theta_{ij,\alpha\beta}\phi\left(\bx\right)_{ij,\alpha\beta} \nonumber \\
& = & \dotprod{\theta}{\phi\left(\bx\right)}
\label{eq:local-overcomplete-energy}
\end{eqnarray}
The overcomplete representation $\phi\left(\bx\right)$ is defined over a discrete set of points.
However, it is useful to consider a relaxation of this set into a convex continuous domains.
The tightest relaxation of the discrete set $\left\{\phi\left(\bx\right) \vert \mathbf{x}\in\left\{1,\ldots,l\right\}^n\right\}$ is the marginal polytop $\marg{\mbox{$\mathcal{E}$}}$:
\begin{definition}[Marginal polytop]
The marginal polytop of $F\left(\bx\right)$ is the convex combination of the vertices $\phi\left(\bx\right)$ for the $l^n$ discrete solutions $\mathbf{x}$. This set is formally defined as:
\begin{equation}
\marg{\mbox{$\mathcal{E}$}} = \left\{\mu\in\mathbb{R}^d \left| \mu=\sum_{\mathbf{x}}p\left(\mathbf{x}\right)\phi\left(\bx\right),\;\mbox{for some distribution $p\left(\cdot\right)$}\right.\right\}
\end{equation}
\label{def:local-marginal-polytop}
\end{definition}
This marginal polytop, $\marg{\mbox{$\mathcal{E}$}}$, is defined by finite, yet exponentially large, number of half-spaces.
Therefore, it is convenient to define a relaxed version of the marginal polytop:
\begin{definition}[Local polytop]
The local polytop of $F\left(\bx\right)$ is the convex set:
\begin{equation}
\local{\mbox{$\mathcal{E}$}} = \left\{\tau\in\mathbb{R}^d\left|
\begin{array}{rl}
\sum_\alpha \tau_{i,\alpha} = 1 & \forall i \\
\sum_{\alpha\beta} \tau_{ij,\alpha\beta} = 1 & \forall ij\in\mbox{$\mathcal{E}$} \\
\sum_{\beta} \tau_{ij,\alpha\beta} = \tau_{i,\alpha} & \forall ij\in\mbox{$\mathcal{E}$},\beta
\end{array}
\right.\right\}
\end{equation}
\label{def:local-local-polytop}
\end{definition}
Unlike the marginal polytop, $\local{\mbox{$\mathcal{E}$}}$ is defined using only polynomial number of half-spaces,
and therefore it admits polynomial time optimization schemes.
In fact, $\local{\mbox{$\mathcal{E}$}}$ is the first order approximation of $\marg{\mbox{$\mathcal{E}$}}$ (\cite{wainwright2008graphicalmodels}).
Note that the geometry of $\marg{\mbox{$\mathcal{E}$}}$ and $\local{\mbox{$\mathcal{E}$}}$ are affected by the number of variables $n$, the number of states $l$ and by the underlying graph structure $\mbox{$\mathcal{E}$}$ defining the interacting pairs of variables.
These polytops are {\em not} affected by the parameters of $\varphi_i$ and $\varphi_{ij}$.
\begin{figure}
\centering
\begin{tabular}{ccc}
\includegraphics[width=.3\linewidth]{intro/marginal_local} &
\includegraphics[width=.3\linewidth]{intro/marginal_local_integral} &
\includegraphics[width=.3\linewidth]{intro/marginal_local_fractional} \\
(a) & (b) & (c)
\end{tabular}
\caption{
{\bf The Marginal and Local polytops:}
{\em An illustration of the marginal, $\marg{\mbox{$\mathcal{E}$}}$, and local, $\local{\mbox{$\mathcal{E}$}}$, polytops.
(a)~$\local{\mbox{$\mathcal{E}$}}$ is an outer bound on the exact constraints of $\marg{\mbox{$\mathcal{E}$}}$. The vertex set of $\local{\mbox{$\mathcal{E}$}}$ includes all integral vertices of $\marg{\mbox{$\mathcal{E}$}}$ (marked in red) . It also includes fractional vertices (marked in black), which are {\em not} vertices of $\marg{\mbox{$\mathcal{E}$}}$. (b),~(c)~Solving an LP with cost vector $\theta$ entails translating a hyperplane with normal $\theta$ until it is tangent to the constraint set. In~(b)~the point of tangency of cost vector $\theta_1$ occurs at an integral vertex. In~(c)~the point of tangency of cost vector $\theta_2$ occurs at a fractional vertex, outside $\marg{\mbox{$\mathcal{E}$}}$. In this case, there is an integrality gap.
(figure taken from \protect\cite{Wainwright2005}).}
}
\label{fig:intro-marginal-local}
\end{figure}
By standard properties of LP, the optimal value is attained at an extreme point of the constraint set (a vertex of the constraints polytop).
The marginal polytop, $\marg{\mbox{$\mathcal{E}$}}$, is a convex set defined by all the possible solutions $\phi\left(\bx\right)$.
Hence, a vertex of $\marg{\mbox{$\mathcal{E}$}}$ corresponds to a vector $\phi\left(\bx\right)$ for some {\em discrete} solution $\mathbf{x}$.
We refer to these vertices as {\em integral} vertices corresponding to integral solutions of the LP.
On the other hand, the local polytop, $\local{\mbox{$\mathcal{E}$}}$, may contain more vertices that do not correspond to any discrete solution, $\mathbf{x}$.
We refer to these vertices as {\em fractional} solutions.
Fig.~\ref{fig:intro-marginal-local} provides an illustration of the marginal and local polytops, the relation between them, and their impact on the optimal solution of LP.
The figure also distinguishes between the integral and fractional vertices of the polytops.
\cite[Example~3]{Wainwright2005} describes in detail a case where fractional solution is optimal.
It is important to note that the parameter vector $\theta$ for which the fractional solution is optimal, in their example, is such that encourages contrast between variables (i.e., $x_i\ne x_j$ for neighboring $i$ and $j$).
The relation between discrete energy minimization (problem~(\ref{eq:intro-optimization-prob})) and convex LP presented in this section lies in the foundation of popular optimization algorithms such as tree-reweighted BP (see Sec.~\ref{sec:intro-trw}).
This relation is also important to illustrate the challenging task of optimizing~(\ref{eq:intro-optimization-prob}):
When $\theta$ represents an energy function that is ``smoothness-encouraging" its optimal value usually corresponds to an integral vertex of $\local{\mbox{$\mathcal{E}$}}$ and thus its optimization can be done {\em exactly} via LP over the relaxed constraint set $\local{\mbox{$\mathcal{E}$}}$.
However, when the energy has contrast-enhancing terms its optimal solution w.r.t $\local{\mbox{$\mathcal{E}$}}$ is {\em fractional} -- the global {\em integral} solution cannot be attained using the relaxed constraints of $\local{\mbox{$\mathcal{E}$}}$.
Therefore, the optimization of energies that have contrast-enhancing terms, is a very challenging task.
This thesis focuses on these energies and proposes methods to cope with this inherent difficulty.
\comment{
In general, the local polytop $\local{\mbox{$\mathcal{E}$}}$ contains fractional vertices that are strictly outside the marginal polytop: $\marg{\mbox{$\mathcal{E}$}}\subset\local{\mbox{$\mathcal{E}$}}$.
It is worthwhile noting, however, that when the underlying graph $\mbox{$\mathcal{E}$}$ has no cycles (i.e., it is a tree $\mbox{$\mathcal{E}$}=\mathcal{T}$) the two polytops are identical: $\marg{\mathcal{T}}=\local{\mathcal{T}}$ \cite[proposition 8.3]{wainwright2008graphicalmodels}.
Using the parametrization $\theta$ and the convex sets $\marg{\mbox{$\mathcal{E}$}}$ and $\local{\mbox{$\mathcal{E}$}}$ defined previously we can write the optimization problem $\mathbf{x}^\star = \arg\min_{\mathbf{x}}F\left(\bx\right)$ as a linear program
\begin{eqnarray}
& \min_\mu & \dotprod{\theta}{\mu} \nonumber \\
& \mbox{s.t.} & \mu \in \marg{\mbox{$\mathcal{E}$}}
\label{eq:local-lp-marg}
\end{eqnarray}
Since the objective is linear, and the constraint set is convex, the optimal solution of the LP~\ref{eq:local-lp-marg} is attained on a vertex of the polytop $\marg{\mbox{$\mathcal{E}$}}$ for which there exist a unique discrete $\mathbf{x}$ s.t. $\mu=\phi\left(\bx\right)$.
Therefore the optimization of $F\left(\bx\right)$ over discrete solutions $\mathbf{x}$ is equivalent to solving LP~(\ref{eq:local-lp-marg}) \cite{Wainwright2005}.
As we mentioned before, solving the LP~(\ref{eq:local-lp-marg}) is not trivial since the constraints defining $\marg{\mbox{$\mathcal{E}$}}$ grows exponentially with $n,l$ and $\mbox{$\mathcal{E}$}$.
Therefore, we resort to a relaxation of the constraints set with the following linear program:
\begin{eqnarray}
& \min_\tau & \dotprod{\theta}{\tau} \nonumber \\
& \mbox{s.t.} & \tau \in \local{\mbox{$\mathcal{E}$}}
\label{eq:local-lp-local}
\end{eqnarray}
Since the constraint set $\local{\mbox{$\mathcal{E}$}}$ has now vertices $\tau$ for which there is no discrete solution $\mathbf{x}$ s.t. $\tau=\phi\left(\bx\right)$,
the optimal solution of LP~(\ref{eq:local-lp-local}) is no longer equivalent to the optimization of $F\left(\bx\right)$.
However, if the optimal solution of LP~\ref{eq:local-lp-local} happens to be $\tau^\prime$ s.t. $\tau^\prime=\phi\left(\mathbf{x}^\prime\right)$ for some integral $\mathbf{x}^\prime$ than $\mathbf{x}^\prime$ is globally optimal for $F\left(\bx\right)$ as well.
The relaxation of the constraint set from $\marg{\mbox{$\mathcal{E}$}}$ in LP~(\ref{eq:local-lp-marg}) to $\local{\mbox{$\mathcal{E}$}}$ in LP~(\ref{eq:local-lp-local}) introduces an {\bf integrality gap}: that is, LP~(\ref{eq:local-lp-local}) may introduce optimal solution that is infeasible for LP~(\ref{eq:local-lp-marg}).
Therefore the optimal solution of LP~(\ref{eq:local-lp-local}) may be strictly lower than the optimal solution of LP~(\ref{eq:local-lp-marg}).
\todo{intuition as to why non-submodularity results with fractional solutions: maybe something to do with uncertainty of the labeling, violated cycles in the graph, etc.}
\todo{relate this to this thesis: we address energies where integral gap exists and is significant. This happens mainly for contrast enhancing energies}
}
\section{Discrete Optimization Algorithms}
In the previous sections we outlined some key properties of the energy function $F\left(\bx\right)$ and their effect on
the minimization process.
We also demonstrated how specific optimization algorithms take advantage of these properties.
In this section we present several prominent optimization algorithms that we refer to later on in this thesis.
These selected representative approaches sketches the main directions at which current discrete optimization research is mainly focused.
\subsection{Iterated Conditional Modes (ICM)}
\label{sec:local-icm}
ICM is a very simple and basic iterative optimization algorithm proposed by \cite{Besag1986}.
It is an approximate method, acting {\em locally} on the variables, suitable for multilabel functions with arbitrary underlying graph and arbitrary $\varphi_{ij}$.
At each iteration ICM visits all the variables sequentially, and choose for each variable the best state (with the lower energy) given the current states of all other variables.
This process can be viewed as a greedy coordinate descend algorithm and it bears some analogy to Gauss-Seidel relaxations of the continuous domain (\cite{Varga1962}).
ICM is a local update process and therefore is prone to getting stuck very fast in local minimum.
It is also extremely sensitive to initialization (see e.g., \cite{Szeliski2008}).
When taking a probabilistic point of view, and considering the energy function as a Gibbs energy, that is representing some measure over all possible solutions:
\begin{equation}
Pr\left(\mathbf{x}\right) \propto \exp \left( - \frac{1}{T} F\left(\bx\right) \right)
\label{eq:intro-gibbs-energy}
\end{equation}
ICM may be viewed as a Gibbs sampler at the temperature limit $T\rightarrow0$.
Therefore, its performance is expected to be inferior to more sophisticated sampling methods such as, e.g., simulated annealing (\cite{Kirkpatrick1983}).
\subsection{Belief Propagation (BP)}
\label{sec:intro-bp}
Belief-propagation is an optimization algorithm based on local updates.
However, in contrast to the hard assignment ICM performs at each update,
BP maintains ``soft" beliefs for each variable and passes messages between neighboring variables according to their current belief.
A message from variable $i$ to its neighbor $j$, $m_{i\rightarrow j}$, is a vector of length $L$.
That is, the message vector encodes how $i$ ``feels" about assigning state $\alpha$ to $j$.
\begin{equation}
m_{i\rightarrow j}\left(\alpha\right) = \min_{\beta} \left\{\varphi_i\left(\beta\right) + \sum_{ki\in\mbox{$\mathcal{E}$},k\ne j}m_{k\rightarrow i}\left(\beta\right) + \varphi_{ij}\left(\beta,\alpha\right) \right\}
\label{eq:intro-bp-message}
\end{equation}
The belief of each variable $i$ is also a vector of length $L$ encoding the tendency of $i$ to be assigned to state $\alpha$:
\begin{equation}
b_i\left(\alpha\right) = \varphi_i\left(\alpha\right) + \sum_{ki\in\mbox{$\mathcal{E}$}}m_{k\rightarrow i}\left(\alpha\right)
\label{eq:intro-bp-belief}
\end{equation}
BP iteratively passes messages and updates the local belief for each variable.
After its final iteration, each variable is assigned the label with the lowest energy, i.e., $x_i = \arg\min_\alpha b_i(\alpha)$.
Originally, BP was used as an inference algorithm in tree-structured graphical models (\cite{Pearl1988,Koller2009}).
Messages were initialized to zero. Then messages were passed from leafs to root and back to the leafs.
This forward-backward message passing converges to the global optimum when $\mbox{$\mathcal{E}$}$ is a tree, regardless of the type of $\varphi_{ij}$ that can be arbitrary.
When the underlying graph $\mbox{$\mathcal{E}$}$ has cycles, BP is no longer guaranteed to converge.
It was proposed to run BP on cyclic graphs, a variant called loopy-BP.
In the loopy case, however, it is not clear how to schedule the messages and how to determine the number of iterations to perform.
Even if the loopy BP converges to some fixed point, it is usually a local optimum with no guarantees on global optimality (\cite{Koller2009,wainwright2008graphicalmodels})
\subsection{Tree-reweighted Belief Propagation (TRW)}
\label{sec:intro-trw}
A significant development of BP was presented in the works of \cite{Wainwright2005,Kolmogorov2006,Werner2007,wainwright2008graphicalmodels,Komodakis2011}.
These works proposed a new interpretation to the basic message passing operation that BP conducts.
They relaxed the discrete minimization of $F\left(\bx\right)$ to form a continuous linear programming (LP), in the same manner that was presented in Sec.~\ref{sec:intro-lp}.
Then they related message passing to the optimization of the resulting LP.
a It was shown that the relaxation of $F\left(\bx\right)$ forms an LP with very specific structure.
This special structure can, in turn, be exploited to devise a specially tailored optimization scheme that uses message passing as a basic operation.
The tree-reweighted BP approach establishes a relation between discrete optimization and continuous convex optimization of LP.
This relation brings forward interesting results and properties from the continuous optimization domain to the discrete one.
For instance, it allows to use Lagrangian multipliers and formulate a Lagrangian dual to the original problem.
The dual representation provides a lower bound to the sought optimal solution.
If a solution $\mathbf{x}^\star$ is found with an energy $F\left(\mathbf{x}^\star\right)$ equals to the lower bound, then a certificate is provided that this $\mathbf{x}^\star$ is a {\em global} minimum.
It was shown (e.g., \cite{Szeliski2008}) that in practice in many computer vision application TRW was able to recover globally optimal solutions.
These results dealt mainly with relaxed metric energies (see Sec.~\ref{sec:intro-multilabel}).
However, there is no general guarantee on TRW and there are cases involving challenging energies, beyond relaxed metric, for which it was shown that an integrality gap exists and TRW can no longer provide a tight approximation (e.g., \cite{Kolmogorov2006,Bagon2012}).
\subsection{Large Move Methods}
As opposed to local methods such as ICM, BP and TRW,
there is the approach of \cite{Boykov2001} that proposes discrete methods based on combinatorial principles.
The basic observation that lies at the heart of the large-move algorithms is that instead of treating the variables locally one at a time, one may affect the labeling of many variables at once by performing large moves.
These large moves are formulated as a binary step, and the difference between the different ``flavors" of the large-move algorithms is the formulation of these binary steps.
What makes these large move effective and efficient is the fact that binary submodular sub-problems can be solved globally and efficiently.
The two basic large move algorithms, $\alpha$-expand and $\alpha\beta$-swap, were already described in Sec.~\ref{sec:intro-multilabel} in the context of relaxed metric and relaxed semi metric energies.
Recently, another large move making algorithm called fusion-moves was proposed (\cite{Lempitsky2007,Lempitsky2010})
At each iteration of the fusion algorithm a discrete solution is proposed.
The proposed solution is fused into the current solution via a binary optimization: each variable can retain its current label ($0$), or switch to the respective label from the proposed solution ($1$).
However, unlike the swap and expand algorithms, the resulting binary optimization of the fusion step is no longer guaranteed to be submodular and highly depends on the types of proposed solutions.
Therefore, it is often the case that QPBO (\cite{Kolmogorov2007}), which is a non-submodular binary approximation algorithm, is used to perform the binary steps of the fusion algorithm.
\
\
To summarize this brief outline of existing approximation algorithms
one may notice that a lot of effort is put in recent years in developing and improving approximate optimization algorithms.
Research is put into both providing better practical results and into exploring the theoretic aspects of the problem.
Approximation methods are derived and inspired by both the continuous optimization domain (e.g., TRW) and the discrete domain (e.g., graph-cuts).
However, these results mainly focus on the minimization of functions $F\left(\bx\right)$ that have some structure to them: either relaxed metric or relaxed semi-metric (see, for example, the survey of \cite{Szeliski2008}).
For these smoothness-encouraging functions current algorithms succeed in providing good approximations in practice, despite their theoretical NP-hardness.
In contrast, when it comes to arbitrary, {\em contrast-enhancing} functions, little is known in terms of approximation and no method currently exists (to the best of our knowledge) that provides satisfying approximations.
This thesis focuses on the optimization of arbitrary, contrast-enhancing functions.
\chapter{Outline of this Thesis}
Discrete energy minimization is a ubiquitous task in computer vision and in other scientific domains.
However, in the previous chapter we saw that the optimization of such discrete energies is known to be NP-hard in most cases (\cite{Boykov2001}).
Despite this theoretical hardness, for many ``smoothness encouraging" energies (relaxed semi-metric), approximate optimization algorithms exist and provide remarkable approximations in practice.
However, as tasks become more sophisticated, the models grow more complex:
From tree structured to cyclic graphs and grids,
and from simple smoothing priors to complex arbitrary pair-wise interactions.
As the energies become less constrained and structured one gains more expressive power for the objective function
at the cost of a significantly more challenging optimization task.
In this work I would like to step outside this ``comfort-zone" of the smoothness-encouraging energies and explore more challenging discrete energies.
This step gives rise to two important questions:
\begin{enumerate}
\item Why bother? Why should one consider energy functions beyond semi-metric? What can be gained (in term of expressive power) considering the significant hardness of the entailed optimization task?
\item In case we decide to embark on this challenging task of approximating arbitrary discrete energy, how can we tackle this problem? Can we propose new approaches and directions for the difficult approximation tasks of discrete energies, beyond semi-metric?
\end{enumerate}
These two research questions provide the road map of this thesis.
Consequently, this thesis revolves around two major axes:
applications and approximations.
\section*{Applications}
The first axis of this thesis involves exploring new applications that require arbitrary, contrast-enhancing energies beyond semi-metrics.
These examples demonstrate how the additional expressive power of arbitrary energies is crucial to derive new applications.
We show how utilizing arbitrary energies gives rise to interesting and desirable behaviors for different applications.
We present these new applications in Part~\ref{part:app} of this thesis.
Chapter~\ref{cp:negaff-sketch} shows an image sketching application that provides a binary sketch from a small collection of images of similar objects.
In this application the binary sketch is described via the interactions between neighboring pixels in the corresponding images.
Neighboring sketch bits corresponding to similar image pixels are encourage to have similar value (i.e., submodular, smoothness-enhancing term).
In contrast, neighboring sketch bits corresponding to {\em dis}similar image pixels are encourage to have {\em different} value (i.e., non-submodular, contrast-enhancing term).
The binary sketch is then the output of the resulting non-submodular energy minimization.
The sketching application may be thought of as a special case of binary image segmentation.
Considering contrast-enhancing objective function for the task of unsupervised segmentation or clustering may introduce a solution not only to the clustering problem, but also may help in determining the underlying number of clusters.
This clustering objective function is commonly known as ``Correlation Clustering" (\cite{Bansal2004}).
Chapter~\ref{cp:negaff-cc} explores the Correlation Clustering functional and its underlying ``model-selection" capability.
Image segmentation and clustering are not the only examples for contrast-enhancing energies.
Chapter~\ref{cp:lighting} describes a 3D surface reconstruction from multiple images under different known lighting.
The reconstruction takes into account the changes in appearance of the surface due to the change in lighting directions.
These changes amounts to an implicit partial differential equation (PDE) that describes the unknown surface.
In this work we propose to pose the solution of the resulting PDE as a discrete optimization task.
Incorporating integrability prior on the unknown surface, the resulting discrete energy has contrast-enhancing terms.
Modeling and prior knowledge are not the only sources for contrast-enhancing terms in energies.
In many cases, the exact parameters of an energy are learned from training data.
Chapter~\ref{cp:dtf} presents Decision Tree Fields (DTF): an example of such a learning scheme.
DTF learns, in a principled manner, a pair-wise energy from labeled training examples.
Since the learning process is not constraint to smoothness-encouraging energies,
it is often the case that the resulting energy has contrast-enhancing terms.
In its training phase DTF seeks parameters that (approximately) maximize the likelihood of the data.
Therefore, the resulting contrast-enhancing terms are better suited to describe the underlying ``behavior" of the data.
Restricting the model to smoothness-encouraging terms only would prohibit DTF from accurately predicting results at test time.
\section*{Approximate Optimization}
The enhancement in descriptive power gained by considering arbitrary energies comes with a price tag:
we no longer have good approximation algorithms at hand.
Therefore, the second axis of this thesis explores possible directions for approximating the resulting challenging arbitrary energies.
In part~\ref{part:approx} of this work I propose practical methods and approaches to approximate the resulting NP-hard optimization problems.
In particular, in Chapter~\ref{cp:CC}, I propose a discrete optimization approach to the aforementioned correlation clustering optimization.
This approach scales gracefully with the number of variables, better than existing approaches (e.g., \cite{Vitaladevuni2010}).
Chapter~\ref{cp:multiscale} concludes this part with a more general perspective on discrete optimization.
This new perspective is inspired by multiscale approaches and suggests to cope with the NP-hardness of discrete optimization using the {\em multiscale landscape} of the energy function.
Defining and observing this multiscale landscape of the energy, I propose methods to explore and exploit it to derive a coarse-to-fine optimization framework.
This new perspective gives rise to a unified multiscale framework for discrete optimization.
Our proposed multiscale approach is applicable to a diversity of discrete energies, both smoothness-encouraging as well as arbitrary, contrast-enhancing functions.
\section{Introduction}
Moving objects change their appearance as they change their orientation with respect to a light source. Such changes make it difficult to identify corresponding points across images, and as a result complicate the task of motion recovery and subsequently also of 3D reconstruction. Existing approaches to motion analysis largely assume either brightness constancy, or rely on extracting distinctive features. These approaches successfully handle both polygonal shapes and objects with noticeable surface markings, but they are less suitable for handling objects with smooth shapes and gradual albedo variations.
In this paper we propose a framework for reconstructing the shape of such objects that explicitly takes lighting effects into account. Our work is based on the observation that the changes in both the location and intensity of a moving object across images are coupled by its motion. Therefore, these two sources of information can be combined to constrain the matching of points across images, allowing us to achieve a veridical reconstruction. A number of previous studies (see Section~\ref{sec:prev-work}) used this observation to propose algorithms for shape recovery either in multi-view settings or in restricted settings (single directional light source and small motion,~\cite{BasriFrolova}) when only two views are available. In this study we propose a method that can work with two or more images and that can deal with general lighting conditions and fairly large motion.
We consider the case of an object that moves rigidly with respect to the camera and the light source. We assume that the object is lambertian, and that both the lighting conditions and the motion parameters are known. We present two algorithms. First we address the problem of shape reconstruction from two views when lighting is composed of a single directional source, and derive a partial differential equation (PDE) that can be solved to recover the shape of the object using continuation (characteristic curve). We further show how we can derive boundary conditions for this method. This formulation extends the work of~\cite{BasriFrolova} to objects undergoing a large motion.
Our second algorithm uses the PDE formulation to construct a cost function on a graph, representing the domain to be reconstructed. The cost function takes the form of a Markov Random Field (MRF). We then use off-the-shelf algorithms to solve for the sought shape. The MRF formulation offers several advantages over previous work. (1) It can work with fairly general reflectance functions. (2) It can be applied to pairs of images as well as to sequences of three or more images. (3) It is more robust to errors than continuation solutions, which can accumulate errors in integration. Finally, (4) prior information can be incorporated; in particular, the method can use boundary conditions, but as we show in our experiments boundary conditions are not essential. Experiments with real smooth objects demonstrate the utility of our formulation.
The paper is divided as follows. Section~\ref{sec:prev-work} provides a brief summary of related work. Section~\ref{sec:continuation} defines the reconstruction problem and derives a continuation solution in the case of large motion. Section~\ref{sec:MRF} casts the problem as an MRF optimization and discusses its solution by discrete techniques. Section~\ref{sec:bc} explains how we compute boundary conditions, and Section~\ref{sec:light-experiments} shows the results of our experiments.
\section{Previous work} \label{sec:prev-work}
The majority of 3D reconstruction techniques use either motion or shading techniques, but rarely combine both cues. Shape from shading~\cite{ShapeFromShadingBOOK} and photometric stereo~\cite{Woodham_1980} utilize shading cues to reconstruct objects from single or multiple images of static objects. Motion is often handled with the assumption of brightness constancy~\cite{Horn:optical-flow,Lucas1981} or by matching sparse feature points (e.g.,~\cite{Crandall}). Some authors proposed to reconstruct static objects by combining shading with stereo cues~\cite{Cryer_Tsai_Shah,FuaLeclerc_StereoShading_IJCV95}. Note that in a stereo setup lambertian objects retain their intensities across images. Another set of studies seek to generalize the brightness constancy assumption to account for local lighting variations~\cite{BlackFleet_ChangesInAppearance_CVIU00,Negahdaripour_OpticalFlow_PAMI1998,HausseckerFleet_OpticalFlow_PAMI2001}.
Studies that use motion and shading cues simultaneously~\cite{CarceroniKutulakos_SurfelSampling_ICCV01,Jin&etal2008,Maki_Watanabe_Wiles,Mukawa_ShapeReflectionIllumination_ICCV90,Simakov_Frolova_Basri,Weber_Blake_Cipolla,Zhang&etal,Joshi_Kriegman_2007,ChenChenHung,Moses_Shimshoni_2006}. typically require a multi-frame setting (typically at least 4 images). These studies also estimate the light source direction along with the shape of the object. An exception is~\cite{BasriFrolova}, which requires only two images, but assumes small motion.
Our work improves over these methods by proposing reconstruction methods that can handle large motion and can work both with image pairs and with larger sequences of images. In addition, our second algorithm allows for general lighting settings, which include point and extended light sources, through the use of spherical harmonic representations~(\cite{BasriJacobsLinSubsp,RamamoorthiHanrahanDetermIllum}).
\section{Problem definition and solution by continuation} \label{sec:continuation}
We consider a pair of images taken by a stationary camera. The images depict a lambertian object moving rigidly in space and illuminated by constant lighting. Denote these images by $I(x,y)$ and $J(x,y)$. Let ${\bf P}=(x,y,z(x,y))$ be a point on the object's surface described in the coordinate frame of $I$, and let ${\bf p}$ and ${\bf q}$ denote its projections onto $I$ and $J$. Assuming a weak perspective projection ${\bf p}=(x,y)$, and ${\bf q}=s R {\bf P} + {\bf t}$ in $J$, where the scale $s>0$, rotation $R$ (represented by a $2 \times 3$ matrix with orthonormal rows) and translation ${\bf t} \in \Re^2$ describe the (known) motion of the object. In general, since the motion is assumed to be known the location of ${\bf q}$ depends on the unknown depth value of ${\bf P}$, $z$. We therefore often emphasize this dependence by writing ${\bf q}(z)$.
Denote the normal to the surface at ${\bf p}$ in $I$ by $\hat {\bf n} \in S^2$,
\[ \hat {\bf n} = \frac{1}{\sqrt{z_x + z_y + 1}} (-z_x, -z_y, 1), \]
where $z_x=\partial z/\partial x$, $z_y=\partial z/\partial y$, and denote the albedo at $p$ by $\rho \in \Re$. Thus, the normal at ${\bf q}(z)$ is given by $R\hat {\bf n}$. We assume that the intensity of ${\bf P}$ in the two images can be expressed as $I({\bf p}) = \rho r(\hat {\bf n})$ and $J({\bf q}) = \rho r(R\hat {\bf n})$, where $r:S^2 \rightarrow \Re$, commonly referred to as the {\em reflectance function}, is a (known) function of the surface normal. This expression can be used to model a variety of materials. In particular, it can model lambertian objects illuminated by a directional (``point'') source by setting $r(\hat {\bf n})=\max({\bf l}^T \hat {\bf n},0)$. Here the direction of the point source is given by ${\bf l}/\|{\bf l}\|$ and its intensity by $\|{\bf l}\|$. Likewise, under more general lighting conditions (that may include multiple point and extended sources) $r(\hat {\bf n})$ can be expressed as an inner product between a vector of coefficients ${\bf b} \in \Re^d$ and the spherical harmonic functions evaluated at $\hat {\bf n}$, denoted $Y(\hat {\bf n}) \in \Re^d$. $d$ is set to either 4 or 9 if respectively the first or second order harmonic approximation is used, see~\cite{BasriJacobsLinSubsp} for details of this model.
Under these assumptions we can eliminate the albedo $\rho$ by taking the ratio between the intensities of corresponding points, namely
\begin{equation} \label{eq:main}
\frac{J({\bf q}(z))}{I({\bf p})} = \frac{r(R\hat {\bf n})}{r(\hat {\bf n})}.
\end{equation}
This PDE captures the relation between the shape of the moving object, its motion and the environment lighting. The shape of the object is captured both implicitly by the correspondence between ${\bf p}$ and ${\bf q}(z)$ (in other words, this equation is implicit in $z$), and explicitly by the surface normal (i.e., the partial derivatives of $z$).
For a lambertian object illuminated by a single directional light source ${\bf l}$ Eq.~\eqref{eq:main} can be written as follows. Let ${\bf n}=(-z_x,-z_y,1)^T$ (so that $\hat{\bf n}={\bf n}/\|{\bf n}\|$) then
\begin{equation} \label{eq:point-source}
\frac{J({\bf q}(z))}{I({\bf p})} = \frac{{\bf l}^T R {\bf n}}{{\bf l}^T {\bf n}}.
\end{equation}
Note that in this equation the nonlinear term $\|{\bf n}\|$ cancels out. Rearranging this equation we obtain
\begin{equation} \label{eq:main-point-source}
{\bf a}^T {\bf n} = 0,
\end{equation}
where ${\bf a}(x,y,z) = J({\bf q}(z)) {\bf l} - I({\bf p}) R^T {\bf l}$.
The equation is quasi-linear (i.e., linear in the derivatives $z_x$ and $z_y$), although implicit in $z$
\cite{BasriFrolova} made this equation explicit in $z$ by using a Taylor approximation for $J({\bf q}(z))$ and used the obtained expression to recover the shape of Lambertian objects illuminated by a single directional source. As their method approximates~\eqref{eq:main} to first order, it can handle only objects undergoing very small motion. Specifically, assuming $I$ and $J$ are rectified such that their epipolar lines are horizontal, they showed that~\eqref{eq:main} can be written as an explicit, quasilinear PDE
\begin{equation} \label{eq:1st-order}
a z_x + b z_y = c,
\end{equation}
where
\begin{eqnarray*} \vspace{-2mm}
a(x,y,z) &=& l_1 (I_\theta - z J_x) - l_3 I\\
b(x,y,z) &=& l_2 (I_\theta - z J_x)\\
c(x,y,z) &=& l_3 (I_\theta - z J_x) + l_1 I.
\end{eqnarray*}
In this equation $J_x=\partial J/\partial x$, $I_\theta=(J-I)/\theta$ and $\theta$ denotes the angle of rotation about the vertical axis. They further described a solution to~\eqref{eq:1st-order} using the method of continuation (characteristic curves) and showed a method to extract Dirichlet boundary conditions along the bounding contour of an object.
Our first contribution is to note that Basri and Frolova's algorithm can be extended also to handle large motion. This is because~\eqref{eq:main-point-source} is quasi-linear even without the Taylor approximation, and so it too can be solved by the method of continuation. Note that, due to~\eqref{eq:main-point-source}, ${\bf a}$ lies on the tangent to the surface $z(x,y)$ at ${\bf p}$. Therefore, any curve $\gamma(t) \subset \Re^3$ whose tangent at every $t \in [0,1]$ is given by ${\bf a}(\gamma(t))$ is characteristic to~\eqref{eq:main-point-source}, and if $\gamma(0)$ happens to lie on $z(x,y)$ the entire curve will lie on $z(x,y)$. To recover $z(x,y)$ the continuation method traces a family of such curves $\{\gamma(t)\}$ by integrating the vectors ${\bf a}(\gamma(t))$ starting from a 1D set of 3D points given as Dirichlet boundary conditions. Unfortunately, as~\eqref{eq:main-point-source} is implicit in $z$, extracting boundary conditions can be complicated; we suggest a method to do this in Section~\ref{sec:bc}.
Finally, we note that the characteristic curves traced with this procedure are all plane curves that lie in parallel planes. This can be readily seen by noticing that ${\bf a}$ is a linear combination of ${\bf l}$ and $R{\bf l}$, and so ${\bf a}^T ({\bf l} \times R{\bf l}) = 0$. in general, unless ${\bf l}$ points in the direction of the axis of rotation (denoted ${\bf u}$) these parallel planes are perpendicular to ${\bf u}$ and are {\em independent} of ${\bf l}$. Note however that generally the planes that contain the characteristics do not coincide with the epipolar planes except when the axis of rotation is parallel to the image plane. In the case that ${\bf l}$ coincides with ${\bf u}$,~\eqref{eq:point-source} becomes degenerate since its right hand side (substituting ${\bf u}$ for ${\bf l}$) is 1. Consequently, $J({\bf q}(z))=I{\bf p}$, and methods that assume constant brightness can be applied. Figure~\ref{fig:cont} shows a reconstruction obtained with our suggested method for continuation. A toy model was rendered using point source light and rotated by angle of about $4^\circ$. The continuation method was applied with exact boundary condition values.
\begin{figure}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
\includegraphics[width=0.25\linewidth]{lighting/Hippo_I.png} &
\includegraphics[width=0.25\linewidth]{lighting/hippo_cont_gt_c_r.pdf} &
\includegraphics[width=0.25\linewidth]{lighting/hippo_cont_imp_c_r.pdf} \\
\includegraphics[width=0.25\linewidth]{lighting/Hippo_J.png} &
\includegraphics[width=0.25\linewidth]{lighting/hippo_cont_gt_map_c_r.pdf}&
\includegraphics[width=0.25\linewidth]{lighting/hippo_cont_imp_map_c_r.pdf} \\
\hline
\end{tabular}
\end{center}
\caption{Reconstructions from rendered images of a toy hippo, using continuation. The left column shows the original images (rotation of $4^\circ$), the middle column shows the 3D model and the right column shows the reconstructed surface. The top row shows the surface (colormap represents depth values) and the bottom row shows the surface painted with intensity values.}
\label{fig:cont}
\end{figure}
\section{Discrete optimization} \label{sec:MRF}
The continuation method suffers from several shortcomings. First, the method accumulates errors in the integration. Moreover,~\eqref{eq:main} is quasi-linear when the lighting is composed of a single directional light source, but is non-linear when more realistic lighting models are used. Continuation can in principle be applied also for non-linear equations, but it is then significantly less robust. In addition, the method relies on boundary conditions that are difficult to compute, and at the same time does not allow for the inclusion of other prior information. Finally, it does not provide a way to integrate information from more than two images when more images of the object are available.
To overcome all of these shortcomings we cast the problem as an MRF optimization and solve it using discrete optimization techniques. Our objective is to find a valid surface $z(x,y)$ that satisfies~\eqref{eq:main}. We therefore define a cost function composed of unary and binary terms. Our MRF is defined over a grid overlaid on the first image $I(x,y)$. Each grid point $(x,y)$ is associated with a state vector determining the parameters of the surface element ({\em surfel}) observed at that pixel. These parameters include both the depth value and the surface normal, i.e., $(z,z_x,z_y)$, of the surfel. Note that such state vectors cannot represent points along the silhouette contour as the derivatives of $z$ at these points diverge. Our cost function combines unary and binary terms over the grid points. The unary term is a function of the residual of~\eqref{eq:main}. Specifically, let
\begin{equation} \label{eq:T}
T = I({\bf p}) r(R \hat {\bf n}) - J({\bf q}) r(\hat {\bf n}).
\end{equation}
We define the unary term as
\begin{equation} \label{eq:light-unary}
E^u_{\bf p} = \exp(\alpha T^2)-1,
\end{equation}
where $\alpha$ is a constant ($\alpha=8$ in our experiments). The binary term is set to prefer integrable surfaces. For ${\bf p}=(x,y)$ we use
\begin{equation} \label{eq:binary}
E^b_{\bf p} = (z(x,y) + z_x(x,y) - z(x+1,y))^2 + (z(x,y) + z_y(x,y) - z(x,y+1))^2.
\end{equation}
Our final energy function is given by
\begin{equation} \label{eq:energy}
E = \sum_{{\bf p} \in \Omega_I} E^u_{\bf p} + \lambda E^b_{\bf p},
\end{equation}
where $\Omega_I$ denotes the silhouette of the object in $I$ and $\lambda$ is constant. (We used values around 0.001.)
In general, when two images are used for reconstruction we need to supply boundary conditions in the form of $z(x,y)$ along a 1D curve $\gamma_0$. In Section~\ref{sec:bc} we describe a technique to estimate the depths near the bounding contours of the object. This procedure provides both the depth values and the normals along a 1D contour in $\Omega_I$. Given these boundary conditions we simply modify the unary cost along this contour to vanish in states $(z,z_x,z_y)$ that coincide with the depth values and normals predicted by the boundary conditions. As the extraction of boundary conditions can be noisy we then let the optimization modify these depth values. When the optimization is applied to three or more images the additional images further constrain the solution, and so boundary conditions are not necessary.
The binary term is clearly non-submodular. We therefore optimize~\eqref{eq:energy} by a sequence of alpha expansion iterations implemented with the QBPO algorithm~\cite{Rother2007}. Each iteration solves a binary decision problem in which each variable is allowed to either change its current state to a fixed state $(z^\alpha,z^\alpha_x,z^\alpha_y)$ or retain its current state. For submodular energies such an iteration is implemented using the st-mincut algorithm~\cite{KolmogorovZabih,Boykov2001}. The QPBO algorithm extends the st-mincut algorithm to handle negative edge capacities arising in the case of non-submodular pairwise terms. The algorithm constructs a graph with two complimentary vertices for each variable, explicitly representing its two possible binary states. By using a redundant variable representation, the resulting graph can be constructed to have positive capacities in all of its edges, and therefore the st-mincut algorithm can be applied to it. The following rule is used to label the variables at the end of the st-mincut step. Each variable for which its two complimentary vertices fall in different sides of the cut is labeled according to the assignment induced by the cut. Variables for which both complimentary vertices fall in the same side of the cut remain unlabeled. As minimizing non-submodular energies generally is NP-hard, QBPO is not guaranteed to label all the variables after each st-mincut step, and some of the variables may remain unlabeled. These unlabeled variables retain their original label from the previous iteration. This procedure is repeated for every possible choice of a label $(z^\alpha,z^\alpha_x,z^\alpha_y)$. The entire batch of iterations is then repeated until convergence. The optimization is initialized by setting the initial state $(z,z_x,z_y)$ for every point to the state that minimizes the unary term~\eqref{eq:light-unary}.
The discrete optimization solver is followed by a continuous quadratic optimization, in order to relax the quantized labeling (discrete values) of the surface. We solve the following quadratic optimization functional, maintaining the integrability constraint while remaining close to the quantized solution:
\begin{equation}
Ec{z} = \sum_{{\bf p} \in \Omega_I} (z(x,y) - \tilde{z}(x,y))^2 + \mu E^b_{\bf p}
\end{equation}
where $\tilde{z}(x,y)$ is the solution obtained by the discrete optimization, an
$\mu = 5$ in our experiments.
Finally, note that this procedure can readily be applied also when the lighting changes between images, by using different reflectance functions for $I$ and $J$ in~\eqref{eq:T}.
\section{Boundary conditions} \label{sec:bc}
Using the differential relation~\eqref{eq:main} to reconstruct a shape from two images requires boundary conditions in the form of depth values -- one depth value is required along each characteristic curve. This is essential for the continuation method and can be useful for the discrete optimization framework to further constrain the solution. Obtaining depth values can be complicated, as they require finding the correspondence between pixels on a smooth objects. This was bypassed in~\cite{BasriFrolova} since under a small motion and with the Taylor expansion~\eqref{eq:1st-order} becomes explicit in $z$, allowing one to compute the depths at the silhouette contours of the object from the intensities along the contour. Below we describe a method that computes depth values near the silhouette contours when $z$ is implicit.
Denote the object silhouette in $I$ and $J$ respectively by $\Omega_I$ and $\Omega_J$. We further denote by $\partial_v \Omega_I$ the portion of the silhouette contour in $I$ that remains visible in $J$ and likewise by $\partial_v \Omega_J$ the portion of the silhouette contour in $J$ that remains visible in $I$. Let ${\bf P}$ be a rim point in the coordinate frame of $I$ such that its projection onto $I$, denoted ${\bf p}$, lies on $\partial_v \Omega_I$. Since ${\bf p}$ lies on the bounding contour of $\Omega_I$ the surface normal at ${\bf P}$ should be parallel to the normal to the occluding contour at ${\bf p}$, and so it can be derived from the image. Let $\hat {\bf n}({\bf p})=(\cos\beta,\sin\beta,0)^T$. Plugging this normal in~\eqref{eq:main} gives us a scalar for the right hand side, and using the known value $I({\bf p})$ we compute the anticipated intensity at the corresponding point $J({\bf q})$. We can then use this intensity to search along the epipolar line on $J$ to determine ${\bf q}$. This for itself can be complicated, since there may be multiple points in along the epipolar line in $J$ with the anticipated intensity, but in addition we face two problems: (1) the intensity values along the bounding contour are unreliable, and (2) neither the continuation nor our discrete optimization scheme can use normals that are parallel to the image plane.
To overcome these problems we move from ${\bf p}$ one pixel in the direction of the normal into a point ${\bf p}'$ inside $\Omega_I$. We assume that the surface normal at ${\bf p}'$ can be expressed by $\hat {\bf n}({\bf p}') = (\cos\beta\cos\psi,\sin\beta\cos\psi,\sin\psi)^T$ for some unknown angle $\psi$. Our aim then is to find a point ${\bf q}'$ in $J$ along the epipolar line of ${\bf p}'$, denoted $e({\bf p}'_i)$, that satisfies~\eqref{eq:main} with this normal. This equation has two unknowns, the corresponding point ${\bf q}'$ in Image $J$ and $\psi$. We therefore regularize the problem by requiring the curve near $\partial_v \Omega_I$ to be mapped to a continuous curve in $J$.
We define the following optimization function.
\begin{eqnarray*}
\min_{\{{\bf q}'_i,\psi_i \}} \sum_{{\bf p}_i \in \partial_v \Omega_I} T({\bf p}'_i,{\bf q}'_i,\psi_i)^2 + \mu_1 \|{\bf q}'_i -{\bf q}_i^0\|^2 + \mu_2 \mathrm{dist}(e({\bf p}'_i),{\bf q}'_i)^2 + \\
\mu_3 \|{\bf q}'_i-{\bf q}_{i-1}'\|^2 + \mu_4 (\cos\psi_i - \cos\psi_{i-1})^2,
\end{eqnarray*}
where $T(.)$ denotes the residual defined in~\eqref{eq:T} for a pair of points ${\bf p}'_i$ and ${\bf q}'_i$ and normal $\hat {\bf n}({\bf p}')$ parameterized by $\psi$, and $\mathrm{dist}(e({\bf p}'_i),{\bf q}'_i)$ denotes the Euclidean distance between the point ${\bf q}'_i$ and the epipolar line $e({\bf p}'_i)$. For ${\bf q}_i^0$ we use the point along $e({\bf p}_i)$ closes to the boundary (at the same side as ${\bf p}$) that has the intensity anticipated from~\eqref{eq:main}. This heuristic is useful with reasonable rotations. To solve this minimization we substitute for $J({\bf q}'_i)$ in $T({\bf q}'_i,\psi_i)$~\eqref{eq:T} its Taylor expansion around ${\bf q}_i^0$. The resulting system of equations is non-linear and we solve it by alternate minimization.
We repeat this procedure for $\partial_v \Omega_J$ to obtain boundary conditions from both sides of the object's silhouette. Figure~\ref{fig:bc} shows an example for boundary conditions obtained with our method.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.45\linewidth]{lighting/Image_I_BC.png}~
\includegraphics[width=0.45\linewidth]{lighting/Image_J_BC.png}
\end{center}
\caption{The figure shows boundary conditions computed for two images of a bear toy. The red curve segments depict boundary points ${\bf p}'$ in the left image and their selected corresponding points in the right image. The yellow curve segments depict correspondences computed in the opposite direction.}
\label{fig:bc}
\end{figure}
\section{Experiments} \label{sec:light-experiments}
We tested our algorithm on two sets of real images and compared them to reconstructions with laser scans. Each image is taken with dark background to allow segmentation of the foreground object. In each image we estimated the motion parameters by manually marking points on the image and the 3D mesh object. As the objects are smooth and contain almost no clear surface markings our motion estimates are not perfect. Subsequently, using the mesh we estimated the environment lighting conditions by calculating the 4 coefficients of the first order harmonic representation of the lighting, representing the ambient lighting and a directional source. The obtained motion and lighting parameters were used as input to our software. The figures below show reconstructions obtained with our optimization algorithm (Section~\ref{sec:MRF}). We measure the quality of the matches obtained using the RMSE error in pixels. The figures show input images of a bear and elephant toys, reconstructions from pairs of images with and without boundary conditions (Section~\ref{sec:bc}), and reconstructions from multiple images. Our experiments cover a range of rotations between $4^\circ$ to about $21^\circ$. It can be seen that for the tested our reconstructions achieve low RMSE values. Interestingly, there is no noticeable advantage to using boundary conditions.
\section{Conclusion}
We described methods for reconstructing the shape of a moving object that accounts for intensities changes due to a change in orientation with respect to the light sources. In particular, we presented a continuation method and a method based on discrete optimization. Our experiments demonstrates the utility of our methods. The setting requires knowledge of the motion and lighting parameters. We plan in the future to seek ways to relax those limiting assumptions.
\begin{figure}
\begin{center}
\includegraphics[width=0.60\linewidth]{lighting/bears_angles.pdf}
\end{center}
\caption{Input images of a bear toy. A laser scan is shown at the center, and relative viewing angles are provided.}
\label{fig:bear}
\end{figure}
\begin{figure}
\begin{tabular}{|c|c|}
\hline
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_03_02_0_gt_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_03_02_0_imp_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_03_02_1_imp_c_r.pdf} &
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_08_07_0_gt_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_08_07_0_imp_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_08_07_1_imp_c_r.pdf} \\
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_03_02_0_gt_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_03_02_0_imp_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_03_02_1_imp_map_c_r.pdf} &
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_08_07_0_gt_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_08_07_0_imp_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear_tiff_08_07_1_imp_map_c_r.pdf} \\
\hline
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_05_06_0_gt_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_05_06_0_imp_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_05_06_1_imp_c_r.pdf} &
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_04_06_0_gt_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_04_06_0_imp_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_04_06_1_imp_c_r.pdf} \\
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_05_06_0_gt_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_05_06_0_imp_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_05_06_1_imp_map_c_r.pdf} &
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_04_06_0_gt_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_04_06_0_imp_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Bear1_tiff_04_06_1_imp_map_c_r.pdf} \\
\hline
\end{tabular}
\caption{Reconstructions from image pairs. Each box shows from left to right the laser scan, reconstruction with and without boundary conditions. The top row in each box shows the surface (colormap represents depth values) and the bottom row shows the surface painted with intensity values. Reconstructions are shown for images e and g ($9.3^\circ$) both achieving RMSE of 1.30 pixels (top left box), images d and f ($10.1^\circ$) both achieving RMSE of 1.52 pixels (top right), a and b ($12.1^\circ$) achieving 2.24 and 2.20 pixels, and c and e ($16.3^\circ$) achieving 3.89 and 3.87.}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=0.60\linewidth]{lighting/elephant_angles.pdf}
\end{center}
\caption{Input images of a elephant toy. A laser scan is shown at the center, and relative viewing angles are provided.}
\end{figure}
\begin{figure}
\begin{tabular}{|c|c|}
\hline
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_02_03_0_gt_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_02_03_0_imp_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_02_03_1_imp_c_r.pdf} &
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_03_02_0_gt_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_03_02_0_imp_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_03_02_1_imp_c_r.pdf} \\
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_02_03_0_gt_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_02_03_0_imp_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_02_03_1_imp_map_c_r.pdf} &
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_03_02_0_gt_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_03_02_0_imp_map_c_r.pdf}
\includegraphics[width=0.15\linewidth]{lighting/Elephant_tiff_03_02_1_imp_map_c_r.pdf} \\
\hline
\end{tabular}
\caption{Reconstructions from image pairs. Each box shows from left to right the laser scan, reconstruction with and without boundary conditions. The top row in each box shows the surface (colormap represents depth values) and the bottom row shows the surface painted with intensity values. Reconstructions are shown for images f and b ($7.5^\circ$) achieving RMSE 2.14 and 2.12 (top left box) and images b and f ($7.5^\circ$) achieving RMSE 1.93 and 1.94 (top right box)}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{c|ccc}
GT (e) & d,e,f & c,d,e,f & c,d,e,f,g \\\hline
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_78__gt_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_78__imp_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_478__imp_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_2478__imp_c_r.pdf}\\
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_78__gt_map_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_78__imp_map_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_478__imp_map_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_2478__imp_map_c_r.pdf}\\\hline
RMSE: & 0.88 & 1.28 & 1.39 \\\hline\hline
GT (e) & d,e,g & c,d,e,g & d,e,f,g \\\hline
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_78__gt_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_28__imp_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_248__imp_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_278__imp_c_r.pdf}\\
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_78__gt_map_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_28__imp_map_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_248__imp_map_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/BearMI_tiff_3_278__imp_map_c_r.pdf}\\\hline
RMSE: & 0.88 & 1.40 & 0.83 \\
\end{tabular}
\caption{Reconstruction from multiple images. Ground truth is shown at left, image labels at the top and RMSE (over all images) at the bottom.}
\label{fig:mi-bear-e2}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{c|ccc}
GT (b) & b,c,f & b,c,d,f & a,b,c,d,e,f \\\hline
\includegraphics[width=.2\linewidth]{lighting/ElephantMI_tiff_3_1256__gt_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/ElephantMI_tiff_3_12__imp_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/ElephantMI_tiff_3_126__imp_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/ElephantMI_tiff_3_12456__imp_c_r.pdf}\\
\includegraphics[width=.2\linewidth]{lighting/ElephantMI_tiff_3_1256__gt_map_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/ElephantMI_tiff_3_12__imp_map_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/ElephantMI_tiff_3_126__imp_map_c_r.pdf}&
\includegraphics[width=.2\linewidth]{lighting/ElephantMI_tiff_3_12456__imp_map_c_r.pdf}\\\hline
Correspondence error:& 1.78 & 1.44 & 1.69\\
\end{tabular}
\caption{Reconstruction from multiple images. Ground truth is shown at left, image labels at the top and RMSE (over all images) at the bottom.}
\label{fig:mi-elephant-b}
\end{figure}
\begin{figure}
\centering
\begin{tabular}{c|cc}
GT (d) & b,c,d,e,f & a,b,c,d,e,f \\\hline
\includegraphics[width=.23\linewidth]{lighting/ElephantMI_tiff_6_12345__gt_c_r.pdf}&
\includegraphics[width=.23\linewidth]{lighting/ElephantMI_tiff_6_1235__imp_c_r.pdf}&
\includegraphics[width=.23\linewidth]{lighting/ElephantMI_tiff_6_12345__imp_c_r.pdf}\\
\includegraphics[width=.23\linewidth]{lighting/ElephantMI_tiff_6_12345__gt_map_c_r.pdf}&
\includegraphics[width=.23\linewidth]{lighting/ElephantMI_tiff_6_1235__imp_map_c_r.pdf}&
\includegraphics[width=.23\linewidth]{lighting/ElephantMI_tiff_6_12345__imp_map_c_r.pdf}\\\hline
Correspondence error:& 0.69 & 1.30 \\
\end{tabular}
\caption{Reconstruction from multiple images. Ground truth is shown at left, image labels at the top and RMSE (over all images) at the bottom.}
\label{fig:mi-elephant-d}
\end{figure}
\section{Introduction}
\begin{figure}[t]
\centering
\framebox[\linewidth][c]{
\parbox{\linewidth}{
\centering
\includegraphics[width=.75\linewidth]{ms/our_multiscale_scheme.pdf}
}}
\vspace*{1mm}
\caption{
{\bf A Unified multiscale framework:}
{\em
We derive multiscale representation of the energy itself = energy pyramid.
Our multiscale framework is unified in the sense that different problems with different energies share the same multiscale scheme, making our framework widely applicable and general.
}}
\label{fig:multiscale-schemes}
\end{figure}
Discrete energy minimization is ubiquitous in computer vision, and spans a variety of problems such as segmentation, denoising, stereo, etc.
Unfortunately, apart from the submodular binary case, minimizing these energies is known to be NP-hard.
A lot of effort is recently put on developing algorithms for approximate discrete optimization for ever more challenging energies: multi-label, non-submodular, etc. (e.g., \cite{Szeliski2008,Kolmogorov2006,Bagon2012}).
Discrete energies may be grossly divided into two categories: submodular (regular) energies and non-submodular energies.
Submodular energies are characterized by smoothness-encouraging pair-wise (and higher order) terms.
These energies reflect the ``piece-wise constant" prior that is very popular and common in computer vision applications.
For this reason most of the effort and research regarding discrete optimization, in the context of computer vision, focuses on these energies with encouraging results.
In practice, despite the NP-hardness of these energies, algorithms were developed that provide solutions with energies close to global optimum (e.g., \cite{Kolmogorov2006,Boykov2001}).
Therefore we consider this type of energies as ``easy to optimize".
In contrast, non-submodular energies are characterized by contrast-encouraging pair wise terms.
These energies may be encountered when the parameters of the energy are learned (e.g., \cite{Nowozin2011}),
or when different functionals are used (e.g., \cite{Bagon2012,Glasner2011}).
When it comes to optimization it is generally considered a more challenging task to optimize a non-submodular energies.
Since these examples of non-submodular energies are only recently explored, their optimization receives less attention, and consequently, the existing optimization methods provide approximations that may be quite unsatisfactory.
We consider these energies as ``hard to optimize".
Algorithms for discrete energy minimization may also be classified into two categories: primal methods and dual methods.
Primal methods act directly on the discrete variables in the label space to minimize the energy (e.g., \cite{Besag1986,Boykov2001}).
In contrast, dual methods formulate a dual problem to the energy and maximize a lower bound to the sought energy (e.g., \cite{Kolmogorov2006}).
Dual methods are recently considered more favorable since they not only provide an approximate solution, but also provide a lower bound on how far this solution is from the global optimum.
Furthermore, if a labeling is found with energy equals to the lower bound a certificate is provided that the global optimum was found.
Since most of the relevant discrete optimization problems are NP-hard, one can only provide an {\em empirical} evaluation of how well a given algorithm approximates representative instances of these energies.
For the submodular, ``easy to optimize", energies it was shown (by \cite{Szeliski2008}) that dual methods tend to provide better approximations with very tight lower bounds.
\parpic[r][r]{\includegraphics[width=.45\linewidth]{ms/multiscale_landscape.pdf}}
But what makes discrete energy minimization such a challenging endeavor?
The fact that this minimization
implies an exploration of an exponentially large search space makes it such a challenging task.
One way to alleviate this difficulty is to use multiscale search.
The illustration on the right shows
a toy ``energy" $E(L)$ at different scales of detail.
Considering only the original scale ($s=0$), it is very difficult to suggest an effective exploration/optimization method.
However, when looking at coarser scales ($s=1,\ldots,3$) of the energy an interesting phenomenon is revealed.
At the coarsest scale ($s=3$) the large basin of attraction emerges, but with very low accuracy.
As the scales become finer ($s=2,\ldots,0$), one ``loses sight" of the large basin, but may now ``sense" more local properties with higher accuracy.
We term this well known phenomenon as the {\em multiscale landscape} of the energy.
This multiscale landscape phenomenon encourages coarse-to-fine exploration strategies:
starting with the large basins that are apparent at coarse scales,
and then gradually and locally refining the search at finer scales.
For more than three decades the vision community has focused on the multiscale pyramid of the {\em image} (e.g., \cite{Lucas1981,Burt1983}).
There is almost no experience and no methods that
apply a multiscale scheme directly to the discrete energy.
In this paper we present a novel unified discrete multiscale optimization scheme that acts {\em directly} on the energy
(Fig.~\ref{fig:multiscale-schemes}).
Our approach allows for an efficient exploration of the discrete solution space through the construction of an energy pyramid.
Moreover, our multiscale framework is application independent:
different problems with different energies {\em share the same} multiscale scheme, making our framework widely applicable and general.
Performing empirical evaluations of non-submodular energies minimization lead us to conclude that when it comes to hard to optimize non-submodular energies, primal methods tend to provide better approximations than dual methods.
Motivated by this observation, we formulate out multiscale framework in the primal space (i.e., expressing it in terms of the variables and labels directly).
Our multiscale framework becomes the core of the optimization process allowing for existing ``off-the-shelf" primal optimization algorithms to efficiently exploit the multiscale landscape of the energy and achieves significantly lower energies faster.
This work makes several contributions:
\renewcommand{\labelenumi}{(\roman{enumi})}
\begin{enumerate}
\item A novel unified multiscale framework for discrete optimization.
A wide variety of optimization problems, including segmentation, stereo, denoising, correlation-clustering, and others share the same multiscale framework.
\item Energy-aware coarsening scheme.
Variable aggregation takes into account the underlying structure of the energy itself, thus efficiently and directly exposes its multiscale landscape.
\item Coarsening the labels.
Our formulation allows for variable aggregation as well as for label coarsening.
This yields an energy pyramid with fewer {\em labels} at the coarser scales.
\item Integrating existing single-scale optimization algorithms into our multiscale framework.
We achieve significantly lower energy assignments on diverse computer vision energies, including challenging non-submodular examples.
\item Optimizing hard non-submodular energies.
Using several classes of non-submodular energies, we empirically exemplify the superiority of primal methods.
We further show how combining in our multiscale framework single-scale primal optimization methods achieve increased optimization performance on these challenging problems.
\end{enumerate}
\renewcommand{\labelenumi}{\arabic{enumi}.}
\subsection{Related work}
There are very few works that apply multiscale schemes directly to the energy.
A prominent example for this approach is that of \cite{Felzenszwalb2006}, that provide a coarse-to-fine belief propagation scheme restricted to regular diadic pyramid.
A more recent work is that of \cite{Komodakis2010} that provides an algebraic multigrid formulation for discrete optimization in the dual space.
However, despite his general formulation \citeauthor{Komodakis2010} only provides examples using regular diadic grids of easy to optimize submodular energies.
The work of \cite{Kim2011} proposes a two-scales scheme mainly aimed at improving run-time of the optimization process.
Their proposed coarsening strategies can be interpreted as special cases of our unified framework.
We analyze their underlying assumptions (Sec.~\ref{sec:local-correlations}), and suggest better methods for efficient exploration of the multiscale landscape of the energy.
A different approach for discrete optimization suggests
large move making algorithms (e.g., \cite{Boykov2001,Swendsen1987}).
We experimentally show how plugging such methods within our multiscale framework improves optimization results.
These methods do not scale gracefully with the number of labels.
\cite{Lempitsky2007} proposed a method to exploit known properties of the metric between the labels to allow for faster minimization of the energy.
However, their method is restricted to energies with clear and known label metrics and requires training.
In contrast, our framework addresses this issue via a principled scheme that builds an energy pyramid with a decreasing number of {\em labels} without prior training and with fewer assumptions on the labels interactions.
\section{Multiscale Energy Pyramid}
\label{sec:unified}
In this work we consider discrete pair-wise minimization problems, defined over a (weighted) graph $\left(\mbox{$\mathcal{V}$}, \mbox{$\mathcal{E}$}\right)$, of the form:
\begin{eqnarray}
E\left(L\right)&=&\sum_{i\in\mbox{$\mathcal{V}$}} \psi_i\left(l_i\right) + \sum_{\left(i,j\right)\in\mbox{$\mathcal{E}$}} w_{ij}\cdot \psi\left(l_i,l_j\right) \label{eq:GenEng}
\end{eqnarray}
where $\mbox{$\mathcal{V}$}$ is the set of variables, $\mbox{$\mathcal{E}$}$ is the set of edges, and the solution is discrete: $L\in\left\{1,\ldots,l\right\}^n$, with $n$ variables taking $l$ possible labels.
Many problems in computer vision are cast in the form of~(\ref{eq:GenEng}) (see \cite{Szeliski2008}).
Furthermore, we do not restrict the energy to be submodular, and our framework is
also applicable to more challenging non-submodular energies.
Our aim is to build an energy pyramid with a decreasing number of degrees of freedom.
The key component in constructing such a pyramid is the interpolation method.
The interpolation maps solutions between levels of the pyramid,
and defines how to approximate the original energy with fewer degrees of freedom.
We propose a novel principled energy aware interpolation method.
The resulting energy pyramid exposes the multiscale landscape of the energy making low energy assignments apparent at coarse levels.
However, it is counter intuitive to directly interpolate discrete values,
since they usually have only semantic interpretation.
Therefore, we substitute an assignment $L$
by a binary matrix $U\in\left\{0,1\right\}^{n\times l}$.
The rows of $U$ correspond to the variables, and the columns corresponds to labels:
$U_{i,\alpha}=1$ iff variable $i$ is labeled ``$\alpha$" ($l_i=\alpha$).
This representation allows us to interpolate discrete solutions, as will be shown in the subsequent sections.
Expressing the energy (\ref{eq:GenEng}) using $U$ yields a relaxed quadratic representation (along the lines of \cite{Anand2000}) that forms the basis for our multiscale framework derivation:
\begin{eqnarray}
E\left(U\right)&=&Tr\left(DU^T+WUVU^T\right) \label{eq:EngU} \\
& \mbox{s.t.} & U\in\left\{0,1\right\}^{n\times l},\ \sum_{\alpha=1}^l U_{i\alpha}=1 \label{eq:const-U}
\end{eqnarray}
where $W$ is sparse with entries $\left\{w_{ij}\right\}$, $D\in\mathbb{R}^{n\times l}$ s.t. $D_{i,\alpha}\mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} \psi_i(\alpha)$, and $V\in\mathbb{R}^{l\times l}$ s.t. $V_{\alpha,\beta}\mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} \psi\left(\alpha,\beta\right)$, $\alpha,\beta\in\left\{1,\ldots,l\right\}$.
A detailed derivation of~(\ref{eq:EngU}) can be found in Sec.~\ref{sec:ms-deriv-U}.
An energy over $n$ variables with $l$ labels is now parameterized by $\left(n, l , D, W, V\right)$.
We first describe the energy pyramid construction for {\em a given} interpolation matrix $P$,
and defer the detailed description of our novel interpolation to Sec.~\ref{sec:matrix-P}.
\subsubsection*{Energy coarsening by variables}
Let $\left(n^f, l, D^f, W^f, V\right)$ be the fine scale energy.
We wish to generate a coarser representation $\left(n^c, l, D^c, W^c, V\right)$ with $n^c<n^f$.
This representation approximates $E\left(U^f\right)$ using fewer {\em variables}: $U^c$ with only $n^c$ rows.
Given an interpolation matrix $P\in\left[0,1\right]^{{n^f}\times{n^c}}$ s.t. $\sum_jP_{ij}=1$ $\forall i$, it maps coarse to fine assignments through:
\begin{eqnarray}
U^f & \approx & PU^c \label{eq:interp}
\end{eqnarray}
For any fine assignment that can be approximated by a coarse assignment $U^c$
we may plug (\ref{eq:interp}) into~(\ref{eq:EngU}) yielding:
\begin{eqnarray}
E\left(U^f\right) & = & Tr\left(D^f{U^f}^T+W^fU^fV{U^f}^T\right) \nonumber \\
& \approx & Tr\left(D^f{U^c}^TP^T+W^fPU^cV{U^c}^TP^T\right) \nonumber \\
& = & Tr\Big(\underbrace{\left(P^TD^f\right)}_{\mbox{\normalsize $ \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} D^c$ }}{U^c}^T + \underbrace{\left(P^TW^fP\right)}_{\mbox{\normalsize $\mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} W^c$}}U^cV{U^c}^T\Big) \nonumber \\
& = & Tr\left(D^c{U^c}^T+W^cU^cV{U^c}^T\right) \nonumber \\
& = & E\left(U^c\right) \label{eq:EngC}
\end{eqnarray}
We have generated a coarse energy $E\left(U^c\right)$
parameterized by $\left(n^c, l, D^c, W^c, V\right)$ that approximates the fine energy $E(U^f)$.
This coarse energy is {\em of the same form} as the original energy allowing us to apply the coarsening procedure recursively to construct an energy pyramid.
\subsubsection*{Energy coarsening by labels}
So far we have explored the reduction of the number of degrees of freedom by reducing the number of {\em variables}.
However, we may just as well look at the problem from a different perspective: reducing the search space by decreasing the number of {\em labels} from $l_f$ to $l_c$ ($l_c<l_f$).
It is a well known fact that optimization algorithms (especially large move making, e.g., \cite{Boykov2001}) suffer from significant degradation in performance as the number of {\em labels} increases (\cite{Bleyer2010}).
Here we propose a novel principled and general framework for reducing the number of labels at each scale.
Let $\left(n, l^f, D^{\hat{f}}, W, V^{\hat{f}}\right)$ be the fine scale energy.
Looking at a different interpolation matrix $\hat{P}\in\left[0,1\right]^{\mbox{$l^f\times l^c$}}$,
we may interpolate a coarse solution by $U^{\hat{f}} \approx U^{\hat{c}}\hat{P}^T$.
This time the interpolation matrix $\hat{P}$ acts on the {\em labels}, i.e., the {\em columns} of $U$.
The coarse labeling matrix $U^{\hat{c}}$ has the same number of rows (variables), but fewer columns (labels).
We use $\hat{\Box}$
notation to emphasize that the coarsening here affects the labels rather than the variables.
Coarsening the labels yields:
\begin{equation}
E\left(U^{\hat{c}}\right) = Tr\left( \left(D^{\hat{f}}\hat{P}\right)\mbox{$U^{\hat{c}}$}^T + WU^{\hat{c}} \left(\hat{P}^TV^{\hat{f}}\hat{P}\right)\mbox{$U^{\hat{c}}$}^T\right)
\label{eq:EngC-V}
\end{equation}
Again, we end up with the same type of energy, but this time it is defined over a smaller number of discrete labels:
$\left(n, l^c, D^{\hat{c}}, W, V^{\hat{c}}\right)$,
where $D^{\hat{c}} \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} D^{\hat{f}}\hat{P}$ and $V^{\hat{c}} \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} \hat{P}^T V^{\hat{f}} \hat{P}$.
\
The main theoretical contribution of this work is encapsulated in the
multiscale ``trick" of equations~(\ref{eq:EngC}) and~(\ref{eq:EngC-V}).
This formulation forms the basis of our unified framework allowing us to coarsen the energy {\em directly} and exploits its multiscale landscape for efficient exploration of the solution space.
This scheme moves the multiscale completely to the optimization side and makes it independent of any specific application.
We can practically now approach a wide and diverse family of energies using {\em the same} multiscale implementation.
The effectiveness of the multiscale approximation of~(\ref{eq:EngC}) and~(\ref{eq:EngC-V}) heavily depends on the interpolation matrix $P$ ($\hat{P}$ resp.).
Poorly constructed interpolation matrices will fail to expose the multiscale landscape of the functional.
In the subsequent section we
describe our principled energy-aware method for computing it.
\section{Energy-aware Interpolation}
\label{sec:matrix-P}
\begin{figure}
\centering
\includegraphics[width=.4\linewidth]{ms/multiscale_P.pdf}
\caption{
{\bf Interpolation as soft variable aggregation:}
{\em {\color{fine}fine} variable {\color{fine}1}, {\color{fine}2} and {\color{fine}4} are aggregated into {\color{coarse}coarse} variable {\color{coarse}1}, while {\color{fine}fine} variables {\color{fine}1},{\color{fine}3} and {\color{fine}4} are aggregated into {\color{coarse}coarse} variable {\color{coarse}2}. Soft aggregation allows for {\color{fine}fine} variables to be influenced by few {\color{coarse}coarse} variables,
e.g.: {\color{fine}fine} variable {\color{fine}1} is a convex combination of $.7$ of {\color{coarse}1} and $.3$ of {\color{coarse}2}.
Hard aggregation is a special case where $P$ is a binary matrix.
In that case each fine variable is influenced by exactly one coarse variable.}
}
\label{fig:multiscale}
\end{figure}
In this section we use terms and notations for variable coarsening ($P$),
however the motivation and methods are applicable for label coarsening ($\hat{P}$) as well due to the similar algebraic structure of~(\ref{eq:EngC}) and~(\ref{eq:EngC-V}).
Our energy pyramid approximates the original energy using a decreasing number of degrees of freedom,
thus excluding some solutions from the original search space at coarser scales.
Which solutions are excluded is determined by the interpolation matrix $P$.
{\bf A desired interpolation does not exclude low energy assignments at coarse levels}.
The matrix $P$ can be interpreted as an operator that aggregates fine-scale variables into coarse ones (Fig.~\ref{fig:multiscale}).
Aggregating fine variables $i$ and $j$ into a coarser one excludes from the search space all assignments for which $l_i\ne l_j$.
This aggregation is undesired if assigning $i$ and $j$ to different labels yields low energy.
However, when variables $i$ and $j$ are {\em strongly correlated} by the energy (i.e., assignments with $l_i=l_j$ yield low energy),
aggregating them together efficiently allows exploration of low energy assignments.
{\bf A desired interpolation aggregates $i$ and $j$ when $i$ and $j$ are strongly correlated by the energy}.
\subsection{Measuring energy-aware correlations}
\label{sec:local-correlations}
We provide two correlations measures, one used in computing variable coarsening ($P$) and the other used for label coarsening ($\hat{P}$).
\noindent{\bf Energy-aware correlations between variables:}
A reliable estimation for the correlations between the variables allows us to construct a desirable $P$ that aggregates strongly correlated variables.
A na\"{\i}ve approach would assume that neighboring variables are correlated (this assumption underlies \cite{Felzenszwalb2006}). This assumption clearly does not hold in general and may lead to an undesired interpolation matrix $P$.
\cite{Kim2011} proposed several ``closed form formulas" for energy-aware variable grouping.
However, their formulas take into account either the unary term or the pair-wise term.
Indeed it is difficult to decide which term dominates and how to fuse these two terms together.
Therefore, there is no ``closed form" method that successfully integrates both of them.
As opposed to these ``closed form" methods, we propose a novel empirical scheme for correlation estimation.
Empirical estimation of the correlations naturally accounts for and integrates the influence of both the unary and the pair-wise terms.
Moreover, our method, inspired by \cite{Ron2011,Livne2011}, extends to all energies (\ref{eq:EngU}): submodular, non-submodular, metric $V$, arbitrary $V$, arbitrary $W$, energies defined over regular grids and arbitrary graphs.
Variables $i$ and $j$ are correlated by the energy when $l_i=l_j$ yields relatively low energy value.
To estimate these correlations we empirically generate several ``locally" low energy assignments,
and measure the label agreement between neighboring variables $i$ and $j$.
We use Iterated Conditional Modes (ICM) of \cite{Besag1986} to obtain locally low energy assignments:
Starting with a random assignment, ICM chooses, at each iteration and for each variable, the label
yielding the largest decrease of the energy function, conditioned on the labels assigned to its neighbors.
Performing $t=10$ ICM iterations for $K=10$ random initializations provides $K$ locally low energy assignments $\left\{L^k\right\}_{k=1}^K$.
Our empirical dissimilarity between $i$ and $j$ is given by $d_{ij}=\frac{1}{K}\sum_k V_{l^k_i,l^k_j}$,
and their correlation is given by $c_{ij}=\exp\left(-\frac{d_{ij}}{\sigma}\right)$,
with $\sigma \propto \max V$.
It is interesting to note that strong correlation between variables $i$ and $j$ usually implies that the pair-wise term binding them together ($\varphi_{ij}$) is a smoothness-preserving type of relation.
We assume that even for challenging energies with many contrast-enhancing pair-wise terms, there are still significant amount of smoothness-preserving terms to allow for effective coarsening.
\noindent{\bf Energy-aware correlations between labels:}
Correlations between labels are easier to estimate, since this information is explicit in the matrix $V$ that encodes the ``cost" (i.e., dissimilarity) between two labels.
Setting
$\hat{c}_{\alpha,\beta}\propto \left(\hat{V}_{\alpha,\beta}\right)^{-1}$,
we get a ``closed-form" expression for the correlations between labels.
\subsection{From correlations to interpolation}
\label{sec:amg-p}
Using our measure for the variable correlations, $c_{ij}$, we follow the Algebraic Multigrid (AMG) method of \cite{Brandt1986} to compute an interpolation matrix $P$ that softly aggregates strongly correlated variables.
We begin by selecting a set of coarse representative variables $\mbox{$\mathcal{V}$}^c\subset \mbox{$\mathcal{V}$}^f$,
such that every variable in $\mbox{$\mathcal{V}$}^f \backslash \mbox{$\mathcal{V}$}^c$ is strongly correlated with $\mbox{$\mathcal{V}$}^c$.
That is, every variable in $\mbox{$\mathcal{V}$}^f$ is either in $\mbox{$\mathcal{V}$}^c$ or is {\em strongly correlated} to other variables in $\mbox{$\mathcal{V}$}^c$.
A variable $i$ is considered strongly correlated to $\mbox{$\mathcal{V}$}^c$ if $\sum_{j\in\mbox{$\mathcal{V}$}^c}c_{ij} \ge \beta \sum_{j\in\mbox{$\mathcal{V}$}^f} c_{ij}$.
$\beta$ affects the coarsening rate, i.e., the ratio $n^c/n^f$,
smaller $\beta$ results in a lower ratio.
We perform this selection greedily and sequentially, starting with $\mbox{$\mathcal{V}$}^c=\emptyset$ adding $i$ to $\mbox{$\mathcal{V}$}^c$ if it is not yet strongly correlated to $\mbox{$\mathcal{V}$}^c$.
Given the selected coarse variables $\mbox{$\mathcal{V}$}^c$,
$I(j)$ maps indices of variables from fine to coarse:
$I(j)$ is the coarse index of the variable whose fine index is $j$ (in Fig.~\ref{fig:multiscale}: $I(2)=1$ and $I(3)=2$).
The interpolation matrix $P$ is defined by:
\begin{equation}
P_{iI(j)} = \left\{
\begin{array}{cl}
c_{ij} & i\in\mbox{$\mathcal{V}$}^f\backslash\mbox{$\mathcal{V}$}^c,\ j\in\mbox{$\mathcal{V}$}^c\\
1 & i\in\mbox{$\mathcal{V}$}^c, j=i\\
0 & \mbox{otherwise}\\
\end{array}
\right. \label{eq:entries-of-P}
\end{equation}
We further prune rows of $P$ leaving only $\delta$ maximal entries.
Each row is then normalized to sum to 1.
Throughout our experiments we use $\beta=0.2$ ($\hat{\beta}=0.75$), $\delta=3$ ($\hat{\delta}=2$) for computing $P$ ($\hat{P}$ resp.).
\section{Unified Discrete Multiscale Framework}
\label{sec:pipeline}
\begin{algorithm}[t]
\caption{Discrete multiscale optimization. \label{alg:multiscale}}
\DontPrintSemicolon
\SetKw{KwInit}{Init}
\SetKw{KwOpt}{Refine}
\SetKw{KwCoarse}{Coarsen}
\KwIn{Energy $\left(\mbox{$\mathcal{V}$}^0, D^0, W^0, V\right)$.}
\KwOut{$U^0$}
\KwInit{$s\leftarrow 0$}\tcp{fine scale}
\tcp{Energy pyramid construction:}
\While{$\abs{\mbox{$\mathcal{V}$}^s} \ge 10$} {
Estimate pair-wise correlations $c_{ij}$ at scale $s$ (Sec.~\ref{sec:local-correlations}).\;
Compute interpolation matrix $P^s$ (Sec.~\ref{sec:amg-p}).\;
Derive coarse energy $\left(\mbox{$\mathcal{V}$}^{s+1}, D^{s+1}, W^{s+1}, V\right)$ (Eq.~\ref{eq:EngC}).\;
$s++$\;
}
\tcp{Coarse-to-fine optimization:}
\While{$s\ge0$} {
$U^s\leftarrow$ \KwOpt{$(\tilde{U}^s)$}\;
$\tilde{U}^{s-1} = P^sU^s$\tcp{interpolate a solution}\label{line:refine}
$s--$\;
}
where \KwOpt{$(\tilde{U}^s)$} uses an ``off-the-shelf" algorithm to optimize the energy $\left(\mbox{$\mathcal{V}$}^{s}, D^{s}, W^{s}, V\right)$ with $\tilde{U}^s$ as an initialization.\;
\end{algorithm}
So far we have described the different components of our multiscale framework.
Alg.~\ref{alg:multiscale} puts them together into a multiscale minimization scheme.
Given an energy $\left(\mbox{$\mathcal{V}$}, D, W, V\right)$,
our framework first works fine-to-coarse to compute interpolation matrices $\left\{P^s\right\}$ that generates an ``energy pyramid".
Typically we end up at the coarsest scale with less than $10$ variables.
As a result, exploring the energy at this scale is robust to the initial assignment of the single-scale method used\footnote{In practice we use ``winner-take-all" initialization as suggested by \cite[\S3.1]{Szeliski2008}.}.
Starting from the coarsest scale, a coarse solution at scale $s$ is interpolated to a finer scale $s-1$.
At the finer scale it serves as a good initialization for an ``off-the-shelf" single-scale optimization that refines this interpolated solution.
These two steps are repeated for all scales from coarse to fine.
The interpolated solution $\tilde{U}^{s-1}$, at each scale,
might not satisfy the binary constraints~(\ref{eq:const-U}).
We round each row of $\tilde{U}^{s-1}$ by setting the maximal element to $1$ and the rest to $0$.
The most computationally intensive modules
of our framework are the empirical estimation of the variable correlations and the single-scale optimization used to refine the interpolated solutions.
The complexity of the correlation estimation is $O\left(\abs{\mbox{$\mathcal{E}$}}\cdot l\right)$, where $\abs{\mbox{$\mathcal{E}$}}$ is the number of non-zero elements in $W$ and $l$ is the number of labels.
However, it is fairly straightforward to parallelize this module.
It is now easy to see how our framework generalizes \cite{Felzenszwalb2006}, \cite{Komodakis2010} and \cite{Kim2011}:
They are restricted to hard aggregation in $P$.
\cite{Felzenszwalb2006} and \cite{Komodakis2010} use a multiscale pyramid, however their variable aggregation is not energy-aware, and is restricted to diadic pyramids.
On the other hand, \cite{Kim2011} have limited energy-aware aggregation, applied to a two level only ``pyramid".
They only optimize at the coarse scale and cannot refine the solution on the fine scale.
\section{Experimental Results}
\label{sec:ms-results}
Our experiments has two main goals: first, to stress the difficulty of approximating non-submodular energies and to show the advantages of primal methods for this type of minimization problems.
The other goal is to demonstrate how our unified multiscale framework improved the performance of existing single-scale primal methods.
We evaluated our multiscale framework on a diversity of discrete optimization tasks\footnote{code available at \url{www.wisdom.weizmann.ac.il/~bagon/matlab.html}.}: ranging from challenging non-submodular synthetic and co-clustering energies to low-level submodular vision energies such as denoising and stereo.
In addition we provide a comparison between the different methods for measuring variable correlations that were presented in Sec.~\ref{sec:local-correlations}. We conclude with a label coarsening experiment.
In all of these experiments we minimize a {\em given} publicly available benchmark energy,
{\em we do not attempt to improve on the energy formulation itself}.
We use ICM (\cite{Besag1986}), $\alpha\beta$-swap and $\alpha$-expansion (large move making algorithms of \cite{Boykov2001}) as representative single-scale ``off-the-shelf" primal optimization algorithms.
To help large move making algorithms to overcome the non-submodularity of some of these energies we augment them with QPBO(I) of \cite{Rother2007}.
We follow the protocol of \cite{Szeliski2008} that uses the {\em lower bound} of TRW-S (\cite{Kolmogorov2006}) as a baseline for comparing the performance of different optimization methods for different energies.
We report how close the results (in percents) to the lower bound: {\bf closer to $100\%$ is better}.
We show a remarkable improvement for ICM combined in our multiscale framework compared with a single-scale scheme.
For the large move making algorithms there is a smaller but consistent improvement of the multiscale over a single scale scheme.
TRW-S is a dual method and is considered state-of-the-art for discrete energy minimization \cite{Szeliski2008}.
However, we show that when it comes to non-submodular energies it struggles behind the large move making algorithms and even ICM.
For these challenging energies, multiscale gives a significant boost in optimization performance.
\begin{table}
\centering
\setlength{\tabcolsep}{1mm}
\begin{tabular}{c||c|c||c|c||c|c||c}
\multirow{3}{*}{$\lambda$} & \multicolumn{2}{c||}{ICM} & \multicolumn{2}{c||}{Swap} & \multicolumn{2}{c||}{Expand} & \multirow{2}{*}{TRW-S} \\
& \multirow{2}{*}{{\color{ours}Ours}} & single & \multirow{2}{*}{{\color{ours}Ours}} &single & \multirow{2}{*}{{\color{ours}Ours}} & single & \\
& & scale & & scale & & scale & \\\hline \hline
$5$ & {\color{ours}$112.6\%$} & $115.9\%$ & {\color{ours}$108.9\%$} & $110.0\%$ & {\color{ours}$110.5\%$} & $110.0\%$ & $116.6\%$ \\
$10$ & {\color{ours}$123.6\%$} & $130.2\%$ & {\color{ours}$118.5\%$} & $120.2\%$ & {\color{ours}$121.5\%$} & $121.0\%$ & $134.6\%$ \\
$15$ & {\color{ours}$127.1\%$} & $135.8\%$ & {\color{ours}$122.1\%$} & $124.1\%$ & {\color{ours}$124.6\%$} & $125.1\%$ & $138.3\%$ \\
\end{tabular}
\caption{ {\bf Synthetic results (energy):}
{\em
Showing percent of achieved energy value relative to the lower bound (closer to $100\%$ is better) for ICM, $\alpha\beta$-swap, $\alpha$-expansion and TRW-S
for varying strengths of the pair-wise term ($\lambda=5,\ldots,15$, stronger $\rightarrow$ harder to optimize.)}
}
\setlength{\tabcolsep}{\origtabcolsep}
\label{tab:res-synthetic}
\end{table}
\subsection{Synthetic}
We begin with synthetic {\em non-submodular} energies defined over a 4-connected grid graph of size $50\times50$ ($n=2500$), and $l=5$ labels.
The unary term $D \sim \mathcal{N}\left(0,1\right)$.
The pair-wise term $V_{\alpha\beta}=V_{\beta\alpha} \sim \mathcal{U}\left(0, 1\right)$ ($V_{\alpha\alpha}=0$) and $w_{ij}=w_{ji} \sim \lambda \cdot \mathcal{U}\left(-1,1\right)$.
The parameter $\lambda$ controls the relative strength of the pair-wise term,
stronger (i.e., larger $\lambda$) results with energies more difficult to optimize (see \cite{Kolmogorov2006}).
Table~\ref{tab:res-synthetic} shows results, averaged over 100 experiments.
The resulting synthetic energies are non-submodular (since $w_{ij}$ may become negative).
For these challenging energies, state-of-the-art dual method (TRW-S) performs rather poorly\footnote{We did not restrict the number of iterations, and let TRW-S run until no further improvement to the lower bound is made.} (worse than single scale ICM) and there is a significant gap between the lower bound and the energy of the actual primal solution provided.
Among the primal methods used,
These results motivate our focusing on primal methods, especially $\alpha\beta$-swap.
\begin{figure}
\centering
\newlength{\cipwidth}
\setlength{\cipwidth}{.11\linewidth}
\setlength{\tabcolsep}{0.5mm}
\begin{tabular}{c|c||c|c|c|c||c|c}
\multirow{3}{*}{GT} & \multirow{3}{*}{Input} & \multicolumn{2}{c|}{ICM} & \multicolumn{2}{c||}{QPBO} &\multirow{3}{*}{TRW-S} & Sim. \\
& & \multirow{2}{*}{{\color{ours} Ours}} & single & \multirow{2}{*}{{\color{ours} Ours}} & single & & Ann.\\
& & & scale & & scale & & \\
\includegraphics[width=\cipwidth]{ms/014-gt-014.png}&
\includegraphics[width=\cipwidth]{ms/test_0014.jpg}&
\includegraphics[width=\cipwidth]{ms/014-icm-ms-014.png}&
\includegraphics[width=\cipwidth]{ms/014-icm-ss-014.png}&
\includegraphics[width=\cipwidth]{ms/014-qpbo-ms-014.png}&
\includegraphics[width=\cipwidth]{ms/014-qpbo-ss-014.png}&
\includegraphics[width=\cipwidth]{ms/014-trws-ss-014.png}&
\includegraphics[width=\cipwidth]{ms/014-ref-ss-014.png}\\
\includegraphics[width=\cipwidth]{ms/016-gt-016.png}&
\includegraphics[width=\cipwidth]{ms/test_0016.jpg}&
\includegraphics[width=\cipwidth]{ms/016-icm-ms-016.png}&
\includegraphics[width=\cipwidth]{ms/016-icm-ss-016.png}&
\includegraphics[width=\cipwidth]{ms/016-qpbo-ms-016.png}&
\includegraphics[width=\cipwidth]{ms/016-qpbo-ss-016.png}&
\includegraphics[width=\cipwidth]{ms/016-trws-ss-016.png}&
\includegraphics[width=\cipwidth]{ms/016-ref-ss-016.png}\\
\includegraphics[width=\cipwidth]{ms/023-gt-023.png}&
\includegraphics[width=\cipwidth]{ms/test_0023.jpg}&
\includegraphics[width=\cipwidth]{ms/023-icm-ms-023.png}&
\includegraphics[width=\cipwidth]{ms/023-icm-ss-023.png}&
\includegraphics[width=\cipwidth]{ms/023-qpbo-ms-023.png}&
\includegraphics[width=\cipwidth]{ms/023-qpbo-ss-023.png}&
\includegraphics[width=\cipwidth]{ms/023-trws-ss-023.png}&
\includegraphics[width=\cipwidth]{ms/023-ref-ss-023.png}\\
\includegraphics[width=\cipwidth]{ms/025-gt-025.png}&
\includegraphics[width=\cipwidth]{ms/test_0025.jpg}&
\includegraphics[width=\cipwidth]{ms/025-icm-ms-025.png}&
\includegraphics[width=\cipwidth]{ms/025-icm-ss-025.png}&
\includegraphics[width=\cipwidth]{ms/025-qpbo-ms-025.png}&
\includegraphics[width=\cipwidth]{ms/025-qpbo-ss-025.png}&
\includegraphics[width=\cipwidth]{ms/025-trws-ss-025.png}&
\includegraphics[width=\cipwidth]{ms/025-ref-ss-025.png}\\
\includegraphics[width=\cipwidth]{ms/033-gt-033.png}&
\includegraphics[width=\cipwidth]{ms/test_0033.jpg}&
\includegraphics[width=\cipwidth]{ms/033-icm-ms-033.png}&
\includegraphics[width=\cipwidth]{ms/033-icm-ss-033.png}&
\includegraphics[width=\cipwidth]{ms/033-qpbo-ms-033.png}&
\includegraphics[width=\cipwidth]{ms/033-qpbo-ss-033.png}&
\includegraphics[width=\cipwidth]{ms/033-trws-ss-033.png}&
\includegraphics[width=\cipwidth]{ms/033-ref-ss-033.png}\\
\includegraphics[width=\cipwidth]{ms/043-gt-043.png}&
\includegraphics[width=\cipwidth]{ms/test_0043.jpg}&
\includegraphics[width=\cipwidth]{ms/043-icm-ms-043.png}&
\includegraphics[width=\cipwidth]{ms/043-icm-ss-043.png}&
\includegraphics[width=\cipwidth]{ms/043-qpbo-ms-043.png}&
\includegraphics[width=\cipwidth]{ms/043-qpbo-ss-043.png}&
\includegraphics[width=\cipwidth]{ms/043-trws-ss-043.png}&
\includegraphics[width=\cipwidth]{ms/043-ref-ss-043.png}
\end{tabular}
\setlength{\tabcolsep}{\origtabcolsep}
\caption{{\bf Chinese characters inpainting:}
{\em Visualizing some of the instances used in our experiments.
Columns are (left to right):
The original character used for testing.
The input partially occluded character.
ICM and QPBO results both our multiscale and single scale results.
Results of TRW-S and results of \cite{Nowozin2011} obtained with a very long run of simulated annealing (using Gibbs sampling inside the annealing).}}
\label{fig:sebastian-dtf}
\end{figure}
\begin{figure}
\centering
\comment{
\begin{tabular}{cc}
\includegraphics[width=.45\linewidth]{ms/binary_dtf_energy_cropped.pdf}&
\includegraphics[width=.45\linewidth]{ms/binary_dtf_runtimes_cropped.pdf}\\
(a) & (b)
\end{tabular}
}
\includegraphics[width=.6\linewidth]{ms/binary_dtf_energy_cropped.pdf}
\caption{ {\bf Energies of Chinese characters inpainting:}
{\em Box plot showing 25\%, median and 75\% of the resulting energies relative to reference energies of \cite{Nowozin2011} (lower than $100\%$ = lower than baseline).
Our multiscale approach combined with QPBO achieves consistently better energies than baseline, with very low variance.
TRW-S improves on only 25\% of the instances with very high variance in the results.}
}
\label{fig:sebastian-dtf-graphs}
\end{figure}
\begin{table}
\centering
\begin{tabular}{c||c|c||c|c||c}
& \multicolumn{2}{c||}{ICM} & \multicolumn{2}{c||}{QPBO} & \multirow{2}{*}{TRW-S} \\
& {\color{ours}Ours} & single-scale & {\color{ours}Ours} & single-scale & \\\hline
(a) & {\color{ours}$114.0\%$} & $114.0\%$ & {\color{ours}$97.8\%$} & $106.2\%$ & $108.6\%$ \\\hline
(b) & {\color{ours}$7.0\%$} & $7.0\%$ &{\color{ours}$77.0\%$} & $34.0\%$ & $25.0\%$ \\\hline
\end{tabular}
\caption{{\bf Energies of Chinese characters inpainting:}
{\em table showing
(a) mean energies for the inpainting experiment relative to baseline of \cite{Nowozin2011} (lower is better, less than $100\%$ = lower than baseline).
(b) percent of instances for which strictly lower energy was achieved.
}}\label{tab:dtf-binary}
\end{table}
\subsection{Chinese character inpainting}
We further experiment with learned binary energies of \cite[\S5.2]{Nowozin2011}\footnote{available at \url{www.nowozin.net/sebastian/papers/DTF_CIP_instances.zip}.}.
These 100 instances of non-submodular pair-wise energies are defined over a 64-connected grid.
These energies were designed and trained to perform the task of learning Chinese calligraphy, represented as a complex, non-local binary pattern.
Despite the very large number of parameters involved in representing such complex energies, learning is conducted very efficiently using Decision Tree Field (DTF).
The main challenge in these models becomes the inference at test time.
Our experiments show how approaching these challenging energies using our unified multiscale framework allows for better approximations.
Table~\ref{tab:dtf-binary} and Fig.~\ref{fig:sebastian-dtf} compare our multiscale framework to single-scale methods acting on the primal binary variables.
Since the energies are binary, we use QPBO instead of large move making algorithms.
We also provide an evaluation of a dual method (TRW-S) on these energies.
In addition to the quantitative results, Fig.~\ref{fig:sebastian-dtf-graphs} provides a visualization of some of the instances of the restored Chinese characters.
These ``real world" energies highlight the advantage primal methods has over dual ones when it comes to challenging non-submodular energies.
It is further clear that significant improvement is made by our multiscale framework.
\begin{table}
\centering
\setlength{\tabcolsep}{.5mm}
\begin{tabular}{c||c|c||c|c||c|c||c}
& \multicolumn{2}{c||}{ICM} & \multicolumn{2}{c||}{Swap} & \multicolumn{2}{c||}{Expand} & TRW-S\\
& \multirow{2}{*}{{\color{ours}Ours}} & single & \multirow{2}{*}{{\color{ours}Ours}} & single & \multirow{2}{*}{{\color{ours}Ours}} & single & \\
& & scale & & scale & & scale & \\
\hline \hline
(a) & {\color{ours}$99.9\%$} & $177.7\%$ & {\color{ours}$99.8\%$} & $101.5\%$ & {\color{ours}$99.8\%$} & $101.6\%$ & $176.2\%$ \\\hline
(b) & {\color{ours}$55.6\%$} & $0.0\%$ & {\color{ours}$71.8\%$} & $15.5\%$ & {\color{ours}$70.8\%$} & $11.6\%$ & $0.5\%$ \\\hline
\comment{(c) & {\color{ours}$68.2$} & $0.8$ & {\color{ours}$1387.6$} & $11787.7$ & {\color{ours}$11676.4$} & $42084.5$ & $84.6$ \\\hline}
\end{tabular}
\caption{ {\bf Co-clustering results: }
{\em
Baseline for comparison are state-of-the-art results of \cite{Glasner2011}.
(a) We report our results as percent of the baseline: smaller is better, lower than $100\%$ even outperforms state-of-the-art.
(b) We also report the fraction of energies for which our multiscale framework outperform state-of-the-art.
\comment{(c) run times. pyramid construction $230.3$ milisec.}
} }
\label{tab:cocluster-res}
\end{table}
\subsection{Co-clustering}
The problem of co-clustering addresses the matching of superpixels within and across frames in a video sequence.
Following \cite[\S6.2]{Bagon2012}, we treat co-clustering as a discrete minimization of {\em non-submodular} Potts energy.
We obtained 77 co-clustering energies, courtesy of \cite{Glasner2011}, used in their experiments.
The number of variables in each energy ranges from 87 to 788.
Their sparsity (percent of non-zero entries in $W$) ranges from $6\%$ to $50\%$,
The resulting energies are non-submodular, have no underlying regular grid, and are very challenging to optimize \cite{Bagon2012}.
Table~\ref{tab:cocluster-res} compares our discrete multiscale framework combined with ICM and $\alpha\beta$-swap.
For these energies we use a different baseline: the state-of-the-art results of \cite{Glasner2011} obtained by applying specially tailored convex relaxation method
(We do not use the lower bound of TRW-S here since it is far from being tight for these challenging energies).
Our multiscale framework improves state-of-the-art for this family of challenging energies.
\subsection{semi-metric energies}
We further applied our multiscale framework to optimize less challenging semi-metric energies.
We use the diverse low-level vision MRF energies from the Middlebury benchmark \cite{Szeliski2008}\footnote{Available at \url{vision.middlebury.edu/MRF/}.}.
For these semi-metric energies, TRW-S (single scale) performs quite well and in fact, if enough iterations are allowed,
its lower bound converges to the global optimum.
As opposed to TRW-S, large move making and ICM do not always converge to the global optimum.
Yet, we are able to show a significant improvement for primal optimization algorithms when used within our multiscale framework.
Tables~\ref{tab:stereo-res} and~\ref{tab:denoise-res}
and
Figs.~\ref{fig:res-stereo} and~\ref{fig:res-denoise}
show our multiscale results for the different submodular energies.
One of the conclusions of the Middlebury challenge was that ICM is no longer a valid candidate for optimization.
Integrating ICM into our multiscale framework puts it back on the right track.
\comment{Table~\ref{tab:variable-runtime} exemplifies how our framework improved running times for two difficult energies (``Penguin" denoising and ``Venus" stereo).}
\begin{table}
\centering
\setlength{\tabcolsep}{.5mm}
\begin{tabular}{c||c|c||c|c||c|c}
& \multicolumn{2}{c||}{ICM} & \multicolumn{2}{c||}{Swap}& \multicolumn{2}{c}{Expand}\\
& {\color{ours}Ours} & single scale & {\color{ours}Ours} & single scale & {\color{ours}Ours} & single scale\\
\hline \hline
Tsukuba & {\color{ours}$102.8\%$} &$653.4\%$ &{\color{ours}$100.2\%$} &$100.5\%$ &{\color{ours}$100.1\%$} &$100.3\%$ \\ \hline
Venus & {\color{ours}$112.3\%$} &$405.1\%$ &{\color{ours}$102.8\%$} &$128.7\%$ &{\color{ours}$102.7\%$} &$102.8\%$ \\ \hline
Teddy & {\color{ours}$102.5\%$} &$234.3\%$ &{\color{ours}$100.4\%$} &$100.8\%$ &{\color{ours}$100.3\%$} &$100.5\%$ \\ \hline
\end{tabular}
\caption{ {\bf Stereo:}
{\em
Showing percent of achieved energy value relative to the lower bound (closer to $100\%$ is better).
Visual results for these experiments are in Fig.~\ref{fig:res-stereo}.
Energies from \cite{Szeliski2008}.}}
\label{tab:stereo-res}
\end{table}
\begin{figure}
\centering
\newlength{\stwidth}
\setlength{\stwidth}{.11\linewidth}
\begin{tabular}{cc||cc||cc||c}
\multicolumn{2}{c||}{ICM} & \multicolumn{2}{c||}{Swap} & \multicolumn{2}{c||}{Expand} & Ground \\
{\color{ours}Ours} & Single scale & {\color{ours}Ours} & Single scale & {\color{ours}Ours} & Single scale & truth \\ \hline
\includegraphics[width=\stwidth]{ms/tsu_icm_0.png}&
\includegraphics[width=\stwidth]{ms/tsu-ICM.png}&
\includegraphics[width=\stwidth]{ms/tsu_swap_0.png}&
\includegraphics[width=\stwidth]{ms/tsu-Swap.png}&
\includegraphics[width=\stwidth]{ms/tsu-EXPAND-9.png}&
\includegraphics[width=\stwidth]{ms/tsu-Expansion.png}&
\includegraphics[width=\stwidth]{ms/tsukuba-truedispL.png}\\
\includegraphics[width=\stwidth]{ms/ven_icm_1.png}&
\includegraphics[width=\stwidth]{ms/ven-ICM.png}&
\includegraphics[width=\stwidth]{ms/ven_swap_0.png}&
\includegraphics[width=\stwidth]{ms/ven-Swap.png}&
\includegraphics[width=\stwidth]{ms/ven-EXPAND-5.png}&
\includegraphics[width=\stwidth]{ms/ven-Expansion.png}&
\includegraphics[width=\stwidth]{ms/venus-truedispL.png}\\
\includegraphics[width=\stwidth]{ms/ted_icm_0.png}&
\includegraphics[width=\stwidth]{ms/ted-ICM.png}&
\includegraphics[width=\stwidth]{ms/ted_swap_0.png}&
\includegraphics[width=\stwidth]{ms/ted-Swap.png}&
\includegraphics[width=\stwidth]{ms/ted-EXPAND-8.png}&
\includegraphics[width=\stwidth]{ms/ted-Expansion.png}&
\includegraphics[width=\stwidth]{ms/teddy-truedispL.png}\\
\end{tabular}
\caption{ {\bf Stereo:}
{\em Note how our multiscale framework drastically improves ICM results.
visible improvement for $\alpha\beta$-swap can also be seen in the middle row (Venus).
Numerical results for these examples are shown in Table~\ref{tab:stereo-res}.
}}
\label{fig:res-stereo}
\end{figure}
\begin{table}
\centering
\setlength{\tabcolsep}{.5mm}
\begin{tabular}{c||c|c||c|c||c|c}
& \multicolumn{2}{c||}{ICM} & \multicolumn{2}{c||}{Swap} & \multicolumn{2}{c}{Expand}\\
& {\color{ours}Ours} & single scale & {\color{ours}Ours} & single scale & {\color{ours}Ours} & single scale\\
\hline \hline
House & {\color{ours}$100.5\%$} &$111.3\%$ &{\color{ours}$100.4\%$} &$100.9\%$ &{\color{ours}$102.3\%$} &$103.4\%$ \\ \hline
Penguin & {\color{ours}$106.9\%$} &$132.9\%$ &{\color{ours}$104.6\%$} &$111.3\%$ &{\color{ours}$104.0\%$} &$103.7\%$ \\ \hline
\end{tabular}
\caption{ {\bf Denoising and inpainting:}
{\em
Showing percent of achieved energy value relative to the lower bound (closer to $100\%$ is better).
Visual results for these experiments are in Fig.~\ref{fig:res-denoise}.
Energies from \cite{Szeliski2008}.} }
\label{tab:denoise-res}
\end{table}
\begin{figure}
\centering
\newlength{\dnwidth}
\setlength{\dnwidth}{.11\linewidth}
\begin{tabular}{c||cc||cc||cc}
Input & \multicolumn{2}{c||}{ICM} & \multicolumn{2}{c}{Swap} & \multicolumn{2}{c}{Expand}\\
& {\color{ours}Ours} & Single scale & {\color{ours}Ours} & Single scale & {\color{ours}Ours} & Single scale\\ \hline
\includegraphics[width=\dnwidth]{ms/house-input.png}&
\includegraphics[width=\dnwidth]{ms/houseM_icm_0.png}&
\includegraphics[width=\dnwidth]{ms/houseM-ICM.png}&
\includegraphics[width=\dnwidth]{ms/houseM_swap_0.png}&
\includegraphics[width=\dnwidth]{ms/houseM-Swap.png}&
\includegraphics[width=\dnwidth]{ms/house-EXPAND-2.png}&
\includegraphics[width=\dnwidth]{ms/houseM-Expansion.png}\\
\includegraphics[width=\dnwidth]{ms/penguin-bar.png}&
\includegraphics[width=\dnwidth]{ms/penguin_icm_0.png}&
\includegraphics[width=\dnwidth]{ms/penguin-ICM.png}&
\includegraphics[width=\dnwidth]{ms/penguin_swap_0.png}&
\includegraphics[width=\dnwidth]{ms/penguin-Swap.png}&
\includegraphics[width=\dnwidth]{ms/penguin-EXPAND-1.png}&
\includegraphics[width=\dnwidth]{ms/penguin-Expansion.png}\\
\end{tabular}
\caption{ {\bf Denoising and inpainting:}
{\em
Single scale ICM is unable to cope with inpainting: performing local steps it is unable to propagate information far enough to fill the missing regions in the images.
On the other hand, our multiscale framework allows ICM to perform large steps at coarse scales and successfully fill the gaps.
Numerical results for these examples are shown in Table~\ref{tab:denoise-res}.
}}
\label{fig:res-denoise}
\end{figure}
\comment{
\begin{table}
\centering
\begin{tabular}{c|c|c||c|c}
Energy & \#variables & \#variables & {\color{ours}Ours} & single \\
& (finest) & (coarsest) & {\color{ours}(multiscale)} & scale \\ \hline
Penguin & $21,838$ & $5$ & {\color{ours}$103.7\%$} & $111.3\%$ \\
& & & {\color{ours}$20.8$ + $95.1$} [sec] & $253.7$ [sec]\\ \hline
Venus & $166,222$ & $6$ & {\color{ours}$102.8\%$} & $128.7\%$ \\
& & & {\color{ours}$54.7$ + $30.7$} [sec] & $130.1$ [sec]\\
\end{tabular}
\caption{{\bf Running times for variable coarsening ($\alpha\beta$-swap):}
{\em Examples of typical running times (in seconds). For multiscale we report the runtime for constructing the pyramid and the overall time it took to optimize coarse-to-fine.
Note that the reported times are of our unoptimized serial Matlab implementation.}}
\label{tab:variable-runtime}
\end{table}
}
\subsection{Comparing variable correlation estimation methods}
As explained in Sec.~\ref{sec:matrix-P} the correlations between the variables are the most crucial component in constructing an effective multiscale scheme.
In this experiment we compare our energy-aware correlation measure (Sec.~\ref{sec:local-correlations}) to three methods proposed by \cite{Kim2011}: ``unary-diff", ``min-unary-diff" and ``mean-compat".
These methods estimate the correlations based either on the unary term or the pair-wise term, but {\em not both}.
We also compare to an energy-agnostic measure, that is $c_{ij}=1$ $\forall i,j$,
this method underlies \cite{Felzenszwalb2006}.
We use ICM within our framework to evaluate the influence these methods have on the resulting multiscale performance for four representative energies.
Fig.~\ref{fig:comapre-weights} shows percent of lower bound for the different energies.
Our measure consistently outperforms all other methods, and successfully balances between the influence of the unary and the pair-wise terms.
\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{ms/compare_weightings_icm_cropped.pdf}
\caption{{\bf Comparing correlation measures:}
{\em
Graphs showing percent of lower bound (closer to $100\%$ is better)
for different methods of computing variable-correlations.
Some of the bars are cropped at $\sim150\%$.
Our energy-aware measure consistently outperforms all other methods.
}}
\label{fig:comapre-weights}
\end{figure}
\subsection{Coarsening labels}
$\alpha\beta$-swap
does not scale gracefully with the number of labels.
Coarsening an energy in the labels domain (i.e., same number of variables, fewer labels) proves to significantly improve performance of $\alpha\beta$-swap, as shown in Table~\ref{tab:coarsening-v}.
For these examples constructing the energy pyramid took only milliseconds, due to the ``closed form" formula for estimating label correlations.
Our principled framework for coarsening labels improves $\alpha\beta$-swap performance for these energies.
\begin{table}
\centering
\begin{tabular}{c|c|c||c|c}
\multirow{2}{*}{Energy} & \#labels & \#labels & \multirow{2}{*}{{\color{ours}Ours}}& single \\
& (finest) & (coarsest) & & scale \\
\hline
Penguin & \multirow{2}{*}{256} & \multirow{2}{*}{67} & {\color{ours}$103.6\%$} & $111.3\%$ \\
(denoising) & & & {\color{ours}$128$} [sec] & $253$ [sec] \\ \hline
Venus & \multirow{2}{*}{20} & \multirow{2}{*}{4} & {\color{ours}$106.0\%$} & $128.7\%$ \\
(stereo) & & & {\color{ours}$100$} [sec] & $130$ [sec]\\
\end{tabular}
\caption{ {\bf Coarsening labels ($\alpha\beta$-swap):}
{\em
Working coarse-to-fine in the labels domain. We use 5 scales
with coarsening rate of $\sim0.7$.
Number of variables is unchanged.
Table shows percent of achieved energy value relative to the lower bound (closer to $100\%$ is better), and running times.}}
\label{tab:coarsening-v}
\end{table}
\section{Conclusion}
This work presents a unified multiscale framework for discrete energy minimization
that allows for efficient and {\em direct} exploration of the multiscale landscape of the energy.
We propose two paths to expose the multiscale landscape of the energy:
one in which coarser scales involve fewer and coarser {\em variables},
and another in which the coarser levels involve fewer {\em labels}.
We also propose adaptive methods for energy-aware interpolation between the scales.
Our multiscale framework significantly improves optimization results for challenging energies.
Our framework provides the mathematical formulation that ``bridges the gap" and relates multiscale discrete optimization and algebraic multiscale methods used in PDE solvers (e.g., \cite{Brandt1986}).
This connection allows for methods and practices developed for numerical solvers to be applied in multiscale discrete optimization as well.
\section*{Appendix}
\begin{subappendices}
\section{Derivation of eq.~(\ref{eq:EngU})}
\label{sec:ms-deriv-U}
In this work we consider discrete pair-wise minimization problems of the form:
~\hfill $E\left(L\right)=\sum_{i\in\mbox{$\mathcal{V}$}} \psi_i\left(l_i\right) + \sum_{\left(i,j\right)\in\mbox{$\mathcal{E}$}} w_{ij}\cdot \psi\left(l_i,l_j\right)$ \hfill (\ref{eq:GenEng})
Using the following parameterizations: $D\in\mathbb{R}^{n\times l}$ s.t. $D_{i,\alpha}\mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} \psi_i(\alpha)$, and $V\in\mathbb{R}^{l\times l}$ s.t. $V_{\alpha,\beta}\mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} \psi\left(\alpha,\beta\right)$
we claim that (\ref{eq:GenEng}) is equivalent to:
~\hfill $E\left(U\right)=Tr\left(DU^T+WUVU^T\right)$ \hfill (\ref{eq:EngU})
~\hfill s.t. $U\in\left\{0,1\right\}^{n\times l},\ \sum_{\alpha=1}^l U_{i\alpha}=1$ \hfill (\ref{eq:const-U})
Assuming both $V$ and $W$ are symmetric\footnote{if they are not we need to be slightly more careful with transposing them, but roughly similar expression can be derived.}.
Looking at the different components in (\ref{eq:EngU}):
\[
\left[VU^T\right]_{\alpha j} = \sum_\beta V_{\alpha \beta}U_{j \beta}
\]
\begin{eqnarray*}
\left[UVU^T\right]_{ij} &=& \sum_\alpha U_{i\alpha}\left[VU^T\right]_{\alpha j} \\
&=& \sum_\alpha U_{i\alpha} \sum_\beta V_{\alpha \beta}U_{j \beta}\\
&=& \sum_{\alpha\beta} V_{\alpha\beta}U_{i\alpha}U_{j\beta} \\
&=& \sum_{\alpha\beta} \psi\left(\alpha,\beta\right)U_{i\alpha}U_{j\beta} \\
&=& \psi\left(l_i, l_j\right)
\end{eqnarray*}
Looking at the trace of the second term:
\begin{eqnarray}
Tr\left(WUVU^T\right) &=& \sum_i \left[WUVU^T\right]_{ii} \nonumber\\
&=& \sum_i \sum_j w_{ij} \left[UVU^T\right]_{ji} \nonumber\\
&=& \sum_i \sum_j w_{ij} \psi\left(l_j, l_i\right) \nonumber\\
&=& \sum_{ij} w_{ij} \psi\left(l_i, l_j\right) \label{eq:pair-wise}
\end{eqnarray}
As for the unary term:
\begin{eqnarray}
Tr\left(DU^T\right) &=& \sum_i \left[DU^T\right]_{ii} \nonumber\\
&=& \sum_i \sum_\alpha D_{i\alpha}U_{i\alpha} \nonumber\\
&=& \sum_i \sum_\alpha \psi_i\left(\alpha\right)U_{i\alpha} \nonumber\\
&=& \sum_i \psi_i\left(l_i\right) \label{eq:ms-unary}
\end{eqnarray}
Putting (\ref{eq:ms-unary}) and (\ref{eq:pair-wise}) together we get:
\begin{eqnarray*}
Tr\left(DU^T+WUVU^T\right)&=&Tr\left(DU^T\right)+Tr\left(WUVU^T\right)\\
&=& \sum_i \psi_i\left(l_i\right) + \sum_{ij} w_{ij} \psi\left(l_i, l_j\right) \quad \square
\end{eqnarray*}
Note that the diagonal of $W$ is assumed to be zero: this is a reasonable assumption as $w_{ii}$ represents an interaction of variable $i$ with itself.
This type of interaction is well represented by the unary term $D_i$.
When coarsening the energy it may happen that $W^c$ will no longer have zeros on the diagonal.
This case may arise when a single coarse variable represents neighboring fine scale variables.
In that case the fine scale pair-wise interaction should be absorbed into the coarse scale unary term.
It is easy to see that the term $w_{ii}V_{\alpha\alpha}$ should be added to the unary term $D_{i\alpha}$, whenever $W$ has non zeros on the diagonal.
After this rectification, the non-zeros entries on the diagonal of $W$ can be set to zero.
\section{More General Energy Functions}
This chapter~(\ref{cp:multiscale}) has focused on the construction of an energy pyramid for energies of the form~(\ref{eq:GenEng}).
However, this form does not cover all possible pair-wise energies.
We have used this slightly restricted form throughout the paper since it emphasizes the simplicity of the algebraic derivation of our multiscale framework.
Nevertheless, our multiscale framework can be as easily applied to more general energies.
\subsection{Pair-wise}
A general from for pair-wise energy over a graph $\mbox{$\mathcal{G}$}=\left(\mbox{$\mathcal{V}$},\mbox{$\mathcal{E}$}\right)$ can be written as
\begin{equation}
E\left(L\right) = \sum_{i\in\mbox{$\mathcal{V}$}} \varphi_i\left(l_i\right) + \sum_{\left(i,j\right)\in\mbox{$\mathcal{E}$}} \varphi_{ij}\left(l_i,l_j\right)
\label{eq:gen-pair-energy}
\end{equation}
In this more general form, the pair-wise term can be entirely different for each pair: $\varphi_{ij}\left(l_i,l_j\right)$.
This is in contrast to the energy~(\ref{eq:GenEng}) where the pair-wise terms differ only by the scaling factor $w_{ij}$ of a {\em single} fixed term $\varphi\left(l_i,l_j\right)$.
The Photomontage energies of \cite[\S4.2]{Szeliski2008} are an example of such general pair-wise energies.
Instead of using a pair of matrices $V$ and $W$ to parameterize the pair-wise terms, we use a collection of matrices $\left\{V^{ij}\right\}_{ij\in\mbox{$\mathcal{E}$}}$ for the same purpose in the general settings.
Each matrix $V^{ij}$ is of size $l\times l$ and is defined as $V^{ij}_{\alpha\beta} \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} \varphi_{ij}\left(\alpha,\beta\right)$.
A general energy is now parameterized by $\left(n, l, D, \left\{V^{ij}\right\}_{ij\in\mbox{$\mathcal{E}$}} \right)$.
\noindent{\bf Coarsening variables:}
The computation of the interpolation matrix $P$ is unchanged.
ICM can be applied to energies of the form~(\ref{eq:gen-pair-energy}) to estimate agreement between neighboring variables.
These agreements are then used to compute the interpolation matrix $P$ (Sec.~\ref{sec:matrix-P}).
In order to write the coarsening of the pair-wise term we need to introduce some notations:
Let $i,j$ be variables at the fine scale, and $I,J$ denote variables in the coarse scale.
An entry, $P_{iI}$, in the interpolation matrix indicates how fine scale variable $i$ is affected by a coarse scale variable $I$.
Given an interpolation matrix $P$ we can coarsen the energy by
\begin{eqnarray}
D^c & \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} & P^T D^f \nonumber \\
\mbox{$V^{IJ}$}_{\alpha\beta} & \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} & \sum_{ij\in\mbox{$\mathcal{E}$}^f} \mbox{$V^{ij}$}_{\alpha\beta} P_{iJ}P_{jJ}
\label{eq:gen-pair-coarse-variables}
\end{eqnarray}
The coarse graph, $\mbox{$\mathcal{E}$}^c$, is defined by all the non-zero pair-wise terms $\mbox{$V^{IJ}$}$.
The coarser energy is now parameterized by
\begin{equation}
\left(n^c, l, D^c, \left\{\mbox{$V^{IJ}$}\right\}_{ij\in\mbox{$\mathcal{E}$}^c}\right) \nonumber
\end{equation}
\noindent{\bf Coarsening labels:}
In this case, the computation of the interpolation matrix $\hat{P}$ is not trivial,
since we no longer have a single matrix $V$ to derive the agreements from.
We leave this issue of deriving an efficient interpolation matrix for the general case for future work.
However, given a matrix $\hat{P}$ the coarsening of labels can be done easily:
\begin{eqnarray}
D^{\hat{c}} &\mbox{$\stackrel{\mbox{\tiny{def}}}{=}$}& D^{\hat{f}}\hat{P} \nonumber \\
\mbox{$V^{ij}$}^{\hat{c}} &\mbox{$\stackrel{\mbox{\tiny{def}}}{=}$}& \hat{P}^T \mbox{$V^{ij}$}^{\hat{f}} \hat{P} \;\; \forall ij\in\mbox{$\mathcal{E}$}
\label{eq:gen-pair-coarse-labels}
\end{eqnarray}
yielding a coarser energy parameterized by
\begin{equation}
\left(n, l^c, D^{\hat{c}}, \left\{\mbox{$V^{ij}$}^{\hat{c}}\right\}_{ij\in\mbox{$\mathcal{E}$}}\right) \nonumber
\end{equation}
with the same number of variables, $n$, but fewer labels $l^c<l^f$.
\
It is fairly straight forward to see that the energy~(\ref{eq:GenEng}) is a special case of the more general form~(\ref{eq:gen-pair-energy}) and so the coarsening of variables and labels of Eq.~(\ref{eq:EngC}) and~(\ref{eq:EngC-V}) can be seen as special cases of Eq.~(\ref{eq:gen-pair-coarse-variables}) and~(\ref{eq:gen-pair-coarse-labels}) resp.
\subsection{High-order energies}
Discrete energies may involve terms that are beyond pair-wise: that is, describe interaction between sets of variables. These energies are often referred to as high-order energies.
Examples of such energies can be found in e.g., \cite{Kohli2009,Rother2009}.
A high order energy is defined over a hyper-graph $\left(\mbox{$\mathcal{V}$},\mbox{$\mathcal{S}$}\right)$ where the hyper-edges are subsets of variables $s\in\mbox{$\mathcal{S}$}$ s.t. $s\subseteq\mbox{$\mathcal{V}$}$.
\begin{equation}
E\left(L\right) = \sum_{i\in\mbox{$\mathcal{V}$}} \varphi_i\left(l_i\right) + \sum_{s\in\mbox{$\mathcal{S}$}} \varphi_{s}\left(\left\{l_i\left|i\in s\right.\right\}\right)
\label{eq:high-ord-energy}
\end{equation}
Where the high-order terms $\varphi_{s}$ are $\abs{s}$-way discrete functions:
\begin{equation}
\varphi_s:\left\{1,\ldots,l\right\}^{\abs{s}}\rightarrow\mathbb{R}
\end{equation}
A high-order energy of the form~(\ref{eq:high-ord-energy}) can be parameterized using tensors.
Each high-order term, $\varphi_s$, is parameterized by a $\abs{s}$-order tensor
\begin{equation}
V^s_{\alpha,\beta,\ldots,\zeta} \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} \varphi_s\left( l_i = \alpha, l_j=\beta,\ldots, l_m = \zeta \left| s=\left\{i,j,\ldots,m\right\} \right. \right)
\end{equation}
A high-order energy~(\ref{eq:high-ord-energy}) is now parameterized by $\left(n, l, D, \left\{V^s\right\}_{s\in\mbox{$\mathcal{S}$}}\right)$.
\noindent {\bf Coarsening variables:}
The computation of the interpolation matrix $P$ is unchanged.
ICM can be applied to energies of the form~(\ref{eq:high-ord-energy}) to estimate agreement between neighboring variables.
These agreements are then used to compute the interpolation matrix $P$ (Sec.~\ref{sec:matrix-P}).
Given an interpolation matrix $P$ we can coarsen the energy by
\begin{eqnarray}
D^c & \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} & P^T D^f \nonumber \\
V^{S=\left\{I,J,\ldots,M\right\}} & \mbox{$\stackrel{\mbox{\tiny{def}}}{=}$} & P_{iI}\cdot P_{jJ}\cdot\ldots V^{\left\{s=i,j,\ldots,m\right\}} \; \; \forall s\in\mbox{$\mathcal{S}$}^f
\end{eqnarray}
For all coarse variables $I,J,\ldots,M$ with non-zero entry $P_{iI}, P_{jJ},\ldots$.
These non-zeros interactions in $P$ defines the coarse scale hyper-edges in $\mbox{$\mathcal{S}$}^c$.
Note that when two (or more) fine-scale variable $i,j,\ldots$ are represented by the same coarse variable $I$, then the size of $S$ (the coarse scale hyper-edge) is reduced relative to the size of $s$ (the fine scale hyper-edge).
\noindent{\bf Coarsening labels:}
In this case, the computation of the interpolation matrix $\hat{P}$ is not trivial,
since we no longer have a clear representation of the interactions between the different labels.
We leave this issue of deriving an efficient interpolation matrix for the general case for future work.
However, given a matrix $\hat{P}$ the coarsening of labels can be done easily.
Using $\boldsymbol{\alpha},\boldsymbol{\beta}$ to denote coarse scale labels
\begin{eqnarray}
D^{\hat{c}} &\mbox{$\stackrel{\mbox{\tiny{def}}}{=}$}& D^{\hat{f}}\hat{P} \nonumber \\
\mbox{$V^{s}$}^{\hat{c}}_{\boldsymbol{\alpha},\boldsymbol{\beta},\ldots} &\mbox{$\stackrel{\mbox{\tiny{def}}}{=}$}& \sum_{\alpha,\beta,\ldots}\mbox{$V^{s}$}^{\hat{f}}_{\alpha,\beta,\ldots} \hat{P}_{\alpha\boldsymbol{\alpha}} \cdot \hat{P}_{\beta\boldsymbol{\beta}} \cdot \ldots
\label{eq:hi-ord-coarse-labels}
\end{eqnarray}
yielding a coarser energy parameterized by
\begin{equation}
\left(n, l^c, D^{\hat{c}}, \left\{\mbox{$V^{s}$}^{\hat{c}}\right\}_{s\in\mbox{$\mathcal{S}$}}\right) \nonumber
\end{equation}
with the same number of variables, $n$, but fewer labels $l^c<l^f$.
\end{subappendices}
\section{Introduction}
\label{sec:sketch-intro}
Given {\em very few images} (e.g., 3-5) containing a common object
of interest, possibly under severe appearance changes, we detect
the common object and provide a simple and compact visual
representation of that object, depicted by a binary sketch (see
Fig.~\ref{fig:hearts}).
The input images may contain additional distracting objects and
clutter, the object of interest is at unknown image locations, and
its appearance may significantly vary across the images (different
colors, different textures, and small non-rigid deformations). We
do assume, however, that the different instances of the object
share a {\em very rough} common geometric shape, of roughly the
same scale ($\pm 20\%$) and orientation ($\pm 15^\circ$). Our
output sketch captures this rough common shape.
The need to extract the common of {\em very few} images
occurs in various application areas, including: \ (i)~object
detection in large digital libraries. For example, a user may
provide very few (e.g., 3) example images containing an object of
interest with varying appearances, and wants to retrieve new
images containing this object from a database, or from the web. \
(ii)~ Co-segmentation of a few images.
\ (iii)~Artistic graphical uses.
\begin{figure}
(a)
\fbox{\includegraphics[width=.7\linewidth]{sketch/new_hearts_inputs}}
\ (b)
\includegraphics[width=.12\linewidth]{sketch/new_hearts_sketch}
\caption{{\bf Detecting and sketching the common:}
{\em (a) The 4 input images provided to the algorithm.
(b) The least trivial common part (the heart) is detected and
sketched by the algorithm. }}
\label{fig:hearts}
\end{figure}
Our method is based on densely computed {\em Local Self-Similarity
Descriptors}~\cite{Shechtman2007}. Our algorithm is composed of
two main steps: (i)~Identify the common object by detecting a
similar (yet ``non-trivial'') {\em ensemble of self-similarity
descriptors}, that is shared by all the input images.
Corresponding descriptors of the common object across the
different images should be similar in their descriptor values, as
well as in their relative positions within the ensemble.
(ii)~Having found such a mutually common ensemble of descriptors,
our method ``inverts'' it to generate a compact binary sketch
which best represents this ensemble. \
It was shown in~\cite{Shechtman2007} that given a {\em single
query image} of an object of interest (with very little background
clutter), it is possible to detect other instances of that object
in other images by densely computing and matching their local
self-similarity descriptors. The query image can be a real or
synthetic image, or even a {\em hand-drawn sketch} of the object.
In this paper we extend the method of~\cite{Shechtman2007} to
handle {\em multiple query images}. Moreover, in our case those
images are not centered around the object of interest (its
position is unknown), and may contain also other objects and
significant background clutter. Our goal is to detect the {\em
``least trivial'' common part} in those query images, and generate
as clean as possible (region-based) sketch of it, while
eliminating the background clutter of the query images. Such clean
sketches can be obtained from {\em very few} query images, and may
be useful for detection, retrieval, recognition, and for artistic
graphical purposes. Some of these applications are illustrated in
our experiments.
Moreover, while~\cite{Shechtman2007} {\em received as an input} a
clean hand-drawn sketch of the object of interest (and used it for
detecting other instances of that object), we {\em produce} a
sketch as one of our {outputs}, thereby also solving the
``inverse'' problem, namely: Given several images of an object, we
can generate its sketch using the self-similarity descriptor.
A closely related research area to the problem we address is that
of 'learning appearance models' of an object category, an area
which has recently received growing attention
(e.g.,~\cite{Chum2007,Chum2009,Ferrari2009,Jojic2005,karlinsky2008,Lee2009,Nguyen2009,Wu2009,Zhu2008},
to name just a few). The goal of these methods is to discover
common object shapes within collections of images. Some methods
assume a single object category
(e.g.,~\cite{Chum2007,Ferrari2009,karlinsky2008,Jojic2005,Nguyen2009,Wu2009,Zhu2008}),
while others assume multiple object categories
(e.g.,~\cite{Chum2009,Lee2009}). These methods, which rely on
weakly supervised learning ({\bf WSL}) techniques, typically
require tens of images in order to learn, detect and represent an
object category. What is unique to the problem we pose and to our
method is the ability to depict the common object from {\em very
few images}, despite the large variability in its appearance. This
is a scenario no WSL method (nor any other method, to our best
knowledge) is able to address. Such a small number of images
(e.g., $3$) does not provide enough 'statistical samples' for WSL
methods. While our method cannot compete with the performance of
WSL methods when many (e.g., tens) of example images are provided,
it outperforms existing methods when only few images with large
variability are available.
We attribute the strength of our method to the use of {\em densely
computed region-based information} (captured by the local
self-similarity descriptors), as opposed to commonly used {\em
sparse and spurious edge-based information} (e.g., gradient-based
features, SIFT descriptors, etc.) Moreover, the sketching step in
our algorithm provides an {\em additional global constraint}.
Another closely related research area to the problem addressed
here is `co-segmentation'
(e.g.,~\cite{Rother2004,Bagon2008,Mukherjee2009}). The aim of
co-segmentation is to segment out an object common to a few
images ($2$ or more), by seeking segments in the different images
that share common properties (colors, textures, etc.) These common
properties are not shared by the remaining backgrounds in the
different images. While co-segmentation methods extract the common
object from {\em very few images}, they usually assume a much
higher degree of similarity in appearance between the different
instances of the object than that assumed here (e.g., they usually
assume similar color distributions, similar textures, etc.)
The rest of the paper is organized as follows:
Sec.~\ref{sec:formulation} formulates the problem and gives an
overview of our approach. Sec.~\ref{sec:detection} describes the
component of our algorithm which detects the `least trivial'
common part in a collection of images, whereas
Sec.~\ref{sec:sketching} describes the sketching component of our
algorithm. Experimental results are presented in
Sec.~\ref{sec:sketch-results}.
\section{Problem Formulation}
\label{sec:formulation}
\begin{figure}
\centering
\includegraphics[width=.8\linewidth]{sketch/self_sim_desc}
\caption{{\bf The Local Self Similarity
Descriptor:} (Figure taken from~\cite{Shechtman2007}.) \
{\it The self-similarity descriptor for any given point
(e.g., the green point in the left image), is computed by
measuring the similarity of a $5 \times 5$ patch around the point
with the surrounding $60 \times 60$ image region. This results in
a `correlation' surface (middle image). The correlation surface is
quantized into a compact log-polar representation of 45 bins (15
angles, 3 radial intervals) to achieve invariance against small
local affine and non-rigid deformations. The maximum value in
each bin constitutes the value at the corresponding descriptor
entry (right most image).
}}
\label{fig:self_sim_desc}
\end{figure}
Let $I_1,...,I_K$ be $K$ input images containing a common object
under widely different appearances. The object may appear in
different colors, different textures, and under small non-rigid
deformations. The backgrounds are arbitrary and contain
distracting clutter. The images may be of different sizes, and the
image locations of the common object are unknown. We do assume,
however, that the different instances of the object share a {\em
very rough} common geometric shape, of roughly the same scale and
orientation. Our output sketch captures this rough common shape.
Our approach is thus based on detecting {\em 'common regions'} (as
opposed to 'common edges'), using densely computed {\em Local
Self-Similarity Descriptors}~\cite{Shechtman2007}. This descriptor
(illustrated in Fig.~\ref{fig:self_sim_desc}) captures local shape
information in the image vicinity where it is computed, while
being invariant to its photometric properties (color, texture,
etc.) Its log-polar representation makes this descriptor {\em
in}sensitive to small affine and non-rigid deformations (up to
$\pm 20\%$ in scale, and $\pm 15^\circ$). It was further shown
by~\cite{Horster2008} that the local self-similarity descriptor
has a strong descriptive power (outperforming SIFT).
The use of local self-similarity descriptors allows our method to
handle much stronger variations in appearance (and in much fewer
images) than those handled by previous methods.
We densely compute the self-similarity descriptors in images
$I_1,...,I_K$ (at every $5$-th pixel). `Common' image parts across
the images will have similar arrangements of self similarity
descriptors.
\begin{figure}
(a)\includegraphics[width=.6\linewidth]{sketch/faces_inputs}
(b)\includegraphics[width=.2\linewidth]{sketch/faces_sketch_1_3_5_6_7}
\caption{{\bf Sketching:} \ {\small \it (a) Five
input images. \ (b) Their joint sketch.}}
\label{fig:faces}
\end{figure}
Let $c_1,...,c_K$ denote the unknown locations of the common
object in the $K$ images. Let $I_k^{c_k}$ denote a $w \times h$
subimage of $I_k$ centered at $c_k$, containing the common object
($k=1,...,K$) (need not be tight). For short, we will denote it by
$\tilde{I_k}$.
The sketch we seek is a binary image $S$ of size $w \times h$
which best captures the rough characteristic shape of the common
object shared by $\tilde{I_1},...,\tilde{I_K}$.
More formally, we seek a binary image $S$ whose {local
self-similarity descriptors} match as best as possible the local
self-similarity descriptors
of $\tilde{I_1},...,\tilde{I_K}$. The descriptors should match in
their {\em descriptor values}, as well as in their {\em relative
positions} with respect to the centers $\{c_k\}$:
\begin{eqnarray}
\label{eq:Score}
Score(S|\tilde{I_1},...,\tilde{I_K}) & = & \sum_{k=1}^{K}match(S,\tilde{I_k}) \\
& = & \sum_{k=1}^{K}\sum_{i=1}^{w \cdot h} sim\left(
d_i^S,d_i^k
\right) \nonumber
\end{eqnarray}
where $d_i^S$ is the $i$-th {self-similarity descriptor} computed
at image location $l_i$ in the sketch image $S$, $d_i^k$ is the
self-similarity descriptor computed {\em at the same relative
position} $l_i$ (up to small shifts) in the $w \times h$ subimage
$\tilde{I_k}$, and $sim(d_1,d_2)=-\parallel d_1-d_2
\parallel_p$ measures how similar two descriptor vectors are (we
experimented with $L_p$ norms for $p=1,2$).
Thus, the {\em binary sketch} we seek is:
\begin{equation}
\label{eq:argmax}
\hat{S}=argmax \{ Score(S|\tilde{I_1},...,\tilde{I_K}) \} \ \
s.t. \ \ S(l)\in\{-1,1\}
\end{equation}
where $S(l)$ is the value of $S$ at pixel $l$. This process is
described in detail in Sec.~\ref{sec:sketching}, and results in a
sketch of the type shown in Fig~\ref{fig:faces}.
While edge-based detection and/or
sketching~\cite{Lee2009,Zhu2008,Ferrari2009} requires many input
images, our region-based detection and sketching can be recovered
from very few images.
Edges tend to be very spurious, and are very prone to clutter
(even sophisticated edge detectors like~\cite{Maire2008} -- see
Fig.~\ref{fig:star}.b). Edge-based approaches thus require a
considerable number of images, to allow for the consistent
edge/gradient features of the object to stand out from the
inconsistent background clutter. In contrast, region-based
information is much less sparse (area vs. line-contour), less
affected by clutter or by misalignments, and is not as sensitive
to the existence of strong clear boundaries. Much larger image
offsets are required to push two corresponding regions out of
alignment than to misalign two thin edges. Thus, region-based cues
require fewer images to detect and represent the common object.
Indeed, our method can provide good sketches from as few as $3$
images. In fact, in some cases our method produces a meaningful
sketch even from a {\em single} image, where edge-based sketching
is impossible to interpret -- see example in Fig.~\ref{fig:star}.
\begin{figure}
\begin{tabular}{ccc}
\includegraphics[width=.28\linewidth]{sketch/syn_pad_06}&
\includegraphics[width=.28\linewidth]{sketch/malik_syn_pad_06}&
\includegraphics[width=.28\linewidth]{sketch/sketch_syn_pad_06}\\
(a) & (b) & (c)
\end{tabular}
\caption{{\bf Regions vs. Edges:}{\small \it \ \
(a) a single input image. \ (b) The edge map generated by the
method of~\cite{Maire2008}. \ (c) The binary sketch generated by
our method when applied to the \underline{single input image}
(using all the self-similarity descriptors densely computed in
that image). \ This illustrates the concept that region-based
information is much richer than sparse edge-based information, and
therefore appears to be more powerful for detection and for
sketching. }}
\label{fig:star}
\end{figure}
In the general case, however, the locations $c_1,...,c_K$ of the
object within the input images $I_1,...,I_K$, are unknown. We seek
a binary image $S$
which sketches the {\em `least trivial'} object (or image
part) that is {\em `most common'} to all those images. The {\em
`most common'} constraint is obvious:
in each image $I_k$ there should be a location $c_k$ for which \
$match \left(S,I_k^{c_k} \right)$ \ is high (where
$\tilde{I_k}=I_k^{c_k}$ is the subimage centered at $c_k$). \
However, there are many image regions that are {\em trivially}
shared by many natural images. For example, {\em uniform regions}
(of uniform color or uniform texture) occur abundantly in natural
images. Such regions share similar self-similarity descriptors,
even if the underlying textures or colors are different (due to
the invariance properties of the self-similarity descriptor).
Similarly, strong vertical or horizontal edges (e.g., at
boundaries between two different uniformly colored/textured
regions) occur abundantly in images.
We do not wish to identify such trivial (insignificant) common
regions in the images as the `common object'.
Luckily, since such regions have good image matches in lots of
locations, the {\em statistical significance}
of their good matches tends to be low (when measured by how many
standard deviations its peak match values are away from its mean
match value in the collection of images).
In contrast, a {\em non-trivial} common part (with non-trivial
structure) should have at least one good match in each input image
(could also have a few matches in an image), but these matches
would be `statistically significant' (i.e., this part would not be
found `at random' in the collection of images).
Thus, in the general case, we seek a {binary sketch} $S$ and
locations $c_1,...,c_K$ in images $I_1,...,I_K$, such that:
\noindent {\bf (i) $S$ is `most common'}, in the sense that it
maximizes $Score(S|I_1^{c_1},..,I_K^{c_K}) =
\sum_{k=1}^{K}match(S,I_k^{c_k})$ of Eq.~(\ref{eq:Score}).
\noindent {\bf (ii) $S$ is `least trivial'}, in the sense that
its matches at $c_1,...,c_K$ are {\em statistically significant},
i.e., it maximizes $\sum_{k=1}^{K} StatSignificance \left(
match(S,I_k^{c_k}) \right)$,\ where the significance of a match
of $S$ is measured by how many standard deviations it is away from
the mean match value of $S$.
Our optimization algorithm may iterate between these two
constraints: \ (i)~Detect the locations $\{c_k\}_{k=1}^K$ of the
least trivial common image part in $\{I_k\}_{k=1}^K$
(Sec.~\ref{sec:detection}). \ (ii)~Sketch the common object given
those image locations (Sec.~\ref{sec:sketching}). \ \ The overall
process results in a sketch image, which provides a simple compact
visual representation of the common object of interest in a set of
query images, while eliminating any distracting background clutter
found in those images.
\section{Detecting the Common}
\label{sec:detection}
We wish to detect image locations $c_1,...,c_K$ in $I_1,...,I_K$,
such that corresponding subimages centered at those locations,
$I_k^{c_k}$, share as many self-similarity descriptors with each
other as possible, yet their matches to each other are non-trivial
(significant). The final sketch $S$ will then be obtained from
those subimages (Sec.~\ref{sec:sketching}).
Let us first assume that the dimension $w \times h$ of the
subimages is given. We will later relax this assumption.
Let $\tilde{I}$ be a $w \times h$ image segment (this could be the
final sketch $S$, or a subimage extracted from one of the $K$
input images in the iterative process). We wish to check if
$\tilde{I}$ has a good match in each of the input images
$I_1,...,I_K$, and also check the statistical significance of its
matches.
We `correlate' $\tilde{I}$ against all the input images (by
measuring the similarity of its underlying self-similarity
descriptors\footnote{We use the same algorithm employed
by~\cite{Shechtman2007} to match ensembles of self-similarity
descriptors, which is a modified version of the efficient
``ensemble matching'' algorithm of~\cite{Boiman2007}. This
algorithm employs a simple probabilistic ``star graph'' model to
capture the relative geometric relations of a large number of
local descriptors, up to small non-rigid deformations.}).
In each image $I_k$ we find the highest match value of
$\tilde{I}$: $maxMatch(\tilde{I},I_k)$. The higher the value, the
stronger the match. However, not every high match value is
statistically significant. The {\em statistical significance}
of $maxMatch(\tilde{I},I_k)$ is measured by how many standard
deviations it is away from the mean match value of $\tilde{I}$ in
the entire collection of images, i.e.,:
\begin{equation*}
\left(maxMatch(\tilde{I},I_k) - avgMatch(\tilde{I}) \right) /
stdMatch(\tilde{I})
\end{equation*}
where $avgMatch(\tilde{I})$ is the mean of all match values of
$\tilde{I}$ in the collection $I_1,...,I_K$, and
$stdMatch(\tilde{I})$ is their standard deviation. We thus define
the `Significance' of a subimage $\tilde{I}$ as:
\begin{equation*}
Significance(\tilde{I}|I_1,...,I_K)=\frac{1}{K} \sum_{k=1}^K
StatSignificance \left( maxMatch(\tilde{I},I_k) \right)
\end{equation*}
Initially, we have no candidate sketch $S$. However, we can
measure how `significantly common' is each $w \times h$ subimage
of $I_1,...,I_K$, when matched against all locations in all the
other $K-1$ images.
We can assign a significance score to each {\em pixel} $p \in I_k$
($k=1,..,K$), according to the `Significance' of its surrounding
$w \times h$ subimage: $Significance(I_k^p|I_1,...,I_K)$.
We set $c_k$ to be the pixel location with the {\em highest}
significance score in image $I_k$, i.e., $c_k = argmax_{p \in I_k}
\{ Significance(I_k^p|I_1,...,I_K) \} $.
The resulting $K$ points (one per image), $c_1,...,c_K$, provide
the centers for $K$ candidates of `non-trivial' common image
parts.
We generate a sketch $S$ from these image parts (using the
algorithm of Sec.~\ref{sec:sketching}).
\begin{figure}
\includegraphics[width=\linewidth]{sketch/ankhs_iterations}
\caption{{\bf Iterations of Detection \&
Sketching:} {\small \it {\bf Left:} The $4$ input images. \ {\bf
Right:} The first iteration of the detection algorithm results in
$4$ detected image regions, of which $3$ are correct and one is an
outlier (marked by red). The resulting sketch produced from these
regions is reasonably good (due to the robustness of the sketching
to outliers -- see Secs.~\ref{sec:sketching}
and~\ref{sec:sketch-experiments}), and is used for refining the detection
in the input images. This results in $4$ correct detections in the
second iteration, and an improved sketch. }}
\label{fig:iterations}
\end{figure}
\begin{figure}
\includegraphics[width=\linewidth]{sketch/new_faces_one_fig}
\caption{{\bf Detecting and sketching the
common:} {\it \small (Left) The input images. \ (Upper-Right)
The detected image regions of the common object, including one
outlier. \ (Lower-Right) The resulting sketch.}}
\label{fig:large-faces}
\end{figure}
We repeat the above process, this time for $\tilde{I}=S$, to
detect its best matches in $I_1,...,I_K$. This should lead to
improved detection and localization of the common object
($c_1,...,c_K$), and accordingly to an improved sketch $S$. This
algorithm can be iterated several times.
In practice, in all our experiments a good sketch $S$ was
recovered already in the first iteration. An additional iteration
was sometimes useful for improving the detection.
Fig.~\ref{fig:iterations} shows two iterations of this process,
applied to $4$ input images. More results of the detection can be
seen in Fig.~\ref{fig:large-faces}.
\noindent {\bf Handling unknown w $\times$ h:}
\ \ In principle, when $w \times h$ is unknown, we can run the
above algorithm {\em ``exhaustively''} for a variety of
$w=w_{min},..,w_{max}$ and $h=h_{min},..,h_{max}$, and choose
``the best'' $w \times h$ (with maximal significance score). In
practice, this is implemented more efficiently using ``integral
images'', by integrating the contributions of individual
self-similarity descriptors into varying window sizes $w \times
h$.
\noindent {\bf Computational Complexity:}
\ \ The detection algorithm is implemented coarse-to-fine. The
first step of the algorithm described above is quadratic in the
size of the input images. However, since the number of images is
typically small (e.g., $3-5$), and since the quadratic step occurs
only in the coarsest/smallest resolutions of the images, this
results in a computationally efficient algorithm.
\section{Sketching the Common}
\label{sec:sketching}
Let $\tilde{I_1},\ldots,\tilde{I_K}$ be the $w \times h$ subimages
centered around the common object (detected and extracted from the
input images using the algorithm of Sec.~\ref{sec:detection}). The
goal of the sketching process is to produce a {binary image} $S$,
which best captures the rough characteristic shape of the object
shared by $\tilde{I_1},...,\tilde{I_K}$, as posed by
Eq.~(\ref{eq:argmax}). Namely, find $S$ whose ensemble of
self-similarity descriptors is as similar as possible to the
ensembles of descriptors extracted from
$\tilde{I_1},\ldots,\tilde{I_K}$. If we were to neglect the binary
constraint $S(l)\in\{-1,1\}$ in Eq.~(\ref{eq:argmax}), and the
requirement for consistency between descriptors of an image, then
the {\em optimal solution} for the collection of self-similarity
descriptors of $S$, \ $\{d_i\}_{i=1}^{w \cdot h}$, \ could be
explicitly computed as:
\begin{eqnarray}
\label{eq:median}
d_i = & \textmd{median}_k \{ d_i^k \} & ~~\mbox{if }L_1\mbox{-norm}\\
d_i = & \textmd{mean}_k \{ d_i^k \} & ~~\mbox{if
}L_2\mbox{-norm} \nonumber
\end{eqnarray}
We use the $L_1$-norm to generate these `combined' descriptors
$\{d_i\}_{i=1}^{w \cdot h}$, because of the inherent robustness of
the median operator to outliers in the descriptors (also confirmed
by our empirical evaluations in Sec~\ref{sec:sketch-results}).
Having recovered such a collection of descriptors for $S$, we
proceed and solve the ``inverse'' problem -- i.e., to generate the
image $S$ from which these descriptors emanated. However, the
collection of descriptors $\{d_i\}_{i=1}^{w \cdot h}$ generated
via a `median' or `average' operations is no longer guaranteed to
be a valid collection of self-similarity descriptors of any real
image (binary or not). We thus proceed to recover the simplest
possible image $S$ whose self-similarity descriptors best
approximate the `combined' descriptors $\{d_i\}_{i=1}^{w \cdot h}$
obtained by Eq.~(\ref{eq:median}).
Self-similarity descriptors cover large image regions, with high
overlaps. As such, the similarity and dissimilarity between two
image locations (pixels) of $S$ are \textit{implicitly} captured
by multiple self-similarity descriptors and in different
descriptor entries. The self-similarity descriptor as defined
in~\cite{Shechtman2007} has values in the range $[0,1]$, where $1$
indicates high resemblance of the central patch to the patches in
the corresponding log-polar bin, while $0$ indicates high {\em
dis}similarity of the central patch to the corresponding log-polar
bin. For our purposes, we stretch the descriptor values to the
range $[-1,1]$, where $1$ signifies ``attraction'' and $-1$
signifies ``repulsion'' between two image locations.
Let $W$ be a $wh \times wh$ matrix capturing the
attraction/repulsion between every two image locations, as induced
by the collection of the `combined' self-similarity descriptors
$\{d_i\}_{i=1}^{w \cdot h}$ of Eq.~(\ref{eq:median}). Entry
$w_{ij}$ in the matrix is the degree of attraction/repulsion
between image locations $l_i$ and $l_j$, determined by the
self-similarity descriptors $d_i$ and $d_j$ centered at those
points. $d_i(l_j)$ is the value of the bin containing location
$l_j$ in descriptor $d_i$ (see
Fig.~\ref{fig:self-sim-correlation}). Similarly, $d_j(l_i)$ is the
value of the bin containing location $l_i$ in descriptor $d_j$.
The entry $w_{ij}$ gets the following value:
\begin{equation}
w_{ij} = \alpha_{ij} \left( d_i(l_j) + d_j(l_i) \right) / 2
\end{equation}
where $\alpha_{ij} = \alpha_{ji}$ is inversely proportional to the
distance $\parallel l_i - l_j
\parallel$ between the two image locations (we give higher weight to bins that are closer to
the center of the descriptor, since they contain more
accurate/reliable information).
\begin{figure}
\centering
\includegraphics[width=.5\linewidth]{sketch/self_sim_correlation_uncropped}
\caption{
{\bf Computing attraction/repulsion matrix $W$:}
{\em The log-polar self-similarity
descriptor $d_i$ is located at $l_i$ (red cross). White bins
signify image areas of high similarity to the central patch, dark
bins signify image areas of dissimilarity to the central patch.
The point $l_j$ (blue cross), which is the center of descriptor
$d_j$ (not drawn), falls in a white bin of descriptor $d_i$ (i.e.,
$0 < d_i(l_j) \leq 1$). The entry $w_{ij}$ in the matrix $W$ is
determined accordingly: $w_{ij} = \alpha_{ij} \left( d_i(l_j) +
d_j(l_i) \right)/2$, where $\alpha_{ij}$ (the certainty assigned
to this entry), is inversely proportional to the distance
$\parallel l_i-l_j \parallel$ (the distance between the red and
blue crosses). Similarly, the point $l_k$ (green cross), which is
the center of another descriptor $d_k$ (also not drawn), falls in
a dark bin of descriptor $d_i$, i.e., $-1 \leq d_i(l_k) < 0$, and
$\alpha_{ik} < \alpha_{ij}$ (because the green cross falls farther
away from the center of $d_i$, hence lower certainty).
}}
\label{fig:self-sim-correlation}
\end{figure}
\begin{figure}
(a) \fbox{\includegraphics[width=.7\linewidth]{sketch/horses_inputs}}
(b) \includegraphics[width=.14\linewidth]{sketch/horses_sketch}
\caption{{\bf Detecting and sketching the
common:} {\it \small (a) Five input images. (b) The resulting
sketch.}}
\label{fig:horses}
\end{figure}
Note that a `pure' attraction/repulsion matrix $W$ of a true
binary image $S$ contains only $3$ types of values $w_{ij}$: $-1,
0, 1$. If $l_i$ and $l_j$ belong to the same region in $S$ (i.e.,
both in foreground or both in background), then $w_{ij}=1$; if
$l_i$ and $l_j$ belong to different regions in $S$, then
$w_{ij}=-1$, and if the points are distant (out of descriptor
range), then $w_{ij}=0$. In the general case, however, the
entries span the range $[-1,1]$, where $1$ stands for ``strong''
attraction, $-1$ for ``strong'' repulsion and $0$ means ``don't
care''. The closer the value of $w_{ij}$ to $0$, the lower its
attraction/repulsion confidence; the closer it is to $\pm 1$, the
higher the attraction/repulsion confidence.
Note that $W$ is different from the classical affinity matrix used
in spectral clustering or in min-cut, which use non-negative
affinities, and their value $0$ is {\em ambiguous} -- it signifies
both {\em high-dissimilarity} as well as {\em low-confidence}. The
distinction between `attraction', `repulsion', and
`low-confidence' is critical in our case, thus we cannot resort
to the max-flow algorithm or to spectral clustering in order to
solve our problem. An affinity matrix with positive and negative
values was used by~\cite{Yu2001} in the context of the normalized-cut
functional. However, their functional is not appropriate for our
problem (and indeed did not yield good results for $S$ when
applied to our $W$). We therefore define a different functional
and optimization algorithm in order to solve for the binary sketch
$S$.
The binary image $S$ which {\em best} approximates the
attraction/repulsion relations captured by $W$,
will minimize the following functional:
\begin{equation}
\label{eq:functional_opt}
\min_S \sum_{i,j} w_{ij} (S(l_i) - S(l_j))^2 ~~~~~\mbox{subject
to}~~ S(l) \in \{-1,1\}
\end{equation}
where $S(l)$ is the value of $S$ at pixel $l$.
Note that for a binary image, the term $(S(l_i) - S(l_j))^2$ can
obtain only one of two values: $0$ (if both pixels belong to
foreground, or both belong to background), {\em or} \ $4$ (if one
belongs to the foreground, and one to the background). Thus, when
$w_{ij}$ is positive (attraction), $S(l_i)$ and $S(l_j)$ should
have the same value (both $1$ or both $-1$), in order to minimize
that term $w_{ij} (S(l_i) - S(l_j))^2$. The larger $w_{ij}$
(stronger confidence), the stronger the incentive for $S(l_i)$ and
$S(l_j)$ to be the same. Similarly, a negative $w_{ij}$
(repulsion) pushes {\em apart} the values $S(l_i)$ and $S(l_j)$.
Thus, $S(l_i)$ and $S(l_j)$ should have opposite signs in order to
minimize that term $w_{ij} (S(l_i) - S(l_j))^2$. When
$w_{ij}\approx 0$ (low confidence), the value of the functional
will not be affected by the values $S(l_i)$ and $S(l_j)$ (i.e.,
``don't care''). It can be shown that in the `ideal' case, i.e.,
when $W$ is generated from a binary image $S$, the global minimum
of Eq.~(\ref{eq:functional_opt}) is obtained at $S$.
\noindent {\bf Solving the constrained optimization problem:} \
The min-cut problem where only non-negative values of $w_{ij}$ are
allowed can be solved by the max-flow algorithm in polynomial
time. However, the weights $w_{ij}$ in the functional of
Eq.~(\ref{eq:functional_opt}) can obtain both positive and
negative values, turning our `cut' problem as posed above into an
NP-hard problem. We therefore {\em approximate}
Eq.~(\ref{eq:functional_opt}) by reposing it as a quadratic
programming problem, while relaxing the binary constraints.
Let $D$ be a diagonal matrix with $D_{ii} = \sum_j w_{ij}$, and
let $L=D-W$ be the graph Laplacian of $W$. Then $\frac{1}{2}
\sum_{i,j} w_{ij} (S(l_i) - S(l_j))^2 = S^T L S$. Thus, our
objective function is a quadratic expression in terms of $S$. The
set of binary constrains are relaxed to the following set of
linear constraints $-1 \leq S(l) \leq 1$, resulting in the
following quadratic programming problem:
\begin{equation}
\hat{S} = \arg \min_S S^T L S \ \ \ \mbox{s.t.} \ \ -1 \leq S(l)
\leq 1
\end{equation}
Since $L$ is not necessarily positive semi-definite, we do not
have a guarantee regarding the approximation quality (i.e., how
far is the achieved numerical solution from the optimal solution).
Still, our empirical tests demonstrate good performance of this
approximation. We use Matlab's optimization toolbox (quadprog) to
solve this optimization problem and obtain a sketch $\hat{S}$. In
principle, this does not yield a binary image. However, in
practice, the resulting sketches look very close to binary images,
and capture well the rough geometric shape of the common objects.
The above sketching algorithm is quite robust to outliers (see
Sec.~\ref{sec:sketch-results}), and obtains good sketches from very few
images. Moreover, if when constructing the attraction/repulsion
matrix $W$ we replace the `combined' descriptors of
Eq.~(\ref{eq:median}) with the self-similarity descriptors of a
{\em single image}, our algorithm will produce `binary' sketches
of a single image (although these may not always be visually
meaningful). An example of a sketch obtained from a single image
(using all its self-similarity descriptors) can be found in
Fig.~\ref{fig:star}.
\section{Experimental Results}
\label{sec:sketch-results} \label{sec:sketch-experiments}
\begin{figure}
\vspace*{1.5mm}
(a) \fbox{\includegraphics[width=.69\linewidth]{sketch/trikona5_inputs}}
(b) \includegraphics[width=.15\linewidth]{sketch/trikona5_sketch}
\caption{{\bf Detecting and sketching the
common:} {\small \it (a) The input images. (b) The resulting
sketch.
}}
\label{fig:trikonasana}
\end{figure}
\begin{figure}
\begin{tabular}{|c|c||c|c|}
\hline
\rule{0cm}{6.5mm}Input images &\parbox[b]{7ex}{Output sketch}& Input images &\parbox[b]{7ex}{Output sketch}\\
\hline
\rule{0cm}{20mm}\includegraphics[width=.25\linewidth]{sketch/ethz3_apples}&
\raisebox{2mm}{\includegraphics[width=.05\linewidth]{sketch/ethz3_sketch_1}}&
\includegraphics[width=.52\linewidth]{sketch/ethz6_apples}&
\raisebox{2mm}{\includegraphics[width=.05\linewidth]{sketch/apple_sketch_14}}\\
\hline
\rule{0cm}{18mm}\includegraphics[width=.25\linewidth]{sketch/ethz3_mugs}&
\raisebox{2mm}{\includegraphics[width=.05\linewidth]{sketch/ethz3_sketch_3}}&
\includegraphics[width=.52\linewidth]{sketch/ethz6_mugs}&
\raisebox{2mm}{\includegraphics[width=.05\linewidth]{sketch/mugs_sketch_5}}\\
\hline
\rule{0cm}{21mm}\includegraphics[width=.25\linewidth]{sketch/ethz3_giraffes}&
\raisebox{2mm}{\includegraphics[width=.06\linewidth]{sketch/ethz3_sketch_10}}&
\includegraphics[width=.52\linewidth]{sketch/ethz6_giraffes}&
\raisebox{2mm}{\includegraphics[width=.06\linewidth]{sketch/giraffes_sketch_3}}\\
\hline
\rule{0cm}{21mm}\includegraphics[width=.25\linewidth]{sketch/ethz3_bottles}&
\raisebox{2mm}{\includegraphics[width=.03\linewidth]{sketch/ethz3_sketch_4}}&
\includegraphics[width=.52\linewidth]{sketch/ethz6_bottles}&
\raisebox{2mm}{\includegraphics[width=.03\linewidth]{sketch/bottles_sketch_5}}\\
\hline
\rule{0cm}{20mm}\includegraphics[width=.25\linewidth]{sketch/ethz3_swans}&
\raisebox{2mm}{\includegraphics[width=.05\linewidth]{sketch/ethz3_sketch_6}}&
\includegraphics[width=.52\linewidth]{sketch/ethz6_swans}&
\raisebox{2mm}{\includegraphics[width=.05\linewidth]{sketch/swan_sketch_12}}\\
\hline
\end{tabular}
\caption{{\bf Sample results on ETHZ
shapes \cite{Ferrari2006} dataset:} \ {\small \it Detection and
sketching using only 3 images (left), and using 6 images
(right).}}
\label{fig:ethz-sketches}
\end{figure}
\begin{figure}
\begin{tabular}{cc}
\includegraphics[width=.45\linewidth]{sketch/graph_eval_sketch_with_detection_up10} &
\parbox[b]{.45\linewidth}{\caption{{\bf
Evaluating sketch quality:} {\em \small Mean values of
$Quality(S)$ as a function of the number of input
images ($K=2,...,10$) randomly sampled from each set of ETHZ shape
dataset~\cite{Ferrari2006}.\vspace*{5mm}}}
\label{fig:graph-sketch-vs-gt}
}
\end{tabular}
\begin{tabular}{cl}
\includegraphics[width=.45\linewidth]{sketch/graph_sketch_with_outliers}&
\parbox[b]{.45\linewidth}{
\caption{{\bf Sketching in
presence of outliers:} {\em \small We ``corrupt" a set of 10
``inlier" with $n$ randomly chosen natural images. Graph shows
mean values of $Quality(S)$ as a function of the percent of
outlier images in the input set, i.e., $n/(10+n)$.}}
\label{fig:graph-sketch-with-outliers}
}
\end{tabular}
\begin{tabular}{cl}
\includegraphics[width=.45\linewidth]{sketch/graph_detect_in_novel_Ori_data_new_up10}&
\parbox[b]{.45\linewidth}{
\caption{{\bf Detection in new
images:} {\em \small We empirically evaluated how well the sketch
generated form very few images ($K=2,..10$) performs in detecting
the common shape in \underline{new} images.\vspace*{5mm}}}
\label{fig:graph-detect-in-novel}}
\end{tabular}
\end{figure}
Figs.~\ref{fig:hearts},\ref{fig:faces},\ref{fig:large-faces},\ref{fig:horses},\ref{fig:trikonasana},\ref{fig:ethz-sketches}
show qualitative results on various image sets. In all of these
examples the number of input images was very small ($3-7$), with
large variability in appearance and background clutter. Our
algorithm was able to detect and produce a compact representation
(a sketch) of the common content.
We further conducted empirical evaluations of the algorithm using
ETHZ shape dataset~\cite{Ferrari2006}. This dataset consists of
five object categories with large variability in appearance:
Applelogos, Bottles, Giraffes, Mugs and Swans (example images can
be seen in Fig.~\ref{fig:ethz-sketches}). There are around $50$
images in each set, with ground-truth information regarding the
location of the object in each image, along with a single
hand-drawn ground truth shape for each category. In order to
assess the quality of our algorithm (which is currently not scale
invariant, although it can handle up to $\pm 20\%$ scale
variation, and $\pm 15^\circ$ rotations), we scaled the images in
each dataset to have {\em roughly} the same object size (but we
have not rotated the images, nor changed their aspect ratios).
\noindent {\bf Sketch quality score:}
Because our sketch $S$ is continuous in the range $\left[-1,\
1\right]$, we stretch the values of the ground-truth sketch
$S_{GT}$ also to this range, and multiply the two sketches
pixel-wise. Our sketch quality score is: \
\mbox{$Quality(S)={<S,S_{GT}>}/{(\#~of~pixels)}.$}
In places where both sketches agree in their sign (either white
regions or black) the pixel-wise product is positive, while in
places where the sketches disagree, the product is negative. This
produces a sketch quality score with values ranging between $-1$
(lowest quality) to $+1$ (highest quality). Note that even if our
sketch displays a perfect shape, its quality will be smaller than
$1$, because it is not a perfect binary image. From our
experience, sketch quality $\geq 0.8$ are usually
excellent-looking sketches.
We first assessed the quality of our algorithm to identify and
sketch the common object correctly, as a function of the number of
input images $K$ ($K=2,3,..,10$). We randomly sampled $K$ images
out of an object category set, applied our detection and sketching
algorithm to that subset, and compared the resulting sketch $S$ to
the ground-truth $S_{GT}$. We repeated this experiment $15$ times
for each $K$, and computed mean sketch quality scores.
Fig.~\ref{fig:graph-sketch-vs-gt} displays plots of the mean
quality score for the $5$ categories. It can be seen that from
relatively few images ($K=3$) we already achieve sketches of good
quality, even for challenging sets such as the giraffes (although,
with the increased number of example images, its legs tend to
disappear from the sketch because of their non-rigid
deformations). Examples for sketching results for some of these
experiments can be seen in Fig.~\ref{fig:ethz-sketches}.
We next evaluated the robustness of the sketching component of our
algorithm to outliers. Such robustness is important, since the
detection algorithm often produces outlier detections (see
Fig.~\ref{fig:iterations}). We used $10$ ``inlier'' images which
alone generate a good sketch with high sketch quality score. We
then added to them $n=1,...,30$ outlier images (cropped at random
from natural images). For every such $10+n$ image set we generated
a sketch, and compared it to the ground-truth. Each experiment was
repeated $15$ times. Fig.~\ref{fig:graph-sketch-with-outliers}
displays plots of sketch quality vs. percent of outliers
$n/(10+n)$. Our sketching method is relatively robust to outliers,
and performs quite well even in presence of $50\%$ outliers (as
expected due to the median operation in Eq.~(\ref{eq:median})).
In addition to sketch quality evaluation we tested the performance
of our algorithm in the scenario described in the Introduction:
given a very small number of example images, how useful is the
output of our automatical detection \& sketching algorithm for
successfully detecting that object in {\em new images}. For
$K=2,3,...,10$, we randomly sampled $K$ images out of an object
category set, applied our detection \& sketching algorithm to that
subset, and used the resulting sketch to detect the object in the
{\em remaining} $50-K$ images of that category set. We consider an
object in image $I_n$ as ``detected'' if the location of
$maxMatch(S,I_n)$ (the detected center $c_n$ of the object) falls
{\em no farther away} than 1/4 of the width or height of the
bounding-box from the ground-truth center.
We repeated each experiment $40$ times and plotted the average
detection rates in Fig.~\ref{fig:graph-detect-in-novel}. For the
Apples, Bottles, and Swans we get high detection rates (for as few
as $K=3$ example images; a scenario no WSL method can handle to
the best of our knowledge). However, our detection rates are not
as good in the Giraffe set, since the giraffes undergo strong
non-rigid deformations (they sometimes tilt their necks down, and
their legs change positions). Our current algorithm cannot handle
such strong non-rigid deformations.
|
1,108,101,565,699 | arxiv | \section*{Executive Summary}
\section*{Executive Summary}
We propose the development of a muon-proton and muon-nucleus
collider, referred to as a ``muon-ion collider'' (MuIC),
that utilizes the existing hadron accelerator facilities at Brookhaven National Laboratory (BNL), Fermilab, or CERN while seeding, or leveraging, the development of a high-energy muon storage ring at the same site. A center-of-mass energy at the TeV scale is achieved when a TeV muon beam is brought into collision with a hadron beam of hundreds of GeV to several TeV.
Muon collider technology has been considered as an avenue toward
reaching the next high energy frontier of particle physics with a relatively compact machine footprint.
Despite its advantages, it remains challenging to achieve a
high energy, high-luminosity muon
collider because of the short muon lifetime.
The efficient muon cooling in six dimensions
(spatial and momentum) and the fast ramping of the muon energy
are among the most critical elements for its realization, necessitating an R\&D program in the U.S and at CERN through the Muon Accelerator Program (MAP) and the international muon collider collaboration (IMCC), respectively.
This proposal enables deep inelastic scattering measurements in new regimes at low parton momentum fraction $x$ and high squared four-momentum transfer $Q^2$, which will further elucidate the structure of the proton and nuclei as well as provide precision QCD and electroweak measurements. We note that these measurements also lay the groundwork for precision measurements made at future high energy hadron colliders, such as the FCC-hh, much as HERA data improved LHC calculations.
The TeV energy scale at the MuIC also allows for the direct production of Standard Model vector gauge bosons, Higgs bosons, and top quarks in a way complementary to hadron colliders, which provides further sensitivity to the electroweak sector.
While similar in scientific potential to the proposed Large Hadron electron Collider (LHeC), the MuIC offers complementary scattering kinematics and complementary sensitivity with a muon beam to beyond Standard Model processes. The MuIC also could provide polarization of both beams (when utilizing the BNL facility) and provide lepton-proton and antilepton-proton collisions with similar luminosity.
The possible configurations and design parameters of muon-ion colliders are explored here along with some representative physics process studies. For muon-proton collisions, a center-of-mass energy of up to 1~TeV at BNL and 6.5~TeV at CERN can be achieved, with a estimate on the achievable luminosity ranging from 10$^{33}$ to 10$^{34}$~$\text{cm}^{-2} \text{s}^{-1}$. This should yield enough integrated luminosity to explore QCD phenomena at extreme parton densities, the electroweak and QCD couplings up to the highest $Q^2$ reach in deep inelastic scattering, and to measured the Higgs boson cross sections in its largest branching fraction decay modes through the vector boson fusion processes. Sensitivity to physics beyond the Standard Model is also feasible, particularly for lepton-flavor violating processes, of which we study leptoquark interactions as a case study.
Experimentally, the final state products other than the scattered lepton tend to be centrally distributed in the experiment at MuIC and LHmuC for processes at the electroweak scale and above, more so than for the same processes at the LHeC. The scattered muon, however, peaks in the muon beam direction for low-$Q^2$ DIS and for vector boson fusion processes, necessitating the need for a muon spectrometer along the beamline. But such a spectrometer design also may prove useful for experiments at a $\mu^+\mu^-$ collider.
In summary, the goals and merits of our proposal are as follow:
\begin{itemize}
\item Open a unique new frontier in particle and nuclear physics, ranging from the partonic structure of matter, precision QCD and electroweak interactions, Higgs bosons, and searches for physics beyond the Standard Model.
\item Serve as a scientific target for a muon collider demonstrator to establish sustained muon collider R\&D, and serve as a stepping stone toward the ultimate ${\cal O}(10+)$ TeV $\mu^+\mu^-$ collider.
\item Provide an affordable option by re-using established infrastructure and leveraging funding resources from both the particle and nuclear physics communities to realize a (first?) muon-based collider in the U.S. and/or CERN in the next 20--25 years.
\end{itemize}
\clearpage
\section{Introduction}
Lepton-hadron (nucleus) deep inelastic scattering (DIS)
has been a powerful tool to understand the fundamental
structure of nucleons and nuclei. Decades of DIS
experiments have revealed the point-like substructure of
quarks and gluons inside the nucleon, and how they share
the longitudinal momentum of a fast-moving nucleon.
To develop a deeper understanding of the quark-gluon
structure and dynamics (especially in three dimensions)
of matter, governed by
quantum chromodynamics (QCD), a high energy and
luminosity polarized electron-ion collider (EIC) has recently
been endorsed to be built at Brookhaven National Laboratory
(BNL) by the late 2020s~\cite{NAP25171} as a high priority
on the agenda of the US nuclear physics community. The EIC
is capable of carrying out deep inelastic electron-proton
and electron-nucleus collisions with polarized beams
at a center-of-mass energy ($\roots$) up to
140~GeV~\cite{Accardi:2012qut,NAP25171}. It will
establish a new QCD frontier to address key open questions
such as the origin of nucleon spin, mass, and the emergence of QCD
many-body phenomena at extreme parton densities. At CERN,
the Large Hadron-electron Collider
(LHeC)~\cite{Agostini:2020fmq} at CERN
has been proposed as a possible
extension to the Large Hadron Collider (LHC)
to explore the TeV energy regime of DIS with high luminosities. As a potential long-term step beyond the LHC at CERN,
the Future Circular Collider (FCC) proposed to be built in a new 100~km tunnel
also includes a mode of electron-hadron collisions
(FCC-he) at $\roots$ = 3.5~TeV~\cite{Abada:2019lih}
by utilizing the LHeC's electron beam.
We propose an alternative approach to achieve the next-generation
lepton-hadron (ion) collider at TeV scales using high-energy
muon beams based on existing hadron collider facilities:
(1) a Muon-Ion Collider (MuIC) at BNL to succeed the EIC after its mission is completed by 2040s; and/or (2) a Large Hadron-Muon Collider (LHmuC) at CERN that can operate concurrently with
hadron collisions at the LHC.
Possibilities of muon-hadron colliders and their scientific
potential have been discussed previously,
for example in Refs.~\cite{Shiltsev:1997pv,Ginzburg:1998yw,doi:10.1063/1.56424,Sultansoy:1999na, Cheung:1999wy, Acar:2016rde,Canbay:2017rbg,Acar:2017eli, Caliskan:2018vep,Ketenoglu:2018fai, Kaya:2019ecf, Ozansoy:2019rmu, Aydin:2021iky, Cheung:2021iev}.
A MuIC at BNL is first proposed in Ref.~\cite{Acosta:2021qpx} by some authors of this paper.
The muon collider proposal has received revived interests
in the particle
physics community in recent years because of
its potential of reaching very high energies
in a compact tunnel (e.g., the size of the LHC)
at relatively low costs. It has
been argued that the discovery potential of certain
unknown hard processes or heavy particles at
a 14~TeV $\mu^{+}\mu^{-}$ collider matches that at
a 100~TeV proton-proton collider
(e.g., FCC-hh)~\cite{Delahaye:2019omf, MCwpPhysics}, as all of the available beam
energy is carried by the interacting muons. While it is in
principle feasible to build the next electron/proton colliders
with more advanced magnets and the
construction of a new, longer {$\mathcal O$}(100)~km tunnel,
the affordability is a concern in terms of both
cost and time. Exploring new accelerator
and collider technology not only provides attractive alternatives
but also will be transformative in bringing the field of
nuclear and particle physics the furthest to new energy frontiers.
Development of muon collider technology is still at the pre-conceptual stage, with many challenges to overcome~\cite{Delahaye:2019omf}. Specifically, muons are short-lived and decay rapidly even when accelerated to TeV
energies. Effective cooling of muon bunches to reduce its
phase space in six dimensions~\cite{MICE:2019jkl} and
subsequent rapid acceleration
are among the crucial elements for realizing a high energy,
high luminosity muon collider.
Beam backgrounds from muon decays also pose challenges
to both the accelerator and the detectors.
Radiation hazards from interacting neutrinos would need
to be mitigated, especially at {$\mathcal O$}(10)~TeV energies.
There is a consensus in the muon collider community that
realizing a smaller-scale muon
collider would be a necessary intermediate step to serve
as a demonstrator before pursing the ultimate
{$\mathcal O$}(10)~TeV $\mu^{+}\mu^{-}$ collider.
Such a demonstrator
still requires tremendous R\&D effort and significant cost, so
a compelling science program that is not
accessible by other proposed facilities is needed. For example,
the physics of a $\mu^{+}\mu^{-}$ collider with $\roots$ of
several hundred GeV to 1 TeV may not be competitive with
an $e^+e^-$ collider proposed with more
established technology, such as the International
Linear Collider (ILC)~\cite{Behnke:2013xla}, the Compact Linear
Collider (CLIC)~\cite{Charles:2018vfv}, the Circular Electron Positron
Collider (CEPC)~\cite{CEPCStudyGroup:2018ghi}, and
the FCC-ee~\cite{Abada:2019lih}.
The international muon collider collaboration (IMCC) was
established recently and is focusing on investigating
the physics potential and design of a 3~TeV $\mu^{+}\mu^{-}$
collider, which is discussed in a separate white paper submitted
to the Snowmass 2021 workshop \cite{MCwp3TeV}.
In this white paper, we focus on the proposal of
muon-hadron (ion) colliders at TeV energies. A MuIC and/or
LHmuC will be simultaneously a discovery machine and a technology
demonstrator to establish a novel muon collider. It has great
potential to attract worldwide interest and funding
resources from both the nuclear and particle physics communities,
and thus can provide a realistic path, or staging option, toward
the realization of an ultimate $\mu^{+}\mu^{-}$ collider at $\roots=10$~TeV and beyond.
\section{Concept and Design Overview}
\label{sec:muic}
\subsection{Collider Configuration Options}
The principal thrust of our proposal is the development of
a muon-ion collider as an upgrade to existing hadron collider
facilities, such as RHIC/EIC and the LHC, although other facilities are not excluded but may require hadron ring development.
These muon-ion collider design scenarios are summarized
in Table~\ref{fig:designtable} along with the expected
maximum possible instantaneous luminosity (see more details
on the luminosity estimation in Section~\ref{sec:lumi}).
\begin{table}[thb]
\caption{Benchmark beam energies for several $\mu p$ collider configurations studied here, the corresponding center-of-mass energy ($\sqrt{s}$), and the expected maximum possible instantaneous luminosity. }
\centering
\includegraphics[width=0.9\linewidth]{Figures/Table1.png}
\label{fig:designtable}
\end{table}
{\bf A Muon-Ion Collider (MuIC) at BNL} would reuse the hadron ring
at the EIC with a maximum proton energy of 275~GeV and replace the
electron storage ring with a high-energy muon one with a maximum
energy of about 1~TeV ($p^{\mu}=0.3Br$), assuming $B=11$~T
dipole bending magnets
(developed for the HL-LHC) and a bending radius of $r=290$~m of the
EIC tunnel. The electron injection
and acceleration chain will be replaced by the proton-driver
muon injection front-end including muon cooling, and a muon
acceleration ring residing either inside the EIC tunnel or
in a separate, larger tunnel.
The MuIC's center-of-mass energy of 1.0~TeV and luminosity of
10$^{33}$--10$^{34}$~cm$^{-2}$s$^{-1}$ is comparable to the
proposed LHeC electron-hadron collider of 1.2~TeV at CERN~\cite{Agostini:2020fmq}
with a 50~GeV electron beam incident on one of the LHC proton rings.
However, if the EIC hadron ring could be similarly upgraded to
1~TeV energy, or if a 1~TeV muon storage ring were added to any
other hadron ring facility with a 1~TeV proton energy
(e.g. the Fermilab Tevatron), an option for doubling the
center-of-mass energy to 2~TeV would be possible, which we
call the {\bf MuIC2} option. Additionally, the MuIC and MuIC2
options at BNL would offer the unique advantage of providing
polarized beams, which is important for understanding the nucleon
spin puzzle, as documented in the EIC white paper~\cite{Accardi:2012qut}.
The ability to run at lower energies (and lower luminosities)
always exists, and may in fact be the starting point in the
development and commissioning of a new
high-energy muon storage ring. Thus the MuIC could run at a
lower center-of-mass energy to facilitate measurements in
overlap with past experiments. For example, the MuIC running
with a ${\sim}100$~GeV muon beam at BNL would have an equivalent
center-of-mass energy to HERA. In fact a beam energy of 65~GeV,
such as might be extracted from a ``Higgs factory'' muon collider
facility, would provide 85\% of the HERA center-of-mass energy
when collided with the BNL hadron beam.
{\bf A Large Hadron-Muon Collider (LHmuC) at CERN} would
achieve an even higher energy, if one of the LHC 7~TeV proton rings
were utilized (as proposed for the LHeC) to collide
with a new TeV-scale muon beam. If we assume that a 3~TeV $\mu^+\mu^-$
collider were to be constructed at CERN as
proposed by the IMCC, one of the 1.5~TeV muon beams
could be brought into collision with one of the LHC 7~TeV hadron rings
to achieve a 6.5~TeV muon-proton center-of-mass energy.
The LHC, LHmuC and
3~TeV $\mu^+\mu^-$ colliders can in principle be operated
concurrently, maximizing the scientific program potential
at CERN. We note that this center-of-mass energy exceeds by a
factor 2 even that possible at an electron-hadron collider facility
where a 50~GeV electron beam is brought into collision with a
50~TeV proton beam at a new large circular ring (FCC-eh), and
avoids the necessity of a new 100~km circular tunnel.
Of course a TeV-scale muon beam brought into collision
with a ${\mathcal O}$(50)~TeV FCC proton ring would reach even higher energies.
The proposed muon-hadron (ion) colliders in this paper can be
compared with other past and proposed future lepton-hadron
facilities in Fig.~\ref{fig:discollider}, which shows the
evolution of the instantaneous luminosity and center-of-mass
energy of the colliders, including multiple operating energies
for some facilities. The muon collider technology, if realized,
provides an alternative way of entering the TeV regime of
DIS physics based on existing facilities and infrastructure,
probing the structure of nucleon and nucleus down to
an unexplored Bjorken-$x$ regime of 10$^{-7}$--10$^{-8}$.
\begin{figure}[thb]
\centering
\includegraphics[width=0.9\linewidth]{Figures/luminosity-energy2.pdf}
\caption{Instantaneous luminosity and $\sqrt{s}$ for various past and
proposed future lepton-hadron colliders. Multiple operating energies
are also shown for some facilities.}
\label{fig:discollider}
\end{figure}
\subsection{Luminosity and Performance Matrix}
\label{sec:lumi}
A recent estimation of the instantaneous luminosity of a muon-proton collider
is discussed in Ref.~\cite{Kaya:2019ecf}, which can be generally expressed as follows:
\begin{equation}
\mathcal{L}_{\mu p} = \frac{N^{\mu}N^{p}}{4\pi\max[\sigma^{\mu}_{x},\sigma^{p}_{x}]\max[\sigma^{\mu}_{y},\sigma^{p}_{y}]}\min[f^{\mu}_{c},f^{p}_{c}]H_{hg},
\end{equation}
\noindent Here, $N^{\mu}$ and $N^{p}$ represent the number of
particles per respective beam bunch. The transverse
RMS beam size in $x$ and $y$ for the muon and proton
beam, $\sigma^{\mu,p}_{x,y}$, is calculated as
\begin{equation}
\sigma^{\mu,p}_{x,y} = \sqrt{\varepsilon_{x,y}^{*}\beta_{x,y}^{*}m^{\mu,p}/E^{\mu,p}}
\end{equation}
where, $\varepsilon^{*}$ is the normalized transverse emittance
and $\beta^{*}$ is the amplitude function at the interaction point.
One can see that the luminosity
is determined by the beam having a larger size.
The $f^{\mu,p}_{c}$ is the bunch frequency. Typically,
the bunch frequency of proton beams is 2--3 orders of magnitude
larger than that of muon beams, so ${L}_{\mu p}$ is largely
determined by $f^{\mu}_{c}$, which is equal to the muon bunch
repetition frequency ($f_{\rm rep}$) multiplied by the number of
cycles ($N_{c}$) muons can make in a circular storage ring
before decaying away (a muon bunch would decay away long before
the next one is injected so there will be just one or one train
of muon bunches in the ring at a time). Each muon bunch will
survive an average of about $300 B$(Tesla) cycles in a ring.
For simplicity, the hour-glass factor, $H_{hg}$, is assumed to be unity.
\begin{table}[t!]
\centering
\caption{Proposed beam parameters and estimates of achievable luminosity
for MuIC at BNL and LHmuC at CERN.}
\vspace{0.5cm}
\includegraphics[width=0.9\linewidth]{Figures/Luminosity.png}
\label{tab:table2}
\end{table}
The proposed parameters of MuIC and LHmuC are listed in
Table~\ref{tab:table2}. For the muon beam,
we use the proposed parameters of the proton driver scheme
from Ref.~\cite{Palmer:2014nza,Delahaye:2019omf}.
The muon bunch repetition frequency is taken to be 12--15~Hz.
The proton beam parameters are assumed to be those
achieved at RHIC~\cite{Aschenauer:2014cki} and the
LHC~\cite{Kaya:2019ecf}, or foreseen to be achieved
at EIC~\cite{eic_cdr}. Assuming the implementation of strong
hadron cooling at EIC, the normalized transverse emittance
of the proton beam is expected to be as small as 0.3$\mu m$.
Note that the proposed muon bunch intensity is an order of magnitude
larger than that of the proton bunch. Such intensity muon
beam will likely disturb the proton beam. To minimize the
beam-beam effect, we propose to split the muon bunch into a
train of 10 bunches. Also note that the EIC design
at BNL adopts a flat transverse beam profile with the horizontal
dimension stretched to be much larger than the vertical
dimension.
At the interaction point, the two beams
would intersect at a finite crossing angle of 25~mrad
in the horizontal plane. The purpose of such a design for EIC is to
maximize the luminosity and, at the same time, fulfill
other requirements such as the possibility of detecting scattered
protons with a transverse momentum as low as 200~MeV by Roman Pots
inside the beam pipe. For simplicity, our estimates assume
round transverse profile of beams. Details of the beam profile
and crossing angle should be further optimized in a conceptual design.
With above parameters, we achieve a peak $\mathcal{L}_{\mu p}$
up to ${\approx}5\times$10$^{33}$~cm$^{-2}$s$^{-1}$ at the highest muon energy for MuIC.
If we assume a running period of 28 weeks per year and a duty cycle
of 0.5, the total delivered integrated luminosity over
10 years are listed in Table~\ref{tab:table2}. A few hundreds of
fb$^{-1}$ can be expected at the highest muon energies.
To put the MuIC and LHmuC luminosity into context, it is
anticipated that the EIC will deliver an integrated luminosity up to about 1.5~fb$^{-1}$/month with
$\mathcal{L}_{ep} \approx 10^{33}$~cm$^{-2}$s$^{-1}$,
and most of science cases studied at the EIC require a total
integrated luminosity of 10~fb$^{-1}$ ($\approx 30$ weeks of operations).
Therefore, even with much less stringent requirements on the
muon beam, e.g., a peak $\mathcal{L}_{\mu p} \approx 10^{32}$~cm$^{-2}$s$^{-1}$ operating for several years,
the MuIC will still be a novel facility that breaks new ground
in high energy nuclear and particle
physics with cutting-edge technology.
\subsection{Design Status}
The design of a muon-ion collider is still at the
very conceptual stage. The design and development of its
muon component, including the muon front-end and the acceleration
and collision rings, almost completely overlaps with that
of $\mu^+\mu^-$ colliders, where the design status, parameters,
and challenges are described in Ref.~\cite{Delahaye:2019omf}.
The Muon Accelerator Program (MAP) collaboration in the U.S. carried out
detailed studies and designs of muon colliders in 2011--2017.
Sketches of possible MuIC and LHmuC designs are shown in Fig.~\ref{fig:muic-sketch}.
For the MuIC at BNL (Fig.~\ref{fig:muic-sketch}a, derived from the EIC design),
essentially all electron-related components will
be replaced by those for muon beams. The proton-driver scheme is
the natural choice at BNL. A dedicated muon front-end is needed to provide the
muon injection, cooling, and initial acceleration. Ideally, both muon
acceleration and storage rings would fit within the existing tunnel to
minimize the infrastructure cost. There are six straight sections of
the BNL tunnel that could be used for the accelerator. On the other hand,
the design by the MAP collaboration favors a large ring with long
straight sections for fast acceleration and a more compact circular ring
for muon storage and collisions. Therefore, it is conceivable that
a new race-track tunnel would be added for muon acceleration, as also illustrated in
Fig.~\ref{fig:muic-sketch}a, which still fits well within the BNL campus.
At CERN, if a 3~TeV $\mu^+\mu^-$ collider is built by the IMCC, it can
be constructed to intersect with one of LHC proton beams (e.g., at IP2 as
in the LHeC proposal, Fig.~\ref{fig:muic-sketch}b). In this way, $\mu^+\mu^-$, $\mu p$, and $pp$ programs
can operate concurrently at CERN, leading to a rich scientific program and
extending the lifetime of the LHC infrastructure. Once the muon collider
technology is established, an upgrade to a ${\cal O}(10+)$~TeV $\mu^+\mu^-$ collider
can be considered using the LHC tunnel as the next stage.
\begin{figure}[thb]
\centering
\includegraphics[width=\linewidth]{Figures/MuIC-sketch.pdf}
\caption{Design sketches for (a) Muon-Ion Collider at BNL and (b) Large Hadron-Muon Collider (LHmuC) at CERN.}
\label{fig:muic-sketch}
\end{figure}
\subsection{Design Challenges and R\&D requirements}
We discuss some of muon-ion collider specific design challenges below:
\begin{itemize}
\item {\bf The muon-ion interaction point} would require dedicated
design and R\&D efforts to maximize the machine's luminosity and
meet all the science requirements. The size and intensity of muon and proton bunches are different, so how to ensure the proper crossing
of two beams without interfering each other would be a topic of
R\&D. It may also be advantageous to separate one high intensity
muon bunch into a train of lower intensity bunches to avoid disturbing the proton beam. Nevertheless, similar challenges
are also present in the design of the electron-ion collider, where
insights will be learned.
\item {\bf Machine-detector interface (MDI):} Because of the
asymmetric colliding configuration, the MDI requires special
consideration somewhat different from that for $\mu^+\mu^-$ colliders. For
example, the shielding tungsten nozzles necessary for protecting detectors against
the secondary particle background from muon decays should only be
necessary on the incoming muon side, instead of both beam directions,
which otherwise restrict the detector acceptance to approximately
$|\eta|<2.4$.
Full simulations are needed to verify the level of beam-induced
background with a single-sided nozzle configuration for muon-ion
collisions. This will have an impact on the available detector
acceptance, which is discussed in Section~\ref{sec:detector}.
\item {\bf Neutrino radiation protection and mitigation:}
One peculiar type of radiation background from muon colliders is
induced by collimated high intensity neutrinos from muon decays, which are mainly
aligned with the plane of the collider. This is commonly
not a concern for proton-proton or e$^{+}$e$^{-}$ colliders.
Here, the main issue is not related to the direct interactions of neutrinos
with human bodies but instead from long-term stationary exposure to secondary particles produced when neutrinos traverse dense materials (soil, buildings, etc.).
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\linewidth]{Figures/neutrino-sketch.png}
\caption{A sketch to illustrate the proposed strategy of mitigating
neutrino radiation backgrounds for MuIC at BNL.}
\label{fig:neutrino-sketch}
\end{figure}
Various approaches for mitigating neutrino radiation backgrounds from multi-TeV $\mu^+\mu^-$ collider are extensively discussed, for example, in Ref.~\cite{NeutrinoMitigate}.
Here, we specifically discuss the possible strategy of mitigating
neutrino backgrounds for the MuIC at BNL. The BNL tunnel is effectively
on the surface located near the tip of Long Island, NY. As long as
we can direct radiated neutrinos to not strike on-site buildings at BNL,
their impact can be kept minimal. A sketch to illustrate our proposed
mitigation strategy is shown in Fig.~\ref{fig:neutrino-sketch}. By tilting
the muon ring (assuming six straight sections as with the existing RHIC tunnel)
by a small angle, e.g., $\theta \lesssim 1^{\circ}$, it would be sufficient
to direct most of neutrinos toward the air in one direction and toward the ground
and sea in the other direction for a given straight section, with little impact on buildings nearby.
\item {\bf Muon beam polarization:} A unique feature of the EIC is its
doubly-polarized beams, which is critical for understanding the nucleon
spin puzzle and also many other physics processes which are spin or helicity
dependent. Therefore, maintaining the capability of muon beam polarization
would be highly beneficial to the science program. The possibility of maintaining
the muon beam polarization has been discussed in Refs.~\cite{Cline:1996yi,Norum:1996mi,Neuffer:1999aw,Ankenbrandt:1999cta}.
Muon beams are produced with about 20\% longitudinal polarization. By extracting
higher energy muons from the injection, the average polarization can be
increased up to 50\% with a compromise in the luminosity. Loss of polarization
after muons go through the ionization cooling stage is estimated to be negligible.
With a series of spin rotators, it is in principle possible to manipulate and
maintain the muon spins. However, significant efforts are needed to investigate
such a feasibility and arrive at a credible design.
\end{itemize}
\subsection{Staging Options and Desirable Demonstrators}
Staging options of MuIC have been indicated in our proposed energy
and luminosity scenarios in Tables~\ref{fig:designtable} and \ref{tab:table2},
and also in Fig.~\ref{fig:discollider}. Starting with a 100~GeV muon beam
at the MuIC would already match the center-of-mass energy and luminosity
of HERA II. A 100~GeV muon beam is much less demanding in muon cooling
and fast-ramping magnet technology, so it serves as a good initial
demonstrator and can be directly compared with HERA data as a calibration.
Next, the muon beam energy can be ramped up to 0.5 TeV and eventually 0.96~TeV
to reach a TeV lepton-hadron collider. If the hadron beam can also be
upgraded to about 1~TeV, the MuIC center-of-mass energy can then be further doubled
(MuIC2).
\section{Synergies with existing facilities in high energy and nuclear physics
communities}
A key merit of the muon-ion collider proposal is the strong synergy
with existing accelerator facilities in the high energy and nuclear physics
communities. Leveraging the established infrastructure, accelerator
expertise, and user community has enormous financial benefits and
is a recipe for success in the evolution of high energy physics research.
The RHIC at BNL and the LHC at CERN are currently the only operating hadron and ion colliders in the world.
At BNL, the realization of the EIC would be nearly impossible if built
from scratch without the existence of RHIC. At CERN, the LHC was constructed
using the LEP tunnel and the existing PS and SPS accelerators. While we are not advocating against constructing more ambitious new infrastructure,
maximizing the utility
of existing ones is something the community should always try to leverage,
especially under the strong financial constraints nowadays.
The muon collider is an attractive but yet unproven concept. The technology
needs to be first demonstrated, and a staged approach is necessary to
achieve the ultimate $\mu^+\mu^-$ collider at 10~TeV and beyond. While it is
ideal to already identify a path toward the ultimate energy, realizing a
muon collider in any form with a strong science program would be a significant
step forward as a technology demonstrator and helps raise the priority of
muon colliders in future planning processes for high energy physics (HEP) collider programs.
The MuIC concept at BNL would serve as such a technology demonstrator, and it has the
potential for the U.S. to take a leadership role in future novel TeV colliders.
The science case of a high energy, high luminosity lepton-ion collider has
already been well established with the EIC, so the MuIC can be considered
as an upgrade to the EIC.
Potential collaboration between HEP and Nuclear Physics (NP) communities (supported by different
offices at funding agencies such as DOE and NSF) could make it more affordable to each community separately,
and helps attract broader interests worldwide. Once the technology is demonstrated,
a higher energy $\mu^+\mu^-$ collider could then be explored at the same site
or at a new, more suitable site. The muon front-end system that is expected to be
costly can even be transferred to another site if necessary and cost effective.
Similarly, the LHmuC concept will significantly broaden and enrich the CERN
program during or beyond the HL-LHC. The LHC tunnel could also be a candidate for
hosting a future 10~TeV $\mu^+\mu^-$ collider.
Our view on a possible road map to a future combined facility for
nuclear physics (NP) and high energy physics
(HEP) based on muon collider technology is presented in Fig.~\ref{fig:timeline}.
Based on the assessment of IMCC, 15--20 years is required to fully
establish the feasibility of a muon collider via intense R\&D
at a testing facility in order to be ready to construct a
muon collider demonstrator. We strongly advocate the HEP and NP
communities to carry out this effort jointly to open new frontiers
in their respective fields. From the NP perspective, the EIC
is the highest priority at present and there is no other program
on the horizon planned beyond the EIC. The next NP long-range planning
process is likely to start in later 2022, so it is an excellent time to
consider the MuIC as a future option and establish a R\&D program. On the
HEP side, the landscape is more complex with many different
proposals of future colliders. In our view, a Higgs factory with $e^{+}e^{-}$
collisions is likely the immediate next step beyond the HL-LHC, because of
its technological maturity and well-defined deliverables of high interest.
If NP and HEP jointly develop the muon collider, a 3~TeV
$\mu^+\mu^-$ collider can be ready for construction by the IMCC at a similar
time scale to the MuIC. With the success of the first muon collider projects,
the community would be ready to pursue a ${\cal O}(10+)$~TeV $\mu^+\mu^-$ collider
(e.g., at CERN) in the 2050s--60s.
\begin{figure}[thb]
\centering
\includegraphics[width=0.95\linewidth]{Figures/timeline.png}
\caption{A possible road map toward realizing future muon colliders by synergizing
efforts of both the nuclear physics and high energy physics communities.}
\label{fig:timeline}
\end{figure}
\section{The Science Case of TeV Muon-Ion Colliders}
The science case for a MuIC using the BNL facility was outlined in our initial concept proposal~\cite{Acosta:2021qpx}, although the scientific
potential of muon-hadron colliders in general have been discussed previously,
for example in Refs.~\cite{Ginzburg:1998yw,doi:10.1063/1.56424, Sultansoy:1999na, Cheung:1999wy, Canbay:2017rbg,
Acar:2017eli, Caliskan:2018vep, Kaya:2019ecf, Ozansoy:2019rmu, Aydin:2021iky,
Cheung:2021iev}.
A large part of the physics case of a TeV collider also overlaps
with that of the proposed LHeC, given the nearly equivalent center-of-mass
energies. This includes structure function measurements, precision electroweak
and QCD measurements, Higgs boson studies, and beyond Standard Model searches.
Therefore, a more comprehensive set of topics is explored in
Ref.~\cite{Agostini:2020fmq}.
However, the initial state muon at the MuIC provides a complementary
sensitivity to any lepton flavor violating processes, and the scattering
kinematics can be quite different than the LHeC, which can lead to
different and sometimes advantageous experimental conditions
(see Ref.~\cite{Acosta:2021qpx} and Appendix~\ref{sec:dis-kin}).
Additionally, the MuIC at BNL also offers the possibility of the
polarization of both beams, and a wide range of ion species for
lepton-ion collisions, enabling a broad nuclear physics program
and detailed studies of the spin structure of nuclei.
Thus the MuIC can extend to new regimes the science program of
the EIC that is documented in the EIC NAS report~\cite{NAP25171}
and Yellow Report~\cite{AbdulKhalek:2021gbh}.
Here we outline the science program of a TeV muon-hadron (ion) collider
and report on a subset of topics with quantitative studies to illustrate
the physics potential, for several target energies and facilities.
\begin{figure}[t!]
\centering
\includegraphics[width=0.7\linewidth]{Figures/Q2xAllColliders-mup.pdf}
\includegraphics[width=0.7\linewidth]{Figures/Q2xAllColliders-muA.pdf}
\caption{Kinematic coverage of $Q^{2}$ and $x$ in deep
inelastic lepton-proton (top) and lepton-nucleus (bottom) scattering
for the muon-ion collider design options presented here and for the
EIC at BNL, HERA at DESY, and the LHeC and FCC-eh options at CERN,
each at their maximum beam energies. The inelasticity ($y$) range
is assumed to be $0.01<y<0.95$ (hatched areas). The long dashed lines
indicate the saturation
scale as a function of $x$ in the proton and the gold ($^{197}$Au)
nucleus from the GBW model~\cite{GolecBiernat:1998js}.}
\label{Q2x-all}
\end{figure}
The physics potential of lepton-hadron (ion) colliders is generally
represented and compared by their kinematic coverage in $Q^{2}$ and $x$,
as shown in Fig.~\ref{Q2x-all} for various past and future colliders.
The MuIC, and equivalently the LHeC, will significantly
extend the kinematic coverage of the EIC to much larger $Q^{2}$ and
smaller $x$ regimes, by an order of magnitude in each compared to the
previous HERA $ep$ collider. In lepton-proton collisions (Fig.~\ref{Q2x-all}, top),
the saturation or non-linear QCD regime parametrized by the GBW model based on the HERA
inclusive data~\cite{GolecBiernat:1998js} is clearly out of the reach for the EIC. However,
it becomes within reach at very small $x$ values at the MuIC and LHeC,
and is opened further by higher energy colliders such as the proposed LHmuC and FCC-eh.
Figure~\ref{Q2x-all} also shows that the LHmuC, by colliding the existing LHC proton beam with a 1.5~TeV muon beam,
will provide even larger coverage in $Q^{2}$ and $x$ than the FCC-eh.
In lepton-nucleus collisions (Fig.~\ref{Q2x-all}, bottom),
choosing the $^{197}$Au nucleus as a representative example,
a factor of 6 enhancement in the saturation scale to the proton
is expected, $Q_{\rm sat}^{2}$($x$, Au$)=A^{1/3}\, Q_{\rm sat}^{2}$($x$, proton).
While the EIC approaches the saturation regime, the MuIC and LHmuC
will bring us well into the domain to explore gluon saturation
and nonlinear QCD phenomena.
\begin{figure}[thb]
\centering
\includegraphics[width=0.7\linewidth]{Figures/muicvsmumu_discovery.pdf}
\caption{The center-of-mass energy of $\mu p$ colliders versus that of
$\mu^+\mu^-$ colliders, which yield equivalent cross sections.
The solid curve corresponds to the $2\rightarrow1$ annihilation process,
while the dashed curve corresponds to the $2\rightarrow2$ scattering process.}
\label{fig:muic-vs-pp}
\end{figure}
To evaluate and compare discovery potentials of new
physics beyond the Standard Model, Ref.~\cite{AlAli:2021let}
argues that a $\mu^+\mu^-$ collider can be competitive
with a $pp$ collider at 5--20 times higher center-of-mass
energy (\roots)
for discoveries via certain processes such as
annihilation ($2\rightarrow1$) and vector
boson fusion ($2\rightarrow2$). This is because a muon carries
100\% of its available momentum in the interaction process, while
only a small fraction of the proton momentum is carried by a parton. We carry out
a similar calculation for the muon-proton collider so that
all three types of colliders can be compared. The parton luminosity
of a $\mu p$ collider is expressed as,
\begin{equation}
\frac{dL_{i}}{d\tau}(\tau,\mu_{f})=\int_{\tau}^{1}\frac{dx}{x}f_{i}(x,\mu_{f}).
\end{equation}
\noindent
Here the $f_{i}(x,\mu_{f})$ are the parton distribution
functions (PDFs) for parton $i$ carrying a
fraction $x$ of the longitudinal momentum, at factorization scale
$\mu_{f}=\sqrt{\hat{s}}/2$, where $\hat{s}$ is the partonic
center-of-mass energy and $\tau=\hat{s}/s$. The resulting
center-of-mass energy of $\mu p$ colliders versus that of
$\mu^+\mu^-$ colliders, which yield equivalent cross sections,
is shown in Fig.~\ref{fig:muic-vs-pp}. The CT18NNLO PDF set is chosen
for this calculation. We find that a $\mu^+\mu^-$ collider is equivalent to
a $\mu p$ collider with 1.5$\times$ higher \roots\ in
terms of its discovery potential. Therefore, the proposed LHmuC with $\roots=6.5$~TeV
matches the potential of a $\mu^+\mu^-$ collider at $\roots=4.3$~TeV,
exceeding the 3~TeV $\mu^+\mu^-$ collider proposed by the IMCC.
This exercise again highlights the unique opportunities by the TeV muon-ion (proton)
collider proposal, taking advantage of existing facilities.
\begin{figure}[thb]
\centering
\includegraphics[width=0.9\linewidth]{Figures/physicslandscape.pdf}
\caption{The landscape of science at lepton-hadron (ion) colliders with
required energy and luminosity.}
\label{fig:muicpotential}
\end{figure}
The landscape of science programs at lepton-hadron (ion) colliders with
their required energy and luminosity is illustrated in Fig.~\ref{fig:muicpotential}.
Higgs boson measurements and beyond the Standard Model searches are the most demanding
in terms of the reach in energy and luminosity (${>}5 \times 10^{33}$cm$^{-2}$s$^{-1}$).
Studies of QCD physics at extreme parton densities generally
prefer high \roots\ in order to probe the smallest possible $x$ regime, but
a luminosity at the level of 10$^{32}$~$\text{cm}^{-2} \text{s}^{-1}$\ is sufficient
because of the large cross sections. Studies of PDFs, including the spin and
flavor structure of nucleons and nuclei, require a moderate luminosity
up to about 10$^{33}$~$\text{cm}^{-2} \text{s}^{-1}$\ (where beam polarization is important
for the spin physics program). With a peak luminosity of
10$^{33}$--10$^{34}$~$\text{cm}^{-2} \text{s}^{-1}$, precision electroweak
physics can be carried out as well as three-dimensional
spatial and momentum imaging and tomography of nucleons and
nuclei. Therefore, even if the maximal proposed luminosity cannot
be achieved, there is still a very rich physics program to explore
at muon-ion colliders.
We discuss the physics potential in detail below, including
quantitative studies on a few topics. We look forward to
engaging the broader nuclear physics community to study
the potential of muon-ion colliders.
\subsection{QCD and Nuclei}
The scientific potential of lepton-hadrons in understanding
the physics of QCD, nucleon and nuclei has been discussed
in detail
in a series of documents on the electron-ion collider to
be constructed at BNL by the 2030s~\cite{Accardi:2012qut,AbdulKhalek:2021gbh,NAP25171}.
We briefly highlight a few selected fundamental questions
in QCD and nuclear physics that the MuIC or LHmuC
will explore to unprecedentedly small-$x$ and large-$Q^{2}$ regimes
as a future extension of the EIC science program.
\subsubsection{Nucleon Spin and 3-D Structure}
The nucleon spin is one of its fundamental properties.
It was found that quark polarization inside a nucleon only
contributes to about 30\% of the total spin. Therefore, the
majority rest of the nucleon spin must be carried by the
gluon polarization and orbital motion of quarks and gluons.
To determine the contribution of gluon polarization,
a measurement of the helicity-dependent gluon distribution function,
$\Delta g(x)$, especially in the small $x$ region, is crucial.
The uncertainty on the overall gluon polarization from RHIC measurements
is still rather large mainly because of the limitation in accessing
the small $x$ region~\cite{Abdallah:2021aut}. For $x<0.01$,
$\Delta g(x)$ is largely unconstrained. With polarized beams,
the EIC is projected to significantly improve the precision
of gluon polarization by accessing $x$ values down to 0.001 at
$Q^{2} \approx 10$~GeV$^{2}$~\cite{Aschenauer:2012ve}, with an
integrated luminosity of 10~fb$^{-1}$. The early phase of MuIC
is already capable of delivering 10~fb$^{-1}$ in one year data taking
at a center-of-mass energy substantially higher than EIC (e.g.,
with a 500~GeV muon beam). Assuming a polarized muon beam, the MuIC
will extend the reach of gluon polarization in $x$ down to $10^{-5}$,
potentially providing a more definitive answer to the gluon spin
contribution. Furthermore, precise measurements
of three-dimensional (3D) parton distribution functions,
generalized parton distributions (GPDs) and
transverse-momentum-dependent (TMD) distributions, over
a much wider range of $x$ values at the MuIC could provide a
complete picture of orbital angular momentum of quarks and
gluons inside the nucleon.
\subsubsection{Gluon Saturation at Extreme Parton
Densities in proton and nucleus}
The gluon density inside the nucleon increases
dramatically toward small $x$ values.
At extreme gluon densities, the nonlinear QCD
process of gluon-gluon fusion will start
playing a key role to limit the divergence of
the gluon density. At a certain dynamic scale of
momentum transfer, known as $Q_{s}$, gluon splitting and fusion
processes reach an equilibrium such that the gluon density
is saturated, resulting in novel universal properties of
hadronic matter. Examples of gluon saturation scales
inferred from fits to HERA data, known as the GBW model,
\cite{GolecBiernat:1998js} are shown Fig.~\ref{Q2x-all}
for muon-hadron and muon-ion colliders. The large $Q_{s}$
scale predicted at small $x$ values, especially in large
nuclei (enhanced by a factor of $A^{1/3}$),
enables perturbative QCD calculations of nuclear
structure functions, as proposed in the color-glass
condensate (CGC) effective field theory~\cite{McLerran:1993ni}.
Predictions of signatures of gluon saturation in large nuclei
at the EIC are presented in Ref.~\cite{Accardi:2012qut}.
As shown in the kinematic coverage of Fig.~\ref{Q2x-all},
the EIC starts entering the domain of gluon saturation in
gold nuclei at $x \approx 10^{-3}$, while the MuIC and LHmuC will
bring us well into the saturation regime.
The MuIC will also probe the saturation
regime in the proton and other light nuclei for the first time,
which is likely not accessible by the EIC.
\subsubsection{QCD Collectivity in Small-System Collisions}
Discovery of QCD collective behavior in high-multiplicity
final states of small-system collisions such as pp~\cite{CMS:2010ifv}
and pPb~\cite{CMS:2012qk} at the LHC has raised fundamental questions
on the initial state and dynamical evolution of non-perturbative
QCD systems. Since the discovery, the origin of observed
QCD collectivity has been intensely debated~\cite{Dusling:2015gta}.
Attempts to search for such collective effects have been
extended to even smaller systems such as $ep$~\cite{ZEUS:2019jya}, $\gamma$A~\cite{ATLAS:2021jhn}, $\gamma$p~\cite{CMS-PAS-HIN-18-008}
and $e^{+}e^{-}$~\cite{Badea:2019vey} collisions. It has also been suggested
that a collective effect can also be developed in a QCD system
as small as a single parton propagating in the vacuum~\cite{Baty:2021ugw}.
The main challenge for observing the collectivity in these
very small systems is that it is difficult to create events
with high multiplicities at energies achieved by previous
colliders. In fact, a lepton-hadron or lepton-ion collider
would be an ideal, clean environment for studying such
collectivity as Q$^{2}$ of the virtual photon can provide
a lever-arm to control
the initial size of the system. The MuIC and LHmuC will
substantially extend the event multiplicity reach of
previous lepton-hadron collisions to a regime that is
comparable to that in $pp$ collisions, providing unique
discovery potential.
\subsection{Standard Model Physics}
\subsubsection{Structure Function and QCD Measurements}
Measurements of the nucleon structure functions form the ``bread and butter'' of DIS experiments, from which information on the PDFs and spin structure (with polarized beams) of the nucleon can be obtained. This information, particularly on the flavor content of the PDFs, is quite independent from the constraints obtained from hadron collider measurements, where such information is less cleanly separated from the hard scattering process.
As already emphasized, the MuIC can perform these measurements on protons and nuclei in new regions at low $x$ and high $Q^2$. Thus, these measurements can be used to reduce the overall PDF uncertainty in calculations for cross sections at future hadron collider facilities, like FCC-hh.
Global fits to all structure function data provide information not only on the PDFs, but simultaneously provide a precise measurement of the QCD coupling parameter $\alpha_{\rm s}(Q^2)$ through the QCD evolution equations. Moreover, as the $Q^2$ reach of the MuIC extends well into the electroweak scale (see Section~\ref{sec:highq2dis}), precise electroweak parameter measurements are also possible from the global fits.
In addition to the inclusive structure function measurements,
direct measurements of multijet production in DIS also allow for a precise determination of $\alpha_{\rm s}$ and its running over a wide $Q^2$ range in a single experiment. The leading-order jet transverse momentum scales approximately as $p_{\rm T} \approx (1-y) \sqrt{Q^2}$. The coverage of the MuIC for jet measurements is very similar to that of the LHeC. The only difference is that the jets tend to be more central, trending toward the muon direction, than the LHeC, where the jets are in the proton direction.
\subsubsection{High $Q^2$ Total and Differential DIS Cross Sections}
\label{sec:highq2dis}
The TeV-scale center-of-mass energy made possible by the muon-ion collider proposal described here leads to a reach in $Q^2$ well beyond that which was achieved by the HERA $e p$ collider, and thus opens sensitivity to possible new particle interactions and substructure. This is illustrated in Fig.~\ref{fig:DiffQ2NC-allColliders}, which shows the calculated neutral-current differential scattering cross section in $Q^2$ compared across different $\mu^- p$ collider options, including at the HERA center-of-mass energy. The cross sections were calculated using Pythia~8 \cite{Sjostrand:2014zea} with the NNPDF2.3 parton density set \cite{NNPDF:2017mvq} for the inelasticity range $0.1 < y < 0.9$. The HERA experiments published differential cross section measurements with polarized and unpolarized lepton beams \cite{H1:2012qti, ZEUS:2012zcp} that were sensitive to cross sections at the level of ${\cal O} (10^{-6})$ pb/GeV$^2$ with individual data samples corresponding to ${\cal O}(100)$~pb$^{-1}$, which translated to a reach in $Q^2$ of approximately $50{\,}000$~GeV$^2$ for $\sqrt{s}=318$~GeV. For a similarly-sized data sample recorded by an experiment at a muon-ion collider, the reach in $Q^2$ would be ${\approx} 200{\,}000$~GeV$^2$ for the MuIC, ${\approx} 400{\,}000$~GeV$^2$ for the MuIC2, and ${\approx} 800{\,}000$~GeV$^2$ for the LHmuC.
However, because of the ${\sim}1/Q^4$ fall-off of the differential cross sections at high $Q^2$, the ability to probe the $Q^2$ scale corresponding to the same high $x\approx 0.5$ as HERA would require 2 (3) orders of magnitude more integrated luminosity for MuIC (MuIC2), and even 4 or more orders of magnitude for LHmuC.
The corresponding target integrated data sample sizes would therefore be 10, 100, and 1000~fb$^{-1}$ for MuIC, MuIC2, and LHmuC, respectively.
While the first two benchmarks are achievable within the first few years of operation at nominal luminosity, the latter is disfavored within 10 years in our estimation given in Section~\ref{sec:lumi}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\linewidth]{Figures/Q2_high_dcs_nc_mu.png}
\caption{The neutral-current differential cross section in $Q^2$ for unpolarized $\mu^- p$ deep inelastic scattering for several collider options: HERA with $\sqrt{s}=0.32$~TeV, MuIC with $\sqrt{s}=1.0$~TeV, MuIC2 with $\sqrt{s}=2.0$~TeV, and LHmuC with $\sqrt{s}=6.5$~TeV. The inelasticity variable is restricted to $0.1<y<0.9$.
}
\label{fig:DiffQ2NC-allColliders}
\end{figure}
The calculated differential cross sections for both neutral-current (NC) and charged-current (CC) DIS and for both muon- and antimuon-proton collisions and for $0.1<y<0.9$ are shown in Fig.~\ref{fig:DiffQ2-CCNC} for center-of-mass energies corresponding to HERA, MuIC, MuIC2, and the LHMuC. The higher energy of a muon-ion collider allows for measurements well into the electroweak unification region at high $Q^2$.
The integrated cross sections above several high $Q^2$ thresholds for neutral-current scattering and charged-current scattering are shown in Tables~\ref{tab:NCxsec} and \ref{tab:CCxsec}, respectively, to compare the production yields at the different collider facilities. The effective total charged-current cross sections (for $Q^2>1$~GeV$^2$) are also shown in Table~\ref{tab:CCxsec}, where one can see that the cross section grows from that at HERA by a factor 3--4 for the MuIC, for example.
\begin{figure}[htb]
\centering
\includegraphics[width=0.45\linewidth]{Figures/Q2_high_dcs_muic_hera.png}
\includegraphics[width=0.45\linewidth]{Figures/Q2_high_dcs_muic.png}
\includegraphics[width=0.45\linewidth]{Figures/Q2_high_dcs_muic2.png}
\includegraphics[width=0.45\linewidth]{Figures/Q2_high_dcs_LHmuC.png}
\caption{The differential cross sections in $Q^2$ for neutral-current and charged-current deep inelastic scattering in unpolarized $\mu^- p$ and $\mu^+ p$ collisions for HERA (top-left), MuIC (top-right), MuIC2 (bottom-left), and LHmuC (bottom-right). The inelasticity variable is restricted to $0.1<y<0.9$}
\label{fig:DiffQ2-CCNC}
\end{figure}
\begin{table}[!htb]
\centering
\caption{Integrated cross sections, in pb, for neutral current scattering in unpolarized $\mu^- p$ and $\mu^+ p$ collisions for various minimum thresholds on $Q^2$ in GeV$^2$ and several machine choices. The inelasticity variable is restricted to $0.1 < y < 0.9$.
\label{tab:NCxsec}
}
\begin{tabular}{l|c|c|c|c|c}
\hline
Machine & $Q^2>3\times 10^4$ & $Q^2>10^5$ & $Q^2>3\times 10^5$ & $Q^2>10^6$ & $Q^2>10^7$ \\ \hline
\multicolumn{6}{c}{$\mu^- p \to \mu^- X$} \\ \hline
HERA & 0.024 & -- & -- & -- & -- \\
MuIC & 3.7 & 0.072 & 0.0028 & -- & -- \\
MuIC2 & 9.8 & 0.59 & 0.12 & -- & -- \\
LHmuC & 37 & 3.4 & 1.1 & 0.060 & 0.012 \\ \hline
\multicolumn{6}{c}{$\mu^+ p \to \mu^+ X$} \\ \hline
HERA & 0.0051 & -- & -- & -- & -- \\
MuIC & 2.1 & 0.020 & 0.0005 & -- & -- \\
MuIC2 & 7.8 & 0.30 & 0.047 & -- & -- \\
LHmuC & 36 & 3.0 & 0.87 & 0.032 & 0.0005 \\
\hline
\end{tabular}
\end{table}
\begin{table}[!htb]
\centering
\caption{Integrated cross sections, in pb, for charged current scattering in unpolarized $\mu^- p$ and $\mu^+ p$ collisions for various minimum thresholds on $Q^2$ in GeV$^2$ and several machine choices. The inelasticity variable is restricted to $0.1 < y < 0.9$.
\label{tab:CCxsec}
}
\begin{tabular}{l|c|c|c|c|c|c}
\hline
Machine & $Q^2>1$ & $Q^2>3\times 10^4$ & $Q^2>10^5$ & $Q^2>3\times 10^5$ & $Q^2>10^6$ & $Q^2>10^7$ \\ \hline
\multicolumn{7}{c}{$\mu^- p \to \nu_\mu X$} \\ \hline
HERA & 68 & 0.038 & -- & -- & -- & -- \\
MuIC & 200 & 5.2 & 0.12 & 0.0053 & -- & -- \\
MuIC2 & 345 & 13 & 0.92 & 0.20 & -- & -- \\
LHmuC & 860 & 43 & 4.6 & 1.6 & 0.098 & 0.020 \\ \hline
\multicolumn{7}{c}{$\mu^+ p \to \overline{\nu}_\mu X$} \\ \hline
HERA & 37 & 0.00095 & -- & -- & -- & -- \\
MuIC & 160 & 1.4 & 0.0090 & -- & -- & -- \\
MuIC2 & 300 & 6.5 & 0.22 & 0.029 & -- & -- \\
LHmuC & 850 & 36 & 3.0 & 0.83 & 0.024 & -- \\
\hline
\end{tabular}
\end{table}
\subsubsection{Standard Model Production Cross Sections}\label{sec:sm_xsec}
Studies of the electroweak production of vector bosons and top quarks are essential ways to measure fundamental SM parameters, such as triple gauge boson couplings and CKM mixing matrix terms involving the top quark.
The production of $W$ bosons was measured at HERA~\cite{ZEUS:2009W,H1:2009W,H1:ZEUS:2010W}
with a total of only 23 identified events.
The MuIC would operate at a much higher center-of-mass energy and with higher luminosity, yielding orders of magnitude more $W$ and $Z$ bosons and top quarks.
This opens additional new opportunities for precision electroweak measurements beyond deep inelastic scattering,
adding to previous combinations based on data collected at LEP, the Tevatron, and the LHC~\cite{GFitter:2018}. The production of Higgs bosons is also very important, and is discussed separately in Section~\ref{sec:higgs_phys}.
The production of $W$ and $Z$ bosons in $\mu^- p$ collisions can be achieved via many diagrams, which can be categorized based on their final states into 5 processes: $Z\mu^-j$, $Z\nu_{\mu}j$, $W^+\mu^-j$, $W^-\mu^-j$, and $W^-\nu_{\mu}j$, where $j$ is the jet from the struck quark.
Figures~\ref{fig:Zmu_diagrams} and~\ref{fig:Znu_diagrams} show the leading order production diagrams of the $Z$ boson, with $\mu^{-}$ or $\nu_{\mu}$ as the scattered lepton, respectively.
Figures~\ref{fig:Wmu_diagrams} and~\ref{fig:Wnu_diagrams} show the leading order production diagrams of the $W$ boson, with $\mu^{-}$ or $\nu_{\mu}$ as the scattered lepton, respectively.
We note that Figs.~\ref{fig:Znu_diagrams}--\ref{fig:Wnu_diagrams} include triple gauge boson terms.
The production diagrams for top quarks are simpler at leading order, where
Fig.~\ref{fig:top_diagrams} shows the leading order production diagrams of single $\bar{t}$ quark and $t\bar{t}$ pair in $\mu^{-}p$ collisions. The corresponding diagrams for $\mu^+ p$ collisions can be obtained through the appropriate application of charge conjugation.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Zmuj1.pdf}
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Zmuj3.pdf}
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Zmuj2.pdf}
\caption{Diagrams of $Z$ boson production in association with $\mu^{-}$ as the scattered lepton in $\mu^{-}p$ collisions.}
\label{fig:Zmu_diagrams}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Znuj1.pdf}
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Znuj2.pdf}
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Znuj3.pdf}
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Znuj4.pdf}
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Znuj5.pdf}
\caption{Diagrams of $Z$ boson production in association with $\nu_{\mu}$ as the scattered lepton in $\mu^{-}p$ collisions.}
\label{fig:Znu_diagrams}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Wmuj1.pdf}
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Wmuj2.pdf}
\includegraphics[width=0.32\textwidth]{Figures/SM_diagrams/Wmuj3.pdf}
\caption{Diagrams of $W^{\pm}$ boson production in association with $\mu^{-}$ as the scattered lepton in $\mu^{-}p$ collisions. Both charges of the $W$ boson can be produced in these modes.}
\label{fig:Wmu_diagrams}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.33\textwidth]{Figures/SM_diagrams/Wnuj1.pdf}
\includegraphics[width=0.33\textwidth]{Figures/SM_diagrams/Wnuj2.pdf}\\
\includegraphics[width=0.33\textwidth]{Figures/SM_diagrams/Wnuj3.pdf}
\includegraphics[width=0.33\textwidth]{Figures/SM_diagrams/Wnuj4.pdf}
\caption{Diagrams of $W^{-}$ boson production in association with $\nu_{\mu}$ as the scattered lepton in $\mu^{-}p$ collisions.
Only negatively (positively) charged $W$ bosons are produced for initial state $\mu^-$ ($\mu^+$) leptons.}
\label{fig:Wnu_diagrams}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.33\textwidth]{Figures/SM_diagrams/single_top.pdf}
\includegraphics[width=0.33\textwidth]{Figures/SM_diagrams/ttbar.pdf}
\caption{Diagrams of single $\bar{t}$ production and $t\bar{t}$ production in $\mu^{-}p$ collisions.}
\label{fig:top_diagrams}
\end{figure}
Cross sections for these SM particle productions are calculated with MadGraph~\cite{Madgraph}, version 3.3.1, using the PDF set NNPDF31\_nlo\_pdfas~\cite{NNPDF:2017mvq} for the proton.
These cross sections are compiled in Tables~\ref{tab:muNeg_p_highE_Znu_xsec} to \ref{tab:muNeg_p_highE_ttbar_lepcut_xsec}.
The production processes for $Z\nu_{\mu}$, $W^-\nu_{\mu}$, and single $\bar{t}$ quark involve a $W$ boson exchange on the muon side, and the outgoing $\nu_{\mu}$ has a well-defined finite $p_{\rm T}$.
Processes for $Z\mu^{-}$, $W^{\pm}\mu^{-}$, and $t\bar{t}$ pair production, on the other hand,
involve a $Z/\gamma$ exchange on the muon side.
Their cross sections increase significantly when the virtual photon mass is close to 0, in which case the $p_{\rm T}$ of the scattered muon also approaches 0.
In total, the inclusive $W^\pm$ production cross section in $\mu^{-}p$ collisions for the MuIC at BNL is 19.4~pb, yielding $2.1\times 10^4$ leptonic $W\to \ell \nu$ decays into each lepton flavor for 10~fb$^{-1}$ of integrated luminosity. This increases by an order of magnitude for the LHmuC configuration.
\begin{table*}[!htb]
\centering
\caption{Cross sections for the $Z\nu_{\mu}$ process in $\mu^{-}p$ collisions for different beam energy configurations. The $\mu^{-}$ beam energy is unpolarized in all cases.}
\begin{tabular}{l|ccc}
\hline
$E_{\mu}\times E_{p}$ (TeV$^{2}$) & $\sigma$ (pb) & Scale unc.&
PDF$\oplus\alpha_{s}$ unc. \\
\hline
$0.96 \times 0.275$ &
0.34 &
${}^{+5.3\%}_{-4.6\%}$ &
${}^{+0.9\%}_{-0.9\%}$ \\
$0.96 \times 0.96$ &
1.16 &
${}^{+2.6\%}_{-2.4\%}$ &
${}^{+0.8\%}_{-0.8\%}$ \\
$1.5 \times 7$ &
6.49 &
${}^{+1.5\%}_{-1.9\%}$ &
${}^{+0.7\%}_{-0.7\%}$ \\
$1.5 \times 13.5$ &
9.51 &
${}^{+2.7\%}_{-3.1\%}$ &
${}^{+0.7\%}_{-0.7\%}$ \\
$1.5 \times 20$ &
11.9&
${}^{+3.4\%}_{-3.9\%}$ &
${}^{+0.7\%}_{-0.7\%}$ \\
$1.5 \times 50$ &
19.3 &
${}^{+5.0\%}_{-5.6\%}$ &
${}^{+0.7\%}_{-0.7\%}$ \\
\hline
\end{tabular}
\label{tab:muNeg_p_highE_Znu_xsec}
\end{table*}
\begin{table*}[!htb]
\centering
\caption{Cross sections for the $W^{-}\nu_{\mu}$ process in $\mu^{-}p$ collisions for different beam energy configurations. The $\mu^{-}$ beam energy is unpolarized in all cases.}
\begin{tabular}{l|ccc}
\hline
$E_{\mu}\times E_{p}$ (TeV$^{2}$) & $\sigma$ (pb) & Scale unc. &
PDF$\oplus\alpha_{s}$ unc. \\
\hline
$0.96 \times 0.275$ &
1.80 &
${}^{+2.8\%}_{-5.6\%}$ &
${}^{+1.4\%}_{-1.4\%}$ \\
$0.96 \times 0.96$ &
7.47 &
${}^{+7.9\%}_{-11\%}$ &
${}^{+1.4\%}_{-1.4\%}$ \\
$1.5 \times 7$ &
52.8 &
${}^{+15\%}_{-17\%}$ &
${}^{+1.3\%}_{-1.3\%}$ \\
$1.5 \times 13.5$ &
79.8 &
${}^{+16\%}_{-18\%}$ &
${}^{+1.2\%}_{-1.2\%}$ \\
$1.5 \times 20$ &
100 &
${}^{+17\%}_{-19\%}$ &
${}^{+1.2\%}_{-1.2\%}$ \\
$1.5 \times 50$ &
167 &
${}^{+19\%}_{-20\%}$ &
${}^{+1.2\%}_{-1.2\%}$ \\
\hline
\end{tabular}
\label{tab:muNeg_p_highE_Wnu_xsec}
\end{table*}
\begin{table*}[!htb]
\centering
\caption{Cross sections for the $Z\mu^{-}$ process in $\mu^{-}p$ collisions for different beam energy configurations and with different cutoffs on the scattered muon $p_{\rm T}$. The listed cross sections are in pb, with scale and PDF$\oplus\alpha_{s}$ uncertainties. The $\mu^{-}$ beam energy is unpolarized in all cases.}
\begin{tabular}{l|c|c|c|c}
\hline
$E_{\mu}\times E_{p}$ (TeV$^{2}$) &
Inclusive &
$p^{\ell}_{\rm T} >$ 1 GeV &
$p^{\ell}_{\rm T} >$ 2 GeV &
$p^{\ell}_{\rm T} >$ 5 GeV \\
\hline
$0.96 \times 0.275$ &
3.33 ${}^{+0\%}_{-0.4\%}$ ${}^{+0.7\%}_{-0.7\%}$ &
0.73 ${}^{+0\%}_{-1.2\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
0.60 ${}^{+0\%}_{-1.3\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
0.44 ${}^{+0\%}_{-1.3\%}$ ${}^{+0.8\%}_{-0.8\%}$ \\
$0.96 \times 0.96$ &
7.44 ${}^{+2.7\%}_{-3.7\%}$ ${}^{+0.7\%}_{-0.7\%}$ &
1.57 ${}^{+2.5\%}_{-4.5\%}$ ${}^{+0.7\%}_{-0.7\%}$ &
1.31 ${}^{+2.7\%}_{-5.0\%}$ ${}^{+0.7\%}_{-0.7\%}$ &
0.97 ${}^{+2.5\%}_{-4.7\%}$ ${}^{+0.7\%}_{-0.7\%}$ \\
$1.5 \times 7$ &
25.8 ${}^{+8.6\%}_{-9.5\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
5.24 ${}^{+9.6\%}_{-11\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
4.34 ${}^{+9.8\%}_{-11\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
3.26 ${}^{+10\%}_{-12\%}$ ${}^{+0.7\%}_{-0.7\%}$ \\
$1.5 \times 13.5$ &
34.5 ${}^{+10\%}_{-11\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
7.00 ${}^{+12\%}_{-12\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
5.81 ${}^{+11\%}_{-13\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
4.34 ${}^{+12\%}_{-13.0\%}$ ${}^{+0.8\%}_{-0.8\%}$ \\
$1.5 \times 20$ &
41.1 ${}^{+11\%}_{-12\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
8.31 ${}^{+12\%}_{-13\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
6.87 ${}^{+13\%}_{-14\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
5.14 ${}^{+13\%}_{-14\%}$ ${}^{+0.8\%}_{-0.8\%}$ \\
$1.5 \times 50$ &
59.9 ${}^{+13\%}_{-14\%}$ ${}^{+1.3\%}_{-1.3\%}$ &
12.1 ${}^{+14\%}_{-15\%}$ ${}^{+1.1\%}_{-1.1\%}$ &
10.1 ${}^{+15\%}_{-15\%}$ ${}^{+1.1\%}_{-1.0\%}$ &
7.50 ${}^{+15\%}_{-16\%}$ ${}^{+1.0\%}_{-1.0\%}$ \\
\hline
\end{tabular}
\label{tab:muNeg_p_highE_Zmu_lepcut_xsec}
\end{table*}
\begin{table*}[!htb]
\centering
\caption{Cross sections for the $W^{+}\mu^{-}$ process in $\mu^{-}p$ collisions for different beam energy configurations and with different cutoffs on the scattered muon $p_{\rm T}$. The listed cross sections are in pb, with scale uncertainties and PDF$\oplus\alpha_{s}$ uncertainties. The $\mu^{-}$ beam energy is unpolarized in all cases.}
\begin{tabular}{l|c|c|c|c}
\hline
$E_{\mu}\times E_{p}$ (TeV$^{2}$) &
Inclusive &
$p^{\ell}_{\rm T} >$ 1 GeV &
$p^{\ell}_{\rm T} >$ 2 GeV &
$p^{\ell}_{\rm T} >$ 5 GeV \\
\hline
$0.96 \times 0.275$ &
8.93 ${}^{+1.0\%}_{-1.2\%}$ ${}^{+0.7\%}_{-0.7\%}$ &
2.29 ${}^{+2.4\%}_{-2.5\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
1.86 ${}^{+2.6\%}_{-2.7\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
1.32 ${}^{+3.2\%}_{-3.1\%}$ ${}^{+0.8\%}_{-0.8\%}$ \\
$0.96 \times 0.96$ &
22.4 ${}^{+1.2\%}_{-1.7\%}$ ${}^{+0.7\%}_{-0.7\%}$ &
6.19 ${}^{+0\%}_{-0.4\%}$ ${}^{+0.7\%}_{-0.7\%}$ &
5.13 ${}^{+0\%}_{-0.3\%}$ ${}^{+0.7\%}_{-0.7\%}$ &
3.77 ${}^{+0.4\%}_{-0.7\%}$ ${}^{+0.7\%}_{-0.7\%}$ \\
$1.5 \times 7$ &
90.1 ${}^{+6.0\%}_{-6.7\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
27.4 ${}^{+4.6\%}_{-5.3\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
23.1 ${}^{+4.3\%}_{-5.0\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
17.6 ${}^{+4.0\%}_{-4.6\%}$ ${}^{+0.8\%}_{-0.8\%}$ \\
$1.5 \times 13.5$ &
124 ${}^{+7.4\%}_{-8.0\%}$ ${}^{+1.1\%}_{-1.1\%}$ &
38.7 ${}^{+5.9\%}_{-6.5\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
32.6 ${}^{+5.6\%}_{-6.3\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
25.0 ${}^{+5.2\%}_{-5.9\%}$ ${}^{+0.8\%}_{-0.8\%}$ \\
$1.5 \times 20$ &
150 ${}^{+8.1\%}_{-8.8\%}$ ${}^{+1.1\%}_{-1.1\%}$ &
47.0 ${}^{+6.6\%}_{-7.3\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
40.0 ${}^{+6.4\%}_{-7.0\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
30.6 ${}^{+5.9\%}_{-6.5\%}$ ${}^{+0.9\%}_{-0.9\%}$ \\
$1.5 \times 50$ &
225 ${}^{+9.9\%}_{-10\%}$ ${}^{+1.3\%}_{-1.3\%}$ &
72.8 ${}^{+8.4\%}_{-8.9\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
61.7 ${}^{+8.2\%}_{-8.7\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
47.8 ${}^{+7.7\%}_{-8.2\%}$ ${}^{+1.0\%}_{-1.0\%}$ \\
\hline
\end{tabular}
\label{tab:muNeg_p_highE_Wpmu_lepcut_xsec}
\end{table*}
\begin{table*}[!htb]
\centering
\caption{Cross sections for the $W^{-}\mu^{-}$ process in $\mu^{-}p$ collisions for different beam energy configurations and with different cutoffs on the scattered muon $p_{\rm T}$. The listed cross sections are in pb, with scale and PDF$\oplus\alpha_{s}$ uncertainties. The $\mu^{-}$ beam energy is unpolarized in all cases.}
\begin{tabular}{l|c|c|c|c}
\hline
$E_{\mu}\times E_{p}$ (TeV$^{2}$) &
Inclusive &
$p^{\ell}_{\rm T} >$ 1 GeV &
$p^{\ell}_{\rm T} >$ 2 GeV &
$p^{\ell}_{\rm T} >$ 5 GeV \\
\hline
$0.96 \times 0.275$ &
8.69 ${}^{+0.7\%}_{-1.0\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
2.10 ${}^{+1.6\%}_{-2.0\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
1.71 ${}^{+1.8\%}_{-2.1\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
1.23 ${}^{+2.4\%}_{-2.4\%}$ ${}^{+0.9\%}_{-0.9\%}$ \\
$0.96 \times 0.96$ &
21.2 ${}^{+1.7\%}_{-2.3\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
5.76 ${}^{+0.7\%}_{-1.4\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
4.79 ${}^{+0.6\%}_{-1.2\%}$ ${}^{+0.8\%}_{-0.8\%}$ &
3.57 ${}^{+0.2\%}_{-0.7\%}$ ${}^{+0.8\%}_{-0.8\%}$ \\
$1.5 \times 7$ &
86.7 ${}^{+6.7\%}_{-7.4\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
26.8 ${}^{+5.5\%}_{-6.3\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
22.8 ${}^{+5.4\%}_{-6.1\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
17.8 ${}^{+5.0\%}_{-5.7\%}$ ${}^{+0.8\%}_{-0.8\%}$ \\
$1.5 \times 13.5$ &
121 ${}^{+7.9\%}_{-8.6\%}$ ${}^{+1.1\%}_{-1.1\%}$ &
38.3 ${}^{+6.8\%}_{-7.6\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
32.6 ${}^{+6.6\%}_{-7.4\%}$ ${}^{+0.9\%}_{-0.9\%}$ &
25.6 ${}^{+6.2\%}_{-6.9\%}$ ${}^{+0.9\%}_{-0.9\%}$ \\
$1.5 \times 20$ &
145 ${}^{+8.6\%}_{-9.3\%}$ ${}^{+1.2\%}_{-1.2\%}$ &
47.0 ${}^{+7.4\%}_{-8.2\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
40.1 ${}^{+7.4\%}_{-8.1\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
31.6 ${}^{+7.0\%}_{-7.7\%}$ ${}^{+0.9\%}_{-0.9\%}$ \\
$1.5 \times 50$ &
221 ${}^{+11\%}_{-11\%}$ ${}^{+1.4\%}_{-1.4\%}$ &
73.6 ${}^{+9.3\%}_{-9.9\%}$ ${}^{+1.1\%}_{-1.1\%}$ &
63.3 ${}^{+9.0\%}_{-9.7\%}$ ${}^{+1.1\%}_{-1.1\%}$ &
50.3 ${}^{+8.6\%}_{-9.3\%}$ ${}^{+1.2\%}_{-1.1\%}$ \\
\hline
\end{tabular}
\label{tab:muNeg_p_highE_Wnmu_lepcut_xsec}
\end{table*}
\begin{table*}[!htb]
\centering
\caption{Cross sections for the $\bar{t}\,\nu_{\mu}$ process in $\mu^{-}p$ collisions for different beam energy configurations. The $\mu^{-}$ beam energy is unpolarized in all cases.}
\begin{tabular}{l|ccc}
\hline
$E_{\mu}\times E_{p}$ (TeV$^{2}$) & $\sigma$ (pb) & Scale unc. &
PDF$\oplus\alpha_{s}$ unc. \\
\hline
$0.96 \times 0.275$ &
0.95 &
${}^{+5.7\%}_{-7.9\%}$ &
${}^{+2.5\%}_{-2.5\%}$ \\
$0.96 \times 0.96$ &
5.44 &
${}^{+8.1\%}_{-10\%}$ &
${}^{+1.8\%}_{-1.8\%}$ \\
$1.5 \times 7$ &
48.1 &
${}^{+12\%}_{-14\%}$ &
${}^{+1.5\%}_{-1.5\%}$ \\
$1.5 \times 13.5$ &
75.0 &
${}^{+13\%}_{-15\%}$ &
${}^{+1.4\%}_{-1.4\%}$ \\
$1.5 \times 20$ &
96.1 &
${}^{+14\%}_{-15\%}$ &
${}^{+1.4\%}_{-1.4\%}$ \\
$1.5 \times 50$ &
164.1 &
${}^{+15\%}_{-16\%}$ &
${}^{+1.4\%}_{-1.4\%}$ \\
\hline
\end{tabular}
\label{tab:muNeg_p_highE_tbar_xsec}
\end{table*}
\begin{table*}[!htb]
\centering
\caption{Cross sections for the $t\bar{t}\,\mu^{-}$ process in $\mu^{-}p$ collisions for different beam energy configurations and with different cutoffs on the scattered muon $p_{\rm T}$. The listed cross sections are in pb, with scale and PDF$\oplus\alpha_{s}$ uncertainties. The $\mu^{-}$ beam energy is unpolarized in all cases.}
\begin{tabular}{l|c|c|c|c}
\hline
$E_{\mu}\times E_{p}$ (TeV$^{2}$) &
Inclusive &
$p^{\ell}_{\rm T} >$ 1 GeV &
$p^{\ell}_{\rm T} >$ 2 GeV &
$p^{\ell}_{\rm T} >$ 5 GeV \\
\hline
$0.96 \times 0.275$ &
0.014 ${}^{+26\%}_{-19\%}$ ${}^{+3.2\%}_{-3.2\%}$ &
0.0056 ${}^{+26\%}_{-19\%}$ ${}^{+3.6\%}_{-3.6\%}$ &
0.0048 ${}^{+26\%}_{-19\%}$ ${}^{+3.6\%}_{-3.6\%}$ &
0.0038 ${}^{+26\%}_{-19\%}$ ${}^{+3.8\%}_{-3.8\%}$ \\
$0.96 \times 0.96$ &
0.221 ${}^{+19\%}_{-15\%}$ ${}^{+1.7\%}_{-1.7\%}$ &
0.099 ${}^{+20\%}_{-15\%}$ ${}^{+1.7\%}_{-1.7\%}$ &
0.087 ${}^{+20\%}_{-15\%}$ ${}^{+1.8\%}_{-1.8\%}$ &
0.072 ${}^{+20\%}_{-15\%}$ ${}^{+1.8\%}_{-1.8\%}$ \\
$1.5 \times 7$ &
4.62 ${}^{+11\%}_{-9.2\%}$ ${}^{+1.1\%}_{-1.1\%}$ &
2.28 ${}^{+11\%}_{-9.5\%}$ ${}^{+1.1\%}_{-1.1\%}$ &
2.05 ${}^{+11\%}_{-9.5\%}$ ${}^{+1.1\%}_{-1.1\%}$ &
1.76 ${}^{+11\%}_{-9.6\%}$ ${}^{+1.1\%}_{-1.1\%}$ \\
$1.5 \times 13.5$ &
8.24 ${}^{+11\%}_{-9.4\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
4.12 ${}^{+11\%}_{-9.1\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
3.72 ${}^{+10\%}_{-9.0\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
3.20 ${}^{+10\%}_{-8.9\%}$ ${}^{+1.0\%}_{-1.0\%}$ \\
$1.5 \times 20$ &
11.3 ${}^{+12\%}_{-10\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
5.70 ${}^{+12\%}_{-10\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
5.16 ${}^{+11\%}_{-9.9\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
4.43 ${}^{+11\%}_{-9.9\%}$ ${}^{+1.0\%}_{-1.0\%}$ \\
$1.5 \times 50$ &
22.2 ${}^{+14\%}_{-12\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
11.3 ${}^{+14\%}_{-12\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
10.3 ${}^{+14\%}_{-12\%}$ ${}^{+1.0\%}_{-1.0\%}$ &
8.90 ${}^{+14\%}_{-12\%}$ ${}^{+1.0\%}_{-1.0\%}$ \\
\hline
\end{tabular}
\label{tab:muNeg_p_highE_ttbar_lepcut_xsec}
\end{table*}
A fiducial selection on the scattered muon is important to tag $Z/\gamma$ exchange diagrams from those with $W$ exchange from the incoming muon, and to separate to some degree the pure photon exchange diagram.
Therefore, for these processes, Tables~\ref{tab:muNeg_p_highE_Zmu_lepcut_xsec}, \ref{tab:muNeg_p_highE_Wpmu_lepcut_xsec}, \ref{tab:muNeg_p_highE_Wnmu_lepcut_xsec}, and \ref{tab:muNeg_p_highE_ttbar_lepcut_xsec} list the inclusive cross sections without any fiducial selection and with selections requiring $p^{\ell}_{\rm T} > $ 1, 2, and 5 GeV, as may be required to scatter into the acceptance of a detector.
As an example, the kinematic profile of $W^{+}\mu^{-}$ events are shown in Fig.~\ref{fig:W_gen_dists}. This process (illustrated in Fig.~\ref{fig:Wmu_diagrams}) can happen with a photon or $Z$ boson exchange from the muon leg. The pseudorapidity distribution of the scattered muon shows a peaking structure at large negative
$\eta$,\footnote{We assume the convention of DIS colliders that the hadron beam defines the $+z$ direction.}
which comes from the diagrams with photon exchange where the virtual photon mass is close to 0. The cutoff towards $\eta=-8$ is because of the cutoff on lepton $p_{\rm T}$. The small bump on the right tail of this $\eta$ distribution shows that diagrams with $Z$ boson exchange start to be the main contribution in this region. The scattered muon, struck parton, and $W$ boson decay products all fall within a central tracking acceptance of $-4 < \eta < 2.4$. The scattered muon with $p^{\ell}_{\rm T} > $ 1 GeV would fall within a muon spectrometer acceptance of $\eta\gtrsim -7$, as would be required for low $Q^2$ DIS measurements as well (see Appendix~\ref{sec:dis-kin}).
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/SM_results/W_lep1GeV_eta_dist.pdf}
\includegraphics[width=0.45\textwidth]{Figures/SM_results/W_lep1GeV_pt_dist.pdf}
\caption{Generator-level kinematic distributions of $\eta$ (left) and $p_{\rm T}$ (right) for different outgoing particles in $W^{+}\mu^{-} \to e^{+}\nu_{e}\mu^{-}$ events at $E_{\mu^{-}} = 960$ GeV and $E_{p} = 275$ GeV collisions. A fiducial cut on the outgoing scattered muon $p^{\ell}_{\rm T} > 1$~GeV is applied to this sample.
}
\label{fig:W_gen_dists}
\end{figure}
\subsubsection{Higgs Boson Production}
\label{sec:higgs_phys}
The prospects for Higgs boson measurements at a lepton-hadron collider are exciting as the experimental environment can be cleaner than at hadron colliders, which suffer from pileup effects and large QCD cross sections. However, the effects of beam-induced backgrounds will need to be assessed experimentally for $\mu p$ collisions. The environment at $e^+e^-$ (or $\mu^+\mu^-$) ``Higgs factories'' would be cleaner still and may offer larger luminosity. Nevertheless, the possibility to measure some decay modes that are extremely difficult at hadron colliders, such as bottom and charm decays and possibly even gluon or light quark decays with sufficient integrated luminosity, should not be discounted.
The main production mechanism of Higgs bosons in $\mu^- p$ collisions, as is also the case in multi-TeV $\mu^+\mu^-$ collisions, is vector boson fusion (VBF) via charged-current or neutral-current exchanges.
Figure~\ref{fig:VBF_diagrams} shows the leading order (LO) diagrams for such production.
The VBF mode here does not consider a top quark in the final state.
The Higgs boson can also be produced in association with a top quark, in the VBF or Higgs-strahlung processes, as shown in Fig.~\ref{fig:tH_diagrams}.
These two modes together are referred to as the $t\,H$ mode in the following.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Higgs/CC_LO_diagram.pdf}
\includegraphics[width=0.45\textwidth]{Figures/Higgs/NC_LO_diagram.pdf}
\caption{Vector boson fusion Higgs production mode via charged current (left) and neutral current (right) exchanges.}
\label{fig:VBF_diagrams}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Figures/Higgs/tH_Hstrahlung_LO_diagram.pdf}
\includegraphics[width=0.45\textwidth]{Figures/Higgs/tH_VBF_LO_diagram.pdf}
\caption{Higgs production associated with a top quark, in the Higgs-strahlung mode (left) and VBF mode (right).}
\label{fig:tH_diagrams}
\end{figure}
The Higgs boson production cross sections at the MuIC have been calculated with MadGraph \cite{Madgraph}, version 3.3.1, using the PDF set PDF4LHC15\_nlo\_mc\_pdfas~\cite{Butterworth:2059563} for the proton. These cross sections are shown in Table~\ref{tab:muNeg_p_xsec} for $\mu^{-}$ beams and Table~\ref{tab:muPos_p_xsec} for $\mu^{+}$ beams using the nominal beam energies of $E_{\mu} = 960$~GeV and $E_{p} = 275$~GeV and for several different muon beam polarization configurations. The total cross section as well as the production process cross sections are reported. For unpolarized beams, the total Higgs boson cross section is 77~fb for $\mu^- p$ collisions at the MuIC. This is comparable to the cross section of 130~fb at the proposed LHeC with a 60~GeV electron beam energy \cite{Agostini:2020fmq}. In fact, as will be noted below, the Higgs decay products are more central in collisions at the MuIC than at the LHeC, and should lead to a relatively larger cross section in the experiment acceptance. The production cross section can be enhanced with appropriate longitudinal polarization of the muon beam.
Scenarios with higher beam energies are also studied. Table~\ref{tab:muNeg_p_highE_xsec} shows the CC and NC Higgs boson production cross sections, along with uncertainties, in several scenarios with different $\mu^{-}$ and proton energies.
The listed uncertainties arise from the renormalization and factorization scale uncertainties ($\mu_{R}$ and $\mu_{F}$, which are varied by a factor of 2 around the $Z$ boson mass), and from PDF set variations including also $\alpha_{s}$ variations.
As shown in the table, the leading uncertainty is the scale uncertainty.
The PDF uncertainty is always around 1\% (smaller than the scale uncertainty), while the $\alpha_{s}$ uncertainty is negligible.
\begin{table}[!htb]
\centering
\caption{Cross sections, in fb, for 125 GeV Higgs boson production in $\mu^{-}p$ scattering. The $\mu^{-}$ beam energy is 960 GeV and the proton beam energy is 275 GeV. P is the polarization of the muon beam.}
\begin{tabular}{l|cccccccc}
\hline
& P = $-40\%$ & P = $-20\%$ & P = $-10\%$ & P = 0 \% &
P = 10\% & P = 20\% & P = 40\% & P = 100\% \\
\hline
$\sigma_{CC}$ & 91.1 & 78.2 & 71.7 & 65.1 &
58.8 & 52.1 & 39.0 & 0 \\
$\sigma_{NC}$ & 12.6 & 12.1 & 11.9 & 11.6 &
11.4 & 11.1 & 10.5 & 8.9 \\
$\sigma_{tH}$ & 0.0224 & 0.0187 & 0.0174 & 0.0158 &
0.0139 & 0.0128 & 0.0096 & 0 \\
\hline
total & 103.7 & 90.3 & 83.6 & 76.7 &
70.2 & 63.2 & 49.5 & 8.9 \\
\hline
\end{tabular}
\label{tab:muNeg_p_xsec}
\end{table}
\begin{table}[!htb]
\centering
\caption{Cross sections, in fb, for 125 GeV Higgs boson production in $\mu^{+}p$ scattering. The $\mu^{+}$ beam energy is 960 GeV and the proton beam energy is 275 GeV. P is the polarization of the muon beam.}
\begin{tabular}{l|cccccccc}
\hline
& P = 40\% & P = 20\% & P = 10\% & P = 0 \% &
P = $-10\%$ & P = $-20\%$ & P = $-40\%$ & P = $-100\%$ \\
\hline
$\sigma_{CC}$ & 45.0 & 38.2 & 35.6 & 32.1 &
28.9 & 25.6 & 19.2 & 0 \\
$\sigma_{NC}$ & 12.4 & 12.0 & 11.7 & 11.6 &
11.3 & 11.0 & 10.6 & 9.1 \\
$\sigma_{tH}$ & 0.0220 & 0.0190 & 0.0173 & 0.0157 &
0.0142 & 0.0127 & 0.0093 & 0 \\
\hline
total & 57.4 & 50.2 & 47.3 & 43.7 &
40.2 & 36.6 & 29.8 & 9.1 \\
\hline
\end{tabular}
\label{tab:muPos_p_xsec}
\end{table}
The $t\,H$ process is subject to significant destructive interference between the Higgs-strahlung and the VBF-tH diagrams. Its cross section is unobservable at low collision energies but may become visible at high energies. The cross section for the $t\,H$ process is summarized in Table~\ref{tab:muNeg_p_highE_tH_xsec}.
\begin{table}[!htb]
\centering
\caption{The Higgs boson cross section for different beam energy scenarios, separately for CC and NC exchange. The $\mu^{-}$ beam energy is unpolarized in all cases. Uncertainties arising from the scale and from the PDF and $\alpha_{s}$ variations are also listed, as discussed in the text.}
\begin{tabular}{l|ccc|ccc}
\hline
$E_{\mu}\times E_{p}$ (TeV$^{2}$) & $\sigma_{CC}$ (fb) & Scale unc. &
PDF$\oplus\alpha_{s}$ unc. & $\sigma_{NC}$ (fb) & Scale unc. & PDF$\oplus\alpha_{s}$ unc. \\
\hline
$0.96 \times 0.275$ &
64.5 &
${}^{+6.5\%}_{-5.5\%}$ &
${}^{+1.3\%}_{-1.3\%}$ &
11.6 &
${}^{+6.1\%}_{-5.2\%}$ &
${}^{+1.2\%}_{-1.2\%}$ \\
$0.96 \times 0.96$ &
235 &
${}^{+3.4\%}_{-3.1\%}$ &
${}^{+1.2\%}_{-1.2\%}$ &
47.9 &
${}^{+2.5\%}_{-2.4\%}$ &
${}^{+1.0\%}_{-1.0\%}$ \\
$1.5 \times 7$ &
1337 &
${}^{+1.5\%}_{-2.0\%}$ &
${}^{+1.1\%}_{-1.1\%}$ &
327 &
${}^{+2.9\%}_{-3.6\%}$ &
${}^{+1.1\%}_{-1.1\%}$ \\
$1.5 \times 13.5$ &
1955 &
${}^{+2.9\%}_{-3.4\%}$ &
${}^{+1.2\%}_{-1.2\%}$ &
496 &
${}^{+4.3\%}_{-5.1\%}$ &
${}^{+1.2\%}_{-1.2\%}$ \\
$1.5 \times 20$ &
2422 &
${}^{+3.6\%}_{-4.2\%}$ &
${}^{+1.2\%}_{-1.2\%}$ &
628 &
${}^{+5.2\%}_{-6.0\%}$ &
${}^{+1.2\%}_{-1.2\%}$ \\
$1.5 \times 50$ &
3883 &
${}^{+5.5\%}_{-6.1\%}$ &
${}^{+1.4\%}_{-1.4\%}$ &
1053 &
${}^{+7.0\%}_{-7.9\%}$ &
${}^{+1.4\%}_{-1.4\%}$ \\
\hline
\end{tabular}
\label{tab:muNeg_p_highE_xsec}
\end{table}
\begin{table}[!htb]
\centering
\caption{tH cross section in different beam energy scenarios. The $\mu^{-}$ beam energy is unpolarized in all cases. Uncertainties arising from the scale and PDF set are also listed, as discussed in the text.}
\begin{tabular}{l|ccc}
\hline
$E_{\mu}\times E_{p}$ (TeV$^{2}$) & $\sigma_{CC}$ (fb) & Scale unc. &
PDF$\oplus\alpha_{s}$ unc. \\
\hline
$0.96 \times 0.275$ &
0.019 &
${}^{+0.8\%}_{-2.7\%}$ &
${}^{+14\%}_{-14\%}$ \\
$0.96 \times 0.96$ &
0.54 &
${}^{+3.8\%}_{-5.8\%}$ &
${}^{+6.3\%}_{-6.3\%}$ \\
$1.5 \times 7$ &
24.1 &
${}^{+7.7\%}_{-9.6\%}$ &
${}^{+3.5\%}_{-3.5\%}$ \\
$1.5 \times 13.5$ &
48.7 &
${}^{+8.7\%}_{-10\%}$ &
${}^{+3.2\%}_{-3.2\%}$ \\
$1.5 \times 20$ &
71.8 &
${}^{+9.2\%}_{-11\%}$ &
${}^{+3.0\%}_{-3.0\%}$ \\
$1.5 \times 50$ &
161 &
${}^{+10\%}_{-12\%}$ &
${}^{+2.6\%}_{-2.6\%}$ \\
\hline
\end{tabular}
\label{tab:muNeg_p_highE_tH_xsec}
\end{table}
A scan of the Higgs production cross section in $\mu^{-}p$ collisions is also performed for different beam energies, as shown in Fig.~\ref{fig:muNeg_xsec}. The left plot shows the cross section as a function of the $\mu^{-}$ beam energy with the proton beam held fixed at the EIC energy of 275~GeV. The right plot shows the cross section as a function of the $\mu^- p$ center-of-mass energy. The muon beam is assumed unpolarized, and only LO diagrams are considered. The cross section exceeds 1~pb for $\sqrt{s} \gtrsim 5$~TeV. In comparison, we note that the cross section for Higgs production at a $\mu^+\mu^-$ collider operating at $\sqrt{s}=3$~TeV center is 500~fb \cite{Costantini:2020stv}. This is actually about 3 times less than the cross section possible when colliding one of the 1.5~TeV muon beams of that collider with a 7~TeV LHC proton beam.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Higgs/muNeg_xsec_eMu.pdf}
\includegraphics[width=0.45\textwidth]{Figures/Higgs/muNeg_xsec_sqrt_s.pdf}
\caption{Left: The Higgs boson cross sections as a function of the $\mu^{-}$ beam energy, with the proton energy fixed at 275~GeV. Right: The Higgs boson cross section scan as a function of the $\mu^{-} p$ center-of-mass energy $\sqrt{s}$. Shaded bands shows the scale uncertainties. Nominal beam energy choices for the machine configurations studied here are indicated by the symbols. }
\label{fig:muNeg_xsec}
\end{figure}
The SM Higgs boson decays dominantly into a $b\bar{b}$ pair\footnote{For single $b$ quarks, charge conjugation is implied hereinafter, unless stated otherwise.} with a branching ratio of 58\%. The observation of this decay mode has been reported by both the ATLAS and CMS Collaborations~\cite{atlas2018hbb, cms2018hbb}. The inclusive NC+CC production yield of $H\to b\bar{b}$ in unpolarized 960~GeV $\mu^- \times$ 275~GeV $p$ collisions at the MuIC for 400~fb$^{-1}$ of integrated luminosity (see Section~\ref{sec:lumi}) is $17\,800$. This number grows by about a factor 4 for 2~TeV collisions (MuIC2) at the same luminosity, and by a factor 13 for 6.5~TeV collisions (LHmuC) at a lower integrated luminosity of 237 fb$^{-1}$.
This opens new opportunities to measure the $Hbb$ coupling strength, and possibly others, at the MuIC.
The kinematics of Higgs boson production and decay at the MuIC have been studied, without losing generality, using simulated $H \to b\bar{b}$ samples. The hard process is generated with MadGraph5~\cite{Madgraph} while the shower activities are simulated with Pythia~8.3 \cite{Sjostrand:2014zea}, with $E_{p} = 275$~GeV and $E_{\mu} = 960$~GeV for the NC and CC collisions.
Figure~\ref{fig:H_gen_dists} shows the generator-level distributions of $\eta$ and $p_{\rm T}$ of the Higgs boson, the Higgs decay products, the struck quark, and the scattered lepton. The Higgs decay products and the scattered quark are quite central. A tracker acceptance spanning the range $-4 < \eta < 2.4$ covers nearly all of the final state particles, and a muon acceptance extended down to $\eta = -7$ in the muon beam direction as would be required for low $Q^2$ DIS measurements at the MuIC (see Appendix~\ref{sec:dis-kin}) easily covers the NC final state muon. This is in contrast to the distributions at the LHeC (see Ref.~\cite{Agostini:2020fmq}), where the Higgs decay products peak in the forward (proton) direction around $\eta \approx 2$ and extend as far forward as $\eta \approx 6$. The struck quark is even more forward peaking at $\eta \approx 5$ and extending further. Thus the Higgs decays are more central in a MuIC detector and should lead to a larger acceptance.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Higgs/eta_dist.pdf}
\includegraphics[width=0.45\textwidth]{Figures/Higgs/pt_dist.pdf}
\caption{Generator-level kinematic distributions of $\eta$ (left) and $p_{\rm T}$ (right) for different final state objects in $H \to b\bar{b}$ events in $\mu^- p$ collisions with $E_{\mu} = 960$ GeV and $E_{p} = 275$ GeV. Nominal tracking and muon acceptance regions as would be required for DIS measurements are indicated by the hatched areas.}
\label{fig:H_gen_dists}
\end{figure}
To estimate the sensitivity for the $Hbb$ coupling measurement at the MuIC, the background processes $Z
\mu^-, Z \to b\bar{b}$ (Z\_NC), $Z\nu_{\mu}, Z \to b\bar{b}$ (Z\_CC), and gluon-initiated $b\bar{b}$ production (DIS\_bb) are also considered. The detector simulation is performed on both the signal and background samples using Delphes~\cite{delphes}, with the detector parameterization described in Appendix~\ref{sec:delphes}. The Higgs and $Z$ boson processes are simulated in their full inclusive phase space. The DIS process, which is by far the dominant background, is generated in a fiducial phase space of 50~GeV~$< m_{b\bar{b}} <$~170~GeV, $p_{\rm T}(b) >$~20~GeV, and $|\eta(b)|<$~6.
For this preliminary estimate, other background processes such as the light-flavor fake backgrounds and multi-jet backgrounds are not considered.
Object and event selections are applied to enhance the $H \to b\bar{b}$ signal over background. Jets are selected with $p_{\rm T} > $~25~GeV and $-5 < \eta < 2.4$, while the b-tagging is only available within the tracker acceptance $-4 < \eta < 2.4$. Muons are selected with $p_{\rm T} >$ 5~GeV in the central muon system $-4 < \eta < 0$, or without $p_{\rm T}$ requirement in the far-backward region $-7 < \eta < -4$. B-jets are rejected if they are within a $\Delta R = 0.4$ cone around any selected muon. The baseline event selection requires 2 selected b-tagged jets in the event. To target the H\_CC process and reduce the large DIS background, an additional set of selections are applied: no selected muon in the event, a light flavor jet (assumed from the struck quark) in addition to the 2 selected b-jets in the event, $p_{\rm T}(H) > 20$~GeV for the dijet Higgs candidate, and missing transverse energy $E_{\rm T}^{\text{miss}} > 30$~GeV.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/Higgs/Hbb_H_mass_preSel.pdf}
\includegraphics[width=0.45\textwidth]{Figures/Higgs/Hbb_H_mass_finalSel.pdf}
\caption{Dijet invariant mass distribution of $H \to b\bar{b}$ events and background processes in $\mu^- p$ collisions with $E_{\mu} = 960$ GeV and $E_{p} = 275$ GeV.
All events are generated with MadGraph5~\cite{Madgraph}, showered with PYTHIA8.3~\cite{Sjostrand:2014zea}, and simulated with Delphes~\cite{delphes} for detector responses.
Events are subject to simple cut-based selection criteria, with the left plot showing distributions before event selection and the right plot after event selection.
Note that the DIS background is generated with a fiducial cut of 50~GeV $< m_{b\bar{b}} <$ 170~GeV. Low and high tails of the DIS background in the left plot are from resolution effects.}
\label{fig:Hbb_mass}
\end{figure}
Figure~\ref{fig:Hbb_mass} shows the distribution of the dijet invariant mass with only the baseline selection (left) and with the full selection criteria (right).
The full selection achieves a very good signal efficiency and reduces most of the DIS and Z\_NC backgrounds. Improvements are expected with more optimized selections and more sophisticated analysis techniques.
In the range of 100~GeV to 130~GeV, the expected signal yield is around 900, with the $S/B$ close to 1. This corresponds to a statistical uncertainty of about 3\%, which is comparable to the envisioned statistical uncertainty of the $H\to b\bar{b}$ analysis in $pp$ collisions with 3000~fb$^{-1}$ of HL-LHC data~\cite{Cepeda:2650162}.
This result is also consistent with that expected at the LHeC~\cite{Agostini:2020fmq}, which has about 1.7 times larger Higgs cross sections, but less acceptance than for the MuIC. The kinematic profile and expected sensitivity for the $H\to b\bar{b}$ measurement is comparable between the LHeC and MuIC. For the case of LHmuC, where we expect 10 times more Higgs events, it is possible to achieve percent or sub-percent level precision on the $Hbb$ coupling measurement.
The $H\to c\bar{c}$ branching ratio is about 20 times smaller than the $H \to b\bar{b}$ decay, while the $Z\to c\bar{c}$ and DIS backgrounds are essentially the same as for the $b\bar{b}$ case. The flavor tagging algorithm envisioned at the LHeC~\cite{Agostini:2020fmq} expects about 28\% efficiency for the $H\to b\bar{b}$ process and about 11\% efficiency for the $H\to c\bar{c}$ process in the $H\to c\bar{c}$ analysis. A similar level of performance can be assumed at the MuIC. This leads to a statistical uncertainty of the $H\to c\bar{c}$ signal at the same order of the signal strength. With the luminosities assumed in Section~\ref{sec:lumi}, the MuIC would not produce enough events for a precision measurement of the $Hcc$ coupling, while the LHMuC would produce 10 times more Higgs events and offer an exciting opportunity for such a measurement.
Some other Higgs channels are also of interest. With 400 fb$^{-1}$ of integrated luminosity at MuIC, a total about 1900 $H\to \tau^-\tau^+$ events and about 2500 $H\to gg$ events are expected. The experimental sensitivity to these channels largely depends on the tagging algorithms and the contamination of DIS backgrounds, which are not elaborated in this paper.
\subsection{Beyond Standard Model Physics}
\subsubsection{Leptoquark Processes and Lepton Flavor Violation}
Leptoquarks (LQ) are hypothetical bosons that are predicted by many theories beyond the Standard Model (SM), such as grand unified theories, technicolor, composite models with quarks and lepton sub-structure,
and R-parity violating supersymmetry. They are color-triplets that carry quantum numbers such as spin and
fractional electric charge, and uniquely carry both baryon and lepton numbers allowing them to mediate quark-lepton transitions.
Historically, LQ searches have only considered models that couple to a lepton and quark of the same generation, motivated by limits placed on flavor changing neutral currents, proton decay, and other rare processes. However, there is growing observational evidence that supports inter-generational mixing in LQ decays. Most notably, a measurement of the ratio $R_K$ by the LHCb collaboration~\cite{lhcblu} hints at a possible breaking of lepton universality, and results of the muon g-2 collaboration~\cite{g-2} see increased tension with the SM in the measured muon magnetic moment. The exchange of a LQ boson can explain either anomaly, assuming that the LQ can couple to quarks and leptons of different generation.
The MuIC will provide unique testing grounds for LQ searches with mixed couplings to second and third generation leptons and quarks. Of particular interests are the t-channel and s-channel production of LQs that couple to a muon and a b-quark or to a muon and a top quark, as well as LQs that can couple both to $\tau$ lepton and a quark, and a muon and a quark. To study these processes, the software toolbox described in~\cite{LQBox} is used to generate, at leading order, muon-proton collisions with specific scalar LQ models that can couple of different quarks and leptons generations (i.e. the R2 and S3 models), and two proton beam energy configurations, 275 GeV and 1 TeV. The Yukawa coupling at the LQ-lepton-quark vertex is set to 0.1 in these simulations.
The cross-section for the production of $\mu$-b via s-channel exchange of a S3 LQ as well as t-channel SM electroweak $\mu$-b production (see Fig.~\ref{fig:Feynman_mub}) is shown in Fig.~\ref{fig:xsec_mub_1} as a function of LQ mass and for 275 GeV and 1 TeV proton beam energies, respectively. Fig.~\ref{fig:xsec_mub_2} shows the LQ production cross-section with SM backgrounds subtracted, and Fig.~\ref{fig:xsec_mub_3} illustrates the kinematic distributions ($p_{\rm T}$ and $\eta$), at generator level, of the outgoing muon and b-quark. Tagging the outgoing muon with a far-backward muon spectrometer, as well as a selection based on the kinematic properties and correlations among the final state muon and b-jet will be instrumental in optimizing the sensitivity for this channel.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.65\textwidth]{Figures/LQ/feynman_LQ_mub.pdf}
\caption{Diagrams of the production of a muon and a b-quark in the final state of a muon-proton collision, via the s-channel production of a S3 LQ, and via SM electroweak t-channel processes.}
\label{fig:Feynman_mub}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/LQ/275_b.pdf}
\includegraphics[width=0.45\textwidth]{Figures/LQ/1000_b.pdf}
\caption{Cross-section of the production of $\mu$p $\rightarrow$ $\mu$b, including s-channel production and decay of a S3 LQ, as a function of LQ mass for a proton beam energy of 275 GeV (left) and 1 TeV (right).}
\label{fig:xsec_mub_1}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/LQ/S3_b_noSM.pdf}
\caption{SM backgrounds-subtracted cross-section of $\mu$p $\rightarrow$ $\mu$b via s-channel production of a S3 LQ, as a function of LQ mass.}
\label{fig:xsec_mub_2}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/LQ/eta_b.pdf}
\includegraphics[width=0.45\textwidth]{Figures/LQ/pT_b.pdf}
\caption{Generator-level kinematic distributions, $\eta$ (left) and $p_{\rm T}$ (right), of the muon and b-quark produced via LQ S3 s-channel exchange, for a LQ mass of 500 GeV, 1 TeV proton and 960 GeV muon collisions. An integrated luminosity of 10 fb$^{-1}$ is assumed.}
\label{fig:xsec_mub_3}
\end{figure}
The kinematic selection used for these LQ studies consists of a 20 GeV threshold on the $p_{\rm T}$ of the outgoing quarks, a 10 GeV threshold on the $p_{\rm T}$ of the outgoing leptons, and a minimum separation in $\eta-\phi$ space of 0.4 between the outgoing lepton and quark.
The production of other third generation quarks and leptons is also investigated. Fig.~\ref{fig:xsec_mutop_1} shows the diagram and cross-section for the t-channel production of a R2 LQ, coupling to the initial muon and either an up or charm quark from the initial proton, and decaying to a muon and a top quark. Fig.~\ref{fig:xsec_mutop_2} shows similar couplings for the s-channel production cross-section of a S3 LQ, decaying to a muon and a top quark.
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/LQ/R2_t.pdf}
\includegraphics[width=0.22\textwidth]{Figures/LQ/feynman-LQ-mutop.pdf}
\caption{(Left) Cross-section for the production of muon and a top quark via a t-channel R2 LQ, as a function of LQ mass, and for two proton beam energies. (Right) R2 LQ t-channel exchange with final states consisting of a muon and a top quark.}
\label{fig:xsec_mutop_1}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/LQ/S3_t.pdf}
\includegraphics[width=0.22\textwidth]{Figures/LQ/feynman-LQ_mutopS3.pdf}
\caption{(Left) Cross-section for the production of muon and a top quark via a s-channel S3 LQ, as a function of LQ mass, and for two proton beam energies. (Right) S3 LQ s-channel exchange with final states consisting of a muon and a top quark.}
\label{fig:xsec_mutop_2}
\end{figure}
The $\eta$ and $p_{\rm T}$ distributions---at generator level---for the final state muon and top quark are shown in Fig~\ref{fig:xsec_mutop_3}. Since there are no competing leading order SM processes with a top quark and a muon produced in the final state, the requirements of a muon tagged in the far-backward spectrometer, the kinematic reconstruction of the top quark from its decay products, and constraints on the reconstructed missing transverse energy in the event should be effective means of reducing other SM backgrounds (e.g. single or pair production of top quarks in association with $\nu_\mu$ or from production of $W$ bosons in association with muons).
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/LQ/eta_t.pdf}
\includegraphics[width=0.45\textwidth]{Figures/LQ/pT_t.pdf}
\caption{Generator-level kinematic distributions, $\eta$ (left) and $p_{\rm T}$ (right), of the muon and top quark produced via LQ S3 s-channel exchange, for a LQ mass of 300 GeV, 1 TeV proton and 960 GeV muon collisions. An integrated luminosity of 10 fb$^{-1}$ is assumed.}
\label{fig:xsec_mutop_3}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.35\textwidth]{Figures/LQ/feynman_LQ_mutau.pdf}
\caption{Diagram of the production of a $\tau$ lepton and a jet in the final state of a muon-proton collision, via the t-channel production of a LQ (either a R2 or a S3).}
\label{fig:Feynman_mutau}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/LQ/R2_ta.pdf}
\includegraphics[width=0.45\textwidth]{Figures/LQ/S3_ta.pdf}
\caption{Cross-section for the production of $\tau$ lepton and a quark via a t-channel R2 LQ (Left) or a t-channel S3 LQ (Right), as a function of LQ mass, and for two proton beam energies.}
\label{fig:xsec_mutau_1}
\end{figure}
\begin{figure}[!htb]
\centering
\includegraphics[width=0.45\textwidth]{Figures/LQ/eta_ta.pdf}
\includegraphics[width=0.45\textwidth]{Figures/LQ/pT_ta.pdf}
\caption{Generator-level kinematic distributions, $\eta$ (left) and $p_{\rm T}$ (right), of the $\tau$ lepton and the quark produced via LQ S3 s-channel exchange, for a LQ mass of 1 TeV, in 1 TeV proton and 960 GeV muon collisions. An integrated luminosity of 10 fb$^{-1}$ is assumed.}
\label{fig:xsec_mutau_2}
\end{figure}
Finally, a striking observation of LFV would be the observation of $\mu$ p $\rightarrow$ $\tau$ jet events, which can occur via the t-channel exchange of a S3 or a R2 LQ, as shown in Fig.~\ref{fig:Feynman_mutau} (such processes can also be mediated by the exchange of a heavy $Z'$ with LFV leptonic decays).
The cross-section for the production of $\mu$ p $\rightarrow$ $\tau$ jet events via the exchange of a R2 or a S3 LQ is shown in Fig.~\ref{fig:xsec_mutau_1}. The generator-level $\eta$ and $p_{\rm T}$ distributions for the final state $\tau$ are shown in Fig.~\ref{fig:xsec_mutau_2} for the case of a 1 TeV S3 LQ. The cross-section for this process is higher than for the previous final states considered, since no specific final state quark flavor is requested. Effective reconstruction of this type of event and background rejection relies on constraining the reconstructed missing transverse energy in the event, on vetoing other final state lepton flavors, and on tagging the final state $\tau$, which is well within the acceptance of the tracking detectors, as Fig.~\ref{fig:xsec_mutau_2} illustrates.
\section{Detector Requirements and Machine-Detector Interface}
\subsection{Detector Design Considerations}
\label{sec:detector}
As mentioned earlier, the detector design at a muon-ion collider shares
a lot in common with a $\mu^{+}\mu^{-}$ collider but it also has its unique
requirements and challenges.
We sketch a conceptual design of a general-purpose
detector at a muon-ion collider in
Fig.~\ref{fig:detector}.
The shielding tungsten nozzle is applied only to the
muon coming side, resulting in central detector
acceptance of $-5<\eta<2.4$. As shown in Appendix~\ref{sec:dis-kin},
the scattered parton (e.g., jets) and muon are mostly going
toward the backward direction. Therefore, the available
central detector acceptance with the single-sided nozzle configuration
is sufficient to meet most of physics requirements at a muon-ion collider.
The shielding nozzle will mainly limit the access to particles
produced by remnants of the proton in the forward direction.
The center detector consists of silicon tracker system with precision
timing information (e.g., 4-D tracking using low gain
avalanche diodes, or LGADs), which is essential
to effectively suppress the muon beam induced background.
The electromagnetic and hadronic calorimeters are needed for
detecting and identifying objects like jets, photons and electrons.
Similarly, emerging technology that embeds silicon sensors inside
calorimeters to provide precision timing and position information
will be particularly beneficial to the detector system at a muon-based
accelerator for suppressing beam induced backgrounds. The particle identification (PID) system for hadrons is crucial for QCD and nuclear
physics at a lepton-hadron collider. Unlike the EIC where the coverage
to the very high momentum (up to 100~GeV) regime by the PID systems
only needed in the forward direction, high momentum hadrons are
produced in both forward and backward directions at the MuIC or LHmuC.
Therefore, ring-imaging Cherenkov (RICH) detectors with gas media
are necessary in both endcaps of the experimental system, which generally
take up to about 1~m in $z$ direction. The gas RICH detectors
have a lower momentum threshold of about 3 GeV. The PID below 3~GeV
can be well covered by the LGADs-based time-of-flight system or 4-D tracker,
as was designed for the EIC.
The far-forward direction ($5<\eta<8$)
should be instrumented with Roman Pots for detecting the scattered
proton in elastic processes. In the far backward region ($-8<\eta<-5$),
a muon spectrometer system is required to detect the scattered muon
and precisely determine its momentum so that the DIS kinematics can be
reconstructed. Considerations and requirements of such a muon spectrometer is discussed
in Section~\ref{sec:muon-spec} below.
The performance of reconstructing DIS kinematic variables, $Q^{2}$ and $x$,
at the MuIC with parameterized detector resolutions is presented in
Ref.~\cite{Acosta:2021qpx}. Detailed simulations of detector system, beam-induced
background, and MDI will further inform the design and R\&D activites.
\begin{figure}[htb]
\centering
\includegraphics[width=\textwidth]{Figures/detector.png}
\caption{A sketch of a possible detector design at muon-proton colliders. The tungsten nozzle is only applied on the muon coming direction.}
\label{fig:detector}
\end{figure}
\subsection{Far-backward Muon Spectrometer}
\label{sec:muon-spec}
A key experimental aspect of a TeV-scale muon-ion collider experiment, as explained in Appendix~\ref{sec:dis-kin}, is the need for a muon spectrometer able to measure muons with reasonable resolution down to very small scattering angles ($\eta \gtrsim -7$) and at momenta up to the beam energy (${\approx }1$~TeV). We note that this need is in common with that for a TeV $\mu^+\mu^-$ collider experiment as well, if the tagging of forward muons from vector-boson fusion processes is considered a physics priority. If one places a measurement plane downstream of the collision point in the muon beam direction at $z=-5$~m (i.e. just beyond the central region of the experiment) with detectors placed outside of the beampipe radius of ${\approx}5$~cm, then this measurement station could cover the pseudorapidity range $-5< \eta < -1$. The radial coverage of the detectors would need to extend to $r\approx 3.5$~m to reach $\eta=-1$. To cover smaller angle scattered muons with $\eta = -7$ and $r>5$~cm requires a station to be placed 30~m or more downstream of the collision point. For example, a measurement station placed at $z=-40$~m, with a radius extending to $r=1$~m, could cover the pseudorapidity range $-7< \eta < -4.5$.
The measurement of the momentum of these small angle muons will require a dedicated magnet spectrometer with the field provided from one or more dipole (or perhaps toroid) magnets. The $\int \vec{B}\cdot d\vec{\ell}$ will need to be substantial for TeV muons, likely of order 10~T$\cdot$m. The measurement of the saggita along with some vertexing capability from the tracking system will require more than the two stations noted above, as more than one measurement of the same muon would be needed. This would argue for a telescope arrangement of measurement stations stretching down the beamline, with a transverse radius of each station that does not need to be so large (of order a meter). However, the detailed design and performance of a plausible muon spectrometer is beyond the scope of this paper.
\bigskip \bigskip \begin{center} \begin{large{}
This work is in part supported by the Department of Energy
grant numbers DE-SC0005131 (W.L.) and DE-SC0010266 (D.A.). N.H. acknowledges support from the Office of Undergraduate Research and Fellowships at Northeastern University.
|
1,108,101,565,700 | arxiv | \section*{Introduction}
\paragraph{}
Observational and numerical data (\citet{ros}) stress the dynamical consideration of spiral arms as for stars have been observed to migrate radially through the spiral arms leaving significant distances between their observed positions and birthregions. It is well known that spiral arms are trajectories of local energetic minima with a very small tangental increase of potential energy thus making it possible to migrate long distances with low effort. This is why local barycenters migrate along these trajectories. The focus of this paper will now be the dynamics of objects in regions of these local barycenters.
\section*{The model}
\paragraph{}
The model contains of two parts: the local gravitational impact and the non-local gravitational impacts. To consider the local impacts I assume the disk to be hot enough in order to take a region with radius $R$ of a dense region or object (a star or planet to be build) so that the region will be nearly homogenous (density $\varrho$, mass $m_2=\frac{4\pi\varrho R^3}{3}$) in all three dimensions. The potential $U$ then integrates
\begin{equation}
\Delta U=4\pi\gamma\varrho
\end{equation}
Obviously this leads to
\begin{equation}
U_i=\frac{2\pi\gamma\varrho}{3}r^2-\frac{\mathcal{A}}{r}+\mathcal{B}
\end{equation}
Demanding $|U_i(r=0)|<\infty$ leads to $\mathcal{A}=0$. Further there shall be
\begin{equation}
U_i(r=R)=-\frac{\gamma m_2}{R}\ \Rightarrow\ \mathcal{B}=-2\pi\gamma\varrho R^2=-\frac{3\gamma m_2}{2R}
\end{equation}
\begin{equation}
\Rightarrow\ U_i=\frac{2\pi\gamma\varrho}{3}r^2-\frac{3\gamma m_2}{2R}=\frac{\gamma m_2}{2R}\left(\frac{r^2}{R^2}-3\right)
\end{equation}
Thus the force will be
\begin{equation}
\Rightarrow\ F_i=-\frac{\gamma m_2}{R^3}r
\end{equation}
Taking other forces into account the force will slightly change, hence will be perturbated to
\begin{equation}
F_i=-\tilde\alpha_2r\qquad,\qquad \tilde\alpha_2>0
\end{equation}
\paragraph{}
Integrating all the non-local impacts you will get a barycentral force
\begin{equation}
F_e=-\frac{\tilde\alpha_1}{r'^2}\qquad,\qquad\tilde\alpha_1>0
\end{equation}
With $\alpha_i:=\frac{-\tilde\alpha_i}{m}$, $m$ being the mass of the dense object, Newton's second law becomes
\begin{equation}
\left(\begin{array}{c}
\ddot x\\ \ddot y
\end{array}
\right)=\left(\begin{array}{c}
\frac{\alpha_1x}{\left(x^2+y^2\right)^{\frac{3}{2}}}+\alpha_2\left(x-\chi\right)\\
\frac{\alpha_1y}{\left(x^2+y^2\right)^{\frac{3}{2}}}+\alpha_2\left(y-\psi\right)
\end{array}
\right)
\end{equation}
Whereas $(\chi;\psi)^T$ is the current position of the local barycenter with respect to the barycenter of the non-local masses and $(x;y)^T$ being the position of the dense object with respect to the barycenter of the non-local masses being in the center.
\paragraph{}
I do not consider any motion normal to the plane of the disk, since it is obvious that it will be just some oscillation. The model also only applies to the middle regions of the disk, since there are non-trivial $z$-components to the force inside the bulge and in the outer regions the considered region will not be as homogenous anymore.
\section*{Stability of trajectories}
\paragraph{}
Now the trajectories can be characterized by
\begin{equation}
f(X):=\left(\begin{array}{c}
\dot x\\
\dot v_x\\
\dot y\\
\dot v_y
\end{array}
\right)= \left(\begin{array}{c}
v_x\\
\frac{\alpha_1x}{\left(x^2+y^2\right)^{\frac{3}{2}}}+\alpha_2\left(x-\chi\right)\\
v_y\\
\frac{\alpha_1y}{\left(x^2+y^2\right)^{\frac{3}{2}}}+\alpha_2\left(y-\psi\right)
\end{array}
\right)
\end{equation}
with $X:=(x,v_x,y,v_y)^T$ and $x^2+y^2\gg R^2$.
Obviously is $f$ in $X$ lipschitz continuous and thus complies with the standard prerequisites. A possible Lyapunovfunction will have to integrate
\begin{equation}
\begin{array}{ll}\label{dV}
0\ge\dot V:=\langle \nabla V|f\rangle=&v_x\partial_xV+\left(\frac{\alpha_1x}{\left(x^2+y^2\right)^{\frac{3}{2}}}+\alpha_2\left(x-\chi\right)\right)\partial_{v_x}V\\
&+v_y\partial_yV+\left(\frac{\alpha_1y}{\left(x^2+y^2\right)^{\frac{3}{2}}}+\alpha_2\left(y-\psi\right)\right)\partial_{v_y}V
\end{array}
\end{equation}
Using the identities $x-\chi=:x'$, $v_x-v_\chi=:v_x'$, $y-\psi=:y'$, $v_y-v_\psi=:v_y'$ and $X':=(x',v_x',y',v_y')^T$
\begin{equation}
V(X'):=\begin{cases}\begin{array}{ll}
\exp\left(-x'^2+v_x'^2-y'^2+v_y'^2\right)-1&,\ x'\ge0,\ v_x'\ge0,\ y'\ge0,\ v_y'\ge0\\
\exp\left(-x'^2+v_x'^2-y'^2-v_y'^2\right)-1&,\ x'\ge0,\ v_x'\ge0,\ y'\ge0,\ v_y'\le0\\
\exp\left(-x'^2+v_x'^2+y'^2+v_y'^2\right)-1&,\ x'\ge0,\ v_x'\ge0,\ y'\le0,\ v_y'\ge0\\
\exp\left(-x'^2+v_x'^2+y'^2-v_y'^2\right)-1&,\ x'\ge0,\ v_x'\ge0,\ y'\le0,\ v_y'\le0\\
\exp\left(-x'^2-v_x'^2-y'^2+v_y'^2\right)-1&,\ x'\ge0,\ v_x'\le0,\ y'\ge0,\ v_y'\ge0\\
\exp\left(-x'^2-v_x'^2-y'^2-v_y'^2\right)-1&,\ x'\ge0,\ v_x'\le0,\ y'\ge0,\ v_y'\le0\\
\exp\left(-x'^2-v_x'^2+y'^2+v_y'^2\right)-1&,\ x'\ge0,\ v_x'\le0,\ y'\le0,\ v_y'\ge0\\
\exp\left(-x'^2-v_x'^2+y'^2-v_y'^2\right)-1&,\ x'\ge0,\ v_x'\le0,\ y'\le0,\ v_y'\le0\\
\exp\left(x'^2+v_x'^2-y'^2+v_y'^2\right)-1&,\ x'\le0,\ v_x'\ge0,\ y'\ge0,\ v_y'\ge0\\
\exp\left(x'^2+v_x'^2-y'^2-v_y'^2\right)-1&,\ x'\le0,\ v_x'\ge0,\ y'\ge0,\ v_y'\le0\\
\exp\left(x'^2+v_x'^2+y'^2+v_y'^2\right)-1&,\ x'\le0,\ v_x'\ge0,\ y'\le0,\ v_y'\ge0\\
\exp\left(x'^2+v_x'^2+y'^2-v_y'^2\right)-1&,\ x'\le0,\ v_x'\ge0,\ y'\le0,\ v_y'\le0\\
\exp\left(x'^2-v_x'^2-y'^2+v_y'^2\right)-1&,\ x'\le0,\ v_x'\le0,\ y'\ge0,\ v_y'\ge0\\
\exp\left(x'^2-v_x'^2-y'^2-v_y'^2\right)-1&,\ x'\le0,\ v_x'\le0,\ y'\ge0,\ v_y'\le0\\
\exp\left(x'^2-v_x'^2+y'^2+v_y'^2\right)-1&,\ x'\le0,\ v_x'\le0,\ y'\le0,\ v_y'\ge0\\
\exp\left(x'^2-v_x'^2+y'^2-v_y'^2\right)-1&,\ x'\le0,\ v_x'\le0,\ y'\le0,\ v_y'\le0\\
\end{array}\end{cases}
\end{equation}
integrates \ref{dV} in the open region
\begin{equation}
\Omega:=\left\{(x,v_x,y,v_y)^T:\ |x-\chi|<\frac{\alpha_1x}{\alpha_2(x^2+y^2)^{\frac{3}{2}}},\ |y-\psi|<\frac{\alpha_1y}{\alpha_2(x^2+y^2)^{\frac{3}{2}}}\right\}
\end{equation}
with $V(\chi,v_\chi,\psi,v_\psi)=0=\dot V(\chi,v_\chi,\psi,v_\psi)$ and coordinates respectively chosen to comply $x,y,v_x,v_y,\chi,\psi>0$, which is perfectly possible since $x,y,\chi,\psi>0$ can be chosen as there is $x^2+y^2\gg R^2\ge (x-\chi)^2+(y-\psi)^2$ and $v_x,v_y>0$ define the surface normal. The implicit function theorem ensures that there will always be a time-interval to locally solve these conditions. Should the Jacobian be singular then simply turn the coordinates a little ($x^2+y^2\gg R^2\ge (x-\chi)^2+(y-\psi)^2$). $\Omega$ herein is the largest possible local region to be observed.
\paragraph{}
Obviously
\begin{equation}
\forall\left(\begin{array}{c}
x\\v_x\\y\\v_y
\end{array}\right)\not=\left(\begin{array}{c}
\chi\\v_\chi\\\psi\\v_\psi
\end{array}\right):\ \exp\left(-x'^2-v_x'^2-y'^2-v_y'^2\right)-1<0
\end{equation}
thus in every region of $\left(\chi,v_\chi,\psi,v_\psi\right)^T$ exists a point $Z$ with $V(Z)<0$. Hence the local region is instable and as a consequence any dense object within the region with leave this in a finite period of time.
\section*{On migration}
\paragraph{}
Since it is known that the local barycenters travel along the spiral arms any dense object in that regions will move with them and this way migrate radially through the disk. On the other hand each object will leave that region after a finite time and hence travel no longer with that local barycenter. Now the object is no more in a local energetic minimum and hence will move back entering another radially migrating region just to soon leave this again and enter right another. Multiple objects of the same birthregion will do so indepenently. Thus the observed galaxies should show a good mixture of young and old stars and stars with different chemical compositions since there are stars from basically every birthregion close another. These are exactly the observed properties in the middle parts of galaxies.
\paragraph{}
It might be interesting to estimate the density of stars in spiral arms for long term effects. In order to migrate outwards for an object it needs energy which has to be taken from other objects for the sum of energies to stay conserved. Since it seems more likely to transfer low amounts of energy I will approximate the energy-transfer-distribution as gaussian with the expected transfer amount $\mu_E=0$. Let then $T$ be the average time between entering two consecutive regions and let $\Lambda$ be the average radial movement, then the energy-transfer-distribution $\varphi_E$ with standard deviation $\sigma_E$ becomes
\begin{equation}
\varphi_E(\Delta E)=\frac{1}{\sigma_E\sqrt{2\pi}}\exp\left(-\frac{\Delta E^2}{2\sigma_E^2}\right)
\end{equation}
In order to estimate $\sigma_E$ I considered the energy $E_g=\frac{\alpha_1m_3}{2r}$ of a mass $m_3$ at radius $r$. As the radius changes by $\Lambda$ the transfered energy is given by
\begin{equation}\label{mu_E}
\mu_{\Delta E}=\frac{1}{2}\left(\frac{|\alpha_1|m_3}{2\left(r+\Lambda\right)}+\frac{|\alpha_1|m_3}{2\left(r-\Lambda\right)}\right)=\frac{|\alpha_1|m_3}{4}\left(\frac{2r}{r^2-\Lambda^2}\right)
\end{equation}
Now I average this over the entire applicable region with inner radius $R_I$ and outter radius $R_O$
\begin{equation}
\overline{\mu}_{\Delta E}=\frac{1}{R_O-R_I}\int_{R_I}^{R_O}\mu_{\Delta E}dr=\frac{|\alpha_1|m_3}{4(R_O-R_I)}\ln\left(\frac{R_O^2-\Lambda^2}{R_I^2-\Lambda^2}\right)
\end{equation}
Now it is possible to approximate $\sigma_E$ by
\begin{equation}
\frac{1}{2}=2\int_0^{\overline{\mu}_{\Delta E}}\varphi_Ed\Delta E
\end{equation}
The energy-density thus will be homogenous. Considering our own galaxy with $R_I=8000ly\ll r\ll 50000ly=R_O$ one finds $E(8000ly)\approx 6\alpha_110^{-21}\frac{Js^2}{m^3}m_3$ as well as $E(50000ly)\approx \alpha_110^{-21}\frac{Js^2}{m^3}m_3$ and thus $\Delta E=5\alpha_110^{-21}\frac{Js^2}{m^3}m_3$ on a distance of $42000ly$ which is nearly homogenous.
\section*{Conclusion}
\paragraph{}
From (\ref{mu_E}) it is easy to see that the energy-transfer enables much larger distances of migration the farther away an object is from the center of the disk. Hence one will more likely find a higher mixture of objects in the outter regions. On the other hand this gives a mechanism preventing too many masses to fall into the interior regions and thus from possibly falling into black holes. Further keeps the disk from accreting too fast making it possible for planets and stars to form.
\paragraph{}
The local instability hence has an impact on the largescale stability of a protoplanitary disks and spiral arms. It does also explain the observed mixture of stars.
|
1,108,101,565,701 | arxiv | \section{Introduction} \label{sec:intro}
Solar filaments/prominences are one of the most common structures in the corona, which may lead to energetic coronal mass ejections (CMEs) and flares when they erupt \citep{che00,for00,lin00,yan13,xue16,yang17}. They appear at the limb as bright features called prominences. In contrast, they appear on the disk as dark filamentary structures called filaments. It is widely accepted that solar filaments are cool, dense plasma structures suspended in the extremely hot solar corona. They often lie above magnetic polarity inversion lines (PILs) on the photosphere and are supported by the local magnetic fields (e.g. magnetic dips or twisted structures) against the gravity \citep{bab55,mar98,yang14}. Generally, according to their locations on the solar disk, filaments can be classified as active region filaments, intermediate filaments, and quiescent filaments \citep{pat02,mac10}. Observationally, active region filaments are lower, smaller and shorter-lived than quiescent or intermediate filaments. However, active region filaments are more likely to erupt than other two types \citep{jin04,par14}.
How cool and dense material is supplied to filaments embedded in the hot and tenuous corona remains an open question. On the one hand, many researchers investigated the formation of filament magnetic field structures and proposed two different efficient ways to form the magnetic field structures of filaments: surface effect \citep{van89,mar01,yan15,yan16,yang16,wan17,xue17,chen18} and subsurface effect \citep{oka08,oka09,lit09,lit10,mact10,yan17}. \cite{mact10} analyzed a set of MHD simulations on filament formation and supported the observation reported by \cite{oka08,oka09} that there was a flux rope emerging under the filament. However, \cite{var12} reanalyzed the data and found that the observational evidence was not enough to support the emergence of a flux rope. It has been reported that the flux cancellation and convergence in the photosphere play an important role in the formation of filament magnetic structures \citep{wan01,wan07}.
Material origination of the filament is another issue for understanding the formation of the filament. It is believed that the material of the large filament must come from the low solar atmosphere (chromosphere), because not enough plasma could be supplied for these large filaments in the corona \citep{pik71,zir94}. Comparing the elemental abundances in situ observations with that of the photosphere, \cite{son17} suggested that the plasma of an active region filament may originate from the lower solar photosphere instead of the corona.
The mechanisms by which low atmospheric material is converted into filament material are not fully understood. According to numerous observations and numerical simulations, three popular models have been proposed by many authors: injection model, levitation model, and evaporation-condensation model \citep{mac10}. The injection model suggests that the cool plasma is forced upward into the filament channels or typical filament heights through sufficient magnetic forces (e.g. magnetic reconnection in the vicinity of PILs or near the filament foot-points) \citep{pri96,lit99,wan99,wan01}. \cite{cha03} reported that the plasma could be ejected into an active-region filament by a successive of jets and small eruptions through magnetic reconnection in the vicinity of the magnetic PIL. \cite{liu05} reported that the formation of two new filaments is closely correlated with surges and the cool material is directly injected into the main axis/channels of filaments by these surges at one foot-point. \cite{zou16} proposed that the cool material was injected in the form of fibrils to replenish the filament and proposed that the magnetic reconnection play an important role in transporting cool material into the filament.
The levitation model proposes that cool plasma is directly lifted by rising magnetic fields at the magnetic PIL. In one scenario of this model, the highly twisted flux rope emerges from the photosphere and brings up cool plasma in the axis of rising twisted flux rope \citep{rus94,gal99,lit05,kuc12}.
Based on the fact that adding heat to a coronal loop increases the density of the corona accompanied by decreasing slightly the chromospheric mass, the evaporation-condensation model has been proposed. According to numerous numerical simulations, many researchers found that the cool material could be condensed at the apex of coronal loop by heating near the foot-points \citep{ant91,kar05,kar06,xia11,xia12,kan17}. Using the observations from SDO/AIA, \cite{liu12} considered that the coronal condensation should be responsible for the formation of a prominence. Although these models can explain some observations of the filament formation, the physical mechanisms of material injection are still hardly to understand fully. Besides, due to the long period of formation process, the entire formation process of filaments, especially quiescent filaments, is difficult to be captured.
Compared with quiescent filaments and intermediate filaments, active region filaments need relatively short time to form. It is about several hours or one day for forming an active region filament in general. In this paper, we investigate a formation process of a filament in active region NOAA 12574 during the period from 2016 August 11 to 12. The whole formation process that the filament was from absent to present can be exhibited clearly, which is a rare case for understanding the formation of the filament. By the ground-based and space-based observations, the material injection of the filament is studied in detail. We investigate jetting events as possible mechanisms for loading mass into the active region filament. We compare the cumulative mass flux supplied by the jets with the estimated mass in the filament. Furthermore, we estimate the magnetic windings injected into the filament system.
\section{Observations and Methods} \label{sec:obser methods}
During the period from 2016 August 11 to 12, a filament formed gradually in active region NOAA 12574, which was located in the northeast hemisphere (eg.about (-375$\arcsec$, 50$\arcsec$)). Observations from the Global Oscillation Network Group \citep{har96,har11} and the \emph{Solar Dynamics Observatory} \citep{pes12} covered the whole process of the filament formation. Combined with the observations by $Hinode$ \citep{kos07}, Goode Solar Telescope \citep{cao10}, Domeless Solar Telescope \citep{nak85} and \emph{Interface Region Imaging Spectrograph} \citep{pon14}, two material injection events (two jets) related to the formation of the filament are studied in detail.
Full disk H$\alpha$ images from six GONG stations are used to monitor filament formation and evolution over two days. The CCD plate-scale of the H$\alpha$ images is about 1$\arcsec$ $\rm pixel^{-1}$ and the cadence is 1 minutes. The data from Atmospheric Imaging Assembly \citep{lem12} and the Helioseismic and Magnetic Imager \citep{scher12,schou12} on board the \emph{SDO} provides full-disk, multi-wavelength, high spatio-temporal resolution observations for this study. The SDO/AIA has seven extreme ultraviolet (EUV) and three ultraviolet-to-visible (UV) channel images with the CCD plate-scale of 0$\farcs6$ $\rm pixel^{-1}$. The cadence of EUV and UV channel images are 12 s and 24 s, respectively. Using the 6173 $\rm \AA$ Fe $\rm I$ absorption line, the SDO/HMI can provide the Doppler shift, line-of-sight magnetic field, continuum intensity and vector magnetic field on the solar photosphere with the CCD plate-scale of 0$\farcs$5 $\rm pixel^{-1}$, with the cadences of the three former channels being about 45s and that of the latter one being about 12 minutes. The SDO/AIA 304 $\rm \AA$ images exhibit the formation process of the active region filament, while the SDO/AIA 1600 $\rm \AA$ images and SDO/HMI line-of-sight magnetic fields are utilized to investigate the physical mechanisms of the material injection related to the filament formation.
TiO (7057 $\rm\AA$) images observed by the Broadband Filter Imagers on the 1.6 m GST at the \emph{Big Bear Solar Observatory} with the CCD plate-scale of 0$\farcs$0342 $\rm pixel^{-1}$ and a cadence of 15 seconds provide important information in the photosphere during a jet at 18:02 UT on August 11. The Spectro-polarimeter instrument \citep{ich08} of the Solar Optical Telescope \citep{tsu08} on board $Hinode$ provides the photospheric vector magnetic field and Doppler shift with the CCD plate-scale of 0$\farcs$16 $\rm pixel^{-1}$ for this study. The vector magnetic fields are derived by performing the Milne-Eddington (M-E) inversion of the spectro-polarimetric profiles of two magnetically Fe lines at 6301.5 $\rm\AA$ and 6302.5 $\rm\AA$ \citep{oro07}. The retrieval of the vector magnetic field is carried out by the data analysis pipeline at HAO/CSAC (Community Spectro-polarimetric Analysis Center of High Altitude Observatory, Boulder). The data are referred to as \emph{Hinode} Level 2 data sets at HAO/CSAC.
Spatially scanned H$\alpha$ and Ca II K spectrums with the Horizontal Spectrograph (HS) obtained simultaneously by DST at Hida Observatory \citep{uen04} are used to investigate a material injection event (Jet B) at 00:42 UT on August 12. The spectral sampling is 0.020 $\rm \AA$ and the angular sampling along the slit is 0$\farcs$24. The full wavelength-range of the obtained spectrum is 16 $\rm \AA$ and it scans the observational region at every 15 s. The spatial scan step is about 0$\farcs$64 and the time interval between two spatial steps is 0.05 s. Based on the H$\alpha$ spectral, we can calculate the Doppler velocity of H$\alpha$ by the follow equation: $v_{dop}=((\lambda_{obs}-\lambda_0)/\lambda_0)*c$, where $\lambda_{obs}$ is the center of observational line, $\lambda_{0}$ is the center of H$\alpha$ line and $c$ is the velocity of light. We use the weight-reverse-intensity method to derive the $\lambda_{obs}$ of H$\alpha$ line \citep{su16}. These velocities represent the ``mean line-of-sight velocity" of the material along the light path, which are justified for estimating direction of the motion (blue shift or red shift) in this study. These velocities are inferred by assuming that the mean velocity in the quiet region is zero. IRIS can provide spectral scan and slit-jaw images (SJIs) at near-ultraviolet (NUV) and far-ultraviolet (FUV) lines simultaneously \citep{pon14}. The 1400 $\rm \AA$ SJI and Si $\rm {IV}$ spectrum of IRIS are used to investigate the jet in this study. The single Gaussian fitting method is used to obtain the center of observational Si $\rm {IV}$ line for calculating the Doppler velocity of Si $\rm {IV}$ line, which allows us to investigate the properties of the jet nearby the one foot-point of the filament.
In order to illustrate the morphological structure of the magnetic fields of the jets and filament, we reconstruct the coronal magnetic field through a nonlinear force-free field (NLFFF) model with vector magnetic field observed by SDO/HMI. NLFFF extrapolation is obtained by using the ``weighted optimization" method \citep{whe00,wie04} after preprocessing the photospheric boundary to meet the force-free condition \citep{wie06}. Before the extrapolation, we use a 2 $\times$ 2 rebinning of the boundary data to 0.72 Mm pixel$^{-1}$ as some authors did \citep{sun12,wie12,liu13}. The extrapolated field derived by the NLFFF model extrapolation is thought to well match the magnetic structures of the observational images \citep{wie05}. From panels (c) and (d) of the Fig.\ref{figure0}, the magnetic field lines produced by the NLFFF model extrapolation are roughly compatible with the EUV loops observed by SDO/AIA 171 \AA. On the other hand, the magnetic helicity injection rate across a surface S can be calculated by using the following equation \citep{ber84}:
\begin{equation}\label{equ3}
\frac{dH}{dt}=-2\int_S(\textbf{A}\cdot \textbf{u})B_n dS,
\end{equation}
in which A is the vector potential of the potential field, $\textbf{u}$ denotes the velocity of the flux tubes on the boundary (the flux transport velocity, $\textbf{u}=\textbf{V}_t - (V_n/B_n)\textbf{B}_t$) \citep{dem03}, and $B_n$ denotes the strength of normal component of the magnetic field. With a acceptable assumption that the solar photosphere S is planar, \citet{par05} proposed the helicity injection rate could be transformed to:
\begin{equation}\label{equ4}
\frac{dH}{dt}=-\frac{1}{2\pi}\int_{S\arcmin} \int_S \frac{d\theta(\bf{r})}{dt}B_nB\arcmin_ndSdS\arcmin,
\end{equation}
and
\begin{equation}\label{equ5}
\frac{d\theta(\bf{r})}{dt}=\frac{1}{r^2}(\bf{r} \it \times \frac{dr}{dt})_n=\frac{1}{r^2}(\bf r \times (u-u\arcmin))\it _n,
\end{equation}
where $ \bf r = x-x\arcmin$ denotes the vector between two photospheric points defined by $x$ and $x\arcmin$, $\textbf{u}$ and $\textbf{u\arcmin}$ are the homologous velocities of two different points. A good proxy of helicity flux density $G_\theta(x)$ can be then defined as:
\begin{equation}\label{equ6}
G_\theta(x)=-\frac{B_n}{2\pi}\int_{S\arcmin}
\frac{d\theta(\bf{r})}{dt}B\arcmin_ndS\arcmin.
\end{equation}
We use the differential affine velocity estimator for vector magnetograms (DAVE4VM) method to compute the velocity of the flux tubes \citep{sch08}. The whole active region of the vector magnetograms (see Fig.\ref{figure0} (b)) are used to compute the helicity flux density. The method is similar to that used by \cite{jin12}. Once the helicity flux density has been determined, the helicity flux can be integrated by the integral region using the equation (\ref{equ4}).
\begin{figure}[ht!]
\figurenum{1}
\plotone{figure0.eps}
\caption{Observations of active region NOAA 12574 at about 12:00 UT on 2016 August 11. (a) Line-of-sight magnetic field from SDO/HMI. White denotes the magnetic field with positive polarity, while black denotes the magnetic field with negative polarity. (b) Photospheric vector magnetogram from SDO/HMI, corresponding to the region marked by the red dotted-dashed box in panel (a). The blue arrows indicate the transverse field. (c) SDO/AIA 171 $\rm\AA$ image. (d) The magnetic field lines derived from the NLFFF model extrapolation. \label{figure0}}
\end{figure}
\section{Results} \label{sec:results}
\subsection{The formation process of the filament} \label{subsec:process}
A filament, located in active region NOAA 12574 on northeast hemisphere (see Fig.\ref{figure0} (a)), formed gradually during the period from 04:00 UT on August 11 to 04:00 UT on August 12, 2016. The entire formation process of the filament was observed by several ground-based and spaced-based telescopes. Fig.\ref{figure1} displays the formation process of the filament in SDO/AIA 304 $\rm\AA$ and H$\alpha$ bands. As is shown by GONG H$\alpha$ observations in panels (a)-(c), the filament was absent at 05:34:34 UT on August 11. The filament formed completely at 03:30:34 UT on August 12. The corresponding SDO/AIA 304 $\rm \AA$ images are shown in panels (d)-(f). Panels (g)-(i) show the photospheric line-of-sight magnetic fields observed by SDO/HMI. The detailed formation process of this filament is shown in the animated version of Fig.\ref{figure1} (a).
\begin{figure}[ht!]
\figurenum{2}
\plotone{figure1.eps}
\caption{Formation process of the filament. (a)-(c) H$\alpha$ images observed by GONG in different moments. (d)-(f) The corresponding SDO/AIA 304 $\rm \AA$ images. (g)-(i) The corresponding line-of-sight magnetic fields from SDO/HMI. White patches denote the magnetic field with positive polarity, while black ones denote the magnetic field with negative polarity. The yellow rectangle in panel (g) indicates the region for calculating different physical quantities in Fig.\ref{figure3}. \label{figure1}}
\end{figure}
According to the animation of Fig.\ref{figure1} and Fig.\ref{figure2}, a series of jets occurring on the western foot-point of the filament during the formation of the filament, injected mass into the coronal height. As cool plasma was injected into the filament by the jets, the dark and elongate filament appeared in the field of view. During the early period from 04:00 UT to 12:30 UT on August 11, most of plasma was lifted to the filament height from the western foot-point by the jets, and then threw down to the other foot-point. There was only a fraction of plasma that could be retained in the filament height. This might be related to the local magnetic structure, which is too little ``dips" or twisted magnetic structure existed in the local corona to capture the injected plasma. This also means that not all plasma injected by the jets can be transformed to the filament material. At a later period, most of the lifted plasma was trapped in the filament and become the material of the filament instead of falling down to the other foot-point. Fig.\ref{figure2} shows serval jets occurring in the vicinity of the western foot-point of the filament. Panels (a1)-(a4) show four jets in different moments in SDO/AIA 304 $\rm \AA$ observations, which are marked by white arrows nearby the western foot-point of the filament. Panels (b1)-(b4) exhibit that the cool material was injected into the filament from low solar atmosphere after each jet, while the brightening in SDO/AIA 1600 $\rm\AA$ could also be identified nearby the western foot-point of the filament during each jet in panels (c1)-(c4). It is found that a number of cool plasma blobs were directly injected into the filament driven by each jet.
\begin{figure}[ht!]
\figurenum{3}
\plotone{figure2.eps}
\caption{A series of typical jets occurred on the western foot-point of the filament, which injected massive plasma into the filament. (a1)-(a4) The jets in SDO/AIA 304 $\rm\AA$ images. The white arrows indicate jets related to the formation of the filament. (b1)-(b4) GONG H$\alpha$ observations after each jet. (c1)-(c4) The corresponding SDO/AIA 1600 $\rm\AA$ images during each jet.\label{figure2}}
\end{figure}
As described in section \ref{sec:obser methods}, we use the NLFFF model to extrapolate the magnetic structures of the jets and the filament. Fig.\ref{figure9} shows the selected magnetic field lines at different moments. Panels in the left column show the magnetic structure seen from top side view, while panels in the right column show the perspective from the left (solar East). Panels (a) \& (b) and panels (c) \& (d) show the magnetic field lines at 04:00:00 UT and 14:00:00 UT on August 11, respectively, while panels (e) \& (f) show the selected magnetic field lines at 22:00:00 UT on August 11. At 04:36:00 UT, there were two flux tubes existing in the field of view before eruption of jets. The small tube corresponds to the magnetic structure of the emerging magnetic field. When the jet erupted, it reconnected with the big flux tube and formed much larger flux tube (see Fig.\ref{figure9} (c) and \ref{figure9} (d)). The repeated process was observed during a series of jet eruptions, and the magnetic structure of the filament formed finally (see Fig.\ref{figure9} (e) \& (f)).
\begin{figure}[ht!]
\figurenum{4}
\plotone{figure9.eps}
\caption{The selected magnetic field lines derived by using NLFFF extrapolation model at different times and different views. The left columns are seen from top side view, while the right columns are seen from left-side view. The blue lines indicate the selected magnetic field lines. The background indicates the radial magnetic field. \label{figure9}}
\end{figure}
In order to understand the physical mechanism of these jets injecting material for the filament, we calculate the magnetic flux, magnetic helicity and intensity of SDO/AIA 304 $\rm \AA$ nearby the western foot-point of the filament (the region marked by the yellow box of the Fig.\ref{figure1} (g)). We only calculate the positive magnetic flux because of the complexity of the negative magnetic flux in this region. Panels (a), (b) and (c) of Fig.\ref{figure3} show the time variations of the positive magnetic flux, magnetic helicity and intensity of SDO/AIA 304 $\rm \AA$ during the period from 04:00 UT on August 11 to 04:00 UT on August 12 in the yellow box of the Fig.\ref{figure1} (g), respectively. The increases of the SDO/AIA 304 $\rm\AA$ intensity during the periods of the jets related to the formation of the filament nearby the western foot-point were identified, which are marked by red arrows in the panel (c). Vertical blue dashed lines indicate the onsets of the jets. In the panel (a), it is found that the positive magnetic flux almost increased to twice during the period from 04:00 UT on August 11 to 00:00 UT on August 12. The increase of the magnetic flux manifests that the magnetic flux emerged in the vicinity of the western foot-point of the filament. It is believed that the flux emergence nearby the western foot-point of the filament played an important role in the occurrences of these jets. On the other hand, the magnetic flux decreased by 5-10$\%$ after each several jets (such as the last four marked jets). Panel (b) shows the evolution of helicity injection rate and accumulated helicity. The helicity injection rate is shown by the solid black line, while the red dotted-dashed line indicates the accumulated helicity. The accumulated helicity is calculated by the time integral of the helicity injection rate and we set it to be zero at 04:00 UT on 2016 August 11. It is found that the negative helicity injection was dominant and the negative accumulated helicity was constantly increasing. It is noted that the negative helicity is injected into the upper atmosphere nearby the western foot-point of the filament. It does not mean that all the injected helicity is stored in the filament. Based on above observations, we propose that these jets were triggered by the magnetic reconnection between closed pre-existing magnetic fields and emerging magnetic fields nearby the western foot-point of the filament. Moreover, these jets injected massive cool plasma into the filament. Some of the lifted plasma could stay in the corona height and became the filament material. However, some of the lifted plasma could not maintain stability in the corona and eventually descended at the other foot-point. Thereafter, the dark and elongated filament appeared. In the meantime, some of the post-reconnected magnetic fields would sink because of the relaxation of magnetic fields after these jets, which corresponded to the \textbf{cancellation} of the magnetic flux in the photosphere.
\begin{figure}[ht!]
\figurenum{5}
\plotone{figure3.eps}
\caption{Time variations of the positive magnetic flux, magnetic helicity and SDO/AIA 304 $\rm\AA$ intensity in the yellow rectangle of Fig.\ref{figure1} (g) during the period from 04:00 UT August 11 to 04:00 UT August 12, 2016. (a) The time variation of the positive magnetic flux. (b) The time variations of the helicity injection rate and accumulated helicity. Helicity injection rate is shown by the solid black line and the red dotted-dashed line indicates the accumulated helicity. (c) The line profile of the time variation of the intensity of SDO/AIA 304 $\rm\AA$. The perpendicular red arrows denote several jets related to the material injection of the filament nearby the western foot-point of the filament. Blue dash lines indicate the onset of each jet.\label{figure3}}
\end{figure}
\subsection{Two material injection events related to the filament formation} \label{jets}
To better investigate the material injection of this active region filament, we present two material injection events (Jet A and Jet B) which are studied in detail as examples, respectively. Jet A occurred at 18:02 UT on August 11, and Jet B occurred at 00:42 UT on August 12. We utilize the data observed by SDO/AIA, GST/BBSO and Hinode/SP to investigate the Jet A, while the data observed by SDO/AIA, DST/Hida and IRIS were used to investigate the Jet B.
\subsubsection{The Jet A at 18:02 UT on August 11 \label{jet1802}}
The Jet A occurred at 18:02 UT on 2016 August 11. Panels (a)-(d) of Fig.\ref{figure4} show the process of the jet in SDO/AIA 304 $\rm\AA$ images. As is shown, the jet occurred nearby the western foot-point of the filament, and then lifted the plasma from the western foot-point of the filament into the filament. In order to investigate the photospheric response of this jet, TiO images observed by GST are utilized to show the change of photosphere during the jet. The field of view of TiO images corresponds to the region marked by the white box in the panel (a). Panels (e)-(g) of Fig.\ref{figure4} exhibit TiO observations at the western foot-point of the filament during the period covering the jet. It is found that some dark threads in the photosphere appeared in the vicinity of the western foot-point after the jet, which was marked by the blue arrow in the panel (g). Panels (h)-(i) of Fig.\ref{figure4} show the vector magnetograms observed by SDO/HMI at 17:48:00 UT and at 18:24:00 UT, which correspond to the times before and after the jet, respectively. Panel (j) of Fig.\ref{figure4} shows the difference of magnetic field inclination at 18:24:00 UT and at 17:48:00 UT. The magnetic field inclination is calculated by the follow formula: $\theta = arctan(\sqrt{bx^2+by^2}/|bz|)$. Thus, the positive value of the difference of magnetic field inclination means that the magnetic field became more horizontal, while the negative value means that the magnetic field became more vertical. It is found that the difference value of magnetic inclination at the location marked by the black circle in the panel (j) of Fig.\ref{figure4} is positive where some dark threads appeared after the jet. This means that the magnetic field became more horizontal after the jet. Therefore, it is reasonable to suspect that the appearance of dark threads would be associated with the change of the magnetic field after the jet. Due to the magnetic reconnection between the pre-existing magnetic field and emerging magnetic field during the jet, the magnetic field became more horizontal with the relaxation or sinking of the post magnetic reconnection field. Therefore, the more dark threads appeared as the magnetic field became more horizontal. In other words, dark threads are the representation of the magnetic field with big inclination in the photosphere.
\begin{figure}[ht!]
\figurenum{6}
\plotone{figure4.eps}
\caption{Evolution of a jet at 18:02 UT on August 11, 2016. (a)-(d): SDO/AIA 304 $\rm \AA$ images. The pink dotted-dashed line in the panel (a) denotes the path of the slice, which the white box denotes the field of view of panels (e)-(j). (e)-(g): Tio images observed by GST/BBSO. The blue box in the panel (e) denotes the field of view of Figs.\ref{figure5} (b) and (c), while the blue arrow in the panel (g) point out the increase of the dark thread after the jet. (h)-(i): Vector magnetograms observed by SDO/HMI. The blue arrows denote the transverse magnetic fields, while the backgrounds denote the vertical magnetic fields. (j) The different magnetic field inclination between 18:24:00 UT and 17:48:00 UT. The red color denotes positive value, while the blue color denotes negative value. Two black circles mark the location of the increase of the dark thread after the jet. \label{figure4}}
\end{figure}
In order to understand the property of the injected plasma, we make a time-distance diagram to estimate the velocity of the injected plasma along the axis of the filament. The path of the time-distance diagram is made by the pink dotted-dashed line in the panel (a) of Fig.\ref{figure4}. Fig.\ref{figure5} (a) shows the time-distance diagram derived from the SDO/AIA 304 $\rm \AA$ observations. According to the time-distance diagram, the projection velocity of the heated plasma along the axis of the filament is about 162.6$\pm$5.4 $km/s$. Figs.\ref{figure5} (b) and \ref{figure5}(c) show the vector magnetic field and Doppler shift on the photosphere nearby the western foot-point of the filament. The field of view of panel (b) of Fig.\ref{figure5} is marked by the blue box in the panel (e) of the Fig.\ref{figure4}. The blue arrows in the panel (b) denote the transverse magnetic fields, while the white and the black patches denote the radial positive and negative magnetic field, respectively. It is found that the western foot-point of the filament rooted in negative magnetic polarity. Therefore, it is easy to derive that the direction of the injected plasma along the axis of the filament was from negative polarity to positive one.
\begin{figure}[ht!]
\figurenum{7}
\plotone{figure5.eps}
\caption{(a) The time-distance diagram along the pink dot-dash line in the panel (a) of Fig.\ref{figure4}. (b) The vector magnetic field observed by the Hinode/SP instrument at the western foot-point of the filament. The background denotes the radial magnetic field while blue arrows denote transverse magnetic field. (c) The doppler shift derived from FI 6302.28 $\rm \AA$ observed by Hinode/SP instrument.\label{figure5}}
\end{figure}
\subsubsection{The Jet B at 00:42 UT on August 12} \label{0042}
The Jet B occurred at 00:42 UT on August 12. This jet also appeared in the vicinity of the western foot-point of the filament at the same place as other jets. This material injection process was captured by DST/Hida, SDO/AIA and IRIS telescopes, simultaneously. Fig.\ref{figure6} shows the process of the jet in different wavelengths and in different moments, respectively. Panels (a)-(c) show the jet acquired at SDO/AIA 304 $\rm \AA$, while Panels (d)-(f) and (g)-(i) are reconstructed images by using the center of the H$\alpha$ 6562.8 $\rm \AA$ and Ca II K 3933 $\rm \AA$ spectrum observed by the DST/Hida, respectively. According to the Fig.\ref{figure6}, massive cool plasma was injected into the filament driven by this jet in different wavelengths. After the jet, the filament became broader and darker. At about 23:47 UT, the dark structure (marked by the white arrow in the panel (d)) filled with massive plasma, existed nearby the western foot-point of the filament. At about 00:59 UT, the plasma in the dark structure started to be injected into the filament driven by the jet at 00:42 UT. At about 01:38 UT, all of the plasma had injected into the filament. And then, this dark structure (marked by the white arrow in the panel (d)) disappeared while the filament became broader and darker.
\begin{figure}[ht!]
\figurenum{8}
\plotone{figure6.eps}
\caption{Observations of a jet at 00:42 UT on August 12. (a)-(c) SDO/AIA 304 $\rm \AA$ images before, during and after the jet. (d)-(f) Corresponding constructed H$\alpha$ images by the center of H$\alpha$ spectrum observed by Hida/DST. The white arrow in the panel (d) indicates the dark structure nearby the western foot-point of the filament. The blue and red plus sign (+) in the panel (e) are the sites of the spectral lines in Fig.\ref{figure7} (a), respectively. (g)-(i) Corresponding constructed Ca II K images by the center of Ca II K spectrum observed by Hida/DST. The blue and red plus sign (+) in the panel (h) are sites of the spectral lines in Fig.\ref{figure7} (b), respectively. \label{figure6}}
\end{figure}
In order to investigate the injected plasma, we analyze the spectrum of H$\alpha$ and Ca II K line during the period of the jet. Fig.\ref{figure7} (a) exhibits the profiles of H$\alpha$ at different sites. The blue line is the profile of H$\alpha$ at the site marked by the blue plus sign (+) in the panel (e) of Fig.\ref{figure6}, while the red line is the profile of H$\alpha$ at the site marked by the red plus sign in the panel (e). The black line indicates the profile of the quiet Sun. The blue line displays blue shift signature while the red one displays red shift signature. Fig.\ref{figure7} (b) shows the line profiles of the Ca II K line at different sites. The blue line shows the profile of Ca II K at the site marked by the blue plus sign in the panel (h) of Fig.\ref{figure6}, while the red line is the one at the site marked by the red plus sign in the panel (h). The black line indicates the profile of the quiet Sun, which is the same as the panel (a). The similar feature as H$\alpha$ profile could be found in Ca II K line, in which the blue and red shifts are also found in the blue and red lines, respectively. These blue and the red shifts manifest that the cool injected plasma displayed different motion states at different site during the jet.
Panel (c) of Fig.\ref{figure7} shows the profiles of Si IV 1402.8 $\rm \AA$ observed by IRIS. The blue line shows the profile of Si IV line at the site marked by the blue plus sign in the panel (d) at 01:02:26 UT on August 12, while the black line shows the profile of Si IV line at the site marked by the black plus sign at 01:06:49 UT on August 12. The panel (d) shows the SJI of the 1400 $\rm \AA$ band nearby the western foot-point of the filament during the jet. With the method described in the section 2, we use a single Gaussian fitting method to derive the Doppler velocities of these two profiles of Si IV lines. The Doppler velocity of the blue line was about -16.11 km/s, while the one of the black line was about 13.81 km/s. This means that the heated plasma driven by the jet at the position of the blue plus sign showed a upward motion, but the one at the position of the black plus sign exhibited downward motion. As is well known, some plasma were injected into the filament, which is manifested as blueshifts of the Si IV line. However, some injected plasma could not remain in the filament owing to the imbalance between magnetic support and gravitation, and fell down to the western foot-point of the filament. Therefore, it showed the red shift in the line profile of Si IV 1402.8 $\rm \AA$ at 01:06:49 UT on August 12.
\begin{figure}[ht!]
\figurenum{9}
\plotone{figure7.eps}
\caption{(a) Line profiles of H$\alpha$ from DST/Hida. The back line denotes the line profile of quiet Sun. The blue one sites in the blue plus sign (+) in the panel (e) of Fig.\ref{figure6}, while the red one sites in the red plus sign (+). (b) Line profiles of Ca II K from DST/Hida. The back line denotes the line profile of quiet Sun. The blue one sites in the blue plus sign in the panel (h) of Fig.\ref{figure6}, while the red one sites in the red plus sign. (c) Line profiles of Si IV 1402.8 $\rm\AA$. The blue line sites at blue plus sign in the panel (d), which the black line is corresponding to the black plus sign in the panel (d). (d) The SJI of 1400 $\rm\AA$ observed from IRIS.\label{figure7}}
\end{figure}
Fig.\ref{figure8} shows the process of the jet in H$\alpha$ line observed by DST/Hida. Reconstructed images at five different moments during the jet are shown. The reconstructed image is constructed from a set of one scan data at the fixed wavelengths. Panels (a1)-(a5) show the jet in the center of the H$\alpha$, while panels (b1)-(b5) and panels (c1)-(c5) are the corresponding images reconstructed by using +0.6 $\rm\AA$ and -0.6 $\rm\AA$ of H$\alpha$ spectrum from DST/Hida, respectively. Panels (d1)-(d5) show the Doppler velocity derived from H$\alpha$ spectral data with the method described at Section 2. It is noted that the maximum of the Doppler velocity is smaller than 15 km/s. In panel (a1), massive dark plasma was driven by the jet to move close to the filament at 00:53:01 UT. In the meanwhile, panels (b1) and (c1) exhibit the corresponding images constructed in H$\alpha$-0.6 $\rm\AA$ and H$\alpha$+0.6 $\rm\AA$. The doppler velocity is shown in the panel (d1). As is shown, the eastern part of the injected plasma exhibited blue shift, while the western part of the injected plasma showed red shift. Combined with the observation of SDO/AIA 304 $\rm\AA$, we infer that the injected plasma was rotating. Based on the location of the filament, it was anticlockwise when viewed from the apex to the western foot-point of the filament.
At 01:03:58 UT on August 12, the jetted plasma began to be injected into the filament. At 01:13:33 UT, the jetted plasma had been injected to the filament and became the material of the filament. At 01:19:57 UT, the filament experienced a little active after the injection of material. At 01:46:19 UT, the filament became more broader and larger after this event. According to the corresponding velocity in the panels (d2)-(d5), the two interesting features were found. Firstly, as the rotated plasma were injected to the filament, the filament also experienced anticlockwise rotation which showed in the panels (d2) and (d3). Secondly, the filament appeared inverse rotation after the anticlockwise rotation. We conjecture that the anticlockwise rotation of the filament was caused by the injected twisted magnetic field structure which was transformed from the jet into the filament. And the reason of the inverse rotation of the filament was due to the back action which balanced the magnetic field configuration after the injection of the twisted magnetic field by the jet. According to these behaviors above, we conclude that the jet caused by the reconnection between the twisted structure magnetic field with massive plasma and the filament magnetic field also injected the magnetic helicity into the filament. In other words, the jet not only forced the plasma into the filament, but also effected the configuration of the filament magnetic field.
\begin{figure}[ht!]
\figurenum{10}
\plotone{figure8.eps}
\caption{Evolution of the jet at 00:42 UT on August 12 observed from DST/Hida. (a1)-(a5) The constructed images of center of H$\alpha$. (b1)-(b5) The constructed images of H$\alpha$+0.6 $\rm\AA$. (c1)-(c5) The constructed images of H$\alpha$-0.6$\rm\AA$. (d1)-(d5) The corresponding Doppler shifts derived by H$\alpha$ spectra. \label{figure8}}
\end{figure}
\section{Estimation of the jets mass, the filament mass and the upper limit of the twist}
With the assumptions that the speed of the injected plasma is constant and the cross-section of the jet is circular, we calculate the total mass carried by the jets. The quality of mass injection can be calculated by the following equation:
\begin{equation}\label{jetsmass}
M_j =m_Hn_H\frac{w^2\pi}{4}vt
\end{equation}
where $m_H$ is the quality of the hydrogen atom, $n_H$ is the total hydrogen number density, $w$ is the jet width, $v$ is the jet speed, $t$ is the jet duration. We assume that the density of the jets equal to that of the chromosphere. The total hydrogen number density $n_H$ is about $(1.71+1.55)\times 10^{10}$ $cm^{-3}$ (the model A from \cite{fon93}), while the mass of the hydrogen atom $m_H$ is about $1.67\times 10^{-24}$ $g$.
\begin{deluxetable*}{cccccc}
\tablecaption{The dynamical parameters of jets}
\tablecolumns{6}
\tablenum{1}
\tablewidth{0pt}
\tablehead{
\colhead{Start Time} &
\colhead{Jet Width($w$)} &
\colhead{Jet Speed($v$)} &
\colhead{Duration($t$)} &
\colhead{Mass($M_j$)} \\
\colhead{(UT)} &
\colhead{(Mm)} &
\colhead{(km/s)} &
\colhead{(minutes)} &
\colhead{($10^{14}g)$}
}
\startdata
2016-Aug-11 10:52 & 0.97 & 73.2 & 11 & 0.02 \\
2016-Aug-11 11:28 & 2.04 & 55.3 & 14 & 0.08 \\
2016-Aug-11 11:57 & 2.70 & 133.2 & 16 & 0.40 \\
2016-Aug-11 13:17 & 6.88 & 174.6 & 33 & 6.99 \\
2016-Aug-11 15:02 & 7.57 & 121.9 & 21 & 3.76 \\
2016-Aug-11 18:02 & 4.94 & 162.6 & 22 & 2.24 \\
2016-Aug-12 00:42 & 5.59 & 88.7 & 35 & 2.49 \\
\enddata
\end{deluxetable*}
Table 1 presents the parameters of the jets. The total mass carried by jets is about $16\times 10^{14}$ $g$. The last four jets supply the most of the mass, which are more than 97$\%$ of the total mass. According to the enhancements of the SDO/AIA 304 $\rm\AA$ intensity during the jets (see Fig.\ref{figure3} (c)), the intensity levels of the last four jets are bigger than the former three jets. Therefore, it is reasonable that the last four jets carried the most of the mass.
Under a simple assumption that the filament is circular column, the quality of the mass in the filament $M_f$ is estimated using the equation: $M_f=n_Hm_H\frac{w^2_f\pi}{4}L_f$, where $n_H$ is the total hydrogen density of the filament, $w_f$ is the filament width, and $L_f$ is the filament length. According to the shape of the filament, we obtain $L_f=56.1$ Mm and $w_f=10.4$ Mm. The total hydrogen number density $n_H$ in the filament is between $3\times 10^{10}$ $cm^{-3}$ \citep{ste97} and $3\times 10^{11}$ $cm^{-3}$ \citep{hir86}. The total filament mass is estimated to be in the range $(2.39-23.9)\times10^{14}$ $g$. Comparing the total mass carried by the jets with the mass of the filament, the total mass carried by the jets is compatible with the range of the filament mass and very closes to the upper value of the filament mass. Thus, the estimated mass loading by the jets is sufficient to account for the mass in the filament.
We estimate the twist injected into the filament system by the formula, $T=H/\phi^2$, where $H$ is the cumulative helicity and $\phi$ is the magnetic flux contained in the eventual filament. $T$ presents the twist in the units of turns. It would provide an upper limit to the number of windings in the filament, if all the helicity injected were stored in the filament. The flux in the filament can be estimate using by the equation: $\phi=\int_s B_tds$, where $B_t$ is the transverse magnetic field strength, $s$ is the cross-section of the filament. The range of the transverse magnetic fields strength in the active-region filament is from 500 to 600 G \citep{kuc09,xu12}. The cross-section of the filament, $s=(w_f/2)^2*\pi$, is about $8.5\times10^{17}$ $cm^2$. Using these values, we derive the magnetic flux in the eventual filament is from $4.25 \times 10^{20}$ to $5.10\times10^{20}$ $Mx$. According to the formulas, the upper limit of the twist $T$ is between 2.28 and 4.11 turns.
\section{Conclusion and discussion}\label{sec:conclusion}
In this study, we examined the formation of a filament in NOAA AR 12574. The material injection of the filament is mainly investigated. This is a rare set of observations. Beginning with the time when the filament did not exist, both ground-based and space borne observatories provided continuous coverage of its formation over the next two days. SDO, GONG and Hida observations covered the entire process of the filament material injection. We focus on the mechanism of the transportation of the filament material. On the one hand, we analyze the variation of the magnetic flux nearby the western foot-point of the filament, which is associated to the material injection events (jets). On the other hand, two mainly material injection events are investigated in detail. One occurred at 18:02 UT on August 11 and the other occurred at 00:42 UT on the next day. The main results are as follows:
1. Material of the filament originate from the low solar atmosphere. The material of the filament are supplied by a series of jets. These jets occurred nearby the western foot-point of the filament and forced massive plasma from low atmosphere to the filament. Jets provide a sufficient and direct way to carry dense and cool plasma from the low atmosphere into high atmosphere.
2. Flux emergence should be responsible for occurrences of these jets. These jets were caused by the magnetic reconnection between the emerging magnetic field and pre-existing magnetic field nearby the western foot-point of the filament.
3. The projection velocity of heated plasma along the filament axis during the Jet A at 18:02 UT on August 11 was about 162.6$\pm$5.4 km/s. For the Jet B on 00:42 UT on August 12, we find the LOS velocity of cool injected plasma derived by H$\alpha$ spectrum was smaller than 15 km/s. Furthermore, Using the Si IV line from IRIS observations, it is found that the blue shift was about 16.11 km/s while the red shift was about 13.81 km/s.
4. Jets not only injected the plasma from the low atmosphere into the filament, but also injected the magnetic helicity into the filament, simultaneously. The rotating injected plasma manifested that the twisted structure had existed in the emerging magnetic field. The twist in the emerging magnetic field could be transformed into the filament when the emerging magnetic field interacted with the filament.
5. The total mass carried by jets is estimated to be about $16\times 10^{14}$ $g$, while the filament mass is estimated to be in the range $(2.39-23.9)\times10^{14}$ $g$. Thus, the mass carried by the jets is sufficient to account for the mass in the filament.
Previous work on quiescent filaments have shown that the mass in such filaments is supplied from the low atmosphere instead of the corona \citep{pik71,zir94,mac10}. The same origin can be also inferred for the active region filament in our study. This study suggests that jets can provide the supply of mass from the lower atmosphere to active region filaments, consistent with previous reports \citep{wan99,cha03,liu05,zou16}. Although the evaporation-condensation model provides an another way to supply the mass for the filaments, which is supported by numerous numerical simulations, but the observation evidences are rare \citep{mac10,liu12,par14}. Besides, there are two different magnetic reconnection sites for injecting material into the filament in the previous studies. One is at the PIL, and the other is nearby the foot-points of the filament \citep{cha03,liu05}. In this study, the magnetic reconnections occurred at the western foot-point of the filament.
In this study, a lot of cool material was carried into the filament by the jets. \cite{tak13} studied the acceleration mechanisms of chromospheric jets associated with emerging flux using a two-dimensional magnetohydrodynamic (MHD) simulation and found that slow-mode waves play key roles in the acceleration mechanisms of chromospheric jets. They mainly investigated two example jets resulting from magnetic reconnection near the photosphere and in the upper chromosphere, and suggested three types of acceleration mechanisms of cool jets: Shock acceleration type, Shock and Whip-like acceleration type, and Whip-like acceleration type \citep{yok96}. Therefore, the magnetic reconnection plays an important role in the transportation of filament material. The jet is a quite sufficient and direct way to carry dense and cool plasma from the lower atmosphere to upper atmosphere.
It is found that the apparent plane-of-sky velocity of the bright material in the SDO/AIA 304 $\rm \AA$ channel is about 160 km/s during the Jet A, while the Doppler velocities derived from the Si IV and H$\alpha$ line diagnostic is 10-20 km/s during the Jet B. For the order of magnitude difference in speed, it might be associated with the topology structure of the magnetic field of the filament. The plasma move along the magnetic field lines due to the low $\beta$ (the ratio of gas to magnetic pressures) in the corona and the magnetic field structure of the filament is almost horizontal (see Fig.\ref{figure9}). Furthermore, the filament is close to the center of the solar disk. Thus, it is reasonable that the plane-of-sky velocity of the bright material is significantly faster than the line of sight Doppler velocity from the Si IV and H$\alpha$ diagnostic.
Observationally, some coronal jets are associated with magnetic flux emergence or cancellation, which are believed as a source of significant mass and energy to input into the upper solar atmosphere \citep{shi98,wan99,liu04,iso07,ada14,rao16,hon17,li17}. However, there are also some studies of coronal jets, which are not associated with clear signatures of flux emergence \citep{moo13,moo15,ste15,ste16,ste17}. In many MHD simulations of coronal jets, the jets can be caused by the magnetic reconnection between an emerging magnetic flux and the pre-existing magnetic field \citep{yok95,yok96,mor08,mor13,che14}. The other signature during the jet is accompanying photospheric flux cancellation, which would be considered as a consequence of the magnetic reconnection after the jets. \cite{shi98} explained that the flux cancellation during the jet is due to the rate of flux emergence is smaller than the rate of photospheric reconnection. Sometimes, magnetic cancellation during jets is also due to the sink of inverse U-loops into the convection zone after the magnetic reconnection. Traditionally, jets are triggered by the magnetic reconnection between small closed magnetic field and adjacent open magnetic field \citep{mor08,che14,rao16}. In addition, some authors also reported that some jets are related to magnetic reconnection between two sets of closed magnetic fields, which are often associated with a fan-spine magnetic topology \citep{che15,wyp16,li17}. The hot or cool plasma are either spurted into outer atmosphere and become coronal mass ejection or fall back to low atmosphere, depending on the releasing energy of the magnetic field associated with jets \citep{yok96,can96}. In our study, we deduce that the magnetic emergence played an important role in producing these jets. Interestingly, the magnetic reconnection in our study was caused by closed and closed magnetic field instead of closed and open magnetic field. Therefore, the plasma jetted from jets could remain in the corona due to the magnetic dips, which usually existed in the closed magnetic field. On the other hand, we find that same dark threads appeared on the photosphere after one jet occurred at 18:02 UT on August 11. This is due to the increase of transverse magnetic field, which may be associated with the sink of small post-jet magnetic filed.
\begin{figure}[ht!]
\figurenum{11}
\plotone{cartoon.eps}
\caption{Cartoon showing the formation of the filament by jets. \label{figure10}}
\end{figure}
The untwisting motion is often found in the jet events, which suggests that magnetic helicity stored in the closed emerging flux is transferred to the outer corona \citep{pik98,ste10,cur12,she11,sch13,moo15}. \cite{shi86} suggested the sudden release of the magnetic twist into an open flux tube is most likely to be due to the reconnection between a twisted loop and the open flux tube during the jet. Furthermore, this model was explained as the untwisting motion during the jet by other authors \citep{can96,moo15}. Similarly, we also find that the injected plasma was rotating during a jet. We deduce that the emerging closed magnetic field was a highly twisted structure. Due to magnetic reconnection between emerging closed magnetic field and closed magnetic field, the twist existing in the emerging closed magnetic field should be transformed to the closed magnetic field of the filament. Therefore, the jets not only injected the cool plasma into the filament height, but also influenced the configuration of the magnetic field of filament. The magnetic field topology of the filament became more and more non-potential after a series of jets, which is due to that the twist was transformed from the emerging magnetic field to the filament during periods of the jets. We draw some cartoons to illustrate this scenario. While the twisted emerging magnetic field reconnected with the pre-existing closed magnetic field, the twist was transformed to the long post-reconnected magnetic field and the material was injected into the long post-reconnected magnetic field (see Fig.\ref{figure10} (a)-(b)). After the magnetic reconnections occurred between emerging magnetic fields and the pre-existing closed magnetic fields for several times, the magnetic field became more and more twisting and the injected material was increasing (see Fig.\ref{figure10} (c)-(d)). The difference from the model suggested by \cite{shi86} is that the twist is transformed to the long post-reconnected magnetic field instead of releasing in the open flux tube. We conjecture that the jetted plasma fell down to other foot-point in early stage easier might be related to the local magnetic structure. At the beginning, less twisted structure existed in the filament magnetic structure. Therefore, the jetted plasma were hardly captured by the local magnetic field. As a series of jets happened and transformed the twist into the filament, the filament magnetic structure became more and more twisting and easily captured the jetted plasma. Thus, it is possible that material injection and magnetic structure of filament should form at the same time, especially for active-region filaments.
\acknowledgments
The authors thank the referee for constructive suggestions and comments that helped to improve this paper. SDO is a mission of NASA's Living With a Star Program and Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in co-operation with ESA and NSC (Norway). The authors are indebted to the SDO, GONG/NSO and Hinode teams for providing the data. This work is supported by the National Science Foundation of China (NSFC) under grant numbers 11603071,11503080, 11633008, 11527804, the Yunnan Talent Science Foundation of China, the Youth Innovation Promotion Association CAS under number 2011056, Yunnan Key Science Foundation of China under number Y8YJ061001, the grant associated with project of the Group for Innovation of Yunnan province, the Joint Research Fund in Astronomy (U1531140) under cooperative agreement between the National Natural Science Foundation of China (NSFC) and Chinese Academy of Sciences (CAS). The BBSO operation is supported by NJIT and US NSF AGS-1821294 grant. GST operation is partly supported by the Korea Astronomy and Space Science Institute and Seoul National University and by the strategic priority research program of CAS with Grant No. XDB09000000.
|
1,108,101,565,702 | arxiv | \section{Introduction}
Consider the space $\ensuremath{\mathbb{F}}^n_2$ of dimension $n$ over
the Galois field $\mathbb{F}_2 = \{0,1\}$. A binary \emph{code} of length $n$ is
a subset of $\ensuremath{\mathbb{F}}^n_2$.
The \emph{(Hamming) distance} $d({\bf x}, {\bf y})$ between two codewords
${\bf x}$, ${\bf y}$ is the number of coordinates in which they differ,
and the \emph{(Hamming) weight} $\ensuremath{\mathrm{wt}}({\bf x})$ is the number of nonzero
coordinates. The \emph{support} of a codeword is the set of nonzero
coordinates, $\ensuremath{\mathrm{supp}}({\bf x}) = \{i : x_i \neq 0\}$. Accordingly,
$d({\bf x}, {\bf y}) = \ensuremath{\mathrm{wt}}({\bf x}-{\bf y}) = |\ensuremath{\mathrm{supp}}({\bf x}-{\bf y})|$.
A code has \emph{minimum distance} $d$ if $d$ is the largest integer
such that the distance between any distinct codewords
is at least $d$. Then the balls of radius $\lfloor (d-1)/2 \rfloor$
centered around the codewords are nonintersecting, and the code is said to be
a $\lfloor (d-1)/2 \rfloor$-error-correcting code. If these balls tile the
whole space, then the code is called \emph{perfect}. The parameters of perfect
codes over an alphabet of prime order are well known~\cite{MacWilliams77},
and perfect binary
codes exist with $d=1$; $d=n$; $d=(n-1)/2$ for odd $n$;
$d=3, n=2^m-1$ for $m\ge 2$; and $d=7, n=23$. The first three types of codes
are called trivial, the fourth has the parameters of Hamming codes,
and the last one is the binary Golay code. A perfect code with minimum distance
$d$ is also called a $\lfloor (d-1)/2 \rfloor$-perfect code.
A binary code with length $n$, minimum distance $d$, and $M$ codewords is called
a $(n, M, d)$ code. In this notation a binary $1$-perfect code is a
$(2^m-1, 2^{2^m-m-1}, 3)$ code. Two related families are the extended
and shortened $1$-perfect codes, which have parameters
$(2^m, 2^{2^m-m-1}, 4)$ and $(2^m-2, 2^{2^m-m-2}, 3)$, respectively.
Existence of binary $1$-perfect codes follows from the existence of
Hamming codes, which are the unique \emph{linear} $1$-perfect codes.
Still constructing all $1$-perfect codes is a longstanding open
problem. It makes sense to approach this issue by considering the
number of \emph{inequivalent} codes (or more formally the number of
equivalence classes). Two codes are said to be
equivalent if one is obtained from the other by permuting coordinates
and adding a constant vector; a formal definition appears in
Section~\ref{sec:prel}.
There is trivially a unique $1$-perfect code of length $3$.
Zaremba~\cite{Zaremba52} showed that also the $1$-perfect code of
length $7$ is unique. However, already the next case of length $15$ has
until this work withstood all attempts of complete classification, although
several constructions of such codes have been published;
see the surveys~\cite{Etzion94, Heden08}.
It turns out that these results were not far from a complete classification
as for the number of codes found.
The growth of the number of $1$-perfect binary codes is double exponential in the length
of the code, see \cite{Krotov08} for a lower bound on this number.
For an in-depth treatment of the topic of classifying
combinatorial objects, see~\cite{KO}.
The aim of the current work is to obtain a complete classification
of inequivalent $1$-perfect binary codes of length $15$.
By computer search it is here shown that their number is
$5\,983$. Also the codes obtained by extending, shortening or extending and
shortening are classified;
the numbers of $(16, 2\,048, 4)$, $(14, 1\,024, 3)$ and $(15, 1\,024, 4)$
codes turn out to be $2\,165$, $38\,408$ and $5\,983$ respectively.
In the rest of the paper we document the classification of the
extended 1-perfect
codes of length $16$, which yields classifications of the 1-perfect codes
of length $15$ and the shortened 1-perfect codes of length $14$.
In Section~\ref{sec:prel} we define some concepts and consider construction
of extended 1-perfect codes via Steiner systems.
In Section~\ref{sec:isom} we present algorithms for detecting and
rejecting equivalent codes, and in
Section~\ref{sec:results}, we take a brief look at
the results; a separate, more detailed study of the
classified codes will appear in a separate paper~\cite{OPP09}.
Finally, in Section~\ref{sec:check} we give a consistency check for
gaining confidence in the computational results.
\section{Preliminaries and Construction}\label{sec:prel}
A permutation $\pi$ of the set $\{1,2,\ldots,n\}$ acts on codewords
by permuting the coordinates: $\pi((c_1, c_2, \ldots, c_{n})) =
(c_{\pi^{-1}(1)}, c_{\pi^{-1}(2)}, \ldots, c_{\pi^{-1}(n)})$.
Pairs $(\pi, {\bf x})$ form the
\emph{wreath product} $S_2\wr S_n$, which acts on codes as
$(\pi, {\bf x})(C) = \pi(C + {\bf x}) = \pi(C) + \pi({\bf x})$.
Two codes, $C_1$ and $C_2$, are \emph{isomorphic}
if $C_1 = \pi(C_2)$ for some $\pi$ and \emph{equivalent}
if $C_1 = \pi(C_2 + {\bf x})$ for some $\pi, {\bf x}$.
The \emph{automorphism group} of a code $C$, $\ensuremath{\mathrm{Aut}}(C)$, is the group
of all pairs $(\pi, {\bf x})$ such that $C = \pi(C + {\bf x})$.
Two important subgroups of $\ensuremath{\mathrm{Aut}}(C)$ are the \emph{group of symmetries},
\[
\ensuremath{\mathrm{Sym}}(C) = \{\pi : \pi(C) = C\}
\]and the \emph{kernel}
\[
\ensuremath{\mathrm{Ker}}(C) = \{{\bf x} : C + {\bf x} = C\}.
\]
If the code contains the all-zero word, ${\bf 0}$, then the elements of
the kernel are codewords.
A \emph{Steiner system} $S(t,k,v)$ can be viewed as a code
$S \subset \mathbb{F}^v_2$ with the property that each codeword of $S$ has weight
$k$, and for any ${\bf y} \in \mathbb{F}_2^v$ with $\ensuremath{\mathrm{wt}}({\bf y}) = t$,
there is a unique ${\bf x} \in S$ such that
$\ensuremath{\mathrm{supp}}({\bf y}) \subseteq \ensuremath{\mathrm{supp}}({\bf x})$. Usually Steiner systems
are defined as set systems rather than codes, but our definition is
more directly applicable for this work.
The parameter $v$ is the \emph{order} of the system.
Steiner systems $S(2,3,v)$ and $S(3,4,v)$, which are called
\emph{Steiner triple systems} and \emph{Steiner quadruple systems},
respectively, are related to 1-perfect codes in the following way.
If $C$ is a 1-perfect
binary code of length $v$ and ${\bf x} \in C$, then the codewords
of $C + {\bf x}$ with weight $3$ form a Steiner triple system of order $v$.
Similarly, if $C$ is an extended 1-perfect binary code and ${\bf x} \in C$, then
the codewords of $C + {\bf x}$ with weight $4$ form a Steiner quadruple system.
These systems are, respectively, the \emph{neighborhood triple system} and
\emph{neighborhood quadruple system} associated with the code and the
codeword.
As a starting point for the classification of the extended
1-perfect binary codes of length 16, we have the classification~\cite{sqs16}
of Steiner quadruple systems of order 16; there are $1\,054\,163$ such
designs. We want to find, for each $S(3,4,16)$, all extended
1-perfect binary codes in which it occurs. This can be done by
puncturing any coordinate, augmenting the resulting code to 1-perfect
codes in all possible ways, and finally extending every resulting code
with a parity bit.
When augmenting a set of codewords to a 1-perfect code, we consider a
1-perfect code as a set of balls with radius one that
form a partition of the ambient space. Accordingly, finding a code
(with specified codewords) is a special case of
the \emph{exact cover problem},
where we are given a set $S$ and a collection $U$ of its subsets, and the
task is to form a partition of $S$ by using sets in $U$.
Let the set $Q$ contain the codewords obtained by puncturing the
all-zero codeword and its neighborhood quadruple system.
In this case we have $S = \ensuremath{\mathbb{F}}^{15}_2 \setminus B(Q)$, and
$U = \{ B({\bf x}) : B({\bf x}) \cap B(Q) = \emptyset\}$, where
$B({\bf x}) = \{ {\bf y} : d({\bf x}, {\bf y}) \le 1\}$
and $B(C) = \{B({\bf x}): {\bf x} \in C\}$.
We use the \emph{libexact} software~\cite{libexact} for solving such
instances of the exact cover problems.
In the search we could in fact have made use of the fact that
all 1-perfect binary codes are self-complementary---in other words, the
all-one word is always in the kernel---but this would not have had any
practical significance as the search was rather fast.
Since the all-zero word and its neighborhood quadruple system contain
$141$ of the $2\,048$ codewords, $1907$ new codewords are needed.
Searching for these was a remarkably easy computational task;
on average the search trees in which codes were found had 1978 nodes
and those in which no codes were found had 3 nodes.
\section{Isomorph rejection}\label{sec:isom}
The general framework by McKay~\cite{McKay98} was used to carry out
isomorph rejection, although a less sophisticated method would
have sufficed in this work.
Recall that we augment a Steiner quadruple system $Q$ to an extended
1-perfect code $C$. We accept $C$ if it passes the following two tests;
otherwise it is rejected. First we require that $C$ shall be the minimum
(with respect to some practically computable total order
of codes) under the action of $\ensuremath{\mathrm{Aut}}(Q)$. Second, we compute the
canonical equivalence class representative $c_E(C)$, consider
$\pi, {\bf x}$ for which $\pi(C + {\bf x}) = c_E(C)$ and require that
${\bf x}$ and ${\bf 0}$ are on the same $\ensuremath{\mathrm{Aut}}(C)$ orbit
(we define $c_E$ so that ${\bf x} \in C$ always holds).
When the extended 1-perfect codes have been classified, classifying
the 1-perfect codes is straightforward. All 1-perfect codes are obtained
by puncturing the extended codes, and the resulting 1-perfect codes
are equivalent if and only if they are obtained by puncturing
the same extended code at coordinates which are in the same
orbit of the automorphism group.
A complete classification of the $(14, 1\,024, 3)$ codes is obtained similarly,
since each such code is obtained by shortening a unique (up to equivalence)
1-perfect code of length $15$; this result was proved by
Blackmore~\cite{Blackmore99}. Although a code can be shortened at any
coordinate in two
ways, by selecting the codewords with $0$ or $1$ in a certain coordinate,
both selections lead to equivalent codes. This follows from the fact
that every 1-perfect binary code is self-complementary.
Furthermore we note that any $(15, 1\,024, 4)$ code is obtained by extending
a $(14, 1\,024, 3)$ code with a parity bit. Hence all such codes are
obtained by shortening and extending a perfect code, or equivalently
removing all words of chosen parity. As the perfect codes are
self-complementary, we get (up to equivalence) same code by chosing
either odd or even parity. As this mapping is reversible, we conclude
that there is one-to-one correspondence between equivalence classes of
$(15, 2\,048, 3)$ codes and equivalence classes of $(15, 1\,024, 4)$
codes, and in both cases their number is $5\,983$.
Let $C$ be a $(15, 1\,024, 4)$ code $C$ and let $C'$ be the corresponding
perfect code. The group $\ensuremath{\mathrm{Aut}}(C')$ contains $\ensuremath{\mathrm{Aut}}(C)$ as a subgroup,
and $\ensuremath{\mathrm{Aut}}(C')$ has one more generator than $\ensuremath{\mathrm{Aut}}(C)$, namely the all-one
codeword. Accordingly $|\ensuremath{\mathrm{Aut}}(C')| = 2|\ensuremath{\mathrm{Aut}}(C)|$.
We still have to describe an algorithm for canonical labeling. The most
straightforward approach of using the general purpose isomorphism
\emph{nauty}~\cite{nug} is rather slow on codes as large and regular
as the $(16, 2\,048, 4)$ codes; this was also noted by
Phelps~\cite{Phelps00}. Hence a tailored approach is necessary.
The method presented below has a lot in common with the one
described in~\cite{Phelps00}. An alternative method based on minimum
distance graphs would also work~\cite{MOPS09}, cf.~\cite{PL99}.
A \emph{triangle} consists of $3$ codewords with mutual distance $4$.
Triangles constitute an easily computable and rather sensitive
invariant of Steiner quadruple systems.
Distinguishing the isomorphism classes of the neighborhood quadruple
systems of a code also constitutes an invariant of the extended 1-perfect codes.
These two invariants turned out to be useful for speeding up our computations.
A canonical isomorphism class representative $c_I(C)$ for a code $C$
can be computed by using \emph{nauty} to label a corresponding graph
canonically. Moreover, \emph{nauty} computes generators of the group $\ensuremath{\mathrm{Sym}}(C)$.
Canonical equivalence class
representative can be defined as
$c_E(C) = \min \{c_I(C + {\bf x}) : {\bf x} \in C\}$,
where the minimum is again taken with respect to some practically computable
total order of codes.
Note that two codewords, ${\bf x}$ and ${\bf y}$, are in the same orbit
of $\ensuremath{\mathrm{Aut}}(C)$ if and only if $c_I(C + {\bf x}) = c_I(C + {\bf y})$.
Because of this, if we know that ${\bf x}$
and ${\bf y}$ are in the same orbit, and $c_I(C + {\bf x})$ has
been computed, then there is no need to compute
$c_I(C + {\bf y})$. Also if $c_I(C + {\bf x}) = c_I(C)$, then \emph{nauty}
yields a permutation $\pi$ such that $\pi(C + {\bf x}) = C$. The
pairs $(\pi, {\bf x})$ are coset representatives of $\ensuremath{\mathrm{Aut}}(C)$ with
respect to $\ensuremath{\mathrm{Sym}}(C)$, so we get generators of the group $\ensuremath{\mathrm{Aut}}(C)$.
\section{Results}\label{sec:results}
There are exactly $2\,165$ inequivalent extended 1-perfect codes of
length $16$, $5\,983$ inequivalent 1-perfect codes of length $15$,
$38\,408$ shortened 1-perfect codes of length $14$ and
$5\,983$ $(15, 1\,024, 4)$ codes.
The orders of the automorphism groups of the codes are presented
in Tables~\ref{tbl:extautorder}, \ref{tbl:perfautorder} and
\ref{tbl:shortautorder}. As noted in Section~\ref{sec:isom}, the
order of the automorphism group of a $(15, 1\,024, 4)$ code is half of the
order of the automorphism group of the corresponding perfect code.
The codes have been made available in electronic form by including them in the source of the arXiv version of this paper. Downloading \url{http://arxiv.org/e-print/0806.2513v3} and uncompressing it with {\tt gunzip} and {\tt tar} yields the files {\tt perfect15} and {\tt extended16}.
Only $15\,590$ of the $1\,054\,163$ nonisomorphic $S(3,4,16)$ can
be augmented to a 1-perfect code, and the total number of extensions
is $22\,814$. The computationally intensive part
of this result was the earlier classification of $S(3,4,16)$,
which required several years of CPU time, while all searches
described in this paper took only a couple of hours of CPU time.
A detailed study of the properties of the classified codes will appear
in a second part of this article~\cite{OPP09}.
\begin{table}[h]
\begin{center}
\caption{Automorphism groups of $(16, 2\,048, 4)$ codes}
\label{tbl:extautorder}
\begin{tabular}{rrrrrr}\hline
$|\ensuremath{\mathrm{Aut}}(C)|$ & \# & $|\ensuremath{\mathrm{Aut}}(C)|$ & \# & $|\ensuremath{\mathrm{Aut}}(C)|$ & \# \\ \hline
128 & 11 & 5\,376 & 1 & 196\,608 & 6 \\
192 & 5 & 6\,144 & 23 & 262\,144 & 3\\
256 & 105 & 8\,192 & 174 & 344\,064 & 1\\
384 & 9 & 10\,752 & 2 & 393\,216 & 3\\
512 & 377 & 12\,288 & 22 & 524\,288 & 2\\
672 & 2 & 16\,384 & 103 & 688\,128 & 1\\
768 & 19 & 24\,576 & 12 & 786\,432 & 2\\
1\,024 & 416 & 32\,768 & 47 & 1\,572\,864 & 3\\
1\,344 & 1 & 43\,008 & 2 & 2\,359\,296 & 1\\
1\,536 & 21 & 49\,152 & 18 & 2\,752\,512 & 1\\
1\,920 & 1 & 61\,440 & 1 & 3\,145\,728 & 1\\
2\,048 & 394 & 65\,536 & 33 & 5\,505\,024 & 2\\
2\,688 & 1 & 86\,016 & 3 & 6\,291\,456 & 1\\
3\,072 & 18 & 98\,304 & 12 & 660\,602\,880 & 1\\
4\,096 & 298 & 131\,072 & 6 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\caption{Automorphism groups of $(15, 2\,048, 3)$ codes}
\label{tbl:perfautorder}
\begin{tabular}{rrrrrr}\hline
$|\ensuremath{\mathrm{Aut}}(C)|$ & \# & $|\ensuremath{\mathrm{Aut}}(C)|$ & \# & $|\ensuremath{\mathrm{Aut}}(C)|$ & \# \\ \hline
8 & 3 & 512 & 1\,017 & 24\,576 & 7 \\
12 & 3 & 672 & 3 & 32\,768 & 8 \\
16 & 5 & 768 & 32 & 43\,008 & 4 \\
24 & 10 & 1\,024 & 697 & 49\,152 & 10 \\
32 & 138 & 1\,536 & 17 & 65\,536 & 5 \\
42 & 2 & 2\,048 & 406 & 98\,304 & 1 \\
48 & 12 & 2\,688 & 1 & 131\,072 & 1 \\
64 & 542 & 3\,072 & 37 & 172\,032 & 1 \\
96 & 22 & 3\,840 & 1 & 196\,608 & 5 \\
120 & 1 & 4\,096 & 202 & 344\,064 & 2 \\
128 & 1\,230 & 5\,376 & 4 & 393\,216 & 2 \\
192 & 18 & 6\,144 & 35 & 589\,824 & 1 \\
256 & 1\,319 & 8\,192 & 94 & 41\,287\,680 & 1 \\
336 & 3 & 12\,288 & 7 \\
384 & 30 & 16\,384 & 44 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[h]
\begin{center}
\caption{Automorphism groups of $(14, 1\,024, 3)$ codes}
\label{tbl:shortautorder}
\begin{tabular}{rrrrrr}\hline
$|\ensuremath{\mathrm{Aut}}(C)|$ & \# & $|\ensuremath{\mathrm{Aut}}(C)|$ & \# & $|\ensuremath{\mathrm{Aut}}(C)|$ & \# \\ \hline
1 & 5 & 168 & 1 & 8\,192 & 80 \\
2 & 75 & 192 & 80 & 12\,288 & 18 \\
3 & 8 & 256 & 4\,392 & 16\,384 & 14 \\
4 & 425 & 336 & 5 & 21\,504 & 1 \\
6 & 39 & 384 & 114 & 24\,576 & 15 \\
8 & 1\,162 & 512 & 2\,469 & 32\,768 & 14 \\
12 & 56 & 768 & 30 & 49\,152 & 1 \\
16 & 3\,465 & 1\,024 & 1\,346 & 65\,536 & 1 \\
21 & 4 & 1\,344 & 1 & 86\,016 & 1 \\
24 & 39 & 1\,536 & 54 & 98\,304 & 2 \\
32 & 7\,311 & 2\,048 & 527 & 172\,032 & 1 \\
48 & 59 & 2\,688 & 6 & 196\,608 & 2 \\
64 & 9\,068 & 3\,072 & 55 & 1\,376\,256 & 1 \\
96 & 49 & 4\,096 & 222 & \\
128 & 7\,172 & 6\,144 & 18 & \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Consistency check}\label{sec:check}
To get confidence in the results, we performed a consistency check
similar to the one used, for example, in~\cite{sqs16}. In this check we count the
total number of codes in two different ways and ensure
that the results agree.
First we consider the set $\mathcal{C}$ of equivalence class representatives
obtained in the classification. By the orbit-stabilizer theorem, the
total number of extended 1-perfect codes is
\[
\sum_{C \in \mathcal{C}} \frac{16! \cdot 2^{16}}{\ensuremath{\mathrm{Aut}}(C)},
\]
where $16! \cdot 2^{16}$ is the order of the wreath product group acting
on the codes.
Let $\mathcal{Q}$ consist of the representative Steiner quadruple systems,
and let $E(Q)$ be the number of all extended 1-perfect codes obtained by
augmenting $Q$. Applying the orbit-stabilizer theorem, we get the expression
\[
\frac{1}{2\,048} \sum_{Q \in \mathcal{Q}} \frac{16! \cdot 2^{16} \cdot E(Q)}
{\ensuremath{\mathrm{Aut}}(Q)},
\]
where the division by $2\,048$ is necessary since each code is counted once
for each codeword. Both formulas yield the same result,
$2\,795\,493\,027\,033\,907\,200$. Similarly we also counted the
1-perfect codes and shortened 1-perfect codes in two different ways;
their number is $1\,397\,746\,513\,516\,953\,600$.
Indeed, there are twice as many extended 1-perfect codes as there are
1-perfect codes, since each 1-perfect code admits two extensions: one with
even parity bit and one with odd. Similarly, we get a bijection from
the 1-perfect codes to shortened 1-perfect codes if we shorten each code
by taking, for instance, the codewords with value $0$
in coordinate $15$ and removing that coordinate. Thus there are equally
many 1-perfect codes and shortened 1-perfect codes.
\def$'${$'$}
|
1,108,101,565,703 | arxiv | \section{Introduction}
Each irreducible representation of a simple Lie algebra is defined by a set of weights which, for rank two algebras, can be conveniently arranged in a two-dimensional weight diagram. These weights result from successive applications of the lowering operators $E_{-\alpha}$ corresponding to the positive roots of the algebra to the highest weight of the representation. As there are, in general, several ways by which a particular weight can be obtained in this form, the weights forming the representation enter in it with some multiplicity. The computation and understanding of weight multiplicities has been a subject of much research along the years \cite{wi37}--\cite{mopa82} and, as it is a rule when dealing with Lie algebra representations, one of the most efficient tools available to address the question is the theory of characters. In a recent paper \cite{nfp14}, we have presented a general method for computing the generating function of the characters of simple Lie algebras which is based on the theory of the quantum trigonometric Calogero-Sutherland system \cite{ca71}--\cite{op76} (see also \cite{ps79,ow07} for other approaches to that problem). In particular, we have applied the method to the cases of the Lie algebras $A_2$ and $C_2$. The aim of this note is to supplement the results of \cite{nfp14} by showing how they can be used to obtain some useful generating functions for weight multiplicities. In doing so, we will specialize to the case of the algebra $C_2$, given that the case of the generating function for multiplicies of $A_2$ has been soundly treated in reference \cite{bgw68}.
Let us recall, to begin with, the way in which characters and weight multiplicities are related. Let ${\cal A}$ be a simple Lie algebra of rank $r$ with fundamental weights
$\lambda_1,\lambda_2,\ldots,\lambda_r$ and let us denote $R_{\lambda}$ the irreducible
representation of ${\cal A}$ with highest weight
$\lambda=p_1 \lambda_1+p_2 \lambda_2+\cdots+p_r \lambda_r$.
The character of this representation is defined as
\bdm
\bchi_{p_1,p_2,\dots,p_r}=\sum_{w} \mu_{w} e(w)
\edm
where the sum extends to all weights $w$ entering in the representation, $\mu_{w}$ is the
mutiplicity of the weight $w$ and, if $w=m_1 \lambda_1+m_2 \lambda_2+\cdots+m_r \lambda_r$,
then $e(w)$ is
\bdm
e(w)=\exp\Big(i\sum_{l=1}^r m_l \varphi_l\Big)=x_1^{m_1} x_2^{m_2}\cdots x_r^{m_r},
\edm
where $\varphi_1,\varphi_2,\ldots,\varphi_r$ are angular coordinates on the maximal torus and
$x_l$ are complex phases, \mbox{$x_l=e^{i \varphi_l}$}. The multiplicity $\mu_{p_1,p_2,\dots,p_r}(m_1,m_2,\dots,m_r)$ of the weight $w\equiv(m_1,m_2,\dots,m_r)$ in the representation $R_\lambda$, $\l\equiv(p_1,p_2,\dots,p_r)$, can be computed as
\beqr
&&\mu_{p_1,p_2,\dots,p_r}(m_1,m_2,\dots,m_r)\nonumber\\
&&\qquad\qquad=\frac{1}{(2\pi)^r}\int_0^{2\pi}
d\varphi_1 e^{-i m_1 \varphi_1}\int_0^{2\pi}d\varphi_2e^{-i m_2 \varphi_2}\ldots\int_0^{2\pi}
d\varphi_r e^{-i m_r \varphi_r}\bchi_{p_1,\dots,p_r}\nonumber\\
[2pt]
&&\qquad\qquad\,=\frac1{(2\pi i)^r}\oint d x_1\oint d x_2\ldots\oint d x_r \frac{\bchi_{p_1,\dots,p_r}}
{x_1^{1+m_1} x_2^{1+m_2}\cdots x_r^{1+m_r}}\, , \label{eq:nmp}
\eeqr
where the integrals in the second line are along the unit circles on the $r$ complex planes
parametrized by the complex coordinates $x_1,x_2,\ldots,x_r$.
In view of (\ref{eq:nmp}), the
generating function for the multiplicities of the weight $w$ in all the representations of ${\cal A}$
\beq
\label{amne}
A_{m_1,m_2,\dots,m_r}(t_1,t_2,\ldots,t_r)=\sum_{p_1=0}^\infty\sum_{p_2=0}^\infty\cdots
\sum_{p_r=0}^\infty t_1^{p_1} t_2^{p_2}\cdots t_r^{p_r}\mu_{p_1,\dots, p_r}(m_1,\dots,m_r)
\eeq
comes from the formula
\beq
\label{amni}
A_{m_1,\dots,m_r}(t_1,t_2,\ldots,t_r)=\frac{1}{(2\pi i)^r}\oint d x_1
\oint d x_2\ldots\oint d x_r \frac{G(t_1,t_2,\ldots,t_r ; z_1,z_2,\ldots,z_r)}
{x_1^{1+m_1} x_2^{1+m_2}\cdots x_r^{1+m_r}} ,\label{eq:igm}
\eeq
where $G(t_1,t_2,\ldots,t_r ; z_1,z_2,\ldots,z_r)$ is the generating function of the characters
\bdm
G(t_1,t_2,\ldots,t_r ; z_1,z_2,\ldots,z_r)=\sum_{p_1=0}^\infty\sum_{p_2=0}^\infty\cdots\sum_{p_r=0}^\infty t_1^{p_1} t_2^{p_2}
\cdots t_r^{p_r} \bchi_{p_1,\dots,p_r}(z_1,z_2,\ldots,z_r) \label{eq:gcar}
\edm
and we have chosen to express the latter by means of a set of variables $z_1,z_2,\ldots,z_r$ which coincide with the characters of the representations corresponding to the fundamental weights. Even for low-rank algebras and small values of the indices $m_j$ the integrand in (\ref{eq:igm}) is a quite complicated rational function but, nevertheless, the integral can be evaluated by iterated application of the Cauchy's residue theorem in each complex plane.
Let us consider, for instance, the case of the generating function of zero weight multiplicities for the Lie algebra $A_2$. According to \cite{ov90}, see also \cite{nfp14}, the fundamental characters are
\bdm
z_1=x_1+\frac1{x_2}+\frac{x_2}{x_1}\,, \quad z_2=x_2+\frac1{x_1}+\frac{x_1}{x_2}\,,
\edm
whereas the generating function $G$ is \cite{nfp14}
\beqrn
G(t_1,t_2;z_1,z_2)&=&\frac{1-t_1 t_2}{(1-t_1 z_1+t_1^2 z_2-t_1^3)(1-t_2 z_2+t_2^2 z_1-t_2^3)} \\
[2pt]
&=&\frac{(1 - t_1 t_2)\, x_1^2\, x_2^2}{
(t_2 - x_1) ( t_1 x_1-1) (t_1 - x_2) (t_2 x_1 - x_2) (t_1 x_2-x_1) (t_2 x_2-1)}.
\eeqrn
Then we have to compute
\bdm
A_{0,0}(t_1,t_2)=\frac{1}{(2\pi i)^2}\oint d x_1 \oint d x_2
\frac{(1 - t_1 t_2) x_1 x_2}{
(t_2 - x_1) ( t_1 x_1-1) (t_1 - x_2) (t_2 x_1 - x_2) (t_1 x_2-x_1) (t_2 x_2-1)}
\edm
and we choose to perform the $x_1$ integral first. As $|x_1|=|x_2|=1$ and $t_1,t_2<1$, there are poles inside
the unit circle for $x_1=t_2$ and $x_1=t_1 x_2$. Thus, by computing the residues, we find
\beqrn
J_1(t_1,t_2;x_2)=\frac{1}{2\pi i}\oint d x_1 \frac{G(t_1,t_2;z_1,z_2)}{x_1 x_2}
=\frac{(1 + t_1 t_2) x_2}{(t_1 - x_2)
(t_2^2 - x_2) (t_1^2 x_2-1) (t_2 x_2-1)}.
\eeqrn
Now, integrating $J_1(t_1,t_2;x_2)$, which has poles inside the $x_2$ unit circle at $x_2=t_1$ and $x_2=t_2^2$,
we finally obtain the generating function for zero weight multiplicities as
\beqrn
A_{0,0}(t_1,t_2)=\frac{1}{2\pi i} \oint d x_2J_1(t_1,t_2,x_2)=\frac{1 - t_1^3\, t_2^3}{(1 - t_1^3) (1 - t_1 t_2)^2 (1 - t_2^3)}\,.
\eeqrn
\section{The generating function $A_{m,n}(t_1,t_2)$ for $C_2$}
In the case of $C_2$, the fundamental characters are \cite{ov90}
\beq
\label{eq:b2}
z_1=x_1 +\frac{1}{x_1}+\frac{x_1}{x_2}+\frac{x_2}{x_1}\,,\quad z_2
=1+x_2+\frac{1}{x_2}+\frac{x_1^2}{x_2}+\frac{x_2}{x_1^2} \,,
\eeq
and the generating function of the characters is \cite{nfp14}
\beqr
&&G(t_1,t_2;z_1,z_2)\nonumber\\
&&=\frac{1+t_2-z_1 t_1 t_2+t_1^2 t_2+t_1^2 t_2^2}{(1-(t_1+t_1^3) z_1+t_1^2 (z_2+1)+t_1^4)(1-(t_2+t_2^3)(z_2-1)+t_2^2(z_1^2-2z_2)+t_2^4)}\nonumber\\
[2pt]
&&=\frac{x_1^3 x_2^2 ((1 + t_2) x_1 x_2 + t_1^2 t_2 (1 + t_2) x_1 x_2 -
t_1 t_2 (1 + x_2) (x_1^2 + x_2))}
{(x_1-t_1) (t_1 x_1-1) (t_1 x_1 - x_2) (t_2 x_1^2 - x_2)
(x_2-t_2) (x_1 - t_1 x_2) (x_1^2 - t_2 x_2) (t_2 x_2-1)}\ . \label{eq:gcb2}
\eeqr
Thus, for $m_1=m_2=0$, the poles of integrand $G/x_1x_2$ in (\ref{eq:igm}) are easy to identify, and going through the steps seen in the previous example, we eventually find that the generating function for zero-weight multiplities for $C_2$ is
\bdm
A_{0,0}(t_1,t_2)=\frac{1 + t_1^2 t_2}
{(1 - t_1^2)^2 (1 - t_2) (1 - t_2^2)}\,.
\edm
The form of $A_{0,0}(t_1,t_2)$ is simple enough to allow us to go one step further. We can expand $A_{0,0}(t_1,t_2)$ as a sum of partial fractions
\[
A_{0,0}(t_1,t_2)=\frac12\left[\frac1{(1-t_1^2)(1-t_2^2)}+\frac{1+t_1^2}{(1-t_1^2)^2(1-t_2)^2}\right]
\]
whose Taylor series are quite simple. Matching coefficients yields the general formula for the multiplicities $\mu_{p,q}(0,0)$ of the zero weight as
\[
\mu_{p,q}(0,0)=\frac12\ve_p[\ve_q+(p+1)(q+1)]
\]
where $\ve_p=1$ for $p$ even, or $\ve=0$ for $p$ odd.
The calculations needed to obtain the generating functions for the multiplicities of other low-lying weights go along the same lines and we list some results in the Appendix. However, using directly formula (\ref{amni}) to find the generating function of the multiplicities of a general weight $(m,n)$ seems to be quite involved. In order to make progress, it it is more convenient to introduce a new generating function $H(t_1,t_2;y_1,y_2)$ defined as
\bdm
H(t_1,t_2;y_1,y_2)=
\sum_{m=0}^\infty\sum_{n=0}^\infty\sum_{p=0}^\infty\sum_{q=0}^\infty \mu_{p,q}(m,n) y_1^m y_2^n t_1^p t_2^q
\edm
which collects the multiplicities $\mu_{p,q}(m,n)$ of all weights $m\lambda_1+n\lambda_2$ in all the
representations $R_{p\lambda_1+q\lambda_2}$ of $C_2$. Expressing $\mu_{p,q}(m,n)$ as was done in (\ref{eq:nmp}), the sums in the indices $m$ and $n$ yield geometric series, leading to the formula
\bdm
H(t_1,t_2;y_1,y_2)=\frac{1}{(2\pi i)^2}\oint d x_1\oint d x_2
\frac{G(t_1,t_2;z_1,z_2)}{(x_1-y_1)(x_2-y_2)}\ ,
\edm
which, after substitution of (\ref{eq:b2}) and (\ref{eq:gcb2}), takes the form of a rational integral to be evaluated by means of Cauchy's theorem as in the previous examples. The result is
\beq
H(t_1,t_2;y_1,y_2)=\frac{a+b_1 y_1+b_2 y_2+c_{1,2}y_1 y_2+d y_1^2 +e y_1^2 y_2}
{(1 - t_1^2)^2(1 - t_2^2)(1 - t_2)(1 - t_1y_1)(1 - t_2^2y_1^2)(1 - t_1^2y_2)(1 - t_2y_2)}
\label{eq:gengen}
\eeq
where
\beqrn
a&=&1+t_1^2 t_2,\hspace{2cm}b_1=t_1 t_2(1-t_1^2),\hspace{1.3cm} b_2=-t_1 t_2(t_1^3+t_1 t_2),\\
c_{1,2}&=&t_1 t_2 (t_1^4-t_1^2),\hspace{1cm} d=-t_1^2 t_2^2(1+t_2),\hspace{1cm}e
=t_1^2 t_2^2(t_1^2+t_1^2 t_2+t_2^2-1).
\eeqrn
Now, trading the factors $(1 - t_1y_1)(1 - t_2^2y_1^2)(1 - t_1^2y_2)(1 - t_2y_2)$ in the denominator by geometric series in $y_1$ and $y_2$, we can rewrite $H(t_1,t_2;y_1,y_2)$ as a series
\bdm
H(t_1,t_2;y_1,y_2)=\sum_{m=0}^\infty\sum_{n=0}^\infty A_{m,n}(t_1,t_2) y_1^m y_2^n
\edm
such that the coefficients
are precisely the generating functions for weight multiplicities which we are seeking for. From (\ref{eq:gengen}), and after some tedious
algebra, one can obtain the explicit form of these generating functions as
\beqr
\label{amn}
&&A_{m,n}(t_1,t_2)\nonumber\\
[2pt]
&&=\frac{t_1^{m+2n+2} (t_1^2-t_2^2)(1-t_2^2)-t_1^{m+2} t_2^{n+1}
(1-t_1^2)(1-t_2^2)-t_2^{m+n+1}(t_1^2-t_2)(1-t_1^2) f(t_1,t_2)}{(1-t_1^2)^2(1-t_2^2)(1-t_2)(t_1^2-t_2^2)(t_1^2-t_2)}
\eeqr
with
\bdm
f(t_1,t_2)=\left\{\begin{array}{ll}t_1^2+t_2\,,& {\rm for}\ \ m\ \
{\rm even}\\t_1(1+t_2)\,,& {\rm for}\ \ m\ \ {\rm odd}\end{array}\right.
\edm
thus generalizing the results for low-lying multiplicities explained before.\footnote{After this work was completed we learned about the very interesting paper by Dokovi\'c \cite{dok} in which he obtains the generating functions for weight multiplicities for the simple Lie algebras of rank 2. Unfortunately, the result quoted in that paper for $B_2\equiv C_2$ and $m$ even is not correct.}
This expression of the generating function differs from the examples of the Appendix by the factors $(t_1^2-t_2^2)(t_1^2-t_2)$ in the denominator. In fact, some further simplification work shows that these factors cancel out, giving
\beqrn
A_{m,n}(t_1,t_2)&=&\frac{1}{D}\Big[(1-t_2^2)\sum_{j=0}^{n-1} t_1^{m+2n-2j} t_2^j+(1-t_1^2+t_2-t_2^3)\sum_{j=0}^{m\over 2} t_1^{2j} t_2^{m+n-2j}\\
&-&(1-t_1^2-t_2^2) t_2^{m+n+1}+t_1^{m+2} t_2^n\Big]
\eeqrn
for even $m$ and
\beqrn
A_{m,n}(t_1,t_2)&=&\frac{1}{D}\Big[(1-t_2^2)\sum_{j=0}^{n-1} t_1^{m+2n-2j} t_2^j
+\,(1+t_2-t_2^2-t_1^2t_2)\sum_{j=0}^{{m-3}\over 2} t_1^{2j+1} t_2^{m+n-2j-1}\\
&+&(1+t_2-t_2^2) t_1^m t_2^n+t_1 t_2^{m+n+1}\Big]
\eeqrn
for odd $m$, with $D=(1-t_1^2)^2(1-t_2^2)(1-t_2)$.
After the generating functions are known some other interesting results come from them. In particular, looking at their form for low $m$ and $n$, one can identify two different recurrence relations among the multiplicities $\mu_{p,q}(m,n)$, and with some additional labour, it is possible to show that these recurrence relations are valid in general. This is described in the next two sections.
\section{The first recurrence relation}
The first recurrence relation is among the multiplicities
of a fixed weight $m\l_1+n\l_2$ in different representations. Let us call
\beq
X_{m,n}(t_1,t_2)=(1-t_1^2)(1-t_2) A_{m,n}(t_1,t_2).
\eeq
From the definition of $A_{m,n}(t_1,t_2)$, one has
\bdm
X_{m,n}(t_1,t_2)=\sum_{p=0}^\infty\sum_{q=0}^\infty [\mu_{p,q} (m,n)-\mu_{p-2,q} (m,n)-\mu_{p,q-1} (m,n)+\mu_{p-2,q-1} (m,n)]t_1^p\, t_2^q
\edm
while the explicit expressions given above yield
\beqrn
X_{m,n}(t_1,t_2)&=&\frac{(1-t_2^2){\displaystyle\sum_{j=0}^{n-1}}\,t_1^{m+2n-2j} t_2^j+(1-t_2^2)\Big[ (1+t_2)\displaystyle\sum_{j=0}^{\frac{m}{2}}\,t_1^{2j}t_2^{m+n-2j}-t_2^{m+n+1}\Big]}{(1-t_1^2)(1-t_2^2)}\\
&+&\frac{t_1^2 t_2^{m+n+1}+t_2^{m+n+2}}{(1-t_1^2)(1-t_2^2)}
\eeqrn
for $m$ even and
\beqrn
X_{m,n}(t_1,t_2)&=&\frac{(1-t_2^2)\displaystyle\sum_{j=0}^{n-1} t_1^{m+2n-2j} t_2^j+(1-t_2^2)(1+t_2)\Big[\sum_{j=0}^{\frac{m-3}{2}}t_1^{2j+1}t_2^{m+n-1-2j}+t_1^mt_2^n\Big]}{(1-t_1^2)(1-t_2^2)}\\
&+&\frac{(1+t_2)t_1 t_2^{m+n+1}}{(1-t_1^2)(1-t_2^2)}
\eeqrn
for $m$ odd.
Let us consider the formula for $m$ even and compare it with the diagram
of Figure 1, which shows all representations $R_{p\l_1+q\l_2}$ with nonzero
multiplicity for a weight $m\l_1+n\l_2$. The labels $(m,n)$, $(p,q)$ etc., represent coordinates in the non-Euclidean $(\l_1$,$\l_2$)-plane. The first term in $X_{m,n}(t_1,t_2)$
can be expanded as a geometric series which contains, always with coefficient equal to one,
all the products $t_1^p t_2^q$ for $p$ and $q$ corresponding to points in the diagonals
beginning in the segment $AB$. In a similar way, the second term in $X_{m,n}(t_1,t_2)$
gives all such products for the points in the diagonals normal to the line from $A$ to $C$,
the third corresponds to the diagonals beginning in $D^\prime, E^\prime, F^\prime,\ldots$,
etc, and the fourth to the diagonals from $D,E,F,\ldots$ etc. From this and an analogous
analysis for $m$ odd, we can finally conclude that
\beq
\label{recrel}
\mu_{p,q} (m,n)-\mu_{p-2,q} (m,n)-\mu_{p,q-1} (m,n)+\mu_{p-2,q-1} (m,n)=y_{p,q}(m,n) \label{eq:rec1}
\eeq
where $y_{p,q}(m,n)=1$ if $(p,q)$ labels a irreducible representation of $C_2$ containing
the weight $m\l_1+n\l_2$ (except for $m$ even, $p=0$ and $q$ of opposite parity to $n$, which gives $y_{p,q}(m,n)=0$) and $y_{p,q}(m,n)=0$ if the weight $m\l_1+n\l_2$
is not in $R_{p\l_1+q\l_2}$.
\medskip
\noindent{\bf Remark.} A geometric interpretation can be given to the function $X_{m,n}(t_1,t_2)$ taking into account that, after direct substitution of the expression (\ref{amne}) of the generating function $A_{m,n}$, we can write it as a combination of sums of infinite geometric series, namely
\beqrn
X_{m,n}(t_1,t_2)&=&\sum_{j,k=0}^\infty t_1^{(m+2n)+2j-2k}t_2^k-\sum_{j,k=0}^\infty t_1^{(m-2)-2j-2k}t_2^{(n+1)+2j+k}\\
& -&\sum_{j,k=0}^\infty t_1^{-2j}t_2^{(m+n+1)+2j+2k}-\sum_{j,k=0}^\infty t_1^{-2-2j}t_2^{(m+n+2)+2j+2k}.
\eeqrn
Each term represents the contribution of the points, with coefficient +1 or $-1$, in a bidimensional lattice obtained after translating a point $(p_0,q_0)$ along some independent directions in the ($\l_1$,$\l_2$)-plane. For instance, the first sum represents the total contribution to $X_{m,n}$ of the lattice generated from the basis $\{2\l_1,-2\l_1+\l_2\}$, with origin the point $(m+2n,0)$ and nonnegative integers $j,k$. All the points thus generated have coefficient +1. With reference to the example in Figure 1, this is the lattice obtained by translating first the point $B(m+2n,0)$ along the $\l_1$-axis with step $2\l_1$; the linear lattice is then translated along $BA$ with step $-2\l_1+\l_2$.
The remaining contributions have the same interpretation, differing only in the fact that the coefficient is now $-1$; this means that points obtained from two contributions of different sign are superposed to give a null contribution. This way of counting gives the same lattice (Figure 1) as before, for both $m$ even or odd.
\begin{figure}
\begin{center}
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\put(0,0){\thicklines\vector(0,1){10}}
\put(0,0){\thicklines\vector(1,1){5}}
\put(0,100){\line(1,-1){30}}
\put(30,70){\line(1,0){40}}
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\multiput(160,160)(0,10){1}{\circle*{2}}
\put(1,10){$\lambda_2$}
\put(5,1.5){$\lambda_1$}
\put(-30,99){\small$C(0,n+m)$}
\put(-6,120){\small$D$}
\put(-6,140){\small$E$}
\put(-6,160){\small$F$}
\put(12,120){\small$D'$}
\put(12,140){\small$E'$}
\put(12,160){\small$F'$}
\put(27,64){\small$A(m,n)$}
\put(69,64){\small$B(m+2n,0)$}
\put(29,26){\small$m$}
\put(-4,39){\small$n$}
\put(60,10){\thinlines\vector(1,0){10}} \put(71,10){$\a_1$}
\put(60,10){\thinlines\vector(-1,1){10}} \put(71,21){$\a_4$}
\put(60,10){\thinlines\vector(0,1){10}} \put(60,21){$\a_3$}
\put(60,10){\thinlines\vector(1,1){10}} \put(48,21){$\a_2$}
\end{picture}
\end{center}
\medskip
{\footnotesize Figure 1. The highest weights (modulo Weyl reflections) corresponding
to representations of $C_2$ containing the weight $m\l_1+n\l_2$ ($m=6$, $n=4$ in the example). Here, $\a_1,\dots,\a_4$ are the positive roots and $\l_1,\l_2$ are the fundamental weights, of magnitude $|\l_1|=1$ and $|\l_2|={\sqrt{2}}$).}
\end{figure}
\section{The second recurrence relation}
There is a second recurrence relation, this time among the multiplicities of weights of the
same representation. Let $P_{m,n}(t_1,t_2)$ be the function defined as
\beq
P_{m,n}(t_1,t_2)=A_{m,n}(t_1,t_2)-A_{m+2,n}(t_1,t_2)-A_{m,n+1}(t_1,t_2)+A_{m+2,n+1}(t_1,t_2).
\eeq
Then, from the previous expressions of $A_{m,n}(t_1,t_2)$, one can obtain
\bdm
P_{m,n}(t_1,t_2)=\frac{1}{1-t_2}\sum_{j=0}^n t_1^{m+2n-2j} t_2^j+\frac{1}{1-t_2}\sum_{j=0}^{\frac{m}{2}-1} t_1^{2j} t_2^{m+n-2j}-\frac{t_2^{m+n+1}}{(1-t_1^2)(1-t_2)}
\edm
for $m$ even and
\bdm
P_{m,n}(t_1,t_2)=\frac{1}{1-t_2}\sum_{j=0}^n t_1^{m+2n-2j} t_2^j+\frac{1}{1-t_2}\sum_{j=0}^{\frac{m-3}{2}} t_1^{2j+1} t_2^{m+n-1-2j}-\frac{t_1 t_2^{m+n+1}}{(1-t_1^2)(1-t_2)}
\edm
for $m$ odd. Expanding the denominators as geometric series and using de definition of $A_{m,n}(t_1,t_2)$,
one finds that for both parities
\beq
\mu_{p,q}(m,n)-\mu_{p,q}(m+2,n)-\mu_{p,q}(m,n+1)+\mu_{p,q}(m+2,n+1)= \varepsilon_{p,q}(m,n) \label{eq:rec2}
\eeq
where the right-hand member is a sum of three terms, $\varepsilon_{p,q}(m,n)=X+Y-Z$,
which are zero except for the cases
\beqrn
X=1\ \ &{\rm when}& m\leq p\ \ {\rm and }\ \ \ p\leq m+2n \leq p+2 q\\
Y=1\ \ &{\rm when}& m\geq p+2\ \ {\rm and }\ \ \ m+n \leq p+q\\
Z=1\ \ &{\rm when}& m+n \leq q-1 .
\eeqrn
As $X=1$ and $Y=1$ do not occur at the same time, $\varepsilon_{p,q}(m,n)$ is always 1, 0 or $-1$. To describe the domains $D1$, $D0$ and $D(-1)$ in which $\varepsilon_{p,q}(m,n)$ takes these values, let us consider the weight diagram of the representation
$p\l_1+q\l_2$ using now Cartesian coordinates, denoted as $[x,y]$ with $x=m$, $y=m+2n$,
instead of the $m$ and $n$ labels of the weights. The weights entering in the diagram form a square lattice with a spacing equal to $|\l_2|$ which includes the highest weight of the
representation and is contained in the polygon of vertices $[0, p+2q]$, $[p,p+2q]$, $[p+q,p+q]$
and $[0,0]$. If we call $P_X$, $P_Y$ and $P_Z$ the regions of the diagram in which,
respectively, $X=1$, $Y=1$ and $Z=1$, it follows that:
\begin{itemize}
\item If $2q-2<p$, $P_X \cup P_Y$ does not intersect $P_Z$. Thus, $D1=P_X\cup P_Y$,
$D(-1)=P_Z$, and $D0$ the remaining weights. So, in this case $D1$ is the upper region
of the diagram, starting from the horizontal line $y=p$, $D0$ is the area below that line
and above the diagonal $y=2q-2-x$, and $D(-1)$ includes that diagonal and the weights below it,
see Figure 2.
\item If $p\leq 2q-2\leq 2p$ the intersection $P_X\cap P_Z$ is the (possibly degenerate)
triangle $T$ of vertices $[0,2q-2]$, $[0,p]$ and $[2q-2-p,p]$. In this case $D1=P_X\cup P_Y-T$,
$D(-1)=P_Z- T$, and $D0$ the remaining weights. Now, as one can see in Figure 3, some weights
above $y=p$ located near the $y$ axis are in D0 instead of in D1.
\item If $2q-2 > 2p$ the intersection $(P_X\cup P_Y)\cap P_Z$ is the cuadrilateral $K$
of vertices $[0,p]$, $[0,2q-2]$, $[q-1,q-1]$ and $[p,p]$. Therefore $D1=P_X\cup P_Y-C$,
$D(-1)=P_Z-C$, and $D0$ the remaining weights, see Figure 4.
\begin{figure}
\begin{center}
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\small
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\qquad\qquad\qquad
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\end{center}
\medskip
{\footnotesize Figure 2. The weights of the representation $R_{10\lambda_1+5\lambda_2}$ in the domains $D1$, $D0$ and $D(-1)$ are marked,
respectively, with black dots, circles and encircled black dots. The region $P_X\cup P_Y$
has perimeter $ABCDEA$, while $P_Z$ is contained in $FGOF$. The number over a weight means
its multiplicity.}
\vskip2mm
{\footnotesize Figure 3. Weights of the representation $R_{10\lambda_1+9\lambda_2}$. The region $P_X\cup P_Y$ is bounded by $AFBCDEGA$, $P_Z$ is
into $FGHOAF$ and the triangle $T$ is $FGAF$.}
\end{figure}
\end{itemize}
\begin{figure}
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\small
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\medskip
{\footnotesize Figure 4. Weights of the representation $P_{5\l_1+10\l_2}$. The region $P_X\cup P_Y$ is bounded by $AFBCDGEAF$, $P_Z$ is into
$FGEOAF$ and the cuadrilateral $K$ is $FGEAF$.}
\medskip
{\footnotesize Figure 5. The multiplicity of $m\l_1+n\l_2$ in the representation
$R_{p\l_1+q\l_2}$ is given by the number of weights marked with a square. By parity, the weight signaled with $X$ is excluded.}
\end{figure}
\section{An application}
The recurrence relations of the previous sections provide useful information on the multiplicities and, in fact,
can be used to devise some simple rules to compute the multiplicity of any desired weight on a given representation. As an example, let us take the case of $m$ even and consider a situation with $(m,n)$ and $(p,q)$ as given in the figure 5. We can write the first recurrence relation (\ref{recrel}) as
\bdm
\mu_{p,q}(m,n)-\mu_{p,q-1}(m,n)=y_{p,q}(m,n)+\mu_{p-2,q}(m,n)-\mu_{p-2,q-1}(m,n)
\edm
and iterating we find
\beqrn
\mu_{p,q}(m,n)&=&y_{p,q}(m,n)+y_{p-2,q}(m,n)+\cdots+y_{2,q}(m,n)+y_{0,q}(m,n)\\
&+&y_{p,q-1}(m,n)+y_{p-2,q-1}(m,n)+\cdots+y_{2,q-1}(m,n)+y_{0,q-1}(m,n)\\
&+&y_{p,q-2}(m,n)+y_{p-2,q-2}(m,n)+\cdots+y_{2,q-2}(m,n)+y_{0,q-2}(m,n)+\cdots
\eeqrn
so that finally
\bdm
\mu_{p,q}(m,n)=\sum_{\beta\in R} y_\beta (m,n)
\edm
where the sum is over to the set $R$ of weights marked in the figure. Thus, $\mu_{p,q}(m,n)$
can be obtained by simply counting the number of points in $R$ except those in the vertical
axis with opposite parity of $q$ and $n$, which have $y_\beta (m,n)=0$. This rule can be used
to obtain some explicit formulae. For instance, for the case $p$ and $q$ even and
$q\leq {p}/{2}$, one finds that the multiplicities of the weights on borders of the diagram
are given by
\beqrn
&&\mu_{p,q}(p+q-2s,0)=(s+1)^2{\hspace{3.5cm}\rm for\ \ \ }0\leq s\leq \frac{q}{2}\\
&&\mu_{p,q}(p-2s,0)=(\frac{q}{2}+1)^2+s(q+1){\hspace{2.2cm}\rm for\ \ \ }0\leq s\leq \frac{p-q}{2}\\
&&\mu_{p,q}(2s,0)=\mu_{p,q}(0,0)-s^2{\hspace{3.9cm}\rm for\ \ \ }0\leq s\leq \frac{q}{2}
\eeqrn
and
\beqrn
&&\mu_{p,q}(0,\frac{p}{2}+q-s)=\frac{(s+1)(s+2)}{2}{\hspace{2.94cm}\rm for\ \ \ }0\leq s\leq q\\
&&\mu_{p,q}(0,\frac{p}{2}-s)=\frac{(s+1)(s+2)}{2}+s(q+1){\hspace{1.7cm}\rm for\ \ \ }0\leq s\leq \frac{p}{2}-q\\
&&\mu_{p,q}(0,s)-\mu_{p,q}(0,s+1)=s+1-\theta(s+1){\hspace{1.35cm}\rm for\ \ \ }0\leq s\leq q-1
\eeqrn
where $\theta(r)$ is one (zero) for $r$ even (odd). The combination of these formulas with the recurrence relation (\ref{eq:rec2})
can be used as another alternative to compute the multiplicities of the inner weights.
\section*{Acknowledgement}
J.F.N. acknowledges financial support from MTM2012-33575 project, SGPI-DGICT(MEC), Spain.
\section*{Appendix}
We list here some generating functions of multiplicities of low-lying weights of $C_2$ up to $m+n=4$ as obtained by computing the corresponding integrals in formula (\ref{amni}).
\beqrn
A_{1,0}(t_1,t_2)&=&\frac{t_1}{(1 - t_1^2)^2 (1 - t_2)^2}\\
[2pt]
A_{0,1}(t_1,t_2)&=&\frac{t_1^2 + t_2}{(1 - t_1^2)^2 (1 - t_2)(1 - t_2^2)}\\
[2pt]
A_{2,0}(t_1,t_2)&=&\frac{t_2^2+t_1^2(1+t_2-t_2^2)}{(1-t_1^2)^2(1-t_2)^2(1+t_2)}\\
[2pt]
A_{1,1}(t_1,t_2)&=&\frac{t_1^3(1-t_2)+t_1t_2}{(1-t_1^2)^2(1-t_2)^2}\\
[2pt]
A_{0,2}(t_1,t_2)&=&\frac{t_1^2t_2+t_2^2+t_1^4(1-t_2^2)}{(1-t_1^2)^2(1-t_2)^2(1+t_2)}\\
[2pt]
A_{3,0}(t_1,t_2)&=&\frac{t_1t_2^2+t_1^3(1-t_2^2)}{(1-t_1^2)^2(1-t_2)^2}\\
[2pt]
A_{2,1}(t_1,t_2)&=&\frac{t_2^3+t_1^4(1-t_2^2)+t_1^2t_2(1+t_2-t_2^2)}{(1-t_1^2)^2(1-t_2)^2(1+t_2)}\\
[2pt]
A_{1,2}(t_1,t_2)&=&\frac{t_1^5(1-t_2)+t_1^3 t_2(1-t_2)+t_1t_2^2}{(1-t_1^2)^2(1-t_2)^2}\\
[2pt]
A_{0,3}(t_1,t_2)&=&\frac{t_1^2t_2^2+t_2^3+t_1^6(1-t_2^2)+t_1^4t_2(1-t_2^2)}{(1-t_1^2)^2(1-t_2)^2(1+t_2)}\\
[2pt]
A_{4,0}(t_1,t_2)&=&\frac{t_2^4 + t_1^4(1 - t_2)(1 + t_2)^2 +
t_1^2t_2^2(1 + t_ 2- t_2^2)}{(1 - t_1^2)^2(1 - t_2)^2(1 + t_2)}\\
[2pt]
A_{3,1}(t_1,t_2)&=&\frac{t_1^5(1 - t_2) + t_1t_2^3 + t_1^3t_2(1 - t_2^2)}{(1-t_1^2)^2(1 - t_2)^2}\\
[2pt]
A_{2,2}(t_1,t_2)&=&\frac{t_2^4+t_1^6(1-t_2^2)+t_1^4t_2(1-t_2^2)+t_1^2t_2^2(1+t_2-t_2^2)} {(1-t_1^2)^2(1-t_2)^2(1+t_2)} \\
[2pt]
A_{1,3}(t_1,t_2)&=&\frac{t_1^7(1 - t_2) + t_1^5t_2(1 - t_2) +t_1^3t_2^2(1 - t_2) +
t_1t_2^3}{(1 - t_1^2)^2(1 - t_2)^2}\\
[2pt]
A_{0,4}(t_1,t_2)&=&\frac{t_1^2t_2^3 + t_2^4 +t_1^8(1 - t_2^2) + t_1^6t_2(1 - t_2^2) +
t_1^4t_2^2(1 - t_2^2)}{(1 - t_1^2)^2(1 - t_2)^2(1 + t_2)} \\
[2pt]
\eeqrn
|
1,108,101,565,704 | arxiv | \section*{Introduction}
\noindent
Let $G$ be a simple algebraic group with Lie algebra $\g$, $\EuScript U(\g)$ the enveloping algebra, and
$\Phi$ the Killing form on $\g$. If $\h\subset\g$ is a reductive subalgebra, then $\Phi\vert_\h$ is
non-degenerate and $\me:=\h^\perp$ is a complementary $\h$-submodule of $\g$, i.e.,
$\g=\h\oplus\me$. Using $\Phi\vert_\h$, one defines
the Casimir element $\EuScript C_\h\in \EuScript U(\h)$, and our goal is to study $\EuScript C_{\h}$-eigenvalues in
$\me$ and related $\h$-modules. In~\cite{jlms01}, we proved that {\sf (i)} the $\EuScript C_\h$-eigenvalues
in $\me$ do not exceed $1/2$ and {\sf (ii)} if $\h$ is the fixed-point subalgebra of an involution, i.e.,
$[\me,\me]\subset\h$, then $\EuScript C_\h$ acts scalarly on $\me$, as $\frac{1}{2}{\sf id_\me}$. First, we
prove a complement to it. Namely, if $\EuScript C_\h$ does have an eigenvalue $1/2$ in $\me$, then
$[\me,\me]\subset\h$ and thereby `$1/2$' is the only $\EuScript C_\h$-eigenvalue on $\me$.
\\ \indent
Then we stick to the case in which $\h$ is a Levi subalgebra of $\g$. Let
$\te\subset\h$ be a Cartan subalgebra and $\Delta$ (resp. $\Delta_\h$) the root system of
$(\g,\te)$ (resp. $(\h,\te)$). Let $\be_\h$ be a Borel subalgebra of $\h$ containing $\te$ and $\be$
a Borel subalgebra of $\g$ such that $\be\cap\h=\be_\h$. This yields the sets of positive roots
$\Delta^+_\h\subset\Delta^+\subset\Delta$ and decomposition $\g=\me^-\oplus\h\oplus\me^+$, where
$\be=\be_\h\oplus\me^+$. Then $\p:=\h\oplus\me^+$ is a standard parabolic subalgebra and
$\Delta^+=\Delta^+_\h\cup \Delta(\me^+)$, where $\Delta(\me^+)$ is the set of $\te$-weights of
$\me^+=\p^{\sf nil}$.
Let $\Pi$ be the set of simple roots in $\Delta^+$ and $\Pi_\h:=\Pi\cap\Delta^+_\h$.
If $k=\#(\Pi\cap \Delta(\me^+))$, then $\g$ is equipped with a natural $\BZ^k$-grading.
While studying $\EuScript C_\h$-eigenvalues in $\me$, one may assume that $\h$ is a maximal Levi,
i.e., $k=1$, see Section~\ref{subs:Z-grad-versus} for details.
For $\Pi\cap \Delta(\me^+)=\{\ap\}$, the corresponding $\BZ$-grading
is called the $(\BZ,\ap)$-{\it grading}. Let $\g=\bigoplus_{i\in\BZ}\g_\ap(i)$ denote this grading, where
$\h=\g_\ap(0)$ and $\me^+=\bigoplus_{i\ge 1}\g_\ap(i)=:\g_\ap({\ge}1)$. In this case, $\ap$ is the lowest
weight of the simple $\g_\ap(0)$-module $\g_\ap(1)$. Moreover, {\bf each} $\g(i)$, $i\ne 0$, is a simple
$\g(0)$-module~\cite[Chap.\,3,\,\S3.5]{t41},\,\cite[Theorem\,0.1]{ko10}. Then we write
$\EuScript C_\ap(0)$, $\be_\ap(0)$, $\p_\ap$ in place of $\EuScript C_{\g_\ap(0)}$, $\be_{\g_\ap(0)}$, $\p$,
respectively.
\\ \indent
Using the partition of $\Delta^+(\me)$ associated with the $(\BZ,\ap)$-grading,
we obtain explicit formulae for the $\EuScript C_\ap(0)$-eigenvalue in any $\g_\ap(i)$, $i\ne 0$.
Let $\vp_\ap$ be the fundamental weight of $\g$ corresponding to $\ap$ and $\Delta_\ap(i)$ the set of roots of $\g_\ap(i)$. The sum of all elements of
$\Delta_\ap(i)$, denoted $|\Delta_\ap(i)|$, is a multiple of $\vp_\ap$, i.e., $|\Delta_\ap(i)|=q_\ap(i)\vp_\ap$ and $q_\ap(i)\in\BN$.
Hence $|\Delta_\ap({\ge}1)|=(\sum_{i\ge 1}q_\ap(i))\vp_\ap=:q_\ap\vp_\ap$ is the sum of all roots in the
nilradical of $\p_\ap$. Set $r_\ap:=\|\theta\|^2/\|\ap\|^2\in \{1,2,3\}$, where $\theta\subset\Delta^+$ is the
highest root, and let $h^*$ be the {\it dual Coxeter number} of $\g$.
Let $\gamma_\ap(k)$ denote the $\EuScript C_\ap(0)$-eigenvalue on $\g_\ap(k)$.
\begin{thm} \label{thm:gamma(k)}
We have $\gamma_\ap(k)=\displaystyle\frac{k}{2h^*r_\ap}\sum_{i\ge 1}q_\ap(ki)$.
\emph{(In particular, $\gamma_\ap(1)
q_\ap/2h^*r_\ap$.)}
\end{thm}
\noindent
We also obtain a series of relations between numbers $\gamma_\ap(i), q_\ap(i), \dim\g_\ap(i)$.
For instance, if $d_\ap=\max\{i\mid \Delta_\ap(i)\ne \varnothing\}$, then
$\gamma_\ap(d_\ap)=1-d_\ap\gamma_\ap(1)$ and $q_\ap+q_\ap(d_\ap)=2h^*r_\ap/d_\ap$.
Let $\delta_\ap(k)$ be the {\bf maximal} $\EuScript C_\ap(0)$-eigenvalue in $\bigwedge^k\g_\ap(1)$, so that
$\delta_\ap(1)=\gamma_\ap(1)$. We relate the values $\{\delta_\ap(i)\mid i=1,2,\dots\}$ to dimensions of
abelian subspaces of $\g_\ap(1)$ as follows.
\begin{thm} \label{thm:delta(k)}
For each $k=1,2.\dots,\dim\g_\ap(1)$, we have $\delta_\ap(k)\le k\gamma_\ap(1)$. This upper bound
is attained for a given $k$ if and only if\/ $\g_\ap(1)$ contains a $k$-dimensional abelian subspace.
\end{thm}
\noindent
Similar results are obtained earlier for abelian subspaces of $\g$~\cite{ko65} and for abelian subspaces
related to certain $\BZ_m$-gradings of $\g$~\cite{jlms01}.
One of the applications is that if $\g_\ap(1)$ is not abelian (which exactly means that $d_\ap>1$)
and $\ah\subset\g_\ap(1)$ is an abelian subspace, then $\dim\ah\le (1/2)\dim\g_\ap(1)$. A related
result is that if there is an abelian subspace $\ah\subset\g_\ap(1)$ of dimension $(1/2)\dim\g_\ap(1)$,
then {\sf\bfseries (1)} $\ah$ has an abelian complement; {\sf\bfseries (2)} all the numbers
$\{\delta_\ap(i)\}$ can explicitly be computed. It appears here that the sequence
$\delta_\ap(1),\dots,\delta_\ap(m)$ has an interesting behaviour that is governed by a relation
between $q_\ap$ and $q_\ap(1)$. We also provide some methods for constructing abelian subspaces
of $\g_\ap(1)$ and point out the maximal dimension of an abelian subspace in $\g_\ap(1)$ for {\bf all}
$(\BZ,\ap)$-gradings. The latter is related to a recent work of Elashvili et al.~\cite{e-j-k}.
For an involution $\sigma$ of $\g$, let $\g=\g_0\oplus\g_1$ be the associated $\BZ_2$-grading and
$\EuScript C_0\in \EuScript U(\g_0)$ the Casimir element defined via $\Phi\vert_{\g_0}$. Then the
$\EuScript C_0$-eigenvalue on $\g_1$ equals $1/2$~\cite{jlms01}. As $\g_1$ is an orthogonal $\g_0$-module,
there is a natural $\g_0$-module $\spin(\g_1)$ related to the exterior algebra of $\g_1$~\cite{tg01},
see Section~\ref{sect:FdV} for details.
Although $\spin(\g_1)$ is often reducible, $\EuScript C_0$ acts scalarly on it, and the corresponding
eigenvalue, $\gamma_{\spin(\g_1)}$, is computed in~\cite[Theorem~7.7]{tg01}, cf. Section~\ref{sect:FdV}. Here we obtain
another uniform expression.
\begin{thm} \label{thm:1/16}
For any involution (=$\BZ_2$-grading) of $\g$, one has $\gamma_{\spin(\g_1)}=(\dim\g_1)/16$.
\end{thm}
\noindent
The {\bf inner} involutions are closely related to $(\BZ,\ap)$-gradings with $d_\ap\le 2$~\cite{kac}, and
in this case we give a proof of Theorem~\ref{thm:1/16} that uses properties of
$\EuScript C_\ap(0)$-eigenvalues. However, the argument that exploits $(\BZ,\ap)$-gradings does not
extend to outer involutions. Our general proof invoke the "strange formula" of Freudenthal--de Vries,
which asserts that $(\rho,\rho)=(\dim\g)/24$~\cite[47.11]{FdV}, where $2\rho=|\Delta^+|$.
On the other hand, the adjoint representation of $\g$ occurs as the isotropy representation related to
the involution $\tau$ of $\g\dotplus\g$ with $\tau(x,y)=(y,x)$. Although $\g\dotplus\g$ is not simple, one
can state an analogue of Theorem~\ref{thm:1/16} for $(\g\dotplus\g,\tau)$, and we prove that that
analogue is equivalent to the "strange formula". It is important here that, for the orthogonal $\g$-module
$\g$, one has $\spin(\g)=2^{\rk\g/2]}\EuScript V_\rho$, where $\EuScript V_\rho$ is the simple $\g$-module with highest weight $\rho$.
This result of Kostant appears in~\cite[p.\,358]{ko61}, cf. also \cite[Sect.\,5]{ko97}. To a great extent,
our general study of `$\spin(V)$' in \cite{tg01} was motivated by that observation.
The paper is structured as follows. In Section~\ref{sect:prelim-Cas}, we recall basic facts on Casimir
elements, the Dynkin index of a simple subalgebra of $\g$, and $\BZ$-gradings. In
Section~\ref{sect:some-prop}, we discuss some properties of $(\BZ,\ap)$-gradings and numbers
$\{q_\ap\}_{\ap\in\Pi}$. Section~\ref{sect:5/2} contains our results
on Theorem~\ref{thm:gamma(k)} and the $\EuScript C_\ap(0)$-eigenvalues in
$\g_\ap(i)$.
In Sections~\ref{sect:3} and \ref{sect:applic}, we study maximal eigenvalues of
$\EuScript C_\ap(0)$ in $\g_\ap(0)$-modules $\bigwedge^i \g_\ap(1)$ ($1\le i\le\dim\g_\ap(1)$) and their relationship to abelian subspaces of
$\g_\ap(1)$. Section~\ref{sect:FdV} is devoted to connections between $\BZ_2$-gradings and
$(\BZ,\ap)$-gradings with $d_\ap\le 2$.
Here we discuss the "strange formula" and a generalisation of it to the $\BZ_2$-graded situation. In Appendix~\ref{sect:App}, we gather the
tables of eigenvalues $\gamma_\ap(i)$ and numbers $q_\ap(i)$ for all $(\BZ,\ap)$-gradings.
\\ \indent
The ground field $\Bbbk$ is algebraically closed and $\cha\Bbbk=0$. We use `$\dotplus$' to denote the direct sum of Lie algebras.
\section{Casimir elements, Levi subalgebras and gradings}
\label{sect:prelim-Cas}
\noindent
Unless otherwise stated, $\g$ is a simple Lie algebra with a fixed triangular decomposition
$\g=\ut\oplus\te\oplus\ut^-$ and $\Phi$ is the Killing form on $\g$. Then $\Delta$ is the root system of
$(\g,\te)$ and $\Delta^+$ is the set of positive roots corresponding to $\be=\te\oplus\ut$. Let
$\Pi=\{\ap_1,\dots,\ap_n\}$ be a set of simple roots in $\Delta^+$, $\{\vp_1,\dots,\vp_n\}$ the
corresponding set of fundamental weights, and $\theta$ the {\it highest root\/} in $\Delta^+$. We also
write $\vp_\ap$ for the fundamental weight corresponding to $\ap\in\Pi$.
\subsection{The Casimir element associated with a reductive subalgebra}
Let $\h$ be a reductive algebraic subalgebra of $\g$. Then $\Phi\vert_\h$ is
non-degenerate~\cite[Chap.\,1,\,\S\,6.3]{t41} and one defines the Casimir element $\EuScript C_\h$.
Namely, if $\{e_i\}$ and $\{e'_i\}$ are the dual bases of $\h$
w.r.t{.} $\Phi\vert_\h$, then $\EuScript C_\h:=\sum_{i=1}^{\dim\h} e'_i e_i\in\EuScript U(\h)$.
As is well known, $\EuScript C_\h$ is a well-defined quadratic element of the centre of $\EuScript U(\g)$ and
the eigenvalues of $\EuScript C_\h$ on finite-dimensional $\h$-modules are non-negative rational numbers, cf.~\cite[Chap.\,3,\S\,2.9]{t41}.
We have $\g=\h\oplus\me$, where $\me=\h^\perp$ is an $\h$-module.
\begin{prop}[cf.\,{\cite[Theorem\,2.3]{jlms01}}] \label{prop:lms01}
{\sf (i)} \ $\tr_\g(\EuScript C_\h)=\dim\h$; \\
{\sf (ii)} \ If $x,y\in\h$, then $\Phi(\EuScript C_\h(x),y)=\tr_\h(\ad(x)\ad(y))$; \\
{\sf (iii)} \ Any $\EuScript C_\h$-eigenvalue in $\me$ is at most $1/2$. Moreover, if this bound is attained and
$\me_{1/2}\ne 0$ is the corresponding eigenspace, then $[\me_{1/2},\me]\subset \h$. \\
{\sf (iv)} \ If\/ $\g=\h\oplus\me$ is a $\BZ_2$-grading (i.e., $[\me,\me]\subset\h$), then $\me=\me_{1/2}$.
\end{prop}
The following is a useful complement to the above properties.
\begin{prop} \label{prop:m(1/2)}
Given $\g=\h\oplus\me$ and $\EuScript C_\h$ as above, suppose that
$\me_{1/2}\ne 0$. Then $\me_{1/2}=\me$ and thereby the decomposition $\g=\h\oplus\me$ is a $\BZ_2$-grading.
\end{prop}
\begin{proof}
Write $\me=\me_{1/2}\oplus \tilde\me$, where $\tilde\me$ is the sum of all other eigenspaces of
$\EuScript C_\h$ in $\me$. One has $\Phi(\EuScript C_\h(x),y)=\Phi(x,\EuScript C_\h(y))$ for all $x,y$. Hence
$\Phi(\me_{1/2},\tilde\me)=0$ and $\Phi$ is non-degenerate on $\tilde \h:=\h\oplus \me_{1/2}$.
Therefore, $\tilde\h$ is reductive and $\tilde\me={\tilde\h}^\perp$ is a $\tilde\h$-module. On the other
hand, $[\me_{1/2},\tilde\me]\subset \h$, see Prop.~\ref{prop:lms01}(iii). Hence $[\me_{1/2},\tilde\me]=0$.
Let $\hat\h$ be the subalgebra of $\tilde\h$ generated by $\me_{1/2}$. Then
$[\h,\hat\h]\subset \hat\h$ and also $[\me_{1/2},\hat\h]\subset\hat\h$, i.e., $\hat\h$ is an ideal of $\tilde\h$.
We can write $\tilde\h=\hat\h\oplus\es$, where $\es$ is a complementary ideal. Then
$\g=\es\oplus\hat\h\oplus\tilde\me$, $[\es,\hat\h]=0$, and $[\hat\h,\tilde\me]=0$. Therefore
$\hat\h$ is an ideal of $\g$. Thus $\hat\h=\g$ and $\es=\tilde\me=0$.
\end{proof}
For $\h=\g$, one obtains the usual Casimir element $\EuScript C=\EuScript C_\g\in \EuScript U(\g)$.
Let $(\ ,\ )$ denote the {\it canonical bilinear form\/} on $\te^*$, i.e.,
one induced by the restriction of $\Phi$ to $\te$, see~\cite[Chap.\,6,\,\S\,1, n$^o$\,12]{bour} for its
properties. If $\EuScript V_\lb$ is a simple $\g$-module with highest weight $\lb$, then $\EuScript C$ acts on
$\EuScript V_\lb$ scalarly with eigenvalue $(\lb,\lb+2\rho)$~\cite[Chap.\,3, Prop.\,2.4]{t41}.
Since $\EuScript C(x)=x$ for any $x\in \g$, this means that $(\theta,\theta+2\rho)=1$. The latter is
equivalent to that $(\theta,\theta)=1/h^*$, where $h^*$ is the dual Coxeter number of $\g$,
cf. e.g.~\cite[1.1]{jlms01}. And the "strange formula" of Freudenthal--de Vries asserts that
$(\rho,\rho)=(\dim\g)/24$, see~\cite[47.11]{FdV}.
\subsection{The transition factor and the Dynkin index}
\label{subs:transit-factor}
Let $\ka\subset\g$ be a {\bf simple} subalgebra and $\Phi_\ka$ the Killing form on $\ka$. Then
$\Phi\vert_\ka$ is proportional to $\Phi_\ka$, i.e., there is $F\in\BQ$ such that
$\Phi(x,x)=F\cdot \Phi_\ka(x,x)$ for any $x\in\ka$. The transition factor $F$ can be expressed via the other known objects. Consider an invariant bilinear form $(\ {|}\ )_\g$ on $\g$, normalised as follows. Let
$\langle\ {,}\ \rangle_\g$ be the induced $W$-invariant bilinear form on $\te^*$. Following Dynkin,
we then require that
$\langle\theta,\theta\rangle_\g=2$; and likewise for $(\ {|}\ )_\ka$ and $\langle\ {,}\ \rangle_\ka$.
\begin{df}[cf.~{\cite[n$^o$\,7]{dy}}] \label{def:D-ind}
The {\it Dynkin index\/} of a simple subalgebra $\ka$ in $\g$ is defined to be
$\ind(\ka\hookrightarrow\g):=\displaystyle\frac{(x|x)_\g}{(x|x)_\ka}$ \ for $x\in\ka$.
\end{df}
The following simple assertion is left to the reader. For a non-degenerate symmetric bilinear form
$\Psi$ on $\BV$, let $\Psi^*$ denote the induced bilinear form on $\BV^*$.
\begin{lm}
\label{lm:dve-formy}
If\/ $\Psi_1$ and $\Psi_2$ are two such forms and $\Psi_1=f\Psi_2$ for some $f\in \Bbbk^\times$,
then $\Psi_2^*=f\Psi_1^*$.
\end{lm}
\noindent Using this, we give a formula for the transition factor $F$
between $\Phi$ and $\Phi_\ka$
or, rather, the transition factor $T$ between the induced canonical bilinear forms $(\ ,\ )$ on $\te^*$ and $(\ ,\ )_\ka$ on $\te^*_\ka$, where $\te_\ka$ is a suitable Cartan subalgebra of $\ka$ and we regard $\te^*_\ka$ as subspace of $\te^*$.
\begin{prop} \label{prop:transition}
{\sf (i)} \ The transition factor between $(\ ,\ )$ and $(\ ,\ )_\ka$ \ is
$T=\displaystyle \frac{1}{F}=\frac{h^*(\ka)}{h^*{\cdot} \ind(\ka\hookrightarrow\g)}$.
\\
{\sf (ii)} \ Furthermore, $\ind(\ka\hookrightarrow\g)=\displaystyle \frac{(\theta,\theta)}{(\ov{\theta},\ov{\theta})}$, where $\ov{\theta}$ is the highest root of\/ $\ka$.
\end{prop}
\begin{proof}
{\sf (i)} Using Lemma~\ref{lm:dve-formy} and Def.~\ref{def:D-ind}, we notice that $T=
\displaystyle\frac{1}{F}$ and
$\displaystyle
\ind(\ka\hookrightarrow\g)=\frac{\langle\nu,\nu\rangle_\ka}{\langle\nu,\nu\rangle_\g}$ for
any $\nu\in\te^*_\ka$.
Since $(\theta,\theta)=1/h^*$ and $\langle\theta,\theta\rangle_\g=2$, we have
$(\ ,\ )=2h^*\langle\ ,\ \rangle_\g$ and likewise for two forms on $\te^*_\ka$. Then for any
$\nu\in\te^*_\ka\subset\te^*$, we obtain
\[
T=\frac{(\nu,\nu)}{(\nu,\nu)_\ka}=\frac{(\nu,\nu)}{\langle\nu,\nu\rangle_\g}{\cdot}
\frac{\langle\nu,\nu\rangle_\g}{\langle\nu,\nu\rangle_\ka}{\cdot}
\frac{\langle\nu,\nu\rangle_\ka}{(\nu,\nu)_\ka}=
\frac{1}{2h^*}\frac{1}{\ind(\ka\hookrightarrow\g)}{\cdot}2h^*(\ka)
=\frac{h^*(\ka)}{h^*{\cdot} \ind(\ka\hookrightarrow\g)}.
\]
{\sf (ii)} \ Taking $\nu=\ov{\theta}$, we obtain
\[
\ind(\ka\hookrightarrow\g)=\frac{\langle\ov{\theta},\ov{\theta}\rangle_\ka}{\langle\ov{\theta},\ov{\theta}\rangle_\g}=\frac{2}{\langle\ov{\theta},\ov{\theta}\rangle_\g}=
\frac{\langle{\theta},{\theta}\rangle_\g}{\langle\ov{\theta},\ov{\theta}\rangle_\g}=
\frac{(\theta,\theta)}{(\ov{\theta},\ov{\theta})}. \qedhere
\]
\end{proof}
\subsection{Levi subalgebras and gradings} \label{subs:Z-grad-versus}
By definition, a Levi subalgebra is the centraliser in $\g$ of a toral subalgebra (i.e., of the Lie algebra of
an algebraic torus). If $\h=\z_\g(\tilde\ce)$ for a toral subalgebra $\tilde\ce$, then $\ce:=\z_\g(\h)$ is
the centre of $\h$ and $\h=\ce\dotplus \es$, where $\es=[\h,\h]$. For $\mu\in\ce^*$, set
$\me(\mu)=\{x\in\me\mid [c,x]=\mu(c)x \ \ \forall c\in \ce\}$. Then $\me(\mu)$ is an $\h$-module. By an
old result of Kostant, $\me(\mu)$ is a {\bf simple} $\h$-module. See~\cite[p.\,136]{ko10}
for a proof and historical remarks. (An alternate independent approach appears
in~\cite[Chap.\,3,\,\S 3.5]{t41}.)
As in the introduction, we assume that $\te\subset\h$ and $\be_\h\subset\be$. This provides the
decomposition $\g=\me^-\oplus\h\oplus\me^+$ and partition $\Delta^+_\h\cup\Delta(\me^+)=\Delta^+$.
If $\dim\ce=k$, then one defines a $\BZ^k$-grading of $\g$ as follows. To simplify notation,
assume that $\Pi\cap\Delta(\me^+)=\{\ap_1,\dots,\ap_k\}$. For $\gamma\in \Delta$, let $\g^\gamma$
denote the corresponding root space. If $\gamma=\sum_{i=1}^na_i\ap_i\in \Delta$, then the
$\ap_i$-{\it height\/} of $\gamma$ is $\hot_{\ap_i}(\gamma)=a_i$ and $\hot(\gamma)=\sum_{i}a_i$ is
the (usual) {\it height} of $\gamma$. For a
$k$-tuple $(j_1,\dots,j_k)\in\BZ^k$, set
\[
\Delta(j_1,\dots,j_k)=\{\gamma\in\Delta\mid \hot_{\ap_i}(\gamma)=j_i, \ 1\le i\le k \} \ \text{ and } \
\g(j_1,\dots,j_k)=\bigoplus_{\gamma\in \Delta(j_1,\dots,j_k)} \g^\gamma .
\]
This yields a $\BZ^k$-grading $\g=\bigoplus_{j_1,\dots,j_k} \g(j_1,\dots,j_k)$
with $\g(0,\dots,0)=\h$. By the above result of Kostant, each $\g(j_1,\dots,j_k)$
with $(j_1,\dots,j_k)\ne (0,\dots,0)$ is a simple $\h$-module. Indeed, if $(\nu_i,\ap_j)=\delta_{ij}$,
$1\le i,j\le k$ and $\mu=\sum_{i=1}^k j_i\nu_i\in\ce^*$, then $\g(j_1,\dots,j_k)=\me(\mu)$.
If $k=1$ and $\Pi\cap\Delta(\me^+)=\{\ap\}$, then $\h$ is a maximal Levi and the corresponding
$\BZ$-grading is called the $(\BZ,\ap)$-{\it grading}. In this case, we write $\g_\ap(j)$ in place of $\g(j)$.
The passage from an arbitrary Levi subalgebra $\h\subset\g$ to a maximal Levi subalgebra of a simple
subalgebra of $\g$ goes as follows. Suppose that we are to compute the $\EuScript C_\h$-eigenvalue on
a simple $\h$-module $V=\g(j_1,\dots,j_k)\subset\me^+$. Here $V^*=\g(-j_1,\dots,-j_k)\subset \me^-$ is
the dual $\h$-module and $\Phi$ is non-degenerate on $V\oplus V^*$. Take
\[
\q=\textstyle \bigoplus_{i\in\BZ}\g(ij_1,\dots,ij_k) \subset \g .
\]
It is a $\BZ$-graded subalgebra of $\g$ with $\q(i)=\g(ij_1,\dots,ij_k)$. Since $\Phi\vert_\q$ is
non-degenerate, $\q$ is reductive.
Furthermore, by~\cite[Sect.\,1]{ko10}, the positive part
$\q({\ge}1)$ is generated by $V=\q(1)$. Since each $\q(i)$, $i\ne 0$, is a simple $\q(0)$-module,
the $\BZ$-grading of $\q$ is determined by a sole simple root of $\q$. Taking the corresponding simple ideal of $\q$, one can write
$\q=\ka\dotplus \el$, where $\el$ is reductive, $\ka$ is simple, and there is a simple root $\beta$ of $\ka$
such that $\q(i)=\ka_\beta(i)$ for $i\ne 0$, while
\beq \label{g(0)-decomp}
\h=\q(0)=\ka_\beta(0)\dotplus \el .
\eeq
Thus, $\ka_\beta(0)$ is a maximal Levi subalgebra of $\ka$ and $V=\ka_\beta(1)$ for the $(\BZ{,}\beta)$-grading of $\ka$. Taking a basis for $\h$ adapted to
the sum in~\eqref{g(0)-decomp}, one can split $\EuScript C_\h$ as
$\EuScript C_{\h}=\tilde{\EuScript C}_{\ka_\beta(0)}+\tilde{\EuScript C}_\el$.
Since $\el$ acts trivially on $V$, the eigenvalues of $\EuScript C_{\h}$ and $\tilde{\EuScript C}_{\ka_\beta(0)}$
on $V$ are the same. Furthermore, if ${\EuScript C}_{\ka_\beta(0)}$ is the true Casimir element associated
with $(\Phi_\ka)\vert_{\ka_\beta(0)}$
and $\Phi\vert_\ka=F{\cdot}\Phi_\ka$ (cf. Section~\ref{subs:transit-factor}),
then $\tilde{\EuScript C}_{\ka_\beta(0)}=F{\cdot}{\EuScript C}_{\ka_\beta(0)}$. Here the factor $F$ comes from the
fact that the dual bases for $\ka_\beta(0)$ required in the Casimir elements
$\EuScript C_{\h}$ (i.e., in $\tilde{\EuScript C}_{\ka_\beta(0)}$) and ${\EuScript C}_{\ka_\beta(0)}$
are being computed via the proportional bilinear forms $\Phi\vert_\ka$ and $\Phi_\ka$, respectively. Thus,
\\ \indent
{\sl for any simple $\h$-module $V\subset\me^+$, there is a simple $\BZ$-graded subalgebra
$\ka\subset \g$ and a simple root $\beta$ of\/ $\ka$ such that $V=\ka_\beta(1)$ and then the
$\EuScript C_{\h}$-eigenvalue in $V$ equals $F$ times the ${\EuScript C}_{\ka_\beta(0)}$-eigenvalue on $V$,
where $\Phi\vert_\ka=F{\cdot}\Phi_\ka$.
}
\\ \indent
For this reason, we restrict ourselves with considering only maximal Levi subalgebras of $\g$ and
the corresponding $(\BZ,\ap)$-gradings.
\section{$(\BZ,\ap)$-gradings and partitions of root systems}
\label{sect:some-prop}
\noindent
If $\Delta$ has two root lengths, then $\Pi_l$ is the set of {\bf long} simple roots and $\theta_s$ stands
for the dominant {\bf short} root. In the simply-laced case, we assume that $\Pi_l=\Pi$ and
$\theta_s=\theta$. Recall that $\hot_{\ap_i}(\gamma)$ is the $\ap_i$-{height} of $\gamma\in\Delta$.
Given $\ap\in\Pi$, set $d_\ap=\hot_\ap(\theta)$,
$\Delta_{\ap}(i)=\{\gamma \mid \hot_{\ap}(\gamma)=i\}$,
$\Delta^+_{\ap}(0)=\Delta^+\cap\Delta_{\ap}(0)$, and
\[
\mathcal R_\ap=\textstyle \bigsqcup_{i=1}^{d_\ap} \Delta_{\ap}(i)=\{\gamma\mid (\gamma,\vp_\ap)>0\} .
\]
Then $ \Delta=\Delta^+_{\ap}(0)\sqcup \mathcal R_\ap$,
and $\Delta_{\ap}(i)$ is the set of $\te$-weights of $\g_\ap(i)$, where
$\g=\bigoplus_{i=-d_\ap}^{d_\ap}\g_\ap(i)$ is the $(\BZ,\ap)$-grading. Since $\g_\ap(i)$ is a simple
$\g_\ap(0)$-module for $i\ne 0$,
$\Delta_\ap(i)$ with $i> 0$ contains a unique minimal root (=\,the lowest weight of $\g_\ap(i)$ w.r.t.
$\Delta_\ap^+(0)$) and a unique maximal root (=\,the highest weight).
As usual, $\gamma^\vee=2\gamma/(\gamma,\gamma)$ and
$\Delta^\vee=\{\gamma^\vee\mid \gamma\in \Delta\}$ is the {\it dual root system}.
The set of simple roots in $(\Delta^+)^\vee$ is $\Pi^\vee$ and notation $\hot(\gamma^\vee)$ refers to
the height of $\gamma^\vee$ in $\Delta^\vee$. The fundamental weight $\vp_\ap$ is {\it minuscule}, if
$(\vp_\ap,\gamma^\vee)\le 1$ for any $\gamma\in \Delta^+$,
i.e., $(\vp_\ap,\theta_s^\vee)=1$; and $\vp_\ap$ is {\it cominuscule}, if
$\hot_{\ap}(\theta)=1$, i.e., $d_\ap=1$.
\un{\it\bfseries Coxeter numbers}. Set $h=h(\g):=\hot(\theta)+1$---the {\it Coxeter number\/} of $\g$ and
$h^*=h^*(\g):=\hot(\theta^\vee)+1$---the {\it dual Coxeter number\/} of $\g$. Since $\theta_s^\vee$ is
the highest root in $\Delta^\vee$, we have $\hot(\theta_s)+1=h^*(\g^\vee)$---the dual Coxeter number of
the Langlands dual Lie algebra $\g^\vee$. Note that $h(\g)=h(\g^\vee)$, hence $h^*\le h$.
However, $h^*(\g)$ and $h^*(\g^\vee)$ can be different.
Thus, there are up to three Coxeter numbers for $(\g,\g^\vee)$, which all coincide in the {\sf\bfseries ADE}-case.
If $M\subset\Delta^+$, then $|M|=\sum_{\gamma\in M}\gamma$, while $\#M$ stands for the cardinality.
As usual, $2\rho=|\Delta^+|$ and hence $(\rho,\gamma^\vee)=\hot(\gamma^\vee)$ for any
$\gamma\in \Delta^+$. The orthogonal projection of $2\rho$ to the edge of the Weyl chamber
corresponding to $\vp_\ap$ can be written as $q_\ap\vp_\ap$ and it is clear that
$\displaystyle q_\ap=\frac{(2\rho,\vp_\ap)}{(\vp_\ap,\vp_\ap)}$. The numbers $\{q_\ap\}_{\ap\in\Pi}$
are needed for the description of the Gorenstein highest weight vector varieties, see~\cite[3.7]{p88},
\cite[Remark~1.5]{ja99}, or
for computing cohomology of invertible sheaves on $G/P_\ap$, where
$P_\ap$ is the maximal parabolic subgroup for $\ap$, see~\cite[4.6]{akh95}.
Let $W_\ap$ be the subgroup of the Weyl group $W$ generated by all {\bf simple} reflections $s_\beta$
with $\beta\in\Pi\setminus \{\ap\}$. Then $W_\ap$ is the stabiliser of $\vp_\ap$ in $W$ and also is the
Weyl group of $\g_\ap(0)$. Write $w_{\ap,0}$ is the longest element in $W_\ap$. Recall that
$w_{\ap,0}^2=1$.
\begin{lm}
\label{prop:edge-labels}
One has $q_\ap\vp_\ap=\vert\mathcal R_\ap\vert$ and $q_\ap\in \BN$.
\end{lm}
\begin{proof}
We have $2\rho=|\Delta^+_\ap(0)|+|\mathcal R_\ap|$ and $(\mu,\vp_\ap)=0$ for
any $\mu\in \Delta^+_\ap(0)$. Hence $(2\rho,\vp_\ap)=(|\mathcal R_\ap|,\vp_\ap)$. Moreover,
$s_\beta(\mathcal R_\ap)=\mathcal R_\ap$ for any $\beta\in\Pi\setminus \{\ap\}$. Therefore,
$|\mathcal R_\ap|$ is proportional to $\vp_\ap$. Clearly, $q_\ap=(|\mathcal R_\ap|,\ap^\vee)$ is an integer.
\end{proof}
\begin{thm} \label{thm:otmetki}
\quad {\sf 1$^o$.} \ For any $\ap\in\Pi$, we have
\begin{itemize}
\item[\sf (i)] \ $q_\ap\le h$; moreover, $q_\ap=h$ if and only if $\vp_\ap$ is minuscule;
\item[\sf (ii)] \ $q_\ap\ge \rk\g+1$ and this minimum is attained for some $\ap$.
\end{itemize}
{\sf 2$^o$}. \ For any $\ap\in\Pi_l$, one has
$q_\ap\le h^*$; moreover, $q_\ap=h^*$ if and only if $\vp_\ap$ is cominuscule.
{\sf 3$^o$}. \ Suppose that $\theta$ is fundamental and $\widehat\ap\in\Pi$ is such that
$(\theta,\widehat\ap)\ne 0$. Then $\widehat\ap\in\Pi_l$, $d_{\widehat\ap}=2$, and
$q_{\widehat\ap}=h^*-1$.
{\sf 4$^o$}. \ If $\theta\ne \theta_s$ and $(\theta_s,\ap)\ne 0$, then $q_{\ap}=h-1$.
\end{thm}
\begin{proof}
Part {\sf 1$^o$(i)} and the first half of {\sf (ii)} are proved in \cite[Appendix]{ja99}. For the sake of
completeness, we provide the full argument.
\\ \indent Clearly, $W_\ap$ preserves each $\Delta_\ap(i)$ and $w_{\ap,0}$ takes the unique
minimal element of each $\Delta_\ap(i)$, $i>0$, to the unique maximal one.
{\sf 1$^o$}. \ We have
$w_{\ap,0}(\mathcal R_\ap )=\mathcal R_\ap $ and
$w_{\ap,0}(\Delta^+_\ap(0))=-\Delta^+_\ap(0)$.
Hence $\rho+w_{\ap,0}\rho=|\mathcal R_\ap |$ and, for any $\gamma\in \mathcal R_\ap $, we have
\[
(\rho,\gamma^\vee)+(\rho,w_{\ap,0}(\gamma^\vee))=
(|\mathcal R_\ap |,\gamma^\vee)=q_\ap(\vp_\ap,\gamma^\vee).
\]
That is, $\hot(\gamma^\vee)+\hot (w_{\ap,0}(\gamma^\vee))=q_\ap(\vp_\ap,\gamma^\vee)$.
Taking $\gamma=\ap$, one obtains
\beq \label{eq:q-ap}
1+\hot (w_{\ap,0}(\ap^\vee))=q_\ap .
\eeq
Since $\hot(\gamma^\vee) \le h-1$ for any $\gamma^\vee\in \Delta^\vee$, we have
$q_\ap\le h$. Furthermore, $q_\ap=h$ if and only if $w_{\ap,0}(\ap^\vee)$ is the
highest root in $\Delta^\vee$. In this case, the equality
$1=(\vp_\ap,\ap^\vee)=(\vp_\ap,w_{\ap,0}(\ap^\vee))$ implies that
$(\vp_\ap,\gamma^\vee)\le 1$ for any $\gamma\in\Delta^+$, i.e.,
$\vp_\ap$ is minuscule. On the other side, $w_{\ap,0}(\ap^\vee)$
is the co-root of maximal height among the roots $\gamma$ such that
$(\vp_\ap,\gamma^\vee)=1$. Since this set contains the co-root
$\sum_{i=1}^n\ap_i^\vee$, we have $\hot (w_{\ap,0}(\ap^\vee))\ge n=\rk\g$.
The existence of $\ap$ such that $q_\ap=\rk\g+1$ can be checked case-by-case. If $\g$ is of type
$\GR{D}{n}$ or $\GR{E}{n}$, then the branching node of the Dynkin diagram will do. For {\sf\bfseries{BCFG}}, one takes the unique long simple
root that is adjacent to a short root. For $\GR{A}{n}$, all simple roots yield $q_\ap=\rk\g+1=h(\g)$.
{\sf 2$^o$}. \ If $\ap\in\Pi_l$, then $\hot(w_{\ap,0}(\ap^\vee))\le \hot(\theta^\vee)=h^*-1$, and the equality occurs if and only if $w_{\ap,0}(\ap)=\theta$. In this case, $\theta\in\Delta_\ap(1)$. Hence
$\hot_\ap(\theta)=\hot_\ap(\ap)=1$, i.e.,
$\vp_\ap$ is comuniscule.
{\sf 3$^o$}. \ Here $(\theta^\vee,\widehat\ap)=(\theta,{\widehat\ap}^\vee)=1$, hence
$\widehat\ap\in\Pi_l$.
Then $2=(\theta,\theta^\vee)=d_{\widehat\ap}(\widehat\ap,\theta^\vee)=d_{\widehat\ap}$. Next,
$w_{\widehat\ap,0}(\widehat\ap)$ is the maximal root whose $\widehat\ap$-height equals $1$, i.e.,
$w_{\widehat\ap,0}(\widehat\ap)=\theta-\widehat\ap$. Then
$\hot(w_{\widehat\ap,0}({\widehat\ap}^\vee))=\hot((\theta-\widehat\ap)^\vee)=
\hot(\theta^\vee-{\widehat\ap}^\vee)=h^*-2$ and it follows from \eqref{eq:q-ap} that
$q_{\widehat\ap}=h^*-1$.
{\sf 4$^o$}. \ Here $\ap\in\Pi_s$ and $\ap^\vee$ is the unique long simple root in $\Pi^\vee$
such that $(\ap^\vee,\theta_s^\vee)\ne 0$. As in the previous part, the $\ap^\vee$-height
of $\theta_s^\vee$ equals $2$ and $w_{\ap,0}(\ap^\vee)=\theta_s^\vee-\ap^\vee$. Since
$\hot(\theta_s^\vee-\ap^\vee)=h-2$, we obtain $q_\ap=\hot(\theta_s^\vee-\ap^\vee)+1=h-1$.
\end{proof}
\begin{ex} \label{ex:spisok-q}
For the reader's convenience, we list the numbers $\{q_\ap\mid \ap\in\Pi\}$, $h$, and $h^*$ for all simple $\g$.
The numbering of simple roots follows~\cite[Table\,1]{t41}; in particular, for $\GR{E}{6}$, $\GR{E}{7}$, and $\GR{E}{8}$, the numbering is
\\[.9ex]
\centerline{
\raisebox{-3.1ex}{\begin{tikzpicture}[scale= .55, transform shape]
\tikzstyle{every node}=[circle, draw,fill=brown!30]
\node (a) at (0,0) {\bf 1};
\node (b) at (1.1,0) {\bf 2};
\node (c) at (2.2,0) {\bf 3};
\node (d) at (3.3,0) {\bf 4};
\node (e) at (4.4,0) {\bf 5};
\node (f) at (2.2,-1.1) {\bf 6};
\foreach \from/\to in {a/b, b/c, c/d, d/e, c/f} \draw[-] (\from) -- (\to);
\end{tikzpicture}}, \ \
\raisebox{-3.1ex}{\begin{tikzpicture}[scale= .55, transform shape]
\tikzstyle{every node}=[circle, draw, fill=brown!30]
\node (a) at (0,0) {\bf 1};
\node (b) at (1.1,0) {\bf 2};
\node (c) at (2.2,0) {\bf 3};
\node (d) at (3.3,0) {\bf 4};
\node (e) at (4.4,0) {\bf 5};
\node (f) at (5.5,0) {\bf 6};
\node (g) at (3.3,-1.1) {\bf 7};
\foreach \from/\to in {a/b, b/c, c/d, d/e, e/f, d/g} \draw[-] (\from) -- (\to);
\end{tikzpicture}}, \ \
and \ \
\raisebox{-3.1ex}{\begin{tikzpicture}[scale= .55, transform shape]
\tikzstyle{every node}=[circle, draw, fill=brown!30]
\node (h) at (-1.1,0) {\bf 1};
\node (a) at (0,0) {\bf 2};
\node (b) at (1.1,0) {\bf 3};
\node (c) at (2.2,0) {\bf 4};
\node (d) at (3.3,0) {\bf 5};
\node (e) at (4.4,0) {\bf 6};
\node (f) at (5.5,0) {\bf 7};
\node (g) at (3.3,-1.1) {\bf 8};
\foreach \from/\to in {h/a, a/b, b/c, c/d, d/e, e/f, d/g} \draw[-] (\from) -- (\to);
\end{tikzpicture}},}
\\ respectively. We also write $q_i$ for $q_{\ap_i}$.
\begin{enumerate}
\item For $\GR{A}{n}$, one has $q_i=n+1=h=h^*$ for all $i$;
\item For $\GR{B}{n}$, one has $q_i=2n-i$ for $1\le i\le n-1$ and $q_n=2n$; here $h=2n, h^*=2n-1$;
\item For $\GR{C}{n}$, one has $q_i=2n-i+1$ for all $i$; here $h=2n, h^*=n+1$;
\item For $\GR{D}{n}$, one has $q_i=2n-i-1$ for $1\le i\le n-2$ and $q_{n-1}=q_n=2n-2=h$.
\item For $\GR{E}{6}$, $h=h^*=12$ and the numbers $\{q_i\}$ are: \quad \begin{E6}{12}{9}{7}{9}{12}{11}\end{E6}
\item For $\GR{E}{7}$, $h=h^*=18$ and the numbers $\{q_i\}$ are: \quad \begin{E7}{18}{13}{10}{8}{11}{17}{14}\end{E7}
\item For $\GR{E}{8}$, $h=h^*=30$ and the numbers $\{q_i\}$ are: \quad \begin{E8}{29}{19}{14}{11}{9}{13}{23}{17}\end{E8}
\item For $\GR{F}{4}$, one has $q_1=11=h-1$, $q_2=7$, $q_3=5$, and $q_4=8=h^*-1$;
\item For $\GR{G}{2}$, one has $q_1=5=h-1$ and $q_2=3=h^*-1$.
\end{enumerate}
\end{ex}
\begin{rmk} \label{rem:refinement-q}
{\sf (1)} Since $\Delta_\ap(i)$ is the set of weights of a $\g_\ap(0)$-module, we have
$|\Delta_\ap(i)|=q_\ap(i)\vp_\ap$, where $\q_\ap(i)>0$ for $i>0$ and $\sum_{i= 1}^{d_\ap} q_\ap(i)=q_\ap$. This provides a refinement
of the numbers $\{q_\ap\mid \ap\in\Pi\}$, which we use in Section~\ref{sect:5/2}.
\\ \indent
{\sf (2)} We frequently use the fact that $\#\{\gamma\in\Delta^+\mid (\theta,\gamma)>0\}=2h^*-3$, see~\cite[Prop.\,1]{suter}.
\end{rmk}
\section{Eigenvalues of Casimir elements associated with $(\BZ,\ap)$-gradings}
\label{sect:5/2}
\noindent
In this section, we fix $\ap\in\Pi$ and work with the $(\BZ,\ap)$-grading
$\g=\bigoplus_{i\in\BZ}\g_\ap(i)$. Recall that the centre of $\g_\ap(0)$ is one-dimensional (and is
spanned by $\vp_\ap$ upon the identification of $\te$ and $\te^*$), each $\g_\ap(i)$, $i\ge 1$, is a
{\bf simple} $\g_\ap(0)$-module, and the set of $\te$-weights of $\g_\ap(i)$ is
$\Delta_\ap(i)$,~cf. Section~\ref{sect:some-prop}. The {\it height\/}
of the $(\BZ,\ap)$-grading is $d_\ap=\max_{j\in\BZ}\{j\mid \g_\ap(j)\ne 0\}$.
The Casimir element in
$\EuScript U(\g_\ap(0))$ corresponding to the restriction of $\Phi$ to $\g_\ap(0)$ is denoted by
$\EuScript C_\ap(0)$.
Write $\gamma_\ap(i)$ for the eigenvalue of $\EuScript C_\ap(0)$ on $\g_\ap(i)$. To keep track of the
length of simple roots, we need $r_\ap=(\theta,\theta)/(\ap,\ap)$. Hence $r_\ap=1$ if and only if
$\ap\in\Pi_l$. Note that $(\ap,\ap)=1/(h^*r_\ap)$ and $(\ap,\vp_\ap)=1/(2h^*r_\ap)$.
In the rest of this section, we write $d$ for $d_\ap=\hot_\ap(\theta)$.
\begin{thm} \label{thm:s-znach-1}
For any $(\BZ{,}\ap)$-grading, we have $\gamma_\ap(1)=\displaystyle\frac{q_\ap}{2h^* r_\ap}$ and
$\gamma_\ap(d )=1-\displaystyle\frac{d q_\ap}{2h^* r_\ap}$.
\end{thm}
\begin{proof}
Set $2\rho_\ap(0)=|\Delta^+_\ap(0)|$. Then
$2\rho=2\rho_\ap(0)+|\mathcal R_\ap|=2\rho_\ap(0)+ q_\ap\vp_\ap$.
By general principle, if $\EuScript V_\lb$ is a simple $\g_\ap(0)$-module with the highest weight $\lb$, then
the $\EuScript C_\ap(0)$-eigenvalue on $\EuScript V_\lb$ equals $(\lb,\lb+2\rho_\ap(0))$,
see~\cite[Ch.\,3, Prop.\,2.4]{t41}.
\\ \noindent
\textbullet \quad In our case, $\ap$ is the lowest weight in $\g_\ap(1)$, hence $w_{\ap,0}(\ap)$ is the
highest weight. Hence
\begin{multline*}
\gamma_\ap(1)=(w_{\ap,0}(\ap),w_{\ap,0}(\ap)+2\rho_\ap(0))=(\ap,\ap-2\rho_\ap(0))=
(\ap,\ap-2\rho)+(\ap, |\mathcal R_\ap|) \\ =(\ap, |\mathcal R_\ap|)
=q_\ap(\ap,\vp_\ap)=\frac{q_\ap}{2}(\ap,\ap)=\frac{q_\ap}{2h^*r_\ap} .
\end{multline*}
\textbullet \quad
Since $\theta$ is the highest weight of the $\g_\ap(0)$-module $\g_\ap(d )$, we obtain
\begin{multline*}
\gamma_\ap(d )=(\theta,\theta+2\rho_\ap(0))=(\theta,\theta+2\rho)-(\theta,q_\ap\vp_\ap)
=1-q_\ap(\theta,\vp_\ap) \\
=1-q_\ap d{\cdot} (\ap,\vp_\ap)=1-\frac{q_\ap d }{2h^*r_\ap} . \qedhere
\end{multline*}
\end{proof}
\begin{cl} \label{cor:2.2}
We have
\begin{itemize}
\item[\sf (i)] \ $d \gamma_\ap(1)+ \gamma_\ap(d )=1$ and hence\/ $1/2d \le \gamma_\ap(1)< 1/d $;
\item[\sf (ii)] \ if\/ $d =1$, i.e., $\vp_\ap$ is cominuscule, then $\gamma_\ap(1)=1/2$;
\item[\sf (iii)] \ if\/ $\theta$ is a multiple of a fundamental weight and $(\widehat\ap,\theta)\ne 0$, then
$\gamma_{\widehat\ap}(1)=(h^*-1)/2h^*$.
\end{itemize}
\end{cl}
\begin{proof}
{\sf (i)} The first equality is clear. Since $\gamma_\ap(d )>0$, one obtains $\gamma_\ap(1)< 1/d $.
On the other hand, any $\EuScript C_\ap(0)$-eigenvalue in $\bigoplus_{i\ne 0}\g_\ap(i)$ is at most
$1/2$,~see Prop.~\ref{prop:lms01}.
Hence $\gamma_\ap(d )\le 1/2$ and then $\gamma_\ap(1)\ge 1/2d $.
{\sf (ii)} This follows from (i) with $d=1$.
{\sf (iii)} If $\theta$ is {\bf fundamental}, then $\widehat\ap\in\Pi_l$ and $q_{\widehat\ap}=h^*-1$, see
Theorem~\ref{thm:otmetki}(3$^o$). Hence the assertion on $\gamma_{\widehat\ap}(1)$. For a
more general situation in which $\theta$ is a multiple of a fundamental weight, we use the fact that
$\Delta_{\widehat\ap}(2)=\{\theta\}$ and
$\Delta_{\widehat\ap}(1)=\{\mu\in\Delta^+\mid (\mu,\theta^\vee)=1\}$. Then
$\#\Delta_{\widehat\ap}(1)=2h^*-4$ (cf. Remark~\ref{rem:refinement-q}(2)) and $\Delta_{\widehat\ap}(1)$ is a union of pairs $\{\mu,\theta-\mu\}$.
Therefore $|\Delta_{\widehat\ap}(1)|=(h^*-2)\theta$ and $|\mathcal R_{\widehat\ap}|=(h^*-1)\theta$. As in
the proof of Theorem~\ref{thm:s-znach-1},
$\gamma_{\widehat\ap}(1)=(\widehat\ap,|\mathcal R_{\widehat\ap}|)=
(h^*-1)(\widehat\ap,\theta)=(h^*-1)(\theta,\theta)(\widehat\ap,\theta^\vee)/2=(h^*-1)/2h^*$.
\end{proof}
\begin{rmk}
If $\g$ is classical, then $d \in\{1,2\}$ for all $\ap\in\Pi$. Therefore, Theorem~\ref{thm:s-znach-1}
describes all eigenvalues of all $\EuScript C_\ap(0)$.
\end{rmk}
To obtain a general formula for any $\gamma_\ap(i)$, we use the refinement
$\{q_\ap(i)\}$ of numbers $\{q_\ap\mid \ap\in\Pi\}$, see Remark~\ref{rem:refinement-q}(1).
Suppose that $1\le k\le d $ and we are going to compute $\gamma_\ap(k)$.
Consider the $\BZ$-graded subalgebra $\g^{[k]}:=\bigoplus_{i\in\BZ}\g_\ap(ki)\subset \g$, i.e.,
$\g^{[k]}(i)=\g_\ap(ki)$. Then
$\g$ and $\g^{[k]}$ share the same $0$-th part and thereby the same Cartan subalgebra $\te\subset \g_\ap(0)$.
\begin{lm} \label{lm:g^k-ss}
$\g^{[k]}$ is semisimple and the root system of\/ $\g^{[k]}$ relative to $\te$ is\/
$\bigsqcup_{i\in\BZ} \Delta_\ap(ki)$.
\end{lm}
\begin{proof}
The centre of $\g^{[k]}$ (if any) belongs to the centre of $\g_\ap(0)$. As the centre of $\g_\ap(0)$ is
one-dimensional and it acts non-trivially on $\g_\ap(k)$, $\g^{[k]}$ must be semisimple. The rest is clear.
\end{proof}
The passage from $\g$ to $\g^{[k]}$ is a particular case of the general construction outlined in
Section~\ref{subs:Z-grad-versus} (a passage from $\g$ to $\q$). Because this time we begin with a
$(\BZ,\ap)$-grading, it is possible to say more on the relevant details and the factor $F$. As a result, we end up with an explicit formula for $\gamma_\ap(k)$.
Each graded part $\g^{[k]}(i)=\g_\ap(ki)$ of $\g^{[k]}$ is a simple $\g_\ap(0)$-module. Therefore, the $\BZ$-grading
of $\g^{[k]}$ is given by a simple root of $\g^{[k]}$. Clearly, this root, say $\beta$, is just the unique
minimal root in $\Delta_\ap(k)$. Although $\g^{[k]}$ is not necessarily simple, one can write
$\g^{[k]}=\ka\dotplus\es$, where $\ka$ is {\bf simple}, $\es$ is semisimple, and $\beta$ is a simple root of $\ka$.
In this case, the whole of $\es$ lies in $\g_\ap(0)$. Therefore $\g_\ap(0)=\ka_\beta(0)\dotplus\es$ and
$\ka_\beta(i)=\g_\ap(ki)$ for $i\ne 0$. Let $\ov{\vp}_\beta$ be the fundamental weight of $\ka$
(= of $\g^{[k]}$) corresponding to $\beta$.
\begin{prop} \label{prop:svyaz}
$\vp_\ap=\displaystyle k\frac{(\ap,\ap)}{(\beta,\beta)}\cdot\ov{\vp}_\beta $.
\end{prop}
\begin{proof}
Since either of the weights $\vp_\ap$ and $\ov{\vp}_\beta$ generates the one-dimensional centre of
$\g_\ap(0)$, these are proportional.
By the assumption, $(\vp_\ap, \ap^\vee)=1$ and $(\ov{\vp}_\beta, \beta^\vee)=1$. On the other
hand, since $\beta$ is a root in $\Delta_\ap(k)$, we have
$(\vp_\ap,\beta^\vee)=k(\vp_\ap,\ap){\cdot} \frac{2}{(\beta,\beta)}=k{\cdot} \frac{(\ap,\ap)}{(\beta,\beta)}$.
Hence $\vp_\ap/\ov{\vp}_\beta=k {\cdot}\frac{(\ap,\ap)}{(\beta,\beta)}$.
\end{proof}
\begin{thm} \label{thm:gen-formula}
For any $\ap\in\Pi$ and $1\le k\le d $, one has
$\gamma_\ap(k)=\displaystyle \frac{k }{2h^* r_\ap}\sum_{i\ge 1} q_\ap(ki)$. In particular,
$\gamma_\ap(d)=\displaystyle \frac{d q_\ap(d) }{2h^* r_\ap}$.
\end{thm}
\begin{proof}
As above, we consider $\g^{[k]}=\ka\dotplus\es$ and the simple root $\beta$ of $\ka$ such that
$\ka_\beta(i)=\g_\ap(ki)$. For the $(\BZ{,}\beta)$-grading of the simple algebra $\ka$, we consider the
same relevant objects as for $(\g,\ap)$. To distinguish them, the former will be marked by `bar'
(cf. $\vp_\ap$ versus $\ov{\vp}_\beta$). This includes $\ov{q}_\beta,\ov{\mathcal R}_\beta, \ov{r}_\beta$,
etc. (see below).
\textbullet \quad Since $\bigsqcup_{i\in\BZ}\Delta_\ap(ki)=\bigsqcup_{i\in\BZ}\Delta_\beta(i)$ is the
partition of the root system of $(\ka,\te)$ corresponding to $\beta$, we have
$|\ov{\mathcal R}_\beta|=\sum_{i\ge 1}|\Delta_\ap(ki)|=\ov{q}_\beta\ov{\vp}_\beta$. On the other hand,
this sum equals $\sum_{i\ge 1}q_\ap(ki)\vp_\ap$. Invoking Proposition~\ref{prop:svyaz},
we obtain
\[
\ov{q}_\beta=k \frac{(\ap,\ap)}{(\beta,\beta)}\sum_{i\ge 1}q_\ap(ki) .
\]
Let $\ov{\EuScript C}_\beta(0)\in\EuScript U(\ka)$ be Casimir element associated with the Levi subalgebra
$\ka_\beta(0)\subset\ka$.
It is important to understand that $\ov{\EuScript C}_\beta(0)$ is defined via the use of the Killing form
$\Phi_\ka$ on $\ka$.
Let $\ov{\gamma}_\beta(i)$ denote the eigenvalue of $\ov{\EuScript C}_\beta(0)$ on $\ka_\beta(i)$.
Set $\ov{r}_\beta=(\ov{\theta},\ov{\theta})/(\beta,\beta)$, where $\ov{\theta}$ is the highest root of $\ka$.
By Theorem~\ref{thm:s-znach-1} applied to $\ka$ and $\beta$, we have
$\ov{\gamma}_\beta(1)=\displaystyle \frac{\ov{q}_\beta}{2h^*(\ka) {\cdot}\ov{r}_\beta}$.
\textbullet \quad Our next step is to compare $\gamma_\ap(k)$ and $\ov{\gamma}_\beta(1)$. Since
$\g_\ap(0)=\ka_\beta(0)\dotplus\es$ and $\es$ acts trivially on each $\g_\ap(ki)$, one can safely remove
from $\EuScript C_\ap(0)$ the summands corresponding to the dual bases for $\es$, while computing
$\gamma_\ap(ki)$. This "almost" yields
$\ov{\EuScript C}_\beta(0)$. The only difference is that the dual bases for $\ka$ occurring in two Casimir elements
are defined via the use of different Killing forms ($\Phi$ and $\Phi_\ka$, respectively). Hence the
eigenvalues of $\EuScript C_\ap(0)$ and $\ov{\EuScript C}_\beta(0)$ on all $\g_\ap(ki)$ are proportional.
More precisely, since the eigenvalues are computed via the use of the canonical bilinear form
on $\te^*$ and $\te^*_\ka$, respectively, the
transition factor equals the ratio of these two canonical forms.
By Proposition~\ref{prop:transition}(i), this factor equals $T=\displaystyle \frac{h^*(\ka)}{h^*{\cdot} \ind (\ka\hookrightarrow \g)}$.
Gathering together previous formulae, we obtain
\beq \label{eq:polovina}
\gamma_\ap(k) =T{\cdot}\ov{\gamma}_\beta(1)=
\frac{h^*(\ka)}{h^*{\cdot} \ind (\ka\hookrightarrow \g)}{\cdot}
\frac{\ov{q}_\beta}{2h^*(\ka) {\cdot}\ov{r}_\beta}=
\frac{k{\cdot}(\ap,\ap)\sum_{i\ge 1}q_\ap(ki)}{2h^*{\cdot} \ind(\ka\hookrightarrow \g){\cdot}\ov{r}_\beta {\cdot}(\beta,\beta)}.
\eeq
Proposition~\ref{prop:transition}(ii) says that $\ind (\ka\hookrightarrow \g)=\displaystyle (\theta,\theta)/(\ov{\theta},\ov{\theta})$. Hence
$\ind (\ka\hookrightarrow \g){\cdot}\ov{r}_\beta{\cdot} (\beta,\beta)=
(\theta,\theta)$, and one simplifies Eq.~\eqref{eq:polovina} to
\[
\frac{k{\cdot}(\ap,\ap)\sum_{i\ge 1}q_\ap(ki)}{2h^* {\cdot}(\theta,\theta)}=
\frac{k}{2h^*r_\ap}\sum_{i\ge 1}q_\ap(ki) ,
\]
as required.
\end{proof}
\begin{cl} \label{cor:sravnenie-2}
For any $\ap\in\Pi$, one has
$d \bigl(q_\ap+q_\ap(d )\bigr)=2h^*r_\ap$
and $2h^*r_\ap/d \in \BN$.
In particular, if $d =2$, then $q_\ap+q_\ap(2)=h^*r_\ap$.
\end{cl}
\begin{proof}
Theorems~\ref{thm:s-znach-1} and \ref{thm:gen-formula} provide two different formulae for $\gamma_\ap(d )$,
which yields everything.
\end{proof}
To apply Theorem~\ref{thm:gen-formula}, one has to know the integers $\{q_\ap(j)\mid 1\le j\le d \}$.
Corollary~\ref{cor:sravnenie-2} allows us to compute $q_\ap(d)$ and thereby settles the problem for $d =2$. For $d>2$, there are some relations between $\{q_\ap(i)\mid i=1,\dots,d\}$, which allows us to solve this problem.
\begin{prop} \label{prop:simmetri-d_ap}
If $d \ge 2$ and $1\le i\le d-1$, then $q_\ap(i)=q_\ap(d-i)$.
\end{prop}
\begin{proof}
Consider $\g^{[d]}=\g_\ap(-d)\oplus\g_\ap(0)\oplus\g_\ap(d)$. Then
$\g^{[d]}$ is the fixed point subalgebra of an automorphism $\psi\in \text{Int}(\g)$ of order $d$. If
$\zeta =\sqrt[d]1$ is primitive and $1\le i\le d-1$, then the eigenspace of $\psi$ corresponding to
$\zeta^i$ is
$\g_i:=\g_\ap(i)\oplus \g_\ap(i-d)$. Since $\g^{[d]}$ is semisimple (Lemma~\ref{lm:g^k-ss}),
the sum of weights of the $\g^{[d]}$-module $\g_i$ equals $0$. That is,
\[
|\Delta_\ap(i)|+ |\Delta_\ap(i-d)|=(q_\ap(i)-q_\ap(d-i))\vp_\ap=0 . \qedhere
\]
\end{proof}
\begin{rmk}
If $d=3$, then $q_\ap(1)=q_\ap(2)$. That is, Corollary~\ref{cor:sravnenie-2} and
Proposition~\ref{prop:simmetri-d_ap} are sufficient for computing the numbers $\{q_\ap(j)\}$.
For $d\ge 4$, one can also consider all $\g^{[k]}$ with $k> d/2$, which yields more relations. For
instance, if $d=4$ and $k=3$, then one get the relation $q_\ap(4)+q_\ap(1)=q_\ap(2)$. All these extra
relations are sufficient for leisure calculations of all $\{q_\ap(j)\}$. Note that the maximal possible value
$d=6$ is attained only for $\GR{E}{8}$ (once).
\end{rmk}
For future use, we record the following by-product of the above theory.
\begin{prop} \label{prop:2gamma>1gamma}
For any $\ap\in\Pi$, we have $2\gamma_\ap(1)>\gamma_\ap(2)$. Moreover, if $d$ is odd, then
$\gamma_\ap(1)>\gamma_\ap(2)$.
\end{prop}
\begin{proof}
We have $\gamma_\ap(1)=\displaystyle \frac{\sum_{i\ge 1} q_\ap(i)}{2h^* r_\ap}=\frac{q_\ap}{2h^* r_\ap}$
and
$\gamma_\ap(2)=\displaystyle \frac{2\sum_{i\ge 1} q_\ap(2i)}{2h^* r_\ap}$, which yields the first inequality.
For $d$ odd, it follows from Proposition~\ref{prop:simmetri-d_ap} that
$2\sum_{i\ge 1} q_\ap(2i)=\sum_{i=1}^{d-1} q_\ap(i)< q_\ap$.
\end{proof}
\begin{ex} \label{ex:inequal}
If $d=d_\ap$ is even, then it can happen that $2\gamma_\ap(1)> \gamma_\ap(2) > \gamma_\ap(1)$.
For instance, look up $(\GR{E}{8},\ap_5)$ or $(\GR{E}{8},\ap_6)$ or $(\GR{F}{4},\ap_2)$ in tables in
Appendix~\ref{sect:App}.
\end{ex}
Another interesting relation is
\begin{prop} \label{prop:k>d/2}
If $k> d/2$ and $\g^{[k]}=\ka\dotplus\es$ as above, then \
$\displaystyle \frac{q_\ap(k)}{\gamma_\ap(k)}=\frac{2h^*}{k}{\cdot}\frac{(\beta,\beta)}{(\ap,\ap)}{\cdot}\ind (\ka\hookrightarrow \g)$.
In particular, for $k=d$, one obtains \ $\displaystyle \frac{q_\ap(d)}{\gamma_\ap(d)}=\frac{2h^*r_\ap}{d}$.
\end{prop}
\begin{proof}
{\sf 1)} \ If $k>d/2$, then $\g^{[k]}=\g_\ap(-k)\oplus\g_\ap(0)\oplus\g_\ap(k)$ has only three summands
and $\g_\ap(k)$ is commutative. That is, $\ov{\vp}_\beta$ is cominuscule and
$|\Delta_\ap(k)|=h^*(\ka){\cdot}\ov{\vp}_\beta$. Hence the eigenvalue of $\ov{\EuScript C}_\beta(0)$ on
$\ka_\beta(1)=\g_\ap(k)$ equals $1/2$, see Corollary~\ref{cor:2.2}. Using the
transition factor $T=\displaystyle \frac{h^*(\ka)}{h^*{\cdot} \ind (\ka\hookrightarrow \g)}$ (cf.
Theorem~\ref{thm:gen-formula}), we obtain
\beq \label{eq:gamma(k)}
\gamma_\ap(k)=\displaystyle \frac{h^*(\ka)}{2h^*{\cdot} \ind (\ka\hookrightarrow \g)} .
\eeq
On the other hand, $|\Delta_\ap(k)|=q_\ap(k)\vp_\ap=h^*(\ka)\ov{\vp}_\beta$.
Hence $h^*(\ka)\ov{\vp}_\beta=q_\ap(k){\cdot}\displaystyle k\frac{(\ap,\ap)}{(\beta,\beta)}\cdot\ov{\vp}_\beta$ and
\beq \label{eq:q(k)}
q_\ap(k)= \frac{h^*(\ka)}{k}{\cdot} \frac{(\beta,\beta)}{(\ap,\ap)} .
\eeq
Combining Eq.~\eqref{eq:gamma(k)} and~\eqref{eq:q(k)} yields the first assertion.
{\sf 2)} \ If $k=d$, then $\beta$ is the minimal root in $\Delta_\ap(d)$, which is $W_\ap$-conjugate to
$\theta$, the
maximal root in $\Delta_\ap(d)$. Hence $\beta$ is long and $\ov{\theta}=\theta$. Therefore,
$(\beta,\beta)/(\ap,\ap)=r_\ap$ and
$\ind (\h\hookrightarrow \g)=1$, cf. Proposition~\ref{prop:transition}.
\end{proof}
{\bf Remark.} Comparing Proposition~\ref{prop:k>d/2} and Corollary~\ref{cor:sravnenie-2}, we see that
$\displaystyle \frac{q_\ap(d)}{\gamma_\ap(d)}=q_\ap+q_\ap(d)$ is an integer.
\begin{ex} \label{ex:primer-E8}
{\sf (1)} \ Consider the $(\BZ{,}\ap_2)$-grading of $\GR{E}{8}$. Here $d=3$ and $q_2=19$ (see
Example~\ref{ex:spisok-q}). By Theorem~\ref{thm:s-znach-1},
$\gamma_{\ap_2}(1)=19/60$ and $\gamma_{\ap_2}(3)=3/60$. Then Corollary~\ref{cor:sravnenie-2}
shows that $q_2(3)=(60/3)-19=1$. Hence $q_2(1)=q_2(2)=9$. Now, using Theorem~\ref{thm:gen-formula}, we compute that $\gamma_{\ap_2}(2)=18/60$.
{\sf (2)} \ Take the $(\BZ{,}\ap_2)$-grading of $\GR{F}{4}$. Here $r_{\ap_2}=2$, $d=4$, and
$q_2=7$. By Theorem~\ref{thm:s-znach-1}, $\gamma_{\ap_2}(1)=7/36$ and
$\gamma_{\ap_2}(4)=1-(4{\cdot}7/36)=8/36$. Then Corollary~\ref{cor:sravnenie-2} shows that
$q_2(4)=(2{\cdot}9{\cdot}2/4)-7=2$. Since $q_2(1)=q_2(3)$ and $q_2(4)+q_2(1)=q_2(2)$, one computes the remaining $q_2(j)$'s. Finally, Theorem~\ref{thm:gen-formula} implies that
$\gamma_{\ap_2}(2)=10/36$ and $\gamma_{\ap_2}(3)=3/36$.
\end{ex}
The complete calculations of the eigenvalues $\{\gamma_\ap(i)\}$ and integers $\{q_\ap(i)\}$ for
all $(\BZ{,}\ap)$-gradings are gathered in Appendix~\ref{sect:App}.
\section{Eigenvalues of $\EuScript C_\ap(0)$ in $\bigwedge^k\g_\ap(1)$ and abelian subspaces of
$\g_\ap(1)$}
\label{sect:3}
\noindent
In this section, we relate eigenvalues of $\EuScript C_\ap(0)$ to dimensions of abelian subspaces
(=\, commutative subalgebras) of $\g_\ap(1)$. The key role is played by the inequality
$\gamma_\ap(2)< 2\gamma_\ap(1)$, see Prop.~\ref{prop:2gamma>1gamma}. As an application of our
theory, we prove that if $d_\ap>1$ and $\ah\subset\g_\ap(1)$ is an abelian subspace, i.e., $[\ah,\ah]=0$,
then $\dim\ah\le \dim\g_\ap(1)/2$. Our results up to Proposition~\ref{prop:mult-free} are parallel to
results of~\cite[Sect.\,4]{jlms01} that concern the case of $\BZ_m$-gradings. Furthermore, most
proofs therein can readily be adapted to the $\BZ$-graded setting.
For this reason, we omit some details.
Let us recall some basic facts on the complexes $(\bigwedge^\bullet\!\g, \partial)$
and $(\bigwedge^\bullet\!\g, \textsl{d})$.
We identify $\g$ with $\g^*$, using $\Phi$, and consider
$\bigwedge^\bullet\g$ with the usual differentials:
\[ \textstyle
\text{$\textsl{d}:\bigwedge^l\g \to \bigwedge^{l+1}\g$ \quad and \quad
$\partial:\bigwedge^l\g \to \bigwedge^{l-1}\g.$}
\]
Here
\[
\partial(x_1\wedge\ldots\wedge x_l):=\sum_{i<j}(-1)^{i+j-1}
[x_i,x_j]\wedge x_1\wedge\ldots \hat{x}_i\dots \hat{x}_j\ldots\wedge x_l
\]
for $l\ge 2$ and $\partial(x_1)=0$. In particular,
$\partial(x_1\wedge x_2)=[x_1,x_2]$.
\\
We regard $\Phi$ as having been extended, in the usual way, via
determinants, from $\g$ to $\bigwedge^\bullet\!\g$.
More precisely, denoting the extension of $\Phi$ to $\bigwedge^l\g$ by
$\Phi^{(l)}$, we have
\[
\Phi^{(l)}(x_1\wedge\ldots\wedge x_l,y_1\wedge\ldots\wedge y_l):=
\det \| \Phi(x_i,y_j)\| \ .
\]
Then $\textsl{d}=-\partial^t$.
For $x\in\g$, let $\esi(x)$ be the exterior product operator and
$i(x)$ the interior product operator in $\bigwedge\g$. That is,
\[
\esi(x){\cdot}x_1\wedge\ldots\wedge x_l:=x\wedge x_1\ldots\wedge x_l,
\]
\[
i(x){\cdot}x_1\wedge\ldots\wedge x_l:=\sum_{i=1}^l(-1)^{i-1}
\Phi(x,x_i)x_1\wedge\ldots \hat{x_i}\ldots \wedge x_l \ .
\]
Let $\vartheta$ denote the natural extension of the adjoint representation
of $\g$ to $\bigwedge^\bullet\!\g$:
\[
\vartheta(x){\cdot}x_1\wedge\ldots\wedge x_l:=\sum_{i=1}^l
x_1\wedge\ldots \wedge [x,x_i]\wedge\ldots \wedge x_l \ .
\]
These operators satisfy the following relations for all $x\in\g$:
\beq \label{eq:3shtuki}
[\textsl{d},\vartheta(x)]=0, \quad
[ \partial , \vartheta (x) ]=0 , \quad
\esi(x)\partial+\partial\esi(x)=\vartheta(x) .
\eeq
Let $e_1,\dots,e_N$ be a basis for $\g$ and $e'_1,\dots,e'_N$ the dual basis.
After Koszul \cite[3.4]{kos}, it is known that
\beq \label{eq:d}
\textsl{d}=\frac{1}{2}\sum_{i=1}^N \esi(e'_i)\vartheta(e_i) \ .
\eeq
Combining Eq.~\eqref{eq:3shtuki} and \eqref{eq:d} yields
\[
2(\textsl{d}\partial+\partial \textsl{d})=\sum_{i=1}^N\vartheta(e'_i)\vartheta(e_i)=\vartheta(\EuScript C) ,
\]
where $\EuScript C$ is the Casimir element for $\g$. In the general $\BZ$-graded situation, we choose a basis
$\mathbb B=(e_1,\dots,e_N)$
compatible with grading, which means that $\mathbb B\cap\g(i)$ is a basis for $\g(i)$ for each $i$.
Let $\mathbb B'=(e'_1,\dots,e'_N)$ be the dual basis. Since $\g(i)^*\simeq \g(-i)$, we have
$(\mathbb B\cap\g(i))'=\mathbb B'\cap\g(-i)$ is a basis for $\g(-i)$. Any compatible basis yields a splitting of the differential:
$\textsl{d}=\sum_{i\in\BZ}\textsl{d}_i$, where
\[
\textsl{d}_i= \frac{1}{2}\sum_{j:\ e_j\in \g(i)} \esi(e'_j)\vartheta(e_j) .
\]
Note that $\textsl{d}_i(\g(j))\subset \begin{cases} \g(i)\otimes\g(j-i), & i\ne j/2 \\
\bigwedge^2\!\g(i), & i=j/2 \end{cases} \ \subset \bigwedge^2\!\g$. In particular, $\textsl{d}_1(\g(2))
\subset \bigwedge^2\!\g(1)$.
The key technical result for $\g(1)$ and $\textsl{d}_1$ is
\begin{prop} \label{prop:d_1}
{\sf (i)} \ Let $(e_1,\dots,e_s)$ be a basis for $\g(1)$ and $(e'_1,\dots,e'_s)$ the dual basis for $\g(-1)$.
For any $y,z\in \g(1)$, we have
\[
\sum_{i=1}^s [e_i,y]\wedge [e'_i,z]=-\textsl{d}_1([y,z]) .
\]
{\sf (ii)} \ For any $x\in\g(2)$ and $u,v\in\g(-1)$, we have
$\Phi^{(2)}(\textsl{d}_1(x), u\wedge v)=-\Phi(x,\partial(u\wedge v))$.
\end{prop}
\begin{proof}
The argument is essentially the same as in the proof of the similar results for $\BZ_m$-gradings,
see Proposition 4.1 and Eq.~(4.4) in~\cite{jlms01}.
\end{proof}
The above assertion holds for any $\BZ$-grading. Below, we again assume that $\g(i)=\g_\ap(i)$ for
some $\ap\in\Pi$ and consider related eigenvalues of $\EuScript C_\ap(0)$.
\begin{prop} \label{prop:wedge-2}
Any $\EuScript C_\ap(0)$-eigenvalue in $\bigwedge^2\!\g_\ap(1)$ is at most $2\gamma_\ap(1)$.
The eigenspace for $2\gamma_\ap(1)$ is spanned by the bi-vectors $y\wedge z$ such that $[y,z]=0$.
\end{prop}
\begin{proof}
By Proposition~\ref{prop:d_1}, we have
\[
\EuScript C_\ap(0)(y\wedge z)=(\EuScript C_\ap(0)y)\wedge z+y\wedge (\EuScript C_\ap(0)z)+
2\sum_{i=1}^s [e_i,y]\wedge [e'_i,z]=2\gamma_\ap(1){\cdot}y\wedge z-2\textsl{d}_1([y,z]) .
\]
By~\cite[Prop.\,4]{ko65} and \cite[Theorem\,3.2]{jlms01}, the maximal eigenvalue of $\EuScript C_\ap(0)$
in $\bigwedge^k\!\g_\ap(1)$ is attained on decomposable
polyvectors. So, we may assume that $y\wedge z$ is a $\EuScript C_\ap(0)$-eigenvector.
\textbullet \quad Assume that $[y,z]\ne 0$. Since $[y,z]=\partial (y\wedge z)$ and $\partial$ is $\g$-equivariant, the $\EuScript C_\ap(0)$-eigenvalues of $y\wedge z$ and $[y,z]$ are equal. As
$[y,z]\in\g_\ap(2)$, its eigenvalue equals $\gamma_\ap(2)$. We also know that $\gamma_\ap(2) <2\gamma_\ap(1)$, see Proposition~\ref{prop:2gamma>1gamma}.
\textbullet \quad If $[y,z]= 0$, then $\EuScript C_\ap(0)(y\wedge z)=2\gamma_\ap(1){\cdot}y\wedge z$.
\end{proof}
Actually, there is a more precise result for $\bigwedge^2\!\g_\ap(1)$.
\begin{thm} \label{thm:wedge-2}
Let $\EuScript A_2=\text{\rm span}\{ y\wedge z\in \bigwedge^2\!\g_\ap(1)\mid [y,z]=0\}$. Then
\begin{itemize}
\item[\sf (i)] \ $\EuScript A_2=\Ker(\partial\vert_{\bigwedge^2\!\g_\ap(1)})$;
\item[\sf (ii)] \ $\bigwedge^2\!\g_\ap(1)=\EuScript A_2\oplus \textsl{d}_1(\g_\ap(2))$; hence
$\EuScript C_\ap(0)$ has at most two eigenvalues in $\bigwedge^2\!\g_\ap(1)$.
\end{itemize}
\end{thm}
\begin{proof}
{\sf (i)} \ If $u=\sum_i y_i\wedge z_i$, then
\[
\EuScript C_\ap(0)u=2\gamma_\ap(1){\cdot}u-
2\sum_i\textsl{d}_1([y_i,z_i])=2\gamma_\ap(1){\cdot}u-2\textsl{d}_1\partial(u).
\]
Therefore, if $\partial(u)=0$, then $u\in \EuScript A_2$ in view of Proposition~\ref{prop:wedge-2}. Hence
$\Ker(\partial\vert_{\bigwedge^2\!\g_\ap(1)})\subset \EuScript A_2$, and the opposite inclusion is obvious.
{\sf (ii)} \ Since $\textsl{d}_1: \g_\ap(2)\to \bigwedge^2\!\g_\ap(1)$ is $\g_\ap(0)$-equivariant,
the eignevalue of $\EuScript C_\ap(0)$ in $\textsl{d}_1(\g_\ap(2))$, which is $\gamma_\ap(2)$, is strictly less than
$2\gamma_\ap(1)$. Therefore, we have $\bigwedge^2\!\g_\ap(1)=\EuScript A_2\oplus \textsl{d}_1(\g_\ap(2))\oplus\mathbb U$
with some $\mathbb U$.
Then it follows from {\sf (i)} that
$\partial(\bigwedge^2\!\g_\ap(1))=\partial(\textsl{d}_1(\g_\ap(2))\oplus\partial\,\mathbb U$
and $\dim \partial(\textsl{d}_1(\g_\ap(2))=\dim (\textsl{d}_1(\g_\ap(2))$. Using Proposition~\ref{prop:d_1}(ii),
we see that $\dim (\textsl{d}_1(\g_\ap(2))=\dim \partial(\bigwedge^2\!\g_\ap(-1))=
\dim (\bigwedge^2\!\g_\ap(1))$. Hence $\partial\,\mathbb U=0$ and then $\mathbb U=0$.
\end{proof}
\begin{rmk} \label{rem:pro-A2}
It follows from Theorem~\ref{thm:wedge-2} that
\[ \textstyle
\bigwedge^2\!\g_\ap(1)\simeq \g_\ap(2) \Longleftrightarrow\ \EuScript A_2=0\ \Longleftrightarrow
\g_\ap(1) \text{ has no 2-dim abelian subspaces}.
\]
This possibility does materialise for the $(\BZ,\ap)$-gradings $(\GR{B}{n}, \ap_n)$ and $(\GR{G}{2},\ap_1)$.
\end{rmk}
The following is a natural generalisation of Proposition~\ref{prop:wedge-2}.
\begin{thm} \label{thm:main3}
For any $k\ge 1$, the maximal eigenvalue of $\EuScript C_\ap(0)$ in $\bigwedge^k\!\g_\ap(1)$ is at most $k\gamma_\ap(1)$.
This bound is attained if and only if $\g_\ap(1)$ contains a $k$-dimensional commutative subalgebra.
In that case, the eigenspace belonging to $k\gamma_\ap(1)$ is spanned by the polyvectors
$\bigwedge^k\!\ah$, where $\ah\subset\g_\ap(1)$ is $k$-dimensional and $[\ah,\ah]=0$.
\end{thm}
\begin{proof}
The proof of the similar result related to $\BZ_m$-gradings goes through {\sl mutatis mutandis\/}
(cf. \cite[Theorem\,4.4]{jlms01}). Following ideas of Kostant~\cite{ko65}, one has to use a compact real
form of $\g$ and a related Hermitian inner product on $\g$.
\end{proof}
Let $\EuScript A_k$ be the subspace of $\bigwedge^k\!\g_\ap(1)$ spanned by the ``$k$-dimensional
commutative subalgebras'', i.e.,
\[
\EuScript A_k=\text{\rm span}\{ y_1\wedge\ldots\wedge y_k\mid y_i\in\g_\ap(1) \ \& \ [y_i,y_j]=0
\ \forall i,j\} .
\]
Then $\EuScript A=\oplus_{k\ge 1}\EuScript A_k$ is a $\g_\ap(0)$-submodule of $\bigwedge^\bullet\!\g_\ap(1)$.
\begin{prop} \label{prop:mult-free}
$\EuScript A$ is a multiplicity-free\/ $\g_\ap(0)$-module.
\end{prop}
\begin{proof}
Set $\be_\ap(0)=\be\cap\g_\ap(0)$. If $\lb$ is a highest weight in $\EuScript A$ w.r.t. $\be_\ap(0)$, then
there is a $\be_\ap(0)$-stable abelian subspace $\ah\subset \g_\ap(1)$ such that the set of $\te$-roots
of $\ah$ is $\Delta_\ah=\{\mu_1,\dots,\mu_l\}$ and $\lb=\sum_{i=1}^l \mu_i$; and vice versa. The
$\be_\ap(0)$-invariance of $\ah$ means that $\Delta_\ah$ is an {\it upper $\Delta^+_\ap(0)$-ideal\/} in
the sense that if $\mu_i+\eta\in\Delta^+$ for some $\eta\in\Delta^+_\ap(0)$, then
$\mu_i+\eta\in\Delta_\ah$.
\\ \indent
Assume that there are two $\be_\ap(0)$-stable commutative subalgebras of $\g_\ap(1)$
whose dominant weights coincide. That is, $\ah\sim \Delta_\ah=\{\mu_1,\dots,\mu_l\}$,
$\ah'\sim \Delta_{\ah'}=\{\nu_1,\dots,\nu_m\}$, and $\sum \mu_i=\sum \nu_j$. Removing the common
elements of these two sets, we have
\[
|\Delta_\ah\setminus \Delta_{\ah'}|= |\Delta_{\ah'}\setminus \Delta_{\ah}|
\]
Hence $(|\Delta_\ah\setminus \Delta_{\ah'}|, |\Delta_{\ah'}\setminus \Delta_{\ah}|)>0$ and there are
$\mu_i\in\Delta_\ah\setminus \Delta_{\ah'} , \nu_j\in \Delta_{\ah'}\setminus \Delta_{\ah}$ such that
$(\mu_i,\nu_j)>0$. Then $\mu_i-\nu_j\in \Delta_\ap(0)$, since $\hot_\ap(\mu_i)=\hot_\ap(\nu_j)$.
If, for instance, $\nu_j-\mu_i$ is positive, then $\nu_j\in \Delta_{\ah}$. A contradiction!
\end{proof}
If $d_\ap=1$, then $\g_\ap(1)$ is commutative. Conversely, if $d_\ap\ge 2$, then $[\g_\ap(1),\g_\ap(1)]=\g_\ap(2)$,
i.e., $\g_\ap(1)$ is {\bf not} commutative. From our theory, we derive a more precise assertion.
\begin{thm} \label{thm:comm-dim}
For any $(\BZ{,}\ap)$-grading of $\g$, one has
\begin{itemize}
\item either\/ $\g_\ap(1)$ is abelian (which happens if and only if $d_\ap=1$);
\item or\/ $\dim\ah\le \frac{1}{2}\dim\g_\ap(1)$ for any abelian subspace $\ah\subset\g_\ap(1)$.
\end{itemize}
\end{thm}
\begin{proof} It suffices to prove that if $\g_\ap(1)$ contains an abelian subspace $\ah$ such that
$\dim\ah> \frac{1}{2}\dim\g_\ap(1)$, then $\g_\ap(1)$ is abelian.
Set $m=\dim\g_\ap(1)$ and let $\delta_\ap(k)$ be the maximal eigenvalue of $\EuScript C_\ap(0)$ on
$\bigwedge^k\!\g_\ap(1)$. Then
$\delta_\ap(1)=\gamma_\ap(1)$ and we have proved that $\delta_\ap(k)\le k\gamma_\ap(1)$ for any $k$.
Note that $\delta_\ap(m)$ is just the eigenvalue on the $1$-dimensional module
$\bigwedge^m\!\g_\ap(1)$.
Write $\g_\ap(0)=\es\oplus \langle h_\ap\rangle$, where $\es$ is semisimple and $h_\ap$ is the element of the centre that has the eigenvalue $k$ on $\g_\ap(k)$. Then $h_\ap$ also has
eigenvalue $k$ on $\bigwedge^k\!\g_\ap(1)$. We have
\beq \label{eq:casimir-summa}
\EuScript C_\ap(0)=\EuScript C_\es+ h_\ap h'_\ap=\EuScript C_\es+ h^2_\ap /(h_\ap,h_\ap) .
\eeq
Since $\bigwedge^k\!\g_\ap(1)$ and $\bigwedge^{m-k}\!\g_\ap(1)$ are isomorphic as $\es$-modules,
their $\EuScript C_\es$-eigenvalues coincide; the difference in $\EuScript C_\ap(0)$-eigenvalues comes the presence of the last summand.
An easy observation is that if the summand $h^2_\ap /(h_\ap,h_\ap)$ has the eigenvalue $\chi$ on
$\g_\ap(1)$, then its eigenvalue on $\bigwedge^k\!\g_\ap(1)$ is $k^2\chi$.
Assume that $k< m/2$ and $\g_\ap(1)$ has a commutative subalgebra of dimension
$m-k$. Then it has a $k$-dimensional commutative subalgebra, too. Hence
$\delta_\ap(k)=k\gamma_\ap(1)$ and $\delta_\ap(m-k)=(m-k)\gamma_\ap(1)$.
Let $F_i$ be the maximal eigenvalue of $\EuScript C_\es$ on $\bigwedge^i\!\g_\ap(1)$. Then
$F_i=F_{m-i}$ and, using
the decomposition in~\eqref{eq:casimir-summa}, we can write \\
\centerline{
$\begin{cases} \delta_\ap(k)& =F_k+k^2\chi=k\gamma_\ap(1), \\
\delta_\ap(m-k)& =F_{k}+(m-k)^2\chi=(m-k)\gamma_\ap(1) . \end{cases}$}
\\[.6ex]
Taking the difference yields $m(m-2k)\chi=(m-2k)\gamma_\ap(1)$. Note also that $F_0=F_m=0$, since
$\bigwedge^m\!\g_\ap(1)$ is a trivial $\es$-module. Hence
$\delta_\ap(m)=m^2\chi=m\gamma_\ap(1)$. By Theorem~\ref{thm:main3}, this means that
$\g_\ap(1)$ is commutative.
\end{proof}
\begin{rmk} \label{rem:pro-alela}
It was recently noticed that, for any nilpotent element $e\in\g$ and the associated Dynkin $\BZ$-grading (so that $e\in\g(2)$), one has $\dim\ah\le (\dim\g(1))/2$ whenever $\ah\subset\g(1)$ and $[\ah,\ah]=0$,
see~\cite[Prop.\,3.1]{e-j-k}. In this case, $\dim\g(1)$ is necessarily even.
However, there are $(\BZ{,}\ap)$-gradings of height $\ge 2$ that are not Dynkin gradings, and it can
also happen that $\dim\g_\ap(1)$ is odd.
\\ \indent
In fact, conversations with A.G.\,Elashvili on results of~\cite{e-j-k} revived my memory of~\cite{jlms01}
and triggered my interest to eigenvalues of the Casimir elements related to Levi subalgebras and
$\BZ$-gradings.
\end{rmk}
\begin{rmk} \label{rem:ne-vsegda-polovina}
For an arbitrary $\BZ$-grading of $\g$, it can happen that $\g(1)$ is not abelian, but
$\dim\ah> \frac{1}{2}\dim\g(1)$ for some abelian $\ah\subset\g(1)$. Suppose that a $\BZ$-grading
$\g=\bigoplus_{i\in\BZ}\g(i)$ is given by a function $\ff:\Pi\to\{0,1\}$, i.e., $\g^\ap\subset \g(\ff(\ap))$, cf.
~\cite[Ch.\,3,\,\S3.5]{t41}. Set $\Pi_1=\{\ap\mid \ff(\ap)=1\}$.
Then $\g(1)=\bigoplus_{\ap\in\Pi_1}\EuScript V(\ap)$, where $\EuScript V(\ap)$ is a simple $\g(0)$-module with
{\bf lowest} weight $\ap$. The set of weights of $\EuScript V(\ap)$ is
\[
\{\gamma\in\Delta^+ \mid \hot_\ap(\gamma)=1 \ \& \ \hot_\beta(\gamma)=0 \ \ \forall \beta\in\Pi_1\setminus\{\ap\}\} .
\]
Take, for instance, $\Pi_1=\{\ap_2,\ap_4\}$ for $\GR{A}{6}$. Then both $\EuScript V(\ap_2)$ and
$\EuScript V(\ap_4)$ are
abelian, of different dimension, but $0\ne [\EuScript V(\ap_2), \EuScript V(\ap_4)]=\EuScript V(\ap_2+\ap_3+\ap_4)\subset \g(2)$. Here
$6=\dim \EuScript V_{\ap_4}> \frac{1}{2}\dim\g(1)=5$.
\end{rmk}
Below, we elaborate on some numerology related to the numbers $q_\ap(i), \dim\g_\ap(i),
(\vp_\ap,\vp_\ap)$, $\delta_\ap(m)$,
etc. Recall that $m=\dim\g_\ap(1)$.
Let $C=(c_{\ap\beta})_{\ap,\beta\in\Pi}$ be the inverse of the Cartan matrix of $\g$. Then $\vp_\ap=\sum_{\beta\in\Pi} c_{\ap\beta}\beta$. Therefore, $(\vp_\ap,\vp_\ap)=c_{\ap\ap}(\vp_\ap,\ap)=
c_{\ap\ap}(\ap,\ap)/2=c_{\ap\ap}/2h^*r_\ap$.
\begin{prop} \label{prop:delta-top}
For any $\ap\in\Pi$, we have $\delta_\ap(m)=q_\ap(1)^2 (\vp_\ap,\vp_\ap)$.
\end{prop}
\begin{proof}
The weight of the $1$-dimensional $\g_\ap(0)$-module $\bigwedge^m\!\g_\ap(1)$ is $|\Delta_\ap(1)|=q_\ap(1)\vp_\ap$. Since
$(\vp_\ap,2\rho)=q_\ap(\vp_\ap,\vp_\ap)$, we obtain
\begin{multline*}
\delta_\ap(m)=(q_\ap(1)\vp_\ap,q_\ap(1)\vp_\ap+2\rho_\ap(0))=
q_\ap(1)(\vp_\ap, |\Delta_\ap(0)|+|\Delta_\ap(1)|)
\\ =q_\ap(1)(\vp_\ap, 2\rho- \sum_{i\ge 2}|\Delta_\ap(i)|)
=q_\ap(1)(\vp_\ap, 2\rho- (q_\ap-q_\ap(1))\vp_\ap)
\\ =q_\ap(1) (\vp_\ap,\vp_\ap){\cdot}\bigl(q_\ap-(q_\ap-q_\ap(1))\bigr)=
q_\ap(1)^2{\cdot} (\vp_\ap,\vp_\ap) . \qedhere
\end{multline*}
\end{proof}
\begin{cl} \label{cor:delta-top-ab} \leavevmode\par
{\sf (i)} \ If\/ $\vp_\ap$ is cominuscule, then $(\vp_\ap,\vp_\ap)=m/2(h^*)^2$, $\delta_\ap(m)=m/2$,
and $m=c_{\ap\ap}h^*$.
\\ \indent
{\sf (ii)} \ If\/ $\theta$ is fundamental and $(\widehat\ap,\theta)\ne 0$, then
$(\vp_{\widehat\ap},\vp_{\widehat\ap})=1/h^*$ and $\delta_{\widehat\ap}(m)=(h^*-2)^2/h^*$.
\end{cl}
\begin{proof}
{\sf (i)} \ Here $r_\ap=1$ and $\g_\ap(1)$ is commutative, hence $q_\ap=q_\ap(1)=h^*$,
$\gamma_\ap(1)=1/2$, and $\delta_\ap(k)=k/2$ for every $k$. Then $(h^*)^2(\vp_\ap,\vp_\ap)=m/2$, and we are done.
\\ \indent
{\sf (ii)} \ Here $q_{\widehat\ap}(1)=h^*-2$ (cf. the proof of Corollary~\ref{cor:2.2}(iii)) and
$\vp_{\widehat\ap}=\theta$, i.e., $(\vp_{\widehat\ap},\vp_{\widehat\ap})=1/h^*$. Note also that here $c_{\widehat\ap\widehat\ap}=\hot_{\widehat\ap}(\theta)=2$.
\end{proof}
\begin{prop} \label{prop:hot_ap-etc}
For any $\ap\in\Pi$ and $k\in\BN$, we have
\[
\hot_\ap(|\Delta_\ap(k)|)=k{\cdot} \dim\g_\ap(k)=c_{\ap\ap} q_\ap(k) .
\]
\end{prop}
\begin{proof}
Since $(\nu,\vp_\ap)=k(\ap,\vp_\ap)$ for any $\nu\in \Delta_\ap(k)$ and $\dim\g_\ap(k)=\#
\Delta_\ap(k)$, we have
$(|\Delta_\ap(k)|,\vp_\ap)=k{\cdot} \dim\g_\ap(k)(\ap,\vp_\ap)$. On the other hand,
\[
(|\Delta_\ap(k)|,\vp_\ap)=\bigl(\hot_\ap(|\Delta_\ap(k)|)\ap+\dots,\vp_\ap\bigr)=
\hot_\ap(|\Delta_\ap(k)|){\cdot}(\ap,\vp_\ap) ,
\]
which gives the first equality. Likewise,
\[
(|\Delta_\ap(k)|,\vp_\ap)=q_\ap(k){\cdot}(\vp_\ap,\vp_\ap)=q_\ap(k){\cdot}c_{\ap\ap}(\ap,\vp_\ap) . \qedhere
\]
\end{proof}
\begin{cl}
The ratio $\bigl(k{\cdot}\dim\g_\ap(k)\bigr)/q_\ap(k)=c_{\ap\ap}$ does not depend on $k$. In particular, any linear
relation between the $q_\ap(i)$'s translates into a linear relation between the $\dim\g_\ap(i)$'s.
\end{cl}
{\bf Example}. By Proposition~\ref{prop:simmetri-d_ap}, one has $q_\ap(d-i)=q_\ap(i)$. Hence
$(d-i){\cdot}\dim\g_\ap(d-i)=i{\cdot}\dim\g_\ap(i)$.
In particular, $\dim\g_\ap(d-1)=\dim\g_\ap(1)/(d-1)$.
\section{Maximal abelian subspaces and applications}
\label{sect:applic}
\noindent
We say that $\g_\ap(1)$ has an {\it abelian subspace of half-dimension}, if there is
$\ah\subset\g_\ap(1)$ such that $[\ah,\ah]=0$ and $\dim\ah=\frac{1}{2}\dim \g_\ap(1)$. First we
discuss some consequences of this property.
\begin{prop} \label{prop:all-sigma(i)}
Suppose that $\g_\ap(1)$ has an abelian subspace of half-dimension, $m=\dim\g_\ap(1)$, and
$k\le m/2$. Then $\delta_\ap(k)=k\gamma_\ap(1)$ and
\beq \label{eq:delta1(m-k)}
\delta_\ap(m-k)=k\gamma_\ap(1)+(m-2k)\frac{q_\ap(1)}{2h^*r_\ap} .
\eeq
\end{prop}
\begin{proof}
The first relation follows from Theorem~\ref{thm:main3}. Next, we know that
$\delta_\ap(k)=F_k+k^2\chi$, see the proof of Theorem~\ref{thm:comm-dim}. Hence
$F_k=k\gamma_\ap(1)-k^2\chi$. Since $F_k=F_{m-k}$, we obtain
\beq \label{eq:delta2(m-k)}
\delta_\ap(m-k)=F_{m-k}+(m-k)^2\chi=F_{k}+(m-k)^2\chi=k\gamma_\ap(1)+m(m-2k)\chi .
\eeq
As $F_0=F_m=0$, we compute $\chi$ and $\delta_\ap(m)$ using the numerology of Section~\ref{sect:3}:
\[
m^2\chi=\delta_\ap(m)=q_\ap(1)^2{\cdot}(\vp_\ap,\vp_\ap)=
q_\ap(1)^2{\cdot}c_{\ap\ap}{\cdot}(\ap,\ap)/2=\frac{m q_\ap(1)}{2h^*r_\ap} .
\]
Here the relation $q_\ap(1){\cdot} c_{\ap\ap}=\dim\g_\ap(1)$ is used, see Prop.~\ref{prop:hot_ap-etc}. Thus,
$m\chi=\displaystyle \frac{q_\ap(1)}{2h^*r_\ap}$ and plugging this into Eq.~\eqref{eq:delta2(m-k)}
yields Eq.~\eqref{eq:delta1(m-k)}.
\end{proof}
{\bf Remark.} We obtain here a formula for $\delta_\ap(i)$ for {\bf all} $i\in\{1,\dots,m\}$. More generally, if $\max(\dim\ah)=r\le m/2$, then the same argument yields $\delta_\ap(i)$ for $i\le r$ and $i\ge m-r$.
\begin{cl} \label{prop:gorka-versus}
Under the above assumptions,
\begin{itemize}
\item[\sf (i)] \ if $q_\ap > 2q_\ap(1)$, then $\max_{i}\{\delta_\ap(i)\}=\delta_\ap(m/2)$ and the sequence
$\{\delta_\ap(i)\}$ is unimodal;
\item[\sf (ii)] \ if $q_\ap= 2q_\ap(1)$, then the sequence $\{\delta_\ap(i)\}$ stabilises after $i=m/2$;
\item[\sf (iii)] \ if $q_\ap < 2q_\ap(1)$, then $\max_{i}\{\delta_\ap(i)\}=\delta_\ap(m)$ and the sequence
$\{\delta_\ap(i)\}$ strictly increases.
\end{itemize}
Furthermore, if $d_\ap\ge 3$, then case {\sf (i)} always occurs.
\end{cl}
\begin{proof}
The sequence $\{\delta_\ap(i)\}$ clearly increases for $1\le i\le m/2$.
By Theorem~\ref{thm:s-znach-1}, one has $\gamma_\ap(1)=q_\ap/2h^*r_\ap$. Hence the coefficient of
$k$ in Eq.~\eqref{eq:delta1(m-k)} equals $(q_\ap-2q_\ap(1))/2h^*r_\ap$. This settles {\sf (i)--(iii)}.
\\
By Proposition~\ref{prop:simmetri-d_ap}, $q_\ap(1)=q_\ap(d_\ap-1)$ for $d_\ap\ge 2$. Hence $q_\ap>2q_\ap(1)$ whenever $d_\ap\ge 3$.
\end{proof}
\begin{ex} \label{ex:starshe-korn}
1) If $d_\ap=2$, then all three possibilities may occur, cf. the good cases $(\GR{D}{n}, \ap_i)$ for
$2\le i\le n-2$ and sufficiently large $n$. (Use data from Table~\ref{table:d=2c}.)
\\ \indent
2) Suppose that $\theta$ is fundamental, i.e., $\theta=\vp_{\widehat\ap}$.
Then $\widehat\ap\in\Pi_l$, $d_{\widehat\ap}=2$, and $\Delta_{\widehat\ap}(2)=\{\theta\}$. Since
$\g_{\widehat\ap}({\ge }1)$ is a Heisenberg Lie algebra, see~\cite[Sect.\,2]{jos76}, $\g_{\widehat\ap}(1)$ has an abelian subspace of half-dimension
and the above computation applies. Here $m=2h^*-4$, $q_{\widehat\ap}=h^*-1$,
and $q_{\widehat\ap}(2)=1$. Since $\theta$ is fundamental,
$h^*\ge 4$ and hence $q_{\widehat\ap} < 2q_{\widehat\ap}(1)$. Then $\gamma_{\widehat\ap}(1)=
(h^*-1)/2h^*$ and $\chi=1/(4h^*)$. Thus, for $k\le m/2=h^*-2$, we obtain
$\delta_{\widehat\ap}(k)=\displaystyle k\frac{(h^*-1)}{2h^*}$ \ and
\[
\delta_{\widehat\ap}(m-k)=k\frac{(h^*-1)}{2h^*}+\frac{m(m-2k)}{4h^*}=
\frac{(h^*-2)^2}{h^*} - k{\cdot}\frac{h^*-3}{2h^*} .
\]
3) For $\GR{C}{n}$ and $n\ge 2$, we have $\theta=2\vp_1$ and $\widehat\ap=\ap_1$ is short. Here
one computes that $2q_{\ap_1}(1)>q_{\ap_1}$ for $n>2$ and $\chi=1/(4h^*)$.
\end{ex}
Another application of our theory, especially of Theorem~\ref{thm:main3}, is the following result.
\begin{thm} \label{thm:abel-complem}
If\/ $\ah\subset\g_\ap(1)$ is an abelian subspace and $\dim\ah=(1/2)\dim\g_\ap(1)$, then there is an
abelian subspace $\tilde\ah$ such that $\ah\oplus\tilde\ah=\g_\ap(1)$.
\end{thm}
\begin{proof}
As above, $m=\dim\g_\ap(1)$ and $\be_\ap(0)=\be\cap\g_\ap(0)$.
\\ \indent
1. Assume that $\ah$ is a $\be_\ap(0)$-{\it stable\/} abelian subspace. In particular, $\ah$ is
$\te$-stable. Therefore, there is a unique $\te$-stable complement $\tilde\ah$ to $\ah$ in $\g_\ap(1)$.
Then $\tilde\ah$ is $\be_\ap(0)^-$-stable.
Choose nonzero (poly)vectors $y\in\bigwedge^{m/2}\ah$ and $\tilde y\in\bigwedge^{m/2}\tilde\ah$.
Then $y$ (resp. $\tilde y$) is a highest (resp. lowest) weight vector in the $\g_\ap(0)$-module
$\bigwedge^{m/2}\g_\ap(1)$.
If $\mathsf{wt}(\cdot)$ stands for the $\te$-weight of a (poly)vector, then
\[
\mathsf{wt}(y) + \mathsf{wt}(\tilde y)=|\Delta_\ap(1)|=q_\ap(1)\vp_\ap .
\]
As was already computed in Proposition~\ref{prop:delta-top},
\[
\delta_\ap(m)=(q_\ap(1)\vp_\ap, q_\ap(1)\vp_\ap+2\rho_\ap(0))=q_\ap(1)^2(\vp_\ap,\vp_\ap)=\frac{mq_\ap(1)}{2h^*r_\ap} .
\]
Since $y$ is a highest weight vector in $\bigwedge^{m/2}\g_\ap(1)$ and $\ah$ is an abelian subspace,
\[
\EuScript C_\ap(0)(y)=(\mathsf{wt}(y),\mathsf{wt}(y)+2\rho_\ap(0)){\cdot}y=\frac{m}{2}\gamma_\ap(1){\cdot}y.
\]
On the other hand, $\mathsf{wt}(\tilde y)$ is anti-dominant w.r.t. $\be_\ap(0)$. Hence the weight
$w_{\ap,0}(\mathsf{wt}(\tilde y))$ is already dominant and the $\EuScript C_\ap(0)$-eigenvalue of
$\tilde y$ equals
\begin{multline*}
\bigl(w_{\ap,0}(\mathsf{wt}(\tilde y)),w_{\ap,0}(\mathsf{wt}(\tilde y))+2\rho_\ap(0)\bigr)=
\bigl(\mathsf{wt}(\tilde y),\mathsf{wt}(\tilde y)-2\rho_\ap(0)\bigr) \\
=(q_\ap(1)\vp_\ap-\mathsf{wt}(y), q_\ap(1)\vp_\ap-\mathsf{wt}(y)-2\rho_\ap(0)) \\
=q_\ap(1)^2(\vp_\ap,\vp_\ap)-2\bigl(q_\ap(1)\vp_\ap, \mathsf{wt}(y)+\rho_\ap(0)\bigr)+\bigl(\mathsf{wt}(y),\mathsf{wt}(y)+2\rho_\ap(0)\bigr)\\
= \frac{mq_\ap(1)}{2h^*r_\ap}-2\bigl(q_\ap(1)\vp_\ap, \mathsf{wt}(y)\bigr)+\frac{m}{2}\gamma_\ap(1)\\
=\frac{mq_\ap(1)}{2h^*r_\ap}-2q_\ap(1){\cdot}\frac{m}{2}{\cdot}(\vp_\ap,\ap)+\frac{m}{2}\gamma_\ap(1)
= \frac{m}{2}\gamma_\ap(1) .
\end{multline*}
Here we used the facts that $(\vp_\ap,\rho_\ap(0))=0$ and
$(\vp_\ap,\gamma)=(\vp_\ap,\ap)=(\ap,\ap)/2$ for any
$\gamma\in\Delta_\ap(1)$. By Theorem~\ref{thm:main3}, the equality
$\EuScript C_\ap(0)(\tilde y)=\frac{m}{2}\gamma_\ap(1){\cdot}\tilde y$ for an $m/2$-vector $\tilde y$
means that the $m/2$-dimensional subspace $\tilde \ah$ is abelian.
\\ \indent
2. If $\ah$ is not $\be_\ap(0)$-stable, then we consider the $B_\ap(0)$-orbit of $\{\ah\}$ in the
Grassmannian of $m/2$-dimensional subspaces of $\g_\ap(1)$. By the Borel fixed-point theorem, the
closure of this orbit contains a $B_\ap(0)$-fixed point, i.e., a $\be_\ap(0)$-stable (abelian) subspace,
say $\ah_1$. If $\tilde\ah_1$ is the complementary abelian subspace for $\ah_1$, as in part~1, then,
by continuity, it is also a complementary subspace for some element of the orbit
$B_\ap(0){\cdot}\{\ah\}$.
\end{proof}
Previous results show that it is helpful to know whether $\g_\ap(1)$ has an abelian
subspace of half-dimension, if $d_\ap{>}1$. We say that $\ap\in\Pi$ is {\it good} if this is the case;
otherwise, $\ap$ is {\it bad}. In many cases, a $(\BZ,\ap)$-grading is also the Dynkin grading associated
with a {\it strictly odd\/} nilpotent element of $\g$, see~\cite[Sect.\,1]{e-j-k}. Then the relevant good cases
have been determined in~\cite{e-j-k}. However, some work is still needed for the $(\BZ,\ap)$-gradings that are not Dynkin. For instance, if $\g$ is exceptional, then one has to handle the
possibilities $(\GR{E}{7}, \ap_3 \text{ or } \ap_7)$ and $(\GR{E}{6}, \ap_2 \text{ or } \ap_4)$. Combining
our computations with \cite{e-j-k}, we describe below the bad cases for all $\g$. For each bad case, the maximal dimension of an abelian subspace, $\dim\ah_{\sf max}$, is given.
Note that in order to compute $\dim\ah_{\sf max}$, it suffices to consider only $\be_\ap(0)$-stable
abelian subspaces of $\g_\ap(1)$, cf. part 2) in the proof of Theorem~\ref{thm:abel-complem}.
\textbullet\ For the classical series, we have $d_\ap\le 2$. If $\g=\spn$ or $\sone$, then
all $\ap\in\Pi$ with $d_\ap=2$ are good. If $\g=\sono$, $n\ge 3$, then the bad cases occur for $\ap_i$ with $3\le i\le n$. Here $\dim\g_{\ap_i}(1)=2i(n-i)+i$
and $\dim\ah_{\sf max}=i(n-i)+1$. Note also that, for $\sone$ and $\sono$, the $(\BZ,\ap_i)$-grading is
Dynkin and associated with a strictly odd nilpotent if and only if $i$ is even (and $d_{\ap_i}=2$).
\textbullet\ For the exceptional algebras, we gather the bad cases in Table~\ref{table:bad-cases},
where we write $m_\ap$ for $\dim\g_\ap(1)$.
\begin{table}[ht]
\caption{Exceptional Lie algebras, the bad cases} \label{table:bad-cases}
\begin{center}
\begin{tabular}{>{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} c ||>{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} >{$}c<{$} c | }
\g & \ap & d_\ap & m_\ap & \dim\ah_{\sf max} & \cite{e-j-k} & \g & \ap & d_\ap & m_\ap & \dim\ah_{\sf max}
& \cite{e-j-k}\\ \hline \hline
\GR{E}{7} & \ap_3 & 3 & 30 &12 & -
& \GR{E}{8} & \ap_3 & 4 & 48 &16 & + \\
& \ap_7 & 2 & 35 &15 & -
& & \ap_4 & 5 & 40 & 16 & + \\ \cline{1-6}
\GR{F}{4} & \ap_1 & 2 & 8 & 2 & +
& & \ap_7 & 2 & 64 &22 & + \\
& \ap_2 & 4 & 6 & 2 & +
& & \ap_8 & 3 & 56 & 21 & +\\ \hline
\end{tabular}
\end{center}
\end{table}
The data in Table~\ref{table:bad-cases} also mean that the non-Dynkin cases
$(\GR{E}{6}, \ap_2 \text{ or } \ap_4)$ are good, cf. also Example~\ref{F4-ap1}(2).
The signs $+/-$ indicate whether that item represents a Dynkin grading (=\,is considered
in~\cite{e-j-k}).
Our methods for constructing abelian subspaces of $\g(1)$, partly for arbitrary $\BZ$-gradings, are
described below. This provides another approach to some of calculations in~\cite{e-j-k} and also
natural descriptions of abelian subspaces of maximal dimension.
\begin{lm} \label{lm:ab-for-d=2}
Let $\g=\bigoplus_{i=-2}^2\g(i)$ be a $\BZ$-grading of height~2 and $\be(0)$ a Borel
subalgebra of\/ $\g(0)$. If\/ $\ah$ is a $\be(0)$-stable abelian subspace of\/ $\g(1)$, then
$\ah\oplus\g(2)$ is an abelian $\be$-ideal of\/ $\g$, where $\be=\be(0)\oplus\g(1)\oplus\g(2)$.
In particular, if\/ $\dim\ah$ is maximal, then $\ah\oplus\g(2)$ is a maximal abelian $\be$-ideal.
\end{lm}
Since the maximal abelian $\be$-ideals are known \cite[Sect.\,4]{adv01}, one readily obtains an upper
bound on $\dim\ah$. Actually, this allows us to determine $\dim\ah_{\sf max}$ for all $\BZ$-gradings of
height~$2$. The next observation applies to $(\BZ,\ap)$-gradings of any height.
\begin{prop} \label{prop:d_ap-&-d_beta}
Given $\ap\in\Pi$, suppose that $\g_\ap(2)\cap \g_\beta(2)=\{0\}$ for some $\beta\in\Pi$. Then
$\ah_{\ap,\beta}:=\g_\ap(1)\cap\g_\beta(1)$ is abelian. Moreover, if also $\g_\ap(1)\cap \g_\beta(2)=\{0\}$,
then $\ah_{\ap,\beta}$ is a $\be_\ap(0)$-stable abelian subspace of $\g_\ap(1)$.
\end{prop}
The proof is straightforward and left to the reader.
\noindent
There are interesting instances of such phenomenon and we provide below some illustrations
to our method. It turns out {\sl a posteriori} that the two assumptions of the above proposition imply
that $d_\beta<d_\ap$. However, even if Proposition~\ref{prop:d_ap-&-d_beta} applies, then the
abelian subspace $\ah_{\ap,\beta}$ does not necessarily have the maximal dimension.
\begin{rmk}
\label{ex:instances}
{\sf (i)} \ Note that $\g_\ap(1)\cap\g_\beta(1)\ne\{0\}$ for {\bf all} pairs
$\{\ap,\beta\}\subset\Pi$. For, take the unique chain in the Dynkin diagram joining $\ap$
and $\beta$. The sum of simple roots in this chain is a root, denoted by $\mu_{\ap,\beta}$, and
it is clear that $\mu_{\ap,\beta}\in \Delta_\ap(1)\cap\Delta_\beta(1)$. Clearly,
$\h=\g_\ap(0)\cap\g_\beta(0)$ is a Levi subalgebra in $\p_\ap\cap \p_\beta$ and the set of simple roots of $\h$ is $\Pi\setminus\{\ap,\beta\}$.
By~\cite[Theorem\,0.1]{ko10} (cf. Section~\ref{subs:Z-grad-versus}),
$\g_\ap(i)\cap\g_\beta(j)$ is a simple $\h$-module
for any $(i,j)$. Obviously, $\mu_{\ap,\beta}$ is the lowest weight of the $\h$-module $\ah_{\ap,\beta}$, so it is an easy task to compute $\dim\ah_{\ap,\beta}$ for any pair $\{\ap,\beta\}\subset\Pi$.
\\ \indent
{\sf (ii)} \ if $d_\beta=1$, then $\g_\beta(1)$ is a maximal abelian $\be$-ideal and the assumptions of
Proposition~\ref{prop:d_ap-&-d_beta} are satisfied.
\\ \indent
{\sf (iii)} \ Another possibility for applying Proposition~\ref{prop:d_ap-&-d_beta} is that in which
$d_\ap\ge 3$ (hence $\g$ is exceptional) and
$\beta=\widehat\ap$ is the unique simple root such that $(\theta,\widehat\ap)\ne 0$. Then
$\Delta_{\widehat\ap}(2)=\{\theta\}$, while $\hot_\ap(\theta)\ge 3$. Hence
$\Delta_{\widehat\ap}(2)\cap(\Delta_\ap(1)\cup\Delta_\ap(2))=\varnothing$.
\end{rmk}
\begin{ex} \label{ex:illustr}
1) Let $\theta$ be a multiple of a fundamental weight (i.e., $\Delta$ is not of type $\GR{A}{n}$,
$n\ge 2$) and, as usual, $(\theta,\widehat\ap)\ne 0$. For the
$(\BZ,\widehat\ap)$-grading, one has $d_{\widehat\ap}=2$,
$\g_{\widehat\ap}(2)=\g^\theta$, and $\g_{\widehat\ap}({\ge}1)$ is a Heisenberg Lie algebra. Here
$\dim\g_{\widehat\ap}(1)=2h^*-4$ and it follows from~\cite[Sect.\,3]{jems} that, for any maximal
abelian $\be$-ideal $\EuScript I$, we have $\dim\bigl(\EuScript I\cap\g_{\widehat\ap}({\ge}1)\bigr)=h^*-1$. Hence
$\dim\bigl(\EuScript I\cap\g_{\widehat\ap}(1)\bigr)=h^*-2=(1/2)\dim\g_{\widehat\ap}(1)$. Thus,
$\g_{\widehat\ap}(1)\cap\EuScript I$ is an abelian $\be_{\widehat\ap}(0)$-stable subspace of $\g_\ap(1)$
of half-dimension for {\bf any} maximal abelian ideal $\EuScript I$. Actually, different $\EuScript I$'s yield
different subspaces $\g_{\widehat\ap}(1)\cap\EuScript I$.
\\ \indent
2) If $\g$ is exceptional, then $\widehat\ap$ is an {\bf extreme} root in the Dynkin diagram. Let $\ap\in\Pi$ be the unique root adjacent to $\widehat\ap$. Then
$1=(\theta,{\widehat\ap}^\vee)=d_{\widehat\ap}(\widehat\ap,{\widehat\ap}^\vee)+d_\ap(\ap,{\widehat\ap}^\vee)=4-d_\ap$.
Hence $d_\ap=3$ and therefore $\ah_{\ap,\widehat\ap}$ is a $\be_\ap(0)$-stable abelian subspace of
$\g_\ap(1)$, cf. Proposition~\ref{prop:d_ap-&-d_beta} and Remark~\ref{ex:instances}(iii). We claim that
$(\g,\ap)$ is a good case.
For, in this case, $\g_\ap(0)'=\tri\dotplus\q$, where $\tri$ corresponds to $\widehat\ap$ and the simple
roots of the semisimple algebra $\q$ are $\Pi\setminus\{\ap,\widehat\ap\}$. Here $\g_\ap(1)\simeq \Bbbk^2\otimes V$ as
$\g_\ap(0)'$-module, where
$\Bbbk^2$ is the standard $\tri$-module and $V$ is a $\q$-module. Therefore, if
$\gamma\in\Delta_\ap(1)$, then $\hot_{\widehat\ap}(\gamma)\in\{0,1\}$; and if
$\hot_{\widehat\ap}(\gamma)=0$, then $\gamma+\widehat\ap\in \Delta_\ap(1)$, and vice versa.
It follows that $\Delta_\ap(1)\cap \Delta_{\widehat\ap}(1)=
\{\gamma\in\Delta_\ap(1)\mid \hot_{\widehat\ap}(\gamma)=1\}$ contains exactly half of the roots in
$\Delta_\ap(1)$. Thus, $\dim\ah_{\ap,\widehat\ap}=\frac{1}{2}\dim\g_\ap(1)$.
\\ \indent
3) For $\GR{E}{n}$, one verifies that if $\beta\in\Pi$ is {\bf any} extreme root of the Dynkin diagram and
$\ap$ is the unique root adjacent to $\beta$, then $\ah_{\ap,\beta}$ is abelian and
$\dim\ah_{\ap,\beta}=\frac{1}{2}\dim\g_\ap(1)$. The last equality is again explained by the fact that here
$\g_{\ap}(0)'\simeq\tri\dotplus\q$ and $\g_\ap(1)\simeq \Bbbk^2\otimes V$.
\end{ex}
\begin{ex} \label{F4-ap1}
1) For $(\GR{F}{4},\ap_1)$, we have $d_{\ap_1}=2$, $\dim\g_{\ap_1}(1)=8$, and $\dim\g_{\ap_1}(2)=7$.
If $\EuScript I$ is an abelian $\be$-ideal, then $\dim \EuScript I\le 9$. Hence $\dim\ah\le 9-7=2$. Actually,
$\dim(\EuScript I\cap\g_{\ap_1}(1))=2$, if $\dim\EuScript I=9$.
\\ \indent
2) For $(\GR{E}{6},\ap_2)$, we have $d_{\ap_2}=2$ and $\dim\g_{\ap_2}(1)=20$. Here $d_{\ap_1}=1$
and hence $\g_{\ap_1}(1)$ is a (maximal) abelian $\be$-ideal. Since
$\dim (\g_{\ap_1}(1)\cap\g_{\ap_2}(1))=10$, this is a good case.
\\ \indent
3) For $(\GR{E}{7},\ap_7)$, we have $d_{\ap_7}=2$, $\dim\g_{\ap_7}(1)=35$, and $\dim\g_{\ap_7}(2)=7$.
Here $d_{\ap_1}=1$ and $\g_{\ap_1}(1)$ is the maximal abelian ideal of maximal dimension $27$. In this
case, $\Pi\setminus \{\ap_1,\ap_7\}$ is the Dynkin diagram of type $\GR{A}{5}$ and
$\g_{\ap_7}(1)\cap\g_{\ap_1}(1)$ is the simple $SL_6$-module $\bigwedge^2(\Bbbk^6)$, of dimension 15.
The minimal (resp. maximal) root in $\Delta_{\ap_7}(1)\cap\Delta_{\ap_1}(1)$ is
\raisebox{-1.7ex}{\begin{tikzpicture}[scale= .7, transform shape]
\node (a) at (0,0) {\bf 1};
\node (b) at (.3,0) {\bf 1};
\node (c) at (.6,0) {\bf 1};
\node (d) at (.9,0) {\bf 1};
\node (e) at (1.2,0) {\bf 0};
\node (f) at (1.5,0) {\bf 0};
\node (g) at (.9,-.5) {\bf 1};
\end{tikzpicture}} (resp.
\raisebox{-1.7ex}{\begin{tikzpicture}[scale= .7, transform shape]
\node (a) at (0,0) {\bf 1};
\node (b) at (.3,0) {\bf 2};
\node (c) at (.6,0) {\bf 3};
\node (d) at (.9,0) {\bf 3};
\node (e) at (1.2,0) {\bf 2};
\node (f) at (1.5,0) {\bf 1};
\node (g) at (.9,-.5) {\bf 1};
\end{tikzpicture}}\!\!).
For the other maximal abelian ideals $\EuScript I$, one obtains
$\dim(\g_{\ap_1}(1)\cap \EuScript I)\le 15$.
\end{ex}
\begin{rmk} \label{rem:vsegda-if-d>3}
If $\g$ is exceptional and $d_\ap\ge 3$, then one can always find $\beta\in\Pi$ such that $d_\beta<d_\ap$, Proposition~\ref{prop:d_ap-&-d_beta} applies, and $\ah_{\ap,\beta}$ has the required dimension,
i.e., $(1/2)\dim\g_\ap(1)$ in the good cases and the numbers $\dim\ah_{\sf max}$ from
Table~\ref{table:bad-cases} in the bad cases.
For instance, one takes
\textbullet\
for $\GR{E}{6}$: \ $\beta=\ap_6$ if $\ap=\ap_3$;
\textbullet\
for $\GR{E}{7}$: \ $\beta=\ap_6$ if $\ap=\ap_3$ or $\ap_5$;
$\beta=\ap_7$ if $\ap=\ap_4$;
\textbullet\
for $\GR{E}{8}$: \ $\beta=\ap_1$ if $\ap=\ap_2$ or $\ap_3$ or $\ap_8$;
$\beta=\ap_7$ if $\ap=\ap_4$ or $\ap_6$;
$\beta=\ap_8$ if $\ap=\ap_5$;
\textbullet\
for $\GR{F}{4}$: \ $\beta=\ap_4$ if $\ap=\ap_2$ or $\ap_3$.
\end{rmk}
\section{Variations on themes of the "strange formula"}
\label{sect:FdV}
\noindent
Let $\g$ be a reductive algebraic Lie algebra.
For any orthogonal $\g$-module $\EuScript V$, there is another $\g$-module, denoted
$\spin(\EuScript V)$. Roughly speaking, one takes the spinor representation of $\sov$ and restricts it to
$\g\subset\sov$.
It has the property that
\[ \textstyle
\bigwedge^\bullet \EuScript V\simeq \begin{cases}
\spin(\EuScript V)\otimes\spin(\EuScript V), & \text{ if $\dim\EuScript V$ is even} \\
2(\spin(\EuScript V)\otimes\spin(\EuScript V)), & \text{ if $\dim\EuScript V$ is odd} \end{cases} \ ,
\]
see~\cite[Section\,2]{tg01}. Moreover, extracting further a numerical factor from the $\g$-module
$\spin(\EuScript V)$, one
can uniformly write $\spin(\EuScript V)=2^{[m(0)/2]}\spin_0(\EuScript V)$ and then \\[.4ex]
\centerline{
$\bigwedge^\bullet \EuScript V\simeq 2^{m(0)}{\cdot}\bigl(\spin_0(\EuScript V)\otimes\spin_0(\EuScript V)\bigr)$,}
\\[.8ex]
where $m(0)$ is the multiplicity of the zero weight in $\EuScript V$. There are only few orthogonal simple
$\g$-modules
$\EuScript V$ such that $\spin_0(\EuScript V)$ is again simple, see~\cite[Section\,3]{tg01}. A notable example is
that $\spin_0(\g)=\EuScript V_\rho$ for any simple Lie algebra $\g$, cf. Introduction.
From now on, $\g$ is again a simple Lie algebra. Let $\g=\g_0\oplus\g_1$ be a $\BZ_2$-grading.
The corresponding involution of $\g$ is denoted by $\sigma$. Write
$\EuScript C_0\in \EuScript U(\g_0)$ for the Casimir element associated with $\Phi\vert_{\g_0}$.
Then the $\EuScript C_0$-eigenvalue on $\g_1$ equals $1/2$, see~\cite{jlms01} and Proposition~\ref{prop:lms01}.
The $\g_0$-module $\g_1$ is orthogonal, and we are interested now in the $\spin$-construction for
$\EuScript V=\g_1$. Then $m(0)=\rk\g-\rk\g_0$, and $m(0)=0$ if and only if $\sigma$ is an inner involution.
Hence $\spin_0(\g_1)=\spin(\g_1)$ whenever $\sigma$ is inner. There is an explicit description of the
irreducible constituents of $\spin_0(\g_1)$ in~\cite[Sections 5,\,6]{tg01}. This also implies that
$\spin_0(\g_1)$ is always reducible if $\sigma$ is inner.
Although $\spin_0(\g_1)$ can be highly reducible, it is proved in~\cite[Theorem\,7.7]{tg01} that
$\EuScript C_0$ acts scalarly on $\spin_0(\g_1)$ and the corresponding eigenvalue is
\[
\gamma_{\spin_0(\g_1)}=(\rho,\rho)-(\rho_0,\rho_0),
\]
where $\rho_0$ is the half-sum of positive roots of
$\g_0$. Of course, we adjust here the Cartan subalgebras, $\te_0\subset \g_0$ and $\te\subset \g$ such that $\te_0\subset\te$. Then we can assume that $\te^*_0\subset \te^*$, etc.
In this section, we show that the $\EuScript C_0$-eigenvalue $\gamma_{\spin_0(\g_1)}$ has another nice uniform expression and that this is related to the ``strange formula of Freudenthal--de Vries'' (={\sf\bfseries sfFdV}).
\begin{thm} \label{thm:FdV-inner}
Let $\sigma$ be an inner involution of $\g$ and $\g=\g_0\oplus\g_1$ the corresponding
$\BZ_2$-grading. Then
\[
\gamma_{\spin_0(\g_1)}=(\dim\g_1)/16 .
\]
\end{thm}
\begin{proof} Our argument relies on the theory developed in Section~\ref{sect:5/2}
and a relationship between involutions (=\, $\BZ_2$-gradings) and
$(\BZ{,}\ap)$-gradings of height at most $2$.
\\ \indent
Suppose that $d_\ap=\hot_\ap(\theta)\le 2$ and let
$\g=\bigoplus_{i=-d}^d\g_\ap(i)$ be the corresponding $\BZ$-grading. Letting $\g_0=\g_\ap(-2)\oplus \g_\ap(0)\oplus\g_\ap(2)$
and $\g_1=\g_\ap(-1)\oplus\g_\ap(1)$, we obtain a $\BZ_2$-grading (obvious). Since $\rk\g=\rk\g_0$,
this involution is inner. The point is that {\bf all inner} involutions of $\g$ are obtained in this way, as
follows from Kac's description in~\cite{kac}, cf. also~\cite[Ch.\,3,\ \S3.7]{t41}.
Different simple
roots $\ap,\beta$ with $d_\ap=d_\beta=2$ lead to ``one and the same'' $\BZ_2$-grading if and only if there is an automorphism of the extended Dynkin diagram of $\g$ that takes
$\ap$ to $\beta$.
If $d_\ap=1$, then $\g_0=\g_\ap(0)$ is not semisimple, whereas $\g_0$ is semisimple for $d_\ap=2$.
The subalgebras $\g_0$ corresponding to $\ap$ with $d_\ap=2$ are indicated in Tables~\ref{table:d=2c}
and \ref{table:d=2e}.
We can express $\rho_0$ and $\rho_1=\rho-\rho_0$ via the data related to the $\BZ$-grading.
That is, $\rho_0=\frac{1}{2}\bigl(|\Delta_\ap^+(0)|+|\Delta_\ap(2)|\bigr)$ and
$\rho_1=\frac{1}{2}|\Delta_\ap(1)|=\frac{1}{2}q_\ap(1)\vp_\ap$. Since $(\vp_\ap,\mu)=0$ for any
$\mu\in \Delta^+_\ap(0)$, we have $(\rho_1,|\Delta_\ap^+(0)|)=0$ and therefore
\[
(\rho,\rho)-(\rho_0,\rho_0)=(\rho_1,2\rho_0+\rho_1)=
(\rho_1, |\Delta_\ap(2)|+\rho_1)=
\frac{1}{4}\bigl(q_\ap(1)\vp_\ap, (q_\ap(1)+2q_\ap(2))\vp_\ap\bigr).
\]
Now, $q_\ap(1)+2q_\ap(2)=q_\ap+q_\ap(2)=h^*r_\ap$, see Corollary~\ref{cor:sravnenie-2} with $d_\ap=2$.
For $d_\ap=1$, one has $r_\ap=1$ and again $q_\ap(1)+2q_\ap(2)=q_\ap=h^*$, see Theorem~\ref{thm:otmetki}(2$^o$). So, if $d_\ap\le 2$, then
\begin{multline*}
\gamma_{\spin_0(\g_1)}=\frac{h^*r_\ap}{4}(|\Delta_\ap(1)|, \vp_\ap)=
\frac{h^*r_\ap}{4}\sum_{\mu\in\Delta_\ap(1)}(\mu,\vp_\ap)=\frac{h^*r_\ap}{4}\sum_{\mu\in\Delta_\ap(1)}
(\ap,\vp_\ap) \\
=\frac{h^*r_\ap}{4}\dim\g_\ap(1)\frac{(\ap,\ap)}{2}=\frac{1}{8}\dim\g_\ap(1)=\frac{1}{16}\dim\g_1 .
\end{multline*}
Here we use the fact that $\hot_\ap(\mu)=1$ for any $\mu\in \Delta_\ap(1)$ and hence
$(\mu,\vp_\ap)=(\ap,\vp_\ap)$.
\end{proof}
Actually, the previous result holds for any involution of $\g$, see below. This can be regarded as both an
application and generalisation of the {\sf\bfseries sfFdV}.
However, whereas the proof of Theorem~\ref{thm:FdV-inner} does not refer to the {\sf\bfseries sfFdV}, the
general argument below, which applies to arbitrary involutions, explicitly relies on the {\sf\bfseries sfFdV}.
\begin{thm} \label{thm:general-g_1}
For any involution of a simple Lie algebra $\g$, we have $\gamma_{\spin_0(\g_1)}=(\dim\g_1)/{16}$.
\end{thm}
\begin{proof}
Write $\g_0=(\bigoplus_i\h_i)\oplus \ce$ as the sum of simple ideals $\{\h_i\}$ and possible centre $\ce$.
To prove the assertion, we need basically the following three facts on $\EuScript C_0=\EuScript C_{\g_0}$:
\begin{itemize}
\item $\tr_\g(\EuScript C_0)=\dim\g_0$, see~\cite{jlms01} and Proposition~\ref{prop:lms01}{\sf (i)};
\item the $\EuScript C_0$-eigenvalue on $\g_1$ equals $1/2$, see Proposition~\ref{prop:lms01}{\sf (iv)};
\item the usual {\sf\bfseries sfFdV} for $\g$ and for the simple ideals of $\g_0$.
\end{itemize}
Let $\EuScript C_{\h_i}$ be the "canonical" Casimir element for $\h_i$. Then $\EuScript C_{\h_i}$ has the eigenvalue $1$ on $\h_i$. Since $\h_i$ is an ideal of $\g_0$,
there is a transition factor, $T_i$, between the eigenvalues of
$\EuScript C_{\h_i}$ and $\EuScript C_0$ on the $\h_i$-modules, cf. the proof of Theorem~\ref{thm:gen-formula}.
Actually, we even know that $T_i=\frac{h^*(\h_i)}{h^*{\cdot}\ind(\h_i\hookrightarrow\g)}$, but this precise
value is of no importance in the rest of the argument.
Since the $\EuScript C_0$-eigenvalue on $\h_i$ is $T_i$, we have
$\tr_{\h_i}{\EuScript C_0}=T_i {\cdot}\dim\h_i$. Hence
\[
\dim\g_0= \tr_\g(\EuScript C_0)=\tr_{\g_1}(\EuScript C_0)+\tr_{\g_0}(\EuScript C_0)=
\frac{1}{2}\dim\g_1+\textstyle \sum_i T_i {\cdot}\dim\h_i .
\]
On the other hand, let $\rho_i$ be the half-sum of the positive roots of $\h_i$.
Then $\rho_0=\sum_i\rho_i$, $(\rho_i,\rho_j)=0$ for $i\ne j$, and
$(\rho_i,\rho_i)=T_i{\cdot} (\dim\h_i/24)$ in view of the {\sf\bfseries sfFdV} for $\h_i$.
Thus,
\begin{multline*}
\gamma_{\spin_0(\g_1)}=(\rho,\rho)-(\rho_0,\rho_0)=
\frac{1}{24}\dim\g- \frac{1}{24}(\textstyle \sum_i T_i\dim\h_i) \displaystyle
\\ = \frac{1}{24}\dim\g- \frac{1}{24} (\dim\g_0-\frac{1}{2}\dim\g_1)=\frac{1}{16}\dim\g_1 .
\qedhere
\end{multline*}
\end{proof}
\begin{rmk} \label{rem:g+g}
The adjoint representation of $\g$ can be regarded as the isotropy representation related to the
permutation, $\tau$, of the summands in $\tilde\g=\g\dotplus\g$. Here $\tilde\g$ is not simple,
but $\tilde\g_0\simeq \g$ is. In this situation, there is an analogue of Theorem~\ref{thm:general-g_1} for
$(\tilde\g, \tau)$, and we demonstrate below that it is equivalent
to the {\sf\bfseries sfFdV} for $\g$. In other words, under proper adjustments of bilinear forms and
$\EuScript C_{\tilde\g_0}$,
the formula of Theorem~\ref{thm:general-g_1} for $(\tilde\g,\tau)$ transforms into
the {\sf\bfseries sfFdV} for $\g$, and vice versa.
One of the main reasons is that, for the orthogonal
$\g$-module $\g$, one has $\spin_0(\g)=\EuScript V_\rho$, see~\cite[(5.9)]{ko61} and~\cite[(2.5)]{tg01}.
\\ \indent
Recall that $\g=\ut\oplus\te\oplus\ut^-$ and $\be=\ut\oplus\te$. Then
$\tilde\be=\be\dotplus\be$ and $\tilde\te=\te\dotplus\te$.
In what follows, various objects related to the two factors of $\tilde\g$ will be marked with the
superscripts `$(1)$' and `$(2)$'. As above, $\Phi$ is the Killing form on $\g$ and $(\ ,\ )$ is the induced
(canonical) bilinear form on $\te^*$.
Let $\tilde\Phi=\Phi^{(1)}\dotplus\Phi^{(2)}$ be the invariant bilinear form on $\tilde\g$. The induced
bilinear form on $\tilde\te^*$ is denoted by $(\ ,\ )^{\sim}$. Then the Casimir element
$\EuScript C_{\tilde\g_0}$ is defined via the restriction of $\tilde\Phi$ to $\tilde\g_0$. We have
$\tilde\rho=\rho^{(1)}+\rho^{(2)}$ and these two summands are orthogonal w.r.t. $(\ ,\ )^{\sim}$; \
hence
$(\tilde\rho ,\tilde\rho )^{\sim}=(\rho^{(1)} ,\rho^{(1)})^{\sim}+
(\rho^{(2)} ,\rho^{(2)})^{\sim}=2(\rho,\rho)$.
The Cartan subalgebra $\tilde\te_0$
of $\tilde\g_0$ is diagonally imbedded in $\tilde\te$, hence so are the roots of $\tilde\g_0$ in
$\tilde\te_0^*$. In particular, both components of $\tilde\rho_0$ are equal to $\rho$.
Identifying ${\tilde\g_0}$ with $\g$ via the projection to the first component, one readily obtains that
$\tilde\Phi\vert_{\tilde\g_0}=2\Phi$. It then follows from Lemma~\ref{lm:dve-formy} that one obtains
the relation with the {\bf inverse} coefficient for the corresponding canonical bilinear forms.
This yields our key equality
\[
(\tilde\rho_0 ,\tilde\rho_0 )^{\sim}=\frac{1}{2}(\rho,\rho) .
\]
Afterwards, using the {\sf\bfseries sfFdV} for $\g$, we obtain
\[
(\tilde\rho ,\tilde\rho )^{\sim}-(\tilde\rho_0 ,\tilde\rho_0 )^{\sim}=2(\rho,\rho)-\frac{1}{2}(\rho,\rho)=
\frac{3}{2}(\rho,\rho)=\frac{1}{16}\dim\g=\frac{1}{16}\dim\tilde\g_1 .
\]
And by \cite[Theorem\,7.7]{tg01}, the $\EuScript C_{\tilde\g_0}$-eigenvalue on $\spin_0(\tilde\g_1)$ equals
$(\tilde\rho ,\tilde\rho )^{\sim}-(\tilde\rho_0 ,\tilde\rho_0 )^{\sim}$. Thus, the equality of
Theorem~\ref{thm:general-g_1} remains true for the involution $\tau$ of $\tilde\g$, and
we have just shown that this equality is equivalent to the {\sf\bfseries sfFdV}.
\end{rmk}
This certainly means that it is of great interest to find a proof of Theorem~\ref{thm:general-g_1} and
its analogue for the semisimple Lie algebra $\tilde\g=\g\dotplus\g$ that is independent of the
{\sf\bfseries sfFdV}.
|
1,108,101,565,705 | arxiv | \section{Introduction}
Shape coexistence is a unique phenomenon of the atomic core in which the nucleus displays intrinsically different shapes within a small energy range. Manifestation of this behaviour has been observed all across the nuclear chart, but the neutron-deficient Pb region (\textit{Z}$\leq$82, $N$<126) is characterized by some of the clearest examples of shape coexistence~\cite{HEY11,AND00,Dracoulis2003,Dracoulis2004,Julin2016,Marsh2018}. The phenomenon was observed in the Pb isotopes using $\alpha$-decay spectroscopy, which found multiple low-lying $0^+$ states in $^{186}$Pb \cite{AND00}. It was shown by Dracoulis \textit{et al.}~\cite{DRA00}, that the high-spin isomeric states in $^{188}$Pb can only be built on unique single-particle configurations of different shape. This clearly demonstrated that three differently shaped potentials (spherical, prolate and oblate) exist in these nuclei.
In the light Hg ($Z$=80) isotopes, this phenomenon was first revealed in optical spectroscopy measurements which identified a large staggering in the isotope shifts between the odd and even Hg isotopes~\cite{BON72}. This isotope shift was interpreted as an alternation between normal and intruder configurations being the ground state with the removal of neutrons. Later laser spectroscopy studies have determined that $^{181}$Hg represents the lighter end of the staggering, and also confirmed the inversion between the ground state and isomeric state in $A$=185~\cite{Marsh2018, Sels2019}. In the even Hg, only recently, a Coulomb excitation study obtained detailed spectroscopic information about shape coexistence for $^{182-188}$Hg~\cite{Bree2014}. By measuring the relative sign of the \textit{E2} matrix elements, Bree \textit{et al.}~\cite{Bree2014} were able to extract information about the different deformations of the $0^+$ states and firmly establish that two different structures coexist at low energies.
Despite these ground-breaking experiments, there is still a significant amount of key information that remains to be measured, especially in the transitional isotopes between the stable $^{200}$Hg and the beginning of the midshell $^{190}$Hg. This experimental data is critical for solidifying our understanding of the region and developing a quantitative understanding of the underlying mechanisms driving these behaviours. The relative energy of the intruder states has a parabola-shape with a minimum observed at $^{182}$Hg. In the heavier transitional isotopes ($190\leq A\leq 200$), the ground and intruder configurations are still sufficiently far apart in energy such that the mixings between the two structures are expected to be significantly reduced. These isotopes thus present a good opportunity to benchmark the normal ground-state configuration without the perturbations (through mixing with the intruder configuration) experienced in the lighter isotopes, thus simplifying the comparison with different theoretical calculations.
One of the main model-independent probes used to study shape coexistence is the measurement of transition strengths, in particular $B(E2)$ and $\rho^2$($E0$) values~\cite{HEY11}. These transition strengths are particularly sensitive to the wavefunctions of the states they connect, and thus are one of the most stringent probes available to test theoretical models used to describe nuclei. With respect to $B(E2)$ values in the transitional Hg, Esmaylzadeh \textit{et al.}~\cite{Esmaylzadeh2018} recently measured the $2^+_1$ lifetimes for $^{190,192,194}$Hg. Due to the isotope production mechanism employed, the experiment suffered from contaminants that significantly limited the precision of the measured half-lives, preventing a meaningful comparison with different theoretical calculations. In the case of $\rho^2$($E0$) values, the excited $0^+$ states have only been identified up to $^{190}$Hg, with the energies of the intruder structures remaining unknown for the heavier isotopes, although some candidate states exist. Theoretical calculations predict an increase in excitation energy for the intruder configuration up to $^{192}$Hg after which a more stable value is maintained~\cite{Nomura2013, Garcia-Ramos2014}.
In order to characterize the evolution from the stable Hg isotopes towards the mid-shell, a systematic study of the decay of the ground- and isomeric states of neutron-deficient $^{188-200}$Tl isotopes into Hg has been performed using the GRIFFIN spectrometer~\cite{GAR19,SVE14,RIZ16} at TRIUMF-ISAC. The high statistics resulting from the measurement of $\gamma$-ray and conversion electrons enable high precision $\gamma-\gamma$ angular correlations and precise branching ratios, which are all important in forming a complete picture of the band structure of these isotopes. In the present article, we focus on the results of the lifetime measurements. Data collected with the ancillary LaBr$_3$(Ce) array have been analyzed using the Generalized Centroid Difference Method (GCDM)~\cite{Regis2013} to precisely measure lifetimes of all the first 2$^+$ and 4$^+$ states of the ground-state bands, as well as some negative-parity and non-yrast states. The extracted $B(E2)$ values are compared with different interacting boson model (IBM) calculations while the negative-parity band is interpreted in comparison with a quasiparticle-rotor model.
\section{Experimental setup\label{sec:experimental_setup}}
The Tl isotopes were produced by a 500-MeV proton beam of 9.8~$\mu$A intensity delivered by the TRIUMF main cyclotron \cite{BYL14} impinging on an uranium carbide (UC$_x$) target. The TRIUMF Resonant Ionization Laser Ion Source (TRILIS)~\cite{LAS06} was tuned to preferentially ionize the 7$^+$ isomeric states in Tl. A small contribution of the $2^-$ ground state Tl was also present in the beam, but no other significant isobaric contaminants were observed. The Tl ion beam was accelerated to 20~kV, mass separated and delivered to the experimental station. The beam intensity was attenuated down to $\sim 10^5$ particles per second for all the masses studied.
The ions were implanted in a mylar tape at the central focus of the GRIFFIN spectrometer~\cite{GAR19,SVE14,RIZ16}. Due to the long half-lives involved in the Hg decay chains and the low-energy $\gamma$-ray transitions of the Hg decay products (all below the energy range of interest in this experiment), the tape remained stationary during the beam delivery. The tape was moved only when changing between beams of different mass in order to remove any remaining longer-lived activity from the previous setting from the chamber. The exception was for the decay of $^{188m}$Tl, where the tape cycling mode was used. The cycle was composed of 1.5~s for the tape movement, 30~s of background measurement, 480~s of the beam being delivered and just 1~s of decay time with the beam blocked. This cycling configuration was designed to maximize the activity of $^{188m}$Tl while suppressing the other decay products present. In order to remeasure the $^{188}$Tl and $^{188m}$Tl half-lives (T$_{1/2}(2^-)$=71(1)~s and T$_{1/2}(7^+)$=71(2)~s, respectively~\cite{Kondev2018}), a small fraction of the data was taken with a different tape cycle; 1.5~s of tape movement, 30~s background measurement, 210~s beam-on and 350~s of decay time with the beam off. The detailed analysis of the decays from these two levels is being prepared for publication \cite{MacLean2020}.
GRIFFIN is an array of up to 16 high-purity germanium (HPGe) clover detectors~\cite{RIZ16} arranged in a rhombicubocatahedral geometry. For this particular experiment, only 15 HPGe-clovers were employed as one must be removed to accommodate the liquid nitrogen dewar of the PACES detector. Seven cylindrical (5.1~cm in diameter by 5.1~cm length) lanthanum bromide crystals doped with a $5\%$ of cerium (LaBr$_3$(Ce)) coupled to a R2083 photomultiplier (PMT) were placed in the ancillary triangular positions of the array (one ancillary position remained empty). Around the implantation point, covering the upstream half of the chamber, a set of five in-vacuum LN$_2$-cooled lithium-drifted silicon detectors (PACES) were used for conversion electron measurements. A fast 1\,mm-thin plastic called Zero Degree Scintillator (ZDS) was placed just a few millimeters behind the ion-deposition point in the tape. The reader is referred to~\cite{GAR19} for further details about the GRIFFIN array and ancillary detector performance.
The energy signals from each detector were digitazed by the GRIFFIN custom-built digital data acquisition system (DAQ)~\cite{GAR17} with a 100~MHz sampling frequency, which, after a digital implementation of a constant-fraction discriminator (CFD) algorithmic interpolation, gives timestamps with a precision down to $\sim 1$~ns. After shaping the signals with a custom made preamplifier, this works well for the timestamps of the HPGe and Si(Li) semiconductor signals, as well as the signals from the bismuth germanate (BGO) shields and SCEPTAR, the thick $\beta$-tagging plastic scintillators (neither of them employed in this experiment). However, it is not sufficient for accurate timing of the fast ZDS and LaBr$_3$(Ce) signals which have a rise time of 0.7~ns Ref.~\cite{Vedia2015}. To make use of the full timing capabilities of these fast scintillators, a hybrid analog-digital electronic timing setup was developed and employed in the present work.
The LaBr$_3$(Ce) energy signal is taken from the last dynode of the PMT and processed by a custom-made preamplifier before being directly provided to the DAQ. The timing signal is taken from the anode and input to an Ortec 935 quad CFD~\cite{ortec935}. External delay cables of 20~ns were employed in order to obtain an uniform time walk over a large energy range while maintaining reasonable timing resolutions. The output of the individual CFDs were fed into Lecroy 429A fan-in/fan-out logic modules~\cite{lecroy429A} in order to obtain all the possible LaBr$_3$(Ce)-LaBr$_3$(Ce) combinations. These logic output signals provide the START and STOP signals to a set of Ortec 566 Time-to-Amplitude Converter (TAC) modules~\cite{ortec566}. These TAC modules delay the output signal by $\sim 2.5 \mu$s which is then digitized by the same 100 MHz ADC described above. . The timestamps were corrected offline to time-match the detector and TAC signals.
Simultaneously the signals from an Ortec 462 Time Calibrator module~\cite{ortec462} operated at a low rate of $\sim 100$~Hz were connected to the TAC modules during the whole data-collecting period. The Time Calibrator has a timing precision of $\sim 10$~ps. This allowed for a precise monitoring of the TAC performance and corrections to any fluctuations due to, for example, temperature changes. These events were easily identified through a lack of LaBr$_3$(Ce) energy coincidence. An offline event-by-event correction of the TACs was performed using this information.
In this configuration the combination of all seven LaBr$_3$(Ce) crystals has a timing resolution of $\sim 330$~ps, with a time-walk of slightly over 100~ps in the 200-1300 keV range, as shown in Fig.~\ref{fig:prd}. Further details on the data shown in this Figure is provided in the following Section.
An additional TAC was set with the ZDS plastic as START and a logic module with an OR of all the LaBr$_3$(Ce) detectors as STOP. Thanks to its reduced thickness of 1~mm, it ensures that charged particles will deposit an approximately constant amount of energy nearly independent of their kinetic energy. This allows for a superior timing resolution and a reduced time-walk when compared to LaBr$_3$(Ce). This comes, however, at the cost of losing all $\beta$-particle energy resolution. The ZDS has an absolute efficiency of $\sim 20\%$, due to its solid angle coverage, which is an order of magnitude higher than the LaBr$_3$(Ce) array.
To reduce the volume and rate of data recorded to disk, the DAQ system employed digital filters. Such events with at least one PACES or one HPGe or two or more LaBr$_3$(Ce) crystals had signals were passed to the data acquisition computer. Any other detector hits that were in temporal coincidence within $2~\mu$s of any one of these conditions were also recorded to disk.
\section{Data analysis\label{sec:data_analysis}}
Data collected with the GRIFFIN array were analyzed using the GRSISort software~\cite{GRSISort} within the ROOT framework~\cite{Brun1997}. General methods for analyzing such experiments are outlined in Ref.~\cite{GAR19}. This experiment focused on measuring lifetimes in the pico- to nanosecond range using the GCDM. This method is an evolution of early electronic fast-timing techniques~\cite{Mach1989,Moszynski1989}, adapted to large arrays of fast inorganic scintillators~\cite{Regis2010,Regis2013}. A detailed explanation of the GCDM can be found in Refs.~\cite{Regis2010,Regis2012,Regis2013,Regis2014,Regis2016}, but a short summary of the method is included here.
\subsection{Centroid difference}
The method uses a TAC to measure the time difference between two transitions in a $\gamma$-ray cascade detected by LaBr$_3$(Ce) crystals. If the decaying $\gamma$-ray is prompt (that is, if the draining transition decays from a nuclear state with a lifetime well below the timing sensitivity of the system, i.e. $\tau < 1$~ps), the TAC spectrum will be a semi-Gaussian distribution centered at zero. When the intermediate level between the two transitions has a mean lifetime in the picosecond or longer range, the resulting distribution centroid will be shifted from zero by an amount equal to $\tau$. Despite the use of analog CFDs in the signal processing, the position of the \textit{prompt} zero-time will depend on the energy of the feeding and decaying transitions. This is known as the time-walk or ``mean prompt response difference'' (PRD(E)) for arrays, and can be calibrated down to 2-5~ps using standard commercially-available radioactive sources such as $^{152}$Eu (all the time distribution centroids must be corrected by the precisely known lifetimes). The curve generated from data collected for the present study is shown in Fig.~\ref{fig:prd}.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, keepaspectratio]{prd.pdf}
\caption{\label{fig:prd} Time-walk response or PRD curve of the whole LaBr$_3$(Ce) array. The curve has been derived from data collected with an offline commercial $^{152}$Eu source and with $^{198m}$Tl online. See text for details.}
\end{center}
\end{figure}
The cables providing signals to the STOP input of the TAC modules are $\sim 25$~ns longer than the ones feeding the START. In practice, this has the effect of shifting the TAC range from between 0 and 50~ns to between $-25$ and 25~ns, allowing for reverse gating of the transitions in the LaBr$_3$(Ce) crystals. In this anti-delayed mode, the decaying transition is the START input for the TAC and the feeding transition is the STOP. The time difference between the signals will therefore be the negative lifetime of the intermediate level. The centroid difference between the direct and reverse gating is equal to twice the lifetime and with this method most systematic errors are suppressed. The correction for the time walk described earlier must still be included and the final expression is of the form:
\begin{equation}
\Delta\text{C} = 2\tau + \text{PRD}(\Delta \text{E}_\gamma) \label{eq:centroid_shift}
\end{equation}
\noindent where $\Delta\text{C}$ is the centroid difference between the two gated spectra, $\tau$ is the sum of all the lifetimes of levels between the two gated transitions, and $\text{PRD}(\Delta \text{E}_\gamma) = \text{PRD}(\text{E}_\text{feed}) - \text{PRD}(\text{E}_\text{decay})$ is the difference in the time response for the energies of the feeding and decaying transitions (see Fig.~\ref{fig:prd}).
For the extraction of nuclear lifetimes, the data were sorted into LaBr$_3$(Ce)-LaBr$_3$(Ce)-TAC-(HPGe) events where the detector name implies the energy from that detector. (The HPGe coincidence was optional but allowed to use the GCDM imposing an additional condition on the high-energy-resolution HPGe detectors to precisely select one specific $\gamma$-ray cascade, if needed). This is especially important because of the very different timing response of the LaBr$_3$(Ce) crystals to full-energy peaks (FEP) and Compton events. This difference in timing response makes it impossible to subtract the nearby background when gating on a peak (as it is done in HPGe-HPGe coincidences, for example), since the difference in energies and type of physics event will yield very different time responses. It is, thus, of paramount importance to reduce the background by other means, like imposing additional coincidence conditions or the use of anti-Compton and background shields. Nevertheless, the time response of the Compton background around the peak is studied and a correction is applied using the following equation:
\begin{equation}
\textrm{A}_\textrm{T} \cdot \textrm{C}_\textrm{T} = \textrm{A}_\textrm{FEP} \cdot \textrm{C}_\textrm{FEP} + \textrm{A}_\textrm{C} \cdot \textrm{C}_\textrm{C} \label{eq:compton_corr}
\end{equation}
\noindent where A stands for area and C for timing-response centroid value. The subscripts refer to full-energy peak ($_\textrm{FEP}$), Compton ($_\textrm{C}$) and the total area of FEP plus Compton ($_\textrm{T}$). The Compton gate is set a few keV above the FEP energy, so, due to the CFD time-walk, C$_\textrm{C}$ must be shifted as a function of energy. Several gates are set on the Compton background around the peak and their centroid values are fitted as a function of the energy. This is shown in Fig~\ref{fig:compton_corr}. For further details on this approach to Compton correction see Refs.~\cite{Mach1989, Regis2013}.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, keepaspectratio]{compton_correction.pdf}
\caption{\label{fig:compton_corr} Compton correction to the $2^+_1$ mean lifetime in $^{194}$Hg ($E_{\gamma}=428$keV). The black squares are the centroid values of the time response for the Compton background at that energy, the blue circle is the centroid of the time response for that FEP and the red line is the quadratic fit to the Compton time-response. The LaBr$_3$(Ce) energy spectrum is superimposed for reference. All spectra were generated with gates in the $4^+_1 \rightarrow 2^+_1$ transition in the START LaBr$_3$(Ce) and $6^+_1 \rightarrow 4^+_1$ in the HPGe array. See text for details.}
\end{center}
\end{figure}
The GRIFFIN array and its ancillary detectors (especially the LaBr$_3$(Ce) array) have a very compact geometry, which causes a significant Compton background. This has been greatly mitigated by the recent addition of BGO active Compton and background suppression shields~\cite{GAR19}. However, since this shielding was not available at the time of the present experiment, an additional coincidence can be set in the HPGe detectors when examining a cascade involving three or more $\gamma$ rays. Alternatively, if the cascade involves only two $\gamma$ rays, anti-coincidence conditions can be imposed with the HPGe data. When $\gamma$ rays involved in cascades of only two $\gamma$ rays are selected in the LaBr$_3$(Ce), it can be assumed that any events in the HPGe detectors will be a Compton or random-background event, and the entire GRIFFIN array can effectively be used as an active suppression shield. This drastically reduced the Compton background, with a minor loss in statistics. During the offline timing calibration, no HPGe data were recorded. This resulted in a peak-to-background ratio in the $^{152}$Eu decay spectra which was much poorer than that achieved in the online data (when the HPGe were active) and in calibrations of subsequent experiments. This has significantly increased the uncertainty in the PRD curve available for this experiment up to $\sim 5$~ps.
To improve the precision of the PRD(E) in the energy range of interest (400 to 650~keV), the $2^+_1\rightarrow0^+_1$ and $4^+_1\rightarrow2^+_1$ transitions from $^{198}$Hg were included in the PRD(E) calibration. The lifetime of the $2^+_1$ state in $^{198}$Hg has been measured in over 10 different experiments using a wide range of techniques, yielding a very accurate evaluated value of $\tau=34.34(25)$~ps~\cite{HUANG2016,PRITYCHENKO2016}. An additional gate on the $5^-_1\rightarrow4^+_1$ transition detected in a HPGe detector was imposed that increased the quality of the LaBr$_3$(Ce) timing spectrum. Due to the improved peak-to-background ratio (now in the 20:1 range) and the abundant statistics, the centroids were measured with a precision of 2~ps, see Fig.~\ref{fig:Hg198_2+_lifetime}.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, keepaspectratio]{plot_lifetime_first2_198Hg.pdf}
\caption{\label{fig:Hg198_2+_lifetime} Centroid shift ($\Delta$C) between the delayed (\textit{black}) and anti-delayed (\textit{red/gray}) time spectra of the $4^+_1 \rightarrow 2^+_1$ and $2^+_1 \rightarrow 0^+_1$ $\gamma$-ray transitions in $^{198}$Hg. The $\Delta$C value must be corrected by the PRD(E) and Compton contributions. Only the PRD(E) corrections derived from the $^{152}$Eu source data were applied here (the precise $^{198}$Hg points were excluded) in the determination of this $\tau(2^+_1)$ value, resulting in a comparatively larger uncertainty than for other masses.}
\end{center}
\end{figure}
\subsection{Deconvolution method}
When the measured lifetime is comparable or longer than the timing resolution of the system (FWHM$\sim 330$~ps for this experiment), the time distribution will present an exponential decay on the delayed part. The lifetime can be extracted directly from the slope of the decay delayed part. This time distribution can be fitted to a Gaussian convoluted with an exponential decay of the form:
\begin{equation}
F(t_j) =\gamma \int_A^{+\infty} e^{- \delta (t_j - t)^2} e^{-\lambda t}dt \label{eq:convolution_method}
\end{equation}
\noindent where $\gamma$ is the normalization factor, $\delta$ is a parameter related to the width of the Gaussian prompt distribution and $A$ is the centroid of said Gaussian, which is related to the position of a prompt transition of the same energy. When needed, additional terms to account for the time-random background can be introduced. Additional details on the method are given in Ref.~\cite{GAR19}.
The ZDS detector used as a TAC start signal is particularly well suited for this convolution method, by giving the time difference between the $\beta^-/\beta^+$ particle and the $\gamma$-ray. Thanks to its reduced timing FWHM, lifetimes will show a slope at shorter lifetime values. Moreover, since it detects charged particles, not $\gamma$-rays, it does not require a transition feeding the excited state of interest, it can be started by the $\beta$ particle directly populating the level. This allows access to levels that are unavailable to LaBr$_3$(Ce)-LaBr$_3$(Ce) coincidences. Lastly, because of its larger efficiency and the fact that it does not require a $\gamma$-ray cascade to operate, in general it will yield far superior statistics. In a similar fashion to the GCDM, additional HPGe coincidences can be imposed to increase selectivity.
\section{Experimental results\label{sec:experimental_results}}
In the present high-statistics study (see Fig.~\ref{fig:energy_spectra} for an example of the quality and quantity of data collected), a large number of lifetimes have been observed across the \textit{n}-deficient Hg isotopic chain. Table~\ref{tab:lifetimes} summarizes all the measured half-lives from this work using the GCDM described in Sec.~\ref{sec:data_analysis} and Ref.~\cite{Regis2013}. E$_\textrm{feeder}$ and E$_\textrm{decay}$ indicate the energies used for the gating transitions in the LaBr$_3$(Ce) crystals. E$_\textrm{HPGe}$ indicates the gated transition in the HPGe array. When previously measured, the literature value is given in the table for comparison. In some cases more than one combination of transition gating could measure the lifetime. In those cases, all employed combinations are described in the table, and as final result the average is given.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, keepaspectratio]{energy_190Hg.pdf}
\caption{\label{fig:energy_spectra} Energy spectra of the LaBr$_3$(Ce) (\textit{red/gray}) and HPGe GRIFFIN (\textit{black}) arrays for the decay of $^{190m}$Tl. This data was taken in LaBr$_3$(Ce)-LaBr$_3$(Ce)-TAC-(HPGe) coincidences mode. Some of the most intense transitions in $^{190}$Hg have been labeled.}
\end{center}
\end{figure}
\begin{table*}
\caption{Summary of the half-lives measured in this experiment with comparison to values in literature. The level and transition energies are taken from Ref.~\cite{NNDC}. E$_\textrm{feeder}$ gives the energy of the feeding transition selected in the LaBr$_3$(Ce) crystals and E$_\textrm{decay}$ to the decaying one. E$_\textrm{HPGe}$ makes reference to the additional condition set in the HPGe detectors. When more than one energy is given in a column, it indicates that different combinations of transitions were used to obtain the lifetime of the level. No significant discrepancies were found between different combinations and the average value is given as the final result.}
\label{tab:lifetimes}
\begin{center}
\begin{tabular}{cccccccccccc}
\hline
Isotope & E$_\textrm{level}$ & $J^\pi$ & E$_\textrm{feeder}$ & $J^\pi_{fi} \rightarrow \text{J}^\pi_{ff}$ & E$_\textrm{decay}$ & $J^\pi_{di} \rightarrow \text{J}^\pi_{df}$ & E$_\textrm{HPGe}$ & $J^\pi_{gi} \rightarrow \text{J}^\pi_{gf}$ & T$_{1/2}$ exp. & T$_{1/2}$ lit. & Ref \\
& (keV) & & (keV) & & (keV) & & (keV) & & (ps) & (ps) & \\ \hline
$^{200}$Hg & 367.9 & $2^+_1$ & 579.3 & $4^+_1 \rightarrow 2^+_1$ & 367.9 & $2^+_1 \rightarrow 0^+_1$ & 828.3 & $3^+_3 \rightarrow 4^+_1$ & 44(3) & 46.4(4) & \cite{PRITYCHENKO2016} \\
& 947.2 & $4^+_1$ & 828.3 & $3^+_3 \rightarrow 4^+_1$ & 579.3 & $4^+_1 \rightarrow 2^+_1$ & 367.9 & $2^+_1 \rightarrow 0^+_1$ & 6(3) & 3.24(5) & \cite{Bockisch1979, Gunther1981} \\
& 1029.3 & $0^+_2$ & 701.6 & $2^+_5 \rightarrow 0^+_2$ & 661.4 & $0^+_2 \rightarrow 2^+_1$ & 367.9 & $2^+_1 \rightarrow 0^+_1$ & 8(4) & & \\
& 1254.1 & $2^+_2$ & 628.8 & $2^+_7 \rightarrow 2^+_2$ & 886.2 & $2^+_2 \rightarrow 2^+_1$ & 367.9 & $2^+_1 \rightarrow 0^+_1$ & 8(6) & 3.5(8) & \cite{Bockisch1979} \\ \hline
$^{198}$Hg & 411.8 & $2^+_1$ & 636.7 & $4^+_1 \rightarrow 2^+_1$ & 411.8 & $2^+_1 \rightarrow 0^+_1$ & 587.2 & $5^-_1 \rightarrow 4^+_1$ & 24(3) & 23.15(28) & \cite{PRITYCHENKO2016} \\
& 1048.5 & $4^+_1$ & 587.2 & $5^-_1 \rightarrow 4^+_1$ & 636.7 & $4^+_1 \rightarrow 2^+_1$ & 411.8 & $2^+_1 \rightarrow 0^+_1$ & <5 & 1.80(8) & \cite{Bockisch1979, Gunther1981} \\
& 1635.7 & $5^-_1$ & 489.6 & $(6,7)^-_1 \rightarrow 5^-_1$ & 587.2 & $5^-_1 \rightarrow 4^+_1$ & 411.8 & $2^+_1 \rightarrow 0^+_1$ & 57(7) & 62(11) & \cite{Beraud1971} \\
& 1683.4 & $7^-_1$ & 226.2 & $6^-_1 \rightarrow 7^-_1$ & 587.2 & $5^-_1 \rightarrow 4^+_1$ & 411.8 & $2^+_1 \rightarrow 0^+_1$ & 6.6(1) ns & 6.9(2) ns & \cite{DULFER1970} \\ \hline
$^{196}$Hg & 426.0 & $2^+_1$ & 635.5 & $4^+_1 \rightarrow 2^+_1$ & 426.0 & $2^+_1 \rightarrow 0^+_1$ & 695.6 & $5^-_1 \rightarrow 4^+_1$ & 16(3) & 17.2(6) & \cite{PRITYCHENKO2016} \\
& 1061.4 & $4^+_1$ & 695.6 & $5^-_1 \rightarrow 4^+_1$ & 635.5 & $4^+_1 \rightarrow 2^+_1$ & 426.0 & $2^+_1 \rightarrow 0^+_1$ & 4(3) & 4(3) & \cite{Esmaylzadeh2018} \\
& 1757.0 & $5^-_1$ & 588.8 & $(5,6,7)^-_1 \rightarrow 5^-_1$ & 695.6 & $5^-_1 \rightarrow 4^+_1$ & 426.0 & $2^+_1 \rightarrow 0^+_1$ & 670(80) & 555(17) & \cite{TON1970} \\
& 1841.3 & $7^-_1$ & 505.2 & $(5,6,7)^-_1 \rightarrow 7^-_1$ & 695.6 & $5^-_1 \rightarrow 4^+_1$ & 426.0 & $2^+_1 \rightarrow 0^+_1$ & 4.8(2) ns & 5.22(16) ns & \cite{TON1970} \\ \hline
$^{194}$Hg & 427.9 & $2^+_1$ & 636.3 & $4^+_1 \rightarrow 2^+_1$ & 427.9 & $2^+_1 \rightarrow 0^+_1$ & 734.8 & $6^+_1 \rightarrow 4^+_1$ & 19(1) & 15(3) & \cite{Esmaylzadeh2018} \\
& & & & & & & 748.9 & $5^-_1 \rightarrow 4^+_1$ & & & \\
& 1064.2 & $4^+_1$ & 734.8 & $6^+_1 \rightarrow 4^+_1$ & 636.3 & $4^+_1 \rightarrow 2^+_1$ & 427.9 & $2^+_1 \rightarrow 0^+_1$ & <3 & 5(3) & \cite{Esmaylzadeh2018} \\
& 1813 & $5^-_1$ & 650.3 & $6^-_2 \rightarrow 5^-_1$ & 748.9 & $5^-_1 \rightarrow 4^+_1$ & 427.9 & $2^+_1 \rightarrow 0^+_1$ & 51(6) & <150 & \cite{TON1970} \\
& 1910.0 & $7^-_1$ & 255.4 & $6^-_1 \rightarrow 7^-_1$ & 734.8 & $6^+_1 \rightarrow 4^+_1$ & 427.9 & $2^+_1 \rightarrow 0^+_1$ & 3.46(3) ns & 3.75(11) ns & \cite{Gunther1977} \\
& & & 208.9 & $(6,7,8)^-_1 \rightarrow 6^-_1$ & 748.9 & $5^-_1 \rightarrow 4^+_1$ & & & & & \\
& & & & & 111.0 & $7^-_1 \rightarrow 6^+_1$ & & & & & \\ \hline
$^{192}$Hg & 422.8 & $2^+_1$ & 634.8 & $4^+_1 \rightarrow 2^+_1$ & 422.8 & $2^+_1 \rightarrow 0^+_1$ & 786.0 & $5^-_1 \rightarrow 4^+_1$ & 12(1) & 15(6) & \cite{Esmaylzadeh2018} \\
& & & & & & & 745.5 & $6^+_1 \rightarrow 4^+_1$ & & & \\
& 1057.6 & $4^+_1$ & 745.5 & $6^+_1 \rightarrow 4^+_1$ & 634.8 & $4^+_1 \rightarrow 2^+_1$ & 422.8 & $2^+_1 \rightarrow 0^+_1$ & 4(3) & 4(3) & \cite{Esmaylzadeh2018} \\
& 1803.0 & $6^+_1$ & 174.0 & $7^-_1 \rightarrow 6^+_1$ & 745.5 & $6^+_1 \rightarrow 4^+_1$ & 634.8 & $4^+_1 \rightarrow 2^+_1$ & 73(10) & & \\
& 1843.9 & $5^-_1$ & 133.1 & $7^-_1 \rightarrow 5^-_1$ & 786.3 & $5^-_1 \rightarrow 4^+_1$ & 634.8 & $4^+_1 \rightarrow 2^+_1$ & 383(14) & & \\
& 1977.0 & $7^-_1$ & 239.2 & $8^-_1 \rightarrow 7^-_1$ & 174.0 & $7^-_1 \rightarrow 6^+_1$ & 422.8 & $2^+_1 \rightarrow 0^+_1$ & 1.03(5) ns & 1.04(6) ns & \cite{MERTIN1978} \\
& & & & & & & 745.5 & $6^+_1 \rightarrow 4^+_1$ & & & \\ \hline
$^{190}$Hg & 416.3 & $2^+_1$ & 625.4 & $4^+_1 \rightarrow 2^+_1$ & 416.4 & $2^+_1 \rightarrow 0^+_1$ & 731.1 & $6^+_1 \rightarrow 4^+_1$ & 15(1) & 15(6) & \cite{Esmaylzadeh2018} \\
& & & & & & & 839.7 & $5^-_1 \rightarrow 4^+_1$ & & & \\
& 1041.8 & $4^+_1$ & 731.1 & $6^+_1 \rightarrow 4^+_1$ & 625.4 & $4^+_1 \rightarrow 2^+_1$ & 416.4 & $2^+_1 \rightarrow 0^+_1$ & 5(4) & <8 & \cite{Esmaylzadeh2018} \\
& & & 839.6 & $5^-_1 \rightarrow 4^+_1$ & & & & & & & \\
& 1772.9 & $6^+_1$ & 305.4 & $7^-_1 \rightarrow 6^+_1$ & 416.4 & $2^+_1 \rightarrow 0^+_1$ & 625.4 & $4^+_1 \rightarrow 2^+_1$ & 7(4) & & \\
& 1881.2 & $5^-_1$ & 370.3 & $(6,7)^-_1 \rightarrow 5^-_1$ & 839.6 & $5^-_1 \rightarrow 4^+_1$ & 625.4 & $4^+_1 \rightarrow 2^+_1$ & <40 & & \\
& & & 196.9 & $7^-_1 \rightarrow 5^-_1$ & & & & & & & \\
& 2078.3 & $7^-_1$ & 240.6 & $8^-_1 \rightarrow 7^-_1$ & 305.4 & $7^-_1 \rightarrow 6^+_1$ & 731.0 & $6^+_1 \rightarrow 4^+_1$ & <200 & & \\ \hline
$^{188}$Hg & 412.9 & $2^+_1$ & 592.1 & $4^+_1 \rightarrow 2^+_1$ & 412.9 & $2^+_1 \rightarrow 0^+_1$ & 504.3 & $6^+_1 \rightarrow 4^+_1$ & 14(3) & 13.1(21) & \cite{Bree2014} \\
& & & & & & & 772.4 & $6^+_2 \rightarrow 4^+_1$ & & & \\
& & & & & & & 904.8 & $5^-_1 \rightarrow 4^+_1$ & & & \\
& 881.1 & $2^+_2$ & 326.9 & $4^+_2 \rightarrow 2^+_2$ & 468.2 & $2^+_2 \rightarrow 2^+_1$ & 412.9 & $2^+_1 \rightarrow 0^+_1$ & <20 & 141(31) & \cite{JOSHI1994} \\
& 1004.9 & $4^+_1$ & 504.3 & $6^+_1 \rightarrow 4^+_1$ & 592.1 & $4^+_1 \rightarrow 2^+_1$ & 412.9 & $2^+_1 \rightarrow 0^+_1$ & <30 & 1.60(12) & \cite{Bree2014} \\
& 1207.9 & $4^+_2$ & 700.1 & $(4,5)^+ \rightarrow 4^+_2$ & 326.9 & $4^+_2 \rightarrow 2^+_2$ & 412.9 & $2^+_1 \rightarrow 0^+_1$ & <40 & & \\
& & & & & 795.2 & $4^+_2 \rightarrow 2^+_1$ & & & & & \\
& 1509.2 & $6^+_1$ & 460.7 & $8^+_1 \rightarrow 6^+_1$ & 504.3 & $6^+_1 \rightarrow 4^+_1$ & 592.1 & $4^+_1 \rightarrow 2^+_1$ & <10 & & \\
& 1777.2 & $6^+_2$ & 424.1 & $7^-_1 \rightarrow 6^+_1$ & 772.4 & $6^+_2 \rightarrow 4^+_1$ & 412.9 & $2^+_1 \rightarrow 0^+_1$ & <30 & & \\
& 1909.7 & $5^-_1$ & 385.8 & $6^-_1 \rightarrow 5^-_1$ & 904.8 & $5^-_1 \rightarrow 4^+_1$ & 592.1 & $4^+_1 \rightarrow 2^+_1$ & 10(9) & & \\ \hline
\end{tabular}
\end{center}
\end{table*}
With the exception of the $7^-_1$ state in $^{190}$Hg (see below), lifetimes given as upper limits are the results of uncertainties larger than the measured values. The main reason for this is a large peak-to-background ratio for weak transitions decaying from a level with a short half-life. When this ratio is $\lesssim 1$, the Compton contribution is substantial and the uncertainty induced by the correction from Eq.~\ref{eq:compton_corr} is generally larger than the lifetime.
For the purpose of this work, lifetimes longer than 100~ps have been obtained by the convolution method, fitting Eq.~\ref{eq:convolution_method} to the time distribution. Every lifetime was measured in delayed (positive lifetime) and anti-delayed (negative lifetime) coincidences. Since these are physically different events, every half-life was effectively measured twice. The result shown in Tab.~\ref{tab:lifetimes} is the average of the two values, which in all cases were in good agreement. Figure~\ref{fig:Hg194_7-_lifetime} shows an example of a lifetime extracted using this method. The $6^-_1 \rightarrow 7^-_1$ transition in $^{194}$Hg is used as START and the $5^-_1 \rightarrow 4^+_1$ as STOP. The $7^-_1$ state decays to the $5^-_1$ state with a 56\% branching ratio and the $5^-_1$ state lifetime is known to be in the picoseconds range (see Table~\ref{tab:lifetimes}), so the long tail shown in Fig.~\ref{fig:Hg194_7-_lifetime} can be unambiguously attributed to the $7^-_1$ state. The time distribution was fitted to a Gaussian function convoluted to a double exponential decay. This second exponential decay is introduced to account for the significant background, which in this case has a much shorter lifetime. An additional constant term was introduced to fit the time-random background.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, keepaspectratio]{194Hg_first7minus.pdf}
\caption{\label{fig:Hg194_7-_lifetime} Lifetime of the $7^-_1$ state in $^{194}$Hg. The fit was performed using a Gaussian convoluted with a double exponential decay, to account for the short lifetime background under the peak, plus a constant term for the time-random background.}
\end{center}
\end{figure}
The lifetime of the $7^-_1$ state in $^{190}$Hg was estimated using a different method. The 305.4-keV $7^-_1 \rightarrow 6^+_1$ decaying transition was visible and could be selected in the LaBr$_3$(Ce) spectrum. An additional transition, 731.1-keV $6^+_1 \rightarrow 4^+_1$, was selected in the HPGe array to reduce the background. Since no discernible feeding transition could be used as the gating transition in the LaBr$_3$(Ce) detectors under these conditions, the lifetime was extracted from ZDS-LaBr$_3$(Ce)-HPGe coincidences. This time difference between the $\beta^+$ particle and the $7^-_1 \rightarrow 6^+_1$ transition was a composition of the lifetimes of the $7^-_1$ and all the levels feeding it from above, which in this case are the $(8^-)$ and $(9^-)$ states~\cite{NNDC}. The resulting TAC spectrum showed no delayed component and thus a conservative upper limit of T$_{1/2}<200$~ps was estimated from the FWHM and the lack of slope. From previous experiments (see Refs.~\cite{Bingham1976, Kortelahti1991, DEL94, NNDC}) it has been established that the decay of $^{190m}$Tl favors the $7^-$ state over the $(8^-)$ and $(9^-)$ levels in $\sim 75\%$ of the decays. For this reason, it cannot be discarded that the $(8^-)$ or $(9^-)$ states have a long lifetime which could not be observed under these conditions, therefore no limits have been deduced for them.
With the exception of the lifetime of $2^+_2$ in $^{188}$Hg (discussed in Sec.~\ref{sec:188Hg-2_2-lifetime}) there is excellent agreement between the results obtained in this work and previous measurements (see Fig.~\ref{fig:be2_2} and Table~\ref{tab:lifetimes}). In particular, excellent agreement is observed with the lifetimes of the $2_1^+$ states of $^{196,198,200}$Hg stable isotopes for which lifetimes have been measured a number of times and are precisely known. This is a strong validation of the quality of the results and the ability of the LaBr$_3$(Ce)-GRIFFIN array to measure lifetimes in the picoseconds range using the GCDM. It should be noted here that the $^{198}$Hg $2_1^+$ state literature lifetime was used to calibrate the time walk of the array and reduce the uncertainty of all the other measured lifetimes, but was not used when determining its own value in this work. The agreement for the $7^-_1$ state lifetimes, measured using the convolution method, is poorer than with the other states. For $^{194,196,198}$Hg these values are $\sim 2\sigma$ lower than previous measurements. No satisfactory explanation for this deviation was found.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, keepaspectratio]{be2_2_plot.pdf}
\caption{\label{fig:be2_2} Comparison of the experimentally deduced B(E2;$2^+_1 \rightarrow 0^+_1$) values from this work and previous literature values. Literature values are taken from the evaluated compilation~\cite{PRITYCHENKO2016}, with the exception of $^{190,192,194}$Hg, which are taken from Ref.~\cite{Esmaylzadeh2018}. Note that the error for the $^{200}$Hg B(E2) value evaluated in~\cite{PRITYCHENKO2016} has one too many digits, making it 10 times larger than it should be. This has been corrected in the present plot.}
\end{center}
\end{figure}
\begin{table*}
\caption{Summary of the deduced reduced probabilities from the results of this experiment. For $\Delta$J=0, $\Delta\pi=0$ transitions, the deduced B(M1) and B(E2) values are given assuming pure multipolarities. The exception is the 886.9-keV $2^+_2 \rightarrow 2^+_1$ transition in $^{200}$Hg for which a $\delta^2=-2.20(10)$ was measured previously and has been used in the calculations~\cite{Breitig1974, Ahmad1989}. Energies and branching ratios are taken from Ref.~\cite{NNDC}. All values have been corrected by the conversion electron coefficient from Ref.~\cite{BRICC}.}
\label{tab:reduced_probabilities}
\begin{center}
\begin{tabular}{ccccccccc}
\hline
Isotope & J$^\pi_i$ & T$_{1/2}$ & J$^\pi_f$ & E$_\gamma$ & $\rho^2$(E0)$\times 10^3$ & B(E1) & B(M1) & B(E2) \\
& & (ps) & & (keV) & & W.u. & W.u. & W.u. \\ \hline
$^{200}$Hg & $2^+_1$ & 44(3) & $0^+_1$ & 367.9 & & & & 26(2) \\
& $4^+_1$ & 6(3) & $2^+_1$ & 579.3 & & & & 20(9) \\
& $0^+_2$ & 8(4) & $2^+_1$ & 661.4 & & & & 8(4) \\
& & & $0^+_1$ & 1029.3 & 0.02(1) & & & \\
& $2^+_2$ & 8(6) & $0^+_2$ & 224.8 & & & & 4(3) \\
& & & $4^+_1$ & 306.9 & & & & 0.6(5) \\
& & & $2^+_1$ & 886.2 & & & $5(4) \times 10^{-4}$ & 1.0(8) \\
& & & $0^+_1$ & 1254.1 & & & & 0.10(8) \\ \hline
$^{198}$Hg & $2^+_1$ & 24(3) & $0^+_1$ & 411.8 & & & & 28(4) \\
& $4^+_1$ & <5 & $2^+_1$ & 636.7 & & & & >16 \\
& $5^-_1$ & 57(7) & $4^+_1$ & 587.2 & & $1.7(2) \times 10^{-5}$ & & \\
& $7^-_1$ & 6.6(1) ns & $5^-_1$ & 47.7 & & & & 29.5(5) \\ \hline
$^{196}$Hg & $2^+_1$ & 16(3) & $0^+_1$ & 426.0 & & & & 36(7) \\
& $4^+_1$ & 4(3) & $2^+_1$ & 635.5 & & & & 20(15) \\
& $5^-_1$ & 670(80) & $4^+_1$ & 695.6 & & $8.9(11) \times 10^{-7}$ & & \\
& $7^-_1$ & 4.8(2) ns & $5^-_1$ & 84.3 & & & & 33(1) \\ \hline
$^{194}$Hg & $2^+_1$ & 19(1) & $0^+_1$ & 427.9 & & & & 30(2) \\
& $4^+_1$ & <3 & $2^+_1$ & 636.3 & & & & >27 \\
& $5^-_1$ & 51(6) & $4^+_1$ & 748.9 & & $4.0(1) \times 10^{-6}$ & & \\
& $7^-_1$ & 3.46(3) ns & $5^-_1$ & 97.0 & & & & 34.4(3) \\
& & & $6^+_1$ & 111.0 & & $1.4(5) \times 10^{-5}$ & & \\ \hline
$^{192}$Hg & $2^+_1$ & 12(1) & $0^+_1$ & 422.8 & & & & 51(4) \\
& $4^+_1$ & 4(3) & $2^+_1$ & 634.8 & & & & 21(15) \\
& $6^+_1$ & 73(10) & $4^+_1$ & 745.5 & & & & 0.51(7) \\
& $5^-_1$ & 383(14) & $4^+_1$ & 786.3 & & $1.7(1) \times 10^{-5}$ & & \\
& $7^-_1$ & 1.03(5) ns & $5^-_1$ & 133.1 & & & & 37(2) \\
& & & $6^+_1$ & 174.0 & & $2.38(3) \times 10^{-5}$ & & \\ \hline
$^{190}$Hg & $2^+_1$ & 15(1) & $0^+_1$ & 416.4 & & & & 45(3) \\
& $4^+_1$ & 5(4) & $2^+_1$ & 625.4 & & & & 18(14) \\
& $6^+_1$ & 7(4) & $4^+_1$ & 731.1 & & & & 6(3) \\
& $5^-_1$ & <40 & $4^+_1$ & 839.6 & & $>6 \times 10^{-6}$ & & \\
& $7^-_1$ & <200 & $5^-_1$ & 196.9 & & & & >30 \\
& & & $6^+_1$ & 305.4 & & $>2.4 \times 10^{-5}$ & & \\ \hline
$^{188}$Hg & $2^+_1$ & 14(3) & $0^+_1$ & 412.9 & & & & 50(11) \\
& $2^+_2$ & <20 & $2^+_1$ & 468.2 & & & $>4 \times 10^{-3}$ & >8 \\
& & & $0^+_1$ & 881.1 & & & & >1 \\
& $4^+_1$ & <30 & $2^+_1$ & 592.1 & & & & >4 \\
& $4^+_2$ & <40 & $4^+_1$ & 203.2 & & & $>3 \times 10^{-3}$ & >33 \\
& & & $2^+_2$ & 326.9 & & & & >26 \\
& & & $2^+_1$ & 795.2 & & & & >0.32 \\
& $6^+_1$ & <10 & $(4^+_4)$ & 269.4 & & & & >110 \\
& & & $4^+_2$ & 301.2 & & & & >250 \\
& & & $4^+_1$ & 504.3 & & & & >90 \\
& $6^+_2$ & <30 & $4^+_2$ & 569.3 & & & & >1.1 \\
& & & $4^+_1$ & 772.4 & & & & >0.8 \\
& $5^-_1$ & 10(9) & $4^+_2$ & 701.7 & & $4.1(37) \times 10^{-6}$ & & \\
& & & $4^+_1$ & 904.8 & & $2.6(23) \times 10^{-5}$ & & \\ \hline
\end{tabular}
\end{center}
\end{table*}
\section{Calculations and discussion}
\subsection{Positive-parity yrast states}
The \textit{Z}$\sim$82, \textit{N}$\sim 104$ mid-shell nuclei are beyond reach of most shell-model calculations, with only the state-of-the-art Monte Carlo shell model (MCSM)~\cite{Otsuka2001} having been recently used to calculate some of the most basic properties of the ground and first excited states of $^{177-186}$Hg~\cite{Marsh2018, Sels2019}. In order to study more complex properties of excited states, such as the B(E2) transition strengths, calculations must be carried out in truncated spaces. In the present study, we turn to results of the Interacting Boson Model (IBM) calculations. Two main sets of theoretical results are available for the B(E2) values of the neutron-deficient Hg isotopes; from the IBM-2 calculations~\cite{Nomura2013} and IBM calculations that incorporate configuration mixing (IBM-CM) ~\cite{Garcia-Ramos2014}.
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, keepaspectratio]{theo_vs_exp_2state.pdf}
\caption{\label{fig:exp_vs_theo_2state} Difference between the B(E2;$2^+_1 \rightarrow 0^+_1$) values measured in this work, literature and different IBM calculations. Literature values are taken from the evaluated compilation~\cite{PRITYCHENKO2016}, with the exception of $^{190,192,194}$Hg, which are taken from Ref.~\cite{Esmaylzadeh2018}. IBM-2 are from Ref.~\cite{Nomura2013} and IBM-CM from~\cite{Garcia-Ramos2014}.}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=\columnwidth, keepaspectratio]{theo_vs_exp_4state.pdf}
\caption{\label{fig:exp_vs_theo_4state} Same as Fig.~\ref{fig:exp_vs_theo_2state}, but for the B(E2;$4^+_1 \rightarrow 2^+_1$) values. }
\end{center}
\end{figure}
While both of them are based on the IBM, there are important differences between the two approaches. The IBM-2 calculations treat protons and neutrons independently, as opposed to the traditional IBM calculations (including IBM-CM), which makes no distinction between the different types of nucleon. Both calculations allow for proton excitations across \textit{Z}=82 and the formation of $4h-2p$ (intruder) states and mixing between the normal and intruder configurations. But while IBM-CM includes cross-shell excitations for the whole isotopic chain, the IBM-2 calculations that were performed in Ref.~\cite{Nomura2013} limits the inclusion of intruder states to the mass $^{172-190}$Hg isotopes, and uses a single configuration elsewhere. The other main difference between the two sets of calculations is that IBM-CM fitted the parameters to the (at the time) measured energies and B(E2) of the $2^+_1$ states, while the IBM-2 did not fit to any of the experimental results but mapped the IBM-2 Hamiltonian to results from a self-consistent mean field calculation using a Gogny-D1M energy-density functional.
Figures~\ref{fig:exp_vs_theo_2state} and~\ref{fig:exp_vs_theo_4state} show the difference between B(E2) values deduced from the lifetimes of the $2^+_1$ and $4^+_1$ states measured in the present work and theoretical calculations. Both the theoretical calculations and the experimental results show a smooth increase in B(E2) as neutrons are removed. In $^{200}$Hg, the B(E2;$2^+_1 \rightarrow 0^+_1$) value is $\sim 25$~W.u., which yields a moderate deformation of $\beta_2 = 0.098(2)$. This deformation increases as the mid-shell $N=104$ is approached, reaching a maximum around $^{182}$Hg with $\beta_2 = 0.147(4)$~\cite{PRITYCHENKO2016}. The new results are fully consistent with this picture but greatly improve the precision of these measurements for $^{190,192,194}$Hg.
Esmaylzadeh and collaborators~\cite{Esmaylzadeh2018} claimed a better agreement between the B(E2;$2^+_1 \rightarrow 0^+_1$) values they measured and the IBM-2 calculations, mainly because of the discrepancy they observed for $^{194}$Hg. The new, more precise values, indicate the opposite conclusion, a significantly better agreement with the IBM-CM is found, rather than with the IBM-2. The agreement between the IBM-CM predictions and the new data are well within one $\sigma$, with the exception of $^{192}$Hg. In this case, the B(E2;$2^+_1 \rightarrow 0^+_1$) value of $^{192}$Hg seems to have been underestimated by the IBM-CM and overestimated by the IBM-2 calculations.
It is significant that the dip experimentally observed in the B(E2;$2^+_1 \rightarrow 0^+_1$) value of $^{194}$Hg is nicely reproduced when the configuration mixing is included in the calculations (IBM-CM clearly reproduces the staggering while IBM-2, which has no configuration mixing for this mass, does not). The IBM calculations do not include any type of sub-shell structure, but they are still able to reproduce this irregularity in the otherwise smooth evolution of the B(E2;$2^+_1 \rightarrow 0^+_1$) values. While subtle effects can, of course, be introduced through the fitting of the IBM parameters to the energies of the states, it is nonetheless remarkable that the staggering of the B(E2) values is reproduced so well. (When the fits in Ref.~\cite{Garcia-Ramos2014} were done, none of the relevant B(E2) values in $^{190-194}$Hg were known.) A similar dip is observed in the evolution of the B(E2;$4^+_1 \rightarrow 2^+_1$) values also for $^{194}$Hg, where the lifetime measured by Esmaylzadeh and collaborators~\cite{Esmaylzadeh2018} certainly hints to the possibility of this staggering being present also for the $4^+_1$ state systematic.
On the other hand, the upper limits obtained for the lifetimes of the $4^+_1$ states (lower limits for the B(E2;$4^+_1 \rightarrow 2^+_1$)), are not stringent enough to distinguish between either of the models. The value for $^{192}$Hg (and maybe $^{196}$Hg) presented in this work and the value for $^{194}$Hg presented in~\cite{Esmaylzadeh2018} seem to favor the results from IBM-CM calculations, but no definitive conclusion can be achieved. More precise measurements of these lifetimes are required for a full description of the nuclei.
It is important to note that when the IBM-CM calculations were performed~\cite{Garcia-Ramos2014}, no lifetime information for $^{190,192,194}$Hg existed, so their B(E2) values were not included in the normalization or constraining of the calculations. It is, thus, remarkable, how the predictions of this set of calculations fit the measured values for $^{190}$Hg and $^{194}$Hg. The IBM-2 is not adjusted to experimental data as it is based on the fully-microscopic energy density functional calculation, so its ability to reproduce the general trend of the B(E2) values is significant. The great predictive power of these IBM calculations seems to validate their results of the minimal mixing between normal and intruder configurations for $^{192}$Hg and heavier isotopes. This confirms the hypothesis of studying these isotopes to benchmark the normal configuration free of perturbations from the intruder one, which in turn should facilitate the study of shape coexistance in the lighter ones.
\subsection{Negative-parity band \label{sec:negative_band}}
The yrast negative-parity band in the Hg isotopes has been successfully explained with a model of two quasi-particles coupled to an oblate rotor. One of the quasi-particles has a high spin ($11/2$ or $13/2$, from the $\pi h_{11/2}$ or the $\nu i_{13/2}$ orbitals, respectively) with a spherical wave function, while the other quasi-particle is of low spin ($\leq 5/2$, from the $pf$ shell) with a deformed Nilson wave function which is the combination of several configurations~\cite{Flaum1974,Neergard1975,Toki1977,Levon2006}.
In contrast, Beuscher \textit{et al.}~\cite{Beuscher1974} concluded that these structures are collective rotational bands involving many particles, not just the suggested two-particle coupling, because states up to $15^-$ were observed. This high spin can not be formed with just two-particles coupled to a core.
The lack of an intense $\gamma$ ray feeding the $7^-$ state in $^{190}$Hg prevented a precise measurement of the lifetime. Only a limit of $<200$~ps was obtained, which fits with the systematic of the chain.
\subsection{Comment on B$_{4/2}$ }
According to the Alaga rules, the ratio B$_{4/2}$ = B(E2;$4^+_1 \rightarrow 2^+_1$)/B(E2;$2^+_1 \rightarrow 0^+_\text{g.s.}$) is strictly larger than one. For an ideal rotor, B$_{4/2} = 10/7\sim1.43$, while for a harmonic vibrator, B$_{4/2}$ has an exact value of~2. In the current description of nuclear structure, B$_{4/2}$ can only have a value lower than 1 for structures conserving seniority (almost only found in semi-magic nuclei) and, in principle nuclei having shape-coexistence, but no example of the latter has been observed so far.
Cakirli and collaborators~\cite{Cakirli2004} carried out an extensive survey and found a few isolated cases with B$_{4/2}<1$ that could not be explained by either seniority or shape-coexistence. Subsequent experiments have re-measured with greater accuracy some of the most relevant of those isotopes and found important discrepancies for B(E2;$4^+_1 \rightarrow 2^+_1$) that made the B$_{4/2}$ values significantly larger than 1~\cite{Williams2006,Radeck2012,Zhu2017}.
Recent works~\cite{Cederwall2018,Esmaylzadeh2018} have suggested that the transitional neutron-deficient Hg isotopes could have B$_{4/2}$ values lower than 1. These suggestions arise from the $4^+_1$ half-life ($T_{1/2}$=7.2(3)~ps) quoted for $^{198}$Hg in the current edition of the Nuclear Data Sheets~\cite{Xialong2009,HUANG2016}, which in turn yields B$_{4/2}$=0.38(14). But the works cited in the compilation measured B(E2;$4^+_1 \rightarrow 2^+_1$) of 0.296(13)~$e^2b^2$~\cite{Bockisch1979} and 0.307(24)~$e^2b^2$~\cite{Gunther1981}, in perfect agreement, yielding $T_{1/2}$=1.80(8)~ps, which returns B$_{4/2}$=1.56(19). Moreover, this is the evaluated value from previous editions of the Nuclear Data Sheets~\cite{CHUNMEI2002}. To the best knowledge of the authors, no new work has been published that supports the 7.2(3)~ps half-life, and thus it should be replaced back in the compilations for the previous $T_{1/2}$=1.80(8)~ps one, value that is in agreement with the upper limit measured in this work. Likewise, all the B$_{4/2}$ values obtained from this work (see Tab.~\ref{tab:lifetimes}), are, within uncertainties, above 1. This includes the T$_{1/2}(4^+_1)$ upper limit of $^{194}$Hg measured in this work, which, as opposed to the results presented in Ref.~\cite{Esmaylzadeh2018}, seems to favour B$_{4/2}>1$. The results presented in this work, thus, negate the hypothetical deviation from the current model of nuclear structure, at least for this isotopic chain.
\subsection{Lifetime of $2_2^+$ in $^{188}$Hg \label{sec:188Hg-2_2-lifetime}}
The literature value for the lifetime of the $2_2^+$ state in $^{188}$Hg was previously measured to be 141(31)~ps by Joshi \textit{et al}~\cite{JOSHI1994}. This value cannot be reconciled with the one observed in this work of $T_{1/2}<20$~ps. They used a different variation of the advanced time-delayed method (Ref.~\cite{Mach1989}), described in Ref.~\cite{Joshi1993}. Their method relies on measuring the time difference between the x-ray created by the electron capture and the one created by the conversion electron, plus the detection of said conversion electron in a Si(Li) detector to select a specific decay branch. Since all x-rays from the same isotope have the same energy, that method does not have the ability to distinguish between delayed and anti-delayed coincidences. Thus, instead of measuring the centroid shift between the delayed and anti-delayed coincidences, that method assumes that an increase of the width of the time difference measured by the TAC is proportional to a lifetime between the two detected x-rays. Since the START signal is given by the x-ray of the electron capture decay, the measurement of a lifetime with that method is susceptible of contribution from higher-lying states that $\gamma$-cascade into the measured one.
The GCDM method described in Section~\ref{sec:data_analysis} and employed in this work involves coincidence gates on specific $\gamma$-rays, not x-rays. This grants it the unambiguous selectivity of measuring the time difference between feeding and decaying $\gamma$-rays of a specific level, thus measuring its lifetime without the contribution of other levels. Moreover, distinguishing the delayed and anti-delayed coincidences allows for a more precise measurement than just the variation of the time difference distribution width. For these reasons, the authors believe the GCDM method to be more reliable than the one described by Joshi \textit{et al.} and the half-life limit presented in this work to be more solid.
\subsection{$\rho^2$ value of the $0_2^+\rightarrow 0_1^+$ in $^{200}$Hg}
The new half life measurement of the first excited $0^+$ state in $^{200}$Hg allows the electric monopole transition strength of the transition to the ground state to be determined for the first time. The $\rho^2$(E0) value of this $0_2^+\rightarrow 0_1^+$ transition is calculated to be $0.02(1)\times 10^{-3}$ based on the new half life of 8(4)\,ps. This is a fairly small value which compares well to others known in the local region which have been reported by the evaluation of Kib\'{e}di~\cite{Kibedi2005}. For example the $0^+ \rightarrow 0^+$ transition in $^{188}_{76}$Os$_{112}$ has a $\rho^2$(E0) value of 0.011(4)\,milliunits, and the $^{194}_{78}$Pt$_{116}$ and $^{196}_{78}$Pt$_{118}$ isotopes have values reported as $<$0.17 and 0.19(10)\,milliunits, respectively.
This new value in $^{200}$Hg is an excellent benchmark of the $E0$ strength in a mercury isotope that is away from the shape coexistence region around the neutron mid-shell of $N \approx 104$. The lighter Hg isotopes display significantly larger $\rho^2$ values where the nature of the first excited 0$^+$ state is significantly different and the energies much closer. This shape coexistence scenario is responsible for driving the large $E0$ strength.
This further supports the configuration mixing scenario discussed in earlier sections. Large $\rho^2$(E0) are indicative of strong mixing between $0^+$ states ~\cite{Kibedi2005}. Both sets of calculations, IBM-2~\cite{Nomura2013} and IBM-CM~\cite{Garcia-Ramos2014}, predicted strong mixing for the lighter Hg isotopes and negligible for the heavier ones (IBM-2 is able to accurately reproduce the B(E2) values for $^{194,196,198,200}$Hg without including any mixing). This is confirmed by small $\rho^2$(E0) measured in this work for $^{200}$Hg, where no mixing is expected, in contrast with the strong $\rho^2$(E0) observed near the \textit{N}=104 mid-shell~\cite{HEY11}, where the mixing is much stronger.
\section{Conclusions}
Using the LaBr$_3$(Ce) detector array of the GRIFFIN spectrometer at the TRIUMF-ISAC facility, we have carried out a systematic study of the transitional even $A=188-200$ mercury isotopes. The present work focused on extracting lifetimes in the pico- to nanosecond range using the GCDM. A total of 33 lifetimes were measured, 10 of them for the first time. Overall, very good agreement was found between the new results and previous measurements, with a significant improvement in precision for many cases.
This increased precision allowed for meaningful comparison with IBM-2 and IBM-CM calculations. There is an excellent agreement between the deduced B(E2;$2^+_1 \rightarrow 0^+_1$) values from this work and the IBM-CM calculation. The lifetimes of the $4^+_1$ states were too short for GCDM, resulting in large relative error bars that prevented comparison to the calculations.
Both IBM studies predicted shape coexistence in light Hg isotopes up to $^{188}$Hg, with maybe a weak effect in $^{190}$Hg. The new, more precise results presented in this work seem to validate this hypothesis, confirming the minimal mixing between normal and intruder structures for $^{192-200}$Hg. The ongoing analysis of the conversion electrons and $\gamma-\gamma$ angular correlations data collected in this experiment will shed more light into the evolution of configuration mixing in the Hg isotopic chain.
\begin{acknowledgments}
We would like to thank the operations and beam delivery staff at TRIUMF for providing the radioactive beam. We are grateful to J.~E.~Garc\'ia-Ramos (IBM-CM~\cite{Garcia-Ramos2014}) and K.~Nomura (IBM-2~\cite{Nomura2013}) for providing the results of their calculations and for fruitful discussions. The GRIFFIN infrastructure has been funded jointly by the Canada Foundation for Innovation, the British Columbia Knowledge Development Fund (BCKDF), the Ontario Ministry of Research and Innovation (ON-MRI), TRIUMF and the University of Guelph. TRIUMF receives funding through a contribution agreement through the National Research Council Canada. C.E.S. acknowledges support from the Canada Research Chairs program. This work was supported by the Natural Sciences and Engineering Research Council of Canada.
\end{acknowledgments}
|
1,108,101,565,706 | arxiv | \section{Introduction}
The modelling of financial markets and other social systems by means
of individual-based simulations has attracted a significant amount of
attention in recent years, both in the economics and socio-economics
community, as well as among statistical physicists
\cite{Axelrod,Gilbert&Troitzsch,LLS, ChalMarsZhanBook,CoolBook,JohnJefHui03,Mantegna&Stanley}. The range of phenomena to which
this approach is applied is broad, and includes not only the modelling
of traders in financial markets, but also opinion formation and
decision dynamics, epidemic spreading, the behaviour of crowds and
pedestrians, as well as vehicular traffic and co-operative behaviour
in animal swarms \cite{helbing1,helbing2,schreck,barabasi,bak}. The
term `individual' in individual-based approaches to the modelling of
such systems may here stand for traders in a financial market, cars on
a highway or companies in an economic network. While practitioners
often make use of sophisticated models with a variety of different
types of agents, statistical physicists have become interested in
minimalist models of such phenomena, which ideally allow for
analytical solutions. While the detail and complexity of more
realistic models often impedes analytical approaches, the reduced
models considered by physicists are often tractable with methods from
statistical physics, and their behaviour can be characterised
analytically for example by exact or approximative calculation of
their phase diagrams. Furthermore the restriction to minimalist models
allows one to focus on the parameters and features of the model which
are truly responsible for its behaviour (e.g. by systematic adding or
removal of individual features), whereas the underlying mechanisms may
be obscured and thus harder to detect in more sophisticated and
detailed models. Apart from the methodology, concepts of physics often
allow one to shed light on phenomena in adjacent fields and to address
them from novel viewpoints. The concepts of self-organised criticality
\cite{bak} or replica symmetry breaking and the corresponding rugged
landscape picture of spin glass physics may here serve as examples
\cite{PaMeVi}. Agent-based models and models from statistical physics
share common features in that in both cases one is interested in
complex co-operative behaviour on a {\em macroscopic} scale emerging
from relatively simple interactions at the {\em microscopic} level,
i.e. between the interacting agents. One may thus expect that
techniques and methods developed originally in the context of physics
may be successfully employed in other disciplines as well in order to
study complex many-agent systems and their emergent collective
phenomena.
This review focuses on some such approaches in the context of the
modelling of financial markets by agent-based trading models. Although
the statistical physics analysis is limited to minimalist models,
several such model systems have been seen to show remarkably complex
behaviour, including phase transitions, non-trivial co-operation and
global adaptation
\cite{ChalMarsZhanBook,CoolBook,JohnJefHui03,Mantegna&Stanley}. From
the point of view of the description of real-world phenomena they also
display what is referred to as `stylised facts' in the economics
literature, namely anomalous non-Gaussian behaviour and fluctuations
in their global observables, similar to statistical features seen in
real market data. This positions the agent-based models considered by
statistical physicists right at the boundary between being
analytically solvable and at the same time sufficiently realistic, and
constitutes some of the appeal of such models.
The objective of this chapter is to review some of the developments in
the study of agent-based models, stressing the emergence of anomalous
behaviour. While some, potentially non-representative focus is given
to Minority Game market models and the authors' own work, we also
present a brief discussion of some other related agent-based models.
\section{Overview and stylised facts in financial markets}
\subsection{Introduction}
`Stylised facts' in the context of finance is a term used by
theoretical economists and by practitioners to summarise statistical
features of time series generated by financial markets. Most notably
financial markets generate non-Gaussian time series with power-law
characteristics, long-range correlations in time, scaling behaviour
and anomalous fluctuations
\cite{Mantegna&Stanley,Bouchaud&Potters}. Non-Gaussian behaviour is
indeed found in many different contexts, already more than 100 years
ago, Vilfredo Pareto \cite{Pareto} pointed out that in numerous
countries and at various points in time the wealth distribution
follows a power-law, i.e. $N\sim w^{-\alpha}$, where $N$ is the number
of people having wealth greater than $w$. Over the past decades
similar power-law behaviour has been detected in numerous other
phenomena, ranging from financial time series, to sand piles, the
distribution of word frequency in texts, citations of scientific
papers, hits on web pages, firm sizes, populations of cities so forth
\cite{bak,newman}.
Anomalous, non-Gaussian fluctuations are not only observed in the
above systems, apparently unrelated to physics, but also in systems of
condensed matter physics, and in models of statistical mechanics.
Anomalous behaviour is here found in a variety of contexts, from
magnetic materials, over solutions of polymers, gels, glasses, fluid-
and superfluid helium, water at its boiling point, and even in
experiments of elementary particle physics and in cosmology
\cite{Kadanoff,Stanley}. The common feature of these systems is that they display phase transitions, i.e. depending on external control parameters (such as
temperature or pressure) the behaviour of one given system can be of
different types. Not only do the characteristics of these systems
change drastically at their transition points, but thermal
fluctuations attain non-Gaussian features, very similar to those
observed in unphysical systems mentioned above. Close to their
critical points systems of condensed matter physics display long-range
correlations, with an algebraic as opposed to an exponential decay of
spatial and temporal correlations, leading to large-scale collective
behaviour, ergodicity breaking and other so-called `anomalous'
characteristics. Power-law correlation functions are here a reflection
of scale-free behaviour of the system, i.e. of fractal properties and
self-similarity. This invariance under re-scaling of units has led to
the development of powerful methods of statistical physics, most
notably of renormalisation group, which allow to address critical
phenomena efficiently \cite{Kadanoff,Stanley,Cardy}.
\subsection{Anomalous fluctuations in financial markets}
Power-law statistics and non-Gaussian behaviour have been detected in
a variety of different markets, and are mostly independent of the
specific trading rules or external circumstances at the given market
place
\cite{Mantegna&Stanley}. For example, if we denote the
price of an asset or an index by $X(t)$, then the log-return at time
scale $\Delta_t$ is defined as
$r_t=\ln\left(X(t)/X(t-\Delta_t)\right)$. Empirical studies show that
the distribution of log-returns has power-law tails with exponents
between $3$ and $5$ \footnote{This result holds for different time
scales $\Delta_t$.}\cite{Gop&al}. Another interesting feature which
is observed in financial markets is the so-called {\em volatility
clustering}: while the correlation of returns decays relatively
quickly in time, the correlation of the absolute log-returns, defined
as $C_{|r|}(\tau)=\left\langle |r(t+\tau)||r(t)|\right\rangle $,
follows a long-range power-law distribution with an exponent ranging
from $0.1$ to $0.3$ \cite{Lo,Liu&al}. Non-Gaussian distributions are
also observed when looking at the trading volume or the number of
trades per time. We illustrate these observations in
Fig. \ref{fig:sptimeseries}, where a financial time-series with a
characteristic non-Gaussian behaviour is shown, along with its
algebraically decaying return distribution.
Given the abundance of financial data physicists have become involved
in the analysis of time series of markets, as well as in research
aiming to model the processes generating these data. This field at the
borderline of statistical physics, econometrics and financial
mathematics is now known as {\em econophysics} \cite{chalweb}. The attempts to model
financial markets here roughly fall into two classes, phenomenological
models and agent-based approaches. In the next section we will briefly
discuss both, and then concentrate on the latter agent-based approach,
as mentioned in the introduction. We will here mostly focus on qualitative aspects, and will not enter mathematical details of the various models. For a recent textbook on mathematical aspects of MGs and related models see also \cite{CoolBook}.
\begin{figure}[t]
\begin{center}
\vspace{10mm}
\includegraphics[width=5cm,angle=270]{ret.eps} ~~~
\includegraphics[width=5cm,angle=270]{pdfreturns.eps}
\end{center}
\caption{{\bf Left:} Sequence of 10$\,$min returns of the S\&P 500 index (top), 1$\,$month returns (middle), and a realisation of a Gaussian random walk for comparison. {\bf Right:} Linear-log plots of the probability distribution for the normalized S\&P500 returns. The solid lines
are power-law fits with exponents $1+ \alpha \approx 4$. Figures taken
from \cite{Gop&al}.\label{fig:sptimeseries}}
\end{figure}
\section{The agent-based approach}
\subsection{Random walks and other phenomenological models}
Many approaches to anomalous statistical properties of financial data
are based on direct modelling of stock market indices as stochastic
processes. Such models do not deal with the market on a microscopic
level, i.e. with individual agents, but with aggregate quantities
only, such as the stock market indices, their return statistics and
fluctuations on a macroscopic level. The very first of such theories
dates back to Bachelier's `Th\'eorie de la Sp\'eculation' from 1900
\cite{bachelier} and describes the stochastic evolution of market
indices as simple Gaussian random walks. While it is now well
established that market data is fundamentally non-Gaussian,
Bachelier's theory of financial markets as log-normal random walks
still forms the basis of many phenomenological approaches to
finance. Most notably, the Black-Scholes theory \cite{BS} commonly
used for the pricing of financial derivatives relies entirely on
Gaussian assumptions, and hence on Bachelier's approach.
More realistic approaches to finance with stochastic processes are of
course no longer based on purely Gaussian processes, but use L\'evy
flights and related processes to model the temporal evolution of
financial indicators \cite{Bouchaud&Potters}. Many different classes
of stochastic processes have here been used, partially relying on the
fact that multiplicative as opposed to additive Gaussian noise
directly leads to power-law behaviour.
\subsection{Agent-based models}
While phenomenological models can describe many features observed in
real markets, they are at least to some extent not fully satisfactory
in that they do not allow for a derivation of global quantities from
the most basic level of description, namely the direct interaction
between agents. These phenomenological and agent-based approaches are,
to some degree, analogous to the description of physical phenomena
through thermodynamics and statistical physics respectively. While
the laws of thermodynamics, such as for example the phenomenological
equation describing an ideal gas, find their justification mostly in
the purely empirical observation, only a direct kinetic theory allows
for a description on the level of individual molecules. During 19th
century such a kinetic theory was being developed, starting from
Newton's laws and using probabilistic arguments, and results from
thermodynamics were re-derived from first principles, marking the
beginning of statistical mechanics.
One can think of phenomenological models of financial markets as the
analogue of the description of physical phenomena on the level of
thermodynamics. Empirical laws like the ideal gas equation here
correspond to the stylised facts empirically found in time series of
financial markets. The aim of phenomenological models is thus, in
essence, to devise and study stochastic processes for macroscopic
observables, without considering the detailed microscopic interaction which bring about the global phenomena.
The counterpart of statistical mechanics would then be given by models
which start from the level of the individual interacting particles of
a financial market, i.e. the traders. While in physics particles can
be considered the basic elements, in markets this role is played by
financial agents. The objective of statistical mechanics approaches
to financial markets is thus to study the interactions between
individual agents, and to derive equations describing global
quantities from these microscopic rules of engagement. Naturally the
analogy between particles and agents is limited, for example because
agents are intelligent and particles are of course not. However,
concepts and techniques from physics can be transferred to the study
of agent-based systems, which mathematically turn out to be
surprisingly similar to models of statistical physics.
The remainder of this review will focus on agent-based models. These
allow one to address fundamental questions for example as to where
anomalous fluctuations may actually come from on the level of the
behaviour of the agents. Agent-based models of financial markets have
attracted substantial attraction, both in the economics and in the
physics communities. They easily accessible by computer simulations,
so that effects of changes in the rules of engagement of the agents,
the ecology of the model-market or variations in external control
parameters are readily examined. The role of physicists has here
mostly been to devise simple minimalist models of financial markets,
which are also accessible analytically and conceptually by tools of
statistical mechanics.
\section{Minority Games and related models}
\subsection{Introduction}
Neoclassical economics usually assumes that agents have perfect, logic
and deductive rationality, as well as full information about the
market \cite{walras,merton,samuelson}. This implies, among other
things, that they are homogeneous and that they act always in the best
possible way. Common sense suggests that this view is quite far from
reality and for different reasons people are neither completely
rational nor deductive and homogeneous. Starting from this
observation, leading economist Brian Arthur introduced a very simple
model, now known as the `El Farol bar problem', to illustrate bounded
rationality and inductive reasoning. In Arthur's model agents have
only limited information and modes of reacting to it at hand, and
learn inductively from past experience. The mathematical abstraction
of this model resulted in the Minority Game (MG) \cite{ChalZhan97},
which now also serves as one of the most popular toy models for
financial markets.
Despite its simplicity, the MG exhibits a very rich behaviour, with
phases in which the resulting model-market behaves fully inefficiently
and others in which the time-series generated by the market still has
some information-content, leading to an effective predictability. As a
function of the model parameters, the agents can either co-operate
successfully and achieve total gains higher than if they were to play
at random, or in other regimes fail to adapt successfully, resulting
in high losses.
In its original formulation the statistics of the MG are mostly
Gaussian (if large populations of agents are considered), and there
are no signs of volatility clustering and other stylised facts in the
time-series generated by the most basic MGs. Small modifications,
however, can produce the fat tailed distributions similar to those
observed in financial markets. Apart from the appeal of the MG as a market model, it has also attracted a significant interest as a model of statistical mechanics, and the methods of theoretical spin glass physics have been seen to provide insight into its phase behaviour, and have revealed new types of global co-operation, complexity and phase transitions, previously unknown in models of disordered systems theory \cite{ChalMarsZhanBook,CoolBook}.
\subsection{Definition of the MG model}
The MG is a mathematical abstraction of the El-Farol bar problem. In
the latter a group of say $100$ agents have to decide whether or not
to attend a bar at a given night every week. In general people will
only enjoy being at the bar if not more than say $60$ people attend
that night, as otherwise the bar will be too crowded. Every agent has
to decide independently every week whether to go or not, and has to
make that decision on the basis of the time-series of previous
attendances. El-Farol agents in Arthur's model are agents of bounded
rationality who learn inductively. Each of them holds a set of
predictors, mapping the past attendances onto a future (predicted) one,
and based on these idiosyncratic predictions the individual agents
decide on whether or not to attend the bar.
The basic MG model describes this systems as a population of $N$
agents (with $N$ an odd integer), in which every player $i$ has to
take a binary decision $b_i(t)\in\{-1,1\}$ at every time-step $t$. The
total attendance is then computed as
\begin{equation}
A(t)=\sum_{i=1}^N b_i(t),
\end{equation}
and agents who take the minority decision are rewarded after each
round, that is if the number of agents playing $1$ is higher than the
number of those playing $-1$ then all $-1$-players are rewarded and
vice versa. In order to take their decisions agents are each equipped
with a pool of $S$ so-called `strategy tables'. Assuming that
agents can remember the correct minority decisions of the last $M$
rounds, that is they can resolve $P=2^M$ different histories of the
game, a strategy $a_{is}$ provides a map from all possible $M$-step
histories onto the binary set $\{-1,1\}$, i. e.
\begin{equation}
a_{is}:\{1,2...,P\}\rightarrow \{-1,1\}.
\end{equation}
In order to take their trading decisions agents follow an inductive
learning dynamics. They are limited to the strategies assigned to them
at the beginning of the game, reflecting the boundedness of their
rationality. These strategy assignments are typically generated at
random, and result in a heterogeneous population of traders. At each
time step each agent has to choose which of his strategy tables to
use, and the agents do so by learning from past experience. A score
$U_{i,s}$ (utility function in terms of economics) is assigned to each
strategy table and updated at each step of the game as follows:
\begin{equation}\label{eq:mgupdate}
U_{i,s}(t+1)=U_{i,s}(t)-a_{i,s}^{\mu(t)}A(t).
\end{equation}
Here $a_{i,s}^{\mu(t)}$ is the action prescribed by the strategy $s$
of agent $i$ when the history $\mu(t)$ occurs. The term
$-a_{i,s}^{\mu(t)}A(t)$ reflects the minority rule: when
$a_{i,s}^{\mu(t)}$ and $A(t)$ have opposite signs, i.e. if the strategy
predicted the correct minority decision, then the strategy score
$U_{i,s}$ is increased. If $a_{i,s}^{\mu(t)}A(t)>0$, the score is
decreased. Each agent then at each time step uses the strategy table
in his pool of strategies with the highest score, i.e. the one that
would have performed best so far, had this particular player always
used the same strategy. In case of equal scores between some
strategies, the agent chooses between those at random.
\subsection{Motivation as market model and price process}
The MG can be understood as a simplified emulation of a financial
market. The binary decisions of the agents are interpreted as `buying'
versus `selling', and hence the aggregate action corresponds to an
`excess demand' in this model market, and leads to movements of the
(logarithmic) price. More precisely, if $p(t)$ is the price of the
good traded on the MG-market, then $\log p(t+1)=\log
p(t)+\frac{A(t)}{\lambda}$, with $A(t)$ the above aggregate action
$A(t)=\sum_{i=1}^N b_i(t)$. $\lambda$ here reflects the liquidity of
the market and represents a proportionality constant. Traders in the
MG take their decisions based on commonly available information, which
may be related to the past price history of the market under
consideration or to information provided externally. The payoff
structure of the MG can here be derived from expectations of the
agents on future price evolution \cite{Marsili}. Minority game agents
effectively correspond to contrarian traders who expect the price to
go down in step $t+1$ if it went up in step $t$. It is straightforward
to extend the model to majority game or mixed minority/majority games
(in which some agents play an minority game, and others a majority
game)
\cite{DeMaGiarMose03,CoolBook}. Majority game agents here
represent so-called `trend-followers' in a real market, their
expectation is that the price movement of time step $t+1$ will be in
the same direction as at time $t$, so that it is favourable to buy
when the majority of players is buying, and to sell whenever the
majority is selling.
\subsection{Behaviour and main feature of the MG}
We will now turn to a brief discussion of the main phenomenology of
the MG. While it follows directly from symmetry arguments that the
time average of total bid vanishes for large system sizes
(i.e. $<A(t)>=0$), the fluctuations of $A(t)$ display a remarkably
complex behaviour when looked at as a function of the memory
capacities of the agents. It was observed in \cite{SaviManuRiol99}
that the ratio $\alpha=P/N$ is the relevant scaling parameter of the
model, i.e. that $\sigma^2/N=\overline{<A(t)^2>}/N$ \footnote{Here
$<\dots>$ stands for a temporal average, while $\overline{\cdots}$ is
the notation used for an average over the quenched disorder, i.e the
strategy assignments.} and other observables depend on the memory
length $M=\log_2 P$ and the system size only through the ratio
$\alpha=P/N$. The variance $\sigma^2$ of $A(t)$, also referred to as
the {\em volatility} is a measure for the efficiency of the game in
terms of co-operation and global gain. The smaller $\sigma^2$ the
smaller is the group of losing agents (i.e. the majority group), with
$A(t)\approx 0$ (i.e. $\sigma^2 \approx 0$) corresponding to a
situation in which close to half of the agents take either trading
decision, i.e. with roughly $N/2$ agents losing and the other $N/2$
agents winners (up to corrections to lower order in $N$). A detailed
analysis shows that the global gains are in fact given by $-\sigma^2$,
due to the inherent negative-sum character of the MG global gains are
always non-positive.
The behaviour of the re-scaled volatility $\sigma^2/N$ as a function
of $\alpha$ is depicted in Fig. \ref{fig:basic}. For large $\alpha$
one finds $\sigma^2/N\approx 1$ which corresponds to the so-called
`random trading limit'. As $\alpha$ is decreased one finds a minimum
of $\sigma^2/N$ at intermediate $\alpha$, and a large volatility
$\sigma^2/N>1$ as $\alpha\to 0$ (for zero initial conditions). In this
regime the co-operation among MG agents is worse than the random
trading limit, mostly due to collective motion of whole crowds of
agents taking the same trading decision (and of the corresponding
anti-crowds) \cite{JohnJefHui03,Jefferies&al,Johnson&al}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=14pc]{h.eps}~~~~~~~~~~~~\includegraphics[width=14pc]{s.eps}
\end{center}
\caption{(Colour on-line) {\bf Left:} Predictability as a function of $\alpha$ for the standard MG for {\em tabula rasa} starts (open markers) and biased initial conditions (solid markers). {\bf Right:} Volatility.}
\label{fig:basic}
\end{figure}
Further insight can be obtained by defining the quantity
\begin{equation}
H=\sum_{\mu=1}^P \left\langle A|\mu \right\rangle^2,
\end{equation}
where the $<A|\mu>$ is the average of $A$ conditional to the occurrence
of a particular history string $\mu$. $H$ is here a measure for the
predictability of the system: if $<A|\mu>\neq 0 $ for at least one
$\mu$ then the system is predictable (in a probabilistic sense). $H=0$ thus corresponds to a fully information-efficient market, in which no exploitable information is contained in the market's time series. If $H>0$, however, statistical predictability is present, and the market is not operating at full information efficiency.
Plotting $H/N$ as a function of $\alpha$ (see Fig. \ref{fig:basic})
one finds that the value $\alpha_c$ marking the minimum of
$\sigma^2/N$ corresponds to a transition point between a predictable
phase at $\alpha>\alpha_c$ and a fully information-efficient one at
$\alpha<\alpha_c$.
At the same time, $\alpha_c$ also marks an ergodic/non-ergodic phase
transition. This can be demonstrated upon plotting the volatility
$\sigma^2/N$ as a function of $\alpha$ obtained from simulations of
the MG learning dynamics started from different initial
conditions. One here distinguishes so-called `tabula rasa' initial
conditions, for which all strategy scores are set to zero at the
beginning of the game, and so-called `biased starts', where some
strategies receive random non-zero score valuations $U_{is}(t=0)=\pm
u_0$ at the beginning (with $u_0\sim{\cal O}(1)$). A corresponding
plot is shown in Fig. \ref{fig:basic}, and demonstrates that the
system is insensitive to initial conditions above $\alpha_c$, but that
the starting point becomes relevant in the information-efficient phase
below the phase transition.
\subsection{The statistical mechanics approach}
Since its mathematical formulation in 1997 \cite{ChalZhan97} the MG
has attracted significant attention in the statistical physics
community. In particular it has been found that the MG is accessible
by the tools of the theory of disordered systems and spin-glasses
\cite{PaMeVi,FiHe}, and that its phase diagram and key observables can
be computed either exactly or in good approximation by methods of
equilibrium and non-equilibrium statistical mechanics. The analogy to
disordered systems, which have studied in the spin-glass and neural
networks communities for $30$ years, is here threefold: firstly, the
random strategy assignments which are drawn at the start of the MG and
through which the agents interact with each other correspond to
`quenched disorder' in statistical physics, that is frozen, fixed
interactions between the microscopic degrees of freedom. In spin
glasses the interactions are specified by the couplings between
magnetic spins, in neural networks they are reflected by fixed
synaptic structures governing the firing patterns of the network of
neurons. Secondly, the MG displays global frustration (not everybody
can win), similar to spin-glass models where not all interactions
between spins can be satisfied. Thirdly, in the MG every agent
interacts with everybody else (through the observation of the
aggregate action $A(t)=\sum_i b_i(t)$ which is used to update strategy
scores). This makes the MG a `mean-field' model in the language of
physics, and tools to address such systems are readily available.
We will here not enter the details of the mathematical analysis of the
MG, but would only like to mention briefly that it can be addressed by both
static methods such as replica techniques \cite{PaMeVi}, and by
dynamical approaches relying on path-integrals and dynamical
generating functionals \cite{CoolBook,HeimCool01}. We will briefly sketch the ideas of both approaches in the following two subsections.
\subsubsection{Statics: replica theory}
The starting point of the static replica approach to the standard MG
is the function $H$ as defined above \cite{ChalMarsZhanBook}. Although
$H$ is not an exact Lyapunov function of the MG learning dynamics, $H$
can be seen as a `pseudo-Hamiltonian', with the ergodic stationary
states of the MG corresponding to extrema of $H$. If one restricts the
discussion to a two-strategy MG (that is each agent $i$ holds only two
strategy tables $a_{is}$ with $s=\pm 1$), then $H$ turns out to be
given by
\begin{equation}
H=\frac{1}{\alpha N}\sum_{\mu=1}^{\alpha N} \left(\sum_i\{\omega_i^\mu+\xi_i^\mu m_i\}\right)^2
\end{equation}
where $\xi_i^\mu=\frac{1}{2}(a_{i,1}^\mu-a_{i,-1}^\mu)$ and
$\omega_i^\mu=\frac{1}{2}(a_{i,1}^\mu+a_{i,-1}^\mu)$. The $\{m_i\}$
are here soft spin variables $m_i\in[-1,1]$ and (up to normalisation)
correspond to the relative frequencies with which each player employs
each of his two strategies.
$H$ as defined above is a random function in that it depends on the
$\{\xi_i^\mu,\omega_i^\mu\}$, which reflect the random strategy
assignments at the beginning of the game. In the language of
spin-glass physics, $H$ contains quenched disorder. Information about
the stationary states of the MG can then be obtained upon minimising
$H$ with respect to the microscopic degrees of freedom $\{m_i\}$. The
starting point is the partition function
\begin{equation}
Z=\int_{-1}^{1}dm_1\cdots dm_N \exp\left(-\beta H(m_1,\dots,m_N)\right)
\end{equation}
where the `annealing temperature' $T=1/\beta$ is taken to zero
eventually to obtain the minima of $H$,
\begin{equation}
\mbox{min}~ H = -\lim_{\beta\to \infty}\frac{1}{\beta}\ln Z.
\end{equation}
The explicit analytical computation of the partition function for any
particular assignment of the strategy vectors this an insurmountable
task, due to complex interaction and global frustration. One hence
restricts to an effective average $\overline{\cdots}$ over the disorder,
and considers typical minima of $H$
\begin{equation}
\overline{\mbox{min}~ H} = -\lim_{\beta\to 0}\frac{1}{\beta}\overline{\ln Z}.
\end{equation}
The disorder-average of the logarithm of the partition function can here be obtained using a replica-approach, as it is standard in spin-glass physics. In particular one has
\begin{equation}
f = -\lim_{n\to 0}\lim_{\beta\to 0}\lim_{N\to\infty}\frac{1}{\beta N n} \left(\overline{Z^n}-1\right)
\end{equation}
for the free energy density in the thermodynamic limit. $Z^n$ here
corresponds to an $n$-fold replicated system, the disorder-average
generates an effective coupling between the different replicas.
The further mathematical steps are tedious, but straightforward, and
we will not report details here
\cite{ChalMarsZhanBook,CoolBook,MarsChalZecc00}, but will only mention that order
parameters such as $\lim_{N\to\infty}H/N$ and
$\lim_{N\to\infty}\sigma^2/N$ in the ergodic phase can be obtained
exactly or in good approximation from this approach. The location of
the phase transition at $\alpha_c=0.3374...$ can be identified exactly
as the point at which the so-called `static susceptibility' diverges,
indicating the breakdown of the ergodic replica-symmetric theory.
\subsubsection{Dynamics: generating functional analysis}
The dynamical approach based on generating functionals deals directly with the learning dynamics of the agents, which can be formulated as follows
\begin{equation}\label{eq:udiff}
q_i(t+1)=q_i(t)-\frac{1}{N}\xi_i^{\mu(t)}\sum_j\left\{\omega_j^{\mu(t)}+\xi_j^{\mu(t)}\mbox{sgn}[q_j(t)]\right\},
\end{equation}
if $q_i(t)$ denotes the score difference
$q_i(t)=\frac{1}{2}\left(u_{i,1}(t)-u_{i,-1}(t)\right)$ of player
$i$'s two strategies at time-step $t$ in a two-strategy MG (a generalisation to $S>2$ has recently been reported in \cite{nima}). Note that
only this difference is relevant for the agent's decision which of his
two look-up tables to use, and that at $t$ he will choose the strategy
with the higher-score, i.e. strategy table $-1$ if $q_i(t)<0$ and
strategy table $+1$ if $q_i(t)>0$. This may be summarised as him
playing strategy $\mbox{sgn}[q_i(t)]\in\{-1,1\}$ at $t$, and hence
taking trading action
$\omega_i^{\mu(t)}+\xi_i^{\mu(t)}\mbox{sgn}[q_i(t)]$
($=a_{i,+1}^{\mu(t)}$ if $q_i(t)>0$ and $=a_{i,-1}^{\mu(t)}$ if
$q_i(t)<0$).
One the defines a dynamical analogue of the partition function as the generating functional \cite{HeimCool01,CoolBook,CoolHeimSher01}
\begin{equation}
\hspace{-2cm}Z[\psi]=\int \left[\prod_{i,t} Dq_i(t)\right] \delta(\mbox{equations of motion})\exp\left(i\sum_{it}\psi_i(t)\mbox{sgn}[q_i(t)]\right),
\end{equation}
where `equations of motion' is an abbreviation for
Eqs. (\ref{eq:udiff}) so that the delta-functions restrict this path
integral to all trajectories of the system allowed by the update
rules. $\psi_i(t)$ is a source term, and allows one to compute correlation
functions upon differentiation with respect to the
$\{\psi_i(t)\}$. Mathematically speaking $Z[\psi]$ is the Fourier
transform of the probability measure in the space of dynamical paths
corresponding to the MG update rules, and contains all relevant
information on the dynamics of the system. Similarly to the replica
approach the evaluation of $Z[\psi]$ is intractable for individual
realisations of the disorder, but is carried out as an average over
all possible strategy assignments. This leads to a set of closed, but
complicated equations for the dynamical order parameters of the
problem (the response and correlation functions), from which one can
proceed to compute stationary order parameters such as the volatility
or the predictability. We will not report these calculations here, but
will only mention that both approaches, the static replica theory and
the dynamical generating functional analysis, lead to identical
expressions for the characteristic observables, and ultimately deliver
the same value for the phase transition point $\alpha_c$
\cite{CoolBook}.
\vspace{1em}
While there are still some mathematical subtleties to be resolved, the
MG is now essentially considered to be solved as its phase diagram
and stationary states can be computed exactly, and as valid and
convincing approximations for the volatility are available at least in
the ergodic phase. Open questions mostly relate to the behaviour in
the non-ergodic phase, where the market is information-efficient
($H=0$). Up to now, no analytical solutions have been found here.
\section{Anomalous fluctuations in Minority Games and other agent-based models}
\subsection{General remarks}\begin{figure}[t]
\begin{center}
\vspace{10mm}
\includegraphics[width=5cm]{hist_smallalpha.eps} ~~~~~~
\includegraphics[width=5cm]{hist_largealpha.eps}
\end{center}
\caption{Return distributions of standard MG, left: symmetric phase ($\alpha\approx0.01$), right asymmetric phase ($\alpha\approx 2$). Red line in right panel represents a fit to a Gaussian distribution.\label{fig:mghist}}
\end{figure}
The MG in its most simple setup does not display stylised facts, such
as anomalous fluctuations or temporal correlations of the market
volatility. On the contrary, return distributions are either Gaussian
or of an unrealistic multi-peak shape (due to global oscillations of
the system), see Fig. \ref{fig:mghist}. Within the minimalist approach
of statistical mechanics it is then natural to ask what features need
to be added to produce more realism. It here turns out that an
evolving trading volume appears to be crucial. While in the standard
MG every agent trades precisely one unit at every time to make the
total trading volume equal to the number $N$ of agents, two approaches
have been pursued which allow for dynamically evolving trading
volumes. These are referred to as grand canonical MGs (GCMG)
\cite{gcmg,mgstylised} and MGs with dynamically evolving capitals
\cite{Challet&al} respectively and will be detailed in the following
section. GCMG here refers to MGs in which each agent is given the
option to abstain at any given trading period. If they decide to trade
they still trade one unit, but due to the number of active agents
fluctuating in time, the total trading volume will evolve
accordingly. The second approach consists in MGs with dynamically
evolving capitals, that is the wealth of each agent evolves in time
according to his success or otherwise in the game, and it is assumed
that the amount traded by a given agent is proportional to his wealth
at the time. Since the wealths change over time so does the trading
volume.
We note at this stage that this tracing back of the emergence of
stylised facts to a modulated total trading volume can be criticized,
as liquidity effects normally also need to be taken into account in
real markets. If one assumes that the total trading action $A(t)$ is
related to the evolution of the (logarithmic) price via $\log
p(t+1)=\log p(t)+\frac{A(t)}{\lambda}$, then the role of the liquidity
$\lambda$ needs to be considered carefully. It controls the impact of
the excess demand $A(t)$ on the price returns, and may well vary in
time as well, i.e. $\lambda=\lambda(t)$. We will here disregard this
fact, and will describe the emergence of stylised facts in GCMGs and
MGs with dynamical capitals in the following section, before we then
turn to a discussion of anomalous fluctuations in agent-based models
different from the MG.
\subsection{Anomalous fluctuations in MGs}
\subsubsection{Grand canonical MGs}
In grand-canonical MGs agents are equipped with $S\geq 1$ `active'
strategies, that is strategy tables with binary entries $\{-1,1\}$ for
each history, prescribing to buy or sell. Furthermore they each hold
one null-strategy, i.e. a trivial strategy which prescribes to abstain
for any given history-string ($a_{is}^\mu\equiv 0$). For each of these
$S+1$ strategies they then keep score values as usual and play the one
with the highest score at any time-step. An additional parameter
$\varepsilon$ is here introduced, and plays the role of a disincentive
for the players to be active. This is implemented by subtracting an
amount $\varepsilon$ of the score of any strategy but the zero one at
any time step, so that the score update rules for the `active'
strategies read
\begin{equation}
U_{is}(t+1)=U_{is}(t)-a_{is}^{\mu(t)}A^{\mu(t)}-\varepsilon.
\end{equation}
The zero-strategy will always carry score zero. Thus agents trade
only if the marginal payoff from trading exceeds a threshold
$\varepsilon$. Modifying $\varepsilon$ thus can be understood as
introducing a trading fee to the model \cite{BiGaMa}.
\begin{figure}[t]
\vspace*{1mm}
\begin{tabular}{c}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~\epsfxsize=80mm \epsffile{phasediagram.eps}
\end{tabular}
\put(0,0){\Huge $\alpha$}
\put(-270,0){\Huge $\varepsilon$}
\put(-50,70){\vector(-2,-1){120}}
\put(-80,73){\small non-ergodic phase of infinite system}
\put(-90,-30){\vector(-4,1){90}}
\put(-140,-35){\small region of anomalous fluctuations for finite system}
\vspace*{4mm} \caption{Phase diagram of the grand-canonical MG. Stylised facts are observed in finite systems in a region around the efficient phase of the infinite-size model.}
\label{fig:pg}
\end{figure}
Since the MG by definition is a negative-sum game (i.e. the sum of all
payoffs is negative), agents loose on average when trading (only a
minority of players wins at every time step). Thus in the long-run all
traders would stop trading in the grand-canonical setup. To maintain
trading activity one usually couples the above group of `speculators'
(that is agents who have the option to abstain) to a background of
so-called `producers'. The latter are endowed with only one non-zero
strategy, and do not have the option not to trade, even if in the long
run they lose money on the MG market. They are considered producers
who make their profits from trading outside the model, and who are
forced to be active on the MG market no matter what
(e.g. internationally operating co-operations who need to be active in
currency markets, but make their profits elsewhere).
The resulting phase diagram of the model at a fixed relative number of
speculators and producers is shown in Fig. \ref{fig:pg} in the
$(\alpha,
\varepsilon)$-plane. For $\varepsilon=0$ one observes the standard
transition of the MG, between an unpredictable phase and a predictable
one. The unpredictable phase (in which $H=0$) is marked by a red line
in Fig. \ref{fig:pg} and extends from some value $\alpha_c$ (which
depends on the composition of the population of agents, i.e. the
relative concentrations of producers and speculators) to smaller
$\alpha$. Here the system is in a non-ergodic state, the so-called
turbulent regime. For $\varepsilon\neq 0$ this transition is absent
and the model market is predictable for all $\alpha$. Generally the
predictability vanishes as $H\sim\varepsilon^2$ as $\varepsilon\to 0$
for $\alpha<\alpha_c$.
\begin{figure}[t]
\vspace*{1mm}
\begin{tabular}{c}
~~~~~~~~~~~~~~~~~~~~~~~~~~~~\epsfxsize=80mm \epsffile{Ns501Np1000.eps}
\end{tabular}
\vspace*{4mm} \caption{Return time series of a grand-canonical MG in the critical region. From \cite{mgstylised}.}
\label{fig:gcmg}
\end{figure}
The above phase diagram is obtained from the statistical mechanics
theory in the thermodynamic limit $N\to\infty$, and anomalous
behaviour can at most be expected for parameters corresponding to the
red line in Fig. \ref{fig:pg} in the infinite system. Simulations show,
however, that anomalous fluctuations are observed in finite system in a
`critical region' around the critical line, as indicated by the gray
area in Fig. \ref{fig:pg}. Examples of corresponding time-series are
in this critical region are shown in Fig. \ref{fig:gcmg}. As the
system size is increased this critical region shrinks, and finally
reduces to the critical line of the infinite system as the number of
players diverges.
\subsubsection{MGs with evolving capitals}
\begin{figure}[t]
\begin{center}
\includegraphics[angle=270, width=6cm]{2strat_h_wealth.eps}
\end{center} \caption{Return fluctuations and total wealth of the population of traders in an MG with dynamical capitals. From \cite{Challet&al}.}\label{fig:dyncap1}
\end{figure}
In MGs with dynamical capitals each agent $i$ in addition to his $S$
strategies (all different from the null strategy) holds a wealth
$c_i(t)$ which evolves in time depending on his success or otherwise
during trading. It is then assumed that he invests a constant fraction
$\gamma$ of this capital $c_i(t)$ at time $t$. The evolution of the
capital of player $i$ is then given by
\begin{equation}
c_i(t+1)=c_i(t)-\gamma c_i(t)b_i\frac{A(t)}{V(t)},
\end{equation}
with $V(t)$ the total trading volume, $A(t)$ the excess demand at time
$t$, and $b_i(t)$ player $i$'s trading decision at time $t$. The minus
sign again reflects the minority game payoff. As in the GCMG one
couples the group of speculative traders (whose volume is taken to
evolve in time according to the above rule) to a group of producers,
who each trade a constant volume. This model was studied in
\cite{Challet&al}, and it was seen that the standard transition of the
MG is preserved when introducing dynamical capitals (as illustrated in
Fig. \ref{fig:dyncap1}). At low $\alpha$ the system is in an efficient
phase, which turns out to be an absorbing state of the dynamics. Above
a critical value of $\alpha$ the dynamics do not approach a fixed
point, and one finds $H>0$, just as in the standard MG with fixed
wealth. The overall wealth of the population attains a maximum at
$\alpha_c$ as shown in Fig. \ref{fig:dyncap1}. We note here that
results of \cite{Challet&al} rely mostly on simulations, as an
analytical solution of MGs with dynamical capitals and $S>1$
strategies per player is tedious due to the presence of fast degrees
of freedom (decisions which strategy to use) as well as slow ones (the
capitals), requiring an adiabatic decoupling. A theory for model with
dynamical capitals and only $S=1$ strategy per player is in progress
\cite{Gallainprogr}, and shows that the standard MG transition is present also in
this simplified system.
Similar to the observations in the GCMG simulations reveal the
emergence of anomalous fluctuations in finite systems in a region
around $\alpha_c$. This is illustrated in Fig. \ref{fig:dyncap2},
where the autocorrelation of absolute returns is shown and seen to
exhibit algebraic decay, indicating long-range volatility
correlations. At the same time return distributions close to the
critical point are strongly non-Gaussian (see right panel of
Fig. \ref{fig:dyncap2}). While these distributions are essentially
Gaussian far away from the critical point, they attain fatter and
fatter tails as $\alpha_c$ is approached (from above), so that the
kurtosis of these distributions may well be seen as measure of the
distance from the critical point.
\begin{figure}[t]
\vspace*{1mm}
\begin{tabular}{cc}
\epsfxsize=60mm \epsffile{2strat_autocorr.eps}~~~&~~~\epsfxsize=60mm \epsffile{2strat_return_pdf.eps}
\end{tabular}
\vspace*{4mm} \caption{{\bf Left:} Correlation function of absolute returns in an MG with dynamical capitals. The phase transition point is located at $\alpha_c\approx 0.6$ for the parameters used, volatility clustering becomes pronounced close to the transition. {\bf Right:} Distribution of returns in an MG with dynamical capitals. With the parameters used one has $\alpha_c\approx 0.32$. Close to the transition (left) returns collapse on non-Gaussian distributions for different time-lags $dt$, far from the transition (right) a cross-over to a Gaussian occurs. Figures taken from \cite{Challet&al}.}
\label{fig:dyncap2}
\end{figure}
\subsection{Other market models}
The MG is only one out of many market agent-based models developed in
recent years, and although this topical review mostly focuses on the
MG, we will provide a brief list of other models, without claiming to
be exhaustive.
We will firstly discuss some close cousins of the MG, then briefly
mention market models which are conceptionally different. We here partly lean on the more extensive reviews \cite{LeBaron2} and \cite{Hommes,LLS}.
While the MG is a `one-step' game, i.e. a model in which all trading
action takes place in a single step, this may seem incompatible with
speculative market which have an intrinsic intertemporal nature:
buying at a price $p(t)$ is only profitable if one is able to sell at
a higher price $p(t_1)$ where $t_1>t$. This lead the authors of
\cite{GiBou,BouGiaMe} as well as Andersen and Sornette
\cite{dollargame} to use a different payoff function,
\begin{equation}
g_i(t+1)=a_i(t)A(t+1).
\label{eq:dollargame}
\end{equation}
The model of \cite{dollargame} is here also known as the `dollar
game'. Payoffs of the form of (\ref{eq:dollargame}) imply that
strategies involve two periods of time. The dynamics induced by this
is characterized by different regimes in which the minority and
majority nature of the interaction alternate \cite{FeMa}. However the
majority rule prevails more often and gives rise to a phenomenon quite
similar to bubble phases in real markets. See also \cite{Challet} for
an extension of MGs to multi-step models.
Agent based models can roughly be divided into two classes: {\em
few-strategy} and {\em many-strategy} models. When they first were
introduced, models of financial markets were intended to recreate the
situation observed in real markets as closely as possible and for
this reason they were called `artificial financial
markets'. Initially the main purpose of these models was to carefully
analyze a small number of strategies used by agents, which is why
they are referred to as few-strategies models. Example of such models
are those by Kirman \cite{Kirman} and by Frankel and Froot \cite{FF}.
Concepts such as chartist and fundamentalist behaviour were first introduced
in the context of agent-based simulation in these models.
In many-strategy models, on the other hand, the dynamic ecology of
strategies is studied with the aim of understanding their co-evolution over
time and to see what strategies will survive and which will die. The
Santa Fe Artificial Stock Market (SF-ASM)
\cite{Arthuretal,LBAP} is probably the most known example of
this type of models. Its main objective is to characterise the
conditions under which the market converges to an equilibrium of
rational expectations. In the Genoa artificial stock market
\cite{Rabertoetal} randomly chosen traders place a limit buy or sell
order (see also \cite{Bouchaud&Potters} for further details) according
their budget limitation, and they show herding behavior similar to the
one studied in the Cont-Bouchaud model \cite{ContBo}. We would also
like to mention the Levy-Levy-Solomon model \cite{LeLeSo,LLS} in which
agents can switch between a risky asset and a riskless bond and do not
use complicated strategies, but try to maximize a one-period utility
function which implies a wealth-dependent impact of agents on prices,
in contrast with the other models mentioned. Strategy switching,
finally is at the centre of the model by Lux and Marchesi
\cite{LuxMarchesi}. In this model a mixed population of noise traders
(who can either be optimistic or pessimistic), and so-called
fundamentalists is studied. Agents can then switch between these
groups dynamically. Finally, there are of course other market models
which deserve mentioning in principle, but which we can not list here due
to space limitations.
With this plethora of models at hand, all or at least most of which in
some regions of their respective parameter spaces reproduce
market-like stylised facts it is of course legitimate to ask for a
justification for this variety of models, and ultimately for a
selection among them. Apart from its application as a model of a
market, the MG presumably has its own right as a spin glass system and
has introduced new types of complexity and phase transitions to the
theory of disordered systems. Its appeal thus rests, as pointed out
already above, in its position at the boundary of a realistic (within
reason), though analytically solvable complex system.
\section{Towards more realism: heterogeneous populations of agents}
\subsection{General remarks}
\begin{figure}[!]
\begin{center}
\subfigure{\scalebox{.30}{\includegraphics{s2f0.000ic.eps}}}
\subfigure{\scalebox{.30}{\includegraphics{s2f0.250ic.eps}}}\\
\subfigure{\scalebox{.30}{\includegraphics{s2f0.750ic.eps}}}
\subfigure{\scalebox{.30}{\includegraphics{s2f1.000ic.eps}}}
\caption{Behavior of $\sigma^2$ and $H$ vs $\alpha$ in the mixed minority/majority game for different proportions of trend-followers and contrarians. $f$ denotes the fraction of majority games players (trend-followers). From \cite{DeMaGiarMose03}.\label{fig:mixed}}
\end{center}
\end{figure}
One of the main drawbacks of the MG as a market model is its simple
set-up. Apart from the random assignments of strategies there is no
heterogeneity in the different types of agents, whereas real markets
are composed of different groups of traders, with different
objectives, impact on the market, and trading behaviour. The two
extensions mentioned above (GCMG and MGs with dynamical capitals)
alleviate these constraints, but more variety and extensions are
possible and have been studied by various authors over the last
years. We summarise some of them in the following sections.
\subsection{Mixed majority and minority games}
Most notably, extensions towards mixed minority and majority games
have been studied in \cite{DeMaGiarMose03}, in order to emulate mixed
populations of trend-followers and contrarians. As mentioned above, MG
mechanisms can be derived from expectations of agents on the future
price movements, assuming contrarian behaviour. MG agents expect the
price to go down in the future if it went up in the past, and vice
versa. Hence they prefer minority trading. Trend-followers on the
other hand assume that the tendency in price movements will continue,
hence if the price is rising due to positive excess demand, they will
buy (and analogously sell when the price is going down), and behave as
majority game players. Pure majority games are trivial in the sense
that all agents agree to either buy collectively at all times or to
sell, so that all agents are frozen. The model itself is closely
related to the Hopfield model of neural networks, and is interesting
from a statistical mechanics point of view
\cite{KozlMars03}. A mixed minority/majority game was studied in
\cite{DeMaGiarMose03}, where a population of $fN$ majority game players and $(1-f)N$ minority game players is considered ($0\leq f\leq 1$). Trend followers are here taken to follow a learning rule
\begin{equation}
U_{i,s}(t+1)=U_{i,s}(t)+a_{i,s}^{\mu(t)}A(t)
\label{eq:payoff_majority}
\end{equation}
where the only difference between (\ref{eq:payoff_majority}) and
(\ref{eq:mgupdate}) is the sign in front of the term
$a_{i,s}^{\mu(t)}A(t)$. The main results regarding this model are the
findings that the presence of trend-followers (a) injects information
into the system (in particular an informationally efficient phase with
$H=0$ is present only for $f<0.5$), and (b) increases the fluctuations
at low $\alpha$ (see curves for $f>1/2$ in Fig. \ref{fig:mixed}. A
further generalization of the mixed-model can be found in
\cite{DeMGiMaTe, TeDeMaGi} where players have a payoff function for
their strategies given by
\begin{equation}
U_{i,s}(t+1)=U_{i,s}(t)+a_{i,s}^{\mu(t)}F(A(t))
\end{equation}
with $F(A)=A-\epsilon A^3$ in \cite{DeMGiMaTe} ($\epsilon$ being a
parameter of the model) and $F(A)=A(t)(\chi(|A|<L)-\chi(|A|>L)$ in
\cite{TeDeMaGi} where $\chi(B)=1$ if $B$ is true (and $0$ otherwise)
and $L$ is a parameter of the model. This kind of choice allows a
player to change his behaviour from contrarian to trend follower and
vice versa in time, depending on the current price movement. In some
phases of these models interesting features are observed (e.g. trends,
bubbles and volatility clustering), we point the reader to the
literature for further details.
\subsection{MG as a platform for simulation of a future market}
The application of the MG as a platform to study ecologies of traders
can be extended to the simulational study of markets. Current work by
two of the authors \cite{GallaZhang,GallaZhang2} for example uses
MG-based simulations to study the influence of different trading
parameters on the overall performance of different groups of traders
in a market of futures. Large scale simulations with a variety of
traders here allow one to study the interaction between different
groups of traders in detail. In particular diverse populations of
short-term and long-term traders are considered, agents with different
weights are taken into account, and a distinction between direct
traders and broker represented agents can be made. In
Fig. \ref{fig:ecology} we show results from simulations of a market
with $200$ tick traders (all contrarian), $50$ long-term speculators,
$50$ broker represented long-term speculators and $1000$ broker
represented (background) traders. While we here depict only results
regarding the influence of a trading fee, the effect of external
market parameters such as so-called `tick size' and `tick value'
parameters and their interdependence can be studied as well. Details
will be reported elsewhere \cite{GallaZhang2}. Note also some recent
work on the effects of Tobin taxes in MG markets
\cite{BiGaMa} in this context.
\begin{figure}[t!]
\centerline{\includegraphics[width=12pc]{act.eps}~~~\includegraphics[width=12pc]{volume_brprod.eps}~~~\includegraphics[width=12pc]{rev.eps}}
\caption{(Colour on-line) {\bf Left:} Participation ratio for tick-traders, long-term speculators and broker represented long-term speculators versus trading fee. {\bf Middle:} Relative percentage of total trading volume attributed to each group. {\bf Right:} revenue for market maker. From \cite{GallaZhang}.\label{fig:ecology}}
\end{figure}
\section{Use of information in Minority Games and related models}
A further unrealistic simplification is the assumption of uniform available information and equal intellectual capacities of all agents, usually made in simple setups of the MG. More accurate models can here be expected to be more diverse and should account for heterogeneity in these aspects. In particular issues of individually different memory capacities and access to information are here to be considered, the latter leading to models with so-called `private information'.
\subsection{An agent-based model with private information}
A model with private information has been considered in \cite{Berg&al}
and \cite{DeGa}. Although the model studied there is not directly
derived from the MG, it shows similar features. The model is based on
the Shapley-Shubik \cite{ShSh} model of markets, and considers
limitation to private information explicitly. As in the MG one assumes
the existence of a global state of the market $\omega(t)$, which takes
one of $P=\alpha N$ values at each time $t$ (with $N$ the number of
traders in the market), but that agents have access to this
information only through individual information filters $k_i^\omega$,
where $i$ labels the individual agents. On the basis of the external
signal $\omega$ agent $i$ receives `filtered' information $k_i^\omega$
which may hence vary across the population of agents. In the simple
setup of
\cite{Berg&al} and \cite{DeGa} the $\{k_i^\omega\}$ take only binary
values $k_i^\omega\in\{-1,1\}$. The state $\omega$ is also assumed to
determine the (quenched random) return $R^\omega$ of an underlying
asset traded on the model market. Thus the vector
$\mathbf{k}_i=(k_i^1,\dots,k_i^P)$ crucially constrains the ability of
agent $i$ to resolve different states of the market. If for example
$k_i^\omega=-1$ for all $\omega$ then agent $i$ is completely blind
with respect to the actual state of the market, he receives
information $-1$ no matter what the actual value of $\omega(t)$ and
hence cannot distinguish any two different information patterns. If
$k_i^\omega$ is highly correlated with $R^\omega$ (e.g $k_i^\omega=1$
whenever the return $R^\omega$ is positive, and $k_i^\omega=-1$ for
all $\omega$ for which $R^\omega$ is negative), then agent $i$ is very
well able to distinguish states of the market, and to make accurate
predictions on the further price movements.
A detailed statistical mechanics analysis shows that the phase
transition of the MG market model is present also in this context, see
Fig. \ref{fig:privateinfo1}. There we plot the resulting
predictability $H$ as a function of $\alpha$ for different choices of
other model parameters (not specified here), note in particular that
the model can again be solved exactly (solid lines in the figure). The
right panel of Fig. \ref{fig:privateinfo1} depicts an order parameter
$q$ which measures the degree to which agents use their privately
obtained information. At the phase transition $q$ is maximal, and the
agents' decisions crucially depend on the information they receive. At
large or low values of $\alpha$ much less use of the available
information is made, and agents essentially trade independently of the
received binary private information.
\begin{figure}[t]
\begin{center}
\includegraphics[width=5cm]{h_vs_alpha_eta_0.eps}~~~\includegraphics[width=5cm]{q_vs_alpha_eta_0.eps}
\end{center}\caption{Predictability $H$ and use of information $q$ in a market model with private information. From \cite{DeGa}.\label{fig:privateinfo1}}
\end{figure}
\subsection{MG with dynamics on heterogeneous time-scales}
\subsubsection{Heterogeneous time-scales}
Agents in real markets can differ in time horizon on which they act
and on which they develop their own strategy. This point is often
neglected, although there is of course some work stressing this
particular aspect of real markets \cite{Olsen}. Very recently LeBaron
pointed out that different time scales can be responsible for the
appearance of different beliefs on the market and the co-existence of
these beliefs helps to generate features across many time scales
through symbiotic effects \cite{LeBaron}. In MGs there are several
ways in which one can implement different time scales among agents,
for example through individual learning rates, score memories or
strategy correlation
\cite{MoChZh}. In 2-strategy MGs one can introduce correlation between the strategies of any given agent by the drawing the strategy tables such that
\begin{equation}
P(a^\mu_{i,1}=a^\mu_{i,-1})=c.
\end{equation}
In this setup, with probability $c$, the two strategies of any given
player will prescribe the same action as response to the appearance of
history $\mu$. Thus, if $c=0$ player $i$'s strategies are fully
anti-correlated, $a_{i,1}^\mu=-a_{i,-1}^\mu$ for all $\mu$. For
$c=1/2$ one recovers the MG with uncorrelated strategies. For $c=1$
each players holds two fully correlated (i.e. identical strategies),
and is an infinitely `slow' player in the sense that his reaction to a
certain information pattern does not change over time. The smaller
$c$ the faster are the agents, at least in the simplified picture of
this model. In this setup, one is particularly interested in the
information ecology of the model, i.e. to understand whether groups of
different strategy correlation exploit each other, and can live in
symbiosis. If two different groups, say fast and slow, are
introduced, with $c_f<c_s$ where $f,s$ stand for fast and slow, the
behaviour is already quite rich and can be studied exactly by means of
replica theory. Fig.
\ref{fig:gain-regions} shows that gains $\gamma_f,
\gamma_s$ of fast and slow agents respectively, depend in a highly non-trivial way on the fraction $\phi_f$ of fast players and the correlation parameters $c_f, c_s$ .
\begin{figure}[t]
\centerline{\includegraphics*[height=5cm]{contour-a2phi0.5.eps}\includegraphics*[height =5cm]{contour-a0.4phi0.01.eps}}
\caption{Regions of relative advantage for $\phi_f=0.5$ (left graph) and $\phi_f=0.01$ (right graph); the region filled corresponds to the symmetric phase, in which the replica calculus is not valid. From \cite{MoChZh}.}
\label{fig:gain-regions}
\end{figure}
Another way to investigate the role of time scales in MGs is to study
the game with just one information pattern (i.e. the limit $P\to 1$)
\cite{Marsili,MC01}. This limiting case is particularly
straightforward to understand analytically. Given $A(t)$ each agent
$i$ receives a payoff $-a_i(t)A(t)$, and keeps a score $\Delta_i(t)$ as follows
\begin{equation}
\Delta_i(t+1)=\Delta_i(t)-\frac{A(t)}{N},
\end{equation}
and his trading action is then determined by the stochastic rule
\begin{equation}
P[a_i(t+1)=1]=\frac{1+\Gamma_i\Delta_i(t)}{2}
\end{equation}
with $\Gamma_i$ a learning rate. Thus high values of $\Delta_i(t)$
favour positive trading action $a_i(t+1)=1$ by player $i$. It is then
possible to consider $G$ groups of $\phi_g N$ agents respectively
(with $g=1,\dots,G$ and $\sum_{g=1}^N \phi_g=1$) with all the agents
belonging group $g$ following a learning rule of rate $\Gamma_g$.
Without going into any detail here, we will just point out that the
relative gains of the respective groups can be obtained as
\begin{equation}
\gamma_g-\gamma= \frac{\Gamma-\Gamma_g}{2(1+\Gamma)}
\label{eq:learning_rates}
\end{equation}
(within some approximation) where $\gamma_g$ is the average gain of
players in group $g$, $\gamma$ the gain averaged over all groups, and
$\Gamma$ the mean learning rate (over all groups). It is interesting
to observe that the smaller $\Gamma_g$ the smaller the loss of group
$g$. For details and other extensions see \cite{MoChZh}.
\subsubsection{Timing of adaptation}
The effects of the timing of adaptation on MGs have been studied for
example in
\cite{SherGall03,SherGall05}. One here distinguishes between so-called
`on-line' and `batch' adaptation of the MG agents. In the more usual
`on-line' games, agents adapt to the external flow of information
instantly, and update their strategy scores at every round of the
game, and then choose the trading strategy to use in the next step. In
batch games, agents adapt only after a large number of rounds has been
played, and the corresponding information regarding the performance of
the strategies has been accumulated. An interpolation between both
cases is possible. Specifically, one allows the agents to update their
scores (and re-adapt their strategy choices) only once every $M$
rounds (with accumulative increments over the past $M$ rounds), i.e.
\begin{equation}
U_{i,s}(t+M)=U_{i,s}(t)-\frac{1}{M}\sum_{\ell=0}^{M-1}a_{i,s}^{\mu(\ell)}A^{\mu(\ell)}[\{U(t)\}].\label{eq:emm}
\end{equation}
Their choice of strategy then remains
fixed in between two updates (as indicated by the dependence of the
total action on the scores at time $t$ in Eq. (\ref{eq:emm})). The
limit $M\to 1$ reproduces the on-line case, the case $M\gg P$ (with $P$
the number of possible values of the information) is referred to as
the `batch' game, in which an effective average over all information
patterns is performed. While the volatility of standard MGs with
uncorrelated strategies is not sensitive to the choice of on-line
versus batch dynamics, qualitative differences are found in MGs with
fully anti-correlated strategy assignments ($c=0$ in the notation of the previous section), see
Fig. \ref{fig:batchonl}. Adaptation at randomly chosen time-intervals
of mean $M$ finally reduces the volatility in the efficient phase of the
market, but not above the phase transition, see Fig. \ref{fig:1_m}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=5cm]{online_rho_0_biased.eps} ~~~\includegraphics[width=5cm]{biased_batch_rho=0_withinset.eps}
\end{center}\caption{On-line versus batch game for fully anti-correlated strategies. {\bf Left:} on-line game, started from different initial conditions. {\bf Right:} batch game. Note the difference in the qualitative behaviour. From \cite{SherGall05}.}\label{fig:batchonl}
\end{figure}
\subsection{Cost of information}
Ideally one would like to consider models in which agents can acquire
information at some cost, i.e in which they pay for the use of
expertise. To the knowledge of the authors, no such attempts have yet
been made in an analytical statistical mechanics approach. One may
also consider models in which agents have both public and private
information at their disposal, and have to decide which to use. One
may think of globally available information through newspapers and
other media, versus the recommendations of a small circle of private
advisors and/or friends.
\section{Summary and outlook on future work}
In summary we have reviewed the recent progress in the description of
financial markets through simple agent-based models, accessible by the
techniques of theoretical statistical mechanics. The MG in particular
can be seen as a minimalist market model, which can be treated
analytically in its basic setup and is now considered to be
essentially fully understood. While in its original setup of each
agent trading one unit at any given time step the MG does not display
anomalous fluctuations and stylised facts as seen in real market data,
only minor modifications are necessary to make the model more
realistic in this sense, without giving up the analytical
tractability. MGs with a finite number of agents and with dynamical
capitals and/or with an option of the agents to abstain have been
shown to display anomalous fluctuations close to their phase
transitions, similarly to systems of statistical mechanics exhibiting
large scale correlations and self-similarity near their critical
points. These observations call for further analysis of such variants
of the MG, for example by means of renormalisation group techniques.
\begin{figure}[t]
\begin{center}
\includegraphics[width=6cm]{vol_1_m_2panels.eps}
\end{center}\caption{Volatility for model with random updates, probability to update $1/M$. Top panel corresponds to MG with uncorrelated strategy assignments, lower panel to the fully anti-correlated case. From \cite{SherGall05}.}\label{fig:1_m}
\end{figure}
The MG can also serve as a platform for more diverse market
simulations, and an ecology of market participants can be studied upon
introducing diversified types of players in the MG. These may include
contrarians and trend-followers respectively, as well as agents
trading on different time-scales and/or with different trading
volumes, and agents of different memory capacities. While it is
presumably unlikely that real market trading decisions can be taken on
the basis of MG simulations, the model as such and its diverse
variants and extensions offer a promising approach to gain a better
understanding of the interplay of market parameters and the ecology of
populations of traders, based on models at the boundary of analytical
solvability. Future research in this direction may thus be of interest
both in an academic environment as well as for practitioners.
\section*{Acknowledgements}
This work was supported by the European Community's Human Potential
Programme under contract HPRN-CT-2002-00319, STIPCO and by EVERGROW,
integrated project No. 1935 in the complex systems initiative of the
Future and Emerging Technologies directorate of the IST Priority, EU
Sixth Framework. The authors would like to acknowledge collaboration
with Damien Challet, Ton Coolen, Andrea De Martino, Irene Giardina,
Matteo Marsili and David Sherrington on some of the material reviewed here.
\section*{References}
|
1,108,101,565,707 | arxiv | \section{Introduction}
\label{secct:intro}
Let $X$ be a geometrically integral smooth projective Fano variety over a number field $F$ and let $\mathcal{L} = \mathcal{O}_{X}(L)$ be an adelically metrized big and nef line bundle on $X$. Manin's Conjecture, first formulated in \cite{FMT89} and \cite{BM}, predicts that the growth in the number of rational points on $X$ of bounded $\mathcal{L}$-height is controlled by two geometric constants $a(X,L)$ and $b(F,X,L)$. These constants are defined for any smooth projective variety $X$ and any big and nef divisor $L$ on $X$ as
\begin{equation*}
a(X,L) = \min \{ t\in {\mathbb R} \mid K_X + tL \in \overline{\mathrm{Eff}}^{1}(X) \}
\end{equation*}
and
\begin{align*}
b(F, X,L) = & \textrm{ the codimension of the minimal supported face} \\
& \textrm{ of } \overline{\mathrm{Eff}}^{1}(X) \textrm{ containing } K_{X} + a(X, L)L
\end{align*}
where $\overline{\mathrm{Eff}}^{1}(X)$ is the pseudo-effective cone of divisors of $X$. If $L$ is nef but not big, we set $a(X,L) =b(F, X, L) = \infty$.
\begin{rema}
One can define the $a$ and $b$ invariants analogously for any big divisor $L$ on $X$, and it is natural to ask whether they still control the behavior of asymptotic point counts for the associated height function. However, in this situation the invariants can exhibit pathological behavior; see Section \ref{sect: nonbigdiv}.
\end{rema}
In the statement of Manin's Conjecture an exceptional set of rational points must be removed in order to obtain the expected growth rate.
For example, it is possible for points to grow more quickly than predicted along certain subvarieties of $X$ and such points should not be counted. More precisely, the following definition identifies the possible geometric obstructions to Manin's Conjecture.
\begin{defi}
Let $X$ be a smooth projective variety over a number field $F$ and let $L$ be a big and nef divisor on $X$. A morphism of smooth projective varieties $f: Y \to X$ is called a breaking thin map if it satisfies the following two conditions:
\begin{enumerate}
\item $f$ is generically finite onto its image, and
\item $(a(Y,f^{*}L),b(F, Y,f^{*}L)) > (a(X,L),b(F, X,L))$ in the lexicographic order.
\end{enumerate}
Note that this definition implicitly depends on the choice of $L$.
\end{defi}
If Manin's Conjecture is self-consistent then the exceptional set should include all subsets of the form $f(Y(F))$ where $f: Y \to X$ is a breaking thin map. However, the point contributions from breaking thin maps need not lie on a Zariski closed proper subset of $X$. \cite{BT-cubic} used this idea to show that the exceptional set in Manin's Conjecture can be Zariski dense, contradicting the original formulation of the conjecture.
\cite{Peyre03} conjectured that Manin's Conjecture should be revised by allowing the exceptional set to be a thin set of points, and this version was subsequently verified in a few examples (\cite{LeRudulier}, \cite{BHB18}).
Our main theorem shows that point contributions from breaking thin maps will always be contained in a thin set. This gives strong support to the conjecture of \cite{Peyre03}: one can never construct a counterexample to the thin set version of Manin's Conjecture using breaking thin maps.
\begin{theo} \label{theo: maintheorem}
Let $X$ be a geometrically uniruled smooth projective variety over a number field $F$ and let $L$ be a big and nef divisor on $X$. As we vary over all breaking thin $F$-maps $f: Y \to X$, the points
\begin{equation*}
\bigcup_{f} f(Y(F))
\end{equation*}
are contained in a thin subset of $X(F)$.
\end{theo}
This theorem generalizes earlier partial results in \cite{BT}, \cite{HTT15}, \cite{LTT14}, \cite{HJ16}, \cite{LTDuke}, and \cite{Sen17}. These papers also establish some practical techniques for computing this thin set.
In fact, we prove a more precise statement (Theorem \ref{theo: precisetheorem}) which also addresses generically finite morphisms $f: Y \to X$ such that $Y$ has the same $a$ and $b$ invariants as $X$. In this situation the rational point contributions from $Y$ can affect the leading constant in Manin's Conjecture, and thus one must decide whether or not $f(Y(F))$ should be included in the exceptional set in order to obtain the constant predicted by \cite{Peyre} and \cite{BT}.
We distinguish the two possibilities using the geometric notion of a face contracting morphism (Definition \ref{defi: facecontraction}).
Finally, we conjecture that the exceptional set in Manin's Conjecture will actually coincide with the geometrically defined subset of $X(F)$ constructed in Theorem \ref{theo: precisetheorem}. In Section \ref{sect: conjecturaldescription} we verify this in many examples where Manin's Conjecture is known to hold.
\subsection{A summary of the proof}
To prove Theorem \ref{theo: maintheorem}, it would suffice to show that there is a finite set of breaking thin maps $\{ f_{i}: Y_{i} \to X \}_{i=1}^{r}$ such that
\begin{equation*}
\bigcup_{f} f(Y(F)) = \bigcup_{i=1}^{r} f_{i}(Y_{i}(F)).
\end{equation*}
In particular, it would suffice to show that there is a finite set of breaking thin maps $\{ f_{i} \}$ such that every breaking thin map $f: Y \to X$ factors through some $f_{i}$. Our proof is built on this idea.
The first step is to prove a factoring result for breaking thin maps over an algebraically closed field of characteristic $0$. However, to obtain a factoring for the map $f: Y \to X$ we will need to allow ourselves to alter the variety $Y$ in the following way.
\begin{defi}
Let $Y$ be a smooth projective variety over a field of characteristic $0$ and let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$. Let $\pi: Y \dashrightarrow W$ be the canonical model associated to $K_{Y} + a(Y,L)L$ and let $U$ be the maximal open subset where $\pi$ is defined. Suppose that $T \to W$ is a dominant morphism of normal projective varieties. Then there is a unique component of $T \times_{W} U$ mapping dominantly to $T$.
Denote by $\widetilde{Y}$ the normalization of the Zariski closure of this component in $T \times Y$.
We call such an $\widetilde{Y}$ an Iitaka base change of $Y$; it is naturally equipped with maps $\widetilde{Y} \to T$ and $\widetilde{Y} \to Y$.
\end{defi}
The following theorem extends earlier results of \cite{HJ16}, \cite{LTDuke}, and \cite{Sen17} to the most general setting.
\begin{theo} \label{theo: mainfiniteness}
Let $X$ be a uniruled smooth projective variety over an algebraically closed field of characteristic $0$. Let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$. There is a finite set of breaking thin maps $\{ f_{i}: Y_{i} \to X\}$ such that for any breaking thin map $f: Y \to X$ either the image of $f : Y \to X$ is contained in the augmented base locus $\mathbf B_+(L)$ or there is an Iitaka base change $\widetilde{Y}$ of $Y$ with respect to $f^{*}L$ such that the induced morphism $\widetilde{f}: \widetilde{Y} \to X$ factors rationally through one of the $f_{i}$.
\end{theo}
The key input is Birkar's solution of the Borisov-Alexeev-Borisov Conjecture (\cite{birkar16} and \cite{birkar16b}). Although the varieties $Y$ in Theorem \ref{theo: mainfiniteness} need not form a bounded family, using Birkar's result we show that their images in $X$ are covered by a set of adjoint rigid subvarieties (Definition \ref{defi: adjointrigid}) which do form a bounded family. Furthermore, \cite{Sen17} controls the possible ramification loci of such morphisms $f$ and their degrees are again bounded by the Borisov-Alexeev-Borisov Conjecture. Thus we can construct the $\{f_{i}\}$ in Theorem \ref{theo: mainfiniteness} using suitably chosen covers over the universal family of adjoint rigid subvarieties. Then the corresponding factoring property follows from the homotopy lifting property of covering spaces.
The second step is to ``descend'' Theorem \ref{theo: mainfiniteness} to a number field $F$. There are two main obstacles. First, infinitely many twists over $F$ can be identified with a single map over the algebraic closure $\overline{F}$. Thus it is more natural to allow ourselves to work with all twists of a finite set of maps $\{ f_{i}: Y_{i} \to X \}$ when proving our factoring result. Second, it is quite difficult to determine when the maps constructed by Theorem \ref{theo: mainfiniteness} descend to the ground field. Even when they do descend, the corresponding homotopy lifting property may not be available over the ground field, and thus it is unclear whether the desired factoring holds. Fortunately, we only care about the situation when our varieties are equipped with a rational point. Using a delicate construction involving the \'etale fundamental group and the homotopy lifting property for a rational basepoint, we prove a factoring result over a number field (Lemma~\ref{lemm: finitelymanycoversovernf}) in the situation when $Y$ admits a rational point.
We have now established that $\cup_{f} f(Y(F))$ is contained in the rational points coming from the twists of a finite set of thin maps. The final step is to show that if we fix a morphism $f_{i}: Y_{i} \to X$, all of its twists which are breaking thin maps will together only contribute a thin set of rational points. The essential ingredient of the following theorem is the Hilbert Irreducibility Theorem proved by Serre.
\begin{theo} \label{theo: mainthinness}
Let $X$ be a geometrically uniruled smooth projective variety over a number field $F$. Suppose that $f: Y \to X$ is a generically finite morphism from a normal projective variety $Y$. As $\sigma$ varies over all $\sigma \in H^1(F, \mathrm{Aut}(Y/X))$ such that
$Y^{\sigma}$ is irreducible and
\begin{equation*}
(a(X, L), b(F, X, L)) < (a(Y^{\sigma}, (f^{\sigma})^{*}L), b(F, Y^\sigma, (f^\sigma)^*L))
\end{equation*}
the set
\begin{equation*}
Z= \bigcup_{\sigma} f^\sigma(Y^\sigma (F)) \subset X(F)
\end{equation*}
is contained in a thin subset of $X(F)$.
\end{theo}
Recall that our main Theorem \ref{theo: precisetheorem} also addresses thin maps $f: Y \to X$ such that the $a$ and $b$ invariants of $Y$ and $X$ are the same. Thus we will actually prove stronger versions of Theorem \ref{theo: mainfiniteness} and \ref{theo: mainthinness} which address this more general situation.
\subsection{Structure of the paper}
Section~\ref{Preliminaries} is devoted to preliminaries. In Section \ref{sect: mmp}, we briefly review some foundational results of the minimal model program from \cite{BCHM}, \cite{birkar16}, and \cite{birkar16b} and derive some consequences in preparation for the rest of the paper.
Section~\ref{sec: geoinv} develops the theory of the geometric invariants $a(X, L), b(F, X, L)$ over an arbitrary field of characteristic $0$. In particular, we construct universal families of adjoint rigid subvarieties in Section \ref{subsec: BAB} and we introduce face contracting morphisms in Section \ref{subsec: facecontracting}.
In Section~\ref{sect: conjecturaldescription}, we propose a conjectural geometric description of the exceptional set in Manin's Conjecture. The main result of this paper (Theorem~\ref{theo: precisetheorem}) is that this proposed set is contained in a thin subset of rational points in accordance with the prediction made by Peyre. We then compare our construction with the exceptional set in various examples and discuss a few counterexamples to possible extensions.
The rest of the paper is devoted to proofs of our main theorems. Section~\ref{sec: twists} is devoted to the study of twists and the relationship with thin sets, proving a stronger version of Theorem~\ref{theo: mainthinness}. In Section~\ref{sect: boundedness} we study the boundedness of breaking thin maps over an algebraically closed field of characteristic $0$ and prove a stronger version of Theorem~\ref{theo: mainfiniteness}. In Section~\ref{sec: thinset}, we work over a number field $F$ and prove our main Theorem \ref{theo: precisetheorem}. The key technical result is Lemma~\ref{lemm: finitelymanycoversovernf}, which constructs universal families for breaking thin maps by combining the results of Section \ref{sect: boundedness} with a careful analysis of \'etale fundamental groups.
\bigskip
\noindent
{\bf Acknowledgments.}
The second author would like to thank his advisor J\'{a}nos Koll\'{a}r for constant support and encouragement. The authors would like to thank Yuri Tschinkel and Anthony V\'arilly-Alvarado for answering our questions about toric varieties and the lifting property and Yoshinori Gongyo for his explanation of Theorem \ref{theo: BAB2}. The authors also would like to thank Tim Browning, Brendan Hassett, and Marta Pieropan for useful comments. Finally, the authors would like to thank an anonymous reviewer for constructive criticisms and detailed suggestions which significantly improved the exposition of the paper.
Brian Lehmann is supported by NSF grant 1600875. Sho Tanimoto is partially supported by MEXT Japan, Leading Initiative for Excellent Young Researchers (LEADER), by Inamori Foundation, and by JSPS KAKENHI Early-Career Scientists Grant numbers 19K14512.
\section{Preliminaries}
\label{Preliminaries}
Let $F$ be a field of characteristic $0$. A variety $X$ defined over $F$ is an integral separated scheme of finite type over $F$. For an extension of fields $F'/F$, we denote the base change of $X$ to $F'$ by $X_{F'}$ and denote the pullback of a $\mathbb{Q}$-Cartier divisor $L$ from $X$ to $X_{F'}$ by $L_{F'}$. For an algebraic closure $\overline{F}/F$, we will also sometimes denote the base change by $\overline{X}$ and the pullback by $\overline{L}$, particularly when $X$ is geometrically integral.
\begin{defi}
Let $X$ and $Y$ be projective varieties. A map $f: Y \to X$ is thin if it is generically finite onto its image and admits no rational section.
\end{defi}
If $F$ is a number field, a thin subset of $X(F)$ is a finite union $\cup_{j} f_{j}(Y_{j}(F))$ where $f_{j}: Y_{j} \to X$ are thin maps over $F$.
\bigskip
Consider a commutative diagram of dominant morphisms of varieties
\begin{equation*}
\xymatrix{ & U \ar[d]\\
T \ar[r] & W}
\end{equation*}
If there is a unique component of $T \times_{W} U$ which dominates $T$ and $U$ under the projection maps, then we call it the ``main component'' of the product.
\bigskip
Let $X$ be a projective variety. A family of subvarieties of $X$ is a diagram
\begin{equation*}
X \times W \supset \xymatrix{ \mathcal{\mathcal{U}} \ar[r]^{s} \ar[d]^{p} & X \\
W & }
\end{equation*}
such that $W$ is a variety, $p$ is projective and flat, and every fiber of $p$ is a subvariety of $X$. A curve $C$ is said to be a movable curve if it is a member of a family of $1$-dimensional subvarieties such that $s$ is dominant.
\bigskip
We next recall several definitions from birational geometry. Suppose that $X$ is a normal projective variety defined over $F$.
We denote the N\'eron-Severi space of $\mathbb{R}$-Cartier divisors up to numerical equivalence by $N^1(X)$ and the space of $\mathbb{R}$-$1$-cycles modulo numerical equivalence by $N_1(X)$. We denote the pseudo-effective cone and the nef cone of divisors by
\[
\overline{\mathrm{Eff}}^1(X), \quad \mathrm{Nef}^1(X)
\]
respectively, and the pseudo-effective cone and the nef cone of curves by
\[
\overline{\mathrm{Eff}}_1(X), \quad \mathrm{Nef}_1(X)
\]
respectively. These are pointed closed convex cones in $N^1(X)$ and $N_1(X)$.
\begin{defi}[\cite{Nakamaye00}, \cite{ELMNP06}]
Let $X$ be a smooth projective variety defined over a field $F$ of characteristic $0$. Let $D$ be a $\mathbb Q$-divisor on $X$. The asymptotic base locus of $D$ is
\begin{equation*}
\mathbf{B}(D) = \bigcap_{m \in \mathbb{Z}_{>0}} \mathrm{Bs}(|mdD|)
\end{equation*}
where $d$ is any positive integer such that $dD$ is Cartier. The definition is independent of the choice of $d$.
The augmented base locus of $D$ is
\begin{equation*}
\mathbf{B}_{+}(D) = \bigcap_{\textrm{ample }\mathbb{Q}\textrm{-div }A} \mathbf{B}(D-A).
\end{equation*}
The augmented base locus is always a closed subset of $X$ by \cite[Proposition 1.5]{ELMNP06}.
\end{defi}
Suppose that $L$ is a big and nef $\mathbb{Q}$-divisor on a smooth variety over an algebraically closed field of characteristic $0$. By \cite[Theorem 0.3]{Nakamaye00} the augmented base locus coincides with the Zariski closure of the set of subvarieties $V \subset X$ such that $L|_{V}$ is not big and nef. In fact, the restriction of $L$ to any component of the augmented base locus fails to be big.
We claim that the same is true for any ground field of characteristic $0$. Indeed, if $X$ is defined over the ground field $F$, then by \cite[Proposition 1.5]{ELMNP06} $\mathbf{B}_{+}(L_{\overline{F}}) = \mathbf{B}(L_{\overline{F}} - A_{\overline{F}})$ for some ample $\mathbb{Q}$-divisor $A$ defined over the ground field. Since the formation of a base locus is compatible with change of base field, we conclude that both sides are defined over the ground field. The fact that $\mathbf{B}_{+}(L)$ is the Zariski closure of the subvarieties $V$ such that $L|_{V}$ is not big and nef can be deduced from the corresponding statement over $\overline{F}$.
Note that if $X$ is defined over a number field and $L$ is a big and nef $\mathbb{Q}$-divisor, the Northcott property for rational points is only guaranteed to hold after removing the points contained in $\mathbf{B}_{+}(L)$. In particular, we will always include $\mathbf{B}_{+}(L)$ in the exceptional set for Manin's Conjecture.
\section{The minimal model program} \label{sect: mmp}
We will use the standard notations of the minimal model program regarding singularities of pairs. We refer to \cite[Definition 2.34]{KM98} for their definitions. We will frequently use the following well-known lemma (see for example \cite[Theorem 2.3]{LTT14}):
\begin{lemm} \label{lemm: terminalpair}
Let $X$ be a smooth projective variety over a field $F$ of characteristic $0$ and let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$. Then there is an effective $\mathbb{Q}$-divisor $\Delta$ and an effective ample $\mathbb{Q}$-divisor $A$ such that $(X,\Delta + A)$ is terminal and $L$ is $\mathbb{Q}$-linearly equivalent to $\Delta + A$.
\end{lemm}
\subsection{Canonical models}
Suppose that $X$ is a smooth projective variety over a field $F$ of characteristic $0$ and that $L$ is a big and nef $\mathbb{Q}$-divisor on $X$. Fix a positive integer $d$ such that $dL$ is Cartier. Note that the base change of the section ring of $dL$ to $\overline{F}$ is isomorphic to the product of the section rings of the pullbacks of $dL$ to all geometric components of $X_{\overline{F}}$.
Thus \cite[Theorem 1.2]{BCHM} (combined with Lemma \ref{lemm: terminalpair}) shows that the section ring
\begin{equation*}
\bigoplus_{m \geq 0} H^{0}(X,\mathcal{O}_{X}(md(K_{X} + L)))
\end{equation*}
is finitely generated. When this ring is non-zero, via the Proj construction we obtain a rational map $\pi: X \dashrightarrow W$ such that $\dim(W) = \kappa(X,K_{X} + L)$. The map $\pi: X \dashrightarrow W$ is known as the canonical model for $(X,L)$, or equivalently, the canonical model for $K_{X} + L$.
\begin{lemm} \label{lemm:birationaltocanonical}
Let $X$ be a geometrically uniruled smooth projective variety and let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$ such that $K_{X} + L$ is pseudo-effective. Suppose that $\psi: X \to W$ is a surjective morphism of projective varieties such that:
\begin{enumerate}
\item the base change of $\psi$ to $\overline{F}$ has connected fibers,
\item $\kappa(X_{w},K_{X_{w}} + L|_{X_{w}}) = 0$ for a general fiber $X_{w}$ over a closed point $w \in W$, and
\item there is an ample $\mathbb{Q}$-divisor $H$ on $W$ such that $K_{X} + L - \psi^{*}H$ is $\mathbb{Q}$-linearly equivalent to an effective divisor.
\end{enumerate}
Then $\psi$ is birationally equivalent to the canonical map for $K_{X} + L$. If the canonical map is a morphism on $X$, then there is a non-empty open subset $W^{\circ} \subset W$ such that on $\psi^{-1}(W^{\circ})$ the canonical map coincides with $\psi$.
\end{lemm}
\begin{proof}
To prove the first statement, we may replace $X$ by a birational model for which the canonical map for $K_{X} + L$ is a morphism and replace $L$ with its pullback to this birational model. Thus it suffices to prove that the last statement holds.
Let $\pi: X \to W'$ denote the canonical map for $K_{X} + L$. By condition (3), there is a rational map $g: W' \dashrightarrow W$ such that $\psi = g \circ \pi$ as rational maps. By condition (2), a general fiber of $\psi$ is contracted by $\pi$. Together with condition (1) these show that $g$ is birational, yielding the desired claim.
\end{proof}
\subsection{Boundedness of singular Fano varieties}
In \cite{birkar16} and \cite{birkar16b} Birkar establishes the Borisov-Alexeev-Borisov Conjecture concerning the boundedness of mildly singular Fano varieties. We will use the following special cases of Birkar's results.
\begin{theo}[\cite{birkar16b} Theorem 1.1]
\label{theo: BAB}
Let $\overline{F}$ be an algebraically closed field of characteristic $0$. Let $d$ be a positive integer and fix $\epsilon > 0$. Then there exists a constant $C_{1} = C_{1}(d,\epsilon) > 0$ such that for any $\epsilon$-lc pair $(X,\Delta)$ such that $X$ has dimension $\leq d$ and $K_{X}+\Delta$ is antiample, we have
\[
(-K_{X}-\Delta)^{\dim X} \leq C_{1}.
\]
\end{theo}
\begin{rema}
This form of \cite[Theorem 1.1]{birkar16b} is not explicitly stated. However, as in the proof of \cite[Theorem 1.1]{birkar16b}, we can write $-\delta K_X \sim_{\mathbb{R}} K_{X} + (1 +\delta)(A + \Delta)$ where $A$ is any ample divisor that is $\mathbb{R}$-linearly equivalent to $-(K_{X} + \Delta)$.
For any fixed $\epsilon' < \epsilon$, we can ensure that the pair $(X,(1+\delta)(A+\Delta))$ is $\epsilon'$-lc by choosing $\delta$ and $A$ appropriately. Thus by running the $(-\delta K_X)$-minimal model program we obtain a rational map $\phi: X \dashrightarrow X'$ where $X'$ is an $\epsilon'$-lc weak Fano variety. In fact, since $X'$ does not depend on $\delta$ and $A$, by taking a limit as $\epsilon' \to \epsilon$ we see that $X'$ is $\epsilon$-lc. Note that
\begin{equation*}
\mathrm{vol}(-K_{X}-\Delta) \leq \mathrm{vol}(-K_{X}' - \phi_{*}\Delta) \leq \mathrm{vol}(-K_{X'}).
\end{equation*}
As $X'$ varies over all $\epsilon$-lc weak Fano varieties \cite[Theorem 2.11]{birkar16b} shows that $\mathrm{vol}(-K_{X'})$ has a universal upper bound depending only on $d$ and $\epsilon$, yielding the desired statement.
\end{rema}
\begin{theo}[\cite{birkar16b}]
\label{theo: BAB2}
Let $\overline{F}$ be an algebraically closed field of characteristic $0$. Let $d$ be a positive integer and fix an $\epsilon > 0$ and a finite set of rational numbers $I \subset [0,1)$. Suppose that
\begin{itemize}
\item $X$ is a projective variety of dimension $\leq d$,
\item $(X,\Delta)$ is a $\epsilon$-lc pair such that the coefficients of $\Delta$ lie in $I$, and
\item $K_{X} + \Delta$ is an antiample $\mathbb{Q}$-Cartier divisor.
\end{itemize}
Then there is a constant $C_{2} = C_{2}(d,\epsilon,I)$ such that that $C_{2}(K_{X} + \Delta)$ is Cartier for all such pairs $(X,\Delta)$.
\end{theo}
\begin{proof}
\cite[Theorem 1.1]{birkar16b} shows that as we vary over all such pairs the set of underlying varieties $X$ is bounded. We show that in fact the set of pairs $(X,\Delta)$ is log bounded. Using the boundedness of the underlying $X$, we can find a family of very ample divisors $A$ on these varieties $X$ such that the space of sections of $A$ is bounded in dimension. Since the coefficient set of $\Delta$ is finite, it suffices to prove that the degree of $\Delta$ against $A$ is bounded. This follows from
\begin{equation*}
\Delta \cdot A^{\dim X-1} < -K_{X} \cdot A^{\dim X-1}
\end{equation*}
and the boundedness of the varieties $X$.
Since the pairs $(X,\Delta)$ are log bounded, by \cite[Lemma 2.24]{birkar16} and the fact that the coefficient set of $\Delta$ is finite we deduce the desired statement on the Cartier index.
\end{proof}
\cite{Araujo10} shows how the Borisov-Alexeev-Borisov Conjecture can be used to deduce a structure theorem for the cone of nef curves. We will give a quick explanation of Araujo's arguments since \cite{Araujo10} only explicitly addresses the case when $\dim(X) = 3$.
\begin{lemm} \label{lemm:intbound}
Let $\overline{F}$ be an algebraically closed field of characteristic $0$. Let $d$ be a positive integer and fix an $\epsilon > 0$ and a finite set of rational numbers $I \subset [0,1)$. Suppose that
\begin{itemize}
\item $X$ is a projective variety of dimension $\leq d$,
\item $(X,\Delta)$ is a $\epsilon$-lc pair such that the coefficients of $\Delta$ lie in $I$, and
\item $K_{X} + \Delta$ is an antiample $\mathbb{Q}$-Cartier divisor.
\end{itemize}
There is a constant $C_{3} = C_{3}(d,\epsilon,I)$ such that for any codimension $2$ set $B \subset X$ there is a movable curve $C$ avoiding $B$ and satisfying
\begin{equation*}
-(K_{X} + \Delta) \cdot C \leq C_{3}.
\end{equation*}
\end{lemm}
\begin{proof}
Theorem \ref{theo: BAB} yields an upper bound $C_{1}$ on $(-K_{X} - \Delta)^{\dim X}$ that only depends on $d$ and $\epsilon$. Choose a positive integer $C_{2}$ as in Theorem \ref{theo: BAB2} so that $-C_{2}(K_{X} + \Delta)$ is a Cartier ample divisor. By \cite[1.1 Theorem and 1.2 Lemma]{Kollar93} there is a positive integer $M = M(d)$ such that $-MC_{2}(K_{X} + \Delta)$ is very ample. Set $C_{3} = C_{1}(C_{2}M)^{d-1}$. Then one can find a suitable curve $C$ by taking intersections of general elements in $|-MC_{2}(K_{X} + \Delta)|$.
\end{proof}
Given a cone $\mathcal{C}$ in $N_{1}(X)$ and an element $\ell \in N^{1}(X)$, we will let $\mathcal{C}_{\ell \geq 0}$ denote the intersection of $\mathcal{C}$ with the half-space of curve classes with non-negative intersection against $\ell$.
\begin{lemm}[\cite{Araujo10}] \label{lemm:conetheorem}
Let $X$ be a smooth projective variety and let $\Delta$ be an effective $\mathbb{Q}$-Cartier divisor on $X$ such that $(X,\Delta)$ is an $\epsilon$-lc pair for some $\epsilon > 0$. Fix an ample $\mathbb{Q}$-Cartier divisor $A$. Then the cone $\overline{\mathrm{Eff}}_{1}(X)_{K_{X} + \Delta \geq 0} + \mathrm{Nef}_{1}(X)$ has only finitely many extremal rays with negative intersection against $K_{X} + \Delta + A$. Furthermore, these rays are generated by the classes of movable curves.
\end{lemm}
\begin{proof}
We first prove the result when the base field is algebraically closed. Let $I$ be the coefficient set of $\Delta$. Suppose that $\phi: X \dashrightarrow X'$ is a run of the $(K_{X} + \Delta)$-minimal model program with scaling resulting in a Mori fiber space $\pi: X' \to Z'$.
Let $Y$ denote a general fiber of $\pi$. Then $\phi_{*}\Delta|_{Y}$ has coefficients in $I$ and $(Y,\phi_{*}\Delta|_{Y})$ is an $\epsilon$-lc Fano pair. Altogether we see that there is a universal bound $C_{3}$ as in Lemma \ref{lemm:intbound} for all such pairs $(Y,\Delta|_{Y})$ obtained in this way.
By \cite[Theorem 1.1]{Araujo10} we have
\begin{equation*}
\overline{\mathrm{Eff}}_{1}(X)_{K_{X} + \Delta \geq 0} + \mathrm{Nef}_{1}(X) = \overline{\mathrm{Eff}}_{1}(X)_{K_{X} + \Delta \geq 0} + \overline{\sum_{\alpha \in \Sigma} \mathbb{R}_{\geq 0}\alpha}
\end{equation*}
where each $\alpha$ is obtained by running a $(K_{X} + \Delta)$-minimal model program with scaling to obtain $\phi: X \dashrightarrow X'$ where $X'$ carries a Mori fiber space structure $\pi: X' \to Z'$ and setting $\alpha$ to be the numerical pullback (as in \cite[Section 4]{Araujo10}) of the class of a curve in a general fiber $Y$ of $\pi$.
As discussed earlier, we know that for any such $Y$ there is a movable curve $C'$ in $Y$ avoiding the $\phi^{-1}$-indeterminacy locus and satisfying $-(K_{X'} + \phi_{*}\Delta) \cdot C' \leq C_{3}$. Since $C'$ avoids the $\phi^{-1}$-indeterminacy locus, its strict transform $C$ in $X$ satisfies $-(K_{X} + \Delta) \cdot C \leq C_{3}$. Note that the numerical pullback of the corresponding class $\alpha$ is represented by the movable curve $C$ because $\rho(X'/Z') =1$.
There are only finitely many classes of curves $C$ which satisfy both $(K_{X} + \Delta + A) \cdot C \leq 0$ and $-(K_{X} + \Delta) \cdot C \leq C_{3}$. This proves that the set of such $\alpha$ as above satisfying $(K_{X} + \Delta + A) \cdot \alpha < 0$ is finite, finishing the proof when the ground field is algebraically closed.
For a general ground field $F$ of characteristic $0$, let $\{\overline{X}_{j}\}_{j=1}^{s}$ denote the components of $X_{\overline{F}/F}$. Note that
\begin{equation*}
N_{1}(X) = (\oplus_{j} N_{1}(\overline{X}_{j}))^{\mathrm{Gal}(\overline{F}/F)}
\end{equation*}
and that the action of $\mathrm{Gal}(\overline{F}/F)$ on $\oplus_{j} N_{1}(\overline{X}_{j})$ factors through a finite group $G$. We can apply the statement of the theorem to each component $\overline{X}_{j}$ and take direct sums to see that $\overline{\mathrm{Eff}}_{1}(X_{\overline{F}})_{K_{X} + \Delta \geq 0} + \mathrm{Nef}_{1}(X_{\overline{F}})$ admits only finitely many $(K_{X} + \Delta + A)$-negative extremal rays generated by numerical classes $\{ \overline{\alpha}_{i} \}_{i=1}^{r}$.
Define $\alpha_{i} = \frac{1}{|G|} \sum_{g \in G} g\overline{\alpha}_{i}$. Since the pseudo-effective and nef cones of curves are preserved by the $G$-action, it is clear that $\overline{\mathrm{Eff}}_{1}(X)_{K_{X} + \Delta \geq 0} + \mathrm{Nef}_{1}(X)$ has only finitely many extremal rays with negative intersection against $K_{X} + \Delta + A$ and these rays are generated by a subset of $\{\alpha_{i}\}_{i=1}^{r}$.
Furthermore, we can construct an irreducible movable curve over the ground field representing some multiple of $\alpha_{i}$ by taking the Galois orbit of an irreducible curve representing $\overline{\alpha}_{i}$ over $\overline{F}$ and descending to $F$.
\end{proof}
\begin{rema}
If $(X,\Delta)$ is a terminal pair, then it is also $\epsilon$-lc for some $\epsilon > 0$. Thus we can apply Lemma \ref{lemm:conetheorem} to terminal pairs $(X,\Delta)$.
\end{rema}
\section{Geometric invariants in Manin's Conjecture}
\label{sec: geoinv}
In this section we study the geometric behavior of the $a$ and $b$ invariants over an arbitrary field $F$ of characteristic $0$.
\subsection{$a$-invariant}
\label{subsec: a-inv}
\begin{defi}
Let $X$ be a smooth projective variety defined over a field $F$ of characteristic $0$. Let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Then we define the {\it Fujita invariant} (or $a$-invariant) by
\[
a(X, L) = \min \{ t \in \mathbb R \mid K_{X} + tL \in \overline{\mathrm{Eff}}^1(X)\}.
\]
When $L$ is nef but not big, we formally set $a(X, L) = +\infty$.
When $X$ is singular, we define the Fujita invariant as the Fujita invariant of the pullback of $L$ to any smooth birational model of $X$. This is well-defined by \cite[Proposition 2.7]{HTT15}.
\end{defi}
We will frequently use the following fundamental properties of the $a$-invariant:
\begin{itemize}
\item If $\phi: X' \to X$ is a birational map then $a(X',\phi^{*}L) = a(X,L)$. (\cite[Proposition 2.7]{HTT15})
\item $a(X, L) > 0$ if and only if $X$ is geometrically uniruled. (\cite[0.3 Corollary]{BDPP})
\item When $a(X, L) > 0$, $a(X, L)$ is always a rational number. (See \cite[Corollary 1.1.7]{BCHM} when $L$ is ample and \cite[Theorem 2.16]{HTT15} when $L$ is big and nef.)
\end{itemize}
It is convenient to concentrate on a certain class of pairs:
\begin{defi} \label{defi: adjointrigid}
Let $X$ be a projective variety and let $L$ be a big and nef divisor on $X$. Let $\phi: X' \to X$ be a resolution of singularities. We say that $(X, L)$ is adjoint rigid if $a(X,L) > 0$ and $\kappa(X',K_{X'} + a(X',\phi^{*}L)\phi^{*}L) = 0$. Note that this definition does not depend on the choice of resolution.
\end{defi}
One should think of adjoint rigid pairs as birational analogues of mildly singular Fano varieties. (See [LTT18, Prop. 2.5] for a precise statement in this direction.) In particular, one can analyze their structure using the recent results of Birkar on the boundedness of singular Fano varieties.
On the other hand, the geometry of arbitrary polarized pairs is controlled by adjoint rigid pairs. Indeed, if $a(X,L)>0$, the following lemma
shows that the closure of a general fiber of the canonical map for $(X,a(X,L)L)$ has the same $a$-value as $X$ and is adjoint rigid.
We will frequently leverage this fact by replacing an arbitrary pair with the fibers of its canonical model map.
\begin{lemm} \label{lemm:ainvandcanonicalfibers}
Let $X$ be a smooth geometrically uniruled projective variety defined over $F$ and $L$ a big and nef $\mathbb Q$-divisor on $X$. Let $\pi: X \dashrightarrow W$ denote the canonical map for $(X,a(X,L)L)$. Let $Y$ be the closure in $X$ of a general fiber of $\pi$. Then $a(Y,L|_{Y}) = a(X,L)$ and $(Y, L|_Y)$ is adjoint rigid.
\end{lemm}
\begin{proof}
By the birational invariance of the $a$-invariant, it suffices to prove this when $\pi$ is a morphism and $Y$ is a smooth fiber of $\pi$. The equality $a(Y,L|_{Y}) = a(X,L)$ follows from the fact that the restriction of $K_{X} + a(X,L)L$ to a general fiber $Y$ has Iitaka dimension $0$, hence lies on the boundary of the pseudo-effective cone.
\end{proof}
We next consider how the $a$-invariant behaves under change of ground field.
\begin{prop} \label{prop: galinvofa}
Let $X$ be a smooth projective variety over a field $F$ of characteristic $0$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$.
We fix an extension $F'/F$ and let $Y \subset X_{F'}$ be an integral subvariety defined over $F'$. Let $\sigma \in \mathrm{Aut}(F'/F)$.
Then we have
\[
a(Y, L_{F'}|_{Y})= a(\sigma(Y), L_{F'}|_{\sigma(Y)})
\]
and the adjoint divisors with respect to $L$ on smooth models of $Y$ and $\sigma (Y)$ have the same Iitaka dimension.
\end{prop}
\begin{proof}
After applying an embedded resolution of singularities, we have a birational model $X'$ over $F'$ such that the strict transform $Y'$ of $Y$ is smooth. Let $\sigma_*X'$ be the pullback of $X' \to \mathrm{Spec} (F')$ by $\sigma^{-1} : \mathrm{Spec} (F') \to \mathrm{Spec} (F')$. We have a natural isomorphism as schemes $\sigma : X' \to \sigma_*X'$ and we denote the image of $Y'$ by $\sigma(Y')$. Then $\sigma(Y')$ is a smooth model of $\sigma(Y)$. Let $K_{Y'}$ be the canonical divisor on $Y'$. Then we have $\sigma(K_{Y'}) = K_{\sigma(Y')}$. Moreover for any integer $m$ such that $ma(Y,L|_{Y})L$ is a Cartier divisor we have
\[
m(K_{Y'} + a(Y,L_{F'}|_{Y})L_{F'}|_{Y'}) \sim D \geq 0 \iff m(K_{\sigma(Y')} + a(Y, L_{F'}|_{Y})L_{F'}|_{\sigma(Y')}) \sim \sigma(D) \geq 0.
\]
This shows that $K_{Y} + a(Y,L_{F'}|_{Y})L_{F'}|_{Y}$ and $K_{\sigma(Y')} + a(Y, L_{F'}|_{Y})L_{F'}|_{\sigma(Y')}$ have the same Iitaka dimension, proving the claims.
\end{proof}
\begin{coro}
\label{coro: flatbasechange}
Let $X$ be a smooth projective variety defined over a field of characteristic $0$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Let $F'/F$ denote a field extension and let $X'$ denote any irreducible component of $X_{F'}$. Then $a(X,L) = a(X',L_{F'}|_{X'})$ and $\kappa(X,K_{X} + a(X,L)L) = \kappa(X', K_{X'} + a(X', L_{F'}|_{X'})L_{F'}|_{X'})$.
\end{coro}
\begin{proof}
It suffices to prove the statement when $F'/F$ is either an algebraic extension or a purely transcendental extension since any extension of fields is a composition of these two types.
Suppose first that $F'/F$ is algebraic. Let $\widetilde{F}/F'$ be an extension such that $\widetilde{F}/F$ is Galois. Then we can prove the desired statement for $F'/F$ by verifying it for both $\widetilde{F}/F$ and $\widetilde{F}/F'$. Thus it suffices to prove the statement under the additional condition that $F'/F$ is Galois.
When the extension is Galois then $\mathrm{Gal}(F'/F)$ acts transitively on the components of $X_{F'}$. By Proposition \ref{prop: galinvofa} we see that every component of $X_{F'}$ has the same $a$-invariant and thus this $a$-invariant is equal to $a(X,L)$.
Furthermore, since base change commutes with taking spaces of sections, we see that for every integer $m$ such that $m(K_{X} + a(X,L)L)$ is a Cartier divisor we have
\begin{equation*}
\dim H^{0}(X,m(K_{X} + a(X,L)L)) = \dim H^{0}(X_{F'},m(K_{X_{F'}} + a(X,L)L_{F'})).
\end{equation*}
Since $X_{F'}$ is a disjoint union of Galois translates of $X'$, we have
\begin{equation*}
\dim H^{0}(X_{F'},m(K_{X_{F'}} + a(X,L)L_{F'})) = r \cdot \dim H^{0}(X',m(K_{X'} + a(X,L)L_{F'}|_{X'})).
\end{equation*}
where $r$ denotes the number of components of $X_{F'}$. Since $r$ does not change as $m$ increases, the Iitaka dimensions coincide.
Now suppose that $F'/F$ is purely transcendental. This implies that $X_{F'}$ is smooth and irreducible. Since sections are preserved by base change we deduce that for any positive rational number $a$ we have $\kappa(X,K_{X} + aL) = \kappa(X_{F'},K_{X_{F'}} + aL_{F'})$. This implies that the $a$-invariants over $F$ and $F'$ also coincide.
\end{proof}
The $a$-invariant is constant for general fibers of a smooth morphism:
\begin{theo} \label{theo: aconstancy}
Let $\pi : \mathcal X \rightarrow W$ be a smooth projective morphism defined over a field $F$ of characteristic $0$ such that every fiber of $\pi$ is integral and geometrically uniruled and let $L$ be a relatively big and nef $\mathbb Q$-divisor on $\mathcal X$. Then there exists a Zariski open subset $W^{\circ} \subset W$ such that the $a$-invariant $a(X_w,L|_{X_w})$ and the Iitaka dimension of $a(X_w, L|_{X_{w}})L|_{X_w} + K_{X_w}$ remain constant as we vary over all closed points $w \in W^{\circ}$.
\end{theo}
\begin{proof}
Over an algebraically closed field this is \cite[Theorem 4.3]{LTDuke}. Over a general $F$, we can take the base change to $\overline{F}$, replace $\mathcal{X}$ with any component of $\mathcal{X}_{\overline{F}}$, and replace the morphism to $W_{\overline{F}}$ with its Stein factorization. Since the space of sections associated to a divisor is stable under flat base change, the result over an arbitrary field of characteristic $0$ follows from the result over its algebraic closure combined with Corollary~\ref{coro: flatbasechange}.
\end{proof}
The following lemmas describe some important geometric properties of the $a$-invariant.
\begin{lemm} \label{lemm: genfinite}
Let $X$ be a projective variety and let $L$ be a big and nef $\mathbb Q$-Cartier divisor on $X$. Suppose that $f : Y \to X$ is a generically finite dominant morphism from a projective variety.
Then we have $a(Y, f^*L) \leq a(X, L)$.
\end{lemm}
\begin{proof}
After birational modifications to $X$ and $Y$ we may assume that $X$ and $Y$ are smooth.
By the ramification formula, we have
\[
a(X, L)f^*L + K_Y = a(X, L)f^*L + f^*K_X + R
\]
where $R \geq 0$ is the ramification divisor. Thus $K_{Y} + a(X, L)f^*L$ is pseudo-effective and our assertion follows.
\end{proof}
\begin{lemm} \label{lemm:ainvdominantfamily}
Let $X$ be a smooth projective variety defined over $F$ and $L$ a big and nef $\mathbb Q$-divisor on $X$. Let $p : \mathcal U \to W$ be a family of subvarieties on $X$ with the evaluation map $s : \mathcal U \rightarrow X$. Suppose that $s$ is dominant. Then for a general member $Y$ of $p$ we have $a(Y, L|_{Y}) \leq a(X, L)$.
\end{lemm}
\begin{proof}
This is stated in \cite[Proposition 4.1]{LTT14} assuming the base field is algebraically closed.
Its generalization to arbitrary fields of characteristic $0$ follows from Corollary~\ref{coro: flatbasechange}.
\end{proof}
\begin{lemm} \label{lemm:dominantequalitycase}
Let $f: Y \to X$ be a dominant generically finite morphism of smooth projective varieties and let $L$ be a big and nef $\mathbb{Q}$-Cartier divisor on $X$. Suppose that $a(Y,f^{*}L) = a(X,L)$. Let $\Gamma$ denote the closure of a general fiber of the canonical map for $(Y,a(Y,f^{*}L)f^{*}L)$ and let $S$ denote its image in $X$. Then $a(S,L|_{S}) = a(X,L)$ and $S$ is adjoint rigid with respect to $L|_{S}$.
\end{lemm}
\begin{proof}
By Lemma \ref{lemm:ainvandcanonicalfibers} we have $a(\Gamma,f^{*}L|_{\Gamma}) = a(Y,f^*L)$ and $\Gamma$ is adjoint rigid with respect to $f^{*}L|_{\Gamma}$. Since $f|_{\Gamma}$ is generically finite for a general $\Gamma$, Lemma \ref{lemm: genfinite} shows that $a(S,L|_{S}) \geq a(\Gamma,f^{*}L|_{\Gamma})$. However as we vary $\Gamma$ the images $S$ form a dominant family of subvarieties of $X$, so by Lemma \ref{lemm:ainvdominantfamily} we must have $a(S,L|_{S}) \leq a(X,L)$. By combining these inequalities with the fact that $a(Y,f^{*}L) = a(X,L)$ we deduce that we have equalities of $a$-values everywhere.
Let $g: \Gamma' \to S'$ be a birational model of $f|_{\Gamma}: \Gamma \to S$ such that $\Gamma'$ and $S'$ are smooth and let $L'$ denote the pullback of $L$ to $S$. By the ramification formula we have
\[
a(S',L')g^*L' + K_{\Gamma'} = a(S', L')g^{*}L' + g^*K_{S'} + R
\]
for some effective divisor $R$. Since the $a$-values of $\Gamma'$ and $S'$ are the same, we deduce that
\begin{equation*}
\kappa(S',K_{S'}+a(S',L')L') \leq \kappa(\Gamma',K_{\Gamma'}+a(\Gamma',g^{*}L')g^{*}L') = 0.
\end{equation*}
Thus $S$ is adjoint rigid with respect to $L|_{S}$.
\end{proof}
\subsection{$b$-invariant}
\label{subsec: b-inv}
\begin{defi} \label{defi:facedefinition}
Let $X$ be a smooth projective variety defined over a field $F$ of characteristic $0$ and let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$.
We let $\mathcal{F}_{X}$ denote the face of $\mathrm{Nef}_{1}(X)$ consisting of classes with vanishing intersection against $K_{X} + a(X,L)L$. We also let $\mathcal{F}^{X} = \mathcal{F}_{X}^{\vee}$ denote the dual face of $\overline{\mathrm{Eff}}^{1}(X)$.
\end{defi}
\begin{defi}
Let $X$ be a smooth projective variety defined over a field $F$ of characteristic $0$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. We define the $b$-invariant by
\begin{align*}
b(F, X, L) = \dim \mathcal \, \mathcal F_X = \mathrm{codim} \, \mathcal F^X.
\end{align*}
When $X$ is singular, we define the $b$-invariant by pulling $L$ back to any smooth birational model of $X$; the result is independent of the choice of model by \cite[Proposition 2.10]{HTT15}. When $L$ is nef but not big, we formally set $b(F, X,L) = \infty$.
\end{defi}
The faces $\mathcal{F}_{X}$ satisfy a natural compatibility under birational transforms.
\begin{lemm} \label{lemm:birfaceinv}
Let $f : X' \to X$ be a generically finite morphism of smooth projective varieties such that every geometric component of $X'$ maps birationally to a geometric component of $X$ via $\overline{f}$. Let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Then the pushforward map $f_{*}: N_{1}(X') \to N_{1}(X)$ induces an isomorphism $f_{*}: \mathcal{F}_{X'} \cong \mathcal{F}_{X}$ where $\mathcal{F}_{X'}$ is the face corresponding to $f^{*}L$.
\end{lemm}
\begin{proof}
For any geometric component of $X'$ the N\'eron-Severi space is spanned by the $\overline{f}$-exceptional divisors and the restriction of $\overline{f}^{*}N^{1}(\overline{X})$ to the component. Since $X'$ is integral we know that $\mathrm{Gal}(\overline{F}/F)$ acts transitively on the geometric components of $X'$. We conclude that $N^{1}(X')$ is spanned by $f$-exceptional divisors and $f^{*}N^{1}(X)$.
We next argue that $f_{*}$ actually maps $\mathcal{F}_{X'}$ to $\mathcal{F}_{X}$. Note that there is an effective $\phi$-exceptional divisor $E$ on $X'$ such that $K_{X'} + a(X',f^{*}L)f^{*}L = f^{*}(K_{X} + a(X,L)L) + E$. Thus any nef curve class with vanishing intersection against $K_{X'} + a(X',f^{*}L)f^{*}L$ will also have vanishing intersection against $f^{*}(K_{X} + a(X,L)L)$ and vanishing intersection against $E$. The former property shows that $f_{*}$ maps $\mathcal{F}_{X'}$ to $\mathcal{F}_{X}$.
Next note that $E$ contains every $f$-exceptional divisor with a positive coefficient. Hence any class in $\mathcal{F}_{X'}$ has zero intersection against every $f$-exceptional divisor. Therefore every curve class in $\mathcal{F}_{X'}$ is pulled back from $\mathcal{F}_{X}$.
Thus the pullback for curve classes induces a map $\frac{1}{\deg(f)} f^* : \mathcal F_X \to \mathcal F_{X'}$ inverse to $f_{*}$ and our assertion follows.
\end{proof}
In contrast to the $a$-value, the $b$-value can change upon field extension, but it can only increase.
\begin{prop}
Let $X$ be a geometrically integral smooth projective variety defined over a field $F$ of characteristic $0$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Let $F'/F$ be a finite extension. Then we have
\[
b(F, X, L) \leq b(F', X_{F'}, L_{F'}).
\]
\end{prop}
\begin{proof}
This follows from the fact that $b(F, X, L)$ is given by
\[
\dim \mathrm{Nef}_1(\overline{X})^{\mathrm{Gal}(\overline{F}/F)} \cap (K_{X} + a(X,L)L)^{\perp}.
\]
\end{proof}
Recall that a face $\mathcal{F}$ of a cone $\mathcal{C}$ is said to be supported if there is a linear functional which is non-negative on $\mathcal{C}$ and vanishes precisely along $\mathcal{F}$. It follows from the theory of dual cones that $\mathcal{F}^{X}$ is the minimal supported face of $\overline{\mathrm{Eff}}^{1}(X)$ containing the class of $K_{X} + a(X,L)L$. The following result shows that $\mathcal{F}^{X}$ satisfies a stronger property; a related statement is given in \cite[Theorem 2.16]{HTT15}.
\begin{lemm} \label{lemm:alternativedescription}
Let $X$ be a geometrically uniruled geometrically integral smooth projective variety and let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$. Then:
\begin{enumerate}
\item $\mathcal{F}^{X}$ is the minimal face of $\overline{\mathrm{Eff}}^{1}(X)$ containing $K_{X} + a(X,L)L$.
\item Let $\mathcal{E}$ denote the set of irreducible divisors $E$ such that for some $c>0$ the divisor $K_{X} + a(X,L)L - cE$ is $\mathbb{Q}$-linearly equivalent to an effective divisor. Then the cone generated by the classes of the divisors in $\mathcal{E}$ contains a relatively open neighborhood of $K_{X} + a(X,L)L$ in $\mathcal{F}^{X}$. In particular, the classes of the divisors in $\mathcal{E}$ span $\mathrm{Span}(\mathcal{F}^{X})$.
\end{enumerate}
\end{lemm}
\begin{proof}
By Lemma \ref{lemm: terminalpair} $a(X,L)L$ is $\mathbb{Q}$-linearly equivalent to a sum $\Delta + A$ where $(X,\Delta+A)$ is terminal and $A$ is ample.
Let $C$ be a compact polyhedral convex hull of a finite set of ample $\mathbb{Q}$-divisors such that its interior contains $A$.
Let $U$ denote the set of classes in $N^{1}(X)$ of the form
\begin{equation*}
U = \left\{ \left. K_{X} + \Delta + \frac{1}{2}A + \lambda D \right| D \in C, \lambda \geq 0 \right\}.
\end{equation*}
In particular $U$ is a closed rational polyhedral set containing $K_{X} + a(X,L)L$ in its interior.
(1) We apply Lemma \ref{lemm:conetheorem} to the pair $(X,\Delta)$ and the ample $\mathbb{Q}$-divisor $\frac{1}{4}A$ to obtain a finite set of $(K_{X} + \Delta+ \frac{1}{4}A)$-extremal rays. Note that these are the only extremal rays of $\mathrm{Nef}_{1}(X)$ which can have vanishing intersection against an element of $U$ that lies on the boundary of the pseudo-effective cone. Thus $\overline{\mathrm{Eff}}^1(X) \cap U$ is a rational polyhedral set, since it is cut out in $U$ by finitely many rational linear inequalities. In particular, every face of $\overline{\mathrm{Eff}}^{1}(X)$ intersecting the interior of $U$ must be a supported face, proving (1).
(2) By \cite[Theorem D]{BCHM} (and the fact that $\mathbb{R}$-linear equivalence can be detected after change of base field), every pseudo-effective class contained in $U$ is represented by an effective divisor. By (1) we know that $K_{X} + a(X,L)L$ is in the relative interior of $\mathcal{F}^{X}$. Thus, if $E$ is an effective divisor representing a class in $\mathcal{F}^{X} \cap U$, there is some positive constant $c$ such that $K_{X} + a(X,L)L - cE$ also has class in $\mathcal{F}^{X} \cap U$ and is thus represented by an effective divisor. The statement follows.
\end{proof}
\cite[Theorem 1.2]{Sengupta17} shows that over an algebraically closed field of characteristic $0$ the $b$-invariant is constant for an open set of fibers in a smooth family of uniruled projective varieties. However, the behavior over a number field is more subtle since the $b$-invariant depends on the splitting behavior of the fibers.
The following useful criterion gives a geometric characterization of the $b$-invariant over an arbitrary field of characteristic $0$. An analogous statement over $\mathbb{C}$ is proved in \cite[Lemma 2.11]{Sengupta17}.
\begin{lemm} \label{lemm: monodromyandbvalue}
Let $X$ be a geometrically uniruled geometrically integral smooth projective variety defined over a field $F$ of characteristic $0$, and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Let $\pi: X \dashrightarrow W$ be the canonical model for $K_{X} + a(X,L)L$. Suppose that there is a non-empty open set $W^{\circ} \subset W$ such that (i) $W^\circ$ is smooth, (ii) $\pi$ is well-defined and smooth over $W^{\circ}$, and (iii) every fiber $X_{w}$ over a closed point $w$ satisfies $a(X_{w},L|_{X_{w}}) = a(X,L)$. Let $w \in W^{\circ}$ be a closed point and fix a geometric point $\overline{w}$ above $w$.
\begin{enumerate}
\item We have
\begin{align*}
b(F,X,L) & = \dim \, \left((N^{1}(X_w)\cap N^1(\overline{X}_{\overline{w}})^{\pi^{\textrm{\'et}}_1(\overline{W^{\circ}},\overline{w})}) / \mathrm{Span}(\{E_{i}\}_{i = 1}^r) \right)
\end{align*}
where $\{ E_{i} \}_{i=1}^r$ is the finite set of irreducible divisors which dominate $W$ under $\pi$ and which satisfy $K_{X} + a(X,L)L - c_iE_{i} \in \overline{\mathrm{Eff}}^{1}(X)$ for some $c_i > 0$. (Here we are identifying $N^1(X_w)$ with its image under the natural embedding into $N^1(\overline{X}_{\overline{w}})$.)
\item Let $i: X_w \hookrightarrow X$ denote the inclusion. Then $i_{*}(\mathcal F_{X_w}) \subset \mathcal F_X$.
\item If $X_{w}$ is a general fiber of $\pi$ then $i_{*}: \mathcal F_{X_w} \to \mathcal F_X$ is a surjection.
\end{enumerate}
\end{lemm}
\begin{proof}
We may resolve $\pi$ to be a morphism without affecting $\pi^{-1}(W^{\circ})$, and since by Lemma \ref{lemm:birfaceinv} blowing-up induces an isomorphism of the faces $\mathcal{F}_{X}$ the statement for the blow-up is equivalent to the statement for the original variety.
(1) We first show that there are only finitely many divisors $\{ E_{i} \}_{i=1}^r$ which dominate $W$ and satisfy $K_{X} + a(X,L)L - c_iE_{i} \in \overline{\mathrm{Eff}}^{1}(X)$ for some $c_i > 0$. The geometric generic fiber $\overline{X}_{\overline{\eta}}$ is adjoint rigid with respect to $L$. If we restrict $E_{i}$ to $\overline{X}_{\overline{\eta}}$, then the support must lie in the unique effective divisor numerically equivalent to $K_{\overline{X}_{\overline{\eta}}} + a(X,L)L|_{\overline{X}_{\overline{\eta}}}$. Thus there are only finitely many divisors $E_{i}$ of this type.
By \cite[Th\'eor\`eme 5.2]{andre96} and \cite[Theorem 1.1]{MP12} we know that for some geometric point $\overline{s} \in \overline{W}^{\circ}$ the specialization map $N^{1}(\overline{X}_{\overline{\eta}}) \to N^{1}(\overline{X}_{\overline{s}})$ is an isomorphism where $\overline{\eta}$ denotes the geometric generic point of $W$. Furthermore by \cite[Proposition 3.3]{MP12} this isomorphism is compatible with restriction from $N^{1}(\overline{X})$ and is a map of $\pi^{\textrm{\'et}}_1(\overline{W^{\circ}},\overline{s})$ modules.
Note that the image of the map
\begin{equation*}
N^1(\overline{X}) \twoheadrightarrow N^{1}(\overline{X}_{\eta}) \to N^{1}(\overline{X}_{\overline{\eta}})
\end{equation*}
is exactly the $\mathrm{Gal}(\overline{\overline{F}(\overline{W})}/\overline{F}(\overline{W}))$ invariant part, or equivalently, the $\pi^{\textrm{\'et}}_1(\overline{W^{\circ}},\overline{s})$ invariant part. Altogether we conclude that under the restriction map $N^{1}(\overline{X})$ will surject onto $N^{1}(\overline{X}_{\overline{s}})^{\pi^{\textrm{\'et}}_1(\overline{W^{\circ}},\overline{s})}$ for a single fiber $\overline{X}_{\overline{s}}$.
This implies that $N^{1}(\overline{X}) \to N^{1}(\overline{X}_{\overline{t}})^{\pi^{\textrm{\'et}}_1(\overline{W^{\circ}},\overline{t})}$ is surjective for every $\overline{t} \in \overline{W}^{\circ}$. Indeed, by \cite[Theorem 10.19]{Voisin07} for every $\overline{t}$ the kernel of $N^{1}(\overline{X}) \to N^{1}(\overline{X}_{\overline{t}})$ is the subspace of $N^{1}(\overline{X})$ spanned by all $\pi$-vertical divisors. Thus it suffices to show that the rank of $N^{1}(\overline{X}_{\overline{t}})^{\pi^{\textrm{\'et}}_1(\overline{W^{\circ}},\overline{t})}$ is constant. Since the generic fiber of $\overline{\pi}$ is rationally connected, the Picard rank of the geometric fibers of $\pi$ is constant. By taking monodromy invariants we obtain the claim.
In particular, we have a surjection $N^{1}(\overline{X}) \to N^{1}(\overline{X}_{\overline{w}})^{\pi^{\textrm{\'et}}_1(\overline{W^{\circ}},\overline{w})}$. Note that the action of $\mathrm{Gal}(\overline{F}/F)$ on $N^{1}(\overline{X})$ and on $N^{1}(\overline{X}_{w})$ factors through a finite group. Thus we see that the restriction map
\begin{equation}\label{equation:monodromy}
N^{1}(\overline{X})^{\mathrm{Gal}(\overline{F}/F)} \to N^{1}(X_w)\cap N^1(\overline{X}_{\overline{w}})^{\pi^{\textrm{\'et}}_1(\overline{W^{\circ}},\overline{w})}
\end{equation}
is surjective.
By Lemma \ref{lemm:alternativedescription}, $b(F,X,L)$ is the dimension of the quotient of $N^{1}(X)$ by all effective irreducible divisors $E$ satisfying $K_{X} + a(X,L)L - c E \in \overline{\mathrm{Eff}}^{1}(X)$ for some $c>0$. Note that $E$ lies in the kernel of the restriction map to $N^{1}(X_{w})$ if and only if $\pi(E) \subsetneq W$ and $E$ satisfies $K_{X} + a(X,L)L - cE \in \overline{\mathrm{Eff}}^{1}(X)$ for some $c > 0$. Thus if we further quotient by the finitely many $E_i$'s which dominate $W$ and satisfy $K_{X} + a(X,L)L - c E \in \overline{\mathrm{Eff}}^{1}(X)$ for some $c>0$, the dimension of the resulting space is $b(F,X,L)$.
(2) When $X_w$ is smooth with trivial normal bundle, \cite[Arxiv version: 6.8 Theorem]{Peternell} shows that after base-changing to $\overline{F}$, pseudo-effective divisors on $X_{\overline{F}}$ restrict to pseudo-effective divisors on $X_{w, \overline{F}}$. This implies that pseudo-effective divisors on $X$ restrict to pseudo-effective divisors on $X_w$, or dually, nef curve classes on $X_w$ push forward to nef curve classes on $X$. Using adjunction and the assumption that $a(X,L) = a(X_w,L|_{X_w})$ we see that $f_*$ maps $\mathcal{F}_{X_w}$ to $\mathcal{F}_X$.
(3) Finally, we show that when $X_{w}$ is general then $i_{*}$ induces a surjection of faces. By Lemma \ref{lemm: terminalpair} $L$ is $\mathbb{Q}$-linearly equivalent to a sum $\Delta + A$ where $A$ is ample and $(X,\Delta + A)$ is terminal. By Lemma \ref{lemm:conetheorem} the face $\mathcal{F}_{X}$ only contains finitely many extremal rays of $\mathrm{Nef}_{1}(X)$ and each such ray is generated by the class of a movable curve $C_{j}$ on $X$. Recall that we have modified $X$ so that the canonical map $\pi$ is a morphism on $X$. Thus there is an ample divisor $H$ on $W$ such that $K_{X} + a(X,L) L - \pi^{*}H$ is $\mathbb{Q}$-linearly equivalent to an effective divisor.
This implies that each $C_{j}$ must be $\pi$-vertical. In particular, for each $C_{j}$, a general fiber of $\pi$ will contain an irreducible deformation of $C_{j}$ which is a movable curve on that fiber. Thus the class of each such curve is nef on $X_{w}$, and in particular, is contained in $\mathcal{F}_{X_{w}}$. This proves that $i_* : \mathcal F_{X_w} \to \mathcal F_X$ is surjective.
\end{proof}
\begin{coro} \label{coro: bvalequality}
Let $X$ be a geometrically uniruled geometrically integral smooth projective variety defined over a field $F$ of characteristic $0$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Suppose that $\pi: X \to W$ is a surjective morphism to a projective variety $W$ which is birationally equivalent to the canonical model map for $K_{X} + a(X,L)L$.
Suppose that there is a non-empty smooth open set $W^{\circ} \subset W$ such that $\pi$ is smooth over $W^{\circ}$ and the geometric monodromy action of $\pi^{\textrm{\'et}}_1(\overline{W^{\circ}},\overline{w})$ on $N^1(\overline{X}_{\overline{w}})$ is trivial for closed points $w \in W^{\circ}$.
Then for a general point $w \in W^{\circ}$ the inclusion $i: X_{w} \to X$ induces an isomorphism $i_{*}: \mathcal{F}_{X_{w}} \to \mathcal{F}_{X}$. In particular we have $b(F,X,L) = b(F,X_{w},L|_{X_{w}})$.
\end{coro}
\begin{proof}
Let $\phi: X' \to X$ be a smooth birational model such that the canonical map for $K_{X'} + a(X',\phi^{*}L)\phi^{*}L$ is a morphism on $X'$. Let $\pi': X' \to W$ be the composition $\pi \circ \phi$. Note that $\pi'$ agrees with the canonical map over an open subset of $W$. We first shrink $W^{\circ}$ to ensure that $\pi'$ coincides with the canonical map over $W^{\circ}$. After shrinking $W^{\circ}$ further we may ensure that $W^{\circ}$ is smooth
and that $\pi'$ is smooth over $W^{\circ}$, and by Lemma \ref{lemm:ainvandcanonicalfibers} we may also ensure that every fiber $X'_{w}$ over a closed point $w \in W^{\circ}$ satisfies $a(X'_{w},\phi^{*}L|_{X'_{w}}) = a(X',\phi^{*}L)$. We may now apply Lemma \ref{lemm: monodromyandbvalue} (3) to deduce that for a general $w \in W^{\circ}$ the map $i_{*}: \mathcal{F}_{X'_{w}} \to \mathcal{F}_{X'}$ is a surjection. By Lemma~\ref{lemm:birfaceinv} the map $i_{*}: \mathcal{F}_{X_{w}} \to \mathcal{F}_{X}$ is also surjective. On the other hand, the monodromy assumption shows that $N^1(X) \rightarrow N^1(X_w)$ is surjective by \eqref{equation:monodromy}. Dually $i_{*}: N_1(X_w) \to N_1(X)$ is an injection, proving the claim.
\end{proof}
\subsection{Boundedness results}
\label{subsec: BAB}
Using the boundedness of singular Fano varieties proved by Birkar in \cite{birkar16} and \cite{birkar16b}, the papers \cite{LTT14}, \cite{HJ16}, \cite{LTDuke}, \cite{Sen17} prove certain types of boundedness results for the $a,b$-invariants over an algebraically closed field. In this section we revisit these results in the setting of an arbitrary field $F$ of characteristic $0$.
The following theorem proves the boundedness of subvarieties with $a$-value at least as large as $X$.
\begin{theo} \label{theo: aconstruction}
Let $X$ be a geometrically uniruled smooth projective variety defined over $F$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. There exists a constructible bounded subset $T \subset \mathrm{Hilb}(X)$, a decomposition $T = \cup T_{i}$ of $T$ into locally closed subvarieties, and smooth projective morphisms $p_{i}: \mathcal U_{i} \to T_{i}$ and morphisms $s_{i}: \mathcal U_{i} \to X$ such that
\begin{itemize}
\item over $\overline{F}$, each fiber of $\overline{p}_i : \overline{\mathcal U}_i \rightarrow \overline{T}_i$ is an integral uniruled variety which is mapped birationally by $\overline{s}_{i}$ onto the subvariety of $\overline{X}$ parametrized by the corresponding point of $\mathrm{Hilb}(\overline{X})$;
\item every fiber $Y$ of $p_{i}$ is a smooth variety satisfying $a(Y,s_{i}^{*}L|_{Y}) \geq a(X,L)$ and is adjoint rigid with respect to $s_{i}^{*}L|_{Y}$;
and
\item for every subvariety $Y \subset X$ not contained in $\mathbf{B}_{+}(L)$ which satisfies $a(Y,L|_{Y}) \geq a(X,L)$ and which is adjoint rigid with respect to $L$, there is some index $i$ such that $Y$ is birational to a fiber of $p_{i}$ under the map $s_{i}$.
\end{itemize}
Furthermore, if $s_{i}: \mathcal U_{i} \to X$ is dominant then $s_i$ must be generically finite.
\end{theo}
Related results have appeared in \cite{HJ16} and \cite{LTDuke}. We will give a quick proof by appealing to \cite{birkar16b}.
\begin{proof}
Suppose that we have a subvariety $\overline{Y} \subset \overline{X}$ which is adjoint rigid with respect to $L$, not contained in $\mathbf{B}_{+}(L_{\overline{X}})$, and has $a$-value at least as large as $a(X,L)$. Let $\psi: \widetilde{\overline{Y}} \to \overline{Y}$ be a resolution. By \cite[Theorem 3.5]{LTDuke}, there is a birational contraction $\phi: \widetilde{\overline{Y}} \dashrightarrow \overline{Y}'$ where $\overline{Y}'$ is a $\mathbb{Q}$-factorial terminal weak Fano variety such that $a(\overline{Y},L|_{\overline{Y}})\phi_{*}\psi^{*}L|_{\overline{Y}} \equiv -K_{\overline{Y}'}$. By \cite[Theorem 2.11]{birkar16b} (whose validity is established in all dimensions by induction), we see that there is some constant $C$ depending only on the dimension of $X$ such that
\begin{align*}
C \geq (-K_{\overline{Y}'})^{\dim \overline{Y}'} & = a(\overline{Y},L|_{\overline{Y}})^{\dim \overline{Y}} \left(\phi_{*}\psi^{*}L|_{\overline{Y}}\right)^{\dim \overline{Y}} \\
& \geq a(\overline{Y},L|_{\overline{Y}})^{\dim \overline{Y}} (\psi^{*}L|_{\overline{Y}})^{\dim \overline{Y}} \\
& = a(\overline{Y},L|_{\overline{Y}})^{\dim \overline{Y}} L|_{\overline{Y}}^{\dim \overline{Y}}.
\end{align*}
Thus we have
\begin{equation*}
L|_{\overline{Y}}^{\dim \overline{Y}} \leq C/a(\overline{Y},L|_{\overline{Y}})^{\dim \overline{Y}} \leq C/a(\overline{X},L)^{\dim \overline{Y}}.
\end{equation*}
By applying \cite[Lemma 4.7]{LTT14} to each irreducible component of $\overline{X}$ we conclude that such subvarieties $\overline{Y}$ are parametrized by a bounded subset of $\mathrm{Hilb}(\overline{X})$.
We let $\overline{M} \subset \mathrm{Hilb}(\overline{X})$ denote a finite union of components which contains this locus (equipped with the reduced structure).
Let $\overline{N} \subset \overline{M}$ denote the constructible subset over which the universal family has irreducible and reduced fibers. By repeatedly resolving singularities in the fibers and stratifying the base, we find a finite union of locally closed subsets $\overline{N}_{i} \subset \overline{N}$ and smooth morphisms $\overline{p}_{i}: \overline{\mathcal U}_{i} \to \overline{N}_{i}$ whose fibers are smooth projective varieties birational to the subvarieties of $\overline{X}$ parametrized by points of $\overline{N}$. We next replace each $\overline{N}_{i}$ by the open subset parametrizing subvarieties not contained in $\mathbf{B}_{+}(L)$; this ensures that $L$ is relatively big and nef on the universal family over $\overline{N}_{i}$. Applying Theorem \ref{theo: aconstancy} and further stratifying the $\overline{N}_{i}$, we may suppose that the $a$-invariant and Iitaka dimension of the adjoint pair is constant in each family. In particular, there is a constructible sublocus $\overline{T} \subset \overline{M}$ parametrizing the subvarieties which are not contained in $\mathbf{B}_{+}(L)$, have $a$-invariant at least $a(X,L)$ and are adjoint rigid with respect to $L$.
Now we note that $\overline{T}$ is defined over the ground field and thus descends to a subset $T$ of $\mathrm{Hilb}(X)$. Indeed, given any subvariety $Y$ parametrized by a point of $\overline{T}$ and any $\sigma \in \mathrm{Gal}(\overline{F}/F)$, by Proposition \ref{prop: galinvofa} $\sigma(Y)$ is also parametrized by $\overline{T}$.
Furthermore $\mathbf{B}_{+}(L_{\overline{X}})$ descends to $\mathbf{B}_{+}(L)$. By repeatedly resolving singularities of fibers of the universal family over $T$ and taking finer stratifications of the base, we construct smooth families $p_{i}: \mathcal U_{i} \to T_{i}$ whose fibers are birational to the subvarieties parametrized by $T$. We can further stratify $T$ so that after base-change to $\overline{F}$ we obtain a substratification of the original stratification of $\overline{T}$; since the $a$-invariant and Iitaka dimension are not affected by base change (see Corollary \ref{coro: flatbasechange}) these families will now have the desired properties. The final statement concerning generic finiteness is proved by \cite[Proposition 4.14]{LTDuke}.
\end{proof}
It will be convenient to work with two variants of Theorem \ref{theo: aconstruction}.
The first focuses on subvarieties with strictly larger $a$-invariant. \cite{HJ16} proves a similar result when $L$ is big and semiample.
\begin{theo}
\label{theo: HJ}
Let $X$ be a geometrically uniruled smooth projective variety and let $L$ be a big and nef $\mathbb Q$-divisor on $X$.
\begin{enumerate}
\item As we vary over all projective subvarieties $Y$ not contained in $\mathbf{B}_{+}(L)$, the set
\begin{equation*}
\{ \, a(Y,L|_{Y}) \, | \, Y \not \subset \mathbf{B}_{+}(L) \} \cap [a(X,L),\infty)
\end{equation*}
is finite.
\item For any fixed $a \geq a(X,L)$, let $V^{a}$ denote the union of all subvarieties $Y$ such that $a(Y, L|_Y) > a$.
Then $V^{a} \subsetneq X$ is a proper closed subset and each component $V^{a}_{d} \subset V^{a}$ satisfies $a(V^{a}_{d}, L|_{V^{a}_{d}}) > a$.
\end{enumerate}
\end{theo}
\begin{proof}
(1) follows immediately from the finiteness of the families in Theorem \ref{theo: aconstruction}. To see (2), we first combine the finiteness of the families in Theorem \ref{theo: aconstruction} with Lemma \ref{lemm:ainvdominantfamily} to see that $V^{a}$ is not Zariski dense in $X$. This argument also shows that each component of the Zariski closure of $V^{a}$ which is not contained in $\mathbf{B}_{+}(L)$ admits a dominant family of subvarieties $Y$ such that $a(Y,L|_{Y}) > a$. By again applying Lemma \ref{lemm:ainvdominantfamily} we see that each component of the Zariski closure of $V^{a}$ which is not contained in $\mathbf{B}_{+}(L)$ has $a$-value larger than $a$. Furthermore \cite{Nakamaye00} shows that each component $B_{d}$ of $\mathbf{B}_{+}(L)$ satisfies $a(B_{d},L|_{B_{d}}) = \infty$. Together these imply that $V^{a}$ is closed and that $a(V^{a}_{d}, L|_{V^{a}_{d}}) > a$ for every component $V^{a}_{d}$.
\end{proof}
The second variant replaces the families in Theorem \ref{theo: aconstruction} with projective closures and replaces the smooth families with the corresponding universal families in $\mathrm{Hilb}(X)$.
\begin{theo}
\label{theo: rigidfamilies}
Let $X$ be a geometrically uniruled smooth projective variety and let $L$ be a big and nef $\mathbb Q$-divisor on $X$.
Then there exist a proper closed subset $V$ and finitely many families $p_i : \mathcal U_i \rightarrow W_i$ of closed subschemes of $X$ where $W_i$ is a projective subscheme of $\mathrm{Hilb}(X)$ such that
\begin{itemize}
\item over $\overline{F}$, $\overline{p}_i : \overline{\mathcal U}_i \rightarrow \overline{W}_i$ generically parametrizes integral uniruled subvarieties of $\overline{X}$;
\item for each $i$, the evaluation map $s_i : \mathcal U_i \rightarrow X$ is generically finite and dominant;
\item for each $i$, a general member $Y$ of $p_i$ is a subvariety of $X$ such that $(Y,L|_{Y})$ is adjoint rigid and $a(X,L) = a(Y,L|_{Y})$; and
\item for any subvariety $Y$ such that $(Y,L|_{Y})$ is adjoint rigid and $a(Y, L|_Y) \geq a(X, L)$, either $Y$ is contained in $V$ or $Y$ is a member of a family $p_i : \mathcal U_i \rightarrow W_i$ for some $i$.
\end{itemize}
\end{theo}
\begin{proof}
First, we replace the loci $T_{i} \subset \mathrm{Hilb}(X)$ in Theorem \ref{theo: aconstruction} with their projective closures $W_{i}$ (equipped with the reduced structure). Let $p_{i}: \mathcal{U}_{i} \to W_{i}$ be the universal family equipped with evaluation maps $s_{i}: \mathcal{U}_{i} \to X$. To start with we set $V = \mathbf{B}_{+}(L)$. Then for each $i$ such that $s_{i}: \mathcal{U}_{i} \to X$ is not dominant, we add the image of $s_{i}$ to $V$ and remove $\mathcal{U}_{i}$ from our set of families. By Theorem \ref{theo: HJ}, any subvariety with $a$-invariant larger than $X$ will be contained in $V$. The desired properties of the $\mathcal{U}_{i}$ then follow directly from Theorem \ref{theo: aconstruction}.
\end{proof}
Finally, we will need two results useful for understanding dominant breaking thin maps. The cited statements are proved over an algebraically closed field but the extension to arbitrary fields of characteristic $0$ is immediate via Corollary \ref{coro: flatbasechange}.
\begin{theo}{\cite[Corollary 2.8]{Sen17}}
\label{theo: akash}
Let $X$ be a geometrically uniruled smooth projective variety and let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$.
Suppose that $f: Y \rightarrow X$ is a dominant generically finite morphism with $Y$ smooth projective and with $a(Y, f^*L)=a(X, L)$. Suppose $R_{i}$ is a component of the ramification divisor $R$ on $Y$ which dominates the base of the canonical model map for $K_{Y} + a(Y,f^{*}L)f^{*}L$ and whose image $B_i$ is a component of the branch divisor $B$ on $X$. Then
\[
a(B_i, L|_{B_i}) > a(X, L).
\]
\end{theo}
\begin{prop}{\cite[Proposition 2.15]{Sen17}}
\label{prop: degree}
Let $X$ be a geometrically uniruled smooth projective variety and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Then there exists a constant $M$ only depending on $\dim X$, $L^{\dim X}$, and $a(X, L)$ such that for any dominant thin map $f: Y \rightarrow X$ such that $Y$ is geometrically integral, $a(Y, f^{*}L) = a(X, L)$, and $(Y, f^*L)$ is adjoint rigid, we have
\[
\deg (f: Y \rightarrow f(Y)) \leq M.
\]
\end{prop}
\subsection{Fiber dimension}
\label{subsec: fiberdim}
As discussed earlier, we will frequently study the geometry of an arbitrary polarized pair $(X,L)$ by replacing it with the fibers of its canonical model. In this section we define and study an invariant which helps us understand this replacement operation.
\begin{defi}
Let $X$ be a smooth projective variety defined over a field $F$ of characteristic $0$. Let $L$ be a big and nef $\mathbb Q$-divisor on $X$. We define
\[
d(X, L) = \dim(X) - \kappa(X,K_{X} + a(X,L)L).
\]
When $X$ is singular, we define $d(X,L)$ by pulling $L$ back to a smooth birational model of $X$. Note that $d(X,L)$ is invariant under extension of the ground field.
\end{defi}
The following lemma shows that any adjoint rigid subvarieties of $X$ with dimension larger than $d(X,L)$ must be contained in a closed subset. We will use it to show that subvarieties of this type can only contribute a thin subset of rational points.
\begin{lemm}
\label{lemm:d(Y)>d(X)}
Let $X$ be a geometrically uniruled smooth projective variety and let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$. There is a proper closed subset $V \subsetneq X$ such that if $f: Y \to X$ is any thin map satisfying $a(Y,f^{*}L) \geq a(X,L)$ and $d(Y,L) > d(X,L)$ then $f(Y) \subset V$.
\end{lemm}
\begin{proof}
Let $V \subset X$ denote the closed subset defined by Theorem \ref{theo: rigidfamilies}. Assume for a contradiction that $f(Y) \not \subset V$. In particular this implies that $a(Y,f^{*}L) = a(f(Y),L|_{f(Y)}) = a(X,L)$. Let $\Gamma$ denote the closure of a general fiber of the canonical model for $(Y,a(Y,f^{*}L)f^{*}L)$ and let $S$ denote its image in $X$. By Lemma \ref{lemm:dominantequalitycase} we see that $S$ has the same $a$-invariant as $X$ and is adjoint rigid with respect to the restriction of $L$. Since the general such $S$ is not contained in $V$, as we vary $\Gamma$ the images $S$ are parametrized by a family $p_{i}: \mathcal{U}_{i} \to W_{i}$ as in Theorem \ref{theo: rigidfamilies} such that the evaluation map $s_{i}$ is dominant (even if $f$ is not dominant).
To summarize this discussion, we will obtain the desired contradiction if we can prove that there is no dominant family of subvarieties $S$ such that $(S,L|_{S})$ is adjoint rigid, $a(S,L|_{S}) = a(X,L)$ and $\dim(S) > d(X,L)$.
To prove this, we may replace $X$ by any birational model. In particular we may suppose that the canonical model map for $K_{X} + a(X,L)L$ is a morphism $\pi: X \to W$. Thus there is an ample $\mathbb{Q}$-divisor $H$ on $W$ and an effective $\mathbb{Q}$-divisor $E$ such that $K_{X} + a(X,L)L$ is $\mathbb{Q}$-linearly equivalent to $\pi^{*}H + E$. Suppose we have a diagram of smooth varieties
\begin{equation*}
\xymatrix{ \mathcal{S} \ar[r]^{g} \ar[d]_{q} & X \\
T & }
\end{equation*}
such that $g$ is generically finite and dominant and the fibers of $q$ are smooth varieties $S_t$ satisfying $a(S_t,g^{*}L|_{S_t}) = a(X,L)$ and $\dim(S_t) > d(X,L)$. We can write
\begin{equation*}
K_{\mathcal{S}} + a(X,L)g^{*}L = g^{*}(K_{X} + a(X,L)L) + R \sim_{\mathbb{Q}} g^{*}\pi^{*}H + (g^{*}E + R)
\end{equation*}
for some effective divisor $R$. Note that the restriction of $g^{*}\pi^{*}H$ to a general fiber of $q$ has Iitaka dimension at least $1$, so that the general fiber of $q$ can not possibly be adjoint rigid with respect to the restriction of $L$. This proves the desired contradiction.
\end{proof}
\subsection{Face contraction}
\label{subsec: facecontracting}
The notion of face contraction refines the $b$-invariant. We will use it to help classify when the point contributions of a dominant map with equal geometric invariants should lie in the exceptional set for Manin's Conjecture.
We will be interested in the following situation:
\begin{assum} \label{assum:fc}
Let $X$ be a geometrically uniruled geometrically integral smooth projective variety defined over a field $F$ of characteristic $0$ and let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$. Let $f: Y \to X$ be a morphism of smooth projective varieties that is generically finite onto its image. Suppose
that either
\begin{enumerate}
\item $f$ is dominant and $a(Y,f^{*}L) = a(X,L)$, or
\item $a(Y,f^{*}L) = a(X,L)$, $d(Y,f^{*}L) = d(X,L)$, and there is a commuting diagram
\begin{equation*}
\xymatrix{ Y \ar[r]^{f} \ar[d]_{\pi_{Y}} & X \ar[d]_{\pi_{X}} \\
T \ar[r] & W}
\end{equation*}
where $\pi_{Y}$ and $\pi_{X}$ are the canonical models for the adjoint pairs and the general fiber of $\pi_{Y}$ maps onto a fiber of $\pi_{X}$ which has the same $a$-value as $X$ and is contained in the locus where $\pi_{X}$ is smooth.
\end{enumerate}
\end{assum}
\begin{lemm} \label{lemm:pushforwardpreservesfaces}
Suppose we are in the situation of Assumption \ref{assum:fc}. Then the pushforward map $f_{*}: N_{1}(Y) \to N_{1}(X)$ satisfies $f_{*}\mathcal{F}_{Y} \subset \mathcal{F}_{X}$.
\end{lemm}
\begin{proof}
In case (1) $f$ is a dominant map, so by the Riemann-Hurwitz formula we have $K_{Y} \geq f^{*}K_{X}$. Since the $a$-invariants are the same we have $K_{Y} + a(Y,f^{*}L)f^{*}L \geq f^{*}(K_{X} + a(X,L)L)$. Thus any nef curve class on $Y$ with vanishing intersection against $K_{Y} + a(Y,f^{*}L)L$ also has vanishing intersection against $f^{*}(K_{X} + a(X,L)L)$, showing that $f_{*}\mathcal{F}_{Y} \subset \mathcal{F}_{X}$.
Suppose that we are in case (2). Fix a general closed point $w$ in the $(\pi_{X}\circ f)$-image of $Y$. By assumption $X_{w}$ is smooth. Let $Y_{t}$ denote the fiber over a closed point $t \in T$ mapping to $w$. Since the fiber $Y_{t}$ is general, Lemma \ref{lemm: monodromyandbvalue} (3) shows that pushforward induces a surjection $\mathcal{F}_{Y_{t}} \to \mathcal{F}_{Y}$. By Lemma \ref{lemm: monodromyandbvalue} (2) pushforward induces a map $\mathcal F_{X_w} \to \mathcal F_{X}$. Thus it suffices to prove that $f|_{Y_{t}*} \mathcal{F}_{Y_{t}} \subset \mathcal{F}_{X_{w}}$.
Since $f|_{Y_{t}}: Y_{t} \to X_{w}$ is a dominant generically finite map of smooth varieties, by the Riemann-Hurwitz formula we have that $K_{Y_{t}} - f|_{Y_{t}}^{*}K_{X_{w}}$ is effective. Since by Lemma \ref{lemm:ainvandcanonicalfibers} the $a$-invariants are the same this implies that $f|_{Y_{t}*}\mathcal{F}_{Y_{t}} \subset \mathcal{F}_{X_{w}}$ by the same argument as before.
\end{proof}
Note that by the birational invariance of faces proved in Lemma \ref{lemm:birfaceinv}, any morphism $f: Y \to X$ with a birational model that satisfies Assumption \ref{assum:fc} will still satisfy $f_{*}\mathcal{F}_{Y} \subset \mathcal{F}_{X}$. This leads us to the following definition.
\begin{defi}{\cite[Definition 3.6]{LT17}} \label{defi: facecontraction}
Let $f: Y \to X$ be a morphism of geometrically integral projective varieties that is generically finite onto its image. Furthermore assume that there is a commuting diagram
\begin{equation*}
\xymatrix{ Y' \ar[r]^{f'} \ar[d]_{\phi_{Y}} & X' \ar[d]^{\phi_{X}} \\
Y \ar[r]^{f} & X}
\end{equation*}
such that $\phi_{Y}$ and $\phi_{X}$ are birational and $f': Y' \to X'$ satisfies Assumption \ref{assum:fc}.
We say that such a morphism $f$ is face contracting if the induced map $f_{*}: \mathcal{F}_{Y} \to \mathcal{F}_{X}$ is not injective.
\end{defi}
Since the dimensions of $\mathcal{F}_{Y}$ and $\mathcal{F}_{X}$ are $b(F,Y,f^{*}L)$ and $b(F,X,L)$ respectively, a dominant breaking thin map is automatically face contracting. However, the converse is not true (see \cite[Example 3.7]{LT17}).
The following lemma shows that for certain families $Y$ of adjoint rigid subvarieties on $X$ the map $f: Y \to X$ must be a face contracting map. We will use it to show that subvarieties of this type can only contribute a thin set of rational points.
\begin{lemm} \label{lemm: facecontractingcondition}
Let $f: Y \to X$ be a dominant generically finite morphism of geometrically integral projective varieties with $X$ geometrically uniruled and fix a big and nef $\mathbb{Q}$-divisor $L$ on $X$. Suppose there is a birational model $f': Y' \to X'$ of $f$ with birational maps $\phi_X: X' \to X$, $\phi_Y : Y' \to Y$ and a diagram
\begin{equation*}
\xymatrix{ Y' \ar[r]^{f'} \ar[d]_{q} & X' \ar[d]_{p} \\
T \ar[r]^{g} & W}
\end{equation*}
satisfying the following conditions:
\begin{enumerate}
\item $X'$ and $Y'$ are smooth and projective,
\item $q$ and $p$ are projective and surjective,
\item $g$ is generically finite and dominant,
\item $a(Y,f^{*}L) = a(X,L)$ and $b(F,Y,f^{*}L) = b(F,X,L)$,
\item $q$ is birationally equivalent to the canonical model for $K_{Y'} + a(Y',f'^{*}\phi_X^{*}L) f'^{*}\phi_X^{*}L$,
\item $\dim(W) > \kappa(X,K_{X} + a(X,L)L)$.\end{enumerate}
Then $f$ is face contracting.
\end{lemm}
\begin{proof}
Note that for any diagram of smooth varieties
\begin{equation*}
\xymatrix{ Y'' \ar[r]^{f''} \ar[d]_{\psi_{Y}} & X'' \ar[d]_{\psi_{X}} \\
Y' \ar[r]^{f'} & X' }
\end{equation*}
with $\psi_{Y}, \psi_{X}$ birational the hypotheses of the theorem still hold for $f''$. Thus by passing to birational models (which for convenience we absorb into the notation) we may assume that $Y' = Y$, $X' = X$, and the canonical model for $K_{X} + a(X,L)L$ is a morphism.
Let $\mathcal{F}_{Y}$ and $\mathcal{F}_{X}$ be the faces as in Definition \ref{defi:facedefinition} with respect to $f^{*}L$ and $L$ respectively. Fix an ample divisor $H$ on $W$. Note that $p^{*}H$ vanishes on every element of $f_{*}\mathcal{F}_{Y}$, since there is some $\epsilon > 0$ such that $K_{Y} + a(Y,f^{*}L)f^{*}L - \epsilon f^{*}p^{*}H$ is pseudo-effective. However, $p^{*}H$ does not vanish on every element of $\mathcal{F}_{X}$. Indeed, Lemma \ref{lemm:alternativedescription} shows that for every divisor $D$ with class in $\mathcal{F}^{X}$ there is some sufficiently small $\epsilon > 0$ such that $K_{X} + a(X,L)L - \epsilon D$ is $\mathbb{Q}$-linearly equivalent to an effective divisor. Since the Iitaka dimension of $p^{*}H$ is greater than $\kappa(X,K_{X} + a(X,L)L)$, we deduce that $p^{*}H \not \in \mathcal{F}^{X}$. Thus $f_{*}\mathcal{F}_{Y} \subsetneq \mathcal{F}_{X}$, and since the $b$-values are equal $f$ must be face contracting.
\end{proof}
\section{A conjectural description of exceptional sets in Manin's Conjecture}
\label{sect: conjecturaldescription}
Let $F$ be a number field and suppose that we have a geometrically rationally connected and geometrically integral smooth projective variety $X$ defined over $F$ carrying a big and nef line bundle $\mathcal L = \mathcal{O}_{X}(L)$ with an adelic metrization on $X$. The adelic metrization defines a height function on the rational points of $X$. Manin's conjecture predicts the asymptotic growth rate of the counting function for rational points of bounded height after removing an exceptional thin set.
Originally \cite{BM} and its refinement \cite{Peyre} predicted that the exceptional set for Manin's Conjecture consisted of points on a proper closed subset. However there are now many counterexamples to these two versions of Manin's Conjecture (\cite{BT-cubic}, \cite{EJ06}, \cite{Els11}, \cite{BL16}, and \cite{LeRudulier}). These counterexamples arise from geometric obstructions; for example, it is possible that as we vary over breaking thin maps $f: Y \to X$ the union of the sets $f(Y(F))$ is Zariski dense. \cite{Peyre03} was the first to modify the conjecture by proposing that the exceptional set in Manin's Conjecture is contained in a thin set.
We next give a conjectural geometric description of the exceptional set in Manin's Conjecture. Suppose that $X$ is a geometrically uniruled and geometrically integral smooth projective variety over $F$ equipped with a big and nef $\mathbb{Q}$-divisor $L$.
When constructing the exceptional set it is harmless to replace $X$ by a birational model, thus we will assume that the canonical model $\pi : X\rightarrow W$ for $K_{X} + a(X, L)L$ is a morphism.
Let $Z_0$ be the set of rational points contained in the union of $\mathbf B_+(L)$ and a proper closed subset $\pi^{-1}V$ where $V \subset W$ is a proper closed subset such that over $W^\circ = W \setminus V$, $\pi$ is smooth.
Note that $Z_0$ consists of points on a proper closed subset of $X$.
Next as $f: Y \rightarrow X$ varies over all $F$-thin maps such that $Y$ is geometrically integral and smooth, $d(Y,f^{*}L) < d(X,L)$ and
\[
(a(X, L), b(F, X, L)) \leq (a(Y, f^*L), b(F, Y, f^*L)),
\]
we define the set $Z_1 \subset X(F)$ by
\[
Z_1 = \bigcup_f f(Y(F)) \subset X(F).
\]
Next as $f: Y \rightarrow X$ varies over all $F$-thin maps such that $Y$ is geometrically integral and smooth, $d(Y,f^{*}L) = d(X,L)$, and either
\[
(a(X, L), b(F, X, L)) < (a(Y, f^*L), b(F, Y, f^*L)),
\]
or the $a$ and $b$ values are equal and $f$ is face contracting, we define the set $Z_2 \subset X(F)$ by
\[
Z_2 = \bigcup_f f(Y(F)) \subset X(F).
\]
Finally, as $f: Y \rightarrow X$ varies over all $F$-thin maps such that $Y$ is geometrically integral and smooth, $d(Y,f^{*}L) > d(X,L)$ and
\[
(a(X, L), b(F, X, L)) \leq (a(Y, f^*L), b(F, Y, f^*L)),
\]
we define the set $Z_3 \subset X(F)$ by
\[
Z_3 = \bigcup_f f(Y(F)) \subset X(F).
\]
By Lemma~\ref{lemm:d(Y)>d(X)}, $Z_{3}$ is contained in a proper closed subset of $X$.
We propose the following refinement of Manin's Conjecture which includes a description of the exceptional thin set. A similar but weaker statement was predicted in \cite{LTDuke}. For any subset $Q \subset X(F)$, we let $N(Q, \mathcal L, T)$ denote the number of rational points on $Q$ whose height associated to $\mathcal L$ is bounded above by $T$.
\begin{conj}[Manin's Conjecture] \label{conj: maninsconjecture}
Let $F$ be a number field. Let $X$ be a geometrically rationally connected and geometrically integral smooth projective variety defined over $F$ and let $\mathcal{L}$ be a big and nef line bundle with an adelic metrization on $X$.
Let $Z$ be the union of $Z_0$, $Z_1$, $Z_2$, and $Z_3$.
Suppose that $X(F)$ is not a thin set.
Then we have
\[
N(X(F) \setminus Z, \mathcal L, T) \sim c(F, Z, L)T^{a(X, L)} \log (T)^{b(F, X, L)-1}
\]
as $T \rightarrow \infty$ where $c(F, Z, L)$ is Peyre-Batyrev-Tschinkel's constant introduced in \cite{Peyre} and \cite{BT}.
\end{conj}
\begin{rema}
Assuming the conjecture of Colliot-Th\'el\`ene that the Brauer-Manin obstructions are the only obstructions to weak approximation for geometrically rationally connected smooth projective varieties, it follows that $X(F)$ is not thin as soon as there is a rational point. See the remark after Conjecture 1.4 in \cite{BL16}.
\end{rema}
\begin{rema}
When $(X, L)$ is adjoint rigid, the constant $c(F,Z,L)$ does not depend on $Z$. But if $(X,L)$ is not adjoint rigid, then the definition of $c(F,Z,L)$ involves a summation of Tamagawa numbers over the base of the canonical map so that we must keep track of which fibers are removed by $Z$ when defining the constant.
\end{rema}
\begin{rema}
By \cite{HM07} a smooth projective variety $X$ is geometrically rationally connected whenever it carries a big and nef $\mathbb{Q}$-divisor $L$ such that $(X,L)$ is adjoint rigid.
(For a careful explanation see \cite[Proof of Theorem 4.5]{LTT14}.)
\end{rema}
\begin{rema}
\cite{Peyre16} formulates an appealing version of Manin's Conjecture using the notion of freeness of a rational point. Peyre's conjecture has some similarities with Conjecture \ref{conj: maninsconjecture}. Let $Z^{f}$ denote the exceptional set as in \cite[Formule empirique 6.13]{Peyre16}. \cite[Proposition 5.8]{Peyre16} shows that $Z^{f}$ includes most points on non-free curves; comparing against \cite[Theorem 1.1 and Proposition 6.14]{LT17} we should expect these points to account for subvarieties $Y$ with $a(Y,L) > a(X,L)$.
Nevertheless, the two proposals for the exceptional set are different. The set $Z^{f}$ may fail to be contained in the union of the $Z_{i}$: a general cubic fourfold has empty $Z_{i}$ but admits non-free lines so that $Z^{f}$ is non-empty by \cite[Proposition 5.8]{Peyre16}. Conversely, the union of the $Z_{i}$ may fail to be contained in $Z^{f}$: in the example of \cite{BT-cubic} the $Z_{i}$ contains every point on a cubic surface fiber with Picard rank $> 1$ while $Z^{f}$ does not (see \cite[Section 8.3]{Peyre16} and particularly \cite[Remarque 8.9]{Peyre16}). However, it is plausible that the difference between the two definitions is negligible when considered against the asymptotic growth rate.
\end{rema}
The main theorem of this paper is the following result:
\begin{theo} \label{theo: precisetheorem}
Let $X$ be a geometrically uniruled geometrically integral smooth projective variety defined over a number field $F$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. The subsets $Z_{0}$, $Z_1$, $Z_2$, and $Z_{3}$ defined above are contained in a thin subset of $X(F)$.
\end{theo}
\begin{rema} \label{rema: computability}
To study Manin's Conjecture in examples Conjecture \ref{conj: maninsconjecture} predicts that one should first calculate the sets $Z_{0}, Z_{1}, Z_{2}, Z_{3}$. Our proof of Theorem \ref{theo: precisetheorem} will show that in principle by the Borisov-Alexeev-Borisov Conjecture (\cite{birkar16}, \cite{birkar16b}) this computation only involves checking the behavior of subvarieties and covers in a finite degree range. However, currently Birkar's result is ineffective and in practice there is room for vast improvement of current computational techniques. For low dimensional examples the framework established by \cite{LTT14}, \cite{LTDuke}, and \cite{Sen17} is often sufficient for calculating these sets.
\end{rema}
Note that in Conjecture \ref{conj: maninsconjecture} we remove point contributions for some thin maps with $a$ and $b$ values equal to $X$. The following examples show that sometimes, but not always, we must discount contributions from such maps in order to obtain the correct leading constant. The face contraction condition is the key criterion for distinguishing the two cases.
\begin{exam}[Peyre's constant]
The papers \cite{EJ06}, \cite{Els11}, \cite{BL16} give many examples of Fano varieties $X$ admitting a Zariski dense set of subvarieties with the same $a$ and $b$ values as $X$ with respect to $-K_{X}$. Suppose that the rational points on these subvarieties grow at the expected rate. If we include these points, \cite[Theorem 1.2]{BL16} shows that Manin's Conjecture with Peyre's constant will be violated for an appropriate choice of anticanonical height function. In order to obtain the correct Peyre's constant we must remove point contributions from all such subvarieties. Theorem \ref{theo: precisetheorem} shows that such points always lie in a thin set, generalizing the examples proved in \cite{BL16}.
\end{exam}
\begin{exam}
As we vary over all dominant generically finite maps $f: Y \rightarrow X$ of degree $\geq 2$ such that
\[
(a(X, L), b(F, X, L)) = (a(Y, f^*L), b(F, Y, f^*L)),
\]
the set $\cup f(Y(F))$ need not lie in a thin set of rational points (see \cite[Example 8.7]{LTDuke}). Thus, in the definition of $Z_{2}$ it is important to only consider contributions from maps which are face contracting.
\end{exam}
Let us compute these exceptional sets for some examples. In all of the following examples $(X,L)$ will be adjoint rigid, so $Z_{0} = \mathbf B_+(L)$ and $Z_{3}$ will be empty and they need not be discussed.
\begin{exam}[Surfaces]
Let $S$ be a geometrically rational geometrically integral smooth projective surface defined over $F$ and let $L$ be a big and nef $\mathbb Q$-divisor on $S$ such that $(S,L)$ is adjoint rigid. For simplicity let us suppose that the Picard rank of $S$ and the geometric Picard rank of $S$ coincide. Then by \cite[Proposition 5.9]{LTT14} and \cite[Theorem 1.8]{LTDuke} $Z_1$ is contained in a proper closed subset and $Z_2$ is empty. Thus we expect that Manin's conjecture should hold after removing points on a closed set. This version of Manin's conjecture for geometrically rational surfaces has been confirmed for many examples, see e.g.~\cite{dBBD07}, \cite{Bro09}, \cite{Bro10}, and \cite{dBBP12}.
\end{exam}
\begin{exam}[Flag varieties]
Let $X$ be a geometrically integral generalized flag variety defined over $F$ with a rational point
and let $L = -K_X$.
Manin's conjecture for flag varieties has been established in \cite{FMT89} with empty exceptional set.
By \cite{Bor96}, the Brauer-Manin obstructions are the only obstructions to weak approximation, so in particular $X(F)$ is not thin.
Hence, $Z_1$ does not cover $X(F)$. Since $X$ is homogeneous, this implies that $Z_1$ must be empty. On the other hand, since there are no subvarieties with higher $a$-value and $X$ is simply connected, there is no dominant morphism $f: Y \to X$ such that $a(Y,-f^{*}K_{X}) = a(X,-K_{X})$ and $(Y,-f^{*}K_{X})$ is adjoint rigid by \cite[Theorem 1.1]{Sen17}. Thus we conclude that $Z_2$ is also empty.
\end{exam}
\begin{exam}[Toric varieties]
Let $X$ be a geometrically integral smooth toric variety defined over a number field $F$ and let $L$ be a big and nef divisor on $X$. Manin's conjecture for such a variety was proved in \cite{BT-general}, \cite{BT-0}, and \cite{Sal98} after removing rational points on the boundary. Suppose that $(X,L)$ is adjoint rigid.
Since any $F$-torus satisfies the weak weak approximation property, by \cite[Theorem 3.5.7]{Serre} we see that $X(F)$ is not thin. Since the torus part is a homogeneous space, we conclude that $Z_1$ is contained in the boundary. By the same reasoning $Z_2$ is also contained in the boundary.
So our refinement is compatible with the above results. A similar proof works for smooth equivariant compactifications of other algebraic groups and Manin's conjecture for such varieties has been established in many cases, see e.g.~\cite{CLT02}, \cite{STBT07} and \cite{ST16}.
\end{exam}
\begin{exam}[Le Rudulier's example]
Let $S$ be the surface $\mathbb P^1 \times \mathbb P^1$ over $\mathbb{Q}$ and set $X = \mathrm{Hilb}^{[2]}(S)$.
\cite{LeRudulier} proved Manin's conjecture for $(X,-K_X)$.
We briefly explain why her result is compatible with our refinement.
We freely use the notations from \cite[Section 9.3]{LTDuke}.
Let $L = H_1[2] + H_2[2]$. Le Rudulier proved Manin's conjecture for $L$ after removing rational points on $D_1, D_2, E$ and $f(W(\mathbb{Q}))$. We denote this exceptional set by $Z'$. The analysis in \cite[Section 9.3]{LTDuke} shows that (i) all subvarieties with higher $a$ values are contained in $D_1, D_2$, or $E$; (ii) the only thin maps $g: Y \rightarrow X$ such that the image is not contained in $D_1\cup D_2\cup E$, $(Y, g^*L)$ is adjoint rigid, $\dim Y < \dim X$, and $(a(X, L), b(\mathbb{Q}, X, L)) \leq (a(Y, g^*L), b(\mathbb{Q}, Y, g^*L))$ are the images of the fibers of one of the projections $\pi_i : W \rightarrow \mathbb P^1$. These imply that $Z_1$ is contained in $Z'$. To analyze $Z_{2}$, we first note that the geometric fundamental group of $X \setminus (D_1 \cup D_2 \cup E)$ is $\mathbb Z/2\mathbb Z$. Thus, over $\overline{\mathbb Q}$ there is only one possible cover $f: W \rightarrow X$ such that $a(W,-f^{*}K_{X}) = a(X,-K_{X})$ and $(W,-f^{*}K_{X})$ is adjoint rigid. On the other hand, by copying the argument of \cite[Example 8.6]{LTDuke} in this setting we see that all nontrivial twists of $f: W \rightarrow X$ have $a, b$ values less than $a, b$ values of $X$. Thus $Z_2 = f(W(\mathbb{Q}))$ is also contained in $Z'$.
Another interesting example of \cite{LeRudulier} is $\mathrm{Hilb}^{[2]}(\mathbb{P}^{2})$ over $\mathbb{Q}$. To obtain the expected growth rate of points of a bounded height, one must remove points from a dominant map $f: W \to \mathrm{Hilb}^{[2]}(\mathbb{P}^{2})$ but not its twists. It turns out that $f$ is face contracting but its twists are not, giving a geometric explanation of this phenomenon; see \cite[Example 8.6]{LTDuke}.
\end{exam}
The circle method has been successfully used to prove Manin's conjecture for low degree complete intersections, e.g., \cite{Bir61} and \cite{BHB17}. Verifying our refinement for this class of varieties is out of reach at this moment. However, based on the properties of rational curves on low degree hypersurfaces proved by \cite{HRS04}, \cite{BK13}, \cite{RY16}, \cite{BV17} and the connection with $a$ and $b$ invariants proved in \cite{LT17}, we expect that $Z_{1}$ and $Z_{2}$ are empty for general smooth hypersurfaces in $\mathbb{P}^{n}$ of degree $\leq n-2$ and for every smooth hypersurface in $\mathbb{P}^{n}$ of degree $\ll \log_2(n)$.
\subsection{Counterexamples to extensions} \label{sect: nonbigdiv}
The Weil height formalism associates a height function to any adelically metrized big line bundle, and it is interesting to study the asymptotic growth rate of points of bounded height in this context as well. It is natural to ask whether the $a,b$-invariants (defined in the analogous way) will still predict the asymptotic growth rate of points. There are a few important classes of varieties for which the $a,b$-invariants for big line bundles do indeed agree with point growth rates: for example, when the ambient variety is a toric variety \cite{BT-general} or an equivariant compactification of a vector group \cite{CLT02}.
In this section we will give a couple of examples which demonstrate the pathological behavior of the $a,b$-invariants when the polarization $L$ is big but not nef. We will concentrate on examples for which $-K_{X}$ is big. One can find many examples of such varieties using the following construction. Let $W$ be any smooth projective variety and choose an ample divisor $H$ on $W$ such that $H-K_{W}$ is ample. Then the projective bundle $X = \mathbb{P}_{W}(\mathcal{O} \oplus \mathcal{O}(-H))$ has big anticanonical divisor. In this situation we have $a(X,-K_{X})=1$ regardless of the choices of $W$ and $H$, while the behavior of rational points depends very heavily on these choices.
\begin{exam}
Let $W$ be the Craighero-Gattazo surface \cite{CG94} defined over $\mathbb{Q}(\zeta)$ where $\zeta^{3} + \zeta^{2} - 1 = 0$. Note that $W$ is a surface of general type with $H^{1}(W,\mathcal{O}_{W}) = H^{2}(W,\mathcal{O}_{W}) = 0$. By \cite[Theorem 6.2]{RTU17} $W$ is simply connected, hence $\mathrm{Pic}(W)_{\mathrm{tor}} = 0$.
Using the construction above we obtain a projective bundle $X$ over $W$ which is ``almost Fano'' in the sense that $-K_{X}$ is big, $H^{i}(X,\mathcal{O}_{X}) = 0$ for $i>0$, and $\mathrm{Pic}(X)$ is torsion free. Equipping $X$ with the anticanonical polarization we have $a(X,-K_{X}) = 1$. However, Lang's conjecture predicts that the set of rational points on $X$ should not be Zariski dense even after a finite extension of the base field.
\end{exam}
\begin{exam}
Suppose that $W$ is the self-product of an elliptic curve without complex multiplication and construct a projective bundle $X$ over $W$ as above. The computations of \cite[Example 1.6]{Cutkosky86} show that if $L$ is a big divisor then $a(X,L)$ can be irrational. (See \cite[Example 2.6]{HTT15} for details.) In this situation the set of rational points on $X$ is thin.
\end{exam}
Even when $X$ is rationally connected, the $a,b$-invariants can exhibit pathological behavior when the polarization $L$ is big.
\begin{exam}
Choose a pencil of cubics on $\mathbb P^2$ defined over $\mathbb Q$ such that the generic fiber has a positive Mordell-Weil rank. (For the existence of such a pencil, see \cite{Shioda92} and \cite{Kurumadani}.) Let $W$ be the blow-up of $\mathbb{P}^{2}$ along the base locus of this pencil.
Then $W$ admits infinitely many $(-1)$-curves which are sections of the elliptic fibration. Choose any ample divisor $H$ on $W$ and set $X = \mathbb{P}(\mathcal{O} \oplus \mathcal{O}(-H))$. If $D$ denotes the rigid section of $\pi: X \to W$, then we have $K_{X} = -2D - H + K_{W}$. Since $-K_{W}$ is effective, $-K_{X}$ is big. We equip $X$ with the anticanonical polarization so that $a(X,-K_{X}) = 1$.
Let $\{ T_{i} \}$ denote the infinite set of $(-1)$-curves on $W$ and let $S_{i} = \pi^{-1}(T_{i})$. Then we have $K_{S_{i}} - K_{X}|_{S_{i}} = \pi^{*}\mathcal{O}_{T_{i}}(-1)$, showing that $a(S_{i},-K_{X}) >1 = a(X,-K_{X})$. Since $S_{i}$ is toric, we know that the $a$-invariant actually does predict the asymptotic growth rate of rational points of bounded height on $S_{i}$. Thus there is a countable union of subvarieties whose rational points grow faster than the expected rate. In particular, for sufficiently small $\epsilon > 0$ there is no open subset $U \subset X$ such that the number of rational points on $U$ of height bounded above by $B$ is $O(B^{a(X,-K_{X})+\epsilon})$.
\end{exam}
\section{Twists}
\label{sec: twists}
In this section we work over a number field. We start with a lemma we will use frequently throughout the paper.
\begin{lemm}[\cite{Cheltsov04}] \label{lemm: birandaut}
Let $f : Y \dashrightarrow X$ be a dominant generically finite rational map between normal projective varieties defined over a number field $F$. Then there exists a birational modification $f': Y' \rightarrow X$ of $f$ such that $Y'$ is smooth and projective and $\mathrm{Bir}(\overline{Y}'/\overline{X}) = \mathrm{Aut}(\overline{Y}'/\overline{X})$.
Furthermore, if we fix a big and nef $\mathbb{Q}$-divisor $L$ on $X$, then we may assume that the canonical model for $K_{Y'} + a(Y',f'^{*}L)f'^{*}L$ is a morphism.
\end{lemm}
In particular, any twist of $f: Y \dashrightarrow X$ is birational to a twist of $f': Y' \to X$.
\begin{proof}
We first replace $Y$ by a normal birational model which admits a morphism to $X$. We then replace $Y$ by its Stein factorization, so we may assume $\mathrm{Bir}(\overline{Y}/\overline{X}) = \mathrm{Aut}(\overline{Y}/\overline{X})$. Let $F'/F$ be a finite Galois extension such that all automorphisms in $G = \mathrm{Aut}(\overline{Y}/\overline{X})$ are defined over $F'$. Then $G \rtimes \mathrm{Gal}(F'/F)$ acts on $Y_{F'}$.
We resolve singularities equivariantly (as in \cite[Theorem 0.1]{AW97}) and take the quotient by the Galois group $\mathrm{Gal}(F'/F)$ to obtain a smooth variety $Y'$ satisfying the desired condition on automorphism groups.
We still must prove the last statement. Let $\pi: Y' \dashrightarrow T$ denote the canonical model for $(Y',a(Y',f'^{*}L)f'^{*}L)$. Choose the same field extension $F'/F$ as before. Then the morphism $\pi_{F'}: Y_{F'} \dashrightarrow T_{F'}$ is equivariant for the group $G \rtimes \mathrm{Gal}(F'/F)$. Thus we may take another equivariant resolution and quotient by the Galois action to ensure that $\pi$ is a morphism.
\end{proof}
We next discuss a lemma encapsulating our application of Hilbert's Irreducibility Theorem.
\begin{lemm} \label{lemm: hilbirrapp}
Let $f: Y \to X$ be a surjective generically finite morphism of geometrically integral normal projective varieties defined over a number field $F$. Suppose that the extension of geometric function fields $\overline{F}(\overline{Y})/\overline{F}(\overline{X})$ is Galois with Galois group $G$. Let $F'/F$ be a finite extension such that $G = \mathrm{Bir}(Y_{F'}/X_{F'})$. There is a thin set of points $Z \subset X(F')$ such that if $x \in X(F') \backslash Z$ then $f^{-1}(x)$ is irreducible and the corresponding extension of residue fields is Galois with Galois group $G$.
\end{lemm}
\begin{proof}
According to Lemma \ref{lemm: birandaut} there is a birational model $f': Y' \to X$ of $f$ such that $\mathrm{Aut}(\overline{Y'}/\overline{X}) = \mathrm{Bir}(\overline{Y'}/\overline{X}) = G$. Since $Y$ and $Y'$ only differ in a closed set, it suffices to prove the statement for $Y'$.
Then our finite field extension $F'/F$ satisfies that $\mathrm{Aut}(\overline{Y}'/\overline{X}) = \mathrm{Aut}(Y'_{F'}/X_{F'})$. Note that $f_{F'}: Y'_{F'} \to X_{F'}$ is a Galois covering over an open subset $X_{F'}^{\circ}$ of $X_{F'}$. Applying the Hilbert Irreducibility Theorem (\cite[Proposition 3.3.1]{Serre}) to this open set and adding on $(X_{F'} \backslash X_{F'}^{\circ})(F')$, we obtain a thin subset $Z' \subset X_{F'}(F')$ satisfying the desired property with respect to $F'$-points.
\end{proof}
Using Hilbert's Irreducibility Theorem, we show that if we fix a thin map $f: Y \to X$ then there is a thin set which contains all point contributions $f^{\sigma}(Y^{\sigma}(F))$ from twists $f^{\sigma}$ that are breaking thin maps.
\begin{theo}
\label{theo:twists}
Let $X$ be a geometrically uniruled smooth projective variety over a number field $F$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Suppose that $f: Y \to X$ is a dominant generically finite morphism from a normal projective variety $Y$. As $\sigma$ varies over all $\sigma \in H^1(F, \mathrm{Aut}(\overline{Y}/\overline{X}))$ such that $Y^{\sigma}$ is irreducible,
\begin{equation*}
(a(X, L), b(F, X, L)) \leq (a(Y^{\sigma}, (f^{\sigma})^*L), b(F, Y^\sigma, (f^\sigma)^*L)),
\end{equation*}
and $f^{\sigma}$ is face contracting the set
\begin{equation*}
Z= \bigcup_{\sigma} f^\sigma(Y^\sigma (F)) \subset X(F)
\end{equation*}
is contained in a thin subset of $X(F)$.
\end{theo}
\begin{proof}
We start with several simplifications. If $X$ is not geometrically integral, then $X(F)$ is empty since $X$ is smooth. So we may suppose $X$ is geometrically integral.
Suppose that $Y$ is not geometrically integral. Then any twist $Y^{\sigma}$ of $Y$ which has a rational point not contained in $\mathrm{Sing}(Y^{\sigma})$ must be reducible. Thus, the set $Z$ is contained in the thin set $(f(\mathrm{Sing}(Y))(F)$. So from now on we assume that $Y$ is geometrically integral.
If $f: Y \to X$ induces an extension of geometric function fields $\overline{F}(\overline{Y})/\overline{F}(\overline{X})$ that is not Galois, then we may conclude by \cite[Proposition 8.2]{LTDuke}.
So we may assume that $\overline{F}(\overline{Y})/\overline{F}(\overline{X})$ is Galois with Galois group $G$.
Suppose that $Y$ is not smooth. Choose a birational model $f': Y' \to X$ as in Lemma \ref{lemm: birandaut}. Note that the statement for $f'$ implies the statement for $f$. Indeed, if $B$ denotes the locus where the rational map $\phi: Y \dashrightarrow Y'$ is not defined then
\begin{equation*}
\bigcup_{\sigma} f^\sigma(Y^\sigma (F)) \subset \bigcup_{\tau} f'^\tau(Y'^\tau(F)) \cup f(B)(F)
\end{equation*}
where $\tau$ varies over all twists of $f': Y' \to X$ as in the statement of the theorem. So from now on we assume that $Y$ is smooth. Similarly, if $Y$ is smooth but $G$ does not coincide with $\mathrm{Aut}(\overline{Y}/\overline{X})$, then we may apply the same construction to reduce to the case when $G =\mathrm{Aut}(\overline{Y}/\overline{X}) = \mathrm{Bir}(\overline{Y}/\overline{X})$.
Since $f$ is dominant Lemma \ref{lemm: genfinite} shows that the only case we need to consider is when $a(Y,f^{*}L) = a(X,L)$.
Suppose that $F_1/F$ is a finite extension so that $N^1(\overline{Y}) = N^1(Y_{F_1})$ and $G = \mathrm{Aut}(Y_{F_1}/X_{F_1})$.
By Lemma \ref{lemm: hilbirrapp} we obtain a thin set $Z'' \subset X(F_1)$ such that for any point $x \in X(F_1)\backslash Z''$ the fiber $f^{-1}(x)$ is irreducible over $F_1$ and the corresponding extension of residue fields is Galois with Galois group $G$. We let $Z'= Z'' \cap X(F)$ which is contained in a thin set by \cite[Proposition 3.2.1]{Serre}.
We prove that if a twist $\sigma$ satisfies $f^{\sigma}(Y^{\sigma}(F)) \not \subset Z'$ then $b(F,Y^{\sigma},f^{\sigma *}L) \leq b(F,X,L)$ and if equality of $b$-invariants is achieved then $f^{\sigma}$ is not face contracting.
We first claim that $N^1(\overline{Y})^G$ is spanned by $N^1(\overline{X})$ and $\overline{f}$-exceptional divisors. To see this, denote the Stein factorization of $\overline{f}: \overline{Y} \to \overline{X}$ by $\overline{g}: \overline{Y} \to \overline{W}$ and $\overline{h}: \overline{W} \to \overline{X}$. Let $D_{\overline{Y}}$ be a $G$-invariant divisor on $\overline{Y}$ and let $D_{\overline{X}} = \overline{f}_{*}D_{\overline{Y}}$. We have $\overline{g}_{*}D_{\overline{Y}} = \frac{1}{|G|}\overline{h}^{*}D_{\overline{X}}$ as $\mathbb{Q}$-Weil divisors on $\overline{W}$. Thus $D_{\overline{Y}} - \frac{1}{|G|}\overline{g}^{*}\overline{h}^{*}D_{\overline{X}}$ is exceptional for $\overline{g}$, and hence for $\overline{f}$. This proves the claim.
Let $\mathcal{F}^{\overline{X}}$ be the minimal face of $\overline{\mathrm{Eff}}^1(\overline{X})$ containing $a(X, L)L + K_X$ and $\mathcal{F}^{\overline{Y}}$ be the minimal face of $\overline{\mathrm{Eff}}^1(\overline{Y})$ containing $a(X, L)f^*L + K_Y$. Since $\mathcal{F}^{\overline{Y}}$ contains all $\overline{f}$-exceptional effective divisors, we conclude that the natural map
\begin{align}
\label{surjection}
N^1(\overline{X})/\mathrm{Span} ( \mathcal{F}^{\overline{X}} ) \rightarrow N^1(\overline{Y})^G/\mathrm{Span} ( \mathcal{F}^{\overline{Y}})^G
\end{align}
is surjective.
Now suppose that $x \in X(F) \backslash Z'$ and that there is a point $y \in Y^{\sigma}(F)$ with $f^{\sigma}(y) = x$. Since the fiber $f^{-1}(x)$ is irreducible over $F_1$ and the corresponding extension of residue fields is Galois with Galois group $G$, the $1$-cycle such that the $G$-torsor $(f^{\sigma})^{-1}(x) \to x$ is trivial induces a surjection from $\mathrm{Gal}(\overline{F}/F_1)$ onto $G$. We claim that this Galois group acts on $N^1(\overline{Y^\sigma})$ at least as $G$ does. Indeed, choose a finite extension $F'/F_{1}$ such that
\[
Y^{\sigma} \otimes_{F} F' \cong Y \otimes_F F'.
\]
Suppose that $\sigma$ is represented by the $1$-cocycle $\mathrm{Gal}(F'/F) \ni s \mapsto \sigma_s \in G = \mathrm{Aut}(Y_{F'}/X_{F'})$. Then $Y^{\sigma}$ is the quotient of $Y \otimes_{F} F'$ by $\mathrm{Gal}(F'/F)$ where the Galois action on $Y \otimes_F F'$ is given by the composition
\[
\sigma_s \circ (\mathrm{id}_Y \otimes s).
\]
By our construction, $\sigma$ induces a surjective homomorphism from $\mathrm{Gal}(F'/F_1)$ to $G$ and $\mathrm{Gal}(F'/F_1)$ acts trivially on $N^1(Y_{F'})$ via $\mathrm{id}_Y \otimes s$. Thus $\mathrm{Gal}(\overline{F}/F)$ also acts on $N^1(\overline{Y^\sigma})$ at least as $G$ does.
Since the action of $\mathrm{Gal}(\overline{F}/F)$ on the N\'eron-Severi space factors through a finite group, by taking the Galois invariant part of (\ref{surjection}) we obtain a surjection
\[
N^1(X)/\mathrm{Span} ( \mathcal{F}^{X} ) \rightarrow (N^1(\overline{Y}^\sigma)^G/\mathrm{Span} ( \mathcal{F}^{\overline{Y}^\sigma})^G)^{\mathrm{Gal}(\overline{F}/F)}.
\]
Then note that since $\mathrm{Gal}(\overline{F}/F)$ at least acts as $G$ on $Y^\sigma$, we have
\begin{align*}
(N^1(\overline{Y}^\sigma)^G/\mathrm{Span} ( \mathcal{F}^{\overline{Y}^\sigma})^G)^{\mathrm{Gal}(\overline{F}/F)} & = (N^1(\overline{Y}^\sigma)/\mathrm{Span} ( \mathcal{F}^{\overline{Y}^\sigma}))^{\mathrm{Gal}(\overline{F}/F)}\\
& = N^1(Y^\sigma)/\mathrm{Span} ( \mathcal{F}^{Y^\sigma}).
\end{align*}
Thus we conclude that $b(F, Y^\sigma, f^{\sigma *}L) \leq b(F, X, L)$. If the equality is achieved, then $\mathrm{Span} (\mathcal{F}_{Y})$ maps isomorphically to $\mathrm{Span} (\mathcal{F}_{X})$ and the cover is not face contracting.
\end{proof}
In the proof of Theorem \ref{theo: precisetheorem} we will often need to replace a thin map $f: Y \to X$ by a map $f': Y' \to X$ constructed using an Iitaka base change. We must show that if $f$ is a breaking thin map then certain twists of $f'$ are also breaking thin maps -- this will ensure that we do not ``lose'' any rational point contributions when performing this replacement.
\begin{lemm} \label{lemm: iitakabasechangeandbvalue}
Let $Y$ be a geometrically uniruled smooth projective variety over a number field $F$ and let $L$ be a big and nef $\mathbb{Q}$-divisor on $Y$. Suppose that $K_{Y}+a(Y,L)L$ has positive Iitaka dimension and let $\pi: Y \dashrightarrow B$
denote the canonical model. Suppose that $h: T \to B$ is any dominant generically finite map from a projective variety $T$.
Let $g: Y' \to Y$ be the Iitaka base change of $Y$ corresponding to $h$.
Then for every twist $g^{\sigma}: Y'^{\sigma} \to Y$ of $g$ with $Y'^{\sigma}$ irreducible, the induced map $g^{\sigma}_{*}: \mathcal{F}_{Y'^{\sigma}} \to \mathcal{F}_{Y}$ is surjective. In particular we have $b(F,Y'^{\sigma},g^{\sigma*}L) \geq b(F,Y,L)$.
\end{lemm}
\begin{proof}
Note that in this situation we have $a(Y'^{\sigma},g^{\sigma*}L) = a(Y,L)$,
since the inequality $\leq$ follows from Lemma \ref{lemm: genfinite} and the inequality $\geq$ follows from Lemmas \ref{lemm:ainvandcanonicalfibers} and \ref{lemm:ainvdominantfamily}. Let $\phi: \widetilde{Y} \to Y$ be a birational morphism resolving the canonical model map $\pi$ such that $\widetilde{Y}$ is smooth and set $\widetilde{\pi} = \pi \circ \phi$. By applying Lemma \ref{lemm: birandaut} to $Y' \dashrightarrow \widetilde{Y}$, we obtain a smooth birational model $\widetilde{Y}'$ of $Y'$ and a morphism $\widetilde{g}: \widetilde{Y}' \to \widetilde{Y}$ such that $\mathrm{Bir}(\overline{\widetilde{Y}'}/\overline{\widetilde{Y}}) = \mathrm{Aut}(\overline{\widetilde{Y}'}/\overline{\widetilde{Y}})$ and
the canonical map for $(\widetilde{Y}'^{\sigma}, a(\widetilde{Y}'^{\sigma},\widetilde{g}^{\sigma*}\phi^{*}L)\widetilde{g}^{\sigma*}\phi^{*}L)$ is a morphism. We claim that the desired statement for $g: Y' \to Y$ follows from the corresponding statement for $\widetilde{g}: \widetilde{Y}' \to \widetilde{Y}$. Indeed, every twist $g^{\sigma}$ of $g$ is birational to some twist $\widetilde{g}^{\tau}$ of $\widetilde{g}$, and by Lemma \ref{lemm:birfaceinv} we can deduce the desired surjection for $g^{\sigma}_{*}$ from the corresponding surjection for $\widetilde{g}^{\tau}_{*}$.
Let $T^\sigma \to B$ be the Stein factorization of $\widetilde{Y}'^\sigma \to \widetilde{Y} \to B$. Since $\pi': \widetilde{Y} \to B$ is the canonical model for $(\widetilde{Y},a(\widetilde{Y},\phi^{*}L)\phi^{*}L)$, there is an ample divisor $H$ on $B$ such that $K_{\widetilde{Y}} + a(\widetilde{Y},\phi^{*}L)\phi^{*}L - \widetilde{\pi}^{*}H$ is $\mathbb{Q}$-linearly equivalent to an effective divisor. Since $T^{\sigma} \to B$ is finite, the pullback of $H$ to $T^{\sigma}$ is still ample, and by the ramification formula we have that $K_{\widetilde{Y}'^{\sigma}} + a(\widetilde{Y}'^{\sigma},\widetilde{g}^{\sigma*}\phi^{*}L)\widetilde{g}^{\sigma*}\phi^{*}L - \widetilde{g}^{\sigma*}\widetilde{\pi}^{*}H$ is $\mathbb{Q}$-linearly equivalent to an effective divisor.
Thus Lemma \ref{lemm:birationaltocanonical} shows that the map $\widetilde{Y}'^{\sigma} \to T^{\sigma}$ agrees with the canonical map for $(\widetilde{Y}'^{\sigma}, a(\widetilde{Y}'^{\sigma},\widetilde{g}^{\sigma*}L)\widetilde{g}^{\sigma*}L)$ over an open subset of $T^{\sigma}$.
For a general closed point in $t \in T^\sigma$ with $b = g^\sigma(t)$, consider the commuting diagram
\begin{equation*}
\xymatrix{
\mathcal F_{\widetilde{Y}'^\sigma_t} \ar@{>}[r] \ar@{>}[d]& \mathcal F_{\widetilde{Y}_b} \ar@{>}[d]\\
\mathcal F_{\widetilde{Y}'^\sigma}. \ar@{>}[r] & F_{\widetilde{Y}}
}.
\end{equation*}
Lemma~\ref{lemm: monodromyandbvalue} (3) shows that the two vertical arrows are surjections. We also know that $\mathcal F_{\widetilde{Y}'^\sigma_t} \to \mathcal F_{\widetilde{Y}_b}$ is an isomorphism by Lemma~\ref{lemm:birfaceinv}. We deduce that the bottom arrow is also a surjection, proving the desired statement.
\end{proof}
\section{The boundedness of breaking thin maps} \label{sect: boundedness}
Our next goal is to prove a boundedness statement for the set of breaking thin maps.
In this section we work over an algebraically closed field $F$ of characteristic $0$.
\begin{defi} \label{defi:goodfamily}
A good family of adjoint rigid varieties is a morphism $p: \mathcal U \to W$ of smooth quasi-projective varieties and a relatively big and nef $\mathbb{Q}$-divisor $L$ on $\mathcal U$ satisfying the following properties:
\begin{enumerate}
\item The map $p$ is projective, surjective, and smooth with irreducible fibers.
\item The $a$-invariant $a(\mathcal{U}_{w},L|_{\mathcal{U}_{w}})$ is constant and positive for the fibers $\mathcal{U}_{w}$ over closed points and $(\mathcal{U}_{w},L|_{\mathcal{U}_{w}})$ is adjoint rigid for each fiber.
\item The $b$-value $b(F,\mathcal{U}_{w},L|_{\mathcal{U}_{w}})$ is constant for the fibers $\mathcal{U}_{w}$ over closed points.
\item Let $Q$ denote the union of all divisors $D$ in fibers $\mathcal{U}_{w}$ such that $a(D,L|_{D}) > a(\mathcal{U}_{w},L|_{\mathcal{U}_{w}})$. Then $Q$ is closed and flat over $W$. Furthermore, if we set $\mathcal{V} := \mathcal U \backslash Q$, there is a projective birational map $\phi: \mathcal{U}' \to \mathcal{U}$ that is an isomorphism over $\mathcal{V}$ such that $\mathcal{U}'$ is smooth over $W$ and $\mathcal{U}' \backslash \mathcal{V}$ is strict normal crossings divisor relative to $W$.
\end{enumerate}
Note that the invariance of the $a$-value implies that the restriction of $L$ to every fiber of $p$ is big and nef.
A base change of a good family is defined to be the good family induced via base change by a map $g: T \to W$.
We say that $p$ has a good section if there is a section $W \to \mathcal U \backslash Q$, i.e.~there is a section avoiding $Q$.
A good morphism of good families is a diagram
\begin{equation*}
\xymatrix{\mathcal Y \ar[r]^{f} \ar[d]_{q}& \mathcal U \ar[d]^{p} \\
T \ar[r]_{g} & W}
\end{equation*}
and a relatively big and nef $\mathbb{Q}$-divisor $L$ on $\mathcal U$ such that $p$ and $q$ are good families of adjoint rigid varieties (with respect to $L$ and $f^{*}L$ respectively), the relative dimensions of $p$ and $q$ are the same, and $a(\mathcal{Y}_{t},f^{*}L|_{\mathcal{Y}_{t}}) = a(\mathcal{U}_{g(t)},L|_{\mathcal{U}_{g(t)}})$ for any point $t \in T$.
\end{defi}
\begin{lemm} \label{lemm:avaluesandfibers}
Let $p: X \to W$ be a surjective morphism of projective varieties and let $L$ be a big and nef $\mathbb{Q}$-Cartier divisor on $X$. Suppose that a component $Y$ of a general fiber of $p$ satisfies $a(Y,L) > 0$. Fix an ample divisor $H$ on $W$. Then there is an open subset $W^{\circ} \subset W$ and a positive integer $m$ such that
\begin{equation*}
a(X,L+mp^{*}H) = a(Y,L|_{Y})
\end{equation*}
for any component $Y$ of a fiber $X_{w}$ over $W^{\circ}$.
\end{lemm}
\begin{proof}
It suffices to prove the statement after replacing $X$ by a smooth birational model. By applying Theorem \ref{theo: aconstancy} to the Stein factorization of $p$
we see there is an open subset $W^{\circ} \subset W$ over which $p$ is a smooth morphism and the $a$-invariant of the components of the fibers with respect to $L$ is constant. For any positive integer $m$ Lemma \ref{lemm:ainvdominantfamily} shows that a component $Y$ of a general fiber $X_{w}$ satisfies
\begin{equation*}
a(X,L+mp^{*}H) \geq a(Y,(L + mp^{*}H)|_{Y}) = a(Y,L|_{Y}).
\end{equation*}
To show the reverse inequality, it suffices to consider the case when $a(Y,L|_{Y}) < a(X,L)$. We first prove that $K_{X} + a(Y,L|_{Y})L + \ell p^{*}H$ is pseudo-effective for some sufficiently large $\ell$. By Lemma \ref{lemm: terminalpair} we can write $a(Y,L|_{Y}) L\sim_{\mathbb Q} \Delta + A$ where $\Delta$ is an effective $\mathbb{Q}$-divisor, $A$ is an ample $\mathbb{Q}$-divisor, and $(X,\Delta+A)$ is a terminal pair. Lemma \ref{lemm:conetheorem} shows there is a finite set of numerical classes $\{ \alpha_{i} \}_{i=1}^{r}$ which generate the extremal rays of $\overline{\mathrm{Eff}}_{1}(X)_{K_{X} + \Delta \geq 0} + \mathrm{Nef}_{1}(X)$ which have negative intersection against $K_{X} + a(Y,L|_{Y})L$.
The following argument of \cite{Peternell} shows that none of these classes $\alpha_{i}$ satisfies $p_{*}\alpha_{i} = 0$. Fix an ample divisor $A'$ on $X$ and fix $\epsilon > 0$. Since $K_{X} + a(Y,L|_{Y})L + \epsilon A'$ is $p$-relatively big, there is some positive $c_{\epsilon}$ such that $K_{X} + a(Y,L|_{Y})L + \epsilon A' + c_{\epsilon}p^{*}H$ is big. In particular, for every $i$
\begin{equation*}
(K_{X} + a(Y,L|_{Y})L + \epsilon A' + c_{\epsilon}p^{*}H) \cdot \alpha_{i} > 0.
\end{equation*}
If $p_{*}\alpha_{i} = 0$ then we obtain $0 < (K_{X} + a(Y,L|_{Y})L + \epsilon A') \cdot \alpha_{i}$ for every $\epsilon > 0$, a contradiction.
Since $p_{*}\alpha \neq 0$ we may choose $\ell$ sufficiently large so that
\begin{equation*}
\ell p^{*}H \cdot \alpha_{i} > -(K_{X} + a(Y,L|_{Y})L) \cdot \alpha_{i}
\end{equation*}
for every $i$. By construction $K_{X} + a(Y,L|_{Y})L + \ell p^{*}H$ has positive intersection against every $\alpha_{i}$. It is also clear that $K_{X} + a(Y,L|_{Y})L + \ell p^{*}H$ has non-negative intersection against any $\beta \in \overline{\mathrm{Eff}}_{1}(X)_{K_{X} + a(Y,L|_{Y})L \geq 0}$.
Together these show that $K_{X} + a(Y,L|_{Y})L + \ell p^{*}H$ is pseudo-effective. Then for any $m \geq \ell/a(Y,L|_{Y})$, we have
\begin{equation*}
a(X,L+mp^{*}H) \leq a(Y,L|_{Y})
\end{equation*}
proving the reverse inequality.
\end{proof}
\begin{lemm} \label{lemm: opensetgoodfamily}
Suppose $p: X \to W$ is a surjective projective morphism of varieties such that $X$ is smooth and $p$ has connected fibers. Let $L$ be a $p$-relatively big $\mathbb{Q}$-Cartier divisor which is the restriction to $X$ of a nef $\mathbb{Q}$-Cartier divisor on some projective compactification of $X$. Suppose furthermore that the general fiber $X_{w}$ of $p$ is adjoint rigid with respect to $L$. Then there is a non-empty open subset $W^{\circ} \subset W$ with preimage $X^{\circ} = p^{-1}(W^{\circ})$ such that $p: X^{\circ} \to W^{\circ}$ is a good family of adjoint rigid varieties.
\end{lemm}
\begin{proof}
Let $W^{\circ}$ denote a smooth open subset of $W$ over which $p$ is smooth. We construct the family by repeatedly shrinking $W^{\circ}$ (and thus also shrinking its $p$-preimage $X^{\circ}$). After shrinking $W^{\circ}$, by Theorem~\ref{theo: aconstancy} we may ensure that the $a$-invariant is constant and positive and that all fibers over $W^{\circ}$ are adjoint rigid with respect to $L$. By \cite[Theorem 1.2]{Sengupta17} after shrinking $W^{\circ}$ again we may ensure that the $b$-invariant is constant.
Let $Q_{+}$ denote the union of all projective subvarieties $Y$ which are contained in some fiber $X_{w}$ over $W^{\circ}$ and which satisfy $a(Y,L) > a(X_{w},L)$. We claim that after shrinking $W^{\circ}$ the set $Q_{+}$ is closed. To verify this, let $p': X' \to W'$ denote a projective morphism of projective varieties such that $X'$ and $W'$ contain $X$ and $W$ as open subsets and $p'$ restricts to $p$ on $X$. After replacing $X'$ by a birational model we may assume that $X'$ is smooth and there is a nef divisor $L'$ on $X'$ that restricts to $L$ on $X$. Since $L'$ is $p'$-relatively big, there is an ample divisor $A'$ on $W'$ such that $L' + p'^{*}A'$ is big and nef. Since the restrictions of $L'$ and $L'+p'^{*}A'$ to the fibers $X_{w}$ are the same divisor, it suffices to prove the theorem with $L' + p'^{*}A'$ in place of $L'$. Thus from now on we assume that $L'$ is a big and nef $\mathbb{Q}$-Cartier divisor.
Fix a very ample divisor $H'$ on $W'$. After possibly shrinking $W^{\circ}$ Lemma \ref{lemm:avaluesandfibers} shows that we may find a positive integer $m$ such that
\begin{equation*}
a(X_{w},L) = a(X',L'+mp^{*}H')
\end{equation*}
for every fiber $X_{w}$ over $W^{\circ}$.
By Theorem \ref{theo: HJ} the union of all subvarieties $Y$ of $X'$ satisfying $a(Y,L'+mp'^{*}H') > a(X',L' + mp'^{*}H')$ is closed. We let $Q_{*}$ denote this closed set. Note that $Q_{+} \subset Q_{*}$. We claim that if we increase $m$ and shrink $W^{\circ}$ further then $Q_{*} \cap X^{\circ}$ coincides with $Q_{+}$. First note that by Noetherian induction $Q_{*}$ must eventually stabilize as $m$ increases. Thus after increasing $m$ we may suppose that $Q_{*}$ does not change upon further increasing the coefficient of $H'$. After shrinking $W^{\circ}$, we may suppose that each component of $Q_{*}$ that intersects $X^{\circ}$ dominates $W'$. By applying Lemma \ref{lemm:avaluesandfibers} to the finitely many components of $Q_{*}$ that surject onto $W'$ and are not contained in $\mathbf{B}_{+}(L')$, we see that if we increase $m$ and shrink $W^{\circ}$ further then $Q_{*} \cap X^{\circ} = Q_{+}$ as claimed. In particular, this implies that $Q_{+}$ is a closed set.
Set $Q$ to be the codimension $1$ components of $Q_{+}$. By shrinking $W^{\circ}$ we may ensure that $p: Q \to W^{\circ}$ is flat. After applying a resolution of singularities and shrinking $W^\circ$ further we may guarantee that the condition (4) of Definition~\ref{defi:goodfamily} is true.
Note that this set now coincides with $Q$ as defined in Definition \ref{defi:goodfamily} and satisfies all the necessary properties.
\end{proof}
Suppose that $p: \mathcal U \to W$ is a good family of adjoint rigid varieties. Let $\mathcal V \subset \mathcal U$ be the complement of the set $Q$ as in Definition \ref{defi:goodfamily}. Suppose that $p$ admits a good section $\zeta$.
The hypotheses of \cite[Expos\'e XIII Exemples 4.4]{Gro} are verified by the existence of the good section along with Definition \ref{defi:goodfamily} (4), showing that
\begin{equation*}
\pi_{1}^\textrm{\'et}(\mathcal V,\zeta(w)) = \pi_{1}^\textrm{\'et}(\mathcal V_{w},\zeta(w)) \rtimes \zeta_* \pi_{1}^\textrm{\'et}(W,w)
\end{equation*}
for every fiber $\mathcal{V}_{w} = \mathcal{V} \cap \mathcal{U}_{w}$ over a closed point $w$.
By \cite[Th\'eor\`eme II.2.3.1]{Gro2}, the \'etale fundamental group of a smooth algebraic variety defined over $F$ is topologically finitely presented. Thus there are only finitely many open subgroups of a given finite index. Hence, for an open finite index subgroup $\Xi \subset \pi_{1}^\textrm{\'et}(\mathcal{V}_{w},\zeta(w))$ its normalizer $N \subset \pi_{1}^\textrm{\'et}(W,w)$ has finite index and the subgroup $\Xi \rtimes \zeta_*N$ has finite index in $\pi_{1}^\textrm{\'et}(\mathcal{V},\zeta(w))$.
\begin{lemm} \label{lemm: finitelymanycoversinduction}
Let $p: \mathcal U \to W$ be a good family of adjoint rigid varieties with a good section $\zeta$. Suppose furthermore that the divisor $L$ on $\mathcal{U}$ is the restriction of a nef $\mathbb{Q}$-Cartier divisor on a projective compactification of $\mathcal{U}$. There is a finite set of dominant generically finite good morphisms of good families $\{ f_{i}: \mathcal Y_{i} \to \mathcal U\}$ with structure maps $q_{i}: \mathcal Y_{i} \to T_{i}$ and a closed proper subset $D \subsetneq W$ such that the following holds. Suppose that $q: \mathcal Y \to T$ is a good family of adjoint rigid varieties admitting a good morphism $f: \mathcal Y \to \mathcal U$. Then either $f(\mathcal{Y})$ is contained in $p^{-1}D$, or there is a base change $\widetilde{q}: \widetilde{\mathcal{Y}} \to \widetilde{T}$ of $q$ by a generically finite surjective morphism $\widetilde{T} \to T$ such that the induced $\widetilde{f}: \widetilde{\mathcal{Y}} \to \mathcal{U}$ factors rationally through the map $f_{j}$ for some $j$.
In the latter case there is an open subset $\widetilde{T}^{\circ} \subset T$ such that every fiber of $\widetilde{\mathcal{Y}} \to \widetilde{T}$ lying over $\widetilde{T}^{\circ}$ is mapped birationally under the map $\widetilde{\mathcal{Y}} \dashrightarrow \mathcal{Y}_{j}$ to a fiber of $q_{j}$.
\end{lemm}
\begin{proof}
Let $\mathcal V$ denote the open subset of $\mathcal U$ given by removing the set $Q$ as in Definition \ref{defi:goodfamily}. By \cite[Expos\'e XIII Exemples 4.4]{Gro}, we know that $\pi_{1}^\textrm{\'et}(\mathcal V,\zeta(w))= \pi_{1}^\textrm{\'et}(\mathcal V_{w},\zeta(w)) \rtimes \zeta_{*}\pi_{1}^\textrm{\'et}(W,w)$. Since $\mathcal{U}_{w}$ is adjoint rigid with respect to the divisor $L$, \cite[Corollary 2.8]{Sen17} shows that for any generically finite cover of $\mathcal{U}_{w}$ which has the same $a$-value and which is adjoint rigid with respect to the pullback of $L$ the divisorial components of the branch locus are supported on the set $Q \cap \mathcal{U}_{w}$. Furthermore, by \cite[Proposition 2.15]{Sen17} there is an upper bound on the degree of such covers depending only on $\dim(\mathcal{U}_{w})$, $a(\mathcal{U}_{w},L)$ and $L|_{\mathcal{U}_{w}}^{\dim (\mathcal{U}_{w})}$.
Altogether there is a finite set of finite index subgroups $\Xi_{i} \subset \pi_{1}(\mathcal{V}_{w},\zeta(w))$ such that for some fiber of $p$ the corresponding \'etale cover has a projective closure which has the same $a$-value as $\mathcal{U}_{w}$ and is adjoint rigid.
For each such $\Xi_{i}$ we use $N_{i}$ to denote its normalizer in $\pi_{1}^\textrm{\'et}(W,w)$; we also denote $\Upsilon_{i} = \Xi_{i} \rtimes \zeta_*N_{i}$. As remarked earlier $\Upsilon_{i}$ will always be a subgroup of $\pi_{1}^\textrm{\'et}(\mathcal{V},\zeta(w))$ of finite index.
Let $\mathcal{E}_{i}$ denote the \'etale cover of $\mathcal{V}$ corresponding to $\Upsilon_{i}$. Note that $\mathcal{E}_{i}$ admits a morphism to the \'etale cover $R_{i} \to W$ defined by $N_{i}$. By construction we know that every fiber of the map $\mathcal{E}_{i} \to R_{i}$ is an \'etale cover of the corresponding fiber of $\mathcal V \to W$, and since these covers are induced by subgroups we deduce that every fiber of $\mathcal{E}_{i} \to R_{i}$ is irreducible. By taking a projective closure of the fibers of $\mathcal{E}_{i}$ over $R_{i}$ and passing to a resolution we obtain a projective family $r_{i}: \widetilde{\mathcal{E}}_{i} \to R_{i}$. By Theorem~\ref{theo: aconstancy} there is an open set $R_{i}^{\circ} \subset R_{i}$ over which $r_{i}$ has smooth irreducible fibers and such that the $a$-value and Iitaka dimension of the fibers over $R_{i}^{\circ}$ is constant. We enlarge $D$ by adding the image of $R_{i} \backslash R_{i}^{\circ}$ for each $i$. First suppose that the remaining fibers $\widetilde{\mathcal{E}}_{i}^{\circ} \to R_{i}^{\circ}$ are adjoint rigid with respect to the pullback of $L$ and have the same $a$-value as the fibers of $p$. Then Lemma \ref{lemm: opensetgoodfamily} shows that $r_{i}$ is a good family of adjoint rigid varieties over an open subset $T_{i}$ of $R_{i}^{\circ}$. (Note that the condition that $L$ be the restriction of a nef divisor from a projective compactification is preserved by arbitrary pullback so the hypotheses of Lemma \ref{lemm: opensetgoodfamily} are satisfied.) In this case we call this good family $q_{i}: \mathcal{Y}_{i} \to T_{i}$ and further enlarge $D$ by adding the image of $R_{i}^{\circ} \backslash T_{i}$. Otherwise, the remaining fibers either fail to be adjoint rigid with respect to the pullback of $L$ or fail to have the same $a$-value as the fibers of $p$. Then we simply ignore the family $\widetilde{\mathcal{E}}_{i}^{\circ} \to R_{i}^{\circ}$.
We have now constructed a finite set $\{ f_{i}: \mathcal Y_{i} \to \mathcal U \}$ of good morphisms of good families and a set $D$. Set $W^\circ = W \setminus D$ and let $(T_i)^\circ$ and $\mathcal{V}^\circ$ denote the preimages of $W^{\circ}$ in $T_i$ and $\mathcal V$. Note that by the construction, the map $g_{i}: (T_i)^\circ \to W^\circ$ is proper \'etale so that for a point $t_{i} \in T_{i}^{\circ}$ we have that $\Xi_{i} \rtimes \zeta_*g_{i*}\pi_1^\textrm{\'et}((T_{i})^\circ,t_{i})$
is a finite index subgroup of $\pi_{1}^\textrm{\'et}(\mathcal{V}^{\circ},g_{i}(t_{i}))$ such that the corresponding \'etale cover is an open subset of $\mathcal{Y}_{i}$.
We next show that the set $\{f_{i}: \mathcal{Y}_{i} \to \mathcal{U} \}$ satisfies the factoring property in the statement of the theorem. Suppose we have a morphism $f: \mathcal Y \to \mathcal U$ as in the statement of the theorem. Let $T'$ be a general intersection of hyperplanes in $\mathcal{Y}$ that maps generically finitely onto $T$ and is not contained in the $f$-preimage of $Q$. We let $q': \mathcal{Y}' \to T'$ denote the base change over $T' \to T$, and let $f': \mathcal{Y}' \to \mathcal{U}$ denote the induced map. After shrinking $T'$ we may ensure that $q': \mathcal{Y}' \to T'$ is a good family and the map $T' \to \mathcal{Y}$ induces a good section $\eta'$ of $q'$ whose $f'$-image in $\mathcal{U}$ is disjoint from $Q$.
Let $\mathcal V_{\mathcal{Y}'}$ denote the open subset obtained by removing the closed subset $Q_{\mathcal{Y}'}$ as in Definition \ref{defi:goodfamily}. Using the good section $\eta'$, we can identify
\begin{equation*}
\pi_{1}^\textrm{\'et}(\mathcal V_{\mathcal{Y}'},\eta'(t'))= \pi_{1}^\textrm{\'et}(\mathcal V_{\mathcal{Y}',t'},\eta'(t')) \rtimes \eta'_*\pi_{1}^\textrm{\'et}(T',t')
\end{equation*}
for every fiber $\mathcal{V}_{\mathcal{Y}',t'}$ of $q'$.
Note however that this semidirect product structure need not be compatible with the semidirect product structure of $\pi_{1}^\textrm{\'et}(\mathcal{V},\zeta(w))$ since there is no relationship between the two sections used in the constructions.
Let $(T')^\circ$ denote the preimage of $W^{\circ}$ in $T'$. Note that $(T')^{\circ}$ is empty if and only if the image of $\mathcal{Y}'$ is contained in $p^{-1}(D)$.
If $(T')^{\circ}$ is not empty, for a general point $t'$ in $T'^{\circ}$ we set $w$ as the image of $t'$ in $W^{\circ}$ via $(T')^\circ \to W^\circ$.
The map $f'|_{f'^{-1}(\mathcal V)_{t'}} : f'^{-1}(\mathcal V)_{t'} \rightarrow \mathcal V_w$ is proper.
Furthermore, \cite[Corollary 2.8]{Sen17} shows the map $f'|_{f'^{-1}(\mathcal V)_{t'}}$ is the composition of a birational map with an \'etale map, so in particular it is \'etale outside of a codimension $\geq 2$ subset in $\mathcal V_w$. We let $\Xi_{j}'$ denote the subgroup defined by the image of $\pi_{1}^\textrm{\'et}(f'^{-1}(\mathcal{V})_{t'},\eta'(t'))$ in $\pi_{1}^\textrm{\'et}(\mathcal{V}_{w},f'\circ \eta' (t'))$.
We claim there is an isomorphism $\Phi: \pi_{1}^\textrm{\'et}(\mathcal{V},f'\circ\eta' (t')) \cong \pi_{1}^\textrm{\'et}(\mathcal{V}, \zeta (w))$ such that $p_*^{f'\circ\eta' (t')} = p_*^{\zeta(w)} \circ \Phi$ where for any point $v \in \mathcal V$ above $w$, $p_*^v$ is the induced map $\pi_1^\textrm{\'et}(\mathcal V, v) \to \pi_1^\textrm{\'et}(W^\circ, w)$ by $p$. Indeed, over $\mathbb{C}$ one can define such an isomorphism of topological fundamental groups $\pi_{1}(\mathcal{V},f'\circ\eta' (t')) \cong \pi_{1}(\mathcal{V}, \zeta (w))$ using a path in the fiber, and the result for \'etale fundamental groups follows from the comparison theorems for \'etale and topological fundamental groups. Over an arbitrary algebraically closed field, one can reduce to the case of $\mathbb{C}$ using \cite[Expos\'e XIII Proposition 4.6]{Gro} which shows that the \'etale fundamental group is unchanged under extensions of algebraically closed fields of characteristic $0$. We fix such an isomorphism $\Phi$.
Under the identification $\Phi$, $\Xi'_{j}$ corresponds to one of the subgroups $\Xi_{j}$ constructed earlier. Furthermore, since the image of $\mathcal{Y}'$ is not contained in $p^{-1}(D)$, the map $f'$ constructs covers of some fibers $\mathcal{U}_{w}$ over $W^{\circ}$ which have the same $a$-value as $\mathcal{U}_{w}$ and which are adjoint rigid with respect to $L$. This means that $\Xi_j$ defines a good family $q_j : \mathcal Y_j \to T_j$ via the construction at the beginning of the proof. In particular, there is a point $t_{j} \in T_{j}^{\circ}$ mapping to $w$ under $g_{j}$.
Since the image of $f'$ is not contained in $p^{-1}(D)$, the group $\pi_{1}^\textrm{\'et}((T')^\circ,t')$ maps into $\pi_{1}^\textrm{\'et}(\mathcal{V}^\circ, f' \circ \eta'(t'))$ by composing $\eta'$ with the map $f'$ (since by construction the $f'$-image of $\eta'$ avoids $Q$). Consider the finite index subgroup $M \subset \pi_{1}^\textrm{\'et}((T')^\circ,t')$ which is the pullback of $\Phi^{-1}(\Xi_{j} \rtimes \zeta_*g_{j*}\pi_1^\textrm{\'et}((T_{j})^\circ,t_{j}))$ by this map.
Let $\widetilde{q}: \widetilde{\mathcal{Y}} \to \widetilde{T}$ be defined as a resolution of the base-change of $q'$ over the cover $\widetilde{T}$ of $(T')^\circ$ defined by $M$ with morphism $\widetilde{f}: \widetilde{\mathcal{Y}} \to \mathcal{U}$. After shrinking $\widetilde{T}$, if necessary, we may ensure that $\widetilde{q} : \widetilde{\mathcal{Y}} \to \widetilde{T}$ is a good family with a good section. Thus by using the section $\widetilde{\eta}$ induced from $\eta'$ by base change and shrinking $\widetilde{T}$ to ensure that the hypotheses of \cite[Expos\'e XIII Exemples 4.4]{Gro} apply to $\widetilde{f}^{-1}(\mathcal{V})$, we have an identification
\begin{equation*}
\pi_{1}^\textrm{\'et}(\widetilde{f}^{-1}(\mathcal{V}),\widetilde{\eta}(\widetilde{t})) = \pi_{1}^\textrm{\'et}(\widetilde{f}^{-1}(\mathcal{V})_{\widetilde{t}},\widetilde{\eta}(\widetilde{t})) \rtimes \widetilde{\eta}_*\pi_{1}^\textrm{\'et}(\widetilde{T},\widetilde{t}).
\end{equation*}
Since $t'$ is general, we can find $\widetilde{t}$ mapping to $t'$.
Then every element in $\widetilde{f}_{*}\pi_{1}^\textrm{\'et}(\widetilde{f}^{-1}(\mathcal{V}),\widetilde{\eta}(\widetilde{t}))$ will be a product of an element in $\Xi_{j}' \rtimes \{ 1\} \subset \pi_{1}^\textrm{\'et}(\mathcal{V}^\circ, f' \circ \eta'(t') )$ with an element in $\widetilde{f}_{*}\widetilde{\eta}_*\pi_{1}^\textrm{\'et}(\widetilde{T},\widetilde{t})$, so by construction this set is contained in $\Phi^{-1}(\Xi_{j} \rtimes \zeta_*g_{j*}\pi_1^\textrm{\'et}((T_{j})^\circ,t_{j}))$. By the lifting property for \'etale fundamental groups as in \cite[Expos\'e V]{Gro} (see \cite{Chen18} for a careful explanation), the map $\widetilde{f}^{-1}(\mathcal{V}) \to \mathcal{V}$ factors through the cover defined by $\Phi^{-1}(\Xi_{j} \rtimes \zeta_*g_{j*}\pi_1^\textrm{\'et}((T_{j})^\circ,t_{j}))$. Hence $\widetilde{\mathcal Y} \rightarrow \mathcal U$ rationally factors through $\mathcal Y_j$.
It only remains to prove the last claim. Let $\widetilde{\mathcal{Y}}'$ be a resolution of the map $\widetilde{\mathcal{Y}} \dashrightarrow \mathcal{Y}_{j}$. Note that we have a morphism $\widetilde{T} \to T_{j}^{\circ}$ induced by the homotopy lifting property via the map $\pi_{1}^\textrm{\'et}(\widetilde{T},\widetilde{t}) \to \pi_{1}^\textrm{\'et}(W^{\circ},w)$ induced by $\Phi$.
Altogether we have a commutative diagram
\begin{equation*}
\xymatrix{ \widetilde{\mathcal{Y}}' \ar[r] \ar[d] & \mathcal{Y}_{j} \ar[d] \\
\widetilde{T} \ar[r] & T_{j}}.
\end{equation*}
By applying Lemma \ref{lemm: opensetgoodfamily} we see there is an open subset $\widetilde{T}^{\circ} \subset \widetilde{T}$ over which $\widetilde{\mathcal{Y}}'$ is a good family. By construction every fiber over $\widetilde{T}^{\circ}$ is mapped birationally under $\widetilde{\mathcal{Y}}' \to \mathcal{Y}_{j}$ to the corresponding fiber of $q_{j}$. After shrinking $\widetilde{T}^{\circ}$, we can also ensure that every fiber of $\widetilde{\mathcal{Y}}'$ over $\widetilde{T}^{\circ}$ is birational to the corresponding fiber of $\widetilde{\mathcal{Y}}$.
\end{proof}
For later use, we note a useful property of the construction of the previous lemma.
\begin{coro}
\label{coro: finitelymanycoversbasechange}
Let $p: \mathcal U \to W$ be a good family of adjoint rigid varieties with a good section. Suppose furthermore that the divisor $L$ on $\mathcal{U}$ is the restriction of a nef $\mathbb{Q}$-Cartier divisor on a projective compactification of $\mathcal{U}$. Consider the good morphisms of good families $\{ f_{i}: \mathcal Y_{i} \to \mathcal U\}$ with family maps $q_{i}: \mathcal Y_{i} \to T_{i}$ and the closed proper subset $D \subsetneq W$ constructed by Lemma \ref{lemm: finitelymanycoversinduction}. Suppose that for each $i$ we fix a projective generically finite dominant map $T_{i}' \to T_{i}$ from a smooth variety $T_{i}'$ and replace $q_{i}: \mathcal{Y}_{i} \to T_{i}$ by the base change $\mathcal Y_{i}' := T_{i}' \times_{T_{i}} \mathcal Y_{i}$ (with the natural induced maps $f_{i}'$ and $q_{i}'$). Construct a closed subset $D' \subsetneq W$ by taking the union of $D$ with the branch locus of each map $T_{i}' \to W$. Then the good families $\mathcal Y_{i}'$ (equipped with the morphisms $f_{i}'$ and $q_{i}'$) and the closed subset $D' \subsetneq W$ again satisfy the conclusion of Lemma \ref{lemm: finitelymanycoversinduction}.
\end{coro}
\begin{proof}
Given a good family of adjoint rigid varieties $q: \mathcal{Y} \to T$ admitting a good morphism $f: \mathcal{Y} \to \mathcal{U}$, Lemma \ref{lemm: finitelymanycoversinduction} shows that there is a base change $\widetilde{q}: \widetilde{\mathcal{Y}} \to \widetilde{T}$ such that the induced $\widetilde{f}: \widetilde{\mathcal{Y}} \to \mathcal{U}$ either has image contained in $p^{-1}(D)$ or factors rationally through some $f_{i}$. In the former case, the image is also contained in $p^{-1}(D')$. In the latter case, let $T_{i}^{\circ}$ denote the complement of the image of the ramification divisor for the map $T_{i}' \to T_{i}$. Let $T_{i}'^{\circ}$ and $\widetilde{T}^{\circ}$ denote the preimages of $T_{i}^{\circ}$. If $\widetilde{T}^{\circ}$ is empty, then the image of $\widetilde{f}$ is contained in $p^{-1}(D')$.
Otherwise, $T_{i}'^{\circ} \times_{T_{i}^{\circ}} \widetilde{T}^{\circ}$ is an \'etale cover of $\widetilde{T}^{\circ}$. Let $R$ be any irreducible component of this product which dominates $\widetilde{T}^{\circ}$. Then we can take a basechange of $\widetilde{q}: \widetilde{\mathcal{Y}} \to \widetilde{T}$ over the map $R \to \widetilde{T}$ to obtain the desired rational factoring.
\end{proof}
Returning to the proof of boundedness, note that we can apply the argument of Lemma \ref{lemm: finitelymanycoversinduction} to the preimage of any component of the closed set $D$ constructed there. Arguing by Noetherian induction, we conclude:
\begin{theo} \label{theo: finitelymanycovers}
Let $p: \mathcal U \to W$ be a good family of adjoint rigid varieties with a good section. Suppose furthermore that the divisor $L$ on $\mathcal{U}$ is the restriction of a nef $\mathbb{Q}$-Cartier divisor on a projective compactification of $\mathcal{U}$. There is a finite set of generically finite good morphisms of good families $\{ f_{i}: \mathcal Y_{i} \to \mathcal U\}$ with family maps $q_{i}: \mathcal Y_{i} \to T_{i}$ such that the following holds. Suppose that $q: \mathcal Y \to T$ is a good family of adjoint rigid varieties admitting a good morphism $f: \mathcal Y \to \mathcal U$. Then there is a base change $\widetilde{q}: \widetilde{\mathcal{Y}} \to \widetilde{T}$ of $q$ such that the induced $\widetilde{f}: \widetilde{\mathcal{Y}} \to \mathcal{U}$ factors rationally through the map $f_{j}$ for some $j$ and a general fiber of $\widetilde{q}$ is birational to a fiber of $q_{j}$.
\end{theo}
As a consequence, we prove a finiteness statement for breaking thin maps.
\begin{theo} \label{theo: thinmapfactoring}
Let $X$ be a uniruled smooth projective variety and let $L$ be a big and nef $\mathbb{Q}$-divisor on $X$. There is a finite set of thin maps $\{ f_{\ell}: Y_{\ell} \to X\}$ with $a(Y_{\ell},f_{\ell}^{*}L) \geq a(X,L)$ satisfying the following property. For any thin map $f: Y \to X$ with $a(Y, f^*L) \geq a(X, L)$, after an Iitaka base change to obtain a variety $\widetilde{Y}$ the induced map $\widetilde{f}: \widetilde{Y} \to X$ will either factor rationally through some $f_{k}$ or will have image contained in $\mathbf{B}_{+}(L)$. Furthermore, in the first case we have
\begin{equation*}
a(Y,f^{*}L) = a(\widetilde{Y}, \widetilde{f}^*L) \leq a(Y_k, f_k^*L)
\end{equation*}
and if equality of $a$-invariants is achieved then
\begin{equation*}
b(F,Y,f^{*}L) \leq b(F,\widetilde{Y},\widetilde{f}^{*}L) \leq b(F,Y_{k},f_{k}^{*}L).
\end{equation*}
\end{theo}
\begin{proof}
Let $p_i : \mathcal U_i \to W_i$ be the families from Theorem~\ref{theo: aconstruction}. If the image of $s_{i}: \mathcal{U}_{i} \to X$ is contained in $\mathbf{B}_{+}(L)$ then we ignore the corresponding family $p_{i}$ from now on. Otherwise, after a finite base change we may ensure each family has a rational section. Since $L$ is a nef divisor on $X$, the hypotheses of Lemma \ref{lemm: opensetgoodfamily} hold and so we can shrink the base to obtain a good family with a good section. By combining a Noetherian induction argument with repeated appeals to Lemma~\ref{lemm: opensetgoodfamily} (and repeated throwing away of families with image in $\mathbf{B}_{+}(L)$), we can repeat the argument along the complement of the open set constructed above to obtain a finite collection of good families with good sections. To each such family we apply Theorem \ref{theo: finitelymanycovers}. The result is a finite collection of good families $\{ q_{i,j}: \mathcal{Y}_{i,j} \to T_{i,j} \}$ with maps $g_{i,j}: \mathcal{Y}_{i,j} \to X$.
We next modify the families $q_{i,j}: \mathcal{Y}_{i,j} \to T_{i,j}$ by performing a couple of base changes over $T_{i,j}$. For notational simplicity, we will use the same notation $q_{i,j}: \mathcal{Y}_{i,j} \to T_{i,j}$ and $g_{i,j}: \mathcal{Y}_{i,j} \to X$ for the results after base change. First, by taking a base change we may ensure that $g_{i,j}$ is not birational. Next, we make a base change to kill the monodromy action of $\pi_{1}^{\textrm{\'et}}(T_{i,j},t_{i,j})$ on the N\'eron-Severi group of a general fiber of $q_{i,j}$. Finally, we take a base change over a cyclic cover of $T_{i,j}$ whose branch divisor is a very ample divisor.
We define the thin maps $\{ f_{\ell}: Y_{\ell} \to X \}$ as follows. For each $\mathcal{Y}_{i,j}$ set $D_{i,j}$ to be the closure of $g_{i,j}(\mathcal{Y}_{i,j})$. By construction $D_{i,j}$ is not contained in $\mathbf{B}_{+}(L)$. If $a(D_{i,j},L|_{D_{i,j}})$ agrees with the $a$-value of the fibers of $q_{i,j}$, then \cite[Proposition 4.14]{LTDuke} shows that $g_{i,j}$ is generically finite.
We differentiate the $\mathcal{Y}_{i,j}$ satisfying this property by calling them ``allowable families.'' If $\mathcal{Y}_{i,j}$ is allowable, we add a variety $Y_{\ell}$ which is a resolution of a projective closure of $\mathcal{Y}_{i,j}$ that comes equipped with a morphism $f_{\ell}: Y_{\ell} \to X$ extending $g_{i,j}$ and a morphism $r_{\ell}: Y_{\ell} \to R_{\ell}$ extending the family map $q_{i,j}$. Since $g_{i,j}$ is not birational, $f_{\ell}$ is a thin map.
We also add a finite collection of inclusions of subvarieties to the list of $f_{\ell}: Y_{\ell} \to X$ as follows. Let $\{a_{c} \}$ denote the finite set of constants constructed by Theorem \ref{theo: HJ} (1). For each $a_{c}$, Theorem \ref{theo: HJ} (2) constructs a proper closed subset $V^{a_{c}} \subsetneq X$. For every component $V^{a_{c}}_{d}$ of $V^{a_{c}}$, we add the inclusion $V^{a_{c}}_{d} \hookrightarrow X$ as one of the $f_{\ell}: Y_{\ell} \to X$. Note that when $\mathcal{Y}_{i,j}$ is not allowable then $D_{i,j}$ will be a subvariety of $V^{a(D_{i,j},L|_{D_{i,j}})}$ and so the map $\mathcal{Y}_{i,j} \to X$ will factor through the map $f_{\ell}$ corresponding to the inclusion of a suitable component of $V^{a(D_{i,j},L|_{D_{i,j}})}$.
Before proving the factoring property of the $f_{\ell}$, we note one additional property that will be needed later. Suppose we constructed $Y_{\ell}$ from an allowable family. Recall that in the construction of $Y_{\ell}$ we took a cyclic cover of the corresponding $T_{i,j}$ branched over a very ample divisor. This guarantees that there is an ample divisor $H$ on $R_{\ell}$ such that $K_{Y_{\ell}} + a(Y_{\ell},f_{\ell}^{*}L)f_{\ell}^{*}L - r_{\ell}^{*}H$ is $\mathbb{Q}$-linearly equivalent to an effective class. Thus by Lemma \ref{lemm:birationaltocanonical} $r_{\ell}$ is birationally equivalent to the canonical map for $K_{Y_{\ell}} + a(Y_{\ell},f_{\ell}^{*}L)f_{i}^{*}L$.
Now suppose $f: Y \to X$ is any thin map satisfying $a(Y,f^{*}L) \geq a(X,L)$ and whose image is not contained in $\mathbf{B}_{+}(L)$. Again we separate into two cases: just as before, we say that $Y$ is allowable if $a(f(Y),L|_{f(Y)}) = a(Y,f^{*}L)$. First suppose that $Y$ is not allowable. Then the map $f: Y \to X$ factors through the inclusion of $V^{a(f(Y),L|_{f(Y)})}$ into $X$, and hence also factors through some $f_{\ell}$. If $Y$ is allowable, then we know that $a(f(Y),L|_{f(Y)}) = a(Y,f^{*}L)$. After resolving we may suppose $Y$ is smooth and admits a morphism $q: Y \to T$ which is the canonical model for $K_{Y} + a(Y,f^{*}L)f^{*}L$. By Lemma \ref{lemm: opensetgoodfamily} there is an non-empty open subset $T^{\circ} \subset T$ such that $q$ is a good family over $T^{\circ}$. By Lemma \ref{lemm:dominantequalitycase} we know that possibly after shrinking $T^{\circ}$ the image of every fiber over $T^{\circ}$ in $X$ will have the same $a$-value as $Y$ does and will be adjoint rigid. Thus, the map $f: Y \to X$ will yield a map $T^{\circ} \to \mathrm{Hilb}(X)$ whose image is contained in the locus parametrizing subvarieties which are adjoint rigid with respect to the pullback of $L$ and have $a$-value equal to $a(Y,f^{*}L)$. Since $T$ is reduced, after taking a base change and shrinking $T^{\circ}$ we obtain a map $T^{\circ} \to W_{i}$ for some $i$. After shrinking $T^{\circ}$ and replacing $Y$ by a birational model yet again, the preimage $Y^{\circ}$ will yield a good map of good families $Y^{\circ} \to \mathcal{U}_{i}$. Following through the construction of the $Y_{\ell}$ above, Theorem \ref{theo: finitelymanycovers} shows that if $\widetilde{Y}$ is the Iitaka base change defined by a suitable cover $\widetilde{T} \to T$, the induced map $\widetilde{Y} \to \mathcal{U}_{i}$ will factor rationally through $\mathcal{Y}_{i,j}$ for some $j$.
We may also ensure that the cover $\widetilde{T} \to T$ is chosen so that the monodromy action on the N\'eron-Severi spaces of a general fiber of $\widetilde{Y} \to \widetilde{T}$ is trivial. Finally, since we are only trying to prove the existence of a rational factoring, we are free to replace $\widetilde{Y}$ by a smooth resolution, which by abuse of notation we will continue to call $\widetilde{Y}$.
Let $\widetilde{f}: \widetilde{Y} \to X$ denote the induced map. If $\mathcal{Y}_{i,j}$ is an allowable family, then $\widetilde{f}$ will factor rationally through the corresponding projective closure $Y_{\ell}$. When $\mathcal{Y}_{i,j}$ is not allowable, then $\widetilde{f}$
factors through the inclusion $D_{i,j} \hookrightarrow X$, and thus (as discussed above) also through some $f_{\ell}$. In sum, in every case there is some index $k$ such that the map $\widetilde{f}$ factors rationally through $f_{k}: Y_{k} \to X$.
We next prove the inequalities for $a$-values. If $Y$ is not allowable, then $Y_{k}$ has $a$-value at least $a(f(Y),L|_{f(Y)}) > a(Y,L)$. Otherwise, recall that by Lemma \ref{lemm:ainvandcanonicalfibers} the $a$-value of a pair is the same as the $a$-value of a general fiber of the canonical map. Since we constructed $\widetilde{Y}$ by a base change over the canonical map, Lemma \ref{lemm:birationaltocanonical} shows that the map $\widetilde{Y} \to \widetilde{T}$ is birational to the canonical map for $K_{\widetilde{Y}} + a(\widetilde{Y},\widetilde{f}^{*}L)\widetilde{f}^{*}L$. Thus the general fibers of the canonical maps for $Y$ and $\widetilde{Y}$ are birational, so we have
\begin{equation*}
a(Y,f^{*}L) = a(\widetilde{Y},\widetilde{f}^{*}L).
\end{equation*}
Next, note that a general fiber of the canonical model for $(\widetilde{Y},a(\widetilde{Y},\widetilde{f}^{*}L)\widetilde{f}^{*}L)$ maps birationally onto a fiber $\mathcal{Y}_{i,j,t_{i,j}}$ of $q_{i,j}: \mathcal{Y}_{i,j} \to T_{i,j}$, and in particular
\begin{equation*}
a(\widetilde{Y},\widetilde{f}^{*}L) = a(\mathcal{Y}_{i,j,t_{i,j}},g_{i,j}^{*}L|_{\mathcal{Y}_{i,j,t_{i,j}}}).
\end{equation*}
Suppose first that $\mathcal{Y}_{i,j}$ is an allowable family. Since the $a$-invariant is constant for the fibers of $q_{i,j}$ (as it is a good family), $Y_{k}$ is dominated by subvarieties with the same $a$-value as $\widetilde{Y}$. By Lemma \ref{lemm:ainvdominantfamily} $a(\widetilde{Y}, \widetilde{f}^*L) \leq a(Y_k, f_k^*L)$. If $\mathcal{Y}_{i,j}$ is not allowable, then as explained before the map $f$ factors through the inclusion $D_{i,j} \hookrightarrow X$ where $D_{i,j}$ has higher $a$-value than the members of the family $\mathcal{Y}_{i,j}$, and thus also higher $a$-value than $\widetilde{Y}$. Thus in either case
\begin{equation*}
a(\widetilde{Y},\widetilde{f}^{*}L) \leq a(Y_{k},f_{k}^{*}L).
\end{equation*}
Finally, we prove the inequalities for $b$-values.
Equality of $a$-values will only occur when $Y$ is allowable and the corresponding $\mathcal{Y}_{i,j}$ is an allowable family. Let $Y_{t}$ denote a general fiber of $q$. Then Lemma \ref{lemm: monodromyandbvalue} (3) shows that
\begin{equation*}
b(F,Y,f^{*}L) \leq b(F,Y_{t},f^{*}L|_{Y_{t}}).
\end{equation*}
Choose a point $\widetilde{t} \in \widetilde{T}$ mapping to $t$, so that $\widetilde{Y}_{\widetilde{t}}$ is birational to $Y_{t}$. Thus $b(F,Y_{t},f^{*}L|_{Y_{t}}) = b(F,\widetilde{Y}_{\widetilde{t}},\widetilde{f}^{*}L|_{\widetilde{Y}_{\widetilde{t}}})$. Since the monodromy action on the N\'eron-Severi spaces of general fibers over $\widetilde{T}$ is trivial and the map $\widetilde{Y} \to \widetilde{T}$ is birational to the canonical map for $K_{\widetilde{Y}} + a(\widetilde{Y},\widetilde{f}^{*}L)\widetilde{f}^{*}L$, by Corollary \ref{coro: bvalequality} we have
\begin{equation*}
b(F,\widetilde{Y}_{\widetilde{t}},\widetilde{f}^{*}L|_{\widetilde{Y}_{\widetilde{t}}}) = b(F,\widetilde{Y},\widetilde{f}^{*}L).
\end{equation*}
Next, note $\widetilde{Y}_{t}$ maps birationally onto a fiber of the map $q_{i,j}$. By constancy of the $b$-invariant for the fibers of $q_{i,j}$ and the fact that $f_{k}$ extends $g_{i,j}$, we see that
\begin{equation*}
b(F,\widetilde{Y}_{t},\widetilde{f}^{*}L|_{\widetilde{Y}_{t}}) = b(F,Y_{k,r},f_{k}^{*}L|_{Y_{k,r}})
\end{equation*}
for a general fiber $Y_{k,r}$ of $r_{k}: Y_{k} \to R_{k}$.
Recall that the geometric monodromy action on the N\'eron-Severi space of the fibers of $r_{k}$ over $T_{i,j}$ is trivial. Since $r_{k}$ is birationally equivalent to the canonical map for $(Y_{k},a(Y_{k},f_{k}^{*}L)f_{k}^{*}L)$, we may apply Corollary \ref{coro: bvalequality} to conclude that
\begin{equation*}
b(F,Y_{k,r},f_{k}^{*}L|_{Y_{k,r}}) = b(F,Y_{k},f_{k}^{*}L).
\end{equation*}
Thus our assertion follows.
\end{proof}
\section{Constructing a thin set}
\label{sec: thinset}
In this section we prove analogues of the results in the previous section over a number field $F$ and use them to prove the main theorem (Theorem~\ref{theo: precisetheorem}). We first discuss some constructions involving the \'etale fundamental group.
\begin{defi}
Let $\overline{T}$ be a smooth variety over an algebraically closed field of characteristic $0$ and let $\overline{t}$ be a geometric point on $\overline{T}$. Let $\Xi$ be a finite index open subgroup of $\pi_1^\textrm{\'et}(\overline{T}, \overline{t})$. We say that $\Xi$ is strongly Galois if it is invariant under $\mathrm{Aut}(\pi_1^\textrm{\'et}(\overline{T}, \overline{t}))$.
More generally, for any finite index open subgroup $\Xi$, we define the strong Galois closure $\Xi'$ of $\Xi$ to be the intersection
\begin{equation*}
\Xi' := \bigcap_{\Phi \in \mathrm{Aut}(\pi_1^\textrm{\'et}(\overline{T}, \overline{t}))} \Phi (\Xi) \subset \pi_1^\textrm{\'et}(\overline{T}, \overline{t})
\end{equation*}
Since the \'etale fundamental group is topologically finitely presented by \cite[Th\'eor\`eme II.2.3.1]{Gro2}, there are only finitely many open subgroups of a fixed finite index, thus this is a finite intersection and $\Xi'$ is also an open subgroup of finite index.
In particular, if $\overline{R} \to \overline{T}$ is the cover defined by a finite index subgroup $\Xi \subset \pi_1^\textrm{\'et}(\overline{T}, \overline{t})$, then the strong Galois closure of $\Xi$ defines an \'etale cover $\overline{R}' \to \overline{T}$. We will call $\overline{R}' \to \overline{T}$ the strong Galois closure of $\overline{R} \to \overline{T}$. Note that this notion is independent of the choice of basepoint used to define the cover.
\end{defi}
This construction is particularly useful in the following situation. Suppose there is a smooth variety $T$ carrying a rational point $t$ defined over the ground field $F$. Then $\mathrm{Gal}(\overline{F}/F)$ acts on $\pi_1^\textrm{\'et}(T, t)$ via the splitting induced by $t$, and hence also acts on $\pi_1^\textrm{\'et}(\overline{T}, \overline{t})$ by conjugation. Thus, for any finite \'etale cover $\overline{R} \to \overline{T}$ the strong Galois closure $\overline{R}' \to \overline{T}$ descends to a morphism $R' \to T$ defined over $F$.
We next define good families of adjoint rigid varieties over $F$.
\begin{defi} \label{defi: goodfamilyovernf}
Fix a number field $F$. A good family of adjoint rigid varieties over $F$ is an $F$-morphism $p: \mathcal{U} \to W$ of smooth quasi-projective varieties and a relatively big and nef $\mathbb{Q}$-divisor $L$ on $\mathcal{U}$ such that the base-change to the algebraic closure is a good family of adjoint rigid varieties over each component of the base.
Let $\overline{Q}$ denote the subset of $\overline{\mathcal{U}}$ as in Definition \ref{defi:goodfamily}. Note that $\overline{Q}$ descends to $F$ by Proposition \ref{prop: galinvofa}. We denote this set by $Q$. A good section of a good family over $F$ is a section avoiding $Q$.
\end{defi}
It is natural to wonder whether one can prove a version of Theorem \ref{theo: thinmapfactoring} over a number field which takes twists into account.
However, it seems quite difficult to prove an analogous factoring result since it is hard to decide whether a cover constructed using an \'etale fundamental group descends to the ground field.
Fortunately, in Theorem~\ref{theo: precisetheorem} we do not care about arbitrary breaking thin maps $f: Y \to X$ but only those which contribute a rational point. By keeping careful track of the rational point we can work with \'etale fundamental groups to construct covers defined over the ground field. In the end we still need a factoring result for twists, but we only prove it when we have a rational point to work with. Thus, in the proof below we will mimic the argument of Lemma \ref{lemm: finitelymanycoversinduction} but with many subtle changes which help us keep track of the behavior of rational points and twists. In particular, in contrast to Section \ref{sect: boundedness} we will need to work with projective varieties throughout.
\begin{lemm} \label{lemm: finitelymanycoversovernf}
Let $X$ be a geometrically uniruled geometrically integral smooth projective variety defined over $F$ and let $L$ be a big and nef $\mathbb Q$-divisor on $X$. Let $p : \mathcal U \rightarrow W$ be a surjective morphism between projective varieties where $\mathcal{U}$ is equipped with a morphism $s : \mathcal U \to X$ which is birational.
Suppose that there exists an open subset $W^\circ \subset W$ such that $p: \mathcal{U}^\circ \to W^\circ$ is a good family of adjoint rigid varieties over $F$ (where $\mathcal{U}^\circ$ denotes the preimage of $W^\circ$) and that any fiber over $W^\circ$ has the same $a$-invariant with respect to $s^{*}L$ as $X$ does with respect to $L$. Then there is a proper closed subset $C \subsetneq X$ and a finite set of dominant generically finite morphisms $\{ f_{j}: \mathcal{Y}_{j} \to \mathcal{U} \}$ defined over $F$ that fit into commutative diagrams
\begin{equation*}
\xymatrix{ \mathcal{Y}_{j} \ar[r]^{f_{j}} \ar[d]_{q_{j}} & \mathcal{U} \ar[d]_{p} \\
T_{j} \ar[r] & W}
\end{equation*}
such that the following holds.
\begin{enumerate}
\item both $\mathcal Y_j$ and $T_j$ are projective and geometrically integral, $\mathcal Y_j$ is smooth, $T_j$ is normal, and $q_{j}: \mathcal{Y}_{j} \to T_{j}$ is generically a good family of adjoint rigid varieties;
\item the canonical model for $a(X, L)f_j^*L + K_{\mathcal Y_j}$ is a morphism and over some open set of $T_j$ this map agrees with $q_j$;
\item $T_j \rightarrow W$ is dominant, generically finite, and Galois;
\item $\mathrm{Bir}(\overline{\mathcal Y}_{j}/\overline{X}) = \mathrm{Aut}(\overline{\mathcal Y}_{j}/\overline{X})$;
\item there is a non-empty Zariski open subset $W' \subset W^\circ$ such that for any $j$, for any twist $\mathcal{Y}_{j}^{\sigma}$ over $X$ and for any closed point $t$ in the preimage $T'^{\sigma}_{j}$ of $W'$ we have an isomorphism $s^\sigma_{j*}: \mathcal F_{\mathcal Y^\sigma_{j, t}} \to \mathcal F_{\mathcal Y^\sigma_j}$ where $s_j : \mathcal{Y}_{j} \rightarrow X$ denotes the composition of $f_j : \mathcal Y_j \rightarrow \mathcal U$ and $s : \mathcal U \to X$ and $\mathcal{Y}^{\sigma}_{j,t}$ denotes the fiber over a closed point $t$ on $T_j'^{\sigma}$;
\item
Suppose that $q: \mathcal{Y} \to T$ is a projective surjective morphism of varieties over $F$ where $\mathcal{Y}$ is smooth and geometrically integral and that we have a diagram
\begin{equation*}
\xymatrix{ \mathcal{Y} \ar[r]^{f} \ar[d]_{q} & \mathcal{U} \ar[d]_{p} \\
T \ar[r]^{g} & W}
\end{equation*}
satisfying the following properties:
\begin{enumerate}
\item There is some open subset $T' \subset T$ such that $\mathcal{Y}$ is a good family of adjoint rigid varieties over $T'$ and the map $f: q^{-1}(T') \to \mathcal{U}$ has image in $\mathcal{U}^{\circ}$ and is a good morphism of good families.
\item There is a rational point $y \in \mathcal{Y}(F)$ such that $s \circ f(y) \not \in C$.
\end{enumerate}
Then for some index $j$ there will be a twist $f_{j}^\sigma : \mathcal Y_j^\sigma \rightarrow \mathcal U$ such that $f(y) \in f_{j}^{\sigma}(\mathcal{Y}_{j}^{\sigma}(F))$. Furthermore, there is a generically finite map $\widetilde{T} \to T$ such that the main component $\widetilde{q}: \widetilde{\mathcal{Y}} \to \widetilde{T}$ of the base change of $q$ by $\widetilde{T} \to T$
satisfies that the induced map $\widetilde{f}: \widetilde{\mathcal{Y}} \to \mathcal{U}$ will factor rationally through $f_{j}^{\sigma}$ and a general geometric fiber $\widetilde{q}$ will map birationally to a geometric fiber of the map $q_{j}^{\sigma}: \mathcal{Y}_{j}^{\sigma} \to T_{j}^{\sigma}$.
\end{enumerate}
\end{lemm}
The most important property is (6), which guarantees that such a morphism $f: \mathcal{Y} \to \mathcal{U}$ must factor rationally through a twist of one of the $\mathcal{Y}_{j}$. The other properties allow us to keep track of twists.
\begin{proof}
Let $Q$ denote the closed subset of $\mathcal{U}^\circ$ as in Definition \ref{defi: goodfamilyovernf} and let $\mathcal{V}$ denote its complement. During the construction we will shrink $W^{\circ}$ several times; when we do this operation, $\mathcal{U}^{\circ}$ will continue to denote its preimage and $\mathcal{V}$ will continue to denote $\mathcal{U}^{\circ} \backslash Q$.
We next construct a finite morphism $W^{\mu} \to W$; during this construction we let $W^{\mu \circ}$ denote the preimage of $W^{\circ}$ (and use the same notation after shrinking $W^{\circ}$). By taking a general complete intersection of hyperplanes in $\mathcal U$ and shrinking $W^\circ$ we obtain a base change $W' \to W$ such that $W'^\circ \to W^\circ$ is proper and \'etale, $\mathcal U'^\circ = \mathcal U^\circ \times_{W^\circ} W'^\circ \to W'^\circ$ is a good family, and it admits a good section $\zeta'$.
After taking the Galois closure $W^\mu \rightarrow W'$, we replace $W^\mu \to W$ by its Stein factorization so that $W^\mu \to W$ is Galois and finite. After shrinking $W^\circ$ again, we may assume that $p^\mu : \mathcal V^{\mu} \rightarrow W^{\mu \circ}$ is a good family and admits a good section $\zeta$ where $\mathcal V^{\mu}$ is $\mathcal V \times_{W^\circ}W^{\mu \circ}$ (and as usual we will use the same notation even after shrinking $W^{\circ}$). We also set $\mathcal{U}^{\mu}$ to be the main component of $\mathcal U \times_{W}W^{\mu}$ with maps $s^{\mu}: \mathcal{U}^{\mu} \to X$ and $p^{\mu}: \mathcal{U}^{\mu} \to W^{\mu}$.
Let $\overline{D} \subset \overline{W}^{\mu \circ}$ be the proper closed subset obtained by applying Lemma~\ref{lemm: finitelymanycoversinduction} over $\overline{F}$. After including its Galois conjugates, we may assume that $D$ is defined over the ground field. Initially we set $C$ to be the closure of $s^\mu((p^{\mu})^{-1}(D\cup R)) \cup s(Q)$ where $R$ is the ramification locus of $W^\mu \to W$; we will increase $C$ later. We also shrink $W^\circ$ so that $W^{\mu \circ} \to W^\circ$ is proper and \'etale.
We next construct the families $\mathcal{Y}_{j}$. We may suppose that $\mathcal{V}$ admits a rational point since otherwise condition (6) is vacuous. Since $W^\mu \to W$ is Galois we may ensure that $\mathcal V^\mu$ admits a rational point after replacing $W^\mu$ by its twist. (Note that after this change the section $\overline{\zeta}$ may not be defined over the ground field but is still defined after base change to $\overline{F}$. The other properties of $W^{\mu}$ are preserved by replacing by a twist.)
Let $w^{\mu}$ denote a rational point on $W^{\mu \circ}$ which is the image of a rational point in $\mathcal{V}^{\mu}$ and define the geometric point $\overline{v}^\mu = \overline{\zeta}(w^\mu)$.
Consider the set of subgroups $\Xi_{j} \subset \pi_1^\textrm{\'et}(\overline{\mathcal V}^\mu \cap \overline{\mathcal U}^{\mu}_{w^\mu}, \overline{v}^\mu)$ constructed as in Lemma \ref{lemm: finitelymanycoversinduction} applied to $\overline{\mathcal{U}}^{\mu \circ} \to \overline{W}^{\mu \circ}$. Each $\Xi_{j}$ yields a normalizer $N_{j} \subset \pi_1^\textrm{\'et}(\overline{W}^{\mu \circ}, w^\mu)$.
Let $(\Xi_{j} \rtimes \overline{\zeta}_*N_j)'$ be the strong Galois closure of $\Xi_{j} \rtimes \overline{\zeta}_*N_j$ in $\pi_1^\textrm{\'et}(\overline{\mathcal V}^\mu, \overline{v}^\mu)$.
We define $\widetilde{N}_{j} \subset N_{j}$ as the preimage of $(\Xi_{j} \rtimes \overline{\zeta}_*N_j)'$ via $\overline{\zeta}_*$. Let $\widetilde{N}_j'$ be the strong Galois closure of $\widetilde{N}_j$ in $\pi_1^\textrm{\'et}(\overline{W}^{\mu \circ}, w^\mu)$.
We define $\widetilde{\Upsilon}_{j} = \Xi_{j} \rtimes \overline{\zeta}_*\widetilde{N}_{j}'$. Each $\widetilde{\Upsilon}_{j}$ defines an \'etale cover $\overline{\mathcal{E}}_{j} \to \overline{\mathcal{V}}^{\mu}$, and by composing with the \'etale map $\overline{\mathcal{V}}^{\mu} \to \overline{\mathcal{V}}$ we obtain an \'etale cover $\overline{\mathcal{E}}_{j} \to \overline{\mathcal{V}}$.
By taking a projective closure of the fibers of $\overline{\mathcal{E}}_{j}$ over $\overline{W}^{\mu \circ}$ and passing to a resolution we obtain a projective family $\overline{\widetilde{\mathcal{E}}}_{j} \to \overline{W}^{\mu \circ}$. Let $\overline{R}^\circ_j \to \overline{W}^{\mu \circ}$ be the Stein factorization of this map; it is defined by $\widetilde{N}_j'$.
Note that by our construction, $\overline{R}_j^\circ \to \overline{W}^{\mu \circ}$ is strongly Galois.
After shrinking $W^\circ$ we may assume that for every $j$ the $a$-values and the Iitaka dimension of any fiber of $\overline{\widetilde{\mathcal{E}}}_{j} \to \overline{T}_j^\circ$ is constant.
If fibers of this family do not have the same $a$-value as $X$ or they are not adjoint rigid, then henceforth we disregard these $j$.
After possibly shrinking $W^\circ$ again we may assume that for every $j$ the family $\overline{\widetilde{\mathcal{E}}}_{j} \to \overline{R}_j^\circ$ is a good family.
We enlarge $C$ by adding $s(p^{-1}(W\setminus W^\circ))$.
We now consider two cases. First suppose that $\overline{\mathcal{E}}_{j} \to \overline{\mathcal{V}}$ fails to descend to $F$ in such a way that $\overline{\mathcal{E}}_{j}$ admits a rational point. At least over $\overline{F}$ we can compactify $\overline{\mathcal{E}}_{j} \to \overline{\mathcal{V}}$ to obtain a morphism of smooth projective varieties $\overline{\mathcal{P}}_{k} \to \overline{\mathcal{U}}$ over $\overline{F}$. We denote by $s_{k}$ the induced map $\overline{\mathcal{P}}_{k} \to \overline{X}$.
We let $\overline{\mathcal{P}}_{k} \to \overline{R}_{k}$ denote the Stein factorization of the map $\overline{\mathcal{P}}_{k} \to \overline{W}$.
Note that by the construction in the previous paragraph $\overline{\mathcal P}_k^\circ \rightarrow \overline{R}^\circ_k$ is a good family where $\overline{\mathcal P}_k^\circ$ is the preimage of $\overline{W}^\circ$.
We enlarge $C$ by adding $s_{k}(\overline{\mathcal P}_{k}\setminus \overline{\mathcal P}_{k}^\circ) \cup s(p^{-1}(\overline{B}_{k})) \cup \overline{E}_{k}$ where $\overline{E}_{k}$ is the branch locus of $s_{k}: \overline{\mathcal P}_k \to \overline{X}$ and $\overline{B}_{k}$ is the branch locus of $\overline{R}_k \to \overline{W}$.
By taking the union with Galois conjugates we may assume that $C$ is defined over the ground field.
For the second case, suppose the map $\overline{\mathcal{E}}_{j} \rightarrow \overline{\mathcal V}$ descends to a morphism $\mathcal{E}_{j} \to \mathcal{V}$ over $F$ in such a way that $\mathcal{E}_{j}$ admits a rational point. Choose one such $F$-model $\mathcal{E}_{j}$ with a rational point.
We then define $\mathcal{Y}_{j}$ over $F$ as a smooth projective compactification of $\mathcal E_j$ with a morphism $\mathcal Y_j \to \mathcal U$ extending $\mathcal E_j \to \mathcal V$ and let $T_{j}$ denote the Stein factorization of the map $\mathcal{Y}_{j} \to W$.
The structure map $q_{j}: \mathcal{Y}_{j} \to T_{j}$ is generically a good family of adjoint rigid varieties.
We let $T_{j}^{\circ}$, $\mathcal{Y}_{j}^{\circ}$ denote the preimages of $W^{\circ}$ and let $\mathcal E_j$ denote the preimage of $\mathcal V$. We will continue to use this notation after shrinking $W^{\circ}$.
We make a few additional changes to the family.
First we apply Lemma~\ref{lemm: birandaut} to $\mathcal{Y}_j/\mathcal U$ and replace $\mathcal{Y}_{j}$ by a birational model to ensure that $\mathrm{Bir}(\overline{\mathcal Y}_{j}/\overline{X}) = \mathrm{Aut}(\overline{\mathcal Y}_{j}/\overline{X})$.
We claim that after shrinking $W^\circ$, $q_{j}$ is a good family of adjoint rigid varieties over $T_{j}^{\circ}$. Indeed, Lemma \ref{lemm: opensetgoodfamily} shows that this is true after base change to $\overline{F}$, and after possibly shrinking further this open set will descend to $F$.
Note that by our convention shrinking $W^{\circ}$ also causes $\mathcal{E}_{j}$ to shrink. After this change if no twist of the $\mathcal{E}_{j}$ contains a rational point, then we add $s_{j}(\mathcal Y_{j}\setminus \mathcal E_{ j}) \cup s(p^{-1}(B_{j})) \cup E_{j}$ to $C$, where $E_{ j}$ is the branch locus of $s_{j}: \mathcal Y_{j}\rightarrow X$ and $B_{j}$ is the branch locus of $T_{j}\rightarrow W$. To distinguish such types of families, we will henceforth relabel them and add them to the list of $q_k : \mathcal P_k \rightarrow R_k$ with the evaluation map $s_k : \mathcal P_k \rightarrow X$. If some twist of $\mathcal{E}_{j}$ does contain a rational point, then we replace our families with this twist and continue the construction. Note that in this process we replace $T_j, W^\mu$ by their twists so that we have morphisms $\mathcal Y_j \to T_j$ and $\mathcal Y_j \to W^\mu$ over the ground field. Indeed, $T_j$ can be defined as the Stein factorization of $\mathcal Y_j \to W$. Since $W^\mu \to W$ is Galois and finite, for some twist $W^{\mu \sigma}$, the image of a rational point on $\mathcal E_j$ is a rational point on $W^{\mu \sigma}$. Then $T_j$ admits a morphism to $W^{\mu \sigma}$.
Since we are in a situation where $T_{j}^{\circ}$ admits a rational point $b_{j}$, there is a fiber of $q_{j}$ over $T_{j}^{\circ}$ defined over the ground field. This implies that we can find a base change defined over $F$ which kills the geometric monodromy action on the N\'eron-Severi space of a general fiber; indeed one can consider a projective closure of the \'etale cover of $T_{j}^{\circ}$ defined by $G \rtimes \mathrm{Gal}(\overline{F}/F) \subset \pi_1^\textrm{\'et}(T_{j}^{\circ}, b_{j})$ where $G$ is the kernel of the geometric monodromy action on the geometric N\'eron-Severi space of the fiber defined over the ground field.
After taking this finite base change (which we continue to represent by the notation $T_{j}$, $\mathcal{Y}_{j}$, etc.~for simplicity), we may assume that the geometric monodromy of $\pi_{1}^{\textrm{\'et}}(\overline{T^{\circ}_{j}},b_{j})$ on the N\'eron-Severi space of a general fiber of $q_{j}$ is trivial. While doing so we keep that $T_j$ is normal and $T_j \to W$ is finite. We take another cyclic cover so that that the ramification locus contains the pullback of an ample divisor on $T_j$ which does not contain $b_j$. After shrinking $W^\circ$ we may guarantee that $T_j^\circ \to W^{\mu \circ}$ is proper and \'etale. After taking a strong Galois closure and descending to $F$,
we may assume that (i) $T_j$ is normal, (ii) $T_{j}/W^\mu$ is finite, and (iii) $T_{j}/W^\mu$ satisfies that the subgroup of $\pi_{1}^{\textrm{\'et}}(\overline{W}^{\mu \circ},w^{\mu})$ defined by $\overline{T}_{j}^\circ$ is strongly Galois. We replace $\mathcal Y_j$ by the main component of the base change.
We then apply Lemma~\ref{lemm: birandaut} to $\mathcal Y_j/\mathcal U$ again and replace $\mathcal{Y}_{j}$ by a birational model to ensure that $\mathrm{Bir}(\overline{\mathcal Y}_{j}/\overline{X}) = \mathrm{Aut}(\overline{\mathcal Y}_{j}/\overline{X})$ and the canonical map for $K_{\mathcal{Y}_{j}} + a(X,L)s_{j}^{*}L$ is a morphism.
We may need to shrink $W^{\circ}$ to preserve the good family structure over its preimage $T_{j}^{\circ}$.
By the construction we have $T_{j}^{\circ}$ is proper and \'etale over the open set $W^{\circ}$.
After possibly shrinking $W^\circ$ we may guarantee that $f_{j} : \mathcal{E}_{j} = f_{j}^{-1}(\mathcal V) \rightarrow \mathcal V$ is \'etale.
If no twist of $\mathcal{E}_{j}$ contains a rational point again, then we add $s_{j}(\mathcal Y_{j}\setminus \mathcal E_{ j}) \cup s(p^{-1}(B_{j})) \cup E_{j}$ to $C$ and then we relabel this family as one of the $q_k : \mathcal P_k \rightarrow R_k$ with the evaluation map $s_k : \mathcal P_k \rightarrow X$. If some twist of $\mathcal{E}_{j}$ does contain a rational point, then we replace our families with this twist.
After all these changes we have a commutative diagram
\begin{equation*}
\xymatrix{
\mathcal Y_{j} \ar@{>}[r]^{f_{j}} \ar@{>}[d]& \mathcal U \ar@{>}[d]\\
T_{j} \ar@{>}[r] & W}
\end{equation*}
We enlarge $C$ by adding $s_{j}(\mathcal Y_{j}\setminus \mathcal E_{ j}) \cup s(p^{-1}(B_{j})) \cup E_{j}$ where $E_{ j}$ is the branch locus of $s_{j}: \mathcal Y_{j}\rightarrow X$ and $B_{j}$ is the branch locus of $T_{j}\rightarrow W$. Note that by the construction we have now verified Lemma \ref{lemm: finitelymanycoversovernf} (1),(3),(4). Recall that during the construction we took a cyclic cover
so that the ramification divisor of $T_j \to W$ contains an ample divisor. Hence there is an ample $\mathbb{Q}$-divisor $H$ on $T_{j}$ such that $K_{\mathcal{Y}_{j}} + a(\mathcal{Y}_{j},s_{j}^{*}L)s_{j}^{*}L - q_{j}^{*}H$ is $\mathbb{Q}$-linearly equivalent to an effective divisor. Thus Lemma \ref{lemm: finitelymanycoversovernf} (2) follows from Lemma \ref{lemm:birationaltocanonical}.
Before continuing, we prove that there is a Zariski open subset $W' \subset W^{\circ}$
such that for any twist $\mathcal{Y}_{j}^{\sigma}$ over $X$ and for any closed point $b \in T'^{\sigma}_{j}$ we have an isomorphism $s^\sigma_{j*}: \mathcal F_{\mathcal Y^\sigma_{j, b}} \to \mathcal F_{\mathcal Y^\sigma_j}$.
By construction $\overline{\mathcal{Y}}^{\sigma}_{j}$ has a birational model with a structure map to $\overline{T}_{j}^{\circ}$ which has a trivial geometric monodromy action. Furthermore, the structure map to $\overline{T}_{j}^{\circ}$ is birational to the canonical map by property (2). Thus Lemma \ref{lemm:birationaltocanonical} verifies the hypotheses of Corollary \ref{coro: bvalequality} on this birational model. By applying Corollary \ref{coro: bvalequality} to this model and using Lemma \ref{lemm:birfaceinv} to transfer the result to $\overline{\mathcal{Y}}_{j}^{\sigma}$, we deduce that for each $j$ there is an isomorphism $\mathcal F_{\overline{\mathcal Y}^\sigma_{j, \overline{b}}} \to \mathcal F_{\overline{\mathcal Y}^\sigma_j}$ for every $\overline{b}$ lying above some open subset of $\overline{W}^{\circ}$.
The desired equality follows by taking the Galois invariant part. Since the index set of $j$ is a finite set we find an open subset $W'$ which works for all $j$ simultaneously. This verifies Lemma \ref{lemm: finitelymanycoversovernf} (5).
Note that the families constructed here are geometrically independent of the initial choice of $\overline{v}^\mu$. Indeed, we only used $\overline{v}^\mu$ to define geometric covers over $\overline{F}$; all the other choices in the construction were obtained intrinsically from the geometry of this finite set of covers. Thus, we can at a later stage choose a (possibly different) basepoint $w^{\mu}$ in a twist of $W^{\mu \circ}$ and pretend that all our constructions were made with respect to this choice all along.
Now we prove the universal property for these families. Assume that $f: \mathcal Y \rightarrow \mathcal U$ is a morphism as in the statement.
Our goal is to show that $f(y) \in f_j^\sigma(\mathcal Y_j^\sigma(F))$ for some twist $\sigma$. We set $t = q(y)$. After resolving $T$ and $\mathcal Y$, we may assume that $t$ is a smooth point of $T$.
First of all, since $W^{\mu}$ is Galois over $W$, after replacing $W^\mu$ by its twist we may assume that it carries a rational point whose image in $W$ is the same as the image of $t \in T$ in $W$. Note that since $s \circ f(y) \not \in C$, the map $T \times_W W^\mu \to T$ is \'etale over an open neighborhood of $t$. Thus there is some component of $T \times_W W^\mu$ which maps dominantly to $T$ and which admits a rational point in its smooth locus mapping to $t$. Let $T^\mu$ be the normalization of this dominant component. Let $\mathcal Y^\mu$ be a smooth resolution of the main component of $\mathcal Y \times_T T^\mu$. Since $\mathcal{Y}^{\mu}$ is \'etale over an open neighborhood of $y$, $\mathcal Y^\mu$ admits a rational point mapping to $y$ which we denote by $y^\mu$. We denote the induced morphism by $q^\mu : \mathcal Y^\mu \to T^\mu$ and let $t^\mu = q^\mu(y^\mu)$.
Next, if we let $T'$ be a general intersection of hyperplanes through $y^\mu$, then for the generically finite surjective base change $T' \rightarrow T^\mu$ there is a rational point $t' \in T'(F)$ mapping to $t^\mu$ such that $T'$ is smooth at $t'$ and the main component $\mathcal Y_{T'}$ admits a rational section $\tau$ such that $(y^\mu, t') = \tau(t')$.
By the generality of $T'$ we may furthermore ensure that the image of $\tau$ intersects the smooth locus of $\mathcal Y_{T'}$. Let $\widetilde{T} \rightarrow T'$ be the blow up at $t'$ and consider the main component $\mathcal Y_{\widetilde{T}}$ of the base change by $\widetilde{T}$. Let $\widetilde{\mathcal Y}_{\widetilde{T}}$ be a resolution of $\mathcal Y_{\widetilde{T}}$ chosen in such a way that $\widetilde{\mathcal Y}_{\widetilde{T}}$ still admits a rational section $\tau$. Note that $\tau$ is well-defined on the generic point of the exceptional divisor lying over $t'$. Thus by taking the image under $\tau$ of a suitable rational point $\widetilde{t}$ in the exceptional divisor we obtain a rational point $y' \in \widetilde{\mathcal Y}_{\widetilde{T}}$ mapping to $y$ and $\widetilde{t}$. Let $v$ be the image of $y'$ in $\mathcal V^\mu$ and set $w^\mu = p^\mu(v)$ and $\overline{v}^\mu = \overline{\zeta}(w^\mu)$.
Recall that the morphism $\mathcal U^{\mu} \to W^{\mu}$ has a geometric section $\overline{\zeta}$. Working over $\overline{F}$, Lemma \ref{lemm: finitelymanycoversinduction} and Corollary \ref{coro: finitelymanycoversbasechange} show that for some Iitaka base change of $\overline{\widetilde{\mathcal Y}}_{\widetilde{T}}$ the induced map to $\overline{\mathcal{U}}^{\mu}$ factors rationally through $\overline{\mathcal Y_{j}}$ for some $j$ or $\overline{\mathcal P}_{k}$ for some $k$. Assume for a contradiction that the map factors rationally through $\overline{\mathcal P}_{k}$.
We claim that if we take the Stein factorization $Y'$ of the map of fibers $\widetilde{\mathcal Y}_{\widetilde{t}} \rightarrow \mathcal U^{\mu}_{w^\mu}$ and then base change to $\overline{F}$ the result is birational to the adjoint rigid variety $\overline{\mathcal{P}}_{k,\overline{r}}$ where $\overline{r}$ is some preimage of $w^\mu$
which we specify later. Note that due to our construction $\overline{\mathcal{P}}_{k,\overline{r}}$ is irreducible with $a$-invariant equal to $a(X, L)$ and it is adjoint rigid.
To see the claim, first choose an open subset $\overline{\widetilde{T}}^{\circ}$ of $\overline{\widetilde{T}}$ that contains $\widetilde{t}$
such that the image of this set in $\overline{W}$ is contained in $\overline{W}^{\circ}$ as defined above and the $\overline{\tau}$-image of this set lies in the preimage of $\overline{\mathcal{V}}$.
Let $\overline{T}^{\nu}$ denote the \'etale cover of $\overline{\widetilde{T}}^\circ$ defined by the finite index subgroup of $\pi_{1}^{\textrm{\'et}}(\overline{\widetilde{T}}^{\circ}, \overline{\widetilde{t}})$ constructed by pulling back under $\overline{\tau}$ the subgroup of $\pi_{1}(\overline{\mathcal{V}}, \overline{v}^\mu)$ corresponding to the \'etale cover defined by $\overline{\mathcal{E}}_{k}$. For the open subset of $\overline{\widetilde{T}}^{\circ}$ over which we have a good family, just as in Lemma \ref{lemm: finitelymanycoversinduction} we know that the main component of the base change $\overline{\widetilde{\mathcal{Y}}}^{\nu}$ over $\overline{T}^{\nu}$ admits a rational map to $\overline{\mathcal{P}}_{k}$. Since the map $\overline{T}^{\nu}$ to $\overline{\widetilde{T}}^\circ$ is \'etale, $\overline{\widetilde{\mathcal Y}}^\nu$ is smooth in a neighborhood of the fiber $\overline{\widetilde{\mathcal Y}}^{\nu}_{\overline{t}^{\nu}}$ where $\overline{t}^\nu$ is a geometric point mapping to $\overline{\widetilde{t}}$.
Let $\overline{\mathcal{Y}}^{*}$ denote a smooth resolution of the rational map to $\overline{\mathcal{P}}_{k}$.
The fiber $\overline{\mathcal{Y}}^{*}_{\overline{t}^{\nu}}$ maps to some fiber $\overline{\mathcal P}_{k,\overline{r}}$. Since the induced map $\overline{\mathcal{Y}}^{*} \to \overline{\mathcal P}_k^\circ\times_{\overline{R}_k^\circ} \overline{T}^{\nu }$ is birational and the target is smooth, this map has connected fibers. In particular the map $\overline{\mathcal{Y}}^{*}_{\overline{t}^{\nu}} \to \overline{\mathcal P}_{k,\overline{r}}$ has connected fibers and we deduce that the Stein factorization of $\overline{\mathcal{Y}}^{*}_{\overline{t}^{\nu}} \to \overline{\mathcal{U}}^{\mu}_{\overline{w^\mu}}$ is birational to ${\overline{\mathcal P}}_{k,\overline{r}}$.
Note that the map $\overline{\mathcal{Y}}^{*}_{\overline{t}^{\nu}} \to \overline{\mathcal{U}}^{\mu}_{\overline{w^\mu}}$ also factors through our original fiber $\overline{\widetilde{\mathcal Y}}_{\overline{\widetilde{t}}}$ and that the first step of this factoring has connected fibers. Thus the Stein factorization of our original fiber over $\overline{F}$ is also birational to $\overline{\mathcal P}_{k,\overline{r}}$. Since Stein factorization commutes with base change to the algebraic closure our assertion follows.
This implies that the subgroup $\Xi_{k}$ which corresponds to the cover $\overline{\mathcal{P}}_{k, \overline{r}} \to \overline{\mathcal U}^{\mu}_{\overline{w^\mu}}$ admits an extension $\widetilde{\Xi}_k \subset \pi_1^\textrm{\'et}(\mathcal V^{\mu}_{w}, \overline{v}^\mu)$ corresponding to the cover $Y' \to \mathcal U^{\mu}_{w^\mu}$ defined over the ground field.
We next show that $f_{k}^{-1}(\overline{\mathcal V})$ must descend to the ground field.
First note that $\pi_1^\textrm{\'et}(\overline{R}_k^\circ, \overline{r}_k)$ is preserved by the conjugation action of the splitting $\epsilon: \mathrm{Gal}(\overline{F}/F) \to \pi_1^\textrm{\'et}(W^{\mu \circ}, w^{\mu})$ coming from $w^\mu$
because there exists some open subset $W^{\mu a}$ containing $W^{\mu \circ}$
such that for the preimage $\overline{R}_k^{a}$ of $\overline{W}^{\mu a}$ in $\overline{R}_k$, $\overline{R}_k^{a} \to \overline{W}^{\mu a}$ is proper, \'etale and strongly Galois. Thus the homomorphism $\widetilde{\Xi}_k \to \pi_1^\textrm{\'et}(W^{\mu \circ}, w^{\mu})$ factors through $\epsilon : \mathrm{Gal}(\overline{F}/F) \to \pi_1^\textrm{\'et}(W^{\mu \circ}, w^{\mu})$ and we take a set-theoretic section of this map $\delta : \mathrm{Gal}(\overline{F}/F) \to \widetilde{\Xi}_k$. Note that every element in $\widetilde{\Xi}_{k}$ is a product of an element of $\Xi_{k}$ and an element in the image of $\delta$.
We claim that $\overline{\zeta}_*(\pi_1^\textrm{\'et}(\overline{R}_k^\circ, \overline{r}_k)) \cdot\widetilde{\Xi}_k = \widetilde{\Xi}_k \cdot \overline{\zeta}_*(\pi_1^\textrm{\'et}(\overline{R}_k^\circ, \overline{r}_k))$ as subsets of $\pi_{1}^{\textrm{\'et}}(\mathcal{V}^{\mu},\overline{v}^{\mu})$.
Pick $\overline{\zeta}_*(\gamma) \in \overline{\zeta}_*(\pi_1^\textrm{\'et}(\overline{R}_k^\circ, \overline{r}_k))$ and $g \in \Xi_k$ and $\sigma \in \mathrm{Gal}(\overline{F}/F)$. Then we have
\[
\overline{\zeta}_*(\gamma) g \delta(\sigma) = \overline{\zeta}_*(\gamma) g \overline{\zeta}_*(\gamma)^{-1} \cdot \overline{\zeta}_*(\gamma) \delta(\sigma)\overline{\zeta}_*(\gamma')^{-1} \cdot \overline{\zeta}_*(\gamma'),
\]
where $\gamma' = \epsilon(\sigma)^{-1}\gamma \epsilon(\sigma)$.
By construction $\overline{\zeta}_{*}(\gamma)$ is contained in the normalizer of $\Xi_{k}$ so we have $\overline{\zeta}_*(\gamma) g \overline{\zeta}_*(\gamma)^{-1} \in \Xi_k$. Thus it suffices to show that $ \overline{\zeta}_*(\gamma) \delta(\sigma)\overline{\zeta}_*(\gamma')^{-1} \in \widetilde{\Xi}_k$. Since $\delta(\sigma) \in \widetilde{\Xi}_k$ we may instead show
\[
\overline{\zeta}_*(\gamma) \delta(\sigma)\overline{\zeta}_*(\gamma')^{-1}\delta(\sigma)^{-1} \in \Xi_k.
\]
Observe that $\overline{\zeta}_*(\gamma) \delta(\sigma)\overline{\zeta}_*(\gamma')^{-1}\delta(\sigma)^{-1}$ maps to the identity under $\pi_1^\textrm{\'et}(\mathcal {V}^{\mu}, \overline{v}^\mu) \to \mathrm{Gal}(\overline{F}/F)$, so that it can be identified as an element in $\pi_1^\textrm{\'et}(\overline{\mathcal V}^{\mu}, \overline{v}^\mu)$. Furthermore it maps to the identity under $\pi_1^\textrm{\'et}(\overline{\mathcal V}^{\mu}, \overline{v}^\mu) \to \pi_1^\textrm{\'et}(\overline{W}^{\mu \circ}, w^\mu)$, so we can identify $\overline{\zeta}_*(\gamma) \delta(\sigma)\overline{\zeta}_*(\gamma')^{-1}\delta(\sigma)^{-1}$ with an element of $\pi_1^\textrm{\'et}(\overline{\mathcal V}^\mu \cap \overline{\mathcal U}^{\mu}_{w^\mu}, \overline{v}^\mu)$.
Since the image of $\pi_1^\textrm{\'et}(\overline{R}_k^\circ, \overline{r}_k)$ via $\overline{\zeta}_*$ is contained in $(\Xi_{k} \rtimes \overline{\zeta}_*N_k)'$, we have $\overline{\zeta}_*(\gamma), \overline{\zeta}_*(\gamma')^{-1} \in (\Xi_{k} \rtimes \overline{\zeta}_*N_k)'$. Then the strong Galois property implies that $\delta(\sigma)\overline{\zeta}_*(\gamma')^{-1}\delta(\sigma)^{-1} \in (\Xi_{k} \rtimes \overline{\zeta}_*N_k)'$.
Thus we have
\[
\overline{\zeta}_*(\gamma) \delta(\sigma)\overline{\zeta}_*(\gamma')^{-1}\delta(\sigma)^{-1} \in \Xi_k\rtimes \overline{\zeta}_*N_k.
\]
However since $ \overline{\zeta}_*(\gamma) \delta(\sigma)\overline{\zeta}_*(\gamma')^{-1}\delta(\sigma)^{-1} \in \pi_1^\textrm{\'et}(\overline{\mathcal V}^\mu \cap \overline{\mathcal U}^{\mu}_{w^\mu}, \overline{v}^\mu)$ our assertion follows.
Thus $\widetilde{\Xi}_k \cdot \overline{\zeta}_*(\pi_1^\textrm{\'et}(\overline{R}_k^\circ, \overline{r}_k))$ is a subgroup of $\pi_{1}^{\textrm{\'et}}(\mathcal{V}^{\mu},\overline{v}^{\mu})$. Since the image of this group in $\mathrm{Gal}(\overline{F}/F)$ is the full group, one may use this group to define an \'etale cover of $\mathcal V^\mu$ that is defined over $F$. Since $\widetilde{\Xi}_k \cdot \overline{\zeta}_*(\pi_1^\textrm{\'et}(\overline{R}_k^\circ, \overline{r}_k))$ is an extension of $\widetilde{\Upsilon}_{k}$, this \'etale cover coincides with $f_{k}^{-1}(\overline{\mathcal V})$ after base change to $\overline{F}$, and thus we will write this cover as $f_{k}^{-1}(\mathcal V) \to \mathcal{V}^{\mu}$.
Moreover we claim that $f_{k}^{-1}(\mathcal{V})$ admits a fiber birational to $Y'$ and this birational map is an isomorphism on an open neighborhood of the image of $y'$. Indeed, let $R_k^\circ$ be the Stein factorization of a compactification of $f_{k}^{-1}(\mathcal V) \rightarrow \mathcal V^\mu \to W^{\mu \circ}$. Then the cover $R_{k}^{\circ} \to W^{\mu \circ}$ corresponds to an extension of $\pi_1^\textrm{\'et}(\overline{R}_k^\circ, \overline{r}_k)$ by $\mathrm{Gal}(\overline{F}/F)$.
Using the splitting $\delta$ constructed above and pushing forward to $R_{k}^{\circ}$, we obtain a
group theoretic splitting $\mathrm{Gal}(\overline{F}/F) \to \pi_1^\textrm{\'et}(R_k^\circ, \overline{r}_k)$ compatible with the splitting $\epsilon: \mathrm{Gal}(\overline{F}/F) \to \pi^\textrm{\'et}_1(W^{\mu \circ}, w^\mu)$. This section is a homomorphism because $\epsilon$ is.
On the other hand since $R_k$ is Galois over $W^\mu$, $R_k^\circ$ admits a twist $R_k^{\sigma \circ}$ with a rational point mapping to $w^\mu$. Moreover the fundamental group of $R_k^{\sigma \circ}$ also has a splitting compatible with the splitting of $\pi^\textrm{\'et}_1(W^{\mu \circ}, w^\mu)$ coming from $w^{\mu}$. Altogether we conclude that $R_k^\circ$ and $R_k^{\sigma \circ}$ must be isomorphic to each other, or in other words, that $R_{k}^{\circ}$ comes with a rational point $r_k$ mapping to $w^\mu$.
By comparing fundamental groups, we see that the fiber over $r_{k}$ is birational to the variety defined by $\widetilde{\Xi}_k$ as claimed. Furthermore, this birational map is an isomorphism on a neighborhood of $y'$ because $y'$ maps to $\mathcal{V}$. We conclude that $f_{k}^{-1}(\mathcal{V})$ admits a rational point coming from $y$.
However, the fact that the geometric model descends to the ground field with a rational point contradicts our definition of the $\overline{\mathcal{P}}_{k}$. We deduce that this case cannot happen; in other words, some base change of $\overline{\widetilde{\mathcal{Y}}}_{\widetilde{T}}$ admits a rational map to $\overline{\mathcal Y}_{j}$ for some $j$. (We do not yet know that this rational factoring can be achieved over the ground field, but we will verify this soon.)
Next we would like to show that some twist of $\mathcal Y_{j}$ contains a rational point $y_j$ mapping to $v$.
We repeat exactly the same construction that we made above for $\overline{\mathcal P}_{k}$. The result is an \'etale cover of $\mathcal{V}^{\mu}$ that admits a rational point mapping to $v$ and after base change to $\overline{F}$ is an open subset of $\overline{\mathcal{Y}}_{j}$. We deduce that there is a twist $\mathcal{Y}_{j}^{\sigma}$ admitting a rational point whose image is $v$. Furthermore, the argument shows that the corresponding fiber $\mathcal{Y}^{\sigma}_{j,t_{j}}$ of $\mathcal{Y}_{j}^{\sigma} \to T_{j}^{\sigma}$ is geometrically birational to $Y'$. Since $\mathcal{Y}^{\sigma}_{j,t_{j}}$ is induced by $\widetilde{\Xi}_j$ corresponding to $Y'$, they are actually birational over the ground field.
To finish the proof of Lemma \ref{lemm: finitelymanycoversovernf} (6) we must prove a factoring property over $F$. Recall that we have constructed a rational point $\widetilde{t} \in \widetilde{T}$ and a rational point $y_{j} \in \mathcal{Y}_{j}^{\sigma}$ such that the image of $y_{j}$ in $\mathcal{U}$ is the same as the image of $\widetilde{t}$ under the rational map $\widetilde{T} \dashrightarrow \mathcal{U}$ given by the composition of the rational section to $\widetilde{\mathcal{Y}}_{\widetilde{T}}$ and the map to $\mathcal{U}$. Let $\widetilde{T}^{\dagger}$ be the open subset where the rational map to $\mathcal{U}$ is defined. Consider the base change
\begin{equation*}
\xymatrix{
\widetilde{T}^{\dagger} \times_{\mathcal U} \mathcal Y_{j}^\sigma \ar@{>}[r]\ar@{>}[d]& \mathcal Y_{ j}^\sigma \ar@{>}[d]\\
\widetilde{T}^{\dagger} \ar@{>}[r] & \mathcal U}.
\end{equation*}
Since $\mathcal{Y}_{j}^{\sigma} \to \mathcal{U}$ is \'etale on a neighborhood of the image of $\widetilde{t}$ and admits a rational point mapping to the image of $\widetilde{t}$, there is a component $T^{\nu}$ of $\widetilde{T} \times_{\mathcal U} \mathcal Y_{j}^\sigma$ which maps dominantly to $\widetilde{T}^{\dagger}$ and admits a rational point $t^{\nu}$ mapping to $\widetilde{t}$. Furthermore, the base change $\widetilde{\mathcal Y}_{T^\nu}$ is smooth at any point of the fiber over $t^\nu$. This construction of a base change is as same as the construction in Lemma~\ref{lemm: finitelymanycoversinduction} and Corollary \ref{coro: finitelymanycoversbasechange}, hence after base changing to $\overline{F}$ the map $\widetilde{\mathcal Y}_{T^\nu}\to X$ factors rationally through the twist $\mathcal Y_{j}^\sigma$.
We claim that the map $\widetilde{\mathcal Y}_{T^\nu}\to X$ factors rationally through $\mathcal Y_{j}^\sigma$ over the ground field. Indeed, by the lifting property over $\overline{F}$ one may find a rational map $\overline{h} : \overline{\widetilde{\mathcal Y}_{T^\nu}} \dashrightarrow \overline{\mathcal Y_{j}^\sigma}$ mapping $(y', t^\nu)$ to the point $y_j$ constructed above.
Let $s$ be an element of the Galois group. Then both $\overline{h}$ and $\overline{h}^s$ are lifts of the same map to $\mathcal U$ and they both map $(y', t^\nu)$ to $y_j$. Thus $\overline{h} = \overline{h}^s$ by the uniqueness of the lift. Thus our assertion follows.
\end{proof}
Finally we prove our main theorem.
\begin{proof}[Proof of Theorem \ref{theo: precisetheorem}:]
As mentioned before $Z_{0}$ and $Z_{3}$ are contained in proper closed subsets of $X$, so it suffices to consider only $Z_{1}$ and $Z_{2}$.
\textbf{Construction of a closed set:}
Let $V$ be the proper closed subset and $p_{i}: \mathcal{U}_{i} \to W_{i}$ be the projective families from Theorem \ref{theo: rigidfamilies} equipped with surjective evaluation maps $s_{i}: \mathcal{U}_{i} \to X$.
Suppose that $\mathcal U_i$ is not geometrically irreducible.
Then the Zariski closure $\overline{s_i(\mathcal U_i(F))}$ is a proper closed subset of $X$ where $s_i : \mathcal U_i \rightarrow X$ is the evaluation map. We enlarge $V$ by adding this proper closed subset to $V$.
Suppose that $\mathcal U_i$ is geometrically irreducible.
Let us further suppose that the evaluation map $s_i : \mathcal U_i \rightarrow X$ is birational.
After applying a resolution, we may assume that $\mathcal U_i$ is smooth.
Let $W_i^\circ$ be a Zariski open locus so that $p_i : p_i^{-1}(W_i^\circ) \rightarrow W_i^\circ$ is a good family of adjoint rigid varieties. Let $Q_i$ be the closed subset associated to this family and define $\mathcal V_i = p_i^{-1}(W_i^\circ) \setminus Q_i$. We enlarge $V$ by adding the proper closed subset $s_i(\mathcal U_i \setminus \mathcal V_i) \cup s_i(E_i)$ where $E_i$ is the ramification divisor of $s_i$. By applying Lemma~\ref{lemm: finitelymanycoversovernf} to $p_i : \mathcal U_i \rightarrow W_i$, we obtain families $q_{i, j} : \mathcal Y_{i, j} \rightarrow T_{i, j}$ with morphisms $s_{i, j} : \mathcal Y_{i, j} \rightarrow X$.
We shrink $W_i^\circ$ if necessary so that Lemma~\ref{lemm: finitelymanycoversovernf} (5) is valid for any closed point on $T_{i, j}^{\circ \sigma}$.
We enlarge $V$ by taking the union with $s_i(p_i^{-1}(W_i\setminus W_i^\circ))$ and $C_{i}$ from Lemma~\ref{lemm: finitelymanycoversovernf} for every $i$.
\textbf{Construction of a thin set:}
We now construct a thin set $Z' \subset X(F)$. The construction involves several steps. First set $Z' = V(F)$.
If $\mathcal{U}_{i}$ is geometrically integral and the evaluation map $s_i : \mathcal U_i \to X$ has degree $> 1$,
then we add $s_i(\mathcal U_i(F))$ to $Z'$.
Suppose that $\mathcal{U}_{i}$ is geometrically integral and $s_i$ is birational.
As $\sigma$ varies over all $\sigma \in H^1(F, \mathrm{Aut}(\overline{\mathcal Y}_{i, j}/\overline{X}))$ such that
\begin{equation*}
(a(X, L), b(X, L)) \leq (a(\mathcal Y_{i, j}^\sigma, (s_{i,j}^\sigma)^*L), b(F, \mathcal Y_{i, j}^\sigma, (s_{i, j}^\sigma)^*L))
\end{equation*}
and the map is face contracting we add the set
\begin{equation*}
\bigcup_{\sigma} s_{i,j}^\sigma(\mathcal Y_{i,j}^\sigma (F)) \subset X(F)
\end{equation*}
to $Z'$. We repeat this process for each of the finitely many $\mathcal{Y}_{i,j}$. Since the $\mathcal{Y}_{i,j}$ are geometrically integral by Lemma~\ref{lemm: finitelymanycoversovernf} (1), $Z'$ is contained in a thin set of $X(F)$ by Theorem~\ref{theo:twists}. We show that $Z_{1}$ and $Z_{2}$ are contained in $Z'$.
\textbf{The set $Z_{1}$:} Assume that $f: Y \rightarrow X$ is a thin map such that $Y$ is smooth and geometrically integral, $d(Y,f^{*}L) < d(X,L)$, and
\begin{equation*}
(a(X,L),b(F,X,L)) \leq (a(Y,f^{*}L),b(F,Y,f^{*}L)).
\end{equation*}
We would like to show that for any rational point $y \in Y(F)$ the image $f(y) \in Z'$. We may assume that $f(y) \not\in V$ since otherwise the statement is clear.
This condition implies that $a(Y,f^{*}L) = a(X,L)$.
We will next perform several constructions improving the properties of $Y$ so that we may apply Lemma \ref{lemm: finitelymanycoversovernf} (6). Let $\phi : Y \dashrightarrow B$ be the canonical map for $K_{Y} + a(Y, f^*L) f^*L$. After replacing $Y$ by a birational model (and taking any preimage of $y$), we may assume that the canonical map is a morphism. If the $a$-values of the images of the general fibers of $\phi$ were larger than $a(X,L)$ then we would have $f(Y) \subset V$, so by assumption we must have an equality instead. Similarly, since the fibers of the canonical map for $Y$ are adjoint rigid with respect to $f^{*}L$, their images also must be adjoint rigid with respect to $L$ by Lemma \ref{lemm:dominantequalitycase}. Thus $B$ admits a rational map $g : B \dashrightarrow W_i$ for some $i$. After some birational modification (and again taking a preimage of $y$), we may assume that this rational map is a morphism. Then $f:Y \rightarrow X$ rationally factors through $f' : Y \dashrightarrow B \times_{W_i} \mathcal U_i \rightarrow \mathcal U_i$.
After again replacing $Y$ by a birational model and replacing $y$ by any preimage, we may suppose that $f'$ is a morphism and is generically a map of good families of adjoint rigid varieties.
We may assume that $\mathcal U_i$ is geometrically irreducible and $s_i : \mathcal U_i \rightarrow X$ is birational as otherwise the desired containment of rational points is clear.
We may now apply Lemma~\ref{lemm: finitelymanycoversovernf} (6) to see that there exists $j$ and a twist $\sigma$ such that
\[
f(y) \in s_{i, j}^\sigma(\mathcal Y_{i, j}^\sigma(F)),
\]
and $f : Y \rightarrow X$ factors through $s_{i,j}^{\sigma}: \mathcal Y_{i, j}^\sigma \to X$ after an Iitaka base change.
It only remains to verify
\begin{equation*}
(a(X, L), b(F,X, L)) \leq (a(\mathcal Y_{i,j}^{\sigma}, (s_{i,j}^{\sigma})^*L), b(F, \mathcal Y_{i,j}^{\sigma}, (s_{i,j}^{\sigma })^*L)),
\end{equation*}
and if equality is achieved then $s_{i,j}^{\sigma}$ is face contracting. By the construction we know that the $a$-values are the same.
We first show that $b(F,Y,f^{*}L) \leq b(F, \mathcal Y_{i,j}^{\sigma}, (s_{i,j}^{\sigma })^*L)$.
Let $c \in B$ be a general closed point. By applying Lemma \ref{lemm: monodromyandbvalue} (3) we obtain
\[
b(F, Y, f^*L) \leq b(F, Y_{c}, f^{*}L).
\]
Since by assumption $f(y) \not \in V$ and $Y_{c}$ is general, $Y_{c}$ will map to a fiber of $\mathcal{Y}_{i,j}^{\sigma}$ lying over $T_{i,j}^{\sigma \circ}$. Let $t \in T_{i, j}^{\sigma \circ}$ denote the image of $c$ and let $\mathcal{Y}^{\sigma}_{i,j,t}$ denote the corresponding fiber of $q_{i,j}$. By construction every geometric component of $Y_{c}$ is birational to a geometric component of $\mathcal{Y}^{\sigma}_{i,j,t}$. Thus Lemma~\ref{lemm:birfaceinv} shows that the $b$-values of these two varieties with respect to $L$ are the same. Finally, by Lemma~\ref{lemm: finitelymanycoversovernf} (5) we have an equality $b(F, \mathcal Y^{\sigma}_{i,j}, s^{\sigma*}_{i, j}L) = b(F, \mathcal Y^{\sigma}_{i,j,t}, s^{\sigma*}_{i, j}L|_{\mathcal{Y}^{\sigma}_{i,j,t}})$. Together these inequalities show the desired statement.
Finally suppose that
\[
(a(X, L), b(F, X, L)) = (a(\mathcal Y_{i,j}^{\sigma}, (s_{i,j}^{\sigma})^*L), b(F, \mathcal Y_{i,j}^{\sigma}, (s_{i,j}^{\sigma })^*L))
\]
holds. Note that we assume $d(\mathcal{Y}^{\sigma}_{i,j},s_{i,j}^{\sigma*}L) = d(Y,f^{*}L) < d(X,L)$. Thus
\begin{align*}
\dim(W_{i}) = \dim(T^{\sigma}_{i,j}) & = \dim(\mathcal{Y}^{\sigma}_{i,j}) - d(\mathcal{Y}^{\sigma}_{i,j},s_{i,j}^{\sigma*}L) \\
& > \dim (X) - d(X,L) = \kappa(X,K_{X} + a(X,L)L).
\end{align*}
By applying Lemma~\ref{lemm: facecontractingcondition} to $s_{i,j}^{\sigma}$ and using the birational equivalence of $X$ and $\mathcal{U}_{i}$ we see that $s_{i,j}^{\sigma}$ is face contracting.
\textbf{The set $Z_{2}$:} Assume that $f: Y \rightarrow X$ is a thin map such that $Y$ is smooth and geometrically integral, $d(Y,f^{*}L) = d(X,L)$, and either
\begin{equation*}
(a(X,L),b(F,X,L)) < (a(Y,f^{*}L),b(F,Y,f^{*}L))
\end{equation*}
or equality is achieved and $f$ is face contracting. We would like to show that $f(Y(F)) \subset Z'$.
The argument is essentially the same as for the set $Z_{1}$. The only difference is the case when the $a$ and $b$ values are equal. In this situation, we must show that if $f: Y \to X$ is face contracting then the map $s_{i,j}^{\sigma}: \mathcal Y_{i,j}^{\sigma} \to X$ is also face contracting. Let $\widetilde{Y}$ be a smooth birational model of an Iitaka base change of $Y$ such that $\widetilde{f} : \widetilde{Y} \rightarrow X$ factors through $s_{i, j}^\sigma : \mathcal Y_{i, j}^\sigma \rightarrow X$. Lemma \ref{lemm: iitakabasechangeandbvalue} gives a surjection $\mathcal{F}_{\widetilde{Y}} \to \mathcal{F}_{Y}$. Thus it suffices to show that the map $\mathcal{F}_{\widetilde{Y}} \to \mathcal{F}_{\mathcal Y_{i,j}^{\sigma}}$ is injective.
As before choose a general fiber $\widetilde{Y}_{c}$ of the canonical map for $\widetilde{Y}$ and let $\mathcal Y_{i,j,t}^{\sigma}$ denote the corresponding fiber of the family map for $\mathcal Y_{i,j}^{\sigma}$. Recall that by assumption $\widetilde{f}(\widetilde{Y})$ is not contained in $V$.
Since $\widetilde{Y}_{c}$ is general, every geometric component of $\widetilde{Y}_{c}$ is birational to a geometric component of $\mathcal{Y}^{\sigma}_{i,j,t}$ and the image $t$ of $c$ is contained in $T_{i,j}^{\sigma \circ}$.
Consider the maps
\begin{equation*}
\xymatrix{
\mathcal{F}_{\widetilde{Y}_{c}} \ar@{>}[r] \ar@{>}[d]& \mathcal{F}_{\widetilde{Y}} \ar@{>}[d]\\
\mathcal{F}_{\mathcal Y_{i,j,t}^{\sigma}} \ar@{>}[r] & \mathcal{F}_{\mathcal Y_{i,j}^{\sigma}}
}
\end{equation*}
The arrow on the left is an isomorphism by Lemma~\ref{lemm:birfaceinv}. By Lemma~\ref{lemm: finitelymanycoversovernf} (5) the arrow on the bottom is an isomorphism and by Lemma \ref{lemm: monodromyandbvalue} (3) the arrow on the top is surjective. This implies that all the arrows are isomorphisms. Thus our assertion follows.
\end{proof}
\bibliographystyle{alpha}
|
1,108,101,565,708 | arxiv |
\section{Introduction}
The neutrino flavor eigenstates are linear combination of the mass eigenstates with the unitary matrix, Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. In the PMNS matrix, the three mixing angles ($\theta_{12}$, $\theta_{23}$, $\theta_{13}$ ) can be measured experimentally. The Daya Bay collaboration has published the first non-zero $\theta_{13}$ results with a significance of 5.2 standard deviations in 2012 \cite{dybPRL, dybCPC}.
The survival probability for electron antineutrino in vacuum with an energy $E$ at a distance $L$ is given by
\begin{equation}
P_{ee} = 1- \cos^4 \theta_{13} \sin^2 2\theta_{12} \sin^2 \Delta_{21} - \sin^2 2\theta_{13}( \cos^2 \theta_{12} \sin^2 \Delta_{31} + \sin^2 \theta_{12} \sin^2 \Delta_{32} ),
\end{equation}
where $ \Delta_{ij} \equiv 1.267 \Delta m^2_{ij}(\rm eV^2)\frac{L(\rm m)}{E( \rm MeV)}$, and $\Delta m^2_{ji}$ is the squared-mass difference between the mass eigenstates.
By measuring $P_{ee}$ from the reactor antineutrinos, the mixing angle $\theta_{13}$ and the oscillation function, defined as $\sin^2 \Delta_{ee} \equiv \cos^2 \theta_{12} \sin^2 \Delta_{31}+\sin^2 \theta_{12} \sin^2 \Delta_{32} $, can be determined.
In this talk, we shall present a precision measurement of $\sin^2 2\theta_{13}$ and the first measurement of $| \Delta m^2_{ee}| $ with 217 days of data at Daya Bay.
\section{Experimental Setup}
The Daya Bay experimental site is located at the southern part of China near the Shenzhen city. The Daya Bay nuclear power complex consists of six reactor cores providing a total of up to 17.4 GW thermal power. There are three underground experimental halls (EHs), two near halls and one far hall. For near halls, each hall contains two detectors for determining the reactor neutrino flux; for far site, there are four detectors for measuring the neutrino oscillations. The baselines had been optimized.
The antineutrino detectors (ADs) are functionally-identical. Each one consists of three zones. Figure~\ref{fig:detectors} (left panel) is a cross-sectional view of an AD. The inner zone is the antineutrino target which contains 20 tons of Gd-doped liquid scintillator. The middle zone contains 20 tons of liquid scintillator for detecting the gammas escaping from the target volume. The outer zone contains 40 tons of minimal oil for shielding the radioactive background. There are 192 PMTs in each AD and three automatic calibration units (ACUs) on the top of the detector. Each ACU has an LED and three sources ( $^{68}Ge$, $^{60}Co$ and $^{241}Am-^{13}C$).
The muon tagging system consists of the water pool and the four-layer RPCs. The antineutrino detectors are merged in the water pools. The efficiency of the muon tagging is greater than 99$\%$. Figure~\ref{fig:detectors} (right panel) is a schematic of a near site experimental hall.
\begin{figure}[htb]
\centering
\subfigure{
\includegraphics[height=2.5in]{AD.png}}
\hspace{0.5 in}
\subfigure{
\includegraphics[height=2.5in]{expHall.png}}
\caption{Left: The cross-sectional view of an antineutrino detector. Right: Schematic of a near site experimental hall.}
\label{fig:detectors}
\end{figure}
\section{Analysis}
\subsection{Event Selection}
First of all, the instrumental background from the spontaneous light emission by PMTs ( ``flasher" ) is removed from the data. We then remove the muon background. The IBD candidates are selected by the prompt-delayed coincident signals. The prompt signals are between 0.7 MeV and 12 MeV and the delayed signals are between 6 MeV and 12 MeV, while the time separation of two signals is between 1 $\mu s$ and 200 $\mu s$. Finally, we apply the multiplicity cut to remove the ambiguities in the IBD pair selection. One of the multiplicity cut used in our analysis requires no additional prompt-like signals 400 $\mu s$ before the delayed event, and no delayed-like signals 200 $\mu s$ after the delayed event. The other produces consistent result.
\subsection{Backgrounds}
We consider five sources of background. In the antineutrino sample, the accidental background is the largest one. The rate and the spectrum of the accidental background have been determined by measuring the singles rates of prompt- and delayed-like signals. It gives 0.3$\%$ relative uncertainty. The other four are all significantly smaller, but they are correlated backgrounds. The fast neutron and $\beta$-n decay of the cosmogenic $^9Li/^8He$ provides the prompt- and delayed-like signals. Besides, the gamma emission from the neutron capture also affects the IBD detections, which can come from the calibration source $^{241}Am-^{13}C$ and the $^{13}C(\alpha, n)^{16}O$ background.
\subsection{Energy Response}
The energy response is non-linear between the particle true energy and the reconstructed energy. This non-linearity relation, $f$, is caused by both scintillator and electronics effects; the former is related to the quenching effect and Cherenkov light emission, the latter is related to the charge collection of the front-end electronics. The energy response model is defined as $f = f_{scint} \times f_{elec}$.
For electrons, the empirical model of scintillator nonlinearity is described by $ f_{scint}(E_{true}) = E_{vis}/E_{true} = (p_0 + p_3 \cdot E_{true})/(1+ p_1 \cdot e^{ - p_2 \cdot E_{true} } ) $. The response models of gamma and positrons are connected to the electron model through the Monte-Carlo method. The electron non-linearity model is an empirical exponential function. There exist different energy response models based on different methodologies. The details are given in \cite{dybArxiv}.
The gamma sources and $^{12}B$ spectrum are used to constrain the non-linearity parameters. Finally, positron energy response models obtained from different methods are consistent with each other to $\sim1.5\%$.
\begin{figure}[htb]
\centering
\subfigure{
\includegraphics[height=1.8in]{RateAndShape.png}}
\hspace{0.1 in}
\subfigure{
\includegraphics[height=1.6in]{LE.png}}
\caption{Left: Allowed regions for the $\sin^2 2\theta_{13}$ and $|\Delta m^2_{ee}|$ at the 68.3$\%$, 95.5$\%$ and 99.7$\%$ confidence levels. Right: The survival probability of the electron antineutrino as a function $L_{eff}/E_\nu$, with the best estimate of the detector response. Here $L_{eff}$ is the effective detector-reactor distance, and $E_{\nu}$ is the neutrino energy inferred from the background-subtracted positron energy spectrum.}
\label{fig:results}
\end{figure}
\section{Conclusion}
The best-fit results for the electron antineutrino oscillation frequency and the value of $\sin^2 2\theta_{13}$ are $| \Delta m^2_{ee}| = 2.59^{+0.19}_{-0.20} \cdot 10^{-3} $eV$^2$ and $\sin^2 2\theta_{13}=0.090^{+0.008}_{-0.009}$ , respectively with a $\chi^2$/NDF of 162.7/153. Under the assumption of normal (inverted) neutrino mass hierarchy, we obtain $| \Delta m^2_{32}| = 2.54^{+0.19}_{-0.20} \cdot 10^{-3} \rm eV^2$ ($| \Delta m^2_{32}| = 2.64^{+0.19}_{-0.20} \cdot 10^{-3} \rm eV^2$). This result is consistent with $| \Delta m^2_{\mu \mu}| $ measured by MINOS \cite{minosPRL}. In Figure~\ref{fig:results}, the left panel shows the allowed regions of 68.3$\%$, 95.5$\%$ and 99.7$\%$ C.L. in the $\sin^2 2\theta_{13}$ vs. $|\Delta m^2_{ee}|$ plane. The right panel shows the comparison of the IBD data with the survival probability by applying the best-fitted values of $\sin^2 2\theta_{13}$ and $|\Delta m^2_{ee}|$ \cite{dybArxiv}.
The total uncertainty is dominated by the statistics. For $\sin^2 2\theta_{13}$, the most significant contributions to the systematic uncertainty are from the reactor, relative detector efficiency and the energy scale. The systematic uncertainty of $|\Delta m^2_{ee}|$ is dominated by the relative energy scale and efficiency. The above results will be improved with the higher statistics and the eight-detector measurement. The measurement of the absolute reactor neutrino flux and other studies taking advantage of Daya Bay detector capabilities will be performed as well.
\bigskip \bigskip \begin{center} \begin{large
The Daya Bay experiment is supported in part by the Ministry of Science and Technology of China, the United States
Department of Energy, the Chinese Academy of Sciences, the National Natural Science Foundation of China, the
Guangdong provincial government, the Shenzhen municipal government, the China Guangdong Nuclear Power Group,
Shanghai Laboratory for Particle Physics and Cosmology, the Research Grants Council of the Hong Kong Special
Administrative Region of China, University Development Fund of the University of Hong Kong, the MOE program
for Research of Excellence at National Taiwan University, National Chiao-Tung University, NSC fund support from
Taiwan, the U.S. National Science Foundation, the Alfred P. Sloan Foundation, the Ministry of Education, Youth and
Sports of the Czech Republic, the Czech Science Foundation, and the Joint Institute of Nuclear Research in Dubna,
Russia. We thank Yellow River Engineering Consulting Co., Ltd. and China railway 15th Bureau Group Co., Ltd.
for building the underground laboratory. We are grateful for the ongoing cooperation from the China Guangdong
Nuclear Power Group and China Light $\&$ Power Company.
|
1,108,101,565,709 | arxiv | \section{Introduction}
Cortical morphology is a useful imaging-based biomarker for a range of applications, including aging and disease. Measures that describe the shape and folding of a brain, such as gyrification and cortical thickness, are being used both over the entire cortex and locally to identify differences in individuals and cohorts of subjects~\cite{Chaudhary2020,Frangou2021,Galovic2019,Libero2019}. Such measures are often studied separately, without taking their interaction into account.
Only recently, a universal scaling law describing the interaction of cortical morphology measures has been proposed~\cite{Mota2015}. This scaling law captures the folding of the cortex as the covariance between the measure of average cortical thickness $T$, total pial surface area $A_t$ and exposed surface area $A_e$ in the equation
\begin{equation}\label{ScalingLaw}
A_t \sqrt{T} = k A_e^{1.25},
\end{equation}
where $k$ is a constant. Empirically, $k$ varies slightly between age groups, and has been interpreted as the tension/pressure applied to the cortical surface ~\cite{Wang2016}. This relation has been shown to hold for different mammalian species~\cite{Mota2015}, in human hemispheres~\cite{Wang2016}, and within human cortices across the lobes~\cite{Wang2019}. It says that the three variables are interdependent. For example, if two brains of a similar age have the same total area, but one is thicker, it will also have more exposed area - and hence less gyrification.
The scaling law can be visualised by the analogy of crumpling a piece of paper into a ball: the thickness of the paper, its total area and the force used to crush it determine the exposed surface area of the ball~\cite{Mota2015}.
The scaling law additionally allows us to derive three linearly independent, interpretable morphological variables as linear combinations of $\log T^2$, $\log A_t$ and $\log A_e$~\cite{Wang2021}:
\begin{equation}\label{K}
K = \log A_t + \frac{1}{4} \log T^2 - \frac{5}{4} \log A_e,
\end{equation}
\begin{equation}\label{I}
I = \log A_t + \log T^2 + \log A_e,
\end{equation}
\begin{equation}\label{S}
S = \frac{3}{2} \log A_t - \frac{9}{4} \log T^2 + \frac{3}{4} \log A_e.
\end{equation}
$K$ is derived directly from Eq.~(\ref{ScalingLaw}) by taking logarithms and arranging for $\log k$. It is a measure of tension/pressure acting on the cortex, or, in the analogy, pressure applied to the paper ball. $I$ describes the overall size of the brain or paper ball. Changes in $I$ correspond to an isometric scaling of the cortex. Lastly, $S$ is the inner product of $K$ and $I$ that contains information about shape; it can be thought of as the ``folding technique'' of the paper ball. Using $K$, $I$, and $S$ avoids covarying variables, whilst keeping them interpretable. Otherwise, analysing cortical thinning without accounting for surface area changes could miss signs of atrophy that can be discovered when accounting for the covariance~\cite{Wang2021}.
However, one distinct challenge remains in regionalising these independent measures of cortical morphology, $K$, $I$, and $S$. It is clear that the scaling law cannot hold for arbitrarily small regions. The goal of this paper is to show that the scaling law still holds for areas smaller than lobes, defined independently of them. We first develop a method to find the raw variables $T$, $A_t$, and $A_e$ in small patches across the cortex, and correct the surface areas so they are independent of the size chosen for patches. This allows us to compare the folding within the regions to that of other partitions of the cortex or even the full hemisphere~\cite{Wang2019}.
If local cortical folding follows the scaling law, it justifies the use of $K$, $I$, and $S$ for further morphological analyses. For example, we will show that they can be used to get a better understanding of regional age-related cortical atrophy.
\section{Methods}
\subsection{MRI data and processing}
To asses the rules of local cortical folding in a cohort of healthy subjects in a large age range, we used the NKI Rockland Sample data (\url{http://fcon_1000.projects.nitrc.org/indi/pro/nki.html} \cite{Nooner2012}).
The MRIs were preprocessed with the FreeSurfer 6.0 recon-all pipeline to obtain the fine, triangular mesh representing the grey matter surface, as well as the grey matter thickness on each point of the mesh. We then ran the local gyrification index processing stream to obtain the outer smooth (exposed) pial surface.
Out of the 929 subjects for whom data was available, 95 subjects were rejected due to inadequate image quality, motion artifacts or parts of their grey matter missing. We performed visual quality inspections on a random sample of 50 subjects and applied manual corrections where needed. We deemed overall image quality and FreeSurfer processing to be adequate. The remaining sample consists of 834 subjects, 508 female and 326 male, in an age range from 6 to 85 years. A full list of subjects used can be found on github (\url{https://github.com/KarolineLeiberg/folding_pointwise}).
\subsection{Extraction of local morphological measures}
The following describes the method in which morphological measures were computed for each point on the pial surface. The full code is available on github: \url{https://github.com/KarolineLeiberg/folding_pointwise}.
The pial surface and smooth pial surface are reduced to 5\% and 10\% of their original resolution respectively, using the downsampling function of the MATLAB iso2mesh toolbox. This retains the main features of cortical folds, but reduces the number of vertices in the pial to about 7000 per hemisphere.
The thickness map is converted from the original pial surface to the downsampled pial by assigning the value of the nearest pial vertex for each downsampled pial point. The pial thickness is subsequently transformed from being a pointwise measure to a measure for each face in the mesh (\url{https://github.com/cnnp-lab/CorticalFoldingAnalysisTools}).
Then, for each point $p$ on the downsampled pial surface, a local patch around $p$ is defined as all vertices within a radius of 25mm, measured as Euclidean distance, that are directly connected to $p$ through neighbouring points which are also in the patch - this is to avoid e.g. disconnected patches across two gyri that are very close to each other, but the connecting sulcus is not included. A visualisation of the process can be found in Figure ~\ref{fig1} a-c. A corresponding smooth patch is defined by first finding the nearest downsampled pial point for each downsampled smooth pial point, and then including smooth pial points if their neighbouring pial point is in the patch. (Fig.~\ref{fig1} d).
\begin{figure}[htp]
\centering
\includegraphics[scale=0.55]{fig2.pdf}
\caption{Defining a patch on the pial around a point. \textbf{a-c} Extracts of the process of adding neighbouring points to the patch. \textbf{c} The final pial surface patch around this point. \textbf{d} The corresponding patch on the smooth pial.} \label{fig1}
\end{figure}
The total surface area $A_t^p$ for the patch around $p$ is then computed as the sum of all triangular faces of the downsampled pial which are contained in $p$'s patch. Similarly, the average cortical thickness $T^p$ is the average thickness across those faces. The exposed surface area $A_e^p$ is computed as the sum of face areas within the patch on the downsampled smooth pial.
\sloppy A convex hull is fitted over $p$'s downsampled smooth pial patch, and the integrated Gaussian Curvature of the patch around $p$ is approximated as the sum of Gaussian curvatures of all points on the convex hull which do not lie on the edge of the smooth patch (\url{https://github.com/cnnp-lab/CorticalFoldingAnalysisTools}).
Having computed the morphological measures for regions around each point on the downsampled pial surface, the data is converted back to the original, full pial, again using values of the nearest neighbour. This is done so the data can later be projected to the FreeSurfer average subject to compare across subjects.
\subsection{Surface area reconstruction}
As shown in Wang \textit{et al.} \cite{Wang2019}, the size of regions into which a cortex is partitioned fully determines its surface areas and thus its location in the space of $A_t$, $A_e$ and $T$, making it impossible to compare patches to each other and infer whether they align on the plane predicted by the scaling law. Each patch's surface areas are thus reconstructed to what they would be if the patch was a full hemisphere of the same gyrification index, using the proportion of curvature it contains, approximated via its convex hull. This correction is done using the formulas~\cite{Wang2019}
\begin{equation}\label{Atdash}
A_t^{\prime p} = A_t^p * \frac{4 \pi}{I_G^p},
\end{equation}
\begin{equation}\label{Aedash}
A_e^{\prime p} = A_e^p * \frac{4 \pi}{I_G^p},
\end{equation}
where $A_t^p$ and $A_e^p$ are the total and exposed surface areas of patch $p$ before correction, and $A_t^{\prime p}$ and $A_e^{\prime p}$ are the values after correction using the integrated Gaussian curvature $I_G^p$ over $p$. This correction of a patch's surface areas by its integrated Gaussian curvature preserves its average cortical thickness, gyrification index, and the average Gaussian curvatures of its total and exposed areas~\cite{Wang2019}.
Points that lie on particularly flat parts of the cortex or have very small surface areas tend to have convex hulls with integrated curvatures close to zero. In such cases, the correction of the surface areas leads to an overcorrection, inflating the surface area to unreasonably large values. For simplicity, we will exclude these points from our analysis, choosing not to investigate if they also obey the scaling law at this stage. All points within 10mm to the left and right of the midline of the brain and all points with a curvature below 0.16 are excluded. We will discuss later on how future work may be able to investigate these more challenging parts of the cortex, where the Gaussian curvature of the convex hull is not a good representation of the proportion of the patch of interest.
\subsection{Fitting the scaling law within subjects}
We assess if local folding follows the scaling law by regressing within each subject over all patches in the two-dimensional projection of the scaling law, where
\begin{equation}\label{X}
X=\log A_e^{\prime p},
\end{equation}
\begin{equation}\label{Y}
Y=\log (A_t^{\prime p}\sqrt{T^p}).
\end{equation}
Here, $A_t^{\prime p}$ and $A_e^{\prime p}$ are the surface areas after correcting by curvature, $T^p$ is the observed thickness. The slope of the regression is a subject-specific estimation of the exponent of $A_e$ in Eq.~(\ref{ScalingLaw}). We verify if each subject's local folding follows the scaling law by seeing how close the slope is to 1.25.
\subsection{Local age effects}
To quantify the effect ageing has on morphology in the raw measures and in the independent variables, we used the FreeSurfer function mri\textunderscore surf2surf to convert all subjects' morphology maps (i.e. pial surfaces with morphological measures computed at each point) to the same surface space, where they can be compared and analysed as a group. We compute the independent variables $K$, $I$, and $S$ on each point of the pial surface. We then fit linear regression models pointwise across all subjects, including both sex and age as covariates. Points represented in fewer than 10 subjects of either sex were excluded to ensure the regression was representative of the population. We then use the coefficient of the ageing covariate at each point and for each variable as an indicator of the local effect of age-related atrophy on that measure.
\section{Results}
\subsection{Surface area reconstruction by Gaussian curvature breaks down in insula and on midline}
We apply Gaussian curvature corrections (Eq.~(\ref{Atdash}),(\ref{Aedash})) to all points to reconstruct surface areas of patches to what they would be for a full hemisphere, but find that some points have approximately zero integrated Gaussian curvature (Fig.~\ref{fig2}a shows the convex hull over one example patch). When we look across all points of the cortex, we observe a distinct subset of points that display zero curvature (Fig.~\ref{fig2}b). These points are usually located in the insula or particularly deep sulci, where the patch has a small, overall convex, exposed surface, or on the midline, where the exposed surface might be large, but very flat. The midline additionally contains points with very large curvature, where the convex hull over a patch can contain an angle of 90° or more.
Correcting points by extremely small curvatures leads to overcorrections (Fig.~\ref{fig2}c): Whilst most points naturally align in the $X$-$Y$-plane (Eq.~(\ref{X}),(\ref{Y})) with a slope of 1.25 after the surface areas are reconstructed, points with curvatures near zero are being overcorrected to the top-right of the plot. For simplicity, we therefore exclude these points from our further analysis, acknowledging that Gaussian curvature of the convex hull is not a good representation of the proportion of those patches. This affects around 20\% of all points on the pial surface.
\begin{figure}[htp]
\centering
\includegraphics[scale=0.5]{fig1.pdf}
\caption{Gaussian curvature and its effect on reconstructing surface areas in one example subject. \textbf{a} Example of patch with zero curvature. \textbf{b} Distribution of curvature across patches. \textbf{c} Sample of 150 patches plotted in the X-Y-plane, both before (round marker) and after (asterisk) surface area corrections. Yellow lines with slope 1 indicate the shift of points caused by the curvature corrections. Green points are included in further analysis, purple points are excluded due to their low curvature or position relative to the midline. The black line is a regression line through the green points after correction.} \label{fig2}
\end{figure}
\subsection{Local folding follows scaling law}
To verify whether the scaling law still applies in small patches of the cortex, meaning if the measures of $T$, $A_t$, and $A_e$ covary locally as predicted, we fit the scaling law within subjects, estimating the slope between points in $X$ and $Y$ as described in Eq.~(\ref{X}) and (\ref{Y}). We find that the slope for each subject, i.e. the subject-specific estimates of the coefficient of $A_e$ in Eq.~(\ref{ScalingLaw}), are distributed around a mean of 1.23 (Fig.~\ref{fig3}), only slightly lower than the observed coefficient of 1.25 for hemispheres and lobes. This shows that the scaling law still holds at this level of patch sizes, meaning that when looking at small, local regions of the cortex with spherical radii of 25mm, the average thickness, total area and exposed area covary predictably according to Eq.~(\ref{ScalingLaw}). Note that the subjects follow the scaling law without any age or sex corrections, because age differences only affect the scaling law intercept ($K$), but not the slope~\cite{Wang2016}.
\begin{figure}[htp]
\centering
\includegraphics[scale=0.6]{fig3.pdf}
\caption{Distribution of slopes observed in subjects when fitting a regression through points across the cortex in the $X$-$Y$-plane. The black line marks the median of observed slopes.} \label{fig3}
\end{figure}
\subsection{$K$, $I$ and $S$ have additional value when observing ageing effects}
As we have seen, the folding follows the scaling law even locally, which implies that $K$, $I$, and $S$ are theoretically independent. We convert to these variables to see if they add insight to the differences in local morphology.
When we look at the effect of healthy ageing on local morphology, we see a decrease in the raw variables $A_t$, $A_e$, and $T$ at varying rates in most areas of the brain (Fig.~\ref{fig4}~a-c). A decrease in surface areas can be interpreted as a flattening of the cortex, since less surface area within a constant radius indicates less folding. The independent variables show an isometric shrinking of most areas, with an overall loss of tension (Fig.~\ref{fig4}~d-e).
In the raw variables, we do not see systematic differences in how the areas around the upper and lower precentral gyrus are affected by atrophy. We do however see such a difference in the shape term $S$ (Fig.~\ref{fig4}f): $S$ decreases in the upper precentral gyrus with ageing, but the lower part is relatively unaffected. This is one example of the added value from switching to independent variables; we gain information otherwise hidden in the covariance of $A_t$, $A_e$, and $T$.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.5]{fig4.pdf}
\caption{Local effects of healthy ageing in the left and right hemisphere. \textbf{a-c} Raw variables $T$, $A_t$, and $A_e$ (logged). \textbf{d-f} Independent variables $K$, $I$, and $S$.} \label{fig4}
\end{figure}
\section{Discussion}
On a big data set covering a large age range, we have demonstrated that the local folding of the brain follows the universal scaling law, meaning the grey matter thickness, total surface area and exposed surface area in small regions covary according to the same rule as the whole cortex. This result extends previous work done in this area, which had shown that morphology in full hemispheres and lobes adheres to the scaling law, by confirming empirically that the minimum size of regions for which the folding rule applies has a radius of under 25mm.
Our method works well for most of the cortex. However, we had to exclude around 20\% of points on the pial surface from our analysis, predominantly located on the midline and in the insula. For these points, we founds that the Gaussian curvature was not an adequate representation of the proportion of the patch. For future iterations of the method, we plan to improve on this. An alternative way to approximate patch curvature would be to compute the integrated curvature of the patch's smooth pial surface, rather than its convex hull. However, this surface may already contain too much local information of folding for our purpose. Another idea is to inflate the pial surface to a sphere and use the proportion of surface area or curvature of the patch's representation on the sphere for corrections.
Another improvement of the method could be achieved by making the radius defining the patch around each point on the pial adaptive depending on the thickness at that point. This flexible way of patch definitions uses the smallest radius required to find enough exposed surface area at each point, making the morphological map as close to pointwise as possible. We expect the ideal patch radii to be around 20-30mm for human brains (spanning at least one sulcus/gyrus), which is why we chose a value in this range for the fixed-size method.
The local applicability of the scaling law allows us to use it in terms of independent components derived from it, to further analyse local morphology. In our results we have shown local differences in how the cortex is affected by the process of ageing in the independent variables, which were not clear from the raw variables alone. Our method could have other practical applications, such as finding local abnormalities in patient groups compared to a control cohort, indicating regional effects of dysfunction. It might also be used to detect subject-specific abnormalities, indicating areas of the brain that fold atypically.
\subsubsection{Acknowledgements.} KL was supported by the Centre for Doctoral Training in Cloud Computing for Big Data (EP/L015358/1). We thank the members of CNNP lab (www.cnnp‐lab.com) and Bruno Mota for discussions of the method and results.
\bibliographystyle{splncs04}
|
1,108,101,565,710 | arxiv | \section*{Abstract}
It is shown that for any $W$ weakly compact set of a real
Banach space $X$, the set $L_{\infty }(\mu ,W)$ is $N$-simultaneously
proximinal in $L_{\infty }(\mu ,X)$ for arbitrary monotonous norm $N$ in
${\R}^n$.
\section{Introduction}
Throughout this paper, $(X,\|\cdot\|)$ is a real Banach space
and $B_{X^*}$ is the closed unit ball of $X^*$, the dual of $X$, with
$\sigma(X^*,X)$-topology. Let
$(\Omega, \Sigma, \mu)$ be a complete probability space
and $L_\infty(\mu,X)$
the Banach space of all $\mu$-measurable and essentially bounded
functions defined on $\Omega$ with values in $X$ endowed with the
usual norm
$$
\|f\|_\infty=\essp_{s\, \in \, \Omega}
\|f(s)\|
$$
for every $f \in L_\infty(\mu,X)$ (see \cite{Diestel}).
Let $n$ be a positive integer. We say that a norm $N$ in ${\R}^n$
is monotonous if
for every $t=(t_i)_{1\leq i\leq n}$,
$s=(s_i)_{1\leq i\leq n} \in {\R}^n$ such that
$ |t_i|\leq |s_i|$ for $i=1,\ldots,n$ we have
$$
N(t)\leq N(s).
$$
Let $Y$ be a subset of $X$, we say that $y_0\in Y$ is a best $N$--{\em simultaneous
approximation from $Y$ of the vectors
$x_1,\ldots,x_n\in X$} if
$$
N\big(\|x_1-y_0\|,\ldots,\|x_n-y_0\|\big) \leq
N\big(\|x_1-y\|,\ldots,\|x_n-y\|\big),
$$
for every $y\in Y$.
If every $n$-tuple of vectors $x_1,\ldots,x_n\in X$ admits
a best $N$--simultaneous approximation from $Y$, then $Y$ is
said to be {\em $N$--simultaneously proximinal in $X$}.
Of course, for $n=1$ the preceding concepts are just best
approximation and proximinality.
When $Y$ is a reflexive subspace of $X$, it was proved in \cite{P1}
that $L_\infty(\mu,Y)$ is $N$--simultaneously proximinal in $L_\infty(\mu,X)$.
We have also
obtained similar results
in the Banach space $L_1(\mu,X)$ of $X$-valued Bochner
\mu $-integrable functions defined on $\Omega $
(see \cite{Khalil}, \cite{MP}).
Our purpose is to study the $N$-simultaneous proximinality of the set
$L_\infty(\mu,X)$ defined by
$$
L_\infty(\mu,W)=\{g\in L_\infty(\mu,X): g(s)\in W \text{ for } \ a.e. \ s\in \Omega\},
$$
for certain subsets $W$ of $X$".
\section{Preliminaries}
By using the theory of lifting (see, \cite[p. 59]{Ionescu}),
the next remark and lemma will explain how we can pass from measurable functions to continuous functions.
Let $\rho$ be a lifting of $\Sigma$ and $\tau_\rho$ is the associated lifting
topology (a base for this topology is $\{\rho(A)\setminus B: A\in \Sigma, \ B$ is a null set$\}$).
\begin{rem}[{\cite[p.1100]{Khurana}}]\label{rem:1}
For any simple function $f=\sum\limits_{i=1}^n\chi_{A_i}x_i$, where
$x_1,\ldots,x_n\in X$ and $A_1,A_2,\ldots,A_n\in\Sigma$, the simple function
$\overline{f}=\sum\limits_{i=1}^n\chi_{\rho({A_i})}x_i$,
$\overline{f}:(\Omega,\tau_\rho)\longrightarrow(X,\|\cdot\|)$, is continuous
and $f=\overline{f}$ except on a null set $A\subset\Omega$, thus
$f:(\Omega\setminus A,\tau_\rho)\longrightarrow(X,\|\cdot\|)$, is continuous.
\end{rem}
\begin{rem}[{\cite[p.1100]{Khurana}}]\label{rem:2}
Let $f:\Omega\longrightarrow X$ be a bounded and $\mu$-measurable function.
Then there is null set $A\subset\Omega$ such that $f:(\Omega\setminus A,\tau_\rho)\longrightarrow(X,\|\cdot\|)$ is continuous. Also
$$
\|f\|_\infty=\essp_{s\in \Omega}\|f(s)\|=\sup_{s\in \Omega\setminus A}\|f(s)\|
$$
and the function
$
\widetilde{f}:(\Omega\setminus A)\times B_{X^*}\longrightarrow\R
$
$($with product topology on $(\Omega\setminus A)\times B_{X^*})$,
defined by $\widetilde{f}(s,x^*)=x^*(f(s))$ for any $(s,x^*)\in(\Omega\setminus A)\times B_{X^*}$,
is continuous.
\end{rem}
We need a definition and a few previous results.
Let $E$ be a topological space and $C_b(E)$ the Banach space of all real-valued continuous
functions on $E$ with norm-sup topology. Let $\N$ be the set of natural numbers
and $\N^*=\N\cup\{\infty\}$.
\begin{lem}[{\cite[Lemma 1]{Khurana}}]\label{lem:1}
Let $E$ be a topological space, $\{g_n: n\in\N^*\}$
a sequence of uniformly bounded, real-valued continuous on $E$ and
$\phi:E\longrightarrow(\R^{\N^*}$, with $\ell_\infty$-norm$)$, $\phi(x)=\big(g_n(x)\big)_{1\leq n\leq\infty}$. Putting $F=\phi(E)$,
$\overline{F}$ its closure in $\R^{\N^*}$, we get a sequence of uniformly bounded, real-valued continuous functions $(h_n)_{1\leq n\leq\infty}$ on the compact space
$\overline{F}$: for any $y=(y_n)_{1\leq n\leq\infty}$, $h_n(y)=y_n$. Then:
$h_n\longrightarrow h_\infty$ pointwise on $\overline{F}$ iff
$g_n\longrightarrow g_\infty$ weakly in the Banach space $C_b(E)$ with
norm-sup topology.
\end{lem}
\begin{defn}
Let $x_1,\ldots,x_n$ be vectors in $X$,
we say that a sequence
$(y_k)_{k\geq1}$ in $Y\subset X$ is $N$-simultaneously approximating
to $x_1,\ldots,x_n$
in $Y$, if
$$
\lim_{k\rightarrow\infty}
N\big(
\big\|x_1-y_k\big\|,\ldots,
\big\|x_n-y_k\big\|
\big)
=
\inf_{z\in \ Y}
N(\|x_1-z\|,\ldots,\|x_n-z\|).
$$
\end{defn}
\begin{lem}[{\cite[Lemma 1]{MP}}]\label{lem:2}
Let $x_1,\ldots,x_n$ be vectors in $X$,
and let
$(y_k)_{k\geq1}$ be a $N$-simultaneously approximating
sequence
to $x_1,\ldots,x_n$
in $Y\subset X$.
Assume that
$(y_k)_{k\geq1}$ is weakly
convergent to $y_0\in Y$. Then $y_0$
is a best $N$-simultaneous approximation from $Y$ of
$x_1,\ldots,x_n$.
\end{lem}
\section{Main result}
We can finally prove the main result.
\begin{thm}
Let $W$ be a weakly compact subset of $X$. Then $L_\infty(\mu,W)$
is $N$--simultaneously proximinal in $L_\infty(\mu,X)$.
\end{thm}
\begin{proof}
Let $f_1,\ldots,f_n$ be functions in $L_\infty(\mu,X)$, and let
$(g_m)_{m\geq1}\subset L_\infty(\mu,W)$ be a
$N$-simultaneously approximating sequence to
$f_1,\ldots,f_n$ in
$L_\infty(\mu,W)$. We have
$$
\lim_{m\rightarrow\infty}
N\big(
\|f_1-g_m\|_\infty,\ldots, \|f_n-g_m\|_\infty
\big)
=
\inf_{h\in L_\infty(\mu,W)}
N(\|f_1-h\|_\infty,\ldots,\|f_n-h\|_\infty).
$$
For a fixed arbitrary $s\in \Omega$, we have a sequence $(g_m(s))_{m\geq1}\subset W$.
Since $W\subset X$ is weakly compact, then $(g_m(s))_{m\geq1}$ has a weakly convergent subsequence, which again denote by
$(g_m(s))_{m\geq1}$. Let us denote $g_\infty(s)$ its weak limit.
Consider the map
$g_\infty:\Omega\longrightarrow W$ defined as
$$g_\infty(s):=w-\lim\big(g_m(s)\big)$$
i.e. $(g_m(s))_{m\geq1}$ converge to $g_\infty(s)$ in the weak topology of $X$.
Therefore for each $x^*\in X^*$ the numerical function $x^*(g_\infty)$ is
$\mu$-measurable. So $g_\infty$ is weakly $\mu$-measurable.
On the other hand for each $m\in\N$, $g_m$ is $\mu$-essentially separably
valued, i.e., there exists $A_m\in \Sigma$ with $\mu(A_m)=0$ and such that
$g_m(\Omega\setminus A_m)$ is a norm separable subset of $X$.
For each $m$ let us pick a dense and
countable subset, $D_m$, of $g_m(\Omega \setminus A_m)$. Then the set
$$
Y=\overline{\co}\bigg(\bigcup_{m=1}^\infty D_m\bigg)
$$
is norm closed and separable. For every $m\in\N$ and $s\in\Omega\setminus A_m$
we have $g_m(s)\in Y$. Since, $g_\infty(s)$ its weak limit of $(g_m(s))_{m\geq1}$ for a.e. $s\in \Omega$, we obtain that $g_\infty(s)\in Y$ for a.e. $s\in \Omega$. Thus
$g_\infty$ is $\mu$-essentially separably valued. Therefore, the
Pettis Measurability Theorem \cite[p. 42]{Diestel} guarantees that the function
$g_\infty:\Omega\longrightarrow X$ is $\mu$-measurable.
Since $(g_m(s))_{m\geq1}$ is weakly convergent to $g_\infty(s)\in W$ for a.e. $s\in \Omega$,
then $(g_m(s))_{m\geq1}$ is bounded and
$$
\|g_\infty(s)\|\leq\liminf_{m\rightarrow\infty}\|g_m(s)\|
$$
for a.e. $s\in \Omega$. Therefor, $g_\infty\in L_\infty(\mu,W)$.
From the above observations (Remarks \ref{rem:1} and \ref{rem:2}), it follows that there is a null set $A\subset \Omega$ such that the
functions $g_m:(\Omega\setminus A,\tau_\rho)\longrightarrow (X,\|\cdot\|)$ are continuous for $1\leq m \leq\infty$. This mean $$\widetilde{g}_m:(\Omega\setminus A)\times B_{X^*}\longrightarrow\R$$ are continuous for
$1\leq m \leq\infty$ and the set $\{\widetilde{g}_m:m\in \N^*\}$ is
uniformly bounded. Putting $E=(\Omega\setminus A)\times B_{X^*}$ (with product topology defined by lifting topology $\tau_\rho$ and weak star topology on the dual $X^*$ of $X$) and
$(\R^{\N^*},\|\cdot\|_{\ell_\infty})$, where
$$\|(x_m)_{1\leq m\leq\infty}\|_{\ell_\infty}:=\sup\limits_{1\leq m\leq\infty}|x_m|,$$
for any $(x_m)_{1\leq m\leq\infty}\in \R^{\N^*}$.
Consider the map
$\phi:E\longrightarrow\R^{\N^*}$ defined by
$$\phi(s,x^*)=\big(\widetilde{g}_m(s,x^*)\big)_{1\leq m\leq\infty}.$$
Let $F=\phi(E)$ and $\overline{F}$ its closure in $(\R^{\N^*},\|\cdot\|_{\ell_\infty})$. We have the same conditions as Lemma \ref{lem:1}, then
we get a sequence of uniformly bounded, real-valued continuous functions $(h_m)_{1\leq m\leq\infty}$ on the compact Hausdorff space
$\overline{F}$: for any $y=(y_m)_{1\leq m\leq\infty}\in \overline{F}$, $h_m(y)=y_m$.
\vspace{5mm}
Finally, we prove that $h_m\xrightarrow[m\rightarrow\infty]{ } h_\infty$ pointwise on $\overline{F}$.
Let $y=(y_m)_{1\leq m\leq\infty}\in \overline{F}$, there is a sequence
$\big((s_k,x^*_k)\big)_{k\geq1}\subset E$ such that
\begin{align*}
\|\phi(s_k,x^*_k)-y\|_{\ell_\infty}&=\|(\widetilde{g}_m(s_k,x^*_k))_{1\leq m\leq\infty}-(y_m)_{1\leq m\leq\infty}\|_{\ell_\infty}\\
&=\sup_{1\leq m\leq\infty}|\widetilde{g}_m(s_k,x^*_k)-y_m|=
\sup_{1\leq m\leq\infty}|x^*_k(g_m(s_k))-y_m| \xrightarrow[k\rightarrow\infty]{ }0.
\end{align*}
This implies that
$$
|x^*_k(g_m(s_k))-y_m| \xrightarrow[k\rightarrow\infty]{ }0, \ \ \text{for all } 1\leq m\leq\infty.
$$
Since, $g_\infty(s)$ its weak limit of $(g_m(s))_{m\geq1}$ for a.e. $s\in \Omega$, we obtain that
$$
|x^*_k(g_m(s_k))-x^*_k(g_\infty(s_k))|\xrightarrow[k,m\rightarrow\infty]{ }0.
$$
Therefor
\begin{align*}
|h_m(y)-h_\infty(y)|
&\leq|y_m-x^*_k(g_m(s_k))|+|x^*_k(g_m(s_k))-x^*_k(g_\infty(s_k))|+\\
&+|x^*_k(g_\infty(s_k))-y_\infty|\xrightarrow[k,m\rightarrow\infty]{ }0.
\end{align*}
So using the result of Lemma \ref{lem:1}, we get $\widetilde{g}_m\longrightarrow\widetilde{g}_\infty$
weakly. Therefor $g_m\longrightarrow g_\infty$
weakly in $L_\infty(\mu,W)$. Therefore by Lemma \ref{lem:2}, we have $g_\infty$
is a best $N$-simultaneous approximation from $L_\infty(\mu,W)$ of
$f_1,\ldots,f_n$.
\end{proof}
\begin{rem}
Observe that the previous theorem, we prove that: let $f_1,\ldots,f_n$ be a functions in $L_\infty(\mu,X)$. Let
$g:\Omega\longrightarrow W$ be a $\mu$-measurable function such
that $g(s)$ is a $N$-best simultaneous approximation of
$\{f_1(s), \ldots,f_n(s)\}\subset X$ in $W$
for almost all $s\in\Omega$. Then
$g$ is a $N$-best simultaneous approximation of
$\{f_1,\ldots, f_n\}\subset L_\infty(\mu, X)$ in $L_\infty(\mu, W)$.
In general if $W$ is not weakly compact subset of $X$, this result is fails.
\end{rem}
|
1,108,101,565,711 | arxiv | \section{Introduction}
{The merging galaxy \object{NGC 6240} has a large
far-infrared (FIR) luminosity of L$_{\rm FIR}$ = 3.5 $\times$
10$^{11}$ \mbox{L$_{\sun}$}\,\citep{yun02} and belongs to the category of luminous
infrared galaxies (LIRGs; L$_{\rm IR}$ = 10$^{11}$$-$10$^{12}$ \mbox{L$_{\sun}$})
\citep[see][for a review]{sand96}}.
{The large luminosity of these LIRGs and Ultra-Luminous Infrared Galaxies
(ULIRGs; L$_{\rm IR}$ = 10$^{12}$$-$10$^{13}$ \mbox{L$_{\sun}$})} is likely to be dominated by starburst
activity induced by galaxy-galaxy interactions, where the UV radiation from
massive star formation is reprocessed to far-infrared radiation in the
dusty environment \citep[e.g.,][]{sand88, sand96, ski97}.
The \object{NGC 6240} merging system has been described at optical
wavelengths \citep{fri83} and is thought to host an active galactic nucleus (AGN) in its
nuclear region \citep[e.g.,][]{dep86}. The two galactic nuclei found in the central
region have a projected separation between
1$\arcsec$.5 and 1$\arcsec$.8 (0.714$-$0.856 kpc) at X-ray and radio wavelengths
\citep{fri83,bes01,kom03,gall04,max07}. Because the northern nucleus, with the highest
absorbing column densities, lies well behind the southern nucleus \citep{baa07}, the
separation varies as the intervening hot (obscuring) dust
becomes optically thinner at longer wavelengths \citep{max07}.
X-ray observations at 2$-$10 keV reveal the presence of iron line emission
at 6.4 keV at both nuclei, which is most prominent in the southern nucleus
and classifies this as a binary AGN \citep{kom03}.
Thermal CO emission reveals a significant mass
concentration that is centered between the two nuclei \citep{tac99,bry99,nak05,dai07}
and largely consists of interstellar gas \citep{sco00}.
This intervening molecular gas concentration results from a superposition of disk gas from the
two merging galaxies, which is confirmed by the structure and peak location of the OH absorption
structure against the extended radio emission of the nuclear region \citep{baa07}.
Radio observations at centimeter wavelengths of the central 2$\arcsec$ region of
\object{NGC 6240} using the NRAO Very Large Array display the two
nuclei and an extended northern structure (N3) that is part of the large-scale
loop structure to the west of the nuclear source \citep{col94,baa07}.
A large scale diffuse component envelops these three structural components.
The nuclear radio components remain unresolved at higher resolution
at 5 GHz using MERLIN \citep{bes01}. The loop structures including N3
have been resolved away at this resolution and the southern nucleus, N1 shows east-west structure
at its lowest contours (PA = 60$\degr$) and a northern extension (PA = -30$\degr$).
At milli-arcsecond resolution using the NRAO Very Long Baseline Array (VLBA),
the northern nucleus N2 also displays a multi-component east-west structure suggesting
a core with a two-sided jet \citep{gall04}.
The southern nucleus has a peak central structure surrounded by weak
additional components possibly related to starformation. Both high-brightness
nuclei show an inverted spectrum at lower frequency
with a spectral turn-over or flatness due to free-free absorption (FFA) or possibly
synchrotron-self absorption (SSA) \citep{gall04}. {Strong H$_{2}$O$\;$ maser emission was discovered
\citep[e.g.,][]{hagi02} at the exact center of the southern nucleus \citep{hagi03,hagi10},
which may be related to a dense molecular gas around a central engine.}
Here, we present European VLBI Network (EVN) observations with increased
surface brightness sensitivity to image the compact sources at increasing
frequencies. New EVN observations at 1.6, 2.2, 5.0 and 8.4 GHz were made to
study the source structure, the overall spectral energy distribution, and to
check for consistency with the earlier VLBA results.
In Section 2 we present the new observations, the data analysis, and the observational
results, {including the identification and morphology of all detected compact radio
sources.} In Section 3 we discuss the
nature of these sources and compare \object{NGC 6240} with other
active galaxies. Finally, in Section 4 we summarize the discussion.
Cosmological parameters of H$_{0}$ = 73
km s$^{-1}$ Mpc$^{-1}$, $\Omega$$_{\Lambda}$ = 0.73, and $\Omega$$_{M}$ =
0.27 are adopted throughout this article, which results in a luminosity distance of
NGC 6240 of 103 Mpc at z = 0.0245 and an angular conversion of 1 arcsec equaling
476 pc in linear scale. The designation of radio components follows that
defined in \cite{col94} with N1 being the southwestern nucleus and N2 being the northeastern nuclei.
The two nuclei were called respectively S and N1 in \cite{gall04}.
\section{Imaging the double nuclei in NGC 6240}
\subsection{Observations and imaging}
The European Very Long Baseline Interferometry Network (EVN) has been used to
observe the nuclear region of \object{NGC 6240} in order to investigate the nature
of the active galactic nuclei and nuclear starbursts in the galaxy.
The stations used for these observations at two epochs in 2003 and
2009 were Effelsberg (Ef), Hartebeesthoek (Hh), the 76-m Lovell
telescope (Jb) at Jodrell Bank, Medicina (Mc), Noto (Nt), Onsala
(On), Shanghai (Sh), Torun (Tr), Urumqi (Ur), and the Westerbork (Wb)
phased-array. A summary of the observational details is presented in Table~\ref{tbl1}.
All observations were conducted in phase-referencing mode at all frequency bands
using the nearest calibration source J1651+0129 (centered at R.A.(J2000)
= 16$^{\rm h}$51$^{\rm m}$03$^{\rm s}$.6620, Dec.(J2000) =
01$\degr$29$\arcmin$23$\arcsec$.458) at
1.01$\degr$ away from \object{NGC 6240}. Only the Lovell telescope did not
participate in the phase-referencing part of the observations due to its
lower slewing rate.
The first epoch observations at 1.6 GHz and 5.0 GHz were made
on 30 October and 10 November, 2003 and utilized a 32 MHz total bandwidth with 2 bit
sampling in dual-circular polarization mode for a total data recording rate
of 256 Mbit s$^{-1}$. Observations of 2.5$-$3 minutes on the target were
interleaved with observations of 1.5$-$2 minutes on the
phase-referencing source, which gave about 4.2 hours total on-source time.
Second epoch observations at 5.0 GHz, 2.2 GHz, and 8.4 GHz were
made on 16, 17 June, 2009. The observations at 5.0 GHz were made in
dual-circular polarization mode utilizing 64 MHz total bandwidth with 2 bit
sampling for a total data recording rate of 512 Mbit s$^{-1}$. Three
minute scans on the target were alternated with one minute scans on
the phase-referencing source, which resulted in 5.7 hours total on-source
time. The observations at 2.2 and 8.4 GHz were made in single polarization
mode with the S/X dual-frequency receiving system with a bandwidth of
128 MHz at each frequency and 2 bit sampling for a total recording rate
of 512 Mbit s$^{-1}$. Three minute scans on the target and one minute
scans on the phase-referencing source were alternated during the
observation, which resulted in 2.9 hrs total on-source time at each frequency.
All observations were done with the same reference position near the southern
nucleus of NGC 6240 (R.A.(J2000) = 16$^{\rm h}$52$^{\rm m}$58$^{\rm s}$.8903,
Dec.(J2000) = 02$\degr$24$\arcmin$03$\arcsec$.339).
The data were correlated using the EVN MkIV correlator at the Joint
Institute for VLBI in Europe (JIVE) and the output visibility data were
{averaged to} 1-2 seconds. Sixteen frequency channels of 500 kHz
were used across each 8 MHz band.
Data calibration and imaging were carried out in the NRAO Astronomical Image
Processing System (AIPS). After initial data editing, the delay error corrections
resulting from ionospheric effects were applied to all visibility data.
A-priori amplitude calibration was performed using the system
temperature and gain information provided for each telescope. The delays
and fringe rates for the calibration sources \object{3C 273} or \object{3C 345}, and
\object{J1651+0129} were applied to the target source, \object{NGC 6240}.
Using the measured flux densities of the phase-referencing source J1651+0129 at 2 epochs
as a basis, the relative accuracy of the target flux densities is estimated to be $\sim$10\%.
This error estimate includes the effects of bandwidth and time smearing and ionospheric delays,
which are estimated to be several percent for these EVN observations.
Cleaned images of 512 $\times$ 512 pixels were produced, one centered
near the southern nucleus and one at the intensity peak
position of the northern nucleus offset by 0$\arcsec$.5051 in
right ascension and 1$\arcsec$.4279 in declination from the phase
center. The noise levels of 1$-$2 times theoretical noise and
synthesized beams from CLEAN-ed images are summarized in
Table~\ref{tbl2}. Self-calibration could not be applied successfully because of the low
signal-to-noise ratio of the images. Final images have been
phase-referenced to the nearby calibrator \object{J1651+0129}.
{Because of residual errors in the calibration of the phase-reference
sources, the associated structure at the lowest contours ($\lesssim$ 3 $\sigma$ or 4 $\sigma$) in the source images may not be reliable.}
The positions of the radio components and their errors at the peak brightness
at each frequency and both epochs are listed inTable~\ref{tbl3}.
These composite position errors have been estimated from
{the theoretical thermal noise errors of interferometer phase} \citep[e.g.,][]{tho86},
the systematic errors of the
phase-referencing VLBI observations ($\sigma_{\rm phr}$), and
the errors of the absolute position of the phase-referencing source ($\sigma_{\rm ref}$), using the quadrature equation:
$\sqrt{(\theta/2SNR)^2 + \sigma_{\rm phr}^2 + \sigma_{\rm ref}^2}$.
Here $\theta$ represents the synthesized beam size and SNR represents
the signal-to-noise ratio. The values of $\sigma_{\rm phr}$ ($\Delta
\alpha \, $cos$\delta$\,=\,0.060 mas, $\Delta$$\delta$\,=\,0.020 mas) are
obtained from the simulated astrometric accuracy of the EVN array
\citep{pra06}, adopting a separation between calibrator and
target of 1.0$\degr$ at position angle 152.2$\degr$ and at an approximate
source declination of 0$\degr$. The value of $\sigma_{\rm ref}$ is 0.58 (mas) from the
VLBA Calibrator List \citep{kova07}. Table~\ref{tbl4} lists the flux
densities of the radio components at each observing frequency,
which also includes two weaker components appearing in the earlier VLBA observations
\citep[see Figures 2, 3, and 4 in][]{gall04}. Source sizes of these components in
Table~\ref{tbl5} are measured from the final images using the AIPS task JMFIT.
Assuming Gaussian distributions for sources, the upper limits of the source sizes have estimated
accuracies of 3$-$30\% depending on observing frequency and epoch.
Table~\ref{tbl6} presents the properties of the
radio sources detected in our observations.
\subsection{Source identification and morphology}
The new VLBI maps obtained from 1.6$-$8.4 GHz display four compact
high-brightness emission components as identified in Figure \ref{fig1}: the
nuclei, N1(south) and N2(north) identified in earlier interferometric
observations \citep{bes01,hagi03} and two new components RS1 and RS2
near the southern nucleus. The radio source RS1 was first identified southwest of
the southern nucleus at 5.0 GHz in October, 2003 and it is detected at 5.0
and 8.4 GHz in June, 2009 (see Figure~\ref{fig2}). The other radio source
RS2 was identified northeast of the southern nucleus at Epoch 1 and was
more prominent at 1.6 GHz than at 5.0 GHz (see Figure~\ref{fig4}).
At Epoch 2 {the source RS2 was clearly detected} at 5.0
GHz but not at 2.2 GHz with an upper limit flux density of 2.75 mJy beam$^{-1}$.
All diffuse emission in the central 2$\arcsec$ region, {observed in earlier
Very Large Array observations \citep[e.g,][]{col94,baa07}}, has been resolved out.
The VLBA observation at 8.4 GHz of August 1999 did not detect
the two nuclei at a 5$\sigma$ upper limit of 0.65 mJy beam$^{-1}$ \citep{gall04}, but
the northern nucleus was detected at 8.4 GHz in June 2009 with similar
sensitivity, which suggests some intensity variability of the northern nucleus.
Both nuclei were detected at 5 GHz at both Epochs. Their peak positions in
the EVN data (Table~\ref{tbl3}) agree at all frequencies with
earlier VLBA measurements using estimated uncertainties of 0.6 mas in
right ascension and 1.5 mas in declination \citep{gall04}.
The 5.0 GHz maps in Figures~\ref{fig2} and \ref{fig3} display the two nuclear
components N1 and N2 obtained at Epoch 1 and
Epoch 2, respectively. The southeastern nucleus, N1 was not detected at the highest
frequency of 8.4 GHz with an upper limit of 0.55 mJy beam$^{-1}$ (Figure~\ref{fig2}). Individual source
maps indicate that {the sources N1 and N2 are resolved but that the minor axis of
RS2 northeast of N1 is unresolved (Figure~\ref{fig4})}. The peak of RS1 is
separated from N1 by 23.3 mas at 5.0 GHz at a position angle of $\sim$35$\degr$,
corresponding to 11.1 pc at the distance to NGC 6240.
At the lowest frequency of 1.6 GHz, N1 and RS1
are not spatially separated. Also, RS1 was not detected at 2.2 GHz
at Epoch 2 possibly because of the lower signal-to-noise ratio.
The source RS2 is located northeast of the southern nucleus in our 1.6 GHz
EVN map of Figure~\ref{fig4}. It appears that the
peak position of RS2 at 1.6 and 5.0 GHz lies close to the compact component S1 at
0$\arcsec$.3 northeast from N1 in the 1.7 and 2.4 GHz VLBA observations
\citep{gall04}. On the other hand, RS2 has an offset of $\approx$ 10$-$15~mas
at 1.6 GHz or 5.0 GHz relative to S1 at 1.7 GHz or 2.4 GHz, which is
not within the estimated positional errors (Table~\ref{tbl3}).
Therefore, RS2 is a different source to S1 in the earlier VLBA data.
We interpret both components to be part of the
circum-nuclear starforming region that extends over 10 mas or $\sim$5 pc around N1.
The structure of the northeastern nucleus N2 at 5.0 GHz (Figure~\ref{fig3})
shows a northwest extension of $\approx$ 0\arcsec.02 ($\approx$ 9.5 pc) that
is consistent with the earlier VLBA maps at 1.7 and 2.4 GHz \citep{gall04}.
Although {this structure at lower contours} may not be reliable because of
incomplete phase and amplitude calibration, an east-west structure may be
explained as a core-jet structure typically seen in high-luminosity Seyfert nuclei.
The northern nucleus itself remains unresolved at both 1.6 and 2.2 GHz.
Only at the highest frequency of 8.4 GHz, N2 has an east-west elongation that
is consistent with the earlier VLBA data (Figure~\ref{fig3}).
\section{Discussion}
Our EVN data present the detection of the double nuclei at higher
frequencies of 5.0 and 8.4 GHz, where earlier VLBA data failed to
detect these nuclei. These results combined with earlier VLBA studies,
allow study of the nature of the nuclei at milliarcsecond-scale structure.
While our study focuses on the physics of the two nuclei, the detection
of two new components is a key to understanding the activity of NGC 6240.
\subsection{The northern nucleus N2}
The northern nucleus N2 shows an inverted spectrum rather than a steep spectrum
that would be more typical for a radio jet (Figure~\ref{fig5}). The structure of N2 at 5.0 GHz is
partially resolved and shows an east-west extension at its lower
contours that still remains questionable because of the insufficient amplitude
calibration (Figure~\ref{fig3}). Changes in source structure may also result
from a different (u,v) coverage for the two epochs. The detection of the X-ray emission
towards N2 \citep{kom03} is less convincing than the definite detection towards N1.
Nevertheless, the radio morphology of N2 resembles that of a core-jet
structure seen in Seyfert nuclei, such as \object{NGC 1068} \citep{gall04b}, but
this requires further verification.
By contrast with \object{Arp 299}, the VLA observations of the H$_{2}$O$\;$ maser in
NGC 6240 have not detected any emission towards N2 during seven years
\citep{hagi03,hagi10}.
\subsection{The southern nucleus N1}
The southern nucleus N1 appears partially resolved in an east-west
direction (Figure~\ref{fig2}b,2c) but its flux density at 5.0 GHz has
not changed significantly during 6 years. Contrary to the spectrum of N2
(see discussion below), the spectrum of N1
with missing Epoch 2 detections at 2.2 and 8.4 GHz may not be inverted
(Figure~\ref{fig5}), although its spectral shape (including
upper limits) strongly indicates a spectral turnover around 2.0 GHz.
The non-detection of N1 at 8.4 GHz suggests a relatively flat AGN-like
spectrum and {a lower limit for the spectral index $\alpha$ $<$ 0.55
between 5.0 and 8.4 GHz, using S$_{\nu}$ $\propto$ $\nu^{\alpha}$ with
S$_{\nu}$ being the flux density at frequency $\nu$}. In addition, the radio brightness
temperature $>$ 10$^6$ K (Table~\ref{tbl6}) and the detection of strong hard
X-ray emission and neutral iron line emission would argue for the presence
of an AGN in N1. Considering the available evidence, the southern
nucleus may host both an AGN and a (circum-)nuclear starburst region.
The H$_{2}$O$\;$ maser features in NGC 6240 nearly coincide with the
continuum peak of the southern nucleus \citep{hagi03,hagi10}.
The narrow H$_{2}$O$\;$ maser lines are redshifted by $\sim$ 200$-$300 km s$^{-1}\;$
with respect to the systemic velocity of N1 \citep[e.g.,][]{hagi10,baa07}
and may originate in the receding side of a compact rotating molecular disk
at the nucleus \citep[e.g.,][]{miyo95}, which would also support the presence
of an AGN. Alternatively, these narrow lines distributed over 120 km s$^{-1}\;$ could
be explained by wind maser emission as observed in the Circinus galaxy
\citep{linc03}.
The southern nucleus N1 of NGC 6240 has characteristics that are comparable
with those of component A of Arp\,299 \citep{nef04,ulv09}.
The 8.4 GHz radio power of N1 of ($P_{N1}$ = 3.7 $\times$ 10$^{22}$ W Hz$^{-1}$) \citep{col94}
is only a factor of two more than the 8.4 GHz
power of nucleus A of Arp\,299 ($P_A$=1.8 $\times$ 10$^{22}$ W
Hz$^{-1}$ and $P_{B1}$=2.9 $\times$ 10$^{21}$ W Hz$^{-1}$) \citep{nef04}.
The compact source A of Arp\,299 has been resolved into discrete sources at
milliarcsecond-scale by recent EVN observations and one of these sources (A1)
is the AGN candidate with a flat spectral index of -0.13 $\pm$ 0.11 \citep{per10}.
Five bright and compact radio sources lie within 10 pc and are candidates of
Type-II young radio supernovae \citep{nef04}. Four of these five
radio sources have flat or inverted radio spectra between 2.2 and 8.4
GHz, very similar to RS1 and RS2 near N1 in NGC 6240.
The luminous H$_{2}$O$\;$ maser emission in Arp\,299 is found towards both nuclei A
and B1 \citep{hen05} and at A it coincides with its continuum peak at VLA resolution
\citep{tar10}. While this spatial coincidence may also suggest an AGN
association \citep[e.g.,][]{hagi07}, the maser in \object{Arp 299} is blueshifted
relative to the systemic velocity and is likely associated with an outflow in a starforming environment.
\subsection{Spectral analysis of N1 and N2}
In order to understand the observed spectra of the southern and northern
nuclei, model fitting may be used to explain the frequency turn-over at lower
frequencies using a power-law plus pure free-free absorption (FFA) or
synchrotron self-absorption (SSA) \citep[e.g.,][]{kam00}. First,
the FFA model for a nuclear spectrum is described by:
\begin{equation}
\label{ffa}
S_\nu=S_{\rm 0}(\nu/1.0)^{\alpha_0} \exp\{{-\tau_{\rm ff}(\nu/1.0)^{-2.1}\}},
\end{equation}
where $\nu$ is the frequency in GHz, $S_{\rm 0}$ is the unobscured
synchrotron flux density in mJy, $\alpha_0$ is the optically thin
non-thermal spectral index, {and $\tau_{\rm ff}$ is the opacity at 1.0 GHz. }
This model describes all FFA in the foreground to the synchrotron
emission source and assumes that all source components are subject
to the same foreground opacity \citep[e.g.,][]{par07}. The second model of
pure SSA is described as follows:
\begin{equation}
\label{ssa}
S_\nu=S_{\rm 0}\nu^{2.5} [1-\exp\{{-\tau_{s}\nu^{\alpha_0-2.5}\}}],
\end{equation}
where $\nu$ is the frequency in GHz, $S_{\rm 0}$$\tau_{s}$ will be
close to the flux density, if the SSA coefficient $\tau_{s}$$<<$ 1, and
$\alpha_0$ is the optically thin non-thermal spectral index.
The parameters of the FFA and SSA spectral fits for N1 and N2 and the
reduced $\chi^2$ values (per degree of freedom) have been summarized
in Table~\ref{tbl7}, and the fitted curves are shown in Figure~\ref{fig5}.
Both the FFA and SSA models produce reliable fits for N1 with low values ($<$ 1.5)
for the reduced $\chi^2$ with the FFA model being slightly worse but
does not make good fits to N2.
However, since there is no compact source in N1 or N2 having a high
brightness temperature, any synchrotron
self-absorption in these nuclei is unlikely \citep{kel69}. Using conventional
energy equipartition arguments at lower radio frequencies, a turn-over
frequency of 2.0 $-$ 3.0 GHz for a self-absorbed source requires an equipartition brightness
temperature of $\sim$10$^{11.95}$ K \citep{rea94}. The observed
brightness temperatures for our sources are only
$\sim$ 10$^{6}$$-$10$^{7}$ K.
Better constraints of the nuclear properties of N1 in terms of model
fitting requires more complete flux density measurements.
While for N2 neither the pure FFA nor the pure SSA model can perfectly explain the spectral
bending at 2$-$3 GHz (see Table~\ref{tbl7} and Figure~\ref{fig5}), the observed
bending does indicate
that free-free self-absorption by ionized foreground gas in a starburst environment
is relevant. Since there is also no hint of self-absorption in the MERLIN
spectrum \citep{bes01}, the size of such an absorbing medium must be very
small and between about 9.5 and 25 pc in linear size \citep{gall04}.
Similarly, the presence of an AGN in N2 cannot be confirmed from our EVN data,
except that the core-jet like structure, as also seen in the 1.7 and 2.4 GHz VLBA
images, could support the presence of an active nucleus.
\subsection{Radio sources near the southern nucleus RS1, RS2}
The radio source RS1 has a radio power at 8.4 GHz of 4.1 $\times$ 10$^{21}$
W Hz$^{-1}$ and a spectral index of $\alpha_{\rm{5.0-8.4}}$ $\approx$ 0.34 in
our Epoch 2 observations. The radio power of RS2 at 5.0 GHz ($\alpha_{\rm{1.6-5.0}}$ $\approx$ -0.91) is 7.5 $\times$ 10$^{20}$ W Hz$^{-1}$.
The radio power of RS2 is comparable with those of the VLBI sources associated
with the nucleus A in \object{Arp 299} while the power of RS1 is nearly a factor
ten higher, which makes it 1250 times the 8.4 GHz radio power of \object{Cas
A} of 6 $\times$ 10$^{17}$ W Hz$^{-1}$ \citep{col94}.
Both RS1 and RS2 display a radio light curve with a long-term rise (Figure~\ref{fig6}).
The {upper} limit for the source size of RS1 at the highest frequency 8.4 GHz is 1.5
$\times$ 0.6 pc (Table~\ref{tbl5}) and the source is not sufficiently
resolved by our EVN synthesized beam. Also, different EVN beams resulting
from different (u,v) coverage at each epoch make it difficult to make a reliable
comparison of the intrinsic source sizes.
Speculation that RS1 could be the true southern nucleus, instead
of N1, may be ruled out because RS1 was not detected in the 8.4
GHz VLBA observations (1999$-$2001) and only appeared in 2003. Also its
location is significantly offset from the positions of N1 in the 2.4 and 1.7 GHz VLBA data.
Similarly, the possibility that RS1 represents ejecta from the active nucleus N1
may be ruled out because its offset (11 pc) from N1 remained unchanged during
6 years and because RS1 is far from the known outflow structures seen in HI and
OH southwest of N1 \citep{baa07}. The positional difference of 10 mas ($\approx$
50 pc) (see Table~$\ref{tbl3}$) between RS2 at 1.6 and 5.0 GHz and the source S1
at 1.7 and 2.3 GHz in VLBA observations at epochs in 2000 and 2001
suggests that they are also different sources.
The radio sources RS1 and RS2 have remained detectable for more than 6 years since
2003. An important question is whether these sources are supernovae (SNe, interacting with the circumstellar matter) or supernova
remnants (SNRs, interacting with the dense ISM).
There is no evidence for expanding supernovae shells with the highest beam size of $\sim$6 mas ($\sim$3 pc) both
in RS1 and RS2 between the two epochs from our data (see Table~$\ref{tbl5}$). The range of radio luminosity
of 10$^{20.8-21.6}$ W Hz$^{-1}$ of RS1 and RS2 is similar to the radio luminosity obtained for observed Type Ib/c or Type II radio supernovae (RSNe) \citep{wei02}. Thus, it is possible that
RS1 and RS2 are SNe or SN-SNR transition objects.
Our data is insufficient to distinguish between the two cases and further sensitive VLBI monitoring would be required.
The VLBA data (1999$-$2001) also displays two similar but weaker
RSN or SNR candidates RS3 and RS4, the latter of which has a spectral index of
$\alpha_{\rm.{1.7-2.4}}$ $\approx$ 0.35.
Many nearby galaxies, such as the LIRG \object{M 82}, and ULIRGs \object{Arp 220} and
\object{Arp 299}, display compact radio sources that are evidence of ongoing
(circum-)nuclear starformation \citep{smi98,mcd01,nef04,par07,con07,ulv09}.
VLBI observations of the merging galaxy Arp\,220 have detected a total of 18
compact radio sources within the western nucleus of the galaxy, over half of
which have radio properties that are consistent with Type-II supernovae
interacting with the surrounding medium \citep{par07}. Likewise, the nuclei in
Arp\,299 show 30 compact radio sources, 25 of which are associated with the
northeastern nucleus A (with an AGN candidate) and are spread over a region
of 30 pc \citep{nef04,ulv09}.
A comparison of the SN$-$SNR sources in NGC\,6240 with those found in other
sources shows that the radio powers of RS1 and RS2 are equivalent to those of
the most powerful sources found in M\,82 \citep[see][]{fenech08},
although they are less powerful than those in Arp\,220. As a result the upper limits for the sizes of RS1 and RS2 lie in the upper range of the general relation between radio luminosity and diameter observed for Galactic and extragalactic SNR sources \citep{ber04,bat10}. This would suggests that the environmental
conditions in the LIRG NGC\, 6240,
and possibly the star formation initial mass function, cannot yet be distinguished from those of the most luminous FIR galaxies.
\section{Summary}
We have conducted multi-frequency EVN observations of the nuclear
region of the merging galaxy NGC 6240 at two epochs. The
new VLBI maps reveal the double radio nuclei of NGC
6240 at milliarcsecond resolution, that are consistent with the
earlier VLBI images obtained with the VLBA \citep{gall04}. {The radio
spectra from both nuclei suggest a spectral turn-over between 2 and 3 GHz. The spectrum of
the southern nucleus N1 may be explained in terms of free-free absorption,
although this explanation is still limited by having source flux densities with insufficient frequency coverage at each epoch. There
is no clear interpretation for the spectrum of the northern nucleus N2. }
Questions still remain about the true nature of the two radio nuclei and whether they
both contain a radio-quiet AGN, a simple starburst, or a composite with an AGN and
a circum-nuclear starburst. We suggest that the southern nucleus hosts
an AGN and a circum-nuclear starburst, as evidenced by the X-ray data and
the radio sources. The association of the H$_{2}$O$\;$ maser with the nuclear source
is still unknown. However, it is not clear that our data, together with
earlier VLBA measurements, confirm the presence of an AGN at the northern
nucleus.
The radio components RS1 and RS2 could be interpreted as long-lived
radio supernovae that result from ongoing circum-nuclear star formation at N1. The radio spectrum of both sources is relatively flat and their location
remains unchanged within error
over about 6 years. More radio supernovae may have been
detected in earlier observations around the active nucleus of N1,
which groups \object{NGC 6240} together with other well-known starburst
nuclei that display RSNe or SNRs in their nuclei, such as \object{M
81}, \object{M82}, \object{Arp 220} and \object{Arp 299} \citep{bar09}. Radio interferometric
observations are a powerful method to identify ongoing nuclear
starformation and detect radio supernovae
and supernova remnants in extragalactic sources.
\acknowledgments We are grateful to Drs. Yoshiharu Asaki and Seiji
Kameno for their helpful suggestions, and we also thank to Dr. Robert
Beswick for providing the MERLIN radio image. The authors thank
Dr. Bob Campbell and other staff members in JIVE for their
assistance in the observations, correlation, and data analysis.
The authors also wish to thank an anonymous referee for suggestions that
improved the manuscript. YH acknowledges Dr. Phil Edwards and
staff members in VLBI Space Observatory Programme 2 (VSOP-2) for
providing valuable advice.
The European VLBI Network is a joint facility of European, Chinese, South
African and other radio astronomy institutes funded by their national
research councils. This work was supported in part by The Graduate
University for Advanced Studies (Sokendai). This research has made
use of NASA's Astrophysics Data System Abstract Service.
This research has made use of the NASA/IPAC Extragalactic Database (NED),
which is operated by the Jet Propulsion Laboratory, California Institute of
Technology, under contract with the National Aeronautics and Space
Administration.
|
1,108,101,565,712 | arxiv | \section{Introduction}
\label{sec:intro}
Mechanism design studies how to achieve a social goal in the presence of
decentralized decision making, and one important subfield is Nash
implementation.\footnote
There are two paradigms in mechanism design: full implementation (e.g., Nash
implementation) and partial implementation (e.g., auction design). The
former requires all solutions deliver the social goal, while the latter
requires one solution only. By adopting different solution concepts, we may
have different full implementation notions, e.g., Nash implementation (i.e.,
adopting Nash equilibria), and rationalizable implementation (i.e., adopting
rationalizability).} \cite{em2, em}\footnote{\cite{em2} is published as \cit
{em}.} propose the famous notion of \emph{Maskin monotonicity}, and prove
that it is necessary for Nash implementation. Given an additional assumption
of \emph{no-veto power}, \cite{em2, em} further prove that Maskin
monotonicity is also sufficient.
The gap between necessity and sufficiency of Nash implementation is finally
eliminated by \cite{jmrr}, \cite{vd} and \cite{ds}, which provide necessary
and sufficient conditions. Throughout the paper, we focus on \cite{jmrr}
\footnote{\cite{ds} focuses on 2-agent environments, and we assume three or
more agents. Our results can be easily extended to 2-agent environments.
\cite{vd}'s full characterization hinges on a domain assumption, while \cit
{jmrr}'s does not, and that is why we focus on the latter only.} \cite{tomas}
provides algorithms to check the conditions in \cite{jmrr}.
Compared to the simple and intuitive Maskin monotonicity, the full
characterization in \cite{jmrr} is complicated and hard to interpret. It is
one of the most celebrated result, and yet, our understanding on it is still
limited.
Most papers in the literature use a canonical mechanism as in \cite{em} to
achieve Nash implementation. There are three cases in the canonical
mechanism: (1) consensus, (2) unilateral deviation, (3) multi-lateral
deviation. The difference between \cite{em} and \cite{jmrr} is how to
eliminate bad equilibria when case (2) (or case (3)) is triggered in the
canonical mechanism. \cite{em} solves this problem by an \emph{exogenous}
condition of no-veto power, which is \emph{essentially} equivalent to
requiring all equilibria in case (2) to be good. Instead, \cite{jmrr}
consider all possible equilibria in case (2), and identify \emph{endogenous
necessary conditions (of Nash implementation), and then embed them into the
canonical mechanism, which achieves Nash implementation.---Therefore, such
conditions are both necessary and sufficient.
From a normative view, the result in \cite{jmrr} is better than that in \cit
{em}, because the former implies the latter. However, the advantage of the
Maskin approach is that the characterization is much simpler and more
intuitive. Furthermore, no-veto power is usually considered as a weak
condition. Thus, from a practical view, Maskin's characterization is usually
considered as an almost full characterization. Because of this, almost all
of the papers in the literature on full implementation follow the Maskin
approach, i.e., identify a Maskin-monotonicity-style necessary condition,
and prove that it is sufficient, given no-veto power, e.g., \cite{cmlr},
\cite{kt2012}.
In this paper, we study Nash implementation by stochastic mechanisms, and
provide a surprisingly simple full characterization. By taking full
advantage of the convexity structure of lotteries, we show that the
complicated Moore-Repullo-style full characterization collapses into a
Maskin-monotonicity-style condition. That is, not only does our simple full
characterization have a form similar to \cite{em}, but also it has an
interpretation parallel to \cite{jmrr}. In this sense, we build a bridge
connecting \cite{em} and \cite{jmrr}.\footnote
The outcome space in \cite{jmrr} is not convex, while it is convex in our
environment, and our full characterization hinges crucially on this
convexity assumption. As a result, the full characterization in \cite{jmrr}
does not imply ours, even though we share the same conceptual idea. See more
discussion in Section \ref{sec:connection}.}
More importantly, our results show a conceptual advantage of the
Moore-Repullo approach, which is not shared by the Maskin approach: the full
characterization rigorously pin down the logical relation between different
notions. For example, the notion of mixed-Nash implementation in \cite{cmlr}
does not require existence of pure-strategy equilibria, but the notion in
\cite{em} does. Both \cite{cmlr} and \cite{cksx2022} argue that this is a
significant difference in their setups. However, how much impact does this
difference induce? With only almost full characterization in both \cite{em}
and \cite{cmlr}, we cannot find an answer for this question. Given the full
characterization in our paper, we prove that this difference alone actually
does not induce any impact (Theorem \ref{thm:pure:SCC}).
In a second example, \cite{bmt} try to compare rationalizable implementation
to mixed-Nash implementation. \cite{bmt} observe that the necessary
condition of the former is stronger than the necessary condition of the
latter, which, clearly, sheds limited light on their rigorous relation,
generally. Only with no-veto power, we can conclude that the former is
stronger than the latter.---This may be misleading. First, as illustrated in
\cite{bmt} and \cite{xiong2022c}, no-veto power has no role in
rationalizable implementation, and hence, there is not much justification to
impose it, when we compare the two implementation notions. Second, no-veto
power is not the reason that rationalizable implementation is stronger than
mixed-Nash implementation, because with our full characterization, we are
able to prove that the former always implies the latter with or without
no-veto power (Theorem \ref{theorem:rationalizable}). The intuition is that
mixed-Nash implementation is fully characterized by \emph{a condition on
agents' modified lower-contour sets} (Theorem \ref{theorem:full:mix:SCC-A}),
which is also shared by rationalizable implementation. The difference
between mixed-Nash equilibria and rationalizability is whether agents have
correct beliefs in the corresponding solutions, and this difference has no
impact on the identified condition on agents' modified lower-contour sets.
\cite{obochet} and \cite{bo2} are the first two papers which propose to use
stochastic mechanisms to achieve Nash implementation. Their results are
orthogonal to ours, because of two differences. First, they impose weaker
assumptions on agents' preference on lotteries, and in this sense, their
results are stronger. Second, they impose exogenous assumptions on agents'
preference on deterministic outcomes, which immediately makes no-veto power
\emph{vacuously} true on non-degenerate lotteries.\footnote
That is, they do not impose no-veto power on deterministic outcomes, but
(implicitly) impose no-veto power on non-degenerate lotteries.} Allowing for
non-degenerate lotteries only in case (2) and case (3) in the canonical
mechanism, \cite{obochet} and \cite{bo2} establish an almost full
characterization as \cite{em} does i.e., \cite{obochet} and \cite{bo2} take
the Maskin approach. In contrast, we establish our full characterization
without any assumption on agents' preference on deterministic outcomes, and
in this sense, our results are stronger. More importantly, we take the
Moore-Repullo approach, i.e., no-veto power may fail for both degenerate and
no-degenerate lotteries, and in spite of this, we find necessary and
sufficient conditions. To the best of our knowledge, we are the first paper
after \cite{jmrr} and \cite{tomas}, which takes the Moore-Repullo approac
\footnote
Here is a difference between the Maskin approach and the Moore-Repullo
approach: there is an exogenously given subset of outcomes that satisfies
no-veto power in the former, but such an exogenous subset does not exist in
the latter.} and establishes a simple full characterization for Nash
implementation.
The remainder of the paper proceeds as follows: we describe the model in
Section \ref{sec:model} and fully characterize Nash implementation for
social choice functions in Section \ref{sec:mix:stochastic}; we compare
pure-Nash and mixed-Nash implementation in Section \ref{sec:pure}; we
compare the ordinal and the cardinal approaches in Section \ref{sec:ordinal
; we focus on social choice correspondences in Section \re
{sec:extension:SCC:A} and study ordinal and rationalizable implementation in
Sections \ref{sec:ordinal:full} and \ref{sec:rationalizable}, respectively;
we establish connection to \cite{jmrr} and \cite{tomas} in Section \re
{sec:connection} and Section \ref{sec:conclude} concludes.
\section{Model}
\label{sec:model}
\subsection{Environment}
\label{sec:environment}
We take a cardinal approach, and a (cardinal) model consists of
\begin{equation}
\left\langle \mathcal{I}=\left\{ 1,..,I\right\} \text{, \ }\Theta \text{, \
Z\text{, }f:\Theta \longrightarrow Z\text{, }Y\equiv \triangle \left(
Z\right) \text{, }\left( u_{i}^{\theta }:Z\longrightarrow
\mathbb{R}
\right) _{\left( i,\theta \right) \in \mathcal{I}\times \Theta
}\right\rangle \text{,} \label{yyr2}
\end{equation
where $\mathcal{I}$ is a finite set of $I$ agents with $I\geq 3$, $\Theta $
a finite or countably-infinite set of possible states, $Z$ a finite set of
pure social outcomes, $f$ a social choice function (hereafter, SCF)\footnote
For simplicity, we focus on social choice functions first. We will introduce
social choice correspondences in Section \ref{sec:pure}, and we extend our
results in Section \ref{sec:extension:SCC:A}.} which maps each state in
\Theta $ to an outcome in $Z$, $Y$ the set of all possible lotteries on $Z$,
$u_{i}$ the Bernoulli utility function of agent $i$ at state $\theta $.
Throughout the paper, we assume that agents have expected utility, i.e.
\begin{equation*}
U_{i}^{\theta }\left( y\right) =\dsum\limits_{z\in Z}y_{z}u_{i}^{\theta
}\left( z\right) \text{, }\forall y\in Y\text{,}
\end{equation*
where $y_{z}$ denotes the probability of $z$ under $y$, and $U_{i}^{\theta
}\left( y\right) $ is agent $i$'s expected utility of $y$ at state $\theta
. Without loss of generality, we impose the following assumption throughout
the paper.
\begin{assum}[non-triviality]
\label{assm:non-trivial} $\left\vert f\left( \Theta \right) \right\vert \geq
2$.\footnote
If $\left\vert f\left( \Theta \right) \right\vert =1$, e.g., $f\left( \Theta
\right) =\left\{ z\right\} $ for some $z\in Z$, the implementation problem
can be solved trivially, i.e., we implement $z$ at every state.}
\end{assum}
For each $z\in Z$, we regard $z$ as a degenerate lottery in $Y$. With abuse
of notation, we write $z\in Z\subset Y$. Throughout the paper, we use $-i$
to denote $\mathcal{I}\backslash \left\{ i\right\} $. For any $\left( \alpha
,i,\theta \right) \in Y\times \mathcal{I}\times \Theta $, defin
\begin{eqnarray*}
\mathcal{L}_{i}^{Y}\left( \alpha ,\theta \right) &\equiv &\left\{ y\in
Y:U_{i}^{\theta }\left( \alpha \right) \geq U_{i}^{\theta }\left( y\right)
\right\} \text{,} \\
\mathcal{L}_{i}^{Z}\left( \alpha ,\theta \right) &\equiv &\left\{ z\in
Z:U_{i}^{\theta }\left( \alpha \right) \geq U_{i}^{\theta }\left( z\right)
\right\} \text{.}
\end{eqnarray*}
For any finite or countably-infinite set $E$, we use $\triangle \left(
E\right) $ to denote the set of probabilities on $E$. For any $\mu \in
\triangle \left( E\right) $, we let SUPP$\left[ \mu \right] $ denote the
support of $\mu $, i.e.
\begin{equation*}
\text{SUPP}\left[ \mu \right] \equiv \left\{ x\in E:\mu \left( x\right)
>0\right\} \text{.}
\end{equation*
Furthermore, defin
\begin{equation*}
\triangle ^{\circ }\left( E\right) \equiv \left\{ \mu \in \triangle \left(
E\right) :\text{SUPP}\left[ \mu \right] =E\right\} \text{.}
\end{equation*
For any finite set $E$, we use UNIF$\left( E\right) $ to denote the uniform
distribution on $E$, and use $\left\vert E\right\vert $ to denote the number
of elements in $E$.
\subsection{Mechanisms and Nash implementation}
\label{sec:model:pure}
A mechanism is a tuple $\mathcal{M}=\left\langle M\equiv \times _{i\in
\mathcal{I}}M_{i}\text{, \ }g:M\longrightarrow Y\right\rangle $, where each
M_{i}$ is a countable set, and it denotes the set of strategies for agent $i$
in $\mathcal{M}$. A profile $\left( m_{i}\right) _{i\in \mathcal{I}}\in
\times _{i\in \mathcal{I}}M_{i}$ is a pure-strategy Nash equilibrium in
\mathcal{M}$ at state $\theta $ if and only i
\begin{equation*}
U_{i}^{\theta }\left[ g\left( m_{i},m_{-i}\right) \right] \geq U_{i}^{\theta
}\left[ g\left( m_{i}^{\prime },m_{-i}\right) \right] \text{, }\forall i\in
\mathcal{I}\text{, }\forall m_{i}^{\prime }\in M_{i}\text{.}
\end{equation*
Let $PNE^{\left( \mathcal{M},\text{ }\theta \right) }$ denote the set of
pure-strategy Nash equilibria in $\mathcal{M}$ at $\theta $. Furthermore, a
profile $\lambda \equiv \left( \lambda _{i}\right) _{i\in \mathcal{I}}\in
\times _{i\in \mathcal{I}}\triangle \left( M_{i}\right) $ is a
mixed-strategy Nash equilibrium in $\mathcal{M}$ at $\theta $ if and only i
\begin{eqnarray*}
&&\Sigma _{m\in M}\left[ \lambda _{i}\left( m_{i}\right) \times \Pi _{j\in
\mathcal{I}\diagdown \left\{ i\right\} }\lambda _{j}\left( m_{j}\right)
\times U_{i}^{\theta }\left[ g\left( m_{i},m_{-i}\right) \right] \right] \\
&\geq &\Sigma _{m\in M}\left[ \lambda _{i}^{\prime }\left( m_{i}\right)
\times \Pi _{j\in \mathcal{I}\diagdown \left\{ i\right\} }\lambda _{j}\left(
m_{j}\right) \times U_{i}^{\theta }\left[ g\left( m_{i},m_{-i}\right) \right]
\right] \text{, }\forall i\in \mathcal{I}\text{, }\forall \lambda
_{i}^{\prime }\in \triangle \left( M_{i}\right) \text{,}
\end{eqnarray*
where $\lambda _{j}\left( m_{j}\right) $ is the probability that $\lambda
_{j}$ assigns to $m_{j}$. Let $MNE^{\left( \mathcal{M},\text{ }\theta
\right) }$ denote the set of mixed-strategy Nash equilibria in $\mathcal{M}$
at state $\theta $.
For any mechanism $\mathcal{M}=\left\langle M\text{, \ }g:M\longrightarrow
Y\right\rangle $, and any $\lambda \equiv \left( \lambda _{i}\right) _{i\in
\mathcal{I}}\in \times _{i\in \mathcal{I}}\triangle \left( M_{i}\right) $,
we use $g\left( \lambda \right) $ to denote the lottery induced by $\lambda
.
\begin{define}[mixed-Nash-implemenation]
\label{def:implementation:mixed}An SCF $f:\Theta \longrightarrow Z$ is
mixed-Nash-implemented by a mechanism $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $ i
\begin{equation*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =\left\{ f\left( \theta
\right) \right\} \text{, }\forall \theta \in \Theta \text{.}
\end{equation*
$f$ is mixed-Nash-implementable if there exists a mechanism that
mixed-Nash-implements $f$.
\end{define}
\begin{define}[pure-Nash-implemenation]
\label{def:implementation:pure}An SCF $f:\Theta \longrightarrow Z$ is
pure-Nash-implemented by a mechanism $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $ i
\begin{equation*}
\dbigcup\limits_{\lambda \in PNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =\left\{ f\left( \theta
\right) \right\} \text{, }\forall \theta \in \Theta \text{.}
\end{equation*
$f$ is pure-Nash-implementable if there exists a mechanism that
pure-Nash-implements $f$.
\end{define}
\section{Mixed-Nash-implementation: a full characterization}
\label{sec:mix:stochastic}
In this section, we focus on mixed-Nash-implementation,\footnote
We will show that mixed-Nash-implementation is equivalent to
pure-Nash-implementation in Theorem \ref{theorem:full:equivalence:double}.}
and provide a surprisingly simple full characterization. As a benchmark, we
first list the theorem of \cite{em} in Section \re
{sec:mix:stochastic:Maskin}. We present the full characterization in Section
\ref{sec:mix:stochastic:full}, which also contains the necessity part of the
proof. The sufficiency part of the proof is more complicated, which is
provided in Section \ref{sec:mix:stochastic:sufficiency}.
\subsection{Maskin's theorem}
\label{sec:mix:stochastic:Maskin}
By applying the ideas in \cite{em} to our environment with stochastic
mechanisms, we adapt Maskin monotonicity as follows.\footnote
With stochastic mechanisms, there are two ways to define Maskin monotonicity
(or related monotnicity conditions): (i) it is defined on the outcome space
of $Y$ (e.g., \cite{bmt} and \cite{cksx2022}) and (ii) it is defined on the
outcome space of $Z$ (e.g., \cite{cmlr}). We follow the tradition of the
former.}
\begin{define}[Maskin monotonicity]
Maskin monotonicity holds i
\begin{equation*}
\left[
\begin{array}{c}
\mathcal{L}_{i}^{Y}\left( f\left( \theta \right) ,\theta \right) \subset
\mathcal{L}_{i}^{Y}\left( f\left( \theta \right) ,\theta ^{\prime }\right)
\text{, } \\
\forall i\in \mathcal{I
\end{array
\right] \text{ }\Longrightarrow \text{ }f\left( \theta \right) =f\left(
\theta ^{\prime }\right) \text{, }\forall \left( \theta ,\theta ^{\prime
}\right) \in \Theta \times \Theta \text{.}
\end{equation*}
\end{define}
\begin{define}[no-veto power]
No-veto power holds if for any $\left( a,\theta \right) \in Z\times \Theta
, we hav
\begin{equation*}
\left\vert \left\{ i\in \mathcal{I}:a\in \arg \max_{z\in Z}u_{i}^{\theta
}\left( z\right) \right\} \right\vert \geq \left\vert \mathcal{I}\right\vert
-1\Longrightarrow f\left( \theta \right) =a\text{.}
\end{equation*}
\end{define}
\begin{theo}[\protect\cite{em}]
\label{theorem:Maskin:sufficient}Maskin monotonicity holds if $f$ is
pure-Nash implementable. Furthermore, $f$ is pure-Nash implementable if
Maskin monotonicity and no-veto power hold.
\end{theo}
\subsection{A simple full characterization}
\label{sec:mix:stochastic:full}
The following easy-to-check notion plays a critical role in our full
characterization.
\begin{define}[$i$-max set]
\label{def:i-max}For any $\left( i,\theta \right) \in \mathcal{I}\times
\Theta $, a set $E\in 2^{Z}\diagdown \left\{ \varnothing \right\} $ is an $i
-$\theta $-max set i
\begin{equation*}
E\subset \arg \max_{z\in E}u_{i}^{\theta }\left( z\right) \text{ and
E\subset \arg \max_{z\in Z}u_{j}^{\theta }\left( z\right) \text{, }\forall
j\in \mathcal{I}\diagdown \left\{ i\right\} \text{.}
\end{equation*
Furthermore, $E$ is an $i$-max set if $E$ is an $i$-$\theta $-max set for
some $\theta \in \Theta $.
\end{define}
This immediately leads to the following lemma, which sheds light on
mechanisms that mixed-Nash-implement $f$. The proof is relegated to Appendix
\ref{sec:lem:mixed:deviation:SCF}.
\begin{lemma}
\label{lem:mixed:deviation:SCF}Suppose that an SCF $f$ is mixed-Nash
implemented by $\mathcal{M}=\left\langle M\text{, \ }g:M\longrightarrow
Y\right\rangle $. For any $\left( i,\theta \right) \in \mathcal{I}\times
\Theta $ and any $\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta \right)
}$, we hav
\begin{equation}
\left(
\begin{array}{c}
f\left( \theta \right) \in \arg \min_{z\in Z}u_{i}^{\theta }\left( z\right)
\text{,} \\
\text{and }\mathcal{L}_{i}^{Z}\left( f\left( \theta \right) ,\theta \right)
\text{ is an }i\text{-max set
\end{array
\right) \Longrightarrow \dbigcup\limits_{m_{i}\in M_{i}}\text{SUPP}\left[
g\left( m_{i},\lambda _{-i}\right) \right] =\left\{ f\left( \theta \right)
\right\} \text{.} \label{iit1}
\end{equation}
\end{lemma}
$\dbigcup\limits_{m_{i}\in M_{i}}$SUPP$\left[ g\left( m_{i},\lambda
_{-i}\right) \right] $ is the set of outcomes that can be induced with
positive probability by $i$'s unilateral deviation from $\lambda $. Lemma
\ref{lem:mixed:deviation:SCF} says that any unilateral deviation of $i$ from
$\lambda $ must induce $f\left( \theta \right) $ if the condition on the
left-hand side of "$\Longrightarrow $" in (\ref{iit1})\ holds. In light of
Lemma \ref{lem:mixed:deviation:SCF}, we refine lower-counter sets as
follows. For each $\left( i,\theta ,a\right) \in \mathcal{I}\times \Theta
\times Z$,
\begin{equation}
\widehat{\mathcal{L}}_{i}^{Y}\left( a,\theta \right) \equiv \left\{
\begin{tabular}{ll}
$\left\{ a\right\} \text{,}$ & if $a=f\left( \theta \right) \in \arg
\min_{z\in Z}u_{i}^{\theta }\left( z\right) \text{ }$ and $\mathcal{L
_{i}^{Z}\left( f\left( \theta \right) ,\theta \right) \text{ is an }i\text
-max set}$, \\
& \\
$\mathcal{L}_{i}^{Y}\left( a,\theta \right) \text{,}$ & otherwis
\end{tabular
\right. \text{.} \label{yjj8}
\end{equation}
\begin{define}[$\protect\widehat{\mathcal{L}}^{Y}$-monotonicity]
\label{def:L-monotonicity-SCF}$\widehat{\mathcal{L}}^{Y}$-monotonicity holds
i
\begin{equation*}
\left[
\begin{array}{c}
\widehat{\mathcal{L}}_{i}^{Y}\left( f\left( \theta \right) ,\theta \right)
\subset \mathcal{L}_{i}^{Y}\left( f\left( \theta \right) ,\theta ^{\prime
}\right) \text{, } \\
\forall i\in \mathcal{I
\end{array
\right] \text{ }\Longrightarrow \text{ }f\left( \theta \right) =f\left(
\theta ^{\prime }\right) \text{, }\forall \left( \theta ,\theta ^{\prime
}\right) \in \Theta \times \Theta \text{.}
\end{equation*}
\end{define}
This leads to a simple full characterization of mixed-Nash-implementation.
\begin{theo}
\label{theorem:full:mix}An SCF $f:\Theta \longrightarrow Z$ is
mixed-Nash-implementable if and only if $\widehat{\mathcal{L}}^{Y}
-monotonicity holds.
\end{theo}
\noindent \textbf{Proof of the "only if" part of Theorem \re
{theorem:full:mix}:} Suppose that $f$ is mixed-Nash-implemented by $\mathcal
M}=\left\langle M\text{, \ }g:M\longrightarrow Y\right\rangle $. Fix any
\left( \theta ,\theta ^{\prime }\right) \in \Theta \times \Theta $ such tha
\begin{equation}
\left[
\begin{array}{c}
\widehat{\mathcal{L}}_{i}^{Y}\left( f\left( \theta \right) ,\theta \right)
\subset \mathcal{L}_{i}^{Y}\left( f\left( \theta \right) ,\theta ^{\prime
}\right) \text{, } \\
\forall i\in \mathcal{I
\end{array
\right] \text{,} \label{hrr1}
\end{equation
and we aim to show $f\left( \theta \right) =f\left( \theta ^{\prime }\right)
$, i.e., $\widehat{\mathcal{L}}^{Y}$-monotonicity holds.
We prove $f\left( \theta \right) =f\left( \theta ^{\prime }\right) $ by
contradiction. Suppose $f\left( \theta \right) \neq f\left( \theta ^{\prime
}\right) $. Pick any $\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta
\right) }$, and we have SUPP$\left[ g\left( \lambda \right) \right] =\left\{
f\left( \theta \right) \right\} $ because $f$ is implemented by $\mathcal{M}
. Since $f\left( \theta \right) \neq f\left( \theta ^{\prime }\right) $, we
have $\lambda \notin MNE^{\left( \mathcal{M},\text{ }\theta ^{\prime
}\right) }$. As a result, there exists an agent $j$ who has a profitable
deviation $m_{j}\in M_{j}$ from $\lambda $ at $\theta ^{\prime }$, i.e.
\begin{equation}
\exists j\in \mathcal{I}\text{, }\exists m_{j}\in M_{j}\text{, }g\left(
m_{j},\lambda _{-j}\right) \in \mathcal{L}_{j}^{Y}\left( f\left( \theta
\right) ,\theta \right) \diagdown \mathcal{L}_{j}^{Y}\left( f\left( \theta
\right) ,\theta ^{\prime }\right) \text{,} \label{ttat1}
\end{equation
where $g\left( m_{j},\lambda _{-j}\right) \in \mathcal{L}_{j}^{Y}\left(
f\left( \theta \right) ,\theta \right) $ and $g\left( m_{j},\lambda
_{-j}\right) \notin \mathcal{L}_{j}^{Y}\left( f\left( \theta \right) ,\theta
^{\prime }\right) $ follow from $\lambda \in MNE^{\left( \mathcal{M},\text{
\theta \right) }$ and $\lambda \notin MNE^{\left( \mathcal{M},\text{ }\theta
^{\prime }\right) }$, respectively.
In particular, $g\left( m_{j},\lambda _{-j}\right) \notin \mathcal{L
_{j}^{Y}\left( f\left( \theta \right) ,\theta ^{\prime }\right) $ implies
g\left( m_{j},\lambda _{-j}\right) \neq f\left( \theta \right) $, and hence,
Lemma \ref{lem:mixed:deviation:SCF} implies failure of the following
condition
\begin{equation*}
\left(
\begin{array}{c}
f\left( \theta \right) \in \arg \min_{z\in Z}u_{j}^{\theta }\left( z\right)
\text{,} \\
\text{and }\mathcal{L}_{j}^{Z}\left( f\left( \theta \right) ,\theta \right)
\text{ is an }j\text{-max set
\end{array
\right) \text{,}
\end{equation*
which, together with (\ref{yjj8}), further implie
\begin{equation}
\widehat{\mathcal{L}}_{j}^{Y}\left( f\left( \theta \right) ,\theta \right)
\mathcal{L}_{j}^{Y}\left( f\left( \theta \right) ,\theta \right) \text{.}
\label{ttat2}
\end{equation
(\ref{ttat1}) and (\ref{ttat2}) impl
\begin{equation*}
g\left( m_{j},\lambda _{-j}\right) \in \widehat{\mathcal{L}}_{j}^{Y}\left(
f\left( \theta \right) ,\theta \right) \diagdown \mathcal{L}_{j}^{Y}\left(
f\left( \theta \right) ,\theta ^{\prime }\right) \text{,}
\end{equation*
contradicting (\ref{hrr1}).$\blacksquare $
\subsection{Sufficiency of $\protect\widehat{\mathcal{L}}^{Y}$-monotonicity}
\label{sec:mix:stochastic:sufficiency}
\subsubsection{Preliminary construction}
In order to build our canonical mechanism to implement $f$, we need to take
three preliminary constructions. First, for each $\left( i,\theta \right)
\in \mathcal{I}\times \Theta $, we defin
\begin{equation}
\widehat{\Gamma }_{i}\left( \theta \right) \equiv \dbigcup\limits_{y\in
\widehat{\mathcal{L}}_{i}^{Y}\left( f\left( \theta \right) ,\theta \right)
\text{SUPP}\left[ y\right] =\left\{ z\in Z
\begin{tabular}{l}
$\exists y\in \widehat{\mathcal{L}}_{i}^{Y}\left( f\left( \theta \right)
,\theta \right) $, \\
$z\in $SUPP$\left[ y\right]
\end{tabular
\right\} \text{,} \label{kkl2}
\end{equation
i.e., $\widehat{\Gamma }_{i}\left( \theta \right) $\ is the set of outcomes
that can be induced with positive probability by lotteries in $\widehat
\mathcal{L}}_{i}^{Y}\left( f\left( \theta \right) ,\theta \right) $. This
leads to the following lemma, and the proof is relegated to Appendix \re
{sec:lem:no-veto:generalized}.
\begin{lemma}
\label{lem:no-veto:generalized}Suppose that $\widehat{\mathcal{L}}^{Y}
-monotonicity holds. Then, $Z$ is not an $i$-max set for any $i\in \mathcal{
}$ an
\begin{equation*}
\left[ \widehat{\Gamma }_{j}\left( \theta \right) \text{ is a }j\text{-
\theta ^{\prime }\text{-max set}\right] \Longrightarrow \widehat{\Gamma
_{j}\left( \theta \right) =\left\{ f\left( \theta ^{\prime }\right) \right\}
\text{, }\forall \left( j,\theta ,\theta ^{\prime }\right) \in \mathcal{I
\times \Theta \times \Theta \text{.}
\end{equation*}
\end{lemma}
Second, for each $\left( \theta ,j\right) \in \Theta \times \mathcal{I}$,
fix any function $\phi _{j}^{\theta }:\Theta \longrightarrow Y$ such that
\begin{equation}
\phi _{j}^{\theta }\left( \theta ^{\prime }\right) \in \left( \arg
\max_{y\in \widehat{\mathcal{L}}_{j}^{Y}\left( f\left( \theta \right)
,\theta \right) }U_{j}^{\theta ^{\prime }}\left[ y\right] \right) \text{,
\forall \theta ^{\prime }\in \Theta \text{,} \label{yuy1a}
\end{equation
Suppose that the true state is $\theta ^{\prime }\in \Theta $. In a
canonical mechanism that implements $f$, when agent $j$ unilaterally
deviates from "all agents reporting $\theta $," we let $j$ choose any
lottery in $\widehat{\mathcal{L}}_{j}^{Y}\left( f\left( \theta \right)
,\theta \right) $, in order to ensure that truthful reporting is always a
Nash equilibrium. Thus, $\phi _{j}^{\theta }\left( \theta ^{\prime }\right) $
in (\ref{yuy1a}) is an optimal lottery in $\widehat{\mathcal{L}
_{j}^{Y}\left( f\left( \theta \right) ,\theta \right) $ for $j$ at $\theta
^{\prime }\in \Theta $.
Furthermore, by (\ref{kkl2}), we hav
\begin{equation}
\phi _{j}^{\theta }\left( \theta ^{\prime }\right) \in \left( \arg
\max_{y\in \widehat{\mathcal{L}}_{j}^{Y}\left( f\left( \theta \right)
,\theta \right) }U_{j}^{\theta ^{\prime }}\left[ y\right] \right) \cap
\triangle \left( \widehat{\Gamma }_{j}\left( \theta \right) \right) \text{,
\forall \theta ^{\prime }\in \Theta \text{.} \label{yuy1}
\end{equation}
Finally, the following lemma completes our third construction, and the proof
is relegated to Appendix \ref{sec:lem:if}.
\begin{lemma}
\label{lem:if}For each $\left( \theta ,j\right) \in \Theta \times \mathcal{I}
$, there exis
\begin{equation*}
\varepsilon _{j}^{\theta }>0\text{ and }y_{j}^{\theta }\in \widehat{\mathcal
L}}_{j}^{Y}\left( f\left( \theta \right) ,\theta \right) \text{,}
\end{equation*
such tha
\begin{equation}
\left[ \varepsilon _{j}^{\theta }\times y+\left( 1-\varepsilon _{j}^{\theta
}\right) \times y_{j}^{\theta }\right] \in \widehat{\mathcal{L}
_{j}^{Y}\left( f\left( \theta \right) ,\theta \right) \text{, }\forall y\in
\triangle \left( \widehat{\Gamma }_{j}\left( \theta \right) \right) \text{.}
\label{yuy2}
\end{equation}
\end{lemma}
\subsubsection{A canonical mechanism}
\label{sec:canonic}
Let $\mathbb{N}$ denote the set of positive integers. We use the mechanism
\mathcal{M}^{\ast }=\left\langle \times _{i\in \mathcal{I}}M_{i}\text{, \
g:M\longrightarrow Y\right\rangle $ defined below to implement $f$. In
particular, we have
\begin{equation}
M_{i}=\left\{ \left( \theta _{i},k_{i}^{2},k_{i}^{3},\gamma
_{i},b_{i}\right) \in \Theta \times \mathbb{N}\times \mathbb{N}\times \left(
Z\right) ^{\left[ 2^{Z}\diagdown \left\{ \varnothing \right\} \right]
}\times Z
\begin{tabular}{l}
$\gamma _{i}\left( E\right) \in E$, \\
$\forall E\in \left[ 2^{Z}\diagdown \left\{ \varnothing \right\} \right]
\end{tabular
\right\} \text{, }\forall i\in \mathcal{I}\text{,} \label{yuy2a}
\end{equation
and $g\left[ m=\left( m_{i}\right) _{i\in \mathcal{I}}\right] $ is defined
in three cases.
\begin{description}
\item[Case (1): consensus] if there exists $\theta \in \Theta $ such tha
\begin{equation*}
\left( \theta _{i},k_{i}^{2}\right) =\left( \theta ,1\right) \text{,
\forall i\in \mathcal{I}\text{,}
\end{equation*
then $g\left[ m\right] =f\left( \theta \right) $;
\item[Case (2), unilateral deviation: ] if there exists $\left( \theta
,j\right) \in \Theta \times \mathcal{I}$ such tha
\begin{equation*}
\left( \theta _{i},k_{i}^{2}\right) =\left( \theta ,1\right) \text{ if and
only if }i\in \mathcal{I}\diagdown \left\{ j\right\} \text{,}
\end{equation*
then
\begin{eqnarray}
g\left[ m\right] &=&\left( 1-\frac{1}{k_{j}^{2}}\right) \times \phi
_{j}^{\theta }\left( \theta _{j}\right) \label{uue1} \\
&&+\frac{1}{k_{j}^{2}}\times \left(
\begin{tabular}{l}
$\varepsilon _{j}^{\theta }\times \left[ \left( 1-\frac{1}{k_{j}^{3}}\right)
\times \gamma _{j}\left( \widehat{\Gamma }_{j}\left( \theta \right) \right)
\frac{1}{k_{j}^{3}}\times \text{UNIF}\left( \widehat{\Gamma }_{j}\left(
\theta \right) \right) \right] $ \\
$+\left( 1-\varepsilon _{j}^{\theta }\right) \times y_{j}^{\theta }
\end{tabular
\right) \text{,} \notag
\end{eqnarray
where $\left( \varepsilon _{j}^{\theta },y_{j}^{\theta }\right) $ are chosen
for each $\left( \theta ,j\right) \in \Theta \times \mathcal{I}$ according
to Lemma \ref{lem:if}. Note that $\gamma _{j}\left( \widehat{\Gamma
_{j}\left( \theta \right) \right) \in \widehat{\Gamma }_{j}\left( \theta
\right) $ by (\ref{yuy2a}) and UNIF$\left( \widehat{\Gamma }_{j}\left(
\theta \right) \right) \in \triangle \left( \widehat{\Gamma }_{j}\left(
\theta \right) \right) $;
\item[Case (3), multi-lateral deviation: ] otherwise,
\begin{equation}
g\left[ m\right] =\left( 1-\frac{1}{k_{j^{\ast }}^{2}}\right) \times
b_{j^{\ast }}+\frac{1}{k_{j^{\ast }}^{2}}\times \text{UNIF}\left( Z\right)
\text{,} \label{uue2}
\end{equation
where $j^{\ast }=\max \left( \arg \max_{i\in \mathcal{I}}k_{i}^{2}\right) $,
i.e., $j^{\ast }$ is the largest-numbered agent who submits the highest
number on the second dimension of the message.
\end{description}
Each agent $i$ uses $k_{i}^{2}$ to show intention to be a whistle-blower,
i.e., $i$ \emph{voluntarily} challenges agents $-i$'s reports if and only if
$k_{i}^{2}>1$. In Case (1), agents reach a consensus, i.e., all agents
report the same state $\theta $, and choose not to challenge voluntarily (or
equivalently, $k_{i}^{2}=1$ for every $i\in \mathcal{I}$). In this case,
f\left( \theta \right) $ is assigned by $g$.
Case (2) is triggered if any agent $j$ unilaterally deviates from Case (1)
(in the first two dimensions of $j$'s message): either $j$ challenges
voluntarily (i.e., $k_{j}^{2}>1$), or $j$ challenges involuntarily (i.e.,
k_{j}^{2}=1$ and $j$ reports a different state $\theta _{j}\left( \neq
\theta \right) $ in the first dimension). In this case, $g$ assigns the
compound lottery in (\ref{uue1}), which is determined by the state $\theta $
being agreed upon by $-j$, and by $\left( \theta
_{j},k_{j}^{2},k_{j}^{3},\gamma _{j}\right) $ of $j$'s message. By (\re
{yuy1a}) and (\ref{yuy2}), the compound lottery in (\ref{uue1}) is an
element in $\widehat{\mathcal{L}}_{j}^{Y}\left( f\left( \theta \right)
,\theta \right) $, which ensures that truth-reporting is always a Nash
equilibrium at each state $\theta \in \Theta $. Note that $k_{j}^{2}$ and
k_{j}^{3}$ determine the probabilities in the compound lottery in (\ref{uue1
). Furthermore, $\left( \theta ,\theta _{j}\right) $ determines $j$'s
challenge scheme $\phi _{j}^{\theta }\left( \theta _{j}\right) $ in (\re
{uue1})---revealing the true state is always $j$'s best challenge scheme due
to (\ref{yuy1a}). Finally, $\theta $ determines the set $\widehat{\Gamma
_{j}\left( \theta \right) $, and $j$ is entitled to pick an optimal outcome
in $\widehat{\Gamma }_{j}\left( \theta \right) $ via $\gamma _{j}$ (i.e.,
the fourth dimensions in $j$'s message): the picked outcome $\gamma
_{j}\left( \widehat{\Gamma }_{j}\left( \theta \right) \right) $ occurs with
probability $\frac{1}{k_{j}^{2}}\times \varepsilon _{j}^{\theta }\times
\left( 1-\frac{1}{k_{j}^{3}}\right) $, and the outcome UNIF$\left( \widehat
\Gamma }_{j}\left( \theta \right) \right) $ occurs with probability $\frac{
}{k_{j}^{2}}\times \varepsilon _{j}^{\theta }\times \frac{1}{k_{j}^{3}}
.---The higher $k_{j}^{3}$, the more probability is shifted from "UNIF
\left( \widehat{\Gamma }_{j}\left( \theta \right) \right) $" to "$\gamma
_{j}\left( \widehat{\Gamma }_{j}\left( \theta \right) \right) $."
Case (3) includes all other scenarios, and as usual, agents compete in an
integer game. The agent $j^{\ast }$ who reports the highest integer in the
second dimension wins, and $j^{\ast }$ is entitled to pick an optimal
outcome $b_{j^{\ast }}$ in $Z$. In this case, $g$ assigns the compound
lottery in (\ref{uue2}): $b_{j^{\ast }}$ occurs with probability $\left( 1
\frac{1}{k_{j^{\ast }}^{2}}\right) $ and UNIF$\left( Z\right) $ occurs with
probability $\frac{1}{k_{j^{\ast }}^{2}}$.---The higher $k_{j^{\ast }}^{2}$,
the more probability is shift from "UNIF$\left( Z\right) $" to "$b_{j^{\ast
}}$."
The following lemma substantially simplifies the analysis of mixed-strategy
Nash equilibria in $\mathcal{M}^{\ast }$, and the proof is relegated to
Appendix \ref{sec:lem:mixed:canonical:pure}.
\begin{lemma}
\label{lem:mixed:canonical:pure}Consider the canonical mechanism $\mathcal{M
^{\ast }$ above. For any $\theta \in \Theta $ and any $\lambda \in
MNE^{\left( \mathcal{M}^{\ast },\text{ }\theta \right) }$, we have SUPP
\left[ \lambda \right] \subset PNE^{\left( \mathcal{M}^{\ast },\text{
\theta \right) }$.
\end{lemma}
Lemma \ref{lem:mixed:canonical:pure} says that every pure-strategy profile
on the support of a mixed-strategy Nash equilibrium in $\mathcal{M}^{\ast }$
at $\theta $ must also be a pure-strategy Nash equilibrium at $\theta $. As
a result, it suffers no loss of generality to focus on pure-strategy Nash
equilibria.
\subsubsection{Proof of "if" part of Theorem \textbf{\protect\re
{theorem:full:mix}}}
\label{sec:canonic:proof}
Suppose that $\widehat{\mathcal{L}}^{Y}$-monotonicity holds. Fix any true
state $\theta ^{\ast }\in \Theta $. We aim to prov
\begin{equation*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M}^{\ast },\text{ }\theta
^{\ast }\right) }}\text{SUPP}\left( g\left[ \lambda \right] \right) =\left\{
f\left( \theta ^{\ast }\right) \right\} \text{.}
\end{equation*
First, truth revealing is a Nash equilibrium, i.e., any pure strategy profil
\begin{equation*}
m^{\ast }=\left( \theta _{i}=\theta ^{\ast },k_{i}^{1}=1,k_{i}^{2},\gamma
_{i},b_{i}\right) _{i\in \mathcal{I}}
\end{equation*
is a Nash equilibrium, which triggers Case (1) and $g\left[ m^{\ast }\right]
=f\left( \theta ^{\ast }\right) $. Any unilateral deviation $\overline{m
_{j}=\left( \overline{\theta _{j}},\overline{k_{j}^{2}},\overline{k_{j}^{3}}
\overline{\gamma _{j}},\overline{b_{j}}\right) $ of agent $j\in \mathcal{I}$
would either still trigger Case (1) and induce $f\left( \theta ^{\ast
}\right) $, or trigger Case (2) and induc
\begin{eqnarray*}
g\left[ \overline{m}_{j},m_{-j}^{\ast }\right] &=&\left( 1-\frac{1}
\overline{k_{j}^{2}}}\right) \times \phi _{j}^{\theta ^{\ast }}\left(
\overline{\theta _{j}}\right) \\
&&+\frac{1}{\overline{k_{j}^{2}}}\times \left(
\begin{tabular}{l}
$\varepsilon _{j}^{\theta ^{\ast }}\times \left[ \left( 1-\frac{1}{\overline
k_{j}^{3}}}\right) \gamma _{j}\left( \widehat{\Gamma }_{j}\left( \theta
^{\ast }\right) \right) +\frac{1}{\overline{k_{j}^{3}}}\times \text{UNIF
\left( \widehat{\Gamma }_{j}\left( \theta ^{\ast }\right) \right) \right] $
\\
$+\left( 1-\varepsilon _{j}^{\theta ^{\ast }}\right) \times y_{j}^{\theta
^{\ast }}
\end{tabular
\right) \text{,}
\end{eqnarray*
and by (\ref{yuy1}) and (\ref{yuy2}), we hav
\begin{equation*}
g\left[ \overline{m}_{j},m_{-j}^{\ast }\right] \in \widehat{\mathcal{L}
_{j}^{Y}\left( f\left( \theta ^{\ast }\right) ,\theta ^{\ast }\right)
\subset \mathcal{L}_{j}^{Y}\left( f\left( \theta ^{\ast }\right) ,\theta
^{\ast }\right) \text{, }\forall \overline{m}_{j}\in M_{j}\text{.}
\end{equation*
Therefore, any $\overline{m}_{j}\in M_{j}$ is not a profitable deviation and
$m^{\ast }$ is a Nash equilibrium which induces $g\left[ m^{\ast }\right]
=f\left( \theta ^{\ast }\right) $.
Second, by Lemma \ref{lem:mixed:canonical:pure}, it suffers no loss of
generality to focus on pure-strategy equilibria. Fix an
\begin{equation*}
\widetilde{m}=\left( \widetilde{\theta _{i}},\widetilde{k_{i}^{2}}
\widetilde{k_{i}^{3}},\widetilde{\gamma _{i}},\widetilde{b_{i}}\right)
_{i\in \mathcal{I}}\in PNE^{\left( \mathcal{M}^{\ast },\text{ }\theta ^{\ast
}\right) }\text{,}
\end{equation*
and we aim to prove $g\left[ \widetilde{m}\right] =f\left( \theta ^{\ast
}\right) $. We first prove that $\widetilde{m}$ does not trigger Case (3).
Suppose otherwise, i.e.,
\begin{equation*}
g\left[ \widetilde{m}\right] =\left( 1-\frac{1}{\widetilde{k_{j^{\ast }}^{2}
}\right) \times \widetilde{b_{j^{\ast }}}+\frac{1}{\widetilde{k_{j^{\ast
}}^{2}}}\times \text{UNIF}\left( Z\right) \text{,}
\end{equation*
where $j^{\ast }=\max \left( \arg \max_{i\in \mathcal{I}}\widetilde{k_{i}^{2
}\right) $. By Lemma \ref{lem:no-veto:generalized}, $Z$ is not an $i$-max
set for any $i\in \mathcal{I}$, and thus,
\begin{equation*}
\exists j\in \mathcal{I}\text{, }\min_{z\in Z}u_{j}^{\theta ^{\ast }}\left(
z\right) <\max_{z\in Z}u_{j}^{\theta ^{\ast }}\left( z\right) \text{,}
\end{equation*
and as a result,
\begin{equation*}
U_{j}^{\theta ^{\ast }}\left( \text{UNIF}\left( Z\right) \right) <\max_{z\in
Z}u_{j}^{\theta ^{\ast }}\left( z\right) \text{,}
\end{equation*
which further implies that agent $j$ finds it strictly profitable to deviate
t
\begin{equation*}
m_{j}=\left( \widetilde{\theta _{j}},k_{j}^{2},\widetilde{k_{j}^{3}}
\widetilde{\gamma _{j}},b_{j}\right) \text{ with }b_{j}\in \arg \max_{z\in
Z}u_{j}^{\theta ^{\ast }}\left( z\right) \text{, for sufficiently large
k_{j}^{2}\text{,}
\end{equation*
contradicting $\widetilde{m}\in PNE^{\left( \mathcal{M}^{\ast },\text{
\theta ^{\ast }\right) }$.
Thus, $\widetilde{m}$ must trigger either Case (1) or Case (2). Suppose
\widetilde{m}$ triggers Case (1), i.e.
\begin{equation*}
\widetilde{m}=\left( \widetilde{\theta _{i}}=\widetilde{\theta },\widetilde
k_{i}^{2}}=1,\widetilde{k_{i}^{3}},\widetilde{\gamma _{i}},\widetilde{b_{i}
\right) _{i\in \mathcal{I}}\text{ for some }\widetilde{\theta }\in \Theta
\text{,}
\end{equation*
and $g\left[ \widetilde{m}\right] =f\left( \widetilde{\theta }\right) $. We
prove $g\left[ \widetilde{m}\right] =f\left( \theta ^{\ast }\right) $ by
contradiction. Suppose $f\left( \widetilde{\theta }\right) \neq f\left(
\theta ^{\ast }\right) $. By $\widehat{\mathcal{L}}^{Y}$-monotonicity, there
exists $j\in \mathcal{I}$ such tha
\begin{equation*}
\exists y^{\ast }\in \widehat{\mathcal{L}}_{j}^{Y}\left[ f\left( \widetilde
\theta }\right) ,\widetilde{\theta }\right] \diagdown \mathcal{L}_{j}^{Y
\left[ f\left( \widetilde{\theta }\right) ,\theta ^{\ast }\right] \text{,}
\end{equation*
which, together with (\ref{yuy1}), implie
\begin{equation*}
U_{j}^{\theta ^{\ast }}\left[ \phi _{j}^{\widetilde{\theta }}\left( \theta
^{\ast }\right) \right] \geq U_{j}^{\theta ^{\ast }}\left[ y^{\ast }\right]
>U_{j}^{\theta ^{\ast }}\left[ f\left( \widetilde{\theta }\right) \right]
\text{.}
\end{equation*
Therefore, it is strictly profitable for agent $j$ to deviate t
\begin{equation*}
m_{j}=\left( \theta ^{\ast },k_{j}^{2},\widetilde{k_{j}^{3}},\widetilde
\gamma _{j}},\widetilde{b_{j}}\right) \text{ for sufficiently large
k_{j}^{2}\text{,}
\end{equation*
contradicting $\widetilde{m}\in PNE^{\left( \mathcal{M}^{\ast },\text{
\theta ^{\ast }\right) }$.
Finally, suppose $\widetilde{m}$ triggers Case (2), i.e., there exists $j\in
\mathcal{I}$ such that
\begin{equation*}
\exists \widetilde{\theta }\in \Theta \text{, }\widetilde{m}_{i}=\left(
\widetilde{\theta _{i}}=\widetilde{\theta },\widetilde{k_{i}^{2}}=1
\widetilde{k_{i}^{3}},\widetilde{\gamma _{i}},\widetilde{b_{i}}\right) \text{
, }\forall i\in \mathcal{I\diagdown }\left\{ j\right\} \text{,}
\end{equation*
an
\begin{eqnarray}
g\left[ \widetilde{m}\right] &=&\left( 1-\frac{1}{\widetilde{k_{j}^{2}}
\right) \times \phi _{j}^{\widetilde{\theta }}\left( \widetilde{\theta _{j}
\right) \label{yuy4} \\
&&+\frac{1}{\widetilde{k_{j}^{2}}}\times \left(
\begin{tabular}{l}
$\varepsilon _{j}^{\widetilde{\theta }}\times \left[ \left( 1-\frac{1}
\widetilde{k_{j}^{3}}}\right) \times \gamma _{j}\left( \widehat{\Gamma
_{j}\left( \widetilde{\theta }\right) \right) +\frac{1}{\widetilde{k_{j}^{3}
}\times \text{UNIF}\left( \widehat{\Gamma }_{j}\left( \widetilde{\theta
\right) \right) \right] $ \\
$+\left( 1-\varepsilon _{j}^{\widetilde{\theta }}\right) \times y_{j}^
\widetilde{\theta }}
\end{tabular
\right) \text{.} \notag
\end{eqnarray
We now prove $g\left[ \widetilde{m}\right] =f\left( \theta ^{\ast }\right)
. By (\ref{yuy1}) and (\ref{yuy2}), we hav
\begin{equation}
g\left[ \widetilde{m}\right] \in \triangle \left[ \widehat{\Gamma
_{j}\left( \widetilde{\theta }\right) \right] \text{.} \label{uiu1}
\end{equation
Since every $i\in \mathcal{I\diagdown }\left\{ j\right\} $ can deviate to
trigger Case (3), and dictate her top outcome in $Z$ with arbitrarily high
probability, $\widetilde{m}\in PNE^{\left( \mathcal{M}^{\ast },\text{
\theta ^{\ast }\right) }$ implie
\begin{equation}
\widehat{\Gamma }_{j}\left( \widetilde{\theta }\right) \subset \arg
\max_{z\in Z}u_{i}^{\theta ^{\ast }}\left( z\right) \text{, }\forall i\in
\mathcal{I}\diagdown \left\{ j\right\} \text{.} \label{uiu2}
\end{equation
Inside the the compound lottery $g\left[ \widetilde{m}\right] $ in (\re
{yuy4}), conditional on an event with probability $\frac{1}{\widetilde
k_{j}^{2}}}\times \varepsilon _{j}^{\widetilde{\theta }}$, we have the
compound lottery
\begin{equation*}
\left[ \left( 1-\frac{1}{\widetilde{k_{j}^{3}}}\right) \times \widetilde
\gamma _{j}}\left( \widehat{\Gamma }_{j}\left( \widetilde{\theta }\right)
\right) +\frac{1}{\widetilde{k_{j}^{3}}}\times \text{UNIF}\left( \widehat
\Gamma }_{j}\left( \widetilde{\theta }\right) \right) \right] \text{,}
\end{equation*
and hence, agent $j$ can always deviate t
\begin{equation*}
m_{j}=\left( \widetilde{\theta _{j}},\widetilde{k_{j}^{2}},k_{j}^{3},\gamma
_{j},\widetilde{b_{j}}\right) _{i\in \mathcal{I\diagdown }\left\{ j\right\}
\text{ with }\gamma _{j}\left( \widehat{\Gamma }_{j}\left( \widetilde{\theta
}\right) \right) \in \arg \max_{z\in \widehat{\Gamma }_{j}\left( \widetilde
\theta }\right) }u_{j}^{\theta ^{\ast }}\left( z\right)
\end{equation*
for sufficiently large $k_{j}^{3}$. Thus, $\widetilde{m}\in PNE^{\left(
\mathcal{M}^{\ast },\text{ }\theta ^{\ast }\right) }$ implie
\begin{equation}
\widehat{\Gamma }_{j}\left( \widetilde{\theta }\right) \subset \arg
\max_{z\in \widehat{\Gamma }_{j}\left( \widetilde{\theta }\right)
}u_{j}^{\theta ^{\ast }}\left( z\right) \text{.} \label{uiu3}
\end{equation
(\ref{uiu2}) and (\ref{uiu3}) imply that $\widehat{\Gamma }_{j}\left(
\widetilde{\theta }\right) $ is a $j$-$\theta ^{\ast }$-max set, which
together Lemma \ref{lem:no-veto:generalized}, further implie
\begin{equation}
\widehat{\Gamma }_{j}\left( \widetilde{\theta }\right) =\left\{ f\left(
\theta ^{\ast }\right) \right\} \text{.} \label{uiu3a}
\end{equation
(\ref{uiu1}) and (\ref{uiu3a}) imply $g\left[ \widetilde{m}\right] =f\left(
\theta ^{\ast }\right) $.$\blacksquare $
\section{Discussion: implementation-in-PNE Vs implementation-in-MNE}
\label{sec:pure}
The current literature has limited understanding on the difference between
implementation in pure Nash equilibria and in mixed Nash equilibria.
Compared to \cite{em}, \cite{cmlr} argue that mixed-Nash implementation
substantially expand the scope of implementation
\begin{equation}
\text{What drives such difference?} \label{ggh1}
\end{equation
Both \cite{cmlr} and \cite{cksx2022} argue that a significant difference
between \cite{em} and \cite{cmlr} is whether we require existence of pure
Nash equilibria in mixed-Nash implementation. That is, \cite{em} actually
adopts the notion of double implementation defined as follows.
\begin{define}[double-Nash-implemenation, \protect\cite{em}]
\label{def:implementation:double}An SCF $f:\Theta \longrightarrow Z$ is
double-Nash-implementable if there exists a mechanism $\mathcal{M
=\left\langle M\text{, \ }g:M\longrightarrow Y\right\rangle $ such tha
\begin{equation*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right)
=\dbigcup\limits_{\lambda \in PNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =\left\{ f\left( \theta
\right) \right\} \text{, }\forall \theta \in \Theta \text{.}
\end{equation*}
\end{define}
The following theorem says that this difference does not answer the question
in (\ref{ggh1}).
\begin{theo}
\label{theorem:full:equivalence:double}Consider any SCF $f:\Theta
\longrightarrow Z$. The following statements are equivalent.
(i) $f$ is pure-Nash-implementable;
(ii) $f$ is mixed-Nash-implementable;
(iii) $f$ is double-Nash-implementable;
(iv) $\widehat{\mathcal{L}}^{Y}$-monotonicity holds.
\end{theo}
The proof in Section \ref{sec:canonic:proof} shows sufficiency of $\widehat
\mathcal{L}}^{Y}$-monotonicity for all of (i), (ii) and (iii) in Theorem \re
{theorem:full:equivalence:double}, while the necessity of $\widehat{\mathcal
L}}^{Y}$-monotonicity is described by (a slightly modified version of) the
proof for the "only if" part of Theorem \ref{theorem:full:mix} in Section
\ref{sec:mix:stochastic:full}.\footnote
We omit the proof Theorem \ref{theorem:full:equivalence:double}, because it
is implied by Theorem \ref{thm:pure:SCC}, which is proved in Appendix \re
{sec:thm:pure:SCC}.}
If we focus on SCFs, Theorem \ref{theorem:full:equivalence:double} implies
that the analysis in \cite{cmlr} remains the same if we replace mixed-Nash
implementation in their setup with pure-Nash-implementation or
double-Nash-implementation. However, \cite{cmlr} considers social choice
correspondences (hereafter, SCC) besides SCFs, and hence, Theorem \re
{theorem:full:equivalence:double} does not provide a full answer for the
question in (\ref{ggh1}). An SCC is a set-valued function $F:\Theta
\longrightarrow 2^{Z}\diagdown \left\{ \varnothing \right\} $, and we thus
extend our definitions to SCCs.\footnote
We will consider six different definitions of mixed-Nash-implementation, and
we call them versions A, B, C, D, E and F.}
\begin{define}[mixed-Nash-A-implemenation, \protect\cite{cmlr}]
\label{sec:mixed:implementation:SCC:A}An SCC $F$ is
mixed-Nash-A-implementable if there exists a mechanism $\mathcal{M
=\left\langle M\text{, \ }g:M\longrightarrow Y\right\rangle $ such tha
\begin{equation*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =F\left( \theta \right)
\text{, }\forall \theta \in \Theta \text{.}
\end{equation*}
\end{define}
\begin{define}
\label{sec:pure:implementation:SCC:A}An SCC $F$ is pure-Nash-implementable
if there exists a mechanism $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $ such tha
\begin{equation*}
\dbigcup\limits_{\lambda \in PNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =F\left( \theta \right)
,\forall \theta \in \Theta \text{.}
\end{equation*}
\end{define}
\begin{define}[mixed-Nash-B-implemenation]
\label{sec:mixed:implementation:SCC:B}An SCC $F$ is
mixed-Nash-B-implementable if there exists a mechanism $\mathcal{M
=\left\langle M\text{, \ }g:M\longrightarrow Y\right\rangle $ such tha
\begin{equation*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right)
=\dbigcup\limits_{\lambda \in PNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =F\left( \theta \right)
,\forall \theta \in \Theta \text{.}
\end{equation*}
\end{define}
\begin{theo}
\label{thm:pure:SCC}Consider any SCC $F:\Theta \longrightarrow
2^{Z}\diagdown \left\{ \varnothing \right\} $. The following statements are
equivalent.
(i) $F$ is pure-Nash-implementable;
(ii) $F$ is mixed-Nash-A-implementable;
(iii) $F$ is mixed-Nash-B-implementable.
\end{theo}
Theorem \ref{theorem:full:mix:SCC-A} will provide a full characterization of
mixed-Nash-A-implemention, which will be used to prove Theorem \re
{thm:pure:SCC}. We will prove Theorem \ref{thm:pure:SCC} in Appendix \re
{sec:thm:pure:SCC},\footnote{\cite{cmlr} observe that
pure-Nash-implementation and mixed-Nash-A-implementation share the same
necessary condition (i.e., set-monotonicity), but do not provide their
relationship. Only with our full characterization, we are able to prove
their equivalence.} after we prove Theorem \ref{theorem:full:mix:SCC-A} in
Appendix \ref{sec:theorem:full:mix:SCC-A}.
Compared to \cite{em}, \cite{cmlr} introduce three new ingredients to the
model: (I) a new class of mechanisms (i.e., stochastic mechanisms), (II) new
solutions (i.e., mixed-strategy Nash equilibria, or pure-strategy Nash
equilibria, or both) and (III) how to interpret "implementing $F\left(
\theta \right) $" (see more discussion in Section \ \ref{sec:extension:SCC:4
). Given SCFs, (III) is the same in both \cite{em} and \cite{cmlr}, and
Theorem \ref{theorem:full:equivalence:double} shows that (II) is also the
same in the two papers, which immediately leads to an answer for the
question in (\ref{ggh1}):\ the difference is solely driven by the new class
of mechanisms, i.e., (I). Given SCCs, Theorem \ref{thm:pure:SCC} shows that
(II) is still the same in the two papers, and hence, the difference must be
driven by (I) and (III).\footnote
Given SCCs, (III) is not the same in the two papers. Maskin's notion
coresponds to mixed-Nash-D-implemention in Definition \re
{def:implementation:mixed:SCC:D}.}
\section{Discussion: cardinal approach Vs ordinal approach}
\label{sec:ordinal}
We take a cardinal approach in this paper, i.e., agents have cardinal
utility functions. However, an ordinal approach is usually adopted in the
literature of implementation (e.g., \cite{cmlr}). We show that our cardinal
approach is more general than the ordinal approach.
Throughout this section, we fix an ordinal model in \cite{cmlr}, which
consists of
\begin{equation}
\left\langle \mathcal{I}=\left\{ 1,..,I\right\} \text{, \ }\Theta ^{\ast
\text{, \ }Z\text{, }f:\Theta ^{\ast }\longrightarrow Z\text{, }Y\equiv
\triangle \left( Z\right) \text{, }\left( \succeq _{i}^{\theta }\right)
_{\left( i,\theta \right) \in \mathcal{I}\times \Theta ^{\ast
}}\right\rangle \text{,} \label{yyr1}
\end{equation
where each ordinal state $\theta \in \Theta ^{\ast }$ determines a profile
of preferences $\left( \succeq _{i}^{\theta }\right) _{i\in \mathcal{I}}$ on
$Z$. This ordinal model differs from our cardinal model on two aspects only.
First, agents have ordinal preference (i.e., $\left( \succeq _{i}^{\theta
}\right) _{\left( i,\theta \right) \in \mathcal{I}\times \Theta ^{\ast }}$),
compared to the cardinal utility functions (i.e., $\left( u_{i}^{\theta
}:Z\longrightarrow
\mathbb{R}
\right) _{\left( i,\theta \right) \in \mathcal{I}\times \Theta }$) in in
Section \ref{sec:environment}. Second, the ordinal state set, denoted by
\Theta ^{\ast }$, is finite, while the cardinal state set, denoted by
\Theta $ in Section \ref{sec:environment}, is either finite or
countably-infinite.
For each ordinal state $\theta \in \Theta ^{\ast }$, we say $u^{\theta
}\equiv \left( u_{i}^{\theta }:Z\longrightarrow
\mathbb{R}
\right) _{i\in \mathcal{I}}$ \ is a cardinal representation of $\succeq
^{\theta }\equiv \left( \succeq _{i}^{\theta }\right) _{i\in \mathcal{I}}$
if and only i
\begin{equation*}
z\succeq _{i}^{\theta }z^{\prime }\Longleftrightarrow u_{i}^{\theta }\left(
z\right) \geq u_{i}^{\theta }\left( z^{\prime }\right) \text{, }\forall
\left( z,z^{\prime },i\right) \in Z\times Z\times \mathcal{I}\text{.}
\end{equation*
Each $\left( u_{i}^{\theta }:Z\longrightarrow
\mathbb{R}
\right) _{i\in \mathcal{I}}$ is called a cardinal state. That is, each
ordinal state $\theta \in \Theta ^{\ast }$ can be represented by a set of
cardinal states defined as follows
\begin{equation*}
\Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }\equiv \left\{ \left( u_{i}^{\theta }:Z\longrightarrow
\mathbb{R}
\right) _{i\in \mathcal{I}}
\begin{tabular}{l}
$z\succeq _{i}^{\theta }z^{\prime }\Longleftrightarrow u_{i}^{\theta }\left(
z\right) \geq u_{i}^{\theta }\left( z^{\prime }\right) \text{,}$ \\
$\forall \left( z,z^{\prime },i\right) \in Z\times Z\times \mathcal{I}\text{
}
\end{tabular
\right\} \subset \left( \left(
\mathbb{R}
\right) ^{Z}\right) ^{\mathcal{I}}\text{.}
\end{equation*
Mixed-Nash ordinal-implementation in \cite{cmlr} requires that $f$ be
implemented by a mechanism under any cardinal representation.
\begin{define}[mixed-Nash-ordinal-implemenation, \protect\cite{cmlr}]
$f\ $is mixed-Nash-ordinally-implementable if there exists a mechanism
\mathcal{M}=\left\langle M\text{, \ }g:M\longrightarrow Y\right\rangle $
such tha
\begin{equation*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M},\text{ }u^{\theta
}\right) }}\text{SUPP}\left( g\left[ \lambda \right] \right) =\left\{
f\left( \theta \right) \right\} \text{, }\forall \theta \in \Theta ^{\ast
\text{, }\forall u^{\theta }\in \Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }\text{.}
\end{equation*}
\end{define}
Since $\Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$ is uncountably infinite, our results do not apply directly.
However, we may consider cardinal utility functions with rational values
only
\begin{equation*}
\Omega ^{\left[ \succeq ^{\theta },\text{ }\mathbb{Q}\right] }\equiv \left\{
\left( u_{i}^{\theta }:Z\longrightarrow \mathbb{Q}\right) _{i\in \mathcal{I
}
\begin{tabular}{l}
$z\succeq _{i}^{\theta }z^{\prime }\Longleftrightarrow u_{i}^{\theta }\left(
z\right) \geq u_{i}^{\theta }\left( z^{\prime }\right) \text{,}$ \\
$\forall \left( z,z^{\prime },i\right) \in Z\times Z\times \mathcal{I}\text{
}
\end{tabular
\right\} \subset \left( \left( \mathbb{Q}\right) ^{Z}\right) ^{\mathcal{I}
\text{.}
\end{equation*
Clearly, $\Omega ^{\left[ \succeq ^{\theta },\text{ }\mathbb{Q}\right]
}\subset \Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$, and $\Omega ^{\left[ \succeq ^{\theta },\text{ }\mathbb{Q}\right]
}$ is countably infinite.
\begin{theo}
\label{thm:mixed:ordinal}$f\ $is mixed-Nash-ordinally-implementable (or
equivalently, $f\ $is mixed-Nash-A-implementable with $\Theta =\cup _{\theta
\in \Theta ^{\ast }}\Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$) if and only if $f\ $is mixed-Nash-A-implementable with $\Theta
=\cup _{\theta \in \Theta ^{\ast }}\Omega ^{\left[ \succeq ^{\theta },\text{
}\mathbb{Q}\right] }$.
\end{theo}
The "only if" part of Theorem \ref{thm:mixed:ordinal} is implied by $\Omega
^{\left[ \succeq ^{\theta },\text{ }\mathbb{Q}\right] }\subset \Omega ^
\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$, and the "if" part is immediately implied by the following lemma.
\begin{lemma}
\label{lem:ordinal}For any $\theta \in \Theta ^{\ast }$ and any $\overline
u^{\theta }}\in \Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$, there exists $\left( \widehat{u^{\theta }},\widetilde{u^{\theta
}\right) \in \Omega ^{\left[ \succeq ^{\theta },\text{ }\mathbb{Q}\right]
}\times \Omega ^{\left[ \succeq ^{\theta },\text{ }\mathbb{Q}\right] }$,
such tha
\begin{equation}
\mathcal{L}_{i}^{Y}\left( z,\widehat{u^{\theta }}\right) \subset \mathcal{L
_{i}^{Y}\left( z,\overline{u^{\theta }}\right) \subset \mathcal{L
_{i}^{Y}\left( z,\widetilde{u^{\theta }}\right) \text{, }\forall \left(
i,z\right) \in \mathcal{I}\times Y\text{.} \label{ggi6}
\end{equation}
\end{lemma}
The proof of Lemma \ref{lem:ordinal} is relegated to Appendix \re
{sec:lem:ordinal}.
\noindent \textbf{Proof of the "if" part of Theorem \ref{thm:mixed:ordinal}:}
We use the canonical mechanism $\mathcal{M}^{\ast }=\left\langle \times
_{i\in \mathcal{I}}M_{i}\text{, \ }g:M\longrightarrow Z\right\rangle $ (with
$\Theta =\cup _{\theta \in \Theta ^{\ast }}\Omega ^{\left[ \succeq ^{\theta
},\text{ }\mathbb{Q}\right] }$) in Section \ref{sec:canonic} to implement $f
, i.e.
\begin{equation}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M}^{\ast },\text{
u^{\theta }\right) }}\text{SUPP}\left( g\left[ \lambda \right] \right)
=\left\{ f\left( \theta \right) \right\} \text{, }\forall \theta \in \Theta
^{\ast }\text{, }\forall u^{\theta }\in \Omega ^{\left[ \succeq ^{\theta }
\text{ }\mathbb{Q}\right] }\text{.} \label{ggi7}
\end{equation
Fix any $\theta \in \Theta ^{\ast }$ and pick any $\overline{u^{\theta }}\in
\Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$. By Lemma \ref{lem:ordinal}, there exists $\left( \widehat
u^{\theta }},\widetilde{u^{\theta }}\right) \in \Omega ^{\left[ \succeq
^{\theta },\text{ }\mathbb{Q}\right] }\times \Omega ^{\left[ \succeq
^{\theta },\text{ }\mathbb{Q}\right] }$, such that (\ref{ggi6})\ holds. In
particular, $\times _{i\in \mathcal{I}}\mathcal{L}_{i}^{Y}\left( f\left(
\theta \right) ,\widehat{u^{\theta }}\right) \subset \times _{i\in \mathcal{
}}\mathcal{L}_{i}^{Y}\left( f\left( \theta \right) ,\overline{u^{\theta }
\right) $ immediately implie
\begin{equation}
MNE^{\left( \mathcal{M},\text{ }\widehat{u^{\theta }}\right) }\subset
MNE^{\left( \mathcal{M},\text{ }\overline{u^{\theta }}\right) }\text{.}
\label{ggi8}
\end{equation
Lemma \ref{lem:no-veto:generalized} implies that $Z$ is not an $i$-max set
for any $i\in \mathcal{I}$. As a result, no equilibrium exists when Case (3)
occurs. In fact, as the proof in Section \ref{sec:canonic:proof} shows that,
at state $\overline{u^{\theta }}\in \Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$, an equilibrium exists in $\mathcal{M}^{\ast }$ only when either
Case (1) occurs, or Case (2) in which agents $-i$ report $\theta ^{\prime }$
with $\widehat{\mathcal{L}}_{i}^{Y}\left( f\left( \theta ^{\prime }\right)
,\theta ^{\prime }\right) =\left\{ f\left( \theta ^{\prime }\right) \right\}
$ occurs. That is, in both cases, we hav
\begin{equation*}
g\left[ \lambda \right] \in Z\text{, }\forall \lambda \in MNE^{\left(
\mathcal{M}^{\ast },\text{ }\overline{u^{\theta }}\right) }\text{,}
\end{equation*
which, together with $\times _{i\in \mathcal{I}}\mathcal{L}_{i}^{Y}\left( z
\overline{u^{\theta }}\right) \subset \times _{i\in \mathcal{I}}\mathcal{L
_{i}^{Y}\left( z,\widetilde{u^{\theta }}\right) $ for every $z\in Z$, implie
\begin{equation}
MNE^{\left( \mathcal{M},\text{ }\overline{u^{\theta }}\right) }\subset
MNE^{\left( \mathcal{M},\text{ }\widetilde{u^{\theta }}\right) }\text{.}
\label{ggi8a}
\end{equation
(\ref{ggi7}), (\ref{ggi8})\ and (\ref{ggi8a}) imply $\dbigcup\limits_
\lambda \in MNE^{\left( \mathcal{M},\text{ }\overline{u^{\theta }}\right) }}
SUPP$\left( g\left[ \lambda \right] \right) =\left\{ f\left( \theta \right)
\right\} $.$\blacksquare $
Theorem \ref{thm:mixed:ordinal} extends to SCCs (see e.g., Theorem \re
{theorem:MR:iff}).
\section{Extension to social choice correspondences}
\label{sec:extension:SCC:A}
\subsection{Four additional definitions}
\label{sec:extension:SCC:4}
Given any solution concept, what does it mean that an SCC $F:\Theta
\longrightarrow 2^{Z}\diagdown \left\{ \varnothing \right\} $ is implemented
in the solution? There are two views in the literature. The first view is
that, at each state $\theta \in \Theta $, each solution must induce a
deterministic outcome and $F\left( \theta \right) $ is the set of all such
deterministic outcomes.--This view is adopted in \cite{ks}. The second view
is that, at each state $\theta \in \Theta $, $F\left( \theta \right) $ is
the set of outcomes that can be induced with positive probability by some
solution--This view is adopted in \cite{cmlr} and \cite{rjain}. Furthermore,
we may or may not require existence of pure Nash equilibria, when we define
mixed-Nash-implementation. Definitions \ref{sec:mixed:implementation:SCC:A}
and \ref{sec:mixed:implementation:SCC:B} follow the second view, while the
former does not require existence of pure Nash equilibria, and the latter
does. Besides these two definitions, we can define four alternative versions
of mixed-Nash-implementation, with different combination of requirements.
For any mechanism $\mathcal{M}=\left\langle M\text{, \ }g:M\longrightarrow
Y\right\rangle $, defin
\begin{equation*}
\Phi ^{\mathcal{M}}\equiv \left\{ \left( \lambda _{i}\right) _{i\in \mathcal
I}}\in \times _{i\in \mathcal{I}}\triangle \left( M_{i}\right) :\left\vert
\text{SUPP}\left( g\left[ \left( \lambda _{i}\right) _{i\in \mathcal{I}
\right] \right) \right\vert =1\right\} \text{,}
\end{equation*
i.e., $\Phi ^{\mathcal{M}}$ is the set of mixed strategy profiles that
induces a unique deterministic outcome.
\begin{define}[mixed-Nash-C-implemenation]
\label{def:implementation:mixed:SCC:C}An SCC $F:\Theta \longrightarrow
2^{Z}\diagdown \left\{ \varnothing \right\} $ is mixed-Nash-C-implementable
if there exists a mechanism $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $ such tha
\begin{equation*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =g\left( MNE^{\left(
\mathcal{M},\text{ }\theta \right) }\cap \Phi ^{\mathcal{M}}\right) =F\left(
\theta \right) \text{, }\forall \theta \in \Theta \text{.}
\end{equation*}
\end{define}
\begin{define}[mixed-Nash-D-implemenation, \protect\cite{em}]
\label{def:implementation:mixed:SCC:D}An SCC $F:\Theta \longrightarrow
2^{Z}\diagdown \left\{ \varnothing \right\} $ is mixed-Nash-D-implementable
if there exists a mechanism $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $ such tha
\begin{equation*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =g\left( PNE^{\mathcal{M
,\text{ }\theta }\right) =F\left( \theta \right) \text{, }\forall \theta \in
\Theta \text{.}
\end{equation*}
\end{define}
\begin{define}[mixed-Nash-E-implemenation]
\label{def:implementation:mixed:SCC:E}An SCC $F:\Theta \longrightarrow
2^{Z}\diagdown \left\{ \varnothing \right\} $ is mixed-Nash-E-implementable
if there exists a mechanism $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $ such tha
\begin{gather*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =F\left( \theta \right)
\text{, }\forall \theta \in \Theta \text{,} \\
\text{and }MNE^{\mathcal{M},\text{ }\theta }\subset \Phi ^{\mathcal{M}}\text
.}
\end{gather*}
\end{define}
\begin{define}[mixed-Nash-F-implemenation]
\label{def:implementation:double:SCC:F}An SCC $F:\Theta \longrightarrow
2^{Z}\diagdown \left\{ \varnothing \right\} $ is mixed-Nash-F-implementable
if there exists a mechanism $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $ such tha
\begin{gather*}
\dbigcup\limits_{\lambda \in MNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right)
=\dbigcup\limits_{\lambda \in PNE^{\left( \mathcal{M},\text{ }\theta \right)
}}\text{SUPP}\left( g\left[ \lambda \right] \right) =F\left( \theta \right)
\text{, }\forall \theta \in \Theta \text{,} \\
\text{and }MNE^{\left( \mathcal{M},\text{ }\theta \right) }\subset \Phi ^
\mathcal{M}}\text{.}
\end{gather*}
\end{define}
\subsection{Version E and version F: full characterization}
It is straightforward to extend Theorem \ref{theorem:full:mix} to
mixed-Nash-E-implementation and Mixed-Nash-F-implementation. Define
\begin{equation}
\widehat{\mathcal{L}}_{i}^{Y}\left( a,\theta \right) \equiv \left\{
\begin{tabular}{ll}
$\left\{ a\right\} \text{,}$ & if $a\in F\left( \theta \right) \cap \arg
\min_{z\in Z}u_{i}^{\theta }\left( z\right) $ and $\mathcal{L}_{i}^{Z}\left(
a,\theta \right) \text{ is an }i\text{-max set}$, \\
& \\
$\mathcal{L}_{i}^{Y}\left( a,\theta \right) \text{,}$ & otherwise
\end{tabular
\right. \label{nnt1}
\end{equation}
\begin{define}[$\protect\widehat{\mathcal{L}}^{Y}$-monotonicity]
\label{def:L-monotonicity-SCC}$\widehat{\mathcal{L}}^{Y}$-monotonicity holds
for an SCC $F$ i
\begin{equation*}
\left[
\begin{array}{c}
a\in F\left( \theta \right) \text{,} \\
\widehat{\mathcal{L}}_{i}^{Y}\left( a,\theta \right) \subset \mathcal{L
_{i}^{Y}\left( a,\theta ^{\prime }\right) \text{, }\forall i\in \mathcal{I
\end{array
\right] \text{ }\Longrightarrow a\in F\left( \theta ^{\prime }\right) \text
, }\forall \left( \theta ,\theta ^{\prime },a\right) \in \Theta \times
\Theta \times Z\text{.}
\end{equation*}
\end{define}
In the degenerate case that $F$ is a social choice function, $\widehat
\mathcal{L}}_{i}^{Y}\left( a,\theta \right) $ in (\ref{nnt1}) becomes
\widehat{\mathcal{L}}_{i}^{Y}\left( a,\theta \right) $ in (\ref{yjj8}), and
\widehat{\mathcal{L}}^{Y}$-monotonicity in Definition \re
{def:L-monotonicity-SCC} becomes $\widehat{\mathcal{L}}^{Y}$-monotonicity in
Definition \ref{def:L-monotonicity-SCF}. Hence, we use the same notation.
\begin{theo}
\label{theorem:full:mix:SCC-E-F}Given an SCC $F:\Theta \longrightarrow
2^{Z}\diagdown \left\{ \varnothing \right\} $, the following three
statements are equivalent.
(i) $F$ is mixed-Nash-E-implementable;
(ii) $F$ is mixed-Nash-F-implementable;
(iii) $Z$ is not an $i$-max set for any $i\in \mathcal{I}$ and $\widehat
\mathcal{L}}^{Y}$-monotonicity holds for $F$.
\end{theo}
It is worth noting that we need the requirement of "$Z$ is not an $i$-max
set for any $i\in \mathcal{I}$" in (iii) of Theorem \re
{theorem:full:mix:SCC-E-F}. We do not need this requirement in Theorem \re
{theorem:full:mix}, because it is implied by $\widehat{\mathcal{L}}^{Y}
-monotonicity when $F$ is a degenerate SCF (see Lemma \re
{lem:no-veto:generalized}).
The proof of Theorem \ref{theorem:full:mix:SCC-E-F} is almost the same as
that of Theorem \ref{theorem:full:mix}, and the detailed proof is relegated
to \cite{sx2022b}.
\subsection{Version A and version B: full characterization}
Defin
\begin{equation}
Z^{\ast }\equiv \left\{
\begin{tabular}{ll}
$\cup _{\theta \in \Theta }F\left( \theta \right) $, & if $Z$ is an $i$-max
set for some $i\in \mathcal{I}$, \\
& \\
$Z\text{,}$ & if $Z$ is not an $i$-max set for any $i\in \mathcal{I}$
\end{tabular
\right. \label{tth2}
\end{equation}
\begin{lemma}
\label{lem:Z-A}Suppose that an SCC $F$ is mixed-Nash-A-implemented by
\mathcal{M}=\left\langle M\text{, \ }g:M\longrightarrow Y\right\rangle $.
Then, we have $g\left( M\right) \subset \triangle \left( Z^{\ast }\right) $.
\end{lemma}
Lemma \ref{lem:Z-A} says that only lotteries in $\triangle \left( Z^{\ast
}\right) $ can be used by a mechanism which mixed-Nash-A-implements an SCC,
and the proof is relegated to Appendix \ref{sec:lem:Z-A}. The implication of
Lemma \ref{lem:Z-A} is that, in order to achieve
mixed-Nash-A-implementation, we should delete $Z\diagdown Z^{\ast }$ from
our model.
In order to accommodate the new implementation notion, we need to further
adapt the notion of $i$-max set as follow.
\begin{define}[$i$-$Z^{\ast }$-$\protect\theta $-max set and $i$-$Z^{\ast }
-max set ]
\label{def:i-Z*-max}For any $\left( i,\theta \right) \in \mathcal{I}\times
\Theta $, a set $E\in 2^{Z^{\ast }}\diagdown \left\{ \varnothing \right\} $
is an $i$-$Z^{\ast }$-$\theta $-max set i
\begin{equation*}
E\subset \arg \max_{z\in E}u_{i}^{\theta }\left( z\right) \text{ and
E\subset \arg \max_{z\in Z^{\ast }}u_{j}^{\theta }\left( z\right) \text{,
\forall j\in \mathcal{I}\diagdown \left\{ i\right\} \text{.}
\end{equation*
Furthermore, $E\in 2^{Z^{\ast }}\diagdown \left\{ \varnothing \right\} $ is
an $i$-$Z^{\ast }$-max set i
\begin{equation}
\Lambda ^{i\text{-}Z^{\ast }\text{-}\Theta }\left( E\right) \equiv \left\{
\theta \in \Theta :E\text{ is an }i\text{-}Z^{\ast }\text{-}\theta \text
-max set}\right\} \neq \varnothing \text{.} \label{grr1}
\end{equation}
\end{define}
For each $E\in 2^{Z}\diagdown \left\{ \varnothing \right\} $, defin
\begin{equation*}
\mathcal{L}_{i}^{Z}\left( E,\theta \right) \equiv \cap _{z\in E}\mathcal{L
_{i}^{Z}\left( z,\theta \right) \text{.}
\end{equation*
For each $\left( i,\theta \right) \in \mathcal{I}\times \Theta $, defin
\begin{equation}
\Theta _{i}^{\theta }\equiv \left\{ \theta ^{\prime }\in \Theta :F\left(
\theta \right) \text{ is an }i\text{-}Z^{\ast }\text{-}\theta ^{\prime
\text{-max set and }F\left( \theta \right) \subset F\left( \theta ^{\prime
}\right) \right\} \text{,} \label{dtta}
\end{equation}
\begin{equation}
\Xi _{i}\left( \theta \right) \equiv \left\{ K\in 2^{\Theta _{i}^{\theta
}}\diagdown \left\{ \varnothing \right\} :\Theta _{i}^{\theta }\cap \left[
\Lambda ^{i\text{-}Z^{\ast }\text{-}\Theta }\left( Z^{\ast }\cap \mathcal{L
_{i}^{Z}\left( F\left( \theta \right) ,\theta \right) \cap \left[
\dbigcap\limits_{\theta ^{\prime }\in K}F\left( \theta ^{\prime }\right)
\right] \right) \right] =K\right\} \text{,} \label{ddttaa}
\end{equation
where $\Lambda ^{i\text{-}Z^{\ast }\text{-}\Theta }\left( \cdot \right) $ is
defined in (\ref{grr1}).
It is worthy of noting that we may replace the definition of $\Theta
_{i}^{\theta }$ in (\ref{dtta}) with $\Theta _{i}^{\theta }\equiv \Theta $,
and use it define $\Xi _{i}\left( \theta \right) $ and $\widehat{\mathcal{L}
_{i}^{Y\text{-}A\text{-}B}\left( \text{UNIF}\left[ F\left( \theta \right)
\right] ,\theta \right) $ in (\ref{ddttaa}) and (\ref{ddtt}), respectively.
With this modification, our full characterization (i.e., Theorem \re
{theorem:full:mix:SCC-A}) still holds. However, since $\Theta _{i}^{\theta }$
is a much smaller set than $\Theta $, our definition of $\Theta _{i}^{\theta
}$ in (\ref{dtta}) is much more computationally efficient, i.e., we need to
check much fewer sets in (\ref{ddttaa}) and (\ref{ddtt}).
The full characterization is established by a monotonicity condition which
is defined on modified lower-contour sets. For each $\theta \in \Theta $,
define
\begin{eqnarray}
&&\widehat{\mathcal{L}}_{i}^{Y\text{-}A\text{-}B}\left( \text{UNIF}\left[
F\left( \theta \right) \right] ,\theta \right) \label{ddtt} \\
&\equiv &\left\{
\begin{tabular}{ll}
$\triangle \left[ Z^{\ast }\cap \mathcal{L}_{i}^{Z}\left( F\left( \theta
\right) ,\theta \right) \cap \left( \dbigcup\limits_{K\in \Xi _{i}\left(
\theta \right) }\dbigcap\limits_{\theta ^{\prime }\in K}F\left( \theta
^{\prime }\right) \right) \right] $, & if $\left(
\begin{tabular}{l}
$F\left( \theta \right) \subset \arg \min_{z\in Z^{\ast }}u_{i}^{\theta
}\left( z\right) $, \\
$\Xi _{i}\left( \theta \right) \neq \varnothing $, \\
and $Z^{\ast }\cap \mathcal{L}_{i}^{Z}\left( F\left( \theta \right) ,\theta
\right) $ \\
is an $i\text{-}Z^{\ast }\text{-max set}
\end{tabular
\right) $, \\
& \\
$\left[ \triangle \left( Z^{\ast }\right) \right] \cap \mathcal{L
_{i}^{Y}\left( \text{UNIF}\left[ F\left( \theta \right) \right] ,\theta
\right) \text{,}$ & otherwis
\end{tabular
\right. \text{.} \notag
\end{eqnarray}
\begin{define}[$\protect\widehat{\mathcal{L}}^{Y\text{-}A\text{-}B}
-uniform-monotonicity]
\label{defin:A-B}$\widehat{\mathcal{L}}^{Y\text{-}A\text{-}B}
-uniform-monotonicity holds for an SCC $F$ i
\begin{equation*}
\left[
\begin{array}{c}
\widehat{\mathcal{L}}_{i}^{Y\text{-}A\text{-}B}\left( \text{UNIF}\left[
F\left( \theta \right) \right] ,\theta \right) \subset \mathcal{L
_{i}^{Y}\left( \text{UNIF}\left[ F\left( \theta \right) \right] ,\theta
^{\prime }\right) \text{, } \\
\forall i\in \mathcal{I
\end{array
\right] \text{ }\Longrightarrow F\left( \theta \right) \subset F\left(
\theta ^{\prime }\right) \text{, }\forall \left( \theta ,\theta ^{\prime
}\right) \in \Theta \times \Theta \text{.}
\end{equation*}
\end{define}
\begin{theo}
\label{theorem:full:mix:SCC-A}Given an SCC $F:\Theta \longrightarrow
2^{Z}\diagdown \left\{ \varnothing \right\} $, the following three
statements are equivalent.
(i) $F$ is mixed-Nash-A-implementable;
(ii) $F$ is mixed-Nash-B-implementable;
(iii) $\widehat{\mathcal{L}}^{Y\text{-}A\text{-}B}$-uniform-monotonicity
holds for $F$.
\end{theo}
The necessity part of $\widehat{\mathcal{L}}^{Y\text{-}A\text{-}B}
-uniform-monotonicity in Theorem \ref{theorem:full:mix:SCC-A} is implied by
the following lemma.
\begin{lemma}
\label{lem:mixed:deviation:SCC}Suppose that an SCC $F$ is
mixed-Nash-A-implemented by $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $. For any $\left( i,\theta \right) \in
\mathcal{I}\times \Theta $ and any $\lambda \in MNE^{\left( \mathcal{M}
\text{ }\theta \right) }$, we hav
\begin{multline*}
\left(
\begin{array}{c}
F\left( \theta \right) \subset \arg \min_{z\in Z^{\ast }}u_{i}^{\theta
}\left( z\right) \text{,} \\
\Xi _{i}\left( \theta \right) \neq \varnothing \text{ and} \\
\text{and }Z^{\ast }\cap \mathcal{L}_{i}^{Z}\left( F\left( \theta \right)
,\theta \right) \text{ is an }i\text{-}Z^{\ast }\text{-max set
\end{array
\right) \\
\Longrightarrow \dbigcup\limits_{m_{i}\in M_{i}}\text{SUPP}\left[ g\left(
m_{i},\lambda _{-i}\right) \right] \subset \left[ Z^{\ast }\cap \mathcal{L
_{i}^{Z}\left( F\left( \theta \right) ,\theta \right) \cap \left(
\dbigcup\limits_{E\in \Xi _{i}\left( \theta \right) }\dbigcap\limits_{\theta
^{\prime }\in E}F\left( \theta ^{\prime }\right) \right) \right] \text{.}
\end{multline*}
\end{lemma}
Like Lemma \ref{lem:mixed:deviation:SCF} for SCFs, Lemma \re
{lem:mixed:deviation:SCC} is the counterpart for SCCs, and the proof of
Lemma \ref{lem:mixed:deviation:SCC} is relegated to Appendix \re
{sec:lem:mixed:deviation:SCC}. The sufficiency part of $\widehat{\mathcal{L}
^{Y\text{-}A\text{-}B}$-uniform-monotonicity in Theorem \re
{theorem:full:mix:SCC-A} is implied by the following lemma.
\begin{lemma}
\label{lem:no-veto:generalized:SCC}Suppose that $\widehat{\mathcal{L}}^{
\text{-}A\text{-}B}$-monotonicity holds. We hav
\begin{equation*}
\left[ Z^{\ast }\text{ is a }j\text{-}Z^{\ast }\text{-}\theta ^{\prime
\text{-max set}\right] \Longrightarrow Z^{\ast }\subset F\left( \theta
^{\prime }\right) \text{, }\forall \left( j,\theta ^{\prime }\right) \in
\mathcal{I}\times \Theta \text{,}
\end{equation*
\begin{equation*}
\text{and }\left[ \widehat{\Gamma }_{j}^{A\text{-}B}\left( \theta \right)
\text{ is a }j\text{-}Z^{\ast }\text{-}\theta ^{\prime }\text{-max set
\right] \Longrightarrow \widehat{\Gamma }_{j}^{A\text{-}B}\left( \theta
\right) \subset F\left( \theta ^{\prime }\right) \text{, }\forall \left(
j,\theta ,\theta ^{\prime }\right) \in \mathcal{I}\times \Theta \times
\Theta \text{,}
\end{equation*
\begin{equation}
\text{where }\widehat{\Gamma }_{j}^{A\text{-}B}\left( \theta \right) \equiv
\dbigcup\limits_{y\in \widehat{\mathcal{L}}_{j}^{Y\text{-}A\text{-}B}\left(
\text{UNIF}\left[ F\left( \theta \right) \right] ,\theta \right) }\text{SUPP
\left[ y\right] \text{.} \label{ddtt4}
\end{equation}
\end{lemma}
Like Lemma \ref{lem:no-veto:generalized} for SCFs, Lemma \re
{lem:no-veto:generalized:SCC} is the counterpart for SCCs, and the proof of
Lemma is relegated to Appendix \ref{sec:lem:no-veto:generalized:SCC}.
The detailed proof of Theorem \ref{theorem:full:mix:SCC-A} is relegated to
Appendix \ref{sec:theorem:full:mix:SCC-A}.
\subsection{Version C and version D: full characterization}
\label{sec:extension:SCC:E}
For each $\left( i,\theta \right) \in \mathcal{I}\times \Theta $, defin
\begin{equation*}
\Theta _{i}^{\theta \text{-}C\text{-}D}\equiv \left\{ \theta ^{\prime }\in
\Theta
\begin{tabular}{l}
$F\left( \theta \right) \cap \arg \min_{z\in Z^{\ast }}u_{i}^{\theta }\left(
z\right) \text{ is an }i\text{-}Z^{\ast }\text{-}\theta ^{\prime }\text{-max
set,}$ \\
$\text{and }F\left( \theta \right) \cap \arg \min_{z\in Z^{\ast
}}u_{i}^{\theta }\left( z\right) \subset F\left( \theta ^{\prime }\right)
\end{tabular
\right\} \text{,}
\end{equation*}
\begin{equation*}
\Xi _{i}^{C\text{-}D}\left( \theta \right) \equiv \left\{ K\in 2^{\Theta
_{i}^{\theta \text{-}C\text{-}D}}\diagdown \left\{ \varnothing \right\}
:\Theta _{i}^{\theta \text{-}C\text{-}D}\cap \left[ \Lambda ^{i\text{-
Z^{\ast }\text{-}\Theta }\left( Z^{\ast }\cap \mathcal{L}_{i}^{Z}\left(
F\left( \theta \right) ,\theta \right) \cap \left[ \dbigcap\limits_{\theta
^{\prime }\in K}F\left( \theta ^{\prime }\right) \right] \right) \right]
=K\right\} \text{.}
\end{equation*
For each $\left( i,\theta ,a\right) \in \mathcal{I}\times \Theta \times Z$,
define
\begin{eqnarray*}
&&\widehat{\mathcal{L}}_{i}^{Y\text{-}C\text{-}D}\left( a,\theta \right) \\
&\equiv &\left\{
\begin{tabular}{ll}
$\triangle \left[ Z^{\ast }\cap \mathcal{L}_{i}^{Z}\left( F\left( \theta
\right) ,\theta \right) \cap \left( \dbigcup\limits_{K\in \Xi _{i}^{C\text{-
D}\left( \theta \right) }\dbigcap\limits_{\theta ^{\prime }\in K}F\left(
\theta ^{\prime }\right) \right) \right] \text{,}$ & if $\left(
\begin{tabular}{l}
$a\in F\left( \theta \right) \cap \arg \min_{z\in Z^{\ast }}u_{i}^{\theta
}\left( z\right) $, \\
$\Xi _{i}^{C\text{-}D}\left( \theta \right) \neq \varnothing $, \\
and $Z^{\ast }\cap \mathcal{L}_{i}^{Z}\left( F\left( \theta \right) ,\theta
\right) $ \\
is an $i\text{-}Z^{\ast }\text{-max set}
\end{tabular
\right) $, \\
& \\
$\triangle \left( Z^{\ast }\right) \cap \mathcal{L}_{i}^{Y}\left( a,\theta
\right) \text{,}$ & otherwis
\end{tabular
\right. \text{.}
\end{eqnarray*}
\begin{define}[$\protect\widehat{\mathcal{L}}^{Y\text{-}C\text{-}D}
-Maskin-monotonicity]
$\widehat{\mathcal{L}}^{Y\text{-}C\text{-}D}$-Maskin-monotonicity holds for
an SCC $F$ i
\begin{equation*}
\left[
\begin{array}{c}
a\in F\left( \theta \right) \text{,} \\
\widehat{\mathcal{L}}_{i}^{Y\text{-}C\text{-}D}\left( a,\theta \right)
\subset \mathcal{L}_{i}^{Y}\left( a,\theta \right) \text{, }\forall i\in
\mathcal{I
\end{array
\right] \text{ }\Longrightarrow a\in F\left( \theta ^{\prime }\right) \text
, }\forall \left( \theta ,\theta ^{\prime },a\right) \in \Theta \times
\Theta \times Z\text{.}
\end{equation*}
\end{define}
\begin{theo}
\label{theorem:full:mix:SCC-C-D}Given an SCC $F:\Theta \longrightarrow
2^{Z}\diagdown \left\{ \varnothing \right\} $, the following three
statements are equivalent.
(i) $F$ is mixed-Nash-C-implementable;
(ii) $F$ is mixed-Nash-D-implementable;
(iii) $\widehat{\mathcal{L}}^{Y\text{-}C\text{-}D}$-Maskin-monotonicity
holds for $F$.
\end{theo}
The proof of Theorem \ref{theorem:full:mix:SCC-C-D} is similar to that of
Theorem \ref{theorem:full:mix:SCC-A}, and it is relegated to \cite{sx2022b}.
\section{Ordinal implementation: full characterization}
\label{sec:ordinal:full}
Throughout this section, we fix an ordinal mode
\begin{equation*}
\left\langle \mathcal{I}=\left\{ 1,..,I\right\} \text{, \ }\Theta ^{\ast
\text{, \ }Z\text{, }F:\Theta ^{\ast }\longrightarrow 2^{Z}\diagdown \left\{
\varnothing \right\} \text{, }Y\equiv \triangle \left( Z\right) \text{,
\left( \succeq _{i}^{\theta }\right) _{\left( i,\theta \right) \in \mathcal{
}\times \Theta ^{\ast }}\right\rangle \text{,}
\end{equation*
and show that it is straightforward to derive full characterization of
ordinal mixed-Nash implementation \emph{\`{a} la} \cite{cmlr}. For any
\left( a,i,\theta \right) \in Z\times \mathcal{I}\times \Theta ^{\ast }$,
conside
\begin{eqnarray*}
\mathcal{L}_{i}^{Z}\left( a,\theta \right) &\equiv &\left\{ z\in Z:a\succeq
_{i}^{\theta }z\right\} \text{,} \\
S\mathcal{L}_{i}^{Z}\left( a,\theta \right) &\equiv &\left\{ z\in Z:a\succ
_{i}^{\theta }z\right\} \text{.}
\end{eqnarray*}
\begin{define}[set-monotonicity, \protect\cite{cmlr}]
\label{def:set-monotone}An SCC $F$ is set-monotonic if for any $\left(
\theta ,\theta ^{\prime }\right) \in \Theta ^{\ast }\times \Theta ^{\ast }$,
we have $F\left( \theta \right) \subset F\left( \theta ^{\prime }\right) $
whenever for any $i\in \mathcal{I}$, one of the following two condition
holds: either (1) $Z\subset \mathcal{L}_{i}^{Z}\left( F\left( \theta \right)
,\theta ^{\prime }\right) $ or (2) for any $a\in F\left( \theta \right) $,
both $\mathcal{L}_{i}^{Z}\left( a,\theta \right) \subset \mathcal{L
_{i}^{Z}\left( a,\theta ^{\prime }\right) $ and $\mathcal{SL}_{i}^{Z}\left(
a,\theta \right) \subset \mathcal{SL}_{i}^{Z}\left( a,\theta ^{\prime
}\right) $ hold.
\end{define}
\cite{cmlr} prove that set-monotonicity is necessary for ordinal
mixed-Nash-A-implementation.
\begin{theo}[\protect\cite{cmlr}]
\label{theorem:MR:necessary}Set-monotonicity holds if $F$ is
mixed-Nash-A-implementable on $\Theta =\cup _{\theta \in \Theta ^{\ast
}}\Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$ (i.e., $F$ is ordinally-mixed-Nash-implementable \emph{\`{a} la}
\cite{cmlr}).
\end{theo}
It is easy to show that $\mathcal{L}^{Y}$-uniform-monotonicity defined below
is necessary condition for mixed-Nash-A-implementation.\footnote
See Lemma \ref{lem:mixed-lottery:lower-contour} and the discussion in
Section \ref{sec:connection:A}.}
\begin{define}[$\mathcal{L}^{Y}$-uniform-monotonicity]
\label{defin:uniform}$\mathcal{L}^{Y}$-uniform-monotonicity holds for an SCC
$F$ i
\begin{gather*}
\left[
\begin{array}{c}
\mathcal{L}_{i}^{Y}\left( \text{UNIF}\left[ F\left( \theta \right) \right]
,\theta \right) \subset \mathcal{L}_{i}^{Y}\left( \text{UNIF}\left[ F\left(
\theta \right) \right] ,\theta ^{\prime }\right) \text{, } \\
\forall i\in \mathcal{I
\end{array
\right] \text{ }\Longrightarrow F\left( \theta \right) \subset F\left(
\theta ^{\prime }\right) \text{, }\forall \left( \theta ,\theta ^{\prime
}\right) \in \Theta \times \Theta \text{,} \\
\text{where }\Theta =\cup _{\theta \in \Theta ^{\ast }}\Omega ^{\left[
\succeq ^{\theta },\text{
\mathbb{R}
\right] }\text{.}
\end{gather*}
\end{define}
\begin{lemma}[\protect\cite{cmlr}, Proposition 1]
\label{lem:set-monotone}The following statements are equivalent.
(i) set-monotonicity holds;
(ii) $\mathcal{L}^{Y}$-uniform-monotonicity holds for $F$ on $\Theta =\cup
_{\theta \in \Theta ^{\ast }}\Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$;
(iii) $\mathcal{L}^{Y}$-uniform-monotonicity holds for $F$ on $\Theta =\cup
_{\theta \in \Theta ^{\ast }}\Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{Q}\right] }$.
\end{lemma}
(i) being equivalent to (ii)\ in Lemma \ref{lem:set-monotone} is Proposition
1 in \cite{cmlr}, which provides its proof. The same argument shows that (i)
is equivalent (iii).
Given Lemma \ref{lem:set-monotone}, the following theorem shows that Theorem
\ref{theorem:MR:necessary} is immediately implied by Theorem \re
{theorem:full:mix:SCC-A}, because $\mathcal{L}^{Y}$-uniform-monotonicity is
immediately implied by $\widehat{\mathcal{L}}^{Y\text{-}A\text{-}B}
-uniform-monotonicity. Theorem \ref{thm:mixed:ordinal:A} is implied by
Theorem \ref{theorem:MR:iff} below.
\begin{theo}
\label{thm:mixed:ordinal:A}An SCC $F\ $is mixed-Nash-A-implementable on
\Theta =\cup _{\theta \in \Theta ^{\ast }}\Omega ^{\left[ \succeq ^{\theta }
\text{ }\mathbb{Q}\right] }$ if and only if $F\ $is
mixed-Nash-A-implementable on $\Theta ^{\prime }=\cup _{\theta \in \Theta
^{\ast }}\Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$.
\end{theo}
In light of Theorem \ref{theorem:full:mix:SCC-A}, defin
\begin{gather*}
\forall \left( i,a,\theta \right) \in Z\times \Theta \text{,} \\
\widehat{\mathcal{L}}_{i}^{Z^{\ast }\text{-}A\text{-}B}\left( a,\theta
\right) \equiv \left\{
\begin{tabular}{ll}
$\left[ Z^{\ast }\cap \mathcal{L}_{i}^{Z}\left( F\left( \theta \right)
,\theta \right) \cap \left( \dbigcup\limits_{K\in \Xi _{i}\left( \theta
\right) }\dbigcap\limits_{\theta ^{\prime }\in K}F\left( \theta ^{\prime
}\right) \right) \right] $, & if $\left(
\begin{tabular}{l}
$F\left( \theta \right) \subset \arg \min_{z\in Z^{\ast }}u_{i}^{\theta
}\left( z\right) $, \\
$\Xi _{i}\left( \theta \right) \neq \varnothing $, \\
and $Z^{\ast }\cap \mathcal{L}_{i}^{Z}\left( F\left( \theta \right) ,\theta
\right) $ \\
is an $i\text{-}Z^{\ast }\text{-max set}
\end{tabular
\right) $, \\
& \\
$Z^{\ast }\cap \mathcal{L}_{i}^{Z}\left( a,\theta \right) \text{,}$ &
otherwis
\end{tabular
\right. \text{,} \\
\\
\widehat{S\mathcal{L}}_{i}^{Z^{\ast }\text{-}A\text{-}B}\left( a,\theta
\right) \equiv \widehat{\mathcal{L}}_{i}^{Z^{\ast }\text{-}A\text{-}B}\left(
a,\theta \right) \cap S\mathcal{L}_{i}^{Z}\left( a,\theta \right) \text{.}
\end{gather*}
\begin{define}[strong set-monotonicity]
\label{def:set-monotone:weak}An SCC $F$ is strongly set-monotonic if for any
$\left( \theta ,\theta ^{\prime }\right) \in \Theta ^{\ast }\times \Theta
^{\ast }$, we have $F\left( \theta \right) \subset F\left( \theta ^{\prime
}\right) $ whenever for any $i\in \mathcal{I}$, one of the following two
condition holds: either (1) $Z^{\ast }\subset \mathcal{L}_{i}^{Z}\left(
F\left( \theta \right) ,\theta ^{\prime }\right) $ or (2) for any $a\in
F\left( \theta \right) $, both $\widehat{\mathcal{L}}_{i}^{Z^{\ast }\text{-}
\text{-}B}\left( a,\theta \right) \subset \mathcal{L}_{i}^{Z}\left( a,\theta
^{\prime }\right) $ and $\widehat{S\mathcal{L}}_{i}^{Z^{\ast }\text{-}A\text
-}B}\left( a,\theta \right) \subset \mathcal{SL}_{i}^{Z}\left( a,\theta
^{\prime }\right) $ hold.
\end{define}
Using a similar argument as in the proof of Lemma \ref{lem:set-monotone} (or
equivalently, Proposition 1 in \cite{cmlr}), it is straightforward to show
the following lemma.
\begin{lemma}
\label{lem:set-monotone:strong}The following statements are equivalent.
(i) strong set-monotonicity holds;
(ii) $\widehat{\mathcal{L}}^{Y\text{-}A\text{-}B}$-uniform-monotonicity
holds for $F$ on $\Theta =\cup _{\theta \in \Theta ^{\ast }}\Omega ^{\left[
\succeq ^{\theta },\text{
\mathbb{R}
\right] }$;
(iii) $\widehat{\mathcal{L}}^{Y\text{-}A\text{-}B}$-uniform-monotonicity
holds for $F$ on $\Theta =\cup _{\theta \in \Theta ^{\ast }}\Omega ^{\left[
\succeq ^{\theta },\text{ }\mathbb{Q}\right] }$.
\end{lemma}
This immediately leads to the following full characterization, and the proof
is relegated to Appendix \ref{sec:theorem:MR:iff}.
\begin{theo}
\label{theorem:MR:iff}The following statements are equivalent.
(i) strong set-monotonicity holds;
(ii) $F$ is mixed-Nash-A-implementable on $\Theta =\cup _{\theta \in \Theta
^{\ast }}\Omega ^{\left[ \succeq ^{\theta },\text{ }\mathbb{Q}\right] }$;
(iii) $F$ is mixed-Nash-A-implementable on $\Theta =\cup _{\theta \in \Theta
^{\ast }}\Omega ^{\left[ \succeq ^{\theta },\text{
\mathbb{R}
\right] }$.
\end{theo}
A prominent class of preferences discussed in \cite{cmlr} is the single-top
preferences. Given single-top preferences, it is straightforward to show
\begin{eqnarray*}
Z &=&Z^{\ast }\text{,} \\
\mathcal{L}_{i}^{Y}\left( \text{UNIF}\left[ F\left( \theta \right) \right]
,\theta \right) &\equiv &\widehat{\mathcal{L}}_{i}^{Y\text{-}A\text{-
B}\left( \text{UNIF}\left[ F\left( \theta \right) \right] ,\theta \right)
\text{,} \\
\text{and set-monotonicity} &\Longleftrightarrow &\text{strong
set-monotonicity.}
\end{eqnarray*
As a result, Theorem \ref{theorem:MR:iff} implies that set-monotonicity
fully characterizes ordinally-mixed-Nash-implementable \emph{\`{a} la} \cit
{cmlr}.
Similarly, we can easy derive full characterization of ordinal
implementation for the other 5 versions of mixed-Nash implementation of SCCs.
\section{Compared to rationalizable implementation}
\label{sec:rationalizable}
Given a mechanism $\mathcal{M}=\left\langle M\equiv \times _{i\in \mathcal{I
}M_{i}\text{, \ }g:M\longrightarrow Y\right\rangle $, define $\mathcal{S
_{i}\equiv 2^{M_{i}}$ and $\mathcal{S}=\times _{i\in \mathcal{I}}\mathcal{S
_{i}$ for each $i\in \mathcal{I}$. For each state $\theta \in \Theta $,
consider an operator $b^{\mathcal{M},\text{ }\theta }:\mathcal{S
\longrightarrow \mathcal{S}$ with $b^{\mathcal{M},\text{ }\theta }\equiv
\left[ b_{i}^{\mathcal{M},\text{ }\theta }:\mathcal{S}\longrightarrow
\mathcal{S}_{i}\right] _{i\in \mathcal{I}}$, where each $b_{i}^{\mathcal{M}
\text{ }\theta }$ is defined as follows. For every $S\in \mathcal{S}$,
\begin{equation*}
b_{i}^{\mathcal{M},\text{ }\theta }\left( S\right) =\left\{ m_{i}\in M_{i}
\begin{tabular}{c}
$\exists \lambda _{-i}\in \triangle \left( M_{-i}\right) $ such that \\
(1) $\lambda _{-i}\left( m_{-i}\right) >0$ implies $m_{-i}\in S_{-i}$, and
\\
(2) $m_{i}\in \arg \max_{m_{i}^{\prime }\in M_{i}}\Sigma _{m_{-i}\in
M_{-i}}\lambda _{-i}\left( m_{-i}\right) u_{i}\left( g\left( m_{i}^{\prime
},m_{-i}\right) ,\theta \right)
\end{tabular
\right\} \text{.}
\end{equation*
Clearly, $\mathcal{S}$ is a lattice with the order of "set inclusion," and
b^{\mathcal{M},\text{ }\theta }$ is monotonically increasing.\footnote
That is, $S\subset S^{\prime }$ implies $b^{\mathcal{M},\text{ }\theta
}\left( S\right) \subset b^{\mathcal{M},\text{ }\theta }\left( S^{\prime
}\right) $.} Thus, Tarski's fixed point theorem implies existence of a
largest fixed point of $b^{\mathcal{M},\text{ }\theta }$, and we denote it
by $S^{\mathcal{M},\text{ }\theta }\equiv \left( S_{i}^{\mathcal{M},\text{
\theta }\right) _{i\in \mathcal{I}}$. We say $m_{i}\in M_{i}$ is
rationalizable in $\mathcal{M}$ at state $\theta $ if and only if $m_{i}\in
S_{i}^{\mathcal{M},\text{ }\theta }$.
We say that $S\in \mathcal{S}$ satisfies the best reply property in
\mathcal{M}$ at $\theta $ if and only if $S\subset b^{\mathcal{M},\text{
\theta }\left( S\right) $. It is straightforward to show that $S\subset S^
\mathcal{M},\text{ }\theta }$ if $S$ satisfies the best reply property.
\begin{define}[\protect\cite{rjain}]
\label{def:rationalizable}An SCC $F:\Theta \longrightarrow 2^{Z}\diagdown
\left\{ \varnothing \right\} $ is rationalizably implementable if there
exists a mechanism $\mathcal{M}=\left\langle M\text{, \ }g:M\longrightarrow
Y\right\rangle $ such tha
\begin{equation*}
\dbigcup\limits_{m\in S^{\mathcal{M},\text{ }\theta }}\text{SUPP}\left[
g\left( m\right) \right] =F\left( \theta \right) \text{, }\forall \theta \in
\Theta \text{.}
\end{equation*}
\end{define}
\begin{theo}
\label{theorem:rationalizable}An SCC $F:\Theta \longrightarrow
2^{Z}\diagdown \left\{ \varnothing \right\} $ is mixed-Nash-A-implementatble
if $F$ is rationalizably-implementable.
\end{theo}
The detailed proof of Theorem \ref{theorem:rationalizable} is relegated to
\cite{sx2022b}.
\section{Connected to \protect\cite{jmrr} and \protect\cite{tomas}}
\label{sec:connection}
In this section, we illustrate \cite{jmrr} and \cite{tomas}. We show that
our full characterization share the same conceptual ideas as those in \cit
{jmrr} and \cite{tomas}, and furthermore, we show why their full
characterization is complicated, and why ours is simple.
\subsection{A common conceptual idea}
\label{sec:connection:concept}
\cite{em} proves that Maskin monotonicity almost fully characterizes Nash
implementation. As being showed in \cite{jmrr} and illustrated in \cit
{tomas}, in order to fully characterize Nash implementation, we need to take
two additional steps before defining Maskin monotonicity. All of these
papers use the canonical mechanism in \cite{em} to achieve Nash
implementation, and the two additional steps corresponds to eliminating bad
equilibria in Case (3) and Case (2) of the canonical mechanism. Roughly, we
have the following two additional steps
\begin{gather}
\text{Step (I): select }\widehat{Z}\in 2^{Z}\diagdown \left\{ \varnothing
\right\} \text{ such that }\cup _{\theta \in \Theta }F\left( \theta \right)
\subset \widehat{Z}\text{,} \notag \\
\text{and }\widehat{Z}\text{ satisfies a unanimity condition:} \notag \\
\forall \left( \theta ^{\ast },y\right) \in \Theta \times \widehat{Z}\text{,
} \notag \\
\left[
\begin{array}{c}
y\in \arg \max_{z\in \widehat{Z}}U_{i}^{\theta ^{\ast }}\left( z\right)
\text{,} \\
\forall i\in \mathcal{I
\end{array
\right] \Longrightarrow \text{"}y\text{ is a good outcome at }\theta ^{\ast
\text{."} \label{tgr1}
\end{gather
where $\widehat{Z}$ is the set of outcomes that agents can choose when Case
(3) is triggered in the canonical mechanism. In particular, "$y$ is a good
outcome at $\theta ^{\ast }$" in (\ref{tgr1}) may have different
formalization in different environments and under different implementation
notions, which will be illustrated in Sections \ref{sec:connection:M-R}, \re
{sec:illustration:SCF} and \ref{sec:connection:A}.
To see the necessity of Step (I), suppose an SCC $F$ is Nash implemented by
\mathcal{M}=\left\langle M\text{, \ }g:M\longrightarrow Y\right\rangle $,
and define $\widehat{Z}=g\left( M\right) $. If $y=g\left( m\right) $
satisfies the left-hand side of (\ref{tgr1}) for some $\theta ^{\ast }\in
\Theta $, then, $m$ must be a Nash equilibrium at $\theta ^{\ast }$, and
hence, $y=g\left( m\right) $ must be a good outcome $\theta ^{\ast }$, i.e.,
the right-hand side of (\ref{tgr1}) holds.
To see the sufficiency of Step (I), suppose the true state is $\theta ^{\ast
}$. Consider any Nash equilibrium in Case (3) of the canonical mechanism,
which induces $y\in \widehat{Z}$. Then, $y$ must be a top outcome in
\widehat{Z}$ for all agents, i.e., the left-hand side of (\ref{tgr1}) holds.
Thus, by (\ref{tgr1}), $y$ is a good outcome at $\theta ^{\ast }$.
\begin{gather}
\text{Step (II): for each }\left( \theta ,i\right) \in \Theta \times
\mathcal{I}\text{ and each }a\in F\left( \theta \right) \text{, select
\widehat{\mathcal{L}}_{i}\left( a,\theta \right) \in 2^{\left[ \widehat{Z
\cap \mathcal{L}_{i}\left( a,\theta \right) \right] }\diagdown \left\{
\varnothing \right\} \text{ } \notag \\
\text{such that }a\in \widehat{\mathcal{L}}_{i}\left( a,\theta \right) \text
,} \notag \\
\text{and }\widehat{\mathcal{L}}_{i}\left( a,\theta \right) \text{ satisfies
a weak no-veto condition:} \notag \\
\forall \left( \theta ^{\ast },y\right) \in \Theta \times \widehat{\mathcal{
}}_{i}\left( a,\theta \right) \text{, } \notag \\
\left[
\begin{array}{c}
y\in \arg \max_{z\in \widehat{Z}}U_{j}^{\theta ^{\ast }}\left( z\right)
\text{, }\forall j\in \mathcal{I}\diagdown \left\{ i\right\} \text{,} \\
y\in \arg \max_{z\in \widehat{\mathcal{L}}_{i}\left( a,\theta \right)
}U_{i}^{\theta ^{\ast }}\left( z\right) \text{,
\end{array
\right] \Longrightarrow \text{"}y\text{ is a good outcome at }\theta ^{\ast
\text{." } \label{tgr2}
\end{gather
where $\widehat{\mathcal{L}}_{i}\left( a,\theta \right) $ is the set of
outcomes that agent $i$ can choose when agent $i$ is the whistle-blower and
agents $-i$ report $\left( a,\theta \right) $, i.e., Case (2) is triggered
in the canonical mechanism.
To see the necessity of Step (II), suppose an SCC $F$ is Nash implemented by
$\mathcal{M}=\left\langle M\text{, \ }g:M\longrightarrow Y\right\rangle $,
and defin
\begin{equation*}
\widehat{\mathcal{L}}_{i}\left( a,\theta \right) =\left\{ g\left(
m_{i},\lambda _{-i}\right)
\begin{tabular}{l}
$m_{i}\in M_{i}\text{,}$ \\
$g\left( \lambda _{i},\lambda _{-i}\right) =a\text{,}$ \\
$\text{and }\left( \lambda _{i},\lambda _{-i}\right) \text{ is a Nash
equilibrium}
\end{tabular
\right\} \text{.}
\end{equation*
If $y=g\left( m_{i},\lambda _{-i}\right) $ satisfies the left-hand side of
\ref{tgr2}) for some $\theta ^{\ast }\in \Theta $, $\left( m_{i},\lambda
_{-i}\right) $ must be a Nash equilibrium at $\theta ^{\ast }$, and
y=g\left( m_{i},\lambda _{-i}\right) $ must be a good outcome at $\theta
^{\ast }$, i.e., the right-hand side of (\ref{tgr2}) holds.
To see the sufficiency of Step (II), suppose the true state is $\theta
^{\ast }$. Consider any Nash equilibrium in Case (2) of the canonical
mechanism such that $i$ is the whistle-blower and it induces $y\in \widehat
\mathcal{L}}_{i}\left( a,\theta \right) $. Then, $y$ must be a top outcome
in $\widehat{Z}$ for agents $-i$ (because they can deviate to Case (3)), and
$y$ must be a top outcome in $\widehat{\mathcal{L}}_{i}\left( a,\theta
\right) $ for agents $i$ (because $i$ can deviate to any outcome in
\widehat{\mathcal{L}}_{i}\left( a,\theta \right) $), i.e., the left-hand
side of (\ref{tgr2}) holds. Thus, by (\ref{tgr2}), $y$ is a good outcome at
\theta ^{\ast }$.
Given Steps (I) and (II), we define a modified Maskin monotonicity as
follows
\begin{gather*}
\widehat{\mathcal{L}}\text{-Maskin-monotonicity holds iff} \\
\text{ }\left[
\begin{array}{c}
a\in F\left( \theta \right) \text{,} \\
\widehat{\mathcal{L}}_{i}\left( a,\theta \right) \subset \mathcal{L
_{i}\left( a,\theta \right) \text{, }\forall i\in \mathcal{I
\end{array
\right] \text{ }\Longrightarrow a\in F\left( \theta ^{\prime }\right) \text
, }\forall \left( \theta ,\theta ^{\prime },a\right) \in \Theta \times
\Theta \times Z\text{.}
\end{gather*
This leads to a full characterization of Nash implementation: an SCC $F$ is
Nash implementable if and only if $\widehat{\mathcal{L}}
-Maskin-monotonicity holds. In particular, in the canonical mechanism, Step
(I) eliminates bad equilibria in Case (3), and Step (II) eliminates bad
equilibria in Case (2), and $\widehat{\mathcal{L}}$-Maskin-monotonicity
eliminates bad equilibria in Case (1).
\subsection{The full characterization in \protect\cite{jmrr} and
\protect\cite{tomas}}
\label{sec:connection:M-R}
In the environment of \cite{jmrr}, the two steps becomes
\begin{gather}
\text{Step (I): select }\widehat{Z}\in 2^{Z}\diagdown \left\{ \varnothing
\right\} \text{ such that }\cup _{\theta \in \Theta }F\left( \theta \right)
\subset \widehat{Z}\text{ and} \notag \\
\forall \left( \theta ^{\ast },y\right) \in \Theta \times \widehat{Z}\text{,
} \notag \\
\left[
\begin{array}{c}
y\in \arg \max_{z\in \widehat{Z}}U_{i}^{\theta ^{\ast }}\left( z\right)
\text{,} \\
\forall i\in \mathcal{I
\end{array
\right] \Longrightarrow y\in F\left( \theta ^{\ast }\right) \text{;}
\label{hhg1}
\end{gather}
\begin{gather}
\text{Step (II): for each }\left( \theta ,i\right) \in \Theta \times
\mathcal{I}\text{ and each }a\in F\left( \theta \right) \text{, select
\widehat{\mathcal{L}}_{i}\left( a,\theta \right) \in 2^{\left[ \widehat{Z
\cap \mathcal{L}_{i}^{Z}\left( a,\theta \right) \right] }\diagdown \left\{
\varnothing \right\} \notag \\
\text{such that }a\in \widehat{\mathcal{L}}_{i}\left( a,\theta \right) \text{
and} \notag \\
\forall \left( \theta ^{\ast },y\right) \in \Theta \times \widehat{\mathcal{
}}_{i}\left( a,\theta \right) \text{, } \notag \\
\left[
\begin{array}{c}
y\in \arg \max_{z\in \widehat{Z}}U_{j}^{\theta ^{\ast }}\left( z\right)
\text{, }\forall j\in \mathcal{I}\diagdown \left\{ i\right\} \text{,} \\
y\in \arg \max_{z\in \widehat{\mathcal{L}}_{i}\left( a,\theta \right)
}U_{i}^{\theta ^{\ast }}\left( z\right) \text{,
\end{array
\right] \Longrightarrow y\in F\left( \theta ^{\ast }\right) \text{ }
\label{hhg2}
\end{gather
This leads to the full characterization in \cite{jmrr}: an SCC $F$ is Nash
implementable if and only if there exists such $\left[ \widehat{Z}\text{,
\left( \widehat{\mathcal{L}}_{i}\left( a,\theta \right) \right) _{i\in
\mathcal{I}\text{, }\theta \in \Theta \text{, }a\in F\left( \theta \right)
\right] $ and $\widehat{\mathcal{L}}$-Maskin-monotonicity hold. However,
\cite{jmrr} is silent regarding how to find such $\left[ \widehat{Z}\text{,
\left( \widehat{\mathcal{L}}_{i}\left( a,\theta \right) \right) _{i\in
\mathcal{I}\text{, }\theta \in \Theta \text{, }a\in F\left( \theta \right)
\right] $, while \cite{tomas} provides an algorithm to find the largest such
sets.\footnote
There may be multiple candidates of $\widehat{Z}$ which satisfies (\ref{hhg1
). The union of these candidates is the largest $\widehat{Z}$ satisfying
\ref{hhg1}), which is identified by \cite{tomas}. Similarly, \cite{tomas}
identifies the largest such $\widehat{\mathcal{L}}_{i}\left( a,\theta
\right) $ for each $i\in \mathcal{I}$, $\theta \in \Theta $ and $a\in
F\left( \theta \right) $.} In order to find $\widehat{Z}$, we need an
iterative process of elimination. At round 1, define $Z^{1}\equiv Z$. If
\widehat{Z}=Z^{1}$ does not satisfy (\ref{hhg1}), we eliminate any $z$ that
is a top outcome in $Z^{1}$ for all agents at some state $\theta $, but
z\notin F\left( \theta \right) $, and let $Z^{2}$ denote the set of outcomes
that survive round 1. We have found the appropriate $\widehat{Z}$ if
\widehat{Z}=Z^{2}$ satisfies (\ref{hhg1}). However, $\widehat{Z}=Z^{2}$ may
not satisfy (\ref{hhg1}), i.e., it may happen that some $z\in Z^{1}$ is not
a top outcome of some agent in $Z^{1}$ at some state $\theta $, but becomes
a top outcome in $Z^{2}$ for all agents at $\theta $, while $z\notin F\left(
\theta \right) $. In this case, we need another round of elimination. At
round 2, if $\widehat{Z}=Z^{2}$ does not satisfy (\ref{hhg1}), we eliminate
any $z$ that is a top outcome in $Z^{2}$ for all agents at some state
\theta $, but $z\notin F\left( \theta \right) $, and let $Z^{3}$ denote the
set of outcomes that survive round 2.... We continue this process until we
find $Z^{n}$ such that $\widehat{Z}=Z^{n}$ satisfies (\ref{hhg1}).
After finding $\widehat{Z}$, we need a similar iterative process to find
each $\widehat{\mathcal{L}}_{i}\left( a,\theta \right) $. At round 1, define
$\mathcal{L}_{i}^{1}\left( a,\theta \right) \equiv \mathcal{L}_{i}\left(
a,\theta \right) $. If $\widehat{\mathcal{L}}_{i}\left( a,\theta \right)
\mathcal{L}_{i}^{1}\left( a,\theta \right) $ does not satisfy (\ref{hhg2}),
we eliminate any $z$ that is a top outcome in $\widehat{Z}$ for agents $-i$
at some state $\theta ^{\ast }$ and a top outcome in $\mathcal{L
_{i}^{1}\left( a,\theta \right) $ for agent $i$ at $\theta ^{\ast }$, but
z\notin F\left( \theta ^{\ast }\right) $. Let $\mathcal{L}_{i}^{2}\left(
a,\theta \right) $ denote the set of outcomes that survive round 1. At round
2, if $\widehat{\mathcal{L}}_{i}\left( a,\theta \right) =\mathcal{L
_{i}^{2}\left( a,\theta \right) $ does not satisfy (\ref{hhg2}), we
eliminate any $z$ that is a top outcome in $\widehat{Z}$ for agents $-i$ at
some state $\theta ^{\ast }$ and a top outcome in $\mathcal{L}_{i}^{2}\left(
a,\theta \right) $ for agent $i$ at $\theta ^{\ast }$, but $z\notin F\left(
\theta ^{\ast }\right) $. Let $\mathcal{L}_{i}^{3}\left( a,\theta \right) $
denote the set of outcomes that survive round 2.... We continue this process
until we find $\mathcal{L}_{i}^{n}\left( a,\theta \right) $ such that
\widehat{\mathcal{L}}_{i}\left( a,\theta \right) =\mathcal{L}_{i}^{n}\left(
a,\theta \right) $ satisfies (\ref{hhg2}).
Clearly, both the existential statement in \cite{jmrr} and the iterative
process in \cite{tomas} make their full characterization complicated.
\subsection{Our full characterization for SCFs}
\label{sec:illustration:SCF}
For simplicity, we focus on SCFs throughout this subsection. Given
stochastic mechanisms, the insight of this paper is that we can easily
select $\left[ \widehat{Z}\text{, }\left( \widehat{\mathcal{L}}_{i}\left(
f\left( \theta \right) ,\theta \right) \right) _{i\in \mathcal{I}\text{,
\theta \in \Theta }\right] $ in Step (I) and Step (II). Taking full
advantage of the convexity structure of lotteries, we assign the following
lottery, when Case (3) is triggered in the canonical mechanism
\begin{equation*}
g\left[ m\right] =\left( 1-\frac{1}{k_{j^{\ast }}^{2}}\right) \times
b_{j^{\ast }}+\frac{1}{k_{j^{\ast }}^{2}}\times \text{UNIF}\left( \widehat{Z
\right) \text{.}
\end{equation*
In particular, the winner of the integer game, i.e., $j^{\ast }$, can
increase the probability of her top outcome $b_{j^{\ast }}$ by increasing
k_{j^{\ast }}^{2}$. As a result, a Nash equilibrium in Case (3) at true
state $\theta ^{\ast }$ must require all agents be indifferent between any
two outcomes in $\widehat{Z}$ at $\theta ^{\ast }$. Given SCFs, this mean
\footnote
Given the canonical mechanism $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $, all agents being indifferent between
any two outcomes in $\widehat{Z}$ at $\theta ^{\ast }$ implies any $m\in M$
is a Nash equilibrium at $\theta ^{\ast }$, i.e., $M\subset MNE^{\left(
\mathcal{M},\text{ }\theta ^{\ast }\right) }$. Since $\cup _{m\in M}$SUPP
\left[ g\left( m\right) \right] =\widehat{Z}$, we conclude that $\widehat{Z
=\cup _{m\in M}$SUPP$\left[ g\left( m\right) \right] =\left\{ f\left( \theta
^{\ast }\right) \right\} $, i.e., $\left\vert \widehat{Z}\right\vert =1$.
\begin{gather*}
\text{Step (I): select }\widehat{Z}\in 2^{Z}\diagdown \left\{ \varnothing
\right\} \text{ such that }f\left( \Theta \right) \subset \widehat{Z}\text{
and} \\
\forall \theta ^{\ast }\in \Theta \text{, } \\
\left[
\begin{array}{c}
\widehat{Z}\subset \arg \max_{z\in \widehat{Z}}U_{i}^{\theta ^{\ast }}\left(
z\right) \text{,} \\
\forall i\in \mathcal{I
\end{array
\right] \Longrightarrow \left(
\begin{array}{c}
\widehat{Z}\subset \left\{ f\left( \theta ^{\ast }\right) \right\} \text{,}
\\
\text{and hence, }\left\vert \widehat{Z}\right\vert =1\text{,
\end{array
\right) \text{.}
\end{gather*
Assumption \ref{assm:non-trivial} and $f\left( \Theta \right) \subset
\widehat{Z}$ imply that $\left\vert \widehat{Z}\right\vert =1$ always fails.
As a result, Step (I)\ become
\begin{gather*}
\text{Step (I): select }\widehat{Z}\in 2^{Z}\diagdown \left\{ \varnothing
\right\} \text{ such that }f\left( \Theta \right) \subset \widehat{Z}\text{
and} \\
\forall \theta ^{\ast }\in \Theta \text{, } \\
\left[
\begin{array}{c}
\widehat{Z}\subset \arg \max_{z\in \widehat{Z}}U_{i}^{\theta ^{\ast }}\left(
z\right) \text{,} \\
\forall i\in \mathcal{I
\end{array
\right] \text{ fails.}
\end{gather*
Since $\widehat{Z}\subset Z$, it is straightforward to sho
\begin{eqnarray*}
\left[
\begin{array}{c}
Z\text{ is not a }i\text{-max set} \\
\forall i\in \mathcal{I
\end{array
\right] &\Longleftrightarrow &\left(
\begin{array}{c}
\left[
\begin{array}{c}
Z\subset \arg \max_{z\in \widehat{Z}}U_{i}^{\theta ^{\ast }}\left( z\right)
\text{,} \\
\forall i\in \mathcal{I
\end{array
\right] \text{ fails} \\
\forall \theta ^{\ast }\in \Theta \text{,
\end{array
\right) \\
&\Longleftarrow &\left(
\begin{array}{c}
\left[
\begin{array}{c}
\widehat{Z}\subset \arg \max_{z\in \widehat{Z}}U_{i}^{\theta ^{\ast }}\left(
z\right) \text{,} \\
\forall i\in \mathcal{I
\end{array
\right] \text{ fails} \\
\forall \theta ^{\ast }\in \Theta \text{,
\end{array
\right) \text{.}
\end{eqnarray*
Therefore, without an iterative process of elimination, we have already
found the largest such $\widehat{Z}$, i.e., $\widehat{Z}=Z$. In particular,
a necessary condition for Nash implementation is: $Z$ is not a $i$-max set
for any $i\in \mathcal{I}$.\footnote
This necessary condition is explicitly stated in (iii) of Theorem \re
{theorem:full:mix:SCC-E-F}, but omitted in Theorem \ref{theorem:full:mix},
because, with SCFs, it is implicitly encoded in $\widehat{\mathcal{L}}^{Y}
-monotonicity\ (see Lemma \ref{lem:no-veto:generalized}).}
Similarly, taking full advantage of the convexity structure of lotteries, we
assign the following lottery, when Case (2) is triggered in the canonical
mechanism
\begin{eqnarray*}
g\left[ m\right] &=&\left( 1-\frac{1}{k_{j}^{2}}\right) \times \phi
_{j}^{\theta }\left( \theta _{j}\right) \\
&&+\frac{1}{k_{j}^{2}}\times \left(
\begin{tabular}{l}
$\varepsilon _{j}^{\theta }\times \left[ \left( 1-\frac{1}{k_{j}^{3}}\right)
\times \gamma _{j}\left( \widehat{\Gamma }_{j}\left( \theta \right) \right)
\frac{1}{k_{j}^{3}}\times \text{UNIF}\left( \widehat{\Gamma }_{j}\left(
\theta \right) \right) \right] $ \\
$+\left( 1-\varepsilon _{j}^{\theta }\right) \times y_{j}^{\theta }
\end{tabular
\right) \text{,}
\end{eqnarray*
In particular, the whistle-blower $j$ can increase the probability of her
top outcome $\gamma _{j}\left( \widehat{\Gamma }_{j}\left( \theta \right)
\right) $ in $\widehat{\Gamma }_{j}\left( \theta \right) $ by increasing
k_{j}^{3}$. As a result, a Nash equilibrium in Case (2) at true state
\theta ^{\ast }$ must require all outcomes in $\widehat{\Gamma }_{j}\left(
\theta \right) $ be top for agents $-j$ at $\theta ^{\ast }$ and agent $j$
be indifferent between any two outcomes in $\widehat{\Gamma }_{j}\left(
\theta \right) $ at $\theta ^{\ast }$. Given SCFs, we thus have
\begin{gather}
\text{Step (II): for each }\left( \theta ,i\right) \in \Theta \times
\mathcal{I}\text{, select }\widehat{\mathcal{L}}_{i}^{Y}\left( f\left(
\theta \right) ,\theta \right) \in 2^{\left[ \mathcal{L}_{i}^{Y}\left(
f\left( \theta \right) ,\theta \right) \right] }\diagdown \left\{
\varnothing \right\} \notag \\
\text{such that }f\left( \theta \right) \in \widehat{\mathcal{L}
_{i}^{Y}\left( f\left( \theta \right) ,\theta \right) \text{ and} \notag \\
\forall \left( \theta ^{\ast },y\right) \in \Theta \times \widehat{\mathcal{
}}_{i}^{Y}\left( f\left( \theta \right) ,\theta \right) \text{, } \notag \\
\left[
\begin{array}{c}
\widehat{\Gamma }_{i}\left( \theta \right) \subset \arg \max_{z\in
Z}u_{j}^{\theta ^{\ast }}\left( z\right) \text{, }\forall j\in \mathcal{I
\diagdown \left\{ i\right\} \text{,} \\
\widehat{\Gamma }_{i}\left( \theta \right) \subset \arg \max_{z\in \widehat
\Gamma }_{i}\left( \theta \right) }u_{i}^{\theta ^{\ast }}\left( z\right)
\text{,
\end{array
\right] \Longrightarrow \left(
\begin{array}{c}
\widehat{\Gamma }_{i}\left( \theta \right) \subset \left\{ f\left( \theta
^{\ast }\right) \right\} \text{,} \\
\text{and hence, }\left\vert \widehat{\Gamma }_{i}\left( \theta \right)
\right\vert =1\text{,
\end{array
\right) \text{,} \notag \\
\text{or equivalently, }\left[ \widehat{\Gamma }_{i}\left( \theta \right)
\text{ is an }i\text{-}\theta ^{\ast }\text{-max set}\right] \Longrightarrow
\left(
\begin{array}{c}
\widehat{\Gamma }_{i}\left( \theta \right) \subset \left\{ f\left( \theta
^{\ast }\right) \right\} \text{,} \\
\text{and hence, }\left\vert \widehat{\Gamma }_{i}\left( \theta \right)
\right\vert =1\text{,
\end{array
\right) \text{,} \label{bbt1} \\
\text{where }\widehat{\Gamma }_{i}\left( \theta \right) \equiv
\dbigcup\limits_{y\in \widehat{\mathcal{L}}_{i}^{Y}\left( f\left( \theta
\right) ,\theta \right) }\text{SUPP}\left[ y\right] \text{.} \notag
\end{gather
Thus, it is straightforward to find the largest such $\widehat{\mathcal{L}
_{i}^{Y}\left( f\left( \theta \right) ,\theta \right) $ by considering three
scenarios
\begin{equation*}
\left(
\begin{array}{c}
\text{scenario (I): }f\left( \theta \right) \in \arg \min_{z\in
Z}u_{i}^{\theta }\left( z\right) \text{ \ and }\mathcal{L}_{i}^{Z}\left(
f\left( \theta \right) ,\theta \right) \text{ is an }i\text{-max set,} \\
\text{scenario (II): }f\left( \theta \right) \in \arg \min_{z\in
Z}u_{i}^{\theta }\left( z\right) \text{ \ and }\mathcal{L}_{i}^{Z}\left(
f\left( \theta \right) ,\theta \right) \text{ is not an }i\text{-max set,}
\\
\text{scenario (III): }f\left( \theta \right) \notin \arg \min_{z\in
Z}u_{i}^{\theta }\left( z\right
\end{array
\right) \text{.}
\end{equation*
In scenario (I), we must have $\widehat{\mathcal{L}}_{i}^{Y}\left( f\left(
\theta \right) ,\theta \right) =\left\{ f\left( \theta \right) \right\} $ by
Lemma \ref{lem:mixed:deviation:SCF}. In scenarios (II) and (III), $\widehat
\mathcal{L}}_{i}^{Y}\left( f\left( \theta \right) ,\theta \right) =\mathcal{
}_{i}^{Y}\left( f\left( \theta \right) ,\theta \right) $ makes (\ref{bbt1})
hold \emph{vacuously}.\footnote
In scenario (III), $f\left( \theta \right) \notin \arg \min_{z\in
Z}u_{j}^{\theta }\left( z\right) $ implies $\widehat{\Gamma }_{j}\left(
\theta \right) =Z$, and $Z$ is not an $i$-$\theta ^{\ast }$-max set for any
i\in \mathcal{I}$, which would be implied by $\widehat{\mathcal{L}}^{Y}
-monotonicity imposed later (see Lemma \ref{lem:no-veto:generalized}).
---This leads to the definition of $\widehat{\mathcal{L}}_{i}^{Y}\left(
f\left( \theta \right) ,\theta \right) $ in (\ref{yjj8}).
\subsection{Our full characterization for mixed-Nash-A-implementation}
\label{sec:connection:A}
We consider SCCs and focus on mixed-Nash-A-implementation throughout this
subsection. We first illustrate why $\widehat{\mathcal{L}}^{Y}
-Maskin-monotonicity is defined on UNIF$\left[ F\left( \theta \right) \right]
$. Suppose that $F$ is mixed-Nash-A-implemented by $\mathcal{M}=\left\langle
M\text{, \ }g:M\longrightarrow Y\right\rangle $, and consider any $\lambda
\in MNE^{\left( \mathcal{M},\text{ }\theta \right) }$. By
mixed-Nash-A-implementation, $g\left( \lambda \right) $ could be any
lotteries in $\triangle \left[ F\left( \theta \right) \right] $, and as a
result, we need to define $\widehat{\mathcal{L}}$-Maskin-monotonicity as
\begin{equation*}
\left[
\begin{array}{c}
\widehat{\mathcal{L}}_{i}^{Y}\left( \eta ,\theta \right) \subset \mathcal{L
_{i}^{Y}\left( \eta ,\theta ^{\prime }\right) \text{, } \\
\forall \eta \in \triangle \left[ F\left( \theta \right) \right] \text{,
\forall i\in \mathcal{I
\end{array
\right] \text{ }\Longrightarrow F\left( \theta \right) \subset F\left(
\theta ^{\prime }\right) \text{, }\forall \left( \theta ,\theta ^{\prime
}\right) \in \Theta \times \Theta \text{.}
\end{equation*
However, the following lemma shows that it suffers no loss of generality to
define $\widehat{\mathcal{L}}$-Maskin-monotonicity on UNIF$\left[ F\left(
\theta \right) \right] $ only (i.e., Definition \ref{defin:A-B}). The proof
of Lemma \ref{lem:mixed-lottery:lower-contour} is relegated to Appendix \re
{sec:lem:mixed-lottery:lower-contour}.
\begin{lemma}
\label{lem:mixed-lottery:lower-contour}For any $E\in 2^{Z}\diagdown \left\{
\varnothing \right\} $ and any $\left( \gamma ,i,\theta ,\theta ^{\prime
}\right) \in \triangle ^{\circ }\left( E\right) \times \mathcal{I}\times
\Theta \times \Theta $, we hav
\begin{equation}
\left(
\begin{array}{c}
\mathcal{L}_{i}^{Y}\left( \eta ,\theta \right) \subset \mathcal{L
_{i}^{Y}\left( \eta ,\theta ^{\prime }\right) \text{,} \\
\forall \eta \in \triangle \left[ E\right
\end{array
\right) \Longleftrightarrow \mathcal{L}_{i}^{Y}\left( \gamma ,\theta \right)
\subset \mathcal{L}_{i}^{Y}\left( \gamma ,\theta ^{\prime }\right) \text{.}
\label{kkg1a}
\end{equation}
\end{lemma}
Second, by a similar argument as in Section \ref{sec:illustration:SCF},
mixed-Nash-A-implementation implies
\begin{gather}
\text{Step (I): select }\widehat{Z}\in 2^{Z}\diagdown \left\{ \varnothing
\right\} \text{ such that }\cup _{\theta \in \Theta }F\left( \theta \right)
\subset \widehat{Z}\text{ and} \notag \\
\forall \theta ^{\ast }\in \Theta \text{, } \notag \\
\left[
\begin{array}{c}
\widehat{Z}\subset \arg \max_{z\in \widehat{Z}}U_{i}^{\theta ^{\ast }}\left(
z\right) \text{,} \\
\forall i\in \mathcal{I
\end{array
\right] \Longrightarrow \widehat{Z}\subset F\left( \theta ^{\ast }\right)
\text{.} \label{hhe1}
\end{gather
Without an iterative process of elimination, it is straightforward to find
the largest such $\widehat{Z}$ by consider two scenarios. If $Z$ is not an
i $-max set for any $i\in \mathcal{I}$, $\widehat{Z}=Z$ makes (\ref{hhe1})
hold \emph{vacuously}. Otherwise, we must have $\widehat{Z}=\cup _{\theta
\in \Theta }F\left( \theta \right) $.\footnote
Given the canonical mechanism $\mathcal{M}=\left\langle M\text{, \
g:M\longrightarrow Y\right\rangle $, all agents being indifferent between
any two outcomes in $\widehat{Z}$ at $\theta ^{\ast }$ implies any $m\in M$
is a Nash equilibrium. As a result, we have $\widehat{Z}=\cup _{m\in M}$SUPP
\left[ g\left( m\right) \right] \subset F\left( \theta ^{\ast }\right) $,
which further implies $\cup _{\theta \in \Theta }F\left( \theta \right)
\subset \widehat{Z}\subset F\left( \theta ^{\ast }\right) \subset \cup
_{\theta \in \Theta }F\left( \theta \right) $, i.e., $\widehat{Z}=\cup
_{\theta \in \Theta }F\left( \theta \right) $.} Therefore, we must consider
Z^{\ast }$ defined in (\ref{hhe1}) for mixed-Nash-A-implementation.
Third, by a similar argument as in Section \ref{sec:illustration:SCF},
mixed-Nash-A-implementation implies
\begin{gather}
\text{Step (II): for each }\left( \theta ,i\right) \in \Theta \times
\mathcal{I}\text{, select }\widehat{\mathcal{L}}_{i}^{Y}\left( \text{UNIF
\left[ F\left( \theta \right) \right] ,\theta \right) \in 2^{\left[
\triangle \left( Z^{\ast }\right) \cap \mathcal{L}_{i}^{Y}\left( \text{UNIF
\left[ F\left( \theta \right) \right] ,\theta \right) \right] }\diagdown
\left\{ \varnothing \right\} \notag \\
\text{such that UNIF}\left[ F\left( \theta \right) \right] \in \widehat
\mathcal{L}}_{i}^{Y}\left( \text{UNIF}\left[ F\left( \theta \right) \right]
,\theta \right) \text{ and} \notag \\
\forall \left( \theta ^{\ast },y\right) \in \Theta \times \widehat{\mathcal{
}}_{i}^{Y}\left( \text{UNIF}\left[ F\left( \theta \right) \right] ,\theta
\right) \text{, } \notag \\
\left[ \widehat{\Gamma }_{i}\left( \theta \right) \text{ is an }i\text{-
Z^{\ast }\text{-}\theta ^{\ast }\text{-max set}\right] \Longrightarrow
\widehat{\Gamma }_{i}\left( \theta \right) \subset F\left( \theta ^{\ast
}\right) \text{.} \label{bbt2}
\end{gather
Thus, we can find the largest such $\widehat{\mathcal{L}}_{i}^{Y}\left(
a,\theta \right) $ by considering three scenarios
\begin{equation*}
\left(
\begin{array}{c}
\text{scenario (I): }F\left( \theta \right) \subset \arg \min_{z\in Z^{\ast
}}u_{i}^{\theta }\left( z\right) \text{ \ and }Z^{\ast }\cap \mathcal{L
_{i}^{Z}\left( F\left( \theta \right) ,\theta \right) \text{ is an }i\text{-
Z^{\ast }\text{-max set,} \\
\text{scenario (II): }F\left( \theta \right) \subset \arg \min_{z\in Z^{\ast
}}u_{i}^{\theta }\left( z\right) \text{ \ and }Z^{\ast }\cap \mathcal{L
_{i}^{Z}\left( F\left( \theta \right) ,\theta \right) \text{ is not an }
\text{-}Z^{\ast }\text{-max set,} \\
\text{scenario (III): }F\left( \theta \right) \diagdown \arg \min_{z\in
Z^{\ast }}u_{i}^{\theta }\left( z\right) \neq \varnothin
\end{array
\right) \text{.}
\end{equation*
In scenario (I), Lemma \ref{lem:mixed:deviation:SCC} implies that we must
hav
\begin{equation*}
\widehat{\mathcal{L}}_{i}^{Y}\left( \text{UNIF}\left[ F\left( \theta \right)
\right] ,\theta \right) =\triangle \left[ Z^{\ast }\cap \mathcal{L
_{i}^{Z}\left( F\left( \theta \right) ,\theta \right) \cap \left(
\dbigcup\limits_{K\in \Xi _{i}\left( \theta \right) }\dbigcap\limits_{\theta
^{\prime }\in K}F\left( \theta ^{\prime }\right) \right) \right] \text{.}
\end{equation*
In scenario (II), $\widehat{\mathcal{L}}_{i}^{Y}\left( \text{UNIF}\left[
F\left( \theta \right) \right] ,\theta \right) =\mathcal{L}_{i}^{Y}\left(
\text{UNIF}\left[ F\left( \theta \right) \right] ,\theta \right) $ makes
\ref{bbt2}) hold \emph{vacuously}. In scenario (III), $\widehat{\mathcal{L}
_{i}^{Y}\left( \text{UNIF}\left[ F\left( \theta \right) \right] ,\theta
\right) =\mathcal{L}_{i}^{Y}\left( \text{UNIF}\left[ F\left( \theta \right)
\right] ,\theta \right) $ implies $\widehat{\Gamma }_{i}\left( \theta
\right) =Z^{\ast }$, and (\ref{bbt2}) holds by Lemma \re
{lem:no-veto:generalized:SCC}.---This leads to the definition of $\widehat
\mathcal{L}}_{i}^{Y\text{-}A\text{-}B}\left( \text{UNIF}\left[ F\left(
\theta \right) \right] ,\theta \right) $ in (\ref{ddtt}).
\section{Conclusion}
\label{sec:conclude}
We study Nash implementation by stochastic mechanisms, and provide a
surprisingly simple full characterization. Even though our full
characterization is of a form similar to Maskin monotonicity \emph{\`{a} la}
\cite{em}, it has an interpretation parallel to \cite{jmrr} and \cite{tomas
. In this sense, we build a bridge between \cite{em} and \cite{jmrr} (as
well as \cite{tomas}).
Furthermore, our full characterization shed light o
\begin{equation*}
\left(
\begin{array}{c}
\text{"mixed-Nash-implementation VS pure-Nash-implementation,"} \\
\text{"ordinal-approach VS cardinal-approach,"} \\
\text{"Nash-implementation VS rationalizable-implementation"
\end{array
\right) \text{.}
\end{equation*}
\bigskip
\newpage
|
1,108,101,565,713 | arxiv | \section{Introduction}
{Recent years have witnessed a spurt of progress in modern machine learning technology, more effective and complex machine learning models can be trained through large-scale distributed training. However, in each iteration of the distributed optimization, information exchange among distributed nodes will incur enormous communication loads due to the large-scale model parameters. }
We focus on federated learning\cite{18}, {which is a distributed learning framework that can effectively help multiple clients perform data usage and machine learning modeling while meeting user privacy protection and data security requirements.} In federated learning clients participating in joint training only need to exchange their own gradients information, without sharing private data. To alleviate the communication bottleneck, gradient compression \cite{1,2,3,4,5,6,7,2020Federated} and efficient mean estimator \cite{8,9,10,11,12,13,14} have been investigated to reduce the communication load. {However, most of the previous works on compression fall to exploit any historical gradients at the server. In fact, the historical gradients can be viewed as side information to compress source information, which has been widely studied in classical information theory. For example, \cite{15} first studied the setting of lossy source compression with side information in the decoder. Channel coding can obtain practical coding for distributed source coding \cite{16,17}, but the main bottleneck lies on the expensive computational complexity of coding and decoding. Recently, \cite{19} considered the correlation between the data for distributed mean estimation, and obtained the mean estimator of the dependent variance through the lattice coding scheme. \cite{20} studied distributed mean estimation with side information at the server, and proposed Wyner-Ziv estimators that require no probabilistic assumption on the clients' data. On the other hand, in convex optimization, the use of historical gradients can accelerate convergence, such as heavy ball method, and Nesterov’s Accelerated gradient Descent (NAG) in \cite{boyd2004convex}. Last but not least, for stochastic gradient descent with variance reduction such as stochastic variance reduced gradient (SVRG)\cite{johnson2013accelerating}, historical gradients can also be used to reduce the variance of stochastic gradients. So an interesting question is whether and how historical gradients can be used for gradient compression.
In this paper, we address this question and propose \emph{practical} gradient compression schemes for federated learning by exploring historical gradients as side information. Our contributions are summerized as follows: \emph{1):} {Motivated by Wyner-Ziv estimator of \cite{20}, we propose a local quantized stochastic gradient descent (LQSGD), which exploits the historical gradients as side information to compress the local gradients. Different from \cite{20}, we focus on federated learning with side information instead of distributed mean estimation. In fact, in the scenario of stochastic gradients descent, gradients between adjacent rounds may have a high correlation since they wish to learn the same model. Therefore, we can use historical gradients to compress gradients.}
\emph{2):} {We establish an upper bound of the average-squared gradients of our quantization methods. Compared with the case of not using any historical gradients, the average-squared gradients may get a smaller upper bound by using the historical gradients. Besides, We also obtain the convergence rate of our scheme under standard assumptions.}
\emph{3):} We implement our gradient quantization method and compare it with quantized stochastic gradients descent (QSGD)\cite{3} and the uncompressed SGD on real dataset. We use these three schemes to train linear regression model on datase \texttt{cpusmall-scale}\cite{cpu1} and the classic neural network ResNet18\cite{he2016deep} on \texttt{cifar10}\cite{krizhevsky2009learning} and \texttt{cifar100}\cite{krizhevsky2009learning}. The results show that our method is effective and has better performance compared with QSGD.}
\section{Problem Setting}
{In this article, we consider the scenario of federated learning in which a set of $N$ clients and one server jointly train a model without exchanging clients' own data, as shown in Fig. 1.} Assume that the clients' communication resources are limited and clients can only communicates with the server. Formally, we deal with the following optimization problem:
\begin{equation}\label{optproblem}
\min_{\boldsymbol{\omega}\in\mathbb{R}^d} f(\boldsymbol{\omega})=\frac{1}{N}\sum_{j=1}^{N}f_j(\boldsymbol{\omega}),
\end{equation}
where $f_j:\mathbb{R}^d\rightarrow\mathbb{R}$ is the local function corresponding to Client $j$ and $\boldsymbol{\omega}$ is the parameter of model. For instance, in statistical machine learning, $f_j$ is usually the local objective function of each Client $j$ which is the expected loss over the set of data points of node $j$, i.e.,
\begin{equation}
f_j(\boldsymbol{\omega})=\mathbb{E}_{\boldsymbol{z}\sim\mathcal{P}_j}l_j(\boldsymbol{\omega},\boldsymbol{z}),
\end{equation}
where $\boldsymbol{z}$ is a random variable with probability distribution $\mathcal{P}_j$ and $l_j(\boldsymbol{\omega},\cdot)$ is loss function measuring the performance of the model with parameter $\boldsymbol{\omega}$. Note that the probability distributions of clients may not be necessarily identical. In this paper, we mainly concentrate on homogeneous scenarios where the probability distributions and loss functions of all clients are the same, i.e. $\mathcal{P}_1=\mathcal{P}_2=\cdots=\mathcal{P}_N$ and $l_1=l_2=\cdots=l_N$. {Heterogeneous federated learning with different data distributions and loss functions, of course, is interesting and important. As we will see, the scheme proposed in this paper is also applicable to heterogeneous federated learning, whose analysis is left as future work.}
\begin{figure}
\centering
\includegraphics[width=8cm,height=6.5cm]{fl.pdf}
\caption{Federated learning}
\label{imgModel}
\end{figure}
\section{Federated Aggregation With Side Information}
In this section, we propose a generalized version of the local quantized stochastic gradient descent (LQSGD) method for federated learning which uses quantized gradients to update model parameters. This method reduces the overall communication overhead by sending fewer bits in each iteration. In Section \ref{fav}, we introduce federated averaging with side information (\texttt{FedSI}) designed for homogeneous settings. Then, in Section \ref{com}, we propose the detailed compression algorithm which uses side information to quantize data. Eventually, we introduce that how to choose the side information used in our method in Section \ref{side}.
\subsection{Federated Averaging with Side Information(\texttt{FedSI})}\label{fav}
In federated learning, {the aggregation algorithm is used to make full use of the gradients information of each client}. Each client sends local gradients information to the server after multiple iterations. The server aggregates the gradients sent by clients and then updates the global model and broadcasts it to each client. {Different from the standard federated average algorithm, in \texttt{FedSI}, not only the client sends quantized gradients, but also the global learning rate and local learning rate can be different.}
Formally, let $R$ be the rounds of communication between server and clients, and $\tau$ be the number of local updates performed between two consecutive communication rounds. Further more, we define $\boldsymbol{\omega}^{(r)}$ as the global model at the master in the $r$-th round of communication. At each round $r$, the server sends the global model $\boldsymbol{\omega}^{(r)}$ to the clients. After that, each Client $j$ computes its local stochastic gradients and updates the model by following the update of SGD
\begin{equation}
\boldsymbol{\omega}^{(c+1,r)}=\boldsymbol{\omega}^{(c,r)}-\eta \tilde{g}_j^{(c,r)},\ \text{for}\ c=0,\cdots, \tau-1,
\end{equation}
where $\tilde{g}_j^{(c,r)}$ is the estimation of the gradient ${g}_j^{(c,r)}\triangleq\nabla f_j(\boldsymbol{\omega}^{(c,r)})$ and $\eta$ is the leaning rate. For example, for statistical machine learning, $\tilde{g}_j^{(c,r)}\triangleq\frac{1}{b_j}\sum_{\boldsymbol{z}\in\mathcal{Z}_j^{(c,r)}} \nabla l(\boldsymbol{\omega}^{(c,r)}, \boldsymbol{z})$, where $\mathcal{Z}_j^{(c,r)}$ is the mini-batch of Client $j$ consisting of $b_j$ samples generated by $\mathcal{P}_j$. Next, each client sends the quantized signal $Q(\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)})$ by applying a compression operator $Q(\cdot)$ defined in the Section \ref{com}. When the server receives the signals from all clients, it updates the global model as follows:
\begin{equation}\label{upd}
\boldsymbol{\omega}^{(r+1)}=\boldsymbol{\omega}^{(r)}-\frac{\eta\gamma}{N}\sum_{j=1}^{N}Q(\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)}),
\end{equation}
where $\gamma$ is the global learing rate. When $\gamma=1$ and all clients send unquantized signals, (\ref{upd}) becomes the standard federated average algorithm.
\subsection{Compression with Side Information}\label{com}
In this section, we introduce our compression operator $Q(\cdot)$. The operator $Q(\cdot)$ contains two parts: the encoder $Q_e(\cdot)$ and the decoder $Q_d(\cdot)$, where $Q_e(\cdot):\mathbb{R}^d\rightarrow \{0,1\}^k,\ k\in\mathbb{N}^+$ and $Q_d(\cdot):\{0,1\}^k\rightarrow \mathbb{R}^d,\ k\in\mathbb{N}^+$.
Now we give a brief introduction about the quantizer $Q_\textnormal{WZ}$\cite{20}, as it is closely related to work. Since all clients use the same quantizer, only the common quantizer is described.
We first describe a modulo quantizer $Q_\textnormal{M}$ for one-dimension input $x\in\mathbb{R}$ with side information $h\in\mathbb{R}$, then present a modulo quantizer $Q_{\textnormal{M},d}(x,h)$ for $d$-dimension data.
\subsubsection{Modulo Quantizer ($Q_\textnormal{M}$)}\label{cq} Given the input $x\in\mathbb{R}$ with side information $h\in\mathbb{R}$, the modulo quantizer $Q_\textnormal{M}$ contains parameters including a distance parameter $\Delta'$ where $|x-h|\leq \Delta'$, a resolution parameter $s\in\mathbb{N}^+$ and a lattice parameter $\epsilon$.
Denote the encoder and decoder of $Q_{\textnormal{M}}$ as $Q^{\textnormal{e}}_{\textnormal{M}}(x)$ and $Q^{\textnormal{d}}_{\textnormal{M}}(Q^{\textnormal{e}}_{\textnormal{M}}(x),h)$, respectively.
The encoder $Q^{\textnormal{e}}_{\textnormal{M}}(x)$ first computes $\lceil x/\epsilon\rceil $ and $\lfloor x/\epsilon\rfloor $, then outputs the message $Q^{\textnormal{e}}_{\textnormal{M}}(x)=m$, where
\begin{equation}
m = \left\{ \begin{array}{llr}
(\lceil x/\epsilon\rceil\mod s), &~\text{w.p.}~x/\epsilon-\lfloor x/\epsilon\rfloor \\
(\lfloor x/\epsilon\rfloor \mod s), & ~\text{w.p.}~\lceil x/\epsilon\rceil- x/\epsilon
\end{array}.
\right.
\end{equation}
The message $m$ has length of $\log s$ bits, which is sent to the decoder. The decoder $Q^{\textnormal{d}}_{\textnormal{M}}$ produces the estimate $\hat{x}$ by finding a point closest to $h$ in the set $\mathbb{Z}_{m,\epsilon}=\{(zs+m)\cdot\epsilon:z\in\mathbb{Z}\}$.
\begin{lemma}{(see \cite{20})}\label{cqpe}
Consider $Q_\textnormal{M}$ with parameter $\epsilon$ set to satisfy
\begin{equation}\label{condk}
s\epsilon\geq2(\epsilon+\Delta').
\end{equation}
Then, for every $x,h\in\mathbb{R}$ such that $|x-h|\leq\Delta'$, the output $Q_\textnormal{M}(x)$ satisfies
\begin{equation*}
\begin{array}{c}
\mathbb{E}[Q_\textnormal{M}(x)]=x,\\
|x-Q_\textnormal{M}(x)|<\epsilon.
\end{array}
\end{equation*}
\end{lemma}
\subsubsection{$d$-dimensional Modulo Quantizer ($Q_{\textnormal{M},d}$)}\label{hcq}
For d-dimensional data, we use $Q_{\textnormal{M}}$ in each dimension. Specifically, given the input $x\in\mathbb{R}^d$ with side information $h\in\mathbb{R}^d$, the modulo quantizer $Q_\textnormal{M,d}$ contains parameters including a distance parameter $\Delta'$ where $||x-h||_{\infty}\leq \Delta'$, a resolution parameter $s\in\mathbb{N}^+$ and a lattice parameter $\epsilon$. For the $i$-th dimension data, set $h_i$ to be the side information of $x_i$, where $h_i$ and $x_i$ are the $i$-th data of $h$ and $x$ respectively. For each dimension, we compress the data into $\log s$ bits. Therefore, we use $d\log s$ bits to represent the $d$-dimensional data $x$.
We can easily extend Lemma \ref{cqpe} to $d$-dimensional case, which is given in the following corollary.
\begin{corollary}\label{co1}
For $d$-dimensional data, we consider $Q_\textnormal{M,d}$ with parameter $\epsilon$ set to satisfy
\begin{equation}
s\epsilon\geq2(\epsilon+\Delta),
\end{equation}
Then, for every $\boldsymbol{x},\boldsymbol{y}\in\mathbb{R}^d$ such that $||\boldsymbol{x}-\boldsymbol{y}||_2\leq\Delta$, the output $Q(\boldsymbol{x})$ satisfies
\begin{equation*}
\begin{array}{c}
\mathbb{E}[Q(\boldsymbol{x})]=\boldsymbol{x},\\
\mathbb{E}||\boldsymbol{x}-Q(\boldsymbol{x})||^2<d\epsilon^2.
\end{array}
\end{equation*}
In addition, if we let $\epsilon=\frac{2\Delta}{s-2}$, we have
\begin{equation}\label{var}
\mathbb{E}||\boldsymbol{x}-Q(\boldsymbol{x})||^2<\frac{4d\Delta^2}{(s-2)^2}.
\end{equation}
\end{corollary}
\begin{Remark}
In the previous works on mean estimation, such as \cite{8} and \cite{20}, by preprocessing the d-dimensional data, we can get a smaller mean square error. At a high-level, we can rotate the $d$-dimensional data $x$ and $h$ at the same time, for example, $x$ and $h$ are multiplied by $W = \frac{1}{\sqrt{d}}HD$, where $D$ is a random diagonal matrix with i.i.d. Rademacher entries($\pm1$ with equal probability) and $H$ is a Walsh-Hadamard matrix \cite{21}. After the data is preprocessed, $||Wx-Wh||_\infty$ is much less than $||Wx-Wh||_2=||x-h||_2$ with a high probability. This means that we can estimate the data more accurately while keeping the Euclidean distance unchanged. However, this method requires additional matrix calculations, and the time complexity is $O(d\log d)$. Because we concern about the role of side information in federated learning, for convenience, our implementation does not preprocess the data.
\end{Remark}
\subsection{Select Side Information}\label{side}
In this part, we discuss which data to be chosen as side information for compressing gradients in federated learning. In convex optimization, historical gradients can accelerate convergence and for stochastic gradient descent with variance reduction, it can also be used to reduce the variance of stochastic gradient. Motivated by the role of historical gradients in accelerating convergence, a natural question is whether the historical gradients can help us compress the current gradients. In fact, gradients between adjacent rounds may have a high correlation since they wish to learn the same model. Therefore, we can regard the historical gradients as the side information of the local gradients.
{We let $N$ clients share the same side information, which can effectively save memory space for storing side information when the dimension $d$ is large.} Since clients upload quantized gradients, the aggregated gradients broadcasted by the server in the last round may differ greatly from the current local gradients of clients. {In this case, using historical gradients as side information would be worse than the standard QSGD quantization scheme. Therefore, we set a threshold $t$ ($0<t\leq1$), and if the ratio of the distance between the current gradients and historical aggregated gradients to the norm of the current gradients is less than $t$, we use historical gradients as side information, otherwise, we do not use any side information.} Formally, at the $r$-th round of communication, Client $j$ obtains the local gradients $\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)}$ {and calculates}
\begin{equation}
D_j^r=\frac{||\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)}-U_q^{r-1}||_2}{||\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)}||_2},
\end{equation}
where $U_q^{r-1}\triangleq\frac{1}{N}\sum_{j=1}^{N}Q(\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r-1)})$. If $D_j^r$ is less than $t$, Client $j$ uses historical gradients $U_q^{r-1}$ to compress $\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)}$, otherwise it will not use any side information to compress gradients. For convenience, we use $\alpha_j^r$ to indicate that whether Client $j$ uses historical gradients $U_q^{r-1}$ as side information to compress local gradients at $r$-th round, i.e.,
\begin{equation}\label{ind}
\alpha_j^r= \left\{ \begin{array}{llr}
1, &D_j^r<t,\\
0, & ~\text{otherwise.}
\end{array}
\right.
\end{equation}
\begin{Remark}
{Sending the bin number $m$ of each coordinate is a natural way of transmitting messages. However, this naive implementation is sub-optimal. In fact, we can encode the transmitted values by using universal compression schemes\cite{elias1975universal},\cite{apostolico1987robust}. For each client, the bin number $m$ decoded by the server is the same, because lossless compression is used to compress the bin number $m$. Therefore, we use the natural transmission method for convenience, which means that, each client simply transmits $d\log s$ bits in each round of communication.}
\end{Remark}
\begin{Remark}
If we set distance parameter $\Delta'$ as $||x||_2$ and side information $h$ as $0$, our quantizer becomes standard QSGD. Therefore, QSGD can be regarded as a quantizer without any side information. Similarly, if we set distance parameter $\Delta'$ as $||x||_\infty$ and side information $h$ as $0$, our quantizer becomes a variant of QSGD\cite{3} (i.e., QSGDinf\cite{3}, for convenience, we still call it QSGD). Since both the client and the server store side information, we can set the distance parameter $\Delta'$ as $||x-h||_2$ or $||x-h||_\infty$, and it is necessary for the client to send $||x-h||_2$ or $||x-h||_\infty$ to the server. Note that when the dimension $d$ is large enough, the communication cost of sending distance parameter is negligible.
\end{Remark}
We formally describe the method in Algorithm~\ref{alg}.
\begin{algorithm}
\caption{Federated averaging with side information (\texttt{FedSI})}\label{alg}
\hspace*{0.02in}{\bf Input:}
Number of communication rounds $R$, number of local updates $\tau$ , learning rates $\gamma$ and $\eta$, initial global model $\omega^{(0)}$, side information $U_q^{-1}=0$ and threshold $t$
\begin{algorithmic}[1]
\For{$0\leq r\leq R-1$}
\For{each client $j\in[N]$}
\State Set $\omega_j^{(0,r)}=\omega^{(r)}$
\For{$0\leq c\leq \tau-1$}
\State Sample a minibatch $\mathcal{Z}_j^{(c,r)}$ and compute $\tilde{g}_j^{(c,r)}$
\State $\boldsymbol{\omega}^{(c+1,r)}=\boldsymbol{\omega}^{(c,r)}-\eta \tilde{g}_j^{(c,r)}$
\EndFor
\State Compute $D_j^r=\frac{||\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)}-U_q^{r-1}||_2}{||\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)}||_2}$
\State Compute $\alpha_j^r$ according to (\ref{ind})
\State Client $j$ sends $Q^{e}(\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)})$, $\alpha_j^r$ and $||\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)}-\alpha_j^r U_q^{r-1}||_2$
\EndFor
\State Server computes $U_q^r=\frac{1}{N}\sum_{j=1}^{N}Q(\sum_{c=0}^{\tau-1}\tilde{g}_j^{(c,r)})$
\State Server updates ${\omega}^{(r+1)}=\boldsymbol{\omega}^{(r)}-\eta\gamma U_q^r$ and broadcasts to all clients
\EndFor
\end{algorithmic}
\end{algorithm}
\section{Convergence Analysis}
Next, we present the convergence analysis of our proposed
method. At the beginning of this section, we state our assumptions which are customary in the analysis of methods with compression (same as \cite{2020Federated}).
\begin{assumption}\label{ass2}(Smoothness and Lower Boundedness).
The local objective function $f_j$ of $j$-th client is differentiable for $j\in[m]$ and $L$-smooth, i.e.,
$||\nabla f_j(\boldsymbol{x})-\nabla f_j(\boldsymbol{y})||\leq L||\boldsymbol{x}-\boldsymbol{y}||$. Moreover,
the optimal value of objective function $f(\boldsymbol{\omega})$ is
bounded below by $f^*=\min_{\boldsymbol{\omega}}f(\boldsymbol{\omega})>-\infty$
\end{assumption}
\begin{assumption}\label{ass3}
For all $j\in[N]$, we can sample an independent mini-batch $\mathcal{Z}_j$ of size $|\mathcal{Z}_j^{(c,r)}|=b_j$ and compute an unbiased stochastic gradient $\tilde{g}_j=\nabla f_j(\boldsymbol{\omega};\mathcal{Z}_j); \mathbb{E}_{\mathcal{Z}_j}(\tilde{g}_j)=\nabla f(\boldsymbol{\omega})=g$. Besides, their variance is bounded above by a constant $\sigma^2$, i.e., $\mathbb{E}||\tilde{g}_j-g||^2\leq \sigma^2$,
\end{assumption}
In the following theorem, we state our main theoretical results for $\texttt{FedSI}$ in the homogeneous setting.
\begin{theorem}\label{main}
For $\texttt{FedSI}$, given the number of communication rounds $R$, the number of local updates $\tau$ , learning rates $\gamma$ and $\eta$, initial global model $\omega^{(0)}$ and threshold $t$, under Assumptions \ref{ass2} and \ref{ass3}, if the learning rate satisfies
\begin{equation}\label{lea}
1\geq \tau^2L^2\eta^2+(\frac{q}{N}+1)\tau\gamma L\eta,
\end{equation}
where $q=\frac{4d}{(s-2)^2}$, then the average-squared gradients after $R$ communication rounds is bounded as follows:
\begin{equation}\label{con}
\begin{aligned}
\frac{1}{R}\sum_{r=0}^{R-1}||\nabla f(\omega^{(r)})||^2&\leq \frac{2(f(\omega^{(0)})-f(\omega^{(*)}))}{\tau\gamma\eta R}\\
&\ \!+\!\frac{\gamma L\eta(q\alpha(t\!-\!1)\!+\!q\!+\!1)}{N}\sigma^2\!+\! \tau L^2\eta^2\sigma^2,
\end{aligned}
\end{equation}
where $\alpha\triangleq\frac{1}{RN}\sum_{r,j}\alpha_j^r$.
\end{theorem}
\begin{proof}
See Appendix \ref{pro}
\end{proof}
\begin{Remark}
Compared with the case where the historical gradients are not used, i.e., $\alpha=0$, our upper bound is reduced by $\frac{\gamma L\eta(q\alpha(1\!-\!t)\!)}{N}\sigma^2$. Obviously, if the learning rates of the two schemes remain the same and $L$, $q$, $\alpha$, and $\sigma^2$ are huge enough, using historical gradients to compress the gradients will converge faster.
\end{Remark}
\begin{Remark}
If we set $\eta\gamma=\Theta(\frac{\sqrt{N}}{\sqrt{R\tau}})$, the upper bound of the average-squared gradients is $O(\frac{1}{\sqrt{NR\tau}}+\frac{1}{R})$. Based on this bound, if we set th communication rounds $R=O(\frac{1}{\delta})$ and the local updates $\tau=O(\frac{1}{N\delta})$, the average-squared gradients can achieve a $\delta$-accurate.
\end{Remark}
\section{Experiments}
For the simulations, we consider three real datasets: \texttt{cpusmall-scale}\cite{cpu1}, \texttt{cifar10}\cite{krizhevsky2009learning} and \texttt{cifar100}\cite{krizhevsky2009learning}. We implement our gradients quantization method, which we call LQSGD for simplicity and compare it with quantized stochastic gradients descent (QSGD)\cite{3} and uncompressed SGD on real dataset. We use these three schemes to train linear regression model on datase \texttt{cpusmall-scale} and the classic neural network ResNet18\cite{he2016deep} on \texttt{cifar10} and \texttt{cifar100}
\subsection{Dataset \texttt{cpusmall-scale}}
The number of samples in the dataset \texttt{cpusmall-scale} is $8192$, and the feature dimension $d$ of each sample is $12$. The classic problem of least-squares is as follows: Some matricex $A$ and target vectors $\boldsymbol{b}$ are given as input and the purpose is to find $\omega^*=\arg\min_{{\omega}}||A\omega-b||_2$. {For convenience, we set the number of local updates $\tau$ as $1$, and simulate the scenarios consisting of eight clients. In order to simulate the scenario where each client's data is sampled from the same distribution, we shuffle the 8192 samples, afterwards assign 1024 samples to each client. We measure the effect on the convergence of the SGD process of the quantization schemes in the next step. After running three algorithms LQSGD, QSGD, and GD, we record the loss and the number of iterations.}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{convergence.pdf}
\caption{Convergence at $2$ bits per coordinate}
\label{lr1}
\end{figure}
As shown in Fig. \ref{lr1}, when the learning rate is set as $0.1$ and the value of each coordinate is compressed into $2$ bits, the loss of GD has converged at 250 iterations, the loss of LQSGD decreases faster than QSGD, and LQSGD converges at 1200 iterations, while QSGD converges at 1700 iterations.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{error.pdf}
\caption{Distance at $2$ bits per coordinate}
\label{lr2}
\end{figure}
{We also compare the distance (Euclidean distance) among the model parameters obtained by these methods and the optimal model parameters. As shown in Fig. \ref{lr2}, the model parameters obtained by LQSGD are closer to the optimal parameters than model parameters obtained by QSGD.
\subsection{Dataset \texttt{cifar10} and \texttt{cifar100}}
\texttt{cifar10} has a total of $60000$ color images of size $32*32$, and these images are divided into $10$ categories, each of them with $6000$ images. There are $50000$ images used for training, forming $5$ training batches with $10000$ images, the other $10000$ are used for testing, forming a single batch. The data of the test batch is taken from each of the $10$ categories, and $1000$ images are randomly taken from each category. The rest is randomly arranged to form a training batch. \texttt{cifar100} is similar to \texttt{cifar10}, except that it has $100$ classes and each class contains $600$ images. Every category has $500$ training images and $100$ test images. We also simulate the scenario of $8$ clients and randomly assign training images to eight clients to ensure that the number of images in each class is the same for each client. We train the classic neural network ResNet18 on datasets. It is impractical to calculate the $2$-norm of the gradients because the number of parameters of ResNet18 is about $33$ million. In practice, we used the $\infty$-norm of gradients in LQSGD and QSGD.}
\begin{table}[h]
\centering
\caption{Test accuracy on \texttt{cifar10} and \texttt{cifar100} using $3$ bits (except for SGD) with $8$ clients}
\begin{tabular}{|p{2.5cm}|p{2.5cm}|p{2.5cm}|}
\hline
\hline
& \texttt{cifar10} & \texttt{cifar100} \\
\hline
SGD & $\bm{94.61\%}$& 76.44\%\\
\hline
QSGD& 94.19\% & 76.03\%\\
\hline
LQSGD &94.53\%&$\bm{76.45\%}$\\
\hline
\hline
\end{tabular}
\label{ta}
\end{table}
We train $100$ epoches of the model ResNet18 and evaluate the models obtained by using LQSGD, QSGD, and SGD on the testset. The test accuracy is shown in Table \ref{ta}. As shown in Table \ref{ta}, on the dataset \texttt{cifar10}, compared with SGD, the test accuracy of QSGD loses $0.42\%$, while the test accuracy of LQSGD loses $0.08\%$. Similarly, on the dataset \texttt{cifar100}, compared with SGD, the test accuracy of QSGD loses $0.41\%$, but the test accuracy of LQSGD is almost the same as that of SGD. It should be noted that the test accuracy of LQSGD is higher than the test accuracy of SGD and this is because our quantization scheme is an unbiased estimate of the gradients, which means that we add additional noise with a mean value of $0$ to the gradients. This may cause our test accuracy to be higher than that of SGD.
We also record the relationship between the test accuracy and the number of epoches. The details are shown in Fig. \ref{nn1} and \ref{nn2}. As shown in Fig. \ref{nn1} and \ref{nn2}, After about $70$ epoches, the test accuracy curves of LQSGD and SGD are almost above the test accuracy curve of QSGD. This means that the performance of our method is better than that of QSGD after about $70$ epoches.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{test_auc.pdf}
\caption{Test accuracy on \texttt{cifar10} for $8$ clients}
\label{nn1}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{test_auc_cifar100.pdf}
\caption{Test accuracy on \texttt{cifar100} for $8$ clients}
\label{nn2}
\end{figure}
\section{Conclusions and Future works}
In this paper, we proposed a gradients quantization method for federated learning, which uses the historical gradients as side information to compress the local gradients. We gave an upper bound of the average-squared gradients of quantization methods and also proved the convergence rate of our scheme under standard assumptions. We not only implemented our gradients quantization method, but also demonstrated the superiority of our scheme over previous work QSGD empirically on deep models and linear regression. In future work, we will extend our scheme to heterogeneous settings and look for ways to use side information more efficiently in federated learning.
\bibliographystyle{IEEEtran}
|
1,108,101,565,714 | arxiv | \section{Introduction}\label{sec: intro}
Optical fibre technology based on silica glass has revolutionised application fields as diverse as telecommunications and manufacturing (e.g. laser cutting and welding) by providing a robust and efficient integrated platform for the generation of visible and near-infrared light. However, for wavelengths longer than about 2.5 $\mu$m, silica fibres become virtually opaque and alternative soft glass materials must be used. Over the past years, fluoride fibre technology based on ZBLAN glass~\cite{POULAIN1973} has shown great promise and has now finally reached a stage of maturity where it is poised to initiate a similar disruption in the mid-infrared~\cite{Jackson2012}. For example, fibre-based supercontinuum sources in the mid-infrared are capable of generating electromagnetic radiation with a wavelength coverage and the brightness of a synchrotron, yet with the footprint of a table-top instrument~\cite{Hudson17}, thus enabling fast spectral mapping with a signal-to-noise ratio (SNR) that surpasses that achievable with a synchrotron source and in a shorter acquisition time~\cite{Borondics18}. However, in order for mid-infrared technology to become a truly disruptive technology, the development of field-deployable systems, i.e. systems that are able to operate under harsh and sometimes even extreme environmental conditions, in stark contrast to purely laboratory-based proof-of-principle instruments, is required. A prerequisite for this is the availability of connectorized (i.e. fiber pigtailed) and thus compact and robust integrated optical components such as splitters, couplers, circulators, and wavelength selective elements, to only name a few. While all these are readily available “off-the-shelf” for silica-glass based systems operating in the near-IR, equivalent components for the mid-infrared are still largely missing due to challenges of high thermal expansion, hygroscopicity and steep viscosity-temperature curve for most mid infrared materials including fluorides. Also, to date, splicing device manufacturers do not offer equipments fully dedicated to soft glasses hence, it is difficult to obtain the high temperature control required around 250-350$^{\circ}$C to process in optimal conditions with fluoride fibers~\cite{Rowe, Schafer18}. In this work, we present a potential solution for this fundamental problem.
\\Ultrafast Laser Inscription (ULI) is a well-studied and utilised technique for the fabrication of buried optical waveguides inside various different glasses~\cite{Gattass}. While the method has the potential to solve the “mid-infrared bottleneck”, standard ZBLAN glass has been shown to respond with only a very limited induced positive/negative index change during ULI, and as such virtually all reported ULI ZBLAN devices are based on a depressed cladding inscription approach, resulting in low NA, large mode area guiding~\cite{Gross15,Berube13}. These structures have been used successfully to produce waveguide lasers in active ZBLAN glasses~\cite{Gross_nanoph}, but are of limited utility in realising other optical components where low loss and mode matching to high NA mid-infrared optical fibres is required. An exhaustive detail of all the techniques used to tailor the refractive index within the ZBLAN glass including slit shaping technique can be found in ref~\cite{Gross2012}.
Other mid-infrared transparent materials like lead-germanates ~\cite{Mamoona2021}, gallo-germanates~\cite{Berube2017}, tellurites~\cite{Smayev2018} and chalcogenides~\cite{Gretzinger2015,Rodenas2012} have been used as substrates for laser inscribed waveguides in this wavelength region. Within this group, gallium lanthanum suphide (GLS) glass is the most attractive glass in demonstrating low-loss waveguiding at longer wavelengths (\textgreater 3 $\mu$m)~\cite{Arriola2017}. But all these glasses have inherently high refractive index which introduces high coupling losses to the lower index fluoride fiber architectures. Even if an intermediate stage is designed to reduce the coupling losses, the large difference in coefficient of thermal expansion (CTE) between the said materials and the fluoride fibers introduces additional thermal management problems for high power applications. The CTE of a ZBLAN glass is $\approx$ 18 $\times$ 10$^{-8}$ K$^{-1}$~\cite{Poulain2010}, whereas for GLS it is more than two orders higher $\approx$ 6 $\times$ 10$^{-6}$ K$^{-1}$~\cite{Hewak2010}.\\
Our goal was thus to develop a mid-infrared compatible glass composition that can produce a smooth, strong and positive refractive index change upon irradiation with ultrafast laser pulses which could then be easily integrated to the existing fluoride fiber architecture. Combined with an optimized inscription strategy, this glass could then be utilized for the inscription of waveguides with mode field diameters and V-numbers (V=$\frac{2\pi}{\lambda}$ a $\times$ NA, where a is the waveguide radius) that perfectly match those of existing optical fluoride fibers, thus enabling the fabrication of fiber-pigtailed integrated components. A recent communication from Heck et.al~\cite{Heck18} reported a positive index change within a fluoride optical fiber upon irradiation with femtosecond laser pulses. Their findings, specifically the significant increase in positive index change when the inscription was carried out at the interface between the core and cladding materials of the fiber was unexplained, yet intriguing as it suggested that ULI in these glasses has the potential to be compositionally tailored. In the case of a ZBLAN optical fiber, a small mol$\%$ of zirconium (Zr) is substituted by hafnium (Hf) in the cladding to lower the refractive index of the glass. One of the conclusions to be drawn from the findings of Heck et al was thus that the compositional variation between zirconium (Zr) and hafnium in the core and cladding was responsible for the unexpected response to femtosecond laser irradiation, giving rise to a small region of positive index change. We have reported in the past that strong thermal and compositional concentration gradients (cf section 5.3 in~\cite{Fernandez2018}) could be two triggering factors for refractive index change upon irradiation with femtosecond laser pulses. We speculate that these also played a role in Heck et al’s findings, as aberrated focusing and thermal profiles~\cite{Fernandez_2015} due to inscribing through the curved fiber along with a step-concentration gradient across the core-clad interface, aided in producing the increase in index change reported.
\\In a recent communication, we had empirically predicted, for other families of glass such as the boroalumino silicates, that if the main glass forming element is accompanied by a second glass forming element that is matched in its polarizability field strength (such as the interplay between aluminium and calcium in a silica glass)~\cite{Fernandez2020}, there is a high probability of obtaining waveguides with an improved positive refractive index contrast under ULI~\cite{Fernandez2020}. In this current work, we have thus explored whether there is a valid analogy in fluoride glasses, by studying ULI in modified ZBLAN compositions containing significant hafnium content. We show that a precise and selective addition of hafnium enables the inscription of optical waveguides with high index-contrast in fluoride glass, while the intrinsically good optical, chemical and mechanical properties are not negatively affected by this compositional re-design.
\section{Results and discussion}\label{sec: R&D}
We fabricated six different glass samples with various hafnium/zirconium content. Further details can be found in the Materials and Methods section. Fig.~\ref{Fig-1} shows that noticeable variations in refractive index (n$_D$), glass transition temperature (T$_g$), Brillouin frequency shift (BFS) and density ($\rho$) were observed in these bulk glasses. It should be noted at this point that there is a large chemical similarity between the hafnium and zirconium atoms and all reported fluorozirconates and fluorohafnates demonstrated strong isomorphism due to same crystalline cell structure and coordination number except for a slightly shorter Hf-F bonds compared to Zr-F bonds. For this reason, hafnium is commonly used for adjusting the refractive indices of the core and cladding glasses in fluoride optical fibers.
\\Raman spectra of all samples and discussions are provided in the supplementary document. The most noticeable variation was found in the distinctive vibrational frequency peak which decreased from 578 $cm^{-1}$ to 574.6 $cm^{-1}$ as the HfF$_4$ content was increased in the composition. As this peak comes from the terminal fluoride vibration due to the heavy stagnant Zr and Hf atoms, a variation in its frequency is believed to be associated with counter cation (Ba, La, Al and Na) re-arrangement~\cite{Phifer91,Gross13}. Understanding the basis of these variations are key in interpreting the origin of index change due to ultrafast laser-matter interaction.
\\The molar substitution of HfF$_4$ for ZrF$_4$ explains the monotonic increase in density due to the heavier Hf atoms. But the counter intuitive monotonic decrease in refractive index with increasing (heavier) hafnium content is due to the lower atomic polarizability~\cite{Ghosh02} as a result of lanthanide contraction. Hafnium, whose inner 4f orbital is large and diffuse, fail to shield the valence shell from the attraction of the nucleus. Such a strong attractive force on the valence electrons causes a contraction in the size of the electron charge cloud reducing its ability to be distorted when interacted with an electromagnetic wave. This further explains the higher glass transition temperatures for higher hafnium content glasses since the bonding between two relatively low polarizable atoms like hafnium and fluorine (lowest among halides) give rise to a greater accumulation of electron density in the bonding region. The attractive forces of the nuclei acting on the bonding electrons are thus stronger, hence requiring a higher temperature/energy to break them. This is directly evident from the measured Brillouin frequency shift (BFS) values (Fig.~\ref{BFS}) that are higher for ZBLAN (17.39 GHz) as compared to HBLAN (15.26 GHz). The isotropic nature of all the custom glasses was confirmed by the existence of single Stokes and anti-Stokes peaks shown in Fig.~\ref{BFS} as expected from amorphous nature of the material with no long-distance order or symmetry in the glass matrix. The BFS~\cite{Palombo19} in a material depends on the refractive index (n), longitudinal modulus (M) and the physical density ($\rho$) and is given by the relation $BFS=\frac{2n}{\lambda} \sqrt \frac{M}{\rho}$. When the measured values of BFS, n and $\rho$ are substituted and $\lambda$ being a constant (660 nm), the longitudinal modulus (Fig.~\ref{BFS} inset) linearly increases when ZrF$_4$ is replaced by HfF$_4$, signifying a more rigid bond for the latter and hence a low polarizability.
\\Utilizing the wide parameter space offered by the ultrafast laser inscription it was found that multiscan waveguides fabricated at lower repetition rates ranging between 5-50 kHz, pulse energies between 200 - 700 nJ and feed rates between 0.02 - 0.5mm/s using a 40$\times$, 0.6 NA focussing objective (Olympus, LUCPlan FL N) were ideal for producing high index contrast waveguides. It was also found that introducing a precise amount of spherical aberration by detuning of the focussing objective collar position between 300 -1500 $\mu$m helped to fine-tune the V-number of the waveguide.
\\Fig.~\ref{Rep_Rate} shows the incremental evolution of positive and negative index changes in all glasses upon fs-laser waveguide inscription, keeping the feed rate a constant at 0.04 mm/s while the energies and the objective collar position were adjusted to obtain a maximum index change. A single laser scan pass produced a strand of waveguide which is approximately 1 $\mu$m wide and 12-14 $\mu$m tall. Hence to produce multiscan waveguides, 13 laterally shifted passes (pitch of 0.55 $\mu$m) were carried out at a depth of 170 $\mu$m below the surface to create a $\approx$7 $\mu$m wide waveguide.
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},width=12.5 cm]{Figure-1.png}\caption{(a) refractive index (n) (left axis), glass transition temperature (Tg) (right axis) (b) Brillouin frequency shift (BFS) (left axis), density ($\rho$) (right axis) and (c) terminal fluorine Raman peak vibrational frequency (left axis), terminal fluorine Raman peak bandwidth (right axis).}
\label{Fig-1}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},width=12 cm]{Figure-2.png}\caption{Brillouin frequency shift spectra for ZBLAN and HBLAN glasses. Inset: The longitudinal modulus for all custom glasses calculated based on Brillouin scattering measurements.}
\label{BFS}
\end{figure}
\begin{figure}[ht]
\vspace{1cm}
\centering
\includegraphics[trim={0 0 0 0},width=10 cm]{Figure-3.png}\caption{Evolution of refractive index as a function of HF content for 4 different repetition rates. Laser energy and focussing objective collar was adjusted to achieve maximum index change. Feed rate was kept constant at 0.04mm/s.}
\label{Rep_Rate}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},width=12 cm]{Figure-4.png}\caption{DIC images of waveguides written in all compositions at 5 kHz, 500 nJ, 0.04 mm/s and focussing objective collar set to 1000 $\mu$m.}
\label{DIC}
\end{figure}
Fig.~\ref{DIC} shows DIC (Differential Interference Contrast) microscope images of the waveguides written at 5 kHz, 500 nJ, 0.04 mm/s feed rate with focusing objective collar set at 1000 $\mu$m in all six glass compositions. Featured generally with strong positive and a negative index changes, an inversion of index change was also observed in pure HBLAN glass compared to the rest. A 1:3 ratio of positive to negative index change area was found in all waveguides, whereas for HBLAN this was inverted too. This indicates a structural or a compositional change rather than inversion of the laser induced thermal profile~\cite{Fernandez_2015,Song2011}. Hybrid glasses that contains both HfF$_4$ and ZrF$_4$ were found to produce higher positive index changes in comparison to pure HBLAN and ZBLAN glasses.
\\A 2.25 $\mu$m laser mode was guided through the 7 $\mu$m waveguide written in the 45HfF$_4$-10ZrF$_4$-45BLAN (mol$\%$) glass which had the highest positive index change. The guided mode profiles along with 2D refractive index profile are provided in Fig.~\ref{2.25mode}. To compare with the most challenging case, the fiber from Le Verre Fluoré that was used in comparison has a standard NA of 0.23 with a core diameter of 6.5 $\mu$m and a single mode cut off at 1.95 $\mu$m wavelength. The 2.25 $\mu$m waveguide mode dimensions were 11.6 $\times$ 15.3 $\mu$m in comparison to the 10.9 $\mu$m input fiber mode, producing a net coupling loss of 0.26 dB/facet including Fresnel losses (0.18 dB/facet). A further optimization was carried out specifically on the 45HfF$_4$-10ZrF$_4$ glass to provide an additional 20\% increase in refractive index contrast by inscribing at a faster feed rate (0.3 mm/s) with objective collar position set at 1500 $\mu$m. This helped to increase the guiding wavelength beyond 3 $\mu$m. A 12 $\mu$m wide waveguide (0.6 $\mu$m pitch) written at a rep. rate of 5 kHz and 700 nJ pulse energy produced an index contrast of 1.2 × 10-2 which is the highest reported value in a fluoride glass till date. The DIC microscope image of this waveguide along with the 2D refractive index profile and the guided mode profile at 3.1 $\mu$m are provided in Fig.~\ref{3.1mode}. The waveguide mode dimensions were 17.5 x 23.2 $\mu$m in comparison to the 11.5 $\mu$m input fiber mode at 3.1 $\mu$m. Considering the refractive index matching between the fiber and the waveguide this produced a net coupling loss of 1.37 dB/facet. The loss figures are best reported till date considering the smallest waveguide mode in the mid infrared with added advantage of waveguide being a tunable type-I and same material as the fiber for integration. By further optimisation of the focusing geometry to increase the vertical size of the positive refractive index region, we believe that it is possible to obtain a nearly perfectly circular mode which should further reduce the coupling loss.
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},width=8 cm]{Figure-5.png}\caption{(a) Refractive index profile (b) 2.25 $\mu$m laser mode and (c) its vertical and horizontal line profiles of the waveguide written in the 45HfF$_4$-10ZrF$_4$-45BLAN glass. DIC image and writing parameters as in Figure 4.}
\label{2.25mode}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},width=8.5 cm]{Figure-6.png}\caption{(a) DIC microscope image (b) Refractive index profile (c) 3.1 $\mu$m mode and (d) its vertical and horizontal line profiles of a 12 $\mu$m wide waveguide (0.6 $\mu$m pitch) written at a repetition rate of 5 kHz and 700 nJ pulse energy in the 45HfF$_4$-10ZrF$_4$ glass. }
\label{3.1mode}
\end{figure}
In order to fine-tune the V-number and thus to further lower the coupling losses, the physical origin of the index change must be identified. A backscattered electron microscope image of the 7 $\mu$m waveguide written at 0.04 mm/s feed-rate is shown in Fig.~\ref{EPMA-BFS}(a) and shows negligible z-contrast, indicating weak/negligible material densification/rarefaction. The formation of nano-voids (observed as dark round spots) were found exclusively in the negative index change region. Nano-voids/nano-pores are a common feature in the athermal regime of ultrafast laser irradiation and are formed due to free carrier accumulation where there is the highest electron density~\cite{Dai2016}. Their presence exclusively in the negative index change region indicates the existence of a strong gradient profile for laser energy deposition within the glass due to the detuning of the focussing objective collar~\cite{Fernandez_2015, Song2011}. An elemental mapping of all the constituent elements within the glass was featureless, indicating that the formation of waveguides is purely caused by a structural reorganization and fixed stoichiometry.
Since fluoride glasses are expected to have a higher ionic character~\cite{REAU1989} which 2-4 orders higher than oxide glasses (depending on alkali free or alkali rich composition) mainly because of the presence of monovalent F- ion, electron beam induced migration~\cite{Jiang2004} should be expected while using any e-beam characterization techniques. Hence, supplementary confirmation was sought through Brillouin scattering measurements across the waveguides as it is a purely light-based probing method of the laser-modified zones. Fig.~\ref{EPMA-BFS}(b) shows the relative Brillouin shift across the laser modified zones with respect to the bulk. Moderate shifts of +172 MHz and \textminus135 MHz were observed in the positive and negative index change regions of the waveguide, respectively. Compared to the measurement uncertainty (10 MHz), estimated based on 10 measurements across three unmodified glasses, this change is statistically significant. It is worth pointing out that the overall difference of the Brillouin frequency shifts in this glass for the whole range of compositions is found to be 4.25 GHz (Figure 2). We could thus confirm the results found from electron probe micro analysis (EPMA) that the waveguide formation is not due to ion migration or any local stoichiometry changes. BFS line scans as shown in Fig.~\ref{BFS-linescans}(a) were carried out from top to bottom along the incoming fs-laser direction in those waveguides whose DIC images are shown in Fig.~\ref{DIC} revealing that the magnitude of Brillouin shift reaches the maximum for glasses modified with 35HfF$_4$-20ZrF$_4$-45BLAN. It is interesting to note that the shift is quite sensitive in the negative index zones whereas the positive index zones are featureless for both pure ZBLAN and HBLAN glasses. We could deduce that the negative index change zone formation is based on a structural modification leading to rarefaction. The formation of nano-voids exclusively at the negative index change zone supports this argument. A comparison of the relative difference observed between the maximum and minimum values of Brillouin frequency shift and the refractive index change measured in the respective waveguides is shown in Fig.~\ref{BFS-linescans}(b). First and last data points of pure HBLAN and ZBLAN glass waveguides indicate that the relative change in BFS and $\Delta$n are the same in the positive index change zone. Since $BFS=\frac{2n}{\lambda} \sqrt \frac{M}{\rho}$ , this substantiates a lack of change in density (supplemented by backscattered electron microscopy (BSE)) and longitudinal modulus in the guiding region during waveguide formation. Whereas both deviates to a maximum value for 35HfF$_4$-20ZrF$_4$-45BLAN composition within the positive index change zone. Given that the density change within the waveguide is negligible, the change in longitudinal modulus must be the responsible factor for this strong deviation. Since the composition contains two glass formers (HfF$_4$ and ZrF$_4$) and the deviations are maximum when the mole fraction (HfF$_4$:ZrF$_4$) is between 0.45-0.82, the responsible factor is believed to be a mixed glass former effect as the rest of the composition (BLAN) is kept constant across all samples. This result is quite surprising because Hf and Zr are highly isomorphic and do not contribute to the glass former effect during the bulk glass formation.
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},width=8 cm]{Figure-7.png}\caption{(a) Backscattered electron microscope image of the 7 $\mu$m wide waveguide written in the 45HfF$_4$-10ZrF$_4$-45BLAN glass. DIC image and writing parameters as in Figure.4. Dark round spots within the negative index change zone are the nano-voids. (b) Brillouin frequency shift mapped across the same waveguide }
\label{EPMA-BFS}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},width=13 cm]{Figure-8.png}
\caption{(a) Brillouin frequency shift line scans of waveguides written in all compositions at 5 kHz, 500 nJ, 0.04 mm/s and focusing objective collar set at 1000 $\mu$m. (b) A comparison of relative difference observed between the maximum and minimum values of Brillouin frequency shift and the refractive index change measured in the same waveguides.}
\label{BFS-linescans}
\end{figure}
Typically, when two isomorphic glass formers are mixed within a glass composition, specifically when one mole of glass former is replaced by another at constant glass modifier composition, the effect is non-linear and non-additive which is reflected in a turning point within its optical, thermal and mechanical properties~\cite{Wang2017}. This effect usually peaks when both species have equal content within a composition because during glass formation, occupiable sites for a particular species are not necessarily energetically or morphologically favourable for a second kind of glass former. In other words, if the three dimensional random network could be considered as a potential energy landscape, two different species will see an entirely different 3D energy profile with entirely different potentially occupiable sites~\cite{Dyre2009}. A 50\% relative substitution between the two different species hence shows a largest site mismatch demonstrating the largest contrast to its properties.
In our case, Hf and Zr being isomorphic can be substituted one for another with linear and additive results as evident from the results of Fig.~\ref{Fig-1}. This possibility makes the HfF$_4$-ZrF$_4$ mixed glass in general difficult to discriminate from each other and identifying the role of each glass former in the overall characteristics of the glass is challenging. However, during waveguide inscription, as it transits through a fast quenching process, the modified zone exhibits a mixed glass former effect, especially for glasses with mole fractions around 0.5.
Raman mapping revealed no changes of the intensity, peak shift and bandwidth of the majority of spectral peaks. The exception was the main peak attributed to the symmetric stretching vibration of the terminal fluorine bonds attached to Zr or Hf atoms. This was the case for all waveguides written across all compositions and inscription parameters. Fig.~\ref{Raman} is a representative image of the 12 $\mu$m waveguide written at 0.3 mm/s translation speeds in the 45HfF$_4$-10ZrF$_4$ glass. It demonstrates the generation of such terminal fluorine bonds and the reduction of its vibration frequency within the positive index zone.
Since there is a high disparity in the atomic polarizability between Hf (4.3 Å) and Zr (170.6 Å$^3$) due to the lanthanide contraction effect in the former, the generation of new terminal fluorines is subjected to strong electron cloud distortion depending on the parent atom to which it is attached. The atomic number of Hf (72) is almost the double of Zr (40) but the ionic radius due to lanthanide contraction in Hf (83 Å) is quite similar to Zr (84 Å). This exhibits how much tighter the nucleus of Hf binds its electrons to itself, so the deformation of its electronic shells under an electric field is more difficult.
Fluorine atoms also possess a low atomic polarizability due to their smaller size. Hence the electron localization when a terminal fluorine gets attached to Hf is very high compared to Zr. The observation of mixed glass former effect during waveguide formation can be explained due to the formation of terminal fluorines attached to glass formers with highly diverse polarizabilities. After the glass undergoes a fast quenching process initiated by the energy of the fs laser pulses, the formation of terminal fluorines are site specific and congested caused by the physical presence of the surrounding energy-diverse ligands within the three dimensional network. Fig.~\ref{Schematic} represent the very simple case demonstrating the effect on the electron cloud of terminal fluoride attached to (a) Zr-Zr (b) Hf-Hf (c) Zr-Hf molecular units. Red arrow indicates the strong electron cloud distortion due to the low polarizable Hf atom whereas the yellow arrow indicates a relatively small distortion due to the highly polarizable Zr atom. Hence the glasses with HfF$_4$:ZrF$_4$ $\approx$ 0.5 shows the maximum effect. It further explains the higher refractive index change induced by the distorted electron clouds in glasses that contain both glass formers and the much lower index change in single component pure ZBLAN and HBLAN glasses.
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},width=8 cm]{Figure-9.png}
\caption{(a) DIC image (b) 575 $cm^{-1}$ Raman peak intensity and (c) its frequency shift of the 12 $\mu$m wide waveguide (0.6 $\mu$m pitch) written at a rep. rate of 5 kHz and 700 nJ pulse energy in the 45HfF$_4$-10ZrF$_4$-45BLAN glass. }
\label{Raman}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[trim={0 0 0 0},width=17 cm]{Figure-10.png}
\caption{Electron cloud of terminal/non-bridging fluorines(F$_{NB}$) experiencing (a) moderate distortion from Zr atoms in pure ZBLAN glasses (b) High distortion from Hf atoms in pure HBLAN glasses (c) terminal fluorines with diverse characteristics due to site specificity depending on its immediate neighbour in hybrid glasses. }
\label{Schematic}
\end{figure}
Terminal fluorine bonds do have a higher polarizability compared to the bridged ones, but since the production is the same across all compositions, it is reasonable to believe that the structural rearrangement obeying the mixed glass former effect induces a higher polarizability due to the strong disparity between Hf and Zr.
In summary, high numerical aperture, highly controllable mid-infrared waveguides were fabricated in a redesigned ZBLAN glass. Our results open up the way to the possibility of pigtailing existing fluoride fibers to integrated functional glass chips, thus enabling a new hybrid architecture for the development of fully integrated field-deployable mid-infrared photonic systems. The generation of highly polarizable terminal fluorides due to the fs-laser inscription was identified as the main mechanism for the high positive index change obtained in the guiding region, whereas the low index change region was characterized by structural modifications with nanovoid formation. Laser induced ion migration or stoichiometry change did not contribute to waveguide formation. The maximum refractive index change can be manipulated by varying the ratio of the two glass formers by controlling electron cloud distortion due to fs-laser induced mixed glass former effect. Hence, this work can also serve as a guideline for materials-based optimization of waveguide fabrication in other glasses and for other wavelengths.
\section*{Materials and methods}
Six different glasses with varying HfF$_4$ content starting with a conventional pure ZBLAN glass having a composition of 55 ZrF$_4$ and the rest 45 mol\% comprised of BaF$_2$, LaF$_3$, AlF$_3$, NaF was used. xHfF$_4$-(55-x)ZrF$_4$-45BLAN where x = 0, 15, 25, 35, 45, 55. The stoichiometry of Ba, La, Al and Na is kept constant for all glasses. Glasses were prepared with the conventional melt quenching technique at the Le Verre Fluore industrial facility. To avoid any experimental or alignment errors during waveguide inscription, all samples were mounted on a float glass substrate, polished down to same thickness and flatness before inscription was carried out at the same time on all samples. A Pharos femtosecond laser system operating at a central wavelength of 1030 nm, pulse duration of 240 fs and a variable repetition rate starting from single pulse up to 1 MHz was used for inscription. We found that lower repetition rates provide ideal inscription windows for multiscan waveguides and hence in this work concentrated on values between 5 - 50 kHz. After inscription, 11 mm long waveguides whose end facets were polished were imaged using differential interference contrast microscopy using an Olympus inverted microscope. The refractive index was profiled using Rinck near field refractometer and mode profiled using a Dataray wincam S-WCD-IR-BB-30 beam profiler. Structural and chemical characterization of the waveguides were carried out using Scanning electron microscope on a JEOL JXA-8500F field-emission EPMA and Micro-Raman spectroscopy with 633 nm excitation wavelength on a Renishaw inVia Raman microscope in confocal mode using a 100×objective (spatial resolution $\approx$0.5 $\mu$m).
Brillouin scattering experiments were carried out to identify differences in the mechanical properties of waveguides and bulk glasses of all compositions. Brillouin spectroscopy is based on inelastic light scattering where photons exchange energy with phonons within the material, leading to a Brillouin frequency shift (BFS, $\Omega$) between the incident and scattered light. This shift is directly proportional to the speed of longitudinal sound waves ($\nu_s$), the refractive index n and inversely proportional to the probing laser wavelength $\lambda$=660 nm, $\Omega$=$\frac{2n}{\lambda}\nu_s$. Thus, Brillouin light scattering directly probes the propagation speed of acoustic phonons together with the material refractive index. The acoustic speed, in turn, is the function of the material mechanical properties, namely its longitudinal elastic modulus M and the material density $\rho$, and given by $\nu_s$=$\sqrt \frac{M}{\rho}$. Spontaneous Brillouin scattering measurements were carried out as a complementary light based technique to electron microscopy as fluoride glasses have a higher ionic character and can be affected by electron beam induced migration of elements within the characterizing region. Brillouin frequency shifts (BFS) were measured using 660 nm single frequency Cobolt Flamenco laser (HÜBNER Photonics) through a confocal microscope (CM1, TableStable Ltd) and the spectra were collected using a 6-pass tandem scanning Fabry-Perot interferometer (TFP1, TableStable Ltd). The backscattering light was collected by an objective lens (20X Mitutoyo Plan Apo infinity corrected objective, NA=0.42, WD = 20 mm) and redirected to the interferometer for analysis. Both line and 2D mappings were carried out by moving the sample on the 3D microscopy stage (SmarAct) along one and two axes of the stage, while keeping the optical system and the objective lens stationary. This experimental apparatus resulted in measurement with spatial resolution of approximately 2 $\mu$m x 2 $\mu$m x 100 $\mu$m in X-Y-Z direction, respectively. The spectral resolution of our instrument is determined by the distance between the mirrors of Fabry-Perot scanning interferometer (3 mm) and the number of acquisition channels (512) to be approximately 276 MHz. The spectral extinction ratio of Fabry-Perot interferometers is above $10^{10}$~\cite{Sandercock1976}.The acquisition time for each point measurement was 20 s to optimise signal-to-noise-ratio and improve the fitting precision. The raw data collected by the spectrometer is fitted using Damped Harmonic Oscillator (DHO) model for each individual Brillouin peak and the centre of these peaks is what determines the Brillouin frequency shift values reported in this manuscript.
\section*{Data availability}
The authors confirm that the data supporting the findings of this study are available within the article [and/or] its supplementary materials.
\section*{Acknowledgements}
This work is funded by the US Air Force Office of Scientific Research under award number FA2386-19-1-4049. This work was performed in-part at the OptoFab node of the Australian National Fabrication Facility, utilising NCRIS \& NSW state government funding. The authors acknowledge the use of facilities supported by Microscopy Australia at the Electron Microscope Unit within the Mark Wainwright Analytical Centre at UNSW Sydney.
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